In this paper we develop a finite volume model to solve the two-dimensional shallow water equations governing the propagation of two superimposed layers, with the upper water layer carrying a dilute sed- iment suspension, and the underlaying layer being a high concentration non-Newtonian fluid mud mix- ture. The model formulation contains non-conservative terms as well as source terms. We propose a scheme able to deal with varying topography and dry areas, providing well-balanced solutions when both water and fluid mud are quiescent. The model is tested against both exact solutions and numerical exam- ples. The results show the ability of the model to deal with wetting and drying of both water and fluid mud layers, providing mass-conservative solutions. Moreover, the model solves discontinuities and steep fronts, computing accurate and oscillation-free solutions.