Journal of Hydrology (2006) 328, 614–619

ava i lab le at www.sc iencedi rec t . com

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An in situ method to measure thelongitudinal and transverse dispersioncoefficients of solute transport in soil

Xiaoxian Zhang a,b,*, Xuebin Qi a,c, Xinguo Zhou a, Hongbin Pang a

a Farmland Irrigation Research Institute, The Chinese Academy of Agricultural Sciences, Xinxiang 453003,Henan Province, Chinab SIMBIOS Centre, University of Abertay Dundee, Bell Street – Kydd Bldg, Dundee DD1 1HG, United Kingdomc Northwest A&F University, Yangling 712100, Shanxi Province, China

Received 2 February 2005; received in revised form 8 September 2005; accepted 7 January 2006

Summary The knowledge of hydraulic conductivity and solute transport parameters of top-soil is important in a variety of fields and their measurement has been an interest in both theoryand practice. In this paper we present an in situ method to measure the longitudinal and trans-verse dispersion coefficients of solute movement by modifying the double-ring infiltrometerinto a triple-ring infiltrometer. Water flow in the apparatus is controlled in one dimensionand solute movement in three dimensions. The solute transport parameters can be measuredsimultaneously with the hydraulic conductivity. Analytical solutions are derived to describethe solute movement, and field experiment was carried out to calculate the solute parametersin homogeneous soil using a simple method developed based on the analytical solutions. Simu-lating results using these estimated parameters predict the observed breakthrough curves rea-sonably well.ª 2006 Elsevier B.V. All rights reserved.

KEYWORDSLongitudinal dispersion;Transverse dispersion;In situ method;Solute transport;Analytical solution;Triple-ring infiltrometer

0d

3

Introduction

Solute transport in top-soil plays an important role in a vari-ety of fields including leaching of agrochemicals to ground-water, nutrient uptake by plants and remediation of

022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reservedoi:10.1016/j.jhydrol.2006.01.004

* Corresponding author. Tel.: +44 1382 308611; fax: +44 138208117/308537.E-mail address: x.zhang@abertay.ac.uk (X. Zhang).

contaminated soils. Its mathematical modelling is usuallybased on the convection–dispersion equation (CDE). InCDE, solute movement is assumed to comprise a convectiveflux and a dispersive flux (Bear, 1979). The convective fluxdescribes the average movement of solute with soil waterand the dispersive flux describes the impact of inherentspatial variation of water velocity induced by soil heteroge-neity at scales that are not resolved in CDE. CDE has beenquestioned to describe chemical transport in natural soils

.

u

Plane view

Cross section

1 12 23 3

4 4

5

Plane view

Cross section

1 12 23 3

4 4

51. Outer ring, 2. Middle ring, 3. Internal ring, 4. Marriott apparatus, 5 Suction cup

Figure 1 Schematic view of the in situ apparatus formeasuring the longitudinal and transverse dispersioncoefficients.

An in situ method to measure the longitudinal and transverse dispersion coefficients 615

characterised by heterogeneity that exists across variousscales. As a result, alternative approaches, ranging fromstochastic model (Gelhar, 1992; Zhang, 2002), continuoustime random walks (Berkowitz et al., 2000; Cortis andBerkowitz, 2004; Dentz et al., 2004), lattice gas-based mod-el (Zhang and Ren, 2003), to fractional convection–disper-sion equation (Benson et al., 2000; Zhang et al., 2005)have been proposed and compared (Bromly and Hinz,2004; Ptak et al., 2004) for modelling solute transport insoils at different scenarios. In practice, however, CDE withconstant or scale-dependent dispersion coefficients still re-mains, and is likely to continue to remain, the main ap-proach for modelling chemical transport in soils andaquifers because of its simplicity and that, in most cases,it describes the chemical transport process satisfactorilywell.

Solving CDE either numerically (Zhang et al., 2002) oranalytically (Leij et al., 1991) needs to know the valuesof solute transport parameters, including the longitudinaldispersion coefficient and the transverse dispersion coeffi-cient. The longitudinal dispersion coefficient measures thespreading of solute along the water flow direction, andthe transverse dispersion coefficient measures the spread-ing of solute in direction perpendicular to the water flowdirection. Their values for a specific soil are generally ob-tained through displacement experiment. There has beenextensive research on the dispersion coefficients, espe-cially the longitudinal dispersion and its dependence onwater velocity and soil heterogeneity, over the past fewdecades in both repacked and undisturbed soil columns(Nielsen and Biggar, 1961; Jardine et al., 1993) and infield soils (Russo, 1989a,b; Butters and Jury, 1989; Butterset al., 1989; Ellsworth and Jury, 1991; Ellsworth et al.,1991).

Compared to the longitudinal dispersion, the transversedispersion is difficult to measure as calculating it needs toknow the solute concentration in the direction perpendic-ular to water flow direction. Like the longitudinal disper-sion, the mechanism of the transverse dispersion is alsodominated by pore structure, grain size and unresolvedheterogeneity by CDE (Grane and Gardner, 1961). A num-ber of indoor studies had been carried out to measurethe dependence of the transverse dispersion coefficienton flow rate (Harleman and Rumer, 1963; Yule and Gard-ner, 1978). Practical applications, however, often needthe parameter to be measured in situ because the unre-solved heterogeneity dominating the dispersion in field soildiffers from the heterogeneity that dominates solutemovement in soil columns.

The purpose of this paper is to present an in situ meth-od to measure the longitudinal and transverse dispersioncoefficients of solute transport in top-soil. The method isto modify the double-ring infiltrometer into a triple-ringinfiltrometer. Water flow is controlled in one dimensionand solute movement in three dimensions. The solutetransport parameters can be measured simultaneously withthe hydraulic conductivity. Analytical solutions for the sol-ute movement in homogenous soil are derived and a simpleanalytical method is presented to calculate the longitudi-nal and transverse dispersion parameters. In situ experi-ments were carried out to measure the two dispersioncoefficients.

Materials and methods

The apparatus

The apparatus designed to measure the longitudinal andtransverse dispersion coefficients is a triple-ring infiltrome-ter as shown in Fig. 1. It comprises an internal ring, a middlering and an outer ring. The diameter of the outer ring is120 cm, and of the middle and internal rings is 100 and10 cm, respectively. Water flow is in saturated conditionand the flow rate is kept a constant during the experimentby controlling the water depth over the soil surface in allthe three rings unchanged using the Marriott apparatusshown in Fig. 1. The tracer is applied to the internal ring,and the lateral movement of the tracer in soil outer the re-gion controlled by the internal ring reveals the significanceof the transverse dispersion. The tracer can be appliedeither as an instant pulse or as a continuous source. For easeof control, we applied the tracer in an instant pulse in thispaper. Continuously supplying tracer may be needed whenstudying the movement of reactive chemicals in soils.

Experiment

Field experiment was carried out in Shangqiu, eastern HenanProvince of China. The soil is loamy sand and the soil profileis relatively uniform until depth of 150 cm, revealed by a soilprofile in a drainage ditch few meters away from the exper-imental site; the soil taken from the augers when installingsuction cups in the site also confirmed this. Prior to theinstallation of the apparatus on the selected site, the top15 cm soil was trimmed off with a spade to make the soil sur-face level. Suction cup samplers were then placed into soil atdifferent positions to measure the breakthrough curves. Thesamplers were assembled by sealing porous ceramic suctioncup samplers (diameter 1.5 cm, length 4 cm) to the end ofplastic pipes of the same diameter. Following carefullydigging holes with augers, the suction cup samplers were

616 X. Zhang et al.

inserted vertically into the soil. Slurry of sieved soil was usedto ensure good contact between the cups and soil. Once thecup samplers were installed, the triple-ring infiltrometerwas pushed into the soil on the site. The site was first irri-gated with water pumped from groundwater from a borehole20 m away from the site in attempts to bring the soil satu-rated and make the initial solute concentration profile uni-form. The water depth over soil surface in all the threerings was kept unchanged during the experiment using twoMarriott apparatuses, one controls the internal ring andthe other controls the water in the outer and middle ringsas shown in Fig. 1. The outer and the middle rings were madehydraulically connected through a hole drilled in the middlering. Measuring the amount of water flowing out the Marriottapparatus that controls the water supply to the internal ringallows the average infiltration rate to be calculated. Oncethe flow rate reached steady state, we measured the waterdepth over the soil surface inside the internal ring and usedthis measurement to calculate the volume of water in theinternal ring (denoted by V) for reason to be seen below.We then switched off the water supply to the internal ringand quickly dried the internal ring using pumps and bowls.After that, 300 g CaCl2 dissolved in water of volume V wasimmediately poured into the internal ring, and the watersupply to the internal ring was then switched on. We appliedtracer in this way in an attempt to minimize the disturbanceto the water velocity field. Soil water samples were takenfrom the cup samplers in a time interval of approximately2 h. The concentration of Cl�1 in the soil water samples weretaken to laboratory for analysis. During the experiment, carewas made to ensure that water depth over soil surface in theinternal, outer and the middle rings was the same. In theexperiment, five suction cups were installed, two were atdepth of 30 and 10 cm from the origin of the internal ring,two were at depth of 50 and 20 cm from the origin of theinternal ring, and one was right through the centre of theinternal ring at depth of 24 cm.

Calculating the transport parameters

In the absences of non-equilibrium physical and chemicalreactions and that the soil profile is uniform where pore-water velocity and solute dispersion coefficients can beapproximated as constants, the solute movement in the soilbeneath the apparatus shown in Fig. 1 can be described bythe following equation:

oc

ot¼ DL

o2c

oz2þ DT

o2c

or2þ 1

r

oc

or

!� u

oc

oz; ð1Þ

where t is time, c is concentration, z and r are coordinateswith z pointing downward and r originating from the centreof the internal ring, DL and DT are the longitudinal and trans-verse dispersion coefficients, respectively, u is pore-watervelocity. Eq. (1) can account equilibrium physical and chem-ical reactions, and in this case the two dispersion coeffi-cients and the average pore water velocity need to bescaled by a retardation factor.

The diameter of the middle ring was made much largerthan the diameter of the internal ring in order to safely as-sume that during the experiment, the tracer in the soil didnot move out the region controlled by the middle ring.

Therefore, the associated initial and boundary conditionswith Eq. (1) for the solute movement in the apparatusshown in Fig. 1 are

cðz; r; 0Þ ¼ 0;

cðz;1; tÞ ¼ 0;

cð1; r; tÞ ¼ 0;

uc� DLoc

oz

� �����z¼0¼ m

R20pn

Hðr � R0ÞdðtÞ;

ð2Þ

where n is soil porosity, m is the mass of tracer being ap-plied into the internal ring, R0 is the radius of the internalring, H(r � R0) is the Heaviside function in that H(r � R0) =1 if r < R0 and H(r � R0) = 1 otherwise, and d(t) is the Diracdelta function.

As shown in Appendix, the analytical solution of Eq. (1)associated with the initial and boundary conditions givenby Eq. (2) is

cðr; z; tÞ ¼ C0Fðz; tÞZ 1

0

J1ðpR0ÞJ0ðprÞ expð�DTp2tÞdp; ð3Þ

where Jm(x) is the Bessel function of the first kind, m is theorder of the Bessel function, C0 = m/R0pn, and

Fðz; tÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiDLptp exp �ðz� utÞ2

4DLt

!� u

2DL

� expuz

DL

� �erfc

zþ ut

2ffiffiffiffiffiffiffiDLtp

� �. ð4Þ

Solution given by Eq. (3) can be further simplified if the ra-dius of the internal ring is much smaller than the diameterof middle ring in that the cross-section of the internal ringcan be approximated as a point. In this limitation, when xis far away from the soil surface. Eq. (3) reduces to

cðr; z; tÞ ¼ m

4nDT

ffiffiffiffiffiDL

pðptÞ3=2

exp �ðz� utÞ2

4DLt� r2

4DTt

!. ð5Þ

There is a very simple way to calculate the transport param-eters using Eq. (5). We denote the peaking concentration ina measured breakthrough curve at location (r, z) by cmax andthe time that the concentration peaks by tmax. Dividing theboth sides of Eq. (5) by cmax yields

c0 ¼ c

cmax

¼ tmax

t

� �1:5

expðz�utmaxÞ2

4DLtmaxþ r2

4DTtmax�ðz�utÞ2

4DLt� r2

4DTt

!.

ð6Þ

Taking logarithm to both sides of Eq. (6) gives

ln c0 ¼ �1:5 ln t0 þ 1

4DLtmaxðz� utmaxÞ2 �

ðz� utmaxt0Þ2

t0

" #

þ 1

4DTtmaxr2 � r2

t0

� �; ð7Þ

where t 0 = t/tmax is a dimensionless time. Eq. (7) can berewritten into the following form:

y ¼ x

4DLþ 1

4DT; ð8Þ

0 20 40 60 800

0.2

0.4

0.6

0.8

1

Time (hours)

'c

R2=0.98

0 20 40 60 800

0.2

0.4

0.6

0.8

1

Time (hours)

'c

R2=0.95

0 20 40 60 800

0.2

0.4

0.6

0.8

1

Time (hours)

'c

(c)

(b)

(a)

R2=0.91

Figure 2 Comparison of the observed (symbols) and simu-lated (solid lines) breakthrough curves for positions: z = 30 cmand r = 10 cm (a, b), and z = 50 cm and r = 20 cm (c).

An in situ method to measure the longitudinal and transverse dispersion coefficients 617

where

y ¼ ½lnðc0Þ þ 1:5 lnðt0Þ�tmax r2 � r2

t0

� ��1;

x ¼ ðz� utmaxÞ2 �ðz� utmaxt

0Þ2

t0

" #r2 � r2

t0

� ��1.

ð9Þ

Eq. (8) reveals that y, which can be calculated usingmeasured data, is a linear function of x that can also be cal-culated using measured data. The reciprocals of the longitu-dinal and transverse dispersion coefficients DL and DT

determine the slope and intercept of the line, respectively.We fitted the experimental data to Eqs. (8) and (9) to esti-mate the values of DL and DT, but the results did not provevery successful. The reason is that when t 0 approaches 1,both x and y defined in Eq. (9) diverge. Therefore, insteadof using Eqs. (8) and (9), we directly applied Eq. (7) to cal-culate the two parameters. We fitted the experimental datato Eq. (7) using the least square method. Since ln(c 0) is a lin-ear function of the reciprocals of the DL and DT, the valuesof DL and DT can be calculated straightforwardly as followsprovided that the pore water velocity is known

DL ¼1

4tmax

PNi¼1

a2iPNi¼1

b2i �PNi¼1

aibi

PNi¼1

aibi

PNi¼1ðaidi þ ai ln c0iÞ

PNi¼1

b2i �

PNi¼1

aibi

PNi¼1ðbidi þ bi ln c0iÞ

;

DT ¼1

4tmax

PNi¼1

a2iPNi¼1

b2i �

PNi¼1

aibi

PNi¼1

aibi

PNi¼1ðbidi þ bi ln c0iÞ

PNi¼1

a2i �PNi¼1

aibi

PNi¼1ðaidi þ ai ln c0iÞ

;

ð10Þ

where N is total number of the measurements taken fromone suction cup and di ¼ 1:5 ln t0i; ai ¼ ðz� utmaxÞ2

�ðz�utmaxt0iÞt0i

2

; bi ¼ r2 � r2

t0i, in which t0i represents the dimen-

sionless time where the ith measurement c0i is taken.The use of Eq. (10) needs to know the pore-water veloc-

ity. Theoretically, the pore-water velocity can be estimatedusing the measured porosity (0.44) and flow rate (0.51cm/h); this gives u = 1.16 cm/h. However, the velocity esti-mated in this way often failed to predict the observedbreakthrough curves in most of our experiments. We there-fore used a method developed by Zhang (1989) to estimatethe pore water velocity from the breakthrough curve ob-tained from a purposely installed suction cup in soil rightthrough the centre of the internal ring at r = 0. From Eq.(5) we know that when r = 0, the change of concentrationat depth of z with time is given by

cðr; z; tÞ ¼ m

4nDT

ffiffiffiffiffiDL

pðptÞ3=2

exp �ðz� utÞ2

4DLt

!. ð11Þ

From Eq. (11) we know that a new variable Y defined asfollows

Yðx; tÞ ¼ cðz; tÞt3=2 ¼ m

4nDT

ffiffiffiffiffiDL

pðpÞ3=2

exp �ðz� utÞ2

4DLt

!ð12Þ

peaks at z = ut, that is, when t* = z/u. The pore water veloc-ity u can therefore be estimated from the peaking time t* of

Y, which can be calculated using the measured data, andthe depth z where the suction cup is located. From thebreakthrough curve measured from the cup located atz = 24 cm and r = 0, the estimated pore-water velocity isapproximately u = 1.4 cm/h.

The estimated pore-water velocity was used to calculatethe two dispersion coefficients using the breakthroughcurves obtained from other suction cups. One cup locatedat depth of 50 cm was fault, we therefore analysed the dataobtained from other three cups, two at position z = 30 cmand r = 10 cm, and one at position of z = 50 cm andr = 20 cm. The estimated values of the DL and DT using Eq.(10) are DL = 4.5 cm2/h DT = 0.87 cm2/h. To evaluate theaccuracy of these estimated parameters, we used them topredict the breakthrough curves obtained from the threesuction cups by solving Eq. (1) with the associated initialand boundary conditions given in Eq. (2) and R0 = 5 cm.Fig. 2 compares the simulations and the measurements.Overall they match well.

Summary

The work reported in this paper presents an in situ methodto measure the longitudinal and transverse dispersion

618 X. Zhang et al.

coefficients of solute transport in soil by modifying thedouble-ring infiltrometer into a triple-ring infiltrometer.Water flow is controlled in one dimension and solute trans-port in three dimensions. An analytical solution is derivedto describe solute transport, and a simple method basedon the analytical solution is proposed to estimate thetwo parameters. Field experiment was carried out to mea-sure the parameters.

The proposed analytical method for calculating the twoparameters assumes that soil is homogenous where thepore-water velocity and the dispersion coefficients can beapproximated as constants. The method is invalid for heter-ogeneous soil where the pore-water velocity and the solutedispersion coefficients may vary spatially. In this situation,it needs numerical solution to solve the transport equation.Also, the apparatus is designed to measure the longitudinaland transverse dispersion coefficients in saturated condi-tions. It might be possible to improve and modify the appa-ratus to measure the two parameters in unsaturatedcondition in a combination with the disc and ring infiltrom-eters (Angulo-Jaramillo et al., 2000). This, however, needsfurther work.

Acknowledgements

We like to thank the two anonymous reviewers for their con-structive comments to improve the manuscript. We alsothank the Chinese Agro-Ecological Experimental Station inShangqiu, Henan, for their financial support to this work.

Appendix

Applying the Laplace transform to Eqs. (1) and (2) with re-spect to t gives

s~c ¼ DLo2~c

oz2þ DT

o2~c

or2þ 1

r

o~c

or

!� u

o~c

oz;

~cðx;1; tÞ ¼ 0;

~cð1; r; tÞ ¼ 0;

u~c� DLo~c

oz

� �����z¼0¼ C0

R0Hðr � R0Þ;

ðA1Þ

where ~c ¼R10 ce�st dt and C0 ¼ m

R0pn.

Applying the Hankel transform to Eq. (A1) with respect tor yields

s�~c ¼ DLo2�~c

oz2� DTp

2�~c� uo�~c

oz;

�~cð1; r; tÞ ¼ 0;

u�~c� DLo�~c

oz

� �����z¼0¼ C0

pJ1ðpR0Þ;

ðA2Þ

where �~c ¼R10 r~cJ0ðrpÞdr. Solving Eq. (A2) for �~c yields

�~c ¼ 2C0

ðuþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ 4DLðDTp2 þ sÞ

pÞJ1ðpR0Þ

p

� expux �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ 4DLðDTp2 þ sÞ

p2DL

!. ðA3Þ

Applying the inverse Laplace transform to Eq. (A3) and usingthe following formula:

1ffiffispþ h

expð�xffiffispÞ ! 1ffiffiffiffiffi

ptp exp � x2

4t

� �� h expðhx þ th2Þ

� erfcx

2ffiffitp þ h

ffiffitp� �

; ðA4Þ

we have

~c ¼ C0J1ðpR0Þ expð�DTp2tÞp

Fðz; tÞ; ðA5Þ

where

Fðz; tÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiDLptp exp �ðz� utÞ2

4DLt

!� u

2DL

� expuz

DL

� �erfc

zþ ut

2ffiffiffiffiffiffiffiDLtp

� �. ðA6Þ

Applying the inverse Hankel transform to Eq. (A6) gives

cðr; z; tÞ ¼ C0Fðz; tÞZ 1

0

J1ðpR0ÞJ0ðprÞ expð�DTp2tÞdp. ðA7Þ

A special case of Eq. (A7) is when the radius of the internalring is small and its cross-section can be seen as a point. Un-der this condition, we have

limR0!0

Z 1

0

C0JðpR0Þ expð�DTp2tÞdp

¼ m

4pnDTtexp � r2

4DTt

� �; ðA8Þ

Eq. (A7) is therefore simplified to

cðr; z; tÞ ¼ m

4nDT

ffiffiffiffiffiDL

pðptÞ3=2

exp �ðz� utÞ2

4DLt� r2

4DTt

!

� um

8pnDTDLtexp

uz

DL� r2

4DTt

� �erfc

zþ ut

2ffiffiffiffiffiffiffiDLtp

� �.

ðA9Þ

Like the analytical solution for solute transport in an infinitedomain under a prescribed concentration, the second termon the right-hand side of Eq. (A9) is smaller than the firstterm when z is far away from soil surface. Under this condi-tion, Eq. (A9) can be further simplified to

cðr;z;tÞ ¼ m

4nDT

ffiffiffiffiffiDL

pðptÞ3=2

exp �ðz�utÞ2

4DLt� r2

4DTt

!. ðA10Þ

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