Environ. Sci. Technol. 2005, 39, 7164-7169
Errors in NAPL Volume Estimates Due to Systematic Measurement Errors during Partitioning Tracer Tests MICHAEL C. BROOKS† AND W I L L I A M R . W I S E * ,‡ U.S. Environmental Protection Agency, Kerr Research Center, P.O. Box 1198, Ada, Oklahoma 74820, and Department of Environmental Engineering Sciences, University of Florida, Gainesville, Florida 32611-6450
During moment-based analyses of partitioning tracer tests, systematic errors in volume and concentration measurements propagate to yield errors in the saturation and volume estimates for nonaqueous phase liquid (NAPL). Derived expressions could be applied to help practitioners bracket their estimates of NAPL saturation and volume obtained from such tests. In practice, many of these effects may be overshadowed by other complications experienced in the field. Errors are propagated for systematic constant (offset) volume, proportional volume, and constant (offset) concentration errors. Previous efforts to quantify the impact of these errors were predicated upon the specific assumption that nonpartitioning and partitioning masses were equal. The current work relaxes that assumption and is therefore more general in scope. Through the use of nondimensional concentration, systematic proportional concentration errors do not affect the accuracy of the method. Specific consideration needs to be given to accurate flow measurements and minimizing baseline concentration errors when performing partitioning tracer tests in order to prevent the propagation of systematic errors.
Introduction The use of partitioning tracers to characterize subsurface nonaqueous phase liquid (NAPL) was introduced in the petroleum industry over three decades ago (1-3), and the technique was later adopted as a contaminant-source-zone characterization technique (4-12). Partitioning tracers are retarded relative to nonpartitioning tracers (or tracers with significantly less partitioning behavior) due to their interaction with NAPL; the NAPL saturation can be estimated based on the extent of retardation. Partitioning tracer tests are based on several assumptions, and they can be summarized broadly as follows: retardation of the partitioning tracer results solely from the NAPL (or at the least, other factors affecting retardation can be accurately quantified), partitioning tracers are in equilibrium contact with all the NAPL within the swept zone, and the partitioning relationship between the NAPL and the tracer can be accurately described by a linear equilibrium relationship (4). Uncertainty in tracer-based * Corresponding author phone: (352)846-1745; fax: (352)392-3076; e-mail:
[email protected]. † U.S. Environmental Protection Agency. ‡ University of Florida. 7164
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 39, NO. 18, 2005
predictions can result when these assumptions are not sufficiently satisfied; this type of uncertainty is based upon the modeling process. A number of papers have been published addressing the validity of these underlying assumptions (12-17). The other type of uncertainty is associated with measured values used in the partitioning tracer technique (i.e., measurement uncertainty). Dwarakanath et al. (14) briefly addressed systematic errors under the restriction that the injected mass of the nonpartitioning and partitioning tracers are equivalent. This paper addresses how systematic measurement errors propagate to yield errors in NAPL volume determination but is not limited to cases in which the injected tracer masses are equal. The authors’ following paper (18) quantifies how random measurement errors affect uncertainty in NAPL volume estimates and builds upon a previous contribution that quantifies how random measurement errors affect uncertainty in the calculated moments of breakthrough curves (19). It is important to document error propagation through methods aimed at quantifying largely inaccessible pollutants so that results from such methods may be properly interpreted.
Theory An experimentally measured breakthrough curve (BTC) is composed of a series of n volume and concentration measurement pairs
V1, ...Vi-1, Vi, Vi+1, ... Vn, and c1d, ...ci-1d, cid, ci+1d, ...cnd (1) where Vi is the ith cumulative volume measurement [L3], and cid is the ith dimensioned concentration measurement [ML-3]. Each dimensioned concentration, cid, may be converted to a dimensionless concentration, ci, by dividing by the tracer injection concentration (cd0 ):
ci )
cdi
(2)
cd0
In general, the absolute kth volumetric moment of the breakthrough curve, mk [L3(k+1)], is defined as
mk )
∫
∞
-∞
cVkdV
(3)
where c is the nondimensional concentration, V is the produced volume [L3], and the kth normalized moment (µk) is defined as the ratio of the absolute kth moment to the zeroth moment. Equation 3 can be approximated using the Trapezoidal rule n-1
mk =
∑∆V V c k
i
i
(4)
i)1
where ∆Vi ) (Vi+1 - Vi) is the change in cumulative volume over the ith interval, and Vkci ) (Vikci + Vi+1kci+1)/2. Note that the forward differencing scheme starts with i ) 1. Haas (20) discussed the difference between approximating moments using averages of Vkci as indicated in eq 4 and the products of the averages of Vki and ci, concluding that eq 4 produced a less biased estimate of the moments and therefore should be used in preference to the alternative formulation. Brooks (21) demonstrated that the difference between the two approaches became more pronounced as fewer data points were used to perform the integration. 10.1021/es048739m CCC: $30.25
2005 American Chemical Society Published on Web 08/17/2005
The pulse-corrected, normalized first moment, µ′1 [L3] is defined as
m 1 Vp µ′1 ) m0 2
(5)
where Vp ) tracer pulse volume [L3]. The normalized first moment for a nonreactive tracer is a measure of the volume through which the tracer was carried. This is generally referred to as the mean residence volume, or for groundwater tracer tests, the swept pore volume. Calculation of NAPL Volume through a Partitioning Tracer Test. For a partitioning tracer, the retardation factor, R, is defined as
R)
µ′R1 µ′NR 1
(6)
R 3 3 where µ′NR 1 [L ] and µ′1 [L ] are pulse-corrected, normalized first moments for the nonreactive and reactive tracers, respectively. The partitioning tracer may be retarded relative to the nonpartitioning tracer due to adsorption onto the aquifer matrix (background retardation). If background retardation, RB, has been measured, it can be taken into account using
R)
µ′R1 µ′NR 1
- (RB - 1)
(7)
where RB is defined as the ratio of the pulse-corrected normalized first moment of the partitioning tracer in the absence of NAPL to the pulse-corrected normalized first moment of the nonpartitioning tracer. Assuming a linear equilibrium partitioning coefficient (KNW) and that the pore space is occupied by water (or air) and NAPL only, the saturation, S, can be calculated from
S)
R-1 R - 1 + KNW
(8)
and the volume of NAPL, VN [L3], is given by
S VN ) µ′NR 1 1-S
t + a) - (V ti + a) ) ∆V ti ∆Vi ) (V i+1
(11)
Consequently, in light of eq 4 with k ) 0:
m0 ) mt0
(12)
Therefore, systematic constant errors in volume measurements have no effect on the absolute zeroth moment. The average volume-concentration product over the ith interval is
1 t Vci ) [(V i+1 + a)ci+1 + (V ti + a)ci] ) Vcti + aci (13) 2 where jci is the average concentration over the ith interval. The absolute first moment is therefore n-1
m1 )
∑∆V (Vc + acj ) ) m t i
i
i
t 1
+ amt0
(14)
i)1
Combining equations 12 and 14 and accounting for the tracer pulse volume gives the pulse-corrected, normalized first moment:
µ′1 )
mt1 + amt0 mt0
-
Vp 2
(15a)
Neglecting any systematic errors in the tracer pore volume results in
µ′1 ) µ′t1 + a
(15b)
The retardation factor can now be expressed as
R(a) )
µ′R,t 1 + a
(16)
µ′NR,t +a 1
Saturation, S, therefore becomes
(9)
The effects of systematic errors can be determined by deriving the moment equations using eq 4, modified to include systematic errors in volume and concentration measurements. The following analyses first examine systematic errors in volume measurements (with no consideration of errors in concentration measurements) and then turn to examine systematic errors in concentration measurements (with no consideration of errors in volume measurements). The partitioning coefficient is assumed to be known without error, and the tracers are considered in context of the local equilibrium assumption. Systematic Constant Errors in Volume Measurements. The first class of errors to be studied is that of a constant offset in volume measurement. Such an error would result from an inaccurate recording of the commencement of the tracer release, for instance. Let “a” represent a systematic constant error in volume measurements, such that
Vi ) V ti + a
ith and (i+1)th measurements is
(10)
where V ti is the true volume. From here, a value superscripted with a “t” will represent a true value. The value of a is constrained such that a g -Vmin, where Vmin is the smallest volume measurement. The change in volume between the
S(a) )
R(a) - 1 R(a) - 1 + KNW
(17)
Likewise the NAPL volume is a function of a
S(a) VN(a) ) (µ′NR,t + a) 1 1 - S(a)
(18)
Systematic Proportional Errors in Volume Measurements. This section examines the impact of systematic proportional errors in volume measurements. Such errors could arise from an improperly calibrated flow meter, for instance. The relationship between a proportional, systematic error in volume measurements, R′, and the true volume measurements can be expressed as
Vi ) V ti + R′V ti ) V ti(1 + R′)
(19a)
For simplicity, let R ) (1+R′), therefore
Vi ) RV ti
(19b)
The difference in the ith and (i+1)th cumulative volume measurement is t ∆Vi ) RV i+1 - RV ti ) R∆V ti VOL. 39, NO. 18, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
(20) 9
7165
and the absolute zeroth moment becomes
m0 )
Rmt0
cdi ) cd,t i + b (21)
Likewise, the injection concentration, cd0 , becomes
The average volume-concentration product over the ith interval is
Vci )
1 t ci+1 + RV tici] ) RVcti [RV i+1 2
(22)
cd0 ) cd,t 0 + b
ci )
n-1
∑
R∆V tiRVcti ) R2mt1
(23)
i)1
R2mt1 Rmt0
-
Vp Rmt1 Vp ) 2 2 mt
(24)
ci ) cti
mt1 mt0
-
Vp 2
(25a)
Vp mt1 ) t - µ′t1 2 m
(25b)
φ1 )
[ ]
(32a)
φ2 )
[
]
(32b)
ci ) φ1 cti + φ2
(33)
Substituting eq 25b into eq 24 and rearranging result in
mt1 (R mt0
- 1) ) µ′t1 +
mt1 R′ mt0
µ′R,t 1
+
R(R′) ) µ′NR,t + 1
mR,t 1 R′ R,t m0 mNR,t 1 R′ mNR,t 0
VN(R′) ) µ′NR,t + 1
)
S(R′) 1 - S(R′)
(34)
The absolute, zeroth moment equation then becomes n-1
∑∆V (φ cj + φ ) i
t 1 i
(35a)
2
i)1
(27)
Assuming no errors in volume measurements, eq 35a becomes
m0 ) φ1mt0 + VTφ2
(28)
(35b)
where VT is the total cumulative volume. Similarly, the absolute first moment is n-1
The NAPL volume can then be expressed as the product of the right-hand side of eq 26 (for the nonreactive tracer) and S(R′)/(1 -S(R′))
mNR,t 1 R′ mNR,t 0
b cd,t 0 + b
cji ) φ1cjti + φ2
m0 )
The saturation, S, then becomes
R(R′) - 1 S(R′) ) R(R′) - 1 + KNW
cd,t 0 + b
Note that φ1 and φ2 are constant for each tracer. Applying eq 33, the average of the ith and (i+1)th concentration is
(26)
Equation 26 neglects a systematic error in the tracer-pulse volume. Using eqs 6 and 26, the retardation factor becomes
cd,t 0
Therefore, eq 31b becomes
0
m1 )
1
∑∆V 2(V i
t i+1(φ1ci+1
+ φ2) + Vi(φ1cti + φ2)) )
i)1
n-1
φ1mt1 + φ2
∑∆V Vh
i i
(36)
i)1
(29)
Systematic Constant Errors in Concentration Measurements. This section addresses constant offset errors in concentration measurements that could arise, for instance, from improper calibration of an analytical instrument or perhaps failure to properly account for a background concentration. The ith measured concentration, cid, in terms of a systematic constant error, b [ML-3], and the true concentration, cid,t, is expressed as 9
(31b)
b cd,t 0 + b
+
cd,t 0 + b
and
which can be rearranged as
7166
[ ][ ]
0
µ′t1 )
(
(31a)
cd,t 0 + b
cd,t 0
For simplicity, let
The true normalized first moment can be expressed as
µ′1 ) µ′t1 +
cd,t i + b
t d,t By using cd,t i ) ci c0 , eq 31a can also be expressed as
Combining equations 5, 21, and 23 and accounting for the tracer-pulse volume give the pulse-corrected, normalized first moment
µ′1 )
(30b)
Note that the value of b is constrained such that b g -cmin, where cmin is the smallest concentration measurement. Therefore, the nondimensional concentration, ci, is
so, the absolute first moment is therefore
m1 )
(30a)
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 39, NO. 18, 2005
where V h i is the average volume of the ith interval. The first normalized moment in terms of the systematic constant concentration error and the true values is n-1
φ1mt1 + φ2 µ′1 )
∑∆V Vh
i i
i)1
φ1mt0 + VTφ2
The retardation factor is
-
Vp 2
(37)
NR R R R(φNR 1 ,φ2 ,φ1 ,φ2 ) ) n-1 R φR1 mt,R 1 + φ2
∑∆V Vh
i i
i)1
µ′R1
Vp 2
R φR1 mt,R 0 + VTφ2
)
µ′NR 1
-
(38)
n-1 t,NR φNR + φNR 1 m1 2
∑∆V Vh
i i
i)1
-
Vp
t,NR φNR + VTφNR 1 m0 2
2
The saturation, S, is NR R R S(φNR 1 ,φ2 ,φ1 ,φ2 ) )
NR R R R(φNR 1 ,φ2 ,φ1 ,φ2 ) - 1 NR R R R(φNR 1 ,φ2 ,φ1 ,φ2 ) - 1 + KNW
(39)
The NAPL volume is obtained by applying eqs 9, 37 (for nonreactive tracer) and 39:
[
NR R R VN(φNR 1 ,φ2 ,φ1 ,φ2 ) ) n-1
t,NR φNR + φNR 1 m1 2
∑∆V Vh
i i
i)1
-
]
Vp
NR R R S(φNR 1 ,φ2 ,φ1 ,φ2 )
2 1 - S(φNR,φNR,φR,φR) 1 2 1 2
t,NR φNR + VTφNR 1 m0 2
(40)
Systematic Proportional Errors in Concentration Measurements. Now the effects of systematic proportional concentration errors are examined. Improper calibration of an analytical instrument or systematic errors in sample handling could be likely causes for such errors. Let β represent a systematic proportional concentration error, such that
FIGURE 1. The effects of systematic errors on retardation (solid line), NAPL saturation (short-dashed line), and NAPL volume (longdashed line) illustrated for the case of (a) systematic constant volume errors, (b) systematic proportional volume errors, and (c) systematic constant concentration errors. The retardation factor was 1.5 in each case, and the BTCs were composed of 100 data points.
cdi ) βcd,t i
represents an analytical solution based on the step change in injected tracer concentration and was used to generate a solution based on a finite pulse of tracer (i.e., c(0, 0 e t e tp) ) c0, c(0, t > tp) ) 0, where tp is the pulse duration [T]) by superposition, lagging one step-input solution by the tracer pulse-input volume and subtracting it from another. The nondimensional pulse length (defined as τp ) vtp/L) was 0.1, the retardation was 1 for the nonpartitioning tracer and 1.5 for the partitioning tracer, and the Peclet number was 10. A total of 100 volume-concentration data points were used to represent the BTCs. Systematic Constant Volume Errors. The impacts of systematic constant errors in volume measurements on retardation, saturation, and NAPL volume are illustrated in Figure 1a. The error shown on the abscissa in Figure 1a is expressed as a percent of the pore volume, as predicted by the nonpartitioning normalized first moment. Systematic constant errors in volume measurements impact the retardation estimate to a lesser extent because the volume error occurs in both the numerator and denominator and the saturation estimate to a greater extent because the magnitude of the partitioning coefficient relative to the error reduces the effect of the error in the denominator of eq 8. The retardation error does grow with increases in the actual value of retardation. The error in saturation is relatively insensitive to retardation with a slight decrease in error for higher retardation values. Interestingly, the final error in NAPL volume is relatively small due to the offsetting errors in saturation and the normalized first moment. This result agrees with that suggested by Dwarakanath et al. (14) but formally reflects that the error in the NAPL volume estimate is a function of the systematic constant volume measurement error, a fact that was summarily dismissed by those authors. Furthermore, the present analysis does not rely on the assumption that nonpartitioning and partitioning tracers are
(41)
The nondimensional concentration with the systematic proportional concentration error then becomes
ci )
βcd,t i
cd,t i ) ) cti d,t d,t βc0 c0
(42)
Therefore, when nondimensional concentrations are used in the moment calculations, systematic proportional errors in concentrations have no effect on the moment calculations nor consequently any effect on NAPL volume estimates.
Results A synthetic data set was used to study error propagation of systematic measurement errors. The synthetic data set was generated using the solution to the one-dimensional advective-dispersive transport equation, subject to the initial condition of c(x,0) ) 0 for x g 0 and the boundary conditions of c(0,t) ) c0 for t g 0 and c(∞,t) ) 0 for t g 0 (22, 23). The nondimensional form of the solution, accounting for retardation, is
c(τ,R,Pe) )
( ){ [x 1 2
erfc
]
Pe (R - τ) + exp(Pe)erfc 4Rτ
[x
]}
Pe (R + τ) 4Rτ (43)
where c is the dimensionless concentration, τ is the dimensionless pore volume (τ ) vt/L, where v ) pore velocity [LT-1], t ) time [T], and L ) linear extent of the flow domain [L]), R ) retardation factor, and Pe ) Peclet number (Pe ) vL/D, where D ) dispersion coefficient [L2 T-1]). Equation 43
VOL. 39, NO. 18, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
7167
injected with similar masses, as did the former study. This difference is important, allowing different classes of compounds to be used, if desired, for the nonpartitioning and partitioning tracers. The NAPL volume error does increase slightly with retardation but is not substantial. Furthermore, as Figure 1a reveals, the NAPL volume error is not pronounced for systematic constant volume errors. Systematic Proportional Volume Errors. The impacts of systematic proportional errors in volume measurements on retardation, saturation, and NAPL volume are illustrated in Figure 1b. The error shown on the abscissa in Figure 1b is the percent volume error. Systematic proportional errors in volume measurements have minimal impact on retardation and saturation estimates, because the error occurs in both the numerator and denominator of those terms. However, this type of systematic error has a larger impact on the NAPL volume estimate because of its impact on the swept volume (see Figure 1b). Systematic Constant Concentration Errors. The range of systematic constant concentration errors is limited to magnitudes equal to or less than method detection limits, based on the assumption that larger errors would be more readily identified by typical quality assurance procedures used in the laboratory. Assuming typical values for alcohol tracers, i.e., injection concentrations on the order of 1000 mg/L and method detection limits on the order of 1 mg/L, dimensionless concentration errors could range from -0.001 to +0.001. The impact of errors in this range on the retardation, saturation, and NAPL volume are shown in Figure 1c. It was assumed that the systematic error was the same for both nonpartitioning and partitioning concentrations. As shown in Figure 1c, this type of error has the largest impact on NAPL volume estimates and smaller but similar impacts on retardation and NAPL saturation estimates. Systematic Proportional Concentration Errors. As shown by eqs 41 and 42, the impact of systematic proportional errors in concentration measurements on the moment calculations is eliminated by using dimensionless concentrations. Therefore, systematic proportional concentration errors do not impact retardation, NAPL saturations, or NAPL volume estimates. Comparison to Observed Field-Scale Variation. In the initial controlled blind release partitioning tracer test performed by Brooks et al. (12), in which the flow patterns were three-dimensional in character, approximately 80% of spilled NAPL was detected by the tracers employed (after quite a bit of tracer data interpretation). However, for three tracers used in conjunction in the lower zone of the test (see ref 12 for details), the coefficient of variation in the NAPL volumes predicted by them was 61%. These tracers were injected, produced, and quantified in the same manner. Consequently, the variability in their NAPL volume estimates must have been due to other sources of error. This type of error and others discussed by refs 12-17 almost certainly dominate those based upon the propagation of systematic errors quantified herein.
Discussion Errors in NAPL volume determination through partitioning tracer tests are readily quantified for both systematic constant and proportional errors for both volume and concentration measurements. While systematic proportional concentration errors are in essence forgiven through the use of a nondimensional concentration, the other three error classes propagate through the processes of determining moments, retardation values, NAPL saturations, and NAPL volumes. The primary purpose of partitioning tracer tests is to determine NAPL volume. The results of this work clearly indicate that it is extremely important to minimize proportional volume errors (see Figure 1b) through the use of an 7168
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 39, NO. 18, 2005
accurate flow measurement method. On the analytical chemistry side of the analysis, methods must accurately correct for any constant errors (see Figure 1c) introduced through a poorly corrected baseline, for instance. These safeguards are a minimum starting point for the proper application of partitioning tracer tests in the field. By not being restricted to having nonpartitioning and partitioning tracer injection masses be equivalent, the present methodology is applicable to conditions where it is desirable to use different classes of compounds for these tracers. However, the authors are not asserting that each partitioning tracer test warrants specific analysis of the propagation of systematic errors. Instead, they hope that the exercise above sheds useful light into the likely bounds of such errors and prescribes generic guidance on how best to minimize them. In practice, performing propagation-of-error analyses on partitioning tracer tests performed in the field would be complicated by further issues addressed by others (12-17), including incomplete breakthrough curves, for instance. The authors chose the well-known solution of the advectiondispersion equation as the base from which to work because it represents a “truth” from which resultant errors may be readily deduced when propagating systematic errors. The following paper (18) quantifies how random measurement errors affect uncertainty in NAPL volume estimates and builds upon a previous contribution that quantifies how random measurement errors affect uncertainty in the calculated moments of breakthrough curves (19).
Nomenclature a
constant volume error [L3]
b
constant concentration error [ML-3]
ci
ith nondimensional concentration (cdi /cd0 )
cdi
ith dimensioned concentration measurement [ML-3]
cd0
dimensioned injection concentration [ML-3]
KNW
equilibrium partitioning coefficient
L
linear extent of the flow domain [L]
mk
absolute kth moment [L3(k+1)]
Pe
Peclet number
R
retardation factor
RB
background retardation factor
S
NAPL saturation
tp
pulse duration [T]
Vi
ith cumulative volume measurement [L3]
VN
NAPL volume [L3]
Vp
tracer pulse volume [L3]
VT
total cumulative volume [L3]
R
proportional volume error factor
R′
relative proportional volume error () R - 1)
β
proportional concentration error factor
∆Vi
volume interval (Vi+1 - Vi) [L3]
φ1
intermediate constant
φ2
intermediate constant
µk
kth normalized moment
µ′NR 1
pulse-corrected, normalized first moments for the nonreactive tracer [L3]
µ′R1
pulse-corrected, normalized first moments for the reactive tracer [L3]
(11)
Acknowledgments The authors would like to thank the reviewers and Editor for their helpful comments. This project was funded by the United States (U.S.) Department of Defense (DOD) Strategic Environmental Research and Development Program, which is a collaborative effort involving the U.S. Environmental Protection Agency, U.S. Department of Energy, and U.S. DOD. This document has not been subjected to peer review within the supporting agencies, and the conclusions stated here do not necessarily reflect the official views of the agencies nor does this document constitute an official endorsement by the agencies.
Literature Cited (1) )Cooke, C. E., Jr. Method for determining residual oil saturation. U.S. Patent No. 3,590, 923, 1971. (2) Dean, H. A. Method for determining fluid saturation in reservoirs. U.S. Patent No. 3,623, 842, 1971. (3) Tang, J. S. Partitioning tracers and in-situ fluid-saturation measurements. SPE Form. Eval. 1995, 10, 33-39. (4) Jin, M.; Delshad, M.; Dwarakanath, V.; McKinney, D. C.; Pope, G. A.; Sepehrnoori, K., Tilburg, C.; Jackson, R. E. Partitioning tracer test for detection, estimation, and remediation performance assessment of subsurface nonaqueous phase liquids. Water Resour. Res. 1995, 31, 1201-1211. (5) Annable, M. D.; Rao, P. S. C.; Hatfield, K.; Graham, W. D.; Wood, A. L.; Enfield, C. G. Partitioning tracers for measuring residual NAPL: field-scale test results. J. Environ. Eng. 1998, 124, 498503. (6) Cain, R. B.; Johnson, G. R.; McCray, J. E.; Blanford, W. J.; Brusseau, M. L. Partitioning tracer tests for evaluating remediation performance. Ground Water 2000, 38, 752-761. (7) Sillan, R. K.; Annable, M. D.; Rao, P. S. C. Evaluation of in-situ DNAPL remediation and innovative site characterization techniques; report submitted to the Florida Center for Solid and Hazardous Waste Management, Gainesville, FL, 1999; 68 pp. (8) Hayden, N. J.; Linnemeyer, H. C. Investigation of partitioning tracers for determining coal tar saturation in soils. In Innovative Subsurface Remediation, Field Testing of Physical, Chemical, and Characterization Technologies; Brusseau, M. L., Sabatini, D. A., Gierke, J. S., Annable, M. D., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999; Vol. 725, pp 226-239. (9) Nelson, N. T.; Brusseau, M. L. 1996. Field study of the partitioning tracer method for detection of dense nonaqueous phase liquid in a trichloroethene-contaminated aquifer. Environ. Sci. Technol. 1996, 30, 2859-2863. (10) Rao, P. S. C.; Annable, M. D.; Kim, H. NAPL source zone characterization and remediation technology performance
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20) (21)
(22)
(23)
assessment: recent developments and applications of tracer techniques. J. Contam. Hydrol. 2000, 45, 63-78. Brusseau, M. L.; Nelson, N. T.; Cain, R. B. The partitioning tracer method for in-situ detection and quantification of immiscible liquids in the subsurface. In Innovative Subsurface Remediation, Field Testing of Physical, Chemical, and Characterization Technologies; Brusseau, M. L., Sabatini, D. A., Gierke, J. S., Annable, M. D., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999; Vol. 725, pp 208-225. Brooks, M. C.; Annable, M. D.; Rao, P. S. C.; Hatfield, K.; Jawitz, J. W.; Wise, W. R.; Wood, A. L.; Enfield, C. G. Controlled release, blind tests of DNAPL characterization using partitioning tracers. J. Contam. Hydrol. 2002, 59, 187-210. Lee, C. M.; Meyers, S. L.; Wright, C. L., Jr.; Coates, J. T.; Haskell, P. A.; Falta R. W., Jr. NAPL compositional changes influence partitioning coefficients. Environ. Sci. Technol. 1998, 32, 35743578. Dwarakanath, V.; Deeds, N.; Pope, G. A. Analysis of partitioning interwell tracer tests. Environ. Sci. Technol. 1999, 33, 38293836. Nelson, N. T.; Oostrom, M.; Wietsma, T. W.; Brusseau, M. L. Partitioning tracer method for the in situ measurement of DNAPL saturation: influence of heterogeneity and sampling method. Environ. Sci. Technol. 1999, 33, 4046-4053. Wise, W. R.; Dai, D.; Fitzpatrick, E. A.; Evans, L. W.; Rao, P. S. C.; Annable, M. D. Nonaqueous phase liquid characterization via partitioning tracer tests: a modified Langmuir relation to describe partitioning nonlinearities. J. Contam. Hydrol. 1999, 36, 153-165. Wise, W. R. NAPL characterization via partitioning tracer tests: quantifying effects of partitioning nonlinearities. J. Contam. Hydrol. 1999, 36, 167-183. Brooks, M. C.; Wise, W. R. Uncertainty in NAPL Volume Estimates due to Random Measurement Errors during Partitioning Tracer Tests. Environ. Sci. Technol. 2005, 39, xxxx-xxxx. Brooks, M. C.; Wise, W. R. Quantifying Uncertainty due to Random Errors for Moment Analyses of Breakthrough Curves. J. Hydrol. 2005, 303, 165-175. Haas, C. N. Moment analysis of tracer experiments. J. Environ. Eng. 1996, 122, 1121-1123. Brooks, M. C. Characterization and Remediation of a Controlled DNAPL Release: Field Study and Uncertainty Analysis, Doctoral Dissertation, University of Florida, 2000, 147 pp. Lapidus, L.; Amundson, N. R. Mathematics of adsorption in beds. VI. The effect of longitudinal diffusion in ion exchange and chromatographic columns. J. Phys. Chem. 1952, 56, 984988. Ogata, A.; Banks, R. B. A solution of the differential equation of longitudinal dispersion in porous media. U.S. Geol. Surv. Professional Pap. 1961, 411-A.
Received for review August 12, 2004. Revised manuscript received June 15, 2005. Accepted June 21, 2005. ES048739M
VOL. 39, NO. 18, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
7169
