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Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328

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Journal of Rock Mechanics andGeotechnical Engineering

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Full length article

Design issues for compressed air energy storage in sealedunderground cavities

P. Perazzelli, G. Anagnostou*

ETH Zurich, Stefano Franscini Platz 5, 8093 Zurich, Switzerland

a r t i c l e i n f o

Article history:Received 2 June 2015Received in revised form22 September 2015Accepted 24 September 2015Available online 28 October 2015

Keywords:Compressed air energy storage (CAES)TunnelsLiningConcrete plugFeasibility assessment

* Corresponding author. Tel.: þ41 (0)44 6333180.E-mail address: georg.anagnostou@igt.baug.ethz.cPeer review under responsibility of Institute o

Chinese Academy of Sciences.

http://dx.doi.org/10.1016/j.jrmge.2015.09.0061674-7755 � 2016 Institute of Rock and Soil MechanicNC-ND license (http://creativecommons.org/licenses/

a b s t r a c t

Compressed air energy storage (CAES) systems represent a new technology for storing very large amountof energy. A peculiarity of the systems is that gas must be stored under a high pressure (p ¼ 10e30 MPa).A lined rock cavern (LRC) in the form of a tunnel or shaft can be used within this pressure range. The rockmass surrounding the opening resists the internal pressure and the lining ensures gas tightness. Thepresent paper investigates the key aspects of technical feasibility of shallow LRC tunnels or shafts under awide range of geotechnical conditions. Results show that the safety with respect to uplift failure of therock mass is a necessary but not a sufficient condition for assessing feasibility. The deformation of therock mass should also be kept sufficiently small to preserve the integrity of the lining and, especially, itstightness. If the rock is not sufficiently stiff, buckling or fatigue failure of the steel lining becomes moredecisive when evaluating the feasible operating air pressure. The design of the concrete plug that sealsthe compressed air stored in the container is another demanding task. Numerical analyses indicate thatin most cases, the stability of the rock mass under the plug loading is not a decisive factor for plug design.� 2016 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

Very large amount of energy can be stored either with pumpedhydroelectric storage (PHS) reservoirs or with compressed air en-ergy storage (CAES) systems. PHS technology is commonly usedand there are several examples in operation, while for CAES onlytwo commercial projects have been undertaken in salt rock(Crotogino et al., 2001; Gardner and Haynes, 2007), as well as onedemonstration project (Mansson and Marion, 2003) and one veri-fication project in granite (Stille et al., 1994).

CAES systems have the peculiarity that gasmust be stored undera high pressure (p ¼ 10e30 MPa) in order to achieve greater effi-ciencies during energy recovery (withdrawal stage). Lined andunlined tunnels, shafts and caverns can all be used within thispressure range. The rock mass surrounding the opening resists theinternal pressure while the lining or the natural hydraulic andgeological conditions ensure gas tightness (Kovári, 1993). A linedrock cavern (LRC) is the most attractive option and the one mostinvestigated over the past 20 years due to its wider application

h (G. Anagnostou).f Rock and Soil Mechanics,

s, Chinese Academy of Sciences. Prby-nc-nd/4.0/).

field, and there is no requirement for particular hydrogeologicalconditions or great depths of cover (Kovári, 1993).

From a geotechnical and structural point of view, the key factorsto be considered in a feasibility assessment of CAES in lined cavitiesare: (1) uplift failure of the overlying rock up to the surface; (2)failure and loss of tightness of the sealing membrane; and (3)shearing of the plug closing the cavern. The loss of tightness of thecavity not only decreases the efficiency of the system, but also mayimpair stability (high air pressures within the overlying rock massincrease uplift risk). The lining concept most investigated for un-derground CAES is a composite structure consisting of an inner thinsteel shell and an outer reinforced concrete shell (see Fig. 1). In thiscase, the sealing membrane is the thin steel shell. It may fail due tothe bending that occurs when it is squeezed into cracks in the outerconcrete lining, buckling during depressurization, the tensile stressdeveloping during cavity expansion or the fatigue induced bycyclical loading (Damjanac et al., 2002; Okuno et al., 2009).

However, few works have analysed these aspects, and in mostcases only for site-specific geotechnical conditions. Recent analysesof the uplift problem include those of Kim et al. (2012), Perazzelliet al. (2014) and Tunsakul et al. (2014). Kim et al. (2012) suggesteda limit equilibrium model assuming that the full shearing re-sistances of the rock mass act along the vertical slip surfaces. Thisassumption is uncertain in viewof the tensile stressfield developing

oduction and hosting by Elsevier B.V. This is an open access article under the CC BY-

Uplift failure

Stability of the concrete plug

Rock deformations

Integrity of the lining

p , T

Key issues

Fig. 1. Key design issues for a lined CAES rock cavity.

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 315

around the expanding cavity. Perazzelli et al. (2014) performedsmall and large strain numerical analyses of a continuum rockmassmodel and showed that the deformations at failure are very large inthe case of weak rocks, thus necessitating a geometrically nonlinearformulation in order to obtain the ultimate uplift pressure. Tunsakulet al. (2014) developed a numerical method based on the element-free Galerkin (EFG)methodwith a cohesive crackmodel to simulatethe fracture propagation patterns in a continuum medium aroundthe pressurised tunnel; the authors found a qualitative agreementbetween physical model tests and numerical results and theyemphasised that the in situ stress ratio has a strong influence onboth the crack initiation location and the propagation path.

Analysis of the rock mass deformations in pressurised linedcavities can be found in Stille et al. (1994), Sofregaz US Inc. and LRC(1999), Brandshaug et al. (2001), Damjanac et al. (2002), Johansson(2003), and Okuno et al. (2009). Stille el al. (1994) and Johansson(2003) presented monitoring results from in situ tests in the Grän-gesberg Pilot Plant (a 9 m high shaft of 4.4 m in diameter, 50m deepin granite). Okuno et al. (2009) presented the results of in situ tests attheGas Storage Pilot Plant in theKamiokamine (a 400mdeep, 6m indiameter tunnel in sedimentary rocks). Both works showed that thecavern diameter increases with the loading cycles. Sofregaz US Inc.and LRC (1999), Brandshaug et al. (2001) and Damjanac et al.(2002) investigated the rock deformations for the Grängesberg Pi-lot Plant and for the Halmstad Demonstration Plant (a 50 m highcavern with 37 m in diameter, 115 m deep in granite, pressurised at20 MPa) by numerical stress analysis of continuum rock massmodels. In these studies, the rock mass is taken as a homogeneous,continuous, linearly elastic-perfectly plastic, no-tension material(st¼ 0) obeying theMohreCoulomb yield criterion, and the effect ofthe cycling loading is not investigated. Damjanac et al. (2002) alsoperformed numerical stress analysis of discontinuum rock massmodels for the Halmstad Demonstration Plant considering thecycling loading. These analyses indicate a small increase in themagnitude of the rock mass displacements with the cycles.

Buckling and fatigue failures of the steel lining in CAES and gasstorage caverns were investigated in very few works. Results ofbuckling analysis were presented by Okuno et al. (2009), but thecomputational model adopted in this work remains unclear. Veri-fications of fatigue can be found in Damjanac et al. (2002).

Concrete plug stability has been investigated mostly in thecontext of PHS systems. Auld (1983) and Ilyushin (1988) describeddifferent types of underground plugs, analysed the factors to beconsidered in design, and suggested a simple design formulaaddressing the possible failure modes in the plug, in the rock massor at the interface e but without analysing the stability of the rockmass explicitly (the equation suggested considers the bearing ca-pacity of the rock as an input parameter). Hökmark (1998)

evaluated the stability of the rock around the concrete plug byintroducing a safety factor based on an elastically computed stressfield and the MohreCoulomb failure criterion. Park et al. (2001)performed a numerical investigation of the mechano-hydraulicbehaviour of concrete plugs taking a fixed air pressure andassumed elastic behaviour for the plug concrete. They alsoconsidered the elastic-perfectly plastic behaviour according to theMohreCoulomb failure criterion for the rock, and elastic-perfectlyplastic MohreCoulomb behaviour for the rockeplug interface.Their computations showed the influence of several factors (e.g. theshape, depth and in situ horizontal stress coefficient K0) on stressesand displacements in the rock mass and the plug.

Studies on the stability of concrete plugs in CAES systems are tobe found in Song and Ryu (2012) and in Pedretti et al. (2013). Theapproach of Song and Ryu (2012) is similar to that of Hökmark(1998). Pedretti et al. (2013) performed numerical stress analyseson the plug of a planned CAES test plant in Switzerland and eval-uated the safety margin against failure by iteratively reducing thestrength parameters of the rock mass.

The present paper investigates the above-mentioned designproblems for underground CAES by means of numerical stress ana-lyses, taking tunnels and shafts above the water table of 4 m indiameter with a thin steel shell under a wide range of geotechnicalconditions. As in Sofregaz US Inc. and LRC (1999), Brandshaug et al.(2001) and Damjanac et al. (2002), we consider the rock mass as ahomogeneous, continuous, linearly elastic-perfectly plastic, no-tensionmaterial (st¼0), obeying theMohreCoulombyield criterion.

We show in detail how the stress field in the surrounding rockand the displacements change during the pressurisation of a CAEScavity and we define an uplift safety criterion based on the exten-sion of the tensile failed zone above the cavity (Section 2).

Rock mass deformations at the walls of the cavity are shown fora wide range of geotechnical conditions and a maximum operatingpressure of 20 MPa (Section 3). These values are computedassuming amonotonic increasing of the air pressure. The behaviourof the adopted rock mass model in the case of loading cycles isdiscussed by a computational example (Section 4).

We show the stress and strain in the steel lining during pres-surisation and depressurisation of the cavity and we clarify whybuckling and fatigue failures can occur (Section 4). Verifications ofthese failures are presented (Section 4). Critical buckling loads arehere computed by means of nonlinear buckling analysis, whilecritical stress ranges are taken from the literature.

We analyse the interaction between a site-specific rock massand plugs of different geometries (Section 5) by means of acomputational model similar to the one of Park et al. (2001). Thestability of the rock around the concrete plug is investigated eval-uating the relation between the pressure in the cavity and thedisplacement of a control point of the plug.

2. Uplift

2.1. Computational model

Uplift failure is investigated by numerical stress analyses usingplane strain and axisymmetric models for the tunnels and shafts,respectively. Fig. 2 shows the computational domains and bound-ary conditions. The analyses were performed using the finite dif-ference code FLAC (Itasca, 2001) under the assumption of smallstrains. The effect of this assumption on the assessment of the limitpressure will be discussed in Section 2.3.

The rock mass is considered to be a linearly elastic-perfectlyplastic, no-tension material obeying the MohreCoulomb yield cri-terion and a non-associated flow rule with dilatancy angle equal tozero. The lining is not introduced into the numerical model because

x

yyx

p = 7 MPa p = 17 MPa

(a) (b)

Cracked zone (tensile failure)

Cracked zone (yield in past)

Crushed zone (shear failure)

Current cracked zone (tensile failure)

Zone cracked in the past

Crushed zone (shear failure)

p = 39 MPa

(c)

Fig. 3. Plastic zones at air pressure p ¼ 7 MPa, 17 MPa and 39 MPa (tunnel of diameterD ¼ 4 m, above the water table, g ¼ 25 kN/m3, f ¼ 30� , sc ¼ 8 MPa, st ¼ 0, j ¼ 0� andK0 ¼ 1).

(a)

6H

H

4H

yD = 4 m

p x

6H

H

4H

L=24 m

y

xp

0.5D = 2 m

(b)

Fig. 2. Computational domains and boundary conditions for the numerical stressanalyses of a CAES tunnel (a) and of a CAES shaft (b).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328316

its stiffness is negligible compared with that of the surroundingrock; the pressurisation of the cavern causes cracking in the con-crete lining and yielding of the steel shell.

The analysis consists of the following steps: (1) initialization ofthe in situ effective stress with a lithostatic stress distribution; (2)deactivation of the grid elements inside the tunnel or shaft andapplication of the average initial effective pressure; and (3)monotonic increase in air pressure up to the value at which anequilibrated solution cannot be found. The effect of the cyclicalloading is not taken into account. Table 1 contains the values of themodel parameters.

2.2. Failure zones and stress fields

In order to illustrate how the stress field in the surrounding rockchanges during pressurisation of a cavity, we consider the exampleof a 50 m deep tunnel of 4 m in diameter. Fig. 3a, b and c shows theextent and type of failure of the rock mass around the tunnel for anair pressure p of 7 MPa, 17 MPa and 39 MPa, respectively. Fig. 4a, band c shows the corresponding distributions of the radial andtangential stresses along the vertical y-axis. Of particular interest isthe development of the tangential (horizontal) stress. The radial(vertical) stress obviously increases during the pressurisation of thecavity.

At a relatively low air pressure p of 7 MPa, the tangential stressdrops to zero close to the tunnel and close to the surface (Fig. 4a),which means that the rock fails in tension (Fig. 3a). The tensilefailure close to the surface is practically irrelevant, because theinitial horizontal stress close to the surface is very low, whichmeans that even an extremely small extensive strain would causecracking.

With increasing air pressure, the boundary of the cracked zonearound the tunnel moves radially away from the tunnel, while thecracked zone at the surface becomes deeper. When the air pressurereaches the uniaxial compressive strength of the rock, shear failureoccurs at the tunnel boundary. Subsequently, a so-called crushedzone of increasing size develops around the cavity.

At an air pressure p of 17 MPa, the two cracked zones jointogether, while the crushed zone extends up to a distance of almost

Table 1Parameter values assumed in the numerical stress analysis of uplift and rock mass defor

Rock mass Uniaxial compressive strength,sc (MPa)

Friction angle, f (�) Tensile

5e60 30e40 0Initial stress field

(lithostatic)Coefficient of lateral stress, K0 Unit weigh, g (kN/m3) Water0.5; 1 25

Note: The texts ‘0.5; 1’ represent K0 ¼ 0.5 and K0 ¼ 1.

D/2 from the tunnel wall (Fig. 3b). Inside the crushed zone, thehorizontal stresses are non-zero along the vertical axis of themodel(they are related to the vertical stresses according to the failurecriterion). Above the crushed zone, however, the horizontalstresses are zero and this continues up to the surface (Fig. 4b),indicating conceptually the presence of continuous cracksthroughout the entire overburden.

At an air pressure p of 39MPa, which is the load found at the lastequilibrated solution, two separated crushed zones are observed:one extending from the tunnel to a shallow depth and another atthe surface. The cracked zones are large. It is not possible torecognise the geometry of the failure mechanism.

As can be seen in Fig. 3b and c, the rock has experienced crackingin some zones in the past, i.e. at lower internal pressures. Under thecurrent pressure, the rock in these zones is subjected only tocompressive stresses. This becomes evident by comparing thedistribution of the horizontal stress sxx in Fig. 4b and c. When theinternal pressure increases from 17 MPa to 39 MPa, the horizontal

mations.

strength, st (MPa) Dilatancy angle,j (�)

Young’s modulus,E (GPa)

Poisson’s ratio, n

0 10 0.3table below the cavity

0

10

20

30

40

50

0 5 10 15 20

y(m

)

(MPa)

yy (radial stress)xx (tangential stress)

xx0= yy0 (initial stress)

(a)p = 7 MPa

0

10

20

30

40

50

0 5 10 15 20

y(m

)

(MPa)

xx

yy

(b)p = 17 MPa

0

10

20

30

40

50

0 5 10 15 20

y (m

)

(MPa)

(c)

xx

yy

p = 39 MPa

Fig. 4. Distribution of radial and tangential stresses along the vertical y-axis atp ¼ 7 MPa, 17 MPa and 39 MPa (tunnel of diameter D ¼ 4 m, above the water table,g ¼ 25 kN/m3, f ¼ 30� , sc ¼ 8 MPa, st ¼ 0, j ¼ 0� and K0 ¼ 1).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 317

stress sxx remains equal to zero only in the vicinity of the surface.When the model exhibits such behaviour, the computed de-formations represent a rough approximation because the adoptedelastoplastic constitutive relationships do not consider the stress-free closing of open cracks. However, such behaviour will notoccur (or is limited to a small part of the model close to the surface)as long as the cracked zone does not reach the surface.

2.3. Safety criterion

A common way to investigate stability by numerical stress an-alyses is to evaluate the relationship between the displacement of acontrol point in the tunnel and the applied pressure. The instabilitymanifests itself as an asymptotic increase in the displacement toinfinity as the pressure approaches the limit value. Fig. 5 shows therelationship between the internal pressure p and the crowndisplacement u for the example introduced in the previous section.According to the above-mentioned failure criterion, the ultimatepressure pul is equal to 39 MPa (i.e. the pressure of the last equili-brated solution). It is important to note that the deformations atfailure are very large and necessitate a geometrically nonlinearformulation. The small strain approach adopted for computingFig. 5 overestimates the uplift pressure (Perazzelli et al., 2014).

A more conservative and intuitively reasonable criterion forsafety against uplift can be derived from the extent of the crackedzone around the tunnel. As discussed previously, at a certainpressure pc, which is lower than the ultimate pressure pul, theentire zone between tunnel and surface becomes cracked (Fig. 3b).Even if the bearing capacity of the rockmass is not fully exploited atthis point, it would be sensible to avoid such a situation. To deter-mine the critical pressure pc, we observe the development of thecracked zone and the distribution of the horizontal stress along thevertical axis with increasing internal pressure. In the previousexample, the critical pressure pc is equal to 17MPa (Figs. 3b and 4b).The deformations at this strain are small (<<20%), which meansthat the small strain formulation is adequate. Consequently, thedetermination of the critical pressure pc, contrary to that of theultimate pressure, avoids the problem of geometric nonlinearity.This provides an additional reason for evaluating safety againstuplift failure from the critical pressure pc, while incorporating an

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 10 20 30 40

u (m

)

p (MPa)

pul

pc

p

u

Fig. 5. Vertical displacement as a function of pressure (parameters are the same asthose in Fig. 3 and Table 1).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328318

adequate safety factor. The parametric study in the next section isbased upon this criterion.

2.4. Parametric study

Figs. 6 and 7 show the critical pressure pc for a 4 m diametertunnel and for a 4 m diameter shaft, respectively, as a function ofthe overburden H for different uniaxial compressive strengths ofthe rock mass and in situ horizontal stresses.

The critical pressure increases almost linearly with the over-burden and nonlinearly with the uniaxial compressive strength(Figs. 6b and 7b). The rock strength sc is important only up to acertain value, which depends on the overburden (Figs. 6b and 7b);for higher rock strengths, it is the tensile failure of the rock massthat governs the critical pressure pc. As expected, the criticalpressure increases with the in situ horizontal stress. Finally, acomparison between Figs. 6 and 7 shows that a shaft is far morefavourable than a tunnel, as an overburden of 60e120 m is neces-sary for ensuring safety against uplift in the case of a 4 m diameterCAES tunnel, while for a CAES shaft of the same diameter, a smalleroverburden of 30e50 m would be sufficient.

0

10

20

30

40

50

60

70

80

40 60 80 100 120 140 160

p c(M

Pa)

H (m)

c (MPa) , ( ) = (5, 30)

(60, 40)

0

10

20

30

40

50

60

70

80

40 60 80 100 120 140 160

p c(M

Pa)

H (m)

c (MPa) , ( ) = (5, 30)

(10, 30)

(15, 35)

(60, 40)

(30, 35)

(b)

(a)K0 = 1K0 = 0.5

H

p

Fig. 6. Pressure pc at which the cracked zone reaches the surface as a function of theoverburden H for a CAES tunnel under different assumptions as to (a) horizontal stressand (b) rock mass strength (parameters: see Table 1).

3. Rock mass deformations

3.1. Introduction

Sufficient safety of the rock mass with respect to uplift failure isa necessary but not a sufficient condition for the feasibility of alined CAES cavity. The deformations in the rockmass also have to besufficiently small to preserve the integrity of the lining and,particularly, its tightness. The present section analyses the problemof tensile failure of the steel membrane due to excessive elongationduring the first pressurisation of the CAES system (cyclical loadingis not considered). We assume a limit tangential deformation of thesteel lining 3tt,s lim of 2%.

The computational models are the same as those used for theuplift analysis and, for the reasons explained in Section 2.1, they donot consider the contribution of lining stiffness to the overallstiffness of the system. In addition, they do not consider the cavityexcavation stage. The tangential strain in the steel is taken equal tothe tangential deformation of the rock mass at the boundary of thecavity, assuming that the excavation-induced rock deformationsare negligible. This simplifying assumption is reasonable, becausethe CAES tunnels and shafts under consideration are located atrelatively shallow depths and, consequently, the response of the

0

10

20

30

40

50

60

70

80

30 40 50 60 70

p c(M

Pa)

H (m)

c (MPa) , ( ) = (5, 30)

(15, 35)

(10, 30)

(60, 40)

(30, 35)

(a)

(b)

0

10

20

30

40

50

60

70

80

30 40 50 60 70

p c (M

Pa)

H (m)

c (MPa) , ( ) = (60, 40)

H

p(5, 30)

K0 = 1K0 = 0.5

Fig. 7. Pressure pc at which the cracked zone reaches the surface as a function of theoverburden H for a CAES shaft under different assumptions about (a) horizontal stressand (b) rock mass strength (parameters: see Table 1).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 319

surrounding rock mass to their excavation will be practicallyelastic.

On account of the small tangential strain limit of the steel lining( 3tt,s lim ¼ 2%), the geometrical nonlinearity is negligible and rockdeformations 3tt are inversely proportional to the Young’s modulusof the rockmass. For this reason, all of the analyses were performedfor a specific value of the Young’s modulus (E ¼ 10 GPa) and theirresults are presented herein in terms of the product 3ttE; the de-formations for other values of the Young’s modulus can be obtainedby simple transformations.

3.2. Parametric study

Fig. 8 shows the maximum tangential strain 3tt of the rock at thetunnel wall (multiplied by the Young’s modulus E) as a function ofthe overburden H for a maximum operating air pressure p of20 MPa. According to Fig. 6, uplift is not a problem for this airpressure with an overburden H > 80 m and the geotechnical con-ditions considered in Fig. 8. Depending on the strength of the rockmass and on the coefficient of lateral stress, the maximumtangential strain 3tt occurs either at the tunnel crown or at the sidewalls (points A and B, respectively, in the inset of Fig. 8). As ex-pected, the tangential strain 3tt decreases nonlinearly withincreasing rock strength sc, but remains constant for rock strengthshigher than the air pressure of 20 MPa (shear failure does not occurin this case). Furthermore, the tangential strain 3tt decreases withincreasing overburden H and increasing coefficient of lateral stressK0 (the solid lines in Fig. 8 are lower than the dotted lines). Thesetwo parameters have a major effect only in the case of a weak rockmass (sc ¼ 5 MPa).

The results of Fig. 8 can be applied to determine the maximumtangential strain 3tt for a given Young’s modulus E. For example, ifH ¼ 80 m, K0 ¼ 1, sc ¼ 5 MPa and E ¼ 500sc ¼ 2.5 GPa, then themaximum tangential strain 3tt ¼ 15% GPa/2.5 GPa ¼ 6%. For a givenYoung’s modulus E and a given limit value of the tangential strain3tt, lim (here 3tt, lim ¼ 3tt,s lim ¼ 2%), the value of 3ttE will be fixed andcan be compared with the curves in Fig. 8 in order to assess the riskof failure of the steel lining under different geotechnical conditions.For the same rockmass considered before (sc¼ 5MPa, E¼ 2.5 GPa),the limit value 3tt, limE ¼ 5% GPa, which is lower than the curves forsc ¼ 5 MPa. This indicates that CAES under a maximum air pressureof 20 MPa is not feasible in 4 m diameter lined tunnels in soft rockmasses (sc � 5 MPa, E ¼ 500sc) at the relatively shallow depthsconsidered (H < 160 m), due to excessive deformations in the steel.On the other hand, in a higher quality rock mass with, e.g.

c = 5 MPa

0

5

10

15

20

25

80 100 120 140 160

ttE

(%G

Pa)

H (m)

p = 20 MPa

= 30

K0 = 1K0 = 0.5

c = 10 MPac > 20 MPa

H

p

A

B

Fig. 8. Maximum tangential strain at the cavity boundary 3tt multiplied by the Young’smodulus E as a function of the overburden H for a CAES tunnel under an air pressure pof 20 MPa (parameters: see Table 1).

sc ¼ 20 MPa and E ¼ 500sc ¼ 10 GPa, the limit value3tt, limE ¼ 20% GPa, which is higher than the curves for sc ¼ 20 MPa,thus indicating that failure of the steel due to excessive strain is notcritical in this case.

Fig. 9 presents (in a similar way to Fig. 8) the results for CAESshafts of 4 m in diameter. Overburdens within the range consideredwill be non-critical with respect to uplift at the maximum oper-ating pressure of 20 MPa. The maximum tangential strain 3tt alwaysoccurs at the mid-height of the shaft (i.e. at y ¼ 0 in the referencesystem of Fig. 2b). For the reasons mentioned above, the uniaxialcompressive strength does not play a role if it is higher than themaximum air pressure. The effect of overburden H and coefficientof lateral stress K0 is very small. According to the results of Fig. 9,CAES under amaximum air pressure of 20MPa is not feasible in 4mdiameter, relatively shallow (H < 70 m) shafts if the rock mass isweak (sc � 5 MPa, E ¼ 500sc). The deformations do not present aproblem, however, in rock masses of medium-good quality(sc � 20 MPa, E ¼ 500sc).

Finally, Fig.10 shows the critical pressure pc against uplift as wellas the air pressure under which the tangential steel strain reachesthe assumed limit value of 2% as a function of the overburden H fortunnels and shafts (solid and dashed lines, respectively) in a weakrock mass (the other parameters are given in the caption of Fig. 10).Safety against uplift failure is critical only for very shallow CAEStunnels (H < 50 m). For CAES tunnels at greater depths and forshafts (independently of the depth), deformations represent thecritical factor and the overburden has only a minor effect. There isno significant difference between shafts and tunnels with respectto deformation. The reason is that the maximum tangential strainin CAES shafts occurs at their mid-height, where there are no three-dimensional effects (the horizontal shaft cross-section is like atunnel of diameter B ¼ 4 m and overburden Hþ0.5L).

4. Buckling and fatigue failures of the steel lining

4.1. Introduction

In CAES cavities, the air pressure varies cyclically between aminimum operating pressure higher than zero and a maximumpressure of 10e30 MPa. The cavity is completely depressurisedduring maintenance work. As explained in Section 4.3, bucklingfailure after the total depressurisation of the cavity and fatiguefailure due to cyclical loading are important hazard scenarios evenif the CAES cavity is above the water table. They are investigated inthe present section by considering a 10 mm thick steel lining and

0

5

10

15

20

25

40 45 50 55 60 65 70

ttE(%

GPa

)

H (m)

c = 5 MPa

p = 20 MPa

= 30

K0 = 1K0 = 0.5

c = 10 MPac > 20 MPa

H

p

C

Fig. 9. Maximum tangential deformation at the cavity boundary 3tt multiplied by theYoung’s modulus E as a function of the overburden H for a CAES shaft at air pressurep ¼ 20 MPa (parameters: see Table 1).

0

5

10

15

20

25

30

35

40

40 50 60 70 80 90 100 110 120

p (M

Pa)

H (m)

pcpc

p ( tt = 2%)p ( tt = 2%)

tunnelshaft

H

p

tt H

p

tt

Fig. 10. Pressure pc at which the cracked zone reaches the surface and pressure cor-responding to a fixed tangential strain as a function of overburden H for a CAES tunnel(solid lines) and a CAES shaft (dashed lines) (f ¼ 30� , sc ¼ 5 MPa, E ¼ 2.5 GPa, K0 ¼ 1,other parameters as in Table 1).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328320

4 m diameter tunnels and shafts located above the water table inrock masses with an isotropic initial stress field (K0 ¼ 1). Bucklingfailure due to the external water pressure, which is relevant fordepressurised CAES cavities beneath the water table, is not inves-tigated here; as the steel lining is very thin, the development ofwater pressuremust anyway be avoided (e.g. bymeans of a suitabledrainage system).

The rock-lining interaction under cyclical loading will be ana-lysed by means of the simple computational model, which is pre-sented in Section 4.2. In order to better understand the interactionproblem and to show certain limitations of the constitutive law ofthe rock mass, we discuss firstly (Section 4.3) the rock behaviouraround an unlined cavity, taking two cases that concern the

tt

rr

0

r

r = a

p

u

Rock mass

Lining

0

300 m

p

a = 2 m

Fig. 11. Computational model adopted for th

uniaxial compressive strength of the rock mass (rock behaviourvaries depending on whether its strength is higher or lower thanthemaximum air pressure). Section 4.4 deals with the actual lining-rock interaction problem along with the possible failure modes ofthe steel lining. Finally, Sections 4.5 and 4.6 assess safety againstbuckling and fatigue, respectively.

4.2. Computational model

We consider a plane strain, axisymmetric model of a verticaltunnel cross-section (or a horizontal shaft cross-section). Fig. 11cshows the computational domain and the assumed boundaryconditions.

The rock mass is taken as an elastic-perfectly plastic, no-tensionmaterial obeying the MohreCoulomb failure criterion with a non-associated flow rule (zero dilatancy angle). This constitutivemodel has been applied in the past to analyse monotonic cavityexpansion in brittle materials (Ladanyi, 1967; Satapathy and Bless,1995). It will be used here only for a preliminary evaluation ofthe above-mentioned hazard scenarios and to illustrate somepotentially important limitations of this model for the cyclicalloading problem.

For simplicity, we neglect the presence of tunnel support (e.g. areinforced shotcrete layer between the steel lining and the rock)and consider the steel lining as being in direct contact with the rockmass. The steel lining is modelled as elastic-perfectly plastic andwith a tensile strength equal to the yield strength of the steel.Yielding in compression is neglected because it is not expected tooccur under pre-buckling loading.

The cyclical variation in the air pressure is considered to bestatic (dynamic effects are not taken into account). All computa-tions assume an initial stress s0 of 2.5MPa, which applies to tunnelsapproximately 100 m deep (or to shafts with H þ 0.5L ¼ 100 m).Table 2 shows the values assumed for the model parameters. Thenumerical analyses were performed with the finite difference codeFLAC (Itasca, 2001).

t

p

pr

tt,s

ar

Lining

u

Rock mass

(a) (b)

0

(c)

1 m

e analysis of the rock-lining interaction.

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 321

4.3. Unlined cavity under cyclical loading

Behaviour under cyclical loading (air pressure variation of 0e20 MPa) will be discussed using computational results for a 100 mdeep CAES tunnel (initial stress s0 ¼ 2.5 MPa). Two cases will beconsidered: in the first case (Fig. 12), the uniaxial compressivestrength, sc, of the rock mass is higher than the maximum airpressure (sc> 20MPa). In the second case (Fig.13), sc is taken equalto 7 MPa, i.e. it is lower than the maximum air pressure, whichmeans that a crushed zone develops around the cavity as the airpressure increases above 7 MPa. In both cases, the response of theground to the cavity excavation is elastic because the rock strengthis twice greater than the initial stress.

Fig. 12a shows the stress path of a point at the cavity wall in thefirst case, i.e. for sc > 20 MPa. Point O shows the initial stress state;whilst path OA is the excavation-induced stress change. Subse-quently, the radial stress at the excavation boundary is always equalto the air pressure p. During the first pressurisation of the cavity,the tangential stress decreases until it becomes equal to zero (pathAB). This occurs at an air pressure p of 2s0 ¼ 5 MPa. Up to point B,the response of the ground is elastic and the displacement of thecavity boundary (Fig. 12c) depends linearly on the air pressure. Atpoint B, the rock fails in tension. During the further pressurisationfrom 5 MPa to 20 MPa, the tangential stress at the cavity boundaryremains equal to zero (path BC), while a cracked zone with anincreasing outer radius develops, with the consequence that thedisplacement of the cavity wall increases super-linearly with airpressure (Fig. 12c). The solid lines in Fig. 12b show the radial dis-tribution of the stresses at the maximum air pressure. The crackedzone clearly extends up to a radius of 8 m at this state.

The ground response following depressurisation is elastic. Thetangential stress at the cavity wall increases by the same amount asthe air pressure decreases (path CD in Fig. 12a) and the displace-ment of the cavity wall decreases linearly with decreasing airpressure. At the end of depressurisation, the tangential stress at thecavity wall is equal to the maximum air pressure (point D inFig. 12a). The end of the first cycle is characterised by a residualpositive displacement, i.e. by an increase in the cavity radius(Fig. 12c). The dashed lines in Fig. 12b show the radial distributionof the stresses after complete depressurisation. The stress state isclearly within the elastic domain everywhere around the cavity.

When the cavity is pressurised and depressurised again, thestress state at the cavity boundary moves along paths CD and DC(Fig. 12a), respectively; the cavity wall displacement increases anddecreases linearly with the air pressure (along path CD in Fig. 12c);and the radial distribution of the stresses around the cavity fluc-tuates between the dashed and solid lines of Fig. 12b. Note that theextent of the cracked zone at the maximum air pressure does notincrease with the number of cycles.

According to the computational results, the cracks that developduring the first pressurisation close during the first depressurisa-tion and they remain closed during the following cycles. The reasonfor this behaviour is that the constitutive model does not accountfor stress-free closing of the open cracks. This can be seen clearly in

Table 2Parameter values assumed in the numerical rock-lining interaction analyses performed f

Rock mass Uniaxial compressivestrength, sc (MPa)

Friction angle, f (�) Tensile strength

�5 30 0Steel lining Thickness, t (mm) Internal radius, a (m) Yield strength,

10 2 355Initial stress field

(homogeneousand isotropic)

Initial stress, s0 (MPa)2.5

Fig. 12a (the tangential stress increases right from the start ofdepressurisation). In reality, the stress acting perpendicularly to acrack can increase only after closure of the crack. Another aspect ofthe model behaviour is that the displacement u of the cavity wall atthe maximum and minimum pressures does not vary with thenumber of cycles. This disagrees with actual rock behaviour; in situtests in CAES and gas storage pilot plants show that the caverndiameter increases with the number of cycles (Stille et al., 1994;Okuno et al., 2009).

We will next discuss the computational results for the case of arock mass with lower uniaxial compressive strength (7 MPa). Up toan air pressure of 7 MPa, the response of the model is exactly thesame as before (path OABC, Fig. 13a). During pressurisation from5 MPa to 7 MPa (path BC), a cracked region develops, inside whichthe stress state is uniaxial. When the air pressure reaches theuniaxial compressive strength, sc, the rock fails in shear (so-calledpassive shear failure). During further pressurisation, the tangentialstress increases with the air pressure, so that the stress state movesalong the failure surface (path CD). Next to the cavity, a crushedzone develops, which is bounded by an outer cracked zone. Thedisplacement of the cavity wall increases super-linearly with the airpressure (Fig. 13c). As can be seen from the radial distributions ofthe stresses (Fig. 13b), at the maximum air pressure of the firstcycle, the crushed zone extends up to a radius of 6.8 m, while thecracked zone extends up to a radius of 9.5 m.

During the first depressurisation of the cavity, the response isinitially elastic (path DE); the tangential stress increases by thesame amount as the radial stress decreases and becomes themaximum principal stress when the air pressure drops below10 MPa. At a certain air pressure p* (point E, 3.3 MPa in the presentexample), the stress state reaches the failure surface (so-calledactive shear failure). Afterwards, the stress state moves along thefailure surface (path EF) and a crushed zone with a graduallyincreasing radius develops around the cavity. The displacement ofthe cavity wall depends initially linearly (path DE, Fig. 13c) andafterwards nonlinearly (path EF, Fig. 13c) on the air pressure. At theend of the cycle, the tangential stress is equal to the uniaxialcompressive strength of the rock (point F). The dashed lines inFig. 13b show the radial distributions of the stresses after completedepressurisation; the peak of tangential stress at r ¼ 2.8 m marksthe boundary of the crushed zone. Again, a residual positivedisplacement can be observed at the end of the first cycle (point F inFig. 13c).

During the next pressurisation, the response is initially elastic(path FG, Fig. 13a). When the air pressure increases above theuniaxial compressive strength of the rock, the stress state moves onthe failure surface (path GD). The stress path during the seconddepressurisation is the same as in the first cycle (path DEF). In allfollowing cycles, the stress state moves along path FGDEF. Thestress distributions of Fig. 13b remain the same for all cycles. Theouter radii of the cracked zone and of the passive shear zone at themaximum air pressure as well as the radius of the active shear zoneafter every complete depressurisation remain constant over thecycles. The same is also true with respect to the displacement of the

or the assessment of buckling and fatigue.

, st (MPa) Dilatancy angle, j (�) Young’s modulus,E (GPa)

Poisson’s ratio, n

0 2e20 0.3sy (MPa) Young’s modulus, Es (GPa) Poisson’s ratio, ns

198 0.3

0

5

10

15

20

25

0 5 10 15 20 25

tt(M

Pa)

rr (MPa)

A

OB C

D

(a)

0

5

10

15

20

25

0 10 20 30 40 50

(MPa

)

r (m)

tt

rr

p = 20 MPa

p = 0

(b)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25

u (m

m)

p (MPa)

AB

C

D

(c)

Fig. 12. Numerical results for an unlined cavity and a rock strength sc > 20 MPa: (a)Stress path of the rock at the cavity boundary during excavation of the cavity andcyclical pressurisation; (b) Radial distribution of the radial (srr) and tangential (stt)stresses at the maximum (solid lines) and minimum (dashed lines) air pressures; (c)Displacement of the cavity boundary as a function of the air pressure p (unlined cavity,cyclical pressure variable between pmax ¼ 20 MPa and pmin ¼ 0; E ¼ 10 GPa, otherparameters as in Table 2).

0

5

10

15

20

25

0 5 10 15 20 25

tt(M

Pa)

rr (MPa)

(a)

A

OB

D

E

F

C, G

0

5

10

15

20

25

0 10 20 30 40 50

(MPa

)

r (m)

tt

rr

(b)p = 20 MPa

p = 0

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25

u (m

m)

p (MPa)

(c)

AB C

D

E

F

G

Fig. 13. Numerical results for an unlined cavity and a rock strength sc ¼ 7 MPa: (a)Stress path of the rock at the cavity boundary during excavation of the cavity andcyclical pressurisation; (b) Radial distribution of the radial (srr) and tangential (stt)stresses at the maximum (solid lines) and minimum (dashed lines) air pressures; (c)Displacement of the cavity boundary as a function of the air pressure p (unlined cavity,cyclical pressure variable between pmax ¼ 20 MPa and pmin ¼ 0; E ¼ 10 GPa, otherparameters as in Table 2).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328322

4

8

12

16

20

p r(M

Pa)

A

B

C

D

(a)

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 323

cavity wall. It increases along path FGD during every pressurisation,but decreases along path DEF during every depressurisation (seeFig. 13c).

Contrary to the in situ observations, the displacements at themaximum and minimum air pressures (points D and F in Fig. 13c)do not change with the number of cycles, although the rock ex-periences failure repeatedly during the cyclical loading. Compara-tive computations show that this behaviour is also related to theassumption of a zero dilatancy angle. In the case of dilatantbehaviour, the displacements at the maximum and minimum airpressures decrease over the cycles. The effect of dilatancy deservesfurther investigation. Another limitation of the constitutive modelis that it does not account for softening behaviour (the failuresurfaces do not change with the cycles). In view of the stress path,softening is probably an important factor.

00 4 8 12 16 20

p (MPa)

O

-200

-100

0

100

200

300

400

0 0.2 0.4 0.6 0.8

tt,s(M

Pa)

tt,s (%)

A B

D

(b)

CO

Fig. 14. Numerical results for a lined cavity and a rock strength sc > 20 MPa: (a)Pressure acting on the rock at the cavity boundary pr as a function of the air pressure p;(b) Stress (stt,s) and strain ( 3tt,s) of the steel during cyclical pressurisation (cyclicalpressure variation between pmax ¼ 20 MPa and pmin ¼ 0; sc > 20 MPa, E ¼ 10 GPa,other parameters as in Table 2).

4.4. Rock-lining interaction under cyclical loading

This interaction between the rock and the lining will be dis-cussed for the two sets of geotechnical conditions, consideringadditionally the constraint imposed by a 10 mm thin steel shell.

Fig. 14 shows the relation between the air pressure p and thepressure pr acting on the rock (upper diagram) as well as thetangential stress stt,s and strain 3tt,s experienced by the steel (lowerdiagram), assuming that the rock strength is higher than themaximum air pressure (sc > 20 MPa).

At the start of pressurisation (curve OA), a small portion of theair pressure p is borne by the lining and the remaining part istransferred to the rock (Fig. 14a). The lining loading (q¼ p�pr) in-duces tangential tensile stress stt,s in the steel (Fig. 14b). At a certainair pressure pA (point A), the steel yields (stt,s ¼ sy). Afterwards (lineAB), the air pressure exceeding pA is transferred to the rock. If themaximum air pressure pB (point B) is too great, then the steel liningfails due to excessive deformation ( 3tt,s ¼ 3tt,s lim). This hazard sce-nario was investigated in the previous section.

During depressurisation (line BCD), both the rock pressure prand the lining loading q decrease. At a certain air pressure, thelining loading q and the tangential stress in the steel become zero(point C). Further depressurisation (line CD) causes compression ofthe lining. If the compression load q is too large, the steel lining failsby buckling. The compression load q and the subsequent risk ofbuckling are the highest after complete depressurisation (point D)and increase with increasing maximum air pressure. Buckling isanalysed in Section 4.5.

During the first depressurisation and during the next cycles, therelationship between the air pressure p and the pressure pr actingon the rock is linear and it does not vary under loading andreloading (line DCB, Fig. 14a). The variations in the tangential stressstt,s and strain 3tt,s during the cycles are also linear and given by theline DCB in Fig. 14b. The maximum compression load (q ¼ 0.68 MPafor p ¼ 0) and the stress range experienced by the steel in a singlecycle Dstt,s (Dstt,s ¼ 504 MPa) do not increase with the number ofcycles. These aspects of the rock-lining interaction are due to thebehaviour of the rock, which was discussed in the previous section(Section 4.3); the stress and strain in a rock mass havingsc > 20 MPa vary elastically during the first depressurisation andduring the next cycles (see Fig. 12).

The repeated loading and unloading of the lining induce the riskof fatigue failure of the steel. Decisive factors in this respect are thenumber of cycles and the stress range in a single cycleDstt,s, and thelatter increases with the maximum air pressure and withdecreasing minimum air pressure. It should be noted that the latterwill be atmospheric only for the performance of maintenancework.During normal CAES operation, the minimum air pressure amounts

to 2e4 MPa (Johansson, 2003). This will be taken into account inthe fatigue assessment (Section 4.6).

The second example (Fig. 15) assumes that the rock strength islower than the maximum air pressure (sc ¼ 7 MPa). Figs. 15b and14b clearly show that the steel experiences a high tangentialstrain in this case. This is due to the passive shear failure of the rock,which reduces the stiffness of the rock and occurs when the pres-sure pr at the rock-lining interface exceeds the uniaxial compressivestrength of the rock. The relationship between the air pressure pand the pressure pr is also slightly nonlinear during the firstdepressurisation (line BCD, Fig. 15a). This is due to the active shearfailure (see the previous section and Fig. 13). The failure-induceddecrease in rock stiffness also leads to a higher compression loadat the end of the first depressurisation (q ¼ 0.89 MPa versus0.68 MPa for p ¼ 0).

Due to the repeated shear failures occurring in loading andunloading (Fig.13), the relationships between the air pressure p andthe pressure pr acting on the rock (linesDEB and BCD during loadingand reloading, respectively) are also slightly nonlinear in the sec-ond and subsequent cycles. The tangential stress stt,s and strain 3tt,sin the steel vary elastically during cyclical loading (line DB inFig. 15b); the minimum and maximum strains 3tt,s and the stressrange Dstt,s remain constant over the cycles. It should be noted thatcomparative analyses, which are not presented here, show that if

0

4

8

12

16

20

0 4 8 12 16 20

p r(M

Pa)

p (MPa)

O

A

B

C

D

E

(a)

-200

-100

0

100

200

300

400

0 0.2 0.4 0.6 0.8

tt,s(M

Pa)

tt,s (%)

A B

D

C, EO

(b)

Fig. 15. Numerical results for a lined cavity and a rock strength sc ¼ 7 MPa: (a)Pressure acting on the rock at the cavity boundary pr as a function of the air pressure p;(b) Stress (stt,s) and strain ( 3tt,s) of the steel during cyclical pressurisation (cyclicalpressure variation between pmax ¼ 20 MPa and pmin ¼ 0, sc > 20 MPa, E ¼ 10 GPa, otherparameters as in Table 2).

ud

q

at

F

u

k

(a) (b)

F

ksp1

Fig. 16. Computational model of the buckling analysis (after Trombetta, 2015).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328324

the rock behaviour is dilatant, then the compression load q in-creases and the minimum and maximum deformations 3tt,sdecrease over the cycles. As mentioned in last section, the effect ofdilatancy is not yet understood and deserves further investigation.

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50

q (M

Pa)

u (mm)

E = 5 GPa (ksp = 320 kN/mm)

E = 0.16 GPa (ksp = 10 kN/mm)

qcr

E = 1.1 GPa (ksp = 70 kN/mm)

q

u

Fig. 17. Relationship between pressure q and displacement u of a control pointaccording to the buckling analysis (a ¼ 2 m, t ¼ 10 mm, Es ¼ 198 MPa, ns ¼ 0.3, n ¼ 0.3)(after Trombetta, 2015).

4.5. Buckling failure

4.5.1. Buckling analysisThe critical buckling load qcr was determined (Trombetta, 2015)

by nonlinear buckling analyses based on the model shown inFig. 16. The rock was handled by introducing discrete normalsprings with nonlinear behaviour, allowing for rock-lining separa-tion (Fig. 16b). The springs provide a reaction only for outwarddisplacements of the ring, i.e. they can bear only compressiveforces. Their stiffness ksp was taken equal to

ksp ¼ Ebd=½ð1� nÞa� (1)

where E and n are the elasticity constants of the rock; d is thespacing of the springs (Fig. 16a, here d ¼ 0.09 m); and b and adenote the width (1 m) and the radius (2 m) of the lining. The latterwas modelled by elastic beam elements having a cross-section of

1 m � 0.01 m and, due to the plane strain conditions, a Young’smodulus, which is written as

Eb ¼ Es.�

1� n2s

�(2)

A global oval-shaped imperfection was assumed (Fig. 16a), cor-responding to the first buckling mode of the linear buckling anal-ysis. A uniform pressure qwas assumed to act over the entire lining.

The analyses were performed (Trombetta, 2015) using thecommercial code Abaqus (Dassault Systèmes, 2010) based upon themodified Riks method. Fig. 17 shows the pressure-deflection curvesobtained for various values of Young’s moduli of the rock. Themaxima of the curves correspond to the critical buckling pressuresqcr. As expected, the buckling pressure qcr decreases with adecreasing Young’s modulus of the rock. The solid line in Fig. 18shows the buckling pressure qcr as a function of the Young’smodulus of the rock. For comparison, the diagram includes twoanalytical solutions: the one of Levy (1884), derived for unconfinedrings, and the one of Glock (1977), derived for rings embedded in arigid medium. With a decreasing Young’s modulus E, the bucklingpressure tends asymptotically to a minimum value, which corre-sponds to the limit case of an unconfined ring and is very close tothe solution of Levy (1884). With an increasing modulus E, thebuckling pressure tends asymptotically to a maximumvalue, whichcorresponds to the limit case of rigid confinement and is close tothe solution of Glock (1977). Note that for the typical values of theYoung’s modulus E (higher than 1 GPa for rocks), the bucklingpressure qcr does not depend significantly on E and is very close tothe buckling pressure of a ring embedded in a rigid medium.

00.10.20.30.40.50.6

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

q cr(M

Pa)

E (GPa)

Nonlinear buckling analysisLevy (1884) (E = 0)Glock (1977) (E )

1 10 –6 1 10 –4 1 10 –2 1 100 1 102

1 10 –5 1 10 –3 1 10 –1 1 101

E (GPa)

Fig. 18. Buckling pressure qcr as a function of the Young’s modulus of the rock (a ¼ 2 m,t ¼ 10 mm, Es ¼ 198 GPa, ns ¼ 0.3, n ¼ 0.3) (after Trombetta, 2015).

0

5

10

15

20

25

30

2 4 6 8 10 12 14 16 18 20

p b(M

Pa)

E (GPa)

c = 5 MPa

> 30 10pmin = 0

= 30qcr = 0.5 MPa

Fig. 20. Pressure pb at which the compression load q at pmin ¼ 0 is equal to thebuckling load qcr as a function of the Young’s modulus E of the rock mass(qcr ¼ 0.5 MPa, other parameters as in Table 2).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 325

4.5.2. AssessmentFor the assessment of safety against buckling, we compare the

buckling load determined above (dashed line in Fig. 19) with theactual load (solid lines in Fig. 19). The latter takes its maximumvalue at the end of every depressurisation and increases with adecreasing strength sc and Young’s modulus E of the rock mass. InFig. 19, it can be observed that the actual load exceeds the criticalload qcr, if the Young’s modulus of the rock is lower than 12e15 GPa(depending on the rock strength).

As explained in Section 4.4, the load developing upon the liningafter complete depressurisation increases with the maximumoperational air pressure. Fig. 20 shows (for the same geotechnicalconditions as Fig. 19) the maximum operational pressure pb thatresults in an actual load equal to the critical load qcr. For everyparameter set, the pressure pb was determined by performing aseries of numerical stress analyses (with the model of Section 4.2)for different, closely spaced values of the maximum air pressure(the minimum air pressure was taken equal to zero). The numericalanalyses provided the relationship between the maximum airpressure and the loading q at zero air pressure. The critical pressurepb, i.e. the one corresponding to the critical load qr, was determinedby linear interpolation.

The results of Fig. 20 can be used to assess the safety againstbuckling for a given maximum air pressure (or to determine a safemaximum operational pressure). In soft rock masses(2 GPa < E < 5 GPa), the maximum air pressure should not exceed

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2 4 6 8 10 12 14 16 18 20

q (M

Pa)

E (GPa)

c = 5 MPa

10> 20 qcr

pmax = 20 MPa

= 30

pmin = 0

Fig. 19. Compression load q on the lining after complete depressurisation of the cavity(pmin ¼ 0) as a function of the Young’s modulus E of the rock mass (pmax ¼ 20 MPa,other parameters as in Table 2).

10 MPa (e.g. p¼ 5 MPa for E¼ 2.5 GPa). This value is lower than thecritical air pressure that was determined from the criterion of steelstrain (14MPa for 100m depth of cover, sc¼ 5MPa and E¼ 2.5 GPa,see Fig. 10). Consequently, safety against buckling is the decisivecriterion for soft rock masses.

4.6. Fatigue failure

The safety against fatigue failure in the steel lining depends, ingeneral, on the number of loading cycles, on the stress range Dstt,sin a single cycle and on the type of welding joints. As suggested invarious technical specifications, the critical stress rangeDsscr can beexpressed as a function of the weld type and the number of cycles.For example, according to BS EN 13445e3 (2009), the critical stressrange Dsscr for 10,000 cycles is equal to 526 MPa for joints of verygood quality (joint class 90) and to 369 MPa for joints of mediumquality (joint class 70).

Fig. 21 shows the actual stress range Dstt,s (solid lines) of thelining as a function of the strength sc and the Young’s modulus E ofthe rock mass for CAES cavities under a cyclical air pressure vari-ation of 2e20 MPa. The diagram was computed with the modelfrom Section 4.2, taking a single loadingeunloading cycle. Thestress range Dstt,s increases with decreasing rock stiffness andstrength. The effect of the strength sc is not, however, significant.For comparison, the diagram also shows the critical stress rangeDsscr for two joint qualities (dashed line). In CAES cavities with a

0

100

200

300

400

500

600

2 4 6 8 10 12 14 16 18 20

tt,s(M

Pa)

E (GPa)

c = 5 MPa> 20

scr (104 cycles; joint class 90)

scr (104 cycles; joint class 70)

pmax = 20 MPa

= 30

pmin = 2 MPa

Fig. 21. Stress range in the steel Dstt,s during cyclical pressurisation as a function of theYoung’s modulus E of the rock mass (pmax ¼ 20 MPa, pmin ¼ 2 MPa, other parameters asin Table 2).

0

5

10

15

20

25

30

2 4 6 8 10 12 14 16 18 20

p f(M

Pa)

E (GPa)

c = 5 MPa

> 20

= 30

pmin = 2 MPa

scr = 369 MPa

Fig. 22. Pressure pf at which the stress range in the steel Dstt,s is equal to the criticalstress range Dsscr for the fatigue as a function of the Young’s modulus E of the rockmass (pmin ¼ 2 MPa, Dsscr ¼ 369 MPa, other parameters as in Table 2).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328326

maximum operating pressure of 20 MPa, the actual stress range ofthe lining exceeds the critical stress range Dsscr, if the Young’smodulus of the rock mass is lower than 8e13 GPa (depending onthe weld type).

The actual stress range experienced by the steel during cyclicalloading increases with the maximum operational air pressure (seeSection 4.4). Fig. 22 shows (for a minimum air pressure of 2 MPa

(a)

H = 80 m

D = 4 m

tp= 0.5 -1.5 m

Lp =3; 6 m

Service tunnel Plug CAES tunnel

50 m

y

(b)

x

45 m 6 m 30 m

0

ptp=1.5 m

2 m

Fig. 23. (a) Geometry of the investigated plugs; (b) Computational domains andboundary conditions adopted in the numerical stress analyses of a specific plug(Lp ¼ 6 m, tp ¼ 1.5 m).

and the same geotechnical conditions as shown in Fig. 21) themaximum operational pressure pf that results in a stress range of369 MPa, which is critical for joint class 70 and 10,000 loadingeunloading cycles. The diagram, which was computed analogouslyin Fig. 20, shows that the maximum air pressure should be lowerthan 10 MPa for 2 GPa < E < 5 GPa, which means that for CAEScavities in soft rocks, fatigue failure is as relevant as buckling.

5. Plug stability

In the section, we consider the example of an 80 m deep CAEStunnel of 4 m diameter located above the water table and accessedby a service tunnel of equal diameter (Fig. 23a). Plugs of differentdimensions (Lp, tp) are studied.

We performed numerical stress analyses (based on theaxisymmetric model in Fig. 23b) in order to estimate the plugloading that would cause collapse of the rock mass. The numericalmodel does not take into account the tunnel linings or the man-hole in the plug. The rock mass is modelled as a continuouslinear elastic-perfectly plastic, no-tension material, obeying theMohreCoulomb yield criterion. A weak rock mass was considered(parameters are given in the caption of Fig. 24). The concrete plug istaken as a linearly elastic (Ecr ¼ 40 GPa, vcr ¼ 0.3), practically rigidmaterial. Practically rigideplastic interface elements (interfaceshear stiffness KsI ¼ 500 GPa/m, interface normal stiffness

0

50

100

150

200

250

300

0.5 1 1.5

p ul(

MPa

)

tp (m)

pul Lp = 6 mLp

tp (a)

Lp = 3 m

0 m

16 m(b)

350

Fig. 24. (a) Ultimate pressure as a function of the dimensions of the plug; (b)Deformed calculation grid for a specific plug (Lp ¼ 6 m, tp ¼ 1.5 m) at the ultimatepressure pul ¼ 300 MPa (rock mass with s0 ¼ 2 MPa, f ¼ 30� , sc ¼ 8 MPa, st ¼ 0, j ¼ 0� ,E ¼ 5 GPa, n ¼ 0.3).

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328 327

KnI¼ 500 GPa/m) with purely frictional strength (fI¼ 25�) are usedto model the contact between the plug and the rock mass.

The analysis consists of the following steps: (1) initialization ofthe in situ stress field, assuming a uniform and isotropic stressdistribution (s0 ¼ g H¼ 2 MPa); (2) simulation of the excavation bydeleting the grid elements inside the CAES tunnel, inside the ser-vice tunnel and inside the plug in a single step without the appli-cation of any support; (3) simulation of plug installation byreactivating the grid elements inside the plug; and (4) applicationof an increasing internal pressure on the plug (no cyclical loading).We applied an internal pressure only on the plug (Fig. 23b) becausethe first analyses showed that the limit load of the rock mass in theplug area is higher than that of the rock mass around the CAEStunnel. In order to estimate the stability of the rock mass subjectedto plug loading, we evaluate the relationship between thedisplacement of a control point (on the wall of the service tunnel)and the pressure applied on the plug. The critical pressure is takenequal to the pressure at the last equilibrated solution.

Fig. 24a shows the critical pressure as a function of the conicitytp and the length Lp of the plug. The critical pressure increases withincreasing plug length and conicity. For typical operational pres-sures of up to 20 MPa, 3e6 m long plugs with a conicity of 0.5e1.5 m are not critical for the stability of the rockmass. Nevertheless,a small conicity (e.g. tp < 1.5 m) is not recommended, because therock mass close to the tunnel may have been disturbed by theexcavation. The safety factor against failure of the concrete isdecisive for selecting the length of the plug.

Fig. 24b shows the deformed calculation grid at the last equili-brated solution. As expected, the plug separates from the rock massalong the interface on the side of the CAES tunnel. This is animportant aspect to be considered in lining design (Okuno et al.,2009). The structural detailing of the lining must be such that itis able to bridge the gap without becoming damaged and, partic-ularly, without losing its tightness.

6. Conclusions

In shallow CAES cavities, uplift failure of the overlying rockmassis an important hazard scenario considering the air pressuresneeded for efficient energy storage (p ¼ 10e30 MPa). For a givenmaximum operational pressure, safety against uplift dependssignificantly on rock strength, depth of cover, horizontal stress andcavity type (tunnel or shaft). The critical overburden for 4 mdiameter CAES tunnels with the maximum air pressure of 20 MPaamounts to 60e120 m, depending on the geotechnical conditions.CAES shafts can be constructed in smaller depths (30e50 m).

The safety of the rock mass against uplift failure is a necessarybut not a sufficient condition for the feasibility of a CAES cavity. Ahigh maximum air pressure may cause tensile, fatigue or bucklingfailure of the steel lining. Safety against these failure types in-creases with rock stiffness, but does not depend on the type ofcavity (tunnel or shaft). The computational results lead to theconclusion that safety against buckling or fatigue failure is thedecisive factor for the maximum air pressure if the rock is soft andthe storage utility is not too shallow. At a depth of 100 m, whichwould not be problematic with respect to uplift safety for CAESfacilities under 20 MPa pressure in tunnels and shafts of 4 mdiameter, safety against fatigue or buckling limits the operationalair pressures to a maximum of 10 MPa, if the rock is softer than5 GPa.

The stability of the rock mass under the plug loading is anotherhazard scenario. It was investigated here assuming specificgeotechnical conditions and plug geometries similar to thoseapplied in previous studies (plug length 6 m, conicity 0.5e1.5 m).

The computational results indicate that rock failure around the plugis not a relevant design factor.

The present paper analysed CAES cavities above the water table.Drainage measures would be essential underneath the water tablein order to avoid buckling of the steel lining in the depressurisationstage; thin, economically viable steel shells may buckle even undera low hydrostatic pressure of about 5 bars in the example of Section4.5.1 (1 bar ¼ 1 � 105 Pa). Groundwater drainage over the entirelifetime of the system may be environmentally unacceptable.Basically, one might switch the drainage system on and off duringthe depressurisation and pressurisation phases, respectively.Intermittent operation of the drainage system, however, wouldreduce the overall reliability of the system and raise questions overthe response time of the ground to drainage in relation to the fre-quency of cycles. These issues may significantly affect the feasibilityof lined CAES systems and they deserve further investigations.

The computational results also illustrate some limitations of thewidely used constitutive models (e.g. their failure to map theclosing of open cracks), which may be very important for studyingthe rock-lining interaction during cyclical loading. In view of theinherent model and parameter uncertainties as well as the lack ofexperience with this type of underground opening, properlymonitored field tests in scaled models will be indispensable forevaluating the feasibility of underground CAES facilities under thespecific geotechnical conditions of a project site.

Conflict of interest

The authors wish to confirm that there are no known conflicts ofinterest associated with this publication and there has been nosignificant financial support for this work that could have influ-enced its outcome.

Acknowledgements

The authors wish to thank Herrenknecht AG, Germany, whichhas supported this research project. Further thanks are due to Dr.Yasunori Ootsuka (OYO Corporation, Japan) for the valuable dis-cussions and suggestions.

Notation

a internal radius of the steel liningb width of the lining (Eq. (1))D diameterd spacing of the springs (Fig. 16a)E Young’s modulus of the rock massEs Young’s modulus of the steelEcr Young’s modulus of the concrete plugF spring reaction (Fig. 16b)H overburdenK0 coefficient of lateral stress at restKsI shear stiffness of the interface elementsKnI normal stiffness of the interface elementsksp stiffness of the springs (Fig. 16b)L height of the shaftLp length of the plug (Fig. 23a)p air pressurepb critical air pressure for bucklingpc critical air pressure for cracking until surfacepf critical air pressure for fatigue failurepr pressure on the rock (Fig. 11b)pu ultimate air pressurepmax maximum air pressurepmin minimum air pressure

P. Perazzelli, G. Anagnostou / Journal of Rock Mechanics and Geotechnical Engineering 8 (2016) 314e328328

p* air pressure at which active shear failure of the rockoccurs during depressurisation

q lining load (Fig. 16a)qcr critical buckling loadr radial coordinatet thickness of the steel liningtp conicity of the plug (Fig. 23a)u radial displacementx horizontal coordinatey vertical coordinate

Greek symbolsg unit weight of the rock massDstt,s stress range in the steel liningDsscr critical stress range for fatigue failure of the steel3tt tangential deformation of the rock mass3tt lim limit tangential deformation of the rock mass3tt,s tangential deformation of the lining3tt,s lim limit tangential deformation of the steel liningn Poisson’s ratio of the rock massns Poisson’s ratio of the steelncr Poisson’s ratio of the concrete plugs normal stresssc uniaxial compressive strength of the rock massst tensile strength of the rock masssy yield strength of the steelsrr radial stress in the rock massstt tangential stress in the rock massstt,s tangential stress in the liningsxx horizontal stresssyy vertical stresssxx0 horizontal initial stresssyy0 vertical initial stresss0 initial homogeneous isotropic stressf friction angle of the rock massfI Friction angle of the plugerock interfacej dilatancy angle of the rock mass

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Dr. Paolo Perazzelli graduated in environmental engi-neering from University of Rome “Sapienza” in 2007. From2007 to 2010 he was a research associate in the Depart-ment of Geotechnical Engineering of “Sapienza” Univer-sity. In 2011 he received his Ph.D. degree with a thesis ontunnelling in soft soils. Subsequently, he transferred toindustry and worked as tunnelling engineer in Astaldi.Since 2012 he is postdoctoral research fellow in theTunnelling Group of the ETH Zurich where he is involvedin research works dealing with deformation and stabilityproblems of underground structures in soil and rock.

Contact information: ETH Zurich, Stefano FransciniPlatz 5, 8093 Zurich, Switzerland.

Tel: þ41-(0)44 6330729

E-mail: paolo.perazzelli@igt.baug.ethz.ch

Dr. G. Anagnostou graduated in civil engineering from theUniversity of Karlsruhe in 1983. From 1983 to 1993 he wasa research associate at the Rock Mechanics and TunnellingGroup of the ETH Zurich. In 1991 he received his Ph.D.degree with a thesis on tunnelling in swelling rock. Sub-sequently, he transferred to industry and worked as aconsulting engineer in Zurich and Athens. He was involvedin several tunnelling projects in Switzerland and abroad. In2003 he was appointed Professor of Tunnelling at ETHZurich. His research focuses on deformation and stabilityproblems of underground structures in soil and rock.

Contact information: ETH Zurich, Stefano FransciniPlatz 5, 8093 Zurich, Switzerland.

Tel: þ41-(0)44 6333180

E-mail: georg.anagnostou@igt.baug.ethz.ch