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Sinusoidal fluctuations of the flux at the soil surface dampen with depth in the vadose zone such that beyond a certain depth, flow may be approximated as steady. To investigate the damping with depth, we developed a new analytic solution for vertical periodic flow in the vadose zone, with a sinusoidal flux specified at the soil surface. The solution is based on the use of the Gardner–Kozeny model for hydraulic conductivity and soil moisture and on a linearization of the diffusive and advective terms in the governing differential equation. A characteristic length is presented for the damping of the flux with depth. At a depth of three times the characteristic length, the amplitude of the flux had reduced to 5% of the value at the surface. We compared the analytic solution to finite-element solutions of the original, nonlinear differential equation for 16 soils based on four reference soils, using a daily sinusoidal cycling of evaporation and infiltration at the soil surface. For the soils and circumstances investigated, the analytic solution produced reasonable values of the damping factor at any depth in the soil profile compared with the finite-element solutions. The solution is more accurate when the fluctuations (both amplitude and period) are smaller. The presented solution may be used for general cases of one-dimensional infiltration when the surface flux is written as a Fourier series.

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 unsaturated 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.