Unsaturated Flow
•
Derivation of Governing Equations
Description of Mass Balance Consideration
Derivation of governing differential equation is obtained
by considering a volumetric element.
The statement of mass balance is:
The difference between the total flux into the element and
the total flux out of the element is equal to the rate of
change of volumetric water content.
The expression for mass balance is
)
1
(
dt
d
z
q
y
q
x
q
z
y
x
θ
=
∂
∂
−
∂
∂
−
∂
∂
−
Assuming D’arcy’s Law
)
2
(
s
H
k
q
∂
∂
−
=
Where
H
z
h
+
=
Unsat71
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Derivation of Flow Equation
Substituting Eqn. 1 into 2 yields
dt
d
z
H
k
z
y
H
k
y
x
H
k
x
z
y
x
θ
θ
θ
θ
=
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
)
(
)
(
)
(
The equation for onedimensional flow can be written as
)
4
(
)
(
t
z
H
k
z
z
∂
∂
=
∂
∂
∂
∂
θ
θ
Rewriting Eqn. 4
)
5
(
1
)
(
t
z
h
k
z
z
∂
∂
=
+
∂
∂
∂
∂
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 Spring '11
 hasan
 Derivative, finite difference, Unsaturated Flow, Unsaturated Flow Solutions

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