Solvability of an Unsaturated Porous Media Flow Problem with Thermomechanical Interaction

Abstract

A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.

A multicomponent flow model in deformable porous media

“…Such problems have been studied by introducing a continuum mechanical model and inserting different parameters for the two branches of the capillary pressure curve in Albers. 1,2 Another approach to the problem consists in modeling hysteresis by an explicit hysteresis operator and solving the resulting operator-PDE system as in Albers and Krejčí 3,4 and Detmann et al 5 The present article follows the structure of the two latter papers. The intention is to describe processes in porous media containing more than two pore fluids.…”

Unsaturated deformable porous media flow with thermal phase transition

“…A model for fluid flow in partially saturated porous media with thermomechanical interaction was proposed and analyzed in [2,5,6]. Here, we extend the model by including the effects of freezing and melting of the fluid in the pores.…”

“…In, e. g., [1], flow in porous media is described in Eulerian coordinates in order to incorporate, for example, the effects of fast convection. Here, instead, we assume that slow diffusion is dominant, and choose the Lagrangian description as in [2,5,28]. The resulting system of coupled ODEs and PDEs then appears to be a nonlinear extension of the linear model in [26], referred to as a simplified Biot system, to the case when also the occurrence of temperature changes and phase transitions is taken into account.…”

“…The resulting system of coupled ODEs and PDEs then appears to be a nonlinear extension of the linear model in [26], referred to as a simplified Biot system, to the case when also the occurrence of temperature changes and phase transitions is taken into account. In addition to the model studied in [5], we include here the effects of freezing and melting. The idea is the following.…”

“…That is, in addition to the other physical quantities like capillary pressure, displacement, and absolute temperature, we need to consider the evolution of a phase parameter χ representing the relative proportion of water in the H 2 O part and its influence on pressure changes due to the different mass densities of water and ice. Unlike in [2,5,6,27], we do not consider hysteresis in the model. We believe that the mathematical results can be extended to the case of capillary hysteresis as in [2,5,6].…”