Nguetseng’s two-scale convergence method for filtration and seismic acoustic problems in elastic porous media

Abstract

Abstract. A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results, we derive Biot's equations of poroelasticity, equations of viscoelasticity, or decoupled system consisting of non-isotropic Lamé's equations and Darcy's system of filtration, depending on ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.

The Muskat problem for a viscoelastic filtration

“…8) where n is the normal vector to the boundary S, and the initial conditions ρ f (x, 0) = ρ 0 f (x), µ(x, 0) = µ 0 (x), x ∈ Ω, (0.9) with discontinuous initial data:…”

“…for the displacement w and the pressure p of continuous medium ( [2], [8], [11], [13]). The microscopic system (0.10), (0.11) describes the joint motion of the viscous liquid in a pore space and of an elastic solid skeleton and is understood in the sense of distributions.…”

“…After a homogenization procedure we arrive at equations of viscoelastic filtration ( [2], [8], [11]), which are the third level approximation of (0.11), (0.12). It is clear that all different asymptotic models of the system (0.11), (0.12) describe the same physical process, but with different degrees of approximation.…”

“…In (0.16), (0.17), m is the porosity of the pore space, ρ = mρ f − (1 − m)ρ s is the density of the mixture of the solid skeleton and the liquid in pores, A s 0 is a symmetric strictly positive definite constant fourth-rank tensor (for the definition of A s 0 and B (f ) see [8]) and the criterion λ 0 is strictly positive and finite.…”

Homogenization of Biomechanical Models for Plant Tissues

“…Formal asymptotic expansion was undertaken by the authors of [5,13,23,42] to derive Biot equations from microscopic description of elastic deformations of a solid matrix and fluid flow in porous space. The rigorous homogenization of the coupled system of equations of linear elasticity for a solid matrix combined with the Stokes or Navier-Stokes equations for the fluid part was conducted in [17,19,24,32] by using the two-scale convergence method. Depending on the ratios between the physical parameters, different macroscopic equations were obtained, e.g., Biot's equations of poroelasticity, the system consisting of the anisotropic Lamé equations for the solid component, and the acoustic equations for the fluid component, the equations of viscoelasticity.…”