Geoscience Reference
In-Depth Information
zone solute concentration (kg/m
3
),
K
x
the longitudinal dispersion coefficient (m
2
/s),
Q
the volumetric flow rate (m
3
/s),
t
the time (
s
),
x
the distance (m), and
is the
a
storage zone exchange coefficient (1/s).
The first equation represents mass balance in a main channel, while the other one
describes exchange of solute between a main channel and a storage zone.
According to the model, mass exchange between a main channel and storage
zones occurs due to difference of concentrations between a main channel and
storage zones, and it is proportional to the difference (Bencala and Walters
1983
).
4 Methods
4.1 Deterministic Calibration of the Dead Zone Model
First of all, calibration was performed to specify parameter values that give the
best fit of the simulated concentrations to the observed ones. Calibration was
performed for flow rates
Q
2and
Q
3 for four sub-reaches separately (Fig.
1
:sub-
reachonebetweenP-1andP-2andsoon). A breakthrough curve from an
upstream cross-section of a sub-reach posed to be a boundary condition, and the
output curve was computed for a downstream cross-section. As a result, eight
calibrated breakthrough curves were obtained for further analyses. Results of
tracer test for
Q
1 were used during verification, which is beyond the scope of
this chapter.
During calibration, for each sub-reach the least square objective function was
applied in the following form:
X
t¼n
2
F
¼
C
o
;
t
C
s
;
t
(3)
t¼
1
where
n
is the number of observed values,
C
o,t
the observed concentration at a time
step
t
, and
C
s,t
is the simulated concentration at a time step
t
.
In the study, the Differential Evolution (DE) method (Price 2005), which is a
global optimization technique, was applied.
Four parameters were estimated in the dead zone model:
A
- channel cross-
sectional area (m
2
),
K
x
- longitudinal dispersion coefficient (m
2
/s),
A
S
- storage
zone cross-sectional area (m
2
), and
- storage zone exchange coefficient (1/s).
Ranges of parameter values for the DE method were chosen as follows:
(0,100) for a channel cross-sectional area (
A
), a longitudinal dispersion coeffi-
cient (
K
x
), and a transient storage zone cross-sectional area (
A
S
)and(0,2)fora
storage zone exchange coefficient (
a
). A starting point was set randomly. Opti-
mal values of parameters are presented in Fig.
2
. Parameter sets obtained by the
DE method result in breakthrough curves that are in a good agreement with
observations (Fig.
3
).
a
