Geoscience Reference
In-Depth Information
Fig. 8.28 Converging section of a duct illustrating inertia effect in
a pore with variable cross section and with the resulting
convective acceleration V / x ; V i and V o are the mean
inflow and outflow velocity, respectively. At higher
velocities this effect becomes more pronounced; the flow
ceases to be creeping and Darcy's equation is no longer
valid.
V i
V o
x
The Reynolds number is a measure of the relative magnitude of the inertial forces over the
viscous forces; this means that Darcy's law cannot be expected to be valid for values of Re p
that are much in excess of order one. It should be emphasized that this initial deviation
from Darcy's law is not caused by the onset of turbulence, but solely by the effects of fluid
accelerations. These accelerations result from the irregular and tortuous flow paths in the
pores, as illustrated in Figure 8.28.
Forchheimer (1930, p. 54) was among the first to analyze experimental data and in 1901
he proposed the following to describe them
|∇ h | = α | q | + β q 2
(8.34)
where α and β are constants for any given soil. Some insight can be gained in the nature of
these constants by dimensional inspection of the Navier-Stokes equation (1.12). The term
on the left and the first term on the right in Equation (8.34) correspond to the three terms
on the right of Equation (1.12); thus
α
represents the effect of viscosity and in terms of the
permeability it is
q 2 , corresponds to the terms
on the left-hand side of (1.12) and therefore, it represents the inertia effects on the flow.
For steady conditions the left-hand side of (1.12) is v ·∇ v . Dimensionally, the two velocity
terms are proportional to q 2 . On the other hand, the dimensions of are [L 1 ]; because k
is the characteristic length scale of the pores, this suggests that β is inversely proportional
to it, or
α = ν/
( gk ). The second term on the right,
β
C
g k
β =
(8.35)
where C is a constant, which can be expected to depend on the geometry and shape of the
pore spaces. Equation (8.35) has been confirmed in numerous experimental investigations.
For instance, Arbhabhirama and Dinoy (1973) have reported values of C ranging between
approximately 0.6 (sand) and 0.2 (angular gravel).
Lower limit
Several experiments with clayey soils have shown that Darcy's law also fails to describe
the measurements when the flow rates become very small (see Miller and Low, 1963;
Swartzendruber, 1968). These experiments indicate that under conditions of low flow rates
(or small h -gradients), the measured specific fluxes q are smaller than k h would require.
The issue has been somewhat controversial (Olsen, 1966) and is still far from resolved
(Neuzil, 1986). Nevertheless, this phenomenon may possibly result from the fact that in the
neighborhood of clay particles, which invariably have electrical charges, the water molecules
may become oriented in a quasi-regular fashion, on account of their dipole characteristics;
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