A problem of significant interest to environmental scientists is the flow of water and solutes through the vadose zone. The partial differential equations which govern this flow are typically time-dependent and nonlinear. Valid solutions to these equations require (1) accurate relationships between various coefficients and variables on which they depend (e.g., coefficient of diffusivity and water content) and (2) sophisticated numerical methods which can handle complexities such as sharp moving fronts. In cases where coefficients are not known explicitly, curve-fitting techniques are needed to smooth out scattered experimental data. Nonlinear coefficients can then be calculated. A constrained least squares spline fit is compared to empirical function fits which have appeared recently. Then, a state-of-the-art numerical technique is used to accurately model transient flow through unsaturated homogeneous soils. The moving finite element method of Miller and colleagues is an adaptive approach in the sense that the grid moves so that nodes are concentrated where they are most needed. As a result, better accuracy is achieved with fewer nodes that are required for standard fixed-grid methods. Petzold's robust Gear-type solver DASSL is used for time-integration. Numerical results are compared to experimental data. Mass balance errors are neglible, and accurate solutions are obtained at all time steps. Though only one-dimensional problems are considered here, the numerical approach generalizes to heterogeneous media and problems in higher dimensions. ¿ American Geophysical Union 1994