Seepage Theory for the Incompressible Fluid

When we solve the diffusion problem which is given by, e.g., (5.29), we encounter a problem that relates to the evaluation of the mean velocity v (note that dca/dt = dcoi/dt + v gradca,). It may be possible to solve a microscale problem based on the Navier-Stokes equation however, in classical soil mechanics we commonly use the seepage equation to determine v. Using the assumption of incompressibility of a fluid, we can derive the seepage equation from (5.18) and (5.37). 

Let us assume that both the mixture fluid and the intrinsic part of solid are incompressible (p = constant, p = constant). Under these incompressible conditions and applying (5.41), we obtain alternative forms of (5.18) and (5.37) as follows  

By substituting (5.54) into (5.51), we finally obtain a seepage equation that includes the volumetric deformation of the solid phase tr Z) as follows  

Using an equation of equilibrium or motion, which determines the deformation of the solid skeleton, and (5.58), a system of differential equations for specifying the mean velocity v (i.e., the conventional consolidation problem) is achieved. Note that in (5.58) p is the reduced Bernoulli potential (i.e., the total head excluding the velocity potential), k is the hydraulic conductivity tensor, p is the pore pressure of the fluid, g is the gravity constant, and is the datum potential. Thus by starting with the mass conservation laws for both fluid and solid phases, we can simultaneously obtain the diffusion equation and the seepage equation which includes a term that accounts for the volumetric deformation of the porous skeleton. 

Note 5.4 (On the permeability andflow in a porous medium). The seepage equation can be obtained by substituting Darcy s law into the mass conservation equations of fluid and solid phases, as described above. The effects of the micro-structure and microscale material property are put into the hydraulic conductivity k, which is fundamentally specified through experiments. It is not possible to specify the true velocity field by this theory, whereas by applying a homogenization technique, we can determine the velocity field that will be affected by the microscale characteristics. In Chap. 8 we will outline the homogenization theory, which is applied to the problem of water flow in a porous medium, where the microscale flow field is specified. 

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