On a Space-Fractional Generalization of the Black Oil Model

Abstract

Classical multiphase filtration models, which are based on Darcy's law, are well studied and actively used in modern oil engineering. However, such models do not allow efficient describing of processes with power-law memory effects or with spatial non-locality effects. In recent years, there has been a significantly increasing interest in models with fractional derivatives and integrals that allow such effects to be taken into account.The article considers the space-fractional generalization of the two-phase Black Oil model. This model is built on the basis of a fractional modification of the Darcy’s law with the Riesz potential, which is one of the possible generalizations of a fractional integral to the case of a multidimensional space. The use of such a modification of Darcy's law allows efficient describing of filtration processes in heterogeneous fractured porous media with the effects of spatial non-locality.For the numerical solution of the system of equations from the obtained model, the use of the IMPES method is proposed. For this purpose, from the presented system of equations, one equation is selected that describes the evolution of pressure. This equation is written provided that the capillary pressure variation within the time step is neglected.A special case of the pressure equation with the Riesz potential of radial function is considered, which describes the pressure variation in the case of a radial plane flow. For this equation, a self-similar solution is constructed using the Mellin integral transform method. A representation of this solution in the form of the Mellin-Barnes contour integral was obtained, which made it possible to write it in terms of the Fox functions. It is shown that in the limiting case of a zero degree of the Riesz potential, this solution coincides with the self-similar solution of the classical heat equation. The constructed self-similar solution can be further used in the software implementation of the numerical solution of the model presentedThe main line of further research is to develop and implement a software computing system based on the proposed fractional generalization of the two-phase Black Oil model.