### The soil moisture velocity equation

Fred L. Ogden, Myron B. Allen, Wencong Lai, Jianting Zhu, Mookwon Seo, Craig C. Douglas, Cary A. Talbot
2017 Journal of Advances in Modeling Earth Systems
Numerical solution of the one-dimensional Richards' equation is the recommended method for coupling groundwater to the atmosphere through the vadose zone in hyperresolution Earth system models, but requires fine spatial discretization, is computationally expensive, and may not converge due to mathematical degeneracy or when sharp wetting fronts occur. We transformed the one-dimensional Richards' equation into a new equation that describes the velocity of moisture content values in an
more » ... soil under the actions of capillarity and gravity. We call this new equation the Soil Moisture Velocity Equation (SMVE). The SMVE consists of two terms: an advection-like term that accounts for gravity and the integrated capillary drive of the wetting front, and a diffusion-like term that describes the flux due to the shape of the wetting front capillarity profile divided by the vertical gradient of the capillary pressure head. The SMVE advection-like term can be converted to a relatively easy to solve ordinary differential equation (ODE) using the method of lines and solved using a finite moisture-content discretization. Comparing against analytical solutions of Richards' equation shows that the SMVE advection-like term is >99% accurate for calculating infiltration fluxes neglecting the diffusion-like term. The ODE solution of the SMVE advection-like term is accurate, computationally efficient and reliable for calculating one-dimensional vadose zone fluxes in Earth system and large-scale coupled models of land-atmosphere interaction. It is also well suited for use in inverse problems such as when repeat remote sensing observations are used to infer soil hydraulic properties or soil moisture. Plain Language Summary Since its original publication in 1922, the so-called Richards' equation has been the only rigorous way to couple groundwater to the land surface through the unsaturated zone that lies between the water table and land surface. The soil moisture distribution and properties of the soil in the unsaturated zone determine how much precipitation becomes runoff or infiltrates into the soil. During nonrainy periods, the soil moisture distribution determines how much water is available for use by plants or for groundwater recharge. Richards' equation is arguably the most difficult equation to accurately and reliably solve in hydrologic science. The first somewhat robust computational solution was not published until 1990. We have converted Richards' equation into a new form that is much simpler to solve and 99% accurate for calculating the vertical flow of water in unsaturated soil in response to rainfall and changes in groundwater levels. Where Richards' equation allows calculation of the change in degree of saturation with time at a point in an unsaturated soil, our simpler equation allows calculation of the speed of travel of specific moisture contents in the soil. For this reason we call this new equation the Soil Moisture Velocity Equation (SMVE). Key Points: We have converted the onedimensional Richards partial differential equation into a new form that is much easier to solve Our new equation is an ordinary differential equation that is accurate, reliable, guaranteed to converge and to conserve mass Neglecting the diffusion term in our new equation has a negligible effect on the calculated flux of water, with errors less than 1% Supporting Information: Supporting Information S1