, ,

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SPE 19784

Interference Testing of Finite Conductivity HydraulicallyFractured WellsD,N, Meehan, Union Pacific Resources/Stanford U,, and R.N. Home and H.J. Ramey Jr., Stanford U. ,

SPE Members T

Copyright 1989, society of Petroleum Englnem, Inc.

Thla paper was prepared for presentationat the S4th Annual T. .+nlcal Conference and Exhlbilionof the Society of PetroleumEn@nearsheld In San Antonio,TX, October S-1 1, 1SSS,

This paper was selected for presentationby an SPE Program Commlttea followingreview of informationcontained in an abetracf eubmltted by the author(a).Contents of the paper,as presented, have notbeen reviewed by the Society of Petroleum Errginseraand are aubjecrto correctionbythe●uthor(s), The material, es presented, does not nemsarlly refieolany positionof the society of Petroleum Englneera, itsoffIcers, or membere, Papers presented at SPE meetlrrgsare subjectto publicationreview by EdltorlalCornmltreeeof the Societyof Patrokum Enginaera.PernrieelorItocrrQYlareebictedtoan abstractof notmorethan300 words.Illuatratlonsmay notbe copbd. The abstractshouldce+rteinconspicuouseck~of where and by whom the paper la presented. Write Publication MunaWr, SPE, P.O. Box 83SS3S, Richardson, TX 7WSS4S3S. Telex, 7S0SS9 SPEDAL

This paper presents techniques for the design and anal- only uniform flux fkacturea; an extension by Cinco-Ley and

ysis of interference tests when both the active and obser- S.arnaniego [3] considered finite conductivity fractures, None

vation well are intercepted by hydraulic fractures. These of these methods work for hydraulically fractured ohserva-

techrdques are based on new mathematical solutions. The tion wells. Abobiae and Tiab [4] and Ekie et al. [5] alsosolution is general for any values of dimensionless fracture used uniform flux models for interference work. Unamb&u-conductivity including infinite conductivity, The solution is ous determination of fracture azimuth requires two obeer-

presented in Laplace spac~ fracture skin, wellbore storage, vation wells located at orientations other than 90 and 180

naturally fissured matrix behavior, etc. are readily included, degrees from the active well.

Any rate or pressure schedule at the active well can be an- Reservoir heterogeneities complicate the determinationalyzed, of fracture ~jmuth fmm jnte~emnce tats, Permeability

Compass orientation of the wells’ hydraulic fractures can variationa in the drainage area of a well are averaged in a

be determined from such an interfer ce test. Relative frac-

%

manner that is not entirely understood. No current modelture lengths, conductivities, and azi uth significantly af- exists to assess the impact of these heterogeneities on inter-fect W+ performance, Reservoir heterogeneities may signif- ference test determination of fracture azimuth,icantly alter interference response, The value of knowing Unfortunately, both high-resolution (high cost) pressurehydraulic fracture azimuth can also be determined. Perfor- tranaduceti and very long teets are required for these inter-mance of a hydraulically fractured well near a large natural ference tests because of the extremely low permeability offracture or another hydraulically fractured well is also fore- most candidate formations, Reservoir heterogeneities, mul-cast, Values of fracture conductivity and fracture length tiple layers, and surface interference can render azimuth de-fer each fracture can not be determined uniquely from the tection by interference testing infeasible, Resolution withinterference response and must be determined separately, respect to azimuth is also low at angles greater than 45°,

Historical Approach Field tests used to evaluate fracture azimuth were reportedGXapplication of interference tests is the determina- bly Frohne and Mercer [6], [7] and Sarda [8] . Elkins and

tion of fracture orientation in hydraulically fractured wells Skov [9] , Komar et a/, [10],[11] , Komar :and Shuck [12] ,

(i.e., wells intersecting a vertical fracture). Pierce et al, [11 and Locke and Sawyer [13] described field experiments thatdescribed a method for determining fracture rizimuth and involved contouring pressures at offset wells during draw-fracture length using pulse testing, Ttis method requires dowm, fracturing operations, and injection,pulse tests before and after the fracturing opeation, and is Historically, most work has used the ‘line source’ ap-

not applicable for very low permeability systems or finite proximation at the observation well, This is not withoutconductivity fractures, Uraiet et al, [2] developed a tech- reason, as numerous interference tests in fields with known

nique for azimuth determination using pressures rccordcd heterogeneities have resulted in tests that match the classi-at an unfractured observation well, This work comidercd cal exponential integral solution, The uniform flux model is

Refererrcea and illuatrationa at end of paper,only reasonable for short fracturea, and in these cases, theobservation well must be close to the active well to dMer-

137

,

2 Interference Teathu of Finite conductivity H~drauliC@ Fractured Wells SPE 19784

entiate between various fracture azimuths. Use of infiniteconductivity fracture models is unrealistic for low perme-abilityy reservoirs where long fractures are createci to iiowcommercial quantities of hydrocarbons,

Fundamental flow equations for Newtonian fluid flowin homogeneous porous media and corresponding assump-tions are well known [14] , Numerous solutions for differentboundary conditions have been published, Many of thesewere direct analogs to solutions of heat conduction prob-lems due to the similarity between the diffusivity equationsin temperature and in pressure [15] . Gringarten popular-ized the use of Source and Green’s functions for solving theseproblems [16] . This section derives the finite conductivityfracture pressure and flux calculations, Laplace space for-mulations for equations are used, simplifying the problemand increasing flexibility of the solution,

Cinco and Samaniego [17] presented a mathematical modelthat has become standard for evaluating @ite conductivity

hydraulic models, The basic procedure is a ‘semi-analytic’one in which the hydraulic fracture is modeled with a largenumber of elements (usually 20 to 40 per wing). Each ele-ment is modeled as having uniform flux; however, flux dis-tribution is not known a priori. Reservoir and fracture flowequations are equated along the fracture and the discretizedsystem is solved for wellbore pressure and flux distribution.

Numerous extensions of the kclmique have been published,Cinco and Meng [18] and van Kruysdijk [19] recently

presented formulations in Laplace space” for finite conduc-tivity fractures Solving the equations in Laplace space hasseveral advantages; viz,,

1. This method is fast using the Stehfest algorithm~20]for rapid inversion to real space. Previous techniquesrequired discretization in both time and space,

2. Addition of wellbore skin effect and wellbore storageare easily obtained. Inclusion of wellbore storage willtypically be import- real well testing problems,

Sandface dimensionless pressure (P,D) is simply con-ventional dimensionless pressure plus skin damage ef-fect (S),

P.D=PD+S (1)

Wellbore storage solutions are obtained by a simplemanipulation of the constant rate Laplace solutionwith skin (p, D(s)):

%D(g) =~,D(s)

1 + cD~2F,D(~)(2)

3. By solving in Lapktce space, constant pressure solu-tions for q~ and QD are easily obtained.

●

?Wf)($)= s [1 + cD&#D($)](3)

Cumulative production for constant pressure produc-tion is simply dimensionless flow rate divided by theLaplace space variable (s), as integrating with respectto time is synonymous to division by 8,

4. The pressure derivative group pj term is obtained an-alytically.

5. Convolution to obtain variable rate solutions is suffi-ciently rapid that computerized automatic type-curveregression is possible,

6. The Laplace space formulation allows immediate solu-tion of transient pressure response for naturally frac-tured cases by substituting the term s~(s) for s inthe wellbore pressure solution terms that result fromthe reservoir flow model. Here, f(s) will be one ofthe dual porosity models, typically either for transientmatrix flow or pseudo-steady state matrix flow. Cinco

and Meng [N presented a formulation that n~glectedcompressible flow in the fracture (fracture linear flow).Cinco [181demonstrated the accuracy of this approx-imation, The van Kruysdijk model [19] included con-

sideration of compressible flow in the fracture.

ofiacfure flow equation8

Consider a fracture to LS a homogeneous slab of uni-form porous media with height h, width b~ and halflength ZJ. Since fracture length is much longer thanfracture width, fluid intlux at the fracture ends m’.ybe neglected, Fluid entem the fracture facea at a rawg(z, t) per unit of fracture length. Unsteady-state flowin the fracture can be described as:

with the following initial and boundary cond~tiorm

apj %@.—-x .=O = 2bfkjh

and:pf(Xjt=O)=pi> 0< W<2!f

By neglecting the fracture compressibility term

LYpf /J !lf(~’>tDzf)~’-~ bf h

Now, defining dimensionless variable as:

noting that:

pfD(*, t) = ‘-PjD

(5)

(6)

(7)

(8)

(9)

(lo)

(11)

(12)

(13)

● ●

Substitution and cancellation leads to

This equation can be integrated twice to yield the pres-sure drop between the wellbore and any point in thefracture:

PwD(~Dzf) – PjD(~D, ~D.f) =

‘{xD;[D~’q,D(stl,t~=,)d~tt~.t}

(16)

This integration uses the no flux boundary conditionat the tips, the known value of dimensionless pressureat the origin, and the total flux condition to evaluatethe constants of integration.

oReservoir jlo w equations

Dimensionless pressure drop at any point in space dueto a plane source of height h, length 221 with fluxdensity g~(z’, tD=J) is:

Equating these two equations for ~D = O and -1<~D < 1 and taking the Laplace transform yields:

Note that for the Laplace transformation, the follow-ing properties are used (reference equation numbers

from Reference [21] are shown in bold face),

29.2.1 t

For a single well in an infinite reservoir, the fracture issymmetric, ~~D(XD, .$) = T~D(-zD, s); the integrationlimits of Equation 18 for the reservoir term may be

changed from [-1 1] to [0 1] by utilizing this symmetry.

This does not usually appll* for interference problems.

● Discretization and matrix formulation

The integral invo~ving the lfo terms can be integratedfor the discretized fluxes as described in Appendix A.Discretization into n equal length fracture segments

I (on each fracture half-length) and the approximationof uniform flux over each–section reduc~- the doubleintegral of the fluxes to

fl~’?,D(z’’@)dz’’d=’= (19)

j-1

{

(A~)2 + &(ZDj – iAz)ZTfLli(3)—i-=1 1+(Ax)2

~~fDj(~)

Subscripts for ZD imply that iocaticms ~Dj BIW mid-~okts of the jth segment. Values for CD, and zDi+~are at the beginning and end of the ith segment, re-spectively. This system of equations can be solved forthe single well problem and has been previously dis-cussed for the real space solution, In the followingsection, the details of solution for interference withtwo finite conductivity hydraulically fkactured wellsis given. .Interference Between Two FYnite Cond ~ctivityHydraulically Fractured Wells

In this section, a solution is generated for the com-

bined interference problem of finite conductivity y frac-tures intersecting both active and observation wells.Formulation is for different length fractures with dif-ferent values of (kjbj)D.

This solution can evaluate behavior of either a linesource well or finite conductivity hydraulically frac-tvred well located near a natural fracture or another

hydraulically fractured well. This method would quan-tify such impacts and predict the behavior of a hy-draulically fractured well in a naturally fractured sys-tem with large, widely spaced fractures. Primary useof the solution is for the well teztirq +.erference prob-lem. Nomenclature is illustrated in . .gure 1.

● Semi-Analytic Solution

Fracture flow equations and reservoir equations arewritten and coupled for each well as in previous deriva-tions, However, in this solution, fluxes from both frac-tures must be specified. In Laplace space, the dimen-sionless pressure drop at the active well is:

+-(:(’?,dz’’,d~’’d=’=‘+(20)

1 ,c~D+l/A@)

~ ~mD-l/A,Ao, ‘jDo(x” a)KO (fi~~) dz’

129

ISimiIarly, for the observation well,

., IJpwD(.) -1 ‘mD+’’A’Ao’qfDo(z’, S)lic,(l x~ - x’ I #i)fzz’

2 =n8D-1/~AO]

+-fl(’~,D(.’’!$)d.’’d.’= (21)’11

L~ *7J,~A(z’,8)Ko(fi& – Zy + d’j’j)dz’

The equation for the observation well only holds forequal fracture conductivities. Modification for differ-ent conductivitiea and permeability anisotropy will bediscussed. If n matrix blocks are used for each fracturehalf length, the above equations constitute 4n equa-

tions for 4n + 2 unknowns. For the remaining twoequations, the flux conditions at each well are incor-porated,

Letting m = 272for the totaI number of fracture blocks,for the active weli:

(22)

for the observation well: Iklo = oi=l

(23)

For the simplified case of equal fracture lengths and

equal fracture conductivities, the matrix formulationis given in Table 1, Aij terms and Dij terms arisefrom the pressure drops at the fractured active andobservation wells due to the fluxes at each of thosewells respectively, For the special ctie of ~fA = z~o

and (kjbf)A = (k@~)o, these two terms will be iden-tical. The Bij and Cij terms are the crow ter~s,

which contribute to the effects of the active well fluxeson the wellbore pressure at the observation well, andvice versa, Equations 20 and 21 are discretized using

the remdts of Equation 19. The counters for the ijterms must be kept consistent with the direction ofdiscretization.

● Modifications for Different Fracture Lengths,

Conductivities

Distances require scaling when fracture lengths or con-ductivities are not equal. The mtio of the active well’shydraulic fracture length to that of the observationwell is given as:

zf,4AIAo] = ~ (24) I

and the relative fracture permeability width product Ias:

(25)

The ratio of the values of (klbj) q for the two wells is I

Because the origin of the observation well is displacedZmD along the x-direction and dD in the y-direction,additional changes to the formulation are also required.Solution consistency requires the active and observa-tion wells to be formulated in their specific coordinatesystems. For example, the integration limits for theAij terms are [-1 1], Correapondlng limits for the Dij

[terms are ~.D - l/AIAO] ZmD + l/AIAOl].

At early times the cross terms (B C) are negligible,

The result is infinite acting behavior at the active welland zero pressure drop at the observation well, At latetimes, the magnitude of the cross terms approachesthat of the main diagonal. In practice, the order ofthe matrix (Table 1) can be reduced, in this case from(m+ 2)” (m+ 2) tom, m. This reduction speeds thematrix inversion and reduces storage. Matrix inver-sion is typically repeated eight times in each time stepin the Stehfest algorithm numerical Laplace transforminvemion. Details of this reduction are presented inAppendix B. Other methods for accelerating the so-lution, both for filling the matrix and inversions, arealso discussed in Appendix B.

● Effects of Azimuth and Spacing

In this section, a series of figures are used to summa-rize solution results. A series of &mea will give thedimensionless pressure (pD) and dimensionless p&s-sure derivatives (p~) at the active and observationwells. Dimemionless pressure derivative groups areuseful for evaluating more subtle characteristics andas a diagnostic tool. These derivativ- are calculatedfrom the Laplace space solutions directly and do notrequire numerical differentiation.

Typically, a group of curves is displayed showing vary-ing fracture azimuths which range from 15-90°. As ac-tive well responses show small variations due to frac-ture azimuth, only the 15 and 90 degree cases are dis-played. C)bservation well figures show responses for

each 15 degree increment, except for cases when allof the responses are spaced very close together. Forthe graphsof pD at the active well, the infinite acting

solution is also displayed. Active well solutions areplotted as a function of tD.jo Invariably, the infiniteacting solution overlays the data at early times. Atlate times, the active well interference solutions showvarying levels of negative skin; During a transition

period of varying length, active well pressures and thepressure derivative groups fall below the correspond-ing values for the irdlnite acting well. During thh time,interference is most pronounced.

For the interference wells, the exponential integral is

plotted, along with ~t) and pi. observation well so-hltions are plotted as a function of tD.f/F~. For rel-atively small valuea of TD, and for low values of azimuth,it is clear that the line source solution is a poor ap-proximation, As T’Dincreases to valuea of four and

above, all of the azimuth solutions collapse to the line

● ✎

5PE 19784 D. N. Meehan, R. N. Horne, and H. J, Ramey, Jr. 1

—

source solution, Figuree 2-9 present type curves for(kfbf)D = r for vahes of r~ varying from 0.5-4.0.

Fracture fluxes at active and observation wells weredifferent for both high and low conductivity cases.Figures 10-14 illustrate early and late time fluxes atthe active and observation wells for both Klgh andlow fracture conductivity csaea and fracture azimuthsvarying from 15-90”.

oPermeabili@ Anisotropy

PermeabWy anisotropy can be handled in the semi-anaiytic solution by appropriate geometric substitu-tions. Approximations for the dimensionless fractureconductivity y are only approximate for very low valuesof conductivity y and very early times. However, sub-stitutions of the values of ~ and x) into the definitions

of pD and tD=f are, for essentially exact. Solutions for

closed boundaries and interference tests require onlythe adjusted geometries. By resealing axes for ~ anclx!, identicai wellbore dimensionless pressures are ob-tained,

● Effects of Different Jkacture Lengths and Con-

ductivities

The effect of the fracture length ratio AIAO1is inves-tigated by keeping the active well at unit length andvarying the length of the observation well. Predictedresponses for

● AIAo] ranging from 0.5-4.0,

● azimuth angles of 15° — 90°, and

● dimensionless conductivities ranging from O.Irr toloom

were evaluated. For each of these examples, the rela-tive fracture conductivities [kfbt]~o are held constant;this implies changing values for the fracture permeabll-ity width product. However, this makea no practicaldifference over the range of interest because sensitiv-

ity to [k@J]Ao is negligible. Values of AIAo] for O = Oand rLI 5 1 + A(AOIwere not considered because thetwo fractures would physically overlay each other, Lowfracture angles in which the two fractures were in closeproximity often required increased numbers of fractureblocks for stabiiity. Uniform flux over a given fractureblock is assumed; fracture blocks were given a maxi-mum size equal to one tenth the distance to the otherwell, dD.

Similarities in responses makes it clear that fracturelengths and conductivities for the two wells cannot bedetermined by a single interference test, and must be

determined independently, This requires tests of suf-ficiently short duration to avoid interference. How-ever, the effect of interference at the active well isnot generally large except for small values of 6 and

rIJ. Comparing the curves for varying vahms of AIAO1shows minimal sensitivity for angles greater than 45°.Caution shouid be exercised in analyzing tests with

iarge vaiues of @due to the ditliculty in differentiatingbetween these angles. Initiai estimates of fracture az-imuth (from other techniques) shouid be used to avoidattempting tests large angles.

At late times, the influence of 8 on observation weilresponse decr~ses and becomes negligible for mostcases at about tr=f/rD2 >10. values of ?’IJ>4.0 aisoshow minimal r~sponse to 6 at ail times, The effect of

finite conductivity at the observation well is minimal,except at extremely early times. Observation well re-sponses show negligible differences for varying valuesof [k~b~]Ao, It is therefore impassible to determine the

value of FCD for either well from interference testing.

The active and observation wells v -ue of FCD and Z!must be determined by independent, active tests.

● Interference With Two Active Wells

By altering the previous matrix formulation, the ef-fects simultaneow production from both weils can bedenmnstrated. Of interest here is the delineation ofwhen the fracture interference between the welis is ofimportance. Figures 15 and 16 compare the perfor-mance of two wells (0 = 15,90, and rD = 1) with lowdimensioniese fractureconductivities with that of twoiine source weiis. By using the efktive wellbore radiusof the pseudodeady state behavior at the iine sourceweii, similar results are obtained. Therefore, solutions

for the finite conductivity hydrauikaily fractured wellin a closed rectangular reservoir require oniy superpesition of modified iine source image welis.

● Interference With a Constant Pressure ActiveWell

Most interference tests are designed for constant ratebehavior at the active weii, Low permeabHity reser-voirs may require weeks or months to obtain the de-sired reservoir information. In practice, maintainingconstant flow rates for low permeddit y wells duringthat length of time is difficult, It is much easier tomaintain constant surface flowing pressure, If pressure

drops are large (as is the case for many low permeabil-ity weiis), bottomhole premum drops wiii vary in timefor constant values of surface flowing pressures, Thesevariations are t ypicaUy small for moderate flow rates,

Verification Comparisons, Example Problem

A series of verifaction runs were used to ensure the ap-

plicability o! these new solutions over a wide range ofvalues. Comparisons with published results for sim-pler cases and with simulation respomes were used,Numericai simulations cordlrmed the semi-anaiytic so-lution presented here.

Figures 17-19 illustrate the automated solutions us-ing non-linear regression, The non-iinear regressionwas used to find a ‘beat fit’ to the modeis providedin the form of tables digitized from the results, The

input data was simuiated Using ~OD = r, @= 45, and

141

.’.

6 hhrferencc T~ting of Fifite Cmductivity HYd~ulically Fnactured Wew SPE 197S4

rLI = 1. Figure 17 M the result of matchingthe simu-lated interference data to the line source solution. Thisis the typical method of analyzing interference welltests. The data match reasonably well at late times,but do not match well at early times. Estimated reser-voir parameters are in error by 10~o for permeability

and 18% for the qtcJJ product. No information is ob-tained about fracture azimuth from this type of anal-ysis. In Mousli et td, ‘u [221solution, the active well isa uniform flux fracture, while the observation well hasan infinite conductivity hydraulic fiwture, For a simi-lar analysis with Mousli et td. ‘Usolution, the estimatederror associated with the non-linear regression is small( < 5% for k,d, and q$cth); however, these parameters

are in error by 470, 9Y0,and 21° respectively. The esti-

mates for permeability and dc~h are fairly good. Thefracture azimuth estimate by this techrique is poor.Varying the initial estimated fracture azimuth did notalter the non-linear digression estimate. Figures 18and 19 illustrate the results for the model developedin this dissertation. Agreement is good, with the esti-

mated values of k, dc~h, and O in error by S1% for allthree caaea. Estimated fracture azimuth was 44.6° andwas independent of the initial parameter estimatea.

Early Behavior

Observation well responses have been plotted as a func-tion of tD=f/r~ where rD is the dktance between the,wellbores of the active and observation well scaled bythe active well fracture length. Other plot ting func-tions were investigated for the time and preesure axea.None of them were completely succesful in collapsingall of the responsea. Approximate reductions are ob-tained for either early or late times,

Figure 20 is the observation well response for r~ = 0.8for 15<8 ~ 90 with the dimensionless pressure (pD)and dimensionless pressure derivative group (p~) plot-ted as a function of tD.f/r& The curves tend to col-lapse at late times; none of the curves matches the Eisolution exactly. All derivativea reach a value of 0,5(corresponding with pseudo radial flow) at approxi-

mately the same value of tD=f/r& Figure 21 showsshifted and scaled pressure and time data. Dimen-sionless pressure is resealed as:

where dD is the vertical distance (normal to the frac-ture direction) as defined in Figure 1. Dimensionlesstime tDr/ is divided by dD rather than by rD, ”

This resealing and reshifting also works for differentvalues of ( kjbi )0 and AIAO1. Unfortunately, it is im-

practical for test analysis since the desired parameter,

fracture azimuth (4) appears in both axes, However,Figure 21 illustrates how this shift is relevant for testdesign and understanding, If an estimated azimuthcan be obtained a priori by another method, the type

of plot given & Figure 21 indkates a minimum teatdesign to dWerentiate fracture behavior.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

1

Vertical finite conductivity hydrauUc fractureaSignificantly effect interference test response, MostSi@ficant r@uh& are for rD <4.

[t is pozsible to simultaneously solve for flux dis-tributions at the active and observation wells togenerate both active and observation well respo-.

The active wel pressure response is not sensitiveto the presence of the observation well for rD 22.

At late times, the presence of either a large natu-ral fracture (or joint ), or hydraulically fracturedwell can be approximated by an additional nega-tive skin at the active well,

Hydraulic fracture conductivity at the observa-tion well is not as important to observation wellresponse as is the fracture conductivity at the ac-tive well. Significant errors in azimuth estimationand test interpretation occur when the active wellconductivity is neglected.

Nothing in the shape of the observation well re-sponse indicates that <ither the active weU or ob-servation well is hydraulically fractured. Thus,fracture length and conductivity can not be de-termined by interference teats. Since both frac-ture lengths are r&-@ad for design or analysis,both wells must be tested independently.

Different fracture lengths at observation wells andactive wells alter theduration and magnitude offracture interference. Limiting responses for largeand small values of A[~ol can be obtained fromthe line source, finite conductivity fracture pairs

as active and observation wells.

For a fixed value of rD, observation well re9pon-sebecomes insensitive tQ azimuth for @ 2 50°.Sensitivity to fracture azimuth is independent offracture conductivity,

Constant pressure responses at the observationwell can be approximated by neglecting the ef-fect of the observation well fracture on the activewell. The Laplace space formulation of the so-lution presented simplifies the required calcula-tions.

SPE 19784 D. N, Meehnn, R, N, Home, ~d H. J. Rarney, Jr. 7

..-. --- —--—.—--

h fracture widthc compressibility yCD dimensionless wellbore storage coefficientd interwell distance (normal to fracture direction)k permeability yK. “modified Bessel function

P pressureflow rate

;D pressure derivative with respect to h of tD#fs Laplace transform variablet~=j dimensionless time, scaled to St

x!. fracture half-lengthXm interwell distance (parallel to fracture direction)L Laplace transform operations skin effect

7 Euler’s constant

u> ( integration variables

AIAOI ratio of active to observation well fracture length

P viscoait y

4 porosity

Diacritical

dimensionless

; fracture

i} j dkxetiaation counters8 skint totalw wellboreif Laplace transform of a .x’, x“ substitutions for x in integration

PI) correlating Substitution for pD

Acknowledgements

This work represents a portion of the first author’sPh. D. research which was supported by the StanfordCenter for Reservoir Forecasting and Union Pacific Re- . ~sources Corporation.

Appendix A

Integrating the Modified Bessel Function

Equation 20 requires evaluating the integral of W I$Dj - Z’ I and & I xDj + d I frOm xDi tO ~Di+l withrespect to z’, Abramowitz and Stegun [21] providea closed form infinite series for a similar integral asfollows:

11*1.9

However, the first equation to be integrated is of the

form: ,S.+Ko(filzDj-Z1l)dz, (28)

aDl

which can immediately separated into two integralsand expressed as .f~i+l - ~~”i of the same function.

Substituting for the argument of the integral:

u=fiI~Dj-~’l (29)

and noting

dz’=-% (30)l/s

the integration limits can be altered accordingly. Re-gardless of relative magnit udea of @Dj and ZDi, theresultant integral can be expressed as follows:

~iKo(fi,.Dj-.’,)=.’=

J@bD,-d -Ko(u) du

fi=D, G

When this integral is combined for both upper and

lower vahwa (~~i and ~Di+\ ), the fouowing expressionis obtained, after some algebra

~i+’‘O(I‘Dj-@’l @dZ’=

[

ZD~+] - xDj]/

b“j-~Di+ll* ~o(U)du

I ‘Di+l ‘~Djl O 77

-[,:::::;,] l’””’-=”i’ti%d’

The second integral is simpler as the term to

(31)

be in-

tegrated is ] ~~j + z’ I V and the relative ~itiom

of the xDj ad ~Di ~ not critical to the fornndation.So, the final expression is:

(32)

JkDj+rDi+ll_ K@)du

‘i o 7

-J o 6

Handling cases for the relative positions of xDi andz Dj was simplified as more general expressions wereintroduced.

hversion Pr ocedure

Unknowns are solved by inverting the left hand sidemat rix (Equation 21), This matrix is represented asA-z = b where A is the known (n+l) by (n+l) coeffi-cient mat rix, x is the unknown flux dist ribut ion vector,and b is the right-hand side vector. Matrix inversionis accomplished by computing the LU factorization ofthe coefficient matrix, Factorization fails if the upper

triangular part of the factorization has a zero diagonalelement, Iterative refinement is performed on the solu-tion vector to improve accuracy. The IMSL subroutineDLSARGwas used for matrix inversions when iterative

refinement was included. The Stehfest algorithm [M]is used to invert Laplace transformed variables intoreal space.

143

Appendix B

Calculation Improvements

● Improving Solution Performance

Seve@ methods can be used to decrease the number ofcomputations for solving the matrix given in Table 1.These include:

(a) Methods to speed the matrix solution.

(b) Methods to reduce the number of calculatio~s re-quired to fill the matrix.

(c) Methods to eliminate matrix calculations entirely.

● Reducing Matrix Order

Most of the matrices have a form which look similartcx

All A12 Ala 1 q~, $DlA21 A22 A23 1 qz . %DzA31 A32 A33 1 qs L?D3

111 0 p 1

For this example case, j = i = 3, and theare used to solve the case of

$WD(s) + ~ Aij “~Dii=l

(33)

equations

(34)

for i = 1,3. The fourth equation arises directly froma flow constraint that

:*,=1 ~ (35)i=l

For clarity, these terms are always separated in themain text. In practice, the matrix order is reduced byincluding the constraint of Equation 35 into the ma-trix. This haa the combined advantages of decreasingstorage requirements, decreasing computation time,

and removing zero (0) terms from the main diagonal.Rewriting Equation 35 for the case of j = 3,

9(3] = 1 -~g(i) = 1 –q(1) – q(2) (36)i= 1

Substituting into the equations and removing extra-neous nomenclature,

Allql + A12q2+ A13(1 –gl –92) + P = 01 (37)

A2191 + A2292 + A33(1 - ql - q2) + P = x2

A31ql + A32q2 + A33(1 - gl - g2) + P = Z3

Gathering like flux terms,

(All - A13)ql + (A12 - A13)g2 + P = c1 - A13(3S)

(A21 - A23)ql + (A22 - 213)q2 + P = Z2 - A23

(A31 - A33)ql + (A32 - A33)q2 + P = Z3 - A33

This can now be expressed in the reduced matrix formas

(All - A13) (A12 - A13) 1 q, ~ol - A13(A21 - A23) (A22 - A23) 1 q, = Z~2 – A13(A31 - A33) (A32 - 433) 1 P ZD3 - A13

(39 1

Interference Teeting of Finite Conductivity Hydraulically Fractured Wells

144

!’

The tinal value of the flux is solved by substitutioninto Equation 35.

● Reducing Matrix Building Requirements

Depending of the value ofs, the time required to cal-culate all of the matrix components in a matrix suchas ~ven in Table 1. may take 20-40 Yo of the total

computation time if done in a ‘brute force’ manner.Numerous simplifications accelerate matrix Ming, in-cluding

●

●

●

●

Many of the terms which are added to the mainAij terms are independent of s. These can becalculated once and added again for each Stehfeststep. This requirea a small increase in the totalstorage requirements.

Integral evaluation for the integrals of the form& Ko(u)du are replaced by 7/2 for values of xgreater than 20.

The n x n matrix components which require inte-gral evaluation actually have only 2n -1 differentintegrals. The tlret column and row contain allthe required values.

Values of [kjbf]~o = 1 or AIAol = 1 reduce the..-required number of calculatio-oa because many in-terference terms becomee identical.

● Early and Late Time Approximation

At very early times, interference terms are negligi-ble, For the interference case with Ilnite conductivitywells, the matrix of Table 1 may be reduced in size by75%. At such early times the active and observationwells behave independently and infinite acting solu-tions apply. Duration of this time may be observedfrom Figures 3-9 to depend on@ and rD. Typicai vrd-

ues for the end of the infde acting period range fromtDzj= 0.01-0.1.

At late times, flux distributions for both wells stabi-lize at dimensionless times on the order of tD=f/r& >10. These flux distributions can be retained to calcu-late wellbore pressures at the active and observationwells directly. Thh may be done either in the Laplaceor real space formulations. At late times, the activewell pressure response may be closely approximated

by a line source well with an effective wellbore radius.The ‘skin’ associated with the effective wellbore radiusarises both from the value of ( ki bj )D and the distanceand angle to the interference well.

References

[1] A. E. Pierce, S, Vela, and K, T. Koonce, Deter-mination of the Compass Orientation and Lengthof Hydraulic Fractures by Pulse Testing. Jowmalof Petroleum Techfiologg, 1433-1438, December1975,

[2] A. Uraiet, R. Raghavan, and G. W, Thomas. De-termination of the Compass Orientation of a Ver-tical Fracture by Interference Tests. Journal ofPetroleum Technology, 73-80, January 1977.

[3] H. Cinco-L. and F. ‘Samaniego-V, Determinationof the Orientation of a Finite Conductivity Verti-cal Fracture by Transient Pressure Analysis. SPE6750, presented at the 52nd Annual Meeting ofthe Society of Petroleum Engineers held in Den-ver, CO. October, 1977.

[4] E, O. Abobise and D. Tiab. Determining Frac-ture Orientation and Formation Permeability

From Pulse Testing. SPE 11027, presented at the57th Annual Meeting of the Society of PetroleumEngineers held in New Orleans, LA. September,1982,

[5] S. Eldest N. Hednoto, and R. Raghavan, Pulse-Testing of Vertically Fractured Wells!. SPE 6751,presented at the 52nd Annual Meeting of the So-ciety of Petroleum Engineers held in Denver, CO.1977,

[61 K. H. Frohne and C. A. Komar. Offiet M“ell Test:An Engineering Study of ‘Devonian Shale Produc-tion Characteristics. Intersociet y Energy Conver-sion Engineering Conference held at Los Angeles,CA, August 1982, 829177.

[7] K. H, Frohne and J. C. Mercer, FracturedShale Gas Reservoir Performance Study - AnOffset Well Interference Field Test. Journal ofPetroleum Technology, 36:2(SPE 11224):291-300,February 1984.

(8] J. P. Sarda. WeU-Linking by Hydraulic Fractur-ing Problem of Hydraulic Fracture Orientation,Bull. SOc, G’eel, (France), 26:5:827-831, 1984,

[91 L. F. Elkins and A. M, Skov. Determination ofFracture Orientation from Pressure Interference,Transactions of the AIME, 219:301-304, 1960,

[10] C. A, Komar, W, K, Overbey Jr,, R, L, Rough,and W, G, Lambert. Factors that Predict Frac-ture Orientation in a Gas Storage Reservoir.Journal ofPetroleun@chnology, 57(5):546-550,1971,

[11] C. A. Komar et al, Delineating a SubsurfaceFracture System in a Petroleum Reservoir—An

Experiment. Journal of Petroleum Technology,531-37, May 1973,

—

. .-

Y

[12] C. A. Komar and L. Z. Shuck. Pressure Re-sponses ,Y’romInduccid Hydraulic Fracturea in Ad- .jacent Wells Within a Petroleum tiervoi~ AnExperiment. Journal of Petroleum Technology,951-53, August 1975.

[13] G. H. Locke and W. K. Sawyer, Constant Pres-sure Injection Test in a Fractured Reservoir-

History Match Using Numerical Simulation andType Curve Analysis. SPE 55Q4, presented at the50th Annual Meeting of the Society of PetroleumEngineers held in Dallas, TX. 1975.

[14] M, Muakat. The Flow of Homogeneous FluidsThrough Porous Media. J. W. Edwards, Inc., AnnArbor, Michigan, 1946.

[15] H. C. Ca@aw and J. C. Jaeger. Conduction ofHeat in Solid$. Clarendon Press, Oxford, 1959.

[16] A, C, Gringarten and H, J. Rarney Jr. The Use ofthe Point Source and Green’s Functions in Solv-ing Unsteady-Flow Problems in Reservoirs. So-ciety of Petroleum Engineers Journal, October1973.

[17] H. Cinco-L. and F. Samaniego-V, Transient Pres-

sure Behavior for a Well with a Finite Conductiv-ity Vertical Fracture. Society of Petroleum Engi.neera Journal, 253-264, August 1978.

[18] H. Cinco-Ley and H, Z. Meng. Pressure Tran-sient Analysis of Wells WMh Finite ConductivityVertical Fractures in Double Porosity Reservoirs.SPE 18172, presented at the 63rd Annual Meet-ing of the Society of Petroleum Engineem held inHouston, TX. October, 1988.

[19] C. P. J. W. van Kr~ysdijk. Sernianalytical Mod-eling of Pressure Tra@~ents in Fractured Reser-voirs. SPE 18169, presented at the 63rd AnnualMeeting of the Society of Petroleum Engineeraheld in Houston, TX. October, 1988.

[20] H, Stehfeat. Algorithm 368: Numerical Inversionof the Laplace Transform, Communication oftheA, C, M,, 1(13), August 1970.

[21] M, Abramowitz and L A. Stegun. Handbookof Mathematical Functions. Dover Publications,ninth edition, 1970.

[22] N, A, Mouali, R. Raghavan, H, C!inco-L., and F,Samaniego-V, The Influence of Vertical FracturesIntercepting Active and Observation Wells on In-terference Tests. Society of Petroleum EngineersJournal, 933-944, December 1980,

Mu

Table 1Matrix formulation for interference solution,

AII A,z 0,.A21 Azz . . .

A~,l A~;2 , .,Ax Ax AxCll C12 , ,,C21 C22 ,.,

cm,l Cm,z ,..000

Al,m/.@m

Am,m

AxC,,mCz,m

c m,mo

1 B,, B,z ,., B1,~ O1 Bz] Bzz ,,, B2,m O

::,, ,,

::,.1 Bm,l000 DIIO Dzl::. .

::. .1 D~,l01

Bm,mo00

L+,m 1D2,m 1

D;m,m

10

TjDAl

?fDA2

~jDAmJWDA($)

~jDol

t7jD02

?jDOm

~wDo($ )

x/)1

$ D2

Xl)n

100

00

...

Figure 1: Illustration of nomenclature for interference testing of hydraulicallyfractured wells

.

10

1

0,1

0.01

Active Well, rD =1,0, Fcd = pi, Theta = 15,90

* Anslylic Soluti4in

— 1S Degrees

. . . . . . . 90 Degrees

().(f)l 0.01 0,1 1 10 100 1000DimensionlessTime, tDxf

Figure 2: Active well solutions for rD = 1 and FCD = n

t

146

Observation Well, rD = 1.0, Fcd - pi, Theta= 15,,,90

z JI 1 1 I J

0,001 0.01 0.1 1 10 100 1000Dimrosionk TMC, tDxf/rfY2

Figure 3: observation well solutions for rD = 1 and J’cD = n

10

0,01

Active Well, rD =2,0, Fcd = pi, Theta = 15,90

* Analytic SoIuli{n

— lst)ef?ecs

. . . . . . . 90 Dcglce$

0,001 0.01 0,1 1 10 100 1000DimensionlessTime, tOxf

Figure 4: .4ctive well solutions for r~ = ~ and ~GD = T

10

0,01

Observation’ M, rD = 2,0, Fcd = pi, Theta= 15,90

0,001 0,01 0,1 1 10 100 1000fxmaluionlessTree, lDxf/rfY2

Figure 5: observation well solutions for rD = 2 and Fc~ = r

w

*E 19784

Active Well, rD =4.0, Fcd = pi, Theta = 15,9010

j

L!31

‘j

j 0,1 ‘,“ ‘* Analytic Sohuk n

!— 15 Dcgrccs

. . . . . . . 90 Degrees

0,010,001 0,01 0,1 1 10 100 1000

DimensionlessTmc, tDxf

Figure 6: Active well solutions for ?’D= 4 and FCD = z

Observation Well, rD = 4,0, Fcd = pi, Theta= 15,9010 [

I I i I t

I I I

!—

I 0,1* Lme Some

&

0,01 ‘O.o111 0,01 0.1 1 10 100 1000

DimensionlessTime, tDxf/rlY2

Figure 7’:Observation WCIIsolutions for rD = 4 ancl Fcu = rr

10 “Active Well, rD =0S. Gcd= pi, Theta= 15,90

1 /

001* AnalyticSohuif n

— IS Degrees

. . . . . . . 90 Degrees

0,010,001 0,01 0,1 1 10 100

DimensionlessTime, lDxf1000

Figurr 8: Active well solutions for rn = 0,5 anti F’cD = m

148

.,, .

10

1

0,1

n ni

Obswation Well, rD = 0.5, Fcd = pi, Theta= 15,90I I I

U$U 1

0.001 0,01 0.1 1 10 100 1000Dimcnsionk.w Tmc, IDxVrfX2

Figure 9: Observation well solutions for rk) = 0.5 and FCD = r

4Active, Observation Well Fluxes, rD = 1,0, FcD = pi, Theta= 15

I I

‘~‘.- . . . ..- . ..-— . . . . . . . . . .

‘.

IDxf=0!1 , II_ ActiveWeU. . . Ots6ervNionWell I ‘$ I

SPE 19?84

-2-1 0 1 2,

Dimensionless Distance, XD= x/xf

Figure lo: Ac.tivc and obscrvatim well fluxes at tlj,,.j= 0.1,F’cI.I= n, 6 = 15,iill(l t’~ s 1

4

2

~

o

-2

4ctive, Observation Well Fluxes, rD = 1.0, FcD = pi, Theta= 45

~~

* . . . . . . . . . . . . . .. w..- ......

‘ ilDXI’ = 0,1

_ Active Well. . . Obwwlion Well I I

.1 0 1 2Dimensionless Distance, XD= x/xf

Figure 11: Active uncl observation WC1lfluxes at t~tf = O,1,Fal) = n, @ = 45,Imd ?’D= 1

140

spE lvl~~

;.

Active, Obser@on “iVellFluxes, rD = 1.0, FcD = pi, Theta= 90

‘~ ‘

2

~ J. .

TLJ

,. . . . ---- -.. . . . . . . . . . . . . . . . . . . . . ------

“L tDxf= 10.0_ Active Well

Obsewntion Well

-2 ‘-”- —-1 -0.5 0 0.5 1

Dimensionless Distance, XD= x/xf

Figure 12: Active and observation well fluxes at tDsj = 10,Fc~ = z, O = !30,and ?’D= 1

Active, Observation Well Fluxes, rD = 1.0, FcD = pi, Theta=4

1’.2A ...-...0 . ..- . . . . -.. ...

.lDxf= 10.0Aclive Well

- Obsefvmion Well 1

-2 ‘“I

-1 0 1 2Dimemionless Distance, XD= x/xf

Figure 13: Active and observation well fluxes at tnr, = lo,~~D = m, 6 = 15,and ?’D= 1

Active. Observation Well Fluxes, rD = 1,0. FcD = 100r)i.Them= 154

2

~

()

-2

t~ L_.J‘.. .

-.. .. . —. . . . . . . . . . . . . . .

tl)xf = 10,0_ Active Well..,, observation Wctl I I

-1 0 1 2Dimensionless Distance, XD= x/xf

Figmv 14: Activo aIMl observation WC1lfluxes at t/J..j= 10,Fc~j = 100~, 6 = 15,[ml rl) = 1

150

1000 i I I

I /

/’(

10 !__._. ------- ~ — .–—--—..J—._.. _— . . .. -

10 100 1000 1=10’Delt.st,hrs

Figllre IQ: I-cnfication plot Of nonlinear regression match of simukttcd test t ~

semi-analytic solution

sN

10

1

0.1

0.01

Observation Well, rD =0.8,

●

●

☛

●

●

Observation Well, rD = 0.8, Fcd = 2000, Theta= 15...9010

1

0.1

n ill“.”.

0.01 0.1 1 10 100 1000

Dimensionless Time, tDxfhDA2

Figure 20: Observation well solution for r~ = 0.S and FCD = T

:cd = 2000, A

15 Degree:

&

h Line Source(W?4’4U8lDxurtY2)

Ie=l 5--90

10 Degrees

0.1 1 10 103 1000

Dimensionless The, tDxfldD**2 1