A theoretical approach for modeling unsaturated flow in spatially variable soils: Effective flow models in finite domains and nonstationarity

Models of unsteady unsaturated flow in large-scale heterogeneous soils are developed considering finite flow domains and nonstationarity of the soil properties and flow characteristics. The problem is cast into a more general stochastic framework than the originally proposed stationary spectral framework (Mantoglou and Gelhar, 1987a, b, c). The methodology considers the three dimensionality of the local governing flow equation, the nonlinear dependence of the local output on the local soil properties, as well as the effect of finite flow domains and nonstationarity of the soil properties and flow characteristics. The large-scale model representation is in the form of a partial differential equation, with large-scale “effective” parameters, subject to a set of initial and boundary conditions. The effective model parameters are related to a set of fluctuation covariance equations obtained by using a linearized fluctuation equation. This set of covariance equations and the corresponding large-scale model of the system are coupled and must be solved simultaneously. Particular cases of interest where stationarity in two or three spatial dimensions occurs are investigated, and, using spectral representations, the dimensionality of the covariance equations is reduced. Simple closed-form and practical expressions for the effective parameters valid in specific situations are presented, and illustrative examples are discussed. The theory and the models presented provide a more complete view of the large-scale unsaturated flow problem and can prove useful for evaluation of unsaturated flow phenomena of paramount importance in practical applications, for example, for predicting the movement of liquid wastes in the unsaturated zone.