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Keywords:

  • dual porosity;
  • fractured media;
  • reactive tracer;
  • sorption;
  • tracer test;
  • tracers

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

[1] A cross-hole tracer test involving the simultaneous injection of two nonsorbing solute tracers with different diffusion coefficients (bromide and pentafluorobenzoate) and a weakly sorbing solute tracer (lithium ion) was conducted in a fractured granite near an underground nuclear test cavity in central Nevada. The test was conducted to (1) test a conceptual radionuclide transport model for the site and (2) obtain transport parameter estimates for predictive modeling. The differences between the responses of the two nonsorbing tracers (when normalized to injection masses) are consistent with a dual-porosity transport system in which matrix diffusion is occurring. The large concentration attenuation of the sorbing tracer relative to the nonsorbing tracers suggests that diffusion occurs primarily into matrix pores, not simply into stagnant water within the fractures. The relative responses of the tracers at late times suggest that the diffusion-accessible matrix pore volume is possibly limited to only half the total volume of the flow system, implying that the effective retardation factor due to matrix diffusion may be as small as 1.5 for nonsorbing solutes in the system. The lower end of the range of possible sorption Kd values deduced from the lithium response is greater than the upper 95% confidence bound of Kd values measured in laboratory sorption tests using crushed granite from the site. This result suggests that the practice of using laboratory sorption data in field-scale transport predictions of cation-exchanging radionuclides, such as 137Cs+ and 90Sr++, should be conservative for the SHOAL site.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

1.1. Objectives

[2] The SHOAL underground nuclear test consisted of a 12-kiloton-yield nuclear detonation on October 26, 1963, in the Sand Springs Range of Churchill County, Nevada, about 50 km southeast of Fallon, Nevada [U.S. Department of Energy, 2000]. Figure 1 shows the location of the SHOAL site and the wells used for tracer testing, which are discussed in more detail later. The nuclear device was emplaced 367 m below land surface, or about 65 m below the water table, in a fractured granite. Details of the geology and hydrogeology at the site are provided elsewhere [Pohll et al., 1998,1999]. The SHOAL test was part of a series of studies to enhance seismic detection of underground nuclear tests, particularly in active earthquake areas.

image

Figure 1. Location of the SHOAL site and the wells used for tracer testing (HC-6 and HC-7).

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[3] Characterization of groundwater contamination resulting from the SHOAL test is being conducted by the U.S. Department of Energy under the Federal Facility Agreement and Consent Order (FFACO) with the State of Nevada Division of Environmental Protection and the U.S. Department of Defense. The objectives of the characterization effort include the evaluation of alternative conceptual radionuclide transport models in the saturated, fractured granite and the estimation of transport parameters for use in radionuclide transport models. To achieve these objectives, a cross-hole tracer test involving the simultaneous injection of both nonsorbing and sorbing solute tracers was conducted at the site in 1999 and 2000. A secondary objective of the tracer test was to determine how well the field-scale transport behavior of a sorbing solute could be predicted based on laboratory-derived sorption parameters. This objective is important because only laboratory sorption measurements are feasible for radionuclides.

1.2. Conceptualization of Flow and Transport at the SHOAL Site

[4] Flow in fractured granitic rocks that have a low-permeability matrix is expected to occur almost exclusively in fractures. However, even relatively low-porosity matrices are capable of storing a significant volume of stagnant or nearly stagnant water compared to fracture volumes. Solute diffusion between flowing water in fractures and stagnant water in the matrix, known as matrix diffusion, can attenuate both concentrations and travel times of contaminants migrating through fractured systems [Freeze and Cherry, 1979; Neretnieks, 1980; Grisak and Pickens, 1980; Tang et al., 1981; Maloszewski and Zuber, 1983, 1985]. We refer to such systems as dual-porosity systems, which represent a subset of the more general dual-permeability classification in that the matrix permeability is smaller by a factor of 100 or more than the bulk fracture permeability, so flow through the matrix can be assumed negligible in transport calculations.

[5] On the basis of available information, the saturated zone at the SHOAL site might be expected to behave as a dual-porosity system. Matrix porosities at the site range from 0.01 to 0.02, as determined from the difference between wet and dry weights of core fragments. Laboratory measurements of matrix diffusion coefficients for bromide in granites from the SHOAL site range from 1.2 × 10−7 to 2 × 10−6 cm2/s (the lower end of the range determined from diffusion cells and the upper end of the range from cores suspended in bromide solutions). Bulk hydraulic conductivities from aquifer tests at the site range from approximately 5 × 10−7 to 8 × 10−4 cm/s [Pohll et al., 1999], which is 5 to 8 orders of magnitude larger than conductivities measured for intact SHOAL granite cores (1 × 10−12 to 3 × 10−12 cm/s) [University of Nevada, 1965].

1.3. Test Approach Using Multiple Tracers

[6] The strategy to accomplish the study objectives involved conducting a cross-hole tracer test in which multiple tracers with different physical and chemical characteristics were simultaneously injected. By dissolving the tracers in the same solution and simultaneously introducing them, it is assured that they all experience the same flow field and hence follow identical flow pathways through the system. This is especially important in field tests where it can be extremely difficult to reproduce flow conditions in different tracer injections because of potential equipment problems and possible irreversible changes in the system (e.g., well development, biofouling, steady drawdown, etc.). Note that although all simultaneously injected tracers should have access to the same pore space in the flow system, they will move through this pore space at different rates due to the differences in their physical and chemical characteristics. These transport differences form the basis for the tracer testing strategy at the SHOAL site.

[7] The rationale for using multiple tracers in a cross-hole test is illustrated in Figure 2. Figure 2a shows hypothetical solute tracer responses (normalized concentration versus time) for a cross-hole tracer test with a short injection pulse in a single-porosity system. Note that there is no distinction between nonsorbing tracers with different diffusion coefficients in this plot because there is no secondary porosity for the tracers to diffuse into and hence no separation of their responses. The sorbing tracer response is delayed in time and is lower in concentration than the nonsorbing tracers.

image

Figure 2. Simulated cross-hole responses of tracers with different physical and chemical characteristics in single- and dual-porosity media, illustrating how multiple tracers can be used to distinguish between the two types of systems. The responses are for a short-duration injection pulse of tracer at time zero into a constant-velocity flow system. For the dual-porosity simulations the matrix is assumed to be semi-infinite.

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[8] Figure 2b shows hypothetical solute tracer responses for a short-residence-time tracer test in a dual-porosity system. In this case, there is a separation between nonsorbing tracers with different diffusion coefficients, with the higher diffusivity tracer exhibiting a lower peak concentration and a longer tail than the lower diffusivity tracer. This separation occurs because the higher diffusivity tracer diffuses more readily into the matrix than the lower diffusivity tracer, resulting in a lower recovery at early times but a longer tail due to subsequent diffusion back out of the matrix after the tracer pulse has passed. Several investigators have exploited this expected behavior of nonsorbing tracers of different diffusivities in tracer experiments in fractured rock systems to identify and parameterize matrix diffusion [Moench, 1995; Sanford et al., 1996; Jardine et al., 1999; Maloszewski et al., 1999; Reimus and Haga, 1999; Becker and Shapiro, 2000; Callahan et al., 2000]. However, it is not possible with nonsorbing tracers alone to unequivocally distinguish between true diffusion into the matrix and diffusion only into stagnant (or near-stagnant) free water in a fracture system. Distinguishing between these two cases is very important for large-scale radionuclide transport predictions because diffusion into the matrix implies that there will be much more stagnant water accessible to radionuclides (and hence longer travel times and lower concentrations) than if diffusion is only into free water. Diffusion into the matrix also implies that there will be much more surface area available for radionuclide sorption.

[9] Figure 2b also shows three possible responses for a sorbing tracer: one with sorption occurring in the matrix, one with sorption occurring in both the fractures and the matrix (if the fractures have sorptive mineral coatings or are filled with sorptive granular material), and one with essentially no sorption because diffusion is assumed to occur primarily into free water where there is minimal surface area available for sorption (curve labeled Sorbing, but Diffusion only into Free Water). In all three cases, it is assumed that the sorbing tracer has a diffusivity that falls between that of the two nonsorbing tracers (as is the case for lithium ion in the SHOAL tracer test). For the matrix-only case, the sorbing tracer response is attenuated in peak concentration but not significantly attenuated in time relative to the nonsorbing tracers, whereas in the fracture-and-matrix case both a concentration and time attenuation are apparent. The minimal time attenuation of the sorbing tracer relative to the nonsorbing tracers in the matrix-only case occurs because the travel time is not long enough for all of the sorbing tracer mass to fully interact with the matrix.

[10] Figure 2c shows three different responses of a sorbing tracer (fracture sorption only, matrix sorption only, and a combination of fracture and matrix sorption) relative to nonsorbing tracers in a dual-porosity system where the residence time is long enough that all of the tracer mass comes into intimate contact with the matrix. In this case, all the sorbing tracer responses are attenuated in time relative to the nonsorbing tracers, but the differences in the responses suggest that it should still be possible to qualitatively determine whether sorption is occurring in the matrix, in fractures, or in both domains. Note that the responses of the nonsorbing tracers with different diffusivities also have noticeably different first and peak arrival times when residence times are longer.

[11] The simulated responses in Figure 2 suggest that a multiple tracer test involving the simultaneous injection of nonsorbing tracers with different diffusion coefficients and a sorbing tracer should allow discrimination between a single-porosity system and a dual-porosity system, as well as discrimination between whether sorption is occurring in the fractures, the matrix, or both (for a dual-porosity system). If a dual-porosity response with matrix sorption is observed, and the relative diffusion coefficients of the two nonsorbing tracers are known, the differences in the responses of the nonsorbing tracers can be used to distinguish between the effects of matrix diffusion and hydrodynamic dispersion. This distinction will allow well-constrained estimates of mean residence times, dispersion coefficients, and matrix diffusion parameters in the tracer test (as opposed to evaluating the response of only a single nonsorbing tracer). Effective sorption parameters associated with the response of a simultaneously injected sorbing tracer can then be estimated by assuming that the sorbing tracer experiences the same mean residence time, longitudinal dispersivity, and matrix diffusion (subject to its diffusion coefficient) as the nonsorbing tracers. In this case, only the sorption parameter(s) need be adjusted to obtain a model fit/match to the sorbing tracer response.

2. Experimental Design and Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

[12] The SHOAL tracer test was conducted by simultaneously injecting two nonsorbing anion tracers with different diffusion coefficients, bromide and pentafluorobenzoate, along with a weakly sorbing cation tracer, lithium ion. The properties of these tracers are summarized in Table 1 along with the injection masses and concentrations used in the test.

Table 1. Tracer Characteristics, Injection Masses, and Injection Concentrations in the Cross-Hole Tracer Testa
TracerDf,b cm2/sSorptionInjection Mass, kgInjection Conc., mg/L
  • a

    Tracers were dissolved in ∼6000 L of groundwater. See Table B1 for groundwater chemistry.

  • b

    Df is free water diffusion coefficient (Callahan et al. [2000] found that diffusion coefficients in rock matrices had the same ratio as free water diffusion coefficients for PFBA and bromide).

  • c
  • d

    From Newman [1973], based on ionic conductances at infinite dilution.

  • e

    Lithium was injected as 22.62 kg LiBr and 91.11 kg LiCl. Note that when lithium bromide and lithium chloride are the dominant salts in solution, these three ions should codiffuse with effectively the same diffusion coefficient.

PFBA7.2 × 10−6cNone2.30400
Bromide2.1 × 10−5dNone20.813600
Lithiume1.0 × 10−5dWeak16.742900

[13] Lithium ion was selected as a sorbing tracer because it has the desirable reactive tracer characteristics of high solubility, low toxicity, low background concentrations, low cost, environmental acceptability, and rapid, yet sufficiently weak sorption so that cross-hole responses can be obtained in reasonable times in field tests. Lithium ion sorbs almost exclusively by cation exchange, which is the dominant sorption mechanism for two relatively abundant radionuclides of concern at the SHOAL site, 137Cs+ and 90Sr++. Thus, although Cs+ and Sr++ should exchange more strongly than Li+, Li+ is a reasonable surrogate for determining the appropriate conceptual transport model to use for these radionuclides at the SHOAL site.

[14] The cross-hole tracer test was conducted between wells HC-6 and HC-7, which are separated by about 30 m and located approximately 350 m southeast of the SHOAL nuclear test cavity (Figure 1). HC-6 was used as the tracer injection well and HC-7 as the production well. Both well completions (Figure 1) consist of a single ∼35-meter-long screened interval extending from approximately 40 to 75 m below the static water table, which is 299 m below ground surface (1386 m above mean sea level). The total depth of the wells is ∼375 m below ground surface, or just below the bottom of the well screens. A submersible pump was set near the bottom of the screen in HC-7 for hydraulic and tracer testing. Pressure transducers were installed to continuously monitor water levels during testing. There were no packers or plugs installed in either of the wells.

[15] Prior to injecting tracers, a weak-dipole flow field was established by pumping HC-7 at a rate of ∼11.3 liters per minute (L/min) while recirculating ∼1.13 L/min of the produced water into HC-6. After establishing a quasi-steady state flow field, as indicated by relatively stable water levels, the recirculation was interrupted, and ∼250 L of a solution containing 500 g of sodium iodide was immediately injected into HC-6 at the same rate as the recirculation flow (∼1.13 L/min). The recirculation of water from HC-7 was then immediately resumed after the tracer solution was injected. Iodide concentrations were monitored in the HC-7 water for about 8 days prior to the injection of all other tracers. The purpose of the sodium iodide injection was to determine if there was a very fast, high-concentration response in HC-7, which would allow tracer injection masses and sample collection intervals to be optimized in the multiple tracer test. Iodide was never detected at HC-7, which made it clear that the multiple tracer test would require large tracer injection masses and a relatively long time to complete.

[16] The multiple tracers were injected into HC-6 in the same manner as the sodium iodide; i.e., with essentially no interruption of the recirculation flow before and after tracer injection. The production and recirculation/injection flow rates were ∼11.3 L/min and ∼1.13 L/min, respectively. The masses and concentrations of tracers in the injection solution are listed in Table 1. The tracers were dissolved in ∼6000 L of HC-7 groundwater, which was injected over a period of ∼90 hrs. The injection well was mixed and sampled during and after tracer injection by circulating water to the surface at ∼1.1 L/min using a Bennett pump.

[17] The multiple tracer injection was initiated on 10 November 1999. The method of tracer injection ensured that weak-dipole flow conditions were maintained with no significant transients before, during, and after tracer injection. The weak dipole was maintained until the last sample was collected from HC-7 on 24 September 2000, so the tracer test was conducted for a total of 319 days or ∼7650 hrs. The production rate from HC-7 decreased gradually over the first 2500 hrs of the test to about 7.5–8.3 L/min, where it remained for the rest of the test. The water level in HC-7 decreased steadily for most of the test, starting at about 12 m below the static water level and ending up almost 60 m below the static water level, or only a few meters above the pump. This steady decrease in water level should not have affected the relative responses of the tracers, which served as the basis for the tracer test interpretations (see section 3).

[18] Water samples were collected at HC-7 either manually or with an automatic sampler throughout the test. The sampling interval was gradually increased as the test progressed. Samples were also collected manually from HC-6 as long as the Bennett pump was used to circulate water (until about 10 days after tracer injection had ceased).

[19] Samples were analyzed for bromide (Br) by liquid chromatography (with a conductivity detector) and for lithium (Li+) by atomic absorption at the Desert Research Institute in Reno, Nevada. Pentafluorobenzoate (or PFBA) was analyzed by high-pressure liquid chromatography (with a UV absorbance detector) at the Harry Reid Center for Environmental Studies of the University of Nevada-Las Vegas in Las Vegas, Nevada.

3. Results and Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

[20] Figure 3 shows the normalized concentrations of the three solute tracers at the production well, HC-7, throughout the test. The concentrations are normalized to the tracer injection masses (units of μg/L-kg injected or L−1 × 109). The fractional recoveries of the tracers were 0.18 for bromide, 0.21 for PFBA, and 0.013 for lithium. Model fits to the tracer data are also shown in Figure 3 (the solid lines); these are discussed below.

image

Figure 3. Normalized concentrations (background subtracted) of tracers in HC-7 water as a function of time since tracer injection into HC-6 started. Concentrations are normalized to injection masses. Also shown are RELAP model fits to the tracer responses.

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[21] It is apparent from Figure 3 that there is considerable separation between the responses of bromide and PFBA, with PFBA having a higher peak concentration and the tails of the responses appearing to converge or cross over at late times. Referring to Figure 2, these features are qualitatively consistent with a dual-porosity transport system. The apparent leveling off or increase in normalized concentrations of bromide and PFBA at late times is an interesting result that hints at the possibility of a bimodal system response. This inflection is apparently real because both tracers behave similarly despite the fact that the tracers were analyzed in two different laboratories. The inflection could be due to a set of very slow fracture flow pathways just beginning to contribute tracer mass to the production well at late times. Alternatively, it may be the result of the declining production rate during the test, which would result in a leveling-off of tracer concentrations due to the fact that tracer diffusion out of the matrix will be into slower moving water. Whatever the reason, the test interpretation procedure (described in section 4) inherently placed greater emphasis on matching the apparent convergence or crossover of the nonsorbing tracer responses than on matching the actual inflection of the responses at late times.

[22] The lithium response is highly attenuated in concentration compared to the nonsorbing tracers, although it is not nearly as attenuated in time. This response is qualitatively consistent with a dual-porosity transport system in which most of the sorption is occurring in the matrix (after diffusive mass transfer from the fractures), with possibly a small amount of sorption also occurring on fracture surfaces.

[23] To obtain estimates of transport parameters in the flow system, the semi-analytical dual-porosity transport model RELAP (reactive transport Laplace transform inversion computer code) was used to simultaneously fit the tracer responses. RELAP is described in detail in Appendix A.

[24] RELAP provides a simultaneous least squares fit of up to four tracer data sets by automatically adjusting the following model parameters (which arise from the dimensionless forms of the governing equations described in Appendix A): the mean fluid residence time in fractures (τ, hr), the Peclet number (Pe = S/α, where S = distance between wells, m, and α = dispersivity in fractures, m), the mass fraction of tracers participating in the test (f), a matrix diffusion mass transfer parameter (equation image), where ϕ = matrix porosity, b = fracture half aperture, cm, and Dm = matrix diffusion coefficient, cm2/s), the ratio of stagnant water volume to flowing water volume (ϕL/2b), the fracture retardation factor (Rf), and the matrix retardation factor (Rm). The fractional mass participation, f, is an adjustable constant (ranging from 0 to 1) that all the simulated tracer response curves are multiplied by to account for an arbitrary fraction of the tracer injection solution that never establishes communication with the production well and therefore doesn't “participate” in the test. Fractional tracer mass participation is frequently observed in field tracer tests in fractured rock [e.g., Reimus and Haga, 1999; Reimus et al., 2003], presumably due to (1) dense tracer solutions “sinking” out of the zone of influence of pumping, (2) a significant volumetric flow of tracer solution into the matrix within the injection wellbore (this tracer mass will not make it to the production well during the tracer test because of the very low flow velocities in the matrix), or (3) the loss of tracer mass due to stagnation points induced either by recirculation or by the superposition of the induced flow field on the ambient flow field. Although these phenomena can affect absolute tracer responses, they should not, in principle, affect the relative responses of different tracers that are injected simultaneously.

[25] The interpretation of the tracer responses involved first fitting the two nonsorbing tracer responses by simultaneously adjusting all of the parameters listed above with the constraint that the matrix diffusion coefficient, Dm, for bromide was 3 times that of PFBA, based on literature data [Newman, 1973; Benson and Bowman, 1994] and the experimental results of Callahan et al. [2000]. Also, Rf and Rm were held equal to 1 for both nonsorbing tracers. Figure 3 shows all of the data points that were fitted for the two tracers. The fits were obtained by minimizing the sum of squares differences between the data and model with no weights assigned to any of the points. However, the sum of squares differences for each data set were divided by the number of points in each set to avoid biasing the fits toward the set with more data.

[26] This fitting procedure implicitly assumed that both tracers had exactly the same mean residence time, Peclet number, mass fraction participation, and characteristic fracture spacing during the tracer test, which is justified because the tracers were injected simultaneously and should thus have experienced the same flow system and same flow conditions. The tracers should have had the same Peclet numbers despite their different diffusion coefficients because Taylor dispersion (dispersion resulting from tracer diffusion across the local parabolic velocity profile in fractures) was estimated to be at least three orders of magnitude less than macrodispersion [Detwiler et al., 2000].

[27] Uniqueness of the RELAP fits was addressed by fixing ϕL/2b and one of the four remaining adjustable parameters (τ, Pe, equation image, and f) while varying the other three to determine a range over which the sum of squares differences between model and data increased by no more than 50% relative to the best fit. A 50% increase resulted in a noticeable degradation of the fits, but the fits were still considered reasonable given the scatter in the tracer data.

[28] After simultaneously fitting the nonsorbing tracer responses, the lithium response was fitted by adjusting Rf and Rm while holding all other parameters equal to the values that provided the best fit to the nonsorbing tracers. However, Dm for lithium was assumed to be two-thirds that of bromide (and ∼2 times that of PFBA). This value was selected because the free-diffusion coefficients of lithium and bromide should differ by about a factor of 0.5 at infinite dilution based on limiting ionic conductances [Newman, 1973], but when lithium and bromide are the dominant ions in solution, they will diffuse together to maintain charge balance. The retardation factors obtained for lithium were not strongly dependent on its assumed diffusion coefficient.

[29] The best RELAP fits to each tracer data set are shown as the solid lines in Figure 3. The model parameters corresponding to these fits are listed in Table 2. Table 2 also provides ranges of parameter values that yielded sum of squares differences between model and data that were within 50% of the best fit. These ranges can be considered a measure of the uncertainty associated with the parameter values obtained from the interpretive procedure. The parameter values yielding good fits are correlated in that higher mean residence times correspond to higher mass fractions and lower Peclet numbers (and vice versa). However, values of equation image are not highly correlated with any of the other model parameters.

Table 2. Best Fitting RELAP Model Parameters Presented as a Range of Values That Provided Good Simultaneous Fits to the Three Tracer Responsesa
ParameterParameter Value
  • a

    The range corresponds to parameter values yielding a sum of squares of differences between model and data within 50% of the best fit. Bold indicates the parameter values associated with the best model fits (shown in Figure 3). The assumption of linear or radial flow affects only the values of τ and Pe.

  • b

    An infinite value for this parameter yielded a sum of squares differences between model and data within 60% of the best fit.

  • c

    The range is 140 to 260 if the fracture retardation factor is set to 1 (i.e., no sorption in fractures). A value of 140 used for the matrix-only fit of Figure 4.

f, mass fraction participation0.65 [0.51–1.0]
τ, linear flow, hr11,800 [8000–40,000]
Pe, linear flow1.9 [0.55–3.0]
τ, radial flow, hr7650 [5800–16,200]
Pe, radial flow3.2 [1.4–4.7]
equation image for Bromide (PFBA), s−1/20.00011 [0.000071–0.00012] 0.000064 [0.000041–0.000071]
ϕL/2b (stagnant water/flowing water)0.5 [0.4–0.6]b
Lithium fracture retardation factor1.5 [1.3–1.8]
Lithium matrix retardation factor110 [90–220]c

[30] The best fit to the lithium data implies very little sorption in fractures but considerable sorption in the matrix (the best fitting fracture retardation coefficient was only 1.5). Figure 4 shows the best fits to the lithium data assuming only matrix sorption, only fracture sorption, and fracture plus matrix sorption (the latter is the lithium curve from Figure 3). The normalized concentration scale is logarithmic in Figure 4 so that the first arrival times and the different shapes of the model curves can be seen more readily. Clearly, the fracture-plus-matrix sorption model offers the best fit to the lithium data and the fracture-only model offers the worst fit. It was not possible to obtain a reasonable fit to the lithium data without assuming a significant amount of matrix sorption.

image

Figure 4. RELAP fits to the normalized lithium response from the SHOAL tracer test assuming different sorption behavior. The fracture plus matrix curve is the same curve as in Figure 3.

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[31] All of the parameters in Table 2 were obtained assuming a constant production rate of 9.5 L/min. Note that separate estimates of τ and Pe are provided depending on whether linear flow or radial flow is assumed (see Appendix A). The quality of the fits is not affected by this assumption, but the best fitting model parameters differ. In a heterogeneous, confined aquifer, the flow velocity to a single production well with no recirculation into an injection well is expected to vary between linear and radial [National Research Council, 1996]. Thus, if it is assumed that the test interval was reasonably confined, presenting the two values of τ and Pe in Table 2 is a rough way of bounding these model parameter estimates as a result of flow field uncertainty. Although the SHOAL system was unconfined (open to the water table), this approach should still yield reasonable bounds for τ and Pe, as the flow velocities in pathways carrying tracers from HC-6 to HC-7 should have started out relatively high due to the recirculation into HC-6, gone through a minimum, and then increased again in the vicinity of HC-7. Thus the weak dipole should have resulted in a flow pattern that was intermediate between linear and radial flow.

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

[32] Estimates of transport parameters that can be used directly in solute transport models were derived from the best fitting model parameters in Table 2. These parameter estimates are presented in Table 3 as ranges of values that are consistent with the tracer test interpretation(s). Additional discussion of these ranges and how they were derived is provided in the following sections. This parameter estimation exercise has several important implications for radionuclide transport at the SHOAL site.

Table 3. Transport Parameter Ranges Derived From the Tracer Test for Use in Radionuclide Transport Modeling at the SHOAL Sitea
ParameterParameter Value
  • a

    See text for discussion.

  • b

    These estimates are probably high (nonconservative) due to flow heterogeneity in the system.

  • c

    Assumes that PFBA and bromide effectively bound sizes of radionuclide solution species.

  • d

    ∞ is listed as an upper limit because RELAP fits assuming a semi-infinite matrix (L = ∞) resulted in a sum of squares differences between model and data that was within 60% of the best fit.

Effective Flow Porosity0.027–0.22b
Longitudinal Dispersivity, m6–80
equation image for Radionuclides, s−1/20.000041–0.00012c
2b, average fracture aperture, cm0.058–1.38
Ratio of stagnant to flowing water volumes0.4–∞d
Lithium Kd value in matrix, cm3/g0.3–3.4

4.1. Conceptual Transport Model

[33] Even without quantitative parameter estimation, it is clear that the tracer responses are consistent with a dual-porosity transport model for the SHOAL site. It is simply not possible to account for the differences in the bromide and PFBA responses, or the relatively small time attenuation but significant concentration attenuation of the lithium response relative to the nonsorbing tracers without invoking diffusion between flowing fractures and stagnant matrix water. Some diffusion into stagnant water within fractures (e.g., dead-end fractures or along rough fracture walls) cannot be ruled out. However, if the stagnant water were primarily in fractures, the surface area for sorption would be small relative to the surface area available in the matrix (per unit volume of water), and it is unlikely that there would be as much concentration attenuation of lithium relative to the nonsorbing solutes as observed in the tracer test. The inability to fit the lithium response without assuming a significant amount of matrix sorption (Figure 4) suggests that much of the accessible stagnant water is in the matrix rather than in fractures.

4.2. Relative Volumes of Stagnant and Flowing Water in System

[34] The fact that the nonsorbing tracer responses were best fit by a relatively small value of the quantity ϕL/2b (the ratio of stagnant to flowing water volume) suggests that the volume of matrix pore water in diffusive communication with flowing fractures may have been relatively small. Sensitivity analyses using RELAP indicated that the best simultaneous fits to the nonsorbing tracer responses were obtained when ϕL/2b ranged from 0.4 to 0.6. However, the sum of squares differences between RELAP simulations when ϕL/2b was set to infinity (infinite fracture spacing L) and the nonsorbing tracer data was only about 60% greater than the differences obtained when a finite fracture spacing was assumed (and the fits looked quite reasonable). Thus the possibility of a significantly greater ratio of stagnant to flowing water volume in the system than that deduced from the best RELAP fits cannot be ruled out.

[35] It is important to emphasize that a small value of ϕL/2b does not necessarily imply small fracture spacings, but rather it more generally implies that a no-flux boundary may be encountered a relatively short distance into the matrix. This is physically reasonable for a competent granite where significant diffusion may be limited to a damaged zone adjacent to the fracture wall [Winberg et al., 2000].

4.3. Effective Retardation Factors Due to Matrix Diffusion

[36] The overall effective retardation factor of a solute in a dual-porosity system due to both sorption and matrix diffusion will be equal to the first moment of the tracer residence time obtained from equation (A6) in Appendix A, which is [Becker, 1996]:

  • equation image

where η is porosity within fractures.

[37] Assuming that L/2 is large compared to bη, equation (1) yields an effective retardation factor for a nonsorbing solute of 1.5 using the best fitting value of ϕL/2b (0.5). However, this value should be considered a lower bound because, as mentioned above, the tracer test data do not rule out the possibility of much larger L values and hence much larger retardation factors.

[38] The estimates of ϕL/2b derived from the RELAP interpretations are very sensitive to the late-time tracer responses; the small ϕL/2b values that result in the best fits are almost entirely due to fitting the late-time convergence or crossover of the bromide and PFBA responses. If the tracer test had been stopped after 5000 hrs, equally good fits would have been obtained assuming ϕL/2b = ∞, which would have led to a very different (and possibly nonconservative) conclusion. This point serves to illustrate the importance of conducting tracer tests for sufficiently long times that the relative tracer tailing behavior can be ascertained.

4.4. Fracture Apertures

[39] An estimate of the average fracture aperture (2b) experienced by the tracers in the tracer test can be obtained from the estimate of the lumped diffusive mass transfer parameter, equation image, provided independent estimates of matrix porosity, ϕ, and matrix diffusion coefficients, Dm, are available. Using laboratory measurements of ϕ (0.01 to 0.02) and Dm for bromide (1.2 × 10−7 to 2 × 10−6 cm2/s) in SHOAL granites, a range of 0.058 to 1.38 cm is obtained for 2b. This large range reflects the positive correlation between matrix porosity and diffusion coefficient observed in laboratory experiments. The upper end of this range seems unrealistically high for a fractured granite; however, it is presented here to emphasize the unconstrained nature of aperture estimates obtained from independent measures of equation image, ϕ, and Dm. These aperture estimates should be distinguished from friction loss or cubic law aperture estimates that are obtained from hydraulic responses [Tsang, 1992].

4.5. Lithium Sorption Behavior

[40] The best fits to the lithium response were obtained with a fracture retardation factor of about 1.5 and a matrix retardation factor ranging from 90 to 200 (depending on the value of ϕL/2b assumed; the retardation factor was lower for larger ϕL/2b values). However, reasonable fits were also obtained with no fracture retardation and matrix retardation factors ranging from about 140 to 260 (again, depending on the value of ϕL/2b assumed). A fracture retardation factor of 1 does not necessarily imply that sorption was not occurring on fracture surfaces, it merely suggests that the majority of the lithium sorption occurred after a diffusive mass transfer step to sorptive surfaces.

[41] A linear partition coefficient, Kd, can be deduced from the fitted matrix retardation factor by simple rearrangement of the expression defining the retardation factor:

  • equation image

Lithium should sorb to SHOAL granite primarily by cation exchange with Na+, Ca++, Mg++, and possibly K+, all of which are present in the granite and the groundwater. Cation exchange normally results in nonlinear sorption isotherms when concentration ranges are large [Anghel et al., 2002; Sullivan et al., 2003]. However, laboratory batch sorption experiments of lithium onto SHOAL granite (Appendix B) indicated that sorption was essentially linear over a concentration range from ∼1 mg/L to ∼1000 mg/L. The lithium concentration injected in the field test was ∼2900 mg/L, although this should have rapidly diluted after injection; and the observed concentrations at the production well were ∼0.1 mg/L (after subtracting background). The production well concentrations, however, should not be taken as a direct indication of concentrations in traced water flowing into the well because significant dilution is normally expected in a pumped well due to mixing of traced and untraced water entering the well. Thus the laboratory range of ∼1–1000 mg/L should have effectively bracketed the range of lithium concentrations in the field.

[42] Assuming that the matrix porosity falls within the range of 0.01 to 0.02, the lithium Kd value in the matrix falls within the range of 0.34 to 2.0 cm3/g. The laboratory batch measurements (Appendix B) indicated a Kd value no greater than ∼0.2 cm3/g at the 95% confidence level. This comparison indicates that the lowest field-derived Kd value is considerably greater than the upper 95% confidence bound of the laboratory values. Thus it can be concluded that the apparent lithium sorption in the field test was greater than in the laboratory tests.

[43] The fact that the field lithium Kd values were greater than the laboratory Kd values suggests that lithium may have come in contact with alteration minerals in the field that were not present in the lab rock samples. The latter came from the “tailings” pile associated with the shaft that was drilled for the original SHOAL nuclear test in the early 1960s. Any loosely adhering fracture alteration minerals (e.g., clays) that may have been present on the original material could have been removed during the original mining operation or washed away during the nearly 40 years of exposure to surface weather conditions in the tailings pile.

4.6. Effective Flow Porosity

[44] Contaminant transport predictions are generally very sensitive to assumed flow porosities because transport rates are directly proportional to the specific discharge divided by flow porosity. Indeed, flow porosity was identified as a very important, though uncertain, parameter for assessing radionuclide transport at the SHOAL site by Pohll et al. [1999].

[45] The effective flow porosity in a cross-hole tracer test without recirculation can be estimated from the following equation, which assumes a confined, homogeneous, isotropic flow system:

  • equation image

where θ is flow porosity, Q is production flow rate (m3/hr), τ is mean residence time (hr), S is distance between wells (m), and T is formation thickness (assumed to be well screen length, 35 m). With recirculation, the situation is complicated by the fact that there is a hypothetical stagnation point, and hence the mean tracer residence time theoretically approaches infinity. However, the interpretive method described in the previous section allows for incomplete tracer mass recoveries that could result from stagnation, so a finite estimate of the mean tracer residence time is always obtained. Guimera and Carrera [2000] discuss an alternative method of estimating effective flow porosity from peak, rather than mean, tracer arrival times in tests with partial recirculation. This method provides what might be considered a lower bound estimate of flow porosity using the following equation [Guimera and Carrera, 2000]:

  • equation image

where Qi is injection flow rate (m3/hr), Qe is production flow rate (m3/hr), tpeak is peak tracer arrival time (hr), and

  • equation image

For the range of mean arrival times (Table 2) and flow conditions in the SHOAL tracer test, equation (3) leads to flow porosity estimates ranging from 0.033 to 0.22, and equation (4) leads to an estimate of 0.027 (assuming an average production rate of 9.5 L/min and a peak arrival time of ∼3750 hrs based on the tracer responses shown in Figure 3). These flow porosities intuitively seem high given our knowledge of the fracture system at the SHOAL site, which is admittedly somewhat limited. Assuming an average fracture aperture of 5 mm (at the upper end of the range in Table 3), flow porosities ranging from 0.027 to 0.22 translate to a flowing fracture spacing of only about 2.5 to 20 cm, which is at odds with borehole information at the site (at least at the lower end of the range). An average fracture aperture of 1 mm (at the lower end of the range in Table 3) results in a fracture spacing of only 0.5 to 4 cm, which is even more at odds with available information.

[46] The seemingly large effective porosity estimates obtained from equations (3) and (4) are most likely due to heterogeneities in the flow field. Reimus [2003] showed that flow heterogeneity in fractured media can result in flow porosity estimates from cross-hole tracer tests that exceed true flow porosity by 1 to 3 orders of magnitude. It is certainly conceivable that high-conductivity features such as large, open fractures or faults could transmit the vast majority of the flow to the production well. If these features do not happen to pass near the injection well, the effective flow rate drawing tracers to the production well will be greatly reduced relative to what would occur in a radial flow field. For these reasons, and because large flow porosities result in nonconservative transport times for a given specific discharge, we do not advocate the use of the flow porosity range derived from equations (3) and (4). Further research is needed to make appropriate corrections to these flow porosities and to assign meaningful uncertainty bounds to them.

4.7. Longitudinal Dispersivity

[47] Longitudinal dispersivity estimates from cross-hole tracer tests generally have considerable uncertainty as a result of (1) uncertainty in the actual tracer transport distance (actual transport pathways are unknown), (2) uncertainty associated with whether the flow field is radial, linear, or some combination, (3) uncertainty in the amount of apparent dispersion caused by nonidealities such as a poorly mixed injection wellbore or density/buoyancy effects, and (4) uncertainty in the amount of apparent dispersion caused by recirculation or the ambient flow field. It is beyond the scope of this paper to address in detail the possible effects of each of these uncertainties on the longitudinal dispersivity estimates provided in Table 3. These estimates can be considered “upper and lower bounds” that were obtained as follows.

[48] 1. The maximum transport distance was assumed to be the distance from the top of one well screen to the bottom of the other (∼47 m), while the minimum transport distance was assumed to be the linear distance between the wells (30 m).

[49] 2. The radial and linear Peclet numbers were both used to obtain estimates of the dispersivity for the two cases above (α = S/Pe), and the most extreme values were used for the upper and lower bounds.

[50] 3. Although the injection wellbore was assumed to be well mixed, the RELAP model accounted for the gradual release of tracer from the borehole to the formation by assuming an exponential decay in tracer concentration in the wellbore. The time constant was determined from measurements of tracer concentrations in samples collected from the injection well during and after injection, as shown in Figure 5. Thus the slow release of tracers from the injection well did not bias the dispersivity (or mean residence time) estimates.

image

Figure 5. Bromide concentrations in the injection well (HC-6) during and after tracer injection. After the injection ceased (∼95 hrs), the concentrations follow an exponential decay with a time constant of ∼13 hrs. This time constant was used to simulate injection wellbore mixing in the RELAP code.

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[51] 4. The dispersion caused by recirculation was “subtracted out” of the overall observed dispersion to obtain the “true” dispersivity (longitudinal) in the aquifer following the procedure of Reimus et al. [2003]. For the ∼10% recirculation in the SHOAL tracer test, this correction was less than 5% of the uncorrected dispersivity.

[52] No attempt was made to account for density/buoyancy effects or the effects of the ambient flow field on the longitudinal dispersivity estimates. The wide range of longitudinal dispersivities in Table 3 indicates that dispersivity is not well constrained by the tracer test interpretations.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

[53] The results and interpretation of the multiple-tracer cross-hole test conducted at the SHOAL site indicate that a dual-porosity conceptual transport model is appropriate for describing dissolved radionuclide transport at the site. Furthermore, the results and interpretation are consistent with stagnant water in the dual-porosity system being primarily in the matrix where there is plenty of surface area for radionuclide sorption, as opposed to being free water in fractures. However, the best fitting model parameters indicate that the ratio of stagnant to flowing water volume in the system may be as small as 0.5, which translates to an effective retardation factor of only 1.5 for nonsorbing solutes. This value should be considered a lower bound because reasonable fits to the tracer responses could also be obtained assuming a large ratio of stagnant to flowing water volume that would result in a much larger retardation factor.

[54] Lithium sorption in the field tracer test was considerably stronger than in laboratory batch sorption experiments, possibly because sorbing mineral phases present in the field were not present in the rock cuttings used in the laboratory tests. This result suggests that the practice of using laboratory sorption data in field-scale transport predictions of cation-exchanging radionuclides, such as 137Cs+ and 90Sr++ (which sorb by the same mechanism as Li+), should be conservative for the SHOAL site.

[55] Estimates of effective fracture apertures, flow porosity, and longitudinal dispersivity deduced from the field tracer test all spanned ranges of an order of magnitude or more. However, the uncertainty in these parameters was less than it would have been if only a single tracer had been used in the tracer test. Effective flow porosity was believed to be significantly overestimated because the estimation method involved assuming a homogeneous, isotropic medium, whereas the actual flow field was undoubtedly highly heterogeneous.

[56] The authors recognize that the tracer test interpretation using a semi-analytical solution that assumes an idealized geometry and a steady flow rate is a considerable simplification of reality. However, the information needed to support more sophisticated representations and models of the flow and transport system is sparse to nonexistent. Furthermore, the agreement between the semi-analytical model and the tracer responses was very good (Figure 3). Although the introduction of additional model complexity could do nothing but improve this agreement, it appears that all of the critical features of the tracer responses have been effectively captured, and the introduction of additional complexity, especially in light of the minimal information to support it, is not justified.

[57] The tracer test results are intended to support predictive calculations that span much larger time and distance scales than represented by the test. From this perspective, we consider it best to capture the important transport processes with as concise a model as possible, and then apply this model on a local scale within a flow model that captures the important hydraulic features of the larger-scale system.

Appendix A:: Description of the RELAP Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

A1. Governing Equations

[58] The RELAP computer model essentially combines the Laplace-domain dual-porosity transport equations derived by Maloszewski and Zuber [1983, 1985] (modified to account for linear sorption) with Laplace-domain transfer functions that describe a finite pulse injection, wellbore mixing, and recirculation (the effect of recirculation was to very slightly raise the tail of the tracer response curves, which had negligible effect on the tracer test interpretations). Similar approaches have been used by others [Moench, 1989, 1995; Becker and Charbeneau, 2000]. Maloszewski and Zuber [1983, 1985] assumed that tracer transport in fractures was described by the 1-D advection-dispersion equation with 1-D diffusion occurring into the surrounding matrix perpendicular to the flow direction in fractures. The simplified flow system geometry assumed by RELAP is shown in Figure A1. The model assumes parallel-plate fractures of constant aperture and constant spacing, no concentration gradients across the fracture aperture, and either linear flow (constant flow velocity between injection and production well) or radial flow (flow velocity inversely proportional to distance from production well) in the fractures. The equations describing dual-porosity transport under these conditions are:Fracture (constant flow velocity)

  • equation image

Fracture (radial flow)

  • equation image

Matrix

  • equation image

subject to the following initial and boundary conditions

  • e quation image
  • equation image
  • equation image
  • equation image
  • equation image
  • equation image

where

Cf

tracer concentration in solution in fractures, mg/L;

Cm

tracer concentration in solution in matrix, mg/L;

vf

fluid velocity in fractures, cm/s;

Df

dispersion coefficient in fractures, cm2/s;

Dm

molecular diffusion coefficient in matrix, cm2/s;

Rf

retardation factor in fractures = 1 + AspkA (or 1 + equation image for parallel-plate fractures);

Rm

retardation factor in matrix = 1+ equation image;

Kd

linear sorption partition coefficient, cm3/g;

kA

Kd/Asp linear surface-based sorption partition coefficient, cm3/cm2;

Asp

surface area per unit mass of material in fractures or on fracture walls, cm2/g;

ρB

bulk density in matrix, g/cm3;

η

porosity within fractures (assumed to be 1.0 in this paper);

ϕ

matrix porosity;

b

fracture half aperture, cm;

L

spacing between centerlines of adjacent fractures, or alternatively, two times the distance into the matrix at which a no-flux boundary is encountered, cm.

Note that although Figure A1 shows that L is the spacing between centerlines of parallel fractures, it can be more generally defined as two times the distance from the fracture centerline to a no-flux boundary in the matrix.

image

Figure A1. System geometry assumed in the RELAP code.

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[59] Equation (A1) and (A2) assume that local sorption equilibrium is established (i.e., rapid sorption kinetics relative to fluid velocities), which should be a valid assumption for Li+ cation exchange over the long timescales of the SHOAL tracer test. RELAP also can accommodate rate-limited sorption, although it is limited to linear sorption isotherms. Note that equations (A1) and (A2) reduce to a single-porosity system if the matrix porosity, ϕ, (or the matrix diffusion coefficient, Dm) is set equal to zero.

A2. Laplace Transform Solutions

[60] The transformation of equations (A1) and (A2) to the Laplace-domain, and their subsequent solution in the Laplace domain and inversion back to the time domain are summarized here; details can be found in Appendix B of Reimus and Haga [1999]. The Laplace transform removes the time derivatives from the governing equations, rendering them ordinary differential equations that can be solved by standard methods. For the constant flow velocity case, the Laplace transforms of equations (A1a) and (A2) are Fractures

  • equation image

Matrix

  • equation image

where s is Laplace transform independent variable (replacing time) and equation image is Laplace transform of dependent variable.

[61] Equations (A3) and (A4) are in the same form as the equations solved by Maloszewski and Zuber [1983]. The only difference is that the fracture and matrix retardation factors did not appear in Maloszewski and Zuber's equations because they only considered nonsorbing solute transport. Referring to Maloszewski and Zuber [1983] for details of the derivation, the final result for transport in the fractures coupled with diffusion into the matrix is a single Laplace-domain equation given by

  • equation image

If we introduce the mean fluid residence time in fractures, equation image, and the dimensionless Peclet number, equation image, which are related to the first and second moments of fluid residence time in the flow system, respectively, equation (A5) can be rewritten as

  • equation image

Equation (A6) is a more natural expression to work with than equation (A5) when considering field data because actual distances and fluid velocities in a field experiment will be dependent on flow pathways in the system, but the concepts of the first and second moments of fluid residence time are less ambiguous. Equation (A6) also clearly delineates the expression equation image as an effective mass transfer coefficient for matrix diffusion. This expression can be further divided into a flow-system-dependent part, equation image, and a tracer-dependent part, equation image. Laplace-domain expressions such as equations (A6) are used extensively in control theory and are referred to as “transfer functions” [Coughanowr and Koppel, 1965].

[62] For the radial flow case, RELAP implements the Laplace-domain solution obtained by Becker and Charbeneau [2000] (with retardation factors added). Without derivation, equation (A6) is replaced by

  • equation image

where rW is production well radius, rLW is production well radius divided by separation between injection and production wells,

  • equation image
  • equation image
  • equation image
  • equation image

and Ai(z) is the Airy function [Spanier and Oldham, 1987).

A3. Convolution of Laplace-Domain Transfer Functions to Obtain Tracer Test Responses

[63] Before a meaningful field-scale transport prediction can be obtained, it is necessary to convolute equation (A6) or (A7) with a realistic tracer injection function. In the time domain, such a convolution is accomplished by a convolution integral, but in the Laplace domain it becomes a simple multiplication [Churchill, 1958]:

  • equation image

where X(t) is time domain function and equation image(s) is Laplace transform of time domain function (i.e., transfer function). This process can be extended to more than two transfer functions by taking the product of all applicable functions. Thus, in a field tracer test, transfer functions for tracer injection, pipeline delays, and storage in the injection and production wellbores can all easily be convoluted with the groundwater system transfer function (e.g., equations (A6) or (A7)) to obtain an overall transfer function for the test. We assume the most practical injection function in a field tracer test: a finite-duration, constant concentration pulse, which has a Laplace transform given by

  • equation image

where Co is concentration of injection pulse (mg/L) and Tp is duration of injection pulse (hr). Wellbore storage is accounted for by assuming that the wellbores are well-mixed regions [Moench, 1989, 1995]. The Laplace domain transfer function for a well-mixed region is given by

  • equation image

where γ is time constant, generally assumed to be the volumetric flow rate divided by the volume of well-mixed region, 1/hr. Pipeline delays can be accounted for by a transfer function of the form

  • equation image

where Td is delay time (T). Reinjection of produced water (i.e., recirculation of tracers) can be accounted for with transfer functions as follows:

  • equation image

where equation imageR(s) is transfer function with recirculation, F(s) is transfer function without recirculation, and ε is recirculation ratio (0 = no recirculation, 1 = full recirculation). Equations (A9) through (A12) are convoluted with (multiplied by) either equation (A6) or (A7) in RELAP to obtain an overall transfer function for tracer transport in the SHOAL tracer test that accounts for all the various subsystem components in the test.

[64] RELAP was tested by (1) inverting several simple Laplace transforms with known time domain solutions to ensure that the Laplace transform inversion algorithm was working properly, and (2) comparing predicted nonsorbing solute breakthrough curves in a single-porosity medium to analytical solutions [Kreft and Zuber, 1978; Levenspiel, 1972]. In all cases, excellent agreement was obtained between the model predictions and the analytical solutions.

A4. Numerical Inversion of Laplace Transforms

[65] The final step in obtaining a transport prediction from the transfer functions is to invert the solution from the Laplace domain to the time domain. This is accomplished in RELAP using a Fourier transform procedure embodied in an algorithm obtained at the MathSoft website (http://www.mathsoft.com/appsindex.html), which is summarized here.

[66] The Laplace transform is defined by

  • equation image

where s is Laplace transform variable, t is time, f(t) is time domain function, and F(s) is Laplace domain function. If we set s = λ + jω, where j = equation image, in equation (A13) and change the lower limit of integration to −∞ (which is permissible for any initial-valued problem because f(t) = 0 for t < 0) we obtain

  • equation image

If λ is a constant, equation (A14) can be rewritten as

  • equation image

where f1(t) = f(t)e−λt. Equation (A15) is the Fourier transform of f1(t), for which very efficient inversion algorithms exist to find f1(t) given F(jω). However, the choice of λ must be such that the integral in equation (A15) converges. This is accomplished by choosing

  • equation image

where Tmax is maximum time at which the function f(t) is to be evaluated. Now, if we have an arbitrary Laplace transform, F(s), we can obtain a spectrum by setting s = λ + jω and computing F(s) for equally spaced values of ω; i.e., for

  • equation image

where i = 0, 1, 2, 3…. The inverse Fourier transform of this spectrum will give the function f(t) in equation (A15), from which the desired function f(t) is easily generated from

  • equation image

The inversion algorithm was tested on equations where time domain solutions were known, and it was found to be very accurate. Accuracy for this work was ensured by using a successively greater number of terms in the Fourier transforms until solutions no longer changed.

Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

B1. Methods

[67] Lithium batch sorption tests were conducted over a range of almost 3 orders of magnitude of lithium starting concentrations (∼1 to ∼1000 mg/L). The starting solutions were prepared by dissolving lithium bromide in SHOAL water collected from well HC-7. This water had a significant lithium background concentration (∼0.3 mg/L) that had to be corrected for the lower starting lithium concentrations. The major ion chemistry of the water is provided in Table B1.

Table B1. Major Ion Chemistry of the HC-7 Water Used in the Batch Sorption Experiments
SpeciesValue
Ca++178 mg/L
Na+77 mg/L
Mg++26.4 mg/L
K+5.2 mg/L
Li+0.3 mg/L
Si26 mg/L
SO42−329 mg/L
Cl189 mg/L
HCO3165 mg/L
pH7.75

[68] All sorption measurements were made in triplicate using a solution-solid ratio of 3 mL/g at each of 7 starting concentrations (∼1, 3, 10, 30, 100, 300, 1000 mg/L). One set of triplicate measurements at 10 ppm Li was conducted using a solution-solid ratio of 10 mL/g to see if changing this ratio had any noticeable effect on the partition coefficient (Kd). Also, one additional set of triplicate measurements at 3 mL/g solid-solution ratio and 10 mg/L Li was conducted using a subset of the rock that had an obvious reddish-brown stain that appeared to be an iron oxide. All of the rock was crushed and dry sieved such that the particles used in batch testing were between 75 and 500 microns nominal diameter. The specific surface areas of two samples of the batch material were 0.44 ± 0.04 m2/g and 0.62 ± 0.03 m2/g, as determined by the Brunauer-Emmet-Teller (BET) condensed nitrogen method. The crushed rock was “preconditioned” by contacting it with the SHOAL water for ∼3 days immediately prior to starting the sorption experiments. All sorption experiments were conducted for ∼3 days. The lithium analysis method was inductively coupled plasma-atomic emission spectrometry (ICP-AES).

B2. Results

[69] Figure B1 shows a log-log plot of the mass of lithium sorbed per unit mass of rock (S, mg/kg) versus final solution concentration (C, mg/L) for all but 6 of the 27 batch sorption trials. The measurements not shown in Figure B1 indicated “negative” sorption (i.e., a higher final concentration than starting concentration). Three of these negative measurements occurred at the second highest starting concentration (∼300 mg/L). This result is relatively common for weakly sorbing ion-exchanging solutes, as sorption is determined by subtracting final concentrations from starting concentrations that are not much larger, and both concentrations have analytical error associated with them. A close look at the raw data suggested that the three negative values for the second highest concentration were almost certainly due to a low measurement of the starting concentration.

image

Figure B1. Log-Log plot of lithium batch sorption results showing best fitting linear and Freundlich isotherms. The open circles indicate the results of two of the three trials conducted at a solution-solid ratio of 10 mL/g (third trial indicated negative sorption).

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[70] The open circles in Figure B1 indicate the results of two of the 3 trials conducted at solution-solid ratio of 10 mL/g (the third indicated negative sorption). Although there were only 3 trials at this ratio, it is apparent that they are more scattered than the other points obtained at the same starting concentration but with a 3 mL/g ratio. This result illustrates why it is always better to use small solution-solid ratios when measuring the sorption of weakly sorbing tracers. The iron-stained rock did not sorb Li any stronger than the other rock.

[71] Figure B1 shows best least squares fits of a linear isotherm and a Freundlich isotherm to the sorption data. The resulting best fitting linear isotherm is S = 0.113 C, where 0.113 is the partition coefficient, or Kd value (cm3/g). The 95% confidence interval for the Kd value is [0.064, 0.199] cm3/g. The best fitting Freundlich isotherm is S = 0.096C1.05, with the 95% confidence intervals of the pre-exponential multiplier and exponent being [0.054, 0.172] and [0.89, 1.22], respectively. A t test [Draper and Smith, 1981] indicated that the Freundlich exponent was indistinguishable from 1.0 at the 90% confidence level, which implies that the lithium sorption isotherm is essentially linear.

B3. Conclusions

[72] For the purposes of this paper, we are primarily concerned with comparing the laboratory-derived lithium Kd values with the Kd values derived from the field tracer test. This comparison indicates that the lowest field-derived Kd value (∼0.3 cm3/g) is approximately equal to the upper 95% confidence bound of the laboratory values (0.2 cm3/g). Thus it must be concluded that the apparent lithium sorption in the field test was greater than in the laboratory tests.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References

[73] The tracer analyses provided by the Harry Reid Center for Environmental Studies (University of Nevada - Las Vegas) and by the Desert Research Institute analytical laboratory are greatly appreciated. Ron Hershey of the Desert Research Institute conducted the matrix porosity measurements and the hanging core diffusion coefficient measurements. This article greatly benefited from the comments of four anonymous reviewers. The work upon which this article is based was supported by the U.S. Department of Energy under contracts DE-AC08-00NV13609 and W-7405-ENG-36.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Design and Methods
  5. 3. Results and Analysis
  6. 4. Discussion
  7. 5. Conclusions
  8. Appendix A:: Description of the RELAP Model
  9. Appendix B:: Laboratory Lithium Batch Sorption Experiments onto SHOAL Granite
  10. Acknowledgments
  11. References
  • Anghel, I., H. J. Turin, and P. W. Reimus, Lithium sorption to Yucca Mountain tuffs, Appl. Geochem., 17, 819824, 2002.
  • Becker, M. W., Tracer tests in a fractured rock and their first passage time mathematical models, Ph.D. dissertation, Univ. of Tex., Austin, 1996.
  • Becker, M. W., and R. J. Charbeneau, First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow, J. Contam. Hydrol., 40, 299310, 2000.
  • Becker, M. W., and A. M. Shapiro, Tracer transport in fractured crystalline rock: Evidence of non-diffusive breakthrough tailing, Water Resour. Res., 36(7), 16771686, 2000.
  • Benson, C. F., and R. S. Bowman, Tri- and tetrafluorobenzoates as nonreactive tracers in soil and groundwater, Soil Sci. Soc. Am. J., 58(4), 11231129, 1994. (Erratum, Soil Sci. Soc. Am. J., 60(6), 1780, 1996.).
  • Callahan, T. J., P. W. Reimus, R. S. Bowman, and M. J. Haga, Using multiple experimental methods to determine fracture/matrix interactions and dispersion of nonreactive solutes in saturated volcanic tuff, Water Resour. Res., 36(12), 35473558, 2000.
  • Churchill, R. V., Operational Mathematics, 2nd ed., McGraw-Hill, New York, 1958.
  • Coughanowr, D. R., and L. B. Koppel, Process Systems Analysis and Control, McGraw-Hill, New York, 1965.
  • Detwiler, R. L., H. Rajaram, and R. J. Glass, Solute transport in variable-aperture fractures: An investigation of the relative importance of Taylor dispersion and macrodispersion, Water Resour. Res., 36(7), 16111625, 2000.
  • Draper, N. R., and H. Smith, Applied Regression Analysis, 2nd ed., John Wiley, Hoboken, N. J., 1981.
  • Freeze, R. A., and J. A. Cherry, Groundwater, Prentice-Hall, Old Tappan, N. J., 1979.
  • Grisak, G. E., and J. F. Pickens, Solute transport through fractured media: 1. The effect of matrix diffusion, Water Resour. Res., 16(4), 719730, 1980.
  • Guimera, J., and J. Carrera, A comparison of hydraulic and transport parameters measured in low-permeability fractured media, J. Contam. Hydrol., 41, 261281, 2000.
  • Jardine, P. M., W. E. Sanford, J. P. Gwo, O. C. Reedy, D. S. Hicks, J. S. Riggs, and W. B. Bailey, Quantifying diffusive mass transfer in fractured shale bedrock, Water Resour. Res., 35(7), 20152030, 1999.
  • Kreft, A., and A. Zuber, On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions, Chem. Eng. Sci., 33, 14711480, 1978.
  • Levenspiel, O., Chemical Reaction Engineering, 2nd ed., John Wiley, Hoboken, N. J., 1972.
  • Maloszewski, P., and A. Zuber, Interpretation of artificial and environmental tracers in fissured rocks with a porous matrix, in Isotope Hydrology, pp. 635651, Int. Atomic Energy Agency, Vienna, 1983.
  • Maloszewski, P., and A. Zuber, On the theory of tracer experiments in fissured rocks with a porous matrix, J. Hydrol., 79, 333358, 1985.
  • Maloszewski, P., A. Herrmann, and A. Zuber, Interpretation of tracer tests performed in a fractured rock of the Lange Bramke Basin, Germany, Hydrogeol. J., 7, 209218, 1999.
  • Moench, A. F., Convergent radial dispersion: A Laplace transform solution for aquifer tracer testing, Water Resour. Res., 25(3), 439447, 1989.
  • Moench, A. F., Convergent radial dispersion in a double-porosity aquifer with fracture skin: Analytical solution and application to a field experiment in fractured chalk, Water Resour. Res., 31(8), 18231835, 1995.
  • National Research Council, Rock Fractures and Fluid Flow:Contemporary Understanding and Applications, Natl. Acad. Press, Washington, D.C., 1996.
  • Neretnieks, I., Diffusion in the rock matrix: An important factor in radionuclide retardation? J. Geophys. Res., 85(B8), 43794397, 1980.
  • Newman, J. S., Electrochemical Systems, Prentice-Hall, Old Tappan, N. J., 1973.
  • Pohll, G., J. Chapman, A. Hassan, C. Papelis, R. Andricevic, and C. Shirley, Evaluation of groundwater flow and transport at the Shoal underground nuclear test: An interim report, DOE/NV/11508-35, 123 pp., Desert Res. Inst., Reno, Nev., 1998.
  • Pohll, G., A. E. Hassan, J. B. Chapman, C. Papelis, and R. Andricevic, Modeling ground water flow and radioactive transport in a fractured aquifer, Ground Water, 37(5), 770784, 1999.
  • Reimus, P. W., A refined approach to estimating effective flow porosity from cross-hole tracer tests in fractured media, paper presented at 10th International High-Level Radioactive Waste Management Conference (IHLRWM), Am. Nucl. Soc., Las Vegas, Nev., 30 March to 2 April 2003.
  • Reimus, P. W., and M. J. Haga, Analysis of tracer responses in the BULLION forced-gradient experiment at Pahute Mesa, Nevada, LA-13615-MS, Los Alamos Natl. Lab., Los Alamos, N. M., 1999.
  • Reimus, P. W., M. J. Haga, A. I. Adams, T. J. Callahan, H. J. Turin, and D. A. Counce, Testing and parameterizing a conceptual solute transport model in saturated fractured tuff using sorbing and nonsorbing tracers in cross-hole tracer tests, J. Contam. Hydrol., 62–63, 613636, 2003.
  • Sanford, W. E., R. G. Shropshire, and D. K. Solomon, Dissolved gas tracers in ground-water: Simplified injection sampling and analysis, Water Resour. Res., 32(6), 16351642, 1996.
  • Spanier, J., and K. B. Oldham, An Atlas of Functions, Taylor and Francis, Philadelphia, Pa., 1987.
  • Sullivan, E. J., P. W. Reimus, and D. A. Counce, Transport of a reactive tracer in saturated alluvium described using a three-component cation-exchange model, J. Contam. Hydrol., 62–63, 675694, 2003.
  • Tang, D. H., E. O. Frind, and E. A. Sudicky, Contaminant transport in a fractured porous media: Analytical solution for a single fracture, Water Resour. Res., 17(3), 555564, 1981.
  • Tsang, Y. W., Usage of “equivalent apertures” for rock fractures as derived from hydraulic and tracer tests, Water Resour. Res., 28(5), 14511455, 1992.
  • University of Nevada, Geological, geophysical, chemical, and hydrological investigations of the Sand Springs Range, Fairview Valley, and Fourmile Flat, Churchill County, Nevada, Vela Uniform Proj. Rep. 1001, Reno, 1965.
  • U.S. Department of Energy, United States Nuclear Tests, July 1945 through September 1992. Rep. DOE/NV-209, rev. 15, Nev. Oper. Off., 2000.
  • Winberg, A., A. B. Conterra, P. Andersson, J. Hermanson, J. Byegard, V. Cvetkovic, and L. Birgersson, Aspo Hard Rock Laboratory: Final report of the first stage of the tracer retention understanding experiments, Rep. TR-00-07, Swed. Nucl. Fuel and Waste Manage. Co., Stockholm, 2000.