Stochastic analysis of unsaturated steady state flow through bounded heterogeneous formations

The effect of heterogeneity in saturated hydraulic conductivity (Ks) and in a pore‐scale distribution parameter (α) on unsaturated steady state flow in bounded domains is studied. The properties are assumed to be random space functions having stationary means and covariances. An analytic model based on small‐perturbation approximation in Ks and α for predicting the spatial moments of the pressure head from the moments of the input variables is developed. The solution applies to the entire unsaturated domain without invoking the unit mean hydraulic gradient assumption. Under conditions of small‐scale variability the solution simplfies considerably resulting in closed‐form expressions for the moments of the pressure head. These expressions are tested in synthetically generated random fields of α and Ks having prescribed statistical properties.