An Upwind Generalized Finite Difference Method for Meshless Solution of Two-phase Porous Flow Equations

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation, and the computational domain is divided into overlapping sub-domains centered on each node. Combining with moving least square approximation and local Taylor expansion, derivatives of oil-phase pressure at the central node are approximated by a generalized difference operator in the local subdomain. By introducing the first-order upwind scheme of phase permeability, and combining the discrete boundary conditions, fully implicit GFDM discrete nonlinear equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation technology, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, and points out the significant effect of the symmetry or uniformity of the node allocation in the node influence domain on the accuracy of the generalized difference operator, and the radius of node influence domain should be as small as possible to achieve high calculation accuracy, which is a significant difference between the studied parabolic two-phase porous flow problem and the elliptic equation previously studied by GFDM. In all, the upwind GFDM with the fully implicit nonlinear solver and related analysis about computational performances given in this work may provide a critical reference for developing a general-purpose meshless numerical simulator for porous flow problems.