Dispersion Paper

8/3/2019 Dispersion Paper

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Transp Porous Med (2009) 80:561579DOI 10.1007/s11242-009-9380-7

Non-Fickian Description of Tracer Transport ThroughHeterogeneous Porous Media

Mostafa Fourar Giovanni Radilla

Received: 8 February 2008 / Accepted: 12 February 2009 / Published online: 1 April 2009 Springer Science+Business Media B.V. 2009

Abstract The porosity and the in situ concentration of tracer testing through differentheterogeneous carbonate cores were performed using X-ray computed tomography. Theresults wereinterpreted usingthree approaches: the convectiondiffusion equation, thearrivaltime moments and the stratied model. The results showed that (i) the Fickian approach ledto a dispersion coefcient varying along each sample (ii) the statistical approach led to apower law of the variance of the arrival time as a function of the distance and (iii) the strat-

ied model allowed quantication of the heterogeneity factor, which also appeared to be apower function of the distance. These data suggest that the temporal moments approach andthe stratied model, but not the classical Fickian approach, are suitable for describing tracertransport through heterogeneous media at the core scale.

Keywords Miscible displacement Non-Fickian transport Heterogeneous porous media Dispersion Heterogeneity

1 Introduction

Modelling tracer transport through porous media is important for understanding and quan-tifying the migration of contaminants in groundwater systems (Dagan and Neuman 1997 ).Tracer tests are also used to characterize petroleum and geothermal reservoirs and aquifers(Moctezuma-B and Fleury 1999 ; Olivier et al. 2004 ). In these two cases, tracer transport isgenerally modelled using the traditional convectiondiffusion equation ( Bear 1972 ), wherethe dispersion coefcient plays a key role. This approach allows determination of tracerconcentration at distance x and time t provided that the ow rate, porosity and dispersion

M. Fourar ( B ) G. RadillaEcole Nationale Suprieure des Mines de Nancy, LEMTA, Parc de Saurupt, 54042 Nancy Cedex, Francee-mail: [email protected]

G. Radillae-mail: [email protected]

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562 M. Fourar, G. Radilla

coefcients of the medium are known.Several studies have demonstrated that the experimen-tally determined dispersion coefcient for any given macroscopic uniform ow conditions ina homogeneous porous medium is constant ( Han et al. 1985 ; Srivastava et al. 1992 ; Sternberg2004 ). However, it is well known that this coefcient has spatial dependency, and therefore

is not constant for heterogeneous media (Domenico and Robbins 1984 ; Gelhar et al. 1992 ;Rajaram and Gelhar 1993 ), even at laboratory scale ( Siddiqui et al. 2000 ; Hidajat et al. 2004 ;Fourar et al. 2005 ; Bauget and Fourar 2008 ). In these cases, breakthrough curves are char-acterized by early breakthrough times and long time-tails. This phenomenon is commonlyreferred to as non-Fickian or anomalous behaviour. Although physical mechanisms of thedispersion process in heterogeneous porous media are well known, there is no theoreticalmodel capable of predicting anomalous breakthrough curves.

The continuous time random walk (CTRW) formulation seems to be a general and effec-tive method for quantifying non-Fickian transport ( Cortis et al. 2004 ; Berkowitz et al. 2006 ).However, this approach assumes statistical homogeneity of the medium and therefore cannotpredict transport in several cases of heterogeneous porous media ( Bauget and Fourar 2008 ).

The dual-porosity concept was proposed to describe tracer behaviour in heterogeneousporous media or media composed of fractures and pores ( Barenblatt et al. 1960 ; Coasts andSmith 1964 ; Gerke and van Genuchten 1993 ; Carlier 2007 ; Aggelopoulos and Tsakiroglou2007 ). In this concept, it is assumed that parts of the porosity of the medium are intercon-nected, which means that it is occupied by mobile uid. The remaining porosity is occupiedby immobile uid. Exchange of tracer between the two domains is attributed to diffusion.However, this approach cannot be applied to heterogeneous porous media in cases where thediffusion is negligible as compared to that of the convection process.

It is possible to characterize non-Fickian displacements in heterogeneous porous mediaby calculating the rst and second temporal moments. Alternatively, the medium can berepresented by an equivalently stratied medium with the same mean and variance of thepermeability. This approach also assumes statistical homogeneity of the medium (i.e. thepermeability of the medium is a probability distribution function) but introduces the hetero-geneity factor as a parameter that evolves along the paths experienced by the tracer.

Theaimof this study was to assessapproaches for interpretingdispersion in heterogeneousmedia. Porosity, tracer transport and ux proles were determined for various carbonate coresamples using computed tomography (CT). We used three different approaches to interpretthe data: the advectiondiffusion equation, the temporal moments approach and the equiv-alent stratied porous medium model. Our data show that the classical approach cannotadequately describe the tracer displacement in the heterogeneous samples; however, the tem-poral moments approach and the stratied model can reliably account for the heterogeneityof the media.

2 Laboratory Experiments

2.1 Core Selection

Four different 38-mm diameter and 80-mm long carbonate samples were selected. Photo-graphsof these carbonate cores are presented in Fig. 1. The samples were characterized by thepresence of oriented shells and fossilized seaweed,which alter the local porosity and the localpermeability. Therefore, the selected carbonate samples present heterogeneous structures atscales much larger than the pore scale, which could affect ow and tracer characterization of core samples. Table 1 gives the porosity and the permeability of the samples at the core scale.

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Non-Fickian Tracer Transport Through Heterogeneous Porous Media 563

Fig. 1 Photographs of the carbonate cores used for tracer tests

Table 1 Properties of thecarbonate cores (core scale)

Sample Mean porosity Core permeability (mD)

1 0.32 562 0.31 3153 0.33 1554 0.27 855

2.2 Porosity Measurements

The porosity of each sample was determined by X-ray CT (Hispeed FX/i medical CT scan-ner; General Electric). The sample was scanned under two different states: fully saturatedwith air (uid 1) and fully saturated with water (uid 2). In each case, the measured CT isthe sum of the CT of the porous matrix (CT pm ) and the CT of the uid (CT uid ), weightedby the porosity:

CT 1 = (1 ) CT pm + CT uid1 and (1)CT 2 = (1 ) CT pm + CT uid2 . (2)

Eliminating the porous matrix CT pm between these equations leads to

=CT 1 CT 2

CT uid1 CT uid2. (3)

For each sample, 1-mm-wide slices were recorded every one millimetre. Each image has aresolution of 512 512 pixels, providing a voxel of 0 .12 0.12 1 mm

3 . Figure 2 containsfour CT cross-sectional images that display the porosity of Sample 1 at 16-mm intervals.Zones of high porosity appear in dark red while zones of low porosity appear in dark blue.The images show the non-uniform porosity distribution for each section. The stack of CTscans allows reconstruction of a three-dimensional (3D) porosity image, as shown in Fig. 3.

The porosity distribution of Sample 1 is also shown in Fig. 3. The sample porosity appearsto be heterogeneous at the core-scale. On the other hand, the heterogeneity of the examinedsamples is illustrated by the porosity proles along each core in Fig. 4, which representscross-sectional mean porosity value as a function of the dimensionless distance from theinlet. Because porosity is measured statically, its error estimates can be very low (previouscalibration tests were performed on reference samples to ensure there were no deviation/dis-persion effects). In our experiments, the maximum absolute error estimate for the porosity is

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Fig. 2 CT images of porosity along Sample 1, obtained at 16-mm intervals

Fig. 3 3D CT porosity image of Sample 1 and the corresponding porosity distribution determined from CTscan measurements

equal to 0.01. This value was used to dene error bars for all porosity data points on Fig. 4.The corresponding relative errors are ranging from 2.8 to 3.8%.

2.3 Tracer Tests

Tracer test experiments were performed using a standard experimental setup (Fig. 5). Thesystem includes a Hassler core-holder, two piston pumps and a conductimeter. The core wasrst saturated with a sodium chloride (NaCl) brine of known concentration ( C o ) using therst pump. Tracer testing was then performed by displacing the resident brine with a solutionof a different concentration ( C 1) using the second pump. The in situ tracer concentration var-iation at several positions along the core was measured by X-ray CT. Each slice was 1-mmwide and the voxel was 0 .12 0.12 1 mm

3 in volume. Theefuent tracer concentration wasmeasured using the conductimeter placed at the outlet of the core-holder. The conductimeter

was calibrated over the range of tracer concentrations used. Experiments were stopped whenthe conductimeter indicated the same concentration as the injected brine. All the experimentswere conducted at a constant room temperature of 21 C.

The injection ow rates were set high enough to make molecular diffusion negligibleas compared to tracer convection. To determine the inuence of the ow rate and brineconcentration, four experiments were performed with Sample 1; ow rates and brine concen-trations used in these experiments are presented in Table 2. For the other samples, tracer tests

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Fig. 4 Porosity proles along each examined core

Fig. 5 Setup of tracer experiments

Table 2 Flow rates and brineconcentrations used in the fourexperiments performed with

Sample 1

Experiment Flow rateQ (cm 3 /h)

Displaced brineconcentration(g/l)

Displacing brineconcentration (g/l)

1 100 10 150

2 100 150 10

3 200 10 150

4 200 150 10

consisted of injecting 150 g/l NaCl brine in the core to displace the resident brine of 10 g/l

concentration at a constant ow rate of 100 cm3

/h.X-ray CT images of the tracer displacement through Sample 1 are shown in Fig. 6. Theycorrespond to four cross sections located at different positions from the core inlet (4, 20, 36and 52 mm) and at different intervals from the beginning of the tracer injection (180, 280 and380 s). A 3D image of the tracer displacement through Sample 1 is shown in Fig. 7. Theseimages clearly show the dispersion of the tracer front. Because of the porosity heterogeneity,the tracer was also dispersed within each cross section. X-ray CT image comparison shows

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Fig. 6 X-ray images of the tracer displacement through Sample 1 for different intervals after tracer injection.The cross sections are located 4, 20, 36 and 52 mm from the core inlet

that there is a link between the porosity distribution and tracer dispersion. Also, Fig. 6 showsclearly that for Sample 1, the upper left part of the cross section at 36 and 52 mm from thecore inlet has a higher permeability. However, comparision between Fig. 3 and Fig. 7 showsthat for the tracer to reach the zones of high porosity, it is necessary for these zones to be con-nected to the inlet of the sample, indicating that the local permeability plays a fundamentalrole in the dispersion processes.

3 Concentration and Flux Proles

3.1 Tracer Concentration Proles

Figure 8 shows typical curves of the average cross-section dimensionless concentration at

different distances as a function of the dimensionless time. The heterogeneity of the cor-responding sample (Sample 3) can be inferred from the fact that the concentration curvesshow a highlydispersive behaviour. For instance,near thecore outlet ( x = 0.95) tracer breaksthrough relativelyearly (before dimensionless time equal to 0.4), and then it takes severalporevolumes for the concentration curve to reach the maximum value ( C = 1). Figure 9 shows acomparison between curves obtained during experiments 1, 2, 3 and 4 at three distances fromthe sample inlet. The four experiments produced almost the same curves. Therefore, we can

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Fig. 7 3D image of the tracerdisplacement through Sample 1

assert that the inuences of the ow rate and the brine concentration in the ranges examinedare negligible. Consequently, molecular diffusion can be assumed negligible as compared tomechanical dispersion. There was good overlap of the curves of tracer ux measured at thecore outlet (Fig. 10). These results suggest that a dead-end pore model, which has been pro-posed for analysing tracer tests conducted on heterogeneous samples ( Hidajat et al. 2004 ), isnot appropriate for interpreting our experimental data. Because this approach is based on thepartitioning of the porous medium into owing and non-owing fractions and on a couplingtermbetween these twofractions(diffusion-like exchange coefcient), the results should have

been sensitive to the ow rate (time to perform the experiment), which was not observed.To verify the accuracy of the in situ X-ray CT tracer concentration measurements, we

determined the total mass balance of tracer at the core-scale. Figure 11 shows the differencebetween the injected mass tracer controlled by the pump and the accumulated mass tracerin the core determined by X-ray CT measurements as a function of the efuent mass tracermeasured by the conductimeter. As can be seen, the total mass balance is well veried forthe four experiments.

3.2 Tracer Flux Proles

The in situ tracer concentration measurements performed at close time intervals allow us tocalculate tracer ux at different cross sections using the mass balance equation:

C t +

1 A

f x = 0, (4)

where A is the cross-sectional area.

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Fig. 8 Average dimensionless tracer concentration as a function of time and distance

Fig. 9 Repeatability of experiments performed on Sample 1

Integrating Eq. 4 from the inlet to a distance x leads to

f ( x, t ) = f 0 A x

0

( x)C ( x, t )

t dx , (5)

where f 0 and f ( x, t ) are the tracer uxes at the inlet and at distance x , respectively.

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Fig. 10 Dimensionless tracer ux at the outlet for Sample 1

Fig. 11 Mass balance of tracer at the whole core scale using the injected mass of tracer (controlled by thepump), the efuent mass (conductimeter) and the local concentration (X-ray)

As previously stated, tracer concentration is known from X-ray CT measurements at

regular time intervals, t i . Therefore, we performed the following approximation:

C ( x, t ) t

C ( x, t i+1) C ( x, t i )t ; (6)

this allows us to rewrite Eq. 5 as

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Fig. 12 Dimensionless tracer ux as function of time at different cross sections along Sample 1

f ( x, t ) = f 0 A

t

x

0

( x) C ( x, t i+1) C ( x, t i ) dx . (7)

The tracer ux at the inlet is given by

f 0 = C 0 Q , (8)

where C 0 is the concentration of the injected brine and Q is the ow rate.To accurately calculate the integral in Eq. 7, we performed piecewise cubic spline inter-

polations on porosity and concentration proles. Accuracy of Eq. 7 combined with the inter-polation of porosity and concentration experimental data was tested on experiments 1, 2, 3and 4, performed with Sample 1. Flux proles were found to be repeatable. Figure 12 showstypical curves of the calculated tracer ux as a function of time at different distances fromthe inlet. Flux proles of Fig. 12 show no overlapping and fairly good smoothness whichconrms that the accuracy of concentration measurements and cubic spline interpolations isvery good.

4 Interpretation

4.1 Dispersion Coefcient

The conventional approach to modelling tracer transport through saturated porous media isto assume that the tracer ux f results from the transport of tracer concentration C at owrate Q , and the dispersion of that concentration by a process similar to molecular diffusion.In other words, it is assumed that for unidirectional ow:

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f = QC A DC x

, (9)

where A is the cross-sectional area of the medium perpendicular to the ow direction, is theporosity and D is the dispersion coefcient. These parameters are constant for homogeneous

samples. If the ow rate is also constant, empirical Eq. 9 associated with the mass balanceEq. 4 leads to the traditional convectiondiffusion equation ( Bear 1972 ), written here in termsof concentration:

C t +

Q A

C x = D

2C x2

. (10)

A similar equation can be obtained for the tracer ux.Equation 10 can be solved to determine the tracer concentration at distance x and time t

if porosity and the dispersion coefcient are constant. In accordance with previous studies,we show that these parameters are spatially dependent. The CT porosity images clearly showthat the porosity distribution is not uniform at the core scale. On the other hand, knowing theconcentration and ux at different positions as functions of time, the local value of the dis-persion coefcient can be calculated from Eq. 9. This method for determining the dispersioncoefcient directly from the differential relationship between the ux and the concentrationavoids the problem of averaging when the integrated solution is used. In addition, the stan-dard boundary conditions of concentration equal to unity at inlet and semi-innite mediumare questionable ( Dauba et al. 1999 ). Figure 13 shows the local dispersion coefcient as afunction of the distance from the inlet for the four samples used in this study. It is shown thatthe local dispersion coefcient D varies along the core samples. The spatial dependence of the dispersion coefcient is the signature of the heterogeneity of the samples. This conrmsthat the classical approach is not suitable for modelling tracer displacement in heterogeneousporous media even at the laboratory scale. Figure 13 also suggests that Sample 1 is the lessheterogeneous while Sample 4 appears to be the most heterogeneous. This may appear tobe in contradiction to Fig. 4 because porosity (i.e. the volume of uid involved in the dis-persion process) of Sample 1 is higher than that of Sample 4. However, this observation canbe explained by the local permeability changes (connectivity between pores) which mightbe higher in Sample 4 than in Sample 1 and which may dominate the dispersion process.Estimates of errors on the local dispersion coefcient have been calculated using error theoryand taking into account the estimates of errors of the concentration measurements which are


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Fig. 13 Dimensionless local dispersion coefcient along each sample

Fig. 14 Tracer-front mean arrival time as function of the dimensionless distance from the core inlet

The plot of < t > as a function of the dimensionless position x is presented in Fig. 14. Forsmall distances, values of the mean arrival time are slightly dispersed. This is probably dueto the lack of accuracy in calculating f ( x, t ) t because of limited experimental data, since thetracer ux quickly reaches its maximum value for small distances. Nevertheless, it is clearthat the mean arrival time of the tracer front is almost equal to the distance from the inlet of the core for all samples. This result is similar to those obtained with homogeneous porousmedia. Consequently, the rst temporal moment does not seem to be affected by the mediumheterogeneity.

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Fig. 15 Variance of the tracer-front arrival time as function of the dimensionless distance from the core inlet

The second temporal moment, dened by

< t 2

> =

0 t 2 f

t dt , (12)

leads to the variance of the arrival time of the tracer front:

2t = < t 2 > < t >

2 . (13)

The plot of 2t as a function of x is presented in Fig. 15. The variance at the same distancefrom the inlet of the core depends on the sample and therefore on the heterogeneity. Severalattempts were made to establish the relationship between the variance of the arrival time of the tracer and the distance from the inlet of the cores. As shown in Fig. 15, the power law

seems to be appropriate:

2t = axb . (14)

A similar result for power law variances has been proposed for probabilistic approaches totracer dispersion in porous media ( Berkowitz and Scher 2001 ). For standard dispersion, anexponent of 1 is observed and for a perfectly layered medium, the variance is proportional tothe square of the distance. Values of a and b for the various samples are presented in Table 3.It should be noted that in our experiments, exponent b varied between 1.43 and 1.7, whichconrmed that the samples were heterogeneous.

4.3 Heterogeneity Factor

The stratied formation constitutes a simple example of a heterogeneous porous mediumand has been intensively investigated (Mercado 1967 ; Marle et al. 1967 ; Gelhar et al. 1979 ;Matheron and de Marsily 1980 ; Communar 1998 ). The leading idea of the stratied modelis to represent the real medium by using an equivalent stratied medium with the same mean

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Table 3 Values of thecoefcients of Eqs. 14 and 17 forall samples

Sample Eq. 14 2t = axb Eq. 17 H = ax

b

a b a b

1 1.15 1.70 0 .37 0.242 1.23 1.70 0 .48 0.393 1.13 1.45 0 .48 0.644 1.56 1.43 0 .63 0.64

and variance of permeability (Fourar 2006 ). Here, the porosity of the medium is assumeduniform, and the permeability of each layer is constant but can differ from one layer toanother. The effects of porescale dispersion and molecular diffusion are assumed negligible

compared to the effects of permeability heterogeneity. It is also assumed that the ow throughthe medium is steady state and parallel to the layers. The tracer convection is then governedby (see Appendix)

C ( x, t ) =12

C 0er f c

ln xV t 1 + H 2

2 ln 1 + H 2

, (15)

where erfc is thecomplementary error function, C 0 the injected tracer concentration, V =Q

A

the mean velocity and H the heterogeneity factor dened as the ratio of the standard deviation K to the mean permeability < K > :

H = K

< K > = 1 + exp 2ln K . (16)

2ln K is the variance of ln K .Equation 15 was used to t the concentration curves of Fig. 8 by optimizing the value of

H for each curve. The results presented in Fig. 16 show that this approach is suitable formodelling our experimental data. Figure 17 shows experimental values of the heterogeneityfactor as a function of the dimensionless length of the samples. It appears that the heteroge-neity factor is a decreasing function of the distance from the medium inlet. The decrease in H is probably related to the fact that the tracer is transported at the inlet as if the medium isstratied (1D ow). As the tracer advances through the medium, the ow becomes 3D andthe stratication effect of the medium decreases. Therefore, the decrease is stronger at theinlet than at the outlet of the medium. It should be noted that the heterogeneity factor tends tobe nearly constant at the outlet of the samples, indicating the existence of a distance beyondwhich the stratication effect on the tracer dispersion is not important.

Curves presented in Figure 17 suggest that the heterogeneity factor can be approximatedby a power function of the dimensionless distance from the medium inlet.

H = axb . (17)

Values of coefcients a and b for each sample are presented in Table 3. By comparing thevalues for a , which correspond to the heterogeneity factor values at the outlet, the samplescan be classied according to their degree of heterogeneity. Sample 1 appears to be the leastheterogeneous, whereas Sample 4 appears to be the most heterogeneous.

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Fig. 16 Typical concentration curves tted to Eq. 15 by optimizing the value of the heterogeneity factor foreach curve

Fig. 17 Heterogeneity factor as function of the dimensionless distance from the core inlet

5 Conclusions

Experiments investigating miscible displacement through different heterogeneous poroussamples were conducted in order to improve interpretation of non-Fickian transport at thecore scale. Several different carbonate cores were used. We found that the porosity of eachsample, according to X-ray CT, is not uniform at the cross-sectional level or at the corescale. Tracer tests were conducted on the cores, using X-ray CT to measure in situ con-centration. In these experiments, ow rates were high enough to make molecular diffusion

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negligible as compared to tracer convection. Data were interpreted using three approaches:the convectiondiffusion equation, the arrival time moments and the stratied model.

We found that the dispersion coefcient was space dependent; therefore, the classicalapproach (Fickian) was not suitable for describing the tracer transport in this study. On the

other hand, the mean arrival time of the tracer front was proportional to the distance fromthe inlet of the cores and the variance of the arrival time was a power law of the distance. Aswell, when the stratied model was used to t the concentration curves, the heterogeneityfactor, which is a key parameter of the stratied model, was a decreasing power law of thedistance.

Therefore, the arrival time moments approach and the heterogeneity factor both accountfor the heterogeneity of the media. These results provide an alternative to the traditionalapproach for interpreting tracer tests at the core scale. Future studies will determine how theheterogeneity factor is related to the macroscopic properties of the medium.

Acknowledgements The experiments presented in this study were conducted at the Institut Franais duPtrole. We thank R. Lenormand, P. Egermann and E. Rosenberg for their helpful discussions and technicalsupport.

6 Appendix

Theanalytical stratiedmodel is presented. We considera porousmedium of a cross-sectionalarea A and a permeability probability distribution function (PDF), G ( K ) . The elementarysection d A of layers of permeability comprised between K and K

+d K is then given by

d A = A G( K )dK . (A1)According to Darcys law, the elementary ow rate through the layers of section d A is:

dq = d AK

P L

, (A2)

where is the dynamic viscosity, L is the length and P is the pressure drop between theinlet and outlet of the medium. The total ow rate Q is obtained by integrating Eq. A2 from

the minimum to the maximum permeability:

Q =K max

K min

dq . (A3)

Taking Eqs. A1 and A2 into account leads to

Q = A

P L

K max

K min

K G ( K )dK . (A4)

By introducing the classic definition of mean permeability, we obtain

< K > =K max

K min

K G ( K )dK , (A5)

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Darcys law of the ow at sample scale is then written:

Q = A< K >

P

L. (A6)

Combining Eqs. A1, A2 and A6 leads to

dq = QK

< K >G ( K )dK . (A7)

If K denotes the permeability of the layer where the tracer front reaches position x attime t , the velocity of the tracer front in this layer is dened by

v = xt

. (A8)

This velocity can also be expressed using Darcys law:

v =1

dqd A =

1

K

P L =

K < K >

Q A

. (A9)

By introducing the mean front velocity,

V =Q

A, (A10)

and using Eq. A8, K is then given by

K =< K > xV t . (A11)The tracer mass in the layers of permeability comprised between K and K +d K is givenby

dm = C 0 dA dx, (A12)where dx is an elementary length in the x direction and C 0 is the injected tracer concentration.Knowing that at time t the tracer has not yet reached the layers of permeability lower thanK , the total mass m of tracer in the pore volume Adx is then obtained by integrating the

previous equation between K and K max :

m =K max

K

C 0 dA dx . (A13)

Inserting Eq. A1 into Eq. A13 and dividing by the pore volume Adx , the equation of the mean concentration for the cross section at position x and time t is

C ( x, t ) = C 0K max

K G ( K )dK . (A14)We assume that the permeability PDF is lognormal:

G ( K ) =1

ln K 2 K exp ( ln K < ln K > )

2

2 2ln K , (A18)

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where < ln K > and 2ln K are the mean and variance of ln K , respectively. These parametersare related to the mean and variance of K as follows:

< K >

=exp < ln K >

+

2ln K

2

and (A19)

2K = exp 2 < ln K > + 2ln K 1 + exp

2ln K . (A20)

The heterogeneity factor is dened as the standard deviation to the mean permeabilityratio:

H = K

< K >. (A21)

Therefore, considering Eqs. A19 and A21 , the heterogeneity factor can be rewritten as

H

= 1

+exp 2

ln K . (A22)

Inserting Eq. A18 into Eq. A14 with K max tending towards innity yields

C ( x, t ) =C 0

ln K 2

K

exp ( ln K < ln K > )

2

2 2ln K

dK K

. (A23)

Substituting in u =ln K < ln K > 2 ln K leads to

C ( x, t ) = C 0

uexp u2 du = C 02 er f c(u), (A24)

where er f c is the complementary error function and u=ln K < ln K > 2 ln K .

Combining Eqs. A11 , A19 , A22 and A23 yields the tracer concentration:

C ( x, t ) =12

C 0 er f c

ln xV t 1 + H 2

2 ln 1

+ H 2

(A25)

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