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An Improved Dilatancy Boundary for Salt Rock

May 2023
Journal of Engineering Mechanics 149(8)

Authors:

Karlsruhe Institute of Technology

Relationship between DP (damage potential) and D 1 based on RD criterion. Solid squares denote triaxial compression; solid circles denote triaxial extension.

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Relationship between DP (damage potential) and D 2 based on RD criterion. Solid squares denote triaxial compression; solid circles denote triaxial extension.

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Relationship between damage potential and n based on RD criterion. Solid squares denote triaxial compression; and solid circles denote triaxial extension.

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Volume loss and volume loss rate of an ellipsoid-shape cavern at 465 m depth in seasonal operation. (Adapted from Habibi 2016.)

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Illustration of stress path based on stress states. (Adapted from De Vries et al. 2005.)

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Content may be subject to copyright.

An Improved Dilatancy Boundary for Salt Rock

Rahim Habibi, S.M.ASCE1

Abstract: Salt layers and domes have been used to store hydrocarbons and to dispose of nuclear wastes. One of the most important

properties of rock salt is its good deformability, which can neutralize the deviatoric stress and heal damage. Structural stability is one crucial

requirement in salt cavern design and construction. Various criteria and methods have been proposed for salt cavern design and stability

analysis. This paper investigated the advantages and limitations of various stability criteria. The results suggest that the stability criterion

proposed by De Vries et al. is the most suitable approach for stability analysis. However, the effects of the stress path and history are not

considered in this criterion. To overcome this shortcoming and to integrate effects of the stress path and history within the criterion,

an updatable dilatancy criterion is proposed. In the proposed dilatancy criterion, a representative parameter is defined based on the healing

and damage that occurred in a previous cycle of loading and unloading, which takes into account the stress path and history and their effects

on the behavior of rock salt in subsequent cycles. The new criterion gives deeper insight into the strength of rock salt based on previous

damage-healing background, and is useful in appropriate design and operation of salt cavern. DOI: 10.1061/JENMDT.EMENG-6980.

Author keywords: Salt cavern; Stability; Dilation-based criteria; Stress path; Stress history.

Introduction

Underground storage of oil and natural gas has been practiced

widely (Lux and Dresen 2012;Djizanne et al. 2012;Wang et al.

2018a,c). One storing method uses cavities leached in salt beds

or domes. Because of special features of rock salt such as low

permeability and self-healing, have these cavities been used not

only for oil or natural gas storage, they also are an option to isolate

nuclear wastes. Design, construction, and operation (short term or

long term) of these cavities and their abandonment relate to the

short- and long-term structural stability, so the integrity of the rock

salt must be satisfied (Wang et al. 2018b,d,2019). Integrity means

that no stored material such as natural gas can flow away through

pre-existing and/or induced fractures. The former rarely have been

reported due to the plastic behavior of salt, which closes the cracks;

however, the latter can take place when stress states around a cav-

ern change (Hou 2003). Based on continuum damage mechanics

(CDM), microfractures can be induced when the damage reaches

a specific magnitude, which depends on mechanical properties

(Lemaitre and Desmorat 2005). To satisfy the stability and damage

suppression of the walls and roof of a cavern, the generation of

microcracks must be avoided. The structural stability of these types

of cavities depends on the hydrogeology of the site; the site’s and

the host rock’s characteristics; the operation method; and the depth,

shape, size, and location of the cavern in the bed or dome (De Vries

et al. 2005) (Fig. 1). Therefore, understanding the time-dependent

and time-independent behavior of rock salt in the complicated

stress state around a cavern is very important. To investigate the

damage using stress analysis, the salt behavior under different

conditions must be determined. However, this is difficult because

of the complicated behavior of rock salt in different stress states.

Moreover, design concepts and construction of salt caverns are very

complex as well. Various methods (analytical and numerical) of

design and stability criteria (stress-based and dilation-based) have

been investigated and applied. They usually are based on laboratory

or in situ investigations, or, preferably, on a combination of both.

Generation and growth of cracks should be limited, especially in

the cavern wall and roof, to preserve the integrity and structural

stability of the cavern, which are related to the increasing of the

deviatoric stress. The deviatoric stress, with respect to the stability,

is defined as the difference between the internal pressure of the

cavern and the geostatic stress. Large deviatoric stress, especially

during the production, causes dilation initiation. If this happens,

it is highly likely that cracks will generate around the cavern which

can be localized and expanded when the cavern experiences low

internal pressure. If this condition continues for a specified period

during which the cracks can grow and connect to each other, the

permeability increases, which increases the risk that the stored ma-

terial will flow into the rock salt around the cavern (De Vries et al.

2005). Therefore, the stress state in which cavern will be structur-

ally stable over its whole lifespan should be determined.

Of the two types of stress (internal and lithostatic stresses)

involved in cavern stability, only internal pressure is easily control-

lable over the cavern lifespan. Therefore, the deviatoric stress

should be kept in a range in which the stability and integrity of the

cavern can be ensured during the lifespan. Habibi et al. (2021) dis-

cussed some methods to address the issue of stress; for example,

in caverns containing natural gas, the large deviatoric stress is neu-

tralized by brine hydrostatic pressure during leaching and by gas

pressure during operation. However, the pressure resulting from

natural gas is not large enough to prevent dilation or fracturing

around the cavern, so generally the internal pressure cannot totally

overcome the lithostatic pressure. Therefore, the maximum pres-

sure (optimized pressure) of the natural gas must be determined and

must be considered in every injection–withdrawal cycle. Based on

modeling of the stress state around the cavern, the dilation stress

and subsequently the fracture stress is a function of pressure

changes, rate of injection and withdrawal, and time-dependent char-

acteristics of the rock salt (Wallner 1988). The operational pressure

1Ph.D. Student, Dept. Geothermal Energy & Reservoir Technology,

Karlsruhe Institute of Technology, Adenauerring 20b, 76131 Karlsruhe,

Germany. Email: R.Habibi12@yahoo.com

Note. This manuscript was submitted on September 27, 2022; approved

on January 9, 2023; published online on May 17, 2023. Discussion period

open until October 17, 2023; separate discussions must be submitted for

individual papers. This paper is part of the Journal of Engineering Me-

chanics, © ASCE, ISSN 0733-9399.

J. Eng. Mech., 2023, 149(8): 04023044

range must be considered as exactly as possible in order to guarantee

the stability of the cavern at minimum and maximum pressure.

Various stability criteria such as dilation boundaries have been

proposed to limit the crack generation and growth around caverns.

Some studies have presented a comprehensive set of parameters that

have been applied in engineering works (De Vries et al. 2003).

This paper discusses the advantages and disadvantages of the

stability criteria including invariant-based and stress-based criteria.

After investigating the criteria, it was concluded that the RESPEC

dilation (RD) criterion proposed by De Vries et al. (2005) gives

reasonable results compared with other criteria. However, because

the criterion does not consider stress path and loading history,

RD was modified in this paper. First, by analyzing the RD criterion

in the compression and extension states, it was shown that the

constant parameters D1,D2, and nvary by stress state in triaxial

compression, meaning that they are stress-dependent. By having

different value of these parameters in similar stress states during

different loading and unloading paths, different dilation boundary

can be generated. In other words, a new dilatancy boundary in

every cycle is introduced because of different input parameters

(D1,D2, and n), whereas the RD gives constant values for the

parameters when the stress state is extensional. Therefore, it was

concluded that RD is suitable only for the extension state.

This paper defines a new concept using instantaneous strength

and damage potential (DP) which not only overcomes the short-

coming, but also carries the effects of the stress state in previous

cycles into the dilation response. Therefore, a representative param-

eter (RP) is needed, which, in addition to addressing the stress

dependency of the parameters, recalls the effects of the stress path

and stress history to integrate the memory effects in RD as well.

Because the material constants relate to the stress by means of po-

tential damage, the representative parameters must be defined using

damage potential. The DP is influenced by the stress state, path, and

history, which are discussed in the section “Modifying the Criterion

to Include Stress Path and Stress History.”The instantaneous

strength and memory effect (Guessous et al. 1987;He et al. 2019)

of salt, which in turn are influenced by the stress state, path, and

history, were investigated to establish a relationship between mate-

rial constants and DP. Based on this relationship, a representative

parameter was defined which takes into account instantaneous

damage and/or healing. An overview of the development of the

representative parameters is given in Fig. 2. Then, a new dilatancy

boundary was introduced for every cycle of loading and unload-

ing which has capability to be updated using material parameters

(D1,D2, and n), because, as mentioned previously, these material

parameters gain new values during every cycle of loading and un-

loading because they experience different stress states. These val-

ues are used to feed the equation of dilation boundary in every cycle

of loading and unloading, so that during every cycle a new dilation

boundary is provided. This kind of updatability carries the memory

effects of rock salt.

Structural Analysis of Salt Cavern

Salt caverns progressively close because salt deforms continuously

(creeps) when subjected to shear stress resulting from the difference

between the cavern pressure acting on the cavern wall and the far

field in situ stress (lithostatic pressure). The shear stresses increase

as the cavern internal pressure decreases. In turn, the rate of creep

closure increases nonlinearly as a power function of the shear stress

(Bérest and Brouard 1998). Creep deformation alone is a constant-

volume process in salt, so that during the secondary stage of creep

no volume change takes place, whereas after the tertiary stage be-

gins, volume changes occurs (Fuenkajorn and Phueakphum 2010).

Similarly, when the internal pressure in a cavern is decreased too

much, the shear stresses in the surrounding salt can exceed the

strength of the salt due the high deviatoric stress, i.e., the surround-

ing salt experiences tertiary creep. Then microfracturing or dilation

Cavern desi gn

parameter

Geome trical

Mechan ical

Thermal

Hydra ul ic

Oper ation

Height; Diameter; Height diameter ratio;

Aver ag e depth ; S ha p e

Density; Poisson's ratio; Young; Shear and Bulk

moduli; Constitutive lo w parameters; Tensile

and compression strength; Stability criterion

parameters

Thermal conductivity; Geo-thermal g radient;

Thermal capacity; Thermal diffusivity; Thermal

expansio n factor

Permeability; Porosity; Effective pressure

gradie nt

Operation program; Injection and production

rate; Surface equipment

Fig. 1. Design parameters of a single cavern located in salt bed or dome. (Data from Habibi 2019.)

J. Eng. Mech., 2023, 149(8): 04023044

takes place in the salt body, which manifests as an increase in vol-

ume due to the creep process, because microfractures and/or

dilation create additional porosity in the salt which quantified as

volume increase. Microfracturing (known as damage) increases

the creep rate because the salt weakens and its strength decreases

mechanically due to microfracturing, which threatens the stability

of the cavern. From a hydraulics point of view, when microfractur-

ing initiates, the permeability of the surrounding salt increases due

to the damage, which in turn threatens the integrity of the cavern.

On the other hand, Düsterloh et al. (2013), using laboratory

measurements, showed that salt rock can experience healing during

unloading, through which the rock strengthens. Because a salt cav-

ern experiences unloading and loading in injection and production

cycles, respectively, the cavern experiences healing and damage

during each operational stage. Describing and considering such

a coupled behavior in the design and operation phases of a cavern

seems crucial because these phenomena are induced by differ-

ences between internal and lithostatic pressure. Moreover, from an

operational prospective, the deliverability of the stored material,

as the main purpose, is controlled directly by the internal pressure.

In some cases, salt caverns are designed for long-lasting cycling

operation. However, in other cases, politics and fluctuating energy

consumption demands cause a cavern to be designed for weekly,

daily, and even hourly cycling operation scenarios. Therefore, the

describing dilation behavior of the salt (which utilized as stability

and integrity criterion) seems challengeable in such short cycling in

which the surrounding rock experiences healing and damage in a

short time during injection and production cycles, respectively.

High deliverability from a salt cavern is controlled by the fol-

lowing geotechnical and operational constraints:

•Operational constraint: among others, cavern pressure and tem-

perature changes control cavern performance (Karimi-Jafari

et al. 2011). For example, the temperature for hydrate formation

depends on the gas pressure and water content.

•Geotechnical constraint: injection and withdrawal cause tempera-

ture changes, resulting in thermal stressesonthewallwhichinturn

may cause microfracturing and damage (Karimi-Jafari et al. 2011).

Literature Review of Stability Analysis Criteria

Similar to creep, the damage resulting from the shear stress differ-

ence also is a progressive process. When the shear stress exceeds

the shear strength of the rock salt, damage occurs. Therefore, in this

condition, during injection and production, which act as unloading

and loading on the cavern wall, it is highly likely that microfractur-

ing will be initiated due to the induced stress state affected by creep

deformation and damage. Initiation and growth of cracks create

weak planes with low strength which have the capability to cleave

the surrounding rock, which may cause the rock salt in the cav-

ern wall and roof to flake and collapse. Thus, a dilatancy criterion

should be developed that represents the dilation of the salt rock

as a sign which takes place before fracturing under different loading

scenarios. Based on the literature review (e.g., Habibi 2019;Bérest

et al. 2006), the most applicable criteria include no or low tensile

zone, no or low effective tensile zone, and a dilation boundary.

Therefore, before beginning the modification procedure, the ad-

vantages and disadvantages of the aforementioned criteria are

discussed.

No or Low Tensile Zone

Generally, the tensile strength of salt rock is low (1.5–2 MPa), and

salt rock behaves like a brittle material in tension. Therefore, ten-

sion must be prevented in cavern design. Karimi-Jafari et al. (2011)

concluded that cavern roof collapse is common in a tension state.

Various factors such as overpressuring, thermal shocks (resulting

from fast and cold withdrawals), and large-span flat roofs can cause

tensile stress around caverns. In turn, large tensile stresses can

cause crack initiation, resulting in fracturing and spalling of the

walls and roof. In general, stresses around a cavern are compres-

sive, but tensile stress might appear around a cavern in the follow-

ing cases:

1. When the cavern pressure is low and the cavern’s profile in-

cludes convex portions [an example of this was described by

Nieland and Ratigan (2006)].

Effect of damage potential changes on material constants

Compression Extension

Effects of stress states Effects of stress path and history

Constants D1,D2,andn

Variants D1,D2,andn

Instantaneous strength

Memory effects of salt

New dilatant boundary

Fig. 2. Determining representative parameters using damage potential.

J. Eng. Mech., 2023, 149(8): 04023044

2. Rapid decrease of the cavern pressure will cause a decrease

of gas temperature due to the thermodynamic response of the

cavern (Bérest et al. 2006). The degree of temperature loss

depends on the production rate and the size and shape of the

cavern. Tensile stresses can be triggered by such temperature

drops (gas cooling) at the cavern wall. When the tensile stress

caused by temperature decrease during rapid injection and pro-

duction is larger than the tensile strength of the rock mass

around the cavern, microcracks are generated (Bauer and

Sobolik 2000;Staudtmeister and Zapf 2015;Karimi-Jafari et al.

2011;Rokahr et al. 2011;Bérest et al. 2012;Leuger et al. 2012,

2000;Lux and Dresen 2012). In most cases, these microcracks

are shallow and they usually are perpendicular to the cavern

wall. However, some of these cracks can propagate deeper when

the conditions are favorable. Because these deep cracks are per-

pendicular, the distances between these cracks and cracks that

have stopped growing increases with time. Therefore, different

cracks with different depths in the cavern wall can cause slabs of

rock salt to detach from the wall. However, these slabs cling

to rock salt because they are thin (Bérest et al. 2012;Pellizzaro

et al. 2011;Ngo and Pellet 2018).

3. Unlike the preceding phenomenon (i.e., a decrease in gas tem-

perature during fast depressurization) which can be measured

during cavern operation, thermally induced cracks cannot be di-

rectly measured. Therefore, studies usually investigate this pro-

cess of salt rock behavior in underground salt mine and extend

the results to caverns. In salt mine wells opening to the air, ther-

mal fracturing commonly occurs in unlined wells (Wallner and

Eickemeier 2001;Zapf et al. 2012). Because wells are cold in

winter, they experience this condition for at least a few months.

Djizanne et al. (2012) proposed a closed-form solution for the

thermoelastic response of salt rock and showed that vertical

stresses at the cavern wall are less compressive than natural geo-

static stresses due to the redistribution of viscoplastic stresses.

In the long term, the difference between these stresses causes

fractures which mainly are horizontal. Additionally, when the

temperature changes are large enough [which commonly occurs

in fast-cycle-operation caverns such as compressed air energy

storage (CAES)], the vertical stress becomes tensile (Bérest

et al. 2012).

No or Low Effective Tensile Stress

Effective tensile stress is related to the porosity of the rock, and is

used progressively in reservoir engineering. It is the actual stress

(with compressive stress defined as negative) plus the fluid pressure

in pores. Because the porosity and permeability of rock salt are low,

defining the effective stress in rock salt remains complex and chal-

lenging. However, the effective tensile stress can be calculated

easily in a cavern wall, because it equals the sum of the actual stress

and the cavern (fluid) pressure (Brouard et al. 2007). Actual stress

in a cavern wall includes normal stress, tangential stress, and cir-

cumferential stress (wall tension).

As an example, in a perfectly cylindrical cavern, the normal

stress is perpendicular to the wall, tangential stress is the vertical

stress, and circumferential stress acts perpendicularly on both.

The effective stress can experience three different status: negative,

meaning that the actual stress is larger than the fluid pressure

(because compressive is defined as negative); positive, meaning

that the fluid pressure is larger than actual stress, resulting in frac-

turing of the cavern wall; and zero, meaning that both are equal.

The goal of this criterion is to prevent the development of

positive effective stress (effective tensile stress) around a cavern,

because, based on the effective stress definition, when it becomes

positive meaning that the fluid pressure is larger than the actual

stress, fracturing can take place. Considering this definition, when

only normal stress (equal to the cavern fluid pressure) is applied to

the cavern wall, effective stress is zero; in other words, only under

this condition, no or less zones affected by effective tensile stresses

(no or less zones can be estimated by this criterion), i.e., under

this specific condition the goal of the criterion can be satisfied.

This means that the actual stress more or less equals the fluid pres-

sure, so no fracturing will occur, due to the equality. However, by

considering a value for other components of the actual stress

(i.e., perpendicular and circumferential stresses), it is highly likely

that the total value of the actual stress may change, and effective

stresses could be nonzero, i.e., negative (compressive stress state)

or positive (tensile stress state). Some authors, such as Bérest et al.

(2001), believe that the effective tensile stress in a cavern wall

must be larger than a given value, such as the tensile strength

of rock salt. Therefore, a condition for no or low effective stress

zone is

σmin þP<Tð1Þ

where T= tensile strength (MPa); P= cavern pressure (MPa); and

σmin = smallest compressive stress (MPa) computed at any point

of the rock mass.

According to Eq. (1), when the cavern experiences large and

fast cycles, an effective tensile stress can develop. For example,

when the cavern experiences rapid production, Pis decreased sig-

nificantly. This causes an increase in σmin, possibly exceeding T.

Consequently, the condition of Eq. (1) is met and fracturing

initiates. Therefore, in salt caverns containing natural gas which

experience large cycles so that cavern experiences large pressure

changes in every injection/production cycle. Moreover, micro-

cracking can be initiated when Eq. (1) is satisfied, resulting in

increased permeability and salt softening (Malinsky 2001;Stormont

2001;Rokhar et al. 1997). Setting T¼0(considering a higher

safety factor), the definition of the criterion becomes simple and

the effective stress no longer appears.

However, because stored material experiences large changes

in temperature, especially in CAES caverns, in which daily cycles

commonly are performed, there is not enough time between two

cycles to reach thermal equilibrium with the rock mass temperature.

This type of thermal unbalance causes thermal expansion or con-

traction in the rock due to the temperature difference, which in turn

induce significant thermal stresses in the cavern wall, which may

lead to spalling or fracturing (Lestringant et al. 2010;Wang et al.

2011). Some field examples of thermally induced fracturing were

given by Sicsic and Berest (2014).

In some cases, due to the thermodynamic evolution such as

rapid temperature changes in operation cycles, thermal stresses,

which commonly are tension stresses, in the cavern wall and roof

are inevitable. However, the aforementioned criteria (i.e., no or low

tensile zone and no or low effective tensile stress) emphasis strictly

to prevent tensile stress around the cavern, i.e., the criteria limits

pressure changes in a range where tensile stress can unlikely be

developed as based on the criteria the zones under compression

have no stability and integrity thread. But as it will be discussed

in the following the stability and integrity can be threatened at com-

pression state as well. Considering the inevitability of the response,

it can occur systematically, i.e., when pressure decreases in the cav-

ern, tensile stress is induced around the cavern. However, based on

the aforementioned criteria, no tensile stress must be induced

around the cavern. To obey the criteria, the rate of injection and

production and the range of working pressure should be decreased

so that no tensile stress is induced around the cavern. In other

J. Eng. Mech., 2023, 149(8): 04023044

words, these criteria require the operator of the cavern to operate

within a pressure range and rate at which no tensile stress is in-

duced, although the range is not optimized economically. This is

a kind of restriction or limitation which is embedded in the nature

of the criteria. Thus, because tensile strength was not integrated in

the criteria, and because commonly induced stress states are not

considered in the criteria, the criteria are not comprehensive and

cannot be applied for practical applications. In other words, these

criteria assess the stability in a strict way so that instead of consid-

ering the tensile stress as a highly likely situation, it is suggested

that a cavern should not be allowed to experience a tension state.

Indeed, any tensile stress means instability according to these cri-

teria. From another point of view, they follow true and false logic

(or a binary system) in such a way that any tensile stress is con-

sidered to be instability, so that any tensile stress is defined as false

and compressive stress is considered as true. Actually, a cavern

could experience instability in zones where a compressive state

is dominant, that is the point referring to incomprehensiveness

of the criteria. As it will be discussed in the following the stability

and integrity can be threatened at compression state while afore-

mentioned criteria assumed that only tension state can threaten

the stability and integrity. Regarding the incomprehensiveness,

these criteria indicate the zones where the tension state is overcome,

without considering zones of a compression state, and it is highly

likely that the microfracturing can initiate in these compressive

zones. Therefore, they face some problems for practical applica-

tions because different zones of the cavern suffer different stress

states; for example, during production, the roof experiences ten-

sion, whereas the wall suffers compressive stresses (Brouard et al.

2007). It would be better to assess the integrity and stability in

terms of dilation rather than stress state. An example follows in

which the generation of thermally induced tensile stress was con-

firmed through modeling and measurements.

Brouard et al. (2007) studied the effect of the fast pressure in-

creases on the stability of caverns which experienced operational

cycles. They applied three constitutive models for a cavern located

1,500 m deep. Their results showed that such pressure changes

induced effective tensile stress in the cavern walls, in which the

dependence of the effective tensile stress on the number of cycles

was greater than on the pressure change rate. As the number of

cycles increased, the effective tensile zone increased as well.

Brouard et al. also investigated the possibility of effective tensile

stress developing in caverns subjected to large pressure changes,

such as natural gas–filled caverns. They concluded that the pos-

sibility of developing an effective tensile stress zone is high at cav-

ern walls because of the large pressure changes. The large changes

of pressure cause stress redistribution around caverns when the

cavern pressure is low enough and the deviatoric stress increases

slowly due to the difference between internal pressure and litho-

static pressure due to the withdrawal. This results in the decrease

of the difference between tangential and radial stress. At the end of

the production and beginning of the injection, when gas is injected

into the cavern, large elastic stresses are induced at the cavern wall,

causing the tangential stress to become larger than the normal

stress. An effective tensile stress is generated at the cavern wall.

However, Brouard et al. (2007) assumed the tensile strength to

be zero and fluid pressure in the rock mass to be equal to the cavern

fluid pressure from a safety viewpoint. Their assumptions seem to

be conservative. Similarly, Djizanne et al. (2012) reported such

thermally induced stress during fast injection and production in

the Etzel K-102 cavern in Germany, which had been in operation

for 20 years. They also concluded that, in some cases, it is highly

likely that the tensile zone may develop only in some local parts.

However, the integrity of a cavern is threatened significantly when

the thickness of the reserved tightness rock salt in the cavern roof is

small (i.e., cavern roof is not properly located from above layers).

Dilation Boundary Criteria

A dilatancy boundary model must be able to determine the dilatant

behavior of salt around salt cavities. The stress state can be deter-

mined using closed-form analytical and numerical methods. Dilat-

ancy surfaces have been studied extensively (Spiers et al. 1988;

Ratigan et al. 1991;Hunsche 1993;Hatzor and Heyman 1997;

Labaune et al. 2018;Rouabhi et al. 2019), and the results indi-

cated damage resulting from the onset of volumetric dilation under

loading. Dilatancy boundary criteria have been proposed in two

main forms: (1) based on invariant stress with experimental fitting

parameters (invariant-based); and (2) using minimum principal

stress and effective stress (stress-based). They have been applied

extensively to identify a condition which causes accumulated

damage in cavern walls (Van Sambeek et al. 1993;Thomas et al.

1999;Chabannes et al. 1999;Ehgartner and Sobolik 2002;Nieland

et al. 2001).

Stress-Based Dilatant Boundary Criteria

These criteria describe a linear or nonlinear dilatancy boundary

distinguishing two zones located above and below the boundary,

which represent dilatant and nondilatant zones, respectively, in the

stress state space. The most well-known and applicable criteria

are given in Table 1. Fig. 3shows the different dilation boundaries

resulting from these stress-based criteria in stress space proposed

by various studies. In addition to having different nonlinearities

in the models, some included intercepts, such as the Karlsruhe

Institute of Technology (KIT) model, and in some criteria the

boundary begins from the origin point, such as the Institut fur

Gebirgsmechanik GmbH Leipzig (IfG) and Gunther–Salzer (G/S)

models. Some models proposed different boundary for extension

and compression, such as the Technische Universitat Clausthal

(TUC) model.

However, these criteria describe the boundary in such a way

that the minimum allowable internal pressure is overestimated in

high figures more than the optimized minimum pressure, resulting

in a decrease in working gas. The internal pressure is generated by

containing gas, which has two portions: a cushion (gas left in the

cavern for stability purposes), and working gas volumes (the vol-

ume injected and withdrawn in each cycle). The economics of salt

caverns largely are dependent on maximizing the ratio between

working gas and cushion gas volumes. This ratio depends directly

on the values of the maximum and minimum gas pressures per-

mitted in the storage cavern. De Vries et al. (2002) performed a

geomechanical study including laboratory testing, theoretical de-

velopment, and analytical and numerical modeling, and showed

that the minimum allowable gas pressure (cushion gas) is almost

10% lower when damage-based criteria are applied. Therefore, it

can be concluded that more cushion gas is needed when the mini-

mum allowable internal pressure is overestimated, meaning less

available working gas and less economic benefit. The overestima-

tion occurs because the stress-based dilation criteria use only the

stress state to represent the dilatation, and do not consider any phe-

nomenological concept or behavior. De Vries et al. (2002) com-

pared the stress-based and damage-based criteria, and showed that

the dilatancy-based criteria enable the dilatancy boundary of rock

salt to be estimated more exactly than do the stress-based criteria.

It also is possible to determine a lower minimum gas pressure using

the damage-based criteria.

J. Eng. Mech., 2023, 149(8): 04023044

Damage-Based Dilatant Boundary Criteria

Damage potential is defined as a ratio of stress invariants ﬃﬃﬃﬃﬃ

p=I1,

where J2is the second invariant of the deviatoric stress (MPa), and

I1is the first stress invariant (MPa). These criteria also are defined

as a function of stress invariants with different intercepts. In recent

decades, some damage-based dilatant boundary criteria have been

developed (Table 1) in which the effects of loading history are not

considered. Fig. 4shows damage-based criteria. These criteria were

developed based on different laboratory investigations of different

rock salts, and various hypotheses resulted in different criteria. For

example, Spiers et al. (1988) used the results of constant-strain-rate

tests that were performed on samples obtained from the Asse salt

mine in Germany. Ratigan et al. (1991) applied the results of volu-

metric strain rate changes of waste isolation pilot plant (WIPP) in

New Mexico and of Avery Island in Louisiana. In these criteria, the

dilation boundary was described linearly by means of damage po-

tential, which was presented in I1−ﬃﬃﬃﬃﬃ

pspace, although, based on

the Mohr–Coulomb dilatant boundary, there is a nonlinear relation-

ship between these invariants. After describing the shortcomings of

old damage-based criteria, this section briefly discusses the RD cri-

terion (De Vries et al. 2005), one of the comprehensive criteria.

During a project on salt rock samples obtained from Cayuta,

New York, performed by RESPEC, De Vries et al. (2005) de-

veloped a comprehensive dilation criterion. It is based on the

Mohr–Coulomb criterion and is applied to evaluate the potential of

microcracking around a cavern. Shortcomings identified for the

previous damage-based criteria are that they do not include (1) a

nonzero intercept, (2) a nonlinear relationship for the dilatancy

boundary in I1−ﬃﬃﬃﬃﬃ

pspace, and (3) coverage of the effects of

Lode angle. Regarding the first shortcoming, there is a need for

criterion with vast intercepts at the vertical axis with respect to

origin, because when the intercept increases, the area of the non-

dilatant zone under the dilatancy boundary increases and covers

more area. As an example, assume that y¼xand y¼xþ1are

functions representing a linear boundary in x−yspace; they have

same slope, but the latter covers more area located under the boun-

dary. By having more area under the boundary, the criterion will be

met in large amounts. To clarify, take x= 1 in two aforementioned

functions, the former reaches the boundary at y= 1 while the latter

reaches the boundary at y= 2. The old damage-based dilation boun-

dary criteria such as Mohr–Coulomb do not consider the effects of

Lode angle changes, so the effects of the mean stress changes are

not included (Fig. 5). However, De Vries et al. (2005) used ana-

lytical interpretation and laboratory measurements to overcome

these shortcomings, and proposed a comprehensive damage-based

dilation criterion

ﬃﬃﬃﬃﬃ

p¼

D1hI1

sgnðI1Þσ0inþT0

ﬃﬃﬃ

pcos ψ−D2sin ψð2Þ

Table 1. Various stability criteria

Concept Model Authors Description

No, or small tensile zone ——Tensile zones must be avoided, because the tensile strength

of salt is low.

No, or limited effective tensile

zone

—Brouard et al. (2007) Effective stress means the actual stress plus the fluid pressure

in the pores of the rock. For a salt cavern, this quantity is

defined easily at the cavern wall, because pore pressure

is simply the cavern fluid pressure.

No, or small

dilatant zone

Stress-based

criteria

Composite dilatancy

model (CDM)

Hampel (2012) In high-stress differences at the boundary, damage and

dilatant changes are modeled as a function of creep strain.

This model enables depicting the effects of transient and

steady creeps.

Gunther and Salzer Gunther and Salzer

(2007)

This is a strain-hardening model in which the total

deformation rate is a function of effective strain hardening.

Minkley and Muhlbauer Minkley and

Muhlbauer (2007)

In this model, the stress–strain relationship is modeled by a

Burgers model in which deformation history is considered

through a state variable. This model includes a damage

module to consider the damage changes, fracture, and

postfailure.

Karlsruhe Institute of

Technology (KIT)

Pudewills (2007) This model applies an elastoviscoplastic context to describe

the total deformation rate.

Lubby2 multimechanism

deformation coupled

fracture (MDCF)

Institut fur

Unterirdisches

Bauen (IUB)

In this model, total inelastic deformation rate in nondilatant

creep and dilatant creep are described by shear deformation

and tensile deformation, respectively.

Hou and Lux Hou and Lux (1999) In this model, inelastic strain rate is considered through

adaption of viscoplastic deformation in creep without

volume changes, damage, and healing resulting from

dilatancy and compression, respectively.

Damage-based

criteria

Spiers et al. Spiers et al. (1988) Damage potential is considered as the dilatant boundary.

Ratigan et al. Ratigan et al. (1991) Very similar to Spiers et al. (1988), but without intersection.

Hunsche Hunsche (1993) A compressibility–dilatancy boundary criterion based on

octahedral stresses was developed.

Hatzor and Heyman Hatzor and Heyman

(1997)

Based on principal stresses and bedding orientation to

consider nonhorizontal salt layers.

RD De Vries et al. (2005) A nonlinear dilatancy boundary is developed based on

Mohr–Coulomb criterion and damage potential.

Source: Data from Habibi (2016).

J. Eng. Mech., 2023, 149(8): 04023044

where nis a power less than or equal to 1; σ0is a dimensional

constant with the same units as I1;D1and D2are material con-

stants; and T0and ψ= tensile strength (MPa) and Lode angle

(degrees), respectively.

Modifying the Criterion to Include Stress Path and

Stress History

The shortcomings of some dilation criteria have been given. This

section discusses more details of the RD criterion for modification

purposes. Based on De Vries et al. (2002), it can be concluded that

the dilatancy-based criteria enable the accurate estimation of the

dilatancy boundary of rock salt in so that the minimum allowable

operating pressure of compressed natural gas storage can be satis-

fied as well. This is because their results showed that the minimum

allowable operating pressure can be increased when the dilatancy-

based criteria were applied (i.e. more natural gas can be stored

under safe condition using these criteria). Dilatancy-based criteria

were developed based on a continuum damage approach and dam-

age potential parameter ( ﬃﬃﬃﬃﬃ

p=I1), which are more suitable for de-

scribing the behavior of rock salt. Moreover, some researchers

(e.g., De Vries et al. 2003) proposed damage-based dilatant criteria

which consider the effects of stress states. Spiers et al. (1986) and

Ratigan et al. (1991) investigated the dilatancy boundary based on a

damage potential parameter. However, the criterion called RD pro-

posed by De Vries et al. (2005) considers not only the stress state

but also the effect of Lode angle. The damage potential parameter

and material constants of the RD are based on the properties of the

rock salt in a given field. RD does not consider the stress path and

history. To overcome these shortcomings, a modification was made

to improve the RD criterion.

De Vries et al. (2005) conducted 23 constant mean stress tests to

determine the volume expansion of the salt rock. This allowed them

to determine the dilation limit, which is applicable for developing

dilatancy criterion. Of these tests, 18 jacketed tests either in exten-

sion or compression were conducted for the virgin condition (those

applied here), and for 5 tests the samples were preconditioned prior

to the testing. During the tests, the mean stress ranged from 6.8 to

20.7 MPa. In the present paper, each test was considered as a stress

state which can be induced at a certain point in the cavern wall or

roof during injection and production, so that in each state the

amount of the damage or healing differs from that in the previous

state.

Triaxial

compression

Triaxial

extension

(2>1>3)

(1>2>3)

(1>3>2)

New criterion

(3>1>2)

Old criterion

(3>2>1)

(2>3>1)

Fig. 5. Original stress-based dilation criterion and the new Mohr–

Coulomb criterion plotted in principal stress space. (Adapted from

De Vries et al. 2005.)

0 5 10 15 20

Minimum principal stress/ MPa

Effective stress/ MPa

IfG-G/S

TUC (TC)

IfG-Mi

IUB(TC)

TUC (TE)

IUB(TE)

Hampel

KIT

Fig. 3. Different dilatant boundary criteria developed through mini-

mum principal stress and effective stress. TC = triaxial compression;

TE = triaxial extension; G/S = Gunther–Salzer model; MI = Minkley

model developed at Institut fur Gebirgsmechanik GmbH Leipzig

(IfG); KIT = Karlsruhe Institute of Technology model developed

by Pudewills (2017); IUB = Institut fur Unterirdisches Bauen; and

TUC = Technische Universitat Clausthal. (Adapted from Hampel

et al. 2012.)

0 -10 -20 -30 -40 -50 -60 -70

J21/2/MPa

I1/MPa

Hatzor and Heyman,

1997

Spiers et al., 1988

Ratigan et al., 1991

Hunsche, 1993

=90

Fig. 4. Salt dilation boundaries developed by different research

organizations. (Adapted from De Vries et al. 2005.)

J. Eng. Mech., 2023, 149(8): 04023044

Material Constants in RD

To investigate the effect of stress state on material constants, Eq. (2)

is solved for the material constants

D1¼ﬃﬃﬃﬃﬃ

pðﬃﬃﬃ

pcos ψ−D2sin ψÞ−T0

hI1

sgnðI1Þσ0inð3Þ

ﬃﬃﬃ

p−D2tan ψ¼

D1hI1

sgnðI1ÞinþT0

ﬃﬃﬃﬃﬃ

pð4Þ

nlogI1

sgnðI1Þ¼logﬃﬃﬃﬃﬃ

pðﬃﬃﬃ

pcos ψ−D2sin ψÞ−T0

D1ð5Þ

The values were calculated and plotted against damage potential

(as stress state) based on laboratory measurements from De Vries

et al. (2005) (Table 2). It was concluded that the material constants

(i.e., D1,D2, and n) in triaxial compression are not constant during

damage potential ( ﬃﬃﬃﬃﬃ

p=I1) changes, meaning that these parameters

are not constant when the stress state changes (Figs. 6–8). The

changes of material constants at different damage potentials take

the form of an ascending curve, and are plotted in Figs. 6–8based

on the data in Table 2and Eqs. (3)–(5) obtained from RD. Because

they are not constant, they have different values in different stress

states. The values of D1,D2, and nchange in every loading cycle

based on the RD criterion, which can be affected by damage or

healing taking place in the rock salt under different stress states.

It can be concluded that the RD criterion is acceptable only in

the triaxial extension state in which the material parameters are

independent of damage potential.

Regarding this dependency of material constants on the damage

potential in the compressional state, presumably the material con-

stants used in the RD criterion depend on stress path and history,

which are not considered in the RD criterion. To validate this claim

(adaptability of criterion RD), the relationship between I1and J2

was investigated versus damage potential. The effects of stress path

and history on dilatancy boundary are discussed in the following

Table 2. Measurement results by De Vries et al. (2005) in triaxial compression and triaxial extension

Loading condition I1ﬃﬃﬃﬃﬃ

pD1D2nﬃﬃﬃﬃﬃ

p=I1(damage

potential)

Representative

parameter

Triaxial

compression

20.4 6.93 0.923039 1.069328 0.833686 0.339706 0

30.9 8.66 0.910207 1.061985 0.810928 0.280259 0.174995

30.99 7.22 0.727305 0.64757 0.746801 0.232978 0.168703

41.4 9.53 0.832694 0.909329 0.777237 0.230193 0.011955

51.6 11.55 0.893222 1.043421 0.789508 0.223837 0.027612

41.4 9.24 0.802859 0.83996 0.767598 0.223188 0.002899

41.4 8.95 0.773025 0.766095 0.757601 0.216184 0.031385

41.4 8.66 0.743191 0.687283 0.747217 0.209179 0.032402

62.1 12.7 0.874948 1.009259 0.779852 0.204509 0.022325

41.4 6.64 0.535379 −0.05266 0.661002 0.160386 0.215748

41.4 4.33 0.297733 −1.74502 0.508681 0.104589 0.347892

Triaxial

extension

15.6 4.33 0.585562 0.262496 0.689397 0.277564 0

21.3 4.62 0.519176 −0.02698 0.649117 0.216901 0.218554

31.8 5.48 0.4995 −0.18497 0.640564 0.172327 0.205505

42.6 7.22 0.583381 0.153187 0.683466 0.169484 0.0165

31.8 5.2 0.464916 −0.36291 0.620505 0.163522 0.035175

53.1 8.37 0.60034 0.205266 0.690271 0.157627 0.03605

63.6 8.95 0.574089 0.0814 0.679273 0.140723 0.107239

0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

R2=0.0385

y=ln(1.9940+1.5824x)

Extension:

Compression

Extension

Fit of extension

Fit of compression

Fit functions

Compression:

y=ln(0.8071+6.1295x)

R2=0.7760

Fig. 6. Relationship between DP (damage potential) and D1based on

RD criterion. Solid squares denote triaxial compression; solid circles

denote triaxial extension.

0.10 0.15 0.20 0.25 0.30 0.35 0.40

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Compression

Extension

Fit of compression

Fit of extension

R2=0.9309

y=ln(-1.5966+16.9519x)

Extension:

Fit functions

Compression:

y=ln(18.8164-34.5023x) R2=0.7760

Fig. 7. Relationship between DP (damage potential) and D2based on

RD criterion. Solid squares denote triaxial compression; solid circles

denote triaxial extension.

J. Eng. Mech., 2023, 149(8): 04023044

section. The coefficients of regressions in the figures are consider-

ably high (average higher than 0.8), so that mathematically they

show a strong relationship between material parameters and dam-

age potential. One can argue that the coefficients should be higher

to verify the relationship; however, the physical concept of these

parameters (D1,D2, and n) has been less studied, so it is highly

likely that other parameters affect the material parameters as well.

Thus, it can be correlated in higher coefficient than those presented

in the figures when the physical concepts of material parameters

and other effective parameters are considered in the relationship.

Effect of Stress Invariants on Damage Potential

In general, based on a continuum damage approach, the damage

potential is a function of ﬃﬃﬃﬃﬃ

p=I1. As the second deviatoric stress

invariant J2decreases, the damage potential decreases as well.

Their values are presented in bold in Table 2. In addition, from a

micromechanical point of view, the damage process is a weakening

process (Kachanov 1958) through which the material loses load-

bearing capacity. Theoretically, it can be determined the magnitude

of the losing once the damage level quantified in term of numbers.

To describe the damage state quantitatively, it should be traced the

kinematics of the deformation during the losing to find out and

quantify the effects of the various process involved. For example,

De Vries et al. (2002) used the dilatant volumetric strain to measure

the damage level during the deformation. On the other hand, from

continuum mechanics, the volumetric strain can be expressed in the

form of stress invariants. Therefore, both volumetric strain and

damage potential can be quantified by stress invariants. Then the

damage potential can be defined as a function of the stress state

by means of stress invariants (I1and J2). In addition, having the

damage potential in stress terms would make it easy to embed the

stress path and history in the dilation boundary, because the dila-

tion boundary is presented in terms of the stress invariant as well.

In other words, by defining the stress path and history in terms of

the stress invariant, they can be embedded into the dilatancy

boundary.

Effect of Stress Path and History on Damage Potential

Habibi (2016), based on numerical modeling of the Nasrabad salt

cavern using LOCAS (Brouard Consulting 2014), concluded that

the cavern volume decreased during the operational period highly

likely in the form of a mathematical series. Generally, a cavern ex-

periences volume loss during its life span; however, the volume loss

rate can be countered or delayed by increasing the internal pressure

during injection. The cavern shrinkage is related to cavern’s initial

volume and depth, the operational pressure, and the thermome-

chanical properties of rock salt. Fig. 9shows the volume variation

and volume loss rate in the Nasrabad salt cavern, in which during

each injection and withdrawal (unloading and loading), the cavern

experiences volume variation so that the overinjection volume loss

is minimum due to the increase of internal pressure. During each

injection, the volume variation increases from about −8;000 m3to

almost −2;000 m3. Although the cavern experiences lower volume

decreases during injection relative to the previous production stage,

the overall volume loss rate is negative, meaning that the conver-

gence takes place throughout the life span of the cavern. Therefore,

some portion of the convergence is offset by loading (injection);

however, a considerable portion remains. In other words, volume

of the cavern decreases over the life span due to the almost plastic

response of the surrounding salt rock. The rate of injection and pro-

duction are a kind of quasi-static unloading and loading from a salt

cavern technology point of view, under which most materials have

an elastic response (Bérest et al. 2006), although salt rock, due to its

plastic nature, has a plastic response.

0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90 Compression

Extension

Fit of compression

Fit of extension

R2=0.7984

y=ln(1.4657+2.9233x)

Extension:

Fit function

Compression:

y=ln(1.8752+0.5841x) R2=0.1087

Fig. 8. Relationship between damage potential and n based on RD

criterion. Solid squares denote triaxial compression; and solid circles

denote triaxial extension.

Volume variation/ m

-2000

-4000

-6000

-8000

-10000

0 500 1000 1500 2000 2500 3000 3500

-1

-2

Volumeloss rate/ %/year

Time/ days

End of leaching

Start of debrining

End of debrining

1234

5678910 11 12

131415 16

171819 20 2122 232425 26 27282930 3132

33 34 35

37 383940

4142 43

Fig. 9. Volume loss and volume loss rate of an ellipsoid-shape cavern at 465 m depth in seasonal operation. (Adapted from Habibi 2016.)

J. Eng. Mech., 2023, 149(8): 04023044

Chan et al. (1996) used the modified multimechanism deforma-

tion coupled fracture (MDCF) model to determine the damage val-

ues of the rock mass around a cavern when the minimum and

maximum pressure were applied to the cavern. They tried to assign

a fixed value to evaluate the damage over all of the lifespan of the

operation. However, the damage changes during every cycle of

operation, resulting in salt strength changes.

In Fig. 10, when rock salt experiences Path ABCD, it first

dilates (i.e., damage increases), and subsequently (during the next

cycle of loading and/or unloading) it heals. The ability to heal

depends on the damage that occurred during the previous stress

state. For example, salt experiences dilatancy in the first cycle

because of withdrawal. In the next cycle, the deviatoric stress

decreases because of injection, and the damage that occurred in

the previous cycle possibly can be healed. Therefore, the general

dilatancy boundary (the relationship between I1and J2) cannot be

applied as the dilatancy boundary in the subsequent cycles. This is

because salt experiences a different stress state, which causes

strength changes, and the changed strength has been saved in its

memory. Under such a condition, the uniaxial compressive strength

decreases. Such changes in strength can be embedded in the dila-

tion response by taking them into account in the dilation criterion.

The strength weakness in I1−ﬃﬃﬃﬃﬃ

pstress space can be taken into

account as a weakening parameter. Considering the reverse path,

when rock salt experiences the DCBA path in Fig. 10, the converse

behavior can be expected (i.e., the salt is damaged at the end of

cycles), which can be embedded in the criterion by an amplifier

parameter, similar to the weakening parameter.

Representative Parameters

As mentioned previously, the RD parameters (i.e., D1,D2, and n)

are not constant during stress state changes. Changing I1and J2

results in damage potential ( ﬃﬃﬃﬃﬃ

p=I1) changes (Figs. 6–8). Based on

the previous section, it is concluded that damage and healing affect

the instantaneous strength (the dilatancy strength of rock salt during

loading and unloading), i.e., damage decreases the strength, and

healing amplifies the strength. It can be concluded that the param-

eters of RD (i.e., D1,D2, and n) depend on the stress state and the

instantaneous strength, which in turn depends on the damage or

healing in every cycle of loading and unloading.

Because the damage or healing function is time-dependent,

a time-dependent dilatancy criterion must be defined including

stress path and history, especially for the triaxial compression.

To consider a weakening (or an amplifier) parameter in the RD cri-

terion, it is necessary to define a healing or damage representative

parameter in RD which is related to I1and J2. First, the effect of

healing or damage on instantaneous strength of rock salt must be

quantified. It is accepted that the damage or healing differences dur-

ing two cycles is interconnected nonlinearly to the strength of rock

salt (Bérest and Brouard 1998). Damage increases at the minimum

operating pressure and it decreases with an increase of the operat-

ing pressure due to increasing internal pressure. When the oper-

ating pressure reaches its maximum, healing of the rock salt takes

place because there is enough time (from end of the injection to

beginning of the withdrawal) during which there is almost hydro-

static pressure around the cavern owing to the low deviatoric stress.

However, the healing cannot offset all the damage that occurred in

the minimum-pressure state (Habibi et al. 2021).

A parameter (representative parameter) is defined that deter-

mines the damage (or healing) based on the first invariant of the

stress tensor (I1) and the second invariant of the deviatoric stress

tensor (J2) in every cycle of loading (because the ratio ﬃﬃﬃﬃﬃ

p=I1is

defined as the damage potential). Eq. (6) determines the value of

damage (weakness) or healing (strengthening) with respect to the

previous cycle. To consider the effect of the RP, the relationship

between RD constants (D1,D2and n) and the representative

parameter must be determined (Figs. 11–13). Using the measure-

ments taken by De Vries et al. (2005) in triaxial compression and

triaxial extension, the relationships between the RP and D1,D2,

and nfor Cayuta rock salt were determined (Table 3). The RP in-

creases linearly with the increase of D1,D2, and nin triaxial com-

pression; in other words, these parameters are RP-dependent in the

compression state, and experience a change of value under different

loading paths. Since, the correlation coefficient is low in the triaxial

extension, meaning that D1,D2, and nare RP-independent. Con-

sidering the independence of these parameters, it can be concluded

that the parameters are independent of stress path and history,

which suggests that the RD is suitable only for the triaxial exten-

sion state. By taking into account the relationships, the values of

0 -10 -20 -30 -40 -50 -60 -70

RD criterion

DP criterion

Compression

Extension

0.18

J21/2/MPa

I1/MPa

Compression

Extension

Fig. 10. Illustration of stress path based on stress states. (Adapted from

De Vries et al. 2005.)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

R2=0.6704

Extension

Compression

Linear fit of extension

Linear fit of compression

y=-0.3573x+0.8586

R2=0.2015

y=-1.3356x+0.8816

Fig. 11. Relationship between D1and RP in triaxial compression

(solid squares) and triaxial extension (solid circles) states.

J. Eng. Mech., 2023, 149(8): 04023044

D1,D2, and nare calculated for every cycle, and then the

updated values of D1,D2, and nare substituted in the place of the

previous ones. The updated D1,D2, and nare applied in RD for

the next cycle, resulting in a new dilatancy boundary, by which the

third cycle can be analyzed using the updated values of D1,D2,

and ncalculated in the second cycle (previous cycle)

RP ¼−currentðDPÞ−priorðDPÞ

priorðDPÞð6Þ

where RP = representative parameter; and current (DP) and prior

(DP) = current value and prior value, respectively, of damage

potential ( ﬃﬃﬃﬃﬃ

p=I1) in every cycle. The value of RP shows the rel-

ative degree of damage or healing compared with the previous state.

The sign of the RP indicates whether rock salt experiences healing

or damage in the current state compared with the previous state

(Table 3). When the RP is positive (regardless of the value), the

rock salt experiences healing in the current state with respect to

previous state, and vice versa (in other words, the damage potential

of the current state becomes lower than the damage potential of the

previous state). From a mathematical point of view, the regression

coefficients in Figs. 11–13 and Table 3indicate a considerable

relationship between the RP and the material parameters. One can

argue that the coefficients are not high enough; however, a simple

linear relationship was considered to define the dependency of

the RP and material parameters. As mentioned in the section

“Modifying the Criterion to Include Stress Path and Stress History,”

the physical concept of the material parameters is not fully under-

stood, and it is highly likely that a more-or-less nonlinear equation

governs the dependency rather than a linear relationship. Therefore,

considering these kind of drawbacks, uncertainties could be ex-

pected in determination.

Because RD constants are changed by the stress state, an

updated dilatancy boundary must be determined in every step

(cycle). Because the updating the dilation boundary needs an ini-

tial dilation boundary, generally the boundary determined from a

mean stress test should be considered as the initial dilation boun-

dary. Therefore, in the first step (first cycle of loading or unload-

ing), the damage potential is set to zero because there is no stress

state change. In this paper, the dilatancy boundary determined

from the measurements in De Vries et al. (2005) (called RD cri-

terion) was used for modification. Therefore, the overall dilatancy

boundary proposed by De Vries et al. (2005) with proposed values

for material constants of RD (i.e., D1,D2, and n) was considered

as the initial dilatancy boundary. To determine the updated boun-

daries in the next steps, the RP resulting from the DP of the pre-

vious steps must be calculated, and it continues in the same way.

Based on the results of De Vries et al. (2005), the updated dilat-

ancy boundaries were plotted for six cycles (six different stress

states) (Fig. 14). Various values of I1and J2resulted from stress

state changes, meaning that every stage produces a certain damage

potential. Having considered various values of damage potential,

the RP changes in turn, and consequently the dilatancy boundary

changes as well. Because during the fifth and sixth cycles the

damage change was low, a very small value was obtained for RP,

and the dilatancy boundary of these cycles was more or less the

same (Fig. 14). However, the updated dilatancy boundary (Fig. 14,

second cycle) moved considerably down in the second cycle rel-

ative to the dilatancy boundary of the first cycle (Fig. 14, first

cycle). This is because of the large RP change. Therefore, the new

stability criterion moves down and up as the stress state changes

to set a new dilatancy boundary. The effects of the stress path and

history are included in development of the new dilatancy boun-

dary in the form of the RP.

As discussed previously, the dilation boundary splits the ﬃﬃﬃﬃﬃ

p=I1

space into a dilatant zone (above the dilation envelope) and a non-

dilatant zone (below the dilation envelope). The area of the zones

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Extension

Compression

Linear fit of extension

Linear fit of compression

=0.6657

y=-6.033x+1.1351

y=-0.3387x+2.541

=0.030

y=-0.3387x+2.541

R2=0.030

y=-6.033x+1.1351

R2=0.6657

Fig. 12. Relationship between D2and RP in triaxial compression

(solid squares) and triaxial extension (solid circles) states.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

Compression

Extension

Linear fit of compression

Linear fit of extension

y=-0.6350x+0.8035

R2=0.6311

y=-0.1323x+0.6965

=0.0361

y=-0.6350x +0.8035

=0.6311

Fig. 13. Relationship between n and RP in triaxial compression (solid

squares) and triaxial extension (solid circles) states.

Table 3. Relationship among D1,D2,n, and RP in triaxial compression and triaxial extension based on measurements performed by De Vries et al. (2005)

Triaxial extension Triaxial compression

Relationship Correlation coefficient Relationship Correlation coefficient

D1¼−0.3573ðRPÞþ0.8586 R2¼0.2015 D1¼−1.3356ðRPÞþ0.8816 R2¼0.6704

D2¼−0.3387ðRPÞþ2.541 R2¼0.03 D2¼−6.0332ðRPÞþ1.1351 R2¼0.6657

n¼−0.1323ðRPÞþ0.6965 R2¼0.0361 n¼−0.6350ðRPÞþ0.8035 R2¼0.6311

J. Eng. Mech., 2023, 149(8): 04023044

is controlled by the slope and intercept of the envelope equation

(RD criterion), which in turn is controlled by the material constants

(i.e., D1,D2, and n). More area below the envelope means that rock

salt will dilate at a higher stress (in other words, it has greater

strength). During updating of the dilation boundary using the pro-

posed method, the boundary moves up and down, so the area below

the boundary (the nondilatant zone) changes in every step. As an

example, Cycle 4 in Fig. 14 moved up considerably with respect to

Cycle 3, providing more area below the boundary. Such a change

shows how much the rock salt healed between the cycles, and

therefore indicates an increase in strength taken into account by

proposed method. Without such a consideration, the healing and

strength increase could be hidden in the determination of the dila-

tion boundary, meaning that the dilation prediction very likely

would be misinterpreted. Therefore, the approach presents a pro-

cedure to prevent such misinterpretation, and increases the ability

to predict the dilatancy behavior of the salt rock.

Conclusions

The following conclusions were reached in this study:

1. The RD criterion proposed by De Vries et al. (2005) considers

the stress state and the effect of Lode angle, and has a relatively

high accuracy in predicting the damage of rock salt. The

material constants of the RD criterion (i.e., D1,D2, and n) vary

in different stress states, especially in triaxial compression.

However, the effect of this variability of the parameters is not

considered in the RD criterion.

2. This study modified the RD stability criterion using damage po-

tential and a representative parameter. The modified RD (MRD)

considers stress path and stress history using a representative

parameter (RP) which is determined by damage potential re-

sulting from current and prior loading or unloading steps. The

MRD proposes an updated dilatancy boundary at every step of

loading, which includes the stress path and history of previous

loading steps.

3. Using the MRD, the damage or/and healing of rock salt that has

occurred in previous steps is considered as an updated dilatancy

boundary. Based on the analyses, the MRD criterion improves

the predictive ability of dilation around salt caverns. The MRD

criterion also provides an improved method for evaluating

cavern design and avoiding dilatant states of stress that would

be detrimental to the long-term stability of the cavern.

4. The correlation performed to determine the relationship between

the material parameters (i.e., D1,D2, and n) and the RP may

carry uncertainty. This is because (1) physical concepts of the

material parameters are not fully understood; and (2) in this

paper, a linear equation was used to correlate the material

parameters with the RP, whereas it is highly likely that a non-

linear equation could govern the relationship.

Data Availability Statement

Some or all data, models, or code that support the findings of this

study are available from the corresponding author upon reasonable

request. In addition to details of the tests performed by De Vries

et al. (2005), all the parameter calculations of the DP and RP are

available. Data analysis, such as the D1,D2, and ncorrelations

against the DP and RP calculations are available. Additionally, the

calculation of the new updatable dilation boundary using D1,D2,n,

DP, and RP is available.

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Conference Paper

Dec 2016

Modeling of hydro-mechanical behavior of rock salt in the near field of repository excavations

Chapter

Dec 2017

Alexandra Pudewills

The CDM constitutive model for the mechanical behavior of rock salt: Recent developments and extensions

Chapter

Jan 2012

A. Hampel

The Composite Dilatancy Model (CDM) describes the deformation of rock salt based on the dominant physical deformation processes and micro structural features. Transient and steadystate creep is modeled as a result of the movement of dislocations and their interactions with each other and with the microstructure. At stresses above the dilatancy boundary, the CDM models the increasing evolution of damage and dilatancy and their influence on the total deformation in dependence of the creep process. In dilatant rock salt, also the humidity influence is taken into account. Recently, the CDM has been extended to calculate the creep failure as a result of increasing damage and dilatancy using an energy criterion. After failure, the approach to residual strength is modeled. The CDM has been checked and compared with other models in three joint projects on the comparison of constitutive models for the mechanical behavior of rock salt (see related contributions in this volume). In this contribution, the current CDM version with a focus on the latest developments and extensions is presented. Some results of 3-D calculations of a mine section will be shown to demonstrate the capabilities of the CDM and its applicability to simulations of underground openings in rock salt.

Geomechanical investigation of roof failure of China's first gas storage salt cavern

Article

Jun 2018
ENG GEOL

JK-A cavern is the first operating cavern of Jintan underground gas storage, which is also the first operating salt cavern gas storage in China. In 2015, the sonar survey carried out under working conditions indicated that a massive roof collapse of the JK-A cavern had taken place. To prevent similar accidents from happening again, finding the causes and the collapse time of JK-A cavern would be valuable. By using the target salt formation information, rock properties, and monitored gas pressure, a 3D geomechanical model of JK-A cavern has been built. The responses of the rock mass around JK-A cavern under leaching, sealing tests, debrining, and gas injection/delivery have been investigated. A safety evaluation system consisting of cavern volume shrinkage, dilatancy safety factor, displacement, vertical stress, and equivalent strain is proposed to find the likely collapse time and the reason for the JK-A cavern roof collapse. Results show that there are two main causes for the cavern roof failure. (1) A large-span flat roof, detrimental for bearing loads. Once local damage takes place, massive collapse may be triggered by the self-weight of the loosened rock. (2) The decrease speed of the internal gas pressure is too fast. The loads applied to the cavern roof cannot be transferred in a timely fashion, which causes a stress concentration zone to form, and local damage develops. The roof collapse of JK-A cavern took place 1.3 years after its gas injection/delivery started. Numerical analysis results show that dilatancy safety factor and vertical stress have high accuracy and sensitivity in the prediction of the cavern roof collapse. There is a high potential for a roof collapse of JK-A cavern to happen again. We propose that the gas pressure and pressure decrease rate are to be controlled strictly in the later operating period to prevent any collapses in the future.

An Improved Dilatancy Boundary for Salt Rock

Figures

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