point of the flow domain, depending on the local values of water content. Therefore, the proportionality factor K becomes a function of volumetric water content 9 and is called the unsaturated hydraulic conductivity function.
According to the Richards approximation and neglecting sinks, sources, and phase changes, equating Darcy's law (5.105) and the equation of continuity (5.106) yields the following partial differential equation governing unsaturated water flow:
An alternative formulation for the flow equation can be obtained by introducing the soil water diffusivity D = Kdh/d9 (dimension of L2/ T and units of m2/s):
where z (m) is the gravitational component of the total soil water-potential head.
The use of 9 as dependent variable seems more effective for solving flow problems through porous media with low water content. However, when the degree of saturation is high and close to unity, employing Eq. (5.112) proves difficult because of the strong dependency of D upon 9. In particular, in the saturated zone or in the capillary fringe region of a rigid porous medium, the term dh/d9 is zero, D goes to infinity, and Eq. (5.112) no longer holds. Also, an unsaturated-flow equation employing 9 as dependent variable hardly helps to model flow processes into spatially nonuniform porous media, in which water content may vary abruptly within the flow domain, thereby resulting in a nonzero gradient V9 at the separation interface between different materials. The selection of h as dependent variable may overcome such difficulties as soil water potential is a continuous function of space coordinates, as well as it yields Eq. (5.111) that is valid under both saturated and unsaturated conditions.
Water transport processes in the unsaturated zone of soil are generally a result of precipitation or irrigation events which are distributed on large surface areas relative to the extent of the soil profile. The dynamics of such processes is driven essentially by gravity and by predominant vertical gradients in flow controlled quantities. These features thus allow us the opportunity to mathematically formulate most practical problems involving flow processes in unsaturated soils as one- dimensional in the vertical direction. The equation governing the vertical, isothermal unsaturated-soil water flow is written traditionally as known as the Richards equation. This equation uses soil water-pressure head h as the dependent variable and usually is referred to as the pressure-based form of the governing unsaturated water-flow equation. In Eq. (5.113), z denotes the vertical space coordinate (m), conveniently taken to be positive downward; t is time (s); K is the unsaturated hydraulic conductivity function (ms-1); and C = d9/dh is the capillary hydraulic storage function (1/m), also termed specific soil water capacity, which can be computed readily by deriving the soil water-retention function 9 (h).
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