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.
Basic equations Definitions of porosity 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 porosity, , which measures the porosity with respect to the current or deformed configuration. Specifically, if represents an infinitesimal volume in the deformed material body, then the pore volume is calculated from . The Lagrangian porosity, , which measures the porosity with respect to the initial or undeformed configuration. In a Lagrangian description of porosity, the pore volume is measured by , where represents an infinitesimal volume of the material in its undeformed state.
The Eulerian and Lagrangian descriptions of porosity are readily related by noting that
where is the Jacobian of the deformation with being the deformation gradient. In a small-strain, linearized theory of deformation, the volume ratio is approximated by , where 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 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 , is related to the deformed and undeformed total material volumes by
where the definition of the Lagrangian porosity further requires . 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:
where is recognized as the linearized volume strain acting in the solid.
Small-strain linear isotropic poroelasticity Considering a fluid saturated, deformable porous solid, we follow an observation frame that moves together with the solid skeleton but allows pore fluid exchange with the surroundings (i.e., an open system). There is no chemical reaction (i.e., mass exchange) between the solid and the fluid phases. Summoning mass balance, momentum balance, the First and the Second laws of thermodynamics of individual phases, one can arrive at the energy balance and entropy imbalance of the overall mixture. By separately discussing the energy dissipation due to mechanical deformation and mass/heat transport, it is possible to arrive at the following free energy imbalance for the porous skeletonwhere is the Second Piola-Kirchoff stress; is the Green-Lagrangian strain; the pore fluid pressure; the Lagrangian porosity; is temperature; and are the entropy and the elastic stored energy (Helmholtz free energy) of the solid skeleton; is the rate of energy dissipation. Assuming small deformation,Small-deformation assumption for porous media implies: (1) As is the requirement for a general continuum solid, displacement gradients should be small, , where is the displacement field of the solid. (2) To further ensure small changes in the solid and pore volumes, should be small in comparison to the characteristic size of the solid matrix or the grain (in case of a granular material), . This is stated as , and implies small changes in the Lagrangian porosity . elastic solid, and isothermal condition, the previous equation simplifies to: where is the infinitesimal strain; and are the displacement and position vectors, respectively; is Cauchy stress. The constitutive equation for an elastic porous media can thus be generally stated as: Let us specify the following quadratic form of :where is the strain deviator; is the initial porosity of the undeformed porous medium; , , , are material constants. We can immediately obtain Biot's linear isotropic poroelasticity in terms of : or more commonly in incremental form: Comparing with the usual linear elasticity equations, one can identify that and are the bulk and shear modulus of the porous material under drained () conditions; , named the Biot's coefficient, is a new property for porous media that relates the change of porosity to the strain variation under drained condition; is a tangent modulus linking the pressure variation and porosity variation under constant volumetric strain ().
The first equation can be rewritten in such a way that the right-hand side is exactly the same as the linear elasticity equation: Term is called the Biot's effective stress that represents the stress transmitted solely through the solid skeleton. Terzaghi's effective stress which is widely used in soil mechanics can be retrieved by setting .
Although the -based formulation is rooted in the free energy balance of the porous skeleton, it is typically difficult to track porosity changes during experiments, making the model calibration/validation inconvenient. In rock mechanics testing, the usual controlled/monitored variables are stress, strain, pore fluid pressure, and the net flux of pore fluid of the test sample. A better external variable in replace of is therefore the variation of fluid content, , defined as the amount of fluid volume entering the solid frame per unit volume of solid frame. Its increment can be written as where is the fluid mass content; is the fluid density. By replacing with , and considering fluid state equation , Biot's theory can be also written in the following -based form: or where is the fluid tangent bulk modulus; is Biot modulus; is the undrained bulk modulus. Note that both and have been called Biot modulus in different literatures. It is expected that would be dependent on the property of the fluid while is a sole property of the porous skeleton. Indeed, in-depth micromechanical analysis permits one to quantitatively connect the macroscopic poroelastic parameters with the intrinsic properties of the constituents of the porous media such as matrix bulk modulus and fluid bulk modulus.
When measuring the linear elastic properties of porous solids, laboratory experiments are typically performed under one of two limit cases: Poroelastic solids are loaded under drained conditions, in which fluid exchange between domains of the porous solid and the exterior occurs rapidly, and the fluid pressure in pore space is held constant, . Such a system is considered to be an open system. Poroelastic solids are loaded under undrained conditions, in which fluid exchange between the porous solid and the exterior is precluded, . A saturated poroelastic solid loaded under undrained conditions typically experiences significant changes in fluid pressure. Such a system is considered to be a closed system.
Governing equations for poromechanical problems The constitutive equation above describes the response of a local porous material in response to stress and fluid pressure changes. The full description of coupled hydromechanical processes relevant for practical applications also requires the complete governing equations and compatible initial and boundary conditions. The governing equations are summarized below:
Balance of linear momentum: (static) or (dynamic)Balance of mass (of the pore fluid):Darcy's law Compatibility of infinitesimal strainswhere is the total density of the porous medium; is the body force; is the relative fluid to solid displacement; and denote solid and fluid displacements, respectively; an overdot denotes a derivative with respect to time; is specific discharge vector; is hydraulic conductivity; is gravitational acceleration. The governing equations (13) combined with the constitutive equations (7) given the previous section provide a total of 20 equations, which correspond to a total of 20 unknowns (). The system is closed and can be solved when supplied with proper initial and boundary conditions.
Solutions to poromechanical problems can be sought analytically or numerically. Numerical techniques including finite difference and finite element methods are frequently invoked in industrial applications. Closed-form analytical solutions have been also developed for many practically relevant problems and are well-documented in textbooks. Analytical solutions are useful for verification of computational codes and developing qualitative intuitions about poromechanical problems.
Historical background General