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DIVISION S-1—SOIL PHYSICS

Scale- and Rate-Dependent Solute Transport within an Unsaturated Sandy Monolith

Dep. of Environmental Sciences and Land Use Planning, Université Catholique de Louvain (UCL), Croix du Sud, 2 Bte 2, B-1348 Louvain-la-Neuve, Belgium

^* Corresponding author (javaux{at}geru.ucl.ac.be).

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
Solute transport in unsaturated soil is a scale- and flow rate-dependent process, and high quality experimental studies are needed to introduce scaling relationships in the governing flow and transport models. Within this paper, we present an experimental study to characterize the solute transport for an unsaturated and undisturbed subsoil monolith sampled in a sandy aquifer. Leaching experiments were performed at different flow rates, and solute breakthrough at different positions in the monolith was measured by time domain reflectometry (TDR) probes. At TDR probe scale, the transport is assessed as being a convective-dispersive process with mixing lengths <0.3 m. At the scale of the monolith, that is, the meso-scale, transport is rather a stochastic-convective process with mixing lengths >1 m. At the bottom of the monolith, the meso-scale dispersivity is at least two times larger than the local-scale dispersivity. In addition, this meso-scale dispersivity can be successfully predicted using a stream tube model in combination with a local scale convective-dispersive process. Important scale effects are also observed in the relationship between the square coefficient of variation of the local velocities and the dispersivity. In contrast to previous studies, we do not observe a significant difference in transport properties for different flow rates (from 1.05 to 54.16 cm d^-1). We attribute this insignificance either to a balance between the decrease of the tortuosity and the increase of the coefficient of variation of the velocity with the flow rate or a non-dependency of both terms on the flow rate.

Abbreviations: BTC, breakthrough curve • CD, convective dispersive • CDE, CD equation • EC, electrical conductivity • superscript [l], local scale • LSM, least square method • superscript [m], meso-scale • MM, moment method • NTC, negative temperature coefficient • PVC, polyvinyl chloride • SC, stochastic convective • TDR, time domain reflectometry

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
THE HYDRODYNAMIC DISPERSIVITY (L) is a key parameter for describing solute transport in natural porous media. However, the link between this parameter and basic soil properties for a given flow regime is still poorly understood. Much evidence shows that hydrodynamic dispersivity depends on the scale and the flow conditions and is therefore not a material constant of the porous medium. Scale-dependent dispersivity has been studied in saturated porous media at the laboratory and the field scale, both in homogeneous as well as in heterogeneous formations (Pickens and Grisak, 1981) and has been related to the variability of the microscale flow field (Huang et al., 1995; Pang and Hunt, 2001). Similarly, scale-dependent dispersivity has been studied in partially saturated media where velocity and water content variations as well as structural variability of the porous medium complicate the study of the scale effect. Diverging scaling behaviors of the hydrodynamic dispersivity were obtained from solute transport experiments, using different experimental soil types, different experimental scales and different boundary conditions. At the scale of an undisturbed soil column (21.8- to 87-cm length, 10- to 20-cm i.d.), Khan and Jury (1990) observed an increasing hydrodynamic dispersivity with the column length when high flow conditions were imposed. In contrast, a constant dispersivity was obtained for lower flow rate or when the material was disturbed. Similar results were obtained by Porro et al. (1993) in uniform and layered soil columns of 0.95-m diam. and 6-m depth. In an undisturbed layered lysimeter (0.8-m i.d., 2-m depth), Vanclooster et al. (1995) observed a nearly linear increase of the hydrodynamic dispersivity with depth. Vanderborght et al. (2001), studying solute transport on small Regosol lysimeters (0.3-m i.d., 0.8-m depth), reported constant dispersivity with depth at low flow rates but depth dependent dispersivity at high flow rates. Studying solute transport through two undisturbed Spodosols monoliths (0.8-m i.d., 1-m depth), Seuntjens et al. (2001) obtained opposite depth scaling behaviors of the dispersivity, and related this to the different morphological characteristics of the two peds.

From a theoretical point of view, the dispersion process in a soil can be viewed as the result of the concomitant movement of a high number of dissolved solute particles, each one having its own velocity and direction, which continuously change along its trajectory. The regime of lateral mixing of solute between vertically oriented stream tubes therefore defines the transport process. To evaluate the mixing process, a mixing time t* has been introduced by different authors (Dagan, 1984; Scheidegger, 1960; Taylor, 1953). Vanderborght et al. (2001) defined t* as "the time interval during which a particle travels with a constant velocity or ‘remembers’ its velocity" (Vanderborght et al., 2001). If the mean solute arrival time is much lower than t*, the solute particles spread longitudinally by virtue of differences in their convective velocities, resulting in an overall dispersion increasing linearly with depth. This process is called stochastic-convective (SC). In contrast, if t* is much lower than the mean solute arrival time, the concentration differences normal to the direction of flow are smoothed out by transverse mixing (Khan and Jury, 1990). In this case, dispersion remains constant with depth and solute transport is convective dispersive (CD).

At a given depth, any medium discontinuity can affect the solute particles velocity distribution at that depth by changing (increasing or decreasing) the transverse mixing between adjacent regions having different velocities. So, transverse layering, structural or textural change of the porous medium along the flow path and non-uniform distribution of the moisture in the partially saturated porous medium may change the lateral mixing regime and hence the longitudinal dispersion (Koch and Flühler, 1994).

In addition, the dispersion process can also be affected by the flow rate. However, two antagonistic effects have to be considered. If flow rate increases, the pores domain encompassed into the transport process becomes larger and therefore the mixing time variance of the solute particles velocity increases. In this case, should increase. This was observed by Vanderborght et al. (1997), amongst others. On the other hand, increase of the soil water content related to a higher flow rate can lead to reduced tortuosity then the flow paths will be shorter and this will narrow the arrival time distribution (Maciejewski, 1993). In that case, diminishes.

Within that context, the experimental protocol or set-up affects the mixing regime of a surface applied solute in a partially saturated soil. Column length, flow rates, and structural heterogeneity of the porous medium within the flow domain influence the solute transport process, and subsequently, the solute transport parameters. It implies that well-conducted experiments are needed to identify the governing transport concept and to assess the solute transport parameters. It also implies that appropriate scaling relationships are needed to introduce solute transport parameters, identified under specific experimental conditions, in predictive solute transport models operational under variable boundary conditions. This is of paramount importance given the influence of the hydrodynamic dispersivity in predicting chemical concentrations in soil and subsoil, especially at the low concentration front end of the pollutant breakthrough curve. In addition, for engineering applications, parameter generation techniques need to be developed which allow the hydrodynamic dispersion to be identified for different porous media in a practical way and this in terms of the scale of application and flow conditions. Unfortunately, the experimental database supporting the development of generic scaling relationships for hydrodynamic dispersion in natural porous media is still small. In addition, the experimental studies reported above have mostly been collected in surface soils and little experimental data are available for the solute transport behavior in subsoil vadose zone. The understanding of the mixing regime in these formations is therefore needed to predict large-scale solute transport in practical applications.

The objective of this study was to investigate the scale- and flow rate-dependency of the solute transport process within an undisturbed sandy soil core at the scale of a TDR sampling window (called ‘local scale’ hereafter) and the scale of the soil column (called ‘meso-scale’ hereafter). We especially focused our attention on the respective contribution of the coefficient of variation of the velocity and the mixing length on the longitudinal apparent dispersivity.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
If the travel time is much greater than the mixing time t*, the vertical transport of inert solutes into a homogeneous partially saturated soil can be modeled by means of convection-dispersion equation (CDE) written as:

[1]

with C (M L^-3) the concentration, D_L (L² T^-1) the longitudinal dispersion coefficient, (L T^-1) the average solute velocity, and t (T), z (L) the time and space coordinates. Neglecting chemical diffusion, the longitudinal hydrodynamic dispersivity _L can be formulated as:

[2]

As pointed out by Vanderborght et al. (2001), _L is a descriptor for the mixing regime in soils. That means that, even if the mixing regime is not CD, _L can be used as an apparent coefficient but its definition of travel-time distribution will be different (see Eq. [9] and [10] hereafter). Mixing regimes can be defined in terms of the autocorrelation of the solute transport velocities in time and the variance of the solute travel distances. Following Scheidegger (1960), the solute mixing time t* (T) is:

[3]

where r_v is the dimensionless coefficient of correlation between the transport velocity v at time t and v at time t + , where is the time lag. The mixing time, also called autocorrelation time (Bear, 1972), is therefore a threshold value discriminating between a process for which the particle velocities at time t and t + are not correlated (if >> t*) and another one for which every particle progresses with its velocity without random effect (when << t*).

By definition, the longitudinal dispersion is defined as

[4]

with ²_z (L²) the variance of solute travel distance. Depending on the travel time, ²_z can be simplified as (Bear, 1972):

[5]

[6]

with ²_v (L² T^-2) the variance of the solute velocities. Actually, in a porous medium, the mixing regime is dependent on the connectivity of the pore space. Thus, it is rather in terms of length than in term of time that we could define a mixing characteristic value. So, we can define a mixing length l* (L) corresponding to the distance traveled during a time t*, assuming that the mixing time t* = l*/ (Skopp and Gardner, 1992; Vanderborght et al., 2001). From Eq. [4], [5], and [6], and given the definition of l*, the following expressions for D_L are obtained:

[7]

[8]

Combining Eq. [7] and [8] with Eq. [2] yields:

[9]

[10]

with CV²_v, the coefficient of variation of solute velocities. The apparent longitudinal hydrodynamic dispersivity can therefore be defined in two ways depending on the mixing regime. Equations [5], [7], and [9] are only suitable if the solute travel depth of the considered transport event is larger than the mixing length. In this case, the travel depth variance grows linearly with time and the dispersivity is independent of depth as long as CV²_v is constant. Considering that the solute travel depth probability density function is normal, this transport process can be modeled with the CDE. In Eq. [10], the _L is related to the depth by the z and the CV_v terms. It can be shown (see Appendix 1) that the condition to have a SC process is that CV_v/ is constant with depth. Furthermore, if is constant, the process can be modeled, for instance, by the convective-lognormal transfer function (CLT) (Jury, 1982). In this case, the _L scales linearly with depth.

Equations [9] and [10] also indicate that, even in a homogeneous soil, flow rate changes can affect _L by modifying the CV_v (Vanderborght et al., 2001). As pointed out in the introduction, this is caused by two antagonistic effects: (i) a change of the tortuosity affecting the length of flow paths and (ii) a modification in the range of pores explored by the flow paths. Moreover, although it is not explicit from the previous equations, it is worth noting that l* can also be influenced by the flow rate or the water content (Matsubayashi et al., 1997). Indeed, when the soil water content increases, flow paths are better connected, which may improve the mixing and reduce the mixing length. However, in the other hand, higher water content may also result in a higher mixing length if continuous macropores are activated.

The spatial scale effect on _L, both horizontally and vertically, is generated by the scale dependence of CV_v and l*. Indeed, if more heterogeneity is encompassed in the flow domain when scale changes, the CV_v will increase because of a broader pore water velocity distribution. In addition, complete mixing between flow paths will take more time if the scale is larger. Consequently, l* should increase with scale.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
Subsoil Sampling and Monolith Equipment
One undisturbed monolith (1-m height by 0.8-m i.d.) was extracted from a sand quarry (Mont-Saint-Guibert, Belgium) at 10-m depth below the ground surface using the method as described by Vanclooster et al. (1995). The subsoil is a Tertiary marine sandy deposit being characteristic for the aquifer material of the region. Since it is the result of a sedimentation process, this subsoil exhibits a series of small horizontal layers (<5 cm thick), inversely size-graded, possibly cross-bedded. Disturbed samples were collected with kopecki rings near the monolith-sampling hole to characterize the grain-size distribution of the material at different depths. As can be seen in Table 1, the texture of the subsoil seems rather homogeneous with roughly 95% of coarse material. It is important to note that some coarse siliceous concretions (max. 100 cm³) were present in the formation, which for obvious reasons, were not considered in the granulometric analysis. The apparent contradiction between the layering and the textural homogeneity is due to kopecki ring volume, which was too large compared with the layer thickness.

View larger version (77K): [in this window] [in a new window]   Fig. 1. Schematic of the equipped column (perspective and top view).

Because of technical problems with the data logger, TDR breakthrough curves (BTCs) could not be recorded for the Exp.1 and outlet BTC for the Exp. 7 (Table 2). They therefore do not appear in the following graphics and tables.

The water content during the leaching experiments was estimated from the TDR readings using the Topp calibration equation (Topp et al., 1980). The steady-state flow rate was estimated both from the tipping bucket measurements at the outlet of the monolith as well as from the measurement of the mean volume of the top reservoir.

Estimating Solute Transport Properties at Local and Meso-Scale
The time-integral normalized resident concentration C^rt*(z,x,t) at time t of a surface applied solute within the TDR sampling volume positioned at location x and at depth z is defined as (Vanderborght et al., 1996):

[11]

By relating C^r(z,x,t) to the impedance of a soil sample Z(z,x,t) () (Eq. [A8]), C^rt*(z,x,t) is directly derived from the TDR readings as follows (Vanderborght et al., 1996):

[12]

with 1/Z*(z,x,t) = 1/Z(z,x,t) - 1/Z₀(z,x), where Z₀(z,x) () is the TDR load at infinity before the solute pulse is added and T_C, the temperature correction factor at 25°C. The T_C is considered uniform at a given depth z and is calculated as (Heimovaara et al., 1995):

[13]

in which T(z) is the temperature (°C) measured at the same depth z at which the bulk electrical conductivity is measured and _C (°C^-1) is a temperature correction coefficient. Mean time-integral normalized resident concentration at each depth z was obtained by averaging time series (see Appendix 2):

[14]

Outlet EC_w(z,t) (µS cm^-1) were transformed to time-integral-normalized flux concentration C^ft*(z,t) by:

[15]

which equals the travel time probability density function f^f(z,t) when the solute input is a narrow pulse when steady-state flow occurs (Jury and Roth, 1990).

Equation [12] defines local scale time series and Eq. [14] and [15] are assumed to define time series representative at the scale of the cross-section of the monolith, called hereafter ‘meso-scale’.

Apparent dispersion and velocity were obtained from the CDE definition through adequate resolution of Eq. [1] in two different ways: (i) fitting the calculated time series of C^rt* and <C^rt*> inferred from the TDR readings to the analytical solution of the CDE in terms of C^rt* by minimizing the mean square error (least squared method, LSM) and (ii) equaling the theoretical and experimental central time moments (method of moments, MM). All calculations were done in a Matlab environment (Matlab 5.3, The Mathworks Inc., Natick, MA). To determine the mixing regime, an apparent longitudinal hydrodynamic dispersivity was obtained by Eq. [2].

The analytical solutions of the CDE and time moments in terms of C^rt* for a Dirac delta input pulse and a semi-infinite medium, assuming that there is no solute present in the system initially, were obtained from Jacques et al. (1998). For the outlet BTC, we used the travel time probability density function for a 3^rd type input function (Jury and Roth, 1990).

Local scale solute transport parameters (at the scale of the TDR sampling window) will be denoted with the superscript l (e.g., ^l_L and ^l for, respectively, apparent dispersivity and velocity) whereas the meso-scale solute transport parameters resulting from the mean of three local BTCs or from the outlet BTCs will be denoted with superscript m.

Statistical Analysis
To discriminate the effect of the location and the depth on the parameters at the local scale (the scale of the TDR sampling window) and meso-scale (the scale of the soil core), analyses of variance (ANOVA1 or ANOVA2) were performed using the statistical toolbox of Matlab. Statistical significance was assessed at the 0.01 and.05 probability levels.

Given the small sampling points, standard deviation estimates were calculated following the equation (Dagnelies, 1998):

[16]

	RESULTS AND ANALYSIS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
Spatial Scale and Flow Rate Dependency of the Local Scale Solute Transport Parameters
Spatial Scale Dependency
For each TDR probe, 11 local scale apparent dispersivities, ^l_L, and velocities, ^l, were estimated by inversion of the CDE, one for each experiment/flow rate. In general, BTCs were unimodal and CDE equations fit the experimental curves pretty well with a determination coefficient (r²) between 0.9 and 1 (see Fig. 2) . At that local scale, the LSM and the MM gave very similar results. To simplify the discussion, we show in the subsequent part only the results obtained by the LSM.

View larger version (34K): [in this window] [in a new window]   Fig. 2. Local scale (top) and meso-scale (bottom) breakthrough curves (BTCs) during Exp. 7 (see Table 2). The six local scale BTCs were measured by time domain reflectometry (TDR) Probes 1 to 6 (see Fig. 1). Gray line shows the fitted BTC.

View larger version (24K): [in this window] [in a new window]   Fig. 3. Local-scale dispersivity as a function of soil depth obtained by using the LSM method (Probes 1 to 6: squares, Probes 7 to 9: circles, and probes 10 to 12: triangles). Error bars show the standard deviation. For improving readability, the L values obtained from time domain reflectometry (TDR)-readings positioned at the same depth have been a slightly shifted on the depth-axis.

We also note a remarkably small variations of ^l_L in terms of flow rate (see further) and in terms of the horizontal position within the monolith. This high uniformity of the local scale dispersivity should be emphasized.

Figure 4 shows the local scale mean solute velocities ^l estimated from the local scale BTCs observed with the three upper TDR probes (TDR6, TDR9, and TDR12). For a given flow rate, ^l differs at each location. However, the velocity pattern seems consistent for the different flow rates. The ^l estimated for TDR9 are systematically lower than those estimated for the other positions. Since the position of the irrigation device was changed for each new experiment, the constant velocity patterns reflect the intrinsic physical behavior of the porous medium. A constant velocity pattern was also observed by Buchter et al. (1995) in a stony soil monolith, when carrying out two consecutive BTC experiments at the same flow rate.

View larger version (25K): [in this window] [in a new window]   Fig. 4. Local-scale velocities estimated at TDR6, TDR9, and TDR12, represented respectively by the first, second, and third bar for each experiment.

View larger version (20K): [in this window] [in a new window]   Fig. 5. Local-scale dispersivities in terms of flow rate. Error bar shows the standard deviation obtained from observations at 12 different probe locations.

Figure 6 shows the observation of ^l in terms of depth and flow rates. As for the horizontal pattern, the vertical pattern of distribution of ^l is consistent for the different flow rates. This elucidates the persistency of the transport process in the different ‘stream tubes’.

View larger version (38K): [in this window] [in a new window]   Fig. 6. Local-scale velocity for TDR1 to 6 for 12 different flow rates. Figures refer to experiments number.

Figure 7 illustrates the depth dependency of the mean and standard deviation of the meso-scale dispersivity ^m_L. We observe that the two methods, the MM and LSM, yield similar results, except for a depth of 0.4 m where MM overestimates ^m_L as compared with LSM. This is probably because the velocity variation at that depth is quite large, which produces multi-modal averaged BTCs from some flow rate. It appears also that the dispersivity value obtained from the outlet time series is remarkably similar to the dispersivity estimates from the meso-scale time series from TDR probes. This confirms the assumption made at the beginning of equivalence of scale between those two types of meso-scale time series.

View larger version (21K): [in this window] [in a new window]   Fig. 7. Meso-scale dispersivity versus depth. Circles represent the dispersivity obtained by the least square method (LSM) while squares those obtained by moment method (MM). Error bars show the standard deviation.

We further observe that ^m_L at 0.75-m depth is at least two times larger than ^l_L. This can be related to the higher heterogeneity encompassed in the solute flow domain at the meso-scale, which increases the CV_v. It will have, of course, significant consequences for the real world predictions of solute transport. From a practical point of view, these experiments justify the use of at least three TDR probes at the same depth to obtain a valuable estimation of the meso-scale dispersivity.

Flow Rate Dependency
Figure 8 shows the estimated ^m_L estimated from <C^rt*(z,t)> at z = 0.75 m and f^f(z,t) at z = 1 m. As confirmed by a ANOVA2 test statistic, the ^m_L remained in the same order of magnitude for all flow rates and are nearly twice as high as ^m_L. No significant differences between MM and LSM estimates could be discerned. In addition, the differences between parameters for the 0.75-m soil depth and the 1-m soil depth are small, showing that flow rate does not affect ^m_L. This could again be attributed the layering, which stabilize the CV_v. To have an estimation of the CV_v, a coefficient of variation of ^l, CV_^l (z,J_w), was calculated at each depth and flow rate (with only three replicates). Indeed, it was observed (not shown here) that CV_V was completely independent of ^l and J_w.

View larger version (21K): [in this window] [in a new window]   Fig. 8. Meso-scale dispersivity at 0.75-m depth estimated by least square method (LSM) (squares), by moment method (MM) (triangles) and dispersivity estimated from outlet breakthrough curves (BTCs) at 1-m depth estimated by LSM (time sign) and MM (plus sign).

Linking the Meso- and Local-Scale Parameters Relationship between CV_^l and ^m_L
To evaluate the contribution of the apparent local velocity variation to the meso-scale apparent dispersivity, we will compare the CV_^l (z,J_w) to ^l_L. For the sake of simplicity, the notation ‘CV_V’ will be used, instead of CV_^l (z,J_w).

Figure 9 shows the ^m_L–CV_V relationship for the ^m_L estimated by the LSM. This figure shows a clear linear and depth dependent relationship. At 0.15- and 0.75-m depth, the scattering of the points is small. Hence, for these depths, the local-scale velocity variations explain the meso-scale dispersivity to a large extent. At 0.45-m depth, however, the meso-scale dispersivity could not be explained by this CV_V (r² = 0.10), probably because velocity variations were too small. The linear regressions between ^m_L and CV_V yields slopes of 11.4 and 18.1 cm for the 0.15- and 0.75-m depth respectively. The increase of this slope parameter with depth is consistent with an identified SC transport process at the meso-scale level.

View larger version (26K): [in this window] [in a new window]   Fig. 9. Meso-scale dispersivity vs. coefficient of variation of the local scale velocities squared. Triangles, squares and plus signs stand respectively for dispersivity estimated at 0.15-, 0.45-, and 0.75-m depth.

It is worth noting that the CV_V is completely independent of ^l (not shown here). This could confirm our assumption that this reflects a intrinsic characteristic of our subsoil sample. However, as already said, our CV_V estimate is may be too rough to show the actual dependence.

Meso-Scale ^m_L Estimated from Lognormal Distribution of Local Scale ^l_L
From our results, it appears that the mixing regime is CD at the local scale and SC at the meso-scale. Therefore, we can model transport at the meso-scale by considering the monolith as an ensemble of parallel stream tubes through which the solute is transported following a CD process. Assuming that ^l and D^l are independent and log-normally distributed, Jury and Roth (see problem 4.7. in Jury and Roth, 1990) defined ^m_L for such a system as:

[17]

with µ the mean and ² the variance of D^l or ^l. We estimated µ_D, ²_D, µ_V and ²_V from the D^l and ^l distribution. Note that, for a log-normal distribution, the term in Eq. [17] is the square root of the coefficient of variation of ln. The second term therefore reflects the variability in the local velocity field and is depth dependent.

Comparison between ^m_L calculated from Eq. [17] and ^m_L estimated from averaged BTC and outflow BTC is given at Fig. 10 for three depths and 11 flow rates. Mean local scale dispersivity values <^l_L> are also given to show the specific contribution of local scale to ^m_L. Note that, from Eq. [17], three outliers came from a bad estimation of the variance. Those are not shown in Fig. 10 and the determination coefficient calculated without those three points was significant (r² = 0.66). This confirms that the stream tube theory allows the scaling of the observed solute transport in the monolith to a reasonable extent.

View larger version (17K): [in this window] [in a new window]   Fig. 10. Meso-scale dispersivity estimated from Eq. [17] (closed circles, right axis) versus meso-scale dispersivity obtained from averaged BTCs. On the left axis, open circles represent the local scale dispersivity.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

1. The mixing length l*, and therefore the apparent transport process, depends on the measurement scale. At the local scale (4 x 10^-2 m²), l* was lower than 0.3 m whereas at the meso-scale (0.5 m²) l* exceeded 1 m. It confirms that both l* and the mixing regime depend on the scale;
   
2. The flow rate does not affect ^m_L or ^l_L. We attribute this insignificance to the slight layering which prevents the formation of preferential flow paths and stabilize both the CV_v and the l*. However, the adopted experimental protocol, in particular the control of the bottom boundary condition during the BTC experiments, did not permit us to explore a large range of water content (see Table 2);
   
3. The local scale dispersivity deviates from the meso-scale ^m_L by a factor of 2 at 0.75-m depth. This can be related to the higher heterogeneity encompassed in the solute flow domain at the meso-scale, which increases the CV_v. Hence several TDR probes at a given horizontal depth need to be installed in large soil columns if a reliable estimate of the meso-scale dispersivity is needed;
   
4. The meso-scale ^m_L can be linearly related to an estimate of the coefficient of variation of the apparent velocity with a slope proportional to the depth. This is characteristic of a SC mixing process;
   
5. The meso-scale solute transport in an undisturbed soil monolith could be reasonably well described by a stream tube model in which transport in each stream tube follows a CD process;
   
6. However quantitative relationships between structure, flow rate, and dispersivity have still to be found and more research is needed to assess the transport regime at a larger scale and relate the mixing length to the scale of interest.
   

	APPENDIX 1

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
Constraints on CV_V(z) for a Stochastic-Convective Process
Considering the spread of arrival time ²_t around the average value E(t) at a given depth z, an alternative definition of the longitudinal dispersion is (Simmons, 1982):

[A1]

where ²_t the variance of the solute travel time pdf. Combining with Eq. [8], it yields:

[A2]

For a SC process, the solute travel time pdf scales with depth as follows (Jury and Roth, 1990):

[A3]

Inserting Eq. [A3] in [A2] yields:

[A4]

and hence:

[A5]

	Appendix 2

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 
Mean Time-Integrated Normalized Resident Concentrations
Large scale resident concentration at a depth z can be obtained by averaging n local BTCs at locations x and depth z as (Vanderborght et al., 1996):

[A6]

By inserting Eq. [A6] in the definition of C^rt* (Eq. [11]), if n = 3, it yields:

[A7]

C^r(z,x,t) is usually defined as (see e.g., in Vanclooster et al., 1995)

[A8]

with K(z) a calibration constant and T_C, the temperature correction factor at 25°C. After plugging Eq. [A8] in [A7], it gives Eq. [14].

Received for publication May 28, 2002.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND ANALYSIS CONCLUSIONS APPENDIX 1 Appendix 2 REFERENCES

 

• Bear, J. 1972. Dynamics of fluids in porous media. Dover publications, New York.
• Beven, K.J., D.E. Henderson, and A.D. Reeves. 1993. Dispersion parameters for undisturbed partially saturated soil. J. Hydrol. (Amsterdam) 143:19–43.
• Buchter, B., C. Hinz, M. Flury, and H. Fluhler. 1995. Heterogeneous flow and solute transport in an unsaturated stony soil monolith. Soil Sci. Soc. Am. J. 59:14–21.[Abstract/Free Full Text]
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