Stochastic analysis of saturated–unsaturated flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method

In this study, a stochastic model for transient saturated–unsaturated flow is developed based on the Karhunen-Loeve expansion of the input random soil properties combined with a perturbation method. The log-transformed saturated hydraulic conductivity f(x) and the soil pore size distribution parameter α(x) are assumed to be normal random functions with known covariances. We decompose f(x) and α(x) as infinite series in a set of orthogonal normal random variables by the Karhunen-Loeve expansion and expand the pressure head as polynomial chaos with the same set of orthogonal random variables. The perfectly correlated and uncorrelated cases between f(x) and α(x) are studied. By using the Karhunen-Loeve expansion of the input random parameters, polynomial chaos decomposition of pressure head, and the perturbation method, the saturated–unsaturated flow equation and the corresponding initial and boundary conditions are represented by a series of partial differential equations in which the dependent variables are the deterministic coefficients of the polynomial chaos expansion. Once the partial differential equations are solved subsequently by a numerical method, the random representation of pressure head is obtained by combining the deterministic coefficients obtained and the random variables from the Karhunen-Loeve expansion of the input random functions. The moments of pressure head and water content are determined directly from the random representation of the pressure head. We demonstrated the applicability of the proposed KL-based stochastic model with some examples of unsaturated and saturated–unsaturated flow in two dimensions, and compared the results with those from the moment-based stochastic model. It is shown that the KL-based models are more computationally efficient than the conventional moment-based models.