Fuzzy structural element method for solving fuzzy dual medium seepage model in reservoir

Abstract

This study primarily introduces fuzzy set theory in the reservoirs modeling to enhance the accuracy of the model. The conventional seepage models of dual medium in reservoirs have several limitations. To be specific, it considers the complexity of the model during the modeling process and idealizes certain reservoir parameters to be constant. By adopting fuzzy set theory to study reservoir seepage theory, on the one hand, the seepage model is capable of fully considering the complex and variability of the reservoir; on the other hand, it can avoid parameter errors attributed to laboratory measurements. A numerical example is given in this study to illustrate the accuracy and superiority of the fuzzy seepage model. Studies suggest that fuzzy set theory is capable of effectively addressing the limitations in conventional seepage models.

Appendices

Transformation of fuzzy structural element

Definition 12

(Zhang et al. 2019d) 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 13

(Zhang et al. 2019d) 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 5

(Zhang et al. 2019d) Let A be a fuzzy subset of X 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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$\lambda *{x}$ is called the point state fuzzy set on X, and its membership function is

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

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Theorem 6

(Zhang et al. 2019d) Considering the common mapping $f:X\rightarrow Y$, the mapping of F(X) to F(Y) can be induced by f

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

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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 7

(Zhang et al. 2019d) Set the fuzzy number $A=a+bE$, a, b as the finite real number,$b>0$. If $\forall \lambda \in [0,1]$, $E_\lambda =[e_\lambda ^L,e_\lambda ^U]$, then the intercepting set of A is

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

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Table 2 Nomenclature

Structural element representation of fuzzy numbers

Theorem 8

(Zhang et al. 2019d) For a given regular fuzzy structure element E and any finite fuzzy number A, there is always a monotone bounded function f on $[-1,1]$ that makes $A=f (E)$.

Theorem 9

(Zhang et al. 2019d) Let f be a monotone bounded function on $[-1,1]$, E is a given fuzzy structural element on R, and a fuzzy number $A=f(E)$. $\forall \lambda \in [0,1]$, the cut set of E is $E_\lambda =[e_\lambda ^L,e_\lambda ^U]$. in which $e_\lambda ^L\in [-1,0]$, $e_\lambda ^U\in [0,1]$. If f is a single increment function on $[-1,1]$, the cut set of the fuzzy number A is closed interval on the R

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

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If f is a single drop function on $[-1,1]$, the cut set of the fuzzy number A is closed interval on the R

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

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Theorem 10

(Zhang et al. 2019d) Let E be a fuzzy structural element, and f and g are two identical monotone functions on $[-1,1]$. There are also $A=f(E)$, $B=g(E)$; then,

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

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And the membership function is

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

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Definition 14

(Zhang et al. 2019d) 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.

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 15

(Zhang et al. 2019d) 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. g(x, E) is called 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 16

(Zhang et al. 2019d) 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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Nomenclature

See Table 2.