The application of the reiterated homogenization method of differential equations to the theory of filtration of compressible liquids in compressible crack-pore media. Part I: The microscopic description

Double porosity models for the liquid filtration in a naturally fructured reservoir are derived from homogenization theory. The basic mathematical model on the microscopic level consists of the stationary Stokes equations for a compressible viscous fluid, occupyinga crack-pore space (liquid domain), and stationary Lame equations for the elastic solid skeleton, coupled with corresponding boundary conditions on the common boundary “solid skeleton–liquid domain”. We suppose, that the liquid domain is a union of two independent systems of crack and pores and that the dimensionless size of pores $\delta$ depends on the dimensionless size $\varepsilon$ of cracks: $\delta=\varepsilon^r$ with $r>1$. Under supposition that the solid skeleton has a periodic structure we suggest the regorous derivation of homogenized equations, based on the G. Allaire and M. Briane method of multiscale convergence as the small parameter tends to zero. For the different combinations of incoming parameters we derive the well-known Biot–Terzaghi system or the anisotropic Lame equations for the mixture of the solid skeleton and liquin in cracks and pores. As a consequence of these results we derive the equations of a filtration the compressible liquid in absolutely rigid crack-pore media, which is the usual Darcy system of filtration. The present publication is the first part of above mentioned investigation, where we consider the microscopic model and derive all nessesary a'priori estimates.