Immiscible Two-phase Darcy Flow in Fractured Porous Media - New Robust Formulation and Application to the Tight Gas Recovery

摘要

Numerical simulations of two-phase Darcy flows in heterogeneous porous media requires choosing an appropriate set of primary unknowns, which may be challenging, especially when dealing with very flat capillary pressure curves, dry regions or saturation jumps at the rock type interfaces. The classical approaches fail to cope with all of those difficulties. In particular the two-pressure formulation allows handling saturation jumps, but beaks down if the capillary pressure doesn't depend on saturation. It also lacks robustness when dealing with nearly residual water saturations. On the other hand, for homogeneous medium, the pressure - saturation formulation is known to be robust when dealing with dry media and can handle vanishing or constant capillar) pressure curves. Unfortunately it is not always possible to extend it to the case of discontinuous capillary pressure curves. In this paper, a new formulation based on parametrization techniques for the capillary pressure monotone graph extension is proposed which handles all the above mentioned difficulties while still using only two unknowns by degree of freedom. We illustrate the efficiency of our approach by numerous numerical experiments dealing with water gas flow in fractured tight gas reservoirs using the data set presented in [2]. Following [1], the fractures are modelled as interfaces of codimension one with continuous pressure at the matrix fracture interfaces. During the injection phase of the simulation, water penetrates only a few tens of centimetres deep in the matrix rock. Therefore, in order to obtain an accurate numerical approximation, an anisotropic refinement of the mesh is used in the neighbourhood of the fractures using prismatic elements. The connection with the surrounding tetrahedral mesh in the matrix domain is achieved using pyramids. Following [1], the model is discretized using the Vertex Approximate Gradient scheme which allows for general polyhedral cells. The numerical performance of the new approach is evaluated for various choices of capillary pressures curves. The comparison with classical formulations shows that the new approach is more efficient both in terms of Newton iterations and CPU tune.