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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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2 J.A. Rushing, A.D. Perego, R.B. Sullivan, and T.A. Blasingame SPE 109625

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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SPE 109625 Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use and Misuse of an Arps Decline Curve Methodology 3

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


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.


(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


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


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


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.


(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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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.


(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


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



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


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


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.


(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.


(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



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



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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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


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


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


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.


(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



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


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


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.


(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


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.


(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


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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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.


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.


(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


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


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.


(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.


(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


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.


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.


(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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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



HFU 1, 40-Ac HFU 2, 40-Ac HFU 3, 40-Ac HFU 4, 40-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





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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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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