Basic Flow Equations

The above conceptualization of a porous medium allows description of water movement through soil, either saturated or unsaturated, by the experimentally derived Darcy's law, q =-K V H, (5.105)

written in vectorial form for a homogeneous isotropic medium under isothermal conditions. This equation relates macroscopically the volumetric flux density (or Darcy velocity) q to the negative vector gradient of the total soil water-potential head H by means of the parameter K, called the soil hydraulic conductivity. For an anisotropic porous medium, this parameter becomes a tensor. The hydraulic conductivity is assumed to be independent of the total potential gradient but may depend on other variables. Because the term VH is dimensionless, both q and K have dimensions of L/ T and generally units of meters per second or centimeters per hour.

Note that Darcy's law can be derived from the Navier-Stokes equations for viscous-flow problems because it practically describes water flow in porous media when inertial forces can be neglected with respect to the viscous forces. Therefore, the range of validity of Darcy's law depends on the occurrence of the above condition; readers wishing further details are directed to the literature [3].

The soil water-flow theory based on the Darcy flux law provides only a first approximation to the understanding and description of water-flow processes in porous media. Apart from the already-cited nonlinear proportionality between the Darcy velocity and the hydraulic potential gradient at high flow velocity due to the increasing weight of the inertial forces with respect to the viscous forces in determining the magnitude of the stresses acting on soil water and the presence of turbulence, allowances are made for possible deviations from Darcy's law even at low flow velocities [16]. Other causes of deviations from the Darcy-based flow theory can be attributed chiefly to the occurrence of macropores (such as earthworm holes, cracks, and fissures), nonisothermal conditions, nonnegligible effects of air pressure differences, and solute-water interactions. However, these causes may become more important when modeling transport processes under field-scale conditions with respect to laboratory-scale situations.

The description of mass conservation is still made using the concept of REV and usually with the assumption of a rigid system. The principle of mass conservation requires that the change with time of mass stored in an elemental soil volume must equal the difference between the inflow- and the outflow-mass rates. Therefore, the basic mass balance for water phase can be written as

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