Poromechanics

Poromechanics is a branch of physics and specifically continuum mechanics that studies the behavior of fluid-saturated porous media.^[1] A porous medium or a porous material is a solid, constituting the matrix, which is 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 constitute two separate, continuously connected domains. An archtypal example of such a porous material is the kitchen sponge, which is formed of two interpenetrating continua. Some porous media has a more complex microstructure in which, for example, the porespace is disconnected. Porespace that is unable to exchange fluid with the exterior is termed occluded porespace. 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,^[2] soils,^[3] biological tissues including heart^[4] and cancellous bone,^[5] and man-made materials such as foams, ceramics, and concrete^[6] can be considered as porous media. Porous media whose solid matrix is elastic and the fluid is viscous are called poroviscoelastic. A poroviscoelastic medium is characterised by its porosity, permeability, and the properties of its constituents - solid matrix and fluid. The distribution of pores, fluid pressure, and stress in the solid matrix gives rise to the viscoelastic behavior of the bulk.^[7] 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.

The concept of a porous medium originally emerged in soil mechanics, and in particular in the works of Karl von Terzaghi, the father of soil mechanics.^[8] However a more general concept of a poroelastic medium, independent of its nature or application, is usually attributed to Maurice Anthony Biot (1905–1985), a Belgian-American engineer. In a series of papers published between 1935 and 1962 Biot developed the theory of dynamic poroelasticity (now known as Biot theory) which gives a complete and general description of the mechanical behaviour of a poroelastic medium.^{[9][10][11][12][13]} Biot's equations of the linear theory of poroelasticity are derived from the equations of linear elasticity for a solid matrix, the Navier–Stokes equations for a viscous fluid, and Darcy's law for a flow of fluid through a porous matrix.

One of the key findings of the theory of poroelasticity is that in poroelastic media, there exist three types of elastic waves: a shear or transverse wave, and two types of longitudinal or compressional waves, which Biot called type I and type II waves. The transverse and type I (or fast) longitudinal waves are similar to the transverse and longitudinal waves in an elastic solid, respectively. The slow compressional wave (Biot’s slow wave) is unique to poroelastic materials. The prediction of Biot’s slow wave generated controversy until Thomas Plona experimentally observed it in 1980.^[14] Other important early contributors to the theory of poroelasticity were Yakov Frenkel and Fritz Gassmann.^[15][16][17]

Energy conversion from fast compressional and shear waves into the highly attenuating slow compressional wave is a significant cause of elastic wave attenuation in porous media.

Recent applications of poroelasticity to biology, such as modeling blood flows through the beating myocardium, have also required an extension of the equations to nonlinear (large deformation) elasticity and the inclusion of inertia forces.

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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:^[1]

• The Eulerian porosity, ${\displaystyle n(\mathbf {x} )}$ , which measures the porosity with respect to the current or deformed configuration. Specifically, if ${\displaystyle \mathrm {d} V_{t}}$ represents an infinitesimal volume in the deformed material body, then the pore volume is calculated from ${\displaystyle n(\mathbf {x} )\mathrm {d} V_{t}}$.
• The Lagrangian porosity, ${\displaystyle \phi (\mathbf {x} )}$, which measures the porosity with respect to the initial or undeformed configuration. In a Lagrangian description of porosity, the pore volume is measured by ${\displaystyle \phi (\mathbf {x} )\mathrm {d} V_{0}}$, where ${\displaystyle \mathrm {d} V_{0}}$ represents an infinitesimal volume of the material in its undeformed state.

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

${\displaystyle \phi (\mathbf {x} )=n(\mathbf {x} ){\frac {\mathrm {d} V_{t}}{\mathrm {d} V_{0}}}=J(\mathbf {x} )n(\mathbf {x} ),}$

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

${\displaystyle e={\frac {n}{1-n}},}$

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

${\displaystyle \mathrm {d} V_{t}^{\mathrm {s} }=(1-n)\mathrm {d} V_{t}=\mathrm {d} V_{t}-\phi \mathrm {d} V_{0}}$

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:

${\displaystyle \epsilon _{\mathrm {v} }=\underbrace {(1-\phi _{0})\epsilon _{\mathrm {s} }} _{\text{solid}}+\underbrace {\phi -\phi _{0}} _{\text{pore}},}$

where ${\displaystyle \epsilon _{\mathrm {s} }=\mathrm {d} V_{t}^{\mathrm {s} }/\mathrm {d} V_{0}^{\mathrm {s} }-1}$ is recognized as the linearized volume strain acting in the solid.

When linearizing the strain in a poroelastic solid body, several conditions should hold true. Firstly, as is the requirement for a general continuum solid, displacement gradients should be small, ${\displaystyle ||\nabla \mathbf {u} ||\ll 1}$. Secondly, to further ensure small changes in the solid and pore volumes, the displacement field of the solid, ${\displaystyle \mathbf {u} _{\mathrm {s} }}$, should be small in comparison to the characteristic length scale defining the grain size (in case of a granular material) or solid matrix (in case of a continuous solid phase), ${\displaystyle L}$. This second requirement is stated as ${\displaystyle ||\mathbf {u} _{\mathrm {s} }/L||\ll 1}$, and implies small changes in the Lagrangian porosity ${\displaystyle (\phi -\phi _{0})/\phi _{0}\ll 1}$.

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 porespace is held constant, ${\displaystyle \mathrm {d} p=0}$. 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; ${\displaystyle \mathrm {d} m_{\mathrm {f} }=0}$, where ${\displaystyle m_{\mathrm {f} }}$ is the local mass of the fluid. 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.

Reinhard Woltman (1757-1837), a German hydraulic and geotechnical engineer, first introduced the concepts of volume fractions and angles of internal friction within porous media in his study on the connection between soil moisture and its apparent cohesion.^[18] His work addressed the calculation of earth pressure against retaining walls. Achille Delesse (1817-1881), a French geologist and mineralogist, reasoned that the volume fraction of voids – otherwise termed the volumetric porosity – equals the surface fraction of voids – otherwise termed the areal porosity – when the size, shape, and orientation of the pores are randomly distributed.^[19] Henry Darcy (1803-1858), a French hydraulic engineer, observed the proportionality between the rate of discharge and the loss of water pressure in tests with natural sand, now known as Darcy’s law.^[20] The first important concept related to saturated, deformable porous solids might be considered the principle of effective stress introduced by Karl von Terzaghi (1883-1963), an Austrian engineer. Terzaghi postulated that the mean effective stress experienced by the solid skeleton of a porous medium with incompressible constituents, ${\displaystyle \sigma '=\sigma -p}$, is the total stress acting on the volume element, ${\displaystyle \sigma }$, subtracted by the pressure of the fluid acting in the pore space, ${\displaystyle p}$.^[21] Terzaghi combined his effective stress concept with Darcy’s law for fluid flow and derived a one-dimensional consolidation theory explaining the time-dependent deformation of soils as the pore fluid drains, which might be the first mathematical treatise on coupled hydromechanical problems in porous media.

• Petrophysics
• Rock physics
• Poroelasticity
• Fluid flow through porous media
• Permeability (materials science)
• Darcy's Law

• Rice JR, Cleary MP (1976). "Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents". Reviews of Geophysics and Space Physics. 14 (2): 227–241. Bibcode:1976RvGSP..14..227R. doi:10.1029/RG014i002p00227.
• Bourbie T, Coussy O, Zinszner B (1987). Acoustics of Porous Media. Houston: Gulf Publication Company.
• Nigmatulin RI (1990). Dynamics of Multiphase Media. Washington, DC: Hemisphere.
• Allard JF (1993). Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials. London: Chapman & Hall.
• Chapelle D, Moireau P (2014). "General coupling of porous flows and hyperelastic formulations: from thermodynamics principles to energy balance and compatible time schemes". European Journal of Mechanics B. 46: 82–96. Bibcode:2014EuJMB..46...82C. doi:10.1016/j.euromechflu.2014.02.009.

• Poronet - PoroMechanics Internet Resources Network
• APMR - Acoustical Porous Material Recipes