Geoscience Reference
In-Depth Information
Equation of continuity
Because q x is a constant along z , it is identical with the average, and the continuity
equation (5.13) derived for free surface open channel flow is also directly applicable here.
In (5.13) the term representing the displacement rate of the free surface
h
/∂
t becomes
∂η/∂
t in the present notation. Because this is an actual velocity of the interface, the
average velocity V in (5.13) must be replaced here by the “true” velocity in the porous
material ( q x /
n e ); similarly, the lateral inflow i must be replaced by a true recharge
velocity ( I
n e ), where I is the recharge rate, representing a source term as a volumetric
flux per unit ground surface area of porous material. Thus one obtains
∂η
/
q x η
n e
I
n e =
t +
0
(10.28)
x
With (10.27) this assumes the form
∂η
cos
η ∂η
α ∂η
k 0
n e
I
n e
t =
α
+
+
sin
(10.29)
x
x
x
in which it is assumed, as is commonly done, that k 0 ,
are constant or can be
treated as effective parameters. In the absence of lateral inflow and for a horizontal
impermeable layer, Equation (10.29) becomes
∂η
n e and
α
k 0
n e
η ∂η
t =
(10.30)
x
x
Both (10.29) and (10.30) are forms of what is usually referred to as the Boussinesq
equation. To repeat, the Boussinesq equation is based on the following assumptions.
(i) The effect of unsaturated flow above the water table is negligible and it can be
parameterized by an effective porosity or specific yield n e ; this is also the basis of the
free surface approach (i.e. the first approximation). (ii) The pressure distribution in the
direction normal to the bed is hydrostatic, which leads to (10.27), which is the basis of
the hydraulic approach (i.e the second approximation).
The derivation of Equations (10.29) and (10.30) is presented here for a two-
dimensional cross section of an unconfined aquifer. It is straightforward to consider
the more general case of three-dimensional flow, with x as the coordinate pointing up
the slope along the impermeable bed, and y as the horizontal lateral or span-wise coor-
dinate, to obtain a more general form of the Boussinesq equation, namely
∂η
cos
k 0
n e
η ∂η
α ∂η
η ∂η
I
n e
t =
α
+
sin
x +
+
(10.31)
x
x
y
y
Mathematically, Equation (10.31) can be characterized as a nonlinear advective diffusion
equation, with a variable (i.e. a function of
) and anisotropic hydraulic (groundwater)
diffusivity, whose two principal components are D h x =
η
k 0 η
cos
α/
n e and D h y =
k 0 η/
n e ,
and with a hydraulic (groundwater) advectivity c h =−
n e .
A basic feature of the hydraulic approach is that two-dimensional flow is represented
by a one-dimensional formulation in Equations (10.29) and (10.30); thus the unknown
hydraulic head h
k 0 sin
α/
=
h ( x , z , t ) is replaced by the unknown position of the water table
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