Poro-Visco-Elastic Compaction in Sedimentary Basins
Eleanor Holland, Ralph E. Showalter · SIAM Journal on Mathematical Analysis · 2018
The porosity of a visco-elastic medium is shown to satisfy a nonlinear pseudoparabolic partial differential equation of the form $ u' + A(u) (\alpha(u) + \eta u' ) = G(t,u)$ in which $u'$ denotes the time derivative, $A(v) = - \boldsymbol{ abla}\cdot \kappa(v) \boldsymbol{ abla}$ is a linear second order elliptic operator in divergence form with coefficient depending on a function $v(x)$, $\alpha(\cdot)$ is affine-bounded and $\alpha(\cdot) + kI$ is monotone for some $k \in \mathbb{R}$, $G(t,u)$ is a linear first order operator in $u$, and $\eta > 0$. The third order nonlinear term $A(u)u'$ distinguishes this equation from the classical porous medium equation. The solvability of an elliptic boundary-value problem for $(I + \eta A(v))u = f$ for $\eta > 0$ and the continuous dependence of the solution $u$ on the function $v$ is used to establish existence of the solution of the initial-boundary-value problem for the pseudoparabolic equation. We establish bounds on the solution that prevent degeneracy of the coefficient $\kappa(\cdot)$ and prove regularity properties of the solution. These results are obtained by methods of monotonicity and compactness.