5.8 Hydraulic Permeability

Hydraulic permeability is a material function needed for biphasic and multiphasic materials.

Constant Isotropic Permeability

When the permeability is isotropic,

\[ \begin{equation} \mathbf{k}=k\,\mathbf{I}\,.\label{eq501} \end{equation} \]

For this material model, \(k\) is constant. Generally, this assumption is only reasonable when strains are small.

Exponential Isotropic Permeability

This isotropic material has a permeability that varies as a function of the determinant \(J\) of the deformation gradient. Its general form is

\[ \begin{equation} \mathbf{k}=k\left(J\right)\mathbf{I}\,,\label{eq:isotropic-permeability} \end{equation} \]

where,

\[ \begin{equation} k\left(J\right)=k_{0}\exp\left(M\frac{J-1}{J-\varphi_{0}}\right)\,.\label{eq:perm-exp-iso} \end{equation} \]

Pore closure occurs as \(J\) approaches \(\varphi_{0}\) from above, in which case the permeability reduces to zero,

\[ \begin{equation} \lim_{J\to\varphi_{0}}k\left(J\right)=0\,.\label{eq:perm-pore-closure} \end{equation} \]

In the special case when \(M=0\), the permeability becomes constant. In the limit of infinitesimal strains, the permeability has the form

\[ \begin{equation} k\left(J\right)=k_{0}\left(1+\frac{M}{1-\varphi_{0}}\left(J-1\right)+\mathcal{O}\left(\left(J-1\right)^{2}\right)\right)\,.\label{eq:perm-expiso-Taylor} \end{equation} \]

For the type of isotropic permeability given in \eqref{eq:isotropic-permeability}, the spatial permeability tangent \(\boldsymbol{\mathcal{K}}\) with respect to the solid matrix strain has the general form

\[ \begin{equation} \boldsymbol{\mathcal{K}}=\left(k+Jk^{\prime}\right)\mathbf{I}\otimes\mathbf{I}-2k\mathbf{I}\odot\mathbf{I}\,,\label{eq:isotropic-perm-tangent} \end{equation} \]

where

\[ \begin{equation} k^{\prime}\left(J\right)=M\frac{1-\varphi_{0}}{\left(J-\varphi_{0}\right)^{2}}k\left(J\right)\label{eq:perm-exp-iso-tangent} \end{equation} \]

in this case.

Holmes-Mow

This isotropic material uses a strain-dependent permeability tensor as formulated by ¹:

\[ \begin{equation} \mathbf{k}=k\left(J\right)\mathbf{I}\,,\label{eq502} \end{equation} \]

where,

\[ \begin{equation} k\left(J\right)=k_{0}\left(\frac{J-\varphi_{0}}{1-\varphi_{0}}\right)^{\alpha}e^{\frac{1}{2}M\left(J^{2}-1\right)}\,.\label{eq503} \end{equation} \]

When \(\alpha>0\), pore closure occurs as \(J\to\varphi_{0}\) from above, in which case \(k\) reduces to \(0\). Setting \(\alpha=0\) and \(M=0\) produces a constant permeability. In the limit of infinitesimal strains,

\[ \begin{equation} k\left(J\right)=k_{0}\left(1+\left(M+\frac{\alpha}{1-\varphi_{0}}\right)\left(J-1\right)+\mathcal{O}\left(\left(J-1\right)^{2}\right)\right)\,.\label{eq:perm-HM-Taylor} \end{equation} \]

The spatial tangent of the permeability tensor with respect to strain may be evaluated from \eqref{eq:isotropic-perm-tangent} using

\[ \begin{equation} k^{\prime}\left(J\right)=\left(J\,M+\frac{\alpha}{J-\varphi_{0}}\right)k\left(J\right)\,.\label{eq:perm-HM-tangent} \end{equation} \]

Referentially Isotropic Permeability

This material uses a strain-dependent permeability tensor that accommodates strain-induced anisotropy ²:

\[ \begin{equation} \mathbf{k}=\left(k_{0r}\mathbf{I}+\frac{k_{1r}}{J^{2}}\mathbf{b}+\frac{k_{2r}}{J^{4}}\mathbf{b}^{2}\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)e^{M\left(J^{2}-1\right)/2}\,,\label{eq504} \end{equation} \]

Note that the permeability in the reference state (\(\mathbf{F}=\mathbf{I})\) is isotropic and given by \(\mathbf{k}=\left(k_{0r}+k_{1r}+k_{2r}\right)\mathbf{I}\).

Referentially Orthotropic Permeability

This material uses a strain-dependent permeability tensor that is orthotropic in the reference configuration, and accommodates strain-induced anisotropy ²:

\[ \begin{equation} \mathbf{k}=k_{0}\mathbf{I}+\sum\limits_{a=1}^{3}k_{1}^{a}\mathbf{m}_{a}+k_{2}^{a}\left(\mathbf{m}_{a}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}_{a}\right)\,,\label{eq505} \end{equation} \]

where,

\[ \begin{equation} \begin{aligned} & k_{0}=k_{0r}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{^{0}}}e^{M_{^{0}}\left(J^{2}-1\right)/2}\,,\\ & k_{1}^{a}=\frac{k_{1r}^{a}}{J^{2}}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{^{a}}}e^{M_{^{a}}\left(J^{2}-1\right)/2}\,,\\ & k_{2}^{a}=\frac{k_{2r}^{a}}{2J^{4}}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{^{a}}}e^{M_{a}\left(J^{2}-1\right)/2}\,. \end{aligned} \quad a=1,2,3\label{eq506} \end{equation} \]

Here, \(\mathbf{m}_{a}\) are second order tensors representing the spatial structural tensors describing the orthogonal planes of symmetry, given by

\[ \begin{equation} \mathbf{m}_{a}=\mathbf{F}\cdot\left(\mathbf{V}_{a}\otimes\mathbf{V}_{a}\right)\cdot\mathbf{F}^{T},\quad a=1-3,\label{eq507} \end{equation} \]

where \(\mathbf{V}_{a}\) are orthonormal vectors normal to the planes of symmetry. Note that the permeability in the reference state (\(\mathbf{F}=\mathbf{I}\)) is given by \(\mathbf{k}=k_{0r}\mathbf{I}+\sum\limits_{a=1}^{3}\left(k_{1r}^{a}+k_{2r}^{a}\right)\mathbf{V}_{a}\otimes\mathbf{V}_{a}\).

Referentially Transversely Isotropic Permeability

This material uses a strain-dependent permeability tensor that is transversely isotropic in the reference configuration, and accommodates strain-induced anisotropy ²:

\[ \begin{equation} \begin{aligned}\mathbf{k} & =k_{0r}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{0}}e^{M_{^{0}}\left(J^{2}-1\right)/2}\mathbf{I}\\ & +\left(\frac{k_{1r}^{T}}{J^{2}}\left(\mathbf{b}-\mathbf{m}\right)+\frac{k_{2r}^{T}}{2J^{4}}\left[2\mathbf{b}^{2}-\left(\mathbf{m}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}\right)\right]\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{T}}e^{M_{T}\left(J^{2}-1\right)/2}\\ & +\left(\frac{1}{J^{2}}k_{1r}^{A}\mathbf{m}+\frac{1}{2J^{4}}k_{2r}^{A}\left(\mathbf{m}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}\right)\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{A}}e^{M_{A}\left(J^{2}-1\right)/2}\,, \end{aligned} \label{eq508} \end{equation} \]

where \(\mathbf{m}\) is a second order tensor representing the spatial structural tensor describing the axial direction, given by

\[ \begin{equation} \mathbf{m}=\mathbf{F}\cdot\left(\mathbf{V}\otimes\mathbf{V}\right)\cdot\mathbf{F}^{T}\,,\label{eq509} \end{equation} \]

and \(\mathbf{V}\) is a unit vector along the axial direction. Note that the permeability in the reference state (\(\mathbf{F}=\mathbf{I}\)) is given by \(\mathbf{k}=\left(k_{0r}+k_{1r}^{T}+k_{2r}^{T}\right)\mathbf{I}+\left(k_{1r}^{A}-k_{1r}^{T}+k_{2r}^{A}-k_{2r}^{T}\right)\left(\mathbf{V}\otimes\mathbf{V}\right)\).

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1. Holmes, M. H.; Mow, V. C.. "The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration." J Biomech, vol. 23, pp. 1145-56 (1990). ↩

2. Ateshian, G. A.; Weiss, J. A.. "Anisotropic hydraulic permeability under finite deformation." Journal of biomechanical engineering, vol. 132, pp. 111004 (2010). ↩↩↩