Supplementary Material
An experimental study on solute transport in one-dimensional clay
soil columns
Muhammad Zaheer1, Zhang Wen2*, Hongbin Zhan3, Xiaolian Chen4, Menggui Jin5
1. Muhammad Zaheer: Ph.D candidate. Affiliation: School of Environmental Studies, China
University of Geosciences, Wuhan, Hubei, 430074, P. R. China. Email: [email protected]
2. Zhang Wen (*Corresponding author): Ph.D. Professor. Affiliation: School of Environmental
Studies, China University of Geosciences, Wuhan, Hubei, 430074, P. R. China. Email:
[email protected].
3. Hongbin Zhan: Professor. Affiliation: a) School of Environmental Studies, China University of
Geosciences, Wuhan, Hubei, 430074, P. R. China; b) Department of Geology and Geophysics,
Texas A & M University, College Station, TX 77843-3115, USA. Email: [email protected]
4. Xiaolian Chen: Master student. Affiliation: School of Environmental Studies, China
University of Geosciences, Wuhan, Hubei, 430074, P. R. China. Email:[email protected]
5. Menggui Jin: Professor. Affiliation: School of Environmental Studies, China University of
Geosciences, Wuhan, Hubei, 430074, P. R. China. Email: [email protected]
1
Mathematical Models
ADE:
Solute transport inhomogeneous medium is expected to satisfy the well-known
Advection-Dispersion Equation (ADE).The one-dimensional ADE for non-reactive solute
transport through homogeneous soil without sink/source is [1, 2]:
C
 2C
C
 D 2 v
t
x
x
(S1)
where C is solute concentration (ML-3), v is the average flow velocity in the column (LT-1)
(which equals the ratio of Darcian velocity over the effective porosity), and D is the
hydrodynamic dispersion coefficient (L2T-1), t is time (T), and x is distance (L). The analytical
solution of ADE for the third-type or flux type inlet boundary condition is[3, 4]
C ( x, t ) 
 ( x  vt ) 2
 x  vt 
C0
v 2t
erfc 
exp 
  C0
 4 Dt
2
D
 2 Dt 

C
 0
2



 vx v 2 t 
 x  vt 
 vx 
 1  
 exp   erfc 

D D 
D
 2 Dt 

(S2)
WhereC0 is the constant source concentration (ML-1) and erfc is the complementary error
function.
FADE:
As pointed out by [5]it is very important to exactly describe both the early arrival time
behavior for contaminants to escape from subsurface waste deposits and the late-time tailing
behavior for ground water remediation problems. The ADE is unable to describe such solute
transport process effectively.Benson [6] and Benson et al.[7, 8] presented the Lévy motion based
theory to define and modeled these anomalous transports, and spatial and temporal spreading of
solute concentration, respectively. For one dimensional non-reactive tracer transport the spatial
Fractional Advection-Dispersion Equation (FADE) [9, 10]can be expressed as:
2

C
C  1    C  1  
 C
 v
   D'     D'

t
x  2 2  x
 2 2   ( x)
(S3)
WhereC is the resident solute concentration (ML-3),v is the average flow velocity in the
column (LT-1), D ' is the dispersion coefficient (LλT-1), λ (1<λ ≤2) is the order of fractional
differentiation and whenλ=2 then FADE collapses to the ADE. It is worthwhile to point out that
the dimension of D ' in FADE is different from that of D in ADE, x is the spatial coordinate,t is
the time (T), γ is the relative weight of solute particle forward versus backward transition
probability, -1≤γ≤1. For backward transition probability is -1≤ γ ≤0, and for forward transition
probability is 0≤ γ ≤1. When γ=0, the transition of solute particles and dispersion in FADE is
symmetric [2]. For a semi-infinite system initially free of solute and a step input at x=0 with
concentration of C0, the analytical solution for Eq. (S3) can be expressed as [9, 11, 2].


x  vt

C  x, t   C0 1  F 

 cos  / 2 D ' t






1/ 


 
 
(S4)
WhereFλ(y) is the symmetric λ-stable probability function[9]:
 1

sign 1    1
F  y   C    
exp
  y U   x  dx

2


0
(S5)
Wherex is the integration variable, C(λ) and Uλ(x) can be expressed as:

1   1
 sin  x / 2   1

and U   x   
C ( )   1

cos  / 2  
 1


2
(S6)
It is interesting to note that the FADE is capable of describing the non-Fickian transport to some
extent, however the dispersion coefficient of FADE was found to be higher in transport process
and lower in the leaching process than those was in ADE.
TRM:
3
In anomalous type of environments, mobile and immobile regions supported the enlargement of
unconventional transport models to the ADE. One of those models is the TRM model proposed
by [12]:
C1
C2
 2C1
C
1
 2
 1 Dm 2  1Vm 1
t
t
x
x
(S7)
Whereθ1 is the water content in the mobile region, θ2 is the water content in the immobile region,
Vmis the mobile pore-water velocity (LT-1), Dm is the dispersion coefficient in mobile region
(L2T-1),C1is the solute concentration in mobile region (ML-3), C2 is the solute concentration in
immobile region (ML-3). The second term expresses the solute mass transfer variation as being
proportional to the concentration difference between these two regions.
C2
   C1  C2 
t
C
C
 2 2    C1  C2   2 2    C1  C2 
t
t
2
(S8)
The total water content is equal to θ as:
θ1 +θ2 = θ
(S9)
where θ is the total water contents and here in the case of fully saturated media equal to porosity
(n).
θ1Vm=q
(S10)
Eq. (S10) showed that the Darcy velocity (LT-1). TRM reduces to the ADE once the water
content of the second region θ2=0 or once the rate of mass transfer is non-existent that is ω=0,
and parameters of the ADE are then v =Vmand D=Dm.The initial of zero concentration and third
type inlet boundary condition for semi-infinite system in Laplace domain, TRM analytical
solution in Laplace domain is [13, 2]:
4
 V  V 2  4D  
m
m
C1  x, u  
exp  m
x


2 Dm
u  u 1  4 Dm / Vm


2C0
C2  x , u  
2C0
  u 2   u  u
1  4 Dm / Vm
In the above equation (S11) the 

(S11)
 V  V 2  4D  
m
m
 exp  m
x


2 Dm




 2

 1 u ,
  m   u 2 

(S12)
where u is the Laplace variable, and
solutions can be inverted analytically [13]or numerically [14].The expected velocity splitting in
the TRM model into mobile and immobile regions is not a precise interpretation of the true
velocity and a single mass transfer rate unable to predict the solute transport with long tails
(e.g.[15, 2, 16 ] ).
CTRW (Based on TPL):
A more effective method for describing the non-Fickian and Fickian transport is
theContinuous Time Random Walk (CTRW) theory. This method provides a quantitative
framework to clarify the Fickian and non-Fickian variety of transport behaviors. For this
framework it is worth mentioning that the CTRW can capture the complete evolution of solute
transport. The main equation concerning solute transport in the CTRW framework is a partial
differential equation[17] and transport equation can be written in the Laplace domain as:
uc  s, u   c0  s    M  u  v c  s, u   D : c  s, u  
(S13)
wherec0 is the initial concentration and u is the Laplace variable, vψtransport velocity, Dψ is
generalized dispersion coefficient (L2T-1) and has a distinct physical explanation from the
hydrodynamic dispersion coefficient (D) used in ADE [18]. Dispersion coefficient (Dψ) of
CTRW varies with transport velocity or flow rate, thus changes the dispersive nature of
transport[17],and M  u  memory function. The transport velocity, generalized dispersion
5
coefficient and memory function in Eq. (S13) can be expressed as v 
1
p( s) sds , is the average
t1 
tracer transport velocity which might be different from the average fluid velocity v, and
D 
1 1
p  s  ssds
t1  2
is dispersion coefficient, p(s) is the probability density function (pdf) of the
transitions lengths, M  u   t1u1
 u 
1   u 
is a memory function and t1 is a characteristics time [19, 17,
16].The probability density function   t   L 1  u  is expressed as the probabilities rate of the
transition time between the sites. The nature of the solute transport was determined with the help
of probability density function, a form of ψ(t), the truncated power law (TPL), can effectively be
applied to a wide range of physical scenarios. The TPL form can be written as [17]:
t
n
e t2
 t  
,0    2
1 
t1 
t
1  
 t1 
where
 t β t1  t  
n=  1  e t 2 Γ  -β, 1  
 t 2 
 t 2  
(S14)
-1
, is a normalized factor, β compute the dispersion and t2 is the cut-off
t
time, as t2 (˃˃t1),   a, x  is the imperfect Gamma function for t1˂˂t˂˂t2  t  ~  
 t1 
1 
, and
t
decreases exponentially as   t  ~ e t in which t ˃˃t2 [16],β is a dispersion related dimensionless
2
parameter, for Fickian transport β≥2 and for non-Fickian solute transport 1˂β˂2, whereas the
smaller value of β is, the more dispersive the transport behavior [18,16].
It should be pointed out that we only provided the solutions for the breakthrough process.
While for the solutions for leaching process, they can be easily obtained by changing the
initialconcentration and the input concentration.
Numerical Simulationof solute transport and leaching processes
6
In this study the CXTFIT 2.1 software package [13]was used to carry out the simulation
analysis of ADE and TRM to various groups of BTCs and leaching data. Two parameters (v, D)
were involved in ADE. For FADE simulation of both BTCs and leaching curves in homogeneous
soils columns, we used FADE Main Fortran code [10, 11]and three parameters (v, D ' , λ) were
estimated, three additional parameters (v, φ, ω) were estimated in TRM, where φ represents the
mobile water fraction (φ =θ1/θ, where θ1 and θ are the mobile water content and total porosity,
respectively) and ω is the mass transfer rate (T-1) between the mobile and immobile domains in
TRM. For the CTRW simulation of both BTCs and leaching curves we used the CTRW Matlab
Toolbox v.3.1(http://wws.weizmann.ac.il/EPS/People/Brian/CTRW/) and five parameters (vψ, Dψ,
β, t1, and t2) were estimated, where vψ is transport velocity (LT-1), Dψ is generalized dispersion
coefficient (L2T-1) and has a distinct physical explanation from the hydrodynamic dispersion
coefficient (D) used in ADE [2]. Dispersion coefficient (Dψ) of CTRW varies with transport
velocity or flow rate, thus changes the dispersive nature of transport [17],β is dispersion related
dimensionless parameter,t1 is a characteristics time (T) and t2is the cut-off time (T). For Fickian
transportβ≥2 and for non-Fickian solute transport 1˂β˂2 [18, 16].
For fitting evaluation, determination of coefficient (R2) and root mean square error (RMSE)
(Table A and Table B), were used to determine the quality of fitwhich can be expressed as [20]:
2
 N

   Cic  Cic  Cim  Cim  

R 2  N i 1
,
N
2
2
  Cic  Cic    Cim  Cim 
i 1
(S15)
i 1
2
RMSE 
1 N
  Cic  Cim  ,
N i 1
(S16)
7
Where N is the number of concentrations, Cic is the estimated concentration (ML-3), Cim is the
measured concentration (ML-3), Cic and Cim are the mean values of Cic and Cim, respectively.
Table S1
Estimated values parameters, Determination coefficient (R²)and root mean square error (RMSE)
of transport process for ADE, FADE, TRM and CTRW (TPL) in three sets of homogeneous soil
columns
ADE
Column length
Column inner
R2
FADE
RMSE
R2
g cm-3
diameter
TRM
RMSE
R2
CTRW
RMSE
g cm-3
R2
RMSE
g cm-3
g cm-3
3 cm with smooth wall
14cm
0.997
0.024
0.989
0.042
0.997
0.024
0.997
0.027
5 cm with smooth wall
14cm
0.995
0.035
0.981
0.056
0.995
0.029
0.995
0.031
8 cm with smooth wall
14cm
0.999
0.016
0.998
0.015
0.999
0.014
0.999
0.016
5 cmwith smooth wall
7 cm
0.999
0.02
0.991
0.032
0.999
0.011
0.999
0.012
5 cmwith smooth wall
9cm
0.998
0.023
0.988
0.041
0.998
0.016
0.998
0.017
5 cmwith smooth wall
10 cm
0.993
0.039
0.992
0.040
0.998
0.013
0.999
0.012
3 cm with roughwall
14 cm
0.996
0.03
0.994
0.032
0.996
0.028
0.995
0.03
5 cm with rough wall
14cm
0.995
0.038
0.985
0.051
0.996
0.03
0.995
0.034
8 cm with rough wall
14cm
0.998
0.021
0.984
0.024
0.999
0.02
0.998
0.022
Table S2
Estimated parameters, Determination coefficient (R²) and root mean square error (RMSE) of
leaching process for ADE, FADE, TRM, and CTRW (TPL) in three sets of homogeneous soil
columns
ADE
Column length
Column inner
R2
diameter
FADE
RMSE
R2
g cm-3
TRM
RMSE
R2
g cm-3
CTRW
RMSE
R2
RMSE
g cm-3
g cm-3
3 cm with smooth wall
14cm
0.9746
0.0411
0.9998
0.0032
0.9925
0.0245
0.9974
0.0137
5 cm with smooth wall
14cm
0.9531
0.0620
0.9973
0.0244
0.9656
0.0692
0.9885
0.0343
8 cm with smooth wall
14cm
0.9888
0.0325
0.9995
0.0082
0.9759
0.0499
0.9983
0.0125
5 cmwith smooth wall
7 cm
0.9896
0.0264
0.9999
0.0042
0.9763
0.0412
0.9982
0.0181
5 cmwith smooth wall
9cm
0.9818
0.0387
0.9988
0.0190
0.9726
0.0469
0.9985
0.0154
5 cmwith smooth wall
10 cm
0.9853
0.0346
0.9997
0.0069
0.9552
0.0837
0.9973
0.0365
3 cm with rough wall
14 cm
0.9837
0.0331
0.9986
0.0118
0.9978
0.0121
0.9953
0.0211
5 cm with rough wall
14cm
0.9713
0.05
0.9986
0.0235
0.9686
0.0525
0.9824
0.0546
8 cm with rough wall
14cm
0.9804
0.05
0.9996
0.0087
0.9839
0.0463
0.9985
0.0191
8
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11