The governing equations that enter into the statement of virtual work are the conservation of linear momentum and the conservation of mass, for the mixture as a whole. Under quasi-static conditions, the conservation of momentum reduces to $$\begin{equation} \divg\boldsymbol{\sigma}+\rho\mathbf{b}=\mathbf{0}\,,\label{eq103} \end{equation}$$ where $\sigma$ is the Cauchy stress for the mixture, $\rho$ is the mixture density and $\mathbf{b}$ is the external mixture body force per mass. Since the mixture is porous, this stress may also be written as $$\begin{equation} \boldsymbol{\sigma}=-p\mathbf{I}+\boldsymbol{\sigma}^{e}\,,\label{eq104} \end{equation}$$ where $p$ is the fluid pressure and $\sigma^{e}$ is the effective or extra stress, resulting from the deformation of the solid matrix. Conservation of mass for the mixture requires that $$\begin{equation} \divg\left(\mathbf{v}^{s}+\mathbf{w}\right)=0\,,\label{eq105} \end{equation}$$ where $\mathbf{v}^{s}=\partial\boldsymbol{\chi}^{s}/\partial t$ is the solid matrix velocity and $\mathbf{w}$ is the flux of the fluid relative to the solid matrix. Let the solid matrix displacement be denoted by $\mathbf{u}$ , then $\mathbf{v}^{s}=\mathbf{\dot{u}}$ .

To relate the relative fluid flux $\mathbf{w}$ to the fluid pressure and solid deformation, it is necessary to employ the equation of conservation of linear momentum for the fluid, $$\begin{equation} -\varphi^{w}\grad p+\rho^{w}\mathbf{b}^{w}+\mathbf{\hat{p}}_{d}^{w}=\mathbf{0}\,,\label{eq106} \end{equation}$$ where $\varphi^{w}$ is the solid matrix porosity, $\rho^{w}=\varphi^{w}\rho_{T}^{w}$ is the apparent fluid density and $\rho_{T}^{w}$ is the true fluid density, $\mathbf{b}^{w}$ is the external body force per mass acting on the fluid, and $\mathbf{\hat{p}}_{d}^{w}$ is the momentum exchange between the solid and fluid constituents, typically representing the frictional interaction between these constituents. This equation neglects the viscous stress of the fluid in comparison to $\mathbf{\hat{p}}_{d}^{w}$ . The most common constitutive relation is $\mathbf{\hat{p}}_{d}^{w}=-\varphi^{w}\mathbf{k}^{-1}\cdot\mathbf{w}$ , where the second order, symmetric tensor $\mathbf{k}$ is the hydraulic permeability of the mixture. When combined with Eq.(2.5.1-5), it produces $$\begin{equation} \mathbf{w}=-\mathbf{k}\cdot\left(\grad p-\rho_{T}^{w}\mathbf{b}^{w}\right)\,,\label{eq107} \end{equation}$$ which is equivalent to Darcy's law. In general, $\mathbf{k}$ may be a function of the deformation.