Explicit analytical solutions for one-dimensional steady state flow in layered, heterogeneous unsaturated soils under random boundary conditions

摘要

In this study, we directly derive first-order analytical solutions to the pressure head moments (mean and variance) for one-dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions. We assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model, and treat the saturated hydraulic conductivity K_s as a random function and the pore size distribution parameter a as a random constant. Unlike the solution given in Lu and Zhang (2004) in which Kirchhoff transformation was used and the solution to pressure head variance was presented as a function of (cross-)covariances related to the intermediate, Kirchhoff-transformed variable, the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities. In addition, we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content. These first-order analytical solutions are compared with those from Monte Carlo simulations. We also investigated the effect of uncertain boundary conditions, the relative contribution of input variabilities to the head variance, and the possible errors introduced by treating the correlated α field as a random constant in the analytical solutions. The results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column. Furthermore, it is found that treating a as a random constant is justified when the correlation length of α is relatively large as compared to the layer thickness.