Qualitative and numerical results on the soil--water redistribution
Authors: Barontini Stefano , Siena Martina , Peli Marco 
Published: February 2023

The soil--water redistribution is an interesting and complex process that takes place after an abundant imbibition of the uppermost soil layers, as after rainfall or irrigation. It is a consequence of the concurrency of other processes, that are downward advection and diffusion, surface evaporation and root water uptake. It is therefore simultaneously characterised by downward and upward water flow and by water extraction, and, as a consequence, it plays a key role at partitioning the water fluxes through the soil, with feedbacks also on the mass fluxes and on the soil layering.Aiming at contributing to better understanding this process, we present a theoretical (qualitative) and a numerical assessment of some properties of the soil--water redistribution, based on the classical framework of the Richards equation.The qualitative analysis focuses on the evolution of the soil--water content of an imbibition front, as a consequence of the onest of a continuous surface evapotranspiration. The process is analogically depicted with the traditional description of the flood--wave propagation in free--surface flow. Particularly we show that, considering an instantaneous water--content wave within the uppermost soil, an observer would meet, from the bottom moving upward, the planes where:The (downward) Darcian velocity q is locally maximum in time, &#8706;q / &#8706;t = 0, where the water content &#952; is in imbibition;
&#952; is locally maximum, &#8706;&#952; / &#8706;t = 0, where q is instantaneously maximum in space, &#8706;q / &#8706;x = 0, and &#952; is still in imbibition;
&#952; is instantaneously maximum in space, &#8706;&#952; / &#8706;x = 0, where the downward flux is purely gravitational, i.e. q = K, being K(&#952;) the hydraulic conductivity;
q = 0, i.e. the zero--flux plane, that separates the downward from the upward flux, where &#952; is instantaneously increasing in space, &#8706;&#952; / &#8706;x > 0;
q is instantaneously minimum in time (i.e. the upward flux is maximum), where&#160;&#8706;&#952; / &#8706;x > 0.
Morevover an observer who follows the peak of water content would see it reducing in space and time, being its total derivative d&#952; / dt < 0, until it vanishes.If the observer stops at fixed depth, these patterns would reflect in a cycle in the (&#952;,q) phase plane, where, starting from initially hydrostatic condition, one would observe the (local) maximum q, the (local) maximum &#952;, the onset of an advective flow q = K when the (spatial) peak of &#952; passes at that depth, the passage of the zero--flux plane, the minimum q and then the hydrostatic condition again.The evapotranspiration is however ruled by diurnal cycles and the soil--water dynamics vary depending on the development of the root apparati. We provide an insight on these aspects by means of the numerical simulation, performed with Hydrus1d. It shows that diurnal evapotranspiration cycles induce fluctuations in the depth of the zero--flux plane and that the root water uptake, which shares the evapotranspirative demand in the whole domain, reduces the uppermost upward flux, thus allowing the zero--flux plane reaching deeper depths.