Soil Science Society of America Journal 66:1087-1094 (2002)
© 2002 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Inspectional Analysis of Convective-Dispersion Equation and Application on Measured Breakthrough Curves
M. K. Shukla*,a,
F. J. Kastanekb and
D. R. Nielsenc
a School of Natural Resources, The Ohio State Univ., 422A Kottman Hall, 2021 Coffey Road, Columbus, OH 43210
b Dep. of Hydraulics, Univ. of Agricultural Sciences, 18-Muthgasse 1190, Vienna, Austria
c Dep. of Land, Air and Water Resources, Univ. of California, 1004 Pine Ln., Davis, CA 95616-1728
* Corresponding author (shukla.9{at}osu.edu)
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ABSTRACT
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Several miscible displacement experiments were carried out in three 10-, 20-, and 30-cm mollisol and antisol filled soil columns. A strong linear relationship between pore-water velocity (v) and apparent diffusion coefficient (D) was obtained for both soil columns (r2 > 0.92). We also derived the nondimensional laws for equilibrium adsorption convective-dispersion equation (CDE) using inspectional analysis, which reduced the physical constants and variables in CDE from seven to four nondimensional-
quantities. After scaling, the times of effluent arrival were nearly the same and all the breakthrough curves (BTCs) coalesced into a very narrow region of scaled time.
Abbreviations: BTC, breakthrough curve • CDE, convective-dispersion equation • D, apparent diffusion coefficient • v, pore-water velocity
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INTRODUCTION
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DIMENSIONAL AND INSPECTIONAL analysis are useful tools for determining the physically significant scale factors. The empirical scale factors can be obtained through functional normalization technique. Although scaling techniques can be applied in many ways, the principle remains the formulation of relevant equations with smallest possible number of variables from the known physical laws and boundary conditions (Simmons et al., 1979; Tillotson and Nielsen, 1984; Sposito and Jury, 1985; Shook et al., 1992).
During the past several decades, large numbers of miscible displacement experiments have been carried out in laboratory soil columns and in fields (Nielsen and Biggar, 1961; Krupp and Elrick, 1968; Gaudet et al., 1977; van Genuchten and Wierenga, 1977; van Gunachten et al., 1977; Rao et al., 1980; Nkedi-Kizza et al., 1984; Smettem, 1984; de Smedt and Wierenga, 1984; de Smedt et al., 1986; Selim et al., 1987; Seyfried and Rao, 1987; Li et al., 1994). Most of the above studies mainly concentrated on v > 0.1 cm h-1. Few studies examine the variation of transport parameters across a wide range of v, from 0.02 to 2.6 cm h-1 using different displacement lengths, and soils. The linear relationship between v and D in saturated soil columns has been reported by de Smedt and Wierenga (1984). We conducted 56 displacement experiments through 10-, 20-, and 30-cm loam and sandy loam soil columns. The purpose of this study was to reduce the one-dimensional CDE and the corresponding initial and boundary conditions to a few nondimensional- terms by inspectional analysis. The second objective was to scale time of effluent arrival of measured BTCs using the nondimensional scale factors or terms. The third objective was to derive relationships between v and D from both loam and sandy loam soil columns.
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THEORY
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Miscible Displacement Equation
Several appropriate one-dimensional miscible displacement equations solved for different boundary conditions have been used to describe the experimental observations of this laboratory study. They included convective-diffusion and mass transfer equations based on adsorbed solutes being in equilibrium and others not in equilibrium with the solid soil phase as well as the inclusion of two-site and mobile-immobile water assumptions. The applicability of such equations has been published elsewhere. For the purpose of this scaling study, we focus our attention on the one-dimensional equilibrium adsorption equation for conditions of steady water flow
 | [1] |
where R is the retardation factor, C is the solution concentration (M L-3); D is the apparent diffusion coefficient (L2 T-1), t is the time (T), v is the average pore-water velocity (L T-1) and x is the distance from the inflow boundary in the direction of flow (L). For linear adsorption, R = 1 + (
KD) 
-1 where is the soil bulk density (M L-3), KD is the distribution coefficient equal to S C-1, S is the adsorbed concentration (M M-1) and is the volumetric soil water content (L3 L-3).
The initial and boundary conditions used for the solution of Eq. [1] are:
 | [2] |
where C0 is the solute concentration at the inlet boundary. For the experimental results of this study, a constant flux boundary condition at the inlet provided nearly identical results.
The solution of Eq. [1] (Lapidus and Amundson, 1952) is
 | [3] |
For a continuous application of C0 at the inlet (t < t0), the last two terms of Eq. [3] are ignored.
Scaling by Inspectional Analysis
Scale factors are simple conversion factors, which relate characteristics of one system to corresponding characteristics of another. Scale factors can be derived from different kinds of dimensional analyses. Here we use inspectional analysis to reduce the miscible displacement equations along with its corresponding initial and boundary conditions to a few nondimensional terms while eliminating as many physical constants and variables as possible. The stepwise procedure for obtaining the nondimensional terms is described by Hellums and Churchill (1961), and Tillotson and Nielsen (1984).
Note that the variables in Eq. [1] are C, x, and t and the parameters or physical constants are KD, , , D, v, and C0. The variables are made nondimensional by dividing by arbitrary reference quantities C0, x0, and t0. Hence, substituting the nondimensional variables (C* = C/C0, x* = x/x0, and t* = t/t0) into Eq. [1] and [2] yields
 | [4] |
and the modified initial and boundary conditions
 | [5] |
Equating the coefficients Dt0/x20 and vt0/x0 to unity, the values of t0 and x0 are obtained as follows.
 | [6] |
and
 | [7] |
Substituting Eq. [6] and [7] into Eq. [4] yields the nondimensional equation
 | [8] |
which is identical to the classical dimensionless equation obtained when a Peclet number P = vx0/D has been substituted into Eq. [1]. From inspection of Eq. [8] and the initial and boundary conditions [5], it can be seen that C* depends only on t* = t/t0 = v2t/D, x* = x/x0 = xv/D, and R. The general form of Eq. [1] becomes
 | [9] |
where G is some function which exactly describes the interrelationship between the terms 1 = C/C0, 2 = v2t/D, 3 = xv/D, and 4 = R. From Eq. [9] it is can be inferred that whenever two or more soil systems have similar values for 2, 3, and 4, they will have similar solute concentration versus soil depth or time curves. The scale factors are obtained by equating corresponding terms. From the original Eq. [1], it can be seen that C is dependent upon the six quantities - t, R, D, x, v, and C0. Equation [9] shows that nondimensional concentration is dependent on three quantities—2, 3, and 4—a result in direct agreement with the Buckingham Pi theorem. The four nondimensional quantities were reduced to four linear equations by taking the logrithm of both sides of the terms. The coefficient matrix for the parameters and variables in four nondimensional groups was obtained in the same manner as Shook et al. (1992). The rank of the coefficient matrix, obtained through elementary column operations (Rawlings, 1988), was four, which was equal to the number of nondimensional scaling factors.
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MATERIALS AND METHODS
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Two soils—a loam and a sandy loam—were selected for experimentation. The loam, identified as an Entisol, was collected from the 0- to 15-cm depth from the Farmers Training Center at Pyhra, Lower Austria, and the sandy loam, identified as a Mollisol, was collected from the 40- to 70-cm depth from the experimental farm of the University of Agricultural Sciences, Vienna located at GrossEnzersdorf, near Vienna. The loam had an average particle diameter, d, of 0.0158 mm and that of the sandy loam was 0.0508 mm. The separate portions of each of the air-dried soils that passed through a 2-mm sieve were prepared into laboratory soil columns and also analyzed for their physical and chemical properties. Soil was packed into 10-cm i.d. acrylic plastic cylinders having lengths of 10, 20, and 30 cm. Care was taken to follow exactly the same procedure for packing all of the soil columns. The porosity and bulk density were quantitatively determined for each soil column. Each soil column was slowly saturated from the bottom with 0.1 M CaBr2. The steady-state flow required for obtaining a given pore-water velocity was adjusted by measuring the effluent volume with respect to time. The effluent solutions were collected at fixed time intervals in small plastic bottles. Displacement experiments using MgCl2 for several different pore-water velocities were performed on each column starting with the lowest pore-water velocity. For a step input, the displacing solution was switched back to the connate solution when no more connate ions were detected in the effluent (Fig. 1)
. For a pulse input, 
300 mL of displacing solution was followed by the connate solution (Fig. 2)
. All the experiments were conducted at a temperature controlled to 20 ± 2°C. Concentrations of Cl- and Br- were determined by titration. An accurate value of soil water content for each soil column was determined gravimetrically at the cessation of each experiment. Fifty-six displacement experiments were conducted using 13 different soil columns. A more elaborate description of the experimental details is available (Shukla, 1998; Shukla et al., 2000).
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Fig. 1. Observed and fitted Cl breakthrough curves using the equilibrium convective-dispersion equation for a slow, medium, and fast pore-water velocity through a 10-cm loam soil column.
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Fig. 2. Observed and fitted Cl breakthrough curves using the equilibrium convective-dispersion equation (a) for a slow, medium, and fast pore-water velocity in 10-cm loam soil columns, and (b) from 10-, 20-, and 30-cm loam soil columns for a pore-water velocity of 0.3 cm h-1.
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RESULTS
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Analysis of Measured Chloride Breakthrough Curves
Mass Recoveries
The amount of Cl pulse applied was independent of column length, pore-water velocities, and soil type; and was 300 mL. The concentration versus time BTCs showed the progressive attenuation of the initial concentration when 300 mL of solute was displaced through greater macroscopic lengths (Fig. 2b). The mass recoveries from both loam and sandy loam soil columns were very high (always >97% of the applied pulse of solute). The experiments for very slow water velocities in loam soil columns for both solute boundary conditions (step and pulse) showed early arrival of Cl in the effluent solution and long flat BTCs (Shukla et al., 2000). These experiments illustrated the contribution of molecular diffusion in the transport process. A detailed theoretical analysis of each experiment and the measured BTC is presented in Shukla et al. (2000).
Parameter Estimation
Measured Cl concentrations in the effluent for each of the 56 experiments plotted as a function of time were fitted to Eq. [3] using program CFITIM (van Genuchten, 1981) to ascertain the values of D and the retardation factor, R. These values together with measured values of soil bulk density (), soil water content (), and average v are given in Table 1 for a step solute input into two 10-cm long columns of loam for a range of v from 0.025 to 2.60 cm h-1. For this range of v, values of D increase approximately one order of magnitude while values of R manifest a slight increasing trend with a mean of 1.05. As illustrated in Fig. 1 for the smallest, intermediate, and largest values of v for Column 2, each of the 17 experimental BTCs were nicely described by Eq. [3] using the values given in Table 1.
Measured and calculated terms for pulse solute inputs into 10-, 20-, and 30-cm columns of the loam and sandy loam soils are given in Tables 2 and 3, respectively. In Table 2, it is evident that the bulk densities of the columns were slightly larger than those in Table 1 yielding slightly smaller corresponding soil water contents. On the other hand, values of R are near unity and not consistently related to either v or . In all cases, theoretical curves matched the experimental results—Fig. 2a shows BTCs for three velocities for a 10-cm column of loam while Fig. 2b shows BTCs for a velocity of 0.3 cm h-1 for three column lengths. The information in Table 3 regarding the sandy loam is similar—values of D are related to v, and slight deviations of soil water content and retardation factor are generally, but not consistently related to soil bulk density. For all sandy loam columns, theoretical curves matched the experimental results—Fig. 3a
shows BTCs for three velocities for a 20-cm column of sandy loam while Fig. 3b shows BTCs for a velocity of 0.3 cm h-1 for the three column lengths of 10, 20, and 30 cm.
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Fig. 3. Observed and fitted Cl breakthrough curves using the equilibrium convective dispersion equation (a) for a slow, medium, and fast pore-water velocity in 20-cm loam soil columns, and (b) from 10-, 20-, and 30-cm sandy loam soil columns for a pore-water velocity of 0.3 cm h-1.
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Relations Between Apparent Diffusion Coefficient and Pore-water Velocity
Within the range of average pore-water velocities studied, molecular diffusion dominates the dispersion of the Cl at smaller displacement velocities and gives way to convective dispersion at greater velocities. Hence, for relatively small average pore-water velocities we expect D to have values close to that of the diffusion coefficient of Cl, Do, in the soil solution, and to be only somewhat dependent of v. At relatively larger velocities, D should be strongly related to v. The results in Table 4 are consistent with those expectations. Linear regressions between D and v derived from the 56 experiments summarized in Table 4 are all highly significant with values of R2 ranging between 0.92 and 0.99. However, not shown in Table 4 for the smaller velocities when diffusion dominates, the regression between D and v using the five values of v < 0.075 cm h-1 (Table 1) was not significant (R2 = 0.49).
The relations between D/Do and Peclet number (vd/Do) given as solid lines in Fig. 4
satisfy the equation
 | [10] |
where the value of Do is estimated from Do = 0.66Dm with Dm being the Cl diffusion coefficient for free solution (2.15 x 10-5 cm2 s-1). The results from the seven loam columns and the six sandy loam columns of this study fall between the earlier reported values derived from measurements made in a natural undisturbed field soil by Biggar and Nielsen (1976) and for laboratory columns of graded sands and other single-grained materials reported by Pfannkuch (1962). Here, we estimated the values of Do for the loam and the sandy loam to be 0.0222 and 0.0216 cm2 h-1, respectively, using the average value of measured for the 35 and 21 experiments of each soil, respectively. In Table 5, it can be seen that values of m increase with decreasing values of average particle diameter, d, while values of n range between 1 and 2. In the loam and sandy loam soils as well as the field soil, decreasing average particle diameter (increasing clay content) is associated with soil structure. The loam has relatively large pores as a result of microaggregates, and the sandy loam, although containing less clay than the loam, nevertheless has large pores also associated with its microaggregates as well as those associated with its greater sand content. The field soil manifests the greatest value of m because of its large pore-size distribution owing to its high clay content, its aggregation, and its natural field structure.
Scaling of Measured Chloride Breakthrough Curves
Having obtained the parameters (Tables 1–3) of the theoretical BTCs that adequately describe the measured BTCs, nondimensional- terms from Eq. [1] were ascertained. Compared with the impact of the large range of pore-water velocities on the positions of the BTCs, we ignore the small variations of retardation factor, R, and consider its value constant for the scaling process. By taking advantage of the strong linear relationships between D and v given in Table 4, the 2 term which contains the time of effluent arrival can be used for scaling the BTCs. Scale factors were obtained for a given displacement length by equating the corresponding 2 terms of each BTC with that of an arbitrary reference curve. For the reference curve for each displacement length, we selected the BTC having a pore-water velocity near the mean pore-water velocity for which displacements were made. From the scale factors the scaled time of effluent arrival was calculated for each BTC.
Measured and scaled BTCs with respect to the time of effluent arrival from the 10-cm columns of loam leached with a step input of Cl are presented in Fig. 5
for the 17 different pore-water velocities. Depending on the pore-water velocity, the times required to measure the entire BTC ranged from as few as 6 h to as many as 780 h. After scaling, the times of arrival are nearly the same for all curves with the 17 BTCs coalesced into a very narrow region of scaled time. From these data using only one soil length, the scaling procedure appears to be reasonably successful.
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Fig. 5. Measured Cl breakthrough curves in 10-cm loam soil columns for step Cl application (a) theoretical and (b) scaled.
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Displacing a pulse input of Cl at different pore-water velocities through progressively longer columns of the loam provides a more critical examination of the scaling process. In Fig. 6a
, we note that each BTC reaches about the same relative maximum for each of the values of v and that all seven curves are coalesced nicely in Fig. 6b. These results appear equivalent and consistent with those in Fig. 5 when a step input of Cl was diplaced through the same length column (10 cm). However, in Fig. 7 and 8
, the effectiveness of the scaling procedure appears less satisfactory. It is apparent that the relative maximum Cl concentration begins to decrease with decreasing values of v for the 20-cm long columns, and is especially noticeable for the 30-cm columns.
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Fig. 6. Measured Cl breakthrough curves in 10-cm loam soil columns for pulse Cl application (a) theoretical and (b) scaled.
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Fig. 7. Measured Cl breakthrough curves in 20-cm loam soil columns for pulse Cl application (a) theoretical and (b) scaled.
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Fig. 8. Measured Cl breakthrough curves in 30-cm loam soil columns for pulse Cl application (a) theoretical and (b) scaled.
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The results of displacing a pulse input of Cl at different pore-water velocities through progressively longer columns of the less structured sandy loam appear more consistent with the proposed scaling process. In Fig. 9, 10, and 11
, we note for each soil length that each BTC reaches about the same relative maximum for each of the values of v and that all seven curves are coalesced to about the same degree. Regardless of the magnitude of v, for each column length, the relative maximum Cl concentration is constant (0.95, 0.9, and 0.8 for x = 10, 20, and 30 cm, respectively).
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Fig. 9. Measured Cl breakthrough curves in 10-cm sandy loam soil columns for pulse Cl application (a) theoretical and (b) scaled.
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Fig. 10. Measured Cl breakthrough curves in 20-cm sandy loam soil columns for pulse Cl application (a) theoretical and (b) scaled.
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Fig. 11. Measured Cl breakthrough curves in 30-cm sandy loam soil columns for pulse Cl application (a) theoretical and (b) scaled.
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CONCLUSIONS
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The equilibrium CDE fitted BTCs matched the measured BTCs very well (r2 always >0.97). The retardation coefficient remained independent of v, , , and soil type and remained close to one. The D remained less dependent on v for lower average pore-water velocities and for relatively larger velocities, D increased linearly with v for both the soils. The nondimensional quantities obtained by inspectional analysis reduced the number of independent variables in CDE from seven to four. The scale factors obtained from 2 terms reduced the times of effluent arrival and the measured BTCs were coalesced into a very small region of scaled time. In general, the proposed scaling process appeared more consistant for BTCs from less structured sandy loam soil columns. The relative maximum Cl concentration was also constant for these sandy loam columns regardless of the magnitude of pore-water velocity.
Received for publication March 22, 2001.
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M. K. Shukla, T. R. Ellsworth, R. J. Hudson, and D. R. Nielsen
Effect of Water Flux on Solute Velocity and Dispersion
Soil Sci. Soc. Am. J.,
March 1, 2003;
67(2):
449 - 457.
[Abstract]
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ABSTRACT
INTRODUCTION
