Multi-scale Asymptotic Analysis of Gas Transport in Shale Matrix

Abstract

Organic-rich resource shales play an important role in global natural gas production. However, many uncertainties exist in an engineering analysis of gas transport and production such as the reservoir-scale flow simulation, history-matching, and optimization. In this work, we introduce a new set of governing equations to describe the characteristic features of porous structures of the organic-rich resource shale. We apply multi-scale analysis to mass balance equations, the equation of state (for free gas), and an adsorption isotherm. Using the macroscopic model, we study gas transport in shales, consisting of nanoporous organic material (kerogen) and the inorganic material. We conclude that both gas in-place and gas production rate depend on the amount of kerogen in the shale matrix. Adsorbed-phase transport by the organic pore walls is responsible for the increase in production rate. We investigate both Henry and Langmuir adsorption as well as different values of length scale ratio and diffusion coefficients.

Appendix: The Mathematical Procedure of the Multiscale Analysis

1. We shall begin with the case $\alpha =1$. The other cases can be studied analogously. We will substitute expansion (10) into Eq. (11) and equate the terms with the same power of $\varepsilon$:

at $\varepsilon ^{-2}$ :

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) =0, \end{aligned}$$

(34)

where $b_{ij}^2$ is defined by (21) (the superscript “2” refers to Case 2).

at $\varepsilon ^{-1}$ :

$$\begin{aligned}&\frac{\partial }{\partial x_i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) +\frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) \nonumber \\&\quad +\,\frac{\partial }{\partial \xi _i }\left( {\phi \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) =0. \end{aligned}$$

(35)

at $\varepsilon ^{0}$ :

$$\begin{aligned} a\left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial t}&= \frac{\partial }{\partial x_i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) \nonumber \\&\quad +\frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial C^{\left( 1 \right) } }{\partial x_j } +\frac{\partial C^{\left( 2 \right) } }{\partial \xi _j }} \right) } \right) \nonumber \\&\quad +\frac{\partial }{\partial x_i }\left( {\phi \, \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) \nonumber \\&\quad +\frac{\partial }{\partial \xi _i }\left( {\phi \, \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) \nonumber \\&\quad +\frac{\partial }{\partial \xi _i }\left( {\phi \, \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\mu }RTC^{\left( 1 \right) } \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) , \end{aligned}$$

(36)

where $a$ is defined by Eq. (12). All equations are equipped with periodic boundary conditions.

It follows that the first term $C^{(0)}$ in a serial expansion for free gas concentration (10) does not depend on the variable $\xi$. This follows from the fact, that $b_{ij}^2$ are components of a positive definite tensor and the gas concentration is a $\xi$–periodic function. Thus,

$$\begin{aligned} C^{\left( 0 \right) } =C^{\left( 0 \right) } \left( {x,t} \right) . \end{aligned}$$

(37)

Taking into account (37), from Eq. (35) we have

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial C^{\left( 0 \right) } \left( {x,t} \right) }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } \left( {x,\xi ,t} \right) }{\partial \xi _j }} \right) } \right) =0. \end{aligned}$$

(38)

This last equation has a $\xi$-periodic solution $C^{(1)}$, which can be represented as

$$\begin{aligned} C^{\left( 1 \right) } \left( {x,\xi ,t} \right) =N_j^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } \left( {x,t} \right) }{\partial x_j }+Q(x,t), \end{aligned}$$

(39)

where $N_j^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right)$ are $\xi$-periodic functions, so that $\left\langle {N_j^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) } \right\rangle _\xi =0$ ($j=1,2,3$), and $Q\left( {x,t} \right)$ is an arbitrary function independent of $\xi$.

We substitute (39) into expression (38) and get

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{lj} } \right) \frac{\partial C^{\left( 0 \right) } \left( {x,t} \right) }{\partial x_l }} \right) =0. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{lj} } \right) } \right) =0,\quad l=1,2,3. \end{aligned}$$

Taking (37) into account, from (36) we obtain

$$\begin{aligned}&a\left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial t}=\frac{\partial }{\partial x_i }\left( b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( \frac{\partial C^{\left( 0 \right) } }{\partial x_j} +\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j } \right) \right) \nonumber \\&+\frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial C^{\left( 1 \right) } }{\partial x_j }+\frac{\partial C^{\left( 2 \right) } }{\partial \xi _j }} \right) } \right) \nonumber \\&+\frac{\partial }{\partial \xi _i }\left( {\phi \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) . \end{aligned}$$

(40)

We then average Eq. (40) over a periodic cell. Due to $\xi$-periodicity of functions $C^{{\left( l \right) }} \left( {x,\xi ,t} \right) (l > 0), N_j^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right)$ and $\frac{\phi \left( {x,\xi } \right) }{\mu }K_{ij} \left( {x,\xi } \right)$, we will get the macroscopic equation in the following form:

$$\begin{aligned} a^{*}\left( {x,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial t}=\frac{\partial }{\partial x_i }\left( {d^{*,2}_{ij} \left( {x,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial x_j }} \right) , i,j=1,2,3, \end{aligned}$$

(19)

where the effective coefficient $a^{*}$ can be defined from relationship (15). Effective coefficients $d^{*,2}_{ij}$ can be computed by averaging the solution of the cell problem (20):

$$\begin{aligned} d^{*,2}_{ij} \left( {x,C^{\left( 0 \right) }} \right) =\left\langle {b_{ij}^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^2 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{ij} } \right) } \right\rangle _\xi . \end{aligned}$$

(41)

2. In a similar way, we can derive the macroscopic Eq.  (14) and cell problem (17) for the case of $\alpha =0$. We again substitute expansion (10) into Eq.  (11) and equate the terms with the same power of $\varepsilon$:

at $\varepsilon ^{-2}$ :

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) =0, \end{aligned}$$

(42)

where $b_{ij}^1$ is defined by (18) (the superscript “1” refers to Case 1).

Just as it was done above for the case $\alpha =1$, we can show that the first term $C^{(0)}$ in serial expansion (10) does not depend on the variable $\xi$, and relationship (37) remains true. Taking (37) into account, we have

at $\varepsilon ^{-1}$ :

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial C^{\left( 0 \right) } \left( {x,t} \right) }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } \left( {x,\xi ,t} \right) }{\partial \xi _j }} \right) } \right) =0, \end{aligned}$$

(43)

at $\varepsilon ^{0}$ :

(44)

where $a$ is defined by Eq. (12).

Equation (43) has a $\xi$-periodic solution $C^{(1)}$, which can be represented as

$$\begin{aligned} C^{\left( 1 \right) } \left( {x,\xi ,t} \right) =N_j^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } \left( {x,t} \right) }{\partial x_j }+Q(x,t), \end{aligned}$$

(45)

where $N_j^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right)$ are $\xi$-periodic functions, so that $\left\langle {N_j^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) } \right\rangle _\xi =0$ ($j=1,2,3$), and $Q\left( {x,t} \right)$ is an arbitrary function independent of $\xi$.

We substitute (45) into expression (43) and get the cell problem (17):

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{lj} } \right) } \right) =0,\quad l=1,2,3. \end{aligned}$$

Note that in this case, both filtration and diffusive terms will appear in the cell problem. They will enter into the equation through the relationship (18) for $b_{ij}^1$.

We then average Eq. (44) over a periodic cell. Due to $\xi$-periodicity of functions $C^{{\left( l \right) }} \left( {x,\xi ,t} \right) (l > 0), N_j^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right)$ and $\frac{\phi \left( {x,\xi } \right) }{\mu }K_{ij} \left( {x,\xi } \right)$, we will get the macroscopic equation in the following form:

$$\begin{aligned} a^{*}\left( {x,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial t}=\frac{\partial }{\partial x_i }\left( {d^{*,1}_{ij} \left( {x,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial x_j }} \right) , i,j=1,2,3, \end{aligned}$$

(14)

where the effective coefficient $a^{*}$ can be defined from relationship (15). Effective coefficients $d^{*,1}_{ij}$ can be computed by averaging the solution of the cell problem (17):

$$\begin{aligned} d^{*,1}_{ij} \left( {x,C^{\left( 0 \right) }} \right) =\left\langle {b_{ij}^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{ij} } \right) } \right\rangle _\xi . \end{aligned}$$

(16)

3. With a view to derive macroscopic equation and cell problem for the case of $\alpha =-1$, we repeat the procedure of substituting the expansion (10) into Eq. (11) and making equal the terms with the same power of $\varepsilon$. We obtain the hierarchy of equations, and the first one looks like:

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {\phi \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \frac{\partial C^{\left( 0 \right) } }{\partial \xi _j }} \right) =0. \end{aligned}$$

(46)

As it previously was done above for cases $\alpha =0$ and $\alpha =1$, we can show that $C^{(0)}$ does not depend on the variable $\xi$, and relationship (37) holds. Taking (37) into account, the next couple of equations in our hierarchy can be written as

$$\begin{aligned}&\frac{\partial }{\partial \xi _i }\left( {\phi \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) =0.\end{aligned}$$

(47)

$$\begin{aligned}&\frac{\partial }{\partial \xi _i }\left( {b_{ij}^2 \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) +\frac{\partial }{\partial x_i }\left( {\phi \left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RTC^{\left( 0 \right) } \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }+\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) } \right) \nonumber \\&\quad +\frac{\partial }{\partial \xi _i }\left( {\phi \!\left( {x,\xi } \right) \frac{K_{ij} \left( {x,\xi } \right) }{\mu }RT\!\left( {C^{\left( 1 \right) } \left( {\frac{\partial C^{\left( 0 \right) } }{\partial x_j }\!+\!\frac{\partial C^{\left( 1 \right) } }{\partial \xi _j }} \right) \!+\!C^{\left( 0 \right) } \left( {\frac{\partial C^{\left( 1 \right) } }{\partial x_j }\!+\!\frac{\partial C^{\left( 2 \right) } }{\partial \xi _j }} \right) } \right) } \right) \!=\!0.\nonumber \\ \end{aligned}$$

(48)

Equation (47) has a $\xi$-periodic solution $C^{(1)}$, which can be represented as

$$\begin{aligned} C^{\left( 1 \right) } \left( {x,\xi ,t} \right) =N_j^3 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } \left( {x,t} \right) }{\partial x_j }+Q(x,t), \end{aligned}$$

(49)

where $N_j^3 \left( {x,\xi ,C^{\left( 0 \right) }} \right)$ are $\xi$-periodic functions, so that $\left\langle {N_j^3 \left( {x,\xi ,C^{\left( 0 \right) }} \right) } \right\rangle _\xi =0$ ($j=1,2,3$), and $Q\left( {x,t} \right)$ is an arbitrary function independent of $\xi$.

We substitute (49) into expression (47) and get the cell problem (24):

$$\begin{aligned} \frac{\partial }{\partial \xi _i }\left( {b_{ij}^3 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^3 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{lj} } \right) } \right) =0,\quad l=1,2,3. \end{aligned}$$

Note that in this case diffusion term does not appear, and filtration term is the only one that enters into relationship (25) for $b_{ij}^3$ and into the cell problem.

We can obtain the homogenized macroscopic Eq. (22) if we substitute (49) into Eq. (48) and average it over the periodic cell. Due to $\xi$-periodicity of functions $C^{{\left( l \right) }} \left( {x,\xi ,t} \right) (l > 0), N_j^3 \left( {x,\xi ,C^{\left( 0 \right) }} \right)$ and $\frac{\phi \left( {x,\xi } \right) }{\mu }K_{ij} \left( {x,\xi } \right)$, we will get the macroscopic equation in the form:

$$\begin{aligned} \frac{\partial }{\partial x_i }\left( {d^{*,3}_{ij} \left( {x,C^{\left( 0 \right) }} \right) \frac{\partial C^{\left( 0 \right) } }{\partial x_j }} \right) , i,j=1,2,3, \end{aligned}$$

(22)

Where effective coefficients $d^{*,3}_{ij}$ can be computed by averaging the solution of the cell problem (24) in accordance with the following expression

$$\begin{aligned} d^{*,1}_{ij} \left( {x,C^{\left( 0 \right) }} \right) =\left\langle {b_{ij}^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) \left( {\frac{\partial N_l^1 \left( {x,\xi ,C^{\left( 0 \right) }} \right) }{\partial \xi _j }+\delta _{ij} } \right) } \right\rangle _\xi \end{aligned}$$

(23)

Note that in this case, homogenized equation is a steady state. That is because of the difference of time scales of diffusion and filtration processes. The case of $\alpha =-1$ corresponds to a large value of the Péclet number. That means that filtration becomes the crucial mechanism of transport, and, due to high permeability of the medium, it goes so rapidly that solution attains the steady state. However, we need to mention that normally permeability of shales is relatively low ($10^{-12}--10^{-9}\,\hbox {D}$). So, the case of $\alpha =-1$ is more interesting from methodological than from a practical point of view.