Page 162 - Analysis and Design of Energy Geostructures

134 Analysis and Design of Energy Geostructures

p ffiffiffiffiffiffiffiffiffiffiffiffi
d p 5 4A=π 5 0:8m
Reynolds number is Re 5 ρ v z d p =μ , hence the density of the fluid is
f
f
μ f 1:2
ρ 5 Re 5 3 6283 5 1346 kg=m 3
f
v z d p 7 3 0:8

qq. Under laminar conditions, Darcy’s law allows expressing a relation
between the hydraulic gradient and the mean seepage flow velocity
under steady conditions. For a porous geomaterial fully saturated with
water, which possesses homogeneity and isotropy with respect to the
mass flow process, Darcy’s law reads

p

v rw;i 52 krh 52 kr z 1 w
γ w
where v rw;i [m/s] is the mean flow velocity, k [m/s] is the hydraulic
conductivity of the geomaterial, r represents the gradient, z [m] is the
elevation of a considered fluid particle above a reference plane, p [Pa]
w
3
is the fluid pressure, γ [N/m ] is the unit weight of the fluid and h
w
[m] is the piezometric head.
The assumption of homogeneity and isotropy indicates that the
thermal conductivity is independent of direction and position and for
this reason appears outside the argument of the gradient. In Darcy’s
law, as above expressed, the minus sign makes the mass flux density a
positive quantity as a consequence of its direction towards decreasing
piezometric head. Some modifications of Darcy’s law are needed for
the analysis of turbulent flow conditions in porous geomaterials, as
well as for the analysis of flows in unsaturated soils.
rr. The parameter governing this aspect is the hydraulic conductivity, k,
which characterises the definition of the fluid velocity with respect to
the solid particles v rw;i in the Darcy’s law.
ss. The hydraulic conductivity of the geomaterial has the dimensions of a
length per unit of time [m/s]. It is a function of both the fluid type and
geomaterial structure. The dependence of the hydraulic conductivity
on the intrinsic permeability can be typically appreciated by expressing
this variable as