Summary: We consider the following problem, arising within a geological model of sedimentation-erosion: For a given vector field \(g\) and a given nonnegative function \(F\) defined on a one- or two-dimensional domain \(\O\), find a vector field under the form \(\tilde g = u g\), with \(0 \leq u(x) \leq 1\) for a.e.\(x\in\O\), such that \(\div\tilde g + F \geq 0\) and \((u- 1)(\div\tilde g + F) = 0\) in \(\O\). We first give a weak formulation of this problem, and we prove a comparison principle on a weak solution of the problem. Thanks to this property, we get the proof of the uniqueness of the weak solution. The existence of a solution results from the proof of the convergence of an original scheme. Numerical examples show the efficiency of this scheme and illustrate its convergence properties.