  Double porosity models for the liquid filtration in a naturally fractured
reservoir is derived from the homogenization theory. The governing equations on
the microscopic level consist of the stationary Stokes system for an
incompressible viscous fluid, occupying a crack-pore space (liquid domain), and
stationary Lame equations for an incompressible 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 cracks (fissures) and pores, and that the dimensionless
size $\delta$ of pores depends on the dimensionless size $\varepsilon$ of
cracks: $\delta=\varepsilon^{r}$ with $r>1$. The rigorous justification is
fulfilled for homogenization procedure as the dimensionless size of the cracks
tends to zero, while the solid body is geometrically periodic. As the result we
derive the well-known Biot -- Terzaghi system of liquid filtration in
poroelastic media, which consists of the usual Darcy law for the liquid in
cracks coupled with anisotropic Lame's equation for the common displacements in
the solid skeleton and in the liquid in pores and a continuity equation for the
velocity of a mixture. The proofs are based on the method of reiterated
homogenization, suggested by G. Allaire and M. Briane. As a consequence of the
main result we derive the double porosity model for the filtration of the
incompressible liquid in an absolutely rigid body.
