Two-dimensional unsaturated flow through a circular inclusion

Abstract

Two-dimensional unsaturated flow is considered through a circular inclusion. The hydraulic conductivity is of the form Kiexp(αh) where the saturated conductivity Ki is different in the main flow regime and the inclusion, α is a constant in the entire flow domain, and h is the pressure head. The problem reduces to the Helmholtz equation as previously used by J. R. Philip and colleagues for solving problems with impermeable regions or cavities in an unsaturated regime. The pressure heads are continuous on the interface of the inclusion. The normal flow velocities at the interface are matched approximately using the analytical element method recently exploited for saturated domains in several studies. Flow enhancement and exclusion through the circular inclusions are dependent on the value of α and the radius of the cylinder but otherwise are similar to that for the saturated case. For example, for ratios of the inclusion to background saturated conductivity of 0.5 the flow is 0.74 of what it would be without the inclusion compared to 0.67 for the saturated case. This was calculated for a dimensionless radius (0.5 α multiplied by the physical radius) of 1. When the ratio of the inclusion saturated conductivity to the background is 5, the comparable value for the unsaturated case is 1.45 and for the saturated case is 1.67.