Equations of liquid filtration in double porosity media as a reiterated homogenization of Stokes equations

An exact double porosity model for the liquid filtration in an absolutely rigid body is derived from homogenization theory. The governing equations of fluid dynamics on the microscopic level consist of the stationary Stokes system for a slightly compressible viscous fluid filling voids in a solid skeleton. In turn, this domain (voids) is a union of two independent periodic systems of cracks (fissures) and pores. We suppose that the dimensionless size $\delta$ of pores depends on the dimensionless size $\varepsilon$ of cracks: $\delta=\varepsilon^r$ with $r>1$. As a result we derive the usual Darcy equations of filtration for the liquid in cracks, while the liquid in pores is blocked and unmoved.