New analytic element solutions are presented for unsaturated, two-dimensional steady flow in vertical planes that include nonoverlapping impermeable elliptical layers and elliptical inhomogeneities. The hydraulic conductivity, which is represented by an exponential function of the pressure head, differs between the inside and outside of an elliptical inhomogeneity; both the saturated hydraulic conductivity and water retention parameters are allowed to differ between the inside and outside. The Richards equation is transformed, through the Kirchhoff transformation and a second standard transformation, into the modified Helmholtz equation. Analytic element solutions are obtained through separation of variables in elliptical coordinates. The resulting equations for the Kirchhoff potential consist of infinite sums of products of exponentials and modified Mathieu functions. In practical applications the series are truncated but still fulfill the differential equation exactly; boundary conditions are met approximately but up to machine accuracy, provided that enough terms are used. The pressure head, saturation, and flow may be computed analytically at any point in the vadose zone. Examples are given of the shadowing effect of an impermeable elliptical layer in a uniform flow field and funnel-type flow between two elliptical inhomogeneities. The presented solutions may be applied to study transport processes in vadose zones containing many impermeable elliptical layers or elliptical inhomogeneities.@en

Natural soils are generally populated with a wide variety of macropores formed from physical processes and/or biological activity. These macropores can have a large influence on the lateral flow of water in hillslope soils even when those macropores are not continuous or connected directly to ponded water. The concept of self-organization of flow paths described by SIDLE et al. (2001) is analyzed through numerical simulation of variably-saturated flow in a large cylinder of soil containing a population of disconnected macropores. It is demonstrated that there is a threshold water pressure at which the macropores will become active, and above this threshold the then active network of macropores significantly increases the effective conductance of the soil volume. In the case examined here the increase exceeded 40%. The analysis presented provides a context for the explanation of soil pipe formation by the process of seepage erosion. An analogy is drawn between percolation theory in porous media and the concept of self-organization of flow pathways at the hillslope scale.@en

The analytic element method, first developed for modeling flow in saturated porous media, is adapted to the solution of the governing equation for steady flow in unsaturated porous media. The governing equation for steady-state unsaturated flow is made amenable to analytic element solution through transformation with the Kirchhoff integral, representation of the hydraulic conductivity by an exponential function of the pressure head, and use of a coordinate transformation. A number of analytic elements are available for the resulting modified Helmholtz equation; analytic element equations are presented for uniform flow, cylindrical drains, and cylindrical inhomogeneities. The analytic element solution allows for the analytic evaluation of the pressure head, saturation, and Darcy flux at any point in the vadose zone. The applicability of the analytic element method to simulate unsatu-rated flow is demonstrated by solving for several cases of steady flow in a region containing arbitrarily located cylindrical inhomogeneities, and for flow in a region containing one cylindrical inhomogeneity and two cylindrical drains.@en