Comprehensive apparent permeability model coupled shale gas transfer mechanisms in natural fractures and matrix

Accepted Manuscript Comprehensive apparent permeability model coupled shale gas transfer mechanisms in natural fractures and matrix Lidong Mi, Hanqiao Jiang, Yang Cao, Bicheng Yan, Cheng An PII:

S0920-4105(18)30753-8

DOI:

10.1016/j.petrol.2018.08.080

Reference:

PETROL 5260

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 31 October 2017 Revised Date:

10 August 2018

Accepted Date: 28 August 2018

Please cite this article as: Mi, L., Jiang, H., Cao, Y., Yan, B., An, C., Comprehensive apparent permeability model coupled shale gas transfer mechanisms in natural fractures and matrix, Journal of Petroleum Science and Engineering (2018), doi: 10.1016/j.petrol.2018.08.080. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

3 4 5 6 7

Lidong Mi1, 2, 3,*, Hanqiao Jiang2, Yang Cao3, Bicheng Yan3, Cheng An3 1 Sinopec Petroleum Exploration and Production Research Institute, Beijing, China 2 China University of Petroleum, Beijing, PR China 3 Texas A&M University, College Station, United States * Corresponding author: [email protected]

RI PT

2

Comprehensive Apparent Permeability Model Coupled Shale gas Transfer Mechanisms in Natural Fractures and Matrix

1

Abstract

9

27

This paper presents a method to quantitatively characterize natural Fractures Intensity (FI) and proposes a Comprehensive Permeability Model (CPM) coupling gas transfer mechanisms in natural fractures and matrix for shale gas reservoirs. Firstly, the Scanning Electron Microscope (SEM) images of natural shale cores are used to quantitatively characterize FI, such as the aperture, length, spacing, general fracture distribution described by Fisher orientation dispersion value (K), and the Angle between the mean pole of Fractures and the Scan-line (AFS). Secondly, the corresponding relationship between the AFS and K, which can be used to evaluate the length of fracture traces per unit area of trace plane and the area of fractures per unit volume of rock, is built based on the numerical simulation results. Then, we can obtain the natural fracture distribution state and quantitatively distinguish the natural fracture volume from the matrix. Finally, the CPM is proposed by combining the FI and matrix pore parameters based on the series-parallel circuit theory. The CPM is not only verified by the numerical simulation in which the Boundary Element Method (BEM) is adopted to simulate the fractures’ influence, but also in good agreement with the lab measurements of shale samples from the Cambrian Niutitang formation in South Sichuan Basin, China. The FI evaluation method and the CPM proposed in this paper provide a new approach to quantitatively understand the natural fracture distribution and the contribution to the shale reservoir percolation capability. In turn, this will enable engineers better understand the shale gas transport mechanisms and develop the simulation in a more practical way.

28

1. Introduction

29

Shale gas formations usually contain nano-pores ranging in size from a few to hundreds of nano-meters[1] and micro-fractures[2], so the physics of gas flow cannot be simply described by regular flow mechanisms such as Darcy’s law [3-5]. Many researchers have indicated that the diameters of matrix pores could vary from 5 to 1000 nanometers and the apertures of natural fractures are approximately in the range of 100 to 8000 nanometers[6-8]. Because of various pore sizes and media properties, the gas storage and transport mechanisms which include viscous flow, Knudsen diffusion, surface diffusion, adsorption-desorption and slip flow are different in the various pore media and fractures[3-5, 9, 10]. Many theoretical models about shale gas transport and apparent permeability have been proposed based on the

14 15 16 17 18 19 20 21 22 23 24 25 26

30 31 32 33 34 35 36 37

M AN U

13

TE D

12

EP

11

AC C

10

SC

8

1

ACCEPTED MANUSCRIPT

73

2. Shale gas apparent permeability

74

Due to the various sizes of pore structures and the different gas storage mechanisms in diverse pores of shale reservoirs, the flow mechanisms of shale gas become really challenging since they could include slip flow, Knudsen diffusion, surface diffusion,

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

75 76

SC

43

M AN U

42

TE D

41

EP

40

AC C

39

RI PT

72

different gas flow phenomena in tiny matrix pores and complex fractures network[11-22]. Javadpour, et al. established an apparent permeability model coupling Knudsen diffusion and Klinkenberg effect in nanoscale pores[11, 12].. Wu, et al. [13] built several gas transport models and apparent permeability models considering the gas transport mechanisms of viscous flow, Knudsen diffusion, surface diffusion and desorption. They also separately investigated their effects on the apparent permeability. Chen and Wu et al. (2015) proposed the fracture apparent permeability models[13, 23], which couple the gas transport mechanisms in fractures and consider the effect of fractures’ shape and tortuosity. They also analyzed the primary influence factors of the gas flow inside the fractures. However, the bulk permeability of shale core samples is generally comprised of the effects of matrix pores and natural fractures, so the bulk permeability models only considering matrix permeability or fracture permeability cannot correctly describe the real gas flow process[2, 24]. In addition, since it very challenging to validate these models by experiments, they are not ready to be directly used for the naturally fractured shale gas reservoirs simulation. On the other hand, since the apertures of many natural fractures are close to the radius of matrix pores [6-8], they can be only qualitatively characterized by Scanning Electron Microscope (SEM) Maps and micro-computed tomography (CT)[2, 25]. Even we assume natural fracture can be quantified, numerous micro natural fractures in shale reservoirs will also bring huge challenges for fracture modeling and reservoir simulation [2629]. In this paper we propose a new comprehensive permeability model based on the principle of equivalent flow resistance and the series-parallel circuit theory[30]. This model not only considers these different gas transport mechanisms in matrix, such as gas diffusion, slip flow, and flow physics in fractures, but also couples the conjunct influence of matrix pores and fractures on the overall permeability. By introducing fracture spacing to calculate the fracture porosity, the natural fractures in shale core samples can be quantitatively characterized and then validated by matching the permeability data from experiments. The objective of this model is to better analyze and calculate the apparent permeability coupling the flow in matrix pores and natural fractures. The CPM is not only validated by the results from a BEM based simulator [31], but also can accurately predict the core experiment results and thus in return improves our understanding of real mechanisms of gas movement in naturally fractured shale reservoirs. Overall, this new CPM provides a novel approach to quantify natural fractures and calculate the apparent permeability for naturally fractured shale reservoirs, and improves fractures quantitative characterization and the availability of naturally fractured shale reservoir modeling and simulation for industrial application.

38

2

ACCEPTED MANUSCRIPT

80 81 82 83 84 85 86 87 88 89

RI PT

79

SC

78

desorption-adsorption[16, 32], and the special flow characteristic inside the fractures[23]. Therefore, to build a shale gas transport model, which both considers these flow processes and is also conveniently applied to reservoir simulation, is attractive to both academia and petroleum industry. Currently, by coupling complex gas flow mechanisms in nano-scale pores and fractures network, many researchers have established lots of theoretical models [11-13, 17-23] about apparent permeability and gas transport as shown in Table 1. However, the apparent permeability measured by Darcy’s law is dependent on the shale rock properties (including fractures and matrix pores), transport conditions, and fluid properties[33]. Thus, these models can only describe the flow mechanisms in matrix pores or fractures and they are difficult to be validated by core experiment data and used in industrial applications. To our current knowledge, no such apparent permeability model has been developed, which can: a) consider these complex flow mechanisms both in matrix pores and natural fractures b) be validated by experiment for industrial applications.

90 91 92

Table 1 Shale gas apparent permeability-gas transport model

M AN U

77

Model

Mathematical Equation(s)

Flow Mechanism(s)

Javadpour [11]

k app − m = c g D µ g + Fk ∞

Slippage and diffusion for straight nanotube.

Civan [17], [18]

4 Kn   k = k ∞ (1 + α Kn ) 1 +   1 − bKn 

Continuum, slippage, transition and free-molecular flow regimes.

 2 −σv k = k l 1 + 4 σv 

kapp, pm =

Darabi, et al. [21]

AC C

Wu, et al. [13]

−

93 94 95 96 97 98 99

)

µM φ Df −2 (δ ′) Dk + kD 1+ b  RTρavg τ  p

EP

(k

Jt = −

Chen, et al. [23]

µ  2 RT ρΛ 

 b  k am = k m 1 + m   Pm 

Azom and Javadpour [20]

Singh [22]

π

TE D

Sakhaee-Pour and Bryant [19]

app, pm eff

 x   1− x   100  ×  k app100  =  k app , pm   , pm     slit  tube 

1 r 2 pM 4 Kn  dp ζ mb (1 + α Kn ) 1 +  (1 + Kn ) 8η RTZ  1 − bKn  dl

1 2 8  D −2  ζ rδ f   RTM π (1 + 1/ Kn ) 3 mb  

k = k0 e

−3c f

(σ

0.5

e −σ e 0

ζ C dp dp − Ds0 ms s dl Mp dl

)

Absorbed layers and slippage. Derived from Javadpour [11] model. Accounts for real gases in porous media. Slippage, surface roughness and Knudsen diffusion in a porous medium. Diffusion and porous medium.

Slip flow, low-density gas flow, Nano-pore structures, stress sensitivity, matrix shrink phenomenon Effects of micro-fractures’ structure and size, continuum flow, slip flow and transient flow

For shale reservoirs containing natural fractures[2, 25], the permeability value measured from experiment is usually the bulk permeability of core samples, which includes the contribution of the matrix permeability and natural fracture permeability. We cannot separately measure the matrix permeability or the fracture permeability by current laboratory equipment. As shown in Fig.1, the fracture permeability (calculated by cubic law) is much larger than the matrix permeability[17] when the pore diameter is equal to the fracture 3

ACCEPTED MANUSCRIPT 100 101 102 103 104 105

aperture[24]. Based on the gray curve, the permeability ratio increases when the pore radius or fracture aperture rises. Therefore, the apparent permeability model which is validated by only adjusting matrix permeability or fracture permeability to match the experiment results cannot represent and reflect the true flow capacity of naturally fractured shale reservoirs. The apparent permeability model simultaneously considering the gas transport mechanisms in matrix pores and fractures is required to be developed.

54

Matrix Permeability Fracture Permeability Ratio

1.60 1.20

SC 18

0.40

0

114 115 116 117 118 119 120 121 122 123 124 125 126 127

90

120

150

Fig. 1 The comparison between matrix permeability and fracture permeability Based on the theory of equivalent flow resistance[30], the fracture volume ratio β is defined as the ratio of effective fractures volume to total rock volume, and we apply the series-parallel circuit theory [30] to calculate the apparent permeability coupling both the matrix permeability and the fracture permeability. This new method is able to not only match the laboratory results, but also provide a new path to quantify the natural fractures for complex fracture modeling, which will definitely improve and optimize the fracture modeling and fluid flow simulation for naturally fractured shale gas reservoirs. 2.1 Matrix apparent permeability In the matrix pores on micro scale or nanoscale and without any fractures, the gas flow is mainly dominated by gas diffusion and slippage flow. The matrix apparent permeability coupled continuum, slip, transition and free-molecular flow regimes could be calculated as the following equation[17, 18]:

TE D

113

60

0

EP

112

30

9

Pore radius/half of fracture aperture (nm)

AC C

111

36 27

0.80

0.00

107 108 109 110

45

M AN U

Permeability (mD)

2.00

Fracture Permeability/Matrix Permeability

RI PT

106

Where,

∞

=

∞

1+

1+

represents the absolute permeability, mD ;

(1) is the rarefaction coefficient,

dimensionless; stands for the slip coefficient, dimensionless; is the Knudsen number. 2.2 Natural fracture apparent permeability Xue, et al. [8] applied BIB-FESEM approach to investigate the Longmaxi Shale in Northwestern Qian Region, and they discovered that there widely exists micro-fractures in 4

ACCEPTED MANUSCRIPT 129 130 131 132

shale, and fractures’ occurrence is greatly variable, but most fracture apertures are less than 400 nm. Chen, et al. [23] established a fracture permeability model considering the effect of fracture density based on the equivalent flow resistance theory and Poiseuille equation. In this work, the permeability of natural fracture in core samples is calculated by the model of Chen, et al. [23]: = 10

133

(2)

134

Where

135

trace plane (fracture area density), µm-1 . The length of traces per unit area is given by:

136

is the fracture aperture, μm;

is the length of fracture traces per unit area of

=

140

which can be written as:

=

145 146 147 148 149

Where & is the fracture spacing as measured on a borehole or scan-line, µm; ,-) is the constant of proportionality between spacing and intensity measure , dimensionless. For simple fracture geometries, ,-) can be obtained by random geometry method[34, 35]; as for arbitrary fracture geometries, Dershowitz et al.[36] used discrete fracture model in Fracman to get the corresponding ,-) through different wellbore orientations in fracture networks, and the relationship is shown as Fig.2. Fisher K value here is used to characterize the dispersion of fracture orientation. Thus the higher the K value is, the denser the fracture orientations will be, and vice versa.

Proportionality Constant CP2

AC C

5

K=50

4

K=20

3

K=5 K=1

2

1

0 0

150

,

(4)

*+

TE D

144

'( )

EP

143

(3)

Dershowitz, et al.[34] defined the transformation between fracture spacing & and

141 142

∑!"# !"$ ! %

M AN U

139

SC

137 138

RI PT

128

20

40

60

80

Angle between mean pole of fracture and scanline for Sf

5

100

ACCEPTED MANUSCRIPT Fig. 2 Proportionality constant ./0 [34, 36] The fracture porosity can be estimated by 1 =

154 155 156 157

Where 1 is the fracture porosity, %;

162 163 164 165

is the area of fractures per unit volume of rock

=

'(

*+

is given by[34]:

Where ,- is the constant of proportionality between spacing and intensity measure

(6) ,

dimensionless. For simple fracture geometries, ,- can be obtained by random geometry method[34, 35];

SC

161

(5)

as for evenly distributed fracture orientations, ,- is usually among 1.0-3.0[37], Dershowitz

M AN U

160

, and can be written as[23]:

∙

(fracture volume density), µm-1 . The area of fractures per unit volume of rock

158 159

and

RI PT

151 152 153

[36] applied the discrete fracture model in Fracman to get the corresponding ,- values through different wellbore orientations in fracture networks, and the relationship is shown as Fig.3.

166

K=50

TE D

4

K=20

3

K=5 K=1

1

EP

2

AC C

Proportionality Constant CP3

5

0

0

167 168 169 170 171 172 173 174 175

20 40 60 80 Angle between mean pole of fracture and scanline for Sf

100

Fig. 3 Proportionality constant ./4 [34, 36]

When both the angles between measurement line and fracture, and the fracture spacing are both measured, fracture length distribution and areal distribution can be calculated based on the equations above. Further, fracture porosity can be obtained through Eq. (5) and the fracture permeability can be calculated by Eq. (2). 2.3 Comprehensive permeability Through FEI Quanta 200F, Yang et al.[6, 7] obtained about 300 SEM images of shale 6

ACCEPTED MANUSCRIPT

179 180 181 182 183 184 185 186 187 188 189 190

RI PT

178

SC

177

samples in Cambrian Niutitang formation and Lower Silurian Longmaxi formation in Sichuan Basin. They found that micro-fractures are widely distributed with average fracture lengths around 5.5 to 12 µm and fracture spacing as high as 50 nm. Basically fractures in those samples seldom extend through the whole slice surface. Fracture lengths are usually in micrometer scale, but core sample length is usually 2.5 cm. Therefore, core sample length is far larger than micro-fracture length. Due to negligible effect of fracture orientation, it is assumed in this work that the ideal matrix and natural fracture geometric model satisfies the following conditions: (1) all the natural fractures distributed in matrix belongs to microfracture type with apertures ranging from 50 to 1000 nm; (2) all of the natural fractures are sparsely distributed in matrix, and there is no individual fracture penetrating through the whole model as shown in Fig.4b; (3) fractures intersecting with either inflow boundary or outflow boundary are permitted as shown in Fig.4d, but fracture settings directly or indirectly connect both boundaries are prohibited as shown in Fig.4c. Let the number of natural fractures in the model be , and the 5 6 fracture’s aperture is 5 = 1, ⋯ , , and thus the summation of all fracture apertures ℎ is: ! ℎ

191

193 194 195

is the aperture summation of those

natural fractures, µm;

the 5 6 natural fracture, µm. The average natural fracture < (=>) aperture is: < =

#+

!

is the aperture of

(8)

197

AC C

EP

196

Where ℎ

(7)

!

TE D

192

= ∑:;

M AN U

176

198 199 200 201 202 203

Fig. 4 matrix-natural fracture ideal geometry model Since the natural fractures and matrix can be tied to each other in both series and parallel modes, and in order to simply the computation. The model in Fig.4a is further equivalently transformed and thus a two dimensional dual-region equivalent model is established, as 7

ACCEPTED MANUSCRIPT 204 205

shown in Fig.5. Assume that all fractures are located in Region I, and no fractures exist in Region II. The following condition is satisfied: < ≤ 10 × ℎ

206

208 209

Where ℎ

A

is model width, cm.

In the model the average length of fractures of Region I is treated as the length of Region I, as shown in Fig.5： C = C 10

210 211

Where C is the length of Region I, cm; C is the average length of fracture, cm.

SC

212

II region

I region L1

· · · · · · wnf n−1 wnf n

213

Fig. 5 Matrix-natural fracture Series Model (whole region) In this model the volumetric coefficient of natural fractures satisfies Eq. (11):

EP

D C +C ℎ

217

219 220 221 222

A

= C < 11

Where D is the effective natural fracture pore proportional coefficient, fraction; C is the length of Region II, cm. Further Region I and II are coupled through series permeability approach[30], and thus the apparent permeability for the whole core sample becomes:

AC C

218

TE D

hm + nf

L2

M AN U

wnf1 wnf 2

214 215 216

9

RI PT

207

A

E$ AE) FGHH

E

E

= F$ + F) 12 $

)

223

Where

224

permeability in Region I, mD; is the permeability in Region II, mD. Assuming that all natural fractures in Region I have the same aperture, and are evenly distributed in the matrix as shown in Fig.6. Thus, through the parallel connection model the matrix-fracture system in Region I has the apparent permeability satisfying Eq. (13):

225 226 227

is the apparent permeability coupling natural fracture and matrix, mD; is the

8

ACCEPTED MANUSCRIPT ℎ

228

230 231

=

ℎ

+

ℎ 13

Where ℎ is the matrix width in Region I, cm. In Region I, the summation of matrix width and natural fracture aperture summation satisfies Eq. (14): ℎ

232

+ℎ =ℎ

A

14

233

I region L1

wnf n−1 wnf n

234

Fig. 6 Matrix-natural fracture Parallel Model (I region)

TE D

Without any fractures in Region II, the apparent permeability in region II is actually it’s matrix apparent permeability, which is, =

239

241 242 243

Substituting Eq. (7), (8), (11), (13), (14), (15) into Equation (12), the shale gas apparent permeability model considering the effect of natural fractures, micro-pores and nano-pores can be described as: =

FGHHMN OFGHHM#+ FGHHMN P R$ SR) OFGHHM#+ FGHHMN P OFGHHM#+ F A Q GHHMN R)

+$

AC C

240

15

EP

238

M AN U

· · · · · ·

hm + nf

235 236 237

SC

wnf1 wnf 2

RI PT

229

A

FGHHMN P

16

244

Where

245

and can be represented by Eq. (1);

246

and can be represented by Eq. (2). Actually the fracture pore volume coefficient is the fracture porosity in the rock, which is D = 1U , and thus Eq. (16) becomes:

247 248 249

250

is the matrix apparent permeability considering diffusion and slippage effect

=

+

is the apparent permeability of natural fracture

FGHHMN OFGHHM#+ FGHHMN P $ R$ SR) OFGHHM#+ FGHHMN P OFGHHM#+ F A V+ GHHMN R)

FGHHMN P

17

Generally, the shale permeability is as low as nano-Darcy and is very difficult to measure 9

ACCEPTED MANUSCRIPT

260

3. Results discussion

261

To show the validity of our model, the computational results of Eq. (17) will be compared with both the numerical simulation result and the physical experiment measurement. The relationship of the apparent permeability to the fracture aperture and the matrix permeability will be disclosed. All the relative errors are calculated from L2-norm in this part. 3.1 Comparison with the numerical simulation result One basic model (1cm×1cm) without any fractures illustrated as in Fig.7(a) and two fractured porous medium models illustrated as in Fig.7(b) and Fig.7(c) have been established. The fractures have no connections with the boundaries in Fig.7 (b), while the connection exists in Fig.7 (c). And the specific fracture parameters can be found in Table 2.

257 258

262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282

SC

256

M AN U

255

TE D

254

EP

253

AC C

252

RI PT

259

through conventional approaches in lab. What’s more, both matrix pore and fracture dimensions are in the magnitude of nano-meter, so currently there is no technical approach to quantitatively distinguish them. However, fracture aperture, length, distribution, spacing and scan-line can be availability obtained by the method explained in the section of natural fracture apparent permeability. Combined with Fig. (2), Fig. (3), Eq. (4) and Eq. (6), fracture areal and volumetric densities are calculated. Through Eq. (5) fracture porosity can be obtained. Matrix and fracture permeability can be obtained by Eq. (1) and Eq. (2) respectively. Ultimately through Eq. (17) a comprehensive apparent permeability considering flow characteristics in both fractures and matrix is calculated.

251

10

ACCEPTED MANUSCRIPT 1

0.8

0.6

0.2

0 0.2

0.4

0.6

(a) Basic case 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0

0.2

0.4

TE D

0.2

0

0

0.2

0.4

0.6

0.8

1

(c) Case II

Table 2 Fractures data

AC C Case II

293

0

1

Fig. 7 fracture models

Case I

292

0.8

(b) Case I

289

290 291

0.6

EP

285 286 287 288

1

M AN U

1

0.8

SC

0

283 284

RI PT

0.4

Fracture Endpoint

X

Y

X

Y

Fracture length

0.1

0.8

0.7

0.8

0.1

0.3

0.3

0.9

0.3

0.3

0.2

0.9

0.9

0.2

0.2

0.4

0.8

1.0

0.8

0.4

0.2

0.5

0.8

0.5

0.2

0.2

0.3

0.8

0.9

0.2

The fracture apparent permeability and the fracture porosity can be calculated through Eq. (2) and Eq. (5) respectively, and the analytical apparent permeability considering the effects of both matrix and fracture is generated through Eq. (17). For calculating the numerical 11

ACCEPTED MANUSCRIPT

297 298 299 300 301 302 303 304 305 306 307 308 309

RI PT

296

SC

295

apparent permeability, a boundary element method[31, 38] based simulator has been adopted.Fig.8 shows the pressure distribution of basic case and case I (where no connection exists between fractures and boundaries), the matrix permeability km =1000nD. Fig.8 (b) and Fig.8 (c) correspond to the results of fracture aperture (w) being 1000nm and 100000nm respectively. Comparing Fig.8 (b), Fig.8(c) with Fig.8 (a), we see that the existence of fractures has no significant influence on the pressure map for fracture aperture (w) being 1000nm, while the pressure distribution is heavily influenced for the case of fracture aperture (w) being 100000nm. Since for Case I the fractures have no connections with the boundaries, no fracture distribution information can be inferred from the analytical method proposed in this paper, the fracture permeability is set to be zero, and the apparent permeability (disregarding the influence of fractures) is actually the matrix permeability. The average relative error between the apparent permeability (matrix permeability) and the numerical result is less than 20% (Fig.9 and Fig.10). If we assume the fracture information is known, the apparent permeability (considering the influence of fractures) calculated through our analytical method is significantly larger than the numerical result, and the average relative error would be 55% (Fig.9 and Fig.10).

M AN U

294

(a) No fracture, km =1000nD

AC C

311 312

EP

TE D

310

313 314

(b) Z=1000nm, km =1000nD

(c) w=100000nm, km =1000nD 12

ACCEPTED MANUSCRIPT Fig. 8 pressure distribution for basic case and case I

317

Fig.9 shows the simulation results for the cases with the same matrix permeability (1000nd) but different fracture apertures. The analytical results (considering the influence of fractures) are significantly larger than the numerical results, and the relative error is about 55%. For the fracture aperture being less than 1000nm, the analytical results (not considering the influence of fractures) are the same as the numerical results, which equal to the matrix permeability. And for fracture aperture being larger than 1000nm, the analytical results (not considering the influence of fractures) are less than the numerical values, and the relative error is less than 20%. The reason is that the pressure propagation will be influenced as the fracture apertures increase to some extent, and the lack of the fracture distribution information causes the analytical results (not considering the influence of fractures) smaller comparing to the numerical values. Fig.10 shows the semi-log curves of the numerical and analytical results of the cases with the same fracture aperture (100000nm) but different matrix permeability. The analytical results (considering the influence of fractures) are larger than the numerical results, with the relative error being 50%. The analytical results (not considering the influence of fractures) are less than the numerical results, with the relative error being less than 20%. Therefore, the following conclusion could be obtained: in the fractured porous medium model where no connection between the fractures and the boundaries exists, the analytical solution without considering the influence of fractures (the matrix permeability) is much closer to the numerical results than the analytical solution containing the fractures influences. Besides, the similar conclusion has been also proved in our another paper[38].

321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336

SC

320

M AN U

319

TE D

318

RI PT

315 316

337 0.004

EP

Relative error ≈ 55 %

Analytical solution (Not include fractures) Analytical solution (Include fractures)

0.0024

Numerical solution

AC C

Apparent permeability (mD)

0.0032

0.0016

0.0008

0

10

338 339 340 341

Relative error < 20%

100

1000

10000

100000

1000000

Fracture aperture (nm)

Fig. 9 comparison between numerical and analytical solution of case I (km=1000nD)

13

ACCEPTED MANUSCRIPT 1 Relative error ≈50%

Analytical solution (Not include fractures)

Numerical solution Relative error < 20%

0.01

0.001

0.0001 0.0001

348 349 350 351 352 353 354 355 356 357 358 359 360 361

SC

Fig.11 shows the pressure distribution of basic case and case II (where connection exists between fractures and the boundaries), the matrix permeability km =1000nD. Fig.11 (b) and Fig.11 (c) correspond to the results of fracture aperture (w) being 1000nm and 100000nm respectively. Comparing Fig.11 (b), Fig.11 (c) with Fig.11 (a), we see that the existence of fractures has no significant influence on the pressure map for fracture aperture (w) being 1000nm, while the pressure propagation is enhanced by the fractures of aperture being 100000nm. Since the fractures intersect with the boundaries for Case II, the fracture distribution information can be acquired through the approach proposed in this paper, and therefore the analytical result is the apparent permeability considering both the influence of the fractures and the matrix.

M AN U

347

1

Fig. 10 comparison between numerical and analytical solution of case I (w=100000nm)

TE D

346

0.1

Matrix permeability (mD)

EP

345

0.01

AC C

342 343 344

0.001

RI PT

Apparent permeability (mD)

Analytical solution (Include fractures) 0.1

14

362 363

370 371 372 373 374 375 376 377 378 379 380 381

Fig.12 shows the simulation results for the cases with the same matrix permeability (1000nd) but different fracture apertures. The analytical results become closer to the numerical results, and the relative error is less than 10%. The relative error decreases as the fracture aperture decreases, and the fracture influence will be negligible for fracture aperture less than 100nm. As the fracture aperture is larger than 10000nm, the apparent permeability will not increase as the fracture aperture increases, this is due to the flow capacity of these cases will be constrained mainly by the values of the matrix permeability now. Fig.13 shows the semi-log curves of the numerical and analytical results of the cases with the same fracture aperture (100000nm) but different matrix permeability. The analytical results from our proposed approach come very close to the numerical results, and the relative error is less than 7%, while the matrix permeability is significantly smaller than the numerical values with the relative error being 60%. Thus, our proposed analytical model considering both the influence of fracture and matrix can reasonably represent the actual flow capacity of the fractured porous medium.

EP

369

(a) w=1000nm, km =1000nD (b) w=100000nm, km =1000nD Fig. 11 pressure distribution for basic case and case II

AC C

364 365 366 367 368

TE D

M AN U

SC

(a) No fracture, km =1000nD

RI PT

ACCEPTED MANUSCRIPT

382

15

ACCEPTED MANUSCRIPT Numerical solution Analytical solution 0.0028

Relative error < 10%

Matrix permeability

0.0021 Relative error ≈ 60% 0.0014

RI PT

Apparent permeability (mD)

0.0035

0.0007

0 10

1000

10000

SC

Relative error < 7%

Analytical solution 0.1

Relative error ≈ 60%

0.01

0.001

393 394 395 396 397 398 399

0.01

0.1

1

Matrix permeability (mD)

Fig. 13 comparison between numerical and analytical solution of case II (w=100000nm)

AC C

392

0.001

EP

0.0001 0.0001

391

Matrix permeability

TE D

Apparent permeability (mD)

Numerical solution

390

1000000

Fig. 12 comparison between numerical and analytical solution of case II (km=1000nD)

1

386 387 388 389

100000

Fracture aperture (nm)

M AN U

383 384 385

100

Overall, by comparing and validating with the numerical simulation results, our proposed apparent permeability model considering both the flow characteristics of matrix and fracture can accurately depict the fluid flow performance in fractured porous medium. 3.2 Comparison with the experiment result Yang, et al. [6] and Yang [7] measured the average pore radius of the shale samples (among which S-3, S-4 and S-5 are cored from Cambrian Niutitang formation of Sichuan Basin, China) by Nitrogen adsorption method, the permeability by the pulse decay method and some other characteristic parameters (Table 3). The existence of the micro-scale fractures has been revealed using the Field Emission Scanning Electron Microscope (FESEM). Xue, et al. [8] pointed out that the aperture of the nano-scale fractures in Longmaxi Formation Shale 16

ACCEPTED MANUSCRIPT 401 402 403 404 405 406

of the Northwest Guizhou, China is less than 400nm. In our study, we take 100 nm as the average aperture of the nano-scale fractures. The matrix apparent permeability (Table 4) is computed through Eq. (1). Comparing results of the pulse decay method permeability, the absolute permeability and the matrix apparent permeability (Fig.14) indicate that the matrix apparent permeability is lower than the Pulse Permeability , which is due to that the measured values result from the influence of both the matrix pores and micro-scale fractures, while the matrix apparent permeability approach only considers the influence of the matrix pores.

407

RI PT

400

Table 3 Pore structure parameters of shale

408 No.

Average Pore Radius/nm

Average Porosity/%

Pulse Permeability /mD

S-3

3.80

8.2

0.000888

S-4

4.67

3.0

S-5

8.11

2.4

SC

0.0004

M AN U

409 410 411 412

0.00055

Table 4 Permeability comparison at different models

413 No.

Pulse Permeability/mD

Absolute Permeability/mD

Apparent Permeability /mD

S-3

0.000888

0.0000024

0.00019

S-4 S-5

0.00055 0.0004

0.0000023 0.0000118

0.00012 0.00018

0.0008

TE D

414

Absolute Permeability Matrix Apparent Permeability

EP

0.0004

Pulse Permeability

0.0002

AC C

Permeability (mD)

0.0006

0

4.5

415 416 417

5.5

6.5 7.5 8.5 pore diameter (nm)

9.5

10.5

Fig. 14 Matrix apparent permeability fitting

421

See the photo from the scanning electron microscope (SEM), such as Fig.15 [7], we observe that the fractures usually extend along one direction and exhibit a small divergence. We get the Fisher K value as 20, the angle between the mean pole of fractures and the scanline (AFS) as 60o from the Fig.14 according to the method introduced in section 2.2, and the

422

fracture spacing as 100nm. Therefore we can calculate Cp2 =1.7，Cp3 =1.7 from Fig.2 and

423

Fig.3, calculate P22 =17，P32 =17 from Eq. (4) and Eq. (6), and calculate fracture porosity as

418 419 420

17

ACCEPTED MANUSCRIPT 425 426 427 428 429 430 431

1.7% from Eq. (5). The results computed from our proposed apparent permeability model (Eq.17), which considers the influence of both the matrix and the natural fractures, can match the measured data with relatively high accuracy as shown in Fig.16. During the comparison, since no accurate experimental parameters are specified in the literature, and the approximate values have been adopted, the computation accuracy is compromised to some extent. If the fractures information, such as the fracture spacing, length, aperture, the general fracture distribution and the matrix pore radius, can be acquired accurately through the SEM or other practical ways, a better match with the experimental data can be expected.

RI PT

424

M AN U

SC

432

433 434 435 436

EP

0.0006

Absolute Permeability Matrix Apparent Permeability Pulse Permeability Apparent Permeability

0.0004 0.0002

AC C

Permeability (mD)

0.0008

TE D

Fig. 15 Micro fractures in shale samples[7]

0

4.5

437 438 439 440 441 442 443

5.5

6.5 7.5 8.5 pore diameter (nm)

9.5

10.5

Fig. 16 Matrix-natural fracture apparent permeability fitting

4. Conclusion (1) A novel method of deriving the fracture porosity from the fracture spacing and aperture has been established, which can quantitatively characterize the natural fractures in shale reservoirs accurately. 18

ACCEPTED MANUSCRIPT (2) By applying the series-parallel circuit theory, the apparent permeability model considering the flow characteristics in the matrix nano-scale pores and the natural fractures has been proposed. Its’ accuracy has been validated by comparing the results with both the numerical simulation and the experimental measurement. (3) The analytical model is fast in computation and the required input parameters are available to acquire. Firstly, the fracture spacing, length, aperture, the general fracture distribution, and the matrix pore radius are acquired through the SEM. Then the fracture area density, fracture volume density, fracture porosity and the core sample apparent permeability can be calculated. (4) The shale apparent permeability model proposed in this paper can predict the permeability of the core sample measurements accurately. A realistic representation of the shale gas flow characteristic influenced by both the matrix pore and natural fractures can also be achieved through this model. (5) The natural fracture quantitative characterization approach and our proposed apparent permeability model improve both the geometric modeling and the fluid flow simulation in shale gas reservoirs embedded with complex fracture network.

444 445 446 447 448 449

RI PT

450 451 452 453 454 456 457 458 459 460

Nomenclature = absolute permeability, mD;

461 ∞

M AN U

SC

455

= rarefaction coefficient, dimensionless. = slip coefficient, dimensionless. = Knudsen number.

465

= fracture aperture, μm;

466

= length of fracture traces per unit area of trace plane (fracture area density), µm-1 . = fracture spacing as measured on a borehole or scan-line, µm; = constant of proportionality between spacing and intensity measure dimensionless. = fracture porosity, %;

EP

469 470

& ,-) 1

AC C

467 468

TE D

462 463 464

= area of fractures per unit volume of rock (fracture volume density), µm-1 .

471 472

,-

473 474

dimensionless. ℎ = aperture summation of those natural fractures, µm;

475 476 477 478

,

= constant of proportionality between spacing and intensity measure

!

ℎ C C

A

= aperture of the 5 6 natural fracture, µm. = model width, cm. = length of Region I, cm; = average length of fracture length, cm. 19

,

ACCEPTED MANUSCRIPT

482 483 484 485

D C ℎ

= effective natural fracture pore proportional coefficient, fraction. = length of Region II, cm. = apparent permeability coupling natural fracture and matrix, mD. = permeability in Region I, mD. = permeability in Region II, mD. = matrix width in Region I, cm. = matrix apparent permeability coupled continuum, slip, transition and free-molecular flow, mD. = apparent permeability of natural fracture, mD.

486 487

RI PT

479 480 481

488 489

Acknowledgement

491 493

This study was supported by the National Key Technology Research and Development Program (973 Program) "The Basic Research of South China Marine Shale Gas Efficient Development "(No. 2013CB228005) and China Scholarship council.

494

Reference

M AN U

[1]

J. Wang, L. Chen, Q. Kang, and S. S. Rahman, "Apparent permeability prediction of organic shale with generalized lattice Boltzmann model considering surface diffusion effect," Fuel, vol. 181, pp. 478-490, 2016.

Y. Ma, Z. Pan, N. Zhong, L. D. Connell, D. I. Down, W. Lin, et al., "Experimental study of anisotropic gas

TE D

[2]

permeability and its relationship with fracture structure of Longmaxi Shales, Sichuan Basin, China," Fuel, vol. 180, pp. 106-115, 2016. [3]

W. Zhu and Q. Qi, "Study on the multi-scale nonlinear flow mechanism and model of shale gas (in Chinese)," Sci Sin Tech, 2016.

[4]

C. Zou, Z. Yang, S. Tao, W. Li, S. Wu, L. Hou, et al., "Nano-hydrocarbon and the accumulation in

EP

495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519

coexisting source and reservoir," Petroleum Exploration and Development, vol. 39, pp. 13-26, 2012. [5]

S. Chen, Y. Zhu, G. Wang, H. Liu, W. Wei, and J. Fang, "Structure characteristics and accumulation significance of nanopores in Longmaxi shale gas reservoir in the southern Sichuan Basin," Journal of

AC C

492

SC

490

the China Coal Society, vol. 37, pp. 438-444, 2012.

[6]

F. Yang, Z. Ning, C. Hu, B. Wang, K. Peng, and H. Liu, "Characterization of microscopic pore structures

in shale reservoirs," Acta Petrolei Sinica, vol. 34, pp. 301-311, 2013-03-25 2013.

[7]

F. Yang, "Pore Structure Characterization and the Adsorption-desorption Mechanism of Shale Gas

Reservoir [D]," PhD, China University of Petroleum (Beijing), 2014.

[8]

B. Xue, J. Zhang, X. Tang, and e. al., "Characteristics of microscopic pore and gas accumulation on shale in Longmaxi Formation [J]," Acta Petrolei Sinica, vol. 36, pp. 138-149,173, 2015-02-25 2015.

[9]

C. An, M. Alfi, B. Yan, and J. E. Killough, "A new study of magnetic nanoparticle transport and quantifying magnetization analysis in fractured shale reservoir using numerical modeling," Journal of Natural Gas Science and Engineering, vol. 28, pp. 502-521, 2016.

[10]

M. Alfi, B. Yan, Y. Cao, C. An, J. E. Killough, and M. A. Barrufet, "Microscale porosity models as powerful tools to analyze hydrocarbon production mechanisms in liquid shale," Journal of Natural Gas Science

20

ACCEPTED MANUSCRIPT and Engineering, vol. 26, pp. 1495-1505, 2015. [11]

F. Javadpour, "Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone)," Journal of Canadian Petroleum Technology, vol. 48, pp. 16-21, 2009.

[12]

F. Javadpour, D. Fisher, and M. Unsworth, "Nanoscale gas flow in shale gas sediments," Journal of Canadian Petroleum Technology, vol. 46, 2007.

[13]

K. Wu, X. Li, C. Wang, Z. Chen, and W. Yu, "A model for gas transport in microfractures of shale and tight gas reservoirs," AIChE Journal, vol. 61, pp. 2079-2088, 2015. K. Wu, Z. Chen, and X. Li, "Real gas transport through nanopores of varying cross-section type and

RI PT

[14]

shape in shale gas reservoirs," Chemical Engineering Journal, vol. 281, pp. 813-825, 2015. [15]

B. Yan, Y. Wang, and J. E. Killough, "Beyond dual-porosity modeling for the simulation of complex flow mechanisms in shale reservoirs," Computational Geosciences, pp. 1-23, 2013.

[16]

W. Song, J. Yao, Y. Li, H. Sun, L. Zhang, Y. Yang, et al., "Apparent gas permeability in an organic-rich shale reservoir," Fuel, vol. 181, pp. 973-984, 2016.

F. Civan, "Effective correlation of apparent gas permeability in tight porous media," Transport in

SC

[17]

porous media, vol. 82, pp. 375-384, 2010. [18]

F. Civan, C. S. Rai, and C. H. Sondergeld, "Shale-gas permeability and diffusivity inferred by improved 925-944, 2011.

[19]

A. Sakhaee-Pour and S. Bryant, "Gas permeability of shale," SPE Reservoir Evaluation & Engineering, vol. 15, pp. 401-409, 2012.

[20]

M AN U

formulation of relevant retention and transport mechanisms," Transport in Porous Media, vol. 86, pp.

P. N. Azom and F. Javadpour, "Dual-continuum modeling of shale and tight gas reservoirs," in SPE annual technical conference and exhibition, 2012.

[21]

H. Darabi, A. Ettehad, F. Javadpour, and K. Sepehrnoori, "Gas flow in ultra-tight shale strata," Journal of

TE D

Fluid Mechanics, vol. 710, pp. 641-658, 2012. [22]

H. Singh, "Nonempirical apparent permeability of shale," 2013.

[23]

D. Chen, Z. Pan, and Z. Ye, "Dependence of gas shale fracture permeability on effective stress and reservoir pressure: model match and insights," Fuel, vol. 139, pp. 383-392, 2015.

[24]

L. Mi, H. Jiang, L. Tang, C. Liu, J. Li, Y. Wang, et al., "A Utility Discrete Fracture Network Model for Field-

EP

Scale Simulation of Naturally Fractured Shale Reservoirs," in SPE Argentina Exploration and Production of Unconventional Resources Symposium, 2016. [25]

W. Li, S. Lu, H. Xue, P. Zhang, and Y. Hu, "Microscopic pore structure in shale reservoir in the argillaceous dolomite from the Jianghan Basin," Fuel, vol. 181, pp. 1041-1049, 2016.

[26]

AC C

520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563

A. Moinfar, A. Varavei, K. Sepehrnoori, and R. T. Johns, "Development of an efficient embedded

discrete fracture model for 3D compositional reservoir simulation in fractured reservoirs," SPE Journal, vol. 19, pp. 289-303, 2014.

[27]

A. Moinfar, "Development of an efficient embedded discrete fracture model for 3D compositional

reservoir simulation in fractured reservoirs," 2013.

[28]

L. Mi, H. Jiang, and J. Li, "The impact of diffusion type on multiscale discrete fracture model numerical simulation for shale gas," Journal of Natural Gas Science and Engineering, vol. 20, pp. 74-81, 2014.

[29]

L. Mi, H. Jiang, J. Li, T. Li, and Y. Tian, "The investigation of fracture aperture effect on shale gas transport using discrete fracture model," Journal of Natural Gas Science and Engineering, vol. 21, pp. 631-635, 2014.

[30]

S. Yang and J. Wei, Fundamentals of Petrophysics. Beijing: Petroleum Industry Press, 2004.

[31]

L. J. Durlofsky, "Upscaling of geocellular models for reservoir flow simulation: a review of recent

21

ACCEPTED MANUSCRIPT progress," presented at the 7th International Forum on Reservoir Simulation Bühl/Baden-Baden, Germany, 2003. [32]

L. Geng, G. Li, P. Zitha, S. Tian, M. Sheng, and X. Fan, "A diffusion–viscous flow model for simulating shaleas transport in nano-pores," Fuel, vol. 181, pp. 887-894, 2016.

[33]

F. Civan, "Improved Data Analysis and Interpretation Method for Laboratory Determination of Crushed-Sample Shale Gas Permeability," 2013. W. S. Dershowitz and H. H. Herda, "Interpretation of fracture spacing and intensity," in The 33th US Symposium on Rock Mechanics (USRMS), 1992.

RI PT

[34] [35]

S. LA, "Integral geometry and geometric probability," Reading Mass, 1976.

[36]

W. Dershowitz, FracMan Version 2.3-Interactive Discrete Feature Data Analysis, Geometric Modeling, and Exploration Simulation: User Documentation: Golder Associates, 1992.

[37]

W. S. Dershowitz, "Rock joint systems," Massachusetts Institute of Technology, 1984.

[38]

L. Mi, C. An, Y. Cao, B. Yan, H. Jiang, Y. Pei, et al., "A Guideline on Optimizing Fracture Modeling for Fractured Reservoir Simulation," in SPE Asia Pacific Hydraulic Fracturing Conference, 2016.

SC

564 565 566 567 568 569 570 571 572 573 574 575 576 577

AC C

EP

TE D

M AN U

578

22

ACCEPTED MANUSCRIPT

Novel gas permeability model for natural fractures and matrix pores. The model is derived according to the series-parallel circuit theory

AC C

EP

TE D

M AN U

SC

RI PT

The model is verified by numerical simulation results and experiments results.