Hysteresis in Filtration through Porous Media

Abstract

We study an evolution problem for filtration through porous media, accounting for hysteresis in the saturation versus pressure constitutive relation. We provide a weak formulation of the problem, assuming that the memory effect in the constitutive relation consists not only of a rate-independent component but also of a rate-dependent one. We prove an existence result, which also applies to the case where the hysteresis operator is of Preisach-type.

Solvability of an Unsaturated Porous Media Flow Problem with Thermomechanical Interaction

“…hold almost everywhere, where · * is a seminorm in R We assume the heat conductivity κ(θ) depending on θ , and as in [5], we obtain the system of momentum balance (1.4), mass balance (1.5), and energy balance equations (1.6) in the form 6) where c 0 > 0 is a constant specific heat, ρ S , ρ L are the mass densities of the solid and liquid, respectively, B is a positive definite viscosity matrix, β ∈ R is the relative thermal expansion coefficient, and g is a given volume force (gravity, e.g. ).…”

“…The boundary ∂Z of Z is the yield surface. The time evolution of ε p is governed by the flow rule ε 5) which implies that ε 6) where M Z * is the Minkowski functional of the polar set Z * to Z . The physical interpretation of (2.5) is the maximal dissipation principle.…”

“…A lot of works have been devoted to this phenomenon, see, e. g., [1,2,3,4,10,11,13]. Mathematical analysis of various mechanical porous media models with capillary hysteresis and without temperature effects has been carried out in [6,7,22,23]. A PDE system for elastoplastic porous media flow with thermal interaction was derived in [5], but the existence of solutions was only proved for the isothermal case.…”

Hysteresis models and gravity fingering in porous media

“…For other numerical approaches and detailed numerical analysis we refer to [12] and the references therein. For the inverted hysteresis relation (1.6), we use a regularized function Ψ δ γ,τ : R → R, 1) where δ > 0 is a regularizing parameter. In all numerical examples, we use the permeabilities…”

“…We investigate here the extended porous media model with static and dynamic hysteresis as it has been introduced in [3] and mention that another hysteresis model has been studied in [1].…”