Geoscience Reference
In-Depth Information
as being reach average values. Solutions to these equations are functions of space
and time:
c
(
x
,
t
) and
s
(
x
,
t
). The following boundary conditions are appropriate for
the scenario in these experiments. At the upstream boundary, the tracer concentra-
tion entering the reach is specified for all time,
t
0to
T
; at the downstream
boundary, a zero diffusive flux is assumed which implies that solute is advected
out of the domain unhindered. For the initial conditions all concentrations are
assumed to be zero at
t
¼
0. The data required to furnish the upstream boundary
condition is supplied by the observed conductivity data from the upper end of the
study reaches.
The model equations are discretised using a control (or finite) volume approach,
evaluating the advection term explicitly in time using a semi-Lagrangian method
(Manson and Wallis
1995
) and by evaluating the dispersion and transient storage
terms implicitly in time using the Crank-Nicolson method, which apportions equal
weight to both present and future values of
c
and
s
(Hoffman
1992
). The solution
consists of estimates for
c
and
s
over some discretised spatial and temporal domain,
i.e.
¼
c
i
s
i
Þ
1to
n
t
, where
n
x
is the number of points in the
spatial domain and
n
t
is the number of points in the temporal domain. Note that the
spatial domain is divided into (
n
x
ð
;
for
i
¼
1to
n
x
and
m
¼
1) cells of size
D
x
and the temporal domain is
divided into (
n
t
1) time steps of size
D
t
. Equation (
1
) is an advection-diffusion-
decay equation and represents a considerable challenge to existing numerical
methods. In order to achieve a satisfactory solution, the DISCUS method (Manson
and Wallis
1995
,
2000
) is adopted. This method employs a conservative semi-
Lagrangian algorithm that combines a control volume discretisation, the method of
characteristics and a flux-based interpolation scheme. The method and its accuracy
portrait is explained in detail elsewhere (Manson et al.
2001
), but note that in this
work, in contrast to earlier applications of this model, the Crank-Nicolson method
is adopted for the dispersion and transient storage terms. The discretised form of
these equations is a coupled pair of equations linking
c
i
and
s
i
to their neighbouring
cells for the whole computational domain at the future time level,
n
þ
1,
a
i
c
nþ
1
þ b
i
c
nþ
1
þ g
i
c
nþ
1
þ d
i
s
nþ
1
¼ e
i
(3)
i
1
i
i
þ
1
i
s
i
c
nþ
1
s
i
s
nþ
1
s
i
b
þ d
¼ e
(4)
i
i
s
i
; d
s
i
; e
s
in which
i
are coefficients related to both physical and numer-
ical parameters. Since there are (
n
x
a
i
; b
i
; g
i
; d
i
; e
i
; b
1) cells, the resulting 2(
n
x
1) equations are
assembled into a matrix and solved to give
c
nþ
1
i
and
s
nþ
i
for each computational
cell. Note that (
4
) may be used to eliminate the transient storage term from (
3
)
before it is solved.
The model prediction for concentration versus time at the downstream end of the
experimental reach was fitted to the observed data at the lower end of the study
reach, which had been collected at
n
t
points in time. A fitting parameter may be
defined as:
