Accurate and Efficient Simulation of Fracture–Matrix Interaction in Shale Gas Reservoirs

Abstract

Gas production from low-permeability shale formations relies on natural or man-made fractures for gas flow pathways to production wells. Shale gas reservoir simulation includes fracture–matrix flow and fracture–matrix interactions as they are key. Attention has focused on modeling fractures in shale gas production, yet there have been few studies carried out to address how to accurately simulate fracture–matrix interaction in unconventional low-permeability gas reservoirs. The classic double porosity model and the MINC method represent studies designed to accurately simulate fracture–matrix interaction; however, methods continue to encounter issues causing them to fall short of the accuracy sought. Applicability of the classic double porosity model with a constant shape factor to low-permeability reservoir simulation is questionable and has not been validated as the required pseudo-steady-state flow condition timing may not sufficiently satisfy shale gas reservoirs. The MINC method treats inter-porosity flow in a fully transient mode by further subdividing individual blocks with a number of 1-D nested meshes. The MINC concept, however, assumes that fracture–matrix flow is controlled only by the distance to the nearest fracture surrounding the matrix block and is shown to be no longer applicable after the early rapid transient period of fracture–matrix flow. A comparative investigation of commonly used fractured reservoir simulation approaches for applicability to fracture–matrix interaction in unconventional reservoirs is presented in this paper. A new nested subdividing concept, referred to as the Schwarz–Christoffel conformal mapping approach, will be introduced. The new method is able to accurately and efficiently simulate the matrix–fracture interaction for the entire transient flow by combining the MINC and Schwarz–Christoffel conformal mapping concepts of gridding inside the matrix. The theoretical development, benchmarking, and application of the new modeling approach explanations follow.

Appendix

1.1 Analytical Solutions of the Gas Flow in a Two-Dimensional System

Flow in a square system can be approximated to a cylindrical flow. The governing equation of a single-phase gas radial flow through a fracture–matrix system can be derived by combining a mass balance on the control volume with the dual-continuum concept (Lai et al. 1983; Wu and Pan 2003). The interface area for any two adjacent control volumes at a distance of $r$ from the center:

$$\begin{aligned} A_r =2\pi rh \end{aligned}$$

(5.1)

where, subscript $r$ is the distance from the block center at radial direction; and $h$ is the formation thickness. Then control volume at radial distance $r$ over distance $\mathrm{d}r$:

$$\begin{aligned} V_r =A_r \cdot \mathrm{d}r=2\pi rh\cdot \mathrm{d}r \end{aligned}$$

(5.2)

Consider an ideal gas flow in a homogenous, isothermal, and incompressible porous media. The gas mass balance equation for a control volume inside the matrix is written as,

$$\begin{aligned} \rho q_r A_r -\left[ {\rho q_r A_r +\frac{\partial }{\partial r}\left( {\rho q_r A_r } \right) \mathrm{d}r} \right] =\frac{\partial }{\partial t}\left[ {V_r \left( {\rho \emptyset _\mathrm{m} +\frac{\alpha P_\mathrm{m} }{P_\mathrm{m} +P_\mathrm{L} }} \right) } \right] \end{aligned}$$

(5.3)

where, $\upalpha$ is the gas adsorption related coefficient,

$$\begin{aligned} \alpha =\rho _\mathrm{k} \rho _\mathrm{g} V_\mathrm{L} \end{aligned}$$

(5.4)

where, $\rho _\mathrm{k}$ and $\rho _\mathrm{g}$ respectively are kerogen density and gas density at standard condition; $V_\mathrm{L}$ and $P_\mathrm{L}$ are the Langmuir volume and Langmuir pressure. Flow rate (Darcy 1856) at surface of any control volume is proportional to the pressure gradient at the surface and the permeability of the matrix, inversely proportional to the fluid viscosity, and with considering the Klinkenberg effect contribution to the gas permeability,

$$\begin{aligned} q_r =-\frac{k_\mathrm{m} (1+\frac{b_\mathrm{m} }{P_\mathrm{m} })}{\mu _\mathrm{g} }\frac{\partial P_\mathrm{m} }{\partial r} \end{aligned}$$

(5.5)

The ideal gas law defines the gas density as a function of pressure with a constant coefficient,

$$\begin{aligned} \rho =\beta P_\mathrm{m} \end{aligned}$$

(5.6)

The unit of the Klinkenberg coefficient is consistent with pressure. Define an equivalent pressure coupling gas pressure and Klinkenberg coefficient as follows,

$$\begin{aligned} P_\mathrm{m,b} =P_\mathrm{m} +b_\mathrm{m} \end{aligned}$$

(5.7)

Substitute (5.4) (5.5) and (5.6) into (5.3),

$$\begin{aligned} \frac{\partial }{\partial r}\left( {\beta P_\mathrm{m} \cdot 2\pi rh\frac{k_\mathrm{m} (1+\frac{b_\mathrm{m} }{P_\mathrm{m} })}{\mu _\mathrm{g} }\frac{\partial P_\mathrm{m} }{\partial r}} \right) \mathrm{d}r=\frac{\partial }{\partial t}\left[ {2\pi rh\cdot \mathrm{d}r\left( {\beta P_\mathrm{m} \emptyset _\mathrm{m} +\frac{\alpha P_\mathrm{m} }{P_\mathrm{m} +P_\mathrm{L} }} \right) } \right] \end{aligned}$$

(5.8)

Use (5.7), leading to,

$$\begin{aligned} \frac{k_\mathrm{m} }{\mu _\mathrm{g} }\frac{1}{r}\frac{\partial }{\partial r}\Bigg [rP_\mathrm{m,b} \frac{\partial P_\mathrm{m} }{\partial r}\Bigg ]=\frac{\partial }{\partial t}\left[ {\emptyset _\mathrm{m} P_\mathrm{m} +\frac{\alpha P_\mathrm{m} }{\beta \left( {P_\mathrm{m} +P_\mathrm{L} } \right) }} \right] \end{aligned}$$

(5.9)

In order to linearization of the above equation, define an averaged value of the gas pressure in the coefficient of RHS,

$$\begin{aligned} \bar{P}_\mathrm{m,b} =\bar{P}_\mathrm{m} +b_\mathrm{m} \end{aligned}$$

(5.10)

In terms of $P_\mathrm{m,b}^2$ and the averaged matrix gas pressure, the governing equation can be linearized as follows

$$\begin{aligned} \frac{k_\mathrm{m} }{\mu _\mathrm{g} }\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial P_\mathrm{m,b}^2 }{\partial r}} \right) =\emptyset _\mathrm{m} \left[ {\frac{1}{\bar{P}_\mathrm{m,b} }+\frac{\alpha P_\mathrm{L} }{\beta \emptyset _\mathrm{m} \bar{P}_\mathrm{m,b} (\bar{P}_\mathrm{m,b} -b_\mathrm{m} +P_\mathrm{L})^{2}}} \right] \frac{\partial P_\mathrm{m,b}^2 }{\partial t} \end{aligned}$$

(5.11)

or,

$$\begin{aligned} \frac{\partial P_\mathrm{m,b}^2 }{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{k_\mathrm{m} }{\emptyset _\mathrm{m} \mu _\mathrm{g} C_\mathrm{t} }\frac{\partial P_\mathrm{m,b}^2 }{\partial r}} \right) \end{aligned}$$

(5.12)

$\hbox {C}_\mathrm{t}$ is a total gas compressibility, which includes the gas sorption term and the gas compressibility.

$$\begin{aligned} C_\mathrm{t} =\frac{1}{\bar{P}_\mathrm{m,b} }+\frac{\alpha P_\mathrm{L} }{\beta \emptyset _\mathrm{m} \bar{P}_\mathrm{m,b} (\bar{P}_\mathrm{m,b} -b+P_\mathrm{L} )^{2}} \end{aligned}$$

(5.13)

The initial matrix pressure is $P_i$; the boundary pressure is constant at $P_\mathrm{f}$. The analytical solution can be expressed as follows (Crank 1975; Lim and Aziz 1995) in term of square pressure,

$$\begin{aligned} \frac{\bar{P}_\mathrm{m,b}^2 -P_i^2 }{P_\mathrm{f,b}^2 -P_i^2 }=1-\sum _{n=1}^\infty \frac{4}{R^{2}\propto _n^2 }\mathrm{Exp}\left[ {\frac{-\propto _n^2 k_\mathrm{m} t}{\emptyset _\mathrm{m} \mu C_\mathrm{t} }} \right] \end{aligned}$$

(5.14)

where, $\propto _{n}$ is the root of

$$\begin{aligned} J_0 (R\cdot \propto _n )=0 \end{aligned}$$

(5.15)

$J_0$ is the Bessel $J$ function of the first kind of order zero.

1.2 Shape Factor Derivation for Gas Flow in a Two-Dimensional System with Lim and Aziz Assumptions (1995)

Lim and Aziz (1995) suggested only using the first term of the infinite summation series to make an approximate solution. In the double porosity model, mass out of the matrix equals mass flowing into the fracture, or vice versa. As the square system for example, the matrix–fracture transfer rate can be expressed relating to the accumulation in the matrix as follows,

$$\begin{aligned} q=\rho \emptyset c_\mathrm{t} \frac{\partial \bar{{P}}_\mathrm{m,b} }{\partial t}V_\mathrm{m} \end{aligned}$$

(5.16)

Use (5.14) and (5.16), leading to,

$$\begin{aligned} q=V_\mathrm{m} \frac{5.78}{R^{2}}\frac{k_\mathrm{m} }{2\mu }\beta (P_\mathrm{m,b}^2 -P_\mathrm{f}^2 ) \end{aligned}$$

(5.17)

Darcy’s law suggests the flow rate from the matrix to the fracture is proportional to the pressure gradient at the matrix–fracture interface,

$$\begin{aligned} q=\rho \frac{k_\mathrm{m} }{\mu }A\frac{\partial \bar{{P}}_\mathrm{m,b} }{\partial x}|_{x=0} \end{aligned}$$

(5.18)

Leads to the matrix–fracture transfer rate as an equation with a shape factor,

$$\begin{aligned} q=\sigma \frac{k_\mathrm{m} }{2\mu }\beta (P_\mathrm{m,b}^2 -P_\mathrm{f}^2 )V_\mathrm{m} \end{aligned}$$

(5.19)

The volume is the basis of equating the square and the cylindrical geometries, leading to $R=0.56L$. $L$ is the fracture dimension of the square system; R is the radius of the equivalent cylindrical system. Therefore, the shape factor can be easily determined

$$\begin{aligned} \sigma =\frac{18.17}{L^{2}} \end{aligned}$$

(5.20)

1.3 Analytical Solutions and Shape Factor for Gas Flow in a Three-Dimensional System

Pressure diffusion in a cubical system surrounded by three sets of fractures can be approximated by that of a spherical system. Follow the same steps above, leading to the pressure diffusion equation in a spherical geometry,

$$\begin{aligned} \frac{\partial ^{2}\partial _{P_\mathrm{m,b} }^2 }{\partial r^{2}}+\frac{2}{r}\frac{\partial P_\mathrm{m,b}^2 }{\partial r}=C_\mathrm{t} \frac{\partial P_\mathrm{m,b}^2 }{\partial t} \end{aligned}$$

(5.21)

The analytical solution gives as follows (Crank 1975; Zimmerman et al. 1992; Lim and Aziz 1995),

$$\begin{aligned} \frac{\bar{{P}}_\mathrm{m,b}^2 -P_i^2 }{P_\mathrm{f,b}^2 -P_i^2 }=1-\frac{6}{\pi ^{2}}\mathop \sum \limits _{n=1}^\infty \frac{1}{n^{2}}Exp\left[ {\frac{-n^{2}\pi ^{2}k_\mathrm{m} t}{\emptyset _\mathrm{m} \mu C_\mathrm{t} R^{2}}} \right] \end{aligned}$$

(5.22)

The equivalent radius equals to 0.62L.With the same assumption and treatment on the analytical solution, the shape factor for a cubical matrix with three sets of fractures can be written as follows,

$$\begin{aligned} \sigma =\frac{25.67}{L^{2}} \end{aligned}$$

(5.23)