Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media

In this work, the monotonicity of the symmetric cell-centred Multi-Point Flux Approximation (MPFA) method on triangular grids is studied for saturated and unsaturated flow in heterogeneous porous media. For the linear saturated flow, we derive necessary conditions on the triangulation to satisfy the Positive Transmissibility (PT) condition which ensures monotonicity of the discrete scheme. For the nonlinear unsaturated flow, an upwind MPFA scheme is developed using the upstream weighting of the hydraulic conductivity. We show analytically that this scheme cannot satisfy the PT condition whatever the kind of the triangulation. In spite of that, numerical experiments in 2D heterogeneous domains and unstructured meshes show that the upwind MPFA scheme avoids spurious oscillations in the numerical solution.