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United States Department of Agriculture

Agricultural Research Service

Title: Generalized Richards' Equation to Simulate Water Transport in Soils


Submitted to: Journal of Hydrology
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: July 8, 2002
Publication Date: March 10, 2003
Citation: Pachepsky, Y.A., Timlin, D.J., Rawls, W.J. 2003. Generalized Richards' Equation to Simulate Water Transport in Soils. Journal of Hydrology. Vol. 272, p. 3-13.

Interpretive Summary: Transport of soil water affects heat and solute transport in soils, defines rates of biological processes in soil and water supply to plants, governs transpiration and ground water replenishment, controls runoff, and has many other important functions in the environment. The Richards' equation is the most often used soil water transport model. However, this equation is not always valid even in simple experiments on water infiltration into horizontal soil column. The objective of this work was to generalize Richards' equation so that it would be applicable when the classic Richards' equation fails. We suggested that a shortcoming of the classic Richards' equation is in its implicit assumption of soil water quasi- particles performing a motion is similar to the motion of particles in free solution. Such motion may not happen in structured limited pore space of soil where water within aggregates may have difficulty moving to intraaggregate space, and long waiting periods are not uncommon for the quasi-particles of water. The modern physics suggests that the transport with long waiting periods can be modeled using fractional calculus. We used it to introduce the generalized Richards' equation that gave an excellent description of experiments on water infiltration into horizontal soil column. It appeared that the use of classical Richards' equation where the generalized equation is actually correct leads to scale-dependence and variability in soil hydraulic conductivity. Using the generalized Richards' equation promises to improve accuracy of simulations of soil water transport in many applications in hydrology, meteorology, agronomy, environmental protection, and other soil-related disciplines

Technical Abstract: Simulations of water transport in soil are ubiquitous, and the Richards' equation introduced in 1931 is the main tool for that purpose. For experiments on water transport in soil horizontal columns, Richards' equation predicts that volumetric water contents should depend solely on the ratio (distance)/(time)q where q=0.5. Substantial experimental evidence eshows that value of n is significantly less than 0.5 in some cases. Donald Nielsen and colleagues in 1962 related values of q < 0.5 to "jerky movements" of the wetting front, i.e. occurrences of rare large movements. The physical model of such transport is the transport of particles being randomly trapped and having a power law distribution of waiting periods. The corresponding mathematical model is a generalized Richards' equation in which the derivative of water content on time is a fractional one with the order equal or less than one. We solved this equation numerically and fitted the solution to data on horizontal water transport. The classical Richards' equation predicted a decrease of the soil water diffusivity as infiltration progressed whereas the generalized Richards equation described all observations well with a single diffusivity function. Validity of the generalized Richards' equation indicates presence of memory effects in soil water transport phenomena and may help to explain scale-dependence and variability in soil hydraulic conductivity encountered by researchers who applied classical Richards' equation. Several errors have been left unnoticed in the manuscript "Generalized Richards' Equation to Simulate Water Transport in Unsaturated Soils" published as a research paper in the Journal of Hydrology, 272:3-13 by Pachepsky, Ya. A., Timlin, and D. J., Rawls, W. J. Correct equations are presented. Fortunately, the errors do not affect main results and conclusions of the paper.

Last Modified: 9/10/2014
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