Geoscience Reference
In-Depth Information
as a result the viscosity of the water may no longer be Newtonian, a necessary condition
for the validity of the Navier-Stokes equations and Darcy's law. Also, in many situations
additional driving mechanisms may be at play, beside the hydraulic gradient
∇
h
; these may
cause flow even when
∇
h
=
0.
Additional driving forces
For most purposes in hydrology it can be assumed that the driving forces are primarily
mechanical in nature, which means that the flow is driven by gravity and by a pressure
gradient in accordance with Darcy's law. However, in general, water transport in a porous
medium can be influenced by several other factors, involving thermal, osmotic and some-
times even electrical effects. For instance, changes in temperature at some point in a partly
saturated soil may result in changes in surface tension which affect the pressure
p
w
for a
given water content, and thus the liquid transport. An input of heat can be accompanied
by local vaporization, which in turn sets up a specific humidity gradient, and thus a water
vapor transport in the air filled pore space; this vapor may then condense further down and
affect the liquid water flux.
On account of the complexity of the various phenomena and their interaction within the
soil, at present there is apparently no theory available, which is generally accepted. Nev-
ertheless, in recent years many problems, involving simultaneous heat and water transport,
have been studied within a framework developed by Philip and DeVries (1957; DeVries,
1958; DeVries and Philip, 1986), but with mixed results (Jackson
et al
., 1974; Kimball
et al
., 1976). Raats (1975) and Nakano and Miyazaki (1979) have explored the theoretical
and practical compatibility of the formulation of Philip and DeVries with concepts of irre-
versible thermodynamics; more recently, Cahill and Parlange (1998) have clarified the role
of the vapor transport. By numerical simulations Milly (1984) has investigated the relative
importance of the temperature gradient on the water transport; he concluded that, for many
practical purposes, it is sufficiently accurate to assume that the water transport is essentially
isothermal and driven only by the hydraulic gradient
∇
h
.
8.3.4
Expressions for the conductivity and the soil water diffusivity functions
Conductivity of saturated materials
In the past, numerous equations have been proposed to predict the hydraulic conductivity
of porous materials, mostly on the basis of measurements of the particle sizes or their dis-
tribution or also of the pore size distribution, as obtained from Equation (8.5). However,
such equations were usually obtained from permeameter measurements of
k
and are there-
fore valid only at the local scale. As noted earlier,
k
tends to be scale dependent, so that
these methods cannot be used when the hydraulic conductivity is to be used at the field
or catchment scale, as is often the case in hydrology. For applications over larger areas it
is therefore advisable to obtain
k
by means of inverse methods with measurements at the
appropriate scale. One such inverse method, based on drought flow recession analysis, will
be considered in Chapter 10.
Conductivity of partly saturated materials
As seen earlier, under partly saturated conditions the determination of
k
(
θ
) is not an easy
task. However, while in many flow calculations the accuracy of
k
at high water contents is
fairly critical, at lower water contents some inaccuracies can be tolerated. Therefore, it has
