Glamheden, Lindblom - 2002 - Thermal and Mechanical Behaviour of Refrigerated Caverns in Hard Rock

Tunnelling and Underground Space Technology 17 (2002) 341353

0886-7798/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0886-7798 02 .00029-9

Thermal and mechanical behaviour of refrigerated caverns in hard rockR. Glamheden *, U. Lindbloma, b

Department of Civil Engineering, Sycon Teknikkonsult AB, SE-205 09 Malmo, Swedena Gecon AB, Viktor Rydbergsgatan 1A, SE-411 34 Goteborg, Swedenb

Received 1 December 2001; received in revised form 15 April 2002; accepted 20 April 2002

Abstract

One key issue in the design of unlined caverns for low temperature products is understanding the disturbance of the rockfracture network. Increasing aperture and extension of fractures inevitably affect the rock mass stability, the heat loss from storedproducts, and the risk of ice growth in cases when water invades the cavern. Consequently, it is essential that the designer of arefrigerated cavern has appropriate knowledge of the coupled thermal and mechanical behaviour of the fractured rock mass.Chalmers University of Technology in Gothenburg, Sweden has for several years, carried out research in the field of mechanicaland physical phenomena of rock masses subjected to low temperatures. The main investigation was performed in a pilot scalecavern in hard rock, constructed as a vertical cylinder with a diameter of 7 m and a height of 15 m. The facility was equippedwith comprehensive instrumentation, including approximately 200 temperature gauges and 140 deformation gauges. In the test,the temperature in the cavern was reduced in a stepwise manner toy40 8C, with comprehensive monitoring of relevant parameterssuch as relative humidity, air and rock temperature, rock strain, rock fracture aperture, cavern convergence and rock massdeformation. Prior to the field test, major efforts were made to predict the field results by analytical and numerical methods. Theessential results of the theoretical analyses and the actual measurements of thermal and mechanical behaviour of the cavern aregiven in this paper. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Igneous rock; Rock cavern; Underground storage; Refrigeration; Thermal strain; Thermal deformation; Thermal properties

*Corresponding author. Tel.: q46-25-56-88; fax: q46-30-46-96.E-mail address: [email protected] (R. Glamheden).

1. Introduction

Cooling down of a rock material makes its mineralmatrix shrink, and this phenomenon has some distincteffects on the rock mechanical situation around a refrig-erated storage cavern:

changes in rock parameters relaxation of compressive rock stresses near the

cavern opening of existing fractures at total relaxation of the

stresses development of new fractures

Unlike rock mechanical tunnel design in general, oneof the key issues in designing unlined caverns for lowtemperature products is an understanding of the distur-bance in the rock fracture network in the surrounding

rock mass caused by the refrigeration. Increasing aper-ture and extension of the fractures, which will follow,inevitably affect the rock mass stability, as does thedevelopment of new fractures. The problem is a coupledthermo-mechanical one.Expansion and development of fractures also increases

the risk for thermal erosion effects and ice growth fromwater invading the cavern (Broms et al., 2001). Toprevent such effects, large provisions of energy to thecavern could be required.Particularly when storing liquefied gas products such

as LPG and LNG, an increase of the fracture porosityaround the cavern augments the risk for boil-off and gasescape through the fracture system. Such phenomenonaat an extended scale may make it difficult and costly tomaintain an appropriate storage temperature (Jacobsson,1977; Lasseter and Witherspoon, 1974).In order to understand the physics that influences the

fracture aperture, the rock mass must be considered asan assembly of discrete blocks. The magnitude of anyindividual fracture aperture is dependent on the defor-

342 R. Glamheden, U. Lindblom / Tunnelling and Underground Space Technology 17 (2002) 341353

Fig. 1. Factors and rock parameters that affect the thermal deformationof a block in the rock mass surrounding a refrigerated cavern. Fig. 2. View of the test facility.

mation of the adjoining blocks, which in turn, is affectedby at least three factors:

thermal strain due to volumetric change of the intactrock material

strain due to stress change strain due to expanding ice in micro-cracks and macro

fractures

These three factors are, furthermore, dependent onthe perturbation of several rock parameters, the cavernshape, etc., as indicated in Fig. 1.It follows that the fracture aperture is dependent on

the block size, since the total block deformation (lengthperturbation) will be greater in a rock mass with widely-spaced fractures than in one with close spacing.Previous research in the area of low temperature

storage has mainly dealt with the temperature depend-ency of rock parameters such as: tensile strength;Youngs modulus; Poissons ratio; coefficient of thermalexpansion; joint stiffness; thermal diffusivity; thermalconductivity and specific heat; (Aoki et al., 1989; Inadaand Yagi, 1980; Lindblom, 1977). Tests have tradition-ally been conducted in the laboratory, with the resultsenlarged to rock mass scale by using the laboratoryvalues as input to numerical models (Ishizuka et al.,1985; Ishizuka and Kinoshita, 1988; Inada et al., 1995,1999). Storage caverns have also been simulated byperforming laboratory experiments on rock blocks witha central hole (see, for example Ishizuka and Kinoshita,1988; Homer et al., 1989).Experience in rock mechanics tells us that it is likely

that the calculated rock mass response, based on labor-atory test data, will diverge from the full scale rockmass response due to scale effects. In addition, manmadedisturbance of the rock and other factors cannot betaken into account in any such model. The predictedrock mass response in a model must consequently, morethan likely be adjusted to match observations in thefield.

Dahlstrom (1992) initiated studies in this area andperformed an introductory refrigeration test in the samepilot scale rock cavern. He compared the results of thefield test with numerical analyses using continuummodels. It was concluded from the study that the rockmass behaviour was too complicated to be describedfully by these types of models.The objective of the present study was to continue

Dahlstroms work in the cavern by employing a modi-fied and more comprehensive instrumentation, as wellas by using extended modelling, including the distinctelement method. The procedure was to use data fromthin sections of intact rock samples to predict thecoefficient of thermal expansion and the thermal prop-erties of the rock material at ambient rock temperature.In the next step, the prediction was to be enlarged torock mass scale using numerical models and input datafrom laboratory tests, in situ stress measurements andstructural geological data. The resulting prediction wasto be compared with the previous full-scale field obser-vations by Dahlstrom (1992). The results of the newinvestigation are found below.

2. Site conditions

2.1. Layout and geometry of the test cavern

Roda Sten Rock Laboratory (RSRL) is a facility inGothenburg, Sweden, operated by Chalmers Universityof Technology. This site was commissioned in 1990,mainly for the purpose of large-scale testing of technol-ogy for underground gas storage. At this site, numerousresearch projects have been carried out, including indus-trial research and doctoral research work (Dahlstrom,1992; Glamheden, 2001; He, 1992; Soder, 1994).RSRL is located approximately 70 m below ground

level and approximately 30 m below the water table ofthe nearby river Gota Alv. Access to the test site isprovided through a tunnel. The site consists of two rock

343R. Glamheden, U. Lindblom / Tunnelling and Underground Space Technology 17 (2002) 341353

Table 1Mechanical properties of the intact rock (modified after Dahlstrom,1992)

Parameter Mean S.D.

Youngs modulus (GPa) 56 5.5Poissons ratio 0.26 0.03Uniaxial compressive strength (MPa) 106 17Tensile strength W. Saturated (MPa) 9.6 1.6

Table 2Evaluated in situ stress field at RSRL, based on seven cells in twoboreholes at level y24.4 m (Dahlstrom and He, 1992)

Principal Mean S.D. Strike Dipstress (MPa) (MPa) (8) (8)

s1 7.2 2.0 102.6 7.2s2 3.4 1.3 193.8 9.2s3 2.0 1.0 335.4 78.3

caverns, A and B, Fig. 2. Cavern A is used for unlinedcompressed gas storage and cavern B for unlined refrig-erated gas storage. In all cases, air was used as the testgas. The cavern described in this paper is cavern B.The test cavern B is constructed as a vertical cylinder

with a diameter of 7"0.5 m and a height of 15"0.5m. The cavern was excavated using conventional shaftsinking methods, i.e. drilling of charge holes, blastingand mucking. The total cavern wall surface was approx-imately 425 m and the cavern volume was approxi-2mately 625 m .3The rock support in the cavern is kept at a minimum

to minimise any influences on the rock mass thermaldeformation. The reinforcement consists of spot boltingcombined with a chain-link mesh in the roof and in theupper, northern part of the cavern. The bolt type installedis cement-grouted dowels with 25 mm diameter andapproximately 3 m in length. Shotcrete and groutingwas not used in the cavern.The drift at the cavern entrance is heavily insulated

by a wooden frame wall with 1 m of mineral woolinsulation. The purpose of the insulation is to limit theheat transfer to the tunnel and to maintain radial heatflow into the cavern.Approximately 7 m away from the cavern entrance, a

concrete plug is constructed. This plug seals off thecavern space from the neighbouring tunnel system. Itsthickness is approximately 1 m with a further 400 mmof thermal insulation on the cavern side.

2.2. Geology

The dominating rock type around cavern B is agranodiorite with gneiss structure and elements of alkaligranite. The grain size is fine to medium and thefoliation direction is approximately N2408y508N. Therock mass is non-weathered and grey to greyish-red.Other rock types in the area are aplite and amphibolite,occurring as dykes. The dyke direction is the same asthe foliation in the main host rock. The mechanicalproperties of the intact rock are presented in Table 1.The joints observed in the cavern can be arranged as

three characteristic joint sets, plus random joints. Thedominating joint set orientation is approximately N1308y808W. The second joint set has the same orientation asthe rock mass foliation at approximately N2408y508N;the least frequent joint set is a sub-horizontal one.

Most of the joints are planar with smooth unalteredsurfaces. Joint coatings and fillings occur as calcite,chlorite and epidote. In some joints the filling materialis altered to clay.Before the rock caverns were constructed, in situ

stress measurements were performed in the body of rockwhere the caverns were to be located (He, 1992). Thestress field was determined by overcoring of nine stresscells type CSIRO HI. The evaluated magnitude of theprincipal stresses and their orientations are listed inTable 2.

2.3. Rock mass mechanical properties

The ability to predict the strength and deformationproperties of jointed rock masses on the basis of directtests is severely limited. Consequently, these parametersare often estimated using relationships coupled to rockmass classification systems (Hoek and Brown, 1980;Serafim and Pereira, 1983; Brady and Brown, 1985;Hoek et al., 1995).In this work, classification of the rock mass around

cavern B was made according to the Rock Mass Rating(RMR) system (Bieniawski, 1989) and the TunnellingQuality Index (Barton et al., 1974). Parameters wereselected based on the geological site investigation, thecharacteristic data for the intact rock and the joint setsand the in situ stresses.For the geological data obtained at the test facility,

the Q-value was estimated to be approximately 15 andthe RMR -value to be approximately 75, both indicating89that the quality of the rock mass around cavern B isgood for tunnel construction. Transferring of the valuesto the Geological Strength Index (GSI), introduced byHoek et al. (1995), yields a GSI of approximately 70.Appendix A presents the classification procedure andAppendix B the rock mass properties coupled to theclassification systems. Rock mass mechanical properties,evaluated at y16 8C, are presented in Table 3.

2.4. Thermo-mechanical and thermo-physical properties

A comprehensive investigation of the thermo-mechan-ical and thermo-physical properties of the host rock wascarried out in this project. Much work was also done inthe previous project in the cavern (Dahlstrom, 1992),and was considered in the project.


Table 3Rock mass mechanical properties evaluated at y16 8C and used inthe prediction

Parameter Value

Rock mass Youngs modulus (GPa) 42Rock mass Poissons ratio 0.23Compressive strength (MPa) 20Cohesion (MPa) 4.5Friction angle (8) 40Dilation angle (8) 10Tensile strength (MPa) 0Joint normal stiffness (GPaym) 76Joint shear stiffness (GPaym) 38Joint cohesion (MPa) 0Joint tensile strength (MPa) 0Joint friction angle (8) 36.5Joint dilation angle (8) 6

Table 4Coefficient of thermal expansion and thermal properties evaluated aty16 8C and used in the prediction

Parameter Mean

Coefficient of thermal expansion (10 y8C)y6 4.28Thermal diffussivity (10 m ys)y6 2 1.65Thermal conductivity (Wym 8C) 2.67Specific heat (Jykg 8C) 602.3Density (kgym )3 2689

According to the results, the tensile strength increaseswith decreasing temperature. At the freezing point,however, the mean values of the tensile strength for air-dry and saturated samples were found to be approxi-mately the same. The results have implications for thefield test, since air-dry rock conditions may occur at thecavern boundary, while the rock outside the cavern iswater-saturated. The mean values of Youngs modulusand Poissons ratio increase only marginally at decreas-ing temperature.The coefficient of thermal expansion at ambient

temperature of the intact rock of samples from RSRLwas calculated by using elastic properties of the rock-forming minerals and the observed mineral compositionof samples from the site. It was estimated to be approx-imately 6=10 y8C.y6In addition, thermal rock properties and the occur-

rence of micro-cracking were investigated. The proper-ties used in the prediction correspond to those evaluatedat y16 8C, see Table 4.

3. Prediction of the rock mass response

Prediction of the expected rock mass response in thelarge field test was made using the theories of conductiveheat transfer and thermo-elasticity in the rock. Thetransient mode in the experiment was solved analyticallyby approximations of the cylindrical geometry to asphere. This simplification allows a relatively clear andsimple solution without introducing too much error.Prediction of the rock mass response was also made bynumerical models; these facilitate the use of a correctgeometry.

3.1. Analytical model

The thermo-elastic solution outside a spherical cavityexposed to a temperature change was developed byClaesson (2001). The analytical solution is divided into

two parts, one for the temperature steps taken duringthe cooling period and the other for the temperaturegain during thawing. A survey of the algorithms usedin the analytical work is presented in Appendix C.The boundary condition during cooling is periodically

constant. During thawing, the boundary conditions arechanged to zero boundary flux. The temperature distri-bution for the number of temperature steps taken in thefield test is obtained by superposition. The analyticalcalculations were performed by two calculation fileswritten in the commercial code Mathcad.

3.2. Numerical models

The numerical codes used in the thermal-mechanicalcalculations were FLAC and UDEC, which are based onthe finite difference method (ITASCA, 2000).

FLAC was used to study the temperature distributionand the thermo-mechanical response assuming the rockmass to be an elastic or elasto-plastic isotropic contin-uum. The elastic model is characterised by deformationsreversible upon unloading; the stressstrain laws arelinear and path independent. This model predicts themaximum thermal stress due to cavern refrigeration.The elasto-plastic models potentially involve some

degree of permanent and path-dependent deformations.The yield function used in the models is the MohrCoulomb failure criterion with a tension cut-off. Theelasto-plastic models can handle the effect of failure inboth tensile and shear modes on the thermaldeformations.

UDEC, a distinct element code, was used to study therock mass as an assembly of discrete blocks in contrac-tion; in this model, existing fractures will open duringrefrigeration. The solid blocks were assigned the prop-erties of an elastic isotropic material. The joint behaviouris linear-elastic in the normal direction and elasto-plasticwith MohrCoulomb slip failure in the shear direction.Two element grids were utilised in the continuum

model. The first grid represented a horizontal plane atthe cavern mid-height and the second grid an axi-symmetric vertical section of the cavern, see Fig. 3.The radius of the horizontal grid was 75 m, and the

width and height of the vertical grid were 75 and 150m, respectively. The cavern radius was set to 3.75 m


Fig. 3. The element grids used in the continuum model. The left graphshows the horizontal plane at the cavern mid-height and the rightgraph the axi-symmetric vertical section of the cavern.

Table 5Designations and characteristics for continuum models

Designation Direction Mechanical Horizontalmodels stresses

Fh1_el Horizontal Elastic s ysH hFv1_el Vertical Elastic sHFv2_el Vertical Elastic shFh1_mo Horizontal MohrCoulomb s ysH hFv1_mo Vertical MohrCoulomb sHFv2_mo Vertical MohrCoulomb sh

Fig. 4. Block grid for the discrete element model in the horizontalplane at the cavern mid-height (Uh1) in a close-up view.

Fig. 5. Block grids for the discrete element model in the verticalsection of the cavern in a close-up view.

Table 6Designations and characteristics for discrete block models

Designation Model Joint Horizontaldirection directions stresses

Uh1 Horizontal N1308y808W s ysH hUv1 N3108yvertical Horizontal 0.75=shUv2 N2208yvertical N1308y808W 0.75=sHUv3 N2208yvertical Horizontal-N 1308y808W 0.75=sHUv4 N3108yvertical Horizontal-N 2458y508N 0.75=sh

and the cavern height to 15.5 m. Designations andcharacteristics of the continuum models are presented inTable 5.The discrete block model consisted of five element

grids, one horizontal plane at the cavern mid-height andfour vertical sections, two perpendicular and two parallelto the dominating joint set, respectively. The dimensionof the horizontal grid was 75=75 m, and the width andheight of the vertical one was 150 m. The geometry of

the horizontal and vertical grids is shown in a close-upview in Figs. 4 and 5, respectively.The symmetry of the cavern geometry could not be

utilised in the discrete block models, since the jointsystems intersect the cavern asymmetrically. The intro-duction of fractures in the model also prevented the useof axi-symmetry. In the vertical discrete block models,the cavern was consequently simulated as an infinitetunnel rather than as a cylinder. This resulted in anoverestimation of both temperature and deformation inthe rock mass. The vertical discrete block models weretherefore mainly used for qualitative studies of thephysics involved in the field test.The joint space and joint length were gradually

increased in the model. The number of joints as well astheir length was limited by construction joints, whichdid not influence the calculations. In the horizontalplane these joints are circular and concentric to thecavern. Designations and characteristics of the discreteblock models are presented in Table 6.


Table 7Boundary conditions

Boundary Mechanical Thermal

Cavern boundary Free Cooling:Periodically constant (T)Thawing:Free temperature (T)

Ground surface Free Zero heat flux

Other boundaries Fixed in Zero heat fluxnormal direction

Table 8Modelling steps in the calculation path

Step Thermal Temp. Commentsno. time (8C)

(days)

1 (q9.25) Setting of initial conditions2 (q9.25) Excavation of the cavern3 000006 q9.25 Start up of the field test4 006090 q0.50 Cooling started5 090104 y7.75 Temperature decrease6 104118 y16.00 Temperature decrease7 118132 y24.75 Temperature decrease8 132160 y32.50 Temperature decrease9 160202 y37.25 Cooling finished10 202370 Increasing Thawing of the cavern

Fig. 6. A photograph of a surface mounted strain gauge together with a surface joint meter.

The boundary conditions assumed in the models aregiven in Table 7. The modelling steps in the calculationpath are presented in Table 8

4. Field test

A major part of the project was the field test in thecavern B at RSRL. A first field test was performed inthis cavern in 1991 and is reported by Dahlstrom (1992).The experience from this test has been an importantbasis for the enlarged, second field test in Cavern B,which is reported here.

4.1. Instrumentation

The facility was equipped with vary comprehensiveinstrumentation, including approximately 200 tempera-ture gauges and 140 deformation gauges. The followingparameters were monitored; relative humidity, air and

rock temperature, rock strain and fracture aperture,cavern convergence and rock mass deformation.Platinum resistance thermometers were installed

inside the cavern on seven horizontal levels in thecavern, in the roof and in several drill holes in the rockmass.The deformation gauges were of the vibrating wire

type. Strain gauges with a length of 150 mm wereinstalled in horizontal and vertical directions on intactrock surfaces in the roof and on the cavern walls.Instruments were installed on blocks parallel and per-pendicular to the dominating joint set, foliation andexisting dikes.Surface joint meters were installed on different levels

in the cavern but were concentrated to the roof, thecavern mid-height and the cavern bottom level. The


Fig. 7. Positions of the extensometers.

Fig. 8. Cavern temperature vs. time at the cavern mid-height.

Fig. 9. Predicted and recorded temperature in the rock mass duringcooling at the cavern mid-height. The model used was the axi-sym-metrical section (Fv1_el). The gauge boreholes strike N220 andN310.

gauge locations were distributed to cross over fracturesof the dominating three joint sets in the rock mass. Aphotograph of a surface strain gauge together with asurface joint meter is shown in Fig. 6.To observe the possible influence of the disturbed

zone on the thermal strain, borehole strain gauges wereinstalled 0.5, 1, 2 and 4 m from the rock surface. Thesegauges were placed in three directions to record theradial, tangential and longitudinal strainHorizontal convergence meters were installed on the

roof in two perpendicular directions, and at the cavernmid-height with a sector interval of 458. A verticalconvergence meter was also installed along the cavernaxis.Extensometers were used to control the radial defor-

mation of the rock mass. As the cavern deformationwas expected to be non-uniform, extensometers wereinstalled in different directions, to observe anisotropydue to the rock fractures. The configuration of theextensometers is shown in Fig. 7. The directions of theextensometers coincide with the direction of the conver-gence meters.

4.2. Cooling procedure

The natural, ambient temperature in the rock masswas approximately q10 8C. In a first step, the caverntemperature was decreased to 0 8C. The temperaturewas held constant at this level for approximately 80days to adjust the instruments. The temperature wasdecreased in four steps of 10 8C from 0 8C to approxi-mately y40 8C. Each step lasted for 2 weeks.On level y40 8C, the temperature was maintained

constant for 40 days, after which the cavern was finallyleft to warm up again. The test cycle lasted a total of415 days, the cooling period comprising approximately200 days and the thawing period approximately 215days. The cavern temperature profile, which was main-tained during the test, is shown in Fig. 8.The lowest air temperature reached in the cavern was

approximately y42.5 8C, while the lowest rock surface

temperature was approximately y40.5 8C. The maintemperature drop between the air and the rock surfaceoccurred within 4 cm of the rock surface.

5. Comparison between predicted and recordedresults

5.1. Temperatures

Fig. 9 present the predicted and the recorded temper-atures in the rock mass at the cavern mid-height during


Fig. 10. Predicted and recorded strain vs. distance in the radial direc-tion at the cavern mid-height. The predicted strains by the elasto-plastic model (Fv2_mo) and recorded by the S22 gauges. Negativevalues correspond to contraction.

cooling. The recorded data are shown in two perpendic-ular directions.The rock temperature was influenced to a distance of

approximately 20 m at the cavern mid-height. The zeroisotherm reached its deepest location, 6.75 m from thecavern rock face, on day 230 during the thawing period.Back-calculation to achieve agreement between pre-

dicted and recorded temperatures during the coolingperiod required a 42.5% higher thermal diffusivity thanpredicted from the laboratory results. A possible sourceof the observed difference is that the laboratory valueswere determined on unloaded, dry rock samples, whereasthe natural rock mass is water saturated and underpressure (Scharli and Ryback, 1984). Variation in themineral composition of the rock is another source ofdifference between the laboratory value and the back-calculated value (Sundberg, 1988).

5.2. Thermal strain

The thermal strain recorded in the cavern appears tobe a good indicator of whether or not a valid materialmodel is used for the rock. For example, in a validelastic model, the thermal strain recorded at the cavernboundary will display a small expansion in the tangentialdirection and a high contraction in the radial direction.Alternatively, if the elasto-plastic model is valid, therecorded strain on the cavern boundary should insteadbe contractile in both tangential and radial directionsduring cooling and expansive in the radial directionduring thawing. The models used also show a sharpdifference in strain vs. depth.In intact rock, the thermal strains observed were

usually contractile at cooling and expansive at thawing,described by an elliptical or drop-shaped curve. Thelargest strain observed in the tangential direction wasy400 mmym and in the radial direction y180 mmym,relative to the axis of the cylindrical cavern. On thecavern wall, the maximum thermal strain was longitu-dinal, y238.1 mmym, equivalent to a linear thermalstrain of 6.2=10 y8C.y6The strain measurements on the cavern boundary and

in drillholes compared best with data from intact rockused in an elasto-plastic model, see Fig. 10. Strainrecords at depth, however, corresponded better to anelastic model. These results probably reflect the occur-rence of a blasting-induced, disturbed rock mass zonearound the cavern; another explanation may be plasticfailure in the rock mass close to the cavern due to theprevious field test.

5.3. Fracture aperture

Two of the key issues when designing caverns forlow temperature products such as LPG and LNG inunlined rock caverns, are; (i) limiting temperature for

fractures to start to open up, and (ii) magnitude offracture separation in the rock mass due to furthercooling of the cavern. These effects have an influenceon the cavern heat loss and on the water inflow to thecavern, but also on the cavern stability.Like instruments in intact rock, most instruments

installed across fractures showed contraction at coolingand expansion at thawing, which implies that most jointsremained closed during the test.The most distinct and largest fracture openings, 868.4

and 637.3 mm, were measured at the end of coolingacross two horizontal fractures just below the cavernmid-height. A sharp change occurred in the deformationrate at y12.5 8C, from "1.5 mmy8C to approximatelyy40 mmy8C, see Fig. 11.The fracture apertures predicted by the discrete block

models were in general greater than the recorded ones.This could partly be the result of variations in theestimated stresses, Youngs modulus or the coefficientof thermal expansion. However, the main reason for thedivergence is believed to be the fact that there are morefractures in the real rock mass than in the model. It isnot possible to include all fractures in a model and thecalculated apertures are divided amongst fewer fractures.The models predicted the fracture opening in the hori-zontal fracture to occur atq0.5 8C; however, it occurredonly at y12.5 8C.The discrete block models indicated a concentration

of the aperture widening to a few fractures, dependingon the fracture location and persistence. However, theconcentration of the opening to a few fractures has not


Fig. 11. Deformation vs. temperature across a horizontal fracture atcavern mid-height. Surface joint meter SJ23. Cooling stage is markedwith boxes and thawing stage with circles. Positive deformation cor-responds to joint opening.

Fig. 12. Horizontal convergence vs. temperature at the cavern mid-height. Convergence meter HC3. The cooling stage is marked withboxes and the thawing stage with circles.

Fig. 13. Displacement vectors in the horizontal section of the discreteblock model (Uh1). The lines of recorded convergence are also indi-cated in the figure. Max vector 0.63 mm.

been proven in the field test, since the number ofinstrumented joints in each joint set was insufficient.5.4. Cavern wall convergence

A fundamental question when modelling a refrigeratedrock cavern is the convergence of the cavern periphery.Are the cavern walls moving inwards or outwards duringcooling and what is the magnitude of the movement?The capability to predict the direction and magnitude ofthe displacement is important, particularly for the designof support and construction elements inside the cavern.Throughout the field test, the cavern walls converged

due to cooling. A maximum convergence of 0.70.8mm was recorded horizontally, in the upper part of thecavern. For convergence meters at the cavern mid-height, the convergence by the end of cooling wasapproximately 0.5 mm, see Fig. 12. The vertical con-vergence along the cavern axis reached a maximum of0.62 mm.Contrary to the elastic model, calculations with the

elasto-plastic and discrete block models confirmed thatconvergence of the cavern could be expected in bothvertical and horizontal directions, see Fig. 13. Again,the results probably reflect the occurrence of a pre-existing, disturbed rock mass zone around the cavern.

5.5. Rock mass deformation

The measurements throughout showed shrinkage ofthe cavern boundary at cooling and expansion duringthawing, both in tangential and vertical directions. Themain part of the rock mass deformation in axial and

radial directions occurred within an 8 m distance fromthe cavern rock wall. A maximum radial deformation of1.05 mm was measured parallel to the main joint set.Contrary to the tangential deformations, the recorded

rock mass deformations in radial direction appear to bebest fitted by an elastic model. However, the calculateddeformations are smaller than the recorded ones andshow a further distinct reduction at depths greater than6 m. The recorded deformations only show a weaktendency towards such reduced deformation at depth,see Fig. 14.


Fig. 14. Predicted and recorded radial deformation vs. distance at thecavern mid-height. Predicted deformation by the elastic model(Fv1_el), and the recorded values by instrument D7, D8 and D13,D14. The deformation is presented with the cavern boundary as ref-erence. Negative values correspond to contraction.

Back-calculation using the elastic isotropic model ordiscrete block model to match the monitored deforma-tion magnitude requires a twofold increase in the coef-ficient of thermal expansion. A higher diffusivity in therock mass than in the model, which the temperaturemeasurements indicated, can explain part of the differ-ence since it gives a more rapid thermal response thanpredicted. However, the magnitude of the divergence issuch that other contributions to the thermal deformationmust also have occurred, which are not considered inthe model.

6. Conclusions

The results demonstrate that among the models used,in general the discrete block models appear best suitedto predict the thermo-mechanical behaviour of a frac-tured rock mass. However, even with this type of model,it is difficult to achieve close agreement between cal-culated and observed values of thermal deformation.The occurrence of a disturbed zone around the cavern

appears to have particular influence on the thermal strainand cavern wall displacement of a refrigerated cavern.This fact should be noted when carrying out thermo-mechanical analyses of refrigerated caverns.Fracture frequency and fracture persistence are two

parameters of great importance to achieve agreementbetween calculated and recorded fracture aperture andtemperature of fracture opening of a cooled rock mass.However, it is difficult to include all fractures in amodel and the calculated apertures are easily divided

amongst fewer fractures. It is also difficult to assign acorrect fracture persistence in a model; a reduced per-sistence of the real, monitored fractures, compared tothe model, will appear as an added resistance to thefracture opening.The thermal response and the modelling technique

can only partly explain the divergence between predictedand monitored deformations in the radial direction.Consequently, there must exist some other source ofrock mass thermal deformation not considered by themodel.

Acknowledgments

The authors wish to acknowledge the assistance by J.Bjork, M. Blomquist and E. Baseri, in the field testoperations. The project was conducted at ChalmersUniversity of Technology (CTH), Department of Geo-technical Engineering, supported by grants from theSwedish Government (NUTEK), and by a number ofindustrial sponsors: Shell; British Gas; Brooklyn UnionGas; Vattenfall; Goteborg Energi; Sydkraft; and NCC.The test site RSRL (Roda Sten) was owned by Goteborg Energi and operated by CTH 19892001.

Appendix A: Classification of the rock mass

RMR -value for the rock mass89

Parameter Value or description Rating

Strength of intact rock, sC 110 MPa 12Drill core quality 90% 18Spacing of discontinuities 0.12 m 15Condition of discontinuity 17Persistence 0.510 m (3)Aperture 02 mm (3)Roughness Roughsmooth (2)Filling Calcite, chlorite, epidote (4)Weathering Slightly weathered (5)

Ground water -10 lymin 12RMR Good rock 74

Q-value for the rock mass

Parameter Description Value

Rock quality designation Good 90Joint set number, Jn Three joint sets plus random 10Joint roughness number, Jr Roughysmooth, planar 1.5Joint alteration number, Ja Unaltered joint wall 1.0Joint water reduction, Jw Minor inflow 1.0Stress reduction factor, SRF Medium stress 1.0Q-Value Good rock 13.5

The results obtained from the two classification sys-tems were compared by transferring the values to the


Geological strength index (GSI), introduced by Hoek etal. (1995):

GSIs9lnQq44

where

GSIsRMR sRMR 9y5 (RMR)23)76 89This corresponds to GSIs6769.

Appendix B: Relationships coupled to rock massclassification systems

B.1. Deformation of the rock massThe in situ deformation modulus of a rock mass is

very difficult to determine in the field and, large-scaleloading tests are few and far between. Because of thisthe deformation modulus of a rock mass is normallyestimated based upon rock mass classifications. From anumber of case histories, relationships have been pro-posed for estimating the rock mass deformation fromRMR and Q-indices (Bieniawski, 1978; Serafim andPereira, 1983; Grimstad and Barton, 1993). The equationused here was the one proposed by Bieniawski:

E s2GSIy100mwhere E srock mass Youngs modulus (GSIsmRMR y5).89In our case, the estimated GSI-value of approximately

70, yields a rock mass deformation modulus of approx-imately 40 GPa at the ambient temperature in the rockmass (q10 8C).Poissons ratio of a rock mass is generally not

investigated. Normally, the value for intact rock is usedor a slightly lower value. Here, the ratio is estimatedusing Youngs moduli of the rock mass and intact rockaccording to the following relationship Brady andBrown, 1985):

Emn snm E

where nsintact rock Poissons ratio, E srock massmYoungs modulus Esintact rock Youngs modulus.Introduction of the actual values into the above

relationship result in a Poissons ratio for the rock massof n s0.18 at the ambient temperature in the rock massm(ns0.26, E s40 GPa, and Es56 MPa). Poissonsmratio and Youngs modulus were then compensatedaccording to the temperature dependence observed inthe laboratory tests.B.2. Strength of the rock massTaking into account the joint systems in a crystalline

rock mass the tensile strength of a rock mass is normallyassumed to be zero. The compressive strength of therock mass can be estimated using the failure criterionof Hoek and Brown (1980):

ys ss Scm cwhere s scompressive strength of the rock mass,cms scompressive strength of intact rock, and Sscon-cstant that depends on rock conditions.For an undisturbed rock the relationship between the

constant S and the geological strength index (GSI) is asfollows (Hoek et al., 1995):

B EGSIy100C FSsexpD G9

A crude estimation of the rock mass compressivestrength can in this way be obtained from the geologicalstrength index. The calculated GSI-value of approxi-mately 70 for the rock mass in Roda Sten gave Ss0.03. According to the relationship above, thecompressive strength of the rock mass is estimated tobe s s20 MPa (s s106 MPa).cm cIn rock mechanics the shear strength of a rock mass

is frequently expressed in terms of the MohrCoulombfailure criteria. By assuming a friction angle the cohesivestrength can be estimated from the compressive strengthby the following relationship:

B E1ysinfC FC ssr cmD G2cosf

where Csintrinsic rock cohesion (MPa), s scom-r cmpressive strength of the rock mass (MPa), and fsintrinsic rock friction angle (8).In our case, the rock friction angle was assumed to

fs408, which yields a rock mass cohesive strength ofapproximately Cs4.5 MPa.rB.3. Joint stiffnessThe joint stiffness values were approximated by back-

calculation from the deformability of the rock mass andthe intact rock, assuming the jointed rock mass to havea deformational response equivalent to an elastic contin-uum (Hart, 1991). For the case of uniaxial loading ofrock containing a single set of uniformly spaced joints,oriented normal to the direction of loading, the followingrelationships apply:

EEmk sn s EyE .mGGmkss s GyG .m

where Esintact rock Youngs modulus E srock massmYoungs modulus; Gsintact rock shear modulus; G smrock mass shear modulus; and ssjoint spacing (m).According to the relationship above the joint normal

stiffness and the joint shear stiffness is estimated to beapproximately k s70 GPaym and ks35 GPaym,n srespectively, at rock mass ambient temperature (ss2m). Since the Poissons ratio and Youngs modulus aretemperature-dependent, the joint stiffness values also


The symbols used in this appendix are not in agreement with1those used in other parts of the paper.

change with temperature. In this manner, the deforma-bility in the continuum and discontinuum models isequalised, which is important when comparing theresults from the respective models.

Appendix C: Survey of formulae

In this section a survey of the main formulas used inthe analytical solution is given. A more detailed account1of the analytical solution is found in Claesson (2001).There is a spherical symmetry and the temperature

depends on the radius and time. The following integralof the temperature is used:

r1 2T r,t s r0 T r0,t dr0 TsT r,t . . . .|3r r0The displacement and the components of strain and

stress are given by the formulae:1qn 1qn

u r,t s arT r,t s aT r,t . . .t1yn 1yn2Ea

s sy T r,t .r 1yn1qn w z

x | s a T r,t y2T r,t . .y ~r 1ynEa w z

x |ss T r,t yT r,t . .y ~t 1ynThe solution for the temperature step is:

B Er ryr0 0T r,t s T yT erfc rGr t)0 . . C Fs 1 0 0r yD G4ator

2T r,t s T yT T9 ryr , 4atyr . . .s 1 0 s 0 0B E1 r9y1

T9 r9,t9 s erfc . C Fs r9 yD Gt0The temperature integral is:

2 T r,t s T yT T9 ryr , 4atyr rGr t)0 . . .s 1 0 s 0 0 0

Sw z B ET1 t9 r9-1 t9 t92 UT9 r9,t9 s r9 y1y erfc q qx | . . C Fs 3 T y2 r9 2 2 p . y ~ yD Gt9V

WT2y r9y1 yt9w z( )

x |X 2y r9q1 e .y ~TY

The solution for the thawing after the temperaturestep is:

2 2T r,t s T yT T9 ryr , 4atyr , 4at yr . . .d 1 0 d 0 0 1 0rGr t)t0 1

w B E1 r9y1 r9y1T9 r9,t9,t9 s erfc ye . C Fxd 1 r9 yD Gt9y` z

ys e f s,t9,t9 ds . |d 1|~r9y1

The corresponding temperature integral is:

2 2 T r,t s T yT T9 ryr , 4atyr , 4at yr . . .d 1 0 d 0 0 1 0rGr t)t0 1

w r9y11 T9 r9,t9,t9 sT9 r9,t9 y sf s,t9,t9 ds . . .xd 1 s d 13 |r9 . y 0

z`r9y1 ysq r9y1 e e f s,t9,t9 ds . . |d 1|

~r9y1

The temperature after N temperature steps at theboundary is:

N2T r,t s T yT T9 ryr , 4a tyt yr . . . .Ns n ny1 s 0 ny1 08

ns1

t -t-tNy1 N

The temperature integral is:N

2 T r,t s T yT T9 ryr , 4a tyt yr . . . .Ns n ny1 s 0 ny1 08

ns1

t -t-tNy1 N

The temperature and temperature integral for thetemperature decline after N temperature steps ( )t)tNare:

N2T r,t s T yT T9 ryr , 4a tyt yr , . . .Nd n ny1 d 0 ny1 08

ns124a t yt yr t)t . .N ny1 0 N

N2

T r,t s T yT T9 ryr , 4a tyt yr , . . .Nd n ny1 d 0 ny1 08ns1

24a t yt yr t)t . .N ny1 0 N

SymbolsRoman letters

a: thermal diffusivity (m ys)2E: Youngs modulus 12)r: radius (m)T: temperature (8C)t: time (s)u: displacement (m)

Greek lettersa: coefficient of linear thermal expansion (10 yy6

8C)


:r radial strain (mmym) :t tangential strainn: Poissons ratios :r radial stress (MPa)s :t tangential stress

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Thermal and mechanical behaviour of refrigerated caverns in hard rockIntroductionSite conditionsLayout and geometry of the test cavernGeologyRock mass mechanical propertiesThermo-mechanical and thermo-physical properties

Prediction of the rock mass responseAnalytical modelNumerical models

Field testInstrumentationCooling procedure

Comparison between predicted and recorded resultsTemperaturesThermal strainFracture apertureCavern wall convergenceRock mass deformation

ConclusionsAcknowledgementsAppendix AClassification of the rock massAppendix BRelationships coupled to rock mass classification systemsAppendix CSurvey of formulaeReferences

Glamheden, Lindblom - 2002 - Thermal and Mechanical Behaviour of Refrigerated Caverns in Hard Rock

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