$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

We model the BFSI domain as an unconstrained mixture of an isothermal, compressible, and viscous fluid and an isothermal, deformable porous solid whose skeleton material is intrinsically incompressible. As usual in mixture theory, each constituent $\alpha$ ($\alpha=s,f$ for solid and fluid, respectively) has its own apparent density $\rho^{\alpha}=dm^{\alpha}/dV$ , which represents the ratio of the elemental mass $dm^{\alpha}$ of constituent $\alpha$ in the elemental mixture volume $dV$ . This apparent density is related to the true density $\rho_{T}^{\alpha}=dm^{\alpha}/dV^{\alpha}$ (mass of $\alpha$ per volume of $\alpha$ ) via $\rho^{\alpha}=\varphi^{\alpha}\rho_{T}^{\alpha}$ , where $\varphi^{\alpha}=dV^{\alpha}/dV$ is the volume fraction of $\alpha$ in the mixture, satisfying $\varphi^{s}+\varphi^{f}=1$ . Since the solid skeleton is intrinsically incompressible, $\rho_{T}^{s}$ is constant. The boundaries of a biphasic domain are defined on the porous solid matrix. The deformation gradient of the solid is denoted by $\mathbf{F}^{s}$ and its determinant, $J^{s}=\det\mathbf{F}^{s}=dV/dV_{r}$ , represents the ratio of the mixture elemental volumes in the current ($dV$ ) and reference ($dV_{r}$ ) configurations. Thus, since the solid skeleton is intrinsically incompressible, $J^{s}$ purely represents the compressibility of the pore volume as fluid enters or leaves the pore space, or as the compressible fluid within the pores changes in volume. The axiom of mass balance for the solid may be integrated in closed form to produce $\rho^{s}=\rho_{r}^{s}/J^{s}$ , where $\rho_{r}^{s}=dm^{s}/dV_{r}$ is the solid apparent density in the mixture reference configuration. Therefore, we may define the referential solid volume fraction as $\varphi_{r}^{s}=\rho_{r}^{s}/\rho_{T}^{s}$ . In the finite element analysis, $\varphi_{r}^{s}=dV^{s}/dV_{r}$ and $\rho_{T}^{s}$ are both specified by the user, then $\rho_{r}^{s}$ , $\rho^{s}$ , $\varphi^{s}$ and $\varphi^{f}$ are evaluated from the above relations, given a solution for $\mathbf{F}^{s}$ (and thus, $J^{s}$ ). The solution for $\mathbf{F}^{s}=\mathbf{I}+\Grad\mathbf{u}$ is obtained by solving for the nodal solid displacement vector $\mathbf{u}$ , where $\Grad\left(\cdot\right)$ represents the gradient operator in the material frame.