In reservoir simulation the pressure is the solution of an elliptic equation. It follows from the maximum principle that this equation satisfies a monotonicity property. This property should be preserved when discretising the pressure equation. If this is not the case, the pressure solution may have false internal oscillations and extrema on no-flow boundaries. Thus, there is a need for sufficient conditions for the discretization methods to be monotone. These conditions will depend on the permeability tensor and the grid. Previously monotonicity for control volume methods has been studied on quadrilateral grids and on hexagonal grids. The former was done for general methods which reproduces linear potential fields, while the latter was done in a Control Volume Finite Element setting. These analyses have given sharp sufficient conditions for the discretisation methods to be monotone. However, for methods whose cell stencils include cells which do not have any edges common with the central cell in the discretisation scheme, no work has been done on unstructured grids. In this work, we study monotonicity on triangular grids for control volume methods which are exact for linear potential fields. We derive sufficient conditions for monotonicity of the MPFA-O and -L methods. The found monotonicity regions for the MPFA methods are also tested numerically. The tests are done both on uniform grids in homogeneous media, and on perturbed grids, which corresponds to heterogeneous media. The investigations are done for single phase flow only. However, the results are relevant for multiphase simulations. The results obtained in this work may be utilised in grid generation. In this way we can construct grids where the discretisation of the pressure equation is guaranteed to be monotone.


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