A dynamic apparent permeability model for shale microfractures: Coupling poromechanics, fluid dynamics, and sorption-induced strain

Journal of Natural Gas Science and Engineering 74 (2020) 103104

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A dynamic apparent permeability model for shale microfractures: Coupling poromechanics, fluid dynamics, and sorption-induced strain Yudan Li a, b, c, Pingchuan Dong a, b, *, Dawei Zhou a, b, d a

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing, 102249, PR China MOE Key Laboratory of Petroleum Engineering, China University of Petroleum (Beijing), Beijing, 102249, PR China c Department of Chemical and Petroleum Engineering, The University of Kansas, Lawrence, KS, 66045, USA d Department of Petroleum Engineering, Colorado School of Mines, Golden, CO, 80401, USA b

A R T I C L E I N F O

A B S T R A C T

Keywords: Microfractures Dynamic fracture aperture Poromechanics Fluid dynamics Sorption-induced strain Fractal theory

The widely distributed microfractures play an important role in shale gas production. However, limited studies focus on gas flow behavior in microfractures, and ignore the complex transport mechanisms, leading to a large error for gas permeability evaluation. In this work, a newly dynamic apparent permeability (AP) model, coupling poromechanics, sorption-induced strain, and gas slippage, has been proposed to effectively reveal the gas flow mechanisms through microfractures of shale. Specifically, a dynamic aperture is innovatively incorporated into the Navier-Stokes (N–S) equation using the second-order slip boundary condition to calculate the gas velocity and volume flux in single microfracture. Based on that, the gas transport model for microfracture networks considering the distributions of aperture and tortuosity is derived using the fractal theory. The newly developed model is verified well with experimental data and network simulation. Results indicate that the gas conductance highly depends on the structure of microfracture networks (i.e., the maximum aperture and fractal dimensions). There are three different AP evolutions under various boundary conditions (i.e., constant confining pressure (Pc ), constant pore pressure (Pp ), and constant effective stress (σeff )) resulting from the coupling transport mecha­ nisms. The AP presents a similar shape of “V” at reservoir conditions (i.e., constant Pc ), indicating the “negative contribution” of poromechanics at an early stage, and the “positive contribution” for both gas slippage and sorption-induced strain at the late stage should be underlined during gas production. Moreover, the “negative factor” of poromechanics is positively correlated with fracture compressibility coefficient but negatively asso­ ciated with Biot’s coefficient at high pressures (>15 [MPa]). Increasing gas desorption capacity, fracture spacing, and internal swelling coefficient can enhance the “positive factor” of sorption-induced strain at low pressures (<15 [MPa]). This work provides a theoretical guidance to develop shale gas effectively.

1. Introduction As shale gas playing an increasingly important role in energy supply around the world, prediction of formation deliverability becomes a decision-making factor for future investment (Davudov and Moghanloo, 2018). Technological advancements in horizontal drilling and multi­ stage hydraulic fracturing make the shale gas development economi­ cally and efficiently. The complex mixture of organic matter (OM), inorganic matrix, natural and hydraulic fractures of shale results in a multiscale gas production process (Wang and Reed, 2009). The OM (i.e., Kerogen) is generally assumed as the gas source and the ideal storage house, while the widely distributed microfracture networks are

considered as the high pathway for fluid flow (Cho et al., 2013; Tao et al., 2019). The classical cubic law, being widely and successfully used in conventional porous media, becomes insufficient for microfractures within shale due to the additional nonlinear-coupled processes including fluid dynamics (gas slippage), poromechanics (stress dependence), and sorption-induced strain (Chen et al., 2019a). Therefore, developing a reliable permeability model to understand the gas flow behavior through microfractures is a key requirement in shale gas production forecasting. Microfractures in shale are generally formed in weak planes or OM edges, owing to local stress variations and hydrothermal fluids pro­ duction in the process of diagenesis and hydrocarbon evolution (Tao

* Corresponding author. State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing, 102249, PR China. E-mail address: [email protected] (P. Dong). https://doi.org/10.1016/j.jngse.2019.103104 Received 5 July 2019; Received in revised form 15 October 2019; Accepted 4 December 2019 Available online 10 December 2019 1875-5100/© 2019 Elsevier B.V. All rights reserved.

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et al., 2019; Cho et al., 2013). The length and aperture of these fractures are usually distributed in the microscale with complex geometries, resulting in some special nanoscale phenomenon (Wu et al., 2015a; Sun et al., 2018). AP related to Knudsen number (Kn ) is frequently used to describe gas transport behavior in unconventional reservoirs. Experi­ mental observations (Moghaddam and Jamiolahmady, 2016) show that continuum, slip, and transition flow regimes can coexist for the transport mechanisms of gas in nanopores under reservoir conditions (2 � 10 4 < Kn < 6). Moreover, the deeply buried shale formation tends to behave high pressure and high temperature, suggesting the “real gas effect” need to be considered (Wu et al., 2017). To better understand the role of gas transport on production from shale reservoirs, many efforts including (1) experimental, (2) analytical, and (3) simulation methods have been made. Some experts use a pulse decay method to investigate both effects of Pc and Pp on the shale permeability (Kwon et al., 2001; Heller et al., 2014; Ma et al., 2016). Some others, however, use crushed shale samples to estimate the matrix permeability by eliminating natural and drilling-induced microfractures (Luffel and Guidry, 1992; Luffel et al., 1993; Cui et al., 2009). Recently, the combined effects of slippage and geomechanics on matrix permeability are analyzed through steady and unsteady state techniques (Moghaddam and Jamiolahmady, 2016). Moreover, an AP model for gas shale by considering the superposed effects of fluid dynamics and poroelastic is established, showing the effective stress coefficient is consistently less than unity (Fink et al., 2017). However, current studies mainly focus on the shale matrix, but information on the AP of microfractures is limited. Besides, compared with adsorptive methane, the non-adsorptive gases (i.e., nitrogen and helium) are commonly used in experiments leading to deviations of gas permeability in shale samples. On the other hand, different simulation methods such as Molecular dynamics (MD), Direct Simulation Monte Carlo (DSMC), and the Lattice-Boltzmann (LBM) are gradually applied to characterize the rarefied gas flow on molecular level (Kazemi and Takbiri-Borujeni, 2015; Wu et al., 2016; Moghaddam and Jamio­ lahmady, 2017). These methods, however, are not suitable for large-scale simulation as they are computationally expensive and time-consuming. In contrast, analytical models with some reasonable assumptions can not only provide instantaneous calculation results but also identify the impact of each key physical parameter (Wang et al., 2018). A broad variety of AP models have been developed to represent the complexity of the multiple physical and chemical processes involved in gas transport through nanochannels. In the latest review (Zeng et al., 2019), these models can be classified into two branches. The first approach is that the slip boundary is modified considering different transmission mechanisms. A unified Hagen-Poiseuille type equation (B-K model) (Beskok and Karniadakis, 1999) is developed to describe all known gas transport mechanisms in a channel. Later, an AP model for OM by applying the second-order slip boundary condition based on N–S equations is proposed (Song et al., 2018). However, a few important mechanisms are ignored in the model. The second approach is to su­ perimpose different transmission mechanisms based on the corre­ sponding contribution weight. Javadpour (2009) firstly proposed the concept of “apparent gas permeability”, and developed an AP model for gas through circular nanotube by linear superposition of slip flow and Knudsen diffusion. Wu et al. (2015a) used the molecular collision fre­ quency as the weighting coefficient to establish a gas transport model for microfractures with coupling slip flow and Knudsen diffusion. However, the variation of fracture aperture with formation pressure depletion is not considered in their model. AP is the resultant of an intrinsic permeability and a correction coefficient (i.e., fluid dynamic). Up to now, gas transport behaviors in nanopores of the matrix have been studied in detail, however, few studies are focused on the AP evolution for microfractures. Moreover, the intrinsic permeability is often assumed as a constant during depressurization, causing large errors for permeability evaluation. The rock intrinsic permeability is a function of aperture and porosity.

The effective stress increases during reservoir depletion, thereby reduces formation porosity and intrinsic permeability (Tan et al., 2018). In conventional reservoirs, effects of poromechanics on rock deformation or permeability are generally neglected (Wu et al., 2014). However, in unconventional shale formations with nanosized pores and micro­ fractures, the poromechanics has a significant impact on the gas flow regime. Experimental works show that the microfractures are more sensitive to effective stress compared with matrix pores during depres­ surization. Cho et al. (2013) indicated that the natural fracture perme­ ability reduction can be up to 80% over practical ranges of pressure. Dong et al. (2010) reported that the permeability of shale varied by two to three orders of magnitude with an increase in Pc due to the existence of microfractures within shale. Song et al. (2016) suggested that a greater permeability decline will be obtained with an increase in the proportion of natural fractures by pressure depletion. Hence, the effects of poromechanics need to be properly considered for gas transport through microfractures. On the other hand, free gas and adsorbed gas present simultaneously in shale formations (Ren et al., 2015). Moreover, it was estimated that 20%–85% of natural gas can be stored as adsorbed gas (Zhang et al., 2018a). Generally, gas adsorption has two major ef­ fects on shale. First, adsorbed gas accounts for a large percentage of original gas in place, which has been extensively studied (Hartman et al., 2011; Leahy-Dios et al., 2011; Ambrose and Hartman, 2012). Second, gas molecules adsorbed on the surface of matrix pores result in pressure and volume (PV) reduction, which implies that gas adsorption affects effective porosity and permeability (Cui et al., 2009; Ambrose and Hartman, 2012; Haghshenas et al., 2014). Many published researches on sorption-induced strain of coalbed show that the porosity and perme­ ability are functions of reservoir pressure (Palmer and Mansoori, 1998; Shi and Durucan, 2003; Tan et al., 2019). Because of the similar adsorption capacity between shale and coalbed, it is necessary to consider the gas-adsorption effect on the characterization of shale-rock properties. More importantly, gas desorption can trigger matrix shrinkage and thereby increase the effective fracture aperture. Initially, a simplified physical model was proposed assuming that the matrix deformation fully contributed to the fracture aperture change (Shi and Durucan, 2003; Robertson, 2005; Robertson and Christiansen, 2006; Haghshenas et al., 2014). Accompanied by further studies on gas sorption-induced strain, the ideal model is inconsistent with experi­ mental data. To address this issue, an improved model (Liu and Rutqvist, 2010) is built by introducing an internal swelling coefficient to revise the fracture-matrix interaction. Their model revealed that only a part of total swelling strain contributes to the change of fracture aperture, and the remaining portion leads to bulk deformation. Typically, Langmuir and BET isotherms are adopted to characterize gas monolayer and multilayer adsorption behaviors in practice (Chen et al., 2019b). Therefore, the conductivity of microfractures is susceptible to the changes of effective stress, as such, incorporating the dynamic aperture into microfracture permeability is significant for assessing the real production potential and the economic outlook of shale reservoirs (Peng et al., 2018). However, desorption-induced shrinkage for shale gas transport through microfractures has not yet been studied before. Moreover, AP models considering the aforementioned influencing fac­ tors are still lacking. Based on previous studies, a powerful AP model, accounting for the combined effects of poromechanics, sorption-induced strain, and fluid dynamics, has been developed for microfractures of shale. Firstly, two fractal parameters (i.e., Df and DT ) are utilized to describe the micro­ structure of fracture networks. Further, a dynamic fracture aperture considering poromechanics and sorption-induced strain is coupled into the N–S equation applying the second-order slip boundary to derive the gas flow model through single microfracture. The model is also upscaled to fracture networks based on the fractal distribution of microfractures. Results of new model-verification are presented. We clearly elucidate the hidden mechanisms of AP evolution under three different boundary conditions (i.e., constant Pc , Pp , and σeff ) using the newly developed 2

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model. After that, the dominated factors and its sensitivity intensity on AP variation with different time scales are discussed specially. Our theoretical results can be well applied to the numerical simulator for shale gas production forecasting. In addition, this work also indicates that more attention needs to be drawn on the gas flow in microfractures.

(1)

NðL � bÞ ¼ ðbmax =bÞDf

where N is the cumulative number of fractures with an aperture L larger than a constant aperture b, bmax is the maximum aperture, and Df is the fractal dimension for FAD, defined as (Zhang et al., 2018b)

2. Mathematical model

Df ¼ dE

Gas production results in a change of the pore pressure and in-situ stresses, which leads to a dynamic vary fracture aperture affecting gas conductance in turn. Moreover, the nanoscale effects including gas slippage and real gas effect should be considered in the microfractures flow. Changes in fracture flow capacity and conductivity are expected to have a significant influence on reservoir performance. Considering the aforementioned coupling effects of poromechanics, gas desorption, and fluid dynamics, we proposed a new dynamic AP model for micro­ fractures in this section. To understand the model well, some basic as­ sumptions are made as follows:

lnφf lnðbmin =bmax Þ

(2)

where dE is the Euclidean dimension, and ¼ dE ¼ 2 in the two di­ mensions, and bmin is the minimum aperture. Differentiating Eq. (1), the fracture numbers whose aperture ranges from b to b þ db can be calculated by (3)

dN ¼ Df bmax Df b ðDf þ1Þ db

Meanwhile, considering the tortuosity property of microfractures at reservoir conditions, the tortuous length is usually larger than the straight one, which follows the fractal scaling law (Li et al., 2016)

● The conductivity of matrix pores within shale is neglected. ● All the microfractures are slit-shape with the same fracture spacing. ● The variation of fractal dimensions and aspect ratio at different stress conditions are ignored. ● All of microfractures with uniform cross-sections. ● The system is isothermal and homogeneous.

Lt ðbÞ ¼ L0 DT bð1

(4)

DT Þ

Where 1 < DT < 2 is the tortuosity fractal dimension in the space of two dimensions, which is defined as (Zhang et al., 2018b) DT ¼ 1 þ

lnτave lnðL0 =bave Þ

(5)

where the detailed calculation process for τave and L0 =bave can refer (Li et al., 2018). In order to hold Eqs. (1)–(5), bmin=b should be satisfied. Never­

2.1. Fractal properties of microfractures

max

theless, a threshold of bmin= < 10 2 usually guarantees the fractal bmax characters of aperture distribution in practice.

There are numerous fractures distributed disorderly within shale at the micro- and nano-scales. It’s difficult and impractical to capture the gas transport behavior through each of the microfractures with complex geometry (Wu et al., 2015b). However, many natural porous medium with nano to micro-meters behave fractal characteristics (Katz and Thompson, 1985; Yu and Liu, 2004). The experimental results show that the fracture aperture distribution (FAD) nearly follows fractal law similar to the fracture length, as shown in Fig. 1. Based on previous studies, the size distribution of fracture aperture can be characterized by the fractal scaling law (Li et al., 2016)

2.2. The dynamic variation of fracture aperture As we mentioned before, the conductivity of microfractures is sus­ ceptible to the change of stress. Therefore, an accurate characterization of the aperture change during depressurization is of great importance to gas production. In this study, we assumed that the aperture is mainly affected by competitions between desorption-induced matrix shrinkage and effective stress transformation (Fig. 2). Some other influencing factors, such as moisture, fracture compressibility coefficient, and sur­ face diffusion will be considered in our future study. 2.2.1. The impact of poromechanics Unlike to the stress insensibility of shale matrix, microfractures distributed within shale are extremely sensitive to effective stress change (Song et al., 2016). A number of empirical and theoretical ex­ pressions in the literature used to describe the relationship between effective stress and fracture aperture (Chen et al., 2015). These models, however, generally ignored the heterogeneous deformation either for single microfracture or multiple microfractures. To address this issue, a new model is proposed (Chen et al., 2012; Liu et al., 2018) that a fracture system consists of two parts, which are subjected to the same stress, but follow different Hooke’s laws: the hard part follows the engineering-strain but the soft part obeys the natural-strain, as shown in Fig. 2. Based on Hooke’s law, the engineering-strain for the hard part can be expressed as (Chen et al., 2012) dσ ¼ Ke ⋅dεbe

(6)

where Ke is the bulk modulus for the hard part, and εbe is the engineering-strain for fracture aperture, defined as (Jaeger et al., 2007) dεbe ¼

Fig. 1. The concept model for microfractures. 3

dbe b0e

(7)

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Journal of Natural Gas Science and Engineering 74 (2020) 103104

Fig. 2. The concept model for the dynamic aperture during depressurization. (a) The impact of poromechanics; (b) The coupling impacts of poromechanics and desorption-induced strain; and (c) Conceptualization of fracture networks characterized with the hard part and soft part.

where b0e is the unstressed aperture for the hard part. Similarly, the deformation for the soft part can be calculated by the natural-strain (Chen et al., 2012)

The bulk modulus for the hard part is much larger than the soft part, which means Ke ≫Kt , thus Eq. (13) can be simplified as � Δbstress ¼ b0e þ ðb0 b0e Þexp cf Δσ e (14)

(8)

From Fig. 2, the hard part (i.e., with high asperities) is only takes a small portion no matter for single microfracture or fracture networks at reservoir conditions. Besides, our attention mainly focused on the effective channel for gas flow. Hence, b0e= and b0e � 0 are obtained, b0 then Eq. (14) can be further expressed as � (15) Δbstress ¼ b0 exp cf Δσ e

dσ ¼ Kt ⋅dεbt

where Kt is the bulk modulus for the soft part, which is the reciprocal of fracture compressibility coefficient. At the same time, dεbt is defined as (Jaeger et al., 2007) dεbt ¼

dbt b0t

(9)

Based on Biot’s law, the effective stress Δσe is defined as (Jaeger et al., 2007)

After some mathematical transformations, the engineering-strain (hard part) and natural-strain (soft part) can be integrated into the following expressions � � Δσ e (10) be ¼ b0e 1 Ke � bt ¼ b0t exp

Δσ e Kt

Δ σ e ¼ Pc

(11)

Based on the above derivations, the total aperture change ​ Δbstress for single microfracture under stress conditions can be calculated by Combining with Eqs. 10–12, Δbstress can be rewritten as � � � � Δσ e Δσ e Δbstress ¼ b0e 1 þ b0t exp Ke Kt

(16)

where af is the Biot’s coefficient and af ¼ 1 for a fracture system based on previous studies (Elsworth and Bai, 1992). Typical Biot’s coefficients are 0.19 for marble, 0.27–0.47 for granite, and 0.64–0.85 for sandstone (Rice and Cleary, 1976; Detournay et al., 1989; Schmitt and Zoback, 1989; Elsworth and Bai, 1992; Hart and Wang, 1995).

�

Δbstress ¼ be þ bt

a f Pp

2.2.2. The impact of sorption-induced strain Shale (i.e., Kerogen) is a kind of natural adsorbent with the property of adsorbing methane. During depressurization, adsorbed gas desorbs from nanopores, resulting in matrix shrinkage and fracture opening. In this study, we assumed the gas desorption as Langmuir isotherm type and then homogenized the sorption-induced volumetric strain as

(12)

(13) 4

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Δεs ¼ εL

Journal of Natural Gas Science and Engineering 74 (2020) 103104

P P þ PL

εL

P0 PL ðP P0 Þ ¼ εL ðP þ PL ÞðP0 þ PL Þ P0 þ PL

(17)

Z

beff 2

q¼2

Moreover, sorption-induced strain partially contributed to the frac­ ture aperture variation (Chen et al., 2019b). As such, the aperture change owing to sorption-induced strain can be calculated as (Liu and Rutqvist, 2010) Δεs Δbdesorption ¼ f ⋅ ⋅s 3

Uy hdy ¼

0

� hb3eff dPp 1 þ 6C1 Kn þ 12C2 K 2n 12μg dy

(28)

Owing to the dynamic variation of fracture aperture, Eqs. (3) and (4) can be rewritten as (29-a)

Df dNeff ¼ Df bmax;eff beff ðDf þ1Þ dbeff

(18)

� Lt beff ¼ LD0 T b1eff DT

(29-b)

where f represents the internal swelling coefficient ranging from 0 to 1.0, and s is the fracture spacing (Fig. 2). Note that the linear strain is one-third of the volumetric mechanic strain (Levine, 1996). Consequently, the aperture for free gas flow accounting for the combined effects of poromechanics and sorption-induced strain is called dynamic aperture, in the form of 2 3 � 1 Δεs 6 7 cf Δσ e (19) beff ¼ b0 4exp f ⋅ ⋅s 5 b0 3 |fflfflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflfflffl } |ffl ffl ffl ffl ffl { zffl ffl ffl ffl ffl } Poromechanics

Combining with Eqs. (28) and (29), we used the fractal theory to get the total volume flux for fracture networks; that is Z bmax;eff hΔPp Df Qeff ¼ qeff dNeff ¼ b2þDT 12μg LD0 T 2 þ DT Df eff ;max bmin;eff 2 � � 3 2 þ DT Df λ 1 δ1þDT Df þ 1 δ2þDT Df þ 6C1 6 7 1 þ DT Df beff ; max 6 7 �6 (30) 7 � � 2 4 5 � 2 þ DT Df λ DT Df 12C2 1 δ beff ; max DT Df

2.3. Development of dynamic AP model

Where δ2þDT Df is far less than one due to δ ¼ bmin;eff =bmax;eff within an order of 10 2 , hence, Eq. (30) can be further simplified as

Sorption induced strain

As we mentioned before, gas flow regimes in unconventional reser­ voirs are divided by the Kn (ratio of a molecular mean free path (MFP) to the average channel size). With considering a dynamic fracture aperture, the effective Kn can be characterized as (Civan et al., 2011) Kn ¼

λ beff

hΔPp Df b2þDT 12μg LD0 T 2 þ DT Df eff ;max 2 2 þ DT Df λ 1 δ1þDT 1 þ 6C1 6 1 þ D D T f beff ; max 6 �6 � �2 4 2 þ DT Df λ 1 δDT 12C2 beff ; max DT Df

Qeff ¼

(20)

hL10 DT Df b2þDT 12A 2 þ DT Df eff ;max 2 2 þ DT Df λ 1 δ1þDT 1 þ 6C1 6 1 þ D D T f beff ; max 6 �6 � �2 4 2 þ DT Df λ 12C2 1 δDT beff ; max DT Df

kaf ¼

where the viscosity for real gas μg and gas compressibility Z are given as (Wang et al., 2018) " � � � �2 � �# Y1 P4r Pr Pr μg ¼ μg0 1 þ 5 20 þ Y3 þ Y2 (22) Tr Tr T r T r þ P4r 2:5Tr

�

P2r

5:524e

� � Pr ¼ Pp Pg ，Tr ¼ T Tg ​

2:5Tr

�

Pr þ 0:044T 2r

0:164Tr þ 1:15

�

1 dPp

(23)

Ujy¼b ¼ 2

∂2 U C2 λ 2 ∂y 2

(32)

kintrinsic

3 � 2 þ D D λ T f 1þD D 6 1þ6C1 1 δ T f 7 7 6 1 þ DT Df beff ; max 7 6 7 6 �6 � �2 7 � 2 þ DT Df λ 7 6 DT Df 5 4 þ12C2 1 δ beff ; max DT Df |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

(33)

kslippage

The Maxwell second-order slip boundary condition is given as (Maurer et al., 2003)

∂U C1 λ ∂y

Df

� 3 þ 7 7 7 � 5

hL1 DT Df kaf ¼ AðζÞ 0 b2þDT 12A 2 þ DT Df eff ;max |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(24)

(25)

μg dy

Df

Furthermore, we introduced a section shape factor AðζÞ to revise the impact of fracture geometry on gas flow, as such, the final expression of AP for microfractures is described as (Wu et al., 2015a)

The N–S equation with second-order velocity slippage boundary condition can be applied to characterize the gas flow behavior in microfractures because the gas flow drops into regions of slip flow and weak transition. The governing equation is expressed as (Song et al., 2019)

∂2 U ¼ ∂y2

(31)

f

Based on the general Darcy’s law, the AP for microfractures can be calculated by

where beff can be found in Eq. (19). Meanwhile, the MFP for real gas is defined as (Song et al., 2019) rffiffiffiffiffiffiffiffiffiffiffi μg πZRT (21) λ¼ 2M Pp

Z ¼ 0:702e

� 3 þ 7 7 7 � 5 D

Df

where AðζÞ iζ, defined as followings (Wu et al., 2015a) ζ¼

(26)

Based on Eqs. (25) and (26), the gas velocity and volume flux for single microfracture calculated as ! beff 2 dPp y2 Uy ¼ (27) 4 2 þ 1 þ 4C1 Kn þ 8C2 Kn 2 8μg dy beff

h b

AðζÞ ¼ 1

(34-a) 192 ζπ 5

∞ X i¼1;3;5;…

tanhðI πζ=2Þ i5

(34-b)

3. Model validation To check the model reliability, the calculated results by the proposed 5

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Journal of Natural Gas Science and Engineering 74 (2020) 103104

model are compared with the experimental data (Cho et al., 2013) and network simulation (Rahmanian et al., 2012). The parameters used for matching are listed in Table 1. From Fig. 3(a), both the effective aperture and apparent permeability decrease with the confining pressure. By comparison, the predicted re­ sults by our model agree well with the experiment data. The shale samples from the middle Bakken formation, cut vertically at the center to simulate the natural fractures, are used to investigate the gas trans­ port behavior (Cho et al., 2013). During the experiment, the pore pressure kept constant, and the confining pressure increased gradually from 1000 to 5000 [psi] by an increment of 1000 [psi] to study the effect of stress on fractures. Furthermore, the pore network simulation (Rah­ manian et al., 2012) is utilized to validate the reliability of our model. As we mentioned before, limited studies focused on gas transport mecha­ nisms for microfractures of shale. Rahmanian et al. (2012) used a network of 50 � 50 � 50 nodes to simulate the effect of slippage on gas flow through slots and microfractures with aperture size ranging from 0.1 to 10 [μm] and average porosity of 3.5%. It can be observed from Fig. 3(b) that the APs from our model are consistent with simulated results when the pore pressure is larger than 0.1 [MPa]. However, when the pore pressure is lower than 0.1 [MPa], the predicted results are al­ ways greater than the simulated data. Such deviation is attributed to the first-order slip boundary condition underestimating the flow capacity especially in the transition regime (high Kn ) in the simulation (Rahma­ nian et al., 2012). In contrast, our model with the second-order slip boundary is more preferable for gas flow behavior characterization through microfractures.

sensitivity intensity at different time scales to better understand the AP evolution well. In this section, the variation of AP at three boundary conditions and its hidden mechanisms are highlighted firstly. Then the effects of poromechanics, sorption-induced strain, fluid dynamics, as well as the microstructure on gas flow are analyzed in detail. Table 2 lists numerical values of different parameters used hereinafter for the pre­ sented analysis. 4.1. The apparent permeability evolution at different boundary conditions In this part, the AP variation with considering different transport mechanisms during the pore pressure depletion stage is analyzed. Later, the impact of complexed transport mechanisms on AP evolution at three boundary conditions is discussed in detail. From Fig. 4(a), for a constant Pp , AP can achieve the maximum considering fluid dynamics and the minimum considering poro­ mechanics, respectively. The range of grey area in Fig. 4(a) is net change of AP due to the coupling impacts of poromechanics, sorption-induced strain, and fluid dynamics. The result indicates that both of the gas slippage and desorption-induced strain improve flow capacity, while the poromechanics significantly weakens the gas conductance. The phe­ nomenon can be well explained by the evolution law of effective fracture aperture owing to its exponentiation correlation with the AP (Eq. (33)). As shown in Fig. 4(b), the effective fracture aperture decreases monot­ onously with Pp decreasing in consideration of poromechanics. The decreasing Pp can increase σeff , which reduces the effective aperture and the flow capacity accordingly. Meanwhile, the fracture aperture can be increased by matrix shrinkage induced by desorption in gas production (Liu and Rutqvist, 2010). Hence, the effective aperture and even AP present a similar shape of “V” by competing between poromechanics and gas desorption during depressurization. Moreover, at a low-pressure range, the MFP increases with the decreasing Pp , which strengthens the slippage effect and thereby enhances gas conductance. Under a constant Pp , the AP decreases with Pc and this trend becomes slowly when Pc is larger than 45 [MPa] (Fig. 5(a)). In this condition, the negative contribution of poromechanics on flow capacity dominates the initial stage of increasing Pc . Meanwhile, the poromechanics can decrease fracture aperture, which strengths the slippage effect and partially offsets the AP loss. Besides, the impact of sorption-induced strain is approximate zero in the condition of constant Pp . Fig. 5(b) shows the AP increases with Pp depletion at a constant σ eff , and the increasing rate goes up suddenly when the Pp reaches 5 [MPa]. In this condition, both gas slippage and sorption-induced strain have a “posi­ tive contribution” on gas flow, especially at low pressures (<5 [MPa]). Fig. 5(c) indicates that the AP shows a similar shape of “V” by Pp decreasing under a constant Pc . However, the “V” shape is not evident at high Pc . This situation induces the most complicated evolution of AP among the three common conditions in laboratory tests. The “negative contribution” of poromechanics dominates at high pressures (>10 [MPa]), which decreases the effective fracture aperture and then con­ tinuum flow rate. However, under low pressures (<10 [MPa]), the “positive contribution” of gas desorption and fluid dynamics prevails to offset the impact of poromechanics by increasing AP considerably. The mentioned mechanisms of AP for different boundary conditions can theoretically explain experimental phenomenon. In the following sec­ tions, we will conduct parameter analysis on the condition of constant Pc , similar to the process of reservoir depletion development.

4. Sensitivity analysis The boundary conditions (i.e., constant Pc , Pp , andσ eff ) commonly used in the experiments lead to varying AP evolution laws. However, the complexed mechanisms behind this phenomenon are still ambiguous. Therefore, it will be attractive to identify the dominant factor and Table 1 Parameters used in the proposed model for validation. Parameters

Value

The maximum fracture aperture, bmax (μm)

2-10 (Cho et al., 2013; Rahmanian et al., 2012) 0.02–0.1

The minimum fracture aperture, bmin (μm)

Ratio of the minimum aperture to the maximum aperture, bmin =bmax (dimensionless) Porosity for fracture networks, φf

0.01 (Zhang et al., 2018b) 0.05 (Cho et al., 2013)

The height of fracture networks, h (μm)

6–30

The fracture spacing, s (μm)

0.85

Aspect ratio, ζ (dimensionless)

3 (Wu et al., 2015a)

Temperature, T (K) Pore pressure, Pp (MPa) Confining pressure, Pc (MPa)

323 0.20–0.35 (Rahmanian et al., 2012; Cho et al., 2013) 7–30

Critical temperature, Tg (K)

190.56 (Wu et al., 2017)

Critical pressure, Pg (MPa)

4.599 (Wu et al., 2017)

The Langmuir pressure, PL (MPa)

1.26 (Liu and Rutqvist, 2010)

The Langmuir volumetric strain, εL (dimensionless)

0.1 (Liu and Rutqvist, 2010)

Ideal gas viscosity, μg0 (Pa⋅ s )

1.49ⅹ10

5

(Wu et al., 2017)

Molecular weight, M (kg/mol) Universal gas constant, R (J/(mol⋅K)) Biot’s coefficient, af (dimensionless)

0.016 8.314 1 (Jaeger et al., 2007)

The first-order slip coefficient, C1

1.47 (Li et al., 2018)

Fitting constant, Y1 (dimensionless)

7.9

Fracture compressibility coefficient, cf (MPa 1) The second-order slip coefficient, C2

Fitting constant, Y2 (dimensionless) Fitting constant, Y3 (dimensionless)

4.2. The impacts of microstructure

0.05

In addition to complex transport mechanisms, the microstructure of fracture networks is also important for gas flow. As shown in Fig. 6, the AP is positively correlated with the maximum

0.78 9:0 � 10

6

​ Wu et al. (2017)

T aperture because the total flow rate is linearly associated with b2þD eff ;max

0.28

based on Eq. (33), suggesting the maximum aperture might be the most 6

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Journal of Natural Gas Science and Engineering 74 (2020) 103104

Fig. 3. Comparisons of the AP calculated by the proposed model with (a) the experimental data (Cho et al., 2013) and (b) numerical simulation (Rahmanian et al., 2012) at different pressure conditions. Note that the red color and blue color represent the different Bakken core samples in Fig. 3(a).

sensitive parameter in the calculation of AP. It’s in accordance with the physical phenomenon that gas flow is typically governed by a few large fractures in natural rock. For the maximum aperture range of 0.1–0.5 [μm], the AP presents a similar shape of “V” with Pp depletion due to the coupling effects of poromechanics and fluid dynamics. It can be observed from Fig. 7 that there always exists a critical pressure (e.g., Pp ¼ 15 ½MPa�). The AP increases with an increasing DT and a decreasing Df when the Pp is larger than the critical pressure, and vice versa. At a high pressure, the gas conductance is mainly controlled

Table 2 Numerical values of different parameters used in the Model’s sensitivity analysis. Parameters

Value

The maximum fracture aperture, bmax (μm)

0.2–1.0 (Wu et al., 2015a) 0.002–0.01

The minimum fracture aperture, bmin (μm)

Ratio of the minimum aperture to the maximum aperture, bmin =bmax (dimensionless)

0.01 (Zhang et al., 2018b)

Porosity for fracture networks, φf The height of the fracture, h (μm)

0.03–0.20 (Wu et al., 2015a) 0.6–3

The fracture spacing, s (μm)

0.2–1.0

Aspect ratio, ζ (dimensionless)

1, 3, 5, 7 (Wu et al., 2015a) 0.3–50

Pore pressure, Pp (MPa)

Confining pressure, Pc (MPa)

The Langmuir pressure, PL (MPa)

The Langmuir volumetric strain, εL (dimensionless) Biot’s coefficient, af (dimensionless) Fracture compressibility coefficient, cf (MPa 1)

T by the continuum flow linearly correlated with b2þD eff ;max (Eq. (33)). Hence,

the larger the DT , the higher the gas conductance is. Besides, the selfpropped tortuous fractures can reduce the pressure dependency to enhance the permeability, which is consistent with the experimental results (Zhou et al., 2016). However, under a low pressure, both gas slippage and desorption-induced strain are the main contributors to gas flow, especially in a rock of large porosity with large Df and small DT . From Fig. 8, the AP increases with an aspect ratio (ζ) increasing under a constant fracture aperture. This is because the larger the ζ, the greater the surface area for gas flow and the higher gas conductance is. It’s observed that the impact of ζ on gas flow can be ignored when the Pp is less than 5 [MPa]. This is because the continuum flow dominates intrinsic permeability, revised by the ζ of fracture geometry (AðζÞ) (Eq. (33) and (34)), in microfractures under high pressures. Moreover, a slight increase (from 1 to 3) of ζ can improve gas permeability dramatically, whereas it’s not sensitive at large ζ (>3). This is closely related to the variation law of AðζÞ with aspect ratio ζ, which is discussed

30-60 (Tan et al., 2018) 1.26–5.44 0.0215–0.1 (Chen et al., 2012) 0.6–1.0 (Chen et al., 2012) 0.05–0.20 (Liu et al., 2018)

Fig. 4. The influence of different transport mechanisms on the gas flow. The variations of (a) AP and (b) fracture aperture with Pp . Note that k0 represents the intrinsic permeability without considering any influencing factors. 7

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Journal of Natural Gas Science and Engineering 74 (2020) 103104

Fig. 5. The AP evolution at constant (a) Pp ; (b) σ eff ; and (c)Pc .

Fig. 6. Effect of the maximum aperture on AP.

in previous study (Wu et al., 2015a).

However, the difference of AP becomes smaller when the Pp is lower than 5 [MPa]. Different from conventional reservoirs, the impact of fracture compressibility on transport capacity in unconventional reserviors need to combine with gas flow regimes. At a high pressure range (i.e., Pp > 5 ½MPa�), the dominated continuum flow is significantly affected by effective stress. A higher cf means the fracture is more easily compressed to reduce gas conductance. However, at a low pressure (i.e., Pp < 5 ½MPa�), a small effective fracture aperture (high Kn ) induced by a

4.3. The impacts of poromechanics Since the poromechanics is the main influencing factor for gas flow in microfractures. In this section, two important parameters (i.e., frac­ ture compressibility coefficient and Biot’s coefficient) are discussed in detail based on Eqs. (15), (16) and (33). As show in Fig. 9, the greater the cf , the smaller the kaf = k0 is. 8

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Journal of Natural Gas Science and Engineering 74 (2020) 103104

Fig. 7. Relationship between the fractal dimensions and AP for microfractures.

Fig. 10. The effect of Biot’s coefficient (af ) on gas conductance.

pressures (Pp < 5 ½MPa�). Although the af is generally considered to be unity for fractures in practice, our theoretical development allows for arbitrary coefficient values. As we discussed before, the poromechanics dominates gas flow at high pressures. Based on Biot’s law, effective stress increases as the af decreases under a constant Pc . As such, a smaller af will result in a larger reduction in fracture aperture to weaken the gas transport capacity. 4.4. The impacts of sorption-induced strain Based on Eq. (33), the effect of desorption-induced matrix shrinkage on the AP for microfractures is studied (Fig. 11). Fig. 11(a) shows the AP increases with internal swelling coefficient (f) ranging from 0 to 1, especially at low pressures. As stated before, under constant Pc , both gas slippage and desorption-induced strain have a positive contribution to gas flow, especially at low Pp . Gas desorption induces matrix shrinkage to enlarge the effective aperture. A larger f means that matrix volume strain induced by gas desorption mostly contribute to fracture aperture. From Fig. 11(b), gas desorption has a significant imapct on kaf =k0 under low pore pressures (i.e., Pp < 4 ½MPa�), which is consistent with the result we discussed before. Moreover, a larger desorption capacity (for example, large εL and small PL ) will produce a greater increase of effective aperture to improve the AP. Fig. 11(c) shows that the smaller the fracture spacing, the larger the permeability reduction. It can be observed from Eq. (18) that the frac­ ture aperture change due to gas desorption is linearly correlated with the fracture spacing. Besides, the AP changes remarkably with fracture spacing at low pressures, which results from the desorption increases as pore pressure decreasing. It can be inferred that the highly fractured shale may experience a larger permeability drop compared to slightly fractured shale under same reservoir and production conditions.

Fig. 8. Influence of aspect ratio on the AP (bmax ¼ 1:0 ½μm�).

5. Conclusions A comprehensive AP model, accounting for the coupling effects (i.e., poromechanics, sorption-induced strain, and fluid dynamics), has been developed to characterize the gas flow behavior through microfractures of shale. To understand the model well, each of the impacts on gas flow is investigated through the parameter analysis, and some conclusions are obtained as followings:

Fig. 9. The variation of gas permeability with Pp at different fracture compressibility coefficients (cf ).

high cf , strengthening the collisions between gas molecules and surface wall, can enhance gas slippage and even flow capacity. Overall, a large cf can significantly weaken the flow capacity, for example the kaf = k0 has a reduction of 100% when the cf is 0.20 [MPa 1] (Fig. 9). Based on Eqs. (16) and (33), Fig. 10 shows the AP evolution with Pp at different af . From Fig. 10, the kaf =k0 decreases with the af decreasing at a constant Pp , especially at high pressures (i.e., Pp > 5 ½MPa�). The difference for AP caused by af variation gradually diminishes under low

(1) The AP evolution varies with three common boundary conditions owing to the complex transport mechanisms. AP decreases with the Pc increasing under constant Pp . AP increases with Pp decreasing under constant σ eff . When the Pc is constant, AP pre­ sents a similar shape of “V” during depressurization. 9

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Journal of Natural Gas Science and Engineering 74 (2020) 103104

Fig. 11. The variation of AP with Pp under different (a) internal swelling coefficients; (b) the Langmuir strain parameters; and (c) the fracture spacing.

(2) The gas conductance is highly depending on the microstructure of fracture networks. The AP presents a positive correlation with the maximum aperture and aspect ratio. Besides, a decreased fractal dimension of FAD and an increased tortuosity fractal dimension increase the AP at high pore pressures, and vice versa. (3) At reservoir conditions (constant Pc ), the AP presents a similar shape of “V” during depressurization. Specifically, the “negative contribution” of poromechanics dominates at high pore pres­ sures, whereas the “positive contribution” for both gas slippage and sorption-induced strain prevails under low pore pressures. (4) The poromechanics is positively correlated with fracture compressibility coefficients but negatively associated with Biot’s coefficients.

(5) Furthermore, a large desorption capacity, fracture spacing, and internal swelling coefficient enhance the influence of sorption-induced strain on AP at low pore pressures. Declaration of competing interest The authors declare that there are no conflicts of interest. Acknowledgements The authors would like to acknowledge the financial support of the China Scholarship Council (File No. 201706440047 and No. 201806440132).

Nomenclature R M Pp Pc

σ eff T P0 PL Pg Tg Tr Pr

μg0 μg λ Z Y1 Y2

Universal gas constant (J/(mol⋅K)) Methane molar mass (kg/mol) Pore pressure (MPa) Confining pressure (MPa) Effective stress (MPa) Temperature (K) The initial pressure (MPa) The Langmuir pressure (MPa) Critical pressure (MPa) Critical temperature (K) Reduced temperature (dimensionless) Reduced Pressure (dimensionless) Viscosity for ideal gas, (Pa⋅s) Viscosity for real gas, (Pa⋅s) Mean free path for real gas (μm) Gas compressibility factor Fitting constant (dimensionless) Fitting constant (dimensionless) 10

Y. Li et al.

Y3 bmax bmin beff δ L0 Lt φf h s ζ AðζÞ Kn Df DT C1 C2 af cf f

εL

q Qeff kaf

Journal of Natural Gas Science and Engineering 74 (2020) 103104

Fitting constant (dimensionless) The maximum fracture aperture (μm) The minimum fracture aperture (μm) The effective fracture aperture (μm) The ratio of minimum aperture to maximum aperture (dimensionless) Characteristic length (μm) Tortuous length for microfracture (μm) Porosity for fracture networks The fracture height (μm) The fracture spacing (μm) The aspect ratio (dimensionless) Fracture geometry (dimensionless) Knudsen number (dimensionless) The fractal dimension for FAD, (dimensionless) The tortuosity fractal dimension, (dimensionless) The first-order slippage coefficient, (dimensionless) The second-order slippage coefficient, (dimensionless) The Biot’s coefficient (dimensionless) Fracture compressibility coefficient (MPa-1) The internal swelling coefficient (dimensionless) The Langmuir volumetric strain (dimensionless) Gas volume flux for single microfracture, (m3) Total gas volume flux for fracture networks, (m3) Apparent gas permeability for fracture networks, (μm2)

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jngse.2019.103104.

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