Darcy’s Law and Unsaturated Flow

AK(y) [3] treat the equation of motion for moisture movement in an unsaturated soil to be effectively the same as Darcy’s Law. How credible is this perception? There are two ways to approach this question. One is to examine the way exThe integral in the denominator on the right-hand side of Eq. [3] represents the resistance to flow. In general, perimental data are mathematically interpreted, and the this integral can be evaluated iteratively if K( ) is known other is to examine the physical processes that distina priori. Two consequences arise because of the need guish saturated and unsaturated flow. First, consider the for this integration. First, even within a uniform column, mathematical framework of Darcy’s Law. Darcy (1856) hydraulic head will not vary linearly between inlet and interpreted his experimental data from a vertical column outlet. The gradients will be gentler at the inlet and of sand with an equation of the form: steeper at the outlet. Second, even in the special case of samples with the same average moisture content, A, Q K hinlet houtlet L A [1] and L, Q will not be directly proportional to (hin hout). Thus, for an unsaturated soil column, the two assumpwhere Q is volumetric flux; K is hydraulic conductivity; tions implicit in Darcy’s Law do not hold. The physical processes and forces governing saturated h is the hydraulic head, combining elevation z of the and unsaturated flow are substantially different: elasticpoint of measurement, and the gage-pressure head as mechanical forces as opposed to surface tension and capilmeasured with a manometer; L is the length between lary forces. Nevertheless, for purposes of mathematical inlet and outlet; and A is the area of cross section. Imanalysis, we are fortunate to be able to write a single plicit in Eq. [1] are the assumptions that in a column of unified equation of motion for both domains since the constant A (i) the hydraulic head varies linearly between water-phase pressure varies continuously and smoothly inlet and outlet and (ii) Q is directly proportional (or between the two states. It is useful for us to be cognizant linearly related) to (hinlet houtlet)/L. If the unsaturated of the fact that in the unified treatment of saturated– flow equation can be mathematically considered an exunsaturated flow, we juxtapose, for mathematical contension of Darcy’s Law, then it must reasonably satisfy venience, dissimilar physical processes and mathematithese two conditions. cal quantities. Following Richards (1931), the equation of motion Should the equation for liquid flux in an unsaturated for an unsaturated soil is written in differential form as soil be given a specific name? Richards (1961) strongly argued for designating it Buckingham’s Law in honor Q K( ) hA [2] of Edgar Buckingham (1907), who was the first to lucidly explain the dependence of hydraulic conductivity on moiswhere is negative in the unsaturated state and is conture content. Swartzendruber (1969) suggested that the tinuous between the saturated and unsaturated zones. unified equation of motion for saturated–unsaturated Because of the dependence of K on , it is not possible flow be referred to as Buckingham–Darcy equation. Both to readily apply Eq. [2] to a soil sample column and of these suggestions seem eminently reasonable. express flux in an explicit form such as Eq. [1]. Instead, one has to use integrals. One possibility is to express REFERENCES flux in the form of Ohm’s Law and write Buckingham, E. 1907. Studies on the movement of soil moisture. Bull. 38, USDA, Bureau of Soils, Washington, DC. Darcy, H. 1856. Dètermination des lois d’ècoulement de l’eau à travers Department of Materials Science and Engineering, Department of le sable. p. 590–594. In Les Fontaines Publiques de la Ville de Dijon. Environmental Science, Policy and Management, 210 Hearst MemoVictor Dalmont, Paris. rial Mining Building, University of California, Berkeley, CA 94720Richards, L.A. 1931. Capillary conduction of liquids through porous 1760. Received 11 Sept. 2004. Letter to the Editor. *Corresponding mediums. Physics 1:318–333. author ([email protected]). Richards, L. A. 1961. Advances in soil physics. p. 67–79. In Seventh Int. Congr. of Soil Sci., Madison, Wisconsin. 1960. Vol. I. Published in Vadose Zone Journal 3:1059 (2004). Swartzendruber, D. 1969. The flow of water in unsaturated soil. © Soil Science Society of America p. 215–291. In R.J.M. deWiest (ed.) Flow through porous media. Academic Press, New York. 677 S. Segoe Rd., Madison, WI 53711 USA