  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. Combining with moving least square
approximation and local Taylor expansion, spatial derivatives of oil-phase
pressure at a node are approximated by generalized difference operators in the
local influence domain of the node. By introducing the first-order upwind
scheme of phase relative permeability, and combining the discrete boundary
conditions, fully-implicit GFDM-based nonlinear discrete 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,
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, roughly
studies the convergence order thus find that the low-order error of GFDM makes
the convergence order of GFDM lower than that of FDM when node spacing is
small, and points out the significant effect of the symmetry or uniformity of
the node collocation in the node influence domain on the accuracy of
generalized difference operators, and the radius of the node influence domain
should be small to achieve high calculation accuracy, which is a significant
difference between the studied hyperbolic two-phase porous flow problem and the
elliptic problems when GFDM is applied.
