Geoscience Reference
In-Depth Information
equations, if some assumptions are made about the nature of the pore space through which the flow is
taking place, and if the flow is slow enough to stay in the laminar regime (e.g. Hassanizadeh, 1986).
This is usually a good assumption for flow in a porous matrix but may break down for flow in soils with
heterogeneous characteristics (where it may be difficult to define a gradient of potential except at very
small scales) and macropores (where flow in large pores and flow in the porous matrix may be responding
to different local gradients). Thus Darcy's law is strictly valid only over a limited range of scales (perhaps,
at least in the unsaturated zone, all smaller than the normal element scale of a rainfall-runoff model,
Beven, 1989, 2006b). There has been an initiative towards the development of appropriate flux equations
to replace Darcy's law for application at larger hillslope element scales (e.g. Reggiani
et al.
, 1999; see
Chapter 9). However, it should be noted that the use of Darcy's law to represent fluxes in large elements
of a distributed model is a gross approximation and may mean that the effective value of the hydraulic
conductivity required may be different from anything that could be measured in the field (see Section 5.2).
It might even not be the most appropriate description locally. There is an argument that the original
laboratory experiments of Richards were not realistic in respect of flows in field soils. Richards used air
pressure to create unsaturated flow conditions (a procedure that became a standard method in laboratory
measurements of the relationships between soil water content and capillary pressure). This has the effect
of precluding flow in larger pores that might occur in field conditions. Such flow, which can be represented
as films of water around particles largely independent of the capillary potentials within the fine pores,
has been described at least since Burdine (1953) using Stokes equation (Gerke
et al.
, 2010). Stokes flow
does not need large continuous macropores to be a feasible description of such
bypassing
flow, and can
be consistent with the type of fingering seen in Figure 1.1. It might well be that we need to revisit the
Richards equation as a basis for physically based rainfall-runoff models in future.
The other important equation in this description of subsurface flow is the continuity or mass balance
equation (see Box 2.3). The combination of Darcy's law with the continuity or mass balance equation
results in a flow equation (generally called the Richards equation) which may be written with capillary
potential
ψ
as the dependent variable as:
C
(
ψ
)
∂ψ
∂K
(
ψ
)
∂z
∂t
=∇
[
K
(
ψ
)
∇
ψ
]
+
−
E
T
(
x, y, z, t
)
(5.3)
where
ψ
is the local capillary potential,
K
(
ψ
) is unsaturated hydraulic conductivity, which is now ex-
pressed as a function of
ψ
rather than
θ
,
C
(
ψ
) is a function of
ψ
defined as the rate of change of moisture
content
θ
with change in
ψ
, called the
specific moisture capacity
, and
E
T
(
x, y, z, t
) is a local uptake
of water by plant roots to satisfy evapotranspiration (see Section 5.1.3). An equivalent equation with
soil moisture content
θ
as the dependent variable can also be written. The derivation of both forms of
the Richards equation is given in Box 5.1. The Richards equation is a partial differential equation (see
Box 2.3) and, because of the nonlinear change of hydraulic conductivity with moisture content, it is
a nonlinear partial differential equation. Such equations tend to be very difficult to solve analytically,
except for some very simple cases of
initial conditions
and
boundary conditions
and simple forms for the
nonlinear relationships relating moisture content, capillary potential and hydraulic conductivity, known
as the
soil moisture characteristic curves
. Some solutions for the special case of infiltration at the soil
surface are given in Box 5.2.
For most cases of interest to hydrologists, it is necessary to use an approximate numerical solution of
the equation. A number of different techniques are available, including finite difference (e.g. the SHE
model), finite element (e.g. IHDM, HYDRUS, InHM, HydroGeoSphere), boundary element, integrated
finite difference (e.g. TOUGH2) and finite volume (e.g. PIHM) techniques. The finite volume method has
become increasingly important, in part because it simplifies the coupling of different process equations
at the discrete boundaries of the finite volumes and is inherently mass conservative. All these methods
involve discretising the flow domain into a network of grids or elements (as shown for a finite element
discretisation in Figure 5.1) and solving for values of moisture content
θ
or capillary potential
ψ
at a
