Environmental Engineering Reference
In-Depth Information
By the application of divergence theorem, we get the local equation:
&&
T
[13.22]
∇
v + C - S - Q = 0
p
p
It expresses that, for an elementary volume, the terms of source in volume
Q
p
are:
- added to the sum of the inflows and outflows by boundaries
q
p
;
- subtracting the mass stored in the fixed water on the matrix or by degradation;
- equal to the mass of pollutant stored in the mobile water.
We must add to the previous balance equations (related to mobile water) the
mass pollutant balances in the other phases considered, (at the surface of the solid
grains and held in the immobile water). Indeed, the storage equations in these two
phases depend on the concentrations in the phases. The corresponding balance
equations are simply obtained by integrating the corresponding storage equations,
giving:
∂
&
&
[13.23]
CS
=
+
S
solid
solid
−
water
solid
−
immobile
∂
t
∂
&
&
[13.24]
C
=
S
−
S
immobile
immobile
−
water
solid
−
immobile
∂
t
Note that equation [13.22] is a partial differential equation describing a field,
while equations [13.23] and [13.24] do not involve any space variable.
The local balances of transport in the aquifer are then formulated. To verify that
the equilibrium is respected throughout the area studied without checking at each
point, the weighted residues method is used. Let us consider the weighting functions
W
. Based on the input weighted form:
δ
∫
=
Q W dV +
q W dA
[13.25]
∫
E
V
A
By using the local balance equations and the divergence theorem, provided the
weight functions can be derived, we can transform this expression according to the
terms of velocity and storage:
∫
&
&
T
⎡
⎤
δ
=
δ
= (C -
) W -
v
∇
(W) dV
[13.26]
WW
S
E
I
⎣
p
⎦
V
