  We study a nonlinear fourth-order extension of Richards' equation that
describes infiltration processes in unsaturated soils. We prove the
well-posedness of the fourth-order equation by first applying Kirchhoff's
transformation to linearize the higher-order terms. The transformed equation is
then discretized in time and space and a set of a priori estimates is
established. These allow, by means of compactness theorems, extracting a unique
weak solution. Finally, we use the inverse of Kirchhoff's transformation to
prove the well-posedness of the original equation.
