Geoscience Reference
In-Depth Information
Solution:
hh
L
−
120 110
2200
−
1
2
Hydraulic gradient,
I
=
=
=
0.0045
For each 1 ft width,
A
= 1 × 40 = 40 ft
2
From Equation 21.6:
Q
= 12 ft/day × 0.0045 × 40 ft
2
) = 2.16 ft
3
/day/ft width
From Equation 21.5:
(
)
=
Kh
−
h
12
×
0 0045
05
.
. 5
1
2
Seepagevelocity,
v
=
= 0.098 ft/day
nL
21.4 GENERAL EQUATIONS OF GROUNDWATER FLOW
The combination of Darcy's law and a statement of mass conservation results in general equations
describing the flow of groundwater through a porous medium. These general equations are partial
differential equations in which the spatial coordinates in all three dimensions,
x
,
y
, and
z
, and the
time are all independent variables. To derive the general equations, Darcy's law and the law of mass
conservation are applied to a small volume of aquifer, the
control volume
. The law of mass conser-
vation is basically an account of all the water that goes into and out of the control volume. That is,
all the water that goes into the control volume has to come out, or there has to be a change in water
storage in the control volume. Applying these two laws to a confined aquifer results in Laplace's
equation, a famous partial differential equation that can also be used to describe many other physi-
cal phenomena—for example, the flow of heat through a solid (Baron, 2003):
2
2
2
∂
∂
h
+
∂
∂
h
y
+
∂
∂
h
z
=
0
2
2
2
x
Applying Darcy's law and the law of mass conservation to two-dimensional flow in an unconfined
aquifer results in the Boussinesq equation:
S
K
∂
∂
h
T
(
)
+∂∂∂∂
(
)
=
y
∂∂ ∂∂
xh hs
xh hy
where
S
y
is the specific yield of the aquifer. If the drawdown in the aquifer is very small compared
with the saturated thickness, the variable thickness,
h
, can be replaced with an average thickness
that is assumed to be constant over the aquifer. The Boussinesq equation can then be simplified to
2
2
∂
∂
h
+
∂
∂
h
y
S
Kb
∂
∂
h
t
y
=
2
2
x
Describing groundwater flow in confined and unconfined aquifers, using the general partial
differential equations, is difficult to solve directly. However, these differential equations can be
simplified to algebraic equations for the solution of simple cases (e.g., one-dimensional flow in a
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