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DIVISION S-1—SOIL PHYSICS

Hydrodynamic Dispersion in an Unsaturated Dune Sand

^a Dep. of Agricultural Sciences, Saga Univ., Saga 840-8502, Japan
^b Arid Land Research Center, Tottori University, Hamasaka 1390, Tottori 680-0001, Japan
^c George E. Brown Jr. Salinity Laboratory, 450 West Big Springs Road, Riverside, CA 92507

^* Corresponding author (nobuo{at}cc.saga-u.ac.jp)

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Solutes spread relative to the mean displacement position during water flow in soils as a result of meandering through the (partially) saturated pore complex. Spreading is characterized by the hydrodynamic dispersion coefficient in the convection-dispersion equation (CDE). This coefficient has been extensively studied for saturated soils. In this study hydrodynamic dispersion coefficients for nonaggregated dune sand were determined as a function of volumetric water contents, , ranging from saturation to 0.08 cm³cm^-3 in columns of 5-cm diam. and 25- to 40-cm length. Unit-gradient flow experiments were conducted to measure solute breakthrough curves (BTCs) using four-electrode salinity probes at several column depths. Transport parameters for the CDE and the mobile-immobile model (MIM) were determined by optimizing analytical solutions to observed BTCs. A maximum dispersivity, , of 0.97 cm was found at = 0.13, whereas for saturated flow 0.1 cm irrespective of pore-water velocity ranging from 208 to 5878 cm d^-1. For the MIM, the mobile water fraction, _m/, gradually deceased from almost unity at saturation to a minimum of 0.85 at = 0.15 followed by a slight increase with further desaturation. The exchange time between the mobile and immobile phases, 1/, was 0.1 to 0.2 d for > 0.15 presumably because of the relatively homogeneous flow with convective solute mixing. For lower , the exchange became much slower since flow predominantly occurs in water films enveloping sand particles. The Peclet number for molecular diffusion, P_e, decreases as the role of transverse diffusion increases at lower because of smaller v and thinner water films while the resistance increases for solute exchange between mobile and immobile phases. These combined effects lead to a maximum dispersivity value at intermediate water contents in the case of the nonaggregated dune sand.

Abbreviations: BTC, breakthrough curve • CDE, convective-dispersive equation • MIM, mobile-immobile model

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
DURING WATER FLOW in porous media, dissolved substances tend to spread because of hydrodynamic dispersion, which includes molecular diffusion and mechanical dispersion (Bear, 1972). Mechanical dispersion occurs because water flow varies in magnitude and direction in soil pores as a result of meandering through the complex pore structure (Perfect and Sukop, 2001). The degree of spreading is related to the distribution of the water velocity at the pore scale, the degree of solute mixing because of convergence and divergence of flow paths, and (transverse) molecular diffusion (Bolt, 1979; Leij and van Genuchten, 2002).

Frequently, the solute flux because of mechanical dispersion is described as a Fickian process. The convenient similarity between diffusion and mechanical dispersion has led to the common practice of combining these processes into a single process of hydrodynamic dispersion (Leij and van Genuchten, 2002). This practice deserves careful scrutiny because mathematical equivalence does not necessarily imply physical similarity. The solute flux, J_S (ML^-2T^-1), may be defined as the following sum of convective and hydrodynamic dispersive fluxes:

[1]

where c is volume-averaged or resident concentration (ML^-3), z is position or depth (L), D is the hydrodynamic dispersion coefficient (L²T^-1), is the volumetric water content (L³ L^-3), and J_w is the Darcy water flux (LT^-1).

In saturated soils, the dispersion coefficient is given by (Bear, 1972):

[2]

where the first term, D_e, is an effective diffusion coefficient (L²T^-1) while the second term describes the coefficient of mechanical dispersion, where is referred to as the dispersivity, v (= J_w/) denotes the pore-water velocity (LT^-1), and n is an empirical constant.

The role of molecular diffusion can be evaluated with the Peclet number of molecular diffusion,

[3]

where d is the mean soil particle size or some other characteristic length of porous medium (Kutílek and Nielsen, 1994). The molecular diffusion term in Eq. [2] is of the same order of magnitude as for the mechanical dispersion term for 0.3 < P_e < 5. As P_e increases (5 < P_e < 20), the contribution of diffusion to hydrodynamic dispersion becomes negligible but transverse diffusion, which is inversely related to mechanical dispersion in the concept of Taylor dispersion (Taylor, 1953), should be considered. Typical values for n are in the range between 1 and 1.2 (Bear, 1972). At higher P_e, the dispersion coefficient, D, exhibits an almost linear increase with pore-water velocity, v (i.e., n = 1) in the case of nonaggregated sands or glass beads (Bear, 1972; Bolt, 1979). The dispersivity, (= D/v), is assumed to be an intrinsic soil property for saturated flow.

Hydrodynamic dispersion in unsaturated soils is more complicated than in saturated soils. As water content decreases, the pore-water velocity decreases and the geometry of the liquid phase in water-conducting pores changes with less opportunity for mixing and increased tortuosity. The dispersion coefficient will depend on both water content and velocity, which may be expressed similar to Eq. [3]:

[4]

In the case of nonaggregated media such as glass beads and sands, greater solute spreading and longer tailing have been observed for BTCs at lower water contents (Gupta et al., 1973; Krupp and Elrick, 1968). Hence, larger values for (= D/v) have been found for unsaturated than for saturated conditions (De Smedt and Wierenga, 1984; Maraqa et al., 1997; Matsubayashi et al., 1997; Padilla et al., 1999). De Smedt and Wierenga (1984) explained greater dispersion in unsaturated glass beads with the MIM. These authors found that the mobile water content, _m, increased linearly with the total water content (_m/ = 0.853), while the coefficient for mass transfer between the mobile and immobile phases, , increased proportionally with the pore-water velocity, v. Maraqa et al. (1997) found greater dispersivity for unsaturated sandy soils, but they did not observe tailing of the BTCs. Padilla et al. (1999) demonstrated that for an unsaturated sand the parameters of the CDE and MIM are not only a function of soil properties but also of water content. Matsubayashi et al. (1997) evaluated the mixing length for the unsaturated dispersion based on the standard deviation of v for different using the capillary retention model.

Despite the aforementioned and other studies, there is a lack of consistent and comprehensive data sets to elucidate unsaturated dispersion. The difficulty to establish uniform unsaturated flow conditions may result in inaccurate estimations of transport parameters. Few data exist for lower water contents because the concomitant low flow rates lead to time-consuming displacement experiments. In most studies effluent concentrations were used to determine BTCs. Apparatus-induced dispersion may result in biased transport parameters (James and Rubin, 1972) while a single BTC for an experiment provides an insufficient basis to assess the transport model.

The main objective of this study is to examine dispersion over a wide range of water contents under unit-gradient flow conditions in a dune sand. For this purpose, we determined in situ BTCs by inferring total resident concentrations from the bulk soil electrical conductivity measured with four-electrode salinity probes at several depths in columns packed with a Tottori dune sand. Values for parameters of CDE and MIM were determined by optimizing analytical solutions of these transport models to the observed BTCs. Solute mixing in the unsaturated sand will be discussed in terms of the behavior of dispersivity as a function of water content.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Displacement Experiment
A well-sorted dune sand from Tottori, Japan, was packed uniformly at a dry bulk density, _b = 1.67 g cm^-3 in columns of 5-cm diam. and 25- to 40-cm length. The dune sand had an average particle size of 0.28 mm, and 95% of the sand consisted of particles with a diameter between 0.149 to 0.5 mm. To minimize filter clogging during the displacement experiments, we reduced fine particles (<0.02 mm) in the sand from 2.5 to 1% by washing the sand with distilled water. The saturated hydraulic conductivity was 550 cm d^-1. Figure 1 shows the water retention curve, (h), measured with a hanging water column starting at saturation. From the curve it is evident that the sand has a low air-entry value and a narrow range in pore sizes.

View larger version (16K): [in this window] [in a new window]   Fig. 1. Water retention curve of Tottori dune sand for drying from saturation. Fitted line is the van Genuchten retention function, ( - r)/(s - r) = [1 + (|h|)n]-1+1/n, with s = 0.35, r = 0.05, = 0.05, and n = 5.2.

View larger version (24K): [in this window] [in a new window]   Fig. 2. Experimental setup for unsaturated displacement experiments.

A similar soil column was used for saturated flow. A fine-mesh screen was used at the bottom instead of the glass filter. After saturating the soil column from the bottom, constant saturated flow rates, ranging from J_w = 73 to 2059 cm d^-1, were established by adjusting the hydraulic head at the top of the column using a Mariotte bottle.

The soil electrical conductivity, EC_a, was measured with four-probe salinity sensors inserted horizontally in the column at three to five depths (Rhoades and Oster, 1986; Inoue et al., 2000). Each probe consists of four stainless steel rods with a 1.6-mm diam. and a 20-mm length. The two inner and two outer rods were spaced at 8 and 16 mm, respectively. The ratio of the electric current flowing through the outer electrodes to the voltage difference between the two inner electrodes was measured using a 21 X data logger with a multiplexer (Campbell Scientific Inc., Logan, UT). The soil water pressure head, h, was monitored with 2-mm diam. and 10-mm long microtensiometers connected to pressure transducers (Fig. 2). After establishing steady-state flow, the influent solution was changed from concentration c₀ to c₁ to allow determination of the BTCs. To minimize concentration effects on water flow, the difference between c₀ and c₁ was relatively small (c₀ = 0.04 or 0.05 M CaCl₂ and c₁ = 0.05 or 0.07 M CaCl₂) while still allowing accurate BTC measurements with the four-probe electrodes. We used slightly higher concentrations for lower water contents (c₀ = 0.1 M CaCl₂) when the four-probe readings tend to be lower (Inoue et al., 2000).

Four-probe EC_a readings are proportional to the electrical conductivity of the soil solution, EC_w, at a particular water content (Rhoades et al., 1989). Since a linear relationship is generally observed between the resident solute concentration of soil water, c, and EC_w, c is also proportional to EC_a as long as is constant:

[5]

where A and B are constants, and t is time (T). When continuous step input (switch from c₀ and c₁) is applied, the two constants of Eq. [5] can be determined by two sets of c and EC_a values corresponding to c₀ and c₁. If the duration time, t₀, of a pulse application (from c₀ and c₁, and then back to c₀) is not long enough to reach a constant EC_a value corresponding to c₁, we assume full mass recovery to determine the constants, that is, the four-probe at depth z detects all the input solute mass to determine the constants (Mallants et al., 1996).

Transport Models
The observed BTCs were analyzed in terms of the conventional CDE and the MIM. We assumed that the four-probe measurements yielded resident-type concentrations. The one-dimensional CDE based on the convective-dispersive flux of Eq. [1] for a nonreactive solute in a homogeneous soil is written as (Kutílek and Nielsen, 1994):

[6]

The MIM assumes that the liquid phase in soil pores can be partitioned into mobile (flowing) and immobile (stagnant) regions; = _m + _im. Solute exchange between the two regions is modeled as a first-order process. The one-dimensional MIM for a nonreactive solute is described as (Coats and Smith, 1964; van Genuchten and Wierenga, 1977):

[7]

[8]

where the subscripts m and im refer to the mobile and immobile regions, respectively, v_mm is equal to the volumetric water flux density J_w (= v), and is a first-order mass transfer coefficient (T^-1) governing the rate of solute exchange between the mobile and immobile regions.

The four-probe sensor presumably measures a total resident concentration, c_T, defined as

[9]

Values for transport parameters were determined with the nonlinear least-squares optimization program CXTFIT2.1 (Toride et al., 1995) using analytical solutions of CDE and MIM subject to the following initial and boundary conditions:

[10]

[11]

[12]

where L is the column length (L), D is equal to D_mm/ for the MIM, and t_o is the duration of the pulse application (T) with t < t_o effectively representing a step application. The use of an infinite outlet condition for a finite soil column is reasonable as long as processes beyond z = L have a negligible effect on the soil concentration at the point of measurement (van Genuchten and Parker, 1984).

If we adopt the common practice of setting n equal to 1 and neglecting the diffusion term in Eq. [2] and [4], the dispersivity can be defined for the CDE as

[13]

The dispersivity can be regarded as a dispersion or mixing length, which is influenced by the microscopic velocity distribution and the opportunity for mixing of solute in different soil pores (Bolt, 1979). The effective overall dispersion coefficient for the MIM, D_MIM, may be defined as (De Smedt and Wierenga, 1984; Parker and Valocchi, 1986):

[14]

The first term on the right-hand side of Eq. [14] represents the contribution from dispersion in the mobile phase. The second term is proportional to v² and includes the contribution of transverse diffusion to solute mixing (Bear, 1972; Bolt, 1979). For sufficiently long travel times, transport may be determined by the CDE using D_MIM according to Eq. [14] (Leij and Toride, 1998; Valocchi, 1985). The effective dispersivity in terms of the MIM, _MIM, may be defined similar to Eq. [15]:

[15]

where _m is the dispersivity in the mobile phase (= D_m/v_m) and the second term is an "immobile" dispersivity or mixing length. When solute mixing between the mobile and immobile phases is the dominant mechanism for dispersion, in other words, transverse diffusion is dominant compared to mechanical dispersion, the dispersivity may increase with the pore-water velocity because of the second term in Eq. [15] (Nielsen et al., 1986).

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Unsaturated Water Flow
Since hydrodynamic dispersion in an unsaturated soil is a function of and v, it is imperative to establish unit-gradient flow to achieve uniform water content throughout the column during the experiments (Maraqa et al., 1997). Figure 3 presents the distribution of and h as a function of depth for one of the experiments using a water flux of 20.2 cm d^-1 and an imposed head (suction) of around -40 cm at the bottom. The water contents were determined gravimetrically afterwards. Values for and h were approximately 0.1 cm³ cm^-3 and -30 cm, respectively, throughout the column, which corresponds to the water retention curve shown in Fig. 1.

View larger version (16K): [in this window] [in a new window]   Fig. 3. Illustration of volumetric water content, , and soil water pressure, h, profiles during unit-gradient flow.

Figure 4 summarizes the v– relationship for all experiments in this study. Note that the water flux, J_w, given by v in Fig. 4 is equal to the unsaturated conductivity because of the unit-gradient flow condition. The water content, as determined by sectioning the soil column, was not perfectly uniform (Fig. 3). Therefore we used the fitted v to infer an average water content for each flow condition based on = J_w/v for the CDE and = J_w/(v_mm/) for the MIM.

View larger version (14K): [in this window] [in a new window]   Fig. 4. Volumetric water content, , as a function of log-scale unsaturated pore-water velocity, v, during unit-gradient flow.

View larger version (40K): [in this window] [in a new window]   Fig. 5. Observed and fitted breakthrough curves at different depths: (a) saturated flow ( = 0.34), (b) unsaturated flow ( = 0.11).

Table 2 provides CDE and MIM parameters for 11 unsaturated experiments obtained from BTCs at center locations along the column. BTCs obtained near the surface (z < 5 cm) may be affected by the more heterogeneous flow conditions because water entered at the center of the soil surface, while BTCs near the bottom are affected by irregular water contents because of possible clogging of the filter. We therefore determined parameter values by simultaneously fitting the solution of the CDE or MIM to the BTCs for all or part of the center locations. Data for Exp. 5 in Table 2, for example, were obtained from BTCs at depths 9.0, 15.1, and 21.1 cm (Table 1).

View larger version (16K): [in this window] [in a new window]   Fig. 6. Dispersivity, , as a function of log-scaled pore-water velocity, v, for saturated and unsaturated flow conditions.

View larger version (17K): [in this window] [in a new window]   Fig. 7. Dispersivity, , as a function of volumetric water content, , for saturated and unsaturated flow conditions. Data from Padilla et al. (1999) for an unsaturated sand with an average particle size of 0.25 mm are also plotted.

Mobile-Immobile Model
The MIM was applied to describe BTCs for both saturated and unsaturated conditions. Because transport in the saturated sand was described well by the CDE, the mobile fraction, _m/, is close to unity (Table 1). Padilla et al. (1999) observed the same for a saturated sand (_m/ = 0.99). Under these conditions, the transfer coefficient, , is no longer relevant and estimates for this value are unstable (Comegna et al., 2001). Since the estimated values are not accurate and the BTC may not be properly evaluated with the analytical solution of the MIM (Maraqa, 2001; Toride et al., 1995), we will not further pursue the description of BTCs in terms of the MIM for saturated conditions.

For unsaturated conditions, the estimated _m/ and values appear reasonable for all cases shown in Tables 1 and 2. In Fig. 8 , we compare the effective dispersivity for the MIM, _MIM, calculated from estimated MIM parameters according to Eq. [15], with for the CDE. Although _MIM was slightly greater for lower water contents, the similarity of the two dispersivities suggests that it is reasonable to partition hydrodynamic dispersion into the two mechanisms of solute mixing defined by Eq. [16] and [17] using the MIM parameters. Figure 8 also includes the dispersivity of the mobile phase, _m (= D_m/v_m), for unsaturated conditions. For < 0.17, _m was around 0.2 to 0.3 cm and between 0.1 to 0.15 cm for > 0.17. The difference between _MIM (or ) and _m, that is, the immobile mixing length, can be regarded as the contribution to mixing because of transverse diffusion between mobile and immobile phases. The greater difference for < 0.17 suggests that the second term in Eq. [15] becomes the dominant mechanism for mixing during unsaturated conditions.

View larger version (17K): [in this window] [in a new window]   Fig. 8. Comparison between the effective dispersivity based on MIM parameters, MIM, according to Eq. [15], and for the CDE for unsaturated flow as a function of volumetric water content, . Dispersivities in the mobile phase, m, for unsaturated conditions are also plotted.

View larger version (18K): [in this window] [in a new window]   Fig. 9. Mobile and immobile fraction, m/ and im/, as a function of volumetric water content, , for unsaturated flow conditions. Note that m/ + im/ = 1. Solid line was fitted by eye based on our measurements and dotted line corresponds to m/ = 0.853 as reported by De Smedt and Wierenga (1984) for unsaturated glass beads.

View larger version (12K): [in this window] [in a new window]   Fig. 10. Time scale for exchange between the mobile and immobile phases, 1/, and the Peclet number of molecular diffusion, Pe, as a function of volumetric water content, , for unsaturated flow conditions.

Unsaturated Flow and Solute Mixing
Examining the water retention curve (Fig. 1), we postulate that most of the soil water is held in soil pores for > 0.15 and that these pores will be empty for < 0.15. Figure 11 provides a schematic illustration. As stated previously, the equilibrium CDE should be applied at saturation and _m/ is close to unity (Table 1). As decreases, air starts entering the pores and _m/ gradually deceases because the flow path becomes more tortuous for lower . In other words, the relative amount of stagnant water will increase as decreases up to around = 0.15. When decreases further ( < 0.15), there will be few pores that are completely filled with water. Hence water flow occurs increasingly in films enveloping the soil particles. As the thickness of this water film decreases, v will decreases exponentially as decreases further. At low values for , the flow will again be more homogeneous in terms of _m/. Figure 9 suggests that the relative amount of stagnant water decreases as v becomes smaller throughout these films.

View larger version (47K): [in this window] [in a new window]   Fig. 11. Schematic illustration of flow domains in a nonaggregated sand for different ranges in volumetric water content, . For > 0.15: relatively homogeneous flow with rapid convective solute mixing in water-filled soil pores occurs. For < 0.15: slow solute mixing primarily by transverse diffusion in water film enveloping the soil particles.

[16]

where D_o is the aqueous diffusion coefficient and is soil porosity (L³ L^-3). The characteristic length, d, in Eq. [3] may be expressed using the concept of the equivalent cylindrical capillary radius (Or and Wraith, 2002):

[17]

where h is the soil pressure head (cm). Table 2 also includes P_e for each experiment using v_m and the value for d according to Eq. [17] for a certain _MIM using the fitted water retention curve shown in Fig. 2. We calculated D_e assuming D_o = 1 cm² d^-1 (Astle and Beyer, 1985) and = 0.35.

Figure 10 includes P_e as a function of . The value for P_e ranges between 10 and 90, and increases with . When 1/ is the greatest, that is, at = 0.08, P_e has a minimum of around 11. As stated for saturated media, the role of transverse diffusion increases as P_e decreases (Bear, 1972). Notice that the dispersion coefficient, D or D_MIM, is still more than one thousand times greater than the diffusion coefficient, D_e in Table 2. As discussed, transverse diffusion becomes more important for spreading at lower and the difference between and _m will increase as is demonstrated for < 0.17 in Fig. 8.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Solute displacement experiments were conducted during unit-gradient flow to determine in situ BTCs for a wide range of water contents in a Tottori dune sand. Total resident concentrations were measured with four-electrode salinity probes at several depths of packed soil columns. Transport parameters for the CDE and MIM were determined by optimizing analytical solutions of these models to measured BTCs. Values for the transport parameters agreed well regardless of the depth at which the BTC was determined. The comparison of parameters at several depths to assess transport model and parameters is an obvious advantage of in situ measurement of BTCs.

For unsaturated conditions, the dispersivity, , had a maximum of 0.97 cm at = 0.13, whereas for saturated flow was around 0.1 cm regardless of the pore-water velocity, v. For the MIM, the difference between and the dispersivity of the mobile phase, _m, increased for < 0.17. This indicates that transverse diffusion becomes the dominant mechanism for solute mixing at lower . The mobile fraction, _m/, gradually decreased from almost unity at saturation to 0.85 at = 0.15, and slightly increased again with further desaturation. The time for solute exchange between the mobile and immobile phases, 1/, was 0.1 to 0.2 d for > 0.15, and increased substantially as decreased further.

For this nonaggregated dune sand, we postulate that the flow conditions are different for higher and lower water contents. As decreases, the pore-water velocity decreases exponentially. For > 0.15, greater _m/ and smaller 1/ occur because of relatively homogeneous flow with rapid convective solute mixing in soil pores. For < 0.15, a greater 1/ was found because the resistance to solute transfer between mobile and immobile phases increases. The Peclet number for molecular diffusion, P_e, decreases as the role of transverse diffusion increases for lower because of smaller v and thinner water films in soil pores. We observed that these effects result in a maximum dispersivity for unsaturated flow in a dune sand at intermediate water contents.

The dependency of solute mixing on water contents will be different for aggregated soils than for the nonaggregated sand used in this study. Further measurement of dispersivity over a range of water contents and for different soils are needed to further explore the effects of flow conditions and soil structure on solute mixing in unsaturated soils. Monitoring in situ concentrations in well-controlled unit-gradient flow conditions, such as used in this study, offers great potential for such investigations.

	ACKNOWLEDGMENTS

 
We thank Dr. Sho Shiozawa of Tokyo University for his technical assistance and Azusa Tsujita for her help with the dispersion experiments. We also appreciate the constructive comments of two anonymous reviewers.

Received for publication February 15, 2002.

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