Philip  was the first to investigate the stability of a wetting front in a stratified soil using rigorous hydrodynamic arguments. He based his analysis on the Green and Ampt  model and treated permeability as a known function of depth. We adopt the same model to develop integro-differential equations for leading statistical moments of wetting front propagation in a three-dimensional, randomly heterogeneous soil. We solve these equations analytically for mean front position and mean pressure head gradient in one spatial dimension, to second order in the standard deviation of log conductivity. We do the same for second moments of front positions and pressure head gradient, which serve as measures of predictive uncertainty. To verify the accuracy of our solution, we compare it with the results of numerical Monte Carlo simulations.