Geoscience Reference
In-Depth Information
consideration in problems involving strong advection, such as the propagation of a sharp
wetting front into the soil, the movement of a steep flood wave down a river or the transport of
a contaminant in a flow with a steep concentration gradient. Numerical solutions using a fixed
nodal grid will inevitably smear out any rapid changes in moisture content or concentration
in such problems. This smearing is called “numerical dispersion” and becomes greater as the
grid gets coarser, even if the solution apparently stays stable. Sharp fronts may also lead to
oscillation in many solution schemes, a product of the approximate solution not the process.
There have been some techniques, such as “upwind” differencing, developed to try to minimise
such problems, especially in one-dimensional advection problems, but the lesson is that these
types of solution must be used with care.
Box 5.4 Soil Moisture Characteristic Functions for Use in the Richards Equation
Use of the Richards equation to predict water flow in unsaturated soil (see Box 5.1) requires
the specification of the nonlinear functions C ( ) and K ( ) if solving for capillary potential ,
or D ( ) and K ( ) with moisture content , as the dependent variable. These functions are time
consuming to measure directly, even in the laboratory, and are generally complicated by being
multi-valued hysteretic functions dependent on the history of wetting and drying. For modelling
purposes, it is often assumed that simpler, single-valued functions can be used. Two sets of
widely used functions are presented here: those suggested by Brooks and Corey (1964) and
those of van Genuchten (1980). Both specify forms for ( ) and K ( ) , from which the specific
moisture capacity, C ( ) , and diffusivity, D ( ) , can be derived. Smith et al. (1993) proposed a
form that is an extension of both the Brooks-Corey and van Genuchten forms but this has not
been widely used. Some theoretical developments have involved developing functional forms
based on assumptions of a fractal pore space (see, for example, Tyler and Wheatcraft, 1992;
Pachepsky and Timlin, 1998). Jaynes (1990) reviews multi-valued hysteretic forms.
B5.4.1 The Brooks-Corey Functions
In the Brooks-Corey functions, moisture content and capillary potential are related as
o
r
r =
(B5.4.1)
s
while hydraulic conductivity and moisture content are related as
3+2
K ( )
K s =
r
(B5.4.2)
s
r
B5.4.2 The van Genuchten Functions
In the van Genuchten functions, moisture content and capillary potential are related as
r
1
+
1
r =
(B5.4.3)
s
1
+{
/ o }
+
1
while hydraulic conductivity and moisture content are related as
+1
1
1
1
1
2
+
+
K ( )
K s =
r
r
(B5.4.4)
s
r
s
r
 
Search WWH ::




Custom Search