Transp Porous Med DOI 10.1007/s11242-009-9380-7
Non-Fickian Description of Tracer Transport Through Heterogeneous Porous Media Mostafa Fourar · Giovanni Radilla
Received: 8 February 2008 / Accepted: 12 February 2009 © Springer Science+Business Media B.V. 2009
Abstract The porosity and the in situ concentration of tracer testing through different heterogeneous carbonate cores were performed using X-ray computed tomography. The results were interpreted using three approaches: the convection–diffusion equation, the arrival time moments and the stratified model. The results showed that (i) the Fickian approach led to a dispersion coefficient varying along each sample (ii) the statistical approach led to a power law of the variance of the arrival time as a function of the distance and (iii) the stratified model allowed quantification of the heterogeneity factor, which also appeared to be a power function of the distance. These data suggest that the temporal moments approach and the stratified model, but not the classical Fickian approach, are suitable for describing tracer transport 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 quantifying 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 is generally modelled using the traditional convection–diffusion equation (Bear 1972), where the dispersion coefficient plays a key role. This approach allows determination of tracer concentration at distance x and time t provided that the flow rate, porosity and dispersion
M. Fourar (B) · G. Radilla Ecole Nationale Supérieure des Mines de Nancy, LEMTA, Parc de Saurupt, 54042 Nancy Cedex, France e-mail:
[email protected] G. Radilla e-mail:
[email protected]
123
M. Fourar, G. Radilla
coefficients of the medium are known. Several studies have demonstrated that the experimentally determined dispersion coefficient for any given macroscopic uniform flow conditions in a homogeneous porous medium is constant (Han et al. 1985; Srivastava et al. 1992; Sternberg 2004). However, it is well known that this coefficient 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 characterized by early breakthrough times and long time-tails. This phenomenon is commonly referred to as non-Fickian or anomalous behaviour. Although physical mechanisms of the dispersion process in heterogeneous porous media are well known, there is no theoretical model capable of predicting anomalous breakthrough curves. The continuous time random walk (CTRW) formulation seems to be a general and effective 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 cannot predict transport in several cases of heterogeneous porous media (Bauget and Fourar 2008). The dual-porosity concept was proposed to describe tracer behaviour in heterogeneous porous media or media composed of fractures and pores (Barenblatt et al. 1960; Coasts and Smith 1964; Gerke and van Genuchten 1993; Carlier 2007; Aggelopoulos and Tsakiroglou 2007). In this concept, it is assumed that parts of the porosity of the medium are interconnected, which means that it is occupied by mobile fluid. The remaining porosity is occupied by immobile fluid. Exchange of tracer between the two domains is attributed to diffusion. However, this approach cannot be applied to heterogeneous porous media in cases where the diffusion is negligible as compared to that of the convection process. It is possible to characterize non-Fickian displacements in heterogeneous porous media by calculating the first and second temporal moments. Alternatively, the medium can be represented by an equivalently stratified medium with the same mean and variance of the permeability. This approach also assumes statistical homogeneity of the medium (i.e. the permeability of the medium is a probability distribution function) but introduces the heterogeneity factor as a parameter that evolves along the paths experienced by the tracer. The aim of this study was to assess approaches for interpreting dispersion in heterogeneous media. Porosity, tracer transport and flux profiles were determined for various carbonate core samples using computed tomography (CT). We used three different approaches to interpret the data: the advection–diffusion equation, the temporal moments approach and the equivalent stratified porous medium model. Our data show that the classical approach cannot adequately describe the tracer displacement in the heterogeneous samples; however, the temporal moments approach and the stratified model can reliably account for the heterogeneity of the media.
2 Laboratory Experiments 2.1 Core Selection Four different 38-mm diameter and 80-mm long carbonate samples were selected. Photographs of these carbonate cores are presented in Fig. 1. The samples were characterized by the presence of oriented shells and fossilized seaweed, which alter the local porosity and the local permeability. Therefore, the selected carbonate samples present heterogeneous structures at scales much larger than the pore scale, which could affect flow and tracer characterization of core samples. Table 1 gives the porosity and the permeability of the samples at the core scale.
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
Fig. 1 Photographs of the carbonate cores used for tracer tests Table 1 Properties of the carbonate cores (core scale)
Sample
Mean porosity
Core permeability (mD)
1 2 3 4
0.32 0.31 0.33 0.27
56 315 155 855
2.2 Porosity Measurements The porosity φ of each sample was determined by X-ray CT (Hispeed FX/i medical CT scanner; General Electric). The sample was scanned under two different states: fully saturated with air (fluid 1) and fully saturated with water (fluid 2). In each case, the measured CT is the sum of the CT of the porous matrix (CTpm ) and the CT of the fluid (CTfluid ), weighted by the porosity: CT1 = (1 − φ)CTpm + φCTfluid1 and
(1)
CT2 = (1 − φ)CTpm + φCTfluid2 .
(2)
Eliminating the porous matrix CTpm between these equations leads to φ=
CT1 − CT2 . CTfluid1 − CTfluid2
(3)
For each sample, 1-mm-wide slices were recorded every one millimetre. Each image has a resolution of 512 × 512 pixels, providing a voxel of 0.12 × 0.12 × 1 mm3 . Figure 2 contains four 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 CT scans 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 appears to be heterogeneous at the core-scale. On the other hand, the heterogeneity of the examined samples is illustrated by the porosity profiles along each core in Fig. 4, which represents cross-sectional mean porosity value as a function of the dimensionless distance from the inlet. Because porosity is measured statically, its error estimates can be very low (previous calibration tests were performed on reference samples to ensure there were no deviation/dispersion effects). In our experiments, the maximum absolute error estimate for the porosity is
123
M. Fourar, G. Radilla
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 CT scan measurements
equal to 0.01. This value was used to define 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). The system includes a Hassler core-holder, two piston pumps and a conductimeter. The core was first saturated with a sodium chloride (NaCl) brine of known concentration (Co ) using the first pump. Tracer testing was then performed by displacing the resident brine with a solution of a different concentration (C1 ) using the second pump. The in situ tracer concentration variation at several positions along the core was measured by X-ray CT. Each slice was 1-mm wide and the voxel was 0.12×0.12×1 mm3 in volume. The effluent tracer concentration was measured 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 when the conductimeter indicated the same concentration as the injected brine. All the experiments were conducted at a constant room temperature of 21◦ C. The injection flow rates were set high enough to make molecular diffusion negligible as compared to tracer convection. To determine the influence of the flow rate and brine concentration, four experiments were performed with Sample 1; flow rates and brine concentrations used in these experiments are presented in Table 2. For the other samples, tracer tests
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
Fig. 4 Porosity profiles along each examined core
Fig. 5 Setup of tracer experiments Table 2 Flow rates and brine concentrations used in the four experiments performed with Sample 1
Experiment
Flow rate Q (cm3 /h)
1
100
10
2
100
150
10
3
200
10
150
4
200
150
10
Displaced brine concentration (g/l)
Displacing brine concentration (g/l)
150
consisted of injecting 150 g/l NaCl brine in the core to displace the resident brine of 10 g/l concentration at a constant flow rate of 100 cm3 /h. X-ray CT images of the tracer displacement through Sample 1 are shown in Fig. 6. They correspond to four cross sections located at different positions from the core inlet (4, 20, 36 and 52 mm) and at different intervals from the beginning of the tracer injection (180, 280 and 380 s). A 3D image of the tracer displacement through Sample 1 is shown in Fig. 7. These images 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
123
M. Fourar, G. Radilla
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 shows clearly that for Sample 1, the upper left part of the cross section at 36 and 52 mm from the core inlet has a higher permeability. However, comparision between Fig. 3 and Fig. 7 shows that for the tracer to reach the zones of high porosity, it is necessary for these zones to be connected to the inlet of the sample, indicating that the local permeability plays a fundamental role in the dispersion processes.
3 Concentration and Flux Profiles 3.1 Tracer Concentration Profiles 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 corresponding sample (Sample 3) can be inferred from the fact that the concentration curves show a highly dispersive behaviour. For instance, near the core outlet (x = 0.95) tracer breaks through relatively early (before dimensionless time equal to 0.4), and then it takes several pore volumes for the concentration curve to reach the maximum value (C = 1). Figure 9 shows a comparison between curves obtained during experiments 1, 2, 3 and 4 at three distances from the sample inlet. The four experiments produced almost the same curves. Therefore, we can
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media Fig. 7 3D image of the tracer displacement through Sample 1
assert that the influences of the flow rate and the brine concentration in the ranges examined are negligible. Consequently, molecular diffusion can be assumed negligible as compared to mechanical dispersion. There was good overlap of the curves of tracer flux measured at the core outlet (Fig. 10). These results suggest that a dead-end pore model, which has been proposed for analysing tracer tests conducted on heterogeneous samples (Hidajat et al. 2004), is not appropriate for interpreting our experimental data. Because this approach is based on the partitioning of the porous medium into flowing and non-flowing fractions and on a coupling term between these two fractions (diffusion-like exchange coefficient), the results should have been sensitive to the flow 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 difference between the injected mass tracer controlled by the pump and the accumulated mass tracer in the core determined by X-ray CT measurements as a function of the effluent mass tracer measured by the conductimeter. As can be seen, the total mass balance is well verified for the four experiments. 3.2 Tracer Flux Profiles The in situ tracer concentration measurements performed at close time intervals allow us to calculate tracer flux at different cross sections using the mass balance equation: ∂C 1 ∂f + = 0, ∂t Aφ ∂ x
(4)
where A is the cross-sectional area.
123
M. Fourar, G. Radilla
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 x f (x, t) = f 0 − A
φ (x) 0
∂C(x, t) d x, ∂t
where f 0 and f (x, t) are the tracer fluxes at the inlet and at distance x, respectively.
123
(5)
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
Fig. 10 Dimensionless tracer flux 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 the pump), the effluent mass (conductimeter) and the local concentration (X-ray)
As previously stated, tracer concentration is known from X-ray CT measurements at regular time intervals, ti . Therefore, we performed the following approximation: ∂C (x, t) C (x, ti+1 ) − C (x, ti ) ≈ ; ∂t t
(6)
this allows us to rewrite Eq. 5 as
123
M. Fourar, G. Radilla
Fig. 12 Dimensionless tracer flux as function of time at different cross sections along Sample 1
A f (x, t) = f 0 − t
x
φ (x) C (x, ti+1 ) − C (x, ti ) d x.
(7)
f 0 = C0 Q,
(8)
0
The tracer flux at the inlet is given by
where C0 is the concentration of the injected brine and Q is the flow rate. To accurately calculate the integral in Eq. 7, we performed piecewise cubic spline interpolations on porosity and concentration profiles. Accuracy of Eq. 7 combined with the interpolation of porosity and concentration experimental data was tested on experiments 1, 2, 3 and 4, performed with Sample 1. Flux profiles were found to be repeatable. Figure 12 shows typical curves of the calculated tracer flux as a function of time at different distances from the inlet. Flux profiles of Fig. 12 show no overlapping and fairly good smoothness which confirms that the accuracy of concentration measurements and cubic spline interpolations is very good.
4 Interpretation 4.1 Dispersion Coefficient The conventional approach to modelling tracer transport through saturated porous media is to assume that the tracer flux f results from the transport of tracer concentration C at flow rate Q, and the dispersion of that concentration by a process similar to molecular diffusion. In other words, it is assumed that for unidirectional flow:
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
∂C , (9) ∂x where A is the cross-sectional area of the medium perpendicular to the flow direction, φ is the porosity and D is the dispersion coefficient. These parameters are constant for homogeneous samples. If the flow rate is also constant, empirical Eq. 9 associated with the mass balance Eq. 4 leads to the traditional convection–diffusion equation (Bear 1972), written here in terms of concentration: f = QC − Aφ D
∂C Q ∂C ∂ 2C + =D 2. ∂t Aφ ∂ x ∂x
(10)
A similar equation can be obtained for the tracer flux. Equation 10 can be solved to determine the tracer concentration at distance x and time t if porosity and the dispersion coefficient are constant. In accordance with previous studies, we show that these parameters are spatially dependent. The CT porosity images clearly show that the porosity distribution is not uniform at the core scale. On the other hand, knowing the concentration and flux at different positions as functions of time, the local value of the dispersion coefficient can be calculated from Eq. 9. This method for determining the dispersion coefficient directly from the differential relationship between the flux and the concentration avoids the problem of averaging when the integrated solution is used. In addition, the standard boundary conditions of concentration equal to unity at inlet and semi-infinite medium are questionable (Dauba et al. 1999). Figure 13 shows the local dispersion coefficient as a function of the distance from the inlet for the four samples used in this study. It is shown that the local dispersion coefficient D varies along the core samples. The spatial dependence of the dispersion coefficient is the signature of the heterogeneity of the samples. This confirms that the classical approach is not suitable for modelling tracer displacement in heterogeneous porous media even at the laboratory scale. Figure 13 also suggests that Sample 1 is the less heterogeneous while Sample 4 appears to be the most heterogeneous. This may appear to be in contradiction to Fig. 4 because porosity (i.e. the volume of fluid involved in the dispersion process) of Sample 1 is higher than that of Sample 4. However, this observation can be explained by the local permeability changes (connectivity between pores) which might be higher in Sample 4 than in Sample 1 and which may dominate the dispersion process. Estimates of errors on the local dispersion coefficient have been calculated using error theory and taking into account the estimates of errors of the concentration measurements which are as a function of the dimensionless position x is presented in Fig. 14. For small distances, values of the mean arrival time are slightly dispersed. This is probably due (x,t) to the lack of accuracy in calculating ∂ f ∂t because of limited experimental data, since the tracer flux quickly reaches its maximum value for small distances. Nevertheless, it is clear that 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 porous media. Consequently, the first temporal moment does not seem to be affected by the medium heterogeneity.
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
Fig. 15 Variance of the tracer-front arrival time as function of the dimensionless distance from the core inlet
The second temporal moment, defined by ∞ < t >= 2
t2 0
∂f dt, ∂t
(12)
leads to the variance of the arrival time of the tracer front: σt2 = < t 2 > − < t >2 .
(13)
The plot of σt2 as a function of x is presented in Fig. 15. The variance at the same distance from the inlet of the core depends on the sample and therefore on the heterogeneity. Several attempts 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: σt2 = ax b .
(14)
A similar result for power law variances has been proposed for probabilistic approaches to tracer dispersion in porous media (Berkowitz and Scher 2001). For standard dispersion, an exponent of 1 is observed and for a perfectly layered medium, the variance is proportional to the 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, which confirmed that the samples were heterogeneous. 4.3 Heterogeneity Factor The stratified formation constitutes a simple example of a heterogeneous porous medium and 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 stratified model is to represent the real medium by using an equivalent stratified medium with the same mean
123
M. Fourar, G. Radilla Table 3 Values of the coefficients of Eqs. 14 and 17 for all samples
Eq. 14 σt2 = ax b
Eq. 17 H = ax b
a
b
a
b
1
1.15
1.70
0.37
−0.24
2
1.23
1.70
0.48
−0.39
3
1.13
1.45
0.48
−0.64
4
1.56
1.43
0.63
−0.64
Sample
and variance of permeability (Fourar 2006). Here, the porosity of the medium is assumed uniform, and the permeability of each layer is constant but can differ from one layer to another. The effects of pore–scale dispersion and molecular diffusion are assumed negligible compared to the effects of permeability heterogeneity. It is also assumed that the flow through the medium is steady state and parallel to the layers. The tracer convection is then governed by (see Appendix) ⎞ ⎛ √ x 2 1 + H ln Vt 1 ⎠ (15) C(x, t) = C0 er f c ⎝
, 2 2 ln 1 + H 2 Q where erfc is the complementary error function, C0 the injected tracer concentration, V = Aφ the mean velocity and H the heterogeneity factor defined as the ratio of the standard deviation σ K to the mean permeability < K >:
σK (16) H= = −1 + exp σln2 K .
σln2 K is the variance of ln K . Equation 15 was used to fit 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 for modelling our experimental data. Figure 17 shows experimental values of the heterogeneity factor as a function of the dimensionless length of the samples. It appears that the heterogeneity 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 is stratified (1D flow). As the tracer advances through the medium, the flow becomes 3D and the stratification effect of the medium decreases. Therefore, the decrease is stronger at the inlet than at the outlet of the medium. It should be noted that the heterogeneity factor tends to be nearly constant at the outlet of the samples, indicating the existence of a distance beyond which the stratification effect on the tracer dispersion is not important. Curves presented in Figure 17 suggest that the heterogeneity factor can be approximated by a power function of the dimensionless distance from the medium inlet. H = ax b .
(17)
Values of coefficients a and b for each sample are presented in Table 3. By comparing the values fora, which correspond to the heterogeneity factor values at the outlet, the samples can be classified according to their degree of heterogeneity. Sample 1 appears to be the least heterogeneous, whereas Sample 4 appears to be the most heterogeneous.
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
Fig. 16 Typical concentration curves fitted to Eq. 15 by optimizing the value of the heterogeneity factor for each curve
Fig. 17 Heterogeneity factor as function of the dimensionless distance from the core inlet
5 Conclusions Experiments investigating miscible displacement through different heterogeneous porous samples were conducted in order to improve interpretation of non-Fickian transport at the core scale. Several different carbonate cores were used. We found that the porosity of each sample, according to X-ray CT, is not uniform at the cross-sectional level or at the core scale. Tracer tests were conducted on the cores, using X-ray CT to measure in situ concentration. In these experiments, flow rates were high enough to make molecular diffusion
123
M. Fourar, G. Radilla
negligible as compared to tracer convection. Data were interpreted using three approaches: the convection–diffusion equation, the arrival time moments and the stratified model. We found that the dispersion coefficient was space dependent; therefore, the classical approach (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 from the inlet of the cores and the variance of the arrival time was a power law of the distance. As well, when the stratified model was used to fit the concentration curves, the heterogeneity factor, which is a key parameter of the stratified model, was a decreasing power law of the distance. Therefore, the arrival time moments approach and the heterogeneity factor both account for the heterogeneity of the media. These results provide an alternative to the traditional approach for interpreting tracer tests at the core scale. Future studies will determine how the heterogeneity factor is related to the macroscopic properties of the medium. Acknowledgements The experiments presented in this study were conducted at the Institut Français du Pétrole. We thank R. Lenormand, P. Egermann and E. Rosenberg for their helpful discussions and technical support.
6 Appendix The analytical stratified model is presented. We consider a porous medium of a cross-sectional area A and a permeability probability distribution function (PDF), G(K ). The elementary section 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 Darcy’s law, the elementary flow rate through the layers of section d A is: dq = d A
K P , µ L
(A2)
where µ is the dynamic viscosity, L is the length and P is the pressure drop between the inlet and outlet of the medium. The total flow rate Q is obtained by integrating Eq. A2 from the minimum to the maximum permeability: K max
Q=
dq.
(A3)
K G(K )dK.
(A4)
K min
Taking Eqs. A1 and A2 into account leads to A P Q= µ L
K max
K min
By introducing the classic definition of mean permeability, we obtain K max
K G(K )dK,
< K >= K min
123
(A5)
Non-Fickian Tracer Transport Through Heterogeneous Porous Media
Darcy’s law of the flow at sample scale is then written: Q=A
< K > P . µ L
(A6)
Combining Eqs. A1, A2 and A6 leads to dq = Q
K G(K )dK.
(A7)
If K ∗ denotes the permeability of the layer where the tracer front reaches position x at time t, the velocity of the tracer front in this layer is defined by v=
x . t
(A8)
This velocity can also be expressed using Darcy’s law: v=
1 dq 1 K ∗ P K∗ Q = = . φ dA φ µ L < K > Aφ
(A9)
By introducing the mean front velocity, V =
Q , Aφ
(A10)
and using Eq. A8, K ∗ is then given by K ∗ =< K >
x . Vt
(A11)
The tracer mass in the layers of permeability comprised between K and K + d K is given by dm = C0 φ dA dx,
(A12)
where d x is an elementary length in the x direction and C0 is the injected tracer concentration. Knowing that at time t the tracer has not yet reached the layers of permeability lower than K ∗ , the total mass m of tracer in the pore volume φ Ad x is then obtained by integrating the previous equation between K ∗ and K max : K max
m=
C0 φ dA d x.
(A13)
K∗
Inserting Eq. A1 into Eq. A13 and dividing by the pore volume φ Ad x, the equation of the mean concentration for the cross section at position x and time t is K max
C(x, t) = C0
G(K )dK.
(A14)
K∗
We assume that the permeability PDF is lognormal:
1 (ln K − < ln K >)2 exp − G(K ) = , √ 2σln2 K σln K 2π K
(A18)
123
M. Fourar, G. Radilla
where < ln K > and σln2 K are the mean and variance of ln K , respectively. These parameters are related to the mean and variance of K as follows:
σln2 K < K > = exp < ln K > + and (A19) 2
(A20) σ K2 = exp 2 < ln K > +σln2 K −1 + exp σln2 K . The heterogeneity factor is defined as the standard deviation to the mean permeability ratio: σK . (A21) H= Therefore, considering Eqs. A19 and A21, the heterogeneity factor can be rewritten as
(A22) H = −1 + exp σln2 K . Inserting Eq. A18 into Eq. A14 with K max tending towards infinity yields
∞ C0 (ln K − < ln K >)2 dK . exp − C(x, t) = √ K 2σln2 K σln K 2π
(A23)
K∗
Substituting in u =
ln K − < ln K > leads to √ 2σln K
C0 C(x, t) = √ π
∞ u∗
C0 exp −u 2 du = er f c(u ∗ ), 2
ln K ∗ − < ln K > . √ 2σln K Combining Eqs. A11, A19, A22 and A23 yields the tracer concentration: ⎞ ⎛ √ x 2 1 + H ln Vt 1 ⎠ C(x, t) = C0 er f c ⎝
2 2 ln 1 + H 2
(A24)
where er f c is the complementary error function and u ∗ =
(A25)
References Aggelopoulos, C.A., Tsakiroglou, C.D.: The longitudinal dispersion coefficient of soils as related to the variability of local permeability. Water Air Soil Pollut. 185, 223–237 (2007). doi:10.1007/ s11270-007-9445-6 Barenblatt, G.E., Zheltov, I.P., Kochina, I.N.: Basic concept of the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. Eng. Transl. 24, 1286–1303 (1960). doi:10.1016/ 0021-8928(60)90107-6 Bauget, F., Fourar, M.: Non-Fickian dispersion in a single fracture. J. Contam. Hydrol. 100, 137–148 (2008). doi:10.1016/j.jconhyd.2008.06.005 Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972) Berkowitz, B., Scher, H.: The role of probalilistic approaches to transport theory in heterogeneous media. Transp. Porous Media 42, 241–263 (2001). doi:10.1023/A:1006785018970 Berkowitz, B., Cortis, A., Dentz, M., Scher, H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003 (2006). doi:10.1029/2005RG000178 Carlier, E.: Solute transport in dual porosity medium and scale effect of dispersivity: proposal of a probabilistic equation. J. Environ. Hydrol. 15(4), 1–8 (2007)
123
Non-Fickian Tracer Transport Through Heterogeneous Porous Media Coasts, K.H., Smith, B.D.: Dead-end pore volume and dispersion in porous media. Soc. Pet. Eng. J. Trans. AIME 231(4), 73–84 (1964) Communar, G.M.: A solute Transport in Stratified Media. Transp. Porous Media 31, 133–134 (1998). doi:10. 1023/A:1006557032656 Cortis, A., Gallo, C., Scher, H., Berkowitz, B.: Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities. Water Resour. Res. 40, W04209 (2004). doi:10.1029/ 2003WR002750 Dauba, C., Hamon, G., Quintard, M., Cherblanc, F.: Identification of parallel heterogeneities with miscible displacement, SCA paper 9933, Golden, USA (1999) Dagan, G.: Flow and Transport in Porous Formations. Springer-Verlag, New York (1989) Dagan, G., Neuman, P.: Subsurface Flow and Transport: A Stochastic Approach. Cambridge University Press, New York (1997) Domenico, P.A., Robbins, G.A.: A dispersion scale effect in model calibration and field tracer experiments. J. Hydrol. (Amst.) 70, 123–132 (1984). doi:10.1016/0022-1694(84)90117-3 Fourar, M., Konan, G., Fichen, C., Rosenberg, E., Egermann, P., Lenormand, R.: Tracer tests for various carbonate cores using X-ray CT, society of core analysis. Annual international symposium, Toronto, Canada, 21–25 Aug 2005 Fourar, M.: Characterization of heterogeneities at the core-scale using the equivalent stratified porous medium approach. In: International symposium of the society of core analysts, Trondheim, Norway, 12–16 Sept 2006 Gerke, H.H., van Genuchten, M.T.: A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res. 29(2), 305–320 (1993). doi:10.1029/ 92WR02339 Gelhar, L.W., Gutjahr, A.L., Naff, R.L.: Stochastic analysis of macrodispersion in a stratified aquifer. Water Resour. Res. 15, 1387–1397 (1979). doi:10.1029/WR015i006p01387 Gelhar, L.W., Welty, C., Rehfeldt, K.R.: A critical review of data on field-scale dispersion in aquifers. Water Resour. Res. 28, 1955–1974 (1992). doi:10.1029/92WR00607 Güven, O., Molz, F.J., Melville, J.G.: An analysis of dispersion in a stratified aquifer. Water Resour. Res. 20, 1337–1354 (1984). doi:10.1029/WR020i010p01337 Han, N., Bhakta, J., Carbonell, R.G.: Longitudinal and lateral dispersion in packed beds: Effect of column length and particle size distribution. AIChE J. 31, 277–288 (1985). doi:10.1002/aic.690310215 Hidajat, I., Mohanty, K.K., Flaum, M., Hirasaki, G.: Study of vuggy carbonates using NMR and X-Ray CT scanning. SPE Reservoir Evaluation and Engineering, October (2004) Lenormand, R.: Determining flow equations from stochastic properties of a permeability field: The MHD model. SPE J. 1, 179–190 (1996) Marle, C.M., Simandoux, P., Pacsirszky, J., Gaulier, C.: Etude du déplacement de fluids miscibles en milieu poreux stratifié. Rev. Inst. Fr. Pet. 22, 272–294 (1967) Matheron, G., de Marsily, G.: Is transport in porous media always diffusive? A counterexample. Water Resour. Res. 16, 901–917 (1980). doi:10.1029/WR016i005p00901 Mercado, A., de Marsily, G.: The spreading pattern of injected waters in a permeability stratified aquifer. IAHS AISH Publ. 72, 23–36 (1967) Moctezuma-B, A., Fleury, M.: Permeability mapping on vuggy core sample using tracer experiments and streamline simulations. SCA paper 9919 Golden, USA (1999) Naff, R.L.: On the nature of the dispersive flux in saturated heterogeneous porous media. Water Resour. Res. 5, 1013–1026 (1990) Olivier, P., Cantegrel, L., Laveissière, J., Guillonneau, N.: Multiphase flow behaviour in vugular carbonates using X-Ray CT, SCA paper 2004-13, Abu Dhabi, UAE (2004) Rajaram, H., Gelhar, L.W.: Plume scale-dependent dispersion in heterogeneous aquifers, 2, Eulerian analysis and three dimensional aquifers. Water Resour. Res. 29, 3261–3276 (1993). doi:10.1029/93WR01068 Siddiqui, S., Funk, J., Khamees, A.: Static and dynamic measurements of resevoir heterogeneities in carbonate reservoirs, SCA paper 2000-06, Abu Dhabi, UAE (2000) Srivastava, R.K., Chatzis, I., Dullien, A.L.: A computerized technique for measuring in-situ concentrations during miscible displacements in porous media. Transp. Porous Media 7, 127–145 (1992). doi:10.1007/ BF00647393 Sternberg, S.P.K.: Dispersion measurements in highly heterogeneous laboratory scale porous media. Transp. Porous Media 54, 107–124 (2004). doi:10.1023/A:1025708313812
123
