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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
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