A new general 1‐D vadose zone flow solution method

Abstract

We have developed an alternative to the one-dimensional partial differential equation (PDE) attributed to Richards (1931) that describes unsaturated porous media flow in homogeneous soil layers. Our solution is a set of three ordinary differential equations (ODEs) derived from unsaturated flux and mass conservation principles. We used a hodograph transformation, the Method of Lines, and a finite water-content discretization to produce ODEs that accurately simulate infiltration, falling slugs, and groundwater table dynamic effects on vadose zone fluxes. This formulation, which we refer to as ''finite water-content'', simulates sharp fronts and is guaranteed to conserve mass using a finite-volume solution. Our ODE solution method is explicitly integrable, does not require iterations and therefore has no convergence limits and is computationally efficient. The method accepts boundary fluxes including arbitrary precipitation, bare soil evaporation, and evapotranspiration. The method can simulate heterogeneous soils using layers. Results are presented in terms of fluxes and water content profiles. Comparing our method against analytical solutions, laboratory data, and the Hydrus-1D solver, we find that predictive performance of our finite water-content ODE method is comparable to or in some cases exceeds that of the solution of Richards' equation, with or without a shallow water table. The presented ODE method is transformative in that it offers accuracy comparable to the Richards (1931) PDE numerical solution, without the numerical complexity, in a form that is robust, continuous, and suitable for use in large watershed and land-atmosphere simulation models, including regional-scale models of coupled climate and hydrology.

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

“…Fluxes calculated using the SMVE advection-like term agree very closely with the numerical solution of Richards' equation (Ogden et al, 2015a), data from column experiments (Ogden et al, 2015b) and exact analytical solutions (Ogden et al, 2017). Results in all cases show that the diffusion-like term in Eq.…”

“…The first term on the righthand side Eq. [20], which is called the "advection-like" term, was derived by Ogden et al (2015a). The second term on the righthand side of Eq.…”

Comment on “Beyond the SCS‐CN method: A theoretical framework for spatially lumped rainfall‐runoff response” by M. S. Bartlett et al.

“…Perform data collection to obtain meaningful parameters from a wide variety of hydro‐geographic settings so that we can reduce reliance on calibration for parameter estimation. Broad application of methods/models developed to simulate infiltration and groundwater‐surface water interactions [ List and Radu , ; Ogden et al ., ] are largely hamstrung by the lack of meaningful soils hydraulic data for assigning parameters and minimizing the need for parameter estimation through calibration. This activity would necessarily include creation of data sets for model testing and help to support the first two objectives in this list.…”

Infiltration from the Pedon to Global Grid Scales: An Overview and Outlook for Land Surface Modeling

“…Conventional solution methods to the highly nonlinear Richards one‐dimensional partial differential equation used in LSMs inevitably lead to numerical and accuracy issues, which impact their hydrological performance. Land surface models may consider the implementation of alternative, one‐dimensional, unsaturated zone flow solution methods (such as those provided by Ogden et al, 2015). The Ogden Soil Moisture Velocity Equation (SMVE) approach uses the hodograph method to transform Richards' equation into a differential equation for a velocity and uses a discretization of the resulting equation in the form of “bins” containing values of the water content.…”

“…The scheme is computationally efficient, although the explicit time steps are limited by stability considerations because there are no convergence limits as imposed by implicit schemes. Ogden et al (2015) considered the transport of three regimes of soil water in detail, namely infiltration, wetting fronts disconnected from the surface, and groundwater recharge. The SMVE method offers accuracy comparable to, or in some cases exceeding, that of the numerical solution of the Richards partial differential equation method but without the numerical complexity and in a form that is robust, continuous, and suitable for use in models of coupled climate and hydrology at a range of scales.…”