You Don't Need Richards'... A New General 1-D Vadose Zone Solution Method that is Reliable

Abstract:

Hydrologic modelers and mathematicians have strived to improve 1-D Richards' equation (RE) solution reliability for predicting vadose zone fluxes. Despite advances in computing power and the numerical solution of partial differential equations since Richards first published the RE in 1931, the solution remains unreliable. That is to say that there is no guarantee that for a particular set of soil constitutive relations, moisture profile conditions, or forcing input that a numerical RE solver will converge to an answer. This risk of non-convergence renders prohibitive the use of RE solvers in hydrological models that need perhaps millions of infiltration solutions. In lieu of using unreliable numerical RE solutions, researchers have developed a wide array of approximate solutions that more-or-less mimic the behavior of the RE, with some notable deficiencies such as parameter insensitivity or divergence over time. The improved Talbot-Ogden (T-O) finite water-content scheme was shown by Ogden et al. (2015) to be an extremely good approximation of the 1-D RE solution, with a difference in cumulative infiltration of only 0.2 percent over an 8 month simulation comparing the improved T-O scheme with a RE numerical solver. The reason is that the newly-derived fundamental flow equation that underpins the improved T-O method is equivalent to the RE minus a term that is equal to the diffusive flux divided by the slope of the wetting front. Because the diffusive flux has zero mean, this term is not important in calculating the mean flux. The wetting front slope is near infinite (sharp) in coarser soils that produce more significant hydrological interactions between surface and ground waters, which also makes this missing term 1) disappear in the limit, and, 2) create stability challenges for the numerical solution of RE. The improved T-O method is a replacement for the 1-D RE in soils that can be simulated as homogeneous layers, where the user is willing to neglect the effects of soil water diffusivity. This presentation emphasizes the transformative nature of the improved T-O finite water-content solution, and highlights the benefits of the methods' reliability in high-resolution large watershed simulations in the high performance computing environment, and discusses coupling of the soil matrix and non-Darcian macropores.