Fuzzy Structure Element Method for Solving Fuzzy Trilinear Seepage Model of Shale Gas Reservoir

Abstract

The seepage of shale gas reservoir is more complicated than that of the conventional energy because of its unique reservoir forming characteristics and physical properties. Accordingly, to achieve the large-scale exploitation of shale gas reservoir, the internal seepage characteristics of the reservoir should be clarified. Shale reservoirs are often accompanied by extreme heterogeneity and responsible variability. This phenomenon reveals that it is not reasonable to idealize the reservoir permeability as a fixed value. However, the conventional percolation model often idealizes the reservoir permeability to a specific amount. This will obviously lead to a reduction in the accuracy of the model. In the present study, the concept of fuzzy permeability was proposed. In other words, a fuzzy number was used to describe the reservoir permeability. Subsequently, the trilinear fuzzy seepage model was built and solved using the fuzzy structure element. To solve the practical application problem, a single numerical solution corresponding to the fuzzy solution set was found using the technique of defuzzification by the obtained fuzzy solution set. Finally, the fuzzy seepage model and the conventional seepage model were compared and analyzed by the numerical simulation.

Appendices

A Fuzzy Structural Element

1.1 A.1 Transformation of Fuzzy Structural Element

Definition A.1

[24] If the membership function of the fuzzy structural element is symmetrical on the axis of the ordinate $x=0$, it is called the symmetric fuzzy structural element.

Definition A.2

[24] If the membership function of a fuzzy structural element has a $E(x)>0$ on the interval $(-1,1)$. And in $(-1,0)$, it is a strictly single increase continuous function, and in the interval (0, 1) it is a strict single drop continuous function. E is called a regular fuzzy structural element.

Theorem A.3

[23] LetAbe a fuzzy subset ofX, and remember that$A\in F(x)$has a membership function. Then

$$\begin{aligned} A=\bigcup _{x\in X}A(x)*\{x\}. \end{aligned}$$

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where$\lambda *{x}$is called the point state fuzzy set onX, and its membership function is

$$\begin{aligned} \begin{aligned}(y)= {\left\{ \begin{array}{ll} \lambda,\quad &{}y=x,\\ 0,\quad &{}y\ne x.\\ \end{array}\right. }\quad \end{aligned} \end{aligned}$$

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Theorem A.4

[23] Considering the common mapping$f:X\rightarrow Y$, the mapping ofF(X) toF(Y) can be induced byf

$$\begin{aligned} f:F(X)\rightarrow F(Y) A\rightarrow f(A)=\bigcup _{x\in X}A(x)*\{f(x)\}=\bigcup _{x\in X}\lambda *f(A_\lambda ). \end{aligned}$$

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The membership function of the fuzzy set$f(A)\in F(Y)$is

$$\begin{aligned}{}[f(A)](y)=\vee _{y=f(x)}A(x). \end{aligned}$$

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Theorem A.5

[23] Set the fuzzy number$A=a+bE$, a, bas the finite real number, $b>0$. if$\forall \lambda \in [0,1]$, $E_\lambda =[e_\lambda ^L,e_\lambda ^U]$, then the intercepting set ofAis

$$\begin{aligned} A_\lambda =\{x\mid A(x)\ge \lambda \}=[a+be_\lambda ^L,a+be_\lambda ^U]. \end{aligned}$$

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1.2 A.2 Structural Element Representation of Fuzzy Numbers

Theorem A.6

[24] For a given regular fuzzy structure elementEand any finite fuzzy numberA, there is always a monotone bounded functionfon$[-1,1]$that makes$A=f (E)$.

Theorem A.7

[24] Letfbe a monotone bounded function on$[-1,1]$, Eis a given fuzzy structural element onR, and a fuzzy number$A=f(E)$. $\forall \lambda \in [0,1]$, the cut set ofEis$E_\lambda =[e_\lambda ^L,e_\lambda ^U]$. in which$e_\lambda ^L\in [-1,0]$, $e_\lambda ^U\in [0,1]$. Iffis a single increment function on$[-1,1]$, the cut set of the fuzzy numberAis closed interval on theR

$$\begin{aligned} A_\lambda =[f(E)]_\lambda =f(E_\lambda )=f[e_\lambda ^L,e_\lambda ^U] =[f(e_\lambda ^L),f(e_\lambda ^U)]. \end{aligned}$$

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Iffis a single drop function on$[-1,1]$, the cut set of the fuzzy numberAis closed interval on theR

$$\begin{aligned} A_\lambda =[f(e_\lambda ^U),f(e_\lambda ^L)]. \end{aligned}$$

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Theorem A.8

[23] LetEbe a fuzzy structural element, andfandgare two identical monotone functions on$[-1,1]$. There is also a fuzzy number$A=f(E)$, $B=g(E)$, then

$$\begin{aligned} A+B =f(E)+g(E)=(f+g)(E). \end{aligned}$$

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The membership function is

$$\begin{aligned} (A+B)(x) =E((f+g)^{-1}(x)). \end{aligned}$$

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Definition A.9

[24] Let A be a finite fuzzy number, if there is a fuzzy structural element E and a finite real number a, r, so that $A=a+rE$(where r>0). It is called A is a fuzzy number generated linearly by the fuzzy structural element E.

1.3 A.3 Structural Element Representation of Fuzzy Valued Function

Let N(R) be a whole of fuzzy numbers on R, X and Y are two real numbers, $D\subseteq X$, and $\widetilde{f}$ is a mapping from D to N(Y). If there is a unique fuzzy number $\widetilde{y}\in N(Y)$ for $\forall x\in D$, remember $\widetilde{y}=\widetilde{f}(x)$, then $\widetilde{f}$ is called the fuzzy value function on D.

Definition A.10

[24] Considering two-dimensional real space $X\times Y$, E is a regular fuzzy structural element on Y. Let g(x, y) be a two element function on $X\times Y$, and for $\forall x\in X$, g(x, y) is a monotone bounded function of y on $[-1,1]$, and $g_x(y)=g(x,y)$. It is known from the principle of fuzzy expansion that $g_x(E)=g(x,E)$ is a finite fuzzy number. Call g(x, E) as the fuzzy value function generated by the fuzzy structural element E on the X, remember to $\widetilde{F}_E=g(x,E)$, or $\widetilde{F}(x)$ for short.

Definition A.11

[24] Let us set $g(x,y)=f(x)+\omega (x)E$, where f(x) and $\omega (x)$ are bounded on X, and $\omega (x)$ is not negative. So the function g(x, y) is the monotone bounded function of y on the interval $[-1,1]$, then

$$\begin{aligned} \widetilde{F}(x)=f(x)+\omega (x)E, \end{aligned}$$

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is a fuzzy value function on X, which is called a fuzzy value function generated linearly by the fuzzy structural element E. Its membership function can be represented by the membership function of the fuzzy structuring element as follows:

$$\begin{aligned}{}[\widetilde{F}](y)=+\omega (x)E\left( \frac{y-f(x)}{\omega (x)}\right), \quad \forall y\in Y. \end{aligned}$$

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B Nomenclature

See Table 2.

Table 2 Nomenclature