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a Soil and Water Science Dep., Univ. of Florida, Gainesville, FL 32611
b Dep. of Crop and Soil Environmental Sciences, Virginia Tech, Blacksburg, VA 24061-0404
* Corresponding author (whuaguo{at}mail.ifas.ufl.edu)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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Abbreviations: 1-D, one dimensional • BTC, breakthrough curve • CDE, convection-dispersion equation • Pe, peclet number
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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This solute transport system has been modeled by specifying appropriate boundary conditions (Cauchy, Dirichlet, etc.) for the solute concentration. The modeled temporal distribution of the tracer concentration at the outlet boundary (the BTC) is generally dependent on the type of boundary condition that is specified (van Genuchten and Alves, 1982; Parker and van Genuchten, 1984; Barry and Sposito, 1988). Similarly, the experimental BTC has also been shown to depend on the application method (Kluitenberg and Horton, 1990). However, the traditional column systems do not simulate miscible displacement scenarios with multiple inputs of the same (or different) solute at several locations in the column. Such scenarios represent miscible displacement with different initial solute distributions, and can arise in studies involving reactive solute transport, designing remediation of contaminated soils and aquifers, and characterizing transport medium heterogeneity. Miscible displacement tests in laboratory columns designed to create different initial solute distributions can benefit and facilitate investigations on these topics.
In one such test, Delay et al. (1997) used a single syringe to directly inject concentrated tracer at the center of the cross-section at an arbitrary location along the length of a Plexiglas column packed with artificial porous media, and used this system to simulate a 1-D initial value solute transport problem with a "solute free" inlet boundary. The transport domain was conceptually discretized into cells and a particle tracking method was developed for solving the governing solute transport equations. The results showed that their injection method was a useful addition to the traditional approach for simulating 1-D solute transport in laboratory columns, especially for scenarios involving transport of solutes from multiple sources in the space domain.
However, several questions arise when applying the injection method of Delay et al. (1997) to column experiments to estimate parameters characterizing solute transport for a given 1-D solute transport model. Some examples are:
The objective of this paper was to seek answers to these questions using both numerical and experimental tests, with special attention to quantifying the influence of initial solute distribution and specification of the outlet boundary condition on observed BTCs and estimated transport parameters. The experimental tests were conducted using a laboratory column system designed to simulate 1-D miscible displacement of an initial solute distribution.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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One syringe of each injection unit was filled with tracer solution, and the other was filled with tracer-free background solution. As shown in Fig. 1, an injection unit consisted of Syringe A, Syringe B, Tubing A, Tubing B, Tubing C, a clamp, and a needle. Assuming Syringe A was filled with a known amount of the tracer solution, and then Syringe B was filled with the background solution.
Before injection of the tracer, the background solution was pumped through the column until it was fully saturated and the required steady fluid flow velocity was achieved. During this time the background solution was allowed to fill the balance tubing, which was then closed with a clamp. At the time of injection, the pump was stopped, and the clamp on the balance tubing was removed. Then the clamp on Tubing C was removed, and tracer solution was injected into the column passing through the Tubing A and C. Some of the background solution in Syringe B was then injected into the column passing through Tubing B and C, to flush any tracer solution remaining in Tubing C and in the needle, into the column. Tubing C was then closed with the clamp. This procedure was repeated as rapidly as possible for all the remaining three injection units. After injection, the balance tube was once again closed, and the pump was restarted. The time when the pump was restarted was taken as zero time for analysis of the initial value solute transport problem.
Any residual tracer solution in Tubing A was flushed back into Syringe A using the background solution remaining in Syringe B. Syringe A was then removed and the contents transferred into a volumetric flask for quantifying the mass of residual tracer in the Syringe A. The solute mass difference relative to the mass before injection was the solute mass injected for that injection unit. These masses for the four injection units were summed to obtain the total mass injected.
Keeping both the outlet and the balance tubing open during injection minimized the variability in the location of the center of mass of the injected solute, increasing the accuracy of taking the plane of injection as the origin of the spatial coordinate for simulating 1-D transport. The need for the balance tubing was confirmed by visually comparing the initial solute distribution after injection with and without use of the balance tubing. Without the balance tubing, the injected solute peak was observed to deviate to the outlet side of the injection location, regardless of whether the discharge tubing was left open or closed during injection.
After tracer was injected into the column, it was assumed to mix completely and instantaneously in the cross-section with the background solution, resulting in an approximate 1-D solute concentration distribution along the length of the column, centered at the plane of injection. To test the effect of solute distribution after injection on the BTCs, the distribution after injection was assumed to be either a normal distribution with a standard deviation of 
, or a uniform distribution with a width of w. A normal distribution approaches a Dirac delta function as 
0.
Initial Value Problem and Solution
The CDE for 1-D conservative solute transport under steady, saturated flow conditions is:
 | [1] |
In Eq. [1] C represents the resident concentration. However, the effluent solution concentration represents the flux concentration rather than the resident concentration at the outlet boundary. The distinction between the flux concentration and the resident concentration is only important when the concentration gradient is relatively large at the outlet or the dimensionless time (= v2t/D and equivalent to the dimensionless Peclet number Pe = vL/D in the column) is low (Parker and van Genuchten, 1984). However, this is not the case for most column experiments. Therefore, the effluent solution concentration was assumed to be the same as the resident concentration at the outlet boundary when applying Eq. [1].
If the outlet boundary condition for Eq. [1] is assumed to be an infinite boundary condition, miscible displacement in the experimental column system can be mathematically described as an initial value solute transport problem in a 1-D infinite domain. Since Eq. [1] is linear, the solute concentration at the outlet boundary (the BTC) can be expressed by the principle of superposition as:
 | [2] |
(x) at l or:
 | [3] |
is the porosity of the medium. When the analytical solution of the CDE is derived for a Dirac delta source initial condition in an infinite 1-D solute transport domain, f(L – l, t) is:
 | [4] |
If the solute distribution after injection is assumed to be a Dirac delta function at the injection location, which is defined as the origin of the coordinate system (l = 0 in Eq. [4]), then Eq. [4] reduces to:
 | [5] |
However, the solute distribution after injection is not a Dirac delta function. The distribution has some width w. If the injection location is used as the coordinate origin and the solute distribution at t = 0 is assumed to be a uniform distribution over the interval [–w/2, w/2]. Equation [2] becomes:
 | [6] |
If the solute distribution at t = 0 is assumed to be a normal distribution with a standard deviation of , Eq. [2] becomes:
 | [7] |
Because the Dirac delta function is defined as a normal distribution in which the standard deviation approaches to zero, Eq. [7] reduces to Eq. [5] as 0.
Numerical Tests
Outlet Boundary Conditions
To quantify the influence of various outlet boundary conditions on BTCs, numerical tests were conducted to compare BTCs of an initial solute distribution taken as a Dirac delta function (Wang, 2002). In these tests, the BTCs were generated using the particle tracking solution of the CDE subject to the zero concentration outlet boundary condition (Barry and Sposito, 1988), the zero gradient finite outlet boundary condition, and the zero gradient infinite boundary condition (Leij and van Genuchten, 2000) for different column Peclet numbers.
Initial Solute Distribution
The influence of the initial solute distribution on BTCs was investigated by assuming that the width w of the uniform distribution, and the interval 2 of the normal distribution, was equal to 5, 10, 20, and 40% of the column length denoted as L and defined as the distance between the injection point to the discharge point. An additional treatment was included with the injected solute distribution taken as a Dirac delta function. In this case, w = 2 
0% of the column length, and this treatment was denoted as 
0%. Solute transport conditions were characterized using column Pe of 1, 10, 100, or 1000. The BTCs were obtained from Eq. [5] for the 0% treatment, Eq. [6] for the uniform distributions, and Eq. [7] for the normal distributions.
Equations [5], [6], and [7] were all based on the solutions of the CDE subject to an infinite outlet boundary condition. To characterize the effects of combining various outlet boundary and initial solute distribution assumptions on BTCs, the particle tracking method (Uffink, 1985; Delay et al., 1997) was used to generate BTCs for solute transport with an initial condition equal to a 1-D normal distribution. The outlet boundary conditions included zero concentration (Barry and Sposito, 1988); zero-gradient, finite length; and zero-gradient, infinite length (Leij and van Genuchten, 2000). The interval 2 of the normal distribution was assumed as 5, 10, 20, and 40% of L and solute transport condition was characterized as Pe = 5. It can be expected that such combined effects, if any, would not be obvious when Pe is large (i.e., >60).
Column Experiments
In the laboratory column system already described (Fig. 1), the distance between the injection point and the discharge outlet of the column was 37.5 cm. The column was uniformly packed with different sizes of glass beads (A-Glass Soda-Lime silica glass beads, Potters Industries Inc., Valley Forge, PA). Diameters of the glass beads were 1.1, 1.6, 1.9, 2.2, 2.6, 3.1, 3.7, and 5.1 mm. The number distribution of different sized glass beads was 49.8, 15, 10.3, 7, 5.2, 3.7, 4.1, and 4.5%, respectively. The fractal dimension of this number distribution, calculated using fragmentation theory (Rieu and Sposito, 1991; Wang, 2002), was 2.02. Porosity of this medium was 0.37.
Aqueous fluorescein (Content >95%; Sigma-Aldrich Co., St. Louis, MO) solution (3 mM fluorescein + 7 mM NaCl) was used as the nonreactive tracer. The tracer mass was approximately 0.0015 mmol for all injections. To exclude light from the column system and prevent potential photobleaching of the fluorescein, both the column and the fraction collector were covered. The outflow samples collected in the fraction collector were kept in the dark and analyzed within 24 h. The background or displacement solution was 10 mM NaCl. Injection volumes were 0.5, 2, 4, and 8 mL. Each experimental test was replicated four times. All experiments were conducted using the same column packed only once with the glass beads. The concentration of fluorescein in the samples was determined using a spectrophotometer (Beckman-Coulter model DU646) at 480 nm. Preliminary tests were made to ensure that light was being adequately excluded, and that there was no photobleaching under the experimental conditions.
An attractive feature of using fluorescein as the tracer and glass beads as the porous medium, was that the width of the solute distribution after injection could be visually estimated. Tests showed that it was very easy to visually distinguish the color difference between fluorescein-free background solution and a 0.001-µM fluorescein solution. The 0.001-µM fluorescein solution was 3000 times lower than the concentration of the fluorescein solution injected for the 0.5-mL treatment. The injection widths for the 0.5-, 2-, 4-, and 8-mL treatments were visually estimated as 1, 2, 3, and 6 cm, respectively. The visually estimated width was taken as equivalent to w if a uniform distribution was assumed, or equivalent to 2 if a normal distribution was assumed. The four estimated widths were about 2.6, 5.3, 8, and 16% of L. The numerical tests discussed above had shown that, for a given value of w or 2, if the solution that was developed using the Dirac delta function for the initial solute distribution after injection was applicable for a high Pe, the same solution would also be applicable for a lower Pe. Consequently, only the results of the numerical tests for Peclet numbers between 100 and 200 were experimentally verified.
The CDE solution given by Eq. [5] for the Dirac delta source initial condition was fitted to the experimental BTCs. The fitted parameters of the solute transport models were estimated using a nonlinear least-squares optimization method based on the Levenberg–Marquardt procedure (Marquardt, 1963). The algorithm used for fitting was taken from Toride et al. (1995), and coded using C++. The fitted parameter used for evaluation of the injection treatments was the dispersivity (
). Because convection was dominant in the experiments, the molecular diffusion was neglected and was calculated as = D/v.
Multiple comparisons among the fitted dispersivities for different injection treatments were conducted using LSD at the 5% significance level.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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Initial Solute Distribution
Results of numerical tests to determine the effect on the BTCs assuming a 1-D normal distribution for the tracer immediately following injection are presented in Fig. 2 for column Peclet numbers of 1, 10, 100, and 1000. The results assuming a uniform distribution were practically identical to those shown in Fig. 2. For both the uniform and normal initial solute distribution, when Pe was equal to 1, the BTCs for treatments 0, 5, 10, 20, and 40% were almost identical. As Pe was increased to 10, the BTC for the 40% treatment was somewhat different from the BTCs for the other four treatments while the BTCs for treatments 0, 5, 10, and 20% were identical. For Pe = 100, the BTCs of the treatments of 0, 5, and 10% were same, but were different than those of the treatments of 20 and 40%. Also, the BTCs for the latter two treatments were clearly different from each other. At Pe = 1000, the BTCs of all five treatments were clearly different from each other.
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is no more than 10% of L, the Dirac delta source solution of the initial value problem (Eq. [5]) is adequate to calculate the BTCs. Also, this solution may provide a good approximation, when the Pe is in the range 100 < Pe < 200 and w or 2 is <5% of L. Although the numerical results assuming uniform and normally distributed initial solute profiles were practically identical, the latter may more closely represent the experimental distribution of the solute mass after injection. Since the solute is not injected instantaneously from the four units, some degree of random mixing of the injected tracer and background solutions would be expected before initiating miscible displacement in the column. This would tend to create a normal rather than a uniform initial distribution.
The BTCs for the 0, 5, 10, 20, and 40% treatments calculated using the particle tracking method for miscible displacement of a 1-D normal initial solute distribution subject to different boundary conditions, showed that the BTCs were different for different boundary conditions (Fig. 3). However, for a given boundary condition the BTC for the 0% treatment was practically identical for cases where 2 
10% of L, regardless of the value of Pe. The BTC calculated using Eq. [5] was identical to the BTC calculated using the particle tracking method for the 0% treatment with a zero gradient infinite boundary condition. This result served to verify the computational procedures used in the particle tracking method.
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) of the initial solute distribution relative to the column length (i.e., distance between injection and outlet points) and Pe.
Column Experiments
The results of the foregoing numerical tests showed that, under certain conditions, the Dirac delta source solution of the 1-D initial value problem was appropriate. These numerical results could only be verified by experiments in which the initial solute distribution after injection could be accurately characterized. However, such characterization was virtually impossible since the solute distribution after injection is not static and any sampling would disturb the distribution. Since it was not possible to directly measure the initial solute distribution after injection, every effort was made to inject the tracer as evenly as possible over a narrow cross-section of the column. Indirect verification of the numerical findings was attempted with miscible displacement experiments in which replicated BTCs were generated with injected tracer volumes of 0.5, 2, 4, and 8 mL. As detailed in the Materials and Methods section above, the use of glass beads and fluorescein in the column experiments facilitated this effort since it permitted visual approximation of the width of the tracer distribution.
If the foregoing numerical results were correct, then experimental BTCs would be almost identical for one or more of the specific initial solute distributions when the Pe was between 100 and 200. To compare the four tracer volumes, the Dirac delta source solution of the 1-D initial value problem (Eq. [5]) was fitted to the experimental BTCs to obtain values of the dispersivity (). The dispersivity values for the 0.5-, 2-, and 4-mL injection volumes were not significantly different from each other at the 5% significance level (Table 1). However, they were all significantly different from that of the 8-mL treatment. For each injection volume, the BTC with fitted closest to the mean value was selected, and these are presented in Fig. 4. These experimental findings supported the findings from the numerical tests. Taken together, these results confirmed that it was appropriate to use the 1-D Dirac delta source solution of the initial value problem when the injection width was no more than 5% of L , and the column Pe number was not too large (<200).
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Nevertheless, it is recognized that the prototype column system does not explicitly verify the accuracy of the 1-D assumption with respect to solute distribution immediately after injection. Therefore, further studies would be useful to examine this issue with respect to the column system described herein. The possibility of a 3-D initial solute distribution might be best addressed by devising some technique for directly sampling of the initial solute distribution. Although practically difficult, one possibility is to immerse the column in liquid nitrogen immediately after injection to halt diffusion, and take samples of the frozen solution. A more practical approach would be to compare BTCs obtained by variations of the injection assembly. For example, BTCs could be compared for each of four combinations of Pe values (10, 100) and injection volumes (0.5 and 4 mL) using one, two, three, and four injection needles. Breakthrough curves that closely match the 1-D Dirac delta source solution of the initial value problem would indicate how well each combination approximated the 1-D initial solute distribution. A third possibility is to use numerical tests designed to compare BTCs generated using 1-D and 3-D assumptions of the initial solute distribution.
Taken in their entirety, the findings of the numerical and experimental tests indicated that the column system devised for this study could be used to simulate miscible displacement of initial solute distributions in laboratory columns. Experimental conditions were defined for which one can use a 1-D Dirac delta source solution to the initial value problem posed by miscible displacement tests in this column system. For a given outlet boundary condition, the injected solute should be distributed over no >5% of the distance between the injection and discharge points, and the column Pe should be <200.
Received for publication February 13, 2003.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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This article has been cited by other articles:
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  H. Wang, N. Persaud, and X. Zhou Specifying Scale-dependent Dispersivity in Numerical Solutions of the Convection-Dispersion Equation Soil Sci. Soc. Am. J., September 20, 2006; 70(6): 1843 - 1850. [Abstract] [Full Text] [PDF]
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