Uncertain soil properties are often modeled as random fields. This renders the unsaturated flow equations stochastic. Determining the statistics of pressure head, ψ, is nontrivial, since the Richards equation is highly nonlinear. The prevalent approach is to linearize relative hydraulic conductivity, K_r(ψ) around the ensemble mean pressure head, ‹ψ›, which often leads to significant errors. Recently, an approach has been proposed to avoid such a linearization for the Gardner model, K_r = exp(αψ); with the soil parameter α being a random variable. We generalize this approach by allowing α to be a random field. This is achieved by means of a partial mean-field approximation with respect to α(x). Using two-dimensional infiltration into a heterogeneous soil with uncertain hydraulic parameters as an example, we demonstrate that our predictions of the mean pressure head and its variance remain accurate for moderately variable α's. The robustness of our solutions increases with the correlation length of α.