Environmental Engineering Reference
In-Depth Information
is the volumetric water content [L
3
L
−
3
],
t
is time [T],
z
is the spatial coor-
dinate [L],
q
is the volumetric flux [LT
−
1
], and
S
is a general sink/source term
[L
3
L
−
3
T
−
1
], for example to account for root water uptake (transpiration).
Equation (
18.8
) is often referred to also as the
mass conservation equation
or
the continuity equation. The mass balance equation in general states that the change
in the water content (storage) in a given volume is due to spatial changes in the
water flux (i.e., fluxes in and out of some small volume of soil) and possible sinks
or sources within that volume. The mass balance equation must be combined with
one or several equations describing the volumetric flux (
q
) to produce the governing
equation for variably saturated flow.
For a soil that can be saturated or unsaturated, the flux is given by the equation:
where
θ
k
(
h
)
∂
h
q
=−
z
+
K
(
h
)
(18.9)
∂
where
K
is the unsaturated hydraulic conductivity [LT
−
1
], and
h
the pressure head.
Eq. (
18.9
) is often referred to as the Darcy-Buckingham equation. The hydraulic
conductivity in this equation is a function of the pressure head,
h
. In saturated sys-
tems, the conductivity becomes independent of the pressure head, in which case the
equation reduces to Darcy law as:
K
s
∂
h
q
=−
z
+
K
s
(18.10)
∂
where
K
s
is the saturated hydraulic conductivity [LT
−
1
]. The Darcy-Buckingham
equation hence is formally similar to Darcy's equation, except that the proportional-
ity constant (i.e., the unsaturated hydraulic conductivity) in the Darcy-Buckingham
equation is a nonlinear function of the pressure head (or water content), while
K
(
h
)
in Darcy's equation is a constant equal to the saturated hydraulic conductivity,
K
s
(e.g., see discussion by Narasimhan
2005
).
Combining the mass balance (Eq. (
18.8
)) with Darcy-Buckingham's law (Eq.
(
18.9
)) leads to
K
(
h
)
∂
K
(
h
)
∂θ
(
h
)
∂
∂
∂
h
=
h
+
−
S
(
h
)
(18.11)
t
z
∂
which was first formulated by Richards (
1931
) and is now generally referred to as
the Richards equation. This partial differential equation is the equation governing
water flow in the variably-saturated vadose zone. Because of its strongly nonlinear
makeup, only a relatively few simplified analytical solutions can be derived. Most
practical applications of Eq. (
18.11
) require a numerical solution, which can be
obtained using a variety of methods such as finite differences or finite elements.
Equation (
18.11
) is generally referred to as the mixed form of the Richards equation
since it contains two dependent variables, i.e., the water content and the pressure
head. Various other formulations of the Richards equation are possible.
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