| Three mathematical models of groundwater flow related to flooding are considered. The first is a model of unsteady filtration in a semi infinite porous stratum with capillary retention. It leads to a free-boundary problem for the generalized porous medium equation, where the location of the boundary of the water mound is determined as part of the solution. The numerical solution is shown to possess a self-similar intermediate asymptotics. Next, assuming self-similarity, a nonlinear eigenvalue value problem is obtained from the generalized porous medium equation. The asymptotic solution is obtained from this eigenvalue problem. The asymptotic solution shows, in particular, that the water mound cannot be extinguished in finite time.; In the second part of the work, the problem of controlling the water mound extension by a forced drainage is considered. The equation and the boundary conditions are derived using a limiting model for the of layered medium. The result is a free-boundary problem with two moving boundaries. The problem is solved numerically. The solution, in contrast with first case, shows that the mound can indeed be extinguished in finite time.; In the third part we account for the effect of fissures, which have been neglected in the first two models. The fissured porous rock is considered as a double porosity medium. The exchange between the pores and the fissures is assumed to be quasi-steady. The result is a system of nonlinear parabolic equations with a nonlinear coupling. We, once again, have to solve a free boundary problem for the new system.; For this system we model the conditions of the flood numerically since the exchange affects the distributions in an essential way from very early in the process. Then we set the boundary conditions corresponding to the natural drainage and let the solution evolve for a long time. While the asymptotic solutions of the decoupled equations are dipole-type, we see that, depending on the relative value of the parameters, the behavior of the coupled system can be drastically different. We compare the extension of the mound, the total mass and the dipole moments of the solutions with those of the solution in a purely porous medium. We see that the presence of fissures, despite their small volume, has a strong effect and should not be neglected. Failure to account for them could lead to incorrect practical conclusions which can be disastrous if the fluid is contaminated. |
