Energetic formulation of large‐deformation poroelasticity

Abstract

The modeling of coupled fluid transport and deformation in a porous medium is essential to predict the various geomechanical process such as CO2 sequestration, hydraulic fracturing, and so on. Current applications of interest, for instance, that include fracturing or damage of the solid phase, require a nonlinear description of the large deformations that can occur. This paper presents a variational energy-based continuum mechanics framework to model large-deformation poroelasticity. The approach begins from the total free energy density that is additively composed of the free energy of the components. A variational procedure then provides the balance of momentum, fluid transport balance, and pressure relations. A numerical approach based on finite elements is applied to analyze the behavior of saturated and unsaturated porous media using a nonlinear constitutive model for the solid skeleton. Examples studied include the Terzaghi and Mandel problems; a gas-liquid phasechanging fluid; multiple immiscible gases; and unsaturated systems where we model injection of fluid into soil. The proposed variational approach can potentially have advantages for numerical methods as well as for combining with data-driven models in a Bayesian framework.

High-dimensional nonlinear Bayesian inference of poroelastic fields from pressure data

“…In this section, we present the forward model, based on a continuum energetic formulation of poroelasticity [49], to predict the water pore pressure p in the soil as a function of the permeability κ and accounting for fluid transport in a deformable medium. Relating this to the notation of Section 2, the observation i is a noisy pressure measure y i = p obs i , and the uncertain field is θ ( x ) = − log e κ ( x ) , where κ has the units m 2 .…”

Deformation Decomposition Versus Energy Decomposition for Chemo- and Poro-Mechanics

“…where J = det F . The referential volume fraction of the solid phase is α, and we assume a single fluid phase; for the case with multiple fluids with the possibility of evolving volume fractions, we refer to [2] and references therein. The form of the fluid contribution JW f J −1 ρ 0 is motivated by the requirement that the energy density W f of a simple fluid depends only on the density in the deformed state, i.e., ρ = J −1 ρ 0 , when we consider the isothermal setting.…”

“…Further, the leading factor of J accounts for the fact that W f is the energy per unit deformed volume, whereas the hyperelastic energy density W is per unit reference volume. An important assumption here is that of affine deformation; i.e., both the solid skeleton and the fluid volume deform under F affinely but this can be relaxed [2].…”

“…There is significant current interest in modeling problems of fluid transport in porous media as well as fluid phase transport in solid materials, i.e., poro-and chemo-mechanics. The motivations range from modeling hydrogels [1], to transport in geological structures [2], to hydrogen embrittlement of metals [3], among various other applications. An approach that has been proposed in the literature to model poro-and chemo-mechanics is to decompose the deformation gradient into an elastic part -that causes stress -and an inelastic part -that accounts for the shape change due to fluid transport; this is the "multiplicative decomposition".…”

Multiscale Mechanical Performance of Wood: From Nano- to Macro-Scale across Structure Hierarchy and Size Effects

“…The softening mechanisms turning on with the increase in both internal and external characteristic size of the system require additional study of interrelations between the multilevel nano-/micro-structure of wood and its physico-mechanical properties. However, there are reasons to suppose that micromechanics of thin filaments, walls, partitions, as well as the macromechanical behavior of wood may have a lot in common with those in other highly porous materials [ 148 , 149 , 150 , 151 , 152 , 153 , 154 ]. Therefore, the general approaches and models developed for the analysis of the latter can be applied to wood as well.…”