Relation between mass balance aperture and hydraulic propertiesfrom field experiments in fractured rock in Sweden

Calle Hjerne & Rune Nordqvist

Abstract Results from tracer tests are often used to inferconnectivity and transport properties in bedrock.However, the amount of site-specific data from tracertests is often very limited, while data from hydraulic testsare more abundant. It is therefore of great interest forpredictive transport modeling to use hydraulic data to infertransport properties. In this study, data from cross-holetracer tests carried out in crystalline bedrock in Swedenwere compiled and analysed. The tests were performedwithin investigations made by the Swedish Nuclear Fueland Waste Management Company (SKB) between 1978and 2009 at five different locations. An empiricalrelationship between mass balance aperture and transmis-sivity was found and quantified by using 74 observations.The empirical relationship deviates considerably from thecubic law aperture, as mass balance aperture is found to beat least one order of magnitude larger than cubic lawaperture. Hence, usage of cubic law aperture, derived fromhydraulic testing, for transport predictions is unsuitable, asthe advective transport time will be considerablyunderestimated. Another result, from the data set studied,is that mass balance aperture appears to correlate better toapparent storativity than to transmissivity.

Keywords Tracer tests . Massbalance aperture . Fracturedrock . Sweden . Nuclear waste

Introduction

The Swedish Nuclear Fuel and Waste ManagementCompany (SKB) has carried out investigations andresearch for the disposal of spent high-level nuclear fuel

in Sweden for several decades. Radionuclide transport inhydrogeological formations is one of the key factors forthe safety analysis of a future deep geological repository.Field tracer tests have been employed extensively withinvarious SKB programmes for investigation and researchconcerning connectivity and transport properties in thebedrock. However, the amount of site-specific data fromtracer tests is often very limited due to practical andeconomical reasons. Instead, hydraulic data, such astransmissivity, are used to infer transport properties inthe safety analysis of a final repository for spent high-levelnuclear fuel and predictive tracer modeling.

This study concerns the relationship between hydraulicand tracer transport properties and comprises a compila-tion and an analysis of results from cross-hole tracer testswith conservative tracers performed within various SKBinvestigations. The specific focus is on the relationshipbetween the tracer mean residence time (mass balanceaperture) and fracture hydraulic parameters.

The geometrical conceptualization of flow and trans-port in rock fractures has evolved from parallel-plate flow(Snow 1965) to more realistic notions of spatial variabilityin fracture aperture due to fracture roughness and asperitycontacts (e.g. Berkowitz 2002). Nevertheless, apertureestimates from hydraulic and transport field experimentsin boreholes are often reported in terms of effectiveparameters based on the parallel-plate model. It is welldocumented that equivalent apertures estimated fromtracer tests are often significantly larger than thoseestimated from hydraulic responses (e.g. Abelin et al.1985; Nowakowski et al. 1985; Tsang and Tsang 1987;Moreno et al. 1988). In true parallel-plate flow, thereshould be no discrepancy between apertures interpretedfrom hydraulic tests and tracer tests, respectively, with thetransmissivity being proportional to the cube of theaperture (Snow 1965), the so-called cubic law. However,in a variable-aperture fracture system, even if the cubiclaw is valid locally, one can argue that hydraulic boreholetests are sensitive to small apertures, while tracer tests aremore sensitive to the arithmetic mean of the apertures(Tsang 1992; Gelhar 1993).

For the hydrogeological site investigations at theSwedish sites, primary field data comprise hydraulic andtransport properties obtained from borehole investigations,usually in isolated sections using multi-packer systems.Transmissivity estimates (T) for borehole sections are the

Received: 14 June 2013 /Accepted: 14 March 2014

* Springer-Verlag Berlin Heidelberg 2014

Electronic supplementary material The online version of this article(doi:10.1007/s10040-014-1132-2) contains supplementary material,which is available to authorized users.

C. Hjerne ()) :R. NordqvistGeosigma AB, Box 894SE-751 08, Uppsala, Swedene-mail: calle.hjerne@geosigma.seTel.: +46-10-4828811

Hydrogeology JournalDOI 10.1007/s10040-014-1132-2

most abundant type of data, while transport properties(including aperture estimates) are available only from arelatively limited number of cross-hole tracer tests. Thus,any empirical relationships between hydraulic data (e.g. T)and transport properties are of interest for parameteriza-tion of hydrogeological models used in the safetyassessment.

Empirical studies of relationships between transportand hydraulic parameters in low-permeability rock arescarce, but a fairly large data set from a number of sites,including some of the experimental data within the presentstudy, was analyzed by Guimerà and Carrera (1997,2000). They found a correlation between hydraulic andtransport properties, although too noisy to be used fordetailed predictions.

This article begins with a brief overview of the conceptof mass balance aperture and continues with a descriptionof field methods used for tracer and hydraulic tests. Datacompiled from cross-hole tracer tests performed within thevarious SKB investigation programmes as well as corre-sponding hydraulic data are then presented and analyzedin order to investigate the relationship between hydraulicparameters and tracer transport parameters. This article isbased on Hjerne et al. (2010) where a more extensivedescription of studied tests, analysis methods, results anddiscussion may be found.

Aperture

The aperture of a single fracture is a measure of thefracture wall separation. The true physical aperture of asingle fracture shows considerable spatial variability.Nevertheless, the aperture of a single fracture is oftendescribed by a single value that is considered to represent,in some sense, the effective or equivalent aperture.

For single fractures, it is reasonable to use the conceptof aperture for describing the water-flowing portion of thefractures involved in advective transport. For fracturezones, the conceptual understanding is often that thefracture zone is made up of several interconnected smallerfractures. In such cases, an alternative is to describe theportion of the fractures involved in advective transport interms of kinematic porosity (also known as effective orflow porosity in literature) together with a specifiedthickness. In this article, for purposes of comparison, asingle value representing the total aperture is usedirrespective of whether the transport path is in a singlefracture or in a zone.

The concept of aperture differs depending on whetherit is considered in terms of advective transport, hydraulicor true physical aspects. Tsang (1992) presents a discus-sion of definitions, notations and usage of the termequivalent aperture. Three definitions of equivalentaperture are discussed: mass balance aperture, frictionalloss aperture and cubic law aperture. Mass balanceaperture relates to transport and residence time and is thefocus of this article. Zheng et al. (2008) showed withlaboratory experiments that mass balance aperture should

be used, rather than the other two apertures, to describetracer transport in a single fracture with variable apertureand that mass balance aperture is a good approximation ofthe arithmetic mean aperture of the fracture.

Mass balance aperture, δm [L], is defined so that it,together with other geometrical assumptions about thefracture, makes up a volume that equals the meanresidence time, tm [T], multiplied by the flow rate Q[L3 T−1]. Assuming a radially converging steady-stateflow field in a homogenous plane-parallel fracture withtransport distance r [L] to a pumping hole with radius rw[L] and a mean residence time tm, the mass balanceaperture is:

δm ¼ Q⋅tmπ r2−r2w� � ð1Þ

Mass balance aperture may be evaluated with othergeometrical assumptions, e.g. linear flow, but the radialcase is by far the dominating assumption for the testsincluded in this study. Alternative names found inliterature for the mass balance aperture include volume-balance aperture, fracture aperture and aperture fromtracer experiment.

For comparison, the cubic law aperture is also used inthis article. Cubic law aperture, δC [L], is derived onlyfrom hydraulic parameters and relates to the transmissiv-ity, T [L2 T−1], and to the fracture aperture, δC [L], in aparallel-plate fracture with smooth laminar flow by:

δc ¼ 12Tμρg

� �1=3 ð2Þ

where ρ is the water density [M L−3 ], μ is the dynamicviscosity [ML−1 T−1] and g is the gravitational accelera-tion [L T−2] (Snow 1965). Cubic law aperture is, inliterature, also sometimes referred to as hydraulic aperture(e.g. Renshaw 1995).

Field methods

Tracer testsThere are two major types of cross-hole tracer tests thatmay be characterized in terms of imposed flow conditions:radial flow and dipole flow. Radial flow is in many casescreated by pumping in a borehole section without otherdisturbances of the flow field. This type of test is called aradially converging tracer test. Radial flow may also becreated by injection and is then known as a radiallydiverging tracer test.

Dipole flow conditions are created by injecting water ina borehole section while pumping from another boreholesection. This is called a dipole tracer test. If the flow ratein the injection section is equal to the flow rate in thepumping section, the test is known as an equal dipole

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tracer test. Dipole tests with a lower injection flow ratethan pumping flow rate are known as weak dipole or anunequal dipole tracer test. A special case of a dipole testis when several borehole sections are used for pumpingand/or injection. In this case, the tracer test is known asmultipolar.

In practice, it is difficult to avoid small hydraulicdisturbances when injecting. The radially converging testmay then more resemble a weak dipole test. Similarly, it isdifficult to inject with the same flow rate as the pumpingrate in an equal dipole test. Some tests may becharacterized as a weak dipole only during the injectionperiod, using a small excess injection pressure, but as aradially converging tracer test during the remainder of thetest. It is therefore somewhat difficult to establish an exactdistinction between the different types of cross-hole tracertests. For evaluation purposes it is common to assumeradial flow for weak dipole tests so that these tests can beevaluated in the same manner as radially converging tests.

Hydraulic testsThere are several methods for evaluation of hydraulicparameters from field tests depending on the boreholeconfiguration (number of boreholes, surface or tunnelboreholes, et.c.) and test equipment. The main objective ofhydraulic tests in fractured rock is often to evaluate thetransmissivity, T.

The two main groups of hydraulic tests are singleborehole test and hydraulic interference test. The defini-tion for single borehole tests in this study is that the dataused for interpretation of the test originates from the sameborehole that is subjected to the hydraulic disturbance(pumping or injection). For hydraulic interference tests,one borehole is disturbed (usually pumping), while levelmeasurements are carried out in other boreholes.

Data

The data included in the study are compiled from testscarried out in crystalline bedrock at five different locationsin Sweden between 1978 and 2009, see Fig. 1. Thepurposes and objectives of the tracer tests have varied.The tests also vary in terms of experimental scale,techniques, tracers, equipment, analyses and interpretationapproaches. However, common features for the testsinclude:

– Carried out within SKB’s investigation programmes orresearch projects

– Performed as cross-hole tracer tests with conservativetracers

– Mass balance aperture is reported (or possible toevaluate from other reported information)

– Transmissivity values and other hydraulic parametersare available

The SKB projects from which data were compiled arelisted in Table 1. In some cases, data were also collectedfrom the SKB data base. The SKB reports are freelyavailable from the SKB web site, www.skb.se, except forGustafsson (1992). Further technical details about thevarious tests are given in Hjerne et al. (2010) and theunderlying reports given in Table 1.

Evaluation of tracer testsThe data presented herein originate from the originalevaluation of the various tracer tests, i.e. no re-interpre-tation has been made. As mentioned in the preceding, nodistinction has been made as to whether the transport pathcan be interpreted to comprise a single fracture, a networkof fractures or a fracture zone, in order to facilitate acomparison of all transport paths. A brief analysis wasmade in Hjerne et al. (2010) to distinguish between singlefractures and fracture zones. While fracture zones tendedto have higher transmissivity and mass balance aperture,the relationship between mass balance aperture andtransmissivity was similar for the two rock formationtypes. The time span during which these tests were carriedout is relatively large (from the early 1980s to present) andavailable techniques and computing resources haveevolved over the years. However, some of the more basicevaluation methods have been rather similar for all of thetracer tests.

For the evaluation of SKB’s tracer tests, a commonapproach has been to use a one-dimensional advection-dispersion (1-D AD) model in order to determine themean residence time and dispersivity. In many cases,alternative transport models were applied as well, forexample models accounting for matrix diffusion ormultiple transport pathways. For the equal dipole testsincluded in this study, a two-dimensional homogenousadvection-dispersion (2-D AD) model were used.

The evaluation was generally performed by fitting asimulated tracer breakthrough curve to the observed tracerbreakthrough curve. The resulting tracer residence timefrom this evaluation was then used in Eq. (1) to obtain anestimate for the mass balance aperture. It is often found,when evaluating the SKB tracer tests, that the 1-D ADmodels for one pathway fit the observed breakthroughcurve fairly well. In such cases, it is difficult to applymodels with a higher degree of freedom without havingmodel parameters too closely correlated to each other;therefore, the evaluation using the 1-D AD model is oftenregarded as the most easy to defend.

Selection, handling and analysis of dataData in this study have generally been collected fromevaluation based on 1-D AD models. This reducessubjectivity in the data selection when comparing datafrom different tracer tests. A cross-hole tracer test includesone or more injection sections and at least one withdrawalsection. The term flow path is used in this article for aparticular combination of injection and withdrawal

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section. For some flow paths, several tracer tests werecarried out. In order to avoid unnecessary bias and toeliminate effects of outliers, the median value of thetransport parameters was considered to be representativefor each flow path.

For most flow paths in this study, several estimates oftransmissivity are available. Transmissivity estimates fromhydraulic interference tests, if available, were consideredbest-choice values and selected for further analysis. If nosuch value was available, a transmissivity estimate from asingle borehole test in the withdrawal section was selectedinstead. Finally, if no other values were available, atransmissivity estimate from the tracer injection section

was selected. The primary reason for this order of priorityis that interference tests are evaluated with considerationto effects both in the pumping hole as well as theobservation hole, just as the cross-hole tracer test. Onemay argue that this could be a very uncertain estimate ifthe hydraulic connection between the two boreholes ispoor. However, the tracer tests considered in this studygenerally resulted in relatively high tracer recovery,indicating good hydraulic connectivity. Transmissivityestimates from the withdrawal section were used as thesecond choice because earlier experience suggests thatthey are often close to transmissivity estimates frominterference tests (see e.g. Lindquist et al. 2008a). If

Fig. 1 Locations in Sweden where studied tracer tests were carried out

Table 1 Tracer tests included in the study

Location Project Reference Subarea notationused in this study

Studsvik Cross-hole tracer test 2 at Studsvik Klockars et al. (1982),Landström et al. (1983)

Studsvik

Finnsjön Cross-hole tracer tests in Finnsjön Gustafsson and Klockars (1981) Fi_G2-G1Finnsjön The fracture zone project phase 3 Andersson et al. (1989a),

Gustafsson and Nordqvist (1993)Fi_Zone2

Stripa Migration in single fracture Abelin et al. (1985) S_Frac2Stripa Small-scale tracer tests at the SGAB tracer test drift Andersson and Klockars (1985) S_SmallSStripa Monitoring of saline tracer transport with borehole radar Zone C Andersson et al. (1989b) S_ZoneCStripa Monitoring of saline tracer transport with borehole radar Zone H Olsson et al. (1991) S_ZoneHÄspö Long term pumping and tracer test - LPT-2 Rhén et al. (1992) Ä_LPT-2Äspö Passage of NE-1 Gustafsson (1992) Ä_NE-1Äspö Tracer Retention Understanding Experiments – TRUE-1 Winberg et al. (2000),

Andersson et al. (2002a)Ä_TRUE-1

Äspö Tracer Retention Understanding Experiments – TRUE-Block Scale Andersson et al. (2002b),Andersson et al. (2004),Andersson et al. (2007)

Ä_TRUE-BS

Forsmark Large-scale tracer test with pumping in KFM02B Lindquist et al. (2008a) Fo_KFM02BForsmark Large-scale tracer test with pumping in HFM14 Lindquist et al. (2008b) Fo_HFM14Laxemar Tracer test with pumping in HLX35 Morosini and Wass (2007) L_HLX35Laxemar Tracer test with pumping in HLX33 Svensson et al. (2008) L_HLX33Laxemar Large-scale tracer test with pumping in HLX27 Lindquist et al. (2008c) L_HLX27Laxemar Large-scale tracer test with pumping in HLX28 Thur et al. (2010) L_HLX28

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several estimates of transmissivity were available for eachtype of test (e.g. interference test), the median wasconsidered to be representative.

Results

Sufficient data for the purpose of this study, i.e. estimatesof transmissivity and mass balance aperture, were foundfor 74 individual flow paths in the references given inTable 1. For 27 of the 74 flow paths, estimates ofhydraulic diffusivity, To/So, and apparent storativity, So,evaluated from hydraulic interference tests, were alsofound. These data are summarized in the electronicsupplementary material (ESM).

In Fig. 2, mass balance aperture is plotted againsttransmissivity for the various flow paths compiled herein.The plot also includes a least-squares linear fit togetherwith a confidence interval for the fit. For comparison, theaperture as predicted by the cubic law (Eq. 2) is shown aswell. The logarithmic residual between data and the fit,log(δm,test) – log(δm,fit), is shown in Fig. 3.

In addition to studying correlation between massbalance aperture and transmissivity, a brief comparisonwith other hydraulic parameters, such as storage andhydraulic diffusivity, were made as well. Previous studieshave indicated that early solute arrival during a tracer testshould be correlated to early hydraulic responses, i.e.apparent storativity or hydraulic diffusivity (e.g. Knudbyand Carrera 2006). On the other hand, it is not obviousthat the mean tracer arrival time, and consequently alsothe mass balance aperture, should be related to storageand/or hydraulic diffusivity (Knudby and Carrera 2005;Trinchero et al. 2008). Nevertheless, as apparentstorativity values are available for many of the transport

paths included in this study, it is interesting to see whetherany empirical relationship exists between mass balanceaperture and storage parameters.

In Figs. 4–6, mass balance aperture is plotted againsthydraulic diffusivity, apparent storativity and transmissiv-ity, respectively, for the 27 flow paths where suchestimates from hydraulic interference tests were available.

Discussion

It is clear that there exists an empirical relationshipbetween mass balance aperture and transmissivity asshown in Fig. 2. Although the plotted data set is noisy,the mass balance aperture is roughly proportional to thecube root of the transmissivity, i.e. analogous to the cubiclaw (Eq. 2). However, the mass balance aperture is one totwo orders of magnitude larger than the aperture predictedby the cubic law. These findings are similar to conclusionspresented in the review by Guimerà and Carrera (2000),who reported a relationship of hydraulic conductivitybeing approximately proportional to the cube of thefracture porosity. That mass balance apertures derivedfrom mean tracer arrival times should differ from aperturevalues derived from hydraulic tests is not surprising and isrelated to the aperture heterogeneity (e.g. Gelhar 1993;Piggott and Elsworth 1993; Moreno et al. 1988).

For the data included in this study, the relationshipbetween hydraulic diffusivity and mass balance aperture isvery poor as seen in Fig. 4. Note that there are fewer datapoints in Fig. 4 than in Fig. 2 as hydraulic diffusivity datawere available for only 27 of the 74 flow paths presentedin Fig. 2. However, due to the poor correlation in Fig. 4, itis not expected that better correlation between mass

Fig. 2 Mass balance aperture plotted against transmissivityincluding a least-square fit with a 95 % confidence interval for thefit. The aperture according to the cubic law is included as reference

Fig. 3 Logarithmic residual between data and the fit in Fig. 2

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balance aperture and hydraulic diffusivity would be foundeven with more data available.

Figure 5 indicates that there exists an empiricalrelationship between apparent storativity and mass balanceaperture. The good fit in Fig. 5 does not seem to be aneffect of a smaller data set used than in Fig. 2 as Fig. 6shows that the fit to the subset of flow paths where valuesof So are available do not deviate considerably from the fitto the entire data set. An interesting difference betweenFigs. 5 and 6 is that the variability in apparent storativityis high compared with the variability in transmissivity for

individual features, e.g. Fi_Zone2, Ä_TRUE-BS andFo_HFM14.

Another observation made is that the flow pathL_HLX28 with a very large value of δm plots fairly closeto the linear fit in Fig. 5 while it deviates considerably inFig. 6. Hence, it seems like that apparent storativity maybe a better indicator than transmissivity of the heteroge-neity of the fracture system in terms of advective transporttime. In contrast to apparent storativity, there appears to beno correlation at all between mass balance aperture andhydraulic diffusivity. It may be possible that hydraulicdiffusivity would be better correlated with the early arrivalof tracer (Knudby and Carrera 2006), rather than the meanarrival time, although early tracer arrival data have notbeen available for the present study.

When discussing the preceding empirical relationshipsand their use for other purposes such as predictivemodeling, it is important to also consider possible sourcesof error and bias in the data that could affect the slope andoffset of the fit as well as deviation between the data andthe fit. Among the more important are:

– Flow paths used for tracer tests are not randomlychosen. On the contrary, in many cases the selectionprocess prior to the tracer test has been rigorous informs of hydraulic testing and geological characteriza-tion in order to choose flow paths where it is likely toobserve tracer breakthrough within a reasonable timeframe.

– Tracer tests where no tracer breakthrough was observedwere not included in this study for the simple reasonthat no estimates of mass balance aperture areavailable. For those cases where tracer breakthrough

Fig. 4 Mass balance aperture plotted against hydraulic diffusivityincluding a least-square fit

Fig. 5 Mass balance aperture plotted against apparent storativityincluding a least-square fit

Fig. 6 Mass balance aperture plotted against transmissivity forflow paths where apparent storativity were available. The figure alsoincludes a least-square fit to the data (red line) and the least-squarefit to all data in Fig. 2 (black line) for comparison

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may have occurred if the test time would have beenlonger, it is likely that the evaluated value of δm wouldplot above the best-fit line in Fig. 2.

– It is often assumed in the evaluation that the transportof tracers occur along a straight line from the injectionand to the withdrawal point. The actual travel distancemay be significantly longer for a number of reasons:e.g. the injection is made in some other fracture thanthe one where pumping takes place, tortuosity effects,or the presence of a region with low transmissivitybetween the injection and pumping points. Regardlessof which, the evaluated mass balance aperture wouldbe larger than the true mean aperture of the flow path.

– Uncertainties in the determination of water residencetime from tracer breakthrough curves may also arisefrom neglecting transport processes that may have asignificant impact on estimated values such as diffusiveprocesses into stagnant parts, multiple pathways (ad-vective heterogeneity) or not accounting for non-linearvelocity distributions between injection and sampling.

Due to the unavoidable bias and uncertainty in the data,it is difficult to establish that the empirical relationshipsbetween mass balance aperture and transmissivity/appar-ent storage are generally valid for fractured rock withinthe studied geographical area. However, the relationshipspresented could nevertheless be useful for modeling asone way to estimate transport times.

Conclusions

Although scattered, an empirical relationship betweenmass balance aperture, δm, and transmissivity, T, havebeen found. The best-fit for the relationship by using 74data was quantified to δm=0.33 ⋅ T0.31. The logarithmicresiduals between data and the fit may be regarded asapproximately normally distributed with a standard devi-ation of 0.5.

The empirical relationship between δm and T deviatesconsiderably from the cubic law aperture, δC. The massbalance aperture δm varies approximately with the cubicroot of T as also does δC. However, δm is found to beapproximately at least one order of magnitude larger thanδC. Hence, usage of cubic law aperture, derived fromhydraulic testing, for transport predictions is unsuitable asthe advective transport time will be considerablyunderestimated.

For the data set studied, the mass balance apertureevaluated from a cross-hole tracer test appears to be bettercorrelated to apparent storativity than to transmissivity.For design of a tracer experiment, and if data from a priorhydraulic interference test is available, apparent storativitymay be more useful than transmissivity for predictingmean tracer arrival.

Acknowledgments This work was supported by the SwedishNuclear Fuel and Waste Management Company (SKB). The authorslike to thank Jan-Olof Selroos, Ingvar Rhén, Sven Follin, Thomas

Doe and Vladimir Cvetkovic for many insightful comments duringthe work with this article.

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Hydrogeology Journal DOI 10.1007/s10040-014-1132-2