Finite analytic numerical method for two-dimensional fluid flow in heterogeneous porous media

Starting from the investigation on the detailed flow pattern, finite analytic numerical method is developed to solve the two-dimensional fluid flows in heterogeneous porous media. It is shown that only for some specific permeability distributions the pressure has the piecewise linear distribution, where harmonic average scheme works very well. In general case, the pressure will have the power-law behavior and its gradient will diverge as approaching the node joining the different permeability areas. The nodal flow effects cause the flow fingering into the high permeability region. It is a challenge problem to numerically describe the nodal fingering effects. With the help of the specific properties of pressure and its gradient around the node, a local analytical nodal solution is derived and then applied to construct a finite analytic numerical scheme. Numerical examples show that the detailed flow pattern can be reconstructed with the proposed numerical scheme under few grid refinements. Only with 2 × 2 or 3 × 3 subdivisions, the proposed numerical scheme can provide rather accurate solutions. The convergent speed of the numerical scheme is independent of the permeability heterogeneity. In contrast, the refinement ratio for the grid cell needs to be increased dramatically to get an accurate result when the traditional numerical method is used for strong heterogeneous cases.