Geoscience Reference
In-Depth Information
Box 5.1 Descriptive Equations for Subsurface Flows
The generally used description for both saturated and unsaturated subsurface flows is based on
Darcy's law (Darcy, 1856) which assumes that the disharge per unit area or Darcian velocity
can be represented as the product of a gradient of hydraulic potential and a scaling constant
called the “hydraulic conductivity”. Thus
K
∂
∂x
v
x
=−
(B5.1.1)
where
v
x
[LT
−1
] is the Darcian velocity in the
x
direction,
K
[LT
−1
] is the hydraulic conductivity,
and
[L] is the total potential (
z
where
[L] is the capillary potential and
z
is the
elevation above some datum). In the case of unsaturated flow, the hydraulic conductivity
changes in a nonlinear way with moisture content so that
=
+
K
=
K
(
)
(B5.1.2)
where
[-] is volumetric moisture content.
Combining Darcy's law with the three-dimensional mass balance equation gives:
∂
∂t
=−
∂v
x
∂x
−
∂v
x
∂x
−
∂v
x
∂x
−
E
T
(
x,y,z,t
)
(B5.1.3)
where
is the density of water (often assumed to be constant) and
E
T
(
x,y,z,t
)[T
−1
]isarateof
evapotranspiration loss expressed as a volume of water per unit volume of soil that may vary
with position and time. Combining Equation (B5.1.1) with Equation (B5.1.3) gives the nonlinear
partial differential equation now known as the Richards equation after L. A. Richards (1931):
∂
∂t
=
K
(
)
∂
∂x
K
(
)
∂
∂y
K
(
)
∂
∂z
∂
∂x
∂
∂y
∂
∂z
+
+
−
E
T
(
x,y,z,t
)
(B5.1.4)
or, remembering that
=
+
z
K
(
)
∂
∂x
K
(
)
∂
∂y
K
(
)
∂
1
∂
∂t
=
∂
∂x
∂
∂y
∂
∂z
+
+
∂z
+
−
E
T
(
x,y,z,t
)
(B5.1.5)
or, in a more concise form using the differential operator
∇
(see Box 2.2),
∂t
=∇
K
(
)
+
∂
∂K
(
)
∂z
∇
−
E
T
(
x,y,z,t
)
(B5.1.6)
This form of the equation assumes that the hydraulic conductivity at a given moisture con-
tent is equal in all flow directions, i.e. that the soil is “isotropic”, but in general the soil or
aquifer may be “anisotropic”, in which case, the hydraulic conductivity will have the form of
a matrix of values. This equation involves two solution variables
and
. It can be modified to
have only one solution variable by making additional assumptions about the nature of the re-
lationship between
and
. For example, defining the
specific moisture capacity
of the soil as
C
(
)
d
d
and assuming that the density of water is constant, then the Richards equation may
be written
=
∂t
=∇
K
(
)
+
C
(
)
∂
∂K
z
(
)
∂z
∇
−
E
T
(
x,y,z,t
)
(B5.1.7)
where
K
(
) and
K
z
(
) now indicate that hydraulic conductivity depends on direction and is
treated as a function of
for the unsaturated case. To solve this flow equation, it is necessary
to define the functions
C
(
) and
K
(
) (see Box 5.4). To make things more complicated, both
