Environ. Sci. Technol. 2005, 39, 7170-7175
Uncertainty in NAPL Volume Estimates Due to Random 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
The uncertainty in NAPL volume estimates obtained through partitioning tracers can be quantified as a function of random errors in volume and concentration measurements when moments are calculated from experimentally measured breakthrough curves using the trapezoidal rule for numerical integration. The methodology is based upon standard stochastic methods for random error propagation. Monte Carlo simulations using a synthetic data set derived from the one-dimensional solution of the advective-dispersive equation serve to verify the process. It is shown that the uncertainty in NAPL volume predictions nonlinearly increases as the retardation factor decreases. An important result of this observation is that there is a large degree of uncertainty in using partitioning tracers to conclude NAPL is absent from the swept zone. Under the conditions investigated, random errors in concentration measurements are shown to have a greater impact on NAPL volume uncertainty than random errors in volume measurements, and it is also shown that uncertainty in NAPL volume decreases as the resolution of the breakthrough curves increases. The impact of uncertainty in background retardation (i.e., sorption of partitioning tracers in the absence of NAPL) was also investigated, and it likewise indicated that the relative uncertainty in NAPL volume estimates increases as the retardation factor decreases.
Introduction The previous paper (1) examined the propagation of systematic errors in volume and concentration measurements to generate errors in the determination of nonaqueous phase liquid (NAPL) saturation and volume through partitioning tracer tests. This paper examines how random errors in volume and concentration measurements generate uncertainty in NAPL saturation and volume estimates and builds upon previous contributions that quantify how random measurement errors affect uncertainty in the calculated moments of breakthrough curves (2) and how errors affect uncertainty in NAPL saturation based on partitioning tracers (4, 5). Few papers have dealt with the direct quantification of uncertainty involved with partitioning tracer technology (3* Corresponding author phone: (352)846-1745; fax: (352)392-3076; e-mail:
[email protected]. † U.S. Environmental Protection Agency. ‡ University of Florida. 7170
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5). Helms (6) compared techniques for estimating moments associated with tracer BTC data sets that were incomplete (i.e., contained gaps or that were truncated). Nonlinear leastsquares regression was found to be an effective method for working with imperfect data; methods to estimate standard deviations and confidence intervals of temporal moments based on a nonlinear regression technique were presented. However, the uncertainty analysis was not extended to NAPLvolume estimates. Jin et al. (3, 5) and Dwarakanath et al. (4) present a series of papers discussing aspects of errors and uncertainty related to partitioning tracer tests. The method presented by Dwarakanath et al. (4) is based on the propagation of random errors in the retardation factor and the partitioning coefficient through to NAPL saturation using the first-order Taylor series expansion for error propagation (Delta method) and assumes that errors in the retardation factor and the partitioning coefficient are independent. The error in the retardation factor was based on the residual error between the measured data and the curve used to fit the data. Random errors in the measurement of the partitioning coefficient were assessed using the standard deviation of the isotherm slope from batch partitioning experiments or by calculating the standard deviation from the results of multiple experiments where the partitioning coefficient was estimated from column experiments. They concluded that random errors in the retardation factor and in the partitioning coefficient result in an uncertainty of approximately 10% in the NAPL saturation when tests yield retardation factors greater than 1.2. Jin et al. (5) made a similar presentation regarding the uncertainty in NAPL saturation as a function of retardation factor and partitioning coefficient uncertainty. They also include a formula for the uncertainty in the retardation factor as a function of the first moments for the nonpartitioning and partitioning tracers and estimate the moment uncertainty based on the residual error between data points and a mathematical curve fit. The limitation in the methods previously presented is that the uncertainty in retardation and partitioning coefficient can only be propagated through to NAPL saturation. The uncertainty in NAPL volume cannot be estimated without the uncertainty in the swept volume (provided by the normalized first moment of the nonpartitioning tracer) and the correlation between the swept volume and saturation. Meinardus et al. (7) provide error estimates on NAPL volume predictions based on partitioning tracer results, but the method used to propagate the error from the NAPL saturation to the NAPL volume was not reported. Furthermore, the uncertainties in the normalized moments and retardation do not explicitly account for measurement uncertainty but is more accurately a measure of how well the curve fits the measured data. Curve-fitting techniques that explicitly include measurement uncertainty could be used with the procedure outlined by Dwarakanath et al. (4) and Jin et al. (5) to better estimate partitioning tracer test uncertainty. The purpose herein is to quantify the uncertainty in NAPL saturation and volume estimates obtained through partitioning tracers when moments are calculated from experimentally measured breakthrough curves using the trapezoidal rule for numerical integration. The methodology is based upon standard stochastic methods for error propagation, and Monte Carlo simulations using a synthetic data set serve to verify the process. General conclusions about NAPL saturation and volume measurement and uncertainty are explored. 10.1021/es048738u CCC: $30.25
2005 American Chemical Society Published on Web 08/17/2005
Theory A brief outline of the equations used to estimate NAPL saturations and volumes from tracer information is presented as an introduction to the uncertainty equations. The retardation factor, R is defined as
R) µ′NR 1
µ′R1
3
µ′R1
(1)
µ′NR 1
3
where [L ] and [L ] are pulse-corrected, normalized first moments for the nonpartitioning (nonretarded) and partitioning (retarded) tracers, respectively. See the companion paper (1) for how they are determined from the raw data. 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 may be accounted for using
R)
µ′R1 µ′NR 1
- (RB - 1)
(2)
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 pore space occupied by water (or air) and NAPL only, the saturation (S) can be calculated from
S)
R-1 R - 1 + KNW
(3)
and the volume of NAPL, VN, is given by
S VN ) µ′NR 1 1-S
1
1
[
∑∑ i)1 j)1
σxi,xj
∂µ′NR ∂µ′R1 1 ∂xi
∂xj
+σVp2
∂µ′NR ∂µ′R1 1 ∂Vp
∂Vp
(5)
where x represents the measurements of volume ({x1,...,xi,...,xn} ) {V1,...,Vi,...,Vn}) and concentration ({xn + 1,...,xi + n,...,x2n} ) {c1,...,ci,...,cn}) upon which are based the normalized first moments, and Vp is the tracer pulse volume. The last term on the right-hand side on eq 5 describes the covariance resulting from a common tracer-pulse volume. It is assumed herein that the tracer-pulse volume uncertainty is negligible due to the controlled conditions generally used in its measurement, and this term will be ignored in subsequent analysis. Since errors in volume and concentration measurements are assumed independent, eq 5 can be written as n
σµ′NR, µ′R = 1
1
∑ i)1
σV2 i
( )( ) ∂µ′NR ∂µ′R1 1 ∂Vi
∂Vi
+
∑σ i)1
( )( ) ∂µ′NR ∂µ′R1 1
n
R cNR i ,ci
mR0
1
∑ i
∂mR1 ∂Vi
-
mR1
∂cNR i
∂cRi
(6)
Expressing the derivatives of the normalized first moments in terms of the zeroth and absolute first moments results in
∂mNR 1
-
∂Vi
mNR 1
∂Vi
] [
]
∂mNR 0 ∂Vi
2 (mNR 0 )
∂mR0
(mR0 )2
mNR 0
n
+
∑σ i
R cNR i ,ci
[
∂mNR 1 ∂cNR i
- mNR 1
∂cNR i
2 (mNR 0 )
mR0
∂mR1
-
∂cRi
mR1
]
∂mNR 0
]
∂mR0 ∂cRi
(mR0 )2
(7)
, R is the covariance between the ith nonpartiwhere σcNR 1 ci tioning and partitioning concentrations. Brooks and Wise (2) report the first derivatives for the zeroth and absolute first moments with respect to both volume and concentration measurements. An effective means of determining the covariance between the ith observations of the nonpartitioning and partitioning concentrations is
(
NR σcNR R = ci i ,ci
)
NR NR ci-1 + cNR + ci+1 i 3 R R R c i-1 + ci + ci+1 cRi , 2 e i e n - 1 (8) 3
(
)
For i ) 1 and i ) n, the concentrations should be equal to or very near zero for well posed tracer tests; so the covariance should be minimal. Using a first-order Delta method approximation to the uncertainty of the ratio of two random variables, the retardation variance, σ2R, is approximated as
σ2R =
( )( )] ( )( )
2n 2n
[
1
σV2 i
(4)
To estimate the uncertainty in R, it is necessary to estimate the covariance between the normalized first moments of the nonpartitioning and partitioning tracers (σµ′NR,µ′R), since they 1 1 are based on the same volume measurements. Furthermore, correlation may exist between the nonpartitioning and partitioning concentrations. The covariance is estimated using a first-order Delta method approximation
σµ′NR, µ′R =
n
σµ′NR, µ′R =
[
mNR 0
( )[ µ′R1
µ′NR 1
2
σ2µ′R1
(µ′R1 )
+ 2
σ2µ′NR 1 2 (µ′NR 1 )
-
]
2σµ′R1 , µ′NR 1 µ′R1 µ′NR 1
(9)
Brooks and Wise (2) present the methodology to determine 2 2 σµ′ NR, and σµ′R based on random errors in volume and 1 1 concentration measurements. Likewise, a first-order Delta method approximation is used to estimate the uncertainty in saturation, as presented by Dwarakanath et al. (4):
σS2 )
KNW2σR2 + (R - 1)2σKNW2
(10)
(R - 1 + KNW)4
Note that eq 10 reflects the assumption that the retardation factor and the partitioning coefficient are independent. It could be argued that R and KNW are correlated since the partitioning coefficient controls the degree of retardation. For this analysis, however, it is assumed that R and KNW are independent because the random errors incurred in measuring either R or KNW are independent, with R being determined in the field (using volume and concentration data) and KNW in the laboratory. The uncertainty of the NAPL volume should account for the correlation between the normalized first moment of the nonpartitioning tracer and the saturation (more formerly the ratio of S to 1-S), since they are both based on the same volume and nonpartitioning concentration measurements. Likewise, it should account for the correlation between nonpartitioning and partitioning concentrations, if present. The covariance between the normalized first moment of the nonpartitioning tracer and the saturation is estimated using the Delta-method approximation: VOL. 39, NO. 18, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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σµ′NR,S/(1-S) = 1
( )( )
2n 2n
1
∑∑
(1 - S)2 i)1 j)1
σxi,xj
NR ∂S ∂µ′1
∂xi
∂xj
(11)
The x values represent the volume and concentration measurements as stated above for eq 5. The derivative of µ′NR 1 with respect to x in terms of the zeroth and absolute first moments is given in eq 7. The derivative of S with respect to measurement xi in terms of the zeroth and absolute first moments is
{[
KNW ∂S ) * µ′NR 1 ∂xi (R - 1 + K )2(µ′NR)2 NW 1
[
mNR 0
µ′R1
mR0
] ]}
∂mR1 ∂mR0 - mR1 ∂xi ∂xi (mR0 )2
∂mNR ∂mNR 1 0 - mNR 1 ∂xi ∂xi 2 (mNR 0 )
-
Applications (12)
The covariance terms σxi,xj in eq 11 include the covariance between the volume and concentration measurements used in S and µ′NR 1 . The covariance between the nonpartitioning and partitioning concentrations can be estimated using eq 8. Finally, the variance in the volume estimate of NAPL is given by
σS2 S 2 2 2 σVN2 ) (µ′NR + σµ′ + (σµ′ NR,S/(1-S))2 + 1 ) j NR 4 1 1 1 S (1 - S) σS2 S 2µ j ′NR σµ′ NR,S/(1-S) + σµj ′ NR2 (13) 1 1 1 1-S (1 - S)4
(
(
)
)
Results A synthetic data set was used to study propagation of random measurement errors and 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 (8, 9). The nondimensional form of the solution, accounting for retardation, is
c(τ,R,Pe) )
( ){ [x 1 2
erfc
] [x
Pe (R - τ) + exp(Pe)erfc 4Rτ Pe (R + τ) 4Rτ
]}
(14)
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 14 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 {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 was varied as appropriate for the partitioning tracer, and the Peclet number was 10. Unless stated otherwise, a total of 100 volume-concentration data points were used to represent the BTCs. As a means to verify error estimates from the errorpropagation equations, a Monte Carlo analysis was conducted in which measurement errors were assumed to be normally distributed random variables with zero-valued means with assumed standard deviations. Concentration measurement 7172
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errors were defined using a coefficient of variation (CV), ranging from 0 to 0.3. Volume measurement errors were defined by assuming the volume standard deviation ranged from zero up to one volume measurement interval. A unique measurement error was applied to each volume and concentration value in the synthetic data set. Moment calculations were then completed on the “measured” BTC. This process was repeated 10 000 times; the means and standard deviations of the retardation, NAPL saturation, and NAPL volume were computed. Convergence of Monte Carlo results was tested by completing three identical simulations, each with 10 000 iterations; the CV for the moments, retardation, NAPL saturation, and NAPL volume differed by no more than 0.0002.
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The retardation, NAPL saturation, and NAPL volume CV are shown in Figure 1a-c, respectively, as functions of retardation for three different sets of dimensionless volume and concentration errors. Dimensionless concentration error is expressed as CV, and dimensionless volume error is expressed as the ratio of the volume error standard deviation to the swept volume. The latter is motivated by the use of cumulative volume as the independent variable of the BTC. Imposing a constant volume CV in this case would imply a monotonically increasing standard deviation for V, which may or may not reflect measurement methods used in the field. It is more appropriate to assign a constant standard deviation to each incremental volume error, resulting in a continually decreasing volume CV. Normalizing the volume measurement error to the swept volume is more robust. The over-sized circles in Figure 1 represent the results from the Monte Carlo analysis. The agreement between the data points and Monte Carlo results demonstrates that the methodology correctly accounts for the uncertainly in volume and concentration measurements, based on the assumption of independence among all measurement uncertainties. As indicated in Figure 1 (parts b and c), the uncertainty in NAPL saturation and volume estimates is high for low retardation values, and the uncertainty decreases as retardation increases. This result agrees with that presented by Jin et al. (10). For reliability, estimates of NAPL saturation and volume should be based on retardation values of 1.2 or greater. The most striking feature of Figure 1 is the higher degree of uncertainty associated with NAPL saturation and volume estimates when the retardation is less than 1.2, which nonlinearly increases as retardation diminishes. This indicates there is a high degree of uncertainty associated with the conclusion that little or no NAPL is present based on small retardation values. Furthermore, this result is based solely on the consideration of random measurement errors and neglects other factors that may further increase the uncertainty. Figure 2 shows the NAPL volume CV as a function of BTC resolution. The data used in the figure are based on a concentration CV of 0.15 and a retardation factor of 1.5. The dimensionless volume error used as the independent variable in the figure was defined as the ratio of the volume standard deviation and the swept volume based on the nonpartitioning pulse-corrected first moment. The volume standard deviation ranged from 0 to ∆V, which was the volume increment associated with a particular BTC resolution, and the swept volume was constant for all cases. It is apparent from the figure that the uncertainty decreases as the resolution increases, which is a reasonable result since the underlying errors shrink with increasing resolution. As indicated by the CV of NAPL volume as shown in Figures 1 and 2, concentration errors have a greater impact on results from partitioning tracer tests than do volume errors.
FIGURE 3. Retardation (triangles), NAPL saturation (squares), and NAPL volume (circles) CV as a function of the concentration detection limit CV. The CV of the maximum concentrations were 15% (open symbols) and 30% (closed symbols). The figure is based on 100 volume-concentration data points, a retardation factor of 1.5, and no volume measurement errors.
FIGURE 1. Coefficient of variation in (a) retardation, (b) NAPL saturation, and (c) NAPL volume as functions of retardation for dimensionless volume and concentration errors of 0.005 and 0.05 (diamonds), respectively; 0.015 and 0.15 (squares), respectively; and 0.03 and 0.30 (triangles), respectively. The dimensionless volume error was defined as the ratio of volume standard deviation to swept volume, and the dimensionless concentration error as the CV. BTCs with 100 data points were used to generate the figure. The oversized circles represent results from the Monte Carlo simulations.
FIGURE 4. Impacts of background-retardation uncertainty. The NAPL volume CV is presented as a function of retardation, for background retardation CVs of 5% (circles), 15% (triangles), and 30% (squares). concentration has less impact than uncertainty in the detection limit concentration. However, even with the detection limit uncertainty set as high as CVDL ) 2.00, the NAPL volume CVs are only approximately 30%. The impact of uncertainty in background retardation on NAPL volume uncertainty is shown in Figure 4. This figure was based on the assumption that all variances are equal to zero except the variance of the background retardation (i.e., from eq 2, σ2R ) σR2 B). Under these conditions, eq 10 can be used to express σS2 as a function of σR2 B, and eq 13 can subsequently be used to express σV2 N as a function of σR2 B. Taking the square root of the ratio of eqs 13 and 4 then yields
σV N FIGURE 2. NAPL volume coefficient of variation as a function of dimensionless volume error deviation for BTCs of 50 (triangles), 100 (squares), and 350 (diamonds) volume-concentration data points. The figure is based on a retardation factor of 1.5. The dimensionless volume error was defined as the ratio of the volume standard deviation to the swept volume, and the concentration CV was set to 15%. Figure 3 shows the impact of variable concentration uncertainty on retardation, NAPL saturation, and NAPL volume. The analysis assumed uncertainty in volume measurements was negligible, concentration uncertainty varied linearly from the uncertainty of the detection limit concentration (a dimensionless detection limit of 0.001 was assumed) to the uncertainty of the peak concentration. The uncertainty of the peak concentration, defined using concentration CV, was 0.15 and 0.30. The uncertainty of the detection limit concentration was varied using CV values ranging from 0.5 to 2.0. As illustrated in Figure 3, the uncertainty in the peak
VN
)
σR B
(15)
(R - 1)
Background retardation uncertainties (defined as CV) of 0.05, 0.15, and 0.30 were used to produce the figure. By comparison to Figure 1, it is evident that NAPL volume uncertainty is more sensitive to background retardation uncertainty in comparison to uncertainty from volume or concentration measurement uncertainty. Equation 15 and Figure 4 further indicate there is a high degree of uncertainty associated with the conclusion that little or no NAPL is present based on small retardation values. As further application of the results of this investigation, a comparison is made to observed field-scale variation in NAPL volume predictions from DNAPL characterization tests completed at the Dover Air Force Base in Dover, DE. Results from four controlled-release blind partitioning tracer tests were reported by Brooks et al. (11) and were referenced in that work as tests 1-4. Each test included a suite of partitioning tracers with a range of water-DNAPL partitioning VOL. 39, NO. 18, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. NAPL Volume Uncertainty Estimates for Partitioning Tracer Demonstrations partitioning tracer
NAPL minimum retardation factora volumeb
CVc
Test 1
n-hexanol 2,4-dimethyl-3-pentanol 2-octanol average standard deviation CV
1.12 1.05 1.10
71 33 22 42 26 61%
13% 30% 16%
1.03 1.01 1.10 1.12
45 12 6 4 17 19 114%
37% ∼50% 16% 13%
1.06 1.06 1.13 1.30
191 78 49 33 88 71 81%
28% 28% 12% 6%
1.07 1.03 1.10 1.25
20 31 29 30 28 5 18%
22% 37% 16% 6%
Test 2
n-hexanol 2,4-dimethyl-3-pentanol 2-octanol 3,5,5-trimethyl-1-hexanol average standard deviation CV Test 3
n-hexanol 2,4-dimethyl-3-pentanol 2-octanol 3,5,5-trimethyl-1-hexanol average standard deviation CV Test 4
n-hexanol 2,4-dimethyl-3-pentanol 2-octanol 3,5,5-trimethyl-1-hexanol average standard deviation CV
a Reference 11, Figure 5. b Reference 11, Tables 3 and 5. c Figure 1c, using dimensionless concentration and volume errors of 0.15 and 0.015.
coefficients (see ref 11 for details). The minimum retardation factor measured for each tracer and the total volume of DNAPL predicted by each tracer for each test is shown in Table 1. The minimum retardation factor for each tracer was used to estimate the maximum CV associated with the total volume of DNAPL using the curve in Figure 1c associated with dimensionless volume and concentration errors of 0.015 and 0.15, respectively (the number of data points used to define the BTCs measured during the tests was on the order of 100). As shown in Table 1, the maximum DNAPL volume CVs based on the minimum retardation factors were 30%, ∼50%, 28%, and 37% for tests 1-4, respectively. However, the precision in the DNAPL volumes predicted by the tracers (i.e., the CV of the DNAPL volume estimates based on all tracers) was 61%, 114%, 81%, and 18% for tests 1-4, respectively. Since these tracers were injected, produced, and quantified in the same manner, this comparison suggests that only for test 4 could the variability in DNAPL volume estimates between tracers be attributed to random errors in volume and concentration measurements. For tests 1-3, the variability in the estimated DNAPL volume must have included factors beyond random error measurements, such as violation of the assumption that the prediction of each tracer could be treated independently. This type of error and others discussed elsewhere (3, 4, 11, 12) almost certainly dominate those based upon the propagation of random errors quantified herein.
Discussion A method was presented for estimating uncertainties associated with partitioning tracer tests. The method differs 7174
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from previous work on measurement uncertainty in that retardation, saturation, and NAPL volume uncertainty are based on the uncertainties (i.e., random error) in volume and concentration measurements, rather than uncertainty based on the difference between measurements and model predictions. Uncertainty in the NAPL volume estimate has also been presented, which has not been elucidated in previous work. The method is equally applicable to volumetric and temporal moments and, in the case of the former, accounts for volume-measurement uncertainty. The conclusion that NAPL is not present based on partitioning tracer test results has a high degree of uncertainty, simply because of measurement uncertainty. This suggests that using partitioning tracers as a means to detect small volumes of NAPL (which results in small retardation values) may not be a reliable technique or, at least if used as such, should be done so with great care. It should be understood that the methods presented herein as well as those presented by Dwarakanath et al. (4) and Jin et al. (5) provide estimates of the uncertainty associated with partitioning tracer tests arising from measurement errors. The ultimate accuracy of partitioning tracer estimates may however be impacted to a larger extent by other factors, such as the validity of underlying assumptions of the partitioning tracer technique (3, 4, 11-15). Fortunately, the method can be considered robust with respect to the type of measurement errors investigated here, the effect of which has been found to be relatively small.
Nomenclature 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]
∆Vi
volume interval (Vi+1 - Vi) [L3]
µk
kth normalized moment
µ′NR 1
pulse-corrected, normalized first moment for the nonpartitioning tracer [L3]
µ′R1
pulse-corrected, normalized first moment for the partitioning tracer [L3]
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) Brooks, M. C.; Wise, W. R. Errors in NAPL Volume Estimates due to Systematic Measurement Errors during Partitioning Tracer Tests. Environ. Sci. Technol. 2005, 39, xxxx-xxxx. (2) Brooks, M. C.; Wise, W. R. Quantifying Uncertainty due to Random Errors for Moment Analyses of Breakthrough Curves. J. Hydrol. 2004, 303, 165-175. (3) Jin, M.; Butler, G. W.; Jackson, R. E.; Mariner, P. E.; Pickens, J. F.; Pope, G. A.; Brown, C. L.; McKinney, D. C. Sensitivity models and design protocols for partitioning tracer tests in alluvial aquifers. Ground Water 1997, 35, 964-972. (4) Dwarakanath, V.; Deeds, N.; Pope, G. A.; Analysis of partitioning interwell tracer tests. Environ. Sci. Technol. 1999, 33, 38293836. (5) Jin, M.; Jackson, R. E.; Pope, G. A. The interpretation and error analysis of PITT data. In Treating Dense Nonaqueous-Phase Liquids (DNAPLs): Remediation of Chlorinated and Recalcitrant Compounds; Wickramanayake, G. B., Gavaskar, A. R., Gupta, N., Eds.; Battelle Press: Columbus, OH, 2000; pp 85-92. (6) Helms, A. D. Moment estimates for imperfect breakthrough data: Theory and application to a field-scale partitioning tracer experiment, Masters Thesis, University of Florida, Gainesville, FL, 1997, 188 pp. (7) Meinardus, H. W.; Dwarakanath, V.; Ewing, J.; Hirasaki, G. J.; Jackson, R. E.; Jin, M.; Ginn, J. S.; Londergan, J. T.; Miller, C. A.; Pope, G. A. Performance assessment of NAPL remediation in heterogeneous alluvium. J. Contam. Hydol. 2002, 54, 173-193. (8) 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.
(9) Ogata, A.; Banks, R. B. A solution of the differential equation of longitudinal dispersion in porous media. U.S. Geol. Sur. Professional Pap. 1961, 411-A. (10) 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. (11) 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. (12) 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. (13) 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. (14) 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. (15) Wise, W. R. NAPL characterization via partitioning tracer tests: quantifying effects of partitioning nonlinearities. J. Contam. Hydrol. 1999, 36, 167-183.
Received for review August 12, 2004. Revised manuscript received June 15, 2005. Accepted June 21, 2005. ES048738U
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