Field Heterogeneity: Some Basic Issues

TLDR: In this article, the authors consider unsaturated and generally unsteady flow in heterogeneous systems and review the mathematical nature of the flow equation, the concept of scale-heterogeneity, analytical and quasi-analytical solutions.

Abstract: Present-day soil-water physics enables useful quantitative predictions in the laboratory and in simple field situations. Difficulties, however, frequently arise for areas of appreciable size in the field. Known and unknown heterogeneities, on many scales, may vitiate predictions based on theory for homogeneous, or very simple heterogeneous, systems. Two types of heterogeneity are distinguished, deterministic and stochastic. The first often demands an extension of established analyses and may involve important phenomena absent from the analogous homogeneous problem. Stochastic heterogeneity may involve many scales and is imperfectly known. The statistical properties may be stationary, but in more complicated cases, randomness may be embedded in (either known or unknown) systematic trends. Some aspects of unsaturated and generally unsteady flow in heterogeneous systems are reviewed: the mathematical nature of the flow equation; the concept of scale-heterogeneity; analytical and quasi-analytical solutions. The enormity of the total problem of unsaturated unsteady flows in stochastic heterogeneous systems is illustrated through a dialectic of eight successive stages of simplification. The concept of the autocorrelation function governing λ, the internal characteristic length, is introduced; and the problem posed in terms involving the distribution and autocorrelation functions of λ, the reduced potential and conductivity functions, and the initial and boundary conditions as the data, from which it is required to establish distribution functions of various descriptors of the flow. The solution to a grossly simplified example of horizontal absorption is presented. Mean apparent sorptivity decreases rapidly to about one fifth of the mean (and about half the minimum) sorptivity of the component soils. Variation about the mean is very great but decreases as absorption proceeds. The example epitomizes the failure of additivity of properties in stochastic heterogeneous media, which arises because spatial textural changes in either sense tend to reduce unsaturated flow rates.