An improved relative permeability model for coal reservoirs

International Journal of Coal Geology 109–110 (2013) 45–57

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International Journal of Coal Geology journal homepage: www.elsevier.com/locate/ijcoalgeo

An improved relative permeability model for coal reservoirs Dong Chen a, b, Zhejun Pan b,⁎, Jishan Liu a, Luke D. Connell b a b

School of Mechanical and Chemical Engineering, The University of Western Australia, WA 6009, Australia CSIRO Earth Science and Resource Engineering, Private Bag 10, Clayton South, VIC 3169, Australia

a r t i c l e

i n f o

Article history: Received 24 September 2012 Received in revised form 5 February 2013 Accepted 5 February 2013 Available online 15 February 2013 Keywords: Relative permeability Coalbed methane Coal seam Swelling Stress

a b s t r a c t In this work, the conventional relative permeability model for two phase flow in porous media is improved to describe the relative permeability for coal. The fracture geometry is considered through applying the matchstick model, instead of the bundle of capillary tubes model which is often used as the conceptual model for conventional porous media, to derive the relative permeability model. The effect of porosity change on relative permeability for coal is taken into account by introducing a residual phase saturation model and a shape factor as functions of permeability ratio. In the improved model, the relative permeability is dependent on both the phase saturation and the porosity (or permeability) change. This improved model shows a strong capability to match the experimental data for different coal relative permeability measurements. Furthermore, we evaluate the relative permeability models as a unary function of wetting phase saturation and as a binary function of wetting phase saturation and permeability ratio in a coupled numerical model for water–gas flow in coal seams. The results illustrate that the relative permeability change due to the porosity change can significantly affect the evolution of wetting phase saturation and the gas production rate. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Coal seams are dual porosity reservoirs that consist of porous matrix and cleat (fracture) network. Most coal seams are saturated with water which may exist in both the matrix and the cleats. The water in the matrix affects the gas transport in coal seams through changing the gas effective diffusivity as well as the coal swelling strain and the water in the cleats affects the gas flow through the relative permeability effect (Chen et al., 2012b). Relative permeability, the ratio of effective permeability to absolute permeability, is an important parameter for primary coalbed methane (CBM) production, since it not only determines whether the commercial gas production rates can be achieved, but also indicates the cost of CBM produced water disposal by knowing the amount of water dewatered (Ham and Kantzas, 2008). The characteristics of relative permeability curves are crucial for laboratory, field, and simulation studies of two phase flow behavior in coal (Clarkson et al., 2011; Shi et al., 2008a). The relative permeability is often expressed as a function of wetting phase saturation, which is normally defined as the relative permeability model, since it is strongly dependent on the phase saturation. A commonly used method to obtain the relative permeability model is through integration from the capillary pressure model (Chen et al., 1999; Li and Horne, 2008). Purcell (1949) proposed a method to calculate the permeability from capillary pressure based

⁎ Corresponding author. Tel.: +61 3 9545 8394; fax: +61 3 9545 8380. E-mail address: [email protected] (Z. Pan). 0166-5162/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coal.2013.02.002

on the bundle of capillary tubes model. This method was extended by Gates and Leitz (1950) to integrate the relative permeability from the capillary pressure which is often named as the Purcell method. Since the capillaries may not always be parallel to each other, but tortuous, Burdine (1953) introduced a tortuosity factor to incorporate the effect of tortuosity into the Purcell method. Besides the Purcell method and the Burdine method, another integration method is derived by Mualem (1976) for homogeneous porous medium. The three methods introduced are among the most commonly used methods for relative permeability integration from capillary pressure data. Many capillary models are available in the literature based on the experimental data of core flooding on different porous media with different fluids (Brooks and Corey, 1966; Brutsaert, 1967; Corey, 1954; Delshad et al., 2003; Gardner, 1958; Huang et al., 1997; Jing and van Wunnik, 1998; Kosugi, 1994, 1996; Lenhard and Oostrom, 1998; Li, 2010; Li and Horne model, 2001; Lomeland and Ebeltoft, 2008; Purcell, 1949; Russo, 1988; Skelt and Harrison, 1995; Thomeer, 1960; van Genuchten, 1980). Among all these capillary pressure models, the Brooks–Corey model (1966) and the van Genuchten model (1980) are two widely used models. The Brooks–Corey model has been widely used for the consolidated porous media while the van Genutchen model has been used for the unconsolidated porous media (Li, 2004). The two commonly used relative permeability models, the BCB model and the VGM model, were obtained through integration from the Brooks–Corey model with the Burdine method (BCB) and from the van Genuchten model with the Mualem method (VGM), respectively (Brooks and Corey, 1966; Chen et al., 1999; van Genuchten, 1980).

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D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

In the past few decades, several experiments have been performed to measure the capillary pressure for different coals (Dabbous et al., 1976; Kissell and Edwards, 1975; Mazumder et al., 2003; Ohen et al., 1991) and the Brooks and Corey model was often used to represent the capillary pressure data as a function of phase saturation. More data were reported for relative permeability of coals (Dabbous et al., 1974, 1976; Durucan and Shi, 2002; Gash, 1991; Hyman et al., 1992; Jones et al., 1988; Meaney and Paterson, 1996; Ohen et al., 1991; Paterson et al., 1992; Puri et al., 1991; Reznik et al., 1974; Shen et al., 2011) and the BCB model was often applied to describe the experimental data. Although the BCB model is able to mathematically fit the relative permeability curves for some coals, the validity of this model for coal reservoirs has not been examined since it was derived from the bundle of capillary tubes model for conventional porous media. Coal seams are fractured and the experimental results showed that the gas flow through fractures and cracks in coal could not be described by the bundle of capillary tubes but by the flow between parallel plates (Harpalani and McPherson, 1986). Based on this concept, Seidle et al. (1992) applied the matchstick geometry to describe the fluid flow in the coal cleat networks. The matchstick model has become the widely accepted conceptual model for coal seams and many absolute permeability models were developed based on this model (Gu and Chalaturnyk, 2010; Seidle and Huitt, 1995; Shi and Durucan, 2004, 2010). Thus, it would be consistent to use the same matchstick conceptual model for both relative permeability model and the absolute permeability model. Moreover, the conventional capillary pressure and relative permeability models are often assumed as unary functions of saturation, because the two variables are strongly dependent on the phase saturation. However the effect of the overburden pressure (which leads to porosity change) on the capillary pressure and the relative permeability change has been observed in the experiments on sandstones (Al-Fossail et al., 1995; Ali et al., 1987; Al-Quraishi and Khairy, 2005). Thus such unary functions may not well represent the capillary pressure and the relative permeability data for some soft rocks such as coals, because the porosity change is another important factor along with the phase saturation. The coal porosity change is mainly controlled by two competing effects of effective stress and coal swelling/ shrinkage. This has been verified by a number of experiments and many models have been proposed to describe the coal porosity change (Liu et al., 2011; Pan and Connell, 2012). Therefore both the capillary pressure and the relative permeability may be affected by either the effective stress or coal swelling. Gray (1987) stated that the opening and closing of the cleats were likely to change the phase relative permeabilities and capillary pressures within the coal. This was supported by the experimental results on a number of Pittsburgh and Pocahontas coals, showing that the capillary pressure and the relative permeability were strongly dependent on the overburden pressure (Dabbous et al., 1974, 1976; Reznik et al., 1974). There is no data available for capillary pressure and relative permeability change due to the coal swelling. However, the coal swelling may also change the capillary pressure and relative permeability due to its impact on the coal porosity change under reservoir conditions. In this work, we improve the relative permeability model for coal considering both the coal fractured macrostructure geometry and the porosity change due to stress change and swelling/shrinkage. Then the model is verified with the experimental data from the literature. Finally, a coupled reservoir simulation model is applied to investigate how the coal porosity change induced relative permeability change affects the CBM production.

were proposed based on the bundle of capillary tubes model as shown in Fig. 1 (a). However, the experimental results showed that the gas flow through fractures and cracks in coal could not be described by the bundle of capillary tubes but by the flow between parallel plates (Harpalani and McPherson, 1986). The matchstick model, as represented by Fig. 1 (b), is a widely used conceptual model for coal. In this work, we implement a similar approach as the Purcell's work (1949) to develop the integration method for coal using the matchstick model, instead of the bundle of capillary tubes model. The details of the derivation are referred to Appendix A. The results are shown in Eqs. (A17) and (A18). Coal seam is not only fractured, but also deformable. Thus its porosity changes during CBM production. As a consequence, the capillary pressure and relative permeability models will need to incorporate both the wetting phase saturation and the effect of porosity change. The effect of porosity change may affect the capillary pressure curve in two ways: (1) changing residual phase saturation; (2) changing curvature. They are incorporated into the capillary pressure and relative permeability as follows. 2.1. Modeling of residual phase saturation change The residual water saturation is defined as Eq. (1). The change of the residual water saturation can be explained as that the void volume increases/decreases as the coal is unloaded/loaded by the decrease/ increase of the overburden pressure. Swr ¼

V wr V void

ð1Þ

where Vwr and Vvoid are the volume of the residual water content and the void volume in the coal cleat, respectively. The residual phase content is the amount of the residual phase contained in the porous media. If the residual water content remains the same with the porosity change, the mass conservation for the residual water content can be expressed as ρw ϕ0 Swr0 V 0 ¼ ρw ϕSwr V

ð2Þ

where ρw is the density of water, ϕ is the porosity, and V is the total volume of the porous media. The subscript “0” indicates that the value is in the initial state. Since the density of water can be regarded as constant, Eq. (2) can be simplified to Eq. (3) by neglecting the total volume change, which is V0 = V.
Swr ¼ Swr0

ϕ ϕ0

−1

ð3Þ

Comparing to the water phase, the gas is compressible and thus the residual gas saturation can be calculated by
Sgr ¼ Sgr0

ϕ ϕ0

−1

ρg ρg0

!−1 ð4Þ

where Sgr is the residual gas saturation and ρg is the density of gas. Since the cubic law relationship between the permeability ratio and the porosity ratio as expressed in Eq. (5) is often used for coals (e.g. Gu, 2009; McKee and Hanson, 1975), Eqs. (3) and (4) can also be written as Eqs. (6) and (7).

2. Relative permeability model development for coal



3 k ϕ ¼ k0 ϕ0

A widely used method to obtain the relative permeability model is to integrate it from the capillary pressure model. The previous integration methods, such as Purcell method and the Burdine method,

Swr ¼ Swr0


−1 3 k k0

ð5Þ

ð6Þ

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

(a) (after Gates and Lietz, 1950)

47

(b) (after Seidle et al., 1992)

Fig. 1. Bundle of capillary tubes model (a) and matchstick model (b).

!−1
−1 3 ρg k Sgr ¼ Sgr0 k0 ρg0

ð7Þ

We applied Eq. (6) to represent the experimental data for Pittsburgh coal on residual water saturation change with the permeability change (Dabbous et al., 1976; Reznik et al., 1974). As shown by the solid line in Fig. 2, Eq. (6) cannot represent the experimental data with accuracy, though it could predict the general trend of the residual water saturation change. One of the reasons for the mismatch may be due to the assumption of the unchanged residual water content with fracture porosity change. In reality, the total amount of residual water may also change with the porosity change as a result of pore size variation, which has been observed in laboratory for unsaturated soils (Chen et al., 2010c). There are two possible reasons for the residual water content change: (1) some residual water in smaller pores may be squeezed out due to the decrease of the pore volume when the coal is compacted (2) capillary force change due to the pore size change by compression may also contribute to the residual water content variation. We assume that the change of the residual water saturation can be expressed as a power law of porosity/permeability ratio. Then this effect can be taken into account by improving Eqs. (6) and (7) to the following forms Swr

ð8Þ

Residual water saturation, Swr

90

Measurement

80

Model Improved model

70 60 50

where nwr and ngr are fitting parameters to determine the relationship between the residual phase saturation and the porosity ratio. The improved residual phase saturation model is more flexible to represent the experimental data. As shown by the dash line in Fig. 2, the improved model is capable of matching the experimental data (nwr = 0.49 for this case). It should be noted that more experimental data are required for a better understanding of the effect of porosity change on the residual water saturation change. The effect of porosity change on the residual phase saturation change can be incorporated into the capillary pressure and relative permeability models by combing the residual phase saturation model and the normalized wetting phase saturation as expressed in Eq. (A16). The updated form of normalized wetting phase saturation is expressed as

 Sw

 −1n wr Sw −Swr0 kk 3 0 ¼  −1n  −1n  −1 : ρg wr gr 1−Swr0 kk 3 −Sgr0 kk 3 ρ 0

0

ð10Þ

g0

Besides the residual phase saturation change, the shape of the capillary pressure curve may change as a result of the porosity change. It is illustrated in Fig. 3 that the shape of the typical capillary pressure curve changes with the permeability (porosity) change according to the Leverett's (1940) J-function. In the Brooks–Corey model, the curvature of the capillary pressure curve is controlled by the cleat size distribution index (λ) and thus it may change with the porosity change. Similarly, the effect of porosity change on the capillary pressure curve change affects the relative permeability curves as well. In order to take into account this effect, a shape parameter, J, is introduced. We use the J factor to correct the cleat size distribution index which controls the shape of the capillary pressure curve. Then the extended capillary pressure model can be written as   −1=ðJ⋅λÞ pc ¼ pe Sw :

40

0

ð9Þ

2.2. Modeling of curvature change


−1n 3 wr k ¼ Swr0 k0

30

!−1
−1n 3 gr ρg k Sgr ¼ Sgr0 k0 ρg0

2

4

6

8

10

Permeability ratio, k/k0 Fig. 2. Matching the experimental data on residual water saturation change as a result of permeability change.

ð11Þ

It should be noted here that the J factor is not a constant, but a function of the porosity/permeability changes. Currently, we simply apply the empirical relationship in this work due to the lack of the literature data. More work is required in the future to investigate the effect of the porosity change on the capillary pressure and relative permeability

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The relation between permeability and effective horizontal stress is expressed as (Seidle et al., 1992): −3cf ðσ −σ 0 Þ

k ¼ k0 e

ð16Þ

where cf is the cleat volume compressibility with respect to changes in the effective horizontal stress normal to the cleats. The residual wetting/nonwetting phase saturation and the normalized wetting phase saturation can be rewritten as Eqs. (17)–(19) by substituting Eq. (16) into Eqs. (8)–(10). Swr ¼ Swr0 e

nwr cf ðσ−σ 0 Þ

ngr cf ðσ−σ 0 Þ

Fig. 3. Effect of permeability magnitude on typical shapes of capillary pressure curves (Purcell, 1949).

change. In addition, the change of the entry capillary pressure due to the porosity change is ignored, since it can be considered as a local effect on the capillary pressure curve at the saturation close to the maximum water saturation which may not significantly affect the general shape of the capillary pressure curve. Similarly, the extended relative permeability model can be expressed as    ηþ1þ2=ðJ⋅λÞ krw ¼ krw Sw

ð12Þ

h   1þ2=ðJ⋅λÞ i    η : krnw ¼ krnw 1−Sw 1− Sw

ð13Þ

At present, the impact of the porosity change on the end-point relative permeability change is ignored because there is lack of experimental data. Moreover, this factor does not have a significant effect on the methane flow rate in general (Karacan, 2008). The change of the shape of relative permeability curves may be more important than the variation of only end-point relative permeability values (Price and Ancell, 1993; Young et al., 1992). 2.3. Linking capillary pressure and relative permeability model to permeability model

υ EΔεs ðp−p0 Þ þ 1−υ 3ð1−υÞ

ð14Þ

where σ is the effective horizontal stress, σ0 is the effective horizontal stress at the initial reservoir pressure, εs is the total volumetric swelling/shrinkage strain change which is calculated by a Langmuir type equation expressed as εs ¼

εL p p þ Pε



Sw ¼

ρg ρg0

!−1 ð18Þ nwr cf ðσ −σ 0 Þ

Sw −Swr0 e

1−Swr0 enwr cf ðσ−σ 0 Þ −Sgr0 engr cf ðσ −σ 0 Þ



 ρg −1 ρg0

ð19Þ

3. Model verification with experimental data Two groups of experimental data are used to validate the model proposed in Section 2. Many relative permeability measurements were carried out under constant stress and sorption conditions, in which the porosity does not change with the two phase flow process. In these experiments, the relative permeability is only affected by phase saturation. For these problems, the simplified relative permeability model as derived in Appendix A can be implemented to describe the relative permeability data. These experimental data are classified as the group under the constant porosity condition. In some other experiments, the relative permeability is measured under different overburden pressure levels. The relative permeability for these tests is not only affected by phase saturation, but also dependent upon the porosity change due to the overburden pressure. They are classified as the group under changing porosity condition. The binary equations as proposed in Section 2 are required to represent the experimental data. 3.1. Model verification with experimental data under constant porosity condition

It has been shown in previous sections that the capillary pressure and the relative permeability models can be expressed as binary equations of both the wetting phase saturation and the absolute permeability. One of the advantages of linking the capillary pressure and the relative permeability models to permeability model is that there have been many permeability models available in the literature (e.g. Liu et al., 2011; Pan and Connell, 2012). In this work, we use one of the most widely used permeability models, the Shi and Durucan model (2004), as the permeability model: σ −σ 0 ¼ −

Sgr ¼ Sgr0 e

ð17Þ

ð15Þ

where εL is the Langmuir volumetric strain constant and Pε is the Langmuir strain constant at 1/2 εL.

Several groups of experimental data on relative permeability of different coals (Conway et al., 1995; Durucan and Shi, 2002; Gash, 1991; Meaney and Paterson, 1996; Paterson et al., 1992; Shen et al., 2011) are used to verify the relative permeability model. In these measurements, the confining pressure and the pore pressure are unchanged or their changes are negligible. The experimental data as shown in Fig. 4 present that the residual water saturation is relatively high for most coal samples. Therefore many relative permeability curves locate in the high water saturation region and this indicates that the relative permeability may change significantly with the decreasing of the water saturation. The high water saturation may be due to the low permeability of coals for which the capillary pressure is high and more water can be trapped as the residual phase. It also shows that the coals are strongly water wet under those circumstances. The sensitivity of the relative permeability to the water saturation change requires an accurate relative permeability model to represent the experimental data. It can be seen in Fig. 4 that the relative permeability model proposed in this work can well represent the experimental data. In addition, Fig. 4 also shows that most of the relative permeability curves to water are concave despite of an UK coal which is nearly a linear as shown in Fig. 4 (g). However, there is no typical shape of the relative permeability curves to gas. It can be linear, concave, or convex. Thus the relative permeability model for gas should be flexible enough

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

b

1.0 Gas and water Bowen Basin (Meany and Patterson, 1996) λ=2; η=2

0.8

Relative permeability

Relative permeability

a

0.6 krw Experimental data

0.4

krg Experimental data Proposed model

0.2

1.0 Gas and water northern Sydney Basin (Meany and Patterson, 1996) λ=7; η=2

0.8 0.6

krw Experimental data

0.4

krg Experimental data Proposed model

0.2

Proposed model

0.0

0

20

Proposed model

40

60

80

0.0

100

0

20

d

Relative permeability

Helium and water Cahn Seam San Juan Basin (Gash, 1991) λ=7; η=1

0.6

krw Experimental data krg Experimental data

0.4

Proposed model

0.2

Relative permeability

c 0.8

0

20

40

60

80

0.8

krw Experimental data

0.4

krg Experimental data

0.2

Proposed model Proposed model

100

0

20

krw Experimental data krg Experimental data Proposed model

f

1.0

Relative permeability

Relative permeability

0.6

0.2

0.8

0

20

40

krw Experimental data

0.4

krg Experimental data Proposed model

0.2

60

80

0.0

100

0

20

h Relative permeability

Relative permeability

Gas and water UK coal (Durucan and Shi, 2002) λ=7; η=0.2 krw Experimental data krg Experimental data Proposed model

0.2

0

20

40

60

80

100

80

100

1.0 Methane and water Wangyun coal Qinshui Basin (Shen et al., 2011) λ=2.6; η=0.3

0.8 0.6

krw Experimental data

0.4

krg Experimental data Proposed model

0.2

Proposed model

0.0

40

Water saturation, Sw (%)

1.0

0.4

100

Proposed model

g

0.6

80

0.6

Water saturation, Sw (%)

0.8

60

Gas and water Black Warrior Basin (Conway et al., 1995) λ=0.25; η=1.8

Proposed model

0.0

40

Water saturation, Sw (%)

Methane and water Bowen Basin (Paterson et al., 1992) λ=7; η=1

0.4

100

0.6

0.0

1.0 0.8

80

Helium and water Seam No. 1 Sundance Pit LaPlata Mine San Juan Basin (Gash, 1991) λ=1.2; η=1

Water saturation, Sw (%)

e

60

1.0

Proposed model

0.0

40

Water saturation, Sw (%)

Water saturation, Sw (%)

1.0

49

Proposed model

60

Water saturation, Sw (%)

80

100

0.0

0

20

40

60

Water saturation, Sw (%)

Fig. 4. Matching relative permeability data on coals from: (a) Bowen Basin and (b) northern Sydney Basin (Meaney and Paterson, 1996); (c) and (d) San Juan Basin (Gash, 1991); (e) Bowen Basin (Paterson et al., 1992); (f) Black Warrior Basin (Conway et al., 1995); (g) UK (Durucan and Shi, 2002); (h) Qinshui Basin (Shen et al., 2011). The relative permeability model is able to well fit the experimental data.

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

to describe this behavior. The results as shown in Fig. 4 demonstrate that the relative permeability model developed in this work is able to achieve a good data matching quality. 3.2. Model verification with experimental data under changing porosity condition The experimental results showed that the capillary pressure and the relative permeability curves of Pittsburgh and Pocahontas coals were shifted toward increasing residual water saturation (smaller cleat size or decreasing porosity) with respect to the increment of overburden pressure (Dabbous et al., 1974, 1976; Reznik et al., 1974). In those experiments, the permeability, the capillary pressure and the relative permeability data were measured under three radial overburden pressure (pov) levels of 200 psig (1.38 MPa), 600 psig (4.14 MPa) and 1000 psig (6.90 MPa), respectively. Fig. 5 shows that the model proposed in this work is capable of representing the experimental data of the capillary pressure data for the Pittsburgh coal as a function of both the wetting phase saturation and the overburden pressure through permeability ratio. In addition, Fig. 6 and Fig. 7 illustrate that the proposed relative permeability model can represent the relative permeability data for two other coal samples. These results validate that the models proposed are able to describe capillary pressure and relative permeability as a function of both the wetting phase saturation and the porosity/permeability ratio. Since the coal porosity change is due to two competing factors of effective stress induced mechanical deformation and gas sorption induced coal swelling/shrinkage (Gray, 1987), the capillary pressure and the relative permeability may also be affected by the coal swelling strain. However, there is lack of experimental data on capillary pressure and relative permeability change due to the porosity change caused by swelling/shrinkage. 4. Application of relative permeability models in reservoir simulation In order to further investigate the effect of different relative permeability models (unary equations and binary equations) on CBM production, two cases are compared based on the reservoir simulation: Case 1, the capillary pressure and relative permeability are input as unary functions of wetting phase saturation as developed in Section 2; Case 2, the capillary pressure and relative permeability input are binary functions of both wetting phase saturation and permeability ratio as developed in Section 3. 4.1. Formulation of reservoir simulation model

Relative permeability of water, krw

50

pc,, MPa

1.38 MPa

0.15

4.14 MPa

0.10 0.05

S wr0 =84%

6.90 MPa

p e=0.006 MPa

Model J=1

λ=0.22

Model J=1.2

n wr =0.49

Model J=4.6

0.00

0

20

40

60

80

100

Sw, % Fig. 5. Data match of capillary pressure of Pittsburgh coal (Dabbous et al., 1974, 1976; Reznik et al., 1974) as a function of wetting phase saturation and permeability ratio.

6.90 MPa Swr=30%

0.6

Swr=52%

0.4 0.2 0.0

0

20

40

60

80

100

Fig. 6. Data match of relative permeability of Pocahontas coal (Dabbous et al., 1974, 1976; Reznik et al., 1974) as a function of wetting phase saturation and permeability ratio.

transport in coal seams. The difference from their model is that the relative permeability model implemented is the proposed model in this work. 4.1.1. Governing equations The mass conservation equations for the water and gas in the coal cleats are defined as Eqs. (20) and (21), respectively.
 ∂ðρw ϕSw Þ kk ¼ ∇⋅ ρw rw ð∇pw −ρw g∇dÞ −qw μw ∂t   ∂ ρg ϕSg ∂t

ð20Þ

!  kkrg  ∇pg −ρg g∇d −qg þ qd ¼ ∇⋅ ρg μg

ð21Þ

where ρw and ρg are the density of water and gas respectively, Sg is the gas saturation, μw and μg are the dynamic viscosity of water and gas in the coal cleats respectively, g is the acceleration of gravity, d is the height of the reservoir, qw and qg are the mass production rates for water and gas respectively, and qd is the gas mass transfer rate between the coal matrix and the coal cleats given by: qd ¼ −ρga ρc

dV dt

ð22Þ

where ρga is the density of gas at standard conditions, ρc is the density of coal and V (m 3/kg) is the average remaining gas content in the coal

Relative permeability of gas, krg

PGH-2

0.20

1.38 MPa

0.8

Sw, %

The two-phase, dual porosity model documented by Shi et al. (2008a, 2008b) is used as the reservoir simulation model for the water and gas

0.25

1.0

1.0 1.38 MPa

0.8 4.14 MPa

0.6

Swr=50% Sgr=5% Swr=65% Sgr=3%

0.4 0.2 0.0

0

20

40

60

80

100

Sw, % Fig. 7. Data match of relative permeability of Pittsburgh coal (Dabbous et al., 1974, 1976; Reznik et al., 1974) as a function of wetting phase saturation and permeability ratio.

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

matrix which is calculated by a quasi steady-state equation for gas diffusion in the coal matrix (King et al., 1986; Shi et al., 2008a,2008b):  i dV 1h ¼ − V−V E pg dt τ

ð23Þ

where τ is a diffusion-time constant of the coal matrix and VE (m 3/kg) is the gas content in equilibrium with cleat gas pressure calculated by:   V L pg V E pg ¼ pg þ P L

ð24Þ

where VL is the Langmuir volume constant and PL is the Langmuir pressure constant at 1/2 VL. The correlation between the water saturation and the gas saturation is expressed as: Sw þ Sg ¼ 1:

ð25Þ

The gas density is a function of gas pressure as expressed in Eq. (26) according to the gas equation of state (EOS). ρg ¼

Mg p ZRT g

ð26Þ

where Mg is the molar mass of gas, nm is the mole amount of the gas in the coal matrix, R is the universal gas constant, T is the temperature and Z is the compressibility factor of gas which is calculated from the NIST webbook at: http://webbook.nist.gov/chemistry/fluid/. 4.1.2. Initial condition The initial pressure of water and gas in the coal cleats is given as follows pw ¼ pw0

ð27Þ

pg ¼ pg0 :

ð28Þ

pg0 : RT

ð29Þ

4.1.3. Boundary condition The well mass production rates for water and gas are computed by (Peaceman, 1978; Wei and Zhang, 2010): Q wwell ¼ ρw

kkrw ðp w −pwf Þ   μ w ln rre − 34 þ S

Table 1 Parameters used in Case 1 and Case 2. Parameters

Values

Young's modulus, E (MPa) Poission's ratio, ν Langmuir volumetric strain constant, εL Langmuir pressure constant at 1/2 εL, Pε (MPa) Initial permeability, k0 (m2) Initial porosity, ϕ0 (%) Coal cleat compressibility, cf (MPa−1) Gas Langmuir volume constant, VL (m3/kg) Gas Langmuir pressure constant at 1/2 VL, PL (MPa) Density of the coal, ρc (kg/m3) Diffusion time, τ (day) Initial residual water saturation, Swr0 (%) Initial residual gas saturation, Sgr0 (%) Tortuosity coefficient, η Cleat size distribution index, λ ⁎ End-point relative permeability of water, krw ⁎ End-point relative permeability of gas, krg

2900 0.35 0.01266 4.31 1.6e−14 0.085 0.29 0.027 2.96 1600 9 84 0 1 0.22 1 1 0.006 305.15 0.1 0.3 0 6.41 95 2 1000 6.5e−4 1.84e−5 0.016 8.314

Entry capillary pressure, pe (MPa) Temperature, T (K) Well radius, rw (m) Bottom hole pressure, pwf (MPa) Skin factor, S Initial water pressure, pw0 (MPa) Initial water satuation, Sw0 (%) Coal seam thickness, h (m) Density of water, ρw (kg/m3) Viscosity of water, μw (Pa·s) Viscosity of gas, μg (Pa·s) Molar mass of gas, Mg (kg/mol) Universal gas constant, R (J/(mol·K))

parameter for residual water saturation change (nwr) uses 0.49 and the shape parameter (J) as expressed in Eqs. (11–13) is input for Case 2 as a function of permeability ratio as shown in Fig. 8. 4.3. Simulation results

The initial gas concentration of the gas in the matrix is c ¼ c0 ¼

51

The evolutions of water saturation of the two cases are compared in Fig. 9. In Case 1, the capillary pressure as well as the relative permeability models are unary functions of water saturation and the residual water saturation is at a constant level of 84%. Fig. 9 shows that the water saturation decreases slowly after 200 days of production. In Case 2, the effect of porosity change is taken into account through defining the capillary pressure and the relative permeability model as binary functions of both water saturation and permeability ratio.

ð30Þ

w

8.0 ð31Þ

w

where pwf is the bottom hole pressure, re is the drainage radius, rw is the wellbore radius and S is the skin factor. No flow condition is applied to other boundaries. All the equations are solved by COMSOL Multiphysics (http://www. comsol.com/). 4.2. Model description The simulation model is a 100 × 100 m reservoir with a corner production well. The parameters used, as listed in Table 1, are obtained from the literature (Palmer and Mansoori, 1998; Pan et al., 2010; Shi and Durucan, 2004, 2008) and the experimental data fit in Section 3. Two more parameters are required for Case 2: the value of the fitting

Shape factor, J

Q gwell

  kkrg p g −pwf  ¼ ρg  μ g ln rre − 34 þ S

6.0

y = 0.1821x 2 - 0.7464x + 1.5643

4.0

2.0

0.0

0

2

4

6

8

Permeability ratio, k/k0 Fig. 8. Regression of shape factor (J), as a function of permeability ratio, from experimental data of Pittsburgh coal (Dabbous et al., 1974, 1976; Reznik et al., 1974) fitting as shown in Fig. 5.

52

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

Case 1: pc, krw, krg = f (Sw)

Sw (%)

93.27

50 days

92.55

92.12

100 days

150 days

Case 2: pc, krw, krg = f (Sw, k / k0 )

92.15

60.32

75.62

50 days

100 days

150 days

Fig. 9. Evolutions of water saturation of the two cases: Case 1, capillary pressure and relative permeability are unary functions of water saturation; Case 2, capillary pressure and relative permeability are binary functions of water saturation and permeability ratio.

5. Discussion 5.1. Effect of wettability In this work, we assume that coal is water wet and the contact angle (θ) of the liquid or gas phase and the solid phase remains constant when we derive our relative permeability model in Appendix A.

Gas production rate, Qg (std m3/day)

The increase of coal permeability during the CBM production tends to decrease the residual water saturation and this effect results in the significant decrease of the water saturation comparing to Case 1. Therefore the water saturation of Case 2 decreases significantly compared to that of Case 1. The high water saturation in Case 1 may hinder the gas production. The low water saturation in Case 2 allows more space for the gas to flow and thereafter may enhance the gas production. It is confirmed by Fig. 10, which shows that the gas production rate in Case 2 is much higher than that in Case 1. The simulation results indicate that the gas production rate may be underestimated if the effect of porosity change on relative permeability is not considered. Furthermore, the sensitivities of the cleat size distribution index (λ) and the tortuosity parameter (η) are studied with the simulation model. The results as shown in Fig. 11 and Fig. 12 illustrate that the gas production is enhanced with the increase (decrease) of the pore size distribution index (tortuosity parameter). It should be noted here that the simulation results obtained are based on the very limited experimental data. However, it suggests that the effect of porosity change on relative permeability change may significantly affect the CBM extraction. Therefore, more experimental and theoretical studies are required in the future to obtain a better understanding of the relative permeability characteristic in fractured and deformable reservoir rocks such as coals and shales. Their consequent effect on CBM production is also required to be investigated with the improvement of our understanding in this behavior.

1800 Case 1

1500

Case 2

1200 900 600 300 0

0

50

100

150

200

250

300

Time, t (day) Fig. 10. Comparison of gas production rates of the two cases: Case 1, capillary pressure and relative permeability are unary function of wetting phase saturation; Case 2, capillary pressure and relative permeability are binary function of wetting phase saturation and permeability ratio.

Gas production rate, Qg (std m3/day)

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

6. Conclusion

1800

λ=0.11 λ=0.22

1500

λ=0.33

1200 900 600 300 0

0

50

100

150

200

250

300

Time, t (day) Fig. 11. Sensitivity study of cleat size distribution index on gas production rate.

However, it should be noted that θ becomes variable when the wettability changes. In the methane–water–coal system as the target of this study, the coal can be considered as water wet and thus wettability can be reasonably assumed as unchanged. However, coal may become gas wet for CO2–water–coal system. Plug et al. (2006) found that the coal is water wet at low pressure condition but CO2 wet at high pressure condition. The wettability change due to the use of CO2 is also observed in other experimental work (Chaturvedi et al., 2009; Sakurovs and Lavrencic, 2011). Therefore, more work on the effect of wettability change on relative permeability in coal is required in the future; this is of great importance for CO2 enhanced coalbed methane recovery.

5.2. Effect of temperature

The improved relative permeability model proposed in this work can be used to represent the experimental data for coal. The model is validated with the experimental data on relative permeability for different coals. The results show that it is capable of matching a variety of relative permeability data for coal. Different to conventional relative permeability model which is often assumed as a unary function of phase saturation, the relative permeability for coal is also affected by the porosity change. To investigate the consequent effect of porosity change induced relative permeability change on CBM production, the relative permeability model is coupled into the reservoir simulation model. The simulation results illustrate that the effect of porosity change can significantly affect the evolutions of wetting phase saturation and then the gas production. However, more experimental and theoretical studies are required to further understand how effective stress and coal swelling affect the relative permeability change. Acknowledgments Financial support from CSC-UWA scholarship, CSIRO Earth Science and Resource Engineering top-up scholarship are acknowledged. Appendix A. Derivation of relative permeability model for matchstick model It is assumed that the flow is parallel to the cleat planes as shown in Fig. 13 (Reiss, 1980). According to the Poiseuille's equation, the flow rate along this single cleat (q1) can be expressed as q1 ¼ −

Although experimental studies have been carried out to investigate the temperature effect on relative permeability in oil/gas/water system for thermal enhanced heavy oil recovery (e.g. Ashrafi et al., 2012; Ayatollahi et al., 2005; Schembre et al., 2005), little has been done on this topic for coals. In this study, our focus is on the modeling of relative permeability in coal for primary CBM process where temperature could be considered as unchanged. However, during enhanced coalbed methane (ECBM) processes, CO2 or N2 or their mixture is often injected at a temperature different to the coal seam. The adsorption induced swelling and thermal expansion due to temperature change will both have impact on cleat size change, thus on the relative permeability behavior. Therefore, as future work, it is important to study the temperature effect on relative permeability for ECBM processes, with the consideration of wettability discussed earlier.

Gas production rate, Qg (std m3/day)

53

1800

η=0.5

b3 l Δp V b2 Δp ¼− 1 2 12μ L 12μL

ðA1Þ

where b, l and L are the geometry of the cleat as shown in Fig. 13. Δp is the pressure drop, μ is the viscosity of the fluid, and V1 = b·l·L is the volume of this single cleat. The negative sign indicates that the flow direction is in the opposite direction of the pressure gradient. Different to the capillary pressure in a single capillary as illustrated in Fig. 14 (a), the sketch of the capillary pressure in a single cleat can be drawn as Fig. 14 (b). Based on this conceptual model, the mechanical balance between the capillary pressure and the interfacial tension for this single cleat can be expressed as pc ⋅ b ⋅ L ¼ 2L ⋅ σ cos θ

ðA2Þ

where σ is the interfacial tension at the interface and θ is the contact angle of the liquid or gas phase and the solid phase at the interface. Substituting Eq. (A2) in Eq. (A1), we have q1 ¼ −

ðσ cos θÞ2 V 1 Δp : 3μL2 ðpc Þ2

ðA3Þ

η=1.0

1500

η=1.5

1200

It is assumed that σ and θ remain constant. Under the assumption of water wet coal, the total flow rate of n cleats can be calculated by

900 qn ¼ −

600

n ðσ cos θÞ2 Δp X V  i 2 3μL2 ð p c Þi i¼1

ðA4Þ

300 0

0

50

100

150

200

250

300

where qn is the total flow rate of n cleats. The flow rate through the cleats is also governed by Darcy's law

Time, t (day) Fig. 12. Sensitivity study of tortuosity parameter on gas production rate.

qn ¼ −

kAΔp μL

ðA5Þ

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D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

The relative permeability to the wetting/nonwetting phase at saturation Sw is therefore dSw ðpc Þ2 1 dS ∫0 w2 ðpc Þ S

krw

k ¼ w¼ k

∫0w

ðA10Þ

dSw ðpc Þ2 1 dSw ∫0 ðpc Þ2 1

krnw Fig. 13. Sketch of fluid flow through a cleat. After Reiss (1980).

where k is the permeability of the cleats and A is the cross-section area. Combining Eqs. (A4) and (A5), it yields k¼

n ðσ cos θÞ2 X V  i 2 : 3AL i¼1 ðpc Þi

ðA6Þ

Eq. (A6) can be further written as k¼

n ðσ cos θÞ2 ϕ X S  i 2 3 i¼1 ðpc Þi

ðA7Þ

where ϕ is the porosity and Si is the percentage of each cleat to the total void volume. The difference between the Purcell method and the current derivation method is summarized in Table 2. It can be seen in Table 2 that the equations derived based on the two different conceptual models can yield the similar form of equations but with different coefficients. This is due to the difference in geometries of the two conceptual models. Eq. (A7) can be further expressed as (Purcell, 1949) k¼

ðσ cos θÞ2 ϕ 1 dS ∫0 : 3 ðpc Þ2

ðA8Þ

When the coal cleats are partial saturated with water at saturation of Sw, the effective permeability of the wetting phase can be calculated by (Gates and Leitz, 1950)

kw ¼

ðσ cos θÞ2 ϕ Sw dSw ∫0 : 3 ðpc Þ2

ðA9Þ

σ

k ¼ nw ¼ k

∫Sw

ðA11Þ

where krw and krnw are the relative permeability of the wetting phase and the nonwetting phase, respectively. Eqs. (A10) and (A11) have the same form of the Purcell method. It verifies that the Purcell's integration equations can be also applied to the matchstick type of reservoirs. There are two sets of cleats in coal seams which are mutually perpendicular and also perpendicular to the bedding (Laubach et al., 1998). The face cleats are more continuous and can be well represented by the matchstick model. However the butt cleats are normally less continuous and end at face cleats. This is not well modeled by the matchstick model in which all the cleats are assumed as straight planes which are parallel to each other. Therefore the tortuosity parameter should be introduced to describe the tortuous characteristics of the coal cleats (Chen et al., 2012a). A tortuosity parameter (η) can be applied as follows (Chen et al., 1999; Mualem, 1976) to improve Eqs. (A10) and (A11) S

2

  η ∫0w dSw =ðpc Þ krw ¼ Sw 1 2 ∫0 dSw =ðpc Þ 1

ðA12Þ

2

  η ∫Sw dSw =ðpc Þ krnw ¼ 1−Sw : 1 2 ∫0 dSw =ðpc Þ

Eqs. (A12) and (A13) can be reduced to the same form as the Burdine method when η equals to 2, the value of which is believed to be true for isotropic and granular porous media (Carman, 1937). In coal seams, the tortuosity parameter may not be a constant, because the tortuosity parameter is not only dependent on the geometry of the coal cleat networks, but also relied on the flow directions. Both the manufacture of the coal core and the selection of the flow directions could affect the tortuosity effect on relative permeability measurement on coals. If the flow direction is toward to the face cleat direction, the tortuosity effect is not obvious. In the opposite,

σ

θ

2r

(a) Capillary pressure in a single capillary

ðA13Þ

(b) Capillary pressure in a single cleat

Fig. 14. Sketch of capillary pressure in microstructures.

D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

1.0

Conceptual model

Current method

Bundle of capillary tubes model Fig. 1 (a) Capillary pressure model Capillary pressure in a single capillary Fig. 14 (a) Capillary pressure pc ⋅ r = 2σ cos θ equation 4 Δp Poiseuille's equation q1 ¼ − πr 8μ L n 2 X Si Permeability model k ¼ ðσ cos2 θÞ ϕ  2 i¼1 ðpc Þi

Matchstick model Fig. 1 (b) Capillary pressure in a single cleat Fig. 14 (b) pc ⋅ b ⋅ L = 2L ⋅ σ cos θ b l Δp q1 ¼ − 12μ L n X Si ðσ cos θÞ2 ϕ k¼  2 3 i¼1 ðpc Þi 3

the tortuosity would take effect once the flow direction is in another extreme that is toward to the butt cleat direction. In order to integrate the relative permeability model with Eqs. (A12) and (A13), we applied the Brooks and Corey (1966) model as the capillary pressure model   −1=λ pc ¼ pe Sw

pc ¼ pnw −pw



ðA15Þ

Sw −Swr 1−Swr −Snwr

ðA16Þ

where pw is the wetting phase pressure, pnw is the nonwetting phase pressure, Sw is the wetting phase saturation, Swr is the residual wetting phase saturation, and Snwr is the residual nonwetting phase saturation. The relative permeability models for fractured reservoir rocks can be obtained, as expressed in Eqs. (A17) and (A18), through incorporating Eq. (A14) into Eqs. (A12) and (A13).    ηþ1þ2=λ krw ¼ krw Sw    krnw ¼ krnw 1−Sw

η=1

0.8 0.6 0.4 0.2 0.0

0

20

40

60

80

100

Water saturation, Sw (%)

krw λ=1

krg λ=1

krw λ=3

krg λ=3

krw λ=7

krg λ=7

Fig. 16. Sensitivity study of cleat size distribution index on relative permeability curves.

ðA14Þ

where pe is the entry capillary pressure, λ is the cleat size distribution index. pc and Sw⁎ are the capillary pressure and the normalized wetting phase saturation which are defined as Eqs. (A15) and (A16), respectively.

Sw ¼

Relative permeability

Table 2 Difference between the Purcell method and the current method. Purcell method

55

η h

ðA17Þ   1þ2=λ 1− Sw

i

ðA18Þ

⁎ is the end-point relative permeability of the wetting phase where krw ⁎ is the end-point relative permeability of the nonwetting phase. and krnw

(a) Effect of cleat size distribution

When η equals to 0, this relative permeability model reduces to the same form of the Purcell model for which the effect of tortuosity is not considered. When η equals to 2, this relative permeability model reduces to the same form of the frequently used BCB model. When both η and λ equal to 2, this relative permeability model further reduces to the same form of the Corey model. There are two parameters for the relative permeability model: the cleat size distribution index (λ) and the tortuosity parameter (η). The cleat size distribution index is a fitting parameter for relative permeability data and thus it is coal dependent. Porous media having primary porosity tends to have large values of cleat size distribution index and porous media that have secondary porosity have small values of cleat size distribution index (Brooks and Corey, 1966). The physical meaning of the two parameters for coal may be demonstrated by Fig. 15. The sizes of the cleats may not be identical and the cleat size distribution index controls the difference in the cleat size. In addition, the cleat is often tortuous but not straight and this effect is controlled by the tortuosity parameter. To further intuitively investigate how the two parameters control the typical shape of the relative permeability curves, two simple hypothetical parameter analyses are presented in Fig. 16 and Fig. 17. Fig. 16 shows that the water relative permeability increases with the increment of the cleats aperture distribution index and the gas relative permeability, in the opposite, decrease with the increment of the pore size distribution index. Fig. 17 illustrates the effect of tortuosity parameter on the relative permeability. When η equals to

(b) Effect of tortuosity

Fig. 15. Effect of cleat size distribution and tortuosity on fluid flow through cleats.

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D. Chen et al. / International Journal of Coal Geology 109–110 (2013) 45–57

1.0

Relative permeability

λ=3

0.8 0.6 0.4 0.2 0.0

0

20

40

60

80

100

Water saturation, Sw (%) krw η=0

krg η=0

krw η=1

krg η=1

krw η=2

krg η=2

Fig. 17. Sensitivity study of tortuosity parameter on relative permeability curves.

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