Equivalent permeability of shale rocks: Simple and accurate empirical coupling of organic and inorganic matter

Journal Pre-proofs Equivalent permeability of shale rocks: Simple and accurate empirical coupling of organic and inorganic matter Jiangfeng Cui, Keliu Wu PII: DOI: Reference:

S0009-2509(20)30023-3 https://doi.org/10.1016/j.ces.2020.115491 CES 115491

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Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

4 November 2019 26 December 2019 10 January 2020

Please cite this article as: J. Cui, K. Wu, Equivalent permeability of shale rocks: Simple and accurate empirical coupling of organic and inorganic matter, Chemical Engineering Science (2020), doi: https://doi.org/10.1016/ j.ces.2020.115491

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Equivalent permeability of shale rocks: Simple and accurate empirical coupling of organic and inorganic matter Jiangfeng Cui*, Keliu Wu State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, China * Corresponding author. E-mail address: [email protected] Abstract Coupling the organic/inorganic permeabilities in shale is fundamental. Simple treatments, including the arithmetic/geometric average methods, are problematic and can lead to large errors. More complicated methods can generate reasonable results but are much more expensive. In this work, a simple and accurate empirical model for the equivalent permeability of shale considering permeability contrast between organic and inorganic matter and TOC is established. Geometric explanation of the parameters in the model is also provided. Afterwards, model validation against numerical data demonstrates the accuracy of the model in all realistic scenarios. Moreover, model comparison with arithmetic/geometric average methods shows the superiority of the model in accurate calculation while keeping the form simple. Finally, the model is applied to liquid flow in shale, and the results are analyzed. The model is highly recommended for theoretical studies in shale gas/oil that need coupling of the organic and inorganic permeabilities, including single-phase and multi-phase flows. Keywords Equivalent permeability, shale, organic, inorganic, couple

1. Introduction The declining production of conventional resources implies the urgent need in the efficient exploration and development of unconventional reservoirs (Yao et al., 2013). Shale gas (Sun et al., 2019, 2018) and shale oil (Cui, 2019a; YANG et al., 2016; Zou et al., 2013) are among the promising unconventional resources (Cui, 2019b), and they have been widely discussed in literature. Shale reserves are commonly rich in organic matter (Ross and Marc Bustin, 2009), which is distinct from other reserves. Therefore, the contribution of organic matter to the equivalent permeability of shale is important to be understood and has been a hot topic. A detailed reference list is given by Table 1. Those references are discussed separately in the following paragraph. Different methods have been used to study the impact of organic matter, as shown in Fig. 1. Arithmetic and geometric average methods have been extensively used to couple the organic and inorganic permeabilities in theoretical studies on shale gas/oil. However, the correctness and accuracy of the models have not been verified, and they potentially can lead to large error in the results. The incomplete layer averaging method combines the advantages of arithmetic, harmonic and geometric mean methods, and is more reliable than each method. Based on the assumption of negligible cross flow, the liquid flow considering 2D heterogeneity is simplified to 1D flow in this method. However, there are still limitations: (1) this method requires much more calculations than the arithmetic and geometric average methods; (2) this method can only calculate the equivalent permeability for a specific distribution, and the result does not have a general sense. As a matter of fact, numerous runs can be generated to arrive at statistical average values. However, even much more calculations are needed for that, and the statistical result might not be reliable due to the approximation made in this method. Several researchers possibly have recognized the difficulty to

easily couple the organic and inorganic permeabilities, and they just arrived at conclusions specific for organic and inorganic pores respectively without combining them in their theoretical research. There are also many research taking the organic matter as continuum media, and studied its effect using multi-continuum numerical simulation. However, many properties of kerogen are not considered explicitly in those papers. There are also other simulation studies that deal with kerogen explicitly, but the approaches are still much more expensive than empirical ones. (Cui et al., 2018) did comprehensive numerical research focusing on the contribution of organic matter to the equivalent permeability of shale. Effects of permeability contrast between organic and inorganic matter, TOC (vol), geometric parameters of organic matter, including the size, shape and connectivity, and inorganic permeability heterogeneity are considered. Each data point is the statistical result from 1000 different random runs. However, a practical empirical method was not proposed, which greatly limits the application of the results. Table 1 Literature review Reference

Fluid

Treatment

(Zhang et al., 2017)

liquid

arithmetic average

(Singh and Cai, 2019)

Gas

geometric average

(Q. Zhang et al., 2018)

Gas

arithmetic average

(Wang et al., 2018)

Gas

arithmetic average

(Wang and Cheng, 2019)

Water

arithmetic average

(T. Zhang et al., 2018)

Liquid

incomplete layer averaging

(Naraghi and Javadpour, 2015)

Gas

incomplete layer averaging

(Javadpour et al., 2015)

Liquid

incomplete layer averaging

(Geng et al., 2016)

Gas

incomplete layer averaging

(Song et al., 2016)

Gas

Respective treatment

(Sun et al., 2017)

Gas

Respective treatment

(Sheng et al., 2018)

Gas

Respective treatment

(Wang et al., 2019)

Liquid

Respective treatment

(Yan et al., 2016)

Gas

numerical simulation

(Alfi et al., 2015)

Two-phase

numerical simulation

(Zhang et al., 2015)

Gas

Continuum media

(Sheng et al., 2015)

Gas

Continuum media

(Sang et al., 2016)

Gas

Continuum media

(Wang et al., 2017)

Gas

Continuum media

(Yan et al., 2018)

Gas

Continuum media

(Akkutlu and Fathi, 2012)

Gas

Continuum media

(J. Wang et al., 2016)

Gas

lattice Boltzmann model

(Chen et al., 2015)

Gas

lattice Boltzmann model

(a)

(b)

(c)

(d)

(e) Fig. 1. Sketch map of available methods. (a) is for arithmetic average. (b) is for continuum modelling. (c) is for flow-based numerical modelling. (d) is for incomplete layer averaging. (e) is for respective treatment. The white grids stand for inorganic matrix. The black grids are organic matter. The blue lines with arrows indicate flow directions. The red lines represent no-flow boundary conditions.

In this paper, a simple and accurate empirical model calculating the equivalent permeability of shale taking organic and inorganic contributions into consideration is established. Afterwards, model validation against available numerical data reveals the wide applicability and high accuracy of the model. Then, model comparison with analytical methods indicates the advantage of the model in simple while accurate calculation. Finally, model application to liquid flow in shale leads to reasonable results, and some new findings as well. The established model is very useful for

theoretical studies in shale gas/oil that need coupling of the organic and inorganic permeabilities. 2. Mathematical model This work aims to establish a simple while accurate empirical model calculating the equivalent permeability of shale based on extracted numerical data from (Cui et al., 2018). The numerical approach is much preferred over the experimental approach on this problem, due to the following reasons (but not limited to): (1) Organic and inorganic permeability heterogeneities are inevitable in cores. However, such permeability heterogeneities are of less research interest than other factors considered in this work, and they are difficult to be determined, but they do have great impact on the permeabilities. (2) Numerical investigation is much cheaper than experimental investigation. 1000 random cases are used in (Cui et al., 2018) for average, because fewer number of cases will inevitably result in large uncertainty in the results. The numerical investigation of 1000 random cases takes several hours at most, while experimental investigation on 1000 real cores is usually impractical: such a large quantity of cores is not easily accessible, and it would take too long time even if so many cores are available. (3) Specially, for the determination of the liquid permeability of shale cores, numerical investigation is much more accurate than experimental investigation. Under current level of experimental techniques, only the magnitude of the liquid permeability of shale cores can be experimentally determined with confidence. Therefore, it is unlikely to accurately observe the relatively slight effects of the factors considered in this work through experiments. To summarize, the numerical approach is inevitable due to the factors listed above. All the data points corresponding to “Sigma=0” (no inorganic permeability heterogeneity) are extracted from

Fig. 4 (a) in (Cui et al., 2018), as shown in Table 3 (see Appendix A) and Fig. 2. The following aspects, which are of most research interest for theoretical investigations, are considered in the data: (1) Permeability contrast between organic and inorganic matrix. To be more exact, it is the ratio of the organic permeability to the inorganic permeability. It is noteworthy that the permeability heterogeneities of organic and inorganic matrix are ignored in this work. Inorganic permeability homogeneity is assured because only points corresponding to “Sigma=0” are extracted from (Cui et al., 2018), while organic permeability homogeneity is assumed in (Cui et al., 2018). Moreover, a wide range of permeability contrast, from infinitesimal to infinity, should be considered, as the ultrafast flow of fluid in nanochannels is recognized (Wu et al., 2017), but kerogen porosity might be totally undeveloped or disconnected (Chen et al., 2013). (2) TOC (vol). Usually, TOC is represented in mass fraction. However, the volume fraction is employed in this paper for numerical convenience. (3) The uniform and random distribution of kerogen particles. To be more exact, each data point is the statistical average from 1000 different random realizations. 1.8

TOC=0.05 TOC=0.10 TOC=0.15 TOC=0.20 TOC=0.25

Normalized equivalent permeability

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 10 -4

10 -2

10 0

10 2

10 4

Permeability contrast

Fig. 2. Relationship among normalized equivalent permeability, permeability contrast and TOC (vol). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article)

As shown in Fig. 2, the relationship between normalized equivalent permeability (defined as

the ratio of the intrinsic equivalent permeability to the inorganic permeability) and permeability contrast presents an “S” curve, regardless of TOC: normalized equivalent permeability first increases very slowly, then much more rapidly, and finally very slowly again with increasing permeability contrast from 0 to infinity. It is noteworthy that the curves are not exactly symmetrical. Moreover, the relationship between normalized equivalent permeability and TOC (vol) is approximately linear. Last but not least, all curves pass the fixed point (1, 1). The following discussion in this section consists of the three steps: (1) Modelling of the relationship between TOC (vol) and maximum/minimum permeabilities. The maximum and minimum permeabilities refer to the normalized equivalent permeabilities when the organic permeability approaches infinity and zero, respectively. (2) Modelling of the relationship between normalized equivalent permeability and permeability contrast; (3) Combination of (1) and (2) leads to the final model. 2.1 Modelling maximum/minimum permeabilities Kerogen particles are finely distributed within shale matrix, and the effect of their existence on the equivalent permeability of shale should be approximately linear when TOC is relatively low. Due to the physical background of the problem, linear fitting and inverse linear fitting are utilized to model the relationship between the maximum/minimum permeabilities and TOC (vol), respectively. It is difficult to tell which fitting is better at this stage, and the choice of fitting depends on the goodness of the fittings. Though there are not many points for fitting (only 5 points), this is already enough due to the aforementioned physical background.

Data point Inverse linear fit 1.6

1.5

1.5

Maximum permeability

Maximum permeability

Data point Linear fit 1.6

1.4

1.3

1.2

1.4

1.3

1.2

1.1

1.1 0.05

0.1

0.15

0.2

0.25

0.05

TOC(vol)

0.1

0.15

0.2

0.25

TOC(vol)

(a)

(b)

Fig. 3. Modelling maximum permeabilities. (a) is linear fitting, and R2=0.9799. (b) is inverse linear fitting, and R2=0.9996.

As demonstrated in Fig. 3, the goodness of inverse linear fitting is much better than the linear fitting. As a result, the inverse linear one is employed to model the maximum permeabilities Kinf, and it can be expressed as:

K inf 

1 1.001  1.591  TOC

(1)

Physically speaking, the first constant 1.001 should be 1. The second constant 1.591 is very close to /21.571. On the other hand, numerical simulation and extraction of data points can result in errors. Therefore, without much loss of accuracy, Eq. (1) can be simplified as:

K inf 

1

 2

1

(2)

 TOC

The introduction of  is well supported by a study on vuggy reservoirs (Neale and Nader, 2007). The vuggy reservoirs can be regarded as the limit when permeability contrast approaches infinity in this work. To be more exact, the kerogen particles with infinite permeability are compared to the infinite conductive vugs.

Data point Inverse linear fit 0.8

0.7

0.7

Minimum permeability

Minimum permeability

Data point Linear fit 0.8

0.6

0.5

0.4

0.6

0.5

0.4

0.3

0.3 0.05

0.1

0.15

0.2

0.25

0.05

TOC(vol)

0.1

0.15

0.2

0.25

TOC(vol)

(a)

(b)

Fig. 4. Modelling minimum permeabilities. (a) is linear fitting, and R2=0.999. (b) is inverse linear fitting, and R2=0.9495.

As illustrated in Fig. 4, the goodness of linear fitting is much better than the inverse linear fitting. Therefore, the linear one is employed to model the minimum permeabilities Kzero, and its expression is:

K zero  0.9866  2.776  TOC

(3)

It is noteworthy that the constant 0.9866 is very close to 1, and it should be 1 from a physical point of view. Moreover, 2.776 is very close to e2.718. Similarly, Eq. (3) is simplified as:

K zero  1  e  TOC

(4)

2.2 Modelling single curves The sigmoid function is one of the most popular S-type functions. However, the x-axis of Fig. 2 is log-transformed. Moreover, the maximum/minimum values are different from those of the sigmoid function. Therefore, the following function is employed to fit the numerical data points:

Kn 

a 1 b Kr

 K zero

(5)

where a and b are fitting parameters, Kr is permeability contrast, and Kn is normalized equivalent permeability. When Kr approaches zero, Kn approaches Kzero. The five fitting curves are shown in

Fig. 5, and the fitting parameters are summarized in Table 2. Based on those results, Eq. (5) can describe the relationship between normalized equivalent permeability and permeability contrast very well. 1.1

1.2

Data point Fitting curve

1.15

1.05

Normalized equivalent permeability

Normalized equivalent permeability

Data point Fitting curve

1

0.95

0.9

1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7

0.85 10 -4

10 -2

10 4

10 2

10 0

10 0

10 -2

10 -4

Permeability contrast

(a)

1.5

Data point Fitting curve

1.2

Normalized equivalent permeability

Normalized equivalent permeability

10 4

(b)

1.3

1.1 1 0.9 0.8 0.7

Data point Fitting curve

1

0.5

0.6 10 -4

10 2

Permeability contrast

10 -2

10 0

10 2

10 4

10 -4

10 -2

10 0

Permeability contrast

10 2

10 4

Permeability contrast

(c)

(d)

Data point Fitting curve

Normalized equivalent permeability

1.6

1.4

1.2

1

0.8

0.6

0.4

10 -4

10 -2

10 0

10 2

10 4

Permeability contrast

(e) Fig. 5. Modelling single curves. (a) is for TOC=0.05. (b) is for TOC=0.10. (c) is for TOC=0.15. (d) is for TOC=0.20.

(e) is for TOC=0.25. Table 2 Fitting parameters TOC

a

b

Kzero

R2

0.05

0.4244

1.841

0.8561

0.9995

0.10

0.8034

1.712

0.7144

0.9986

0.15

1.134

1.577

0.5752

0.9985

0.20

1.32

1.328

0.4484

0.9975

0.25

1.403

1.085

0.3362

0.9959

2.3 Final model The exact expressions for a and b are derived in this subsection. When Kr approaches infinity, Kn approaches Kinf:

Kn 

a  K zero  K inf b

(6)

a  K zero  1 b 1

(7)

When Kr=1, Kn should also be 1:

Kn 

Based on Eqs. (6) and (7), a and b can be expressed by Kzero and Kinf:

a  b( K inf  K zero )

b

1  K zero K inf  1

(8)

(9)

To summarize, the ultimate mathematical model for the normalized equivalent permeability of shale as a function of TOC (vol) and permeability contrast is:

Kn 

a 1 b Kr

 K zero

(10)

where

a  b( K inf  K zero ), b 

1  K zero K inf  1

(11)

where

K zero  1  e  TOC , K inf 

1

 2

1

(12)

 TOC

Fig. 6. Geometric explanation of a and b in Eq. (10).

Interestingly, a and b are related to the geometric characteristics of the “S” curves, as demonstrated in Fig. 6: b characterizes the extent of the deviations of the curves from ideal symmetry, and this parameter decreases with increasing TOC (vol); a is the product of b and the difference between maximum and minimum permeabilities, and this parameter increases with increasing TOC (vol). It should be noted that the form of Eq. (10) is presumably general. In other words, the parameters 1, e, /2 can be adjusted for more complicated scenarios (Cui et al., 2018). For example, consideration of the geometric characteristics of kerogen, i.e., the connectivity, size and shape of kerogen, and inorganic permeability heterogeneity.

3. Model evaluation 3.1 Model validation The established mathematical model (Eqs. (10), (11) and (12)) is validated against the numerical data extracted from (Cui et al., 2018), as shown in Fig. 7. Results show that the model can accurately calculate the equivalent permeability in most scenarios. Slight deviations are observed for intermediate permeability contrast and relatively larger TOC. This section might seem repetitive to Sec 2.2. However, a and b are directly fitted parameters in Fig. 5, while in this section they are determined by Eq. (11). Therefore, the good fitness of Fig. 4 does not necessarily mean the accuracy of the established model (Eqs. (10)-(12)). All in all, the model is very simple in form, while it can result in very accurate results. Therefore, the model is strongly recommended to calculate the equivalent permeability of shale considering contributions from organic and inorganic matrix. 1.8

Normalized equivalent permeability

1.6 1.4 1.2 1 Data points(TOC=0.05) Data points(TOC=0.10) Data points(TOC=0.15) Data points(TOC=0.20) Data points(TOC=0.25) Fitting curve(TOC=0.05) Fitting curve(TOC=0.10) Fitting curve(TOC=0.15) Fitting curve(TOC=0.20) Fitting curve(TOC=0.25)

0.8 0.6 0.4 0.2 10 -4

10 -2

10 0

10 2

10 4

Permeability contrast

Fig. 7. Model validation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article)

3.2 Model comparison The results from the arithmetic and geometric average methods are compared against the numerical data, as shown in Fig. 8. Results show that the arithmetic average method is quite

problematic, and it can lead to orders of magnitude error in the results. This method is always inaccurate unless Kr=1. The geometric average method is relatively much more reliable, but it can also result in deviation by several times. This method is recommended for 10-1.51 and underestimation if Kr<1. Comparably speaking, the newly established model in this paper is only a bit more complicated in form but can generate much more accurate results. Therefore, the new model is strongly recommended to calculate the equivalent permeability of shale gas/oil considering the effect of

10

4

10

3

10

2

Data points(TOC=0.05) Data points(TOC=0.10) Data points(TOC=0.15) Data points(TOC=0.20) Data points(TOC=0.25) Arithmetic average(TOC=0.05) Arithmetic average(TOC=0.10) Arithmetic average(TOC=0.15) Arithmetic average(TOC=0.20) Arithmetic average(TOC=0.25)

10 1

10

0

10 -1 10 -4

10 -2

10 0

Permeability contrast

(a)

10 2

10 4

10

Normalized equivalent permeability

Normalized equivalent permeability

kerogen. 1

10 0 Data points(TOC=0.05) Data points(TOC=0.10) Data points(TOC=0.15) Data points(TOC=0.20) Data points(TOC=0.25) Geometric average(TOC=0.05) Geometric average(TOC=0.10) Geometric average(TOC=0.15) Geometric average(TOC=0.20) Geometric average(TOC=0.25)

10 -1 10 -4

10 -2

10 0

10 2

10 4

Permeability contrast

(b)

Fig. 8. Validation of analytical methods against numerical data. (a) is for the arithmetic average method (Wang and Cheng, 2019; Wang et al., 2018; Q. Zhang et al., 2018; Zhang et al., 2017). (b) is for the geometric average method (Singh and Cai, 2019). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article)

4. Model application In this section, the newly established model is applied to liquid flow in shale rocks to reexamine the effect of organic matter on the equivalent permeability of shale. Oil and water both

belong to common reservoir liquids, and they are both of research interest and discussed in this section. The inorganic matrix is characterized by equivalent pore radius Rin, and it usually lies in the range of tens of nanometers. Similarly, the organic matter is depicted by equivalent pore radius Ror, and it is mostly several nanometers. The slip length ls in organic matter is non-negligible and can be as large as hundreds of nanometers or even several micrometers (Javadpour et al., 2015; S. Wang et al., 2016a), while that in inorganic matrix is only about one nanometer and thus trivial (S. Wang et al., 2016b). Therefore, the organic permeability kor and inorganic permeability kin can be expressed as:

 4l  1 kor  Ro2r 1  s  8  Ror 

(13)

1 kin  Rin2 8

(14)

Consequently, the expression for permeability contrast Kr is:

 4l  Ror2 1  s   Ror  k K r  or  kin Rin2

(15)

It should be noted that the porosity and tortuosity differences of inorganic matrix and organic matter should be considered in the above expression. However, it is assumed that they are the same here, as they are out of the interest of this paper. As an alternative, the pore radius can be adjusted to accommodate the differences. Moreover, the effect of oil adsorption in shale has been demonstrated to be trivial (Cui et al., 2017; Yang et al., 2019). For water adsorption in shale, the viscosity variation is greater, but its effect is still far less than the factors considered above. Therefore, liquid adsorption in shale is ignored here. The basic parameters are set as (ls=1000 nm, Ror=5 nm, Rin=50 nm, TOC=0.15) for the following sensitivity analysis. Effects of slip length,

organic pore radius, inorganic pore radius, and TOC (vol) on the normalized equivalent permeability

1.3

1.35

1.25

1.3

Normalized equivalent permeability

Normalized equivalent permeability

are shown in Fig. 9.

1.2

1.15

1.1

1.05

1

2

3

4

5

6

7

8

9

1.25

1.2

1.15

1.1 10

10

20

Organic pore radius/nm

30

40

50

60

70

80

90

100

Inorganic pore radius/nm

(a)

(b)

1.35

1.6

1.3

Normalized equivalent permeability

Normalized equivalent permeability

1.5 1.25 1.2 1.15 1.1 1.05

1.4

1.3

1.2

1.1

1 0.95 10 2

10 3

10 4

1

Slip length/nm

(c)

0

0.05

0.1

0.15

0.2

0.25

TOC(vol)

(d)

Fig. 9. Sensitivity analysis. (a) is the analysis for organic pore radius. (b) is the analysis for inorganic pore radius. (c) is the analysis for slip length. (d) is the analysis for TOC (vol).

As shown in Fig. 9 (a), the normalized equivalent permeability first increases very rapidly and then more and more slowly increasing with organic pore radius. The turning boundary is about 4 to 5 nm. Based on Fig. 9 (b), with the increase of inorganic pore radius, the normalized equivalent permeability decreases steadily. Fig. 9 (c) demonstrates the increasing trend of the normalized equivalent permeability with increasing slip length. Moreover, the contribution of organic matter is negligible if slip length is about 100 to 200 nm. This conclusion is consistant with (Javadpour et al.,

2015). On the other hand, the effect of slip length changes little if it is on the order of thousands of nanometers or even greater. The measured slip length of water flow in organic matter is 250 nm (Javadpour et al., 2015), therefore the contribution of organic matter to the equivalent permeability of water flow in shale is most likely to be trivial. On the contrary, the calculated slip length of oil flow in organic matter can be several micrometers (S. Wang et al., 2016a), thus the contribution of organic matter to the equivalent permeability of oil flow in shale is most likely to be non-trivial. Fig. 9 (d) conveys the information discussed before: the effect of TOC (vol) on the normalized permeability is approximately linear. 5. Conclusions In this work, a simple and accurate empirical model for the equivalent permeability of shale considering permeability contrast between organic and inorganic matrix and TOC (vol) is derived based on the extracted numerical data points from (Cui et al., 2018). The model is accurate in most cases, while there are slight deviations for intermediate permeability contrast and comparably larger TOC (vol). Comparison with the arithmetic and geometric average methods demonstrates the superiority of the established model in accurately calculating the equivalent permeability without introducing much more complexity. Model application to liquid flow in shale rocks results in reasonable results and new findings. The derived model is strongly recommended for theoretical studies in shale gas/oil that need coupling of the organic and inorganic permeabilities, including single-phase and multi-phase flows. This research will shed light on the efficient exploration and development of shale reservoirs. Acknowledgements The authors would like to acknowledge the financial support of Beijing Natural Science

Foundation of China (No. 2204093). Nomenclature Kr

Permeability contrast between organic and inorganic matter

Kn

Normalized equivalent permeability

TOC

Total Organic Carbon

Kinf

Maximum permeability

Kzero

Minimum permeability

a, b

Fitting parameters

Rin

Equivalent pore radius of inorganic matrix

Ror

Equivalent pore radius of organic matter

kin

Permeability of inorganic matrix

kor

Permeability of organic matter

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Appendix A. Extracted data points from (Cui et al., 2018) Table 3 Extracted data points from (Cui et al., 2018) Kn Kr TOC=0.05

TOC=0.10

TOC=0.15

TOC=0.20

TOC=0.25

0.0001

0.856

0.704

0.562

0.430

0.299

0.0003

0.856

0.709

0.565

0.433

0.306

0.001

0.856

0.711

0.572

0.435

0.309

0.003

0.858

0.714

0.575

0.438

0.319

0.01

0.858

0.727

0.590

0.463

0.357

0.03

0.868

0.744

0.613

0.501

0.400

0.1

0.894

0.795

0.694

0.608

0.537

0.3

0.942

0.886

0.830

0.770

0.716

1

1

1

1

1

1

3

1.053

1.096

1.144

1.200

1.261

10

1.073

1.147

1.241

1.344

1.466

30

1.083

1.170

1.273

1.410

1.572

100

1.083

1.182

1.291

1.441

1.628

300

1.083

1.187

1.301

1.451

1.643

1000

1.089

1.190

1.301

1.453

1.651

3000

1.089

1.190

1.301

1.458

1.651

10000

1.089

1.190

1.306

1.461

1.661

Highlights 1. A new permeability model of shale coupling organic/inorganic matrix is established. 2. Effects of permeability contrast, TOC (vol) and random distribution are considered. 3. The model is simple but accurate for all contrast and TOC (vol) after validation.

Credit author statement Jiangfeng Cui: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Resources; Software; Validation; Visualization; Roles/Writing - original draft; Keliu Wu: Funding acquisition; Supervision; Writing - review & editing.