Evaluation of single-well injection-withdrawal tests in Swedish crystalline rock using the Lagrangian travel time approach



[1] A series of 10 single-well injection-withdrawal (SWIW) tests are evaluated with two tracers each: uranine and cesium (Cs). An evaluation tool for SWIW tests in crystalline rock is presented on the basis of the Lagrangian travel time approach, whereby probabilities of tracer particle residence times are computed for key stages of the test cycle. Calibration results for three transport parameters and each breakthrough curve are presented. We show that estimates of the controlling retention parameter group equation imageequation image are robust for Cs but highly uncertain for uranine. The estimated retention for Cs is larger for the Laxemar-Simpevarp site compared to the Forsmark site. Deviations from the −3/2 asymptotic breakthrough curve slope observed in a few of the tests at Forsmark are possibly due to a thin fracture coating that has been identified in mineralogical studies at some locations of the site.

1. Introduction

[2] Tracer experiments are an integral part of fractured rock characterization, which still represents a serious challenge in hydrogeology [Neuman, 2005]. A typical tracer test configuration consists of at least two boreholes, where a tracer is injected into one and extracted (detected) from the other. A significantly simpler alternative to multiple borehole tracer tests is a single-well injection-withdraw (SWIW) test, where only one borehole is required. Potential advantages and disadvantages of different test configurations for inferring retention properties of rock fractures were discussed by Tsang [1995].

[3] SWIW tests have been performed in different subsurface environments for a variety of purposes and have been modeled using various approaches. Analytical solutions for inferring a radial macrodispersion coefficient for a SWIW (or a “recharge-discharge” cycle) test configuration were presented by Gelhar and Collins [1971]. Their solution was implemented by Guven et al. [1985] in a stratified aquifer with known advective properties of the stratifications. The work of Guven et al. [1985] showed that provided some prior knowledge of the heterogeneous structure, the SWIW test may yield information on the effect of pore-scale dispersion. Novakowski et al. [1998] considered a SWIW test in a single fracture of a shale and limestone formation, where a numerical simulation tool was developed for test result evaluation and interpretation. The results of Novakowski et al. [1998] indicate that diffusive mass transfer properties of the rock can be inferred with the SWIW test, when compared with independent estimates. An evaluation methodology of breakthrough curve (BTC) asymptotic tailing has been presented [Haggerty et al., 1998, 2001] and implemented for inferring multirate parameters from SWIW tests in Culebra dolomite reported by Meigs and Beauheim [2001]. A total of five breakthrough curves (BTCs) were analyzed by Haggerty et al. [2001], with four calibration parameters in each one; the results indicate that exchange is controlled by multiple rates varying over a wide range, attributed to the internal structure of the rock. Becker and Shapiro [2003] provided an analytical solution for a SWIW test with nonreactive tracers and illustrated basic sensitivities. More recently, Gouze et al. [2008] evaluated a SWIW test result and assessed the specific impact of immobile zone internal structure on the asymptotic tailing of two BTCs.

[4] The SWIW tests so far reported in the literature have been limited to essentially nonsorbing tracers, revealing, to some extent, effects of diffusive mass transfer. However, SWIW test results for nonsorbing tracers will generally be ambiguous for inferring retention properties as it is at best difficult and at worst impossible to distinguish between slow advection and diffusive mass transfer [Lessoff and Konikow, 1997]. Inference of retention properties of fractured rock is highly uncertain even for two-well tracer tests when using nonsorbing or weakly sorbing tracers only [Cvetkovic, 2010b]. Simultaneous injection of multiple tracers with sufficiently contrasting sorption properties may reduce this uncertainty considerably when assessing diffusion-controlled mass transfer on the field scale; such assessment has not yet been made using SWIW tests.

[5] In this work, we analyze the results from a series of SWIW tests conducted at two crystalline rock sites, Forsmark and Laxemar-Simpevarp in eastern Sweden [Nordqvist, 2008]. These tests were conducted with a cocktail of one nonsorbing tracer (uranine) and two sorbing tracers (rubidium and cesium), providing a unique opportunity to better understand the possibilities and limitations of SWIW tests as a potentially routine characterization method for retention properties of crystalline rock.

2. Trajectory Formulation of the Problem

[6] In a crystalline rock formation, a SWIW test will typically be applied in a conducting fracture, or fracture zone. A SWIW test cycle may be sketched, as in Figure 1, where establishing a steady state recharge flow field is the first step (Figure 1a). A cloud of tracer particles enters the fracture from a specified packed-off section injection volume V, in the interval t = 0 to t = t1 (Figure 1b), and is then “chased” deeper into the fracture until t = t2 (Figure 1c) by an additional injected volume of water. When the flow is reversed at t = t3, particles move back toward the borehole. Because of retention (matrix diffusion and sorption), particles are delayed in the fracture, gradually returning to the well (Figure 1d). If pure advection is assumed, all particles that enter the fracture at a given time will return to the well at the same time. In reality, transverse diffusion and dispersion in the mobile phase of the fracture will lead to exchange between advection trajectories and ultimately to some macroscale dispersion.

Figure 1.

Sketch of the single-well injection-withdrawal (SWIW) test cycle in a single fracture: (a) steady state water injection, (b) tracer injection during time interval t1 < t < t2, (c) tracer chasing during time interval t2 < t < t3, and (d) tracer withdrawal until test termination.

[7] The principle stages (or phases) of a SWIW test cycle are (1) injection, (2) chasing, (3) waiting, and (4) withdrawal; the injection and pumping rates, as well as the duration of each test phase, are design parameters. The SWIW cycle can be described following particle transport along trajectories. In an idealized situation of pure advection, a single tracer particle is advected along a given trajectory X(t). The first particle to leave the borehole along X(t) (extending in the radial direction from center of the well) is shown in Figure 2. At time t1 the injection is terminated, but water injection continues (“chasing”), whereby the particle moves further along X(t). At time t2 the injection (chasing) is terminated, and the trajectory radial extent is at a maximum. The last particle to enter the fracture along X(t) is shown in blue and reaches a shorter distance; this defines an unknown spatial interval within which all particles along X(t) are located. At time t3, withdrawal (pumping) is initiated; the particles then move back toward the well along the same trajectory X(t) following the principle “first particle in is the last particle out, and last particle in is the first particle out.”

Figure 2.

Tracer particle trajectory conceptualization of the SWIW test cycle with the key test times noted.

[8] Since water flow is the main driver of transport (by advection), a tracer particle will follow the same steps as the noninteracting particle but will additionally be subject to matrix diffusion-sorption that in physical terms implies its temporary trapping in the immobile matrix. As a result, a tracer will be transported a shorter distance than the injected water, depending on its sorption properties; the stronger the sorption affinity, the shorter the distance the tracer will enter the fracture following some irregular shape of the injection front that depends on fracture heterogeneity (Figure 2).

[9] The prime objective of the SWIW tests analyzed here is to extract and infer controlling retention parameters that can, in principle, be verified using independent information. Our working hypothesis is that a single retention parameter (referred to as equation imageequation image) is dominant for sorbing tracers and essentially sufficient to quantify retention in addition to the test design parameters; this hypothesis will be tested by seeking the minimum number of parameters required for reproducing reasonably well the measured BTCs. The main assumptions of this investigation are as follows.

[10] 1. SWIW tests to be considered are designed primarily as transport experiments, which means that the waiting phase (t3t2) in Figure 2 normalized with t2 is considerably smaller than 1. As a consequence, the SWIW cycle reduces to two dominant transport regimes (stages): injection and chasing and withdrawal.

[11] 2. Transport during withdrawal is modeled as advection coupled with diffusion-sorption of tracer “parcels” that are initially spatially distributed. Prior to the start of withdrawal, each parcel is partitioned between the fracture and matrix following the injection-chasing phase. The observed BTC is interpreted as an aggregation of point sources (tracer parcels), first along each trajectory and then from all trajectories.

[12] 3. Transport during injection and chasing (which includes borehole mixing) sets the initial condition for transport during the withdrawal phase and is simplified by combining peak displacement (subject to retention) and an average effect of rate limitations quantified by the return time density from the matrix. The most important component is the injection function as it will define the distribution of water residence times for the withdrawal phase.

[13] In this paper, we first present an evaluation tool for SWIW tracer tests that accounts for key physical processes on the basis of the assumptions above; conditional and unconditional probabilities of particle residence times are to be computed. Next, the implementation of the proposed tool is illustrated on a series of SWIW tests carried out at two entirely different crystalline rock sites. Finally, a few basic sensitivities and dependencies will be discussed.

3. Transport Model

[14] Modeling a SWIW tracer test with diffusion-controlled retention is a relatively complex transport problem that strictly requires numerical solutions. In fact, most SWIW test evaluation tools reported in the literature rest on numerical solutions [e.g., Lessoff and Konikow, 1997; Novakowski et al., 1998]. By limiting the evaluation to asymptotic tails of BTCs, the modeling can be simplified [Haggerty et al., 1998], but a more detailed analysis of tails may still require combined numerical and semianalytical approaches [Haggerty et al., 2001].

[15] Although numerical approaches are useful for fundamental analysis, routine SWIW test applications would ideally require that (1) the method or tool is simple and portable, with as few parameters to calibrate as possible, (2) the method or tool captures key processes as accurately as possible, relying on relevant test information, and (3) parameterization of retention processes in the modeling is such that independent verification using alternative tests in situ or in the laboratory is possible, at least in principle. In order to provide an operational tool for SWIW test evaluation in line with these premises, the Lagrangian travel time approach for sorbing tracers [Cvetkovic, 1991; Cvetkovic and Dagan, 1994] is to be implemented.

3.1. Tracer Discharge

[16] Equation (A4) of Appendix A provides the basis of our SWIW test evaluation. The three components of the model are as follows: (1) Function equation image accounts for the fact that when injection and chasing are terminated, the tracer is partitioned between the fracture and matrix; tracer release during the withdrawal phase is diffusion controlled and quantified by this function. (2) Function equation image is the water residence time density for the withdrawal phase quantifying advection and small-scale dispersion; its main parameter is the mean water residence time, which will be deduced from the injection function controlled by test design parameters. (3) Component equation image is the usual memory function [Cvetkovic and Dagan, 1994; Cvetkovic et al., 1998], in this case for Fickian diffusion coupled with equilibrium sorption in the rock matrix. We now proceed to specify these three functions.

3.2. Injection Rate

[17] Tracer injection rate is equal to the output or discharge from the borehole volume (viewed as a “continuously stirred flow reactor”) defined by

equation image
equation image

where the turnover rate is

equation image

where V is the injection volume (exposed borehole), Q [L3/T] is the injection flow rate, equation image [L] is the surface sorption coefficient on the borehole wall, and A is the exposed borehole surface area. For a nonsorbing tracer, equation image.

3.3. Noninteracting Particle Residence Time

[18] Water residence time for pure (plug flow) advection during withdrawal is to be defined as a “mirror image” of the injection rate. Our principle for the noninteracting particle (NIP) is simple: the first particle injected is the last particle withdrawn at t4 for pure advection, which yields

equation image


equation image

and zero otherwise.

[19] Clearly, macrodispersive effects will be present to some degree, depending on the heterogeneity of the flow and the rate of small-scale diffusive-dispersive exchange, similar to a stratified medium [Guven et al., 1985; Dagan, 1989]. Dispersion (i.e., deviations from ideal plug flow) will be included using a conditional NIP residence time density, where conditioning is on the advective travel time, applicable under idealized “plug flow” conditions. Assuming macrodispersion controlled by the advection-dispersion equation [Gelhar and Collins, 1971], we write, in the Laplace domain, the conditional NIP residence time fd (t) that accounts for dispersion as

equation image

where equation image is the water residence time coefficient of variation. Note that (5) is a conditional inverse Gaussian density.

[20] The unconditional NIP residence time density for withdrawal is then

equation image

which can be used for computing the unconditional interacting particle residence time h (equation (A4)). In (6), equation image is the random variable (NIP residence time), which now includes dispersion, and equation image is the NIP residence time for pure advection obtained from the injection function (4). Equation (6) is consistent with the time domain random walk formulation of particle transport as presented, for instance, by Painter et al. [2008].

[21] Applying equation image to any tracer for computing h (equation (A4)) during withdrawal would imply that NIP residence time for both sorbing and nonsorbing tracers during withdrawal is the same for a given test. Clearly, nonsorbing tracers would penetrate the fracture farther than a sorbing one, during injection and chasing, whereby the NIP for the two (as applicable during withdrawal) should be different to some degree. We can account for this difference in equation image by introducing an additional parameter that quantifies a delay and is, in effect, an equilibrium approximation of retention during the injection-chasing phase. Specifically, equation image (equation (6)) is modified with a new parameter (a type of injection phase “retardation coefficient”) denoted by R* to obtain the final form of equation image as follows:

equation image

[22] If the tracer is injected over a relatively short time (t1 is small), the withdrawal NIP residence time (without dispersion) will be close to a pulse. Under these conditions we can simplify equation image, where equation image is the Dirac delta function, whereby

equation image

and fd is defined in (5). The physical significance of the R* and its relationship to the inferred retention parameters from the withdrawal phase will be discussed in section 6.2.

[23] The time t4 will depend on the ratio between the injection and withdrawal flow rates, Q/Qw, where Qw is the pumping rate during withdrawal. For Q/Qw ≈ 1, we have t4t3 + t2; in the general case, (t4t3)/t2 will be a function of Q/Qw.

3.4. Retention

[24] A Fickian diffusion model in an unbounded matrix [e.g., Neretnieks, 1980; Cvetkovic, 1991] is obtained with equation image in (A3), yielding an asymptotic slope of the breakthrough as −3/2; the two-well “tracer retention understanding experiments” (TRUE) tracer test results, for instance, exhibit tailing with a tendency toward a −3/2 slope [Cvetkovic et al., 2007, 2010]. Under the assumption that the same model applies here, we have equation image, where equation image [L/T1/2], with equation image being the matrix porosity, De [L2/T] being effective diffusivity, and R being the retardation coefficient in the matrix. For a nonsorbing tracer, equation image. The effective diffusion coefficient can be written De = FDw, where F is the formation factor and Dw is diffusivity in pure water.

[25] Perfect correlation is assumed between the advective parameters equation image and equation image [T/L] (with b [L] being the spatially dependent fracture half aperture), as equation image, where sf [1/L] is the active specific surface area [e.g., Cvetkovic et al., 1999; Cvetkovic and Frampton, 2010]. For a fracture network, the simplest estimate of sf is obtained using the specific surface area of fractures and flow porosity [Shapiro, 2001; Cvetkovic et al., 2004]. With equation image, the characteristic retention time is

equation image


equation image

with equation image; Kd [L3/M] is the sorption coefficient in the matrix and equation image [M/L3] is the rock density.

[26] The retention function equation image can now be written in the Laplace domain as

equation image


equation image

3.5. Tracer Release Rate From Matrix

[27] Consider a tracer particle following a trajectory during injection and chasing up to t = t3 (including the brief waiting phase). The particle can enter the matrix and is likely to be in the matrix at t = t3. When the flow is reversed at t = t3, particles in the matrix have a finite probability of returning to the fracture. We interpret the function equation image in (A1) or (A4) as the density of return times from the matrix during withdrawal, having entered the matrix during the injection and waiting phases. Thus, equation image provides the coupling between injection and withdrawal in an approximate manner.

[28] It is shown in Appendix B that equation image can be expressed as

equation image

[29] Note that no process parameters appear in (13), only the test design parameter t3. The particle that enters the matrix first will come out last and vice versa. In other words, during the injection-chasing time, particles diffuse far if diffusivity is large but also return faster if diffusivity is large and vice versa. Thus, (13) is a robust representation, with the key approximation being that longitudinal variation is neglected; (13) quantifies an integrated effect along the trajectory up to the time t = t3 (Appendix B).

3.6. Two Limiting Cases

[30] The transport model (A4) has three principal components (equation image, f, and equation image), which depend on various parameters. A number of these are test design parameters (such as injection and withdrawal volume and flow rate and times t1, t2, and t3), and three are process parameters (equation image, equation image, and R*) that need to be either calibrated or inferred from independent information.

[31] As part of our SWIW test evaluation tool, we consider two limiting cases (or models) for h (equation (A4)) as follows.

[32] For model 1, rate limitations of matrix release of tracer accumulated during the injection-chasing phase are included in the transport during withdrawal approximately with equation image(13):

equation image

[33] For model 2, matrix release of tracer accumulated during the injection-chasing phase is treated as a pulse in the withdrawal phase, i.e., equation image or equation image, whereby

equation image

[34] In both cases, we have approximated ffd (equation (8)) and equation image, and equation image is defined in (12). Since R* and equation image both quantify retention along the same fracture area (R* during the injection-chasing phase and equation image during withdrawal phase), they should be correlated to some degree; this point will be elaborated in section 6.2. Note that model 2 ignores only rate-limiting effects from the injection phase; any retention occurring during injection and chasing is included in R*.

4. Summary of SWIW Tests

[35] A long history of investigations within the Swedish high-level nuclear waste isolation program have led to two candidate sites, at Forsmark and Laxemar-Simpevarp. A characterization program was launched at the two sites with the main objective of locating a suitable rock volume for a repository at approximately 500 m depth, which fulfils prescribed safety requirements. Both sites are composed of Paleoproterozoic rocks of the western part of the Fennoscandian Shield [Milnes et al., 2008].

[36] As part of the site characterization, a suite of SWIW tests were performed at Forsmark and Laxemar-Simpevarp sites, shown in Figure 3. Table 1 summarizes 10 tests, indicating the borehole section, flow rates, and times t1, t2, and t3. Prefix KFM refers to the tests performed at the Forsmark site, KLX refers to the tests performed at the Laxemar site, and KSH refers to the tests performed at the Simpevarp site, which geologically is related to the Laxemar site. The detailed experimental procedures are described by Nordqvist [2008].

Figure 3.

Plan view of the two sites, Forsmark and Laxemar-Simpevarp (Sweden). The location of the test wells subject to SWIW tests considered in this work are indicated in red.

Table 1. Basic Data for 10 SWIW Tests at Forsmark and Laxemar-Simpevarp Sites, Eastern Swedena
TestSection (m)t1 (h)t2 (h)t3 (h)Q (×103 m3/h)V (×103 m3)A (×103 m2)Uranine (%)Cs (%)
  • a

    The boreholes starting with KFM are at the Forsmark site, and boreholes starting with KLX are at the Laxemar site; an exception is KSH02B, which is at the Simpevarp site located near the Laxemar site. Thus, the first five single-well injection-withdrawal (SWIW) tests are from the Forsmark site, and the last five are from the Laxemar-Simpevarp site. A is the packed borehole section area, and V is the packed borehole section volume. The uranine and Cs columns summarize recovery for the two tracers. In all cases, the withdrawal (discharge) volumetric flow rate is approximately equal to the injection (recharge) volumetric flow rate, denoted by Q.


[37] In the following sections we will refer to the tracer tests using their borehole section labels. If two sections of the same borehole were used for the tracer tests, we distinguish between them by using numbers 1 and 2 as suffixes. For example, KFM01D1 refers to the tracer tests performed in the borehole KFM01D section 377.4–378.4 m, and KFM01D2 refers to the tracer tests performed in the borehole KFM01D section 431–432 m. Cesium (Cs) was used as a sorbing tracer in all tests, and uranine was used as a conservative tracer (relative to Cs); rubidium (Rb) was also used as an additional sorbing tracer in all tests except KSH02, KFM02A, and KFM03A.

[38] The experimental results are given in the form of the flux-averaged concentration [M/L3] as a function of time from the recovery phase; for the purpose of evaluation, we use the normalized tracer discharge [1/T] (or BTCs). Tracer recovery for each tracer and test is given in Table 1.

5. Results

[39] In this analysis we shall focus on the tracers uranine and Cs. The evaluation will be done with both models, i.e., with model 1 (equation (14)) and model 2 (equation (15)). For each test and tracer, three parameters are calibrated: equation image, equation image, and R*. Since uranine and Cs in the same test “see” different parts of the fracture area because of retardation, each tracer has to be treated as a separate test; only the test design parameters are the same for both tracers in each test (Table 1).

[40] The measured BTCs in terms of the normalized discharge as a function of time are plotted in Figures 4 and 5, together with modeled BTCs. The difference of time ranges and peak concentrations in the different tests is readily identified.

Figure 4.

Comparison between modeled and measured breakthrough curves (BTCs) for uranine and Cs for the five SWIW tests at the Forsmark site. The blue symbols are measured BTCs for uranine, and red symbols are for Cs. The thick solid lines are injection curves, and the thin solid lines are obtained from equation (3). The dashed line is obtained with model 1, and the dotted line is obtained with model 2.

Figure 5.

Comparison between modeled and measured breakthrough curves for uranine and Cs for the five SWIW tests at the Laxemar-Simpevarp site. The blue symbols are measured BTCs for uranine, and red symbols are for Cs. The thick solid lines are injection curves, and the thin solid lines are obtained from equation (3). The dashed line is obtained with model 1, and the dotted line is obtained with model 2.

[41] The injection phase input is shown with thick solid lines for the two tracers, always at the start of the test. The NIP residence time density (6) with (4) is shown with thin solid lines for the two tracers, which are spiked around t = t4. These spikes are then scaled with the parameter R* as t4/R* to account for tracer retention during injection and chasing, as a starting point for the withdrawal phase.

[42] The “best fit” model curves are by no means always a close fit, depending on the model used. The purpose, however, is not to have a detailed reproduction of all BTCs with the two relatively simple models but to capture main qualitative features in all tests and at the same time obtain modeled BTCs that are as close to measured BTCs as possible.

[43] In all cases, the BTCs for the two tracers have a relatively steep, almost overlapping early arrivals. The peaks of the BTCs for the sorbing tracer are consistently lower compared to the corresponding BTC peaks of uranine. In some cases, the asymptotic part of the BTC attains a −3/2 slope for both uranine and Cs, whereas in a number of cases the tail of uranine decreases more rapidly than for Cs. Although there are obvious deviations in the tailing from the −3/2 asymptotic slope (which will be discussed in section 7.3), it is clear that to the extent measured the tailing of the BTCs is roughly consistent with the −3/2 slope (as always maintained by the modeled curves).

[44] The two sites show the same features with no qualitative difference. In all but two tests (KFM03a and KLX11a1) the Cs BTC is closely reproduced, apart from some deviation in the tails. For uranine, the modeled curves are also relatively close to measured curves in most cases. The curves obtained by models 1 and 2 are almost the same for Cs (albeit with different parameters) but not for uranine. Clearly, the BTCs for uranine are less sensitive to equation image, and hence, its calibration is considerably more uncertain since retention is weak [Cvetkovic and Frampton, 2010]. For uranine, in all but two cases (KFM01D1 and KFM01D2), models 1 and 2 yield comparable BTCs, whereas in the KFM01D1 and KFM01D2 tests, model 1 yields a closer representation of the measured data, indicating that equation image (equation (13)) overestimates the rate limitations during the injection-chasing phase.

[45] The calibrated parameters equation image, R*, and equation image for the 10 SWIW tests and both tracers, using models 1 and 2, are all summarized in Table 2. Numerical Laplace inversion was carried out by the Stehfest method [Stehfest, 1970] as part of the Mathematica standard Laplace inversion package.

Table 2. Calibrated Parameters ψ, ζ, and R* for All 10 SWIW Tests and the Two Tracers
ψ (1/equation image)R*ζψ (1/equation image)R*ζ

6. Retention Parameters

[46] The two calibration parameters related to retention are equation image and R*. From the application viewpoint, the parameter of main interest is equation image. In fact, SWIW test implementation for site characterization would be done mainly for the purpose of inferring in situ equation image.

6.1. Parameter Group ψ

[47] The parameter equation image for uranine varies significantly between tests as well as between models 1 and 2 (Table 2). Parameter equation image for Cs varies between models 1 and 2, but much less compared to that of uranine. Calibration of equation image should depend on whether transport during withdrawal assumes instantaneous release of a tracer parcel as in model 2 with equation image or whether transport during withdrawal accounts for rate limitations in the tracer release due to diffusive partitioning in the injection-chasing phase with equation image (equation (13), model 1). For model 1, the calibrated equation image is lower since including equation image (equation (13)) yields an effect comparable to that of higher equation image; for model 2, equation image is then higher, more notably so for uranine than for Cs (Table 2).

[48] As already shown elsewhere [Cvetkovic et al., 2010], calibration of equation image for nonsorbing and weakly sorbing tracers is significantly more uncertain compared to sorbing tracers since the BTC are considerably less sensitive to the relatively low value of equation image. In a few tests with uranine (in particular, for the KFM01D1 and KFM01D2 tests), the approximation of rate-limiting effects in the tracer release during withdrawal with equation image (equation (13)) seems to overestimate retention effects; hence, we anticipate a relatively low equation image, and we set equation imageequation image as the upper bound for uranine in these tests.

[49] In our earlier evaluation of two-well TRUE tests [Cvetkovic et al., 2007, 2010], a clear dependence of equation image on the sorption coefficient Kd was identified and exploited when estimating individual retention parameters that constitute the group equation image. Figure 6 illustrates equation image as a function of the dimensionless equation image, where Kd is a selected reference value obtained in the laboratory for the tracer and rock in question. Here a reference Kd is adopted as obtained from through-diffusion tests on generic, unaltered samples of the so-called “Äspö diorite” [Byegård et al., 1998; Widestrand et al., 2007]; for Cs it is Kd = 0.006 m3/kg. The lines represent realistic values as inferred from laboratory tests on generic (unaltered) samples combined with numerical simulations of sf [e.g., Cvetkovic and Frampton, 2010]; details are provided in the Figure 6 caption.

Figure 6.

Parameter group equation image as a function of equation image. The retention parameters for obtaining the lines with equation (10) use equation image kg/m3, Dw = 7 × 10−6 m2/h, and the following values: for the solid curve, equation image, F = 0.00014, and sf = 4000 1/m; for the top dashed curve, equation image, F = 0.0016, and sf = 10000 1/m; and for the bottom dashed curve, equation image, F = 0.000004, and sf = 2000 1/m. Small symbols are obtained from calibrated equation image using model 1, medium-sized symbols are obtained from calibrated equation image using model 2, and large symbols are obtained from all calibrated parameters using both models. Red symbols are for the Forsmark site, and blue symbols are for the Laxemar-Simpevarp site. Circles are for Cs, and triangles are for uranine, which is included with an insignificant but finite value of Kd for ease of log-log illustration.

[50] Harmonic means of equation image using values in Table 3 are included in Figure 6. (We prefer to use the harmonic mean since it gives more weight to low data points compared to, e.g., the arithmetic mean, which gives high data points greater weights; from the application viewpoint, greater weight to higher data points is nonconservative.) The wide range of calibrated values of equation image for uranine is most apparent in Figure 6: The difference in the harmonic means is almost two orders of magnitude. The corresponding difference for Cs is significantly smaller, around a factor 2, indicating that inference of equation image for Cs is much more robust than for uranine. There is a systematic difference in the mean values of equation image for the two sites, for both models and tracers: The retention appears to be somewhat stronger at the Laxemar-Simpevarp site relative to the Forsmark site (compare blue and red symbols in Figure 6).

Table 3. Arithmetic Mean and Coefficient of Variation (CV) of ζ and R* for Different Cases
 CV (%)8202642
 CV (%)823638
 CV (%)8232039

[51] The harmonic mean of equation image for uranine inferred by the two models sets the range 0.0012–0.0546 1/equation image. If we consider an aperture of, say, 0.4 mm, then the matrix porosity would be in the range 0.0015–0.022, with Dw = 5 × 10−6 m2/h, and equation image assuming Archie's law is applicable [Clennell, 1997], with a cementation exponent consistent with an unaltered rock matrix [Cvetkovic et al., 2007].

6.2. Parameter R*

[52] The parameter R* is best described in physical terms as the delay (retardation) of the peak occurring during the injection-chasing phase due to matrix diffusion and sorption; it is a key parameter in setting the initial condition for withdrawal and, in particular, for estimating the water residence time equation image (equation (8)) used in models 1 and 2.

[53] Calibrated R* varies in a relatively narrow range for both uranine and Cs compared to equation image (Table 2). In Table 3 we summarize mean R* and the coefficient of variation for the two sites and two models.

[54] Parameters R* and equation image both reflect retention and should therefore be correlated. An expression of this correlation is a simple model of the peak travel time for the case of pure advection with matrix diffusion and sorption. In this case, the peak arrival time during injection is equation image, where equation image is the NIP residence time [Cvetkovic et al., 2004]. If we set equation image, a relationship between the “retardation coefficient” R* and retention parameter group equation image is

equation image

where a is a parameter. Theoretically, a = 1; however, we anticipate effects of nonideal conditions such that a ≠ 1. In Figure 7 we plot modeled and calibrated R* for the two sites. (By “modeled” we mean using the model (16) and calibrating only one parameter a, whereas “calibrated” refers to R* as calibrated in each test, without relation to (16).) A reasonable representation of calibrated values by the model (16) with a = 1.65 using (14) is found. This, in effect, means that a single parameter a = 1.65 can replace the 20 calibrated values of R* for Cs (Table 2). The fact that R* is proportional to equation image reasserts the physical basis of both equation image and R*.

Figure 7.

Correlation between calibrated R* and the model value obtained with (16). Blue dashed line is for model 1, and red dashed line is for model 2; a 1:1 (solid) line is also included.

7. Discussion

7.1. Advection and Dispersion

[55] Our Lagrangian travel time conceptualization is based on a conditional residence time density (“reaction function”; equation (11)) that quantifies retention processes given the NIP residence time equation image; thus, we require the density equation image to compute the unconditional IP residence time h (equation (A4)). The two mechanisms of advection and retention are clearly distinguished and parameterized separately. In Figure 8 the blue dashed line is the (nonmeasurable) residence time density of the “ideal” tracer particle that is used to compute the BTC for the sorbing tracer with (A4). The actual steps for obtaining the blue curve, which includes the approximation (8), are as follows.

Figure 8.

Example of the KFM01D1 test to illustrate different conditional and unconditional densities in the modeling. The black solid line is the input equation image ((1) and (2)), the solid blue line is f ((3) with (4)) as the “mirror image” of equation image applicable to pure (plug flow) advection, the blue dashed line is equation image ((6) with (4) and (5)) where dispersion is accounted for, the blue dotted line is the approximate expression for equation image ((8), with R* = 1), the red dashed line is the scaled equation image ((8), with R* given in Table 2 for Cs), the red solid line is the modeled normalized Cs discharge that is equivalent to the unconditional tracer residence time density h (equation (14)), and symbols are measured values.

[56] Consider as an example the test KFM01D1 with parameters given in Table 1. At the test onset, tracer injection rate is the black line in Figure 8 following equation image ((1) and (2)), a pulse-type input of approximately 1 h duration. The tracer pulse is chased for approximately 6.5 h with a waiting time of 1.3 h. Then pumping starts, and the tracer arrival under idealized conditions of no retention or dispersion is given by the blue solid line as a “mirror image” of the input ((3) with (4)), i.e., a pulse-type BTC that would be measured at around t4 = t3 + t2, where t3 includes the waiting time; for the KFM01D1 test, t2 = 7.3 h and t3 = 8.5 h (Table 1), whereby t4 = 16 h.

[57] Equation (6) with (5) and (4) yields the noninteracting tracer residence time (subject to advection and dispersion), shown as the blue dashed line; expression (8) yields the blue dotted line (with R* = 1), a rather close approximation. If the tracer is not subject to any retention during injection, the blue dotted or dashed curves are the NIP residence time densities during pumping.

[58] The interacting (sorbing) tracer Cs is subject to retention during injection as quantified by R* and is not displaced in the fracture as far as a less sorbing (or nonsorbing) tracer. When the withdrawal starts at t3, the NIP residence time density (red dashed curve in Figure 8) is scaled as t4/R* for Cs (blue dashed curve in Figure 8), precisely to ensure that the noninteracting and Cs tracer parcels coincide (at unknown locations) at the start of withdrawal. In other words, the blue dashed line is the NIP residence time density equation image that accounts for dispersion and is used for computing the unconditional IP residence time density h (equation (14)) for Cs (red line in Figure 8). It is important to emphasize that R* has been found to be correlated with equation image, consistent with the simple theoretical relationship (16) (Figure 7).

7.2. Dispersivity Estimates

[59] Pure advection in a SWIW test implies that each tracer particle returns to the borehole at the same time after withdrawal and along the same trajectory. The effect of pore-scale dispersion, however, results in some dispersion being measured after withdrawal. This process was analyzed for stratified media by Guven et al. [1985]. Clearly, there is natural variation in the velocities between flow paths and trajectories, resulting in tracer particles penetrating the fracture at varying (undefined) distances following injection. As in a stratified medium, diffusion and small-scale dispersion are at work within the mobile phase of the fracture, resulting in some transfer of particles between advective trajectories that is manifested as dispersion during withdrawal. In our framework, equation image quantifies the macrodispersion; the mean and coefficient of variation equation image for the evaluated SWIW tests are given in Table 3. Although there is variation among tests and tracers, equation image is rather small, between 6% and 26% for uranine.

[60] An estimate of the longitudinal (radial) macro dispersivity equation image can be obtained using the simple expression

equation image

where x is the (unknown) transport distance.

[61] Considering, for instance, tests KFM01D1 and KFM01D2, the injected volume is obtained from test design parameters in Table 1 around 0.1 m3. If we assume a factor 4 aperture range 0.000125–0.0005 m, then the extent of the injected plume is 8–16 m, which on the basis of (17) and values of equation image given in Table 2 (for uranine), would imply equation image m for these tests. Coincidently, equation image m is the best estimate from a SWIW test reported by Novakowski et al. [1998], whereas Becker and Shapiro [2003] report equation image m also from a SWIW test in a crystalline fracture at the Mirror Lake site.

[62] A more detailed quantitative analysis of dispersion in a SWIW test could be done by simulations; however, such analysis requires data on hydraulic properties of fractures that are typically hard to obtain. Fortunately, equation image has a relatively small impact on evaluation of retention parameters in SWIW tests, as has also been found in earlier works [e.g., Haggerty et al., 2001]; our results iterate this point even more strongly for estimates of equation image for Cs, which are not sensitive to equation image for the considered SWIW tests.

7.3. Asymptotic Slope

[63] Although modeled curves reproduce the measured BTCs relatively well in all cases, there are clear deviations. A particular deviation addressed here is the departure of the tail slope from the −3/2 line in the KFM01D1 and KFM01D2 tests. The Cs BTC tail slope increases, whereas the uranine BTC tail slope decreases in both tests, relative to the modeled −3/2.

[64] One possible explanation for these deviations is the microstructure of the rim zone, where, say, a thin coating is present adjacent to the fracture [Sandström et al., 2008; Drake et al., 2008]. To illustrate such a possible effect, consider a thin layer of 0.15 mm and 1% porosity, beyond which is an infinite matrix with 0.1% porosity. With assumed retention parameters as shown in Figure 9, we see that, indeed, the qualitative behavior observed in the asymptotic part of the BTC can be reproduced closely as an accumulation of the tracer in the layer (Appendix C). It can be shown that for similar conditions, but with a varying coating thickness, the reduction of the uranine tail slope as observed in the data (Figure 9) can also be qualitatively reproduced; a motivation for including the coating spatial variability for uranine would be that it is exposed to a larger area in the fracture compared to Cs. A more detailed, quantitative analysis of the impact of the rim zone structure on the outcome of SWIW tests requires analysis of associated microstructural data and is beyond the scope of this work.

Figure 9.

Possible interpretation of the deviation in the tail from the −3/2 slope in the KFM01D1 and KFM01D2 tests as the presence of a thin layer (coating) in the rim zone of the matrix. The solid curve is obtained using (14) with (12), whereas the dashed curve is also obtained with (14) but with (C1). Index 1 refers to the first layer, and index 2 refers to the second (essentially infinite) layer (see Appendix C).

7.4. A Single Parameter Model

[65] Since we find a relatively narrow range of equation image and a reasonable predictability of R* using (16), the robustness of equation image estimates in Table 2 can be assessed in a simple manner by assuming a single value of equation image and using (16). Consider the measured normalized discharge of Cs in the Forsmark tests. Let equation image take a constant value of 0.1 and R* be computed from the deterministic relationship (16) with the best estimate a = 1.65. With calibrated equation image parameters using model 1 (Table 2), five BTCs for Cs at the Forsmark site are plotted in Figure 10, comparing measured BTCs (circles) to the modeled BTCs (lines), with each color corresponding to the same test.

Figure 10.

Comparison between the measured and modeled BTCs for Cs in the Forsmark SWIW tests using a single value for the coefficient of variation equation image and (16) for R*. See text for further details.

[66] Although there is clearly more deviation between measured and modeled BTCs compared to Figure 4, the reproduction of the measured BTCs is remarkably close. Thus, the proposed SWIW tests evaluation tool is robust for inferring in situ equation image for Cs and could be used as a single-parameter model. The comparison in Figure 10 demonstrates that equation image is the single most dominant transport-retention parameter for the sorbing tracer, thereby verifying our working hypothesis.

8. Conclusions

[67] In this paper, a Lagrangian travel time approach for evaluating single-well injection-withdrawal tests in crystalline rock has been presented and applied for 10 SWIW tests from the Laxemar-Simpevarp and Forsmark crystalline rock sites (Sweden). The novel aspect of these tests is the simultaneous use of nonsorbing and sorbing tracers. The methodology is based on a semianalytical stochastic description of tracer particle residence time densities, which is well suited for the analysis of SWIW tests. The main feature of a SWIW test cycle is precisely that given a borehole location, we can monitor flux-averaged concentration changes in time, which directly relates to tracer particle residence time. A modeling approach for evaluating SWIW tests with nonreactive tracer that takes advantage of the travel time formulation was presented by Becker and Shapiro [2003].

[68] Our general conclusion is that SWIW tests of the type analyzed here can provide useful information about the retention properties of the matrix, in particular, estimates of the parameter group equation image, which for a sorbing tracer, is the controlling parameter. An important next step would be to compare SWIW estimates of equation image with estimates obtained in the same fractures using alternative methods, such as two-well tracer tests and/or core sampling, in order to assess possible biases and better understand the reliability of field-scale measurements of retention properties in the crystalline rock.

[69] Specific conclusions of this work are summarized as follows.

[70] 1. The key step when implementing the Lagrangian travel time approach is to define the noninteracting particle residence time equation image, which coincides with the water residence time if the effect of pore-scale dispersion and diffusion is neglected; in the proposed SWIW evaluation tool, the dominant component of the noninteracting particle (or NIP) residence time for withdrawal is defined as a temporal “mirror image” of the tracer input equation image (equation (4)). The retention processes during withdrawal are controlled by the parameter equation image assuming diffusion-sorption in an infinite matrix. The proposed evaluation tool seems to strike the right balance between complexity of processes and the need for robustness and operational simplicity.

[71] 2. An approximate means for including the matrix diffusion effect during injection with the function equation image (equation (13)) and the parameter R* has been proposed. The function equation image quantifies an integrated (or average) effect of diffusion during injection in (A4) and depends on test design parameter t3. The parameter R* (defined as the retardation of the peak in (16)) accounts in a simple manner for the fact that sorbing tracers are also retarded during injection, and hence, their NIP travel times during withdrawal need to be scaled by factor R* relative to those of a nonsorbing tracer in the same test.

[72] 3. The calibrated values of equation image seem reasonable; as expected, the estimates of equation image for the nonsorbing tracer (uranine) are considerably more uncertain than for the sorbing tracer (Cs). As indicated by equation image inferred from using the limiting models 1 and 2, for uranine the range is almost 2 orders of magnitude, whereas for Cs the difference is around a factor of 2 (Figure 6).

[73] 4. It has been shown that equation image is the key controlling parameter for evaluating sorbing tracer BTCs from SWIW tests, thereby verifying our working hypothesis; the other two parameters (equation image and R*) vary little among tests (Tables 2 and 3). R* can be relatively well predicted with the simple formula (16) because of its correlation with equation image. Thus, we find that for the given sites, equation image can be used as the single calibration parameter when evaluating BTCs of Cs, with constant equation image and R* computed from (16), which is consistent with the physical basis of equation image and R*. The parameter equation image is also correlated with Kd, as expressed by (10) (Figure 6), consistent with the findings in TRUE [e.g., Cvetkovic et al., 2010].

[74] 5. The two limiting models 1 (equation (14)) and 2 (equation (15)) set bounds on equation image as part of the proposed SWIW test evaluation methodology. An average between the calibrated values of equation image obtained with models 1 and 2 is believed to provide a reasonable estimate in view of the approximations involved in the two limiting models. A harmonic mean appears to be best suited for the lower-value range and, in this sense, is conservative. However, other averages may also be used.

[75] 6. The calibrated equation image for the Forsmark site is consistently lower compared to the Laxemar-Simpevarp site, irrespective of the model assumed, indicating weaker retention at the Forsmark site. The harmonic mean of equation image for all tests and both models is 0.35 1/equation image.

[76] This work further demonstrates the fact that using sorbing tracers is a necessary condition for characterizing field-scale retention processes in fracture formations. Using only nonsorbing or weakly sorbing tracers makes distinguishing effects of diffusion in the immobile matrix from effects of low velocities in the flow field highly uncertain [Lessoff and Konikow, 1997; Becker and Shapiro, 2000; Shapiro, 2001]. Multiple tracers with a wide range of sorption affinities provide the best means for reducing uncertainty when assessing retention properties [Cvetkovic, 2010b]. For the particular sites considered here, further analysis work on core samples could reveal direct information on the microstructural properties of the rim zone. Moreover, laboratory measurements of Kd for Cs, as well as porosity and diffusivity using core samples, could be used to better understand the limitations of equation image inferred from SWIW tests. Studying matrix diffusion during injection further could also provide more accurate alternatives to equation image.

Appendix A:: Theoretical Framework

[77] Let h(t; L) denote the unconditional density of particle residence time. In the general case, the particle residence time is measured between two distinct locations (two boreholes, or the injection and detection planes); for SWIW tests the two locations coincide, i.e., tracer particle residence time is monitored from and to a single borehole, over a SWIW test cycle. A tracer particle is subject to hydrodynamic transport (advection and dispersion) and to retention because of some type of exchange with the immobile phase; it is referred to as an “interacting particle” (IP for short). The residence time density for an IP will be conditioned on the residence time of an “noninteracting particle” (NIP for short) [Cvetkovic and Haggerty, 2002]. For pure advection, NIP and water transport are equivalent.

[78] A general expression for an IP residence time density is

equation image

where equation image [1/T] is the normalized particle injection rate (or a probability density function [1/T] of particle release), equation image is the “reaction function” quantifying, in this case, reversible retention processes, and an asterisk denotes convolution. The vector P{P1, P2,…} denotes parameters that control retention and, in general, are spatially variable [Cvetkovic et al., 1998]. For low tracer concentrations, retention is linear, and the reaction function equation image can be written in a general form:

equation image

where equation image denotes Laplace transform, s [1/T] is the Laplace transform variable, g*(t) (dimensionless) is a memory function containing information on the exchange processes, and equation image [T] is the NIP residence time.

[79] A particularly useful formulation of an effective memory function g [T] is [Cvetkovic and Haggerty, 2002]:

equation image

where equation image is an exponent and T0 [T] is a characteristic retention time for the matrix depending on retention parameters P and on equation image. For simplicity (and lack of information) we assume that retention parameters are uniform (effective values), whereby T0 = const and the conditioning simplifies as equation image; spatially variable retention parameters in both the longitudinal direction and depthwise trends in the rim zone have been considered elsewhere [Cvetkovic et al., 1999; Cvetkovic and Cheng, 2008; Cvetkovic, 2010a]. Note that the power law memory function in (A3) is equivalent to a multirate model with a power law distribution of rates [Cvetkovic and Haggerty, 2002]; its application to TRUE tests with different values of equation image and a discussion on inferring T0 is given by Cvetkovic and Cheng [2008]. Inserting (A3) into (A2), we get from (A1)

equation image

where equation image is the NIP residence time density. In other words, equation image quantifies hydrodynamic transport, either as pure advective transport or as combined advection and longitudinal macrodispersion.

Appendix B:: Release Rate From Matrix ϕ (t)

[80] During injection, we can write the concentration in the fracture Cf and the matrix Cm in the Laplace domain as (written in the t, equation image, z domain and neglecting dispersion [Cvetkovic et al., 1999])

equation image

where equation image is given in (11) with (12) and z [L] is the spatial coordinate perpendicular to the fracture with the origin at the fracture matrix interface. Equation (B1) is applicable for pulse injection of mass M0, and Gequation image is the controlling parameter that can be expressed in terms of matrix retention properties [Cvetkovic et al., 1999].

[81] We have a space-time one-to-one correspondence, following equation image, where X(t) is the trajectory and v is the nonstationary radial component of velocity vector, i.e., spatially dependent, random, and deterministic. To compute the density of particle position along a trajectory in an average sense, we take the following steps.

[82] 1. Integrate equation image (equation (B1)) along the trajectory between 0 and equation image; this gives an integrated concentration equation image [M/L2] as mass in fluid phase of matrix.

[83] 2. Compute the integrated concentration equation image per unit bulk area, i.e., over an element equation image irrespective of whether it is in the solid or aqueous phase of matrix as equation image.

[84] 3. Normalize equation image with the total mass that at a time t is in the rock matrix.

[85] The result is an “effective” density of particle penetration depth z at a given time t for a specified equation image defined in the Laplace domain as

equation image


equation image

is the normalized mass in the fracture at time t and for a given equation image (integrated from 0 to equation image).

[86] A useful approximation to (B2) derives from a Gaussian distribution which quantifies pure matrix diffusion [Carslaw and Jaeger, 1959]

equation image

with equation image. A comparison between the exact and approximate f(z) is illustrated in Figure B1 for a typical parameter set and at a few specified times.

Figure B1.

Comparison between the approximate and exact expressions (B2) and (B4) for two values of the parameter G: (a) G = 3420 equation image and (b) G = 65,440 equation image.

[87] Next, we consider the density of release times from the matrix for a tracer particle starting at z subject to pure diffusion, assuming zero initial condition in the fracture (“clean” water due to withdrawal). Pure diffusion concentration for instantaneous point injection in the matrix at z = Z is

equation image

whereby the discharge (flux) from the fracture into the matrix at z = 0 reads

equation image

which is here interpreted as conditional density for a particle injected at z = Z to enter the fracture at z = 0 at time t.

[88] The function equation image is the release density from the matrix after withdrawal, which we can now compute by combining (B4) and (B6). During injection and chasing up to t = t2t3, the density of tracer particle penetration depth to z = Z in an average (integrated) sense along the fracture is given by (B4) with equation image (assuming that waiting time t3t2 is relatively short). The conditional density of particle release time from the matrix starting from z = Z in the matrix is given by (B6). The function equation image is then the unconditional density of particle release from the matrix defined in the Laplace domain as

equation image


equation image

is the Laplace transform of (B6).

[89] Integration in (B7) with (B8) yields (13), where we have taken into account that the penetration takes place in either half of the entire matrix (not just half of it).

Appendix C:: Rim Zone Effects

[90] Assume that the rim zone where retention takes place consists of two layers with distinct physical and sorption properties. The tracer particle residence time can be computed from (14) or (15), with the memory function defined by Cvetkovic [2010a]:

equation image


equation image
equation image

where equation image and equation image are defined from (10) with parameter values for respective layers 1 and 2 and z = d1 and z = d2 are distances to the end of respective layers.


[91] This work was supported by the Swedish Nuclear Fuel and Waste Management Co. (SKB). The authors are grateful to Anders Winberg (Conterra AB, Sweden) and Jan-Olof Selroos (SKB, Sweden) for their valuable comments which helped improve the presentation of the results. We are also very grateful for thorough and constructive comments by three anonymous reviewers that helped improve the original version of the manuscript.