Geoscience Reference
In-Depth Information
increased by ploughing, harrowing and other agricultural operations. But the concept
has its limitations, and should be used with caution, especially in the case of clayey
soils. Depending on the type of clay, the pore structure of some soils may be sensitive to
the type of electrolytes or salts that are in solution in the water; for instance, sodium is
notorious in this regard. This means that it is not always possible to separate the effects
of the fluid on the hydraulic conductivity completely from those of the porous matrix.
It is of interest to observe that in the form of Equation (8.24), Darcy's law can be
considered as a generalization of several well-known creeping flow equations, obtainable
for certain flow regions with a regular geometry, to the case of a totally irregular pore
geometry resulting from the random assemblage of the particles of soils and other gran-
ular materials. For instance, in the case of flow through a straight pipe of circular cross
section, creeping flow is described by the Hagen-Poiseuille equation; this is equivalent
with (8.24) if
q
i
is taken to represent the average velocity in the pipe,
R
e
is the radius of
the pipe and Ge
8). (Tubes with other cross-sectional shapes have been analyzed by,
among others, Boussinesq (1868; 1914) and Graetz (1880).) It is also easy to show (see,
for example, Lamb, 1932, p. 582) that Equation (8.24) can describe the flow between
parallel plates (as used in the Hele-Shaw model) by putting Ge
=
(1
/
3) and
R
e
as half
the spacing of the plates, again if
q
i
is made to represent the average velocity. Similarly,
Equation (8.24) can be used to describe the average velocity of flow down a plane as in
(5.32), provided Ge
=
(1
/
3,
R
e
is taken as the depth of flow
h
and the hydraulic gradient
is the slope of the plane
S
0
. All three expressions just mentioned are exact solutions of
the Navier-Stokes equations (1.12) for creeping flow, and can be found in elementary
textbooks in fluid mechanics. (Recall that creeping flow is flow with a very low Reynolds
number so that (
D
v
=
1
/
Dt
)becomes negligible.) In the literature there have been numerous
attempts to derive Darcy's law from the Navier-Stokes equations, mostly by analogy
with these exact solutions. On account of the irregular geometry of the pores resulting
from random packings of particles, to arrive at the desired result any such derivation must
involve some kind of ensemble averaging and other stochastic assumptions, which may
not always be valid. But regardless of such considerations, pragmatically it is probably
preferable to adopt Darcy's law simply as it is, that is as an experimentally obtained and
verified relationship, in which
k
or
k
is best obtained from measurements.
/
True velocity
As defined in Darcy's equation,
q
or
q
i
is the volumetric rate of flow per unit bulk area
of porous material. Thus even though it has the basic dimensions of [L T
−
1
] it does not
represent the average velocity of the fluid particles. The “true” average velocity inside
the pores is usually assumed to be given by (
q
i
/
n
0
) under fully saturated conditions, and
by (
q
i
/θ
) under partly saturated conditions.
Anistropy
As formulated in Equation (8.19) (or (8.24)),
q
i
and
x
i
are vectors pointing in the
same direction and
k
(or
k
) is a scalar quantity, that is independent of direction. Porous
materials in which this holds true are referred to as
isotropic
. A material is said to be
anisotropic
when its properties, such as the hydraulic conductivity or the permeability,
−
∂
h
/∂
