Research ArticlePressure Transient Behavior of Horizontal Well withTime-Dependent Fracture Conductivity in Tight Oil Reservoirs
Qihong Feng1 Tian Xia1 Sen Wang1 and Harpreet Singh2
1School of Petroleum Engineering China University of Petroleum (East China) Qingdao 266580 China2Department of Petroleum and Geosystems Engineering The University of Texas at Austin Austin TX USA
Correspondence should be addressed to Sen Wang fwforestgmailcom
Received 9 June 2017 Revised 29 July 2017 Accepted 7 August 2017 Published 28 September 2017
Academic Editor Jianchao Cai
Copyright copy 2017 Qihong Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This work presents a discussion on the pressure transient response of multistage fractured horizontal well in tight oil reservoirsBased on Greenrsquos function a semianalytical model is put forward to obtain the behavior Our proposed model accounts for fluidflow in four contiguous regions of the tight formation by using pressure continuity and mass conservation The time-dependentconductivity of hydraulic fractures which is ignored in previous models but highlighted by recent experiments is also taken intoaccount in our proposedmodelWe also include the effect of pressure drop along a horizontal wellboreWe substantiate the validityof our model and analyze the different flow regimes as well as the effects of initial conductivity fracture distribution and geometryon the pressure transient behavior Our results suggest that the decrease of fracture conductivity has a tremendous effect on thewell performance Finally we compare our model results with the field data from a multistage fractured horizontal well in Jimsarsag Xinjiang oilfield and a good agreement is obtained
1 Introduction
Unconventional resources are playing increasingly importantroles in the energy industry [1 2] We have witnessed a greatsuccess in North America [3 4] Owing to its extremelylow permeability [5] tight oil could not be economicallydeveloped via conventional technology [6ndash8] Multistagefractured horizontal well (MFHW) is an efficient techniquein the development of unconventional reserves [9] Howeveranalyzing the pressure response of multifractured horizontalwells is challenging because several factors for examplefracture conductivity fracture geometry skew angle betweenfracture and horizontal well are responsible for the pressuretransient behavior Therefore the variation of pressure as afunction of these factors is still ambiguous
Pressure transient analysis is an important tool to esti-mate the formation characteristics Significant efforts havebeen dedicated in proposing models to describe the processGringarten and Ramey Jr analyzed the transient behaviorof uniform-flux fracture and infinite-conductivity fracturewell through source function method [10] Cinco-Ley and
Samaniego developed a mathematical model for finite-conductivity fracture [11]ThenCinco-Ley andMeng gave thesolution forwells with finite-conductivity fractures in Laplacedomain [12] Ozkan and Raghavan employed point-sourcesolution method to get transient pressure solutions under avariety of conditions [13] Although these studies only dealtwith a single fracture they have laid a solid foundation forthe analysis of pressure behavior in multiple fractures
The tools that are commonly used to make the transientpressure analysis of multiple fractures can be divided intothree categories analytical semianalytical and numericalmethod Guo et al developed an analytical method forhorizontal well intersecting multiple fractures [14] Howeverthe interference of the fractures was neglected Wan and Azizderived an analytical 3D solution for horizontal well withmultiple random fractures by using Fourier analysis to a 2Dsolution [15] Ozkan et al proposed a trilinear flow model inwhich the linear flow in outer reservoir inner reservoir andhydraulic fractures are included [16] Brown et al improvedthis model to simulate the pressure transient and productionbehaviors of fractured horizontal wells in unconventional
HindawiGeofluidsVolume 2017 Article ID 5279792 19 pageshttpsdoiorg10115520175279792
2 Geofluids
Table 1 Background information for the experimental studies on the variation of fracture conductivity with time
Author Material Confining stressMcDaniel (1986) Sand resin coated sand and three ceramic proppants 8000 psiCobb and Farrell (1986) Ceramic proppants 10000 psi and 5000 psiHandren and Palisch (2007) Sand and resin coated sand 6000 psi
0 270240210180150120906030Time (d)
Relat
ive f
ract
ure c
ondu
ctiv
ity
00
02
04
06
08
10
Figure 1 Schematic showing the variation of relative fractureconductivity with time (Montgomery 1984)
shale reservoirs [17] Al Rbeawi and Djebbar introduced anew analytical model that can be used to investigate thepressure behavior and flow regimes of a horizontal well withmultiple inclined hydraulic fractures and applied it in typecurve matching [18]
Semianalytical approaches are another important wayin analyzing the transient behavior Horne and Temengconsidered the interference among the fractures via thesuperposition of influence functions [19] Zerzar and Bet-tam combined the boundary element method and Laplacetransformation to deal with interaction of reservoir flow andfracture flow [20] Yao et al presented a method based onGreenrsquos functions and the sourcesink method to obtain thetransient pressure response for a multifractured horizontalwell in a closed box-shaped reservoir [21] Zhou et alproposed a semianalytical model to simulate the pressuretransient behavior in complex hydraulic fracture networks[22] Yu combined gas desorption and Zhou et alrsquos model tosolve the gas production problem in shale gas reservoir [23]Jia et al presented a model to solve the transient behavior incomplex fracture networks with deep consideration for flowin fractures [24]
Numerical approaches overcome many limitations inanalytical and semianalytical method in studying uncon-ventional reservoirs Al-Kobaisi et al presents a hybridnumerical-analytical model for the pressure transient re-sponse of a finite-conductivity fracture intercepted by ahorizontal well [25] Freeman et al used numerical sensitivitystudies to show the effect of mechanisms and factors on
the performance of multifractured horizontal well [26ndash28]Olorode et al employed numerical method to study the effectof fracture angularity and nonplanar fracture configurationson well performance [29] Yu et al conducted numerical sim-ulation to investigate the impact of fracture patterns matrixpermeability cluster spacing and fracture conductivity[30]
In previous studies the conductivity of hydraulic frac-tures was often assumed to be uniformly distributed andremained constant with time However this assumptioncontradicted field practice After hydraulic fracturing thefractures close rapidly In order to mitigate the productiondecrease proppants are added in the fracturing fluid to propthe fracture and maintain the productivity However thisoperation can only alleviate the decrease rate of the fractureconductivity The conductivity will eventually decrease [31ndash34] Previous studies suggest that the fracture conductivitydecreases rapidly during the first couple of days and forthe rest of the time the decline degree remains very small(Figure 1)
Table 1 shows some experiments that studied the variationof fracture conductivity with time McDaniel reported thatthe sand lost 80 of the conductivity within 15 days amongwhich the resin coated sand lost 55 and ceramic proppantslost 25 to 30 [35] Tests by Cobb and Farrell showedthat ceramic proppants lost sim20 of conductivity in 70 dayswhen confined at 10000 psi and the sand lost over 30 whenconfined at 5000 psi [36] Handren and Palisch reporteddecline in the conductivity with sands losing 55 and resincoated sand losing 25 to 30 [37] Because the productionrate is strongly dependent on the fracture conductivity itsvariation with time must be taken into account to accuratelypredict the well performance
Contrary to the reported observations many studies haveassumed horizontal well to be of infinite conductivity [1624 38] This assumption is not reliable as it cannot reflectthe radial influx frictional and acceleration effects It isnecessary to examine the effect of pressure drop within thewellbore on the production performance
The objective of this study is to examine the effect of time-dependent fracture conductivity on the transient behavior ofMFHWAs shown in Figure 2(a) somehydraulic fractures arenot perpendicular to the horizontal wellbore which contra-dicts the common assumption in the analytical models Thatis the pressure transient behavior of this complex fracturenetwork is not readily to be analytically accounted forThere-fore we present a semianalytical model to take into accountthe time-dependent fracture conductivity and the pres-sure drop along a horizontal wellbore as well as the complexfracture networks
Figure 2 (a)Microseismic data showing the complex fractures in Jimsar sag Xinjiang oilfield (b) Schematic of a fracture networkThe blackred and blue lines represent the wellbore hydraulic fractures and natural fractures respectively
2 Methodology
21 Mathematical Model The reservoir after fracturingincludes four regions the matrix the natural fracture net-work hydraulic fractures and the horizontal wellboreThere-fore the fluid flow in the reservoir consists of (1) fluid flowfrom the reservoir to the hydraulic fractures ((i) in Fig-ure 2(b)) (2) fluid flow from the reservoir to the naturalfractures ((ii) in Figure 2(b)) (3) fluid flow from natural frac-ture to hydraulic fractures ((iii) in Figure 2(b)) (4) fluid flowamong natural fractures ((iv) in Figure 2(b)) (5) fluid formhydraulic fractures to the horizontal wellbore ((v) in Fig-ure 2(b)) (6) fluid flow in the horizontal wellbore ((vi) in Fig-ure 2(b))These six types of flow can be categorized into threegroups reservoir flow ((i) and (ii) in Figure 2(b)) fractureflow ((iii) (iv) and (v) in Figure 2(b)) and wellbore flow ((vi)in Figure 2(b))
Following assumptions are made
(1) Reservoir is isotropic homogeneous box-shapedand of uniform thickness with impermeable bound-aries
(2) Fluid in the reservoir is single-phase and slightly com-pressible and its compressibility and viscosity are con-stant
(3) Fractures are rectangular and vertical The flux rate isuniform along the fracture
(4) Horizontal well is parallel to the upper and lowerboundary of the reservoir
(5) Effect of gravity is neglected
First we define some dimensionless parameters for gen-erality The dimensionless pressure and time can be definedas
119901119863 = 2120587120588119896ℎ119876120583 (119901119894 minus 119901) 119905119863 = 1198961199051206011205831198621199051198712
(1)
The dimensionless flow rate and influx rate can be defined as
211 Reservoir Flow We use Greenrsquos function that has beenfrequently applied to solve problems of transient flow sinceits usage was first explored in well testing by Gringartenand Ramey Jr [10] Based on the Newman product methodinstantaneous source functions of fracture panels can beobtained therefore the pressure response at any point in thereservoir from one fracture panel can be expressed as follows(see Appendix for further details)
Δ119901 (119909 119910 119911 119905) = 119901119894 minus 119901 (119909 119910 119911 119905)= 1120601119862119905 int
where 119878119895(119909 119910 119911 120591) is the instantaneous plane source functionof the jth panel and 119901119894 is the initial reservoir pressure whichis assumed to be uniformly distributed in the reservoir
For fracture 119895 rotated at any horizontal angle to the wellthe plane source function is
exp(minus119899212058721205781199091199051199092119890 ) cos 1198991205871199091015840119909119890 cos 119899120587119909119909119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199101199051199102119890 ) cos 1198991205871199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890]
4 Geofluids
sdot [1 + 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(5)
By using the superposition principle pressure at any point inthe reservoir at a given time can be given by
Therefore pressure map can be obtained at a given time
212 Fracture Flow We assume that each hydraulic fractureis of finite-conductivity and fluid flow inside the fractureis one dimensional [17 39ndash41] We use Darcyrsquos equation todescribe the fluid flow in fractures For the jth panel thepressure at any point in the fracture is (see Appendix forfurther details)
1199011198951 minus 119901119895119898= int1199091198951198981199091198951
( 120583120588119896119891119887119891ℎ)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(8)
Then the dimensionless form is
119901119895119898119863 minus 1199011198951119863= 2120587119896119871119896119891119887119891 int
119909119895119898119863
1199091198951119863
[1199021198951119863 + 119902119891119895119863 (119909119863 minus 1199091198951119863)] 119889119909119863 (9)
Equations (8) and (9) can be applied to both hydraulicfractures and natural fractures
It is worth noting that fluid flow from the fracturesto the horizontal wellbore is radial in the near-well region(Figure 3) In order to minimize the error caused by linearflow approximation a choke flow factor is introduced [42]
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (10)
where 119903119908 is the radius of the horizontal well and 119887119891 is thehydraulic fracture width
213 Time-Dependent Conductivity Proppants are oftenpumped into the formation to maintain the fracture con-ductivity However the proppant particles usually break and
Figure 3 Sketch showing the change of flow pattern from linear toradial
embed which causes the fracture conductivity to reduce untilequilibrium is established Montgomery and Steanson sug-gested that there is a logarithmic relation between hydraulicfracture conductivity and time for 1020 Sand and 2040 Sand(Figure 4(a)) [43] Other tests also show logarithmic relationbetween fracture conductivity and time including Lanzhousand from China with diameter of 045sim090mm [44] andsand from Shanshan oil field in China (Figure 4(c)) [45]Other kinds of proppants such as ceramic proppants withdiameter of 045sim09mm and Lanzhou sand with diameterof 09sim125mm show similar correlations but the slope of thecurve also referred to as the decline coefficient for each prop-pant is differentThe decline coefficient decreases in the orderof Lanzhou sand with diameter of 090sim125mm the ceramicproppant and the Lanzhou sand with smaller diameter
Based on the trend of observations in Figure 4 we usethe following model to describe the variation of fractureconductivity as a function of time
119862119891 = 1198621198910 (1 minus 120573 lg 1199051199050) (11)
where 119862119891 is the fracture conductivity at time 119905 1198621198910 is theinitial conductivity and 120573 is the decline coefficient 119862119891 in(11) is substituted by 119896119891 sdot 119887119891 in (8) and (9) It is assumedthat when the effect of hydraulic fracture disappears totallythe fracture conductivity is equal to the product of formationpermeability and fracture width
214 Wellbore Flow Previous studies assumed horizontalwellbore as infinite-conductivity pipe [15 16] which implic-itly assumes no pressure drop along the horizontal wellbecause of uniform pressure distribution along the wellboreTo examine the validity of this approximation we include thewellbore pressure drop in the present model The pressuredecrease along the wellbore consists of frictional losses andacceleration losses (Figure 5) Based on the Darcy-WeisbachEquation [46] the fictional pressure drop can be expressed as
where 119891119894 is the frictional coefficient and Δ119897119894 is the length ofthe wellbore segment For each segment of the wellbore fluidflow from the hydraulic fractures to the wellbore will cause an
Geofluids 5
1 10 100
200
300
02
04
06
08
10
Relat
ive f
ract
ure c
ondu
ctiv
ity
Time (d)Experiment data case 1Fitting curve case 1Experiment data case 2Fitting curve case 2
(a)
01 1 10 100
02
04
06
08
10
Time (d)Re
lativ
e fra
ctur
e con
duct
ivity
Experiment dataFitting curve
(b)
1 10 100
1000
00
02
04
06
08
10
Time (d)Field dataFitting curve
Relat
ive f
ract
ure c
ondu
ctiv
ity
(c)
Figure 4 Relation between relative fracture conductivity and time (a) Montgomery and Steanson (1985) tested two samples with differentkind of proppants for 9 months case 1 denotes 1020 Sand at 250∘F and case 2 denotes 2040 Sand at 75∘F (b) Experiment conducted by Yu(1987) (c) Field data from Shanshan oilfield
Flow direction
pＩＯＮAi pＣＨAi
iDiΔli
Figure 5 Sketch of microsegment in horizontal well
increase of the flow rate This will result in the change of themomentum of the fluid which leads to the acceleration pres-sure drop
Δ119901acce = 120588119894 (V2out minus V2in) (13)
Dimensionless form of the pressure drop along the well boreis given by (see Appendix for further details)
119901119863119894 minus 119901119863119894+1= 2120587119896ℎ1198761205831198602 (1199022119901119863119895 minus 1199022119901119863119895+1 minus 119891119895Δ1198971198952119863 1199022119901119863119895) (14)
215 Coupling Relationship Due to the pressure continuityat the center of each fracture the pressure response obtainedfrom the reservoir flow should be consistent with the fractureflow Therefore we have
1199011198951198881198631 and1199011198951198881198632 are given by (7) and (9) respectivelyThemassbalance is applied to both the intersection nodes between
fractures and the intersection nodes of fractures and wellbore(Figure 6) Therefore for each node the inflow of fluid mustbe equal to the outflow of the fluid
119902in119863 = 119902out119863 (16)
For the intersection nodes of fractures and wellbore theinflow and outflow are taken into account between the frac-tures and between the fractures and the wellbore pipe
22 Computational Approach From (17) we obtain 119899V equa-tions at the nodes for mass balance 119899119901 equations for pressuredrop along the fractures and 119899119901 equations for pressure con-tinuity in the panel centers resulting in 119899V + 2119899119901 nonlinearequations that need to be solved Newtonrsquos method has beenwidely used to solve systems of equations because of its quickconvergence However it requires the inverse of Hessianmatrix at each iteration and the convergence may not bereached if the Hessian matrix is ill-conditioned or nonposi-tive definiteTherefore we useGauss-Newtonmethod whichis an improved version of Newtonrsquos method for finding aminimum of a function [47] The basic idea is to use theTaylor series expansion to approximate nonlinear regressionmodel and correct the solution through iteration This algo-rithm is robust and has a good convergence rateThe iterationequation of Gauss-Newton method can be expressed as
Pressure at each node flow rate inside the fracture and fluxesalong the fractures can be obtained through iteration Thusdimensionless pressure at any point in the reservoir can becalculated via (7) If the bottom-hole pressure is given we canget the flow rate in the same way The flow rate behavior indifferent conditions can be obtained consequently
6 Geofluids
qm+11
qm2
qpi
qpi+1
(a)
qm+11
qm2
(b)
Figure 6 Sketch of mass balance at intersection nodes (a) Intersection node between fractures and wellbore (b) Intersection node betweenfractures (Black line denotes wellbore red lines represent hydraulic fracture and blue line is natural fracture)
Pressure derivativePressure derivativePressure derivative m = 3
Figure 7 Effect of discretized number per fracture on the simula-tion resultm is the number of discrete panels in each fracture
23 Model Validation While solving for the numerical solu-tion fractures are discretized into several panels and toensure that we obtain correct numerical solution it is nec-essary to probe the grid independence with respect to thenumber of panels used Figure 7 shows the grid independenceanalysis for a horizontal well with three transverse hydraulicfractures where 119898 is the number of panels that a fracture isdiscretized into We can conclude that the result we obtainedfrom our model is independent of segment when119898 is greaterthan 1
CMG a commercial reservoir numerical simulator [48]was employed to validate ourmodel In this article the IMEXmodule in CMG which is a conventional three-phase black-oil simulator is utilized to make the comparison Interestedreaders may find the governing equations of this modulefrom the textbooks on reservoir numerical simulation Theconventional Cartesian grids are employed and the totalnumber of cells is 121 times 121 times 5 and the basic parameters used
12
14
16
18
20
22
24
26
28
30
CMGOur model
100 101 102 103
t (d)
pwf
(MPa
)
Figure 8 Bottom-hole pressure obtained using our proposedmodeland CMG
in the validation are listed in Table 2 The outer boundaryof our simulation domain is impermeable Meanwhile to beconsistent with the assumptions of our proposed model wemaintain the oil production rate constant in the simulationFigure 8 compares the bottom-hole pressure obtained fromCMG simulator and our model and it shows a good matchduring initial stage Figure 9 shows the comparison ofreservoir pressure from CMGwith our model and except forthe pressures near fractures we see a good match betweenthe two The reason for this slight difference around thefractures is because in CMG the fluid flows directly fromthe reservoir to the horizontal well however many studies[27 49 50] have reported that the correct representationof fluid flow must consider intersection and activation ofpreexisting natural fractures with hydraulic fractures whichwill create a complex fracture networkThis representation ofcomplex fracture network can be readily incorporated in ourmodel however it is an extremely challenging task to do thatin CMG
Geofluids 7
140
156
172
188
204
220
236
252
268
284
300
(MPa)
(a)14
156
172
188
204
22
236
252
268
284
30
(MPa)
(b)
Figure 9 Pressure distribution of the reservoir (a) CMG simulator (b) Our model The horizontal line represents the well
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Table 1 Background information for the experimental studies on the variation of fracture conductivity with time
Author Material Confining stressMcDaniel (1986) Sand resin coated sand and three ceramic proppants 8000 psiCobb and Farrell (1986) Ceramic proppants 10000 psi and 5000 psiHandren and Palisch (2007) Sand and resin coated sand 6000 psi
0 270240210180150120906030Time (d)
Relat
ive f
ract
ure c
ondu
ctiv
ity
00
02
04
06
08
10
Figure 1 Schematic showing the variation of relative fractureconductivity with time (Montgomery 1984)
shale reservoirs [17] Al Rbeawi and Djebbar introduced anew analytical model that can be used to investigate thepressure behavior and flow regimes of a horizontal well withmultiple inclined hydraulic fractures and applied it in typecurve matching [18]
Semianalytical approaches are another important wayin analyzing the transient behavior Horne and Temengconsidered the interference among the fractures via thesuperposition of influence functions [19] Zerzar and Bet-tam combined the boundary element method and Laplacetransformation to deal with interaction of reservoir flow andfracture flow [20] Yao et al presented a method based onGreenrsquos functions and the sourcesink method to obtain thetransient pressure response for a multifractured horizontalwell in a closed box-shaped reservoir [21] Zhou et alproposed a semianalytical model to simulate the pressuretransient behavior in complex hydraulic fracture networks[22] Yu combined gas desorption and Zhou et alrsquos model tosolve the gas production problem in shale gas reservoir [23]Jia et al presented a model to solve the transient behavior incomplex fracture networks with deep consideration for flowin fractures [24]
Numerical approaches overcome many limitations inanalytical and semianalytical method in studying uncon-ventional reservoirs Al-Kobaisi et al presents a hybridnumerical-analytical model for the pressure transient re-sponse of a finite-conductivity fracture intercepted by ahorizontal well [25] Freeman et al used numerical sensitivitystudies to show the effect of mechanisms and factors on
the performance of multifractured horizontal well [26ndash28]Olorode et al employed numerical method to study the effectof fracture angularity and nonplanar fracture configurationson well performance [29] Yu et al conducted numerical sim-ulation to investigate the impact of fracture patterns matrixpermeability cluster spacing and fracture conductivity[30]
In previous studies the conductivity of hydraulic frac-tures was often assumed to be uniformly distributed andremained constant with time However this assumptioncontradicted field practice After hydraulic fracturing thefractures close rapidly In order to mitigate the productiondecrease proppants are added in the fracturing fluid to propthe fracture and maintain the productivity However thisoperation can only alleviate the decrease rate of the fractureconductivity The conductivity will eventually decrease [31ndash34] Previous studies suggest that the fracture conductivitydecreases rapidly during the first couple of days and forthe rest of the time the decline degree remains very small(Figure 1)
Table 1 shows some experiments that studied the variationof fracture conductivity with time McDaniel reported thatthe sand lost 80 of the conductivity within 15 days amongwhich the resin coated sand lost 55 and ceramic proppantslost 25 to 30 [35] Tests by Cobb and Farrell showedthat ceramic proppants lost sim20 of conductivity in 70 dayswhen confined at 10000 psi and the sand lost over 30 whenconfined at 5000 psi [36] Handren and Palisch reporteddecline in the conductivity with sands losing 55 and resincoated sand losing 25 to 30 [37] Because the productionrate is strongly dependent on the fracture conductivity itsvariation with time must be taken into account to accuratelypredict the well performance
Contrary to the reported observations many studies haveassumed horizontal well to be of infinite conductivity [1624 38] This assumption is not reliable as it cannot reflectthe radial influx frictional and acceleration effects It isnecessary to examine the effect of pressure drop within thewellbore on the production performance
The objective of this study is to examine the effect of time-dependent fracture conductivity on the transient behavior ofMFHWAs shown in Figure 2(a) somehydraulic fractures arenot perpendicular to the horizontal wellbore which contra-dicts the common assumption in the analytical models Thatis the pressure transient behavior of this complex fracturenetwork is not readily to be analytically accounted forThere-fore we present a semianalytical model to take into accountthe time-dependent fracture conductivity and the pres-sure drop along a horizontal wellbore as well as the complexfracture networks
Figure 2 (a)Microseismic data showing the complex fractures in Jimsar sag Xinjiang oilfield (b) Schematic of a fracture networkThe blackred and blue lines represent the wellbore hydraulic fractures and natural fractures respectively
2 Methodology
21 Mathematical Model The reservoir after fracturingincludes four regions the matrix the natural fracture net-work hydraulic fractures and the horizontal wellboreThere-fore the fluid flow in the reservoir consists of (1) fluid flowfrom the reservoir to the hydraulic fractures ((i) in Fig-ure 2(b)) (2) fluid flow from the reservoir to the naturalfractures ((ii) in Figure 2(b)) (3) fluid flow from natural frac-ture to hydraulic fractures ((iii) in Figure 2(b)) (4) fluid flowamong natural fractures ((iv) in Figure 2(b)) (5) fluid formhydraulic fractures to the horizontal wellbore ((v) in Fig-ure 2(b)) (6) fluid flow in the horizontal wellbore ((vi) in Fig-ure 2(b))These six types of flow can be categorized into threegroups reservoir flow ((i) and (ii) in Figure 2(b)) fractureflow ((iii) (iv) and (v) in Figure 2(b)) and wellbore flow ((vi)in Figure 2(b))
Following assumptions are made
(1) Reservoir is isotropic homogeneous box-shapedand of uniform thickness with impermeable bound-aries
(2) Fluid in the reservoir is single-phase and slightly com-pressible and its compressibility and viscosity are con-stant
(3) Fractures are rectangular and vertical The flux rate isuniform along the fracture
(4) Horizontal well is parallel to the upper and lowerboundary of the reservoir
(5) Effect of gravity is neglected
First we define some dimensionless parameters for gen-erality The dimensionless pressure and time can be definedas
119901119863 = 2120587120588119896ℎ119876120583 (119901119894 minus 119901) 119905119863 = 1198961199051206011205831198621199051198712
(1)
The dimensionless flow rate and influx rate can be defined as
211 Reservoir Flow We use Greenrsquos function that has beenfrequently applied to solve problems of transient flow sinceits usage was first explored in well testing by Gringartenand Ramey Jr [10] Based on the Newman product methodinstantaneous source functions of fracture panels can beobtained therefore the pressure response at any point in thereservoir from one fracture panel can be expressed as follows(see Appendix for further details)
Δ119901 (119909 119910 119911 119905) = 119901119894 minus 119901 (119909 119910 119911 119905)= 1120601119862119905 int
where 119878119895(119909 119910 119911 120591) is the instantaneous plane source functionof the jth panel and 119901119894 is the initial reservoir pressure whichis assumed to be uniformly distributed in the reservoir
For fracture 119895 rotated at any horizontal angle to the wellthe plane source function is
exp(minus119899212058721205781199091199051199092119890 ) cos 1198991205871199091015840119909119890 cos 119899120587119909119909119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199101199051199102119890 ) cos 1198991205871199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890]
4 Geofluids
sdot [1 + 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(5)
By using the superposition principle pressure at any point inthe reservoir at a given time can be given by
Therefore pressure map can be obtained at a given time
212 Fracture Flow We assume that each hydraulic fractureis of finite-conductivity and fluid flow inside the fractureis one dimensional [17 39ndash41] We use Darcyrsquos equation todescribe the fluid flow in fractures For the jth panel thepressure at any point in the fracture is (see Appendix forfurther details)
1199011198951 minus 119901119895119898= int1199091198951198981199091198951
( 120583120588119896119891119887119891ℎ)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(8)
Then the dimensionless form is
119901119895119898119863 minus 1199011198951119863= 2120587119896119871119896119891119887119891 int
119909119895119898119863
1199091198951119863
[1199021198951119863 + 119902119891119895119863 (119909119863 minus 1199091198951119863)] 119889119909119863 (9)
Equations (8) and (9) can be applied to both hydraulicfractures and natural fractures
It is worth noting that fluid flow from the fracturesto the horizontal wellbore is radial in the near-well region(Figure 3) In order to minimize the error caused by linearflow approximation a choke flow factor is introduced [42]
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (10)
where 119903119908 is the radius of the horizontal well and 119887119891 is thehydraulic fracture width
213 Time-Dependent Conductivity Proppants are oftenpumped into the formation to maintain the fracture con-ductivity However the proppant particles usually break and
Figure 3 Sketch showing the change of flow pattern from linear toradial
embed which causes the fracture conductivity to reduce untilequilibrium is established Montgomery and Steanson sug-gested that there is a logarithmic relation between hydraulicfracture conductivity and time for 1020 Sand and 2040 Sand(Figure 4(a)) [43] Other tests also show logarithmic relationbetween fracture conductivity and time including Lanzhousand from China with diameter of 045sim090mm [44] andsand from Shanshan oil field in China (Figure 4(c)) [45]Other kinds of proppants such as ceramic proppants withdiameter of 045sim09mm and Lanzhou sand with diameterof 09sim125mm show similar correlations but the slope of thecurve also referred to as the decline coefficient for each prop-pant is differentThe decline coefficient decreases in the orderof Lanzhou sand with diameter of 090sim125mm the ceramicproppant and the Lanzhou sand with smaller diameter
Based on the trend of observations in Figure 4 we usethe following model to describe the variation of fractureconductivity as a function of time
119862119891 = 1198621198910 (1 minus 120573 lg 1199051199050) (11)
where 119862119891 is the fracture conductivity at time 119905 1198621198910 is theinitial conductivity and 120573 is the decline coefficient 119862119891 in(11) is substituted by 119896119891 sdot 119887119891 in (8) and (9) It is assumedthat when the effect of hydraulic fracture disappears totallythe fracture conductivity is equal to the product of formationpermeability and fracture width
214 Wellbore Flow Previous studies assumed horizontalwellbore as infinite-conductivity pipe [15 16] which implic-itly assumes no pressure drop along the horizontal wellbecause of uniform pressure distribution along the wellboreTo examine the validity of this approximation we include thewellbore pressure drop in the present model The pressuredecrease along the wellbore consists of frictional losses andacceleration losses (Figure 5) Based on the Darcy-WeisbachEquation [46] the fictional pressure drop can be expressed as
where 119891119894 is the frictional coefficient and Δ119897119894 is the length ofthe wellbore segment For each segment of the wellbore fluidflow from the hydraulic fractures to the wellbore will cause an
Geofluids 5
1 10 100
200
300
02
04
06
08
10
Relat
ive f
ract
ure c
ondu
ctiv
ity
Time (d)Experiment data case 1Fitting curve case 1Experiment data case 2Fitting curve case 2
(a)
01 1 10 100
02
04
06
08
10
Time (d)Re
lativ
e fra
ctur
e con
duct
ivity
Experiment dataFitting curve
(b)
1 10 100
1000
00
02
04
06
08
10
Time (d)Field dataFitting curve
Relat
ive f
ract
ure c
ondu
ctiv
ity
(c)
Figure 4 Relation between relative fracture conductivity and time (a) Montgomery and Steanson (1985) tested two samples with differentkind of proppants for 9 months case 1 denotes 1020 Sand at 250∘F and case 2 denotes 2040 Sand at 75∘F (b) Experiment conducted by Yu(1987) (c) Field data from Shanshan oilfield
Flow direction
pＩＯＮAi pＣＨAi
iDiΔli
Figure 5 Sketch of microsegment in horizontal well
increase of the flow rate This will result in the change of themomentum of the fluid which leads to the acceleration pres-sure drop
Δ119901acce = 120588119894 (V2out minus V2in) (13)
Dimensionless form of the pressure drop along the well boreis given by (see Appendix for further details)
119901119863119894 minus 119901119863119894+1= 2120587119896ℎ1198761205831198602 (1199022119901119863119895 minus 1199022119901119863119895+1 minus 119891119895Δ1198971198952119863 1199022119901119863119895) (14)
215 Coupling Relationship Due to the pressure continuityat the center of each fracture the pressure response obtainedfrom the reservoir flow should be consistent with the fractureflow Therefore we have
1199011198951198881198631 and1199011198951198881198632 are given by (7) and (9) respectivelyThemassbalance is applied to both the intersection nodes between
fractures and the intersection nodes of fractures and wellbore(Figure 6) Therefore for each node the inflow of fluid mustbe equal to the outflow of the fluid
119902in119863 = 119902out119863 (16)
For the intersection nodes of fractures and wellbore theinflow and outflow are taken into account between the frac-tures and between the fractures and the wellbore pipe
22 Computational Approach From (17) we obtain 119899V equa-tions at the nodes for mass balance 119899119901 equations for pressuredrop along the fractures and 119899119901 equations for pressure con-tinuity in the panel centers resulting in 119899V + 2119899119901 nonlinearequations that need to be solved Newtonrsquos method has beenwidely used to solve systems of equations because of its quickconvergence However it requires the inverse of Hessianmatrix at each iteration and the convergence may not bereached if the Hessian matrix is ill-conditioned or nonposi-tive definiteTherefore we useGauss-Newtonmethod whichis an improved version of Newtonrsquos method for finding aminimum of a function [47] The basic idea is to use theTaylor series expansion to approximate nonlinear regressionmodel and correct the solution through iteration This algo-rithm is robust and has a good convergence rateThe iterationequation of Gauss-Newton method can be expressed as
Pressure at each node flow rate inside the fracture and fluxesalong the fractures can be obtained through iteration Thusdimensionless pressure at any point in the reservoir can becalculated via (7) If the bottom-hole pressure is given we canget the flow rate in the same way The flow rate behavior indifferent conditions can be obtained consequently
6 Geofluids
qm+11
qm2
qpi
qpi+1
(a)
qm+11
qm2
(b)
Figure 6 Sketch of mass balance at intersection nodes (a) Intersection node between fractures and wellbore (b) Intersection node betweenfractures (Black line denotes wellbore red lines represent hydraulic fracture and blue line is natural fracture)
Pressure derivativePressure derivativePressure derivative m = 3
Figure 7 Effect of discretized number per fracture on the simula-tion resultm is the number of discrete panels in each fracture
23 Model Validation While solving for the numerical solu-tion fractures are discretized into several panels and toensure that we obtain correct numerical solution it is nec-essary to probe the grid independence with respect to thenumber of panels used Figure 7 shows the grid independenceanalysis for a horizontal well with three transverse hydraulicfractures where 119898 is the number of panels that a fracture isdiscretized into We can conclude that the result we obtainedfrom our model is independent of segment when119898 is greaterthan 1
CMG a commercial reservoir numerical simulator [48]was employed to validate ourmodel In this article the IMEXmodule in CMG which is a conventional three-phase black-oil simulator is utilized to make the comparison Interestedreaders may find the governing equations of this modulefrom the textbooks on reservoir numerical simulation Theconventional Cartesian grids are employed and the totalnumber of cells is 121 times 121 times 5 and the basic parameters used
12
14
16
18
20
22
24
26
28
30
CMGOur model
100 101 102 103
t (d)
pwf
(MPa
)
Figure 8 Bottom-hole pressure obtained using our proposedmodeland CMG
in the validation are listed in Table 2 The outer boundaryof our simulation domain is impermeable Meanwhile to beconsistent with the assumptions of our proposed model wemaintain the oil production rate constant in the simulationFigure 8 compares the bottom-hole pressure obtained fromCMG simulator and our model and it shows a good matchduring initial stage Figure 9 shows the comparison ofreservoir pressure from CMGwith our model and except forthe pressures near fractures we see a good match betweenthe two The reason for this slight difference around thefractures is because in CMG the fluid flows directly fromthe reservoir to the horizontal well however many studies[27 49 50] have reported that the correct representationof fluid flow must consider intersection and activation ofpreexisting natural fractures with hydraulic fractures whichwill create a complex fracture networkThis representation ofcomplex fracture network can be readily incorporated in ourmodel however it is an extremely challenging task to do thatin CMG
Geofluids 7
140
156
172
188
204
220
236
252
268
284
300
(MPa)
(a)14
156
172
188
204
22
236
252
268
284
30
(MPa)
(b)
Figure 9 Pressure distribution of the reservoir (a) CMG simulator (b) Our model The horizontal line represents the well
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 2 (a)Microseismic data showing the complex fractures in Jimsar sag Xinjiang oilfield (b) Schematic of a fracture networkThe blackred and blue lines represent the wellbore hydraulic fractures and natural fractures respectively
2 Methodology
21 Mathematical Model The reservoir after fracturingincludes four regions the matrix the natural fracture net-work hydraulic fractures and the horizontal wellboreThere-fore the fluid flow in the reservoir consists of (1) fluid flowfrom the reservoir to the hydraulic fractures ((i) in Fig-ure 2(b)) (2) fluid flow from the reservoir to the naturalfractures ((ii) in Figure 2(b)) (3) fluid flow from natural frac-ture to hydraulic fractures ((iii) in Figure 2(b)) (4) fluid flowamong natural fractures ((iv) in Figure 2(b)) (5) fluid formhydraulic fractures to the horizontal wellbore ((v) in Fig-ure 2(b)) (6) fluid flow in the horizontal wellbore ((vi) in Fig-ure 2(b))These six types of flow can be categorized into threegroups reservoir flow ((i) and (ii) in Figure 2(b)) fractureflow ((iii) (iv) and (v) in Figure 2(b)) and wellbore flow ((vi)in Figure 2(b))
Following assumptions are made
(1) Reservoir is isotropic homogeneous box-shapedand of uniform thickness with impermeable bound-aries
(2) Fluid in the reservoir is single-phase and slightly com-pressible and its compressibility and viscosity are con-stant
(3) Fractures are rectangular and vertical The flux rate isuniform along the fracture
(4) Horizontal well is parallel to the upper and lowerboundary of the reservoir
(5) Effect of gravity is neglected
First we define some dimensionless parameters for gen-erality The dimensionless pressure and time can be definedas
119901119863 = 2120587120588119896ℎ119876120583 (119901119894 minus 119901) 119905119863 = 1198961199051206011205831198621199051198712
(1)
The dimensionless flow rate and influx rate can be defined as
211 Reservoir Flow We use Greenrsquos function that has beenfrequently applied to solve problems of transient flow sinceits usage was first explored in well testing by Gringartenand Ramey Jr [10] Based on the Newman product methodinstantaneous source functions of fracture panels can beobtained therefore the pressure response at any point in thereservoir from one fracture panel can be expressed as follows(see Appendix for further details)
Δ119901 (119909 119910 119911 119905) = 119901119894 minus 119901 (119909 119910 119911 119905)= 1120601119862119905 int
where 119878119895(119909 119910 119911 120591) is the instantaneous plane source functionof the jth panel and 119901119894 is the initial reservoir pressure whichis assumed to be uniformly distributed in the reservoir
For fracture 119895 rotated at any horizontal angle to the wellthe plane source function is
exp(minus119899212058721205781199091199051199092119890 ) cos 1198991205871199091015840119909119890 cos 119899120587119909119909119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199101199051199102119890 ) cos 1198991205871199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890]
4 Geofluids
sdot [1 + 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(5)
By using the superposition principle pressure at any point inthe reservoir at a given time can be given by
Therefore pressure map can be obtained at a given time
212 Fracture Flow We assume that each hydraulic fractureis of finite-conductivity and fluid flow inside the fractureis one dimensional [17 39ndash41] We use Darcyrsquos equation todescribe the fluid flow in fractures For the jth panel thepressure at any point in the fracture is (see Appendix forfurther details)
1199011198951 minus 119901119895119898= int1199091198951198981199091198951
( 120583120588119896119891119887119891ℎ)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(8)
Then the dimensionless form is
119901119895119898119863 minus 1199011198951119863= 2120587119896119871119896119891119887119891 int
119909119895119898119863
1199091198951119863
[1199021198951119863 + 119902119891119895119863 (119909119863 minus 1199091198951119863)] 119889119909119863 (9)
Equations (8) and (9) can be applied to both hydraulicfractures and natural fractures
It is worth noting that fluid flow from the fracturesto the horizontal wellbore is radial in the near-well region(Figure 3) In order to minimize the error caused by linearflow approximation a choke flow factor is introduced [42]
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (10)
where 119903119908 is the radius of the horizontal well and 119887119891 is thehydraulic fracture width
213 Time-Dependent Conductivity Proppants are oftenpumped into the formation to maintain the fracture con-ductivity However the proppant particles usually break and
Figure 3 Sketch showing the change of flow pattern from linear toradial
embed which causes the fracture conductivity to reduce untilequilibrium is established Montgomery and Steanson sug-gested that there is a logarithmic relation between hydraulicfracture conductivity and time for 1020 Sand and 2040 Sand(Figure 4(a)) [43] Other tests also show logarithmic relationbetween fracture conductivity and time including Lanzhousand from China with diameter of 045sim090mm [44] andsand from Shanshan oil field in China (Figure 4(c)) [45]Other kinds of proppants such as ceramic proppants withdiameter of 045sim09mm and Lanzhou sand with diameterof 09sim125mm show similar correlations but the slope of thecurve also referred to as the decline coefficient for each prop-pant is differentThe decline coefficient decreases in the orderof Lanzhou sand with diameter of 090sim125mm the ceramicproppant and the Lanzhou sand with smaller diameter
Based on the trend of observations in Figure 4 we usethe following model to describe the variation of fractureconductivity as a function of time
119862119891 = 1198621198910 (1 minus 120573 lg 1199051199050) (11)
where 119862119891 is the fracture conductivity at time 119905 1198621198910 is theinitial conductivity and 120573 is the decline coefficient 119862119891 in(11) is substituted by 119896119891 sdot 119887119891 in (8) and (9) It is assumedthat when the effect of hydraulic fracture disappears totallythe fracture conductivity is equal to the product of formationpermeability and fracture width
214 Wellbore Flow Previous studies assumed horizontalwellbore as infinite-conductivity pipe [15 16] which implic-itly assumes no pressure drop along the horizontal wellbecause of uniform pressure distribution along the wellboreTo examine the validity of this approximation we include thewellbore pressure drop in the present model The pressuredecrease along the wellbore consists of frictional losses andacceleration losses (Figure 5) Based on the Darcy-WeisbachEquation [46] the fictional pressure drop can be expressed as
where 119891119894 is the frictional coefficient and Δ119897119894 is the length ofthe wellbore segment For each segment of the wellbore fluidflow from the hydraulic fractures to the wellbore will cause an
Geofluids 5
1 10 100
200
300
02
04
06
08
10
Relat
ive f
ract
ure c
ondu
ctiv
ity
Time (d)Experiment data case 1Fitting curve case 1Experiment data case 2Fitting curve case 2
(a)
01 1 10 100
02
04
06
08
10
Time (d)Re
lativ
e fra
ctur
e con
duct
ivity
Experiment dataFitting curve
(b)
1 10 100
1000
00
02
04
06
08
10
Time (d)Field dataFitting curve
Relat
ive f
ract
ure c
ondu
ctiv
ity
(c)
Figure 4 Relation between relative fracture conductivity and time (a) Montgomery and Steanson (1985) tested two samples with differentkind of proppants for 9 months case 1 denotes 1020 Sand at 250∘F and case 2 denotes 2040 Sand at 75∘F (b) Experiment conducted by Yu(1987) (c) Field data from Shanshan oilfield
Flow direction
pＩＯＮAi pＣＨAi
iDiΔli
Figure 5 Sketch of microsegment in horizontal well
increase of the flow rate This will result in the change of themomentum of the fluid which leads to the acceleration pres-sure drop
Δ119901acce = 120588119894 (V2out minus V2in) (13)
Dimensionless form of the pressure drop along the well boreis given by (see Appendix for further details)
119901119863119894 minus 119901119863119894+1= 2120587119896ℎ1198761205831198602 (1199022119901119863119895 minus 1199022119901119863119895+1 minus 119891119895Δ1198971198952119863 1199022119901119863119895) (14)
215 Coupling Relationship Due to the pressure continuityat the center of each fracture the pressure response obtainedfrom the reservoir flow should be consistent with the fractureflow Therefore we have
1199011198951198881198631 and1199011198951198881198632 are given by (7) and (9) respectivelyThemassbalance is applied to both the intersection nodes between
fractures and the intersection nodes of fractures and wellbore(Figure 6) Therefore for each node the inflow of fluid mustbe equal to the outflow of the fluid
119902in119863 = 119902out119863 (16)
For the intersection nodes of fractures and wellbore theinflow and outflow are taken into account between the frac-tures and between the fractures and the wellbore pipe
22 Computational Approach From (17) we obtain 119899V equa-tions at the nodes for mass balance 119899119901 equations for pressuredrop along the fractures and 119899119901 equations for pressure con-tinuity in the panel centers resulting in 119899V + 2119899119901 nonlinearequations that need to be solved Newtonrsquos method has beenwidely used to solve systems of equations because of its quickconvergence However it requires the inverse of Hessianmatrix at each iteration and the convergence may not bereached if the Hessian matrix is ill-conditioned or nonposi-tive definiteTherefore we useGauss-Newtonmethod whichis an improved version of Newtonrsquos method for finding aminimum of a function [47] The basic idea is to use theTaylor series expansion to approximate nonlinear regressionmodel and correct the solution through iteration This algo-rithm is robust and has a good convergence rateThe iterationequation of Gauss-Newton method can be expressed as
Pressure at each node flow rate inside the fracture and fluxesalong the fractures can be obtained through iteration Thusdimensionless pressure at any point in the reservoir can becalculated via (7) If the bottom-hole pressure is given we canget the flow rate in the same way The flow rate behavior indifferent conditions can be obtained consequently
6 Geofluids
qm+11
qm2
qpi
qpi+1
(a)
qm+11
qm2
(b)
Figure 6 Sketch of mass balance at intersection nodes (a) Intersection node between fractures and wellbore (b) Intersection node betweenfractures (Black line denotes wellbore red lines represent hydraulic fracture and blue line is natural fracture)
Pressure derivativePressure derivativePressure derivative m = 3
Figure 7 Effect of discretized number per fracture on the simula-tion resultm is the number of discrete panels in each fracture
23 Model Validation While solving for the numerical solu-tion fractures are discretized into several panels and toensure that we obtain correct numerical solution it is nec-essary to probe the grid independence with respect to thenumber of panels used Figure 7 shows the grid independenceanalysis for a horizontal well with three transverse hydraulicfractures where 119898 is the number of panels that a fracture isdiscretized into We can conclude that the result we obtainedfrom our model is independent of segment when119898 is greaterthan 1
CMG a commercial reservoir numerical simulator [48]was employed to validate ourmodel In this article the IMEXmodule in CMG which is a conventional three-phase black-oil simulator is utilized to make the comparison Interestedreaders may find the governing equations of this modulefrom the textbooks on reservoir numerical simulation Theconventional Cartesian grids are employed and the totalnumber of cells is 121 times 121 times 5 and the basic parameters used
12
14
16
18
20
22
24
26
28
30
CMGOur model
100 101 102 103
t (d)
pwf
(MPa
)
Figure 8 Bottom-hole pressure obtained using our proposedmodeland CMG
in the validation are listed in Table 2 The outer boundaryof our simulation domain is impermeable Meanwhile to beconsistent with the assumptions of our proposed model wemaintain the oil production rate constant in the simulationFigure 8 compares the bottom-hole pressure obtained fromCMG simulator and our model and it shows a good matchduring initial stage Figure 9 shows the comparison ofreservoir pressure from CMGwith our model and except forthe pressures near fractures we see a good match betweenthe two The reason for this slight difference around thefractures is because in CMG the fluid flows directly fromthe reservoir to the horizontal well however many studies[27 49 50] have reported that the correct representationof fluid flow must consider intersection and activation ofpreexisting natural fractures with hydraulic fractures whichwill create a complex fracture networkThis representation ofcomplex fracture network can be readily incorporated in ourmodel however it is an extremely challenging task to do thatin CMG
Geofluids 7
140
156
172
188
204
220
236
252
268
284
300
(MPa)
(a)14
156
172
188
204
22
236
252
268
284
30
(MPa)
(b)
Figure 9 Pressure distribution of the reservoir (a) CMG simulator (b) Our model The horizontal line represents the well
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(5)
By using the superposition principle pressure at any point inthe reservoir at a given time can be given by
Therefore pressure map can be obtained at a given time
212 Fracture Flow We assume that each hydraulic fractureis of finite-conductivity and fluid flow inside the fractureis one dimensional [17 39ndash41] We use Darcyrsquos equation todescribe the fluid flow in fractures For the jth panel thepressure at any point in the fracture is (see Appendix forfurther details)
1199011198951 minus 119901119895119898= int1199091198951198981199091198951
( 120583120588119896119891119887119891ℎ)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(8)
Then the dimensionless form is
119901119895119898119863 minus 1199011198951119863= 2120587119896119871119896119891119887119891 int
119909119895119898119863
1199091198951119863
[1199021198951119863 + 119902119891119895119863 (119909119863 minus 1199091198951119863)] 119889119909119863 (9)
Equations (8) and (9) can be applied to both hydraulicfractures and natural fractures
It is worth noting that fluid flow from the fracturesto the horizontal wellbore is radial in the near-well region(Figure 3) In order to minimize the error caused by linearflow approximation a choke flow factor is introduced [42]
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (10)
where 119903119908 is the radius of the horizontal well and 119887119891 is thehydraulic fracture width
213 Time-Dependent Conductivity Proppants are oftenpumped into the formation to maintain the fracture con-ductivity However the proppant particles usually break and
Figure 3 Sketch showing the change of flow pattern from linear toradial
embed which causes the fracture conductivity to reduce untilequilibrium is established Montgomery and Steanson sug-gested that there is a logarithmic relation between hydraulicfracture conductivity and time for 1020 Sand and 2040 Sand(Figure 4(a)) [43] Other tests also show logarithmic relationbetween fracture conductivity and time including Lanzhousand from China with diameter of 045sim090mm [44] andsand from Shanshan oil field in China (Figure 4(c)) [45]Other kinds of proppants such as ceramic proppants withdiameter of 045sim09mm and Lanzhou sand with diameterof 09sim125mm show similar correlations but the slope of thecurve also referred to as the decline coefficient for each prop-pant is differentThe decline coefficient decreases in the orderof Lanzhou sand with diameter of 090sim125mm the ceramicproppant and the Lanzhou sand with smaller diameter
Based on the trend of observations in Figure 4 we usethe following model to describe the variation of fractureconductivity as a function of time
119862119891 = 1198621198910 (1 minus 120573 lg 1199051199050) (11)
where 119862119891 is the fracture conductivity at time 119905 1198621198910 is theinitial conductivity and 120573 is the decline coefficient 119862119891 in(11) is substituted by 119896119891 sdot 119887119891 in (8) and (9) It is assumedthat when the effect of hydraulic fracture disappears totallythe fracture conductivity is equal to the product of formationpermeability and fracture width
214 Wellbore Flow Previous studies assumed horizontalwellbore as infinite-conductivity pipe [15 16] which implic-itly assumes no pressure drop along the horizontal wellbecause of uniform pressure distribution along the wellboreTo examine the validity of this approximation we include thewellbore pressure drop in the present model The pressuredecrease along the wellbore consists of frictional losses andacceleration losses (Figure 5) Based on the Darcy-WeisbachEquation [46] the fictional pressure drop can be expressed as
where 119891119894 is the frictional coefficient and Δ119897119894 is the length ofthe wellbore segment For each segment of the wellbore fluidflow from the hydraulic fractures to the wellbore will cause an
Geofluids 5
1 10 100
200
300
02
04
06
08
10
Relat
ive f
ract
ure c
ondu
ctiv
ity
Time (d)Experiment data case 1Fitting curve case 1Experiment data case 2Fitting curve case 2
(a)
01 1 10 100
02
04
06
08
10
Time (d)Re
lativ
e fra
ctur
e con
duct
ivity
Experiment dataFitting curve
(b)
1 10 100
1000
00
02
04
06
08
10
Time (d)Field dataFitting curve
Relat
ive f
ract
ure c
ondu
ctiv
ity
(c)
Figure 4 Relation between relative fracture conductivity and time (a) Montgomery and Steanson (1985) tested two samples with differentkind of proppants for 9 months case 1 denotes 1020 Sand at 250∘F and case 2 denotes 2040 Sand at 75∘F (b) Experiment conducted by Yu(1987) (c) Field data from Shanshan oilfield
Flow direction
pＩＯＮAi pＣＨAi
iDiΔli
Figure 5 Sketch of microsegment in horizontal well
increase of the flow rate This will result in the change of themomentum of the fluid which leads to the acceleration pres-sure drop
Δ119901acce = 120588119894 (V2out minus V2in) (13)
Dimensionless form of the pressure drop along the well boreis given by (see Appendix for further details)
119901119863119894 minus 119901119863119894+1= 2120587119896ℎ1198761205831198602 (1199022119901119863119895 minus 1199022119901119863119895+1 minus 119891119895Δ1198971198952119863 1199022119901119863119895) (14)
215 Coupling Relationship Due to the pressure continuityat the center of each fracture the pressure response obtainedfrom the reservoir flow should be consistent with the fractureflow Therefore we have
1199011198951198881198631 and1199011198951198881198632 are given by (7) and (9) respectivelyThemassbalance is applied to both the intersection nodes between
fractures and the intersection nodes of fractures and wellbore(Figure 6) Therefore for each node the inflow of fluid mustbe equal to the outflow of the fluid
119902in119863 = 119902out119863 (16)
For the intersection nodes of fractures and wellbore theinflow and outflow are taken into account between the frac-tures and between the fractures and the wellbore pipe
22 Computational Approach From (17) we obtain 119899V equa-tions at the nodes for mass balance 119899119901 equations for pressuredrop along the fractures and 119899119901 equations for pressure con-tinuity in the panel centers resulting in 119899V + 2119899119901 nonlinearequations that need to be solved Newtonrsquos method has beenwidely used to solve systems of equations because of its quickconvergence However it requires the inverse of Hessianmatrix at each iteration and the convergence may not bereached if the Hessian matrix is ill-conditioned or nonposi-tive definiteTherefore we useGauss-Newtonmethod whichis an improved version of Newtonrsquos method for finding aminimum of a function [47] The basic idea is to use theTaylor series expansion to approximate nonlinear regressionmodel and correct the solution through iteration This algo-rithm is robust and has a good convergence rateThe iterationequation of Gauss-Newton method can be expressed as
Pressure at each node flow rate inside the fracture and fluxesalong the fractures can be obtained through iteration Thusdimensionless pressure at any point in the reservoir can becalculated via (7) If the bottom-hole pressure is given we canget the flow rate in the same way The flow rate behavior indifferent conditions can be obtained consequently
6 Geofluids
qm+11
qm2
qpi
qpi+1
(a)
qm+11
qm2
(b)
Figure 6 Sketch of mass balance at intersection nodes (a) Intersection node between fractures and wellbore (b) Intersection node betweenfractures (Black line denotes wellbore red lines represent hydraulic fracture and blue line is natural fracture)
Pressure derivativePressure derivativePressure derivative m = 3
Figure 7 Effect of discretized number per fracture on the simula-tion resultm is the number of discrete panels in each fracture
23 Model Validation While solving for the numerical solu-tion fractures are discretized into several panels and toensure that we obtain correct numerical solution it is nec-essary to probe the grid independence with respect to thenumber of panels used Figure 7 shows the grid independenceanalysis for a horizontal well with three transverse hydraulicfractures where 119898 is the number of panels that a fracture isdiscretized into We can conclude that the result we obtainedfrom our model is independent of segment when119898 is greaterthan 1
CMG a commercial reservoir numerical simulator [48]was employed to validate ourmodel In this article the IMEXmodule in CMG which is a conventional three-phase black-oil simulator is utilized to make the comparison Interestedreaders may find the governing equations of this modulefrom the textbooks on reservoir numerical simulation Theconventional Cartesian grids are employed and the totalnumber of cells is 121 times 121 times 5 and the basic parameters used
12
14
16
18
20
22
24
26
28
30
CMGOur model
100 101 102 103
t (d)
pwf
(MPa
)
Figure 8 Bottom-hole pressure obtained using our proposedmodeland CMG
in the validation are listed in Table 2 The outer boundaryof our simulation domain is impermeable Meanwhile to beconsistent with the assumptions of our proposed model wemaintain the oil production rate constant in the simulationFigure 8 compares the bottom-hole pressure obtained fromCMG simulator and our model and it shows a good matchduring initial stage Figure 9 shows the comparison ofreservoir pressure from CMGwith our model and except forthe pressures near fractures we see a good match betweenthe two The reason for this slight difference around thefractures is because in CMG the fluid flows directly fromthe reservoir to the horizontal well however many studies[27 49 50] have reported that the correct representationof fluid flow must consider intersection and activation ofpreexisting natural fractures with hydraulic fractures whichwill create a complex fracture networkThis representation ofcomplex fracture network can be readily incorporated in ourmodel however it is an extremely challenging task to do thatin CMG
Geofluids 7
140
156
172
188
204
220
236
252
268
284
300
(MPa)
(a)14
156
172
188
204
22
236
252
268
284
30
(MPa)
(b)
Figure 9 Pressure distribution of the reservoir (a) CMG simulator (b) Our model The horizontal line represents the well
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Time (d)Experiment data case 1Fitting curve case 1Experiment data case 2Fitting curve case 2
(a)
01 1 10 100
02
04
06
08
10
Time (d)Re
lativ
e fra
ctur
e con
duct
ivity
Experiment dataFitting curve
(b)
1 10 100
1000
00
02
04
06
08
10
Time (d)Field dataFitting curve
Relat
ive f
ract
ure c
ondu
ctiv
ity
(c)
Figure 4 Relation between relative fracture conductivity and time (a) Montgomery and Steanson (1985) tested two samples with differentkind of proppants for 9 months case 1 denotes 1020 Sand at 250∘F and case 2 denotes 2040 Sand at 75∘F (b) Experiment conducted by Yu(1987) (c) Field data from Shanshan oilfield
Flow direction
pＩＯＮAi pＣＨAi
iDiΔli
Figure 5 Sketch of microsegment in horizontal well
increase of the flow rate This will result in the change of themomentum of the fluid which leads to the acceleration pres-sure drop
Δ119901acce = 120588119894 (V2out minus V2in) (13)
Dimensionless form of the pressure drop along the well boreis given by (see Appendix for further details)
119901119863119894 minus 119901119863119894+1= 2120587119896ℎ1198761205831198602 (1199022119901119863119895 minus 1199022119901119863119895+1 minus 119891119895Δ1198971198952119863 1199022119901119863119895) (14)
215 Coupling Relationship Due to the pressure continuityat the center of each fracture the pressure response obtainedfrom the reservoir flow should be consistent with the fractureflow Therefore we have
1199011198951198881198631 and1199011198951198881198632 are given by (7) and (9) respectivelyThemassbalance is applied to both the intersection nodes between
fractures and the intersection nodes of fractures and wellbore(Figure 6) Therefore for each node the inflow of fluid mustbe equal to the outflow of the fluid
119902in119863 = 119902out119863 (16)
For the intersection nodes of fractures and wellbore theinflow and outflow are taken into account between the frac-tures and between the fractures and the wellbore pipe
22 Computational Approach From (17) we obtain 119899V equa-tions at the nodes for mass balance 119899119901 equations for pressuredrop along the fractures and 119899119901 equations for pressure con-tinuity in the panel centers resulting in 119899V + 2119899119901 nonlinearequations that need to be solved Newtonrsquos method has beenwidely used to solve systems of equations because of its quickconvergence However it requires the inverse of Hessianmatrix at each iteration and the convergence may not bereached if the Hessian matrix is ill-conditioned or nonposi-tive definiteTherefore we useGauss-Newtonmethod whichis an improved version of Newtonrsquos method for finding aminimum of a function [47] The basic idea is to use theTaylor series expansion to approximate nonlinear regressionmodel and correct the solution through iteration This algo-rithm is robust and has a good convergence rateThe iterationequation of Gauss-Newton method can be expressed as
Pressure at each node flow rate inside the fracture and fluxesalong the fractures can be obtained through iteration Thusdimensionless pressure at any point in the reservoir can becalculated via (7) If the bottom-hole pressure is given we canget the flow rate in the same way The flow rate behavior indifferent conditions can be obtained consequently
6 Geofluids
qm+11
qm2
qpi
qpi+1
(a)
qm+11
qm2
(b)
Figure 6 Sketch of mass balance at intersection nodes (a) Intersection node between fractures and wellbore (b) Intersection node betweenfractures (Black line denotes wellbore red lines represent hydraulic fracture and blue line is natural fracture)
Pressure derivativePressure derivativePressure derivative m = 3
Figure 7 Effect of discretized number per fracture on the simula-tion resultm is the number of discrete panels in each fracture
23 Model Validation While solving for the numerical solu-tion fractures are discretized into several panels and toensure that we obtain correct numerical solution it is nec-essary to probe the grid independence with respect to thenumber of panels used Figure 7 shows the grid independenceanalysis for a horizontal well with three transverse hydraulicfractures where 119898 is the number of panels that a fracture isdiscretized into We can conclude that the result we obtainedfrom our model is independent of segment when119898 is greaterthan 1
CMG a commercial reservoir numerical simulator [48]was employed to validate ourmodel In this article the IMEXmodule in CMG which is a conventional three-phase black-oil simulator is utilized to make the comparison Interestedreaders may find the governing equations of this modulefrom the textbooks on reservoir numerical simulation Theconventional Cartesian grids are employed and the totalnumber of cells is 121 times 121 times 5 and the basic parameters used
12
14
16
18
20
22
24
26
28
30
CMGOur model
100 101 102 103
t (d)
pwf
(MPa
)
Figure 8 Bottom-hole pressure obtained using our proposedmodeland CMG
in the validation are listed in Table 2 The outer boundaryof our simulation domain is impermeable Meanwhile to beconsistent with the assumptions of our proposed model wemaintain the oil production rate constant in the simulationFigure 8 compares the bottom-hole pressure obtained fromCMG simulator and our model and it shows a good matchduring initial stage Figure 9 shows the comparison ofreservoir pressure from CMGwith our model and except forthe pressures near fractures we see a good match betweenthe two The reason for this slight difference around thefractures is because in CMG the fluid flows directly fromthe reservoir to the horizontal well however many studies[27 49 50] have reported that the correct representationof fluid flow must consider intersection and activation ofpreexisting natural fractures with hydraulic fractures whichwill create a complex fracture networkThis representation ofcomplex fracture network can be readily incorporated in ourmodel however it is an extremely challenging task to do thatin CMG
Geofluids 7
140
156
172
188
204
220
236
252
268
284
300
(MPa)
(a)14
156
172
188
204
22
236
252
268
284
30
(MPa)
(b)
Figure 9 Pressure distribution of the reservoir (a) CMG simulator (b) Our model The horizontal line represents the well
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 6 Sketch of mass balance at intersection nodes (a) Intersection node between fractures and wellbore (b) Intersection node betweenfractures (Black line denotes wellbore red lines represent hydraulic fracture and blue line is natural fracture)
Pressure derivativePressure derivativePressure derivative m = 3
Figure 7 Effect of discretized number per fracture on the simula-tion resultm is the number of discrete panels in each fracture
23 Model Validation While solving for the numerical solu-tion fractures are discretized into several panels and toensure that we obtain correct numerical solution it is nec-essary to probe the grid independence with respect to thenumber of panels used Figure 7 shows the grid independenceanalysis for a horizontal well with three transverse hydraulicfractures where 119898 is the number of panels that a fracture isdiscretized into We can conclude that the result we obtainedfrom our model is independent of segment when119898 is greaterthan 1
CMG a commercial reservoir numerical simulator [48]was employed to validate ourmodel In this article the IMEXmodule in CMG which is a conventional three-phase black-oil simulator is utilized to make the comparison Interestedreaders may find the governing equations of this modulefrom the textbooks on reservoir numerical simulation Theconventional Cartesian grids are employed and the totalnumber of cells is 121 times 121 times 5 and the basic parameters used
12
14
16
18
20
22
24
26
28
30
CMGOur model
100 101 102 103
t (d)
pwf
(MPa
)
Figure 8 Bottom-hole pressure obtained using our proposedmodeland CMG
in the validation are listed in Table 2 The outer boundaryof our simulation domain is impermeable Meanwhile to beconsistent with the assumptions of our proposed model wemaintain the oil production rate constant in the simulationFigure 8 compares the bottom-hole pressure obtained fromCMG simulator and our model and it shows a good matchduring initial stage Figure 9 shows the comparison ofreservoir pressure from CMGwith our model and except forthe pressures near fractures we see a good match betweenthe two The reason for this slight difference around thefractures is because in CMG the fluid flows directly fromthe reservoir to the horizontal well however many studies[27 49 50] have reported that the correct representationof fluid flow must consider intersection and activation ofpreexisting natural fractures with hydraulic fractures whichwill create a complex fracture networkThis representation ofcomplex fracture network can be readily incorporated in ourmodel however it is an extremely challenging task to do thatin CMG
Geofluids 7
140
156
172
188
204
220
236
252
268
284
300
(MPa)
(a)14
156
172
188
204
22
236
252
268
284
30
(MPa)
(b)
Figure 9 Pressure distribution of the reservoir (a) CMG simulator (b) Our model The horizontal line represents the well
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 10 Pressure response of multistage fractured horizontal well(fracture number 119899 = 3 dimensionless fracture half-length 119909119891119863 = 1dimensionless fracture spacing 119889119863 = 8 and dimensionless reservoirlength 119909119890119863 = 240)
3 Results and Discussions
31 Flow Regime Analysis The transient behavior can beshown by type curves which is employed to obtain the char-acteristic of formation and the reservoir fluids and to figureout different flow regimes Figure 10 shows the dimensionlesswell pressure and its derivative for a horizontal well with threefractures The transient behavior can be divided into severalflow periods
(1) Linear Flow (Figure 11(a)) Fluid flows linearly from thereservoir to the fracture and each fracture behaves inde-pendently Both the slope of dimensionless pressure andderivative is 12 in this stage
Table 2 Basic parameters for model validation
Properties ValueReservoir permeability 119896 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length 119909119890 m 600Formation thickness ℎ m 20Total compressibility 119862119905 MPaminus1 4 times 10minus3
Initial pressure pi MPa 30Fluid viscosity 120583 mPasdots 20Fluid density 120588 kgm3 900Fracture conductivity 119862119889119891 120583m2sdotm 05Fracture half-length 119909119891 m 75Fracture spacing 119889 m 100Production rate 119876 td 432
(2) Early Radial Flow (Figure 11(b)) An early radial flowoccurs around each fracture after the linear flow This periodmainly depends on the fracture length and fracture spacingBesides in this period fractures still behave independentlyThe characteristic of this stage is a horizontal line of 1(2119873)in pressure derivative curve (119873 is the fracture stage) Wecan figure out that the value of the horizontal line plateau is16
(3) Biradial Flow (Figure 11(c)) Fractures interact with eachother and flow becomes elliptical to the wellbore The slopeof pressure derivative is 036
(4) Pseudo Radial Flow (Figure 11(d)) Fluid flows to thefracture-well system appear to be radial and flow acrossthe outermost elements plays important part The pressurederivative curve demonstrates a horizontal line of 05
8 Geofluids
(a) (b) (c)
(d)
Figure 11 Flow regimes for a multistaged fractured horizontal well (a) linear flow (b) early radial flow (c) biradial flow and (d) pseudoradial flow
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
kf middot w = 10 Ｇ2middotcmkf middot w = 25 Ｇ2middotcm
kf middot w = 35 Ｇ2middotcmkf middot w = 50 Ｇ2middotcm
(b)
Figure 12 Effect of initial fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracture half-length119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 and dimensionless reservoir length 119909119890119863 = 120) (a) Pressure and derivative curve (b)Dimensionless production rate
(5) Boundary Dominated Flow In the closed system theflow will reach pseudo-steady state The pressure curve andderivate tend to merge and the slope equals 1
32 Effect of Initial Fracture Conductivity Figure 12 illustratesthe effect of initial conductivity on the pressure transient
response and production rateThis figure shows that increas-ing hydraulic fracture conductivity results in an increase inwell productivity however the incremental benefit decreasesas the fracture conductivity increases Figure 12 also showsthat variations in the dimensionless pressure and the pro-duction rate tend to disappear as the flow regime approaches
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 13 Effect of time-dependent fracture conductivity on pressure and rate behavior (fracture number 119899 = 2 dimensionless fracturehalf-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 and decline coefficient 120573 = 011)(a) Dimensionless pressure and derivative curve (b) Dimensionless production rate
pseudo radial flow The reason why the variations in thesetwo parameters disappear is because the flow at late times isdominated by the outer reservoir and therefore the effect offracture conductivity will be extremely small Therefore toutilize the potential of the hydraulic fractures we must delaythe occurrence of pseudo radial flow and that can be done byoptimal well placement
Although it is well documented that the fracture conduc-tivity decreases rapidly at first and tends to converge asym-ptotically [31ndash34 37] previous studies always assumed thathydraulic fractures do not vary with time Below we analyzethe difference between time-independent and time-depend-ent fractures As discussed earlier the conductivity and hencethe productivity of the horizontal well decreases with timeTherefore the pressure response of the well is larger thanthe time-independent situation as suggested by Figure 13(a)The flow rate of a horizontal well with time-independentconductivity fractures exceeds that with time-dependentconductivity fractures in the initial stage (Figure 13(b)) andthat could be attributed to the dominant role of fracturesDuring late times of production the fluid flow in the outerreservoir becomes dominant and negligible difference canbe observed between time-dependent and time-independentcases Therefore for the optimal exploitation of an uncon-ventional reservoir it is necessary to maintain the fractureconductivity especially in the initial stage of the production
Figure 14 shows the effect of decline coefficient on theproduction performance of the horizontal well where a largerdecline coefficient results in smaller production rate and if thedecline coefficient is too large the fracture permeability tendsto quickly approach the formation permeability which resultsin sharp decline in the production rate
Figure 14 Effect of decline coefficient on the production perfor-mance of the horizontal well (fracture number 119899 = 2 dimensionlessfracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 =067 and dimensionless reservoir length 119909119890119863 = 120)
33 Effect of Fracture Distribution The pressure differencecreated during hydraulic fracturing allows the proppants tostay in the fractures away from the horizontal well heel [51]and as a result the fracture conductivities are different thatmay further have an impact on the pressure behavior Twoscenarios of even and uneven proppant distribution can be
10 Geofluids
Heel Toe
1 2 3
(a)
Heel Toe
1 2 3
(b)
Figure 15 Sketch of proppant distribution in different fracture (a) uneven distribution (b) even distribution
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 16 Effect of unevenly distributed fracture conductivity on the pressure and production behavior (fracture number 119899 = 3dimensionless fracture half-length 119909119891119863 = 1 dimensionless fracture spacing 119889119863 = 067 dimensionless reservoir length 119909119890119863 = 120 Theconductivity of evenly distributed fracture is 0375 120583m2sdotm for unevenly distributed fracture the conductivities are 025120583m2sdotm 0375 120583m2sdotmand 05 120583m2sdotm) (a) Dimensionless pressure curve (b) Dimensionless production rate curve
seen as schematics in Figures 15(a) and 15(b) respectivelywhich are used to study the effect of proppant distribution
For unevenly distributed fractures the fracture nearestto the well toe has the largest conductivity whereas theconductivity of the intermediate fracture equals that of theevenly distributed fractures
Figure 16(a) shows the dimensionless pressure drop of ahorizontal well (at constant production rate multistage frac-tured well) with evenly distributed and unevenly distributedfractures The pressure response with unevenly distributedproppants is slightly larger than that with evenly distributedproppants at initial period but the gap disappears afterthat Result of Figure 16(b) suggests that horizontal wellwith evenly distributed proppants produces with higher rateinitially than with unevenly distributed proppantsThereforethese two results indicate that well with evenly distributedproppants performs better than that with unevenly dis-tributed proppants
Figure 17 shows the dimensionless flux rate of three frac-tures that have evenly and unevenly distributed proppants
When the fracture conductivities (or proppant distribution)are evenly distributed the dimensionless flux rate of Fractures1 and 3 increases with time but the flux rate of Fracture 2decreases sharply with time At the initial stage each fracturebehaves independently however with time the fracturesbegin to interfere with each other and the flux rate ofFractures 1 and 3 tends to increase whereas owing to the sym-metry the flux rate of intermediate fracture Fracture 2 is hin-dered If the fracture conductivities are unevenly distributedthe flux rate of the third fracture will first decrease andthen increase with the time However production rate of thefirst fracture increases steadily with time and it crosses theconductivity of the second fracture at some point
34 Effect of Fracture Geometry In hydraulic fracturing theexistent propped fractures result in the redistribution of localearth stresses Moreover microseismic measurements haveproved that there is mechanical-stress interference betweenmultiple transverse fractures The stress-shadow effect canrestrict the growth of the fracture in the middle section while
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 17 Dimensionless flux rate of each fracture (a) Fracture conductivity evenly distributed (b) Fracture conductivity unevenlydistributed
(a) (b) (c)
Figure 18 Sketch of different fracture geometries (a) Equilong type (b) Spindle-shaped type (c) Dumbbell-shaped type
promoting the growth of the fractures at the heel or the toe[49 52] therefore different fracture geometry may exist inthe reservoir Three geometries shown in Figure 18 equilongtype spindle-shaped type and dumbbell-shaped type arestudied
Figure 19 shows the dimensionless flux rate of three frac-tures from three fracture geometries of Figure 18 respectivelyFor spindle-shaped geometry the flux rate of the first andthird fracture will first decrease and then slightly increasebefore levelling off at long time (Figure 19(b)) However theflux of the second fracture always remains smaller than theother two fractures although it has a larger fracture lengthFigure 19(c) shows that for dumbbell-shaped geometry theflux rate of fractures near the heel and toewill slightly increasewith time whereas the flux of the intermediate fracture willsharply decrease with time From these results it is evidentthat no matter what the fracture geometry is the flux of theintermediate fracture always decreases with time Figure 20suggests that if the bottom-hole pressure is constant theproduction rate of equilong fracture type is higher than theothers because it has the longest effective interference fracturelength Apparently this is the reason why equilong fracturetype is usually preferred over other fracture geometries
35 Effect of Horizontal Wellbore Pressure Drop Many previ-ous studies considered the horizontal wellbore as an infinitewellbore however it has been shown that wellbore pressuredrop exists in the production [53 54] We investigatedthe effect of horizontal wellbore pressure drop on pressurebehavior as shown in Figure 21 which shows that the effectof considering wellbore pressure drop is negligible Thisobservation can be explained by the low permeability andlow flow rate in ultratight reservoir Therefore we concludethat the horizontal wellbore pressure drop can be neglectedin ultratight reservoir
36 Complex Fracture Network It is inappropriate to sim-ulate the production with biwing fracture when the char-acteristic of complex fracture network is evident [55ndash57]Figure 22 shows the schematic of a complex fracture networkthe red lines represent the hydraulic fractures the blue linesare natural fractures and black line denotes horizontal wellParameters of the reservoir fluid and the fractures are shownin Table 3 Figure 23 shows the pressure distribution of thecomplex fracture network at different times With increasingtime the area that contributes to production as well as theelastic production in a specific volume increases
12 Geofluids
006008010012014016018020022
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(a)
016018020022024026028030032034
Fractures 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(b)
005010015020025030035040
Fracture 1 and 3Fracture 2
10minus2
10minus1
100
101
102
103
104
106
10minus3
105
tD
qfD
(c)
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 19 Dimensionless flux rate of each fracture (a) Equilong type 1199091198911 = 1199091198912 = 1199091198913 = 100m (b) Spindle-shaped type 1199091198911 = 75m1199091198912 = 150m and 1199091198913 = 75m (c) Dumbbell-shaped type 1199091198911 = 120m 1199091198912 = 60m and 1199091198913 = 120m
010 100 1000 10000 1000001
5
10
15
20
tEquilongSpindle-shapedDumbbell-shaped
qD
Figure 20 Dimensionless production rate for different fractureconfigurations
4 Field Example
Well J172-H is amultistage fractured horizontal well in Jimsarsag which is a typical tight oil region located in the southeastof the Junggar Basin of China Tight oil in the sag is mainlyin Permian Lucaogou Formation which is divided into thefirst member and the second member from bottom to top(Figure 24) Sweet spots are developed in the lower parts ofboth members that is the upper sweet spot and the lowersweet spot J172-H is placed in the upper sweet spot andthe reservoir is characterized by alternating layers of finedolarenite dolomitic siltstone and fine dolomitic mudstonein the vertical direction [58] The interpretation of nuclearmagnetic log shows that the porosity of the upper sweet spotlies between 0061 and 0258 with an average of 0101 whereas
Table 3 Basic parameters for complex fracture network case
Properties ValueReservoir permeability k 120583m2 1 times 10minus4
Reservoir porosity 120601 10Reservoir length xe m 600Formation thickness h m 20Total compressibility Ct MPaminus1 276 times 10minus3
the permeability of the upper sweet spot is in the range of0001 times 10minus3 120583m2 to 0284 times 10minus3 120583m2 with an average of0012 times 10minus3 120583m2 The permeability of over 90 percent of thesamples is less than 01 times 10minus3 120583m2
J172-H went into production in September of 2012 andduring the initial stage the production rate of J172-H wasapproximately 15 times that of the adjacent vertical well Abuild-upwell test was conducted inMay of 2013 and Figure 25shows the pressure and pressure derivative data of that build-up test which was conducted for a short period Thus it canbe concluded that the data shown in Figure 25 comes fromvery beginning of the build-up test
Due to the short test and ultralow reservoir permeabilitysome characteristic responses in the middle and late timeperiod corresponding to the biradial flow and boundarydominated flow regimes featured by the slope of 036 and 1in the pressure derivative curve cannot be found from thewell test data In the regimes of wellbore storage and earlylinear flow the slopes of pressure derivative curves are equalto 1 and 05 respectivelyWematched the curves with straightlines of slope 1 and 05 and identified the flow regime from thedata as wellbore storage period and linear flow period Note
Geofluids 13
Considering pressure dropWithout considering pressure drop
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Considering pressure dropWithout considering pressure drop
tD
pD
(b)
Figure 21 Effect of horizontal wellbore pressure drop on pressure behavior (a) Pressure curve of multistage fractured horizontal well (b)An enlarged view of pressure curve within green circle
Figure 22 Schematic of a complex fracture network The blackred and blue lines represent the wellbore hydraulic fractures andnatural fractures respectively
that the effect of wellbore storage is not taken into accountin this study because our model is proposed on the basis ofGreenrsquos function in real domain and it is difficult to obtainthe derivative of pressure with respect to time in the innerboundary condition However the wellbore storage effectcan be accounted for by transforming the equations fromreal domain to the Laplace domain by using the followingequation
where 119901119908119863storageskin stands for the dimensionless pressureincluding wellbore storage and skin effects (in Laplace space)119901119908119863 is the dimensionless pressure without these effects (inLaplace space) 119904 is the Laplace variable and 119862119863 and 119878 arethe wellbore storage coefficient and skin factor respectivelyThis is the subject of future studies Therefore data from thelinear flow period is employed in our analysis A physicalexperiment was conducted to estimate the decline coefficientof hydraulic fractures and the resulting value is 0106Figure 26 shows the match for linear flow regime from
Table 4 Comparison between fitting parameter and the field data
Parameter Our model Field dataFracture half-length m 148 110sim230Fracture spacing m 81 785 (in average)Formation permeability 10minus3 120583m2 00107 0001sim0284
which we estimate the fracture and reservoir parameters InTable 4 we compare the fitting parameters fromour proposedmodel and the field data frommicroseismic monitoring Ourestimated average half-length of these fractures approximatesto 148m which lies in the range of 110ndash230m The fracturespacing obtained from our match 81m approximates to theaverage value from microseismic map Because the reservoirheterogeneity is not taken into account in our proposedmodel we only get the mean permeability of the formationwhich is fairly comparable to the well logging results
5 Conclusions
A semianalytical model was proposed to analyze the pressurebehavior of multistage fractured horizontal well in tight oilreservoirs Factors that influence the pressure behavior ofmultistage fractured horizontal well were analyzed to providea deep understanding of the pressure transient behaviorFollowing conclusions were reached
(1) Higher hydraulic fracture conductivity will result inan increase in well productivityTherefore hydraulic fractureconductivities should be optimized according to the wellperformance and investment
(2) The stimulated region has more significant impor-tance on the transient pressure and rate behavior than theouter reservoir region In order to make use of the hydraulicfractures the existence of pseudo radial flowmust be delayedthrough optimal well placement
14 Geofluids
100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(a)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(b)100 150 200 250 300 350 400 450 500
100
150
200
250
300
350
400
450
500
2252323524245252552626527275282852929530
(c)
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Figure 23 Pressure distribution of complex fracture network at different production time (a) 119905 = 100 d (b) 119905 = 500 d and (c) 119905 = 1000 d
0 4(km)
minus3000
Fault
Pinchout boundary of Lucaogou Formation
Structural contour of top boundary of the first member Lucaogou Formation (m)
minus4000 minus
3000
minus20
00
minus1000
Well Ji1 South No 1 fault
Xidi fault
Santai fault
Well Ji7 South fault
Houbaozi fault
Jimsar fault
(a)
Stra
ta
CALI
GR
SP
166
0 150
20
Dep
th (m
)
RI
RT
RXO
DEN
AC
CNL
0 1000
0 1000
0 1000
19 29
150 50
45
4050
4100
4150
4200
4250
4300
4350
minus80
Lithology
minus15
０3７Ｎ
０2）21
０2）22
０2）11
０2）12
(b)
Figure 24 Tectonic location and comprehensive logging evaluation of Lucaogou Formation in Jimsar sag
Geofluids 15
PressurePressure derivative
10minus3
10minus2
10minus1
100
101
10minus3 10minus2 10minus1 100 101 102 103
t (hour)
Δp
(MPa
)
Figure 25 Well build-up test data of J172-H
Pressure our modelPressure derivative our model
Pressure field dataPressure derivative field data
100 101 102 10310minus2
10minus1
100
101
Δp
(MPa
)
t (hour)
Figure 26 Curve match for linear flow
qj1
pj1
qj2
pj2
qfj
Figure 27 Fluid flow in panel 119895
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
(3) The comparison among different factors suggests thatfracture conductivity is the most important factor whenexploiting the full potential of a horizontal wellThedecline offracture conductivity has a tremendous influence on the wellperformance Therefore the proppant selection should begiven the first priority to maintain the fracture conductivity
(4) The interference among fractures of equilong type isthe strongestThis geometry is favorable for both fracture fluxrate and total production rate
(5) Due to the low permeability and flow rate of ultratightreservoir the effect of wellbore pressure drop is negligibletherefore the horizontal wellbore pressure drop can beignored in ultratight reservoir
Appendix
Here we provide the detailed derivation of equations thatdescribe fluid flow in reservoir fracture and wellbore
Reservoir Flow For fractures rotated at any horizontal angleto the well instantaneous point function can be obtainedthrough the superposition of source function in three dimen-sions Then the instantaneous plane function of the fracturepanel can be calculated through the integration on thefracture panel The infinite plane source function in slabreservoir with no-flow boundary is
VII (119909) = 1119909119890 [1
+ 2infinsum119899=1
exp(minus119899212058721205781199091199051199092119890 ) cos 119899120587119909119908119909119890 cos119899120587119909119909119890 ]
(A1)
where 119909119908 denotes the location of the plane source and 119909 is thevalue in 119909-direction of an arbitrary point in the reservoir Byapplying Newman method the instantaneous point functionis described as
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
sdot cos 1198991205871199101198951 + 1199091015840 tan 120579119895119910119890 cos 119899120587 119910119910119890] sdot [1
+ 2infinsum119899=1
exp(minus119899212058721205781199111199051199112119890 ) cos 1198991205871199111015840119911119890sdot cos 119899120587119911119911119890 ]119889119909
10158401198891199111015840(A3)
Fractures perpendicular to the horizontal well can be treatedas a special case of the above situation Complex double inte-gration is no longer needed in calculating the source functionof the fracture panelsThe instantaneous plane function of thefracture panel can be directly obtained through the superpo-sition of infinite slab source function and infinite plane sourcefunction in slab reservoir with no-flow boundaries
The infinite slab source function in slab reservoir with no-flow boundary is
X (119909) = 119909119891119909119890 [1 +4119909119890120587119909119891infinsum119899=1
In fracture flows fluid flow is considered to be one-dimensional flow The panel can be illustrated as Figure 27The flow rate at any point in the panel can be expressed by
119902119895 (119909) = 1199021198951 + 119902119891119895 (119909 minus 1199091198951) (A8)
By substituting (A8) to (A7) we obtain
1199011198951 minus 1199011198952= int11990911989521199091198951
( 120583120588119896119891119887119891119889)119895 [1199021198951 + 119902119891119895 (119909 minus 1199091198951)] 119889119909(A9)
The choke flow factor results from the radial flow near thewellbore entry point An additional pressure drop is takeninto account when calculating the pressure in the wellboreFor a horizontal well in the midplane of the reservoir thepressure drop is
where ℎ is the fracture height and 119896119891 and 119887119891 denote fracturepermeability and width respectivelyWhen the flow from thefracture to the reservoir is treated as linear flow the pressuredrop is
Therefore the pressure difference between radial and linearflow can be obtained by
Δ119901119904 = Δ119901119903 minus Δ119901119897 (A12)
and the choke flow factor is
119904119888 = ( ℎ2120587119909119890)[ln(ℎ2119903119908) minus
1205872 ] (A13)
Wellbore Flow Due to the influx flow to the wellbore thevelocity profile is modified The inflow expands and liftsthe boundary layer causing an increase of axial velocitybeyond the layer The axial velocity near the pipe walldecreases consequently As a consequence using no-wall-flow frictional factormay cause inaccuracy in calculating fric-tional losses In order to take the effect of wall roughness andfluid mixing into account the frictional factor corrected byOuyang et al is employed [59]
For laminar flow the frictional factor is
119891119894 = 64Re119894
(1 + 004303Re06142119908119894 ) (A14)
Geofluids 17
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
where Re119908119894 is the inflow Reynolds number which can bedescribed by Re119908119894 = 119902119904119894120588119894120587120583119894 119902119904119894 is the inflow rate per unitlength
For turbulent flow the frictional factor can be calculatedby
119891119894 = 1198910119894 (1 minus 00153Re03978119908119894 ) (A15)
where 1198910119894 is the no-wall-flow frictional factor and can beobtained by Colebrook-White correlation
1radic1198910119894 = 4 log 1120576119863 + 228 minus 4 log[
467radic1198910119894Re 120576119863 + 1] (A16)
However the no-wall-flow frictional factor is hard to getexplicitly by (A16) and iteration is needed With the aimof simplicity the equation was approximated [60] Theapproximate equation can be expressed as follows
1radic1198910119894 = 114 minus 2 log(120576119863 + 2125
Re09) (A17)
The inflow pressure is 119901in119860 119894 the outflow pressure is119901out119860 119894 and the shear force by the pipewall surface is 120591119894120587119863119894Δ119897119894the following equation is obtained consequently
119901in119860 119894 minus 119901out119860 119894 minus 120591119894120587119863119894Δ119897119894= 120588out119902outVout minus 120588in119902inVin (A18)
Equation (A18) can be rewritten as
119901in minus 119901out minus 120591119894120587119863119894119860 119894Δ119897119894 = 120588outV2out minus 120588inV2in (A19)
The right side of (A19) denotes the accelerational pressurelosses namely
Δ119901acce = 120588119894 (V2out minus V2in) (A20)
Therefore the pressure drop in the wellbore can be expressedby
119889 Fracture spacing m119891119894 Fraction coefficient dimensionlessℎ Reservoir thickness m119896 Formation permeability 120583m2119896119891 Fracture permeability 120583m2119871 Reference length m119898 Number of panels that a fracture is discretized into119899 Fracture number119873119901 Number of fracture panels119901119863 Dimensionless pressure119901119894 Initial pressure MPa1199011198951 Pressure at one end of 119895 panel MPa119901119895119888 Pressure at the center of 119895 panel MPa119876 Flow rate td119902119863 Dimensionless flow rate dimensionless119902119891119863 Dimensionless flux rate dimensionless119903119908 Well radius m119904119888 Chock flow factor dimensionless119905119863 Dimensionless time dimensionless1199050 Initial time s119909119890 Reservoir length m119910119890 Reservoir width m
119863 Dimensionless119891 Fracture119888 Center119908 Well
Disclosure
The present address for Harpreet Singh is National EnergyTechnology Laboratory Morgantown WV USA
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
Qihong Feng and Tian Xia conceived and designed thesimulations Tian Xia and Sen Wang wrote the simulationprograms Qihong Feng and Sen Wang analyzed the simu-lation results Tian Xia SenWang and Harpreet Singh wrotethe paper
Acknowledgments
This work is supported by the National Program for Fund-amental Research and Development of China(2015CB250905) the State Major Science and Technology
18 Geofluids
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
Special Projects during the 13th Five-Year Plan(2017ZX05071007) the Program for Changjiang Scholarsand Innovative Research Team in University (IRT1294)the National Postdoctoral Program for Innovative Talents(BX201600153) China Postdoctoral Science Foundation(2016M600571) and Qingdao Postdoctoral Applied ResearchProject (2016218)
References
[1] K ChengWWu S A Holditch W B Ayers and D A McVayldquoAssessment of the Distribution of Technically RecoverableResources in North American Basinsrdquo in Proceedings of theCanadian Unconventional Resources and International Petro-leum Conference Calgary Alberta Canada October 2010
[2] J D Hughes ldquoEnergy A reality check on the shale revolutionrdquoNature vol 494 no 7437 pp 307-308 2013
[3] S A Cox D M Cook K Dunek R Daniels C Jump andB Barree ldquoUnconventional resource play evaluation A look atthe bakken shale play of North Dakotardquo in Proceedings of theUnconventional Reservoirs Conference 2008 pp 204ndash217 Key-stone Colo USA February 2008
[4] JMason ldquoOil production potential of the north dakota bakkenrdquoOil amp Gas Journal vol 10 2012
[5] X Wang and J J Sheng ldquoEffect of low-velocity non-Darcy flowon well production performance in shale and tight oil reser-voirsrdquo Fuel vol 190 pp 41ndash46 2017
[6] R Liu B Li Y Jiang and N Huang ldquoReview Mathematicalexpressions for estimating equivalent permeability of rockfracture networksrdquo Hydrogeology Journal vol 24 no 7 pp1623ndash1649 2016
[7] A Javaheri H Dehghanpour and J MWood ldquoTight rock wet-tability and its relationship to other petrophysical properties AMontney case studyrdquo Journal of Earth Science vol 28 no 2 pp381ndash390 2017
[8] H Singh ldquoRepresentative elementary volume (REV) in spatio-temporal domain a method to find REV for dynamic poresrdquoJournal of Earth Science vol 28 no 2 pp 391ndash403 2017
[9] C R Clarkson ldquoProduction data analysis of unconventional gaswells review of theory and best practicesrdquo International Journalof Coal Geology vol 109-110 pp 101ndash146 2013
[10] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquoSociety of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973
[11] H Cinco-Ley and F Samaniego ldquo Transient pressure analysisfor fractured wellsrdquo Journal of Petroleum Technology vol 33 no9 pp 1749ndash1766 1981
[12] H Cinco-Ley and H-Z Meng ldquoPressure transient analysisof wells with finite conductivity vertical fractures in doubleporosity reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE-18172-MS Society of PetroleumEngineers Houston Tex USA October 1988
[13] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991
[14] GGuo R Evans andMMChang ldquoPressure-Transient Behav-ior for a Horizontal Well Intersecting Multiple Random Dis-crete Fracturesrdquo inProceedings of the SPEAnnual Technical Con-ference and Exhibition New Orleans La USA September 1994
[15] JWan andK Aziz ldquoMultiple Hydraulic Fractures inHorizontalWellsrdquo in Proceedings of the SPE Western Regional MeetingAnchorage Alaska USA May 1999
[16] E Ozkan M L Brown R S Raghavan and H Kazemi ldquoCom-parison of Fractured Horizontal-Well Performance in Conven-tional andUnconventional Reservoirsrdquo inProceedings of the SPEWestern Regional Meeting San Jose Calif USA March 2009
[17] M Brown E Ozkan R Raghavan and H Kazemi ldquoPracticalsolutions for pressure-transient responses of fractured hori-zontal wells in unconventional shale reservoirsrdquo SPE ReservoirEvaluation and Engineering vol 14 no 6 pp 663ndash676 2011
[18] S J Al Rbeawi and T Djebbar ldquoTransient Pressure Analysisof a Horizontal Well With Multiple Inclined Hydraulic Frac-tures Using Type-Curve Matchingrdquo in Proceedings of the SPEInternational Symposium and Exhibition on Formation DamageControl Lafayette La USA March 2012
[19] R N Horne and K O Temeng ldquoRelative productivities andpressure transient modeling of horizontal wells with multiplefracturesrdquo in Proceedings of the 9th Middle East Oil Show ampConference Part 2 (of 2) pp 563ndash574 Bahrain March 1995
[20] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal well in closed systemsrdquo in Proceedingsof the Canadian International Petroleum Conference PETSOC-2004-027 Alberta Canada June 2004
[21] S Yao F Zeng H Liu and G Zhao ldquoA semi-analytical modelformulti-stage fractured horizontal wellsrdquo Journal of Hydrologyvol 507 pp 201ndash212 2013
[22] W Zhou R Banerjee B Poe J Spath andMThambynayagamldquoSemianalytical production simulation of complex hydraulic-fracture networksrdquo SPE Journal vol 19 no 1 pp 6ndash18 2014
[23] W Yu ldquoDevelopment of A Semi-Analytical Model for Simula-tion of Gas Production in Shale Gas Reservoirsrdquo in Proceedingsof the Unconventional Resources Technology Conference DenverColo USA August 2014
[24] P Jia L Cheng S Huang and H Liu ldquoTransient behavior ofcomplex fracture networksrdquo Journal of Petroleum Science andEngineering vol 132 pp 1ndash17 2015
[25] M Al-Kobaisi E Ozkan andH Kazemi ldquoAHybridNumerical-Analytical Model of Finite-Conductivity Vertical FracturesIntercepted by a Horizontal Wellrdquo in Proceedings of the SPEInternational PetroleumConference inMexico Puebla Pue NMUSA November 2004
[26] C M Freeman G J Moridis D Ilk and T A Blasingame ldquoANumerical Study of Performance for Tight Gas and Shale GasReservoir Systemsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition New Orleans La USA October2009
[27] C L Cipolla E Lolon J Erdle and V S Tathed ldquoModelingWell Performance in Shale-Gas Reservoirsrdquo in Proceedings ofthe SPEEAGE Reservoir Characterization and Simulation Con-ference Abu Dhabi UAE October 2009
[28] G J Moridis T A Blasingame and C M Freeman ldquoAnalysisof mechanisms of flow in fractured tight-gas and shale-gasreservoirsrdquo in Proceedings of the Latin American and CaribbeanPetroleum Engineering Conference 2010 LACPEC 10 pp 1310ndash1331 Lima Pe USA December 2010
[29] O Olorode C M Freeman G Moridis and T A BlasingameldquoHigh-Resolution Numerical Modeling of Complex and Irreg-ular Fracture Patterns in Shale-Gas Reservoirs and Tight GasReservoirsrdquo SPE Reservoir Evaluation amp Engineering vol 16 no04 pp 443ndash455 2013
Geofluids 19
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
[30] W Yu B Gao and K Sepehrnoori ldquoNumerical Study of theImpact of Complex Fracture Patterns on Well Performance inShale Gas Reservoirsrdquo Journal of Petroleum Science Researchvol 3 no 2 p 83 2014
[31] S N Shah M C Vincent R X Rodriquez and T T PalischldquoFracture Orientation And Proppant Selection For OptimizingProduction In Horizontal Wellsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition Mumbai IndiaJanuary 2010
[32] J Weaver M Parker D van Batenburg and P NguyenldquoFracture-related diagenesis may impact conductivityrdquo SPEJournal vol 12 no 3 pp 272ndash281 2007
[33] M C Vincent ldquoRefracs - Why do they work and why do theyfail in 100 published field studiesrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition 2010 ATCE 2010pp 2237ndash2281 Florence Italy September 2010
[34] M C Vincent ldquoRestimulation of unconventional reservoirsWhen are refracs beneficialrdquo Journal of Canadian PetroleumTechnology vol 50 no 6 pp 36ndash52 2011
[35] B McDaniel ldquoConductivity Testing of Proppants at HighTemperature and Stressrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA April 1986
[36] S Cobb and J Farrell ldquoEvaluation of Long-Term Proppant Sta-bilityrdquo in Proceedings of the International Meeting on PetroleumEngineering Beijing China March 1986
[37] P Handren and T Palisch ldquoSuccessful hybrid slickwater-frac-ture design evolution an east texas cotton valley taylor case his-toryrdquo SPE Production andOperations vol 24 no 3 pp 415ndash4242009
[38] Y-L Zhao L-H Zhang J-X Luo and B-N Zhang ldquoPerfor-mance of fractured horizontal well with stimulated reservoirvolume in unconventional gas reservoirrdquo Journal of Hydrologyvol 512 pp 447ndash456 2014
[39] E Stalgorova andLMattar ldquoPractical AnalyticalModel To Sim-ulate Production of Horizontal Wells With Branch Fracturesrdquoin Proceedings of the SPE Canadian Unconventional ResourcesConference Calgary Alberta Canada November 2012
[40] W Luo C Tang and X Wang ldquoPressure transient analysis ofa horizontal well intercepted by multiple non-planar verticalfracturesrdquo Journal of PetroleumScience andEngineering vol 124pp 232ndash242 2014
[41] C Guo J Xu MWei and R Jiang ldquoPressure transient and ratedecline analysis for hydraulic fractured vertical wells with finiteconductivity in shale gas reservoirsrdquo Journal of PetroleumExplo-ration and Production Technology vol 5 no 4 pp 435ndash4432015
[42] H Mukherjee and M J Economides ldquoParametric comparisonof horizontal and vertical well performancerdquo SPE FormationEvaluation vol 6 no 2 pp 209ndash216 1991
[43] C T Montgomery and R Steanson ldquoProppant selection thekey to successful fracture stimulationrdquo Journal of PetroleumTechnology vol 37 no 12 pp 2163ndash2172 1985
[44] S C Yu ldquoEvaluation of long term fracture conductivity forceramic proppant and Lanzhou sandrdquoOil Drilling amp ProductionTechnology vol 11 pp 1ndash7 1987
[45] C Y Jiao S L He H J Zhang andH X Liu ldquoSouthwest Petro-leum Universityrdquo in Southwest Petroleum University vol 33 pp107ndash110 2011
[46] G O Brown ldquoThe History of the Darcy-Weisbach Equationfor Pipe Flow Resistancerdquo in Proceedings of the Environmentaland Water Resources History Sessions at ASCE Civil Engineering
Conference and Exposition 2002 pp 34ndash43 Washington DCUSA 2003
[47] H S Bae S Pyun W Chung S-G Kang and C ShinldquoFrequency-domain acoustic-elastic coupled waveform inver-sion using the Gauss-Newton conjugate gradient methodrdquoGeo-physical Prospecting vol 60 no 3 pp 413ndash432 2012
[49] M Fisher J Heinze C Harris B Davidson C Wright and KDunn ldquoOptimizing Horizontal Completion Techniques in theBarnett Shale Using Microseismic Fracture Mappingrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionHouston Tex USA September 2004
[50] C L Cipolla X Weng M G Mack et al ldquoIntegrating Micro-seismic Mapping and Complex Fracture Modeling to Charac-terize Hydraulic Fracture Complexityrdquo in Proceedings of the SPEHydraulic Fracturing Technology Conference The WoodlandsTex USA January 2011
[51] S Jain M Soliman A Bokane et al ldquoProppant Distributionin Multistage Hydraulic Fractured Wells A Large-Scale Inside-Casing Investigationrdquo in Proceedings of the SPE HydraulicFracturing Technology Conference The Woodlands Tex USAFebruary 2013
[52] N P Roussel andMM Sharma ldquoOptimizing Fracture Spacingand Sequencing inHorizontal-Well Fracturingrdquo SPE Productionamp Operations vol 26 no 02 pp 173ndash184 2011
[53] V R Penmatcha and K Aziz ldquoComprehensive reservoirwellbore model for horizontal wellsrdquo in Proceedings of the 1998SPE India Oil and Gas Conference and Exhibition pp 191ndash204New Delhi India April 1998
[54] A H Kabir and J A Vargas ldquoAccurate Inflow Profile Predictionof Horizontal Wells Using a Newly Developed Coupled Reser-voir andWellbore Analytical Modelsrdquo in Proceedings of the SPEOil and Gas India Conference and Exhibition The WoodlandsTex USA February 2009
[55] J Cai WWei X Hu R Liu and J Wang ldquoFractal characteriza-tion of dynamic fracture network extension in porous mediardquoFractals vol 25 no 02 2017
[56] J Cai and S Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013
[57] J Cai S Guo L You and X Hu ldquoFractal analysis of sponta-neous imbibition mechanism in fractured-porous dual mediareservoirrdquo Acta Physica Sinica vol 62 2013
[58] XWang L Sun R Zhu et al ldquoApplication of charging effects inevaluating storage space of tight reservoirs A case study fromPermian Lucaogou Formation in Jimusar sag Junggar BasinNW Chinardquo Petroleum Exploration and Development vol 42no 4 pp 516ndash524 2015
[59] L-B Ouyang S Arbabi andK Aziz ldquoA Single-PhaseWellbore-Flow Model for Horizontal Vertical and Slanted Wellsrdquo SPEJournal vol 3 no 2 pp 124ndash133 1998
[60] P K Swamee and A K Jain ldquoExplicit equations for pipe-flowproblemsrdquo Journal of the Hydraulics Division vol 102 no 5 pp657ndash664 1976
![Pressure-Transient Responses of Horizontal and …s-skj/CoNing/SPE-84378-PA-P[1].pdfPressure-Transient Responses of Horizontal and Curved Wells in Anticlines ... observations from](https://static.documents.pub/doc/80x56/5aa2cea97f8b9ac67a8d9160/pressure-transient-responses-of-horizontal-and-s-skjconingspe-84378-pa-p1pdfpressure-transient.jpg)
