SPE 114514 - Petroleum Engineering - Texas A&M University

SPE 114514 Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery of Coalbed Methane Wells J.A. Rushing, SPE, Anadarko Petroleum Corp., A.D. Perego, SPE, Anadarko Petroleum Corp., and T.A. Blasingame, SPE, Texas A&M University

Copyright 2008, Society of Petroleum Engineers This paper was prepared for presentation at the CIPC/SPE Gas Technology Symposium 2008 Joint Conference held in Calgary, Alberta, Canada, 16–19 June 2008. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract This paper presents results of a simulation study designed to evaluate the applicability of Arps' [1945] decline curve methodology for assessing reserves in coalbed methane reservoirs. We simulated various coal properties and well/operational conditions to determine their impact on the production decline behavior as quantified by the Arps decline curve exponent, b. We then evaluated the simulated production with Arps' rate-time equations at specific time periods during the well's production decline period and compared estimated reserves to the "true" value (defined in this paper as the 30-year cumulative production volume). To satisfy requirements for using Arps' models, all simulations were conducted using a constant bottomhole flowing pressure condition in the wellbore. The significant results from our study include: All of the computed values of the long-term decline exponents were within the limits originally defined by Arps, i.e., 0.0 < b < 1.0. Agreement between Arps' recommended b-exponent range and our results using simulated performance data also suggests that, if applied under the correct conditions, the Arps rate-time models are appropriate for assessing reserves in coalbed methane reservoirs; The Arps b-exponents were not constant during the production decline period. For many simulated cases, the early decline behavior (within a few years after reaching the peak production rate) appeared to have exponential decline but eventually became more hyperbolic later in the well's life. Use of Arps' exponential model early in the production history in those wells with long-term hyperbolic decline behavior tended to underestimate gas reserves; The largest reserve estimate errors typically occurred during the first few years after reaching the peak production rate and during the initial production decline period. For those wells exhibiting long-term hyperbolic behavior, the initial reserve estimate errors underestimated reserves by as much as 20 to 30 percent; Heterogeneities in coal properties cause the production declines to deviate from exponential to hyperbolic. Properties having the largest impact on the production decline behavior include the shape of the adsorption isotherm, cleat permeability anisotropies, the shape of cleat gas-water relative permeability curves, stress-dependent cleat permeability and porosity, and layered coal seams with differences in initial reservoir pressures; We also observed a strong influence of well flowing pressure conditions as modeled with a bottomhole flowing pressure constraint. For all other properties and conditions being equal, wells with lower bottomhole flowing pressures exhibited more long-term hyperbolic behavior as defined by higher Arps b-exponents. Introduction Unconventional natural gas resources — tight gas sands, naturally-fractured gas shales, coalbed methane, and deep basincentered gas systems — comprise a significant percentage of our domestic natural gas resource base identified to date and represent an important source for future natural gas production and reserve growth. According to Kawata and Fujita [2001], the coalbed methane (CBM) resource-in-place in North America is estimated to total more than 3,000 Tcf. While the resource base is large, the unique gas storage and flow properties characteristic of CBM reservoirs make efficient and effective gas recovery technically difficult. Of the total resource in place, the total technically recoverable gas is estimated to be 98 Tcf. Those same unique coal properties also cause CBM production profiles to differ in shape from the production profiles for common, more conventional reservoirs. And, these differences in production profiles present unique challenges for the

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accurate evaluation of reserves. The typical CBM production profile is an initial inclining production period to some peak or maximum production rate and followed by a declining rate profile. Consequently, assessing CBM reserves using the traditional Arps decline curve models and methodology is only viable during the period when the gas production is declining. However, CBM reserves may be estimated prior to the onset of a declining production profile using various reservoir models as well as numerical simulation. Regardless of the challenges posed by a typical production profile, CBM reserves are routinely assessed using the traditional Arps decline curve methodology and models. The Arps decline curve evaluation methodology consists of plotting the logarithm of production rate against time, history matching the production data using Arps' rate-time equations — i.e., the Arps exponential, hyperbolic or harmonic decline models — and extrapolating the established trend into the future. The long-term decline behavior of the extrapolated trend is typically quantified using Arps' decline exponent, b. The original Arps paper [1945], which was developed and initially applied to conventional reservoirs, indicated the b-exponent should fall between 0 and 1.0 on a semilog production plot. However, the correct b-exponent in unconventional resources like CBM reservoirs is difficult to identify correctly, particularly during the early decline period after reaching the maximum production rate. And, selection of the incorrect b-exponent will have a tremendous impact on the accuracy of CBM reserve estimates. To address these problems with reserve evaluations using the Arps models in CBM reservoirs, we have conducted a series of single-well, parametric simulation studies or "experiments" to develop a better understanding of both the short- and long-term production decline behavior and to identify those parameters affecting the production decline. For this work, our specific study objectives are: To validate the practicality and utility of the Arps rate-time relationships and decline curve methodology for estimating reserves accurately in coalbed methane reservoirs; To develop physical interpretations of Arps' b-exponents in CBM reservoirs — i.e., to determine what coal properties and operational conditions most affect the value of b; and To provide guidelines for the applicable range of b-exponents for reserve evaluations. Description of CBM Model We developed a three-dimensional, two-phase (gas-water) finite-difference model using the Computer Modeling Group's GEM (Generalized Equation-of-State Model) simulator [Reference 9]. GEM is a three-phase, multi-component compositional equation-of-state model that has been modified and adapted to capture all of the storage and flow phenomena characteristic of coalbed methane reservoirs. The model also has the capability of incorporating stress-dependent and sorption-controlled changes in coal porosity and permeability during the gas and water production process. All simulations were conducted for a single-well on a spacing of 80 aces per well. The grid system was constructed with 1521 grids (39 grids in both the x- and y-directions). We employed a Cartesian grid geometry so that we could model linear flow geometry and any permeability anisotropy associated with the natural fracture or cleat system in coals. Grid dimensions in the x- and y-directions were smaller immediately around the wellbore but increased geometrically away from the wellbore. Dimensions in the vertical direction ranged from three grids in the single-layer case to fifteen grids in the five-layer cases. We should note that we developed a reservoir model that addressed reservoir inflow performance, and other than using a bottomhole flowing pressure constraint, we did not attempt to model well outflow performance. Productive coals are characterized by an extensive, orthogonal set of natural fractures or cleats as illustrated schematically in Fig. 1. The primary cleat system is often referred to as the "face" cleats, while the orthogonal cleats are called "butt" cleats. Typically, the face cleats are better connected and more continuous, while the butt cleats are less well connected and more discontinuous. Spacing between face cleats ranges from tenths of an inch to several inches. Interactions between the natural fracture or cleat system and the coal matrix are modeled with the dual-porosity system (Fig. 1) developed by Gilman and Kazemi [1983], where this model is modified to include all coal storage and flow processes.

Actual Coal System

Idealized Coal Model Face Cleat

Coal Matrix Butt Butt Cleat Cleat Fig. 1 — Schematic diagram comparing actual coal cleat and matrix system with idealized dual-porosity model used in study.

The majority of coal gas is stored by adsorption (i.e., gas molecules that are physically attached to the coal surfaces) rather than by "free" or unattached gas molecules stored in a matrix porosity structure similar to conventional sandstone or carbonate rocks. Most coal porosity is a combination of a micro-pore structure (pore diameters less than 2 nm) and a meso-pore structure (pore diameters between 2 and 50 nm). Because of the large surface area of the coal particles, significant volumes of gas may be stored in the adsorbed state in the micro- and meso-pore systems. Although it is usually not considered to be a significant contributor to either gas storage (as "free" gas that is not adsorbed)) or production (by Darcy flow), the matrix does

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

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have a limited macro-porosity that is characterized by pore diameters greater than 50 nm. Typically, lower rank coals have a proportionately greater volume of macro-pores than higher rank coals. Gas adsorption is a surface phenomenon and is predominantly a physical bond caused by the inter-molecular attractive forces (i.e., van der Waals forces) between the hydrocarbon and coal molecules. Because physical adsorption is a weak bond due to low activation energy required for gas molecules to adsorb to the coal surfaces, it can be reversed relatively easily by increasing temperature or reducing pressure. The sorption process (either adsorption or desorption) is usually modeled with a Langmuir [1918, 1919] isotherm given by VL p ...............................................................................................................................................................................(1) Gc pL p where VL = pL = p =

Langmuir volume defined as the maximum volume of gas that can be adsorbed at an infinite pressure, scf/ton Langmuir pressure defined as the pressure at which half of VL is achieved, psia Pressure in the CBM reservoir, psia

Note that Eq. (1) models the adsorption characteristics for a pure coal; however, the adsorption capacity of a coal will be reduced by the presence of ash and/or moisture in the coal matrix. For our simulation study, we assume that the ash and moisture contents are negligible.

Desorption from Coal Surfaces

Diffusion Through Coal Matrix

Fluid Flow in Coal Cleat Network

As illustrated by the schematic diagram in Fig. 2, gas production from coals is a complex, threestage process beginning with desorption of gas molecules from the coal surfaces immediately MicroMesoMacroMicro-Porosity Meso-Porosity Macro-Porosity adjacent to the cleats. The desorption process (d < 2 nm) (2 nm < d < 50 nm) (d > 50 nm) Coal Coal begins when the cleat system has been Particle Matrix sufficiently dewatered and the cleat pressure has Fig. 2 — Schematic diagram illustrating three-stage coal gas production been reduced to the critical desorption pressure. process and interaction among mechanisms (adapted from Reference 3). Movement of the desorbed or detached gas molecules in the cleat system occurs as two-phase (gas-water) flow and results from a pressure gradient as governed by Darcy's Law. Gas production occurs when the gas accumulation in the cleat system reaches the critical gas saturation. When a sufficient volume of gas molecules have been desorbed from coal surfaces in the region in and near the cleat system and as pressures continue to decrease in the coal matrix, more gas molecules desorb. However, because the coal matrix is composed primarily of a very fine micro- and meso-pore structure, it has a very low flow capacity with absolute permeabilities in the micro-darcy range. Consequently, movement of gas molecules in the coal matrix occurs by diffusion [Thimons and Kissell, 1973; Crank, 1975; Smith and Williams, 1984] caused by a concentration gradient, rather than a pressure gradient. Diffusion is a very slow process and typically controls the production rates, ultimate recovery, and recovery efficiency. We should note that the GEM model does not allow gas flow through or from the macro-pores to the cleat system by Darcy flow. Decline Curve Analysis of Simulated CBM Production All simulated CBM production profiles or declines were evaluated using the industry-standard or "traditional" Arps decline curve methodology which consists of plotting production rate against time, history matching the production data using one of several industry-standard models — i.e., the Arps exponential, hyperbolic or harmonic decline models — and extrapolating the established trend into the future. Decline curve analysis as commonly practiced by the industry is quite simply a curve fitting process which does not necessarily have a theoretical basis. An exception to this statement is the exponential decline case which can be derived from a single-well model producing at a constant bottomhole flowing pressure during boundarydominated flow conditions. We evaluated the simulated production using common industry practices — i.e., assuming the Arps rate-time equations are applicable. Our objectives were to not only quantify the Arps decline curve parameters (i.e., the initial rate, qi; the initial decline rate, Di; and the decline exponent, b), but to also assess the reserves at various times during the well's productive life. Reserve estimates were obtained by extrapolating the best-fit Arps model through the simulated production. Reserve estimate errors were computed by comparing those estimated reserves to the "true" value — where for this paper, we define the "true" estimated ultimate recovery (EUR) or reserve volumes to be the 30-year cumulative production volume. Although the exponential and hyperbolic relations (as provided by Arps) are empirical, the Arps decline curves are excellent tools for estimating reserves when applied under the correct conditions. Unfortunately, many of us in the industry either have

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forgotten or have chosen to ignore the conditions under which use of the Arps decline curves are appropriate. Application of a decline curve methodology using the Arps models implicitly assumes the following [Lee and Wattenbarger, 1993]: Extrapolation of the best-fit curve through the current or historical production data is an accurate model for future production trends; There will be no significant changes in current operating conditions or field development that might affect the curve fit and the subsequent extrapolation into the future; The well is producing against a constant bottomhole flowing pressure; 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 form of the Arps rate-time (hyperbolic) decline equation: q(t )

qi [1 bDi t ] 1/b

....................................................................................................................................................................(2)

where Di is the initial decline rate, qi is the initial gas flowrate, and b is Arps decline curve constant (or decline exponent). Note that the units of these three variables must be consistent. Equation 2 has three different forms — exponential, harmonic, and hyperbolic — depending on the value of the b-exponent. Each of these equations exhibits a different shape on typical production plots: Cartesian format: gas production rate versus cumulative gas production, and Semilog format: logarithm of gas production rate versus time. The exponential or constant-percentage decline case is a special case of Eq. 2 where the b-exponent is zero, and is characterized by a decrease in production rate per unit of time that is directly proportional to the production rate. The exponential decline equation (b-exponent of zero) is written as: q(t ) qi exp[ Di t ] ......................................................................................................................................................................(3) Similarly, the harmonic decline is also a special case of Eq. 2 (b-exponent equals one), and is written as: q(t )

qi

.......................................................................................................................................................................(4)

[1 bDi t ]

The Arps hyperbolic decline is given by the general form (Eq. 2) and is "valid" for any condition where the b-exponent varies between 0 and 1.0. We should note that the value of the b-exponent determines the degree of curvature of the semilog decline plot (log(q) versus t), ranging from a straight line with b=0 to increasing curvature as b increases. Although Arps offered no theoretical basis, Arps did indicate that the b-exponent should lie between 0 and 1.0 — but he provided no justification of the possibility that b might be greater than one. Therefore, variations in the computed b-exponents outside of the expected range suggests the Arps' rate-time relationships may not be valid for modeling the decline behavior of coalbed methane reservoirs. Effect of Coal Properties on Production Decline Behavior We first evaluated the effects of various coal properties and heterogeneities — including multiple-seam or layered coals, effective cleat porosity, absolute cleat permeability, cleat permeability anisotropy, various Langmuir sorption isotherm curve shapes, shapes of the cleat gas-water relative permeability curves, and stress-dependent and sorption-controlled changes in coal properties — on the production decline behavior and reserve estimate errors. The CBM production decline characteristics were quantified and compared using primarily the Arps decline exponent, b. We evaluated the simulated production with Arps' rate-time equations at specific time periods during the well's production decline period and compared estimated reserves to the "true" value (i.e., defined in this paper as the 30-year cumulative production volume). Negative differences or errors represent an underestimation of reserves, while positive errors are overestimates.

Table 1 — Summary of coal properties and operational conditions used to simulate single-layer base case. Reservoir Property/Operational Condition

Value

Well spacing Depth to top of coal Initial bottomhole (cleat) pressure Bottomhole temperature Coal thickness Coal density Absolute cleat permeability Effective cleat porosity Cleat compressibility Cleat spacing Initial cleat water saturation Initial coal ash and moisture contents Langmuir pressure Langmuir volume Critical desorption pressure Initial gas content Desorption time Minimum bottomhole flowing pressure Maximum water production rate Stress- or desorption-dependent properties

80 acres/well 1,000 ft 461.25 psia o 90 F 50 ft 3 1.30 g/cm 50 md 3% -1 1.0e-05 psi 0.50 in 100% 0% 100 psia 300 scf/ton 361.25 psia 235 scf/ton 5 days 30 psia 500 bbl/day No

Coal Properties, Single-Layer Base Case. All simulated production decline cases are compared to either a single-layer or a five-layer base case. We begin with the single-layer base case described by the coal properties summarized in Table 1. We assumed a hydrocarbon gas that was pure methane with no nonhydrocarbon contaminants commonly found in produced coal gases, i.e., no carbon dioxide, nitrogen, helium, etc.

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Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

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Except for an investigation of the effects of well flowing pressure on the production decline, all cases were simulated using a minimum bottomhole flowing pressure constraint of 30 psia. We should note that the actual bottomhole flowing pressure was typically greater than the minimum constraint during much of the initial dewatering period and before reaching the maximum gas production rate; however, the bottomhole flowing pressure was essentially constant (pwf = 30 psia) during most of the decline period. We also assumed an initial, maximum water production rate of 500 bbl/day. The hypothetical adsorption isotherm used to simulate the single-layer base case (shown in Fig. 3) was computed with Eq. (1) and using the Langmuir constants shown in Table 1. We assumed the cleat system was initially undersaturated with a gas content of 235 scf/ton and a critical desorption pressure is 361.25 psia (i.e., the critical desorption pressure differential, pcdp = 100 psi). 1.00

300

Initial Reservoir Pressure = 461.25 psia

200 Critical Desorption Pressure = 361.25 psia

150

Relative Permeability, frac.

Adsorbed Gas Content, scf/ton

0.90

250

100

50

0

100

200

300

400

0.60 0.50 0.40 0.30 0.20

pL = 100 psia

0.10

500

600

700

Pressure, psia

Fig. 3 — Hypothetical gas adsorption isotherm (VL = 300 scf/ton, pL = 100 psia) used for base case.

Gas

0.70

VL = 300 scf/ton

0

Water

0.80

0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Water Saturation, frac.

Fig. 4 — Hypothetical cleat gas-water relative permeability used to simulated base case.

The hypothetical cleat gas-water relative permeability curves (shown in Fig. 4) for the single-layer base case were generated using the equations developed by Brooks and Corey [1964, 1966] and as summarized by Mavor [1996]. The effective gas and water permeabilities are given by Eqs. (5) and (6), respectively, as k rg

* ) n ......................................................................................................................................................................(5) k gr (1 S w

k rw

* m ...............................................................................................................................................................................(6) (S w )

where k rg = k gr

=

relative permeability to gas, fraction gas relative permeability coefficient, dimensionless

k rw =

relative permeability to water, fraction

k wr

water relative permeability coefficient, dimensionless gas relative permeability exponent, dimensionless water relative permeability exponent, dimensionless normalized water saturation, fraction

= n = m = S *w =

The normalized water saturation is defined by S w S wi ............................................................................................................................................................................(7) S *w 1 S wi where Sw = S wi =

water saturation, fraction irreducible water saturation, fraction.

Decline Curve Analysis, Single-Layer Base Case. The simulated gas production history for the single-layer base case is shown in Fig. 5. The production profile has the characteristic CBM shape with an initial inclining production period beginning at about 30 days after commencing water production. The decline period begins at 1.75 years after initial gas production and following a peak rate of 340 Mscf/day. Although the initial decline appears to be almost exponential, most of the decline behavior seems to be slightly hyperbolic.

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These observations are confirmed by the computed Arps decline b-exponents (summarized in Table 2) which were determined from the best-fit Arps model through the simulated production and at specific time periods during the production (decline) history. Rather than assuming any knowledge of decline behavior from field analogs, we simply history matched the simulated production and reported the b-exponent from the best-fit Arps model. We evaluated the production decline profiles at 5, 10, 20 and 30 years. Note that, except for the 30-year cumulative production (i.e., our surrogate for "exact" reserves), all production time periods are referenced to the date at which the peak rate occurs (i.e., the beginning of the gas flowrate decline period), rather than the start of production. For example, the five-year time period actually refers to the decline at 6.75 years after first gas production. As an example of our decline curve analysis, consider the five- and ten-year periods shown in Fig. 5. An evaluation of the five-year production period (Fig. 6) results in a b-exponent of 0.0 indicating exponential decline. However, the computed bexponents for the remaining time periods (see ten-year time period evaluation, Fig. 7) indicate the rate profile changes back to hyperbolic decline and becomes even more hyperbolic (as indicated by increasing b-exponents) during the last 20 years of production. Variations in the b-exponents with time probably reflect the relative, temporal impact of various coal properties during the production decline period. 1,000

Gas Production Rate, Mscf/day

Gas Production Rate, Mscf/day

1,000

100

10

Exponential Decline Arps b-exponent = 0.0 EURcomputed = 795.1 MMscf 30-Year Gp = 983.2 MMscf

100

10

1

1 0

3

6

9

12

15

18

21

24

27

0

30

3

6

9

Fig. 5 — Simulated production history and decline behavior for the single-layer base case.

18

21

24

27

30

Gas Production Rate, Mscf/day

1,000

*Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

We have also computed and tabulated the reservoir estimate errors associated with each of the decline b-exponents shown in Table 2. The percent reserve estimate errors were computed by EURtrue

15

Fig. 6 — Decline curve analysis of simulated data for the single-layer base case at the five-year time period.

Table 2 — Computed Arps Decline Exponents and Reserve Estimate Errors, Single-Layer Base Case Producing Arps Decline Reserve Time Period b-Exponent Estimate (years)* (dimensionless) Errors (%) 5 0.00 -19.1 10 0.48 -0.5 20 0.51 -0.4 30 0.59 -

Reserve Estimate Error 100

12

Producing Time, years

Producing Time, years

EURcomputed

EURtrue

... (8)

100

Hyperbolic Decline Arps b-exponent = 0.48 EURcomputed = 977.9 MMscf 30-Year Gp = 983.2 MMscf

10

1 0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

Fig. 7 — Decline curve analysis of the simulated data for the single-layer base case at the ten-year time period.

where EURtrue is the "true" estimated ultimate recovery (EUR) or reserve which we define in this paper as the 30-year cumulative production volume. Further, EURcomputed is the estimated ultimate recovery computed from the best-ft Arps model through the simulated production and extrapolated to the 30-year time period. Since we honor the production decline behavior for each of the producing time period segments, variations in the reserve estimates should reflect differences in the decline behavior. As we mentioned previously, positive errors indicate an overestimation, while negative errors indicate an underestimation. Results for the five-year time period suggest that we would significantly underestimate reserves if we had honored the decline behavior exhibited at that time period. Note that, after the rate profile starts to become more hyperbolic, the reserve estimate errors are essentially negligible for the last twenty years of production. Effects of Langmuir Pressure and Volume. Two previous studies by Aminian, et al. [2004] and Okuszko, et al. [2007] suggested the shape of the Langmuir isotherm will have an effect on the long-term decline behavior in CBM wells. We have also made an evaluation of the isotherm curve shape, or more specifically, we have examined the effects of the Langmuir

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

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constants, VL and pL, on the decline behavior. First, we examined variations in the Langmuir pressure, pL. Figure 8 shows three different Langmuir adsorption isotherms for pL = 100, 300, and 1,000 psia (corresponding to gas contents of 235, 164, and 80 scf/ton, respectively) but all generated using Eq. (1) and with the same value of VL = 300 scf/ton. All three cases assume an initial undersaturated condition with an initial cleat pressure of 461.25 psia and critical desorption pressure of 361.25 psia. Note that the adsorption isotherm shown in the solid red line is the single-layer base case we presented previously. A comparison of the simulated isotherms shows that, for the same value of VL, decreasing values of pL causes a much more non-linear shape, especially at the lower pressures. The simulated gas production profiles for the three isotherms are plotted in Fig. 9. Other than differences in the Langmuir pressure and initial gas content of the three isotherms, all coal properties and operational conditions listed in Table 1 were used to simulate the gas production histories. Inspection of the simulated data in Fig. 9 shows that lower values of pL (more nonlinearity in the adsorption isotherms) cause more hyperbolic decline behavior for most of the decline portion of the production profile. During the first few years of the initial decline period, the production profile generated using the isotherm with pL = 1,000 psia seems to exhibit a much steeper decline than the other isotherms. Similarly, the long-term decline behavior for the production profiles simulated using the isotherms with pL = 100 and 300 psia appear to have a more hyperbolic decline. 1,000

400

V L = 300 scf/ton, pL = 100 psia, G c = 235 scf/ton

VL = 300 scf/ton, pL = 100 psia

V L = 300 scf/ton, pL = 300 psia, G c = 164 scf/ton

Gas Production Rate, Mscf/day

Adsorbed Gas Content, scf/ton

350

V L = 300 scf/ton, pL = 1,000 psia, G c = 80 scf/ton

300 250 200 150 100

VL = 300 scf/ton, pL = 300 psia VL = 300 scf/ton, pL = 1,000 psia 100

10

50 1

0 0

100

200

300

400

500

600

700

800

900

1,000

Pressure, psia

0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

Fig. 8 — Langmuir adsorption isotherms generated using VL = 300 scf/ton and pL = 100, 300, and 1000 psia.

Fig. 9 — Comparison of simulated gas production profiles for VL = 300 scf/ton and pL = 100, 300, and 1000 psia.

The computed Arps decline b-exponents for the three simulated production profiles are shown in Table 3a for time periods of 5, 10, 20 and 30 years. All three curves exhibit an initial exponential decline at the five-year time period followed by long-term hyperbolic decline which we believe is a manifestation of the increasing non-linear curve shape of the isotherms at the lower reservoir pressures.

Table 3a — Comparison of Arps decline b-exponents for CBM production profiles simulated with isotherms in Fig. 9 Producing Time Period VL = 300 scf/ton, VL = 300 scf/ton, VL = 300 scf/ton, (years)* pL = 100 psia pL = 300 psia pL = 1,000 psia 5 0.00 0.00 0.00 10 0.48 0.34 0.24 20 0.51 0.35 0.27 30 0.59 0.36 0.23

As we observed from the simulated gas profiles in Fig. 9, increasing values of pL cause a much steeper long-term decline as verified by larger values of b. The simulated production for the two cases with pL = 100 and 300 psia show increasing b-exponents with time. The case for pL = 1,000 psia shows an increasing value of b between 10 and 20 years; however, we see a slight decrease at the 30-year time period. We believe this is related to the low gas flow rates relative to the bottomhole flowing pressure constraint.

Table 3b — Comparison of reserve estimate errors (%) for CBM production profiles simulated with isotherms in Fig. 9 Producing Time Period VL = 300 scf/ton, VL = 300 scf/ton, VL = 300 scf/ton, (years)* pL = 100 psia pL = 300 psia pL = 1,000 psia 5 -19.1 -13.9 -9.6 10 -0.5 -2.1 -1.7 20 -0.4 -2.4 -4.5 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

The computed reserve estimate errors are summarized in Table 3b for the same time periods shown in Table 3a. For all values of pL, we note that we will significantly underestimate reserves (errors ranging from almost 10 to 20 percent) if we honor the production trend at the 5-year time period using exponential decline. We also note the reserve estimate errors tend to decrease as the rate profile becomes more hyperbolic. Finally, the increasing reserve estimate error at the 20-year time period for the case of pL = 1000 psia is again related to the very low gas flow rates relative to the bottomhole flowing pressure constraint. Next, we examined variations in the Langmuir volume, VL, on the production decline behavior. Figure 10 shows three different Langmuir adsorption isotherms for VL = 400, 200, and 100 scf/ton (corresponding to gas contents of 251, 129, and 64 scf/ton, respectively) but all generated using Eq. (1) and with the same value of pL = 200 psia. Similar to the previous isotherms, the three cases assume an initial undersaturated condition with an initial cleat pressure of 461.25 psia and critical desorption pressure of 361.25 psia. A comparison of the simulated isotherms indicates that, for the same value of pL, changes

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in values of VL causes similar non-linear shapes at lower pressures. We also observe greater changes in gas content with increasing pressure at higher pressures for isotherms with higher values of VL. On the basis of the curve shapes alone, we would expect to see differences in the early decline but very similar decline behavior at later times. 1,000

500

400

V L = 400 scf/ton, pL = 200 psia, G c = 251 scf/ton

VL = 400 scf/ton, pL = 200 psia

V L = 200 scf/ton, pL = 200 psia, G c = 129 scf/ton

VL = 200 scf/ton, pL = 200 psia

Gas Production Rate, Mscf/day

Adsorbed Gas Content, scf/ton

450

V L = 100 scf/ton, pL = 200 psia, G c = 64 scf/ton 350 300 250 200 150 100

VL = 100 scf/ton, pL = 200 psia 100

10

50 1

0 0

100

200

300

400

500

600

700

800

900

1,000

Pressure, psia

Fig. 10 — Langmuir adsorption isotherms generated using pL = 200 psia and VL = 100, 200, and 400 scf/ton.

The simulated gas production profiles for the three isotherms shown in Fig. 10 are plotted in Fig. 11. Note that, other than differences in the Langmuir volume and gas content of the three isotherms, all coal properties and operational conditions listed in Table 1 were used to simulate the gas production histories. Our preliminary conclusions about the effects of VL on the gas rate curve shapes were correct. Inspection of the simulated data in Fig. 11 shows that lower values of VL seem to cause much steeper declines during the initial decline period (i.e., related to the isotherm curve shape at higher pressures). Conversely, the decline behavior for all isotherms appears to be similar for producing times greater than 10 years.

0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

Fig. 11 — Comparison of simulated gas production profiles for pL = 200 psia and VL = 100, 200, and 400 scf/ton. Table 4a — Summary of Arps decline b-exponents for CBM production profiles simulated with isotherms in Fig. 11 Producing Time Period VL = 400 scf/ton, VL = 200 scf/ton, VL = 100 scf/ton, (years)* pL = 200 psia pL = 200 psia pL = 200 psia 5 0.00 0.01 0.09 10 0.33 0.40 0.38 20 0.40 0.35 0.35 30 0.42 0.35 0.32 Table 4b — Summary of reserve estimate errors (%) for CBM production profiles simulated with isotherms in Fig. 11 Producing Time Period VL = 400 scf/ton, VL = 200 scf/ton, VL = 100 scf/ton, (years)* pL = 200 psia pL = 200 psia pL = 200 psia 5 -13.7 -17.6 -15.0 10 -1.0 -2.4 -3.0 20 -0.1 -2.6 -3.4

Our observations about the decline behavior are confirmed by the computed Arps b-exponents shown in Table 4a for *Note: All time periods (except 30-year time) are referenced to the the three simulated production profiles for time periods of occurrence of the peak rate and the beginning of the decline period. 5, 10, 20 and 30 years. All three curves exhibit essentially an initial exponential decline at the five-year time period followed by long-term hyperbolic decline behavior over the next 25 years. On the basis of the values investigated in our study, larger values of VL cause greater hyperbolic decline behavior (as suggested by slightly greater values of the b-exponent). In general, however, variations in the Langmuir volume have much less effect on the decline behavior than the Langmuir pressure. The computed reserve estimate errors are summarized in Table 4b for the same time periods shown in Table 4a. For all values of VL, we note that we will again significantly underestimate reserves (errors between 13% and 17%) if we honor the production trend and use exponential decline. We also note that, other than slight increases for the cases with VL = 100 and 200 scf/ton, the reserve estimate errors tend to decrease as the rate profile becomes more hyperbolic. Effects of Initial Gas Content and Sorption Time. The previous studies by Aminian, et al. [2004] and Okuszko, et al. [2007] indicated neither initial gas content nor sorption time affected the long-term decline behavior. The results of our simulations study confirm these observations. As an example, consider two of the decline profiles discussed in the previous section. A comparison of the long-term decline behavior for the case with VL = 200 scf/ton and pL = 200 psia with that for the case with VL = 100 scf/ton and pL = 200 psia have different initial gas contents (i.e., Gc = 129 and 64 scf/ton, respectively) but have similar 20-year and 30-year Arps decline b-exponents (i.e., b = 0.35). The results of our simulation study suggest the shape of the sorption isotherm rather than the initial gas content has a much greater impact on the decline behavior. We also evaluated a range of desorption times on decline behavior. The gas sorption time, , is determined from whole core canister desorption tests and is defined as the time required to desorb 63.2 percent of the initial gas volume [Mavor, 1996]. The sorption time is also related to the coal diffusion coefficient, D, by 1/ D ......................................................................................................................................................................................(9) where is the coal matrix shape factor. We evaluated sorption times of 0.5, 5, and 50 days and, other than minor differences in the incline and initial decline profile, we observed no significant differences in the long-term decline behavior.

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

9

Effects of Cleat Spacing. According to an extensive set of measurements by Law [2003], spacing between cleats (both face and butt cleats) ranges from tenths of an inch to several inches. The single-layer base case was generated using a cleat spacing of 0.50 inches, but we also evaluated cases with 0.10, 12 and 24 inches and using the other coal properties summarized in Table 1. We also assumed the face and butt cleat spacings were the same (i.e., no heterogeneity in cleat spacing). Although not shown, we did see some differences in rate behavior during the inclining and the early decline periods, but the long-term decline profiles were quite similar. Effects of Cleat Porosity. The cleat system in coals represents the primary "conventional" pore system since most coals are characterized by a secondary or matrix pore system that is a combination of a micro-pore structure (pore diameters less than 2 nm) and a meso-pore structure (pore diameters between 2 and 50 nm). Although it is usually not a significant contributor to either gas storage (as "free" gas that is not adsorbed) or production (by Darcy flow), the matrix does have a limited macroporosity that is characterized by pore diameters greater than 50 nm. Typically, lower rank coals have a proportionately greater volume of macro-pores than higher rank coals. For this study, we modeled a dual porosity system in which gas is stored in the coal matrix as adsorbed gas only while the cleat system has a finite pore volume similar to conventional reservoirs. Cleat porosities vary depending on the coal rank, maturity, and in-situ stress conditions [Nelson, 2000]. Estimates of cleat porosity in the literature range from approximately 0.10 to 0.60 percent for coals from the San Juan Basin [Gash, et al., 1993; Pyrak-Nolte, et al. 1993] to more than 6 percent for coals from the Powder River Basin [Mavor, et al.,2003]. The porosities reported for coals in the Powder River Basin are probably a combination of cleat and macro-pores. For this study, we simulated gas production for cleat porosities of 0.50, 1.5, 3.0 and 6.0 percent (using other coal properties as summarized in Table 1). Note that the single-layer base case assumed no stress-dependent properties with a cleat compressibility of 1.0 x 10-5 psi-1, i.e., a relatively incompressible system. Similar to the conclusions offered by two similar studies by Aminian, et al. [2004] and Okuszko, et al. [2007], we did not observe any significant differences in long-term decline behavior caused by varying the effective cleat porosities. We should note that measured cleat compressibilities published in the literature indicate coals are actually more compressible than we initially modeled with compressibilities ranging from 1.3 x 10-3 to 7.70 x 10-4 psi-1 [McKee, et al., 1988; Roberston and Christiansen, 2005, 2006]. In fact, coal porosity may be affected by changes in the effective stresses which in turn may be caused either by reductions in pore or cleat pressures [Rose and Foh, 1984; Koenig and Stubbs, 1996; McKee, et al., 1988; Seidle, et al., 1992; Palmer and Mansoori, 1996; Massarotto, et al., 2003] or desorption-controlled shrinkage in the coals [Harpalani and Schraufnagel, 1990; Harpalani and Chen, 1993; Seidle and Huitt, 1995; Pekot and Reeves, 2003; Palmer, et al., 2005] or both. Consequently, we repeated our simulations with the same range of effective cleat porosities but with a cleat compressibility of 1.0 x 10-4 psi-1 and incorporating stress-dependent changes in porosity using the model developed by Palmer and Mansoori [1996]. The form of the Palmer-Mansoori model used in our study is that given in Reference 12 as follows:

i

1 cf (p

pi )

L 1 i

pi M

pi

pL

p

p pL

........................................................................................................(10)

where = effective porosity, fraction = initial effective porosity, fraction i cf = cleat compressibility, psia-1 = maximum strain at infinite pressure, dimensionless L = bulk modulus, psi M = constrained axial modulus, psi p = pressure, psia pi = initial reservoir pressure, psia pL = Langmuir pressure, psia The ratio of bulk to constrained axial modulus in Eq. (10) is related to Poisson's ratio ( ) in the coal by M

1 1 3 1

..........................................................................................................................................................................(11)

Similarly, the constrained axial modulus is related Poisson's ratio by M

E

(1

1 ............................................................................................................................................................(12) )(1 2 )

J.A. Rushing, A.D. Perego, and T.A. Blasingame

Table 5 — Summary of coal mechanical properties used to simulate gas production profiles in Figs. 12 and 13 Reservoir Property Value -4 -1 Cleat compressibility 1.0 x 10 psi Maximum strain 0.008 in/in Young's modulus 600,000 psi Poisson's ratio 0.30 Langmuir pressure 100 psia Initial porosity 0.5%, 1.5%, 3%, 6%

The coal mechanical properties used to evaluate the effects of stress-dependent cleat porosities on the gas production profile are summarized in Table 5. The range of typical values for many mechanical properties was taken from Robertson and Christiansen [2006]. We also used other coal properties and operational conditions from Table 1 for the single-layer base case.

SPE 114514

1,000

Gas Production Rate, Mscf/day

10

f

= 6%

f

= 3%

f

= 1.5%

f

= 0.5%

100

10 0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

Fig. 12 — Comparison of simulated production profiles for effective cleat porosities, f = 0.5%, 1.5%, 3% and 6%.

The results shown in Fig. 12 indicate the cleat porosities do have an impact on the long-term decline behavior, particularly when stress-dependencies are included. As expected, coals with lower cleat porosities will "dewater" sooner. Under these conditions, initial gas production as well as the peak gas production rate will occur sooner. And, as illustrated in Fig. 12, both the incline as well as the initial decline will be steeper with lower effective cleat porosities. It also appears that the long-term decline will be more hyperbolic. Our general observations are confirmed by the computed Arps b-exponents shown in Table 6a. Neither of the two cases with the lowest cleat porosities (i.e., f = 0.5 and 1.5 percent) have an initial exponential decline period but do exhibit hyperbolic decline for most of the early- and late-time decline period. We also observe that the late-time decline behavior tends to be more hyperbolic as the effective cleat porosity decreases (i.e., b-exponents increase as f decreases). However, it also appears as if the decline curves for f = 0.5 and 1.5 percent have flattened as indicated by the relatively constant values of b at the 20year and 30-year time periods, while the curves for f = 3.0 and 6.0 percent appear to continue to flatten since the b-exponents continue to increase. The reserve estimate errors corresponding to the computed Arps decline b-exponents given in Table 6a are listed in Table 6b. Again, the largest errors occur during the first 5 years but generally decrease over the remaining well life. Because the case with the lowest porosity exhibits more hyperbolic behavior than the other cases, we tend to slightly overestimate the reserves. Table 6a — Summary of Arps decline b-exponents for CBM production profiles with f = 0.5%, 1.5%, 3% and 6% Producing Time Period (years)* f = 0.5% f = 1.5% f = 3.0% f = 6.0% 5 0.61 0.31 0.00 0.00 10 0.75 0.48 0.48 0.17 20 0.64 0.55 0.51 0.43 30 0.64 0.56 0.59 0.47 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Table 6b — Summary of reserve estimate errors (%) for CBM production profiles with f = 0.5%, 1.5%, 3% and 6% Producing Time Period (years)* f = 0.5% f = 1.5% f = 3.0% f = 6.0% 5 -2.0 -17.7 -12.4 -12.4 10 3.8 -2.9 -6.1 -6.1 20 0.8 -0.2 -0.6 -0.6 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Effects of Absolute Cleat Permeability. Although the previous studies by Aminian, et al. [2004] and Okuszko, et al. [2007] indicated that the value of the absolute cleat permeability, kf, has no effect on the decline behavior of wells producing from CBM reservoirs, we conducted our own evaluation with values ranging from 10 md to 500 md. All other coal properties and operational conditions are summarized in Table 1 for the single-layer base case we used in the models. While the behavior of the inclining and early declining simulated production were different for the range of kf values investigated, we observed very little or no differences between the long-term rate declines. We note that the single-layer base case again incorporated no stress-dependent properties and assumed a cleat compressibility of 1.0 x 10-5 psi-1, i.e., a relatively incompressible system. However, measured cleat compressibilities published in the literature indicate that coals are actually more compressible than we initially modeled, so we repeated our simulations with the same range of absolute cleat permeablities but with a cleat compressibility of 1.0 x 10-4 psi-1. Similar to the previous evaluation of stress-dependent cleat porosities, we also included stress-dependent permeabilities using the Palmer and Mansoori [1996] model which relates changes in absolute permeability to changes in the effective porosity by k ki

3

................................................................................................................................................................................(13) i

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

11

Where the effective porosity ratio ( / i) is computed from Eq. (12) and ki is the initial absolute permeability. We evaluated initial absolute cleat permeabilities of 10 md, 50 md, and 200 md using the coal mechanical properties shown in Table 5 and the single-layer base case properties given in Table 1. 1,000

kf = 200 md

Gas Production Rate, Mscf/day

The results shown in Fig. 13 indicate that the stress-dependent cleat permeability does affect long-term decline behavior much more severely than does effective cleat porosity. As expected, coals with higher cleat permeabilities will "dewater" more effectively, thereby lowering cleat pressure sooner which results in an initial gas production as well as the peak gas production rate occurring sooner than the other cases with lower permeabilities. And, as illustrated in Fig. 13, both the incline and initial decline will be steeper with higher absolute cleat permeabilities. Moreover, the long-term decline will be less hyperbolic (i.e., lower b-exponents). The converse is true for the case with lower cleat permeability.

kf = 50 md kf = 10 md 100

10

1 0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

The Arps decline b-exponents for the simulated production Fig. 13 — Comparison of simulated production profiles profiles shown in Fig. 13 are summarized in Table 6a for time for absolute cleat permeabilities, kf = 10, 50, and 200 md. periods of 5, 10, 20, and 30 years, while the associated reserve estimate errors are given in Table 6b. All curves exhibit an exponential decline during the initial 5-year time period which causes us to underestimate gas reserves. The errors for the two higher permeability cases are much greater since the initial decline periods are much steeper than the low permeability case. We also note the long-term decline for the low-permeability case is very flat as indicated by b-exponents approaching 0.80. Finally, the long-term decline behavior for the 50-md and 200md cases are very similar with b-exponents between 0.48 and 0.50. This indicates there is some permeability threshold below which the effects of stress-dependent properties will significantly affect productivity.

*Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Effects of Cleat Permeability Anisotropy. Many coals exhibit anisotropy in the absolute cleat permeability [Gash, et al., 1993; Pyrak-Nolte, et al., 1993; Li, et al., 2004]. Typically, the face cleats are better connected and more continuous, while the butt cleats are less well connected and more discontinuous. Published data show that the ratio of the face to butt cleat absolute permeabilities (kx/ky) is quite variable — with ratios ranging from 1.7 to 4.9 [Koenig and Stubbs, 1986; McKee, et al., 1988; Massarotto, et al., 2003].

Table 7b — Summary of reserve estimate errors (%) for CBM production profiles with kf = 10, 50, and 200 md Producing Time Period (years)* kf = 10 md kf = 50 md kf = 200 md 5 -6.2 -17.7 -22.3 10 -3.0 -2.9 0.7 20 0.0 -0.2 0.1 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

1,000

kx/ky = 1

Gas Production Rate, Mscf/day

Table 7a — Summary of Arps decline b-exponents for CBM production profiles with kf = 10, 50, and 200 md Producing Time Period (years)* kf = 10 md kf = 50 md kf = 200 md 5 0.00 0.00 0.00 10 0.25 0.39 0.52 20 0.73 0.48 0.50 30 0.78 0.49 0.50

kx/ky = 2 kx/ky = 10

100

For our study, we evaluated ratios of 1, 2, and 10. Other than differences in the cleat permeabilities in the x- and y-directions, 10 the coal properties and operational conditions listed in Table 1 for 0 3 6 9 12 15 18 21 24 27 30 the single-layer base case were used to simulate the gas productProducing Time, years ion histories shown in Fig. 14. Inspection of the decline behavior Fig. 14 — Comparison of simulated production profiles shows larger contrasts between the face and butt cleat for cleat permeability anisotropies, kx/ky = 1, 2, and 10. permeabilities causes much flatter decline over most of the production history. The production profile for kx/ky = 10 exhibits essentially exponential decline for both the 5-year and 10year production periods. Moreover, larger values of kx/ky not only delay the onset of initial gas production and the peak gas rate, but also decrease the peak rate. We should note that we did not include stress-dependent properties in these cases since we wanted to isolate the impact of cleat permeability anisotropy. However, given the results from the previous two sections, we would expect inclusion of stress-dependent properties to exacerbate the effects of cleat permeability anisotropy.

12

J.A. Rushing, A.D. Perego, and T.A. Blasingame

SPE 114514

The Arps decline b-exponents for the simulated production profiles shown in Fig. 14 are summarized in Table 8a for time periods of 5, 10, 20, and 30 years, while the associated reserve estimate errors are given in Table 8b. All cases again exhibit an exponential decline during the initial 5-year time period which causes us to underestimate gas reserves as indicated by the large negative errors in Table 8b. Although the simulated production decline for kx/ky = 10 is still exponential at the 10-year time period, the other two cases are starting to exhibit hyperbolic decline. After about 20 years of production, all declines are hyperbolic with Arps b-exponents ranging from 0.51 to 0.53 for the kx/ky = 1 and 2 cases while the kx/ky = 10 case has bexponents ranging from 0.57 to 0.64. We note that all reserve estimate errors are again largest during the early decline period (underestimating reserves), but are essentially zero after the 20-year time period. Table 8a — Summary of Arps decline b-exponents for CBM production profiles with kx / ky= 1, 2, and 10 Producing Time Period (years)* k x / ky = 1 kx / ky = 2 kx / ky = 10 5 0.00 0.00 0.00 10 0.48 0.35 0.00 20 0.51 0.52 0.57 30 0.53 0.53 0.64 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Table 8b — Summary of reserve estimate errors (%) for CBM production profiles with kx / ky= 1, 2, and 10 Producing Time Period (years)* kx / ky = 1 kx / ky = 2 kx / ky = 10 5 -19.1 -16.4 -2.8 10 -0.5 -4.2 -4.8 20 -0.4 -0.2 -0.2 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Effects of Cleat Gas-Water Relative Permeability Curve Shape. In this section, we examine the effects of the cleat gas-water relative permeability curve shape on the gas production decline behavior. Similar to more conventional reservoirs with twophase flow through the pore system, the effective permeabilities to gas and water in the cleat system are a function of the relative fluid saturations. Effective gas-water permeabilities in coals are difficult to measure, but the limited published data suggests the shapes of the relative permeability curves are primarily non-linear [Dabbous, et al., 1974; Reznik, et al., 1974; Puri, et al., 1991; Gash, et al., 1991 and 1993]. Recall that the single-base case was simulated using the cleat gas-water relative permeability curves shown in Fig. 4. These hypothetical curves were generated with the Brooks and Corey [1964, 1966] equations given by Eqs. (5), (6) and (7). We generated a range of other cleat gas-water relative permeability curves by adjusting the various parameters in the Brooks and Corey equations. Figure 15 shows three (non-linear) gas-water relative permeability curves computed with Eqs. (5)-(7) and one linear curve set. Although we realize the linear curve set is unrealistic, this case does represent an "end-point" for comparison with the more realistic, non-linear relative permeability sets. 1.00

1,000

krg 1

krw 1 - krg 1

krw 2

krg 2

krw 2 – krg 2

0.80

krw 3

krg 3

0.70

krw 4

krg 4

Relative Permeability, frac.

Gas Production Rate, Mscf/day

krw 1

0.90

0.60 0.50 0.40 0.30 0.20

krw 3 - krg 3 krw 4 – krg 4

100

0.10 0.00 0.00

10 0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Water Saturation, frac.

Fig. 15 — Cleat gas-water relative permeability curves used to evaluate curve shape on gas decline behavior.

0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

Fig. 16 — Comparison of simulated production profiles for various cleat gas-water relative permeability curves.

The gas production profiles shown in Fig. 16 were simulated with the cleat gas-water relative permeability curves plotted in Fig. 15 and using the coal properties and operational conditions summarized in Table 1 for the single-layer base case. As we would expect, the linear curves (krw4 – krg4) allow more efficient and effective "dewatering" of the cleat system (i.e., higher relative permeability to water over the entire water saturation range) and the associated reduction in cleat pressure. Consequently, we see that the initial gas production and the peak gas production rate occur earlier than for the non-linear relative permeability sets. Conversely, the non-linear relative permeability set krw2 – krg2 are the least efficient as indicated by a delay in both initial gas production and the occurrence of the peak rate. It also appears that the simulated gas production for curve sets krw3 – krg3 and krw4 – krg4 transition from an early exponential decline to hyperbolic decline much sooner than the other simulated data. We should note that we did not include stress-dependent properties in these cases since we wanted to isolate the impact of the cleat

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

13

gas-water relative permeability shape. However, given the results from the previous two sections, we would expect inclusion of stress-dependent properties to exacerbate the effects of two-phase flow characteristics in the cleat system. All of these observations are confirmed by the computed Arps b-exponents summarized in Table 9a. Those relative permeability curve sets with lower relative permeabilities to water (i.e., curve sets krw1 – krg1 and krw2 – krg2) exhibit exponential decline at the five-year producing time period, while the other two curve sets (i.e., krw3 – krg3 and krw4 – krg4) have already transitioned to hyperbolic decline at that same relative time period. Regardless of the differences in early time behavior, all gas production rate profiles display consistent and similar long-term hyperbolic decline as indicated by b-exponents ranging from 0.50 to 0.56 at the 20-year and 30-year time periods. The reserves estimate errors shown in Table 9b also reflect differences in the early-time decline behavior. The largest errors occur for curve sets krw1 – krg1 and krw2 – krg2 which both have exponential decline during the five-year time period. All errors are (however) less than 2 percent for all curve sets after the 10-year time period indicating similar decline behavior. Table 9a — Summary of Arps decline b-exponents for CBM production profiles with cleat relative permeability curves Producing Time Period (years)* krg1 – krw1 krg2 – krw2 krg3 – krw3 krg4 – krw4 5 0.00 0.00 0.29 0.48 10 0.48 0.29 0.53 0.58 20 0.51 0.50 0.56 0.60 30 0.53 0.53 0.56 0.53

Table 9b — Summary of reserve estimate errors (%) for CBM production profiles with cleat relative permeability curves Producing Time Period (years)* krg1 – krw1 krg2 – krw2 krg3 – krw3 krg4 – krw4 5 -19.1 -15.7 -12.0 -5.3 10 -0.5 -0.4 -0.1 0.2 20 -0.4 0.0 0.1 1.6 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

*Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Coal Properties, Five-Layer Base Case. All of the previous simulated cases addressed the impact of coal properties on the production decline behavior for single-layer cases with the coal properties and operational conditions summarized in Table 1. However, we also evaluated several multiple seam or multilayer cases. To illustrate the effects of multiple layers on CBM production and decline behavior, we present a five-layer coal with the basic properties listed in Table 10. In order for us to make relevant comparisons to the singlelayer base case, all of the five-layer cases maintained many of the same properties as the single-layer base case, including a total coal thickness of 50 ft (not necessarily the same thickness in each layer), a total thickness-averaged flow capacity of 2500 md-ft (again, not necessarily the same cleat permeability in each layer), and an average initial cleat pressure (evaluated at the mid-point of the total coal thickness) of 461.25 psia.

Table 10 — Summary of coal properties and operational conditions used to simulate five-layer base case Reservoir Property/Operational Condition Well spacing Depth to top of coal Initial bottomhole (cleat) pressure at mid-point Bottomhole temperature (at mid-point) Total Coal thickness Coal density Absolute cleat permeability Effective cleat porosity Cleat compressibility Cleat spacing Initial cleat water saturation Initial coal ash and moisture contents Langmuir pressure Langmuir volume Critical desorption pressure Initial gas content Desorption time Min. bottomhole flowing pressure (at mid-point) Maximum water production rate Stress- or desorption-dependent properties

Value 80 acres/well 999 ft 461.25 psia o 90 F 50 ft 3 1.30 g/cm 50 md 3% -1 1.0e-05 psi 0.50 in 100% 0% 100 psia 300 scf/ton 361.25 psia 235 scf/ton 5 days 30 psia 500 bbl/day No

Homogeneous, Isotropic 5-Layer System. The first five-layer case is a homogeneous, isotropic system in which each coal seam has the same thickness (10 ft), the same cleat permeability (50 md) and the same cleat porosity (3 percent). Each coal seam also has the same Langmuir volume and pressure (Table 10) but has slightly different initial cleat pressures (normal pressure gradient), critical desorption pressures, and gas contents (Table 11) selected in order to maintain the same constant critical desorption pressure differential of 100 psia as the single-layer cases. The hypothetical adsorption isotherm used to simulate the five-layer is shown in Fig. 3, while other coal properties are listed in Table 10. The simulated gas production profile for the five-layer case is plotted in Fig. 17a. We also plot the simulated rate profile for the single-layer base case (previously shown in Fig. 5) for comparison. Even though most coal properties and operational conditions are essentially the same, we still see significant differences in both the incline and decline behavior over the entire time period.

Table 11 — Layer properties used to simulate CBM profiles shown in Figs. 17a-c

Depth to Mid-Layer Layer (ft) 1 1,004.0 2 1,014.5 3 (mid-point) 1,025.0 4 1,035.5 5 1,046.0 Avg./Total

Coal Thickness (ft) 10 10 10 10 10 50

Initial Cleat Pressure (psia) 451.80 456.53 461.25 465.98 470.70 461.25

Critical Desorption Pressure (psia) 351.80 356.53 361.25 365.98 370.70 361.25

Critical Desorption Pressure Differential (psia) 100 100 100 100 100

Gas Content (scf/ton) 233.7 234.4 235.0 235.7 236.3 235.0

14

J.A. Rushing, A.D. Perego, and T.A. Blasingame

SPE 114514

Quite surprisingly, both the initial gas production and the peak gas rate occur (slightly) sooner for the single-layer case. An inspection of Fig. 17a also suggests single-layer case has a slightly steeper initial decline period and slightly more long-term hyperbolic behavior than the five-layer case. Since the minimum bottomhole flowing pressure constraint was set at the midpoint of the coals (i.e., mid-point of layer 3), the bottom two layers were probably producing against a higher pressure than the two upper layers. And, as we will illustrate in a later section, small changes in bottomhole pressures have a significant impact on gas productivity. Consequently, we attribute the differences in decline behavior to different bottomhole pressure constraints even though the initial cleat pressures in the bottom layers are slightly higher.

*Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

1,000 Single-Layer Base Case

Gas Production Rate, Mscf/day

Table 12 — Comparison of Arps decline b-exponents and reserve estimate errors for single- & five-layer base cases Producing Single-Layer Base Case Five-Layer Base Case Time Period (years)* b-Exponents Errors b-Exponents Errors 5 0.00 -19.1 0.00 -8.9 10 0.48 -0.5 0.36 1.6 20 0.51 -0.4 0.31 0.9 30 0.53 0.30 -

Five-Layer Base Case

100

10

The computed Arps decline b-exponents shown in Table 12 confirm our observations. Although both simulated cases exhibit 1 an initial exponential decline, the reserve estimate error 0 3 6 9 12 15 18 21 24 27 30 (underestimating reserves) for the single-layer case is more than Producing Time, years twice as large as that for the five-layer case since it has a steeper Fig. 17a — Comparison of simulated production initial decline. Moreover, the long-term decline behavior of the profiles for single-layer and five-layer base cases. single-layer case is more hyperbolic as indicated by b-exponents ranging from 0.51 to 0.53 as compared to values of 0.30 to 0.31 for the five-layer case. Both the decline behavior and the reserve estimate errors are illustrated in Figs. 17b and 17c for the five- and ten-year time periods, respectively. As we described, the initial decline for the single-layer case is much steeper which results in a much larger reserve estimate error. Similarly, the decline behavior for the ten-year period is more hyperbolic for the single-layer period. We also made comparisons of the single-layer cases with VL = 300 scf/ton, pL = 300 psia and VL = 300 scf/ton, pL = 1000 psia (see adsorption isotherms, Fig. 8) with their respective five-layer cases. Again, the five-layer cases were homogeneous, isotropic systems in which each coal seam had the same total thickness (10 ft), the same cleat permeability (50 md) and the same cleat porosity (3 percent). Each coal seam also had the same Langmuir volume and pressure as their respective singlelayer case but had slightly different initial cleat pressures, critical desorption pressures, and gas contents. Both five-layer cases maintained the same constant critical desorption pressure differentials of 100 psia in all layers. Other coal properties used in the simulations are listed in Table 10. Although not shown, the comparative single- and five-layer production profiles were similar as that shown previously, i.e., both the initial gas production and the peak gas rate occurred sooner for the single-layer case, and the single-layer cases had a slightly steeper initial decline period and slightly more long-term hyperbolic behavior than the five-layer case. 1,000

Five-Layer Case Exponential Decline Arps b-exponent = 0.0 EUR computed = 732.9 MMscf 30-Year Gp = 804.5 MMscf

100

Gas Production Rate, Mscf/day

Gas Production Rate, Mscf/day

1,000

10 Single-Layer Case Exponential Decline Arps b-exponent = 0.0 EUR computed = 795.1 MMscf 30-Year Gp = 983.2 MMscf 1

Single-Layer Case Exponential Decline Arps b-exponent = 0.48 EURcomputed = 977.9 MMscf 30-Year Gp = 983.2 MMscf

100

Five-Layer Case Hyperbolic Decline Arps b-exponent = 0.36 EURcomputed = 817.4 MMscf 30-Year Gp = 804.5 MMscf

10

1 0

3

6

9

12

15

18

21

24

27

Producing Time, years

Fig. 17b — Comparison of decline curve analyses of the simulated data for the single-layer and five-layer base cases at the five-year time period.

30

0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

Fig. 17c — Comparison of decline curve analyses of the simulated data for the single-layer and five-layer base cases at the ten-year time period.

Heterogeneous, Isotropic 5-Layer System. Next, we evaluated the impact of layer heterogeneities in which we varied both coal thickness and absolute cleat permeability in each of the five-layers (i.e, a heterogeneous, isotropic layered system). However, we still maintained the same total overall coal thickness (50 ft) and flow capacity (2500 md-ft) as in the single-layer

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

15

and five-layer base cases so that we could make appropriate comparisons in the production decline behavior. Tables 13a and 13b list the individual layer properties used to simulate the two heterogeneous, isotropic five-layer systems. For one case (Table 13a), we systematically decreased coal thickness while proportionately increasing cleat permeability in the deeper coals. For the other case (Table 13b), we did the opposite — increasing coal thickness while decreasing cleat permeability with depth. Note that the initial cleat pressures, critical desorption pressures, critical desorption pressure differential, and initial gas contents are the same as for the five-layer base case (i.e., the homogeneous, isotropic case).

Table 13a — Layer properties used to simulate CBM profiles for heterogeneous, isotropic five-layer case no. 1 Absolute Absolute Initial Critical Coal Cleat Cleat Flow Cleat Desorption Layer Thickness Permeability Capacity Pressure Pressure No. (ft) (md) (md-ft) (psia) (psia) 1 5.0 100.0 500 451.80 351.80 2 7.5 66.7 500 456.53 356.53 3 (mid-point) 10.0 50.0 500 461.25 361.25 4 12.5 40.0 500 465.98 365.98 5 15.0 33.3 500 470.70 370.70 Avg./Total 50.0 2500 461.25 361.25

Gas Content (scf/ton) 233.7 234.4 235.0 235.7 236.3 235.0

Table 13b — Layer properties used to simulate CBM profiles for heterogeneous, isotropic five-layer case no. 2 Absolute Absolute Initial Critical Coal Cleat Cleat Flow Cleat Desorption Layer Thickness Permeability Capacity Pressure Pressure No. (ft) (md) (md-ft) (psia) (psia) 1 15.0 33.3 500 451.80 351.80 2 12.5 40.0 500 456.53 356.53 3 (mid-point) 10.0 50.0 500 461.25 361.25 4 7.5 66.7 500 465.98 365.98 5 5.0 100.0 500 470.70 370.70 Avg./Total 50.0 2500 461.25 361.25

Gas Content (scf/ton) 233.7 234.4 235.0 235.7 236.3 235.0

Although not shown, we compared the simulated gas production profiles for the two heterogeneous, isotropic cases with the five-layer homogeneous, isotropic cases. The comparisons showed some minor differences during the inclining and the initial decline periods. However, the long-term decline behavior was essentially identical for all three cases. These results are consistent with our previous assessment of the effects of absolute cleat permeability in the single-layer case. We should also note that, none of the cases included the effects of stress-dependent cleat permeability or porosity. However, previous results have demonstrated the effect of stress-dependent properties in several single-layer cases, and we would expect the similar effects in a multi-seam case. Effects of Operational Conditions on Production Decline Behavior In this section, we examine the effects of operational conditions on the production decline behavior of CBM reservoirs. More specifically, we evaluate the effects of bottomhole flowing pressure constraints on the gas rate profiles and long-term decline behavior. Results from several five-layer cases shown previously suggested that a bottomhole flowing pressure constraint, pwf, may also affect the CBM production profile. All of the previous cases were simulated using pwf = 30 psia, but we have also assessed pwf = 15 and 60 psia. To satisfy requirements for using Arps' models, all simulations were conducted using a constant bottomhole flowing pressure condition in the wellbore.

We should note that the actual bottomhole flowing pressure was typically greater than the minimum constraint during much of the initial dewatering period and before reaching the maximum gas production rate; however, the bottomhole flowing pressure was essentially constant during most of the decline period.

1,000

pwf = 15 psia

Gas Production Rate, Mscf/day

We used the single-layer, base case with the properties shown in Table 1 to evaluate pwf. The model was also constructed with the adsorption isotherm shown in Fig. 3 and the cleat gas-water relative permeability curves in Fig. 4. Since stress-dependent properties are pressure-dependent and will affect the decline behavior, we have incorporated the Palmer-Mansoori model [1996] given by Eqs. (10) and (13). Coal mechanical properties given in Table 5 were used in all simulations.

pwf = 30 psia pwf = 60 psia 100

10

1 0

3

6

9

12

15

18

21

24

27

30

Producing Time, years

The simulated gas production profiles for pwf = 15, 30, and 60 Fig. 18 — Comparison of simulated gas production profiles for pwf = 15, 30, and 60 psia. psia are shown in Fig. 18. These results suggest that the minimum bottomhole flowing pressure constraints evaluated in this study have negligible effect on the early production behavior, in particular the inclining production period, as well as the timing of the initial gas production and the peak production rate. We do, however, see differences in the long-term production decline behavior. The Arps decline b-exponents for the simulated production profiles shown in Fig. 18 are summarized in Table 14a for time periods of 5, 10, 20, and 30 years, while the associated reserve estimate errors are given in Table 14b. As indicated by b = 0

16

J.A. Rushing, A.D. Perego, and T.A. Blasingame

SPE 114514

results in Table 14a, all curves again exhibit exponential decline during the initial 5-year time period which causes us to underestimate gas reserves as indicated by the large negative errors in Table 8b. The magnitude of the errors also suggests higher bottomhole flowing pressure constraints cause steeper initial exponential declines. All curves transition to hyperbolic decline between the 5-year and 10-year time periods. The results also suggest higher bottomhole flowing pressure constraints will create more long-term hyperbolic decline behavior. Table 14a — Summary of Arps decline b-exponents for CBM production profiles with pwf = 15, 30, and 60 psia Producing Time Period (years)* pwf = 15 psia pwf = 30 psia pwf = 60 psia 5 0.00 0.00 0.00 10 0.30 0.39 0.43 20 0.36 0.48 0.53 30 0.35 0.49 0.55 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Table 14b — Summary of reserve estimate errors (%) for CBM production profiles with pwf = 15, 30, and 60 psia Producing Time Period (years)* pwf = 15 psia pwf = 30 psia pwf = 60 psia 5 -14.2 -17.7 -19.4 10 -2.0 -2.9 -3.2 20 0.2 -0.2 -0.5 *Note: All time periods (except 30-year time) are referenced to the occurrence of the peak rate and the beginning of the decline period.

Summary and Conclusions This paper presents results of a series of single-well, parametric simulation studies or "experiments" designed to evaluate the applicability of Arps' models and decline curve methodology for assessing reserves in coalbed methane reservoirs. We simulated various coal properties and well operational conditions to determine their impact on both the short- and long-term the production decline behavior as quantified by the Arps decline curve exponent, b. We then evaluated the simulated production with Arps' rate-time equations at specific time periods during the well's production decline period and compared estimated reserves to the "true" value (i.e., defined in this paper as the 30-year cumulative production volume). Based on the results of our simulation study, we offer the following conclusions about use of an Arps decline curve methodology for evaluating reserves in coalbed methane (CBM) reservoirs: 1. The computed values of the long-term Arps decline b-exponents from an evaluation of the simulated production ranged between 0.20 < b < 0.80 which is within the limits originally defined by Arps (i.e., 0.0 < b < 1.0). Agreement between Arps' recommended b-exponent range and our results using simulated performance data also suggests the Arps rate-time models are appropriate for assessing reserves in coalbed methane reservoirs. 2. The Arps b-exponents were not constant during the production decline period. The early decline behavior for many cases often appeared to have exponential decline but eventually became more hyperbolic later in the well's life. Use of the Arps' exponential model early in the production history in those wells with long-term hyperbolic decline behavior tended to underestimate gas reserves. The largest reserve estimate errors typically occurred during the first few years after reaching the peak production rate and during the initial production decline period. 3. None of the simulated cases exhibited long-term exponential decline behavior. We attribute this to the non-linear relationship between key coal properties and either pressure or saturation. Coal properties having the largest impact on the production decline behavior include: the shape of the adsorption isotherm, cleat permeability anisotropies, the shape of cleat gas-water relative permeability curves, and stress-dependent cleat permeability and porosity. 4. We also observed a strong influence of the flowing pressure conditions as modeled using a bottomhole flowing pressure constraint. Considering all other properties and conditions to be equal, wells with higher bottomhole flowing pressures exhibited longer-term hyperbolic behavior as defined by higher Arps b-exponents. Nomenclature b cf D Di E Gc

k ki kf krg

= = = = = = = = = =

Arps' decline exponent, dimensionless cleat compressibility, psia-1 rate-dependent skin factor, (Mscf/d)-1 initial decline rate, (days)-1 Young's modulus, psia adsorbed gas content, scf/ton permeability, md initial permeability, md Cleat permeability, md relative permeability to gas, fraction

SPE 114514

Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery in Coalbed Methane Wells

k gr

=

gas relative permeability coefficient, dimensionless

krw

=

relative permeability to water, fraction

k wr

=

water relative permeability coefficient, dimensionless

M m n p pcdp pL pwf qi Sw Swi

= = = = = = = = = =

constrained axial modulus, psi water relative permeability exponent, dimensionless gas relative permeability exponent, dimensionless pressure, psia critical desorption pressure, psia Langmuir pressure, psia bottomhole flowing pressure, psia initial gas production rate, Mscf/day water saturation, fraction initial water saturation, fraction

S *w

=

normalized water saturation, fraction

VL

=

Langmuir volume, scf/ton

17

Greek Symbols = maximum strain at infinite pressure, dimensionless L = effective cleat porosity, fraction = initial effective cleat porosity, fraction i = bulk modulus, psi = Poisson's ratio, dimensionless

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

15. 16. 17. 18. 19. 20. 21.

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