l ¼ l
1
þ r
b
k
1
C
n ¹ 1
v l
2
14c where C ¼ contaminant concentration; D ¼ dispersion
coefficient neglecting molecular diffusion; e ¼ dispersivity; R ¼ retardation coefficient; l ¼ overall first order decay
coefficient; l
1
¼ dissolved phase, first order, decay coeffi-
cient; l
2
¼ adsorbed phase, first order, decay coefficient;
r
b
¼ soil bulk density; k
1
¼ Freundlich coefficient; and n ¼
Freundlich exponent.
24
The soil profile is initially contami- nant free. The boundary conditions correspond to a steady
inflow of a TCE dissolved water pulse from the profile top at C0,t ¼ C
until t ¼ t followed by fresh water injection.
The profile base is assumed to be under zero dispersive flux condition. These initial and boundary conditions can be
written as C
z ,
t ¼ 0 ¼
0 z L
c
15a C
, t
¼ C
t t 15b
¼ t . t
]C ]z
z ¼ L
¼ 15c
5 RESEARCH APPROACH
In this paper, the focus is to study the six research areas identified in a previous section related to the objectives, and
the study is performed using simulation of a GFCT model related to flow and transport through a one-dimensional
vertical soil profile. Once the base case scenario and related properties are defined, simulations are performed to gener-
ate the breakthrough concentration at the profile base for different scenarios, and these simulations are performed
using the computer code, HYDRUS.
24
Using a selected number of key features of the breakthrough curve BTC,
useful ANN concepts related to architecture, sampling, training, and multiple function approximation are studied,
and the potentials of BPA and GA in ANN training are compared. Finally, a general guideline for simulating
GFCT responses using ANN is suggested.
5.1 Base problem: unsaturated zone flow and single species mass transport
The base problem consists of a vertical sandy clay loam profile of 500 cm with a steady water flux of q ¼
10 cm day
¹ 1
at the profile top and a steady water table at the profile base. The flow parameters are a ¼ 0.05 cm
¹ 1
; b ¼ 2; v
s
¼ 0.4; v
r
¼ 0.1; and K
s
¼ 30 cm day
¹ 1
, and the TCE slug consists of a uniform concentration of C
¼ 15 mg L
¹ 1
for t ¼
15 days. The remaining properties are e ¼ 1.6 cm; r
b
¼ 1.6 gm cm
¹ 3
; l
1
¼ 0.001 per day; l
2
¼ 0;
n ¼
1 linear
Freundlich isotherm;
and k
1
¼ 0.293 cm
3
gm
¹ 1
k
1
¼ k
d
, the distribution coefficient, for n ¼ 1. Also, to ensure numerical stability and minimal
numerical dispersion, Dx ¼ 2 cm Dx ¼ space increment and Dt ¼ 0.1 days Dt ¼ time step were used.
The base problem is solved using HYDRUS for 160 days
for n ¼ {0.5, 1, 2}, where the base value is n ¼ 1. Fig. 2 shows the three BTCs at the profile base, and it also shows
the line corresponding to the maximum contaminant level MCL of 5 mg L
¹ 1
for TCE. Four important features of the BTC are noted. First, as t increases from 0 to 160 days,
the concentration increase from zero to maximum to zero. Second, as n increases from 0.5 to 1 to 2, the breakthrough
time varies approximately from 23 to 32 to 20 days. Third, as n increases from 0.5 to 1, the time to exceed the MCL of
5 mg L
¹ 1
varies approximately from 24 to 42 days. When n ¼
2, the concentrations remained less than the MCL. Fourth, as n increases from 0.5 to 1 to 2, the maximum concentra-
tion varies approximately from 14.5 to 12.5 to 2.5 mg L
¹ 1
. As such, the BTC is a nonlinear nonmonotonic function of t
and n, and the problem appears to be well-defined to pursue the research objective.
5.2 Breakthrough concentration
ANN development for simulating the BTC requires precise identification of the input and output vectors. From eqn 10,
the pressure head, h, can be expressed as
h ¼ h z ,
L
c
, a
, b
, v
s
, v
r
, K
s
, q
ÿ 16
and from eqn 13, the concentration, C, can be expressed as
C ¼ C v ,
t ,
q ,
r
b
, e
, l
1
, l
2
, k
1
, n
, C
, t
ÿ 17
From eqns 16 and 17, the breakthrough concentration, C
, at the profile base may be expressed as C p ¼ C p L
c
, a
, b
, v
s
, v
r
, K
s
, q
, t
, r
b
, e
, l
1
, l
2
, k
1
, n
, C
, t
ÿ 18
In order to reduce the domain, L
c
; b, v
s
, v
r
, r
b
, l
2
, C , and t
16 12
8
2
Concentration µ
gL
20 40
60 80
100 120
140 160
Time days n = 1
2 0.5
14 10
4 6
MCL
Fig. 2. Sensitivity of the breakthrough curve to nonlinear adsorp-
tion isotherm parameter, n
150 J. Morshed, J. J. Kaluarachchi
are held constant at the base values, and eqn 18 is reduced to
C p ¼ C p F
, t
, T
19a where
F ¼ {a ,
K
s
; q}
19b T ¼ {q
, e
, l
1
, k
1
, n}
19c where F ¼ subset of flow parameters; and T ¼ subset of
transport parameters. It should be noted that when eqn 18 is simplified to eqn 19, a number of flow and transport
parameters, L; b; v
s
; v
r
, r
b
, l
2
, C , and t
, are kept constant. This simplification is made to simplify studying the applic-
ability of ANN while considering the most significant flow and transport parameters such as a or K
s
. Also, the simpli- fication helps to make conclusive remarks on the overall
applicability by considering the effects of the few most significant parameters instead of all parameters. In the
next section, the applicability of ANN in simulating eqn 19 is assessed.
5.3 Training and testing subset
