swell with fluid influx, $\mathrm{det}\bF
= \mathrm{det}\bF^{\mathrm{g}^\mathrm{f}} > 1$. A compressible fluid may have $\mathrm{det}\bF^{\mathrm{e}^\mathrm{f}} < 1$ allowing $\mathrm{det}\bF < 1$ even with $\mathrm{det}\bF^{\mathrm{g}^\mathrm{f}} >1$. Even in this case, however, in the stress-free state there will be swelling.

Therefore, for the fluid phase, the isotropic swelling law can be extended to the unsaturated case by introducing a degree of saturation, $\tilde{v}^\iota$, defined in the current configuration, $\Omega_t$. We have $\tilde{v}^\iota =
\rho^\iota/\tilde{\rho}^\iota$, where $\tilde{\rho}^\iota$ is the intrinsic density in $\Omega_t$ and is given by $\tilde{\rho}^\iota =
\tilde{\rho}^\iota_0/\mathrm{det}\bF$. Note that the intrinsic reference density, $\tilde{\rho}^\iota_0$, is a material property. Upon solution of the mass balance equation (\[massbalcurr\]) for $\rho^\iota$, the species volume fractions, $\tilde{v}^\iota$, can therefore be computed in a straightforward fashion. The sum of these volume fractions is our required measure of saturation defined in $\Omega_t$. Also, recognizing that for the dilute solutions obtained with physiologically-relevant solute concentrations, the saturation condition is very well approximated by $\tilde{v}^\mathrm{f} +
\tilde{v}^\mathrm{s} = 1$, we proceed to redefine the fluid growth-induced component of the pore deformation gradient tensor as follows:

$$\bF^{\mathrm{g}^\mathrm{f}} = \left\{ \begin{array}{ll}  \left(
\frac{\rho_0^\mathrm{f}}{\rho_{0_{\mathrm{sat}}}^\mathrm{f}} \right)^
{\frac{1}{3}}{\bf 1},&
\tilde{v}^\mathrm{f} + \tilde{v}^\mathrm{s} = 1 \\ {\bf 1},& \mathrm{otherwise.}
\end{array}\right.
\label{saturation}$$

In (\[saturation\]) $\rho_{0_{\mathrm{sat}}}^\mathrm{f}$ 