Conceptual Modeling of Contaminated Solute Transport Based on Stream Tube Model

doi:10.4236/aces.2012.24059

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Advances in Chemical Engineering and Science, 2012, 2, 481-489

http://dx.doi.org/10.4236/aces.2012.24059 Published Online October 2012 (http://www.SciRP.org/journal/aces)

Conceptual Modeling of Contaminated Solute Transport

Based on Stream Tube Model

Seung-Gun Chung1,2, Soon-Jae Lee1,2, Dong-Ju Kim2, Sang-Hyup Lee1, Jae-Woo Choi1*

1Center for Water Resource Cycle Research, Korea Institute of Science and Technology, Seoul, South Korea

2Department of Earth and Environmental Sciences, College of Science, Korea University, Seoul, South Korea

Email: *[email protected]

Received July 28, 2012; revised August 30, 2012; accepted September 9, 2012

ABSTRACT

In this study, we performed a conceptual modeling on solute transport based on theoretical stream tube model (STM)

with various travel time distributions assuming a pure convective flow through each tube in order to investigate how the

lengths and distributions of solute travel time through STM affect the breakthrough curves at the end mixing surface.

The conceptual modeling revealed that 1) the shape of breakthrough curve (BTC) at the mixing surface was determined

by not only input travel time distributions but also solute injection mode such as sampling time and pulse lengths; 2) the

increase of pulse length resulted in the linear increase of the first time moment (mean travel time) and quadratic in-

crease of the second time moment (variance of travel time) leading to more spreading of solute, however, the second

time moment was not affected by travel time distributions; and 3) for a given input distributions the increase in travel

distance resulted in more dispersion with the quadratic increase of travel time variance. This indicates that stream tube

model obeying strictly pure convective flow follows the concept of convective-lognormal transport (CLT) model re-

gardless the input travel time distributions.

Keywords: Conceptual Modeling; Solute Transport; Pure Convective Flow; Stream Tube Model; Travel Time

Distribution

1. Introduction

At macroscopic scale, there are three basic causes of ve-

locity difference in pore-water that are friction loss, dif-

ferences in pore size and path length, all or each of which

leads to mixing of solute and thereby brings about me-

chanical dispersion [1]. This mechanical dispersion is

considered as a key parameter responsible for mixing or

dilution of solutes during transport through soils and dis-

tinguished from molecular diffusion arising from density

or concentration differences. Combination of mechanical

dispersion with molecular diffusion is termed hydrody-

namic dispersion which constitutes major transport pa-

rameters of solute transport equation together with con-

vection. For instance, the convection-dispersion equation

(CDE) of solute transport for nonreactive solute is de-

scribed by only these two parameters. It is, however,

based on a set of assumptions that are likely to be valid

only under certain conditions in soil or aquifer systems.

The CDE makes the hypothesis that the dispersion proc-

ess is formally equivalent to diffusion even though dis-

persion is a convective transport process.

Because the CDE is valid only after sufficient time has

elapsed to smooth out transport by convective velocity

differences along the direction of motion, it cannot be

valid at early times. For this reason, Jury [2] developed

an alternative formulation for solute transport, called the

transfer function or convective lognormal transfer func-

tion (CLT) model, which does not require the restrictive

assumptions of the CDE and can be applied to problems

at different scales. This model relates an input condition

to the output variable of interest based on the assumption

that the velocity of a solute particle remains constant

along its travel path through the soil. It further assumes

that there is no exchange of solute particles between

pores with different water velocities. In this regard, sol-

ute particle arrival times are perfectly correlated with

lognormal distribution of pore water velocities and solute

travel time variance increases quadratically with depth.

Recently Bardsley [3] studied temporal moments of a

tracer pulse in a perfectly parallel flow system by parti-

tioning water flow into isolated one-dimensional stream

tubes which maintain spatial correlation of all properties

in the direction of flow. The individual flow tubes in the

parallel flow model are referred to as channels for frac-

ture-flow and layers in the context of perfectly stratified

aquifer systems.

*Corresponding author.

S.-G. CHUNG ET AL.

482

Tracer transport in parallel systems provides a non-

diffusive mechanism contributing to dispersion [4-8].

This stream tube effect has been referred to as a convec-

tive-dispersive process [9,10], causing tracer arrival-time

variance to increase as a quadratic function of travel dis-

tance for convective flow. Since within a single stream

tube all hydraulic properties remain constant, the distri-

bution of travel time or flow velocities between stream

tubes will be the utmost important factors determining

the dispersive process at the outflow mixing surface. In

the previous studies of dispersion, the effect of normal

distributions [11,12], normal or lognormal arrival times

[10,13,14] between the stream tubes on solute transport

were investigated. However, to date there has been no

systematic study on both a pure convective process

within a stream tube and the effect of normal or log-

normal distributions of arrival times between stream

tubes on solute transport. In addition to the travel time

distributions, there would be some other factors such as

sampling time and pulse duration which influence the

dispersive process for pure convective flow.

The objective of this study is (1) to generate the break-

through curves (BTC) of nonreactive solute using a con-

ceptual stream tube model (STM) by assuming pure

convective process and (2) to investigate how the pulse

conditions and various travel time distributions affect the

transport concept of stream tube model using time mo-

ment analysis.

2. Material and Methods

2.1. Modeling of Conceptual Stream Tube

The conceptual stream tube model can be constructed in

two different ways. The first is a diameter-based type to

assemble a finite number of tubes with equal lengths but

different cross-sectional areas or diameters so that dif-

ferent average flow velocities are achieved by allowing

equal flow rate to each tube. The other is a length-based

type to construct the tubes with equal diameter but dif-

ferent lengths in order for the stream tubes to have nor-

mal or lognormal distributions of travel time when an

equal amount of discharge is imposed to each tube. In

this study, the latter case is adopted.

Following assumptions should be made for the con-

ceptual stream tube model:

1) The stream tube model is constructed with a finite

number of bundles of one-dimensional stream tubes with

equal diameter but different lengths.

2) Flow velocity within a single tube is steady and in-

variant.

3) Tracer solution is injected uniformly at the inlet

boundary of each tube.

4) Tracer transport in each tube obeys the convective

flow.

5) Molecular diffusion during pulse input is neglected.

6) Dispersive mixing occurs due to velocity differences

between tubes at the outflow mixing surface by instanta-

neous and complete mixing.

7) Effluents are collected at one common mixing sur-

face regardless the differences in tube lengths.

8) Velocity differences can be realized by normal or

lognormal distributions of travel length or time.

Consider a set of stream tubes which has a normal dis-

tribution of tube length as shown in Figure 1 where each

bundle consisting of the same tube length has a different

tube number. Let the length of ith tube bundle be Li, the

number of ith tube bundle Ni. Then the total number of

max

NTT N





. Let the diameter and cross- tubes is

sectional area of each tube be d and A, and injection flow

rate at the inlet boundary Q, injection concentration Co,

pulse duration To. Then the flow velocity and flow rate in

each tube is





vQNTTA



qQNTT

LvT and pulse

length 0p





. If a STM consists of five different

lengths (30, 40, 50, 60, 70 cm) and sampling time inter-

val Lvt



, then the sampling length interval





and number of pulse length p. Let the

number of sampling J and then sampling time will be

NPL LL





TJ t





L. For instance, the number of times of



which can have the effluent concentration for ith

bundle ii

TLLL, and the number of times of





required for breakthrough of ith tube ii

TT .

Then the number of tubes with tracer effluent at sampling

time J would be the following:

T NPL



where

iii

NTENIi TJTT







(1)

Therefore the effluent concentration at J will be

qtNTE NTE

CC C

Qt NTT

 





(2)

In order to investigate the effect of travel time distri-

bution on the convective-dispersive process, we gener-

ated three different input probability density functions of

travel times or tube lengths (60, 80, 100, 120, 140 cm) as

was shown in Figure 2 and used them for simulation of

BTCs at the mixing surface based on pure convective

flow within stream tubes. Sensitivity analysis was also

performed to see the effect of sampling time (



= 10,

20 cm) and pulse length (NPL = 1 to 6) on the BTCs.

2.2. Time Moment Analysis

In order to investigate the effect of sampling time, pulse

duration and input distribution of travel length on the

dispersive process along stream tubes, time moment

analysis was performed. The simulated concentrations in

time were first converted to probability density function

(pdf) as:

S.-G. CHUNG ET AL.

483

Figure 1. Examplary case of effluent concentration for the normal distribution of STM shown in Figure 2.

Length of tube (cm)

406080100 120 140 160

Number of tubes

Normal distribution

Right skewed distribution

Left skewed distribution

Figure 2. Three different input probability density functions (pdf) of travel times or tube lengths (60, 80, 100, 120, 140 cm)

used for simulation of BTCs.















Ttftt





21TTT





(3)

The nth order time moment is given by:

N (4)

The variance of travel time is given by:

(5)

3. Results and Discussion

For a given sampling time or length ( = 10 cm), the

effect of pulse length on the BTC is given in Figure 3 for

three different travel time distributions. The general

shape of BTC coincided with the input distribution of

travel as evidenced in Figures 3(a)-(c) for normal, right

skewed, left skewed distributions. For a given distribu-

tion, more dispersion was generated as pulse length in-

creased. It is noted that multi-peaks were due to the small

sampling time (



= 10 cm) compared to the relatively

large difference (20 cm) in length between tubes regard-

less the input travel time distributions. Thus shape of

BTC was dependent on both pulse length and travel time

distribution. Figure 4 shows the BTCs of three input pdfs

of travel time when the sampling time or length increased

to L



= 20 cm. Increased sampling time led to the

smoothing of BTC and increased dispersion. The output

BTC is identical to input BTC when NPL = 1. Tailing

was evident for the skewed pdfs with the fact that the

S.-G. CHUNG ET AL.

484

Time (min)

02468101214

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 NPL = 1

NPL = 2

NPL = 3

NPL = 4

NPL = 5

NPL = 6

406080100 120140 160

(a)

Time (min)

02468101214

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 NPL = 1

NPL = 2

NPL = 3

NPL = 4

NPL = 5

NPL = 6

406080100 120 140160

(b)

Time (min)

02468101214

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 NPL = 1

NPL = 2

NPL = 3

NPL = 4

NPL = 5

NPL = 6

406080100 120 140 160

(c)

Figure 3. Simulated BTCs for ΔL = 10 cm: (a) Normal distribution; (b) Lognormal distribution (right-skewed); (c) Log-

normal distribution (left-skewed).

S.-G. CHUNG ET AL. 485

Time (min)

02468101214

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 NPL = 1

NPL = 2

NPL = 3

NPL = 4

NPL = 5

NPL = 6

406080100120 140 160

(a)

Time (min)

0 2 4 6 8101214

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 NPL = 1

NPL = 2

NPL = 3

NPL = 4

NPL = 5

NPL = 6

406080100 120 140 160

(b)

Time (min)

0 2 4 6 8101214

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 NPL = 1

NPL = 2

NPL = 3

NPL = 4

NPL = 5

NPL = 6

406080100 120140 160

(c)

Figure 4. Simulated BTCs for ΔL = 20 cm: (a) Normal distribution; (b) Lognormal distribution (right-skewed); (c) Log-

normal distribution (left-skewed).

S.-G. CHUNG ET AL.

486

L

general shape follows the input distribution of travel time

or length. The relationship between the first time moment

(T1) of BTCs and NPL is given in Figure 5 for two dif-

ferent sampling time ( = 10, 20 cm). The first mo-

ment increased linearly with NPL. The highest T1 was

obtained for the left-skewed one among three pdfs. The

higher T1 was obtained for sampling time = 20 cm

(Figure 5). This indicates that the mean travel time of

solute along the STM with the assumption of pure con-

vective flow is affected by both sampling time and pulse

length. The variance of travel time of BTCs (Var) is

plotted as a function of NPL in Figure 6. The Var in-

creased quadratically with NPL showing the substantial

increase of Var for = 10 cm. It is noted that there

was negligible effect of input pdf on Var. The effect of

travel distance or time on the BTC is demonstrated in

Figure 7 where the travel distance increased from one-

fold to four-fold of the initial lengths (60, 80, 100, 120,

140 cm) of STM with three input pdfs for a given



20 cm and NPL = 8. It is noted that the number of

multi-peaks increased as the travel length increased. It is

evident that more dispersion occurred as the solute pulse

traveled along longer pathways. The relationship be-

tween T1 with travel distance is given in Figure 8. It was

found that T1 increased linearly with travel distance and

the input pdfs did not significantly affect T1. The rela-

tionship between Var and travel distance is given in Fig-

ure 9 where a quadratic increase of Var with travel dis-

tance was observed regardless the different input pdfs.

This implies that transport properties such as mean travel

time and variance of solute are not dependent on the

physical nature of the STM and agree with those of the

CLT model concept which assumes a correlated flow

without lateral mixing.

12 SciRes.

NPL

L = 10cm (Normal)

9L = 20cm (Normal)

L = 10cm (Lognormal - Right)

L = 20cm (Lognormal - Right)

L = 10cm (Lognormal - Left)

L = 20cm (Lognormal - Left)

123456

Figure 5. First time moment as a function of NPL for the simulated BTCs.

NPL

123456

Var

L = 10cm (Normal)

L = 20cm (Normal)

L = 10cm (Lognormal - Right)

L = 20cm (Lognormal - Right)

L = 10cm (Lognormal - Left)

L = 20cm (Lognormal - Left)

Y = 0.0



64x

2.13

+ 1.17

L = 20cm

Y = 0.015x

2.14

+ 1.1

L = 10cm

Figure 6. Second time moment as a function of NPL for the simulated BTCs.

S.-G. CHUNG ET AL. 487

Time (min)

0 10203040

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 x 1

x 1.5

x 2

x 3

x 4

(a)

Time (min)

0 10203040

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 x 1

x 1.5

x 2

x 3

x 4

(b)

Time (min)

0 10203040

Relative concentration

0.0

0.2

0.4

0.6

0.8

1.0 x 1

x 1.5

x 2

x 3

x 4

(c)

Figure 7. The effect of travel distance on the simulated BTCs: (a) Normal distribution; (b) Lognormal distribution (right-

skewed); (c) Lognormal distribution (left-skewed).

S.-G. CHUNG ET AL.

488

Distances (cm)

306090120 150 180 210 240 270

Normal

Lognormal - Right

Lognormal - Left

Figure 8. First time moment as a function of travel distance for the simulated BTCs for ΔL = 20 cm.

Distances (cm)

306090120 150 180 210 240 270

Var

Normal

Lognormal - Right

Lognormal - Left

x2.01

Figure 9. Second time moment as a function of travel distance for the simulated BTCs for ΔL = 20 cm.

4. Conclusion tance, indicating that STM with pure convective flow

follows the concept of CLT model. However the impact

of travel distance was shown to be independent on the

input pdfs. We further found that the input pdfs influence

only the shape of BTC at the mixing surface and is not

related with the transport process of solute.

In order to investigate the effect of sampling time, pulse

duration and input travel time distribution (normal, log-

normal) on the transport processes of a stream tube

model, we performed conceptual modeling on stream

tube model assuming a pure convective flow within each

stream tube. Sampling time and pulse length has a great

impact on BTC shape: 1) linear increase of T1 with num-

ber of pulse length (NPL); 2) quadratic increase of Var

with NPL; and 3) output BTC becomes identical to input

BTC when = NPL. Travel distance has also a great

impact on the transport processes such that variance of

travel time showed a quadratic increase with travel dis-

REFERENCES

[1] J. Bear, “Dynamics of Fluids in Porous Media,” Elsevier,

Amsterdam, 1972.

L[2] W. A. Jury, “Simulation of Solute Transport Using a Trans-

fer Function Model,” Water Resources Research, Vol. 18,

No. 2, 1982, pp. 363-368.

S.-G. CHUNG ET AL. 489

doi:10.1029/WR018i002p00363

[3] W. E. Bardsley, “Temporal Moments of a Tracer Pulse in

a Perfectly Parallel Flow System,” Advances in Water Re-

sources, Vol. 26, No. 6, 2003, pp. 599-607.

doi:10.1016/S0309-1708(03)00047-2

[4] L. W. Gelhar, A. L. Gutjahr and R. L. Naff, “Stochastic

Aquifer Analysis of Macrodispersion in a Stratified Aqui-

fer,” Water Resources Research, Vol. 15, No. 6, 1979, pp.

1387-1397. doi:10.1029/WR015i006p01387

[5] O. Güven, F. J. Molz and J. G. Melville, “An Analysis of

Dispersion in a Stratified Aquifer,” Water Resources Re-

search, Vol. 20, No. 10, 1984, pp. 1337-1354.

doi:10.1029/WR020i010p01337

[6] G. Matheron and G. de Marsily, “Is Transport in Porous

Media Always Diffusive? A Counter Example,” Water

Resources Research, Vol. 16, No. 5, 1980, pp. 901-917.

doi:10.1029/WR016i005p00901

[7] J. F. Pickens and G. E. Grisak, “Scale-Dependent Disper-

sion in a Stratified Granular Aquifer,” Water Resources

Research, Vol. 17, No. 4, 1981, pp. 1191-1211.

doi:10.1029/WR017i004p01191

[8] K. W. Thorbjarnarson and D. M. Mackay, “A Field-Test

of Tracer Transport and Organic Contaminant Elution in a

Stratified Aquifer at the Rocky-Mountain Arsenal (Den-

ver, Colorado, USA),” Journal of Contaminant Hydrol-

ogy, Vol. 24, No. 3-4, 1997, pp. 287-312.

doi:10.1016/S0169-7722(96)00015-0

[9] W. A. Jury and K. Roth, “Transfer Functions and Solute

Movement Through Soil,” Birkhauser, Boston, 1990.

[10] C. S. Simmons, “A Stochastic-Convective Transport Rep-

resentation of Dispersion in One-Dimensional Porous Me-

dia Systems,” Water Resources Research, Vol. 18, No. 4,

1982, pp. 1193-1214. doi:10.1029/WR018i004p01193

[11] A. Mercado, “The Spreading Pattern of Injected Water in

a Permeability Stratified Aquifer,” In: Artificial Recharge

and Management of Aquifers. Symposium of Haifa, March

1967, Organized in the Framework of the IHD, IASH

Publication, Gentbrugge, 1967, pp. 23-26.

[12] D. Grecov and J. R. Clermont, “Numerical Simulations of

Non-Newtonian Flows between Eccentric Cylinders by

Domain Decomposition and Stream-Tube Method,” Jour-

nal of Non-Newtonian Fluid Mechanics, Vol. 126, No.

2-3, 2005, pp. 175-185. doi:10.1016/j.jnnfm.2004.10.004

[13] G. Demmy, S. Berglund and W. Graham, “Injection Mode

Implications for Solute Transport in Porous-Media-Ana-

lysis in a Stochastic Lagrangian Framework,” Water Re-

sources Research, Vol. 35, No. 7, 1999, pp. 1965-1973.

doi:10.1029/1999WR900027

[14] J. M. Köhne, S. Köhne and J. Šimůnek, “A Review of

Model Applications for Structured Soils: a) Water Flow

and Tracer Transport,” Journal of Contaminant Hydrol-

ogy, Vol. 104, No. 1-4, 2009, pp. 4-35.

doi:10.1016/j.jconhyd.2008.10.002