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SPE 109625
Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use andMisuse of an Arps Decline Curve MethodologyJ.A. Rushing, SPE, Anadarko Petroleum Corp., A.D. Perego, SPE, Anadarko Petroleum Corp., R.B. Sullivan, SPE,Anadarko Petroleum Corp., and T.A. Blasingame, SPE, Texas A&M University
Copyright 2007, Society of Petroleum Engineers
This paper was prepared for presentation at the 2007 SPE Annual Technical Conference andExhibition held in Anaheim, California, U.S.A., 1114 November 2007.
This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435.
Abst ractThis paper presents the results of a simulation study designedto evaluate the applicability of an Arps
1decline curve method-
ology for assessing reserves in hydraulically-fractured wellscompleted in tight gas sands at high-pressure/high-temperature(HP/HT) reservoir conditions. We simulated various reservoirand hydraulic-fracture properties to determine their impact on
the production decline behavior as quantified by the Arpsdecline curve exponent, b. We then evaluated the simulatedproduction with Arps' rate-time equations at specific timeperiods during the well's productive life and comparedestimated reserves to the true value. To satisfy requirementsfor using Arps' models, all simulations were conducted using a
specified constant bottomhole flowing pressure condition inthe wellbore.
Our study indicates that the largest error source is incorrectapplication of Arps' decline curves during either transientflowor the transitional period between the end of transient andonset of boundary-dominated flow. During both of theseperiods (principally the transient period), we observed b-
exponents greater than one and corresponding reserve estimateerrors exceeding 100 percent. The b-exponents generallyapproached values between 0.5 and 1.0 as flow conditionsapproached true boundary-dominated flow. Agreement be-tween Arps' suggested b-exponent range and our results usingsimulated performance data also indicates that, if applied
under the correct conditions, the Arps rate-time models areappropriate for assessing reserves in tight gas sands at HP/HTreservoir conditions.
IntroductionTight gas sands constitute a significant percentage of thedomestic natural gas resource base and offer tremendous
potential for future reserve and production growth. According
to a recent study by the Gas Technology Institute (GTI),2tightgas sands in the US comprise 69 percent of gas productionfrom all unconventional natural gas resources and account for
19 percent of total gas production from both conventional andunconventional sources. The same study
2 estimates total
domestic producible tight gas sand resources exceed 600 Tcf,while economically recoverable gas reserves are 185 Tcf.
Most of the resources assessed in the 2001 GTI study were atdepths less than 15,000 ft, yet the natural gas industry
continues to extend exploration and development activities tomuch greater depths. In some geologic basins, those depthsare approaching 20,000 to 25,000 ft. Many of these deepnatural gas resources are not only characterized by low-permeability, low-porosity reservoir properties, but thesereservoirs also exhibit abnormally high initial pore pressureand temperature gradients i.e. high-pressure/high-
temperature (HP/HT) reservoir conditions.
Similar to conventional natural gas resources, tight gas sand
reserves are routinely assessed with Arps decline curvetechniques. The original Arps1 paper suggested the declinecurve exponent, b,should fall between 0 and 1.0 on a semilogplot. However, we often observe values much greater than
1.0, particularly in tight gas sands at HP/HT reservoirconditions. Deviations in observed b-exponents from the
expected range suggest Arps' rate-time relationships may not
be valid for modeling the decline behavior of tight gas sands
at HP/HT conditions. More importantly, inappropriate use of
the Arps models may cause significant reserve estimate errors
in these unconventional natural gas resources.
Since these depths and extreme reservoir conditions require
wells that are very expensive to drill, complete and operate; itis imperative that we understand both the well productivityand production decline behavior. We also need to determinethe applicability of the Arps rate-time equations for assessingreserves. To address these concerns, we have conducted a
series of single-well simulation studies to develop a betterunderstanding of both the short- and long-term production
decline behavior and to identify those parameters affecting theproduction decline. In this study we simulated a range ofreservoir and hydraulic fracture properties, including:
Vertical heterogeneity from layering, permeability contrast
among layers, horizontal permeability anisotropy, and stress-dependent reservoir properties;
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Variable effective fracture conductivities and lengths, unequalfracture wing lengths, two-phase and non-Darcy flow, andstress-dependent fracture properties; and
Reservoir temperatures ranging from 300 to 400oF and initialpore pressure gradients ranging from 0.60 to 0.90 psi/ft.
We evaluated the simulated production with the Arps1 rate-
time equations. Reserve estimates were obtained at varioustime periods during the wells productive life by extrapolatingthe best-fit Arps model through the simulated production. Our
assumed economic conditions for estimating reserves wereeither a rate of 50 Mscf/d or a producing time period of 50years, whichever came first. Reserve estimate errors werecomputed by comparing those estimated reserves to the true
value. For this paper, we define the true estimated ultimaterecovery (EUR) to be the 50-year cumulative productionvolume. For reference, we also summarize the Arps rate-timeequations in Table 1, given below:
Table 1 Summary of the Arps' rate-time relations (Ref. 1)
Case Rate Relation
Hyperbolic:(0
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properties and their distribution would be similar for a deeperreservoir. A significant part of the core evaluation programincluded classification of hydraulic rock types.13 When
described on the basis of the dominant pore throat diameterdetermined from high-pressure, mercury capillary pressuredata, we observed distinct groupings of rocks having similarflow and storage properties, i.e., hydraulic rock types (HRT).
Figure2is a semilog plot showing the general region of eachhydraulic rock type in porosity-permeability space andgrouped according to empirical relationships developed byPittman.
14
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
Effective Porosity, %
AbsoluteKlinkenberg-CorrectedPermeability,md
HRT 1
HRT 2
HRT 3
HRT 4
0.30 microns
0.035 microns
0.075 microns
0.15 microns
0.015 microns
PoreSize,microns
Fig. 2 Pittman14
plot showing hydraulic rock types (HRT)as a function of permeability, porosity anddominant pore throat size.
To capture the range of permeability and porosity exhibited bythe four hydraulic rock types, we developed a four-layer case
where each layer represents a different hydraulic rock typewith average properties from Fig. 2. The layers can be
considered to be hydraulic flow units (HFU) which are definedsimply as "groupings of rocks with similar properties."13-15
For modeling purposes, we have assumed the relativethickness of each HFU is directly proportional to the
percentage of rock types shown in Fig.2. For example, HRT1 has the fewest measured points from the coring program andwill represent the thinnest layer. Conversely, HRT 3 has themost measured data points and will be the thickest layer. InTable 3we summarize the properties for each HFU (or layer)
in a reservoir with a gross sand thickness of 200 ft. We shouldnote that the average properties are "thickness-averaged"values.
Table 3 Summary of hydraul ic flow units (HFU) andaverage rock properties, four-layer case.
HFU HRTh
(ft)kg
(md) (%)
Sw(%)
1 1 10 0.07291 8.43 10.0
2 2 30 0.01752 7.45 20.0
3 3 120 0.00428 6.57 40.0
4 4 40 0.00086 5.65 50.0
Avg./Total 200 0.0090 6.61 36.4
We also evaluated reservoirs with one, eight and sixteenlayers. For the single-layer case, we assumed a homogeneous,isotropic system with average effective porosity and absolute
horizontal permeability equal to the thickness-averaged values
shown in Table 3. For the eight- and sixteen-layer cases, weincorporated more contrast among layer properties by utilizinga greater range of permeabilities and porosities as shown in
Fig. 2. For example, the permeabilities and porosities given inTable 4 for the eight-layer case represent the average + onestandard deviation for each HRT.
Table 4 Summary of hydraul ic flow unit s (HFU) andaverage rock propert ies, eight-layer case.
HFU HRTh
(ft)kg
(md) (%)
Sw(%)
1 1 5 0.05209 7.77 10.0
2 2 15 0.01191 6.73 20.0
3 3 60 0.00292 5.87 40.0
4 4 20 0.00053 4.94 50.0
5 1 5 0.09372 9.10 10.0
6 2 15 0.02314 8.16 20.0
7 3 60 0.00564 7.26 40.0
8 4 20 0.00119 6.35 50.0
Avg./Total 200 0.0090 6.61 36.4
Similarly, the permeabilities and porosities listed in Table 5for the sixteen-layer case range from the average + onestandard deviation to the average + two standard deviations.Note also that thickness-averaged properties shown in Tables4 and 5 are the same as those for the four-layer case
summarized in Table3. Maintaining these equalities allowsus to make direct comparisons among the various cases.
Table 5 Summary of hydraul ic flow unit s (HFU) andaverage rock propert ies, sixteen-layer case.
HFU HRTh
(ft)kg
(md) (%)
Sw(%)
1 1 2.5 0.05209 7.77 10.0
2 2 7.5 0.01191 6.73 20.0
3 3 30 0.00292 5.87 40.0
4 4 10 0.00053 4.94 50.0
5 1 2.5 0.09372 9.10 10.0
6 2 7.5 0.02314 8.16 20.0
7 3 30 0.00564 7.26 40.0
8 4 10 0.00119 6.35 50.0
9 1 2.5 0.03128 7.10 10.0
10 2 7.5 0.00629 6.02 20.0
11 3 30 0.00156 5.18 40.0
12 4 10 0.00021 4.24 50.0
13 1 2.5 0.11454 9.76 10.0
14 2 7.5 0.02876 8.88 20.0
15 3 30 0.00700 7.95 40.0
16 4 10 0.00152 7.06 50.0
Avg./Total 200 0.0090 6.61 36.4
To illustrate the comparative heterogeneity for each layeredcase, we constructed a Modified Stratigraphic Lorenz Plot
(Fig. 3)15-17
which is simply a plot of the normalizedcumulative flow capacity against normalized cumulativestorage capacity from the top of the shallowest layer to thebottom of the deepest layer. Note that the single-layer case,
which forms a 45-degree line, represents a homogeneous, iso-tropic system. Deviations from this line indicate hetero-geneity. Surprisingly, the four-layer case displays much moredeviation than the other two multi-layer cases. In contrast, thesixteen-layer case shows the least deviation of all multi-layer
cases. As we will demonstrate later, more heterogeneity is
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manifested by hyperbolic decline behavior with larger Arps b-exponents than cases with less heterogeneity. In fact, theheterogeneous cases will display more of the hyperbolic
decline shape characteristic of wells completed in tight gassands.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Normalized Cumulative Storage Capacity, frac.
NormalizedCumulativeFlowCapacity,
frac
1-Layer Case
4-Layer Case
8-Layer Case
16-Layer Case
Fig. 3 Modified Stratigraphic Lorenz15-17 plot showingrelative heterogeneity for hydraulic flow units (HFU)for all l ayered cases.
Stress-Dependent Reservoir Properties. Although all rockshave some stress-dependent properties, tight gas sands (inparticular), display both stress-dependent porosity and per-meability characteristics.18 Observed changes in porosity,however, are usually much less significant than changes in
permeability. We also note that the magnitudes of the stressdependencies are functions of hydraulic rock type. Generally,the lower-permeability, lower-porosity rock types will exhibitmuch larger relative changes in stress-dependent properties.
For our simulation study, we have re-scaled laboratorymeasurements from the shallower Lower Cotton Valley sandsto reflect the higher pressures in our hypothetical deep gas
reservoirs. Stress-dependent effective porosity and absolutepermeability for each of the four hydraulic rock types used inour simulation study are illustrated in Figs. 4 and 5,respectively. Note that the stress measurements have beenconverted to multipliers representing a fraction of the originalproperty value as a function of reservoir pore pressure.
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000
Reservoir Pore Pressure, psia
FractionOriginalPorosity,
frac.
HRT 1
HRT 2
HRT 3
HRT 4
Fig. 4 Stress-dependent effective poros ity relationshipsfor various hydraulic roc k types (HRT).
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000
Reservoir Pore Pressure, psia
FractionOriginalPermeability,
frac.
HRT 1
HRT 2
HRT 3
HRT 4
Fig. 5 Stress-dependent, absolute Klinkenberg-correctedpermeability relationships for various hydraulicrock types (HRT).
Gas-Water Capillary Pressure Curves. To model the verticaldistribution of fluids in our hypothetical HP/HT tight gas
sands, we also employed gas-water capillary pressure curveswhich were measured using core samples from shallower
Cotton Valley Sands. Since no single conventional measure-ment technique can achieve the low water saturations (andassociated high capillary pressures) in low-permeability rocks,we used data generated using a hybrid vapor desorption/high-
pressure porous plate method.19,20
We also have capillarypressure curves for each HRT. Moreover, we converted the
data to a Leverett J-function21
(Fig.6) for use in the simulator.
0.01
0.1
1
10
100
0.00 0.20 0.40 0.60 0.80 1.00
Water Saturation, fraction
LeverettJ-Function,dimensionless
HRT 1
HRT 2
HRT 3
HRT 4
Fig. 6 Gas-water Leverett J-Functions21
for various hy-draulic rock types (HRT).
Gas-Water Relative Permeability Curves. Although we aremodeling a basin-centered tight gas sand6,7 in which the
connate water saturations are relatively immobile, we stillincorporated gas-water relative permeability data in oursimulation study. The gas relative permeability curves, shownin Fig.7for each HRT from the Lower Cotton Valley Sands,
were measured using an incremental phase trappingtechnique.22 Further, the data have been corrected for gas
slippage or Klinkenberg effects.23
Because of the low per-meability, we were able to measure the effective permeabilityto water for HRT 1 and 2 only. The data shown for HRT 3and 4 were derived from single, end-point measurements withsynthetic curves drawn to mimic the general curve shape for
HRT 1 and 2.
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0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Water Saturation, %
RelativeP
ermeability,
frac.
HRT 1 Gas
HRT 2 Gas
HRT 3 Gas
HRT 4 Gas
HRT 1 & 2 Water
HRT 3 Water
HRT 4 Water
Fig. 7 Gas-water relative permeabili ty curves for varioushydraulic rock types (HRT).
Hydraulic Fracture Model. We modeled a vertical fractureextending over the entire sand thickness. Although some ofour simulation cases evaluated unequal fracture lengths, all
cases assumed each fracture half-length (or fracture "wing")extended equally in the vertical direction in each HFU orlayer. As mentioned previously, the grid system wasconstructed using a Cartesian rather than radial grid geometryso that we could model the linear flow patterns characteristic
of hydraulically-fractured wells producing from tight gassands.
We also modeled several types of stimulation treatments andproperties. Specifically, we investigated fractures with threedifferent types of generic proppants, each having differentcrushing and embedment characteristics. We evaluated thefollowing range of fracture heterogeneities and the impact ofthese heterogeneities on well productivity and performance:
Effective fracture half-length (equal wing lengths) of 50, 100,
200, 300, 400, and 500 ft;
Unequal effective fracture half-lengths, including wing lengthratios of 6:1, 3:1 and 1.5:1;
A range of fracture conductivities (0.1 FCD < 200) includingabsolute fracture permeabilities of 1, 10, 100, and 1000 md
(constant throughout entire fracture);
Variable fracture permeabilities within the fracture, includingthe choked fracture case (lower fracture permeability near the
wellbore);
Two-phase (gas-water) flow in the fracture using non-linearfracture relative permeabilities (Fig. 8);24
Stress-dependent fracture conductivity for three generic typesof proppants (Fig. 9).
Simulated Operational Conditions. To better representoperational conditions, we implemented several flowingbottomhole pressure constraints in the simulation process toensure that we modeled field operations as realistically as
possible. For example, we modeled a well that is choked backduring the initial flow back for the first few weeks during theclean up period, but is allowed to reach line pressure at the endof that time period. Specifically, we evaluated the effect ofthe maximum pressure drawdown on long-term production
decline behavior, and we observed little differences forpressure drawdown constraints from 2000 psia to 8000 psia.For all cases, we used a constant bottomhole flowing pressure
of 3,000 psia after the initial clean-up period and for theremaining productive life of the well.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Water Saturation, %
RelativePermeability,frac.
Gas
Water
Fig. 8 Gas-water fracture relative permeability profi lesused in th is study (ref. 24).
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000
Fracture Closure Pressure, psia
FractionOriginalFractureConductivity,frac.
Proppant No. 1
Proppant No. 2
Proppant No. 3
Fig. 9 Stress-dependent fracture conducti viti es for various
types of generic proppants modeled in stud y.
Decline Curve Analysis o f Simulated ProductionTraditional decline curve analysis consists of plottingproduction rate against time, history matching the productiondata using one of several industry-standard models i.e., theArps1exponential, hyperbolic or harmonic decline models and extrapolating the established trend into the future. Decline
curve analysis is quite simply a curve fitting process which
does not necessarily have a theoretical basis. An exception tothis statement is the exponential decline case which can bederived from a single-well model producing at a constant
bottomhole flowing pressure during boundary-dominated flowconditions.
We evaluated the simulated production using common in-
dustry practices i.e., assuming the Arps rate-time equationsare applicable. Our objectives were to not only quantify theArps decline curve parameters (i.e., initial decline rate, Di,decline exponent, b, etc.), but to also assess reserves at various
times during the wells productive life. Reserve estimateswere obtained by extrapolating the best-fit Arps model
through the simulated production. Our assumed economicconditions for estimating reserves were either a rate of 50Mscf/d or a time period of 50 years, whichever came first.
Reserve estimate errors were computed by comparing thoseestimated reserves to the true value. For this paper, we
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define the true estimated ultimate recovery (EUR) orreserves volumes to be the 50-year cumulative productionvolume.
Even though the exponential and hyperbolic relations (as
provided by Arps) are empirically derived, the Arps declinecurves are excellent tools for estimating reserves when applied
under the correct conditions. Unfortunately, many of us in theindustry either have forgotten or have chosen to ignore the
conditions under which use of the Arps decline curves areappropriate. Application of a decline curve methodology
using the Arps models implicitlyassumes the following:25
Extrapolation of the best-fit curve through the current orhistorical production data is an accurate model for futureproduction trends;
There will be no significant changes in current operatingconditions or field development that might affect the curve fitand the subsequent extrapolation into thefuture;
The well is producing against a constant bottomhole flowingpressure; and
Well production is from an unchanging drainage area with no-flow boundaries (i.e., the well is in boundary-dominated(stabilized) flow).
Traditional decline curve analysis is based on the general formof the Arps1rate-time (hyperbolic) decline equation:
[ ] bi
i
tbD
qtq
/11
)(+
= .........................................................(1)
whereDiis the initial decline rate, qiis the gas flow rate, and bis Arps decline curve constant (or decline exponent). Notethat the units of these three variables must be consistent.Equation 1 has three different forms exponential, harmonic,
and hyperbolic depending on the value of the b-exponent.Each of these equations has a different shape on Cartesian and
semilog graphs of gas production rate versus time andcumulative gas production.
The exponential or constant-percentage decline case is aspecial case of Eq. 1 where the b-exponent is zero, and ischaracterized by a decrease in production rate per unit of timethat is directly proportional to the production rate. The ex-ponential decline equation (b-exponent of zero) is written as:
)exp()( tDqtq ii = ...........................................................(2)
Similarly, the harmonic decline is also a special case of Eq. 1
(b-exponent equals one), and is written as:
[ ]1)(
tD
qtq
i
i
+
= .............................................................(3)
The Arps hyperbolic decline is given by the general form (Eq.1) and is "valid" for any condition where the b-exponent variesbetween 0 and 1.0. We should note that the value of the b-exponent determines the degree of curvature of the semilogdecline plot (log(q) versus t), ranging from a straight line withb=0 to increasing curvature as b increases. Although there isno theoretical basis, Arps
1indicated the b-exponent should lie
between 0 and 1.0 but he offered no justification of thepossibility that b might be greater than one. Therefore,
variations in the computed b-exponents outside of theexpected range suggests the Arps' rate-time relationships may
not be valid for modeling the decline behavior of tight gassands at HP/HT conditions.
Effect of Reservoir Properties on Production DeclineWe first evaluated the effects of various reservoir properties
and heterogeneities including vertical heterogeneity fromlayering, permeability contrast among layers, horizontal
permeability anisotropy, and stress-dependent reservoirproperties on the production decline behavior. Declinecharacteristics were quantified and compared using primarilythe Arps decline exponent, b.
Layering and Permeability Contrast Among Layers. Thesimulated short- and long-term production profiles for the
single- and multi-layer cases are shown in Figs. 10 and 11,respectively. Recall that the single-layer case assumes ahomogeneous, isotropic reservoir with absolute permeabilityand effective porosity equal to the "thickness-averaged" valuesof the multi-layer cases (Tables 3-5). Other reservoir and
hydraulic-fracture input properties used to simulate theproduction profiles are summarized in Table6.
Table 6 Summary of reservoi r and hydrauli c-fracture inputproperties used to simulate production profiles.
Reservoir Property Value
Well spacing 80 acres/wellInitial pore pressure gradient 0.90 psi/ftInitial bottomhole pressure 16,200 psiaBottomhole temperature 400
oF
Thickness-averaged absolute permeability 0.0090 mdThickness-averaged effective porosity 6.61%Thickness-averaged water saturation 36.4%Vertical to horizontal permeability ratio 0.001Gross sand thickness 200 ftS
tress-dependent properties No
Hydraulic Fracture Property Value
Effective half-length 300 ftAbsolute permeability 100 mdNumerical dimensionless fracture conductivity 18.5Vertical to horizontal permeability ratio 1.0Equal wing half-length YesStress-dependent properties No
Inspection of the simulated production profiles in Figs. 10and11shows that the single-layer case has a much sharper initialdecline (Dei= 71 days
-1) than the 4-, 8- and 16-layer cases (Dei
= 69, 67 and 66 days-1, respectively). Figure10 also showsthat, after the initial steep decline, the single-layer case has theflattest production profile during the first 5 years ofproduction. However, the multilayer cases begin to flatten
substantially after that period. In fact, the multilayer casesdisplay more of the long-term, hyperbolic decline shapecharacteristic of tight gas sands (Fig. 11).
All of these observations concerning the decline behavior areconfirmed by the b-exponents (Table7) which were computedfrom the best-fit Arps1 model through the simulated
production for producing times of 1, 5, 10, 20 and 50 years.Surprisingly, all b-exponents lie approximately in the range
suggested by Arps1 (i.e., 0 < b
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is not necessarily sufficient to generate large b-exponentsfrequently observed in tight gas sands. Recall the ModifiedStratigraphic Lorenz plot shown in Fig. 3which indicates that
the 4- and 8-layer cases are the most heterogeneous based ontheir deviations from the 45-degree line (representing a homo-geneous system). We also see that these two cases have thelargest b-exponents during the last 30 years of production.
Conversely, both the computed b-exponents and the ModifiedStratigraphic Lorenz plot suggest that, even though it has morelayers, the 16-layer case "behaves" more like the single-layercase than either of the other multi-layer cases.
100
1,000
10,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasP
roductionRate,Mscf/day
1-Layer Case
4-Layer Case
8-Layer Case
16-Layer Case
Fig. 10 Simulated shor t-term producti on pr ofiles for 1-, 4-,8-, and 16-layer cases.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProductionRate,Mscf/day
1-Layer Case
4-Layer Case
8-Layer Case
16-Layer Case
Fig. 11 Simulated long-term produc tion profi les for 1-, 4-, 8,and 16-layer cases
Table 7 Compu ted b-exponents for 1-, 4-, 8-, and 16-layer
cases.
ProducingTime Period
(years)1-Layer
Case4-Layer
Case8-Layer
Case16-Layer
Case
1 3.62 2.78 2.89 2.975 2.95 1.35 1.30 1.39
10 1.48 1.04 1.18 1.2620 0.58 1.01 1.04 0.9650 0.44 1.00 1.02 0.82
We also estimated reserves from an extrapolation of the best-fit Arps1model through the simulated production and at eachof the time periods shown in Table 7. We then comparedthose estimates to the "true" reserve value for each respective
multi-layer case. Not unexpectedly, the differences or errors
(Table 8)between the estimates and "true value" generallydecline as more production is available for the history match.
Table 8 Computed reserve estimate errors fo r 1-, 4-, 8-,and 16-layer case.
ProducingTime Period
(years)
1-LayerCase(%)
4-LayerCase(%)
8-LayerCase(%)
16-LayerCase(%)
1 131.6 109.6 117.2 127.85 58.0 21.7 11.1 20.4
10 19.2 7.8 8.6 10.920 0.1 7.3 7.2 2.7
Additionally, we generally found a strong correlation betweenthe computed b-exponents and reserve estimate errors i.e.,larger reserve estimate errors corresponding to larger b-
exponents. This correlation suggests that computed b-exponents which lie significantly outside the range suggestedby Arps1 (i.e., 0 < b
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hydraulic fracture axis (i.e., monitoring point no. 2, Fig. 1).Figure13 shows that HFU 1 and 2 again exhibit the largestand earliest pressure reduction. However, unlike pressure
monitoring point no. 1 at which we saw measurable pressurereductions in the first six months, we observed no significantpressure changes in HFU 3 for the first two years. Mostsignificantly, we see very little change in reservoir pressure in
HFU 4 for the first 25 years of production.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HFU 1 HFU 2 HFU 3 HFU 4
Fig. 13 Layer pressu res at moni toring po int no. 2 for t he 4-layer case.
Comparison of the layer pressure responses in Figs. 12and 13demonstrates several points. First, even in reservoirs with nopermeability anisotropy, we observe an elliptical flowgeometry (with the major ellipse axis centered along the
fracture axis). Secondly, variations in layer flow capacitiescause unequal growth of the elliptical drainage patterns.Finally, the pressure responses in HFU 4 suggest that the welldid not reach trueboundary-dominated flow but was in eithertransientor transitionalflow for more than 20 years.
Since HFU 4 does not represent a significant percentage(approximately 13 percent) of the total hydrocarbon pore
volume, it apparently does not affect the overall declinebehavior as significantly as HFUs 1-3. Furthermore, the lackof boundary-dominated flow in HFU 4 does not prevent theoverallwell production decline from "behaving" as if it werein true boundary-dominated flow conditions after 20 years ofproduction. We observed similar pressure distributions for the
8- and 16-layer cases.
Vertical-to-Horizontal Permeability Ratio. All of thesimulated production profiles shown in Figs. 10and 11weregenerated with a vertical-to-horizontal permeability ratio, kv/kh
= 0.001. In this section, we investigate the effects of othervalues of kv/kh. Figures 14 and 15 compare the short- andlong-term production profiles, respectively, for the 4-layercase but with kv/khvalues of 0.001, 0.01, and 0.1. Other thanvariations in kv/kh, reservoir and hydraulic fracture propertiesshown in Table6 were used as additional input for all cases.
All of the short-term production profiles (Fig. 14) have very
similar initial decline rates (Di=68.8, 66.6, and 60.8 day-1
forkv/kh = 0.001, 0.01 and 0.1, respectively). However, the
production profile for kv/kh= 0.1 is much flatter than the othertwo cases after the initial decline and during the first five years
of production. Vertical flow from less permeable layers into
HFU 1 and 2 provides pressure support and helps maintainhigher rates as well as the shallower decline profile during thisearly time period. However; after 15 to 20 years of
production, the two cases with lower kv/kh values begin toflatten more than the kv/kh = 0.1 case. Moreover, the lowerkv/kh cases display more of the long-term hyperbolic shapecharacteristic of tight gas sands (Fig. 15).
100
1,000
10,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasProductionRate,
Mscf/day
kv/kh=0.001
kv/kh=0.01
kv/kh=0.1
Fig. 14 Simulated short-term producti on profiles for kv/kh=0.001, 0.01 and 0.10.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
Gas
ProductionRate,
Mscf/day
kv/kh=0.001
kv/kh=0.01
kv/kh=0.1
Fig. 15 Simulated long-term p roduc tion profi les for kv/kh =0.001, 0.01 and 0.10.
The observed decline characteristics are reflected by the
computed b-exponents (Table 9). The b-exponents for thekv/kh= 0.001 and 0.01 cases are very similar for the entire timeperiod evaluated. However, b-exponents for the kv/kh = 0.1cases are generally larger for the first ten years, but begin to
decrease more rapidly than the other cases after that timeperiod. Again, we observe that the long-term b-exponents areapproaching values between approximately 0.5 and 1.0.
Table 9 Com puted b-exponents for kv/kh= 0.001, 0.01 and0.10.
ProducingTime Period
(years)kv/kh
= 0.001kv/kh
= 0.01kv/kh= 0.1
1 2.78 2.95 3.065 1.35 1.37 1.4310 1.04 1.19 0.7620 1.01 1.06 0.6650 1.00 0.95 0.51
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SPE 109625 Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use and Misuse of an Arps Decline Curve Methodology 9
The differences (or errors) between the 50-year cumulativeproduction volume and the reserves estimated fromextrapolation of the best-fit Arps decline curve through the
simulated production data are summarized in Table 10.Similar to the results shown in Table 8, we note a generalcorrelation between b-exponents and reserve estimate errors i.e., the largest errors are associated with the largest b-
exponents. We also observe that the largest errors occur forcases where there is less than 10 years of production.
Table 10 Computed reserve estim ate errors for kv/kh =0.001, 0.01 and 0.10.
ProducingTime Period
(years)
kv/kh= 0.001
(%)
kv/kh= 0.01
(%)
kv/kh= 0.1(%)
1 109.6 128.5 130.65 21.7 22.8 30.8
10 7.8 8.4 6.820 7.3 7.9 3.8
The differences in decline behavior for the various values ofkv/khcan again be explained by the simulated layer pressures.
Figure16shows the individual layer pressures at monitoringpoint no. 1 for kv/kh= 0.001 (color-filled triangles) and kv/kh=0.1 (color-filled circles). Larger values of kv/kh allow moreeffective drainage of all layers resulting in larger pressurereductions throughout the entire drainage area, including HFU
4 (the least permeable layer). Figure17shows similar resultsat monitoring point no. 2.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholeP
ressure,psia
HFU 1, kv/kh= 0.1 HFU 2, kv/kh= 0.1 HFU 3, kv/kh= 0.1 HFU 4 , kv/kh= 0.1
HFU 1, kv/kh=0.001 HFU 2, kv/kh=0.001 HFU 3, kv/kh=0.001 HFU 4, kv/kh=0.001
Fig. 16 Comparison of layer p ressures fo r 4-layer case atmonitoring point no. 1 for kv/kh= 0.001 and 0.10.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HFU 1, kv/kh=0.1 HFU 2, kv/kh= 0.1 HFU 3, kv/kh=0.1 HFU 4, kv/kh=0.1
HFU 1, kv/kh=0.001 HFU 2, kv/kh=0.001 HFU 3, kv/kh=0.001 HFU 4, kv/kh=0.001
Fig. 17 Comparison of layer p ressures fo r 4-layer case atmonitoring point no. 2 for kv/kh= 0.001 and 0.10.
The pressure behavior from both Figs. 16and 17suggests thekv/kh = 0.1 case may be in or is approaching boundary-dominated flow after less than five years of production, while
the lower value of kv/kh delays the onset of true boundary-dominated flow conditions for more than 20 years.Differences between the duration of transient and transitionalflow periods helps to explain both the smaller reserve estimate
errors as well as the smaller values of the computed b-exponent for the kv/kh= 0.1 case.
Horizontal Permeability Anisotropy. All of the previoussimulated performance cases addressed the impact of vertical
heterogeneities (i.e., layering, permeability contrast amonglayers, and kv/kh) on the production decline in these cases
no horizontal permeability anisotropy was considered (i.e., kx= ky). In this section we evaluate the effects of horizontalpermeability anisotropy as quantified by various values ofky/kx. Specifically, we evaluated ky/kx = 0.1, 1.0 and 10.
Except for variations in ky/kx, all reservoir and hydraulicfracture properties shown in Table6 were used as additional
input for the simulated cases.In previous cases which considered vertical heterogeneity, weobserved much less pressure reduction in the layers with thelowest permeability in a direction perpendicular (y-direction)to the hydraulic fracture axis, so we would expect variations inky/kx to impact the production decline and pressure depletion
even more significantly. Figures 18 and 19 compare thesimulated short- and long-term production profiles,respectively, for the 4-layer case and with ky/kxvalues of 0.1,1.0, and 10. Smaller values of ky/kx result in higher initialdecline rates (i.e., Dei= 76, 69, and 63 days
-1for ky/kxof 0.1,1.0, and 10, respectively). Conversely, larger values of ky/kxcause much sharper declines throughout most of the
productive life. This decline behavior is confirmed by thecomputed b-exponents (Table 11) which are consistentlylarger for the cases simulated with smaller ky/kx values. Infact, we see b-exponents greater than two for the first ten yearsof production for ky/kx= 0.1.
10
100
1,000
10,000
100,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasProductionRate,
Mscf/day
ky/kx=0.1
ky/kx=1.0
ky/kx=10
Fig. 18 Simulated short-term production profiles for 4-layercase and for ky/kx= 0.10, 1.0 and 10.
Differences between the 50-year cumulative productionvolume and the reserves estimated from the best-fit Arpsdecline curve through the simulated data for ky/kx values of
0.1, 1.0 and 10 are summarized in Table12. Again, we note
that there is a direct association between the largest reserve
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estimate errors and the largest computed b-exponents.Further, the largest reserve estimate errors generally occurwhen there is less than 10 years of production and before
reaching boundary-dominated flow conditions.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProductionRate,
Mscf/day
ky/kx=0.1
ky/kx=1.0
ky/kx=10
Fig. 19 Simulated long-term p roduc tion profi les fo r 4-layer
case and for ky/kx= 0.10, 1.0 and 10.
Table 11 Computed b-exponents for ky/kx = 0.10, 1.0and 10.
ProducingTime Period
(years)ky/kx= 0.1
ky/kx= 1.0
ky/kx= 10
1 3.00 2.78 2.175 2.75 1.35 1.06
10 2.26 1.04 0.9620 1.33 1.01 0.9250 1.13 1.00 0.76
Table 12 Computed reserve estim ate errors for ky/kx= 0.10, 1.0 and 10.
ProducingTime Period
(years)
ky/k
x= 0.1(%)
ky/k
x= 1.0(%)
ky/k
x= 10(%)
1 134.6 109.6 102.05 53.1 21.7 11.1
10 34.9 7.8 7.520 18.9 7.3 7.1
These observations are confirmed by the simulated layer
pressure responses shown in Figs. 20 and 21 at monitoringpoints 1 and 2, respectively. The curves compare layerpressures for ky/kx= 0.1 (color-filled triangles) and ky/kx = 1(color-filled circles). The pressure responses demonstrate thatlower values of ky/kx delay the onset of boundary-dominatedflow even more than in the isotropic case. In fact, neither
HFU 3 nor 4 exhibit measurable pressure reductions atpressure monitoring point no. 2 for at least the first 20 years ofproduction for the ky/kx = 0.1 case. We should note that theabsence of boundary-dominated flow conditions during thefirst 30 years of production for ky/kx= 0.1 also corresponds tothe larger errors shown in Table12.
Stress-Dependent Reservoir Properties. All tight gas sands
exhibit some stress-dependent characteristics. Typically, re-ductions in permeability are much greater than that forporosity. For this paper, we evaluated the impact of stress-dependent reservoir properties on the production decline usingthe functions shown in Figs. 4and 5. Except for the inclusionof stress-dependent reservoir properties, we used the reservoir
and hydraulic fracture properties shown in Table6as input forthe simulation.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HF U 1 , ky/ kx= 0.1 HF U 2 , ky/ kx= 0. 1 HF U 3 , ky/ kx= 0.1 HF U 4 , ky/ kx= 0. 1
HFU 1, ky/kx=1 HFU 2, ky/kx=1 HFU 3, ky/kx= 1 HFU 4, ky/kx=1
Fig. 20 Comparison of layer pressures at moni toring po intno. 1 for ky/kx= 0.10 and 1.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HFU 1 , ky/kx=0.1 HFU 2, ky /kx=0.1 HFU 3 , ky/kx=0.1 HFU 4, ky /kx=0.1
HFU 1, ky/kx=1 HFU 2, ky/kx=1 HFU 3, ky/kx=1 HFU 4, ky/kx=1
Fig. 21 Comparison of layer pressures at moni toring po intno. 2 for ky/kx= 0.10 and 1.
We compared simulated production profiles for all multi-layered cases with and without stress-dependent permeabil-ity and porosity and we saw little or no differences in the
decline behavior. Apparently, the vertical heterogeneitycaused by reservoir layering and permeability contrast amonglayers has a much larger impact on the decline behavior.These observations are based on results generated using the
functions shown in Figs. 4 and 5, so we should caution thatresults may be different with other stress-dependent functions.
The greatest impact occurred in the single-layer, homogen-
eous, and isotropic case. Following very similar initialdeclines, the inclusion of stress-dependent properties causedthe production profile (Fig. 22) to be much flatter during thefirst ten years of production, but to decline much faster afterthat time period. Although not shown, the computed b-
exponents match our observations. We also computed largerreserve estimate errors than those summarized in Table 7during the first ten years of production when stress-dependentproperties were included.
Lateral Rock Heterogeneity. We frequently encounter
variations in rock properties in low-permeability reservoirs inthe lateral direction (i.e., x- and y-directions) caused by
differential diagenetic events following deposition. Dia-
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SPE 109625 Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use and Misuse of an Arps Decline Curve Methodology 11
genesis defined as any physical or chemical processcausing changes in initial rock properties is the principalsource of reductions in both permeability and porosity in tight
gas sands. The primary diagenetic processes typically seen intight gas sands are mechanical and chemical compaction,quartz and other mineral cementation, mineral dissolution, andclay genesis. Diagenesis may affect the rock properties
sufficiently enough to cause the rock to act as flow baffles,and in extreme cases, may create flow barriers.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProductionRate,
Mscf/day
with stress-dependent properties
w/o stress-dependent properties
Fig. 22 Effects o f st ress-dependent reservoi r properties onlong-term production decline behavior for thesingle-layer case.
To evaluate the effects of diagenesis, we randomly populatedthe model grid cells in each layer with variable effectiveporosity and absolute horizontal permeability values. Eachrandom distribution was constrained by the maximum and
minimum range of values observed for each HRT shown inFig. 2. Moreover, the random property distributions for each
layer or HFU were generated so that the averages equaled theindividual layer values shown in Tables3-5for the 4-, 8, and16-layer cases, respectively. Maintaining these averageproperties throughout this work allowed us to make relevant
comparisons among the various cases.
Except for the inclusion of the heterogeneous distribution of
properties, we used the reservoir and hydraulic fractureproperties shown in Table6 as input for the simulation. Wecompared simulated production profiles for all layered cases with and without heterogeneity and we observed nosignificant differences. Similar to the evaluation of stress-dependent reservoir properties, it appears as if the vertical
heterogeneity caused by reservoir layering and permeabilitycontrast among layers has a much bigger effect on the declinebehavior than does the lateral heterogeneity.
Effect of Fracture Properties on Production DeclineIn this section, we evaluate the effects of various hydraulicfracture properties and heterogeneities including variableeffective fracture lengths and conductivities, unequal fracturelengths ("wings"), stress-dependent fracture properties, and
two-phase and non-Darcy flow on the production declinebehavior. Decline characteristics were again quantified andcompared using the Arps decline exponent, b. Except forvariations in specific hydraulic fracture properties beingevaluated, all reservoir and hydraulic fracture properties
summarized in Table6were used as input for the simulationcases.
Effective Fracture Half-Length. We first examined theinfluence of effective or propped fracture half-length on pro-duction decline behavior. All of the previous simulationstudies were generated with an effective hydraulic fracture
half-length of 300 ft, but we also evaluated effective half-lengths,Lf= 50, 100, and 500 ft in our study.
The simulated short- and long-term production declines for the4-layer case with Lf= 50, 100, 300 and 500 ft are shown inFigs. 23 and 24, respectively. As expected, the cases withshorter effective fracture half-lengths exhibit steeper initialdeclines but are followed by flatter profiles during the earlyyears of production. Conversely, cases with longer effective
fracture half-lengths display a more gradual initial decline thatis followed by sharper production decline profile.
100
1,000
10,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasProductionRate,
Mscf/day
Lf=50 ft
Lf=100 ftLf=300 ft
Lf=500 ft
Fig. 23 Simulated shor t-term production prof iles for L f= 50,
100, 300 and 500 ft.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProductionRate,
Mscf/day
Lf=50 ft
Lf=100 ft
Lf=300 ft
Lf=500 ft
Fig. 24 Simulated long-term producti on profiles for L f= 50,100, 300 and 500 ft.
Although the initial declines are much steeper, the long-termdeclines (as illustrated by Fig. 24 for the shorter effectivehydraulic half-lengths) seem to be slightly flatter for most ofthe productive period. Differences between decline profiles
after about ten years of production are, however, very small.
Our observations for the decline behavior are validated by
examination of the computed b-exponents (see Table13). Al-though the b-exponents are consistently larger for cases with
shorter fracture half-lengths, all b-exponents approach a value
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of one after 50 years of production. Unlike previous cases, wealso observe b-exponents much greater than 3.0 for fracturehalf-lengths of 50 and 100 ft during the first year of
production.
Table 13 Com puted b-exponents for Lf= 50, 100, 300 and500 ft.
ProducingTime Period(years)
L f=50 ft
L f=100 ft
L f=300 ft
L f=500 ft
1 4.01 3.60 2.78 1.915 1.53 1.44 1.35 1.20
10 1.10 1.08 1.04 1.0320 1.07 1.06 1.01 0.9650 1.02 1.01 1.00 0.97
Reserve estimate errors for various time periods and computed
from the best-fit Arps1 decline curve through the simulated
data forLfvalues of 50, 100, 300 and 500 ft are summarized inTable14. We again see that there is a correlation between thelargest reserve estimate errors and the largest computed b-exponents shown in Table 13. Interestingly, the reserve
estimate errors are less than 10 percent after ten years ofproduction. These small errors suggest that each case may beapproaching the "true" b-exponent for all fracture half-lengthsinvestigated in this study.
Table 14 Computed reserve estimate errors for L f = 50,100, 300 and 500 f t.
ProducingTime Period
(years)
L f=50 ft(%)
L f=100 ft(%)
L f=300 ft(%)
L f=500 ft(%)
1 144.6 139.3 109.6 78.05 33.8 24.5 21.7 12.0
10 14.2 9.7 7.8 6.520 11.3 9.3 7.3 6.2
Figures 25 and 26 show the simulated layer pressures atmonitoring points no. 1 and 2, respectively, for Lf = 50 ft(color-filled circles) andLf= 500 ft (color-filled triangles). At
pressure monitoring point no. 1 (Fig. 25), we see that (asexpected) the wells with longer effective fracture half-lengths
will recover gas more effectively in all layers. We even seesignificant pressure reductions in the least permeable layer(HFU 4) after less than 2 years of production for theLf= 500ft case. Even though it is delayed for almost 20 years, we alsosee small but measurable pressure reductions in HFU 4 for the
Lf= 50 ft case.
Similar to the results from our investigation of reservoir pro-
perties on production decline behavior, we also see evidence
of an elliptical flow geometry from the pressure responsesshown at pressure monitoring point no. 2 in Fig. 26. Althoughthe layer pressure behavior in HFU 1 and 2 are comparable,
we see no measurable pressure reduction in HFU 4 for eitherthe Lf = 50 or 500 ft case. More significantly, we see
essentially no pressure reduction for either fracture half-lengthcase after almost 30 years of production.
Based on the pressure responses shown in Figs. 25and 26, weoffer several conclusions. First, we observe an elliptical flowgeometry for a wide range of effective fracture half-lengths.The flow duration and geometrical parameters may be
different, but all cases exhibit elliptical flow. Secondly, the
pressure responses in HFU 4 for both the Lf = 50 or 500 ft
cases suggest that both cases were either in transient ortransitional flow during the first 30 years of production. Itdoes appear, however, that both cases had either reached or
were approaching boundary-dominated flow. Again, sinceHFU 4 does not represent a significant percentage of the totalhydrocarbon pore volume, it apparently does not affect theoverall decline behavior as significantly as HFUs 1-3.
Consequently, the lack of boundary-dominated flow in HFU 4does not prevent the well from "behaving" as if it were in trueboundary-dominated flow conditions after 30 years.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HFU 1, Lf = 50 ft HFU 2, Lf = 50 ft HFU 3, Lf = 50 ft HFU 4, Lf = 50 ft
HFU 1 , L f = 5 00 ft HFU 2 , L f = 5 00 ft HFU 3 , L f = 5 00 ft HFU 4 , L f = 5 00 f t
Fig. 25 Comparison of layer pressures at pressure moni tor-ing point no. 1 for L f= 50 and 500 ft.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
Bottomhole
Pressure,psia
HFU 1, Lf = 50 ft HFU 2, Lf = 50 ft HFU 3, Lf = 50 ft HFU 4, Lf = 50 ft
HFU 1, L f = 5 00 ft HF U 2 , Lf = 50 0 ft HF U 3, Lf = 500 ft HF U 4, L f = 5 00 ft
Fig. 26 Comparison of layer pressures at pressure moni tor-ing point no. 2 for L f= 50 and 500 ft.
Absolute Fracture Conductivity. We next evaluated the
effects of absolute fracture conductivity, wfkf, on the
production decline behavior. All of the previous simulatedperformance cases were generated using an absolute fractureconductivity, wfkf= 50 md-ft (i.e., a numerical FCD= 18.5).
Figures 27 and 28 compare the short- and long-termproduction decline profiles, respectively, for wfkf= 0.5, 5.0, 50and 500 md-ft (i.e., numerical FCD = 0.185, 1.85, 18.5, and
185, respectively). Cases with lower effective fracture con-ductivities exhibit much steeper initial declines followed byflatter profiles during the early years of production, whilecases with higher effective fracture conductivities display a
gradual initial decline followed by a sharper long-termproduction decline profile.
We also note that cases with numerical conductivities of 50and 500 md-ft are almost identical. These similarities suggest
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the decline behavior for all values of infinitely conductivefractures will be similar, while the production profiles forfractures of (low) finite conductivity will vary depending on
the effective conductivity.
Fig. 27 Simulated short-term produc tion profi les fo r wfk f =
0.5, 5, 50 and 500 md-ft.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProductionRate,
Mscf/day
wfkf=0.5 md-ft
wfkf=5 md-ft
wfkf=50 md-ft
wfkf=500 md-ft
Fig. 28 Simulated long-term production p rofi les for wfk f =0.5, 5, 50 and 500 md-ft.
The computed b-exponents (Table 15) reflect the declinebehavior shown in Figs. 25 and 26. Similar to those caseswith short effective fracture half-lengths shown previously inTable 13, cases with low fracture conductivities have b-
exponents greater than 3 for the first year of production. Wealso note that all b-exponents equal or are approaching oneafter 50 years of production. In fact, the computed b-exponents change little after about ten years of production,
which suggests those cases are approaching boundary-dominated flow conditions.
Table 15 Com puted b-exponents for wfk f = 0.5, 5, 50 and500 md-ft.
ProducingTime Period
(years)wfkf=
0.5 md-ftwfk f=
5 md-ftwfk f=
50 md-ftwfkf=
500 md-ft1 4.34 3.71 2.78 2.345 1.87 1.58 1.35 1.14
10 1.29 1.07 1.04 1.0220 1.14 1.04 1.01 1.0150 1.05 1.02 1.00 1.00
Reserve estimate errors for various time periods are sum-
marized in Table16. We again see a correlation between the
largest reserve estimate errors and the largest computed b-exponents shown in Table 15. We also note the reserveestimate errors are less than 10 percent after ten years of
production. These small errors also suggest that each casemay be approaching the "correct" b-exponent for all fractureconductivities investigated in this study.
100
1,000
10,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasProductionRate,
Mscf/day
wfkf=0.5 md-ft
wfkf=5 md-ftwfkf=50 md-ft
wfkf=500 md-ft
Table 16 Computed reserve estimate errors for wfk f= 0.5,5, 50 and 500 md-ft.
ProducingTime Period
(years)
wfk f=0.5 md-ft
(%)
wfk f=5 md-ft
(%)
wfkf=50 md-ft
(%)
wfk f=500 md-ft
(%)
1 138.6 135.2 109.6 107.85 41.8 38.1 21.7 10.6
10 10.5 8.2 7.8 7.620 8.2 7.7 7.3 7.2
Although not shown, we also monitored the simulated layerpressures for each of the fracture conductivities that weremodeled. We observed similar pressure responses as thoseshown in Figs.25 and 26 for various effective fracture half-
lengths. In general, the pressure responses for fracture cases
with higher conductivities behaved similarly to those forlonger, effective fracture half-lengths, while fractures withlower fracture conductivities behaved like wells with short,
effective fracture half-lengths.
Unequal Fracture Wing Lengths. In this section, weevaluated the effects of unequal hydraulic fracture half-lengthson the production decline behavior. All previous simulationswere conducted assuming that the fracture "wings" extended
equal distances on either side of the wellbore; however,heterogeneities in rock properties may cause fractures to growunequally during the stimulation treatment.
For this study, we modeled wing length ratios ranging from
1.0 (i.e., equal lengths), 1.5, 3 and 6.0. Results from our studyindicate that variations in fracture wing ratios do not signifi-cantly affect long-termproduction profiles but this variablewill cause variations in the short-term decline behavior.
Figure 29 illustrates the decline behavior for the first fiveyears. In general, larger wing length ratios exhibit steeperinitial declines and flatter profiles during this time period.
100
1,000
10,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasProductionRate,
Mscf/day
Lf1/Lf2=6
Lf1/Lf2=3
Lf1/Lf2=1.5
Lf1/Lf2=1
Fig. 29 Simulated short-term production profiles for L f1/L f2= 1.0, 1.5, 3.0 and 6.0.
Computed b-exponents are summarized in Table 17 for the
same wing length ratios shown in Fig. 29. We see larger wing
length ratios result in larger b-exponents for the first five years
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14 J.A. Rushing, A.D. Perego, R.B. Sullivan, and T.A. Blasingame SPE 109625
of production. However, we also notice that the b-exponentsfor all wing length ratios start to approach a value of 1.0 afterten years of production. In fact, the values change little for the
last 30 years of production.
Table 17 Com puted b-exponents for hydraulic fracturewing length ratios of 1, 1.5, 3 and 6.
ProducingTime Period(years)
L f1/Lf2= 1.0
L f1/L f2= 1.5
L f1/L f2= 3.0
L f1/Lf2= 6.0
1 2.78 2.91 3.05 3.235 1.35 1.37 1.38 1.40
10 1.04 1.11 1.12 1.1220 1.01 1.01 1.03 1.0350 1.00 1.01 1.03 1.03
The computed reserve estimate errors are compiled in Table
18. We observe the same relationships that we have seen inall of our previous results between the computed b-exponentsand reserve estimate errors. The largest reserve estimateerrors occur during the first five years of production when thewells are experiencing either transient or transitional flow
conditions.Table 18 Computed reserve estimate errors for L f1/Lf2= 1,
1.5, 3 and 6.
ProducingTime Period
(years)
L f1/Lf2= 1.0(%)
L f1/L f2= 1.5(%)
L f1/L f2= 3.0(%)
L f1/Lf2= 6.0(%)
1 109.6 133.9 129.4 135.65 21.7 22.9 23.4 23.6
10 7.8 10.0 10.1 10.120 7.3 7.3 7.6 7.7
Although not shown, the layer responses at pressure moni-toring points 1 and 2 confirm that HFUs 1-3 have experienced
significant pressure depletion during the first 20 years of
production. Similar to previous results, HFU 4 does notexperience much pressure reduction for the first 20 years(especially at monitoring point no. 2). But, since HFU 4
represents a much smaller percentage of the total connectedhydrocarbon pore volume than HFUs 1-3, the overall welldecline stabilizes and behaves as if it were in true boundary-dominated flow. This explains why the computed b-exponentsshown in Table 17appear to be stabilizing after 20 years of
production.
Variable Fracture Conductivity (Choked Fracture Case).Next, we considered the effects of variable fractureconductivities, or more specifically, the "choked" fracture casein which the fracture conductivity near the wellbore is lowerthan in the fracture towards the tip. The variable fracture con-
ductivity is quantified by the ratio of fracture conductivities,kf1/kf2, where kf1and kf2are the near (wellbore) and far (field)
fracture conductivities, respectively. Figures30and 31pre-sent the short- and long-term production profiles, respectively,for kf1/kf2 = 0.01, 0.10 and 1.0. As illustrated in Fig. 30, theeffect of lower fracture conductivity in the near-wellbore area
causes steep initial declines in the production rates followedby relatively flat profiles for the first five years. Long-termproduction profiles also tend to remain flat for the entireproducing period shown in Fig. 31.
Computed b-exponents for the production profiles shown in
Figs. 30 and 31 are summarized in Table 19. Both of the
choked fracture cases with kf1/kf2 < 1 have b-exponentssignificantly greater than 3.0 during the first year. As wewould expect, the corresponding reserve estimate errors
(Table 20) exceed 100 percent. We also note that the b-exponents for the kf1/kf2 = 1 case tend to "stabilize" after about10 years of production, while the choked fracture casescontinue to change after that time period. All cases, however,
equal or approach 1.0 after 50 years of production.
100
1,000
10,000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Producing Time, years
GasProductionRate,
Mscf/day
kf1/kf2=0.01
kf1/kf2=0.10
kf1/kf2=1.0
Fig. 30 Simulated shor t-term production prof iles for kf1/k f2=0.01, 0.10 and 1.0.
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProductionRate,
Mscf/day
kf1/kf2=0.01
kf1/kf2=0.10
kf1/kf2=1.0
Fig. 31 Simulated long-term production profil es for k f1/kf2=0.01, 0.10 and 1.0.
Table 19 Com puted b-exponents for k f1/kf2 = 0.01,0.10 and 1.0.
ProducingTime Period
(years)
kf1/kf2
= 0.01
k f1/kf2
= 0.10
k f1/k f2
= 1.01 4.73 4.20 2.785 1.69 1.54 1.3510 1.22 1.14 1.0420 1.15 1.06 1.0150 1.01 1.00 1.00
Table 20 Computed reserve estimate errors for k f1/kf2= 0.01, 0.10 and 1.0.
ProducingTime Period
(years)
kf1/kf2= 0.01
(%)
k f1/kf2= 0.10
(%)
k f1/k f2= 1.0(%)
1 115.2 114.9 109.65 26.1 23.0 21.710 10.1 8.7 7.820 7.9 7.6 7.3
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Stress-Dependent Fracture Conductivity. Most hydraulicfractures, particularly those in reservoirs with high initialstress conditions (similar to HP/HT gas reservoirs), experience
some reduction in conductivity created during the initialstimulation treatment. The most common causes of con-ductivity reduction are proppant crushing and/or embedmentinto the reservoir rock. Some proppants (i.e., resin-coated
ceramic, bauxite, and resin-coated sands) are designed tominimize conductivity reductions in high-stress reservoirs.
For this paper, we evaluated the impact of stress-dependent
reservoir properties on the production decline using thefunctions shown in Fig. 9. These curves represent the fractionof original fracture conductivity as a function of the netfracture pressure for various types of generic proppants.
Except for the inclusion of stress-dependent reservoirproperties, we used the reservoir and hydraulic fracture
properties shown in Table6as input for the simulation.
We compared simulated production profiles for all multi-
layered cases with and without stress-dependent permeabil-
ity and porosity and we saw little or no differences in thedecline behavior. Again, the vertical heterogeneity caused byreservoir layering and permeability contrast among layers hasa much more significant effect on the decline behavior than
the stress-dependent fracture conductivity.
Two-Phase and Non-Darcy Fracture Flow. The finalfracture characteristics evaluated in our study were two-phaseand non-Darcy flow phenomena. Two-phase flow conditions
typically occur primarily early in the production historyimmediately following the stimulation treatment and duringthe fracture clean-up period. Non-Darcy flow occurs duringany phase of the production history and is dependent on the
flowrate and pressure drawdown.We modeled two-phase flow during the fracture cleanupperiod using a method similar to that described in Reference
24. Two-phase (gas-water) flow in the fracture was accountedfor using the relative permeability curves shown in Fig.8. Wealso varied fracture conductivity from 50 to 500 md-ft.Simulation results show that two-phase flow primarily affectsvery earlyproduction declines, but has little to no impact onthe long-termdecline behavior.
We also evaluated non-Darcy flow in the fracture. For thisstudy, we modeled non-Darcy flow using a rate-dependent
skin,D.3-5
Similar to the two-phase flow simulations, we also
varied fracture conductivity from 50 to 500 md-ft. Again, thesimulation results suggest that non-Darcy flow affects pri-marily the earlydecline behavior but has no substantial impacton the long-termproduction profiles. For both the two-phase
and non-Darcy flow simulations, vertical heterogeneity fromlayering and permeability contrast among layers had a muchlarger effect on the long-term production decline.
Effect of Well Spacing on Product ion DeclineAll of the previous results evaluating the effects of reservoirand hydraulic fracture properties were generated for a welldrilled on an 80-acre spacing. In this section, we evaluate theimpact of other well spacings on the production decline
behavior. Except for variations in well spacing, all reservoir
and hydraulic fracture properties shown in Table6were usedas input for the simulation cases.
Figure 32 compares the simulated long-term productionprofiles for well spacings of 40, 80 and 160 acres per well. As
expected, we observe much sharper initial decline rates as wellas much steeper long-term profiles for wells drilled on closer
spacing. The computed b-exponents for each well spacingalso reflect our observations concerning both short- and long-
term decline characteristics. As shown in Table21, computedb-exponents are consistently lower for smaller well spacings
(for the time periods shown). Surprisingly, we also note thatall b-exponents, regardless of the spacing, lie approximately inthe range 0.50 < b< 1.0 after 50 years of production.
1
10
100
1,000
10,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
GasProduction
Rate,
Mscf/day
40-Ac Spacing
80-Ac Spacing
160-Ac Spacing
Fig. 32 Simulated long-term production profiles for wellspacings of 40, 80 and 160 acres per well.
Table 21 Compu ted b-exponents for well spacings of 40,80 and 160 acres per well.
ProducingTime Period(years)
40 AcresperWell
80 AcresperWell
160 AcresperWell
1 2.11 2.78 2.985 1.15 1.35 1.96
10 1.02 1.04 1.0920 0.97 1.01 1.0650 0.84 1.00 1.05
The reserve estimate errors for the three well spacings con-sidered in our study are shown in Table 22. Similar to theprevious results for various reservoir and hydraulic fractureproperties, we see a direct correlation between the computedb-exponent and reserve estimate error. The results also showthat the largest errors for the 40-acre and 80-acre well spacing
cases occur with less than five years of production, while theerrors for the 160-acre spacing case are still quite significant(greater than 10%) for the first 20 years of production.
Table 22 Computed reserve estimate errors for wellspacings of 40, 80 and 160 acres per w ell.
ProducingTime Period
(years)
40 Acresper Well
(%)
80 Acresper Well
(%)
160 Acresper Well
(%)
1 67.0 109.6 122.25 10.5 21.7 25.9
10 7.2 7.8 17.420 5.5 7.3 14.7
Figures 33 and 34 present the simulated layer pressures at
monitoring points no. 1 and 2, respectively, for well spacings
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16 J.A. Rushing, A.D. Perego, R.B. Sullivan, and T.A. Blasingame SPE 109625
of 40-acres per well (color-filled circles) and 80-acres perwells (color-filled triangles). At pressure monitoring point 1(Fig. 33), we see that all HFUs for both well spacings
experience some pressure depletion within the first year ofproduction. As expected, the largest pressure reductions occurfor the 40-acre spacing case.
Figure34shows the pressure response at pressure monitoringpoint no. 2. Unlike most of the previous cases in which wesaw little or no pressure change for the entire 50-yearproducing period, we see significant pressure depletion in
HFU 4 for the 40-acre spacing case after 20 years ofproduction. Based on the pressure responses shown in Figs.33and 34, we conclude that the well is in boundary-dominatedflow after 20 years of production. And, we would expect the
reserve estimate errors to be significantly lower after that timeperiod.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HFU 1, 40-Ac HFU 2, 40-Ac HFU 3, 40-Ac HFU 4, 40-Ac
HFU 1, 80-Ac HFU 2, 80-Ac HFU 3, 80-Ac HFU 4, 80-Ac
Fig. 33 Comparison of layer pressures at pressure moni tor-ing point no. 1 for well spacings of 40- and 80-acres
per well.
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5 10 15 20 25 30 35 40 45 50
Producing Time, years
BottomholePressure,psia
HFU 1, 40-Ac HFU 2, 40-Ac HFU 3, 40-Ac HFU 4, 40-Ac
HFU 1, 80-Ac HFU 2, 80-Ac HFU 3, 80-Ac HFU 4, 80-Ac
Fig. 34 Comparison of layer pressures at pressure moni tor-ing point no. 2 for well spacings of 40- and 80-acresper well.
Effect of Reservoir Pressure and Temperature onProduction DeclineThe final phase of our simulation studies addressed the effectsof both reservoir pressure and temperature conditions. Recallthat all of the previous results were generated with an initialreservoir pressure of 16,200 psia (i.e., initial pore pressuregradient of 0.90 psi/ft at an average well depth of 18,000 ft)
and a bottomhole reservoir temperature of 400o
F.
To assess the impact of pressure, we also simulated severalcases with initial reservoir pressures of 10,800, 12,600, and14,400 psia corresponding to pore pressure gradients of 0.60,
0.70 and 0.80 psi/ft, respectively. All simulated cases used toevaluate the effects of reservoir pressure were generated witha bottomhole temperature of 400
oF. Although not shown, the
simulated production profiles generally exhibited lower initial
production rates and sharper initial declines for lower initialreservoir pressures. However, both the intermediate- and
long-term decline behavior were very similar for all pressures.Moreover, differences between the computed b-exponents andreservoir estimate errors were also very minor.
We also compared the production decline characteristics forreservoir temperatures of 300oF and 400oF for the same rangeof initial reservoir pressures. Again, we observed very littledifferences in the results generated using these two tem-
peratures for the complete range of pressures considered.
ConclusionsBased on the results of our simulation study, we offer thefollowing conclusions about use of an Arps decline curvemethodology for evaluating reserves in tight gas sands atHP/HT reservoir conditions:
1. The most significant error source for reserve evaluations
using a traditional Arps decline curve methodology isincorrect application of Arps' decline curves during eithertransient flow or the transitionalperiod between the end of
transient and onset of boundary-dominated flow. Duringboth of these periods (principally the transient period), we
observed b-exponents greater than one and correspondingreserve estimate errors exceeding 100 percent.
2. Although only a few of the simulated cases evaluated in our
study reached true boundary-dominated flow during the first
50 years of production, we found that the reserve estimateerrors were quite often less than 10 percent when the well
was in the "late" transitional flow period. It appears thatreserve estimate errors using the Arps models will be
negligible if the least permeable layers contributing to flowrepresent a small percentage of the overall contacted pore
volume.
3. b-exponents computed from the long-term (i.e, 50 years)
decline characteristics generally approached values between0.5 and 1.0 for the range of reservoir and hydraulic fracture
properties and heterogeneities investigated in our study.Agreement between Arps' suggested b-exponent range and
our simulated evaluations also indicates that, if appliedunder the appropriate conditions, the Arps rate-time models
are appropriate for assessing reserves in tight gas sands atHP/HT reservoir conditions.
4. Hyperbolic decline behavior commonly observed in wellsproducing from tight gas sands is caused by various types ofboth reservoir and hydraulic fracture heterogeneities. Our
simulated results demonstrate that low permeability alone ina homogeneous system is not sufficient to generate large b-
exponents and flat declines.
5. Although not shown, terminal decline rates for the simulatedproduction ranged from about 1.5 to 5.0 percent for the
ranges of reservoir and hydraulic fracture properties andheterogeneities evaluated in our study. These rates are basedon our reservoir inflow model and do not account for
wellbore problems that might affect well outflowperformance. These results also suggest that much larger
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SPE 109625 Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use and Misuse of an Arps Decline Curve Methodology 17
terminal decline rates may not be attributable to reservoirphenomena, but may be caused by operational problems
(e.g., liquid loading in the wellbore, loss of fractureconductivity, plugging or closure of perforations, etc.).
AcknowledgmentsWe would like to express our thanks to Anadarko Petroleum
Corp. for permission to publish this paper.
Nomenclatureb = Arps' decline exponent, dimensionlessD = rate-dependent skin factor, (MMscf/d)-1
Di = initial decline rate, (days)-1
qi = initial gas production rate, Mscf/day
h = gross sand thickness, ftHFU = hydraulic flow unit
HRT = hydraulic rock typeFCD = dimensionless fracture conductivity = wfkf/kgLfkg = absolute Klinkenberg-corrected permeability, md
kv = absolute vertical permeability, mdkh = absolute horizontal permeability, md
kx = absolute horizontal permeability in thex-direction, mdky = absolute horizontal permeability in they-direction, mdLf = effective fracture half-length, ft
wfkf = fracture conductivity, md-ftkf = absolute fracture permeability, md
pi = initial bottomhole reservoir pressure, psia
= effective porosity, frac.g = gas specific gravity, dimensionless
HC = gas hydrocarbon component specific gravity,
dimensionless
Sw = water saturation, frac.
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