Numerical Investigation of a Dipole Type Solution for Unsteady Groundwater Flow with Capillary Retention and Forced Drainage

TL;DR

The paper addresses unsteady groundwater seepage in a horizontal porous layer with capillary retention by formulating a generalized porous medium equation with discontinuous coefficients $\kappa_1$ and $\kappa_2$ that depend on the sign of $\partial_t h$, leading to a free boundary problem for a dipole-type mound and a paired forced-drainage control problem. It demonstrates that the dipole-type case exhibits self-similar intermediate asymptotics of the second kind, derived via a nonlinear eigenvalue problem that yields the scaling exponent $\beta(\kappa_1/\kappa_2)$ and matches PDE solutions. A robust numerical framework—a moving-boundary PDE solver and a shooting-based eigenproblem solver—confirms the self-similar predictions and elucidates the influence of capillary retention on mound extent. The study also shows that introducing a controlled drainage flux $q(t)$ can extinguish the mound in finite time, providing a practical approach to contain contamination and guide groundwater-management strategies. Overall, the work extends classical dipole-type filtration analyses by incorporating capillary retention and nonlinear self-similarity, and offers computational tools for planning drainage-based mound control.

Abstract

A model of unsteady filtration (seepage) in a porous medium with capillary retention is considered. It leads to a free boundary problem for a generalized porous medium equation where the location of the boundary of the water mound is determined as part of the solution. The numerical solution of the free boundary problem is shown to possess self-similar intermediate asymptotics. On the other hand, the asymptotic solution can be obtained from a non-linear boundary value problem. Numerical solution of the resulting eigenvalue problem agrees with the solution of the partial differential equation for intermediate times. In the second part of the work, we consider the problem of control of the water mound extension by a forced drainage.

Figures (12)

• Figure 1: Groundwater dome extension in a porous medium.
• Figure 2: Evolution in time of the partial differential equation solution with $\frac{\kappa_1}{\kappa_2}=.5$ for time interval t=[0,18].
• Figure 3: Example of the time evolution for the position $x_r$ of the free boundary. Time interval $t=[0,450]$. For this plot $\frac{\kappa_1}{\kappa_2}=.5$.
• Figure 4: Plot of $\log(\max_x u(x,t))$ vs. $\log(t)$ with $\frac{\kappa_1}{\kappa_2}=.5$. The straight line shows a linear fit to the straight part of the graph.
• Figure 5: Plot of the dependence of $\beta$ on $\frac{\kappa_1}{\kappa_2}$. The solid line is obtained from graph \ref{['fig:fig4']} for different values of $\frac{\kappa_1}{\kappa_2}$, by applying least-squares fit to the straight part of the graph and then using equation (\ref{['eq20']}). The dashed line is obtained from graphs \ref{['fig:fig3']}, by applying the procedure above and then (\ref{['eq21']}). The dotted line is obtained from the solution of the nonlinear eigenvalue problem given by equation (\ref{['eq12']}) with initial conditions \ref{['eq14']} for different values of $\frac{\kappa_1}{\kappa_2}$.