Meshless Multi-Point Flux Approximation of Fluid Flow in Porous Media

Abstract

Abstract In the last decade, the multi-point flux approximations have found significant interest. However, non-physical oscillations can appear in the developed multi-point flux approximations when the anisotropy is really strong. It has been found that the oscillations are closely related to the poor approximation of pressure gradient in the flux computation. In this paper, the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media is proposed. The MMPFA is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems and can be extended to include higher-order terms in the appropriate Taylor series. The proposed MMPFA is combined with the mixed corrections which ensure linear completeness. The mixed correction utilizes Shepard Functions in combination with a correction to derivative approximations. Incompleteness of the kernel support combined with the lack of consistency of the kernel interpolation in conventional meshless method results in fuzzy boundaries. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The resulting normalized and corrected MMPFA scheme not only ensures first order consistency O(h) but also alleviates the particledeficiency (kernel support incompleteness) problem. Furthermore, a number of improvements to the kernel derivative approximation are proposed. The primary attraction of the present method is that it provides a weak formulation for Darcy's law which can be of use in further development of meshless methods. The SPH model can be used to model three-dimensional miscible flow and transport in porous media with complex geometry, and for large (field) scale simulation of transport in porous media with general permeability distributions. To illustrate the performance of the MMPFA, a modelling of a single phase fluid flow in fully anisotropic porous media is presented.