Geoscience Reference
In-Depth Information
Fig. 8.27 Soil water diffusivity as a function
of water content,
D
w
=
D
w
(
θ
)
during desorption for the same
soils as in Figure 8.26. (After
Gardner and Miklich, 1962.)
1.E+05
D
w
1
2
1.E+04
3
5
(cm
2
d
−
1
)
1.E+03
4
1.E+02
1.E+01
1.E+00
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
θ
Water content,
by defining (Klute, 1952) the soil water diffusivity
D
w
(
θ
)by
k
dH
d
D
w
=−
(8.32)
θ
or also
D
w
=
). From the physical point of view, Equation (8.31) does not
contain any more information than (8.19), and in many practical simulations there may
be no significant advantage in this diffusion formulation. In fact, when part of the flow
domain is saturated, (8.31) with (8.32) may present some difficulties on account of the
singular nature of
D
w
when
k
(
d
ψ
w
/
d
θ
is constant; these are avoidable with (8.19). Nevertheless,
as will be seen in the next chapter, the diffusion formulation continues to be of interest
because it has greatly simplified the analytical treatment of a number of important soil
water problems. Some examples of the dependence of soil water diffusivity on water
content are shown in Figure 8.27.
θ
8.3.3
Limitations of Darcy's law
Upper limit
In light of the analogy between Darcy's law and other creeping flow equations of fluid
mechanics, it should not be surprising that experiments have shown that, as the Reynolds
number increases beyond a certain limit, the specific flux
q
gradually deviates from its
linear proportionality with the hydraulic gradient
∇
h
. Indeed, by definition, creeping flow
is flow for which the appropriate Reynolds number is sufficiently small, so that the accel-
eration terms, both temporal and advective, in the Navier-Stokes equations are negligible.
Any Reynolds number definition requires the adoption of a characteristic velocity and
of a characteristic length of the flow geometry. The definition of the permeability
k
in
Equation (8.23) indicates that it has the basic dimensions of [L
2
] and that it can be con-
sidered proportional to a characteristic or typical cross-sectional area of flow; thus, since
the specific flux has the basic dimensions [L T
−
1
], it is convenient to define the Reynolds
number for flow in a porous medium as follows
|
q
|
√
k
ν
Re
p
=
(8.33)
