analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

11
HYDROLOGICAL PROCESSES Hydrol. Process. 21, 2526–2536 (2007) Published online 28 February 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.6496 Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion Jui-Sheng Chen, 1 * Chia-Shyun Chen 1 and Chih Yu Chen 2 1 Institute of Applied Geology, National Central University, Jhongli City, Taoyuan 320, Taiwan, ROC, China 2 Department of Environmental Engineering and Science, Fooyin University, Kaohsiung 831, Taiwan, ROC, China Abstract: It has been known for many years that dispersivities increase with solute displacement distance in a subsurface. The increase of dispersivities with solute travel distance results from significant variation in hydraulic properties of porous media and was identified in the literature as scale-dependent dispersion. In this study, Laplace-transformed analytical solutions to advection-dispersion equations in cylindrical coordinates are derived for interpreting a divergent flow tracer test with a constant dispersivity and with a linear scale-dependent dispersivity. Breakthrough curves obtained using the scale-dependent dispersivity model are compared to breakthrough curves obtained from the constant dispersivity model to illustrate the salient features of scale-dependent dispersion in a divergent flow tracer test. The analytical results reveal that the breakthrough curves at the specific location for the constant dispersivity model can produce the same shape as those from the scale-dependent dispersivity model. This correspondence in curve shape between these two models occurs when the local dispersivity at an observation well in the scale-dependent dispersivity model is 1Ð3 times greater than the constant dispersivity in the constant dispersivity model. To confirm this finding, a set of previously reported data is interpreted using both the scale-dependent dispersivity model and the constant dispersivity model to distinguish the differences in scale dependence of estimated dispersivity from these two models. The analytical result reveals that previously reported dispersivity/distance ratios from the constant dispersivity model should be revised by multiplying these values by a factor of 1Ð3 for the scale-dependent dispersion model if the dispersion process is more accurately characterized by scale-dependent dispersion. Copyright 2007 John Wiley & Sons, Ltd. KEY WORDS tracer test; analytical solution; scale-dependent dispersion; divergent flow Received 16 January 2006; Accepted 16 June 2006 INTRODUCTION Predicting the fate and transport of contaminants in the subsurface environment is crucial to subsequent con- trol and remediation of contaminants. Dispersivity is the primary input parameter for the advection-dispersion equation (ADE) that has been widely applied for describ- ing mathematically the transport and fate of a solute in the subsurface through time and space. It is generally accepted that the tracer test is an efficient method of determining dispersivity values. In typical field settings, forced-gradient tracer tests are preferred over natural gra- dient experiments. In forced-gradient tracer tests, the flow conditions are well controlled and the duration of the test is reduced. Forced-gradient tracer tests can be conducted in numerous ways by using a few injection and extrac- tion wells. Traditional forced-gradient tracer tests include convergent and divergent flow tracer tests. The advan- tage of the convergent flow tracer tests is the possibility of achieving high tracer mass recovery. However, unless multiple tracers are employed, only one or two sources and withdrawal wells can be used during a given experi- ment (Novakowski, 1992). In a divergent flow tracer test, numerous observation wells can be employed using a * Correspondence to: Jui-Sheng Chen, Institute of Applied Geology, National Central University, Jhongli City, Taoyuan 320, Taiwan, ROC, China. E-mail: [email protected] single tracer. A number of analytical and numerical solu- tions for the advection-dispersion equations in cylindrical coordinates have been previously developed for interpret- ing the divergent flow tracer test in the literature (Chen, 1985, 1986, 1987; Valocchi, 1986; Wang and Crampon, 1995; Sauty, 1977, 1980; Tomasko et al., 2001). These models were traditionally based on the assumption that dispersivity is spatially independent. However, results of observations in the field tests have shown that dis- persivities generally increase with tracer displacement distances during forced-gradient tracer tests. Examples of scale-dependent dispersion include convergent flow tracer tests at the Corbas, France (Sauty, 1977) and WIPP (McKenna et al., 2001) sites, and a divergent flow test at Horkheimer Insel, Germany (Ptak and Teutsch, 1994). The increase in dispersivity with tracer travel distance is due to significant variation in hydraulic properties of porous media and was identified as scale-dependent dis- persion (Pickens and Grisak, 1981a). It is important to quantify the scale dependence of dispersivity on solute transport distance. The values of the dispersivity/distance ratio that characterize the scale-dependent process have previously been reported in a few studies (Pickens and Grisak, 1981a,b; Gelhar et al., 1992; Schulz-Makuch, 2005). However, values of dispersivity/distance ratio are predominately derived using constant dispersivity models (CDMs) and are not entirely comparable with the values Copyright 2007 John Wiley & Sons, Ltd.

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Page 1: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

HYDROLOGICAL PROCESSESHydrol. Process. 21, 2526–2536 (2007)Published online 28 February 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.6496

Analysis of solute transport in a divergent flow tracer testwith scale-dependent dispersion

Jui-Sheng Chen,1* Chia-Shyun Chen1 and Chih Yu Chen2

1 Institute of Applied Geology, National Central University, Jhongli City, Taoyuan 320, Taiwan, ROC, China2 Department of Environmental Engineering and Science, Fooyin University, Kaohsiung 831, Taiwan, ROC, China

Abstract:

It has been known for many years that dispersivities increase with solute displacement distance in a subsurface. The increaseof dispersivities with solute travel distance results from significant variation in hydraulic properties of porous media andwas identified in the literature as scale-dependent dispersion. In this study, Laplace-transformed analytical solutions toadvection-dispersion equations in cylindrical coordinates are derived for interpreting a divergent flow tracer test with a constantdispersivity and with a linear scale-dependent dispersivity. Breakthrough curves obtained using the scale-dependent dispersivitymodel are compared to breakthrough curves obtained from the constant dispersivity model to illustrate the salient features ofscale-dependent dispersion in a divergent flow tracer test. The analytical results reveal that the breakthrough curves at thespecific location for the constant dispersivity model can produce the same shape as those from the scale-dependent dispersivitymodel. This correspondence in curve shape between these two models occurs when the local dispersivity at an observation wellin the scale-dependent dispersivity model is 1Ð3 times greater than the constant dispersivity in the constant dispersivity model.To confirm this finding, a set of previously reported data is interpreted using both the scale-dependent dispersivity model andthe constant dispersivity model to distinguish the differences in scale dependence of estimated dispersivity from these twomodels. The analytical result reveals that previously reported dispersivity/distance ratios from the constant dispersivity modelshould be revised by multiplying these values by a factor of 1Ð3 for the scale-dependent dispersion model if the dispersionprocess is more accurately characterized by scale-dependent dispersion. Copyright 2007 John Wiley & Sons, Ltd.

KEY WORDS tracer test; analytical solution; scale-dependent dispersion; divergent flow

Received 16 January 2006; Accepted 16 June 2006

INTRODUCTION

Predicting the fate and transport of contaminants in thesubsurface environment is crucial to subsequent con-trol and remediation of contaminants. Dispersivity isthe primary input parameter for the advection-dispersionequation (ADE) that has been widely applied for describ-ing mathematically the transport and fate of a solute inthe subsurface through time and space. It is generallyaccepted that the tracer test is an efficient method ofdetermining dispersivity values. In typical field settings,forced-gradient tracer tests are preferred over natural gra-dient experiments. In forced-gradient tracer tests, the flowconditions are well controlled and the duration of the testis reduced. Forced-gradient tracer tests can be conductedin numerous ways by using a few injection and extrac-tion wells. Traditional forced-gradient tracer tests includeconvergent and divergent flow tracer tests. The advan-tage of the convergent flow tracer tests is the possibilityof achieving high tracer mass recovery. However, unlessmultiple tracers are employed, only one or two sourcesand withdrawal wells can be used during a given experi-ment (Novakowski, 1992). In a divergent flow tracer test,numerous observation wells can be employed using a

* Correspondence to: Jui-Sheng Chen, Institute of Applied Geology,National Central University, Jhongli City, Taoyuan 320, Taiwan, ROC,China. E-mail: [email protected]

single tracer. A number of analytical and numerical solu-tions for the advection-dispersion equations in cylindricalcoordinates have been previously developed for interpret-ing the divergent flow tracer test in the literature (Chen,1985, 1986, 1987; Valocchi, 1986; Wang and Crampon,1995; Sauty, 1977, 1980; Tomasko et al., 2001). Thesemodels were traditionally based on the assumption thatdispersivity is spatially independent. However, resultsof observations in the field tests have shown that dis-persivities generally increase with tracer displacementdistances during forced-gradient tracer tests. Examplesof scale-dependent dispersion include convergent flowtracer tests at the Corbas, France (Sauty, 1977) and WIPP(McKenna et al., 2001) sites, and a divergent flow test atHorkheimer Insel, Germany (Ptak and Teutsch, 1994).The increase in dispersivity with tracer travel distanceis due to significant variation in hydraulic properties ofporous media and was identified as scale-dependent dis-persion (Pickens and Grisak, 1981a). It is important toquantify the scale dependence of dispersivity on solutetransport distance. The values of the dispersivity/distanceratio that characterize the scale-dependent process havepreviously been reported in a few studies (Pickens andGrisak, 1981a,b; Gelhar et al., 1992; Schulz-Makuch,2005). However, values of dispersivity/distance ratio arepredominately derived using constant dispersivity models(CDMs) and are not entirely comparable with the values

Copyright 2007 John Wiley & Sons, Ltd.

Page 2: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

ANALYSIS OF TRANSPORT TEST WITH SCALE-DEPENDENT DISPERSION. 2527

derived using scale-dependent models (SDMs). It is morereasonable to derive the actual dispersivity/distance ratiousing an appropriate SDM for a scale-dependent dis-persion problem. Studies have also pointed out that theactual value of the dispersivity/distance ratio derivedfrom SDM is 2 times greater than that estimated fromthe CDM in a one-dimensional flow tracer test (Pangand Hunt, 2002) and 4 times greater than that estimatedfrom the CDM in a convergent flow tracer test (Chenet al., 2003). A few studies developed the solutions to theadvection-dispersion equations in cylindrical coordinatesthat account for scale-dependent dispersion to interpretthe results of the divergent flow tracer tests. Welty andGelhar (1994) gave approximate analytical solutions fora divergent flow tracer test with a constant dispersivityand the case with a linearly scale-dependent dispersiv-ity. The approximate solutions presented by Welty andGelhar (Pang and Hunt, 2002) are based on the stream-tube modelling approach presented by Gelhar and Collins(1971) who applied a two-coordinate integral transform toreduce the formulation to a diffusion equation. Althoughtheir solution is explicit and can be applied with littlecomputational effort, it is only effective for interpret-ing tracer tests under advection-dominated conditionsand encounters some difficulties in interpreting diver-gent flow tracer tests under certain conditions when thedispersion dominates. Additionally, Kocabas and Islam(2000) classified in detail the solutions of advection-dispersion equations in cylindrical coordinates based onthe different scale-dependent forms of velocity and dis-persion coefficient. Kocabas and Islam (2000) presentedan analytical solution in the Laplace domain to advection-dispersion equations in cylindrical coordinates assumingspatial velocity and scale dependence of dispersivity ina divergent flow field. However, their solution is onlyapplicable for a divergent flow tracer test with a contin-uous input source. In a divergent flow tracer test, tracercontrol may be considered to be easier for an instan-taneous slug input source than for a continuous inputsource, as the latter requires that a continuously constantinput concentration is monitored and maintained duringthe entire duration of the experiment (Welty and Gelhar,1994). Additionally, the adequacy of Kocabas and Islam’ssolution has not been assessed in a field case. To ourknowledge, no current analytical solution is applicable,nor currently available, for the full range of dispersiv-ity for describing scale-dependent solute transport in adivergent tracer test with an instantaneous slug source.This study derived the analytical solutions for the CDMand SDM to interpret the divergent flow tracer test withan instantaneous slug input source. The derived analyticalsolutions for CDM and SDM are compared to the approx-imate solutions obtained by Welty and Gelhar (1994)to determine under what conditions the latter solutionscan be applied. Furthermore, comparisons between break-through curves obtained from CDM and SDM assist inelucidating the salient features of scale-dependent disper-sion in a divergent flow tracer test. The resulting solutions

for the CDM and SDM are applied to a set of previ-ously reported field data to determine the differences inthe scale dependence of dispersivity estimated from theCDM and SDM in a divergent flow tracer test.

MATHEMATICAL MODEL

A divergent flow tracer test problem is studied as shownin Figure 1. The tracer test is initiated with the injectionof a constant flow rate, Q, in the fully penetrating well.When the groundwater flow field reaches the pseudo-steady state condition, a slug of conservative tracer withknown mass M, is quickly introduced in the injectionwell. Since the volume of injected tracer solution issmall relative to Q, the flow field is not affected by thetracer injection and the groundwater flow velocity Vr�r�remains as

Vr�r� D Q

2�b�rD A

r�1�

where b is the thickness of the aquifer, � representsthe effective porosity, and A D Q/2�b�. The advection-dispersion equation in cylindrical coordinates describingthe transport of solute in a radial flow field, is

1

r

∂r

(rDL

∂C�r, t�

∂r

)� Vr

∂C�r, t�

∂rD R

∂C�r, t�

∂t�2�

where C�r, t� is the concentration, DL represents the lon-gitudinal dispersion coefficient, and R is the retardationfactor. In Equation (2), molecular diffusion is negligiblerelative to mechanical dispersion, while the longitudi-nal dispersion coefficient, DL, is assumed to be linearlydependent on the seepage velocity as

DL D aL,cjVj �3�

where aL,c is the constant longitudinal dispersivity.Accordingly, Equation (2) is reduced to the advection-dispersion equation in the cylindrical coordinates for theCDM as

aL,cA

r

∂2C�r, t�

∂r2 � A

r

∂C�r, t�

∂rD R

∂C�r, t�

∂t�4�

However, there are studies (e.g. Yates, 1990; Weltyand Gelhar, 1994; Hunt, 1998, 2002; Pang and Hunt,

observation wellinjection well

r = r0 r = 0

2 r1

b

mass input, M injection rate, Q

divergent flow divergent flow

Figure 1. Schematic diagram of a divergent flow tracer test

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 3: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

2528 J.-S. CHEN, C.-S. CHEN AND C. Y. CHEN

2002 and Chen et al., 2003) that suggest the dispersivitycan increase with the solute travel distance as

aL,s�r� D eL Ð r �5�

where aL,s�r� is scale-dependent dispersivity at a spe-cific location r; eL is the dispersivity/distance ratio.When this linear relationship is substituted for aL,c inEquations (3), (5) is changed to

1

r

∂r

(reLA

∂C�r, t�

∂r

)� A

r

∂C�r, t�

∂rD R

∂C�r, t�

∂t�6�

which can be rewritten as

eLA∂2C�r, t�

∂r2 C A

r�eL � 1�

∂C�r, t�

∂rD R

∂C�r, t�

∂t�7�

Equation (6) or (7) is the governing equation for SDM.The initial tracer concentration in the aquifer is assumedto be zero before starting the test:

C�r, t D 0� D 0 �8�

The inlet boundary condition, which is used to describethe instantaneous slug injection of the tracer and transportof the tracer across the well screen into the adjacentaquifer, can be derived from mass balance and presentedas

Mυ�t� D 2�rb�

[A

rC�r, t�� aL,cA

r

∂C�r, t�

∂r

] ∣∣∣rDrI

CDM

�9�

Mυ�t� D 2�rb�

[A

rC�r, t� �eL Ð A

∂C�r, t�

∂r

]∣∣∣rDrI

SDM

�10�

where υ�� is Dirac delta function.Another boundary condition required for obtaining a

unique solution to both the governing Equations (4) and(7) is imposed at infinity by stating

C�r ! 1, t� D 0 �11�

Table I presents dimensionless variables. Transposingthese definitions into Equations (4) and (7) obtains thedimensionless governing equation in the following form:

2R

�1 � r2ID�

∂C

∂tDD 1

Pe

1

rD

∂2C

∂r2D

� 1

rD

∂C�rD, tD�

∂rDfor CDM

�12�

2R

�1 � r2ID�

∂C�rD, tD�

∂tDD eL

∂2C�rD, tD�

∂r2D

C �eL � 1�

rD

∂C�rD, tD�

∂rDfor SDM �13�

Table I. Dimensionless parameters

Dimensionless quantity Expression

Timea tD D tta

Distance rD D rro

Injection well radius rID D rIro

Peclet number Pe D roaL,c

a Here, ta D �b��r2o � r2

I �/Q.

Consequently, the initial condition and boundary con-ditions shown in Equations (8–11) become

C�rD, tD D 0� D 0 �14�

CIυ�tD� D[C�rD, tD� � 1

Pe

∂C�rD, tD�

∂rD

]∣∣∣∣rDDrID

CDM

�15�

CIυ�tD� D[C�rD, tD� � eLrD

∂C�rD, tD�

∂rD

]∣∣∣∣rDDrID

SDM

�16�

C�rD ! 1, tD� D 0 �17�

where CI D M�b��r2

0 � r2I �

ANALYTICAL SOLUTIONS IN THE LAPLACEDOMAIN

Solution for the CDM

In this study the governing Equations (12) and (14) aresolved using the Laplace transform approach to eliminatetemporal derivatives. By performing the Laplace trans-form with Equations (12), (13) and the associated bound-ary conditions, Equations (15) and (17) with respect totD yield

1

Pe

1

rD

d2C�rD, s�

dr2D

� 1

rD

dC�rD, s�

drD

� 2Rs

�1 � r2ID�

C�rD, s� D 0 �18�

Boundary conditions Equations (15) and (17) become

CI D[

C�rD, s� � 1

Pe

dC�rD, s�

drD

]∣∣∣∣∣rDrID

�19�

C�rD ! 1, s� D 0 �20�

where s denotes the Laplace transform parameter andC�rD, s� represents the Laplace transform of C�rD, tD�,as defined by

C�rD, s� D∫ 1

0C�rD, tD�e�stD dtD �21�

Chen (1987) employed the Airy function, Ai��(Abramowitz and Stegun, 1970), to derive the solution for

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 4: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

ANALYSIS OF TRANSPORT TEST WITH SCALE-DEPENDENT DISPERSION. 2529

a similar problem with a continuous input source. Usingthis approach, the solution to Equation (18) subject to theboundary conditions Equations (19) and (20) is derivedas shown in Appendix A. As a result, the CDM solutionin the Laplace domain is

C DCI exp

(y � yI

2

)Ai��

1/3y�

1

2Ai��1/3yI� � �1/3Ai0��1/3yI�

�22�

where Ai0�� is the first derivative of Ai�� and

yI D rIDPe C 1

4�y D rDPe C 1

4�� D 2Rs

Pe2�1 � r2ID�

Solution for the SDM

The application of the Laplace transform with respectto tD to Equations (13), (14), (16), and (17) results in

eLd2C�rD, s�

dr2D

C �eL � 1�

rD

∂C�rD, s�

∂rD

� 2R

�1 � r2ID�

sC�rD, s� D 0 �23�

CI D[

C�rD, t� � eLrDdC�rD, s�

drD

]∣∣∣∣∣rDrID

�24�

C�rD ! 1, s� D 0 �25�

Equation (23) is in the general form of the modifiedBessel function (Aparci, 1966), and its solution subjectto the boundary conditions Equations (24) and (25) isderived as shown in Appendix B. As a result, the SDMsolution in the Laplace domain is

C�rD, s� D CIr�DK��qrD�

r�ID[K��qrID� � eLrIDqKv�1�qrID�]

�26�

Note that v, the order of the modified Bessel functionK��Ð�, becomes large when eL is small. For a large v (say,v > 40 ¾ 50), it is recommended that K��Ð� and K��1�Ð�in Equation (26) be evaluated using the appropriate large-order asymptotic formula (Abramowitz and Stegun, 1970;p. 378) in order to avoid possible numerical difficulties.For v less than 40, K��Ð� and K��1�Ð� can be evaluatedwithout difficulty using the appropriate subroutines inIMSL. The Laplace inversion of Equations (22) and (26)gives the temporal tracer concentrations of CDM andSDM, respectively. We use the De Hoog et al. (1982)algorithm to carry out the inversion.

RESULTS AND DISCUSSION

Comparisons of the developed analytical solution andapproximate solution

The applicable range of the approximate solutionsgiven by Welty and Gelhar (1994) is now determined bycomparison to the developed solutions. The dimensional

Time [min]

Con

cent

ratio

n [k

g/m

3 ]

this studyWelty andGelhar (1994)

Pe=100

50

20

10

0

0.04

0.08

0.12

0.16

0

0.04

0.08

0.12

0.160 50 100 150 200

0 50 100 150 200

(a)

this studyWelty andGelhar (1994)

eL=0.01

0.02

0.05

0.1

0 40 80 120 200160

0 40 80 120 2001600

0.04

0.08

0.12

0.16

0.2

0

0.04

0.08

0.12

0.16

0.2

Con

cent

ratio

n [k

g/m

3 ]

Time [hr]

(b)

Figure 2. Comparison of breakthrough curves at the observation wellbetween the developed analytical solutions and those obtained by Welty

and Gelhar (1994): (a) CDM; (b) SDM

CDM and SDM solutions given by Welty and Gelhar(1994), Equations (17) and (19)) are

C D M

2�b�r2L

(4

3�

1

Pe

)1/2 �t3/2��1/2

ð exp

∣∣∣∣∣∣∣��1 � t�2

16

3

1

Pet3/2

∣∣∣∣∣∣∣ CDM �27�

C D M

2�b�r2L

(2

3�eL

)1/2

(3

2

)�1/2

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 5: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

2530 J.-S. CHEN, C.-S. CHEN AND C. Y. CHEN

ð t�1 exp

∣∣∣∣∣��1 � t�2

4eLt2

∣∣∣∣∣ SDM �28�

where t D ttm , tm D r2

L2A and all other variables are previ-

ously defined.A hypothetical tracer test is considered, and Table II

summarizes the simulation conditions and the trans-port parameters for comparisons. Figure 2(a–b) show thebreakthrough curves at ro D 5 m obtained from the devel-oped analytical solutions and the approximate solutionsobtained by Welty and Gelhar (1994) under various Pecletnumbers (Pe� for the CDM or under various dispersiv-ity/distance ratios (eL� for the SDM, respectively. Thedifferences are noticeable between the derived analyticalsolutions and the approximate solutions when Pe � 100for the CDM or when eL ½ 0Ð01 for the SDM. Butthe discrepancy is negligible when Pe ½ 100 for the

CDM or when eL � 0Ð01 for the SDM. In the mean-time, the accuracy of the developed solutions is checkedagainst the numerical solutions to the Equations (12) and(13) obtained with the Laplace transform finite differ-ence technique (LTFD) (Moridis and Reddel, 1991). Thenumerical solutions coincide with the developed analyti-cal solutions for both the SDM and CDM under a widerange of transport conditions (not shown here). Thesecomparisons reveal that the developed analytical solu-tion is accurate for a full range of transport parameters,whereas the Welty and Gelhar solution is approximatewhen Pe ½ 100 for the CDM or when eL � 0Ð01 for theSDM. Therefore, the proposed analytical solutions aresuperior to the approximate solution derived by Weltyand Gelhar (1994).

Analysis of scale-dependent solute transport behaviourThe SDM and CDM solutions are compared with each

other to illustrate how the scale-dependent dispersion

0 40 80 120 200160

0 40 80 120 200160

Time [hr]

0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

Con

cent

ratio

n [k

g/m

3 ]

aL,c=1.25m

aL,c=0.96m CDMSDM

(a)

0 50 100 200150

0 50 100 200150

0

0.02

0.04

0.06

0.08

0

0.02

0.04

0.06

Time [min]

Con

cent

ratio

n [k

g/m

3 ]

aL,c=0.5m

aL,c=0.38m

CDMSDM

(b)

0 50 100 200150

0 50 100 200150

Time [min]

0

0.04

0.08

0.12

0

0.04

0.08

0.12

Con

cent

ratio

n [k

g/m

3 ]

aL,c=0.125m

aL,c=0.96m

CDMSDM

(c)

Figure 3. Comparison of breakthrough curves at the observation well between the SDM and CDM. -The solid circle results from the SDM with(a) eL D 0Ð25, (b) eL D 0Ð1 and (c) eL D 0Ð025

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 6: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

ANALYSIS OF TRANSPORT TEST WITH SCALE-DEPENDENT DISPERSION. 2531

Table II. Descriptive simulation conditions and transportparameters of the hypothetical tracer test

Parameter Test 1

Injection volume rate(Q�, m3 min�1

2

Aquifer thickness (b�, m 10Effective porosity (��,

dimensionless0Ð2

Radius of injection well (rI�, m 0Ð05Observation distance (ro�, m 5Injected mass (M�, Kg 10Dispersivity/distance ratio

(SDM) (eL�, dimensionless0Ð25, 0Ð1, 0Ð025

Constant dispersivity (CDM)(aL,c�, m

1Ð25, 0Ð5, 0Ð125

influences solute transport during a divergent flow tracertest. The input parameters are the same as those usedin the previous comparisons. In Figure 3(a–c) the break-through curves at rO D 5 m of the SDM using eL D 0Ð25,0Ð1, and 0Ð025, are compared with the breakthroughcurves at rO D 5 m of the CDM usingaL,c D 1Ð25, 0Ð5,and 0Ð125 m. Note that aL,c D aL,s�rO� D eLÐrO is usedin the comparison. The peak concentrations, Cpeak , ofthe SDM breakthrough curves arrive later than those ofthe CDM for the three cases. Moreover, the spreading ofthe CDM breakthrough curves exceeds that of the SDMbreakthrough curves. In general, a reduction of aL,c cancause the CDM breakthrough curves to merge with the

SDM breakthrough curves, as shown in Figure 3(a–c),where aL,c is reduced from 0Ð125 to 0Ð96 m, from 0Ð5 to0Ð38 m, and from 0Ð0125 to 0Ð096 m. For aL,c D 0Ð38 and0Ð096 m, the CDM breakthrough curves coincide with theSDM breakthrough curves of eL D 0Ð1 and 0Ð025, respec-tively, for the complete test durations. However, there isnoticeable discrepancy existing at large times between theCDM breakthrough curve of aL,c D 0Ð96 m and the SDMbreakthrough curve of eL D 0Ð25. The values of aL,c usedin the CDM are approximately 1/1Ð3 of aL,s�rO� used inthe SDM.

For data analysis, dimensionless type curves areuseful in the determination of transport parameters.They are prepared by plotting normalized concentration,C�rO, t�/Cpeak , against dimensionless time, tD, on log-arithmic paper. The dimensional breakthrough curvesexhibited in Figure 3 are transformed to the dimen-sionless type curves as shown in Figure 4, where eL DaL,s�rO�

rOD 0Ð25, 0Ð1 and 0Ð025 in the SDM and Pe D

rOaL,c

D 4, 20 and 80 in the CDM. It is seen that the peakconcentration arrival times of the CDM and the SDMtype curves are about the same in the three cases, but therising limbs of the SDM type curves with eL D 0Ð25 and0Ð1 are noticeably different from the rising limbs of theirCDM counterparts with Pe D 5Ð2 and 13. The productsof eL and Pe of the three cases approximately equal 1Ð3.As a matter of fact, we have done the analysis at the othertwo observation wells of rO D 10 m and rO D 15 m. Theresults (not shown here) are similar to what are shown in

0 50 100 150 200

0 50 100 150 200

Dimensionless time

0

0.5

1

0

0.5

1

Dim

ensi

onle

ss c

once

ntra

tion

SDMCDM

Pe=5.2, eL=0.25Pe=13, eL=0.1

Pe=52, eL=0.025

Figure 4. Comparison of type curves at the extraction well between the CDM and SDM for eL D 0Ð25, 0Ð1 and 0Ð025

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 7: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

2532 J.-S. CHEN, C.-S. CHEN AND C. Y. CHEN

C/C

0

field data

CD

type curve

Pe=72

1×100

tD

1×10−1

1×10−2

1×100

1×10−3

1×10−1

1×10−1

1×100

1×100

1×10−1

1×10−1

1×100

1×100

1×101

1×101

t [hr]1×102

1×10−3

1×10−1

1×10−2

1×10−1

1×1021×101

1×101

1×10−2

(a)

Figure 5. Breakthrough concentration data at ro D 7Ð6 m from a divergent flow tracer test conducted at the Palo Alto Baylands field site, superimposedon the types of curves from the developed analytical solutions (a) CDM; (b) SDM

Figures 3(a–c) of rO D 5 m, and it was found that thiscorrelative relationship (Pe Ð eL ³ 1Ð3� to be also valid atro D 10 and 15 m.

Nevertheless, the value of Pe Ð eL depends on geomet-rical configuration of the flow field. For example, Chenet al. (2003) reported a value of 4 for a convergent flowtracer test, while Pang and Hunt (2002) reported a valueof 2 for a uniform flow tracer test. Such a difference canbe explained by using the concept of ‘equivalent disper-sivity’, aL,c, as being representative of an integrated mea-sure of the variable scale-dependent dispersivities fromthe injection well up to a specific location (an observa-tion well). That is, aL,c of the CDM breakthrough curvematching the SDM breakthrough curve can be viewed asaL,c. In this regard, aL,c can be approximated by averag-ing aL,s�r� from the injection well up to an observationwell; that is,

aL,c D

∫ r0

rI

aL,s�r� Ð 2�rdr

��r2O � r2

I ��29�

Substituting Equation (5) into Equation (29) yields

aL,c D 2

3

eL�r2O C rOrI C r2

I �

�rO C rI��30�

For r0 × rI, Equation (30) can be approximated as

aL,c ³ 2

3eL Ð rO �31�

which givesPe Ð eL ³ 1Ð5 �32�

It is observed that Equation (32) is close to the fittedresult of Pe Ð eL ³ 1Ð3. In a similar manner, the productsof Pe and eL were determined for a convergent flowfield (Pe Ð eL ³ 3� and a uniform flow field (Pe Ð eL ³ 2).Application of the constant dispersivity obtained usingthe CDM to the relationships of Pe and eL allowsthe determination of the dispersivity/distance ratio forthe SDM.

Model application

Both the CDM and SDM solutions are applied, usingthe field data presented by Duffy et al. (1981), to esti-mate the constant and the scale-dependent dispersivities,respectively. The divergent flow tracer test was carriedout in a sand and gravel aquifer with thickness approx-imately 1 ¾ 2 m at the Palo Alto Baylands field site(Hoehn and Roberts, 1982). A constant flow rate of2Ð5 m3/h was applied in this tracer test to establish the

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 8: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

ANALYSIS OF TRANSPORT TEST WITH SCALE-DEPENDENT DISPERSION. 2533

C/C

0

field data

CD

type curve

eL=0.018

1×100

tD

1×10−1

1×10−2

1×100

1×10−3

1×10−1

1×10−2

1×100

1×100

1×10−2

1×10−1

1×100

1×100

1×101

1×101

t [hr]1×102

1×10−3

1×10−1

1×10−2

1×10−1

1×1021×101

1×101

1×10−2

(b)

Figure 5. (Continued )

divergent flow field. After the flow field reached a steadystate condition, a slug of tracer mass Br82 and 3H wasinstantaneously introduced into the injection well, andtheir concentration data were collected at two observationwells, one at ro D 7Ð6 m and the other at ro D 16Ð8 m.These two sets of data are conducive to the study of scale-dependent dispersivity. Because the transport behaviourof both tracers agreed closely with each other, the fol-lowing analysis evaluates the tritium data only.

The data analysis employs the curve matching tech-nique, where the dimensionless type curves are preparedusing the CDM and SDM, respectively. The CDM typecurves are used to determine Pe while the SDM typecurves for eL. Curve matching of dimensionless typecurves against the measured concentrations consists of thefollowing steps: (1) Dimensionless type curves obtainedfrom the SDM and CDM are plotted logarithmically forvarious eL and Pe values, respectively. (2) The measuredconcentrations at one observation well are plotted againstreal time on logarithmic paper with the same scale as thetype curves. (3) The field data paper is superimposed onthe type curves with the axes sets kept parallel. (4) Thevalues of eL and Pe are respectively determined using theSDM and CDM type curves that match most field data.

Figures 5 and 6 show the fitting results at ro D7Ð6 m and 16Ð8 m. At ro D 7Ð6 m, neither the SDM northe CDM can fit the field data well at most times.Welty and Gelhar (1994) ascribed this phenomenon tonear-field, non-Fickian effects that cause unaccounted-for complications in breakthrough data, suggesting thatonly the rising limb and peak of breakthrough dataare used for fitting. Fitting the rising limb and peakof breakthrough data to both types of curves yieldsPe D 72, and an approximate constant dispersivity valueof 0Ð106 m for the CDM and eL D 0Ð018, yieldingan approximate scale-dependent dispersivity value of0Ð137 m, for the SDM. For ro D 16Ð8 m, the data fit toboth types of curves exhibits a better match than that forro D 7Ð6, including the spreading tails of breakthroughconcentration. The Pe and eL values from curve-fittingusing the CDM and SDM were estimated to be 72 and0Ð18, determining an approximate constant dispersivityvalue of 0Ð233 m for the CDM and scale-dependentdispersivity value of 0Ð302 m, respectively. The productsof Pe and eL from curve-fitting at ro D 7Ð6 and 16Ð8 mboth equal 1Ð3, consistent with the proposed relationshipin the previous section. Additionally, the estimated scale-dependent dispersivity values obtained from the SDM are

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 9: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

2534 J.-S. CHEN, C.-S. CHEN AND C. Y. CHEN

1×10−1

1×10−2

1×10−1

1×10−1

1×10−1

1×10−4

1×10−3

1×10−2

1×10−3

1×1011×100 1×102

1×101 1×103

1×100

1×100

1×1021×101

1×1001×10−2

1×101 1×103

1×10−21×10−4

Pe=72field data

type curve

tDt [hr]

C/C

0

CD

(a)

1×10−1

1×10−2

1×10−1

1×10−1

1×10−1

1×10−4

1×10−3

1×10−2

1×10−3

1×1011×100 1×102

1×101 1×103

1×100

1×100

1×1021×101

1×1001×10−2

1×101 1×103

1×10−21×10−4

eL=0.018field data

type curve

tDt [hr]

C/C

0

CD

(b)

Figure 6. Breakthrough concentration data at ro D 16Ð8 m from a divergent flow tracer test conducted at the Palo Alto Baylands field site, superimposedon the types of curves from the developed analytical solution (a) CDM; (b) SDM

1Ð3 times greater than the constant dispersvity obtainedusing the CDM at the two observation wells.

Figure 7 plots the dispersivity values for the CDM andSDM versus interwell (transport) distance to examinelinear increasing dispersivity assumption. It is evidentfrom this analysis that both the constant dispersivity forthe CDM and scale-dependent dispersivty for the SDMincrease with solute travel distance, indicating that thesolute transport process at this test site conforms to thelinear scale-dependent assumption. However, it shouldbe noted that the dispersivity/distance ratio derivedusing the CDM is lower than that derived with the

SDM by a factor of 1Ð3. Previous studies pointedout that the actual dispersivity/distance ratio valuesderived using appropriate SDMs are more reasonablethan those obtained by CDMs for a scale-dependentproblem. Therefore, the analytical result means that theCDM underestimates dispersivity/distance ratios by afactor of 1Ð3. Notably, when applying the previouslyreported dispersivity/distance ratio derived from theCDM in a divergent radial flow test into a scale-dependent dispersion problem, dispersivity/distance ratiofrom the CDM should be revised by multiplying reporteddispersivity/distance ratio by a factor of 1Ð3 for the SDM.

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp

Page 10: Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion

ANALYSIS OF TRANSPORT TEST WITH SCALE-DEPENDENT DISPERSION. 2535

rO [m]

a L [m

]

dispersivity from SDMdispersivity from CDMfitted line equation

aL=0.018 rO

aL=0.0138 rO

00

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4

5 10 15

0 5 10 15

Figure 7. Dispersivity (aL� and observation well distance (ro� relationshipfor CDM and SDM

CONCLUSION

This study presents the Laplace transform analytical solu-tions for describing the solute transport in a divergentflow tracer test. The case with a linearly increasing scale-dependent dispersivity and that with a constant disper-sivity are considered, respectively. The derived analyt-ical solution is compared to the approximate solutionobtained by Welty and Gelhar (Yates, 1990) to deter-mine the applicable range of the latter solution. Com-parison of the breakthrough curves obtained from thescale-dependent dispersivity model and the constant dis-persivity model reveals that given appropriately cho-sen parameters, the solutions obtained with the scale-dependent dispersivity model and the constant dispersiv-ity model produce similar breakthrough curves. Notably,the product of the Peclet number used in the constantdispersivity model and dispersivity/distance ratio usedin the scale-dependent dispersivity model for the sametype of curve is approximately 1Ð3. The developed solu-tions for the constant dispersivity model and the scale-dependent dispersivity model are both applied to a fieldcase to determine the scale-dependent dispersivity for thescale-dependent dispersivity model and constant disper-sivity for the constant dispersivity model in a divergentflow tracer test. Analytical results indicate that the solutetransport process at this test site conforms to the lin-ear scale-dependent assumption. The previously reporteddispersivity/distance ratios derived from the constant dis-persivity model reported previously should be revised bymultiplying these dispersivity/distance ratios by a factorof 1Ð3 for SDM in a divergent flow tracer test.

ACKNOWLEDGEMENT

The authors would like to thank the National ScienceCouncil of the Republic of China, Taiwan, for financially

supporting this research under Contract No. NSC 94-2313-B-008-001.

REFERENCES

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APPENDIX A

The solution to the governing Equation (18), subject toboundary conditions Equations (19) and (20), is derived.

By successive application of the following relation-ships

y D rDPe C 1

4��A1�

x D �1/3y �A2�

G D Cey/2 �A3�

where

� D 2Rs

Pe2�1 � r2ID�

to Equation (21), it can be transformed to

d2G

dx2 D xG �A4�

which is the standard form of Airy equations (Abramo-witz and Stegun, 1970). The general solution to Equation(A4) is (Abramowitz and Stegun, 1970)

G D C1Ai�x� C C2Bi�x� �A5�

or

C D C1 exp(y

2

)Ai��1/3y� C C2 exp

(y

2

)Bi��1/3y�

�A6�where Ai�� and Bi�� are linearly independent Airyfunctions; C1 and C2 are two coefficients dependent onthe prescribed boundary conditions Equations (19) and(20).

Since Bi�x� approaches infinity when x is large,C2 must be zero in order to satisfy Equation (20) and,thus, the general solution is reduced to

C�rD, s� D C1 exp(y

2

)Ai��1/3y� �A7�

In a straightforward manner, C1 can be determinedwith the help of Equation (19) as

C1 DCI exp

(�yI

2

)1

2Ai��1/3yI� � �1/3Ai0��1/3yI�

�A8�

where

yI D rIDPe C 1

4�.

Thus the complete solution to Equation (18) subject toEquations (19) and (20) is

C DCI exp

(y � yI

2

)Ai��1/3�

1

2Ai��1/3yI� � �1/3Ai0��1/3yI�

�A9�

which is presented as Equation (24) in the text.

APPENDIX B

The solution to the governing Equation (23), subject toboundary conditions Equations (24) and (25) is derived.

By successively performing the following variablechanges

x D qrD �B1�

W D xnuC �B2�

where

q D√

2Rs

eL�1 � r2ID�

� D 1

2eL

to Equation (25), and it is rearranged to give

x2 d2W

dx2 C xdW

dx� �x2 � �2�W D 0 �B3�

which is the general form of modified Bessel equationswith order � (Abramowitz and Stegun, 1970). Thismodified Bessel equation has the general solution

W D C1I��x� C C2K��x� �B4�

or the general solution to Equation (25) then becomes

C D C1rnuD I��qrD� C C2rnu

D K��qrD� �B5�

where I��� and K��� are modified Bessel functions ofthe first and second kind of order �; C1 and C2 aretwo unknown constants that should be determined by theprescribed boundary conditions Equations (26) and (27).

Using boundary condition Equation (27), C2 is forcedto be zero since K��x� approaches infinity when x is large;and the general solution is then reduced to

C D C1rnuD I��qrD� C C2rnu

D K��qrD� �B6�

By direct substitution of Equation (B6) into the bound-ary condition Equation (26), C1 can be determined as

C�rD, s� D CI

rnuID[K��qrID� � eLrIDqKv�1�qrID�]

�B7�

Thus, the particular solution to Equation (25) subjectto Equations (26) and (27), is

C�rD, s� D CIrnuD K��qrD�

rnuID[K��qrID� � eLrIDqKv�1�qrID�]

�B8�

which is presented as Equation (28) in the text.

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 21, 2526–2536 (2007)DOI: 10.1002/hyp