What is the spatial profile of fluid infiltration within an anisotropic medium?

Question

Starting at time $t = 0$, a unit of compressible fluid of volume $v$ and density $\rho_f$ is pumped into a medium at some location $\mathbf{r}_0$. Initially, the density $\rho_m(0)$ and pressure $P_m(0)$ of the medium are constant everywhere; furthermore, $\rho_f = a\rho_m (0)$ and $P_f = bP_m(0)$ where $a >>1$ and $b << 1$. The medium is anisotropic, such that the fluid is $x$ times more likely to diffuse along $\hat{\mathbf{z}}$ than along all other directions. The time rate of fluid delivery decreases with the amount of stress exerted by fluid already delivered into the surrounding medium. [1]. After how much time does fluid delivery end? [2]. Over what total volume $V$ has the fluid diffused? [3]. What is the shape of $V$?