Investigation on Nonlinear Behaviors of Seepage in Deep Shale Gas Reservoir with Viscoelasticity

Abstract

The nonlinear behaviors in deep shale gas seepage are investigated, involving the non-Darcy effect, desorption, and viscoelasticity. The seepage model accounts for the nonlinear compressibility factor and gas viscosity due to their stronger non-linearity at a high pressure and temperature. The viscoelastic behavior in deep shales, including matrix deformation and proppant embedment, is quantified, and the evolution of the time-varying and pressure-dependent porosity and permeability is derived. A semi-analytical approach with explicit iteration schemes is developed to solve the pressure field. The proposed model and method are verified by comparing the simulation results with the field data. The results show that the gas production contributed by the non-Darcy effect and desorption is much higher in deep shale than in shallow shale. However, Darcy flow contributes 85% of the total gas production of deep shales. If the effect of viscoelastic behavior is neglected, the accumulative gas production would be overestimated by 18.2% when the confining pressure is 80 MPa. Due to the higher pressure and temperature, the accumulative gas production in deep shale is 150% higher than that in shallow shale. This investigation helps to clarify the performance of the non-Darcy effect, desorption, and viscoelastic behavior in deep shales, and the proposed model and approach can facilitate the optimization simulations for hydraulic fracturing strategy and production system due to its high efficiency.

1. Introduction

As conventional oil and gas reservoirs gradually deplete, shale gas is attracting increasing attention as an alternative energy source due to its cleanness, efficiency, and abundant reserves [1]. Its large-scale development can alleviate the pressure to reduce emissions, optimize the energy structure and inevitably reshape the global energy landscape [2]. The nanoscale pore structure in the original shale gas reservoir results in extremely low porosity and permeability [3]. Therefore, hydraulic fracturing technology is applied to create complex artificial fracture networks in shale formation and to increase production capacity [4]. For long-term stable production, it is necessary to add proppants to the fracturing fluid to prevent fracture closure under high confining pressure [5]. To date, China has achieved the commercial development of shallow shale gas, but deep shale gas reservoirs still have huge resource potential and promising exploitation prospects. For example, deep shale gas resources account for 84% of the total gas resources in the southern Sichuan Basin [6]. However, high pressure and high temperatures pose new challenges to developing deep shale gas reservoirs with increasing burial depths.

Free gas and absorbed gas coexist in the shale formation, and the former occupies the fractures and pores, while the latter adheres to the solid surfaces. According to the Knudsen number, fluid flow in porous media involves multiple regimes, such as continuous flow, slip, transition flow, and free molecule flow [7,8]. The first is Darcy flow, so the other three are known as the non-Darcy effect in this paper. As pressure drops, absorbed gas desorbs and becomes free gas; this process is known as desorption. Research shows that the non-Darcy and desorption effects, behaving non-linearly with the pressure, play a significant role in shallow shale gas reservoirs [9], especially the desorption effect, since absorbed gas accounts for from 30 to 80 percent of the gas in place [10]. However, the high pressure and temperature in deep shales would strengthen the non-linearity of the non-Darcy and desorption effects and inevitably influence their behavior in deep shales.

The compressibility factor and gas viscosity are important parameters due to their influence on gas density, gas transport regime, and reservoir simulation. For convenience, they are usually taken as constants in shallow shale gas simulations [11,12]. Shang et al. have verified the feasibility of this popular approximation technique by comparing the difference between various pressure-dependent compressibility and viscosity models and constants [13]. Research shows that the two parameters are strongly non-linear functions of pressure and temperature [14,15]. In deep shales, the increasing pressure and temperature would magnify the non-linearity of the compressibility factor and gas viscosity and further influence the gas density and the whole seepage process. Therefore, it is necessary to re-investigate the impact of the non-linearity of the non-Darcy effect, compressibility factor, and gas viscosity on the deep shale gas seepage.

Unlike shallow shales, deep shales usually exhibit viscoelastic behavior due to the increasing clay and organic content [16,17,18]. For example, Shi et al. found that clay-rich or quartz-rich shales creep at the nanoscale through the nanoindentation test [19]. Song et al. found that the fracture fluid would soften the shale, and the weakening of the cementation bond could lead to a higher creep rate [20]. The creep of shales would increase the matrix strain and proppant embedment depth [21], and it is necessary to clarify the evolution of porosity and permeability in deep shales. Generally, there are two ways to quantify the effect that viscoelastic behavior has on permeability [22,23]. Experiments can directly obtain the permeability values at different pressures, and then the formula between the permeability and the pressure can be derived by data fitting [24,25,26]. However, maintaining the accuracy of the experimental results is challenging, as the experimental conditions do not fully match the actual reservoirs. Theoretical modeling is an indirect method: derive the strain/deformation under the stress condition, then calculate the porosity variation according to the definition, and finally obtain the permeability evolution through the porosity–permeability relationship [27,28,29,30]. Ding et al. derived the relationship between the proppant viscoelastic embedment and fracture permeability and investigated its influence on tight gas production [31]. Chen et al. characterized matrix elastic deformation and proppant viscoelastic embedment in the evaluation of geothermal development [32]. However, in addition to proppant embedment, the viscoelastic behavior in deep shales also contains viscoelastic deformation in the matrix. Investigations of the co-influence of the non-linear seepage and the viscoelastic behavior on deep shale gas production are still lacking.

This study focuses on the performance of the non-Darcy effect, desorption, and viscoelastic behavior in deep shale gas reservoirs, as the high pressure and temperature in formation strengthen the non-linearity of the non-Darcy effect, desorption, compressibility factor, and gas viscosity. First, the nonlinear seepage model, involving the non-Darcy effect and desorption, is presented, which accounts for the nonlinear variation in the compressibility factor and gas viscosity. Then, the viscoelastic behavior in deep shales, including matrix deformation and proppant embedment, is quantified, and the evolution of time-varying and pressure-dependent porosity and permeability is derived. Next, the pressure field is solved using an efficient semi-analytical approach through explicit iteration. Finally, the influence of the non-Darcy effect, desorption, and viscoelastic behavior on deep shale gas production is analyzed.

2. Mathematical Modeling of Nonlinear Seepage

The mass conversation equation for shale gas is given by:

$$\frac{\partial }{\partial t}\left({\rho }_g\varphi \left(x,p,t\right)+q_a\right)+\nabla \cdot ({\rho }_{g}v)=0$$

(1)

where ${\rho }_g$ is the gas density, kg/m³; $\varphi \left(x,p,t\right)$ is the time-varying and pressure-dependent inhomogeneous porosity discussed in detail in Section 3.4; $x$ is the space vector, m; $p$ is the pressure, Pa; $t$ is the time, s; $q_a$ is the mass of the absorbed gas, kg/m³; and $v$ is the velocity vector, m/s.

The gas density ${\rho }_g$ is given by:

$${\rho }_g\left(p\right)={\rho }_{gsc}\frac{T_{sc}{Z}_{sc}}{TZ}\frac{p}{p_{sc}}$$

(2)

where $T$ is the temperature, K; $Z$ is the compressibility factor; and ${\rho }_{gsc}$, $p_{sc}$, $T_{sc}$, and $Z_{sc}$ are the gas density, pressure, temperature, and compressibility factor under the standard condition, respectively.

The non-linear compressibility factor $Z$ is calculated by the formula developed by Mahmoud [33]:

$$Z\left(p\right)=0.702{\left(\frac{p}{p_c}\right)}^2{e}^{-\frac{2.5T}{T_c}}-5.524\frac{p}{p_c}{e}^{-\frac{2.5T}{T_c}}+0.044{\left(\frac{T}{T_c}\right)}^2-0.164\frac{T}{T_c}+1.15$$

(3)

where $p_c$ and $T_c$ are the critical pressure and temperature of the shale gas, respectively.

The unified apparent permeability model proposed by Beskok and Karniadakis is introduced to characterize the nonlinear seepage of the shale gas, which accounts for both Darcy flow and the non-Darcy effect [34]:

$$v=\frac{K_0\left(x,p,t\right)f(K_n)}{\mu }\nabla p$$

(4)

where $K_0\left(x,p,t\right)$ is the time-varying and pressure-dependent inhomogeneous intrinsic permeability discussed in Section 3.4, mD; $f(K_n)$ is the dimensionless permeability correlation, and $\mu$ is the gas viscosity, mPa∙s.

The non-linear gas viscosity $\mu$ is calculated by the correlation developed by Lee [35]:

$$\mu \left(p\right)=c_1\exp \left(c_2\cdot {\left({\rho }_g\times {10}^{-3}\right)}^{c_3}\right)\times {10}^{-4}$$

(5)

where $c_1=\frac{\left(9.4+0.02M_w\right){\left(1.8T\right)}^{1.5}}{209+19M_w+1.8T}$; $c_2=3.5+\frac{986}{1.8T}+0.01M_w$; $c_3=2.4-0.2c_2$; and $M_w$ is the gas molecular weight, kg/kmol.

To illustrate the difference in gas parameters between shallow and deep shales, Figure 1 shows the variations in the compressibility factor, gas viscosity, and gas density under different pressures and temperatures, and the parameters used are listed as follows: ${\rho }_{gsc}$ = 0.78 kg/m³, $Z_{sc}$ = 1, $p_{sc}$ = 1.013 MPa, $T_{sc}$ = 293.15 K, $p_c$ = 4.59 MPa, $T_c$ = 190.55 K, $M_w$ = 19.45 kg/kmol. The pressure ranges in shallow and deep shales are approximately 0~40 MPa and 0~100 MPa, respectively, and the corresponding temperature values were selected as 333.15 K and 393.15 K, respectively. As shown in Figure 1, in shallow shale, the compressibility factor changes slightly and remains close to 1, while the gas viscosity and density increase approximately linearly with pressure. In deep shale, however, the three parameters change significantly with pressure. For example, the maximum values of the compressibility factor and gas viscosity are more than twice their corresponding minimum values as the pressure increases from 0 MPa to 100 MPa. Meanwhile, the compressibility factor, gas viscosity, and gas density all have strong non-linear relationships with pressure in deep shale. This illustrates that the non-linearity of gas seepage is more significant in deep shale than in shallow shale.

The permeability correlation $f(K_n)$ in Equation (4) is given by [34]:

$$f(K_n)=\left(1+\alpha K_n\right)\left(1+\frac{4K_n}{1-b{K}_n}\right),\alpha =\frac{128}{15{\pi }^2}{\tan }^{-1}\left(4K_n^{0.4}\right)$$

(6)

where $b$ is the slip coefficient, usually taken as −1; and $K_n$ is the Knudsen number, defined as [36]:

$$Kn=\sqrt{\frac{\pi RT}{2M_w}}\frac{\mu \left(p\right)}{r_{p}p}$$

(7)

where $R$ is the universal gas constant, 8314.41 J∙K⁻¹∙kmol⁻¹; and $r_p$ is the average pore radius, nm.

The mass of the absorbed gas $q_a$ is characterized by the Langmuir isothermal adsorption model [37]:

$$q_a={\rho }_{gsc}{\rho }_s\cdot \frac{V_{L}p}{p_L+p}$$

(8)

where ${\rho }_s$ is the shale density, kg/m³; $V_L$ is the Langmuir volume, m³/kg; and $p_L$ is the Langmuir pressure, Pa.

Combining Equations (2), (4) and (8) with Equation (1), the following governing equation for non-linear seepage is derived:

$$\nabla \cdot \left(G\left(p,t\right)\nabla p\right)=\frac{\partial F(p,t)}{\partial t}$$

(9)

where the coefficient $G\left(p,t\right)=\frac{K_0(x,p,t)f\left(K_n\right)p}{\mu \left(p\right)Z\left(p\right)}$ and the mass source function $F(p,t)=\frac{p\varphi \left(x,p,t\right)}{Z\left(p\right)}+\frac{p_{sc}T{\rho }_s}{T_{sc}{Z}_{sc}}\frac{V_{L}p}{p_L+p}$.

3. Porosity and Permeability Model Considering Viscoelastic Behavior

The viscoelastic behavior in deep shale includes matrix deformation and proppant embedment in the fracture surface. The viscoelasticity of the shale is characterized by the generalized Kelvin model, whose three-dimensional differential constitutive equation is derived in Section 3.1. The matrix strain and the proppant embedment depth in viscoelastic deep shale are obtained in Section 3.2 and Section 3.3, respectively. In Section 3.4, the time-varying and pressure-dependent porosity and permeability model is proposed.

3.1. Three-Dimensional Viscoelastic Constitutive Equation

The creep compliance of the generalized Kelvin model is given by:

$$J(t)=\frac{1}{E_1}+\frac{1}{E_2}-\frac{1}{E_2}{e}^{-\frac{E_2}{\eta }t}$$

(10)

where $E_1$ and $E_2$ are the elastic moduli, GPa; and $\eta$ is the shale viscosity, GPa∙h.

According to the genetic integral, the three-dimensional integral constitutive equations of the generalized kelvin model are given by:

$${\epsilon }_m\left(t\right)=\left(1-2\nu \right){\sigma }_m\left(t\right)J\left(0\right)+\left(1-2\nu \right){\displaystyle {\int }_0^t\frac{dJ\left(t-\tau \right)}{d\left(t-\tau \right)}{\sigma }_m\left(\tau \right)d\tau }$$

(11)

$$e_{ij}\left(t\right)=\left(1+\nu \right)S_{ij}\left(t\right)J\left(0\right)+\left(1+\nu \right){\displaystyle {\int }_0^t\frac{dJ\left(t-\tau \right)}{d\left(t-\tau \right)}{S}_{ij}\left(\tau \right)d\tau }$$

(12)

where ${\epsilon }_m$ is the volumetric strain; ${\sigma }_m$ is the volumetric stress; $\nu$ is the Poisson’s ratio; $e_{ij}$ is the deviatoric strain $e_{ij}\left(t\right)={\epsilon }_{ij}\left(t\right)-{\delta }_{ij}{\epsilon }_m\left(t\right)/3$; ${\epsilon }_{ij}$ is the strain; $S_{ij}$ is the deviatoric stress $S_{ij}\left(t\right)={\sigma }_{ij}\left(t\right)-{\delta }_{ij}{\sigma }_m\left(t\right)/3$; ${\sigma }_{ij}$ is the stress; and ${\delta }_{ij}$ is the Kronecker delta.

Then, the corresponding differential constitutive equation of the generalized kelvin model can be easily derived:

$${\dot{\sigma }}_{ij}+\frac{E_1+E_2}{\eta }{\sigma }_{ij}={\delta }_{ij}\frac{\nu E_1}{\left(1-2\nu \right)\left(1+\nu \right)}\left({\dot{\epsilon }}_m+\frac{E_2}{\eta }{\epsilon }_m\right)+\frac{E_1}{1+\nu }\left({\dot{\epsilon }}_{ij}+\frac{E_2}{\eta }{\epsilon }_{ij}\right)$$

(13)

3.2. Viscoelastic Deformation in Matrix

The equilibrium equation of the shale matrix is given by:

$${\sigma }_{ij,j}+\beta p_{,i}=0$$

(14)

where $\beta$ is the Biot coefficient, which is dimensionless.

The relationship between the strain and the displacement is:

$${\epsilon }_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right)$$

(15)

where $u_i$ is the displacement in the i-direction.

Substituting Equations (13) and (15) into Equation (14) and combining them with the formula ${\nabla }^{2}u=\nabla \left(\nabla \cdot u\right)-\nabla \times \left(\nabla \times u\right)$ gives:

$$\begin{array}{l}\frac{\left(1-\nu \right)E_1}{\left(1-2\nu \right)\left(1+\nu \right)}\left[\nabla \left(\nabla \cdot \dot{u}\left(p,t\right)\right)+\frac{E_2}{\eta }\nabla \left(\nabla \cdot u\left(p,t\right)\right)\right]\\ -\frac{E_1}{2\left(1+\nu \right)}\left[\nabla \times \left(\nabla \times \dot{u}\left(p,t\right)\right){\nabla }^2+\frac{E_2}{\eta }\nabla \times \left(\nabla \times u\left(p,t\right)\right)\right]+\beta \left(\nabla \dot{p}+\frac{E_1+E_2}{\eta }\nabla p\right)=0\end{array}$$

(16)

Since the rotation of the matrix microelement is small enough to be neglected, namely $\nabla \times u\left(p,t\right)=0$, Equation (16) is simplified as:

$$\frac{\left(1-\nu \right)E_1}{\left(1-2\nu \right)\left(1+\nu \right)}\left(\nabla \dot{\theta }\left(p,t\right)+\frac{E_2}{\eta }\nabla \theta \left(p,t\right)\right)+\beta \left(\nabla \dot{p}\left(x,t\right)+\frac{E_1+E_2}{\eta }\nabla p\left(x,t\right)\right)=0$$

(17)

where the volumetric strain $\theta \left(p,t\right)=\nabla \cdot u\left(p,t\right)$ and satisfies the requirement that $\theta \left(p,0\right)=0$.

The Laplace transform of the function $\psi \left(t\right)$ is defined as:

$$\tilde{\psi }\left(s\right)=L\left[\psi \left(t\right)\right]={\displaystyle {\int }_0^{\infty }\psi \left(t\right)e^{-st}dt}$$

(18)

Applying the Laplace transform to Equation (17) gives:

$$\nabla \left[\frac{\left(1-\nu \right)E_1}{\left(1-2\nu \right)\left(1+\nu \right){\beta }_B}\tilde{\theta }\left(p,s\right)+\frac{s+\left(E_1+E_2\right)/\eta }{s\left(s+E_2/\eta \right)}\left[\left(s\tilde{p}\left(s\right)-p\left(0\right)\right)+p\left(0\right)\right]\right]=0$$

(19)

where $\tilde{\theta }\left(p,s\right)=L\left[\theta \left(p,t\right)\right]$, and $\tilde{p}\left(s\right)=L\left[p\left(t\right)\right]$.

Then, by applying the inverse Laplace transform to Equation (19), the following equation is derived:

$$\nabla \left[\frac{\left(1-\nu \right)E_1}{\left(1-2\nu \right)\left(1+\nu \right)\beta }\theta \left(p,t\right)+J\left(0\right)p\left(x,t\right)-{\displaystyle {\int }_0^{t}p\left(x,\tau \right)\frac{dJ\left(t-\tau \right)}{d\tau }d\tau }\right]=0$$

(20)

Since the matrix pressure varies little with time, the integral in Equation (20) can be approximated as:

$${\displaystyle {\int }_0^{t}p\left(x,\tau \right)\frac{dJ\left(t-\tau \right)}{d\tau }d\tau }=p\left(x,t\right){\displaystyle {\int }_0^t\frac{dJ\left(t-\tau \right)}{d\tau }d\tau }=p\left(x,t\right)\left[J\left(0\right)-J\left(t\right)\right]$$

(21)

Then, substituting Equation (21) into Equation (20) yields:

$$\frac{\left(1-\nu \right)E_1}{\left(1-2\nu \right)\left(1+\nu \right)\beta }\theta \left(p,t\right)+p\left(x,t\right)J\left(t\right)=g\left(t\right)$$

(22)

where $g\left(t\right)$ is an undetermined function.

To determine the function $g\left(t\right)$, the supply boundary condition of the matrix is introduced: there is no strain at any time if the pressure remains the same as the initial pressure $p_e$, namely $\theta \left(p_e,t\right)=0$. Substituting this condition into Equation (22), the volumetric strain $\theta \left(p,t\right)$ is derived:

$$\theta \left(p,t\right)=\frac{\left(1-2\nu \right)\left(1+\nu \right)\beta }{\left(1-\nu \right)}J\left(t\right)\left[p_e-p\left(x,t\right)\right]$$

(23)

In particular, the volumetric strain $\theta \left(p,t\right)$ reduces to elastic strain when $t=0$.

3.3. Proppant Embedment in the Viscoelastic Fracture Surface

In this part, the Boussinesq solution of the semi-infinite body subjected to a normal force on its surface is generalized to viscoelastic materials, and the embedment depth of the proppant in the viscoelastic fracture surface is derived in terms of Hertzian contact theory.

Without the body force, the displacement equilibrium equations of the spatial axial symmetry problems in elastic materials are [38]:

$$\frac{1}{1-2\nu }\frac{\partial {\epsilon }_m}{\partial \rho }+{\nabla }^2{u}_{\rho }-\frac{u_{\rho }}{{\rho }^2}=0, \frac{1}{1-2\nu }\frac{\partial {\epsilon }_m}{\partial z}+{\nabla }^{2}w=0$$

(24)

where the volumetric strain ${\epsilon }_m=\partial u_{\rho }/\partial \rho +u_{\rho }/\rho +\partial w/\partial z$; the differential operator ${\nabla }^2={\partial }^2/\partial {\rho }^2+\partial /\left(\rho \partial \rho \right)+{\partial }^2/\partial z^2$; $u_{\rho }$ is the radial displacement, m; and $w$ is the displacement in the z-direction, m.

The boundary conditions of the semi-infinite body subjected to a normal force on its surface are listed as follows:

$${\left.{\tau }_{z\rho }\right|}_{z=0,\rho \ne 0}=0, {\left.{\sigma }_z\right|}_{z=0,\rho \ne 0}=0$$

(25)

$${\displaystyle {\int }_0^{\infty }2\pi \rho d\rho \cdot \left(\frac{\nu E}{\left(1-2\nu \right)\left(1+\nu \right)}{\epsilon }_m+\frac{E}{1+\nu }{\epsilon }_z\right)}+F=0$$

(26)

where ${\tau }_{z\rho }$ is the shear stress; ${\sigma }_z$ and ${\epsilon }_z$ are the stress and strain in the z-direction, respectively; and $F$ is the normal force.

Then, the Boussinesq solution of Equations (24)–(26) is given by [38]:

$$w=\frac{\left(1+\nu \right)F}{2\pi E\sqrt{{\rho }^2+z^2}}\left[2\left(1-\nu \right)+\frac{z^2}{{\rho }^2+z^2}\right]$$

(27)

The next step is to generalize the Boussinesq solution to viscoelastic materials. Similar to Equation (24), the displacement equilibrium equations based on the viscoelastic constitutive Equation (13) are expressed as:

$$\left(\eta \frac{\partial }{\partial t}+E_2\right)\left(\frac{1}{1-2\nu }\frac{\partial {\epsilon }_m}{\partial \rho }+{\nabla }^2{u}_{\rho }-\frac{u_{\rho }}{{\rho }^2}\right)=0, \left(\eta \frac{\partial }{\partial t}+E_2\right)\left(\frac{1}{1-2\nu }\frac{\partial {\epsilon }_m}{\partial z}+{\nabla }^{2}w\right)=0$$

(28)

Applying the Laplace transform to Equation (28) yields:

$$\frac{1}{1-2\nu }\frac{\partial {\tilde{\epsilon }}_m}{\partial \rho }+{\nabla }^2{\tilde{u}}_{\rho }-\frac{{\tilde{u}}_{\rho }}{{\rho }^2}=0, \frac{1}{1-2\nu }\frac{\partial {\tilde{\epsilon }}_m}{\partial z}+{\nabla }^2\tilde{w}=0$$

(29)

where ${\tilde{\epsilon }}_m=L\left[{\epsilon }_m\right]$; ${\tilde{u}}_{\rho }=L\left[u_{\rho }\right]$; and $\tilde{w}=L\left[w\right]$.

Applying the Laplace transform to boundary conditions (25) and (26) yields:

$${\left.{\tilde{\tau }}_{z\rho }\right|}_{z=0,\rho \ne 0}=0, {\left.{\tilde{\sigma }}_z\right|}_{z=0,\rho \ne 0}=0$$

(30)

$${\displaystyle {\int }_0^{\infty }2\pi \rho d\rho \cdot \left[\frac{\nu E_1}{\left(1-2\nu \right)\left(1+\nu \right)}{\tilde{\epsilon }}_m+\frac{E_1}{1+\nu }{\tilde{\epsilon }}_z\right]}+\frac{s+\left(E_1+E_2\right)/\eta }{s\left(s+E_2/\eta \right)}F=0$$

(31)

where ${\tilde{\sigma }}_z=L\left[{\sigma }_z\right]$; and ${\tilde{\tau }}_{z\rho }=L\left[{\tau }_{z\rho }\right]$.

Comparing the displacement equilibrium in Equations (24) and (29) and boundary conditions in (25) and (26) with (30) and (31), the frequency domain solution of Equations (29)–(31) can be expressed as:

$$\tilde{w}=\frac{\left(1+\nu \right)}{2\pi E_1\sqrt{{\rho }^2+z^2}}\frac{s+\left(E_1+E_2\right)/\eta }{s\left(s+E_2/\eta \right)}F\left[2\left(1-\nu \right)+\frac{z^2}{{\rho }^2+z^2}\right]$$

(32)

Applying the inverse Laplace transform to Equation (32), the surface settlement of the viscoelastic semi-infinite body subjected to a normal force on its surface is obtained:

$${\left.w\left(t\right)\right|}_{z=0}=\frac{\left(1-{\nu }^2\right)F}{\pi \rho }J\left(t\right)$$

(33)

Then, the surface settlement Equation (33) is used in the contact problem between the viscoelastic fracture surface and the incompressible spherical proppant shown in Figure 2.

As shown in Figure 2, the proppant radius is $R_p$, the distance between points $M_1$ and $M_2$ is $z_1$, and the distance between point $M_2$ and the common normal is $r$. They satisfy the geometric relationship $z_1=r^2/\left(2R_p\right)$.

With the proppant subjected to the normal force $F$, local deformation occurs near the contact point $O$, and a contact area $\mathsf{\Omega }$ with a circular boundary appears. Since the proppant is incompressible, the displacement of point $O$ along the z-direction equals the embedment depth of the proppant in the fracture surface, denoted by $d$. The displacement of point $M_1$ along the z-direction can be denoted as $w_1$. When points $M_1$ and $M_2$ coincide, the following relationship can be derived:

$$d-w_1=z_1$$

(34)

where $w_1$ is calculated using Equation (33):

$$w_1=\frac{1-{\nu }^2}{\pi }J\left(t\right){\displaystyle \underset{\mathsf{\Omega }}{\iint }qdS}$$

(35)

According to Hertzian contact theory, the pressure field in the contact area has a hemisphere-shape distribution, and the pressure value is proportional to the height of the spherical surface [38]. Denote the maximum height of the spherical surface as $a$ and the corresponding pressure value as $q_0$. Substituting Equation (35) into Equation (34) and then integrating them gives:

$$\frac{\pi q_0}{4a}\left(1-{\nu }^2\right)J\left(t\right)\left(2a^2-r^2\right)=d-\frac{r^2}{2R_p}$$

(36)

By comparing the coefficients of Equation (36), the following equations are derived:

$$\frac{\pi q_{0}a}{2}\left(1-{\nu }^2\right)J\left(t\right)=d, \frac{\pi q_0}{4a}\left(1-{\nu }^2\right)J\left(t\right)=\frac{1}{2R_p}$$

(37)

The height $a$ and the pressure $q_0$ satisfy the following boundary condition:

$$\frac{q_0}{a}\cdot \frac{2\pi }{3}{a}^3=F$$

(38)

Combining Equations (37) and (38) yields:

$$d=\sqrt[3]{\frac{9F^2}{16R_p}{\left[\left(1-{\nu }^2\right)J\left(t\right)\right]}^2}$$

(39)

It is worth noting that Equation (39) can reduce to the classical Hertzian solution if time $t=0$ [38].

When the confining pressure is $p_e$, the normal force acting on the proppant particle is $F=2\sqrt{3}{R}_p^2{p}_e$, and the proppant embedment depth in the viscoelastic fracture surface is:

$$d=\frac{3}{\sqrt[3]{4}}{R}_p{\left[p_e\left(1-{\nu }^2\right)J\left(t\right)\right]}^{\frac{2}{3}}$$

(40)

3.4. Time-Varying and Pressure-Dependent Porosity and Permeability

As in our previous work [39], the initial porosity ${\varphi }_0$ and the intrinsic permeability $K_0$ are assumed to follow Gaussian-like distributions along the coordinate axe:

$${\varphi }_0\left(x,y,z\right)={\varphi }_{m0}+\left({\varphi }_{f0}-{\varphi }_{m0}\right)\exp \left(-{\left(x/x_w\right)}^{n_x}/{\sigma }^2(y)-A{\left(y/L_f\right)}^{n_y}-{\left(z/z_w\right)}^{n_z}\right)$$

(41)

$$K_0\left(x,y,z\right)=K_{m0}+\left(K_{f0}-K_{m0}\right)\exp \left(-{\left(x/x_w\right)}^{n_x}/{\sigma }^2(y)-B{\left(y/L_f\right)}^{n_y}-{\left(z/z_w\right)}^{n_z}\right)$$

(42)

$$\sigma (y)=x_w\left(1-\frac{y}{y_c}\right)\exp \left(-\frac{y^2}{2y_c^2}\right)+x_c\frac{y}{y_c}\exp \left(\frac{1}{2}\left(1-\frac{y^2}{y_c^2}\right)\right)$$

(43)

where $A=\ln [({\varphi }_{f0}-{\varphi }_{m0})/({\varphi }_{mf0}-{\varphi }_{m0})]$; $B=\ln [(K_{f0}-K_{m0})/(K_{mf0}-K_{m0})]$; ${\varphi }_{c0}$ and $K_{c0}$ are the average initial porosity and permeability of the c zone (subscripts m, mf, and f denote matrix, micro-fracture, and macro-fracture, respectively); $n_i(i=x,y,z)$ controls the attenuation in the i-direction; $x_w$ and $z_w$ are the widths of the gas outlet in the x- and z-directions, respectively; $L_f$ is the half-length of the hydraulic fracture; $x_c$ and $y_c$ control the horizontal range of the macro-fracture zone.

The initial fracture width ${\lambda }_0$ is assumed to obey the following Gaussian-like distribution:

$${\lambda }_0\left(x,y,z\right)={\lambda }_{f0}\exp \left(-{\left(x/x_w\right)}^{n_x}/{\sigma }^2(y)-\ln \left({\lambda }_{f0}/{\lambda }_{mf0}\right){\left(y/L_f\right)}^{n_y}-{\left(z/z_w\right)}^{n_z}\right)$$

(44)

where ${\lambda }_{f0}$ and ${\lambda }_{mf0}$ are the average fracture widths of the macro-fracture and micro-fracture zones, respectively. In particular, there is no fracture in the matrix as the matrix is not disturbed by hydraulic fracturing.

Matrix deformation and proppant embedment usually occur under different conditions. If the spherical proppant can enter the fracture, the proppant will embed in the fracture surface. Otherwise, matrix deformation will occur. There are four kinds of proppant with different radii, so it is necessary to specify the proppant radius when calculating the embedment depth. As we know, the larger the proppant radius, the better the support effect. Therefore, for a fracture of a certain width, the proppant radius used in the calculation is the largest radius of all the proppants that can enter that fracture. For example, the four proppant radii are $r_1$ = 0.85 mm, $r_2$ = 0.425 mm, $r_3$ = 0.212 mm, and $r_4$ = 0.106 mm. If the fracture width is 1mm, the proppant radius is 0.425 mm. If the fracture width is 0.2 mm, no proppant can enter this fracture, and matrix deformation will occur. Therefore, the proppant radius $R_p$ is given by:

$$R_p=\left\{\begin{array}{l}{r}_1\left(w_0>2r_1\right)\\ r_2\left(2r_2\le w_0<2r_1\right)\\ r_3\left(2r_3\le w_0<2r_2\right)\\ r_4\left(2r_4\le w_0<2r_3\right)\end{array}\right.$$

(45)

where $H$ is the Heaviside function; and $r_1$, $r_2$, $r_3$, and $r_4$ are the proppant radii mentioned above.

The matrix deformation is assumed to be caused only by the recombination of solid particles; then, the porosity in the matrix zone $\varphi \left({\lambda }_0\le 2r_4\right)$ can be derived from its definition:

$$\varphi =\frac{{\varphi }_0-\theta \left(p,t\right)}{1-\theta \left(p,t\right)}\approx {\varphi }_0-\theta \left(p,t\right)\left({\lambda }_0\le 2r_4\right)$$

(46)

After the proppant embeds in the fracture surface, the fracture width $\lambda$ is expressed as:

$$\lambda ={\lambda }_0-2d$$

(47)

Then, the porosity in the fracture zone $\varphi \left({\lambda }_0>2r_4\right)$ can be given by:

$$\varphi ={\varphi }_0\frac{\lambda }{{\lambda }_0}={\varphi }_0\left(1-2d/{\lambda }_0\right)\left({\lambda }_0>2r_4\right)$$

(48)

Therefore, the time-varying and pressure-dependent porosity $\varphi$ can be expressed as:

$$\varphi =\left\{\begin{array}{ll}{\varphi }_0-\theta \left(p,t\right)& \left({\lambda }_0\le 2r_4\right)\\ {\varphi }_0\left(1-2d/{\lambda }_0\right)& \left({\lambda }_0>2r_4\right)\end{array}\right.$$

(49)

According to the Kozeny–Carman model, the time-varying and pressure-dependent intrinsic permeability $K$ is obtained by the cubic law:

$$K=K_0\frac{{\varphi }^3}{{\varphi }_0^3}=\left\{\begin{array}{ll}{K}_0{\left[1-\theta \left(p,t\right)/{\varphi }_0\right]}^3& \left({\lambda }_0\le 2r_4\right)\\ K_0{\left(1-2d/{\lambda }_0\right)}^3& \left({\lambda }_0>2r_4\right)\end{array}\right.$$

(50)

To illustrate the effect of viscoelastic behavior on the initial porosity and intrinsic permeability, Figure 3a shows the percentage drops of the matrix porosity and permeability at different pore pressures. Figure 3b shows the percentage drops of the fracture porosity and permeability under different confining pressures. The parameters used for Figure 3 are: ${\varphi }_{f0}$ = 0.4, ${\varphi }_{mf0}$ = 0.2, ${\varphi }_{m0}$ = 0.05, $x_w$ = 1.5 m, $L_f$ = 50 m, $z_w$ = 6 m, $y_c$ = 50 m, $x_c$ = 5 m, $n_x$= 2, $n_y$ = 6.5, $n_z$ = 2, $K_{f0}$ = 200 mD, $K_{mf0}$ = 50 mD, $K_{m0}$ = 0.0005 mD, $E_1$ = 10 GPa, $E_2$ = 30 GPa, $\eta$ = 2.16 × 104 GPa∙h, $\nu$ = 0.2, $\beta$ = 0.7. The fracture width and the proppant radius are 2 mm and 0.85 mm, respectively. As we can see, the matrix porosity and permeability decrease approximately linearly as pore pressure drops. As the pore pressure decreases from 80 MPa to 0 MPa, the matrix porosity and permeability drop by 13.4% and 35%, respectively. The higher the confining pressure, the lower the fracture porosity and permeability. The fracture permeability drops by 21% when the confining pressure is 80 MPa.

4. Semi-Analytical Solution

Generally, hydraulic fracturing creates multiple horizontal periodic segments in the shale reservoir, so it is reasonable to select only a representative one for convenient numerical simulations. Since it is feasible to calculate the pressure field by interpolation with pressure values at finite nodes, several typical flow paths are selected to significantly reduce the node number. Meanwhile, to improve the reliability of the simulation results, the flow paths are set to extend approximately along the gradient directions of the permeability. Figure 4a,c show sketches of the permeability distributions and corresponding finite flow paths in the xy- and yz-planes of the selected representative segment.

To further facilitate numerical iteration, smooth spatial flow paths are approximated with line segments parallel to one of the three coordinate axes. For example, as shown in Figure 4c, the yellow flow path is replaced with several green broken line segments according to the principle of proximity.

The first step to developing a semi-analytical approach is the application of the Boltzmann transformation to the ith line segment $\left(x_i,y_i,z_i\right)\to \left(x_{i+1},y_{i+1},z_{i+1}\right)$ [39,40]:

$$u=\frac{{\xi }^2}{4t},\xi =\left\{\begin{array}{l}x\left(x_i\ne x_{i+1}\right)\\ y\left(y_i\ne y_{i+1}\right)\\ z\left(z_i\ne z_{i+1}\right)\end{array}\right.$$

(51)

where $u\left(u_i\le u\le u_{i+1}\right)$ is the new variable with two endpoints $u_i={\xi }_i^2/\left(4t\right)$ and $u_{i+1}={\xi }_{i+1}^2/\left(4t\right)$.

This reduces the number of arguments from 4 to 1, allowing us to convert Equation (9) into an ordinary differential equation. During this, the following pseudo-pressure function is defined in the same line segment:

$$m_i\left(p\left(u\right)\right)=={\displaystyle {\int }_{u_i}^{u}G\left(p\left(u\right)\right)}dp\left(u\right), \left(u_i\le u\le u_{i+1}\right)$$

(52)

Substituting Equations (51) and (52) into Equation (9) and then integration derives

$$\exp \left({\displaystyle {\int }_{u_i}^{u_{i+1}}\frac{\mu \left(u\right)z\left(u\right)}{K_0\left(u\right)f\left(K_n\left(p\right)\right)p\left(u\right)}\frac{d{F}_i}{dp}du}\right){\left.\frac{d{m}_i}{du}\right|}_{u=u_{i+1}}={\left.\sqrt{\frac{u_i}{u_{i+1}}}\frac{d{m}_i}{du}\right|}_{u=u_i}$$

(53)

Approximating the definite integral of Equation (53) with the left rectangle formula provides the iteration scheme for the pseudo-pressure gradient:

$$\begin{array}{l}{C}_{i+1}=\sqrt{\frac{u_i}{u_{i+1}}}\exp \left(\frac{-\mu \left(u_i\right)\left(u_{i+1}-u_i\right)}{K_0\left(u_i\right)f\left(K_n\left(p_i\right)\right)p_i}\left(\varphi \left(u_i\right)-\frac{p_i\varphi \left(u_i\right)}{Z\left(u_i\right)}\frac{dZ}{dp}+p_i\frac{d\varphi }{dp}+\frac{p_{sc}TZ\left(u_i\right){\rho }_s{V}_L}{T_{sc}{Z}_{sc}{\left(p_L+p_i\right)}^2}\right)\right)C_i \\ \left(i=1,2,\cdots ,N-1\right)\end{array}$$

(54)

where $C_i={\left.d{m}_i/du\right|}_{u=u_i}={\left.G\left(u_i\right)dp/du\right|}_{u=u_i}$ is the pseudo-pressure gradient to be determined.

The pressure iteration scheme is obtained after forward differentiation on the derivative in C above:

$$p_{i+1}=\sqrt{p_i^2+C_i\frac{2\mu \left(u_i\right)Z\left(u_i\right)\left(u_{i+1}-u_i\right)}{K_0\left(u_i\right)f\left(K_n\left(p_i\right)\right)}},p_1=p_w \left(i=1,2,\cdots N-1\right)$$

(55)

Iteration Schemes (54) and (55) are explicit. Furthermore, as mentioned above, the number of pressure nodes is quite limited, so this semi-analytical approach can significantly improve the efficiency of numerical simulations [39].

5. Results and Discussion

5.1. Model Validation

In order to validate the proposed model and approach, the simulation results are compared with the field data from Marcellus Shale and history match results in the reference [41].

Table 1. Parameters of Marcellus Shale.

Parameter	Value	Unit
Langmuir pressure $p_L$	3.46	MPa
Langmuir volume $V_L$	0.0057	m3/kg
Fracture half-length $L_f$	97.54	m
Matrix porosity ${\varphi }_{m0}$	0.065	1
Reservoir temperature $T$	352.59	K
Bottom hole pressure $p_w$	3.69	MPa
Wellbore radius $r_w$	0.1	m
Fracture number	6	1
Matrix permeability $K_{m0}$	0.0006	mD
Initial reservoir pressure $p_e$	32.58	MPa

shows the reservoir parameters of Marcellus Shale [41], and the other simulation parameters are: ${\rho }_{gsc}$ = 0.78 kg/m³, $Z_{sc}$ = 1, $p_{sc}$ = 1.013 MPa, $T_{sc}$ = 293.15 K, $p_c$ = 4.59 MPa, $T_c$ = 190.55 K, $M_w$ = 19.45 kg/kmol, ${\rho }_s$ = 2600 kg/m³, $r_p$ = 20 nm, ${\varphi }_{f0}$ = 0.4, ${\varphi }_{mf0}$ = 0.2, $K_{f0}$ = 100 mD, $K_{mf0}$ = 20 mD, ${\lambda }_{f0}$ = 3 mm, ${\lambda }_{mf0}$ = 1.5 mm, $x_w$ = 1.5 m, $z_w$ = 6 m, $y_c$ = 45 m, $x_c$ = 3.8 m, $n_x$ = 2, $n_y$ = 6.5, $n_z$ = 2, $E_1$ = 10 GPa, $E_2$ = 30 GPa, $\eta$ = 2.16 × 10⁴ GPa∙h, $\nu$ = 0.2, $\beta$ = 0.7.

Figure 5 compares the simulation results with the field data from Marcellus Shale and history match results in the reference. Although lower than the field data for the first 20 days, the simulation results are still in good overall agreement with the field data and history match results, validating the present model and method. The underestimation of the production rate may be due to the neglect of the high-velocity turbulent flow in propped fractures with high conductivity in the initial production stage.

5.2. Influence of Confining Pressure on Nonlinear Seepage

The parameters used in the following simulations are: $T$ = 393.15 K, $r_p$ = 15 nm, $p_w$ = 4 MPa, $p_e$ = 80 MPa, ${\phi }_{m0}$ = 0.05, $K_{f0}$ = 200 mD, $K_{mf0}$ = 50 mD, $K_{m0}$ = 0.0005 mD, $L_f$ = 50 m, $y_c$ = 50 m, $x_c$ = 5 m, and the fracture number = 10. Other unlisted parameters are the same as those used in Section 5.1. Meanwhile, non-Darcy flow means the combination of Darcy flow and the Darcy effect, and nonlinear effects refer to the non-Darcy effect and desorption in this paper.

Figure 6a shows the curves in accumulative gas production from Darcy flow and non-Darcy flow under different confining pressures. Figure 6b,c show the gas production contributed only by the non-Darcy effect and its ratio to total production, respectively. Simulation results show that non-Darcy flow leads to higher gas production than Darcy flow, and their production difference increases with the confining pressure. Meanwhile, as shown in Figure 6b, the gas production contributed by the non-Darcy effect increases stably over time. However, its ratio to total production drops from 14% to 7.8% as the confining pressure increases from 20 MPa to 80 MPa. Compared to shallow shale, the non-Darcy effect contributes more in terms of gas production in deep shale but with a lower proportion.

The non-Darcy effect plays a less dominant role at higher pressure, and complies with the physical mechanism. Physically, gas seepage results from collisions between gas molecules and the solid surface or between the gas molecules themselves. The first kind results in the non-Darcy effect, while the other is Darcy flow. Their collision frequencies are in the same order of magnitude at a lower pressure. Although the two collision frequencies increase with pressure, the frequency of Darcy flow grows much faster than the non-Darcy effect.

Figure 7a shows the curves of accumulative gas production with and without desorption under different confining pressures. Figure 7b,c show the gas production contributed only by desorption and its ratio to total production, respectively. As shown, desorption can improve gas production, and the improvement increases with the confining pressure. The gas production contributed by desorption increases little when the confining pressure increases from 50 MPa to 80 MPa. This illustrates that the amount of the absorbed gas is close to the storage limit when the confining pressure is 50 MPa. Although the gas production contributed by desorption increases with time, its growth rate decreases sharply, and 60% of the absorbed gas is desorbed in the first five years. After 25 years of production, the desorption effect has exhausted the absorbed gas, and its contribution to gas production reaches its limit. On the other hand, as the confining pressure increases from 20 MPa to 80 MPa, the ratio of the absorbed gas to total production drops from 25% to 5.4%. Compared to shallow shale, the desorption effect contributes more absorbed gas in deep shale, but the proportion of the absorbed gas drops sharply.

Figure 8a shows accumulative gas production with and without nonlinear effects under different confining pressures. Figure 8b,c show the gas production contributed only by nonlinear effects and its ratio to total production, respectively. As we can see, the gas production contributed by nonlinear effects increases with the confining pressure, but its ratio to total production decreases. For example, when confining pressure increases from 20MPa to 80 MPa, the ratio decreases from 40% to 13.3%. Nonlinear effects play a significant role in shallow shale gas seepage. However, the pressure-driven Darcy flow contributes 85% of the gas production in deep shale.

5.3. Influence of Viscoelastic Behavior on Shale on Nonlinear Seepage

Figure 9 shows the curves of accumulative gas production from Darcy flow with and without viscoelastic behavior under different confining pressures. As shown, viscoelastic behavior has little impact on gas production when confining pressure is 20 MPa. However, as the confining pressure rises to 80 MPa, accumulative gas production drops by 18.6% due to the viscoelastic behavior. The gas seepage problems in deep shale should account for the viscoelastic behavior.

Figure 10a shows the curves of accumulative gas production of non-Darcy flow with and without viscoelastic behavior under different confining pressures. Figure 10b shows the gas production contributed only by the non-Darcy effect in these two cases. As shown in Figure 10b, when the confining pressure is 80 MPa, viscoelastic behavior makes the gas production contributed by the non-Darcy effect drop by 25.3%, higher than the production drop rates of Darcy flow and non-Darcy flow (18.6% and 19%, respectively). This illustrates that compared to Darcy flow, the non-Darcy effect is more sensitive to viscoelastic behavior, and the non-Darcy effect would strengthen the impact of viscoelastic behavior.

Figure 11a shows accumulative gas production from Darcy flow and desorption with and without viscoelastic behavior under different confining pressures. Figure 11b shows the gas production contributed only by desorption in these two cases. When the confining pressure is 80 MPa, the difference in gas production contributed by desorption between the two cases is only 3.4%, illustrating that viscoelastic behavior has little impact on desorption. Meanwhile, as shown in Figure 11a, the gas production contributed by Darcy flow and desorption drops by 17.7% due to viscoelastic behavior, lower than Darcy flow (18.6%). This illustrates that the desorption effect would weaken the impact of viscoelastic behavior. The larger the ratio of the absorbed gas, the weaker the effect of viscoelastic behavior.

Figure 12 shows accumulative gas production from non-Darcy flow and desorption with and without viscoelastic behavior under different confining pressures. As we can see, the impact of viscoelastic behavior increases with confining pressure and time. When the confining pressure is 80 MPa, accumulative gas production of the first ten years drops by 17% due to viscoelastic behavior, and neglecting the viscoelastic behavior will lead to an 18.2% overestimate of the accumulative gas production for 25 years.

Figure 13 compares the gas production from shallow and deep shales. The confining pressure and temperature of the shallow shale are 25 MPa and 333.15 K, respectively, and the corresponding parameters of the deep shale are 80 MPa and 393.15 K, respectively. The production difference between shallow and deep shales increases with time. For example, the relative difference between the production rates increases from 179% to 210% from the 10th to 25th years. After 25 years, the accumulative gas production in deep shale is 150% higher than in shallow shale.

5.4. Parameter Sensitivity Analysis of Elastic Modulus, Biot Coefficient and Temperature

Figure 14 shows accumulative gas production with different elastic moduli $E_1$. As shown in Figure 14, gas production increases with the elastic modulus. As the elastic modulus increases from 8 GPa to 12 GPa, the drop rate of gas production due to viscoelastic behavior decreases from 22% to 18.2%.

Figure 15 displays accumulative gas production with different Biot coefficients. As shown, accumulative gas production drops with the Biot coefficient. When Biot coefficient increases from 0.5 to 0.9, the production drop rate increases from 18.2% to 23.9%. The bigger the Biot coefficient, the stronger the effect of viscoelastic geomechanics. Figure 15 shows accumulative gas production with different Biot coefficients. As shown, accumulative gas production drops with the Biot coefficient. When the Biot coefficient increases from 0.5 to 0.9, the drop rate of gas production increases from 18.2% to 23.9%.

Figure 16 shows accumulative gas production with and without viscoelastic behavior under different temperatures. As can be seen, when the temperature rises from 363.15 K to 423.15 K, the gas production without viscoelastic behavior drops only by 6%. While with viscoelastic behavior, the drop rates of gas production differ slightly in the two cases (19.2% and 17.3%, respectively). Therefore, the temperature has little effect on gas seepage and viscoelastic behavior. It is advisable to take the formation temperature as a constant for simplicity.

6. Conclusions

Due to the increasing burial depth, deep shales exhibit viscoelastic behavior, and high pressure and temperature would strengthen the non-linearity of the non-Darcy effect, desorption, compressibility factor, and gas viscosity during the shale gas seepage process. This study investigated the performance of the non-Darcy effect, desorption effect, and viscoelastic behavior in deep shale gas seepage problems. The model of shale gas seepage, including non-Darcy flow and desorption, was proposed, considering the nonlinear compressibility factor and gas viscosity. The matrix strain and the embedment depth of the proppant in the viscoelastic deep shales were derived, and the time-varying and pressure-dependent porosity and permeability model was established. The pressure field was solved efficiently by a developed semi-analytical approach through explicit numerical iteration. The agreement between the simulation results and the field data from Marcellus Shale verified the proposed model and method. The main conclusions are as follows.

The non-Darcy effect and desorption contribute more gas production in deep shale than in shallow shale but with a lower proportion. Pressure-driven Darcy flow contributes 85% of total gas production in deep shales. The non-Darcy effect would strengthen the impact of viscoelastic behavior, while desorption does the opposite. When the confining pressure is 80 MPa, neglecting the viscoelastic behavior will lead to an 18.2% overestimate of the accumulative gas production for 25 years. Due to the higher pressure and temperature, the accumulative gas production in deep shale is 150% higher than that in shallow shale. The temperature has little effect on gas seepage and viscoelastic behavior. The research could provide a reference for the production prediction of deep shales with viscoelasticity, and the proposed approach is efficient and practical for application in optimization simulations such as hydraulic fracturing strategy and production system.

This study has not considered uncertainty in the permeability distribution due to hydraulic fracture deviation, and further research will investigate the effect of the uncertain permeability distribution on gas production in deep shales with viscoelasticity.