Double Porosity Models for Absolutely Rigid Body via Reiterated Homogenization

TL;DR

This paper derives double porosity models for liquid filtration in rigid bodies using reiterated homogenization, revealing different behaviors for long- and short-time processes through rigorous mathematical analysis.

Contribution

It introduces a homogenization approach for double porosity models in rigid bodies, accounting for pore-crack size relations and deriving effective equations for various time scales.

Findings

Long-time filtration follows Darcy's law with blocked pores.

Short-time dynamics are described by acoustic equations with two velocities.

Rigorous justification of homogenization as crack size tends to zero.

Abstract

Double porosity models for the liquid filtration in an absolutely rigid body is derived from homogenization theory. The governing equations of the fluid dynamics on the microscopic level consist of the Stokes system for a slightly compressible viscous fluid, occupying a crack -- pore space. In turn, this domain is a union of two independent 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 $r>1$. 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 a result, for the long-time process we derive the usual Darcy equations of filtration for the liquid in cracks, while the liquid in pores is blocked and unmoved. For the short-time processes the homogenized system consists of acoustic equations, describing a two-velocity continuum with three independent parameters: the liquid velocity in pores, the liquid velocity in cracks and the common pressure. The proofs are based on the method of reiterated homogenization, suggested by G. Allaire and M. Briane.