Poromechanics

Poromechanics is a branch of physics and specifically continuum mechanics that studies the behavior of fluid-saturated porous media. A porous medium or a porous material is a solid (referred to as matrix) permeated by an interconnected network of pores or voids filled with a fluid. In general, the fluid may be composed of liquid or gas phases or both. In the simplest case, both the solid matrix and the pore space occupy two separate, continuously connected domains, such as in a kitchen sponge. Some porous media has a more complex microstructure in which, for example, the pore space is disconnected. Pore space that is unable to exchange fluid with the exterior is termed occluded pore space. Alternatively, in the case of granular porous media, the solid phase may constitute disconnected domains, termed the "grains", which are load-bearing under compression, though can flow when sheared.

Natural substances including rocks, soils, biological tissues including plants, heart, and cancellous bone, and man-made materials such as foams, gels, ceramics, and concrete can be considered as porous media. Porous materials share common coupled processes such as diffusion and consolidation, hydration and swelling, drying and shrinkage, heating and build-up of pore pressure, freezing and spalling, capillarity and cracking. Porous media whose solid matrix is elastic and the fluid is viscous are called poroelastic. The structural properties of a porous medium is characterized by its porosity, pore size and shape, connectivity, and specific surface area. The physical (mechanical, hydraulic, thermal) properties of a porous media are determined by its microstructure as well as the properties of its constituents (solid matrix and fluid). Porous media whose pore space is filled with a single fluid phase, typically a liquid, is considered to be saturated. Porous media whose pore space is only partially fluid is a fluid is known to be unsaturated.

Poromechanics relates the loading of solid and fluid phases within a porous body to the deformation of the solid skeleton and pore space. A representative elementary volume (REV) of a porous medium and the superposition of the domains of the skeleton and connected pores is shown in Fig. 1. In tracking the material deformation, one must be careful to properly apportion sub-volumes that correspond to the solid matrix and pore space. To do this, it is often convenient to introduce a porosity, which measures the fraction of the REV that constitutes pore space. To keep track of the porosity in a deforming material volume, mechanicians consider two descriptions, namely:

The Eulerian and Lagrangian descriptions of porosity are readily related by noting that

where ${\displaystyle J=\det(\mathbf {F} )}$ is the Jacobian of the deformation with ${\displaystyle \mathbf {F} }$ being the deformation gradient. In a small-strain, linearized theory of deformation, the volume ratio is approximated by ${\displaystyle J\simeq (1+\epsilon _{\mathrm {v} })}$, where ${\displaystyle \epsilon _{\mathrm {v} }}$ is the infinitesimal volume strain. Another useful descriptor of the REV's pore space is the void ratio, which compares the current volume of the pores to the current volume of the solid matrix. As such, the void ratio takes definition in an Eulerian frame of reference and is calculated as

where ${\displaystyle 1-n}$ measures the fraction of the volume occupied by the solid skeleton.

When a material element of a porous medium undergoes a deformation, the porosity changes due to i) the material's observable macroscopic dilation and ii) the volume dilation of the material's solid skeleton. The latter cannot be assess from experiments on the material's bulk structure. The volume of the solid skeleton in an infinitesimal material element, which is denoted by ${\displaystyle \mathrm {d} V_{t}^{\mathrm {s} }}$, is related to the deformed and undeformed total material volumes by

where the definition of the Lagrangian porosity further requires ${\displaystyle 1-\phi _{0}=\mathrm {d} V_{0}^{\mathrm {s} }/\mathrm {d} V_{0}}$. Thus, under the assumption of infinitesimal strain theory, the total volumetric strain of a material element can be separated into strain contributions of the solid matrix and pore space as follows: