Page 136 - Analysis and Design of Energy Geostructures

108 Analysis and Design of Energy Geostructures

3.11 Seepage flow
3.11.1 Physical phenomenon and governing equation
In the context of the analysis of problems involving groundwater seepage under lami-
nar conditions, Darcy’s law allows the expression of a relation between the hydraulic
gradient and the mean flow velocity under steady conditions. Considering the flow of
groundwater across a geomaterial that possesses homogeneity and isotropy with respect
to the mass transfer phenomenon, Darcy’s law reads

v rw;i 52 krh 52 kr z 1 p w ð3:35Þ
γ
w
where k is the hydraulic conductivity of the geomaterial, p w is the pore water pressure
and γ is the unit weight of water. The assumption of homogeneity and isotropy indi-
w
cates that the hydraulic conductivity is independent of direction and position and for
this reason appears outside the argument of the gradient.
In Eq. (3.35) the minus sign makes the mass flux density a positive quantity as a
consequence of its direction towards decreasing piezometric head. The law expressed
in Eq. (3.35) was first stated based on experimental evidence by Darcy (1856).It
represents a particular expression of the Navier Stokes equations (Whitaker, 1986)
(cf. Section 3.15).
Some modifications of Darcy’s law are needed for the analysis of turbulent flow
conditions in porous geomaterials (Khalifa et al., 2002), as well as for the analysis of
flows in unsaturated soils (Mitchell and Soga, 2005) and in fissured rocks (Vulliet
et al., 2016). These problems are not treated herein.
If, instead of establishing a relationship between the apparent flow velocity, v rw;i ,
and the hydraulic gradient, rh, via the hydraulic conductivity, k, a relationship
(i.e. the rate of fluid volume, V, transferred
between the mass flux density _q
D;i
through a given surface, A, per unit time, t) and the hydraulic gradient is considered,
it is found that the rate equation describing mass transfer in the considered case is
ð3:36Þ
_ q 52 krh
D;i
Eq. (3.36) is analogous to Eqs (3.2), (3.5) and (3.11).
Darcy’s law can be markedly simplified for the situation of a plane geometry in
steady-state conditions. In this case the hydraulic head distribution across a surface is
linear and the mass flux in the direction of the flow, x, reads

dh
ð3:37Þ
_ q 52 k
D;x
dx