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LETTER TO THE EDITOR

Darcy's Law and Unsaturated Flow

Department of Materials Science and Engineering, Department of Environmental Science, Policy and Management, 210 Hearst Memorial Mining Building, University of California, Berkeley, CA 94720-1760.
^* Corresponding author (tnnarasimhan{at}LBL.gov)

Received 11 September 2004.

IT IS COMMON in hydrogeology and soils literature to 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 experimental data are mathematically interpreted, and the other is to examine the physical processes that distinguish saturated and unsaturated flow. First, consider the mathematical framework of Darcy's Law. Darcy (1856) interpreted his experimental data from a vertical column of sand with an equation of the form:

[1]

where Q is volumetric flux; K is hydraulic conductivity; h is the hydraulic head, combining elevation z of the point of measurement, and the gage-pressure head as measured with a manometer; L is the length between inlet and outlet; and A is the area of cross section. Implicit in Eq. [1] are the assumptions that in a column of constant A (i) the hydraulic head varies linearly between inlet and outlet and (ii) Q is directly proportional (or linearly related) to (h_inlet – h_outlet)/L. If the unsaturated flow equation can be mathematically considered an extension of Darcy's Law, then it must reasonably satisfy these two conditions.

Following Richards (1931), the equation of motion for an unsaturated soil is written in differential form as

[2]

where is negative in the unsaturated state and is continuous between the saturated and unsaturated zones. Because of the dependence of K on , it is not possible to readily apply Eq. [2] to a soil sample column and express flux in an explicit form such as Eq. [1]. Instead, one has to use integrals. One possibility is to express flux in the form of Ohm's Law and write

[3]

The integral in the denominator on the right-hand side of Eq. [3] represents the resistance to flow. In general, this integral can be evaluated iteratively if K() is known a priori. Two consequences arise because of the need for this integration. First, even within a uniform column, hydraulic head will not vary linearly between inlet and outlet. The gradients will be gentler at the inlet and steeper at the outlet. Second, even in the special case of samples with the same average moisture content, A, and L, Q will not be directly proportional to (h_in – h_out). Thus, for an unsaturated soil column, the two assumptions implicit in Darcy's Law do not hold.

The physical processes and forces governing saturated and unsaturated flow are substantially different: elastic-mechanical forces as opposed to surface tension and capillary forces. Nevertheless, for purposes of mathematical analysis, we are fortunate to be able to write a single unified equation of motion for both domains since the water-phase pressure varies continuously and smoothly between the two states. It is useful for us to be cognizant of the fact that in the unified treatment of saturated–unsaturated flow, we juxtapose, for mathematical convenience, dissimilar physical processes and mathematical quantities.

Should the equation for liquid flux in an unsaturated soil be given a specific name? Richards (1961) strongly argued for designating it Buckingham's Law in honor of Edgar Buckingham (1907), who was the first to lucidly explain the dependence of hydraulic conductivity on moisture content. Swartzendruber (1969) suggested that the unified equation of motion for saturated–unsaturated flow be referred to as Buckingham–Darcy equation. Both of these suggestions seem eminently reasonable.

REFERENCES

• 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 le sable. p. 590–594. In Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris.
• Richards, L.A. 1931. Capillary conduction of liquids through porous mediums. Physics 1:318–333.
• Richards, L. A. 1961. Advances in soil physics. p. 67–79. In Seventh Int. Congr. of Soil Sci., Madison, Wisconsin. 1960. Vol. I.
• Swartzendruber, D. 1969. The flow of water in unsaturated soil. p. 215–291. In R.J.M. deWiest (ed.) Flow through porous media. Academic Press, New York.

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