Analytical decomposition of the nonlinear unsaturated flow equation

Abstract

An analytical procedure for the solution of the nonlinear unsaturated flow equation is proposed. The horizontal and vertical flow equations subject to constant boundary conditions and continuous wetting are approximated using analytical decomposition series. The results are verified with the Philip [1955] and the Parlange [1971] classical solutions and with experimental data. A convergence theorem with proof and discussion on the numerical accuracy of decomposition series are included. Applications to the modeling of infiltration subject to hysteresis in the soil-water functional relationships, soil heterogeneity, and time variable point rainfall are also discussed. Other potential applications in water management models under more realistic conditions are possible. The methodology offers advantages over the common small-perturbation solutions: the possibility to study nonstationarity in the random quantities; statistical separability from the physics of the problem (i.e., independence between the system parameters and the input quantities), rather than forced through closure approximations; and the construction of an analytical series solution that converges uniformly to the true nonlinear solution. Finally, because of stability and computational economy with respect to linearized numerical solutions, an analytical solution appears promising.