Unsaturated flow through a spherical inclusion

Unsaturated flow is considered through a spherical inclusion. The hydraulic conductivity is of the form K_iexp(αh), where the saturated conductivity K_i is different in the main flow regime, the inclusion α is a constant in the entire flow domain, and h is the pressure head. The solution technique is analogous to that used by the authors previously to analyze flow through circular inclusions for two-dimensional flow by reducing Richards' equation to the Helmholtz equation and applying the analytic element method. Comparisons show differences between the two-dimensional case (circles) and three-dimensional case (spheres) in terms of flow enhancement and exclusion through the inclusions. When the inclusion permeability is less than the background conductivity, a lesser fraction of flow occurs through the three-dimensional case than for the two-dimensional case; conversely, when the permeability is higher within the inclusion, there is a higher enhancement of flow through that region in the case of the three-dimensional inclusion.