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, n ( x ) {\displaystyle n(\mathbf {x} )} , which measures the porosity with respect to the current or deformed configuration. Specifically, if d V t {\displaystyle \mathrm {d} V_{t}} represents an infinitesimal volume in the deformed material body, then the pore volume is calculated from n ( x ) d V t {\displaystyle n(\mathbf {x} )\mathrm {d} V_{t}} . The Lagrangian porosity, ϕ ( x ) {\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 ϕ ( x ) d V 0 {\displaystyle \phi (\mathbf {x} )\mathrm {d} V_{0}} , where d V 0 {\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
ϕ ( x ) = n ( x ) d V t d V 0 = J ( x ) n ( x ) , {\displaystyle \phi (\mathbf {x} )=n(\mathbf {x} ){\frac {\mathrm {d} V_{t}}{\mathrm {d} V_{0}}}=J(\mathbf {x} )n(\mathbf {x} ),}
where J = det ( F ) {\displaystyle J=\det(\mathbf {F} )} is the Jacobian of the deformation with F {\displaystyle \mathbf {F} } being the deformation gradient. In a small-strain, linearized theory of deformation, the volume ratio is approximated by J ≃ ( 1 + ϵ v ) {\displaystyle J\simeq (1+\epsilon _{\mathrm {v} })} , where ϵ v {\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
e = n 1 − n , {\displaystyle e={\frac {n}{1-n}},}
where 1 − n {\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 d V t s {\displaystyle \mathrm {d} V_{t}^{\mathrm {s} }} , is related to the deformed and undeformed total material volumes by
d V t s = ( 1 − n ) d V t = d V t − ϕ d V 0 {\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 1 − ϕ 0 = d V 0 s / d V 0 {\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:
ϵ v = ( 1 − ϕ 0 ) ϵ s ⏟ solid + ϕ − ϕ 0 ⏟ pore , {\displaystyle \epsilon _{\mathrm {v} }=\underbrace {(1-\phi _{0})\epsilon _{\mathrm {s} }} _{\text{solid}}+\underbrace {\phi -\phi _{0}} _{\text{pore}},}
where ϵ s = d V t s / d V 0 s − 1 {\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.
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 skeleton Φ s = S i j d E i j + p d ϕ − S s d T − d Ψ s ≥ 0 {\displaystyle {\Phi ^{s}}={S_{ij}}d{E_{ij}}+pd\phi -{S^{s}}dT-d{\Psi ^{s}}\geq 0} where S i j {\displaystyle {S_{ij}}} is the Second Piola-Kirchoff stress; E i j {\displaystyle E_{ij}} is the Green-Lagrangian strain; p {\displaystyle p} the pore fluid pressure; ϕ {\displaystyle \phi } the Lagrangian porosity; T {\displaystyle T} is temperature; S s {\displaystyle S^{s}} and Ψ s {\displaystyle \Psi ^{s}} are the entropy and the elastic stored energy (Helmholtz free energy) of the solid skeleton; Φ s {\displaystyle \Phi ^{s}} is the rate of energy dissipation. Assuming small deformation, elastic solid, and isothermal condition, the previous equation simplifies to:
d Ψ s = σ i j d ε i j + p d ϕ {\displaystyle d{\Psi _{s}}={\sigma _{ij}}d{\varepsilon _{ij}}+pd\phi }
where ε i j = 1 2 ( ∂ u i ∂ x j + ∂ u j ∂ x i ) {\displaystyle {\varepsilon _{ij}}={\frac {1}{2}}\left({{\frac {\partial {u_{i}}}{\partial {x_{j}}}}+{\frac {\partial {u_{j}}}{\partial {x_{i}}}}}\right)} is the infinitesimal strain; u i {\displaystyle u_{i}} and x i {\displaystyle x_{i}} are the displacement and position vectors, respectively; σ i j {\displaystyle \sigma _{ij}} is Cauchy stress. The constitutive equation for an elastic porous media can thus be generally stated as:
σ i j = ∂ Ψ s ∂ ε i j ; p = ∂ Ψ s ∂ ϕ {\displaystyle {\sigma _{ij}}={\frac {\partial {\Psi _{s}}}{\partial {\varepsilon _{ij}}}};p={\frac {\partial {\Psi _{s}}}{\partial \phi }}}
Let us specify the following quadratic form of Ψ s ( ε i j , ϕ ) {\displaystyle {\Psi _{s}}\left({{\varepsilon _{ij}},\phi }\right)} :
Ψ s ( ε i j , ϕ ) = 1 2 ( K − 2 3 G + α 2 N ) ε k k 2 + G ε i j ε i j − α N ε k k ( ϕ − ϕ 0 ) + 1 2 N ( ϕ − ϕ 0 ) 2 {\displaystyle {\Psi _{s}}\left({{\varepsilon _{ij}},\phi }\right)={\frac {1}{2}}\left({K-{\frac {2}{3}}G+{\alpha ^{2}}N}\right){\varepsilon _{kk}}^{2}+G{\varepsilon _{ij}}{\varepsilon _{ij}}-\alpha N{\varepsilon _{kk}}\left({\phi -{\phi _{0}}}\right)+{\frac {1}{2}}N{\left({\phi -{\phi _{0}}}\right)^{2}}} where e i j = ε i j − δ i j ε k k / 3 {\displaystyle {e_{ij}}={\varepsilon _{ij}}-{\delta _{ij}}{\varepsilon _{kk}}/3} is the strain deviator; ϕ 0 {\displaystyle {\phi _{0}}} is the initial porosity of the undeformed porous medium; K {\displaystyle K} , G {\displaystyle G} , α {\displaystyle \alpha } , N {\displaystyle N} are material constants. We can immediately obtain Biot's linear isotropic poroelasticity in terms of ϕ {\displaystyle {\phi }} :
{ σ i j = ∂ Ψ s ∂ ε i j = ( K + α 2 N − 2 3 G ) δ i j ε k k + 2 G ε i j − α N δ i j ( ϕ − ϕ 0 ) p = ∂ Ψ s ∂ ϕ = − α N ε k k + N ( ϕ − ϕ 0 ) {\displaystyle \left\{{\begin{array}{l}{\sigma _{ij}}={\frac {\partial {\Psi _{s}}}{\partial {\varepsilon _{ij}}}}=\left({K+{\alpha ^{2}}N-{\frac {2}{3}}G}\right){\delta _{ij}}{\varepsilon _{kk}}+2G{\varepsilon _{ij}}-\alpha N{\delta _{ij}}\left({\phi -{\phi _{0}}}\right)\\p={\frac {\partial {\Psi _{s}}}{\partial \phi }}=-\alpha N{\varepsilon _{kk}}+N\left({\phi -{\phi _{0}}}\right)\end{array}}\right.}
or more commonly in incremental form:
{ d σ i j = ( K − 2 3 G ) δ i j d ε k k + 2 G d ε i j − α δ i j d p d ϕ = 1 N d p + α d ε k k {\displaystyle \left\{{\begin{array}{l}d{\sigma _{ij}}=\left({K-{\frac {2}{3}}G}\right){\delta _{ij}}d{\varepsilon _{kk}}+2Gd{\varepsilon _{ij}}-\alpha {\delta _{ij}}dp\\d\phi ={\frac {1}{N}}dp+\alpha d{\varepsilon _{kk}}\end{array}}\right.}
Comparing with the usual linear elasticity equations, one can identify that K {\displaystyle K} and G {\displaystyle G} are the bulk and shear modulus of the porous material under drained ( d p = 0 {\displaystyle dp=0} ) conditions; α {\displaystyle \alpha } , 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; N {\displaystyle N} is a tangent modulus linking the pressure variation and porosity variation under constant volumetric strain ( d ε k k = 0 {\displaystyle d\varepsilon _{kk}=0} ).
The first equation can be rewritten in such a way that the right-hand side is exactly the same as the linear elasticity equation:
σ i j ″ = σ i j + α p δ i j = ( K − 2 3 G ) δ i j ε k k + 2 G ε i j {\displaystyle {\sigma ''_{ij}}={\sigma _{ij}}+\alpha p{\delta _{ij}}=\left({K-{\frac {2}{3}}G}\right){\delta _{ij}}{\varepsilon _{kk}}+2G{\varepsilon _{ij}}}
Term σ i j ″ {\displaystyle {\sigma ''_{ij}}} 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 α = 1 {\displaystyle \alpha =1} .
Although the ϕ {\displaystyle {\phi }} -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 ϕ {\displaystyle {\phi }} is therefore the variation of fluid content, ζ {\displaystyle \zeta } , defined as the amount of fluid volume entering the solid frame per unit volume of solid frame. Its increment can be written as
d ζ = d m f ρ f = d ( ρ f ϕ ) ρ f = d ϕ + ϕ d ρ f ρ f {\displaystyle d\zeta ={\frac {dm_{f}}{\rho _{f}}}={\frac {d\left({{\rho _{f}}\phi }\right)}{\rho _{f}}}=d\phi +\phi {\frac {d{\rho _{f}}}{\rho _{f}}}}
where m f {\displaystyle m_{f}} is the fluid mass content; ρ f {\displaystyle \rho _{f}} is the fluid density. By replacing d ϕ {\displaystyle d\phi } with d ζ {\displaystyle d\zeta } , and considering fluid state equation d ρ f / ρ f = d p / K f {\displaystyle d{\rho _{f}}/{\rho _{f}}=dp/K_{f}} , Biot's theory can be also written in the following ζ {\displaystyle \zeta } -based form:
{ d σ i j = ( K − 2 3 G ) δ i j d ε k k + 2 G d ε i j − α δ i j d p d ζ = 1 M d p + α d ε k k {\displaystyle \left\{{\begin{array}{l}d{\sigma _{ij}}=\left({K-{\frac {2}{3}}G}\right){\delta _{ij}}d{\varepsilon _{kk}}+2Gd{\varepsilon _{ij}}-\alpha {\delta _{ij}}dp\\d\zeta ={\frac {1}{M}}dp+\alpha d{\varepsilon _{kk}}\end{array}}\right.}
or
{ d σ i j = ( K u − 2 3 G ) δ i j d ε k k + 2 G d ε i j − α M δ i j d ζ d p = M ( d ζ − α d ε k k ) {\displaystyle \left\{{\begin{array}{l}d{\sigma _{ij}}=\left({{K_{u}}-{\frac {2}{3}}G}\right){\delta _{ij}}d{\varepsilon _{kk}}+2Gd{\varepsilon _{ij}}-\alpha M{\delta _{ij}}d\zeta \\dp=M\left({d\zeta -\alpha d{\varepsilon _{kk}}}\right)\end{array}}\right.}
where