## Homogenization of a model of compressible miscible flow in porous media.(English)Zbl 0727.76093

Let $$\Omega \subset R^ 1$$ be a region of the flow of two miscible fluids in a heterogeneous porous medium. The porosity $$\phi$$ and permeability k of the rock are oscillating functions of the spatial variable x and depend on a parameter $$\epsilon >0$$ connected with microstructure. If the functions $$p^{\epsilon}$$, $$q^{\epsilon}$$ and $$u^{\epsilon}$$ stand for pressure, rate of flow and concentration of mass respectively, the flow is governed by the following system of parabolic-hyperbolic type: $$\phi^{\epsilon}(x)\frac{\partial}{\partial t}p^{\epsilon}(t,x)+\frac{\partial}{\partial x}q^{\epsilon}(t,x)=0$$, $$q^{\epsilon}(t,x)=-k^{\epsilon}(x)\frac{\partial}{\partial x}p^{\epsilon}(t,x)$$; $$\phi^{\epsilon}(x)\frac{\partial}{\partial t}u^{\epsilon}(t,x)+q^{\epsilon}(t,x)\frac{\partial}{\partial x}u^{\epsilon}(t,x)=0$$, $$0\leq u^{\epsilon}(t,x)\leq 1$$; $$q^{\epsilon}(t,0)=q_ 1^{\epsilon}(t)$$, $$q^{\epsilon}(t,1)=q_ 2^{\epsilon}(t)$$, $$p^{\epsilon}(x,0)=p_ 0^{\epsilon}(x)$$, $$u^{\epsilon}(t,0)=u_ 1^{\epsilon}(t)$$, $$u^{\epsilon}(0,x)=u_ 0^{\epsilon}(x)$$. Authors assume, that $$\phi^{\epsilon}(x)=\phi (x,x/\epsilon)$$, $$k^{\epsilon}(x)=k(x,x/\epsilon)$$, where the functions $$\phi$$ (x,y) and k(x,y) are 1-periodic with respect to the variable y. They derive the equations obtained asymptotically ($$\epsilon\to 0)$$ for the pressure and concentration (in weak sense).

### MSC:

 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76S05 Flows in porous media; filtration; seepage 76T99 Multiphase and multicomponent flows

### Keywords:

heterogeneous porous medium; microstructure