This article is devoted to the study of the equation \(\partial\theta/\partial t=\text{div}({\mathcal R}\nabla p_t)+f\), where \(\theta(t,x)\) is the saturation, \(\mathcal R\) is the matrix \((a_{ij})=(\rho/\mu)\kappa_{ij}\), where the \(n\times n\)-matrix \((\kappa_{ij}(x))\) describes the permeability of the medium, \(\mu\) is the viscosity of the fluid and \(p_t(x)\) is the pressure of the water. The author defines in a precise way the notion of a solution of the moving boundary problem formulated in this paper, by introducing a concept of ``classical solution''. He also introduces a concept of ``weak solution'' and under corresponding assumptions on the permeability, proves that a classical solution is a weak solution. The method used is that of transforming the problem into a series of elliptic variational inequalities. From this the main result, the existence and uniqueness of a weak solution for an arbitrary given initial domain as well as regularity and some monotonicity properties of the solution follows. A series of weakenings of the concept of solution is performed: classical solution \(\Rightarrow\) weak solution \(\Rightarrow\) solution of complementarity problems \(\Leftrightarrow\) solution of variational inequalities. Solution of complementarity problems \(\Rightarrow\) weak solution.