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and from solute movement in the
z
direction is
−
∂
∂
J
z
z
xyz
(3.7)
Adding Equations 3.5, 3.6, and 3.7 yields the net mass of solute (inflow-
outflow) per unit time for the entire volume element as a result of solute
movement in the
x
,
y
, and
z
directions.
∂
∂
J
y
∂
∂
J
x
∂
∂
J
z
y
Netmasstransport
=
−
x
+
+
z
xyz
(3.8)
This net rate of solute flow represents the amount of mass of solute gained
or lost within the volume element per unit time. This is often called the
rate of solute accumulation. Now we assume that the solute species con-
sidered, in our example, is of the nonreactive type, that is, the solute is not
adsorbed or retained by the soil matrix. Therefore, we can further assume
that for a nonreactive solute
S
in Equation 3.1 is always zero and the solute is
only present in the soil solution phase having a concentration
C
. Moreover,
if Θ is the volumetric soil water content, that is, the volume of water per
unit volume of bulk soil, then Θ Δ
x
Δ
y
Δ
z
is the total volume of water in the
volume element shown in Figure 3.1. At any time
t
, the total solute mass
in the volume element is Θ
C
Δ
x
Δ
y
Δ
z
. Therefore, based on the principle of
mass conservation, the rate of solute accumulation, that is, the rate of gain
or loss (ΜΘ
C
/Μ
t
) Δ
x
Δ
y
Δ
z
, is equivalent to the net rate of mass flow (inflow-
outflow). That is:
∂
∂
J
y
∂Θ
∂
C
t
∂
∂
J
x
∂
∂
J
z
y
xyz=
−
x
+
+
z
xyz
(3.9)
By dividing both sides of Equation 3.9 by the volume element Δ
x
Δ
y
Δ
z
,
we have
∂
∂
J
y
∂Θ
∂
C
t
∂
∂
J
x
∂
∂
J
z
y
=
−
x
+
+
z
=
−
div
J
(3.10)
which is called the solute continuity equation for nonreactive solutes. For
the general case where the solute is of the reactive type, we can denote the
extent of solute reaction in terms of the amount retained on the soil matrix
S
as described in Equation (3.1). Therefore, the rate of change of the total mass
χ for the
i
ith species with time may be represented by the following general
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