Effective parameter functions for the Richards equation in layered porous media

Upscaling of the Richards equation is important for large-scale modeling of water flow in the unsaturated zone. We derive an upscaled model for Richards' equation in a layered porous medium. Homogenization theory is applied to derive the upscaled equations for slow flow processes. Two flow regimes are compared. The first flow regime is quantified by small Bond numbers, meaning forces due to pressure gradients are dominant on the small scale. The second flow regime is quantified by large Bond numbers, meaning that forces due to pressure gradients and gravity contribute equally on the small scale, while the large scale is dominated by gravity forces. The case of intermediate Bond numbers is also addressed. We compare the effective relative permeability function and the effective capillary pressure head-saturation function for both flow regimes. In the case of small Bond numbers the effective curves can be derived based on capillary equilibrium. The procedure to calculate effective curves is then quite simple. In the case of large Bond numbers the procedure is more complex. However, comparing effective curves of different test cases showed that the difference between the effective curves for different Bond numbers is not that large for moderate parameter contrasts and when the gradients of the capillary-saturation curve do not become too small. The effective parameter functions calculated with a capillary equilibrium assumption are therefore often a good estimate.