We consider transient flow in unsaturated heterogeneous porous media with uncertain hydraulic parameters. Our aim is to provide unbiased predictions (estimates) of system states, such as pressure head, water content, and fluxes, and to quantify the uncertainty associated with such predictions. We achieve this goal by treating hydraulic parameters as random fields, and the corresponding flow equations as stochastic. Current stochastic analyses of transient flow in partially saturated soils require linearization of the constitutive relations, which may lead to significant inaccuracies when these relations are highly nonlinear. If relative conductivity and saturation vary exponentially with pressure and the corresponding scaling parameters are random variables, the transient Richards equation is mapped onto a linear equation by means of the Kirchhoff transformation. This allows us to develop deterministic differential equations for the first and second ensemble moments of pressure and saturation. We solve these equations analytically, for vertical infiltration, and compare them with direct Monte Carlo simulations.