Multiple Fuzzy Parameters Nonlinear Seepage model for Shale Gas Reservoirs

Abstract

This paper focuses on improving the accuracy of seepage models for shale gas reservoirs on the basis of the fuzzy set theory. First, the fuzzy numbers are used to describe the permeability, formation thickness, initial reservoir pressure, gas viscosity, limiting adsorbed mount, and compression factor of shale gas reservoirs. Second, based on the conventional seepage model, single fuzzy parameter seepage models and the multiple fuzzy parameter seepage model for shale gas reservoirs are set up in turn. Solutions to the fuzzy seepage model are handled by means of the fuzzy structural element method. Finally, with the assistance of the centroid method, we work out a single numerical solution in accordance with the fuzzy solution set. In contrast with the conventional seepage models of shale gas reservoirs, the fuzzy seepage model in this paper makes use of the fuzzy numbers to describe the parameters with uncertainty, which also completely considers the in-homogeneity and complex variability of shale gas reservoirs. A numerical simulation example is presented to show the efficiency and development of the fuzzy seepage model. On the basis of the fuzzy seepage model, the sensitivity analysis of relevant main control parameters is carried out to conduct a further study on relevant seepage laws of shale gas reservoirs.

Fuzzy Formation Thickness Model

The derivative of the Eq. (6) about h is

$$\begin{aligned} \frac{D_2}{2\sqrt{\frac{A_2^2}{k_0^2} +\frac{C_2+D_2h}{k_0}} + E_2}, \end{aligned}$$

(32)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} A_2 = \frac{3\pi \mu aD_k}{16},\\ B_2 = \frac{P_{sc}T\mu Z \pi \left(r_e^2 - r_w^2\right)\left[\rho _c V_m\left(\frac{P_e}{P_e+P_L} - \frac{P}{P+P_L}\right) - \phi _m\right]}{T_{sc}Z_{sc}\rho _{gsc}},\\ C_2 = -2A_2ln\frac{r_e}{r}(P_e -P_w) + 2A_2P_e,\\ D_2 = \frac{B}{2}\left[(r_e^2 - r^2) - \frac{ln\frac{r_e}{r}(r_e^2 - r_w^2)}{ln\frac{r_e}{r_w}}\right],\\ E_2 = P_e^2 - ln\frac{r_e}{r}(P_e^2 - P_w^2). \end{array}\right. }\quad \end{aligned}$$

The value of Eq. (32) is positive. Therefore, the solution of the fuzzy defined differential equation Eq. (11) is piecewise representation, and the solution is Eq. (12). Let us denote the function as follows:

$$\begin{aligned} f_2(h)= \sqrt{\frac{A_2^2}{k_0^2}+\frac{C_2 +D_2h}{k_0} +E_2}-\frac{A_2}{K_0}, \end{aligned}$$

(33)

then

$$\begin{aligned} g_2(x)= \sqrt{\frac{A_2^2}{k_0^2}+\frac{C_2 +D_2(a_2 + b_2. x)}{k_0} +E_2}-\frac{A_2}{K_0}, \end{aligned}$$

where $g_2(x)$ is monotonous and bounded in $[-1,1]$, $f_2(\widetilde{h}) = g_2(E)$. According to the local mapping theorem (see [23]), we obtain

$$\begin{aligned} s_2= g_2^{-1}(x)=-\frac{k_0[E_2-(x+\frac{A_2}{K_0})^2 + \frac{A_2^2}{k_0^2} + \frac{C_2+D_2a_2}{K_0}]}{D_2b_2}, \end{aligned}$$

we have

$$\begin{aligned}\mu _{f_2(\widetilde{h})}(x)=\mu _{g_2(E)}(x)= E(g_2^{-1}(x))\\&\quad ={\left\{ \begin{array}{ll} 1-\frac{k_0[E_2-(x+\frac{A_2}{K_0})^2 + \frac{A_2^2}{k_0^2} + \frac{C_2+D_2a_2}{K_0}]}{D_2b_2}\ &{} x\in [\alpha _1,\alpha _2],\\ 1+\frac{k_0[E_2-(x+\frac{A_2}{K_0})^2 + \frac{A_2^2}{k_0^2} + \frac{C_2+D_2a_2}{K_0}]}{D_2b_2}\ &{} x\in [\alpha _2,\alpha _3],\\ 0\ &{}\text{ other }. \end{array}\right. }\quad \end{aligned}$$

with

$$\begin{aligned} \alpha _1&=\sqrt{\frac{A_2^2}{k_0^2}+\frac{C_2 +D_2(a_2 - b_2 )}{k_0} +E_2}-\frac{A_2}{K_0},\\ \alpha _2&=\sqrt{\frac{A_2^2}{k_0^2}+\frac{C_2 +D_2a_2}{k_0} +E_2}-\frac{A_2}{K_0},\\ \alpha _3&= \sqrt{\frac{A_2^2}{k_0^2}+\frac{C_2 +D_2(a_2 + b_2 )}{k_0} +E_2}-\frac{A_2}{K_0}. \end{aligned}$$

Using the centroid method to defuzzify, we get Eq. (13).