Non-linear homogenization of Forchheimer law for porous media with cylindrical and spherical impervious inclusions

Abstract

This paper investigates, with the help of micromechanics, the overall nonlinear filtration law of a porous solid containing impervious inclusions. At the microscopic scale, we assume that the fluid flow in the porous region is described by the Forchheimer law. More specifically, the nonlinear variational homogenization approach is considered to derive closed-form bounds for the macroscopic filtration law, which is based on velocity or pressure gradient-dependent potentials applied to a unit cell with uniform boundary conditions. The approach is developed in the case of a porous solid with impervious cylinders or spheres, for which we derive explicit analytic upper and lower bounds, considering appropriate simple trial fields. The closed-form expressions of the non-linear bounds are provided and compared to FFT solutions for various microstructures.

A Derivation of analytic bounds for the spherical inclusion

1.1 A.1 The lower bound

The values of $\alpha ^{hom}$ and $\gamma ^{hom}$ are given by (introducing the variable change $u=a^3/r^3$):

$$\begin{aligned} \alpha ^{hom}= & {} \frac{\alpha f}{2(1-f)^2}\int _{u=f}^{u=1}\int _{\varphi =0}^{\varphi =\pi }\frac{R(u)}{u^2}\sin (\varphi )dud\varphi \end{aligned}$$

(64)

$$\begin{aligned} \gamma ^{hom}= & {} \frac{\alpha f}{2(1-f)^3}\int _{u=f}^{u=1}\int _{\varphi =0}^{\varphi =\pi }\frac{R(u)^{3/2}}{u^2}\sin (\varphi )dud\varphi \end{aligned}$$

(65)

with:

$$\begin{aligned} R(u)=\frac{1}{4}[u^2+4u+4+3u(u-4)\cos ^2(\varphi )] \end{aligned}$$

(66)

The calculation of $\gamma ^{hom}$ is performed by considering a Taylor expansion of $R(u)^{3/2}$ at $u=0$:

$$\begin{aligned} R(u)^{3/2}\simeq \frac{1}{8}\left[ 1+\frac{3}{2}(1-3\cos ^2(\varphi ))u+\frac{3}{8}(2-3\cos ^2(\varphi )+9\cos ^4(\varphi ))u^2\right] \end{aligned}$$

(67)

By keeping only the first two terms of the Taylor series, we obtain for $\gamma ^{hom}$ the expression given in Eq. (43).

1.2 A.2 The upper bound

Considering the trial field (44) in Eq. (9), we obtain the following dual potential:

$$\begin{aligned} U_D^+({\varvec{J}})=\frac{\alpha ^3 f}{12\gamma ^2}\int _{u=f}^{u=1}\frac{P^*(u)}{u^2}du \end{aligned}$$

(68)

where $P^*(u)$ is defined as:

$$\begin{aligned} P^*(u)=\frac{1}{2}\int _{\varphi =0}^{\varphi =\pi }\left\{ \Big [1+QR(u)^{1/2}\Big ]^{3/2}-\frac{3Q}{2}R(u)^{1/2}-1\right\} \sin (\varphi )d\varphi \end{aligned}$$

(69)

and the coefficient Q is given by:

$$\begin{aligned} Q=\frac{8\gamma J_3}{\alpha ^2(2+f)} \end{aligned}$$

(70)

Exact closed-form expression for $U_D^+({\varvec{J}})$ cannot be derived and thus an approximation is needed. To this end, we use a Taylor series to approximate $P^*(u)$:

$$\begin{aligned} P^*(u)=(1+Q)^{3/2}-\frac{3Q}{2}-1+\frac{9Q}{40}\left[ \frac{3+4Q}{3\sqrt{1+Q}}-1\right] u^2+O(u^4) \end{aligned}$$

(71)

which leads to the following expression for $U_D^+({\varvec{J}})$:

$$\begin{aligned} U_D^+({\varvec{J}})=\frac{\alpha ^3(1-f)}{12\gamma ^2}\left\{ (1+Q)^{3/2}-\frac{3Q}{2}-1+f\frac{9Q}{40} \left[ \frac{3+4Q}{3\sqrt{1+Q}}-1\right] \right\} \end{aligned}$$

(72)