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Abstract

Robust techniques for analysis of production data of single porosity gas reservoirs have been developed and widely used for many years. These methods range from the traditional Arp’s decline method to modern type-curve matching. The more recent techniques are based on the use of pseudo-time for linearization of gas-flow equations, and material balance time to account for variable operating conditions.

Whereas the governing equation for gas flow in a single porosity reservoir has been successfully linearized using pseudo-time evaluated at average reservoir pressure, the linearization of the governing equation in a double porosity reservoir is problematic due to pressure differences between the matrix and fracture systems. Similarly, the suitability of the material balance time concept, which has been successfully applied to a single porosity gas reservoir, has not been demonstrated for a double porosity gas reservoir. The purpose of this work is to suggest an appropriate pseudo-time and material-balance time functions when gas is produced from a naturally fractured gas reservoir. For this purpose, the resulting equations describing the matrix-fracture flow are cast in a form similar to those proposed by Warren and Root. These are then linearized, using appropriately-defined pseudo-time and material-balance pseudo-time functions. The results are compared against those of a commercial numerical simulator for two production scenarios, including constant rate and constant pressure production over some range of reservoir parameters. The results show that pseudo-time and material balance pseudo-time allow the accurate use of traditional double-porosity type-curves for naturally fractured gas reservoirs provided that the gas properties are evaluated at

average reservoir pressure as determined from the material balance equation for double porosity systems.

Introduction

Type-curves are routinely used by engineers to estimate initial hydrocarbon-in-place and hydrocarbon reserve s at some abandonment conditions, as well as flowing characteristics of individual wells such as permeability and skin.

Type-curves are plots of the theoretical solution to the governing flow equation for constant-rate production, or constant-pressure production, from a well in any kind of reservoir model. Generally, the operating conditions during production from a well are not constant. Hence, to analyze real production data, one needs to develop a robust methodology to account for these changes. This problem was solved by the use of the material balance time concept1. Blasingame and Lee1 showed that, with use of this concept, one could have a single solution to the governing flow equation for both types (i.e., constant-rate and constant-pressure) of boundary conditions. It was shown that the same solution applies to cases where both rate and pressure are smoothly changing with time.

Whereas the governing flow equation for a slightly compressible fluid is linear, the governing flow equation for a compressible gas is non-linear due to the pressure dependency of gas properties. By applying the pseudo-pressure2 and pseudo-time3 transformations, the non-linear governing gas flow equation may be linearalized allowing one to use the slightly compressible liquid solution for a gas system.

Generally, the methodology for type-curve analysis of production data under variable operating conditions requires: (i) an equation to obtain the average reservoir pressure — this is generally a material balance equation; (ii) evaluation of the material balance time at average reservoir pressure; and (iii) performing traditional well test (constant rate) analysis, by transformation of real time (or pseudo-time) to material balance time (or material balance pseudo-time) and use of rate normalized pressure-drop. Similar analysis could be performed Using production decline analysis techniques.

IPTC-11278-PP

Decline Curve Analysis for Naturally Fractured Gas Reservoirs: A Study on the Applicability of “Pseudo-time” and “Material Balance Pseudo-time” Shahab Gerami*, Mehran Pooladi-Darvish / University of Calgary and Huifang Hong / Fekete Associates Inc. *Now at National Iranian Oil Company (NIOC)

2 [IPTC-11278-PP]

A review of literature on type-curve analysis of gas reservoirs reveals that there is no specific attention paid to type-curves of naturally fractured gas reservoirs. Due to the double porosity behavior in a naturally fractured gas reservoir and the importance of fracture and matrix compressibilities, the evaluation of type-curve analysis components — such as average reservoir pressure, total compressibility, pseudo-time, and pseudo material balance time — requires special attention. Therefore, the purpose of this study is to present just such a production analysis model for naturally fractured gas reservoirs.

In the following, we start with a short review of the most commonly used mathematical models of naturally fractured reservoirs. Next, a production analysis model consisting of (i) a material balance equation, (ii) pseudo-time functions, and (iii) a governing equation and the corresponding solutions for the wellbore boundary condition, subjected to constant-rate and constant-pressure productions, is developed. Then, the assumptions made in the development of the production model is verified over some range of reservoir parameters using a commercial numerical simulator.

Background

Naturally fractured reservoirs have been studied intensively for several decades in the geologic and engineering fields. Transient pressure analysis has received particular attention. Based on the theory of fluid flow in naturally fractured porous media developed in the 1960's by Barenblatt et al.4, Warren and Root5 introduced the concept of dual-porosity models into petroleum reservoir engineering. Their idealized model consisting of a highly interconnected set of fractures, which is supplied by fluids from numerous small matrix blocks, is shown in Figure 1.

Whereas the matrix permeability is much smaller than the fracture permeability, the fracture porosity of a particular class of naturally fractured reservoirs seldom exceeds 1.5% or 2%, and usually falls below 1%.6 The high permeability of a fracture results in a high diffusivity of the pressure propagation pulse along the fracture; however, due to significant contrast between matrix and fracture permeabilities, the matrix has a “delayed” response to pressure changes that occur in the surrounding fractures. Such a non-concurrent response induces matrix-to-fracture cross-flow.

In theory, double-porosity behavior yields two parallel straight lines on a semi-log plot, provided there is no wellbore nor outer boundary effects. The semi-log plot consists of three sections: (i) the first straight line, which represents the homogeneous behavior of the naturally fractured medium before the matrix medium starts to respond (transient radial flow) — the slope of this line gives the fracture permeability; (ii) a transition section (between two straight lines), which corresponds to the onset of inter-porosity flow; and (iii) the second semi-log straight line, which represents the homogeneous behavior of composite media (fracture permeability with the sum of matrix and fracture storages)

when recharge from the matrix medium is fully established.

The nature of matrix and fracture interaction is manifested during this transitional period of matrix-to-fracture fluid transfer7. The important features of the most common models are listed in Table 1. Accordingly, the Warren and Root5 model is known as the pseudo-steady state model, since the matrix-fracture transfer term is expressed using a tank-tyope formulation. Streltsova’s model7 is known as the pressure gradient model, since, according to her assumption, the matrix-fracture flow rate is controlled by the average pressure gradient in the matrix. There unsteady state models are developed by Kazemi8, de Swaan9, and Najurieta10 models, where matrix-to-fracture flow is described by an unsteady state flow equation.

Considering the Warren and Root5 model, the general assumptions in well test analysis of naturally fractured reservoirs are: (i) pseudo-steady state matrix flow; (ii) the matrix consists of a set of porous rock systems that are not connected to each other, have a low transmissibility, and high storage capacity; (iii) the fracture system has low storage capacity, high transmissibility, and it interconnects the porous media; and (iv) the matrix supplies the fluids to the fractures, and the fractures transport the fluids to the well (i.e., the matrix does not provide fluid directly to the well).

Warren and Root5 characterized the naturally fractured porous medium by two parameters: storativity ratio (ω ) and inter-porosity flow parameter ( λ ). The parameter ω is a dimensionless quantity relating the fluid capacitance of the fractures to that of combined system. The parameter λ is proportional to the ratio of matrix permeability to fracture permeability. Typical values of λ range from 310 − to 910− , where low values of λ indicate low fluid transfer between matrix and fractures. A homogeneously distributed porosity is considered as the limiting case in the model. It happens when

1=ω or ∞=λ .

Physical Model

In this study, a cylindrical naturally fractured gas reservoir is considered where gas production and corresponding pressure decline in the fracture leads to gas flow from the matrix into the fracture system (see Figure 2). Thus, the fracture system acts both as a sink to the matrix system and as a conduit to production wells.

Production Analysis Model

In the following, the production analysis model consisting of (i) a material balance equation, (ii) pseudo-time functions, and (iii) the governing equation for gas production from a naturally fractured gas reservoir is presented. The following assumptions are considered in the production analysis model.

1. Single-phase, compressible gas, and Darcy flow.

[IPTC-11278-PP] 3

2. Gravitational forces are negligible and pressure gradient is small.

3. The porosity of either medium (fracture or matrix) is a function of pressure in that medium alone. The compressibility of the matrix, the fracture and the water are small and constant.

4. Flow through the wellbore is via fractures; the matrix acts as a source.

5. Constant permeability.

Material Balance Equation

In order to develop a general material balance equation, all sources of expansion, such as water influx from an aquifer, formation expansion, and connate water expansion, must be considered. For the volumetric naturally fractured gas reservoir described in this paper, the familiar Zp equation can be represented as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

GG

Zp

Zp p

i

i 1**

(1)

Equation (1) is the gas phase mass balance over the total reservoir system including matrix and fracture systems. Note that it is assumed that the initial water saturation in the fracture system is zero. In Equation (1), *Z is the modified compressibility factor, accounting for the effect of matrix, fracture, and connate water compressibilities, and can be represented as:

( )ppcZZ

ie −−=

1* (2)

where ec is the cumulative effective compressibility and is expressed by:

( )( ) 21

2211

1 φφφφ

+−++

=wi

wiwe S

ccScc (3)

Using Equation (1), the plot of *Zp versus pG gives a straight line with the intercept of total initial gas-in-place. The group ( )ppc ie − , can be regarded as the dimensionless effect of expansion of the connate water and matrix-fracture towards pressure maintenance in the system.

Pseudo-time Functions

A well produced at a constant rate exhibits a varying bottomhole flowing pressure, whereas a well produced at a constant bottomhole pressure exhibits a varying rate curve. The material balance time concept was first developed by Blasingame and Lee1 to match the variable flowing pressure data on a Fetkovich11 type curve, which is essentially developed for constant flowing pressure production data. Later, Agarwal et al.12 demonstrated that material balance time converts the constant pressure solution into the widely used

constant rate solution. Due to the varying PVT properties of gas, the material balance time for gas reservoirs was developed in terms of pseudo-pressure and pseudo-time. 1

To apply the concept of material balance time to a volumetric naturally fractured gas reservoir, one needs to consider the material balance equation, *Zp , and pseudo-time, at , which is defined later. As shown in Appendix A, the material balance pseudo-time can be obtained from:

( )( )

a

t

t

itica dt

ctq

tqct

a

∫=0

*

**

µµ (4)

where *tc is defined by:

nggt ccc +=* (5) and ngc is related to the compressibility of matrix pore volume, fracture pore volume, and connate water (non-gas components) compressibilities and is defined as:

( )[ ]ppccc igeng −−= 1 (6) In analogy with homogeneous gas reservoirs, we use pseudo-time given by:

∫=t

ttiia c

dtct0

**

µµ (7)

Governing Equation — Warren and Root5 Approach

As shown in Appendix B, the governing equations describing the flow of gas through a fracture system and transfer of gas from matrix to fracture can be expressed by Equations (8) and (9), respectively:

( )D

Dm

D

Df

D

DfD

DD ttrr

rr ∂∆∂

−+∂∆∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∆∂

∂∂ ψω

ψω

ψ11 (8)

( ) ( )DmDfD

Dm

tψψλ

ψω ∆−∆=

∂∆∂

−1 (9)

where ( )

( )( ) ( )2211

22

1 ccccScc

ggwi

g

+++−

+=

φφφ

ω (10)

2w

f

m rkkα

λ = (11)

Other parameters are defined in Table 2 and in the Nomenclature.

As discussed in Appendix B, Equations (8) and (9) are derived using gas properties evaluated at the average reservoir pressure (one pressure point). This is the first step in dealing with the non-linearity of the flow equations in the naturally fractured gas reservoir described earlier. However, Equations (8) and (9) are still non-linear due to pressure dependency of the storativity ratio (ω ). Unlike naturally fractured oil reservoir models, the storativity ratio for a naturally fractured

4 [IPTC-11278-PP]

gas reservoir is not constant and changes with gas compressibility (see Equation (10)). To remedy this problem, we assume that the storativity ratio evaluated at average reservoir pressure ( ( )pω ) can be used in the available solutions6, developed for the flow of a slightly compressible fluid (subject to constant rate and constant pressure production scenarios) listed in Table 3. Later on, we shall examine the appropriateness of this assumption.

Summary of Production Analysis Model

As summarized in Tables 2 and 3, the production analysis model developed here consists of a set of equations including modified forms of the Zp material balance equation, pseudo-time, and material balance pseudo-time functions, as well as the matrix and fracture governing equations in terms of pseudo-pressure and pseudo-time. The average reservoir pressure is obtained as a function of cumulative production and hence as a function of production time. This equation is a key component in the production model, because it affects the evaluation of all parameters listed in Tables 2 and 3. For example, total compressibility and transformation of real time to pseudo-time are based on the average reservoir pressure. In addition, average reservoir pressure affects the calculation of material balance pseudo-time.

Verification

As mentioned earlier, the flow governing equations (Equations (8) and (9)) are linearized on the basis of two important assumptions. First, the gas properties for both the matrix and fracture systems are evaluated at average reservoir pressure as determined from the material balance equation. Second, the time-varying parameter, ( )pω , is replaced with the constant parameter, ω , in the solutions developed for the flow equations of a slightly compressible oil in a double porosity medium6 (see Table 3). Theoretically, both of these assumptions would represent flaws in the mathematical description of the problem; however, in this section, the validity of these assumptions for practical purposes is examined with a commercial numerical reservoir simulator (CMG-IMEX13), which does not make any of the above simplifying assumptions. First, we investigate the effectiveness of pseudo-time, at , for linearizing the governing equations, Equations (8) and (9). For this purpose, the wellbore pseudo-pressure as a function of time is compared between the two models (analytical and numerical). Then, the applicability of material balance pseudo-time is investigated. For this purpose, it is shown how the simulator results for a constant pressure scenario resemble those of a constant rate scenario when the qψ∆ results for both models are plotted versus the material balance pseudo-time, cat .

A hypothetical cylindrical naturally fractured gas reservoir with a well located at the centre of the drainage area is considered. The reservoir radius and thickness are 1000 m and 100 m, respectively. The initial reservoir temperature is 60 OC

(140 OF) and the initial pressure is 28 MPa (4060 psia). The matrix and fracture porosities are 15% and 1%, respectively, and the matrix and fracture permeabilities are 5 md and 50 md, respectively. The reservoir performance is studied under two operating conditions, namely (i) constant rate (q=2 ×106 std. m3/day) and (ii) constant wellbore pressure (pwf =3.5 MPa). Other relevant physical properties of the reservoir and the different cases studied are given in Table 4.

For simulation, the single layer hypothetical reservoir is divided into 60 radially distributed sections (i.e., annular rings). For the sake of consistency, the value of fracture spacing in the numerical simulator is adjusted in such a manner so as to obtain the same shape factor as the analytical model. To compare the simulator results with those of the analytical model, the calculated pressure, rate, and time information are transformed into the appropriate pseudo-values as required by the solution presented in this paper.

Figures 4 and 5 show the results corresponding to the Base Case with the specifications listed in Table 4. Figure 4(a) shows that, for pressure values less than 28,000 kPa, the difference between the total compressibility, *

tc , and gas compressibility, gc , is negligible. Figure 4(b) shows that the storativity ratio, ( )pω , significantly changes with pressure. Figures 5(a) and 5(b), respectively, show the computed values of pseudo-time as a function of real time for constant rate and constant pressure production scenarios. The pseudo-time (Equation (7)) for each scenario is evaluated at average reservoir pressure (Equation (1)). Figures 5(c) and 5(d) show good agreement between the results of the analytical and numerical (simulator) models, indicating that the shrinkage of real time, using the pseudo-time transformation as shown in 5(a) and 5(b) has alleviated the non-linearity of the governing equation. Figure 5(c) shows that, at constant rate production, the flowing bottomhole pseudo-pressure vs. pseudo-time exhibits linear-decline behavior during the boundary-dominated flow regime (straight line on a Cartesian graph), while Figure 5(d) shows that, at constant bottomhole pressure, the production rate vs. pseudo-time exhibits exponential-decline behavior (straight line on a semi-log graph). Figure 5(e) shows a comparison of average reservoir pressure and flowing wellbore pressure between the analytical and numerical models; good agreement between the two models is evident. Figure 5(e) also demonstrates that the differences between the average reservoir pressure and flowing wellbore pressure are small. This is due to the relatively high fracture flow-capacity ( mdk f 50= ). Figure 5(f) shows that the calculated production data for both cases of constant pressure and constant-rate fall on the analytically calculated pressure-derivative type-curve. The agreement is excellent for infinite-acting (zero slope) and boundary-dominated (unit slope) flow regimes.

The Base Case results (analytical model) were obtained using a time-varying storativity ratio evaluated at average reservoir pressure ( ( )pω ). To examine the effectiveness of this

[IPTC-11278-PP] 5

assumption on the behavior of the solution, we used a constant value of storativity ratio evaluated at initial reservoir pressure ( ( )ipω ) for calculation of both ( )Sf (Equation(24) in Table 3) and Dt (Equation (B-21)). Figure 6(a) indicates that, for such a constant storativity ratio assumption, the wellbore performance predicted by the analytical model does not agree with the corresponding numerical results at late times. The predicted values of wellbore pseudo-pressure (analytical solution) of the double porosity reservoir and the corresponding single porosity reservoir are compared on Figure 6(b). This shows that, by using ( )pω , the behavior of the double porosity model is similar to that of a single porosity model during the boundary-dominated flow regime. Note that, in the single porosity reservoir, all properties are the same as those of the double porosity reservoir, except that the permeability is equal to the fracture permeability ( mdk 50= ) and the porosity is the sum of matrix and fracture porosities ( 16.0=φ ).

Cases 2 to 5 consider different reservoir properties including changes in matrix and fracture permeability, thickness, porosity and operating conditions. The results shown in Figure 7 shows good agreement between the analytical and numerical models. This agreement supports the application of the assumptions used in the production analysis model developed here.

Summary and Conclusions

An analytical production model including pressure transient analysis and production data analysis is presented for the analysis of production data of a naturally fractured gas reservoir. The analytical model is validated against a numerical reservoir simulator13 with excellent agreement. On the basis of the results presented, the following conclusions are drawn: 1. The pseudo-time calculated at average reservoir pressure

successfully linearizes the governing flow equation of the naturally fractured gas reservoir, provided that the time-varying storativity ratio, ( )pω , is used in the available solution6 of the Warren and Root5 model.

2. The concept of material balance pseudo-time developed for conventional gas reservoirs can be applied successfully to a naturally fractured gas reservoir, using pseudo-time evaluated at average reservoir pressure.

Acknowledgements The authors acknowledge Computer Modeling Group for the use of CMG-IMEX. The study program of the first author in the University of Calgary was also supported by the National Iranian Oil Company (NIOC). This support is gratefully acknowledged. Nomenclature

1c Matrix pore volume compressibility, ( )( )pc iii ∂∂= 111 φφ , kPa-1

2c Fracture pore volume compressibility, ( )( )pc ii ∂∂= 222 1 φφ , kPa-1

ec Effective compressibility, kPa-1

gc Gas compressibility, kPa-1

ngc Non-gas compressibility, kPa-1 *tc Modified total compressibility, kPa-1

cw Water compressibility, kPa-1 h Total formation thickness, m k Fracture permeability, m2 G Initial gas-in-place, std. m3 Gp Cumulative gas production, std. m3 M Molecular weight of methane, kg/kmol p Pressure, kPa p Average reservoir pressure, kPa

bp Optional base pressure, kPa psc Standard pressure, kPa q Production rate at standard conditions, m3/s

*gq Gas discharge rate, kg/ m3s

r Radius, m er Reservoir radius, m

wr Wellbore radius, m S Laplace parameter

wiS Initial water saturation in matrix, dimensionless t Time, s

at Pseudo-time, s

ct Material balance time, s

cat Material balance pseudo-time, s

T Reservoir temperature, K Tsc Standard temperature, K

gv Gas velocity, m/s Z Compressibility factor, dimensionless

*Z Modified compressibility factor, dimensionless

α Shape factor, 1/m2 1φ Matrix void space to bulk volume, dimensionless

2φ Fracture void space to bulk volume, dimensionless ρ Gas Density, kg/m3 µ Gas viscosity, kPa.s µ Gas viscosity, ( )pµ , kPa.s ψ Pseudo-pressure, kPa/s ψ Pseudo-pressure, ( )pψ , calculated at average reservoir

pressure, kPa/s λ Inter-porosity flow parameter, dimensionless ω Storativity ratio, dimensionless ω Storativity ratio, ( )pω , dimensionless Subscripts 1 Matrix 2 Fracture D Dimensionless f Fracture medium i Initial conditions

6 [IPTC-11278-PP]

m Matrix medium sc Standard conditions wf Flowing wellbore pressure References 1. Blasingame, T.A., and Lee, W.J.: “Variable Rate Reservoir

Limits Testing,” Paper SPE 15028 presented at the Permian Basin Oil and Gas Recovery Conference, Midland, TX, p. 1314 (March 1986).

2. Al-Hussainy, R., Ramey, H.J., and Crawford, P.B.: “The Flow of Real Gases through Porous Media,” Journal of Petroleum Technology (JPT), 18, pp. 626-636 (1966).

3. Fraim, M.L., and Wattenbarger, R.A.: “Gas Reservoir Decline-Curve Analysis Using Type Curves with Real Gas Pseudo-pressure and Normalized Time,” SPE Reservoir Evaluation and Engineering, pp.671-682 (Dec. 1987).

4. Barenblatt, G.I., and Zheltov, Yu.P.: ”Fundamental Equations of Filtration of Homogeneous Liquids in Fissured Rocks,” Soviet Physics, Doklady 5, p. 522 (1960).

5. Warren, J.E., and Root, P.J.: “The Behavior of Naturally Fractured Reservoirs,” SPEJ, pp. 245-55, Trans. AIME, p. 228 (Sept. 1963).

6. Sabet, M.A.: Well Test Analysis, Gulf Publishing Company (1991).

7. Streltsova, T.D.: “Well Pressure Behavior of a Naturally Fractured Reservoir,” pp. 769-80 (Oct. 1983).

8. Kazemi, H: “Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution,” SPEJ, pp. 451-62, Trans. AIME, p. 261 (Dec. 1969).

9. de Swaan, A.: “Analytical Solution for Determining Naturally Fractured Reservoir Properties by Well Testing,” SPEJ, pp.117- 22, Trans. AIME, p. 228 (June 1976).

10. Najurieta, H.L,: “A Theory for Pressure Transient Analysis in Naturally Fractured Reservoirs,” JPT, pp. 124-50 (July 1980).

11. Fetkovich, M.J.: “Decline Curve Analysis Using Type-Curves,” JPT, p. 1065 (June 1980).

12. Agarwal, R.G., Gardner, D.C., Kleinsteiber, S.W., and Fussell, D.D.: “Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts,” SPE 57916, SPE Reservoir Evaluation and Engineering, pp. 478-486 (Oct. 1999).

13. Da Prat, G.: Well Test Analysis for Fractured Reservoir Evaluation, Elsevier Science Ltd. (Dec. 1990).

14. Computer Modeling Group Inc.: CMG-IMEX User Manual (2007).

Appendix A: Pseudo-time and Material Balance Pseudo-time

This appendix presents the derivation of the material balance pseudo-time for the naturally fractured gas reservoir described earlier. For the sake of completeness, all equations in the main text are also shown here. To obtain the material balance pseudo-time for a naturally fractured gas reservoir, we follow Faraim and Wattenbarger’s approach for single-porosity

reservoirs3 and start with the material balance equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

GG

Zp

Zp p

i

i 1** (A-1)

where

( )ppcZZ

ie −−=

1* (A-2)

( ) dpZpp

p

pb

∫=µ

ψ 2 (A-3)

dtd

dpd

Zp

pdd

Zp

dtd ψ

ψ××⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

** (A-4)

Re-arranging Equation (A-4) yields:

⎟⎠⎞

⎜⎝⎛

×⎟⎠⎞

⎜⎝⎛

=

*

*

Zp

pdd

pdd

Zp

dtd

dtd

ψψ (A-5)

The first derivative in the numerator of Equation (A-5) is simply the derivative of Equation (A-1) with respect to time.

( )tqGZp

Zp

dtd

i

i**

−=⎟

⎠⎞

⎜⎝⎛ (A-6)

The second derivative in the numerator of Equation (A-5) is the derivative of Equation (A-3) with respect to p . This yields:

Zp

pdd

µψ 2

= (A-7)

Expanding the derivative of the denominator of Equation (A-5), it simplifies to:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=⎟⎠⎞

⎜⎝⎛

pZ

ZpZp

Zp

pdd *

***

11 (A-8)

Combining Equations (A-6), (A-7), and (A-8), one obtains:

( )*

*

2

t

i

i

c

tqGZp

dtd µψ

−

= (A-9)

where *tc is the modified total compressibility given by:

( )[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−−−=p

ZZp

ppcc iet

*

** 111 (A-10)

pZ ∂∂ * can be expressed in terms of fluid and reservoir properties as:

( )[ ] ( )[ ]

( )[ ]2*

1

11

ppc

ppcp

ZppcpZ

pZ

ie

ieie

−−

−−∂∂

−−−∂∂

=∂∂ (A-11)

Expanding the derivatives gives:

( )ppcc

pZ

ZpZ

Z ie

e

−−−

∂∂

=∂∂

111 *

*

Substituting into Equation (A-10), the modified total compressibility can be obtained as:

nggt ccc +=* (A-12) where ngc is non-gas compressibility, which is expressed by:

( )[ ]ppccc igeng −−= 1 (A-13) Therefore, Equation (A-9) becomes:

[IPTC-11278-PP] 7

( )tqGcZ

pdt

d

ti

i**

2µ

ψ −= (A-14)

Integrating Equation (A-14) within the appropriate limits and dividing the result by ( )tq , gives:

( ) ( ) dtc

qtqGZ

ptq

t

ti

ii ∫=−

0**

2µ

ψψ (A-15)

Similar to homogeneous gas reservoirs, we define a modified pseudo-time as:

∫=t

ttiia c

dtct0

**

µµ (A-16)

Now we will define the material balance pseudo-time, cat , as:

( )( )

a

t

t

a

a

itica dt

ctq

tqct

a

∫=0

*

*

µµ (A-17)

Combining Equations (A-15) and (A-17), we finally obtain:

caiiti

ii tGZc

pq **

2µ

ψψ=

− (A-18)

Appendix B: Governing Equation-Warren and Root5 Approach

With the assumptions mentioned in the main body of the paper, this section presents the derivation of the governing equation for the flow of gas in a radial naturally fractured gas reservoir, along with the corresponding solutions under constant rate and constant pressure production scenarios. For the sake of completeness, all equations in the main text are also shown here. A mass balance on the elemental volume shown in Figure 3 can be expressed as:

( )t

qr

pkr

rrr

fg

f

f

ff

∂

∂=+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂

∂

∂∂ 2*1 φρ

µρ

(B-1)

The source term in Equation B-1, ( )trq g ,* is the mass flow rate of gas released from the matrix into the fractures per unit bulk volume and is expressed by:

( )[ ]t

Sq wim

g ∂−∂

−=11* φρ (B-2)

where mρ is the gas density evaluated at matrix pressure.

Using the Al-Hussainy and Ramey2 pseudo-pressure, Equation (B-1) can be written in terms of pseudo-pressure:

( ) ( )

( )tk

cc

tkccS

rr

rr

f

f

gff

m

f

gmmwif

∂∂+

=∂∂+−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

ψµφ

ψµφψ

22

11 11

(B-3) where 1c and 2c are matrix and fracture pore volume compressibilities, respectively, and can be expressed by:

pc i

jj ∂

∂=

φφ1 j = (1, 2) (B-4)

Equation (B-3) has two unknowns. In this paper, an equation similar to that introduced by Warren and Root is used to calculate the matrix-fracture transfer rate. This relation in terms of pseudo-pressure can be stated as follows:

( )( ) ( )mfm

mmgmwi

kdt

dccS ψψµαψφ −=+− 11 1 (B-5)

The parameter α is a shape factor having the dimensions of reciprocal area. It reflects the shape and size of the matrix blocks and controls the flow between the fracture and matrix. For matrix blocks in the shape of slab this is expressed as:

212 mh=α (B-6)

In homogeneous gas reservoirs, the use of pseudo-time, Equation (4), evaluated at average reservoir pressure, was successfully used to “linearize” the governing equation. However, the definition of pseudo-time in a naturally fractured gas reservoir is not clear due to the fact that, for a naturally fractured gas reservoir, two average pressures may be defined one for the matrix where most of the gas resides and one for the fracture where reservoir-wide flow occurs. We have used both of these pressures without satisfactory results. In the hopes of achieving linearization of the governing equations of the naturally fractured gas reservoir, we use the average reservoir pressure of the “whole” reservoir ( mf ppp << ) as determined from the material balance Equation (A-1). On this basis, Equations (B-3) and (B-5) can be respectively simplified to:

( )( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∂

∂++

∂∂

+−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

∂∂

tcc

tccS

krr

rr fg

mgwi

f

f

ψφ

ψφµψ

22

11 11 (B-7)

( )( ) ( )mfm

m

gwi

dtd

kccS

ψψψ

αµφ

−=+− 11 1

(B-8)

Note that the term ( )( )µ11 ccS gwi +− in the right-hand side of Equation (B-7) and left-hand side of Equation (B-8) is a function of average reservoir pressure which changes with time. Rewriting Equations (B-7) and (B-8) in terms of the pseudo-time defined by Equation (7) yields:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

∂∂

a

fc

a

mc

f

tiif

tN

tN

kc

rr

rrψψµψ

21

*1 (B-9)

( )mfa

mc

m

tii

tN

kc

ψψψ

αµ

−=∂∂

1

*

(B-10)

where ( )( ) *

111 1 tgwic cccSN +−= φ (B-11)

8 [IPTC-11278-PP]

( ) *222 tgc cccN += φ (B-12)

For the case of negligible matrix and fracture pore volume compressibilities and assuming that 0=wiS , the total compressibility becomes equal to the gas compressibility ( gt cc =* ). As a result, the parameters 1cN and 2cN become equal to 1φ and 2φ , respectively.

Equations (B-7) and (B-8) are the governing differential equations. Analogous to the naturally fractured oil reservoir situation6, one can obtain the dimensionless form of these equations as:

( )D

Dm

D

Df

D

DfD

DD ttrr

rr ∂∆∂

−+∂∆∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∆∂

∂∂ ψω

ψω

ψ11 (B-13)

( ) ( )DmDfD

Dm

tψψλψω ∆−∆=

∂∂

−1 (B-14)

where

21

2

cc

c

NNN+

=ω (B-15)

2w

f

m rkkα

λ = (B-16)

wD rrr = (B-17) ( )

TpqThk

sc

miscfDm

ψψπψ

−= (B-18)

( )Tpq

Thk

sc

fiscfDf

ψψπψ

−= (B-19)

( ) qhkT

Tpq

wisc

scD ψψπ −= (B-20)

( ) acwtii

fa

cwtii

fD t

Nrck

tNrc

kt ω

µω

µ−== 1

12*

22*

(B-21)

Unlike double porosity oil reservoirs, the time domain solution for Equations (B-13) and (B-14) cannot be obtained using Laplace transformations. This is because the storativity ratio, ( )pω , is a time-varying function. However, if we assume that ( )pω can be treated as a constant, the solution of the flow

equations for slightly compressible oil in a double porosity medium6, (see Table 3) may be applicable for double porosity gas reservoirs. The implication of this assumption is discussed in the body of the text.

Table 1: Important Features of Some Mathematical Models Describing Flow from Matrix to Fracture Warren and Root (1965)5

Analytical model. Pseudo-steady state model. Matrix flux is independent of a spatial position and is

proportional to the pressure difference between matrix and fracture.

Simplifying the mathematical analysis of the flow problem.

S-shaped transitional curve with an inflection point. The separation of the two parallel lines allows

calculation of the storativity ratio.

Kazemi (1969)8

Numerical model. Unsteady state model. Linear transitional curve with no inflection point.

de Swaan (1976)9

Analytical model. Unsteady state model. A convolution theorem gives the relationship between

the source term and the pressure in the fracture medium. Linear transitional curve with no inflection point.

Najurieta (1980)10

Analytical model. Unsteady state model. Approximate solution to de Swaan model. Only applicable for transient period (no boundary

dominated period). Linear transitional curve with no inflection point.

Streltsova (1983)7

Analytical model. Pressure gradient model. Matrix flux is proportional to the averaged pressure

gradient throughout the matrix block. Linear transitional curve with no inflection point.

[IPTC-11278-PP] 9

Table 2: Summary of Equations and Relations

Description Equation Material balance equation

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

GG

Zp

Zp p

i

i 1**

( )[ ]ppcZZ ie −−= 1* ( )( ) 21

2211

1 φφφφ

+−++

=wi

wiwe S

ccScc

(A-1) (A-2) (3)

Pseudo functions

( ) dpZpp

p

pb

∫=µ

ψ 2

( ) ( ) ( ) ∫=−=t

ptptiii

i

iitia c

dtcpGZ

tqcpt

0 )()(

**

2 µµψψµ

( ) ( ) ( )( )

( ) ( )

*

0*

**

2 a

t

ptp

itii

i

iitica dt

ctq

tqc

pGZ

tqct ∫=−=

µµψψµ

nggt ccc +=* ( )[ ]ppccc igeng −−= 1

(A-3) (A-16) (A-17) (A-12) (A-13)

Dimensionless variables/parameters wD rrr =

( )ψψπ

ψ −= isc

scD qTp

khT

( ) qhkT

Tpq

wisc

scD ψψπ −=

( ) acwtii

fa

cwtii

fD t

Nrck

tNrc

kt ω

µω

µ−== 1

12*

22*

( ) ( )( )( ) ( )2211

22

1 ccccScc

pggwi

g

+++−

+==

φφφ

ωω

2w

f

m rkkα

λ =

( )( ) *111 1 tgwic cccSN +−= φ

( ) *222 tgc cccN += φ

(B-17) (B-18,19) (B-20) (B-21) (10) (11) (B-11) (B-12)

Matrix and fracture flow governing equations

( )D

Dm

D

Df

D

DfD

DD ttrr

rr ∂∆∂

−+∂∆∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∆∂

∂∂ ψω

ψω

ψ11

( ) ( )DmDfD

Dm

tψψλ

ψω ∆−∆=

∂∂

−1

(B-13) (B-14)

Table 3: Solution for Bounded Reservoir

Constant rate Laplace space solution ( )

( )( ) ( )( ) ( )( ) ( )( )[ ]( ) ( )( ) ( )( ) ( )( ) ( )( )[ ]SSfISSfrKSSfrISSfKSSfS

SSfrKSSfrISSfrISSfrK

Sp

DeDe

DDeDDe

wD

1111

0101

−

+

=

(B-22) Constant pressure Laplace space solution

( )( ) ( )( ) ( )( ) ( )( ) ( )( )[ ]

( )( ) ( )( ) ( )( ) ( )( )[ ]SSfrISSfKSSfISSfrKS

SSfKSSfrISSfISSfrKSSf

Sq

DeDe

DeDe

D

1001

1111

+

−

=

(B-23)

( ) ( )( ) λω

λωω+−+−

=SS

Sf11

(B-24)

Table 4(a): Reservoir Properties for the Base Case*

Parameter Case 1 (Base)

q (std. m3/day) (for CR) 2x106

pw (MPa) (for CP) 3.5

pi(MPa) 28

T(K) 333.15 h(m) 100

rw(m) 0.1

re(m) 1000

mk (md) 5

fk (md) 50

1φ 0.15

2φ 0.01

wiS 0.2

wc (kPa-1) 4.35e-7

1c (kPa-1) 4.35e-7

2c (kPa-1) 4.35e-6

* CR: Constant rate / CP: Constant pressure

Table 4(b): Parameters that Change for Other Cases.

Case 2 Case 3 Case 4 Case 5

mdkm 5= mdk f 200=

mh 10=

mdkm 10= mdk f 200=

005.02 =φ

mdk m 1= mdk f 200=

MPapi 5= 5102×=q

MPapwf 5.0=

10 [IPTC-11278-PP]

Figure 1: Warren and Root’s idealization of naturally fractured reservoir (after Warren and Root5, 1963).

Figure 2: Hypothetical radial-cylindrical naturally fractured gas reservoir.

( )rrgg vA

∆+ρ( )

rgg Avρ

∆

( )trp ,h

r

( )trqg ,

Figure 3: Elemental volume in naturally fractured gas reservoir.

Figure 4: Variation of the compressibilities and storativity ratio over a broad range of reservoir pressure.

Produced Gas

( )[ ]ppcccc igegt −−+= 1*

( )( ) 21

2211

1 φφφφ

+−++

=wi

wiwe S

ccScc

( )( )( ) ( )2211

22

1 ccccScc

ggwi

g

+++−

+=

φφφ

ω

[IPTC-11278-PP] 11

Figure 5: Comparison between analytical and numerical models: Base Case.

∫=t

ttiia c

dtct0

**

µµ ∫=

t

ttiia c

dtct0

**

µµ

12 [IPTC-11278-PP]

Figure 6: Effect of a constant storativity ratio on behavior of the solution.

Figure 7: Comparison between the analytical and numerical models (a): Case 2, (b): Case 3, (c):Case 4, (d): Case 5.

1 Blasingte Reservoir Limits Testing,” Paper SPE 15028 presented at the Permian Basin Oil and Gas Recovery Conference, Midland, TX, March 1314,(1986). 2 Al-Hussainy, R., Ramey, H.J., and Crawford, P.B.: “The Flow of Real Gases through Porous Media,” Journal of Petroleum Technology (JPT), 18, pp. 626-636, (1966). 3 Fraim, M. L., and Wattenbarger, R.A.: “Gas Reservoir Decline-Curve Analysis Using Type Curves with Real Gas Pseudo-pressure and Normalized time,” SPE Reservoir Evaluation and Engineering, Dec. pp.671-682, (1987). 4 Barenblat, G.I. and Zheltov. Yu. P.: ‘Fundamental Equations of Filtration of Homogeneous Liquids in Fissured Rocks,” Soviet Physics, Doklady (1960) 5, 522, (1960). 5 Warren, J.E. and Root, P.J.: “The Behavior of Naturally Fractured Reservoirs,” SPEJ (Sep. 1963) 245-55: Tram AIME. 228, (1963). 6 Sabet, M.A.: “Well Test Analysis,” Gulf Publishing Company, (1991). 7 Streltsova, T.D.: “Well Pressure Behavior of a Naturally Fractured Reservoir,” SPEJ (Oct. 1983) 769-80,( 1983). 8 Kazemi. H: “Pressure Transient Analysis of Naturally Fractured Reservoir with Uniform Fracture Distribution,” SPEJ (Dec. 1969) 451-62 Trans. AIME 261, (1969). 9 de Swaan-O., A.: “Analytical Solution for Determining Naturally Fractured Reservoir Properties by Well Testing,” SPEJ (June) pp.117- 22, Trans. , AIME, 228, (1976). 10 Najurieta, H.L,: “A Theory for Pressure Transient Analysis in Naturally Fractured Reservoirs,”, JPT (July 1980) 124-50. 11 Fetkovich, M.J.: “Decline Curve Analysis Using Type-Curves,” JPT (June) 1065, (1980). 12 Agarwal, R.G., Gardner, D.C., Kleinsteiber, S.W., and Fussell, D.D.: “Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts,” SPE 57916, SPE Reservoir Evaluation and Engineering, October, (1999).

13 Computer Modeling Group Inc.: “CMG-IMEX User Manual,” (2007).