Environmental Engineering Reference
In-Depth Information
By application of the chain rule results:
"
!
!
#
0
1
2
2
exp
t
2
x
vt
2
v
D
x
x
þ
vt
2
p
p
p
exp
exp
@
A
p
Dt
Dt
Dt
@
c
@v
ðx; tÞ¼c
in
erfc
x
2
D
v
D
x
x
þ
vt
2
þ
exp
p
Dt
(10.24)
The derived equations are implemented in the following subroutine. As in the
examples mentioned, the measurements are represented in the two vectors
xfit
and
cfit
. They are assumed to be measured at time
T
after the start of migration.
D
denotes the relevant diffusivity,
c0
is the initial and
c1
the inflow concentration.
function
f = myfun(v);
global
xfit cfit T D c0 c1
e=diag(eye(size(xfit,2))); h=1./(2.*sqrt(D*T));
arg1 = h*(xfit-v*T*e'); arg2 = h*(xfit+v*T*e'); arg3 = (v/D)*xfit;
% solve advection diffusion equation for c with c(t=0)=c0 and c(x=0)=c1
c = c0 + 0.5*c1*(erfc(arg1)+(exp(arg3).*erfc(arg2)));
% compute derivative of solution due to v
cv = c1*((T*h/sqrt(pi))*(exp(-arg1.*arg1)-exp(arg3).*…
exp(-arg2.*arg2))+0.5*(xfit/D).*exp(arg3).*erfc(arg2));
% specify function f to vanish
f = 2*(c-cfit)*cv';
The algorithm is demonstrated for an example of chloride concentrations
measured in a sediment core of the Marmara Sea (Pekdeger
2006
). It is assumed
that a concentration profile, as it is observed today, results from two interacting
processes. Sedimentation compounds the sediment layer. For the sake of simplicity
a constant sedimentation rate is assumed which corresponds to a sedimentation
velocity
v
. The second process is diffusion.
When there are no horizontal differences in variables or parameters, a 1D
description can be used. The sediment water interface reduces to a single position.
One may choose the origin to be located at the sediment water interface for the
complete simulation. Note that this point moves in a coordinate system that is fixed
to the earth, but that does not change the validity of the differential equations. The
change of the concentration profile can be described by the usual transport equation
with effective diffusivity
D
and positive sedimentation velocity
v
.
