Gas and Water Relative Permeability in Different Coals

International Journal of Coal Geology 122 (2014) 37–49

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Gas and water relative permeability in different coals: Model match and new insights Dong Chen, Ji-Quan Shi ⁎, Sevket Durucan, Anna Korre Department of Earth Science and Engineering, Royal School of Mines, Imperial College London, London SW7 2BP, UK

a r t i c l e

i n f o

Article history: Received 11 October 2013 Received in revised form 5 December 2013 Accepted 5 December 2013 Available online 14 December 2013 Keywords: Relative permeability Coal Coal rank Model match Cleat size distribution Tortuosity

a b s t r a c t A recently derived relative permeability model for coal reservoirs has been applied to fit the published gas–water relative permeability data for different coals from Europe, China, Australia and the U.S., which exhibit a myriad of shapes and curvatures. The two-parameter model is shown to be capable of describing a total of 32 sets of data, including those history-matched from field production and laboratory core flooding tests as well as laboratorymeasured ones. The fitted values of the two model parameters, namely cleat tortuosity parameter (η) and cleat size distribution index (λ), fall in the range between 0 and 2, and 0.3 and 8.8, respectively. For the European and Chinese coals whose rank information is available, there is tentative evidence that a U shape correlation between λ and coal rank exists, whereas no discernible trend is observed for η. This U shape dependency on coal rank has also been reported for some other coal properties such as total porosity. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Coal seams are naturally fractured reservoirs consisting of matrix blocks, where most gas is adsorbed on coal surfaces, and a network of cleats, which provide the main flow paths for gas and water flow in coal seams. The cleats are usually saturated with water at the in-situ state, so water is produced first (dewatering) from the coal seams in commercial coalbed methane (CBM) production through pressure depletion. This is followed by a two phase flow regime where water and gas production is primarily governed by the characteristics of gas– water relative permeability of the cleats. Knowledge of relative permeability of coals is important for simulation studies of two phase flow behaviour in coal (Clarkson et al., 2011; Ham and Kantzas, 2008). Coal is evolved from peat, which is formed after organic material is buried, compressed, and dewatered. As peat is buried more deeply over geological time, heat and pressure progressively drive off more and more water and volatiles. This process is referred to as coalification, whereby the carbon content of the coal is gradually increased through develoatilization. During coalification the rank of coal increases accordingly from lignite to sub-bituminous, bituminous, and anthracite. The rank of a coal is defined by its physical and chemical properties such as vitrinite reflectance, fixed carbon content, volatile matter content,

⁎ Corresponding author. Tel.: +44 20 7594 7374. E-mail address: [email protected] (J.-Q. Shi). 0166-5162/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coal.2013.12.002

moisture content (ASTM standards: http://www.astm.org/), which change progressively with coalification. It has been reported that a number of coal properties, such as pore surface area (Williams et al., 2001), gas sorption capacity (Ahsan, 2006; Levine, 1996; Yao et al., 2011; Yee et al., 1993; Zhang et al., 2011) and total porosity (Mares et al., 2009; Rodrigues and Lemos de Sousa, 2002), are closely related to its rank in the form of a U shape curve (Fig. 1). This U-shape trend was first reported by King and Wilkins (1944). There is also evidence that cleat spacing in coal is rank-dependent (Law, 1993). Recently Yao et al. (2009) investigated the correlation between fractal dimension and coal rank for a number of Chinese coals with rank from sub-bituminous right up to anthracite (vitrinite reflectance between 0.4 and 4.2). Since 1970s a number of relative permeability experiments have been conducted on coal samples (Dabbous et al., 1974; Gash, 1991; Gash et al., 1993; Meaney and Paterson, 1996; Paterson et al., 1992; Puri et al., 1991; Reznik et al., 1974). However, very few attempts have been made to relate the measured relative permeability curves of coals to their ranks. This is partly due to the lack of coal rank data in the publications. Ahsan (2006) measured gas (helium) and water relative permeability for several European coals of various ranks (from high vol. bituminous B to anthracite). Shen et al. (2011) investigated the influence of coal properties such as coal rank and maceral composition on the relative permeabilities of gas and water for different rank coals selected from South Qinshui Basin, China. In this study the relative permeability equations derived specifically for coal by one of the authors (Chen et al., 2013) are used to fit the experimental data for both Chinese and European coals covering a wide

38

D. Chen et al. / International Journal of Coal Geology 122 (2014) 37–49

h 1þ2=λ i η krnw ¼ krnw 1−Sw 1− Sw

ð2Þ

where k∗rw is the end-point relative permeability of the wetting phase, η is the cleat tortuosity parameter, λ is the cleat size distribution index analogous to the pore size distribution index used for porous media, k∗rnw is the end-point relative permeability of the nonwetting phase, and S∗w is the normalized wetting phase saturation given by:

Sw ¼

Fig. 1. Relationship between coal porosity and rank (after Rodrigues and Lemos de Sousa, 2002).

range of ranks. An attempt has been made to correlate the two parameters defining the characteristics of relative permeability to coal rank. In addition, published relative permeability data for Australian and the U.S. coals, whose rank information is unknown, has also been fitted with the same relative permeability model to obtain the range of the parameters, and towards building a database of laboratory measured relative permeability data together with fitted parameters.

2. Relative permeability model for coals Most conventional relative permeability models are derived based on the bundle of capillary tubes model representation of porous media (Fig. 2a). Recognising that the cleat network in coals is better represented by a matchstick model (Fig. 2b), Chen et al. (2013) recently derived the following equations to describe the gas and water relative permeability in coal: ηþ1þ2=λ krw ¼ krw Sw

ð1Þ

Sw −Swr 1−Swr −Snwr

ð3Þ

where Sw is the wetting phase saturation, Swr is the residual wetting phase saturation, and Snwr is the residual nonwetting phase saturation. In this study, coal is assumed as water wet and thus the gas is the nonwetting phase. It can be seen that the general shape of relative permeability in coals is determined by two parameters, namely cleat tortuosity parameter (η) and cleat size distribution index (λ). Note that Eqs. (1) and (2) have very similar forms as Brooks and Corey (1966) equations. Indeed, they reduce to the latter when η equals to 2. If η is set to 0, then Eqs. (1) and (2) reduce to the Purcell (1949) equations. The cleat tortuosity parameter (η) is introduced to account for the fact that actual cleat structure in coal (Fig. 3a) is often much more complex than the simple matchstick model. Its value reflects the degree of tortuosity of the flow path formed by the connecting cleats. The value of η obtained from fitting the experimental data in the current study falls in the range between 0 and 2 (see Section 3). For comparison, isotropic and granular porous media usually have a tortuosity of 2 (Carman, 1937). It might be expected that an increase in tortuosity would lead to a reduction in both gas and water relative permeability. This is clearly illustrated in Fig. 4, which compares the computed relative permeabilities for different tortuosity parameter values between 0 (Purcell model) and 2 (Brooks and Corey model) for two representative values of λ. It is noted that the curvature of the relative permeability to gas undergoes a transition from being convex to concave as η is increased from 0 to 2. Both shapes have been observed in the relative permeability of coals measured in the laboratory. It shall be seen that this flexibility of the model allows it to match relative permeability curves of different shapes and curvatures. It is further noted that the impact appears to be more pronounced for gas than water phase, especially when the gas saturation is low. The asymmetric impact of tortuosity on the non-

Fig. 2. (a) bundle of capillary tubes model (after Gates and Leitz, 1950); (b) Matchstick model (after Seidle et al., 1992).


39

Fig. 3. Schematic illustration of coal cleat structure (after Laubach et al., 1998), underlining the need to introduce tortuosity factor and cleat size distribution index in the relative permeability equations.

wetting phase has also been observed in conventional porous media (Li and Horne, 2002). The following explanation has been proposed by Li and Horne. Unlike the wetting phase which exists on the rock surface in the form of continuous film during drainage, the non-wetting phase initially emerges in the form of discontinuous droplets at low gas

1.0 Purcell model

Relative permeability

0.8 0.6 0.4

Brooks and Corey model

0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

a) λ = 1.0 1.0 Purcell model


0.8 0.6 0.4 Brooks and Corey model

0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

b)λ = 3.0 Fig. 4. The impact of cleat tortuosity parameter η on relative permeability curves for two different values of λ.

saturation. Consequently, tortuosity has a more pronounced effect on slowing down the non-wetting phase flow until a continuous flow path is established. The cleat size distribution index (λ) is used to account for the fact that there generally exists a hierarchy of cleat aperture within the cleat network in coal (Fig. 3b). The cleat size distribution index decreases with increasing degree of heterogeneity in the cleat size (aperture) distribution, as with the pore size distribution index defined for conventional porous media. The value of λ obtained from fitting the experimental data falls in the range between 0.3 and 8.8 (see Section 3). Fig. 5 shows the influence of λ on relative permeability curves for four different values spanning the range of η. The following observations can be made: 1) The relative permeability curves tend to be less sensitive to λ with increasing η (the cleat network becomes more tortuous); 2) The influence of λ on the relative permeability curves becomes less pronounced when λ exceeds 3; 3) As λ is progressively reduced, the gas and water curves move in opposite directions, with the relative permeability to gas in ascending whereas the relative permeability to water in descending. 3. Model match of relative permeability data for different coals The Chen et al. (2013) model (Eqs. (1)–(3)) has been applied to fit the published relative permeability data for different coals of both known ranks (European and Chinese), and unknown ranks (Australian and the U.S.). The relative permeability data were digitised from the original publications. For the European coals, individual data points, which show some degree of scattering, are also available. For each set of relative permeability data, an initial estimate of the two parameters (η and λ) was firstly made by fitting the data manually. Following Brown (2001), the generalized reduced gradient (GRG) method in Microsoft Excel Solver, which employs an iterative least squares fitting routine to produce the optimal goodness of fit between data and function, was then used to yield η and λ. To ensure that a true minimum is reached during regression, different initial guess of the parameters η and λ were also tested. For a few sets of data when η exceeds 2, manual adjustment was applied by setting an upper limit of 2 for η. For the European coals, a measure of the goodness of fit was evaluated and the 95% confidence intervals around the fitted curve were presented. 3.1. European coals The rank of selected European coals collected from opencast and underground coal mines in the UK, France and Germany (Ahsan,

40



1.0

Table 1 The rank indicators for the European coals and relative permeability model match results.

0.8

Coal seam

Fixed carbon (d.a.f.) (%)

Vitrinite reflectance (%)

k∗rw

k∗rnw

Swr

Snwr

η

λ

R2

Schwalbach Warndt– Luisenthal no. 1 Splint Tupton Dora Selar 9 ft Tower 7 ft

56.4 58.4

0.79 0.71

0.11 0.53

0.91 0.92

0.05 0.19

0.27 0.32

0.4 0.0

6.0 4.5

0.93 0.95

59.8 64.7 83.5 89.2 90.9

0.55 0.49 0.71 2.41 2.28

0.55 1 0.22 0.97 0.28

0.64 1 0.85 1 0.91

0.68 0.36 0.07 0.17 0.34

0.15 0.23 0.33 0.4 0.25

0.0 0.8 0.3 1.2 0.7

7.0 1.7 1.1 2.8 7.8

0.74 0.94 0.95 0.95 0.95

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

a) η = 0.0 (Purcell model) Relative permeability

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

b) η = 0.5 (current model)


1.0 0.8 0.6 0.4 0.2

2006; Durucan et al., 2012) varied from high volatile bituminous to anthracite (Table 1). The unsteady state method was used to measure gas–water relative permeabilities on cylindrical coal samples due to its operational simplicity. The impact of factors such as wettability, absolute permeability and overburden pressure on coal relative permeability was analysed. Considerable variation in the shapes of the relative permeability curves for different rank coals was observed, which was attributed to the heterogeneous nature of coal. Irreducible water saturations (S wr ) ranged from 0.15 to 0.40 for all coal types except for Splint, which was over 0.65. Critical gas saturations (S nwr ) appear to be spread out over a broad range of saturations but are generally found to lie in the 0.15– 0.35 band. Using the parameter values in Table 1, good match to the experimental data was obtained, as illustrated in Fig. 6. The fitted values of the two parameters and the coefficient of determination (R2), which gives a measure of the goodness of fit, are summarised in the last three columns of Table 1. The cleat tortuosity parameter (η) falls in the range between 0 and 1.2 and the cleat size distribution index (λ) in the range between 1.1 and 7.8. It is seen that, with the exception of Splint (R2 = 0.74), the coefficients of determination for the European coals are all greater than 0.9. The 95% confidence intervals of the datafit were computed for all the samples and the results are presented in Appendix B.

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Watersaturation

c) η = 1.0 (current model)


1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

3.2. Chinese coals The coal samples, which cover a wide range of ranks, were collected from the South Qinshui basin (Shen et al., 2011), one of the hottest spots for CBM exploration and production in China (Table 2). The cylindrical sample size was about 50 mm in diameter and 100 mm in length. The unsteady state method was used to measure gas–water relative permeability. A key characteristic of the Chinese coals tested is that they have high irreducible water saturation (about 0.7), and low end-point relative permeability values to both water and gas (generally less than 0.3). Using the reported end-point saturation and relative permeability values listed in Table 2, good match to the experimental data was obtained, as illustrated in Fig. 7. The fitted values of the two parameters are summarised in the last two columns of Table 2. The cleat tortuosity parameter (η) falls in the range between 0 and 1.3 and the cleat size distribution index (λ) in the range between 0.7 and 3.5.

d) η = 2.0 (Brooks and Corey model) 3.3. Australian coals Fig. 5. The impact of cleat size distribution index on relative permeability curves for four different values of η.

The coal samples used for unsteady state gas–water relative permeability measurements were from Bowen and northern


laboratory tests, relative permeability curves obtained from history matching the field production data in the two basins were also reported.

Sydney basins (Meaney and Paterson, 1996). Similar to the Chinese coals, the Australian coals tested had high irreducible water saturations (S wr ) for most coal samples. As well as

1.0

1.0

krw Experiment 0.8



41

krg Experiment krw Model

0.6

krg Model 0.4 0.2 0.0

krw Experiment krg Experiment

0.8

krw Model 0.6

krg Model 0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

Water saturation

0.6

0.8

1.0

Water saturation

a) Schwalbach - Sample 4

b) Warndt - Luisenthal-Sample1 1.0



1.0 0.8 0.6


0.4

krw Model

0.2

0.8 krw Experiment

0.6

krg Experiment

0.4

krw Model

0.2

krg Model

krg Model

0.0

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

Water saturation

0.6

0.8

1.0

Water saturation

c) Splint - Sample 2

d) Tupton - Sample 2

1.0 1.0



0.6

krg Model 0.4 0.2 0.0

0.8 0.6

krw Experiment

0.4


0.2

krg Model

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

Water saturation

0.6

0.8

Water saturation

e) Dora - Sample 10

f) Selar 9ft - Sample 2

1.0



krw Experiment 0.8


0.8

krw Model

0.6

krg Model 0.4

0.2 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

g) Tower 7ft Fig. 6. Relative permeability curves measured in the laboratory for different ranks of European coals and model match.

1.0

42


Table 2 The rank indicators for the Chinese coals and relative permeability model match results. Source location

Fixed carbon (d.a.f.) (%)

Vitrinite reflectance (%)

k∗rw

k∗rnw

Swr

Snwr

η

λ

Lijiacun Changcun Wangyun Chengzhuang

67.9 85.9 89.2 92.7

0.89 2.1 2.17 2.87

0.14 0.25 0.13 0.09

0.30 0.08 0.22 0.11

0.69 0.70 0.74 0.71

0.05 0.02 0.04 0.07

1.3 0.26 0.3 0

0.9 3.3 3.5 0.7

As shown in Figs. A-1 and A-2 in Appendix A, both the laboratorymeasured and history-matched relative permeability data are well fitted with the relative permeability model. The parameter values obtained are listed in Table 3 (laboratory data) and Table 4 (field data), respectively. Note that for ease of reference, the relative permeability curves appear in exactly the same order as in the original publication. The cleat tortuosity parameter (η) falls in the range between 0 and 2 and the cleat size distribution index (λ) in the range between 1.2 and 8.8.

permeability curves for the U.S. coal curves cover a wide range of water saturation, as the European coals described above. Fig. A-3 in Appendix A shows that the relative permeability data are well matched with the model and parameters used are listed in Table 5. The cleat tortuosity parameter (η) falls in the range between 0.7 and 2.0 and the cleat size distribution index (λ) in the range between 0.3 and 5.5.

4. Discussion 3.4. U.S. coals Gash (1991) measured the helium and water relative permeabilities for four coal samples from the San Juan and Black Warrior basins. Ohen et al. (1991) obtained their relative permeability curves from matching the production data in core flooding experiments on core samples from the Black Warrior basin. The relative

4.1. Range and distribution of cleat tortuosity parameter and cleat size distribution index It has been shown that the published gas–water relative permeability data for different coals from four continents can be fitted well using the model developed by Chen et al. (2013). The fitted values of the two

1.0

1.0

0.8

krw Experiment

krg Experiment



krw Experiment

krw Model 0.6

krg Model 0.4 0.2 0.0

0.8


0.6

krg Model 0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

Water saturation

a) Lijiacun

0.6

0.8

1.0

0.8

1.0

b) Chengzhuang 1.0

1.0

krw Experiment

krw Experiment 0.8

krg Experiment



0.4

Water saturation

krw Model

0.6

krg Model 0.4

0.2

0.8


0.6

krg Model 0.4 0.2

0.0

0.0 0.0

0.2

0.4

0.6

Water saturation

c) Wangyun

0.8

1.0

0.0

0.2

0.4

0.6

Water saturation

d) Changcun

Fig. 7. Relative permeability curves measured in the laboratory for different ranks of Chinese coals and model match.

D. Chen et al. / International Journal of Coal Geology 122 (2014) 37–49 Table 3 Relative permeability model match results for the Australian coals (laboratory data). k∗rw

k∗rnw

Swr

Snwr

η

λ

(a) Bowen (b) Northern Sydney (c) Bowen (e) Bowen (f) Northern Sydney (g) Northern Sydney (h) Bowen (i) Bowen (j) Bowen

0.86 0.47 0.21 0.78 1.00 0.82 0.42 0.28 0.19

0.42 0.11 0.36 0.19 0.11 0.21 0.42 0.18 0.55

0.77 0.55 0.85 0.59 0.79 0.85 0.93 0.92 0.68

0.00 0.11 0.05 0.14 0.05 0.03 0.02 0.03 0.01

2.0 0.4 0.6 0.0 0.6 0.1 0.8 0.6 0.5

3.1 1.2 3.4 1.6 8.8 2.9 7.0 6.5 3.0

14 13

12

Number of values

Source basin

15

43

9

6

3 3

2

0

Table 4 Relative permeability model match results for the Australian coals (field data). k∗rw

k∗rnw

Swr

Snwr

η

λ

(i) Bowen (ii) Bowen (iii) Northern Sydney (iv) Northern Sydney (v) Northern Sydney (vi) Northern Sydney

0.67 1.00 1.00 1.00 1.00 1.00

0.32 0.70 0.93 1.00 0.96 0.71

0.76 0.58 0.57 0.84 0.60 0.75

0.08 0.01 0.05 0.01 0.03 0.00

0.5 0.5 0.9 0.6 1.0 0.9

2.8 2.8 1.6 7.0 6.0 1.4

model parameters, namely cleat tortuosity parameter (η) and cleat size distribution index (λ), for a total of 32 sets of data, including those history-matched from field production and laboratory core flooding tests as well as laboratory-measured ones, fall in the range between 0 and 2, and 0.3 and 8.8, respectively. Examination of the breakdown of η values (Fig. 8a) shows that the overwhelming majority (84%) is less than 1, and half lie in the range between 0.5 and 1.5. This observation highlights the complex and unique nature of the cleat network in coals, which generally cannot be adequately described by either the Purcell or the Brook and Corey models. As discussed in Section 2, the relative permeability curves become less sensitive to cleat size distribution index (λ) when it exceeds 3. It can be seen from Fig. 8b that about half of the λ values are larger than 3, and the rest are evenly distributed in the range between 0 and 3. Cross-reference with descriptions of visual observation of the samples in the relevant publications is made, where available, and the results are summarised in Table 6. It is noted that those core samples which have one or more sets of large/obvious fractures (cleats) tend to have a cleat size distribution index greater than 3. This suggests that the two-phase flow is dominated by the large fractures (or primary porosity) during the tests. Small λ values (b 3) tend to be associated with coal samples with no visible/dominant fractures or with many mineralised fractures (italicised entries in Table 6).

a) η 15

16

Number of values

Source basin

12

7

8

5

5

4

0

b) λ Fig. 8. Distribution of the model-fitted values of the two parameters.

4.2. Correlation with coal rank—European and Chinese coals As already stated, rank dependence of several properties of coal, e.g. total porosity and surface area, has previously been reported (see Fig. 1). Generally, the coal rank increases with both fixed carbon content and vitrinite reflectance. The fixed carbon is the solid combustible residue that remains after the coal is heated and the volatile matter is expelled. While vitrinite is one of the primary components of coal and it has a shiny appearance resembling glass. The vitrinite reflectance indicates the thermal maturity or rank of coals. It is the measured percentage of reflected light from a sample which is immersed in oil and normally represented by Ro that stands for “reflectance in oil”.

Table 5 Relative permeability model match results for the U.S. coals. Reference

Source location

k∗rw

k∗rnw

Swr

Snwr

η

λ

Gash (1991)

Core A, Cahn seam, San Juan Basin Core B, Ignacio seam, San Juan Basin Core C, LaPlata mine, San Juan Basin Core D, Blue Creek seam, Black Warrior Basin Sample 1, Black Warrior Basin Sample 2, Black Warrior Basin

1.00

0.52

0.20

0.00

1.0

5.5

0.61

0.98

0.04

0.16

0.8

3.3

0.82

0.71

0.17

0.06

0.7

1.0

1.00

0.76

0.21

0.00

1.0

0.7

1.00 1.00

1.00 1.00

0.00 0.00

0.00 0.00

2.0 2.0

1.2 0.3

Ohen et al. (1991)

44


Table 6 Relationship between cleat size distribution index and coal cleat structure. Coal

λ

Description

European coals (Table 1)

Splint

7

Australian coals (Table 3)

Sample (a) Sample (c) Sample (f) (g) (h) (i) Sample (b) Sample (e) Core A

3.1 3.4 8.8 2.9 7.0 6.5 1.2 1.6 5.5

Heavily fractured with large visible fractures that were responsible for the channelling of gas and water at high flow rates Extensively fractured With a well-developed fracture network With obvious fractures parallel to the flow direction. Samples (h) and (i) had a particularly significant fracture parallel to the flow

Core B

3.3

Core C; Core D

1.0 0.7

U.S. coals (Table 5)

As shown in Fig. 9, the cleat size distribution index λ for the European and Chinese coals exhibits a familiar U type correlation with the fixed carbon content/vitrinite reflectance (Ro). An attempt was made to fit the cleat size distribution index data as a parabola equation of the fixed carbon content C (in % daf) (Fig. 9a). The following correlation was obtained (R2 = 0.57) 2

λ ¼ 0:0184ðC−75:8Þ :

λ = 0.0184(C– 75.8)2 R2 = 0.57

RP not sensitive to λ

70

RP sensitive to λ

ð4Þ

Cleat size distribution index, λ

8

λ European coals λ Chinese coals Parabola equation

7 6 5 4 3 2 1 0 50

60

80

90

100

Fixed carbon, C, d.a.f. (%)

With many mineralised fractures No visible fractures A single large fracture across the entire diameter of the core and extends halfway through the length of the core. A cleat network is present in the other half The cleat structure is staggered at the bedding planes, which occur at 1/ 8 in. to 1/4 in. intervals Many individual cleats extend the entire length of each core

On the other hand, no discernible relationship is observed between the cleat tortuosity η and coal rank for the two sets of relative permeability data, as illustrated in Fig. 10.

5. Conclusions The two-parameter model of Chen et al. (2013) has been shown to be capable of describing the published relative permeability data for different coals from Europe, China, Australia, and the U.S., which exhibit a myriad of shapes and curvatures. The fitted values of the two model parameters, namely cleat tortuosity parameter (η) and cleat size distribution index (λ), fall in the range between 0 and 2, and 0.3 and 8.8, respectively. For the European and Chinese coals whose rank information is available, there is tentative evidence that a U shape correlation between λ and coal rank exists, whereas no discernible trend is observed for η. This U shape dependency on coal rank has also been reported for some other coal properties such as total porosity. More work is warranted to validate this Ushape trend for other coals.

Acknowledgement The constructive comments by the two anonymous reviewers have helped improve the quality of this paper.

RP not sensitive to λ

8

λ European coals λ Chinese coals

6

Parabola equation 4

2

0 0

0.5

1

1.5

2

2.5

3

RP sensitive to λ

Cleat size distribution index, λ

a) fixed carbon content

2.0

European coals 1.6

Chinese coals

1.2 0.8 0.4

Vitrinite reflectance, Ro (%)

b) vitrinite reflectance Fig. 9. Correlation between cleat size distribution index and coal rank. The relative permeability curves are sensitive to λ in Zone I (λ b =3), and much less sensitive in Zone II (λ N 3).

0.0 50

60

70

80

90

100

Fixed carbon, d.a.f. (%) Fig. 10. No correlation between cleat tortuosity parameter and the coal rank.


45

Appendix A. Model match of relative permeability data for Australian and the U.S. coals 1.0

0.8

λ=3.1 η=2.0

0.6

krw Experiment

0.4

krg Experiment



1.0

krw Model 0.2 0.0 0.0

0.8 0.6 0.4

0.4

0.6

0.8

0.0 0.0

1.0

krw Experiment krg Experiment krw Model

0.2

krg Model 0.2

λ =1.2 η=0.4

krg Model

0.2

Water saturation

a) Bowen Basin

0.4

krg Experiment



krw Experiment

krw Model

0.0 0.0

krg Model

0.2

0.4

0.6

0.8

1.0



krg Model

0.2

0.4

0.6

0.8

0.8 0.6 0.4

krg Model

0.2

0.4

0.6

0.8

1.0

0.8

1.0

λ =2.9 η=0.1


0.2 0.0 0.0

1.0

krg Model

0.2

0.4

0.6

Water saturation

Water saturation

f) Northern Sydney Basin

g) Northern Sydney Basin

1.0

1.0 λ=7.0 η=0.8

0.6

krw Experiment

0.4

krg Experiment



krw Model

0.2

1.0 λ =8.8 η=0.1

krw Model

krw Model

0.0 0.0

krg Experiment

e) Bowen Basin

krg Experiment

0.2

0.4

krw Experiment

c) Bowen Basin

0.4

0.8

0.6

Water saturation

krw Experiment

0.0 0.0

1.0

λ =1.6 η=0.0

Water saturation

0.6

0.2

0.8

0.0 0.0

1.0 0.8

0.8

1.0 λ =3.4 η=0.6

0.6

0.2

0.6

b) Northern Sydney Basin

1.0 0.8

0.4

Water saturation

krg Model

0.2

0.4

0.6

0.8

1.0

0.8 0.6 0.4

λ=6.5 η=0.6


0.2 0.0 0.0

krg Model

0.2

0.4

0.6

Water saturation

Water saturation

(h) Bowen Basin

i) Bowen Basin

Fig. A-1. Model match to relative permeability data measured in the laboratory for the Australian coals.

0.8

1.0

46



1.0 0.8 0.6 0.4

λ=3.0 η=0.5


0.2

krg Model

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Water saturation

j) Bowen Basin Fig. A-1 (continued).

0.6

krw Experiment

0.4

krg Experiment λ=2.8 η=0.5

krw Model

0.2


0.8

0.0 0.0

krg Model

0.2

0.4

0.6 0.4

krw Experiment krg Experiment λ=2.8 η=0.5

krw Model

0.2

0.6

0.8

0.0 0.0

1.0

0.2

0.4

0.6

0.8

Water saturation

i) Bowen Basin

ii) Bowen Basin

krw Experiment

0.4


λ=1.6 η=0.9

krg Model

0.2

0.4

0.6

0.8

1.0

1.0

Two phase region

0.6

0.0 0.0

0.8

Water saturation

0.8

0.2

Two phase region

krg Model

1.0


1.0

Two phase region



1.0

0.8 0.6 0.4

krw Experiment krg Experiment λ=7.0 η=0.6

krw Model

0.2 0.0 0.0

1.0

Two phase region

krg Model

0.2

0.4

0.6

0.8

Water saturation

Water saturation

iii) Northern Sydney Basin

iv) Northern Sydney Basin

1.0

1.0

0.6

krw Experiment

0.4


krw Model

0.2 0.0 0.0

krg Model

0.2

0.4

0.6

0.8

Water saturation

v) Northern Sydney Basin

Two phase region

1.0



Two phase region

0.8

1.0

0.8 0.6

krw Experiment

0.4


krw Model 0.2 0.0 0.0

krg Model 0.2

0.4

0.6

0.8

Water saturation

vi) Northern Sydney Basin

Fig. A-2. Model match to relative permeability data determined from matching the field production data for the Australian coals.

1.0


krw Model

0.6

krg Model λ=5.5 η=1.0

0.4 0.2

0.2

0.4

0.6

0.8


krg Model λ=1.0 η=0.7

0.2

0.4

0.6

0.8

0.4

0.6

krw Model 0.6 0.4

krg Model

λ=0.7 η=1.0

0.2

0.2

0.4

0.6

Core C

Core D

0.8

1.0

krw Model

0.6

krg Model

0.4 λ=1.2 η=2.0

0.4

0.6

0.8

1.0

1.0

krg Experiment

Water saturation

krg Experiment

0.8

krw Experiment

0.8

0.0 0.0

1.0

krw Experiment

0.2

0.2

Water saturation

1.0

0.0 0.0

0.2

1.0

krg Experiment

0.6

0.2

λ=3.3 η=0.8

0.4

Core B

krw Experiment

0.2

krg Model

Core A

krw Model

0.0 0.0

krw Model 0.6

Water saturation

0.8

0.4

krg Experiment

0.8

0.0 0.0

1.0

krw Experiment

Water saturation

1.0



krg Experiment

0.8

0.0 0.0


1.0

krw Experiment



1.0

47

0.8

1.0


0.8

krw Model

0.6

krg Model

0.4 0.2 0.0 0.0

λ=0.3 η=2.0

0.2

0.4

0.6

Water saturation

Water saturation

Sample 1

Sample 2

Fig. A-3. Model match to relative permeability data determined for the U.S. coals.

0.8

1.0

48


1.0



1.0 0.8 0.6

krg

0.4 0.2

0.8 0.6

krg

0.4 0.2

krw

krw 0.0 0.0

0.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

Water saturation

a) Schwalbach - Sample 4

0.6

0.8

1.0

b) W-L No. 1

1.0

1.0



0.4

Water saturation

0.8 0.6

krg

0.4 0.2

0.8

krw

krg

0.6 0.4 0.2

krw 0.0 0.0

0.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

Water saturation

c) Splint – Sample 2

0.6

0.8

1.0

0.8

1.0

d) Tupton Sample 2

1.0

1.0


0.8

krg

0.6 0.4 0.2

0.8 0.6

krg

krw

0.4 0.2

krw 0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.0

0.2

Water saturation

0.4

f) Selar 9ft – Sample 2

1.0 0.8

krg

0.6 0.4

krw 0.2 0.0 0.0

0.6

Water saturation

e) Dora – Sample 10



0.4

Water saturation

0.2

0.4

0.6

0.8

1.0

Water saturation

g) Tower 7ft Fig. B-1. 95% confidence intervals (dotted line) around the fit for European coals.


Appendix B. Confidence intervals of the data-fit for the European coals Following Brown (2001), the following steps were used to compute 95% confidence intervals of the fitted data. (1) Calculate standard error (σ) of the kr values: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 uX u kr −krfit t σ¼ df

ðB 1Þ

where df is the degrees of freedom that is defined as the number of data point minus the number of parameters in the relative permeability model (df = n − 2). (2) Use the TINV function in Excel to return the two-tailed inverse of the Student's t-distribution. The syntax is TINV (probability, deg_freedom). The critical t value at a significance level of 95% is calculated by entering the following formula: Ct ¼ tinvð0:05; df Þ:

ðB 2Þ

(3) The upper and lower confidence intervals (95%) of the fit can be obtained as: krup ¼ krfit þ Ct σ

ðB 3Þ

krlow ¼ krfit −Ct σ :

ðB 4Þ

The resulting intervals for the European coals are plotted in Fig. B-1. References Ahsan, M., 2006. Gas Flow and Retention Characteristics of Coal Seams for ECBM Recovery and CO2 Storage. (PhD Dissertation) Department of Earth Science and Engineering, Imperial College London. Brooks, R.H., Corey, A.T., 1966. Properties of porous media affecting fluid flow. J. Irrig. Drain. Eng. 92 (2), 61–90. Brown, A.M., 2001. A step-by-step guide to non-linear regression analysis of experimental data using a Microsoft Excel spreadsheet. Comput. Methods Prog. Biomed. 65, 191–200. Carman, P.C., 1937. Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166. Chen, D., Pan, Z., Liu, J., Connell, L.D., 2013. An improved relative permeability model for coal reservoirs. Int. J. Coal Geol. 109–110, 45–57. Clarkson, C.R., Rahmanian, M., Kantzas, A., Morad, K., 2011. Relative permeability of CBM reservoirs: controls on curve shape. Int. J. Coal Geol. 88, 204–217. Dabbous, M.K., Reznik, A.A., Taber, J.J., Fulton, P.F., 1974. The permeability of coal to gas and water. SPE J. 14 (6), 563–572. Durucan, S., Ahsan, M., Syed, A., Shi, J.Q., Korre, A., 2012. Two phase relative permeability of gas and water in coal for enhanced coalbed methane recovery and CO2 storage. Presented at the GHGT-11 Conference, 18–22 November, Kyoto International Conference Center, Japan. Gash, B.W., 1991. Measurement of “rock properties” in coal for coalbed methane production. SPE Annual Technical Conference and Exhibition, Dallas, Texas, SPE 22909, October.

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