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).


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