This chapter addresses one-dimensional infiltration and vertical flow problems. Traditionally, infiltration has received more attention than other unsaturated flow procedures, both for empirical formulations and for applications of Richards’ equation. Rarely is infiltration the only process of interest, and from an overall point of view it is only one example of soil water dynamics. Here, we will first emphasize systems for which analytical (or quasi-analytical) solutions can be found. These include the Green and Ampt solution (1911), which adds gravity to the simplified analysis discussed in chapter 4. Then a linearized form of Richards’ equation will be examined, followed by the perturbation of the horizontal problem of Philip leading to his famous series solution. Although the closed-form and quasi-analytical solutions are convenient for calculations and discussing the physical principles, generally, the nonlinearity of Richards’ equation precludes such convenient forms. However, numerical approximations can be used. The conventional numerical methods applied in water and solute transport are based on finite differences and finite elements. Because of its greater simplicity, we will emphasize finite differences and build on the methodology from the saturated-flow example in chapter 3. Richards’ equation is a parabolic partial differential equation reducing to an elliptical form for steady-state cases. The analyses and methods parallel developments for techniques developed primarily for the linear diffusion equation. Many texts exist for numerical methods; one to which we refer is by Smith (1985). Ideally, numerical methods give solutions that are as accurate as the input warrants or as necessary for application. In some cases, results may be easier or more accurate than the evaluation of a complex analytical expression. Clearly, infiltration is of limited duration, with drainage and redistribution occurring over much longer time frames. We will visit briefly some steady-state examples, including layered profile and upward flow from a shallow water table. Other examples include modeling plant water uptake from the profile and drainage of initially wet profiles. The rapid increase in computational power and availability of computers make solutions feasible and routine for problems that were very tedious or time consuming only a few years ago. This is particularly true of the one-dimensional numerical solutions.
Hyperpolarizabilities for the one-dimensional infinite single-electron periodic systems. I. Analytical solutions under dipole-dipole correlations