The upstream mobility scheme is widely used to solve hyperbolic conservation equations numerically. When heterogeneities in the porous medium are introduced, the flux function in these equations attains a spatial discontinuity. In many systems the upstream mobility scheme approximates a solution close to the solution produced by solving the associated Riemann problem, but in the case of a heterogeneous system there exists no convergence analysis.

In this work the upstream mobility scheme is applied to a counter-current flow in a reservoir where discontinuities in the flux function are introduced through the permeability. Examples of such counter-current flow systems in heterogeneous reservoirs are Water-Alternating-Gas (WAG), Steam-Assisted Gravity Drainage (SAGD) and flow of CO2 and brine. The flux functions in these systems involve both advection and gravity segregation components. In this work we show that the upstream mobility scheme may exhibit large errors compared to the physically relevant solution for some combinations of flux functions in such a system. We show that a small perturbation of the relative permeability values can lead to a large difference in the solution produced by the upstream mobility scheme.

We look at the one-dimensional case since the Riemann solutions are mostly unknown for more than one dimension. However, in the multidimensional case, many numerical methods use one-dimensional calculations for the flux in the direction normal to the boundaries of the discretization cell. Not only does the scheme encounter large errors compared to the physically relevant solution, but the solution also lacks entropy consistency. Studies of these errors and the convergence performance of the scheme are important due to the extensive use of the upstream mobility scheme in the reservoir simulation community.

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