Geoscience Reference
In-Depth Information
Figure 5.1
Finite element discretisation of a vertical slice through a hillslope using a mixed grid of triangular
and quadrilateral elements with a typical specification of boundary conditions for the flow domain; the shaded
area represents the saturated zone which has risen to intersect the soil surface on the lower part of the slope.
large number of nodes, either on the edges or at the centre of the elements. More details of these solution
techniques are given in Box 5.3. Note that it is important to make a distinction between these solutions
of the continuum partial differential equations and the semi-distributed models discussed in Chapter 6
where the fluxes between elements are based on the current states of storages in the element. In general,
the latter are based on solutions of ordinary differential equations and have often been less careful about
solution methodologies (see Clark and Kavetski, 2010).
Distributed models of this type are very demanding in their data requirements. Model parameters must
be provided for every grid element in the flow domain and boundary conditions must be specified for ev-
ery discrete length or area of the domain. Figure 5.1 shows a two-dimensional section through a hillslope
with a discretisation into a finite element grid and an indication of the boundary conditions that might
be applied. Specified flux boundaries are called “Cauchy-type” boundary conditions; zero flux (imper-
meable) boundaries are called “Neumann-type” boundary conditions; specified pressure boundaries are
called “Dirichlet-type” boundary conditions.
Along the boundaries AD and BC, a symmetry boundary is usually implemented under the assumption
that flow conditions are identical for the hillslope at the other side of the divide at A or the channel at D. A
symmetry boundary is equivalent to having a no-flow condition along a direction normal to the boundary.
Along boundary CD, a no-flow boundary is also generally assumed on the basis that the hillslope is
underlain by an impermeable layer or
aquiclude
. The boundary condition along AB may be time variable.
While it is raining and the soil surface has not ponded, there will be a specified flow rate equal to the net
rainfall rate at the ground surface. If the soil surface reaches saturation and infiltration into the soil starts
to fall below the rainfall rate then part of this boundary may be controlled as a fixed head boundary, with
a potential equal to the depth of ponded water. Under dry conditions, again there may be a specified flow
rate equal to the estimated loss of water from the surface as evaporation. Time-varying solutions also
require the initial conditions at the start of the simulation period to be specified. The initial conditions
are values of
θ
or
ψ
for every node in the flow domain at the start of the simulation.
One major problem in applying the Richards equation is specifying the nonlinear soil moisture
characteristics curves for a particular location or solution grid element. Most models of this type use func-
tional relationships between the moisture content, capillary potential and hydraulic conductivity variables
