4.7 Buckingham-Darcy Law
In 1902, a physicist named Edgar Buckingham was hired by the US Department of Agriculture to work in its newly formed Bureau of Soils [12]. He stayed there for only four years, but his studies during that time led to a conceptual breakthrough that would change the course of soil physics and hydrology. Drawing on earlier work by John Maxwell and Lyman Briggs, Buckingham reasoned that in unsaturated soil the water was attracted to and held by the surfaces of the soil solids by what he called “capillary potential”, i.e. matric potential [13]. He saw that spatial gradients in this potential would act as a force to drive soil water flow and that the resulting flow would be proportional to and in the opposite direction of that gradient. Buckingham also correctly predicted that in unsaturated soil the “capillary conductance”, i.e. the hydraulic conductivity, would be largely dependent on the soil water content. And, Buckingham measured the variation of the equilibrium soil water content with height above a water table in long laboratory soil columns, resulting in the first published soil water retention curves relating water content and “capillary potential”.
Because of Buckingham’s careful experimental work and perceptive theoretical insights, subsequent soil physicists were able to formulate a single equation which describes both saturated and unsaturated water flow in soil and other porous media:
(Eq. 4-5)
where K(θ) indicates that the hydraulic conductivity is a function of the soil volumetric water content and d(ψp + ψg)/dz indicates the gradient of the total water potential in the z direction. Osmotic potential can also be included with the other potentials if necessary. This equation can be viewed as a generalization of Darcy’s Law and can be referred to as the Buckingham-Darcy Law.
To understand unsaturated water flow in soil, we need to appreciate at least three key differences between Darcy’s Law and the Buckingham-Darcy Law. First, we need to understand that the hydraulic conductivity depends strongly on the soil water content in unsaturated soil and that water content can vary in space and time. For example, the hydraulic conductivity of Grenoble sand is near 15 cm h-1 when the water content is 0.30 cm3 cm-3, but the hydraulic conductivity drops more than one order of magnitude, to approximately 1 cm h-1 when the water content decreases to 0.20 cm3 cm-3 (Fig. 4‑7, [14]). The cause of this dramatic sensitivity to water content is linked to what we learned previously about the soil water retention curve, capillary rise, and Poiseuille’s Law.
From the soil water retention curve, we know that soil water content is positively related to soil matric potential, so if the soil water content decreases from 0.30 to 0.20 cm3 cm-3, a corresponding decrease in matric potential must occur. From the capillary rise equation, we can infer that as matric potential decreases (analogous to an increase in capillary rise), water is held i