A new exact upscaling formula and investigations of numerical errors in simulation of single- and two-phase Darcy flow in barrier geometries

The pressure solution to a class of boundary value problems relevant for 2D Darcy flow in barrier geometries is derived using theta functions and elliptic integrals. From this solution an exact upscaling formula is established giving the effective permeability for flow through a homogenous layer bounded by two infinitely thin barriers with a single gap in each barrier. Comparisons with the exact pressure solution shows that the standard first order finite difference scheme underpredicts the volumetric flux in such barrier geometries when a given pressure drop is applied. Actually, using a coarse grid just fine enough to catch the description of the barrier geometry, the numerical scheme (the 'pressure solver') may underpredict the flux by more than a factor four. This numerical error is also present for two phase flow in the same type of geometry, and can be estimated from the corresponding single phase simulation. The errors observed for increasingly refined grids are tabulated for a few single and two-phase cases. Also exact pressure solutions for two examples are tabulated for benchmarking purposes.