We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definition of entropy solution of the model in the sense of Kru z ˇ \check {\mbox {z}} kov and prove convergence of the scheme to the unique B V BV entropy solution of the problem, up to satisfaction of one of the boundary conditions.