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Title: A new shape design method of salt cavern used as underground gas storage
Author: Tongtao Wang

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Applied Energy 104 (2013) 50–61

Contents lists available at SciVerse ScienceDirect

Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

A new shape design method of salt cavern used as underground gas storage
Tongtao Wang a,b, Xiangzhen Yan a,b,c,⇑, Henglin Yang d, Xiujuan Yang b,c, Tingting Jiang b, Shuai Zhao b
a

State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Qingdao 266580, Shandong, China
College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, Shandong, China
c
Key Laboratory of CNPC Underground Storage Drilling and Production Engineering, China University of Petroleum, Qingdao 266580, Shandong, China
d
CNPC Drilling Research Institute, Beijing 100097, China
b

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

" We propose a new model to design

Safety factor contours of four salt cavern gas storages after running 10 years.
-900

-900

-950

-950

-1000

-1000
Depth / m

Depth / m

the shape of salt cavern gas storage.
" The concepts of slope instability and
pressure arch are introduced into the
shape design.
" The max. gas pressure determines
the shapes and dimensions of cavern
lower structure.
" The min. gas pressure decides the
shapes and dimensions of cavern
upper structure.

-1050

-1150

-1150

-1200
-120

-1200
-120

-80

-40

0
X/m

40

80

120

(a) Enlarged bottom cavern

-950

-1000
Depth / m

Depth / m

-950

-1050

-1100

Keywords:
Energy storage
Underground salt cavern gas storage
Shape design
Slope instability
Pressure arch
Numerical simulation

40

80

120

-1050

-1150

-80

-40

0
X/m

40

80

(c) Enlarged top cavern

Article history:
Received 30 July 2012
Received in revised form 16 October 2012
Accepted 17 November 2012
Available online 14 December 2012

0
X/m

-1100

-1150

i n f o

-40

-900

-1000

-1200
-120

-80

(b) Enlarged middle cavern

-900

a r t i c l e

-1050

-1100

-1100

120

-1200
-120

-80

-40

0
X/m

40

80

120

(d) Proposed cavern

a b s t r a c t
A new model used to design the shape and dimension of salt cavern gas storage is proposed in the paper.
In the new model, the cavern is divided into two parts, namely the lower and upper structures, to design.
The concepts of slope instability and pressure arch are introduced into the shape design of the lower and
upper structures respectively. Calculating models are established according to the concepts. Field salt
cavern gas storage in China is simulated as examples, and its shape and dimension are proposed. The
effects of gas pressure, friction angle and cohesion of rock salt on the cavern stability are discussed. Moreover, the volume convergence, displacement, plastic volume rate, safety factor, and effective strain are
compared with that of three other existing shapes salt caverns to validate the performance of newly proposed cavern. The results show that the max. gas pressure determines the shape and dimension of cavern
lower structure, while the min. gas pressure decides that of cavern upper structure. With the increase of
friction angle and cohesion of rock salt, the stability of salt cavern is increased. The newly proposed salt
cavern gas storage has more notable advantages than the existing shapes of salt cavern in volume convergence, displacement, plastic volume rate, safety factor, and effective strain under the same conditions.
Ó 2012 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, Shandong, China. Tel.: +86 532 86981231; fax: +86
532 86983097.
E-mail address: yanxzh@163.com (X. Yan).
0306-2619/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.apenergy.2012.11.037

51

T. Wang et al. / Applied Energy 104 (2013) 50–61

other significantly, indicating the shapes and dimensions design
theories and methods of salt cavern gas storages have not formed
a unified view in the academic and engineering fields.
Many scholars studied the problems in relative fields, e.g., Dennis [20] had built up the shape model of salt cavern gas storage
based on the property parameters of rock salt in Alberta of Canada.
He thought the salt cavern with a shape of ellipsoid was more conducive to keep its stability and to reduce volume shrinkage than
the caverns with other shapes. However, he neither gave the mechanism of cavern with ellipsoid shape nor optimized its dimensions.
Staudtmeister and Rokahr [21] divided the design process of salt
cavern gas storage into four steps and gave the corresponding design criteria. They thought the salt cavern with a shape of slender
cylinder was reasonable in keeping stability. Sobolik and Ehgartner
[22] studied the safety factor, volume shrinkage, displacement, and
ground subsidence of salt cavern gas storages with shapes of cylinder, enlarged top, enlarged middle, and enlarged bottom by
numerical simulations respectively. They pointed out the salt cavern with a shape of enlarged bottom performing the worst of them.
Wang et al. [23] studied the max. displacement, plastic volume,
pillar width of salt cavern gas storages with shapes of ellipsoid
and pear-like under the long-term running conditions respectively.
They considered the ellipsoidal cavern performed much better
than pear-like cavern. Meanwhile, the ratio of long and short axes
of ellipsoidal cavern was proposed as 7/3. Heusermann [24] discussed the design process of salt cavern gas storage by numerical
simulations. Moreover, he studied the equivalent stresses of salt
caverns under different gas pressures based on LUBBY2 rock salt
creep law, and pointed out the cavern stability should be checked
both under gas pressures and long-term creep loads. Cristescu and
Paraschiv [25] simplified the design of underground structure in
continuous rock masses as a plane strain problem, and discussed
the effects of corner radius and ratio of length/width on the stability of rectangular-like cavern. The optimized dimensions of rectangular cavern were proposed.
As described above, the shape and dimension design of salt
cavern gas storage was still a difficult problem, and a unified

1. Introduction
Underground salt cavern is recognized as one of the best places
to storage the power sources, such as oil, gas, and compressed air
[1–6]. Main reasons are that rock salt comparing to other rock
materials has four aspects of advantages [7–12]. (i) Low permeability. The permeability of rock salt is about 10 21–10 24 m2, which
can ensure the excellent sealing of salt cavern. (ii) Good mechanical properties. Damage self-recovery capability of rock salt can ensure the safety of salt cavern with frequent changes of gas
pressure. (iii) Solution in water. Rock salt is easily dissolved into
water, which facilitates the construction and shape control of salt
cavern. (iv) Abundant resources. Rock salt resource is a very rich
mineral resource with wide distributions and large reserves.
Therefore the site used to construct salt cavern gas storage near
the oil and gas consumption area can be found conveniently. However, volume shrinkage, excessive displacement, ground subsidence, and even collapse of salt cavern gas storage, etc., have
become challengeable problems to the engineers for the typical
creep and rheology of salt rock. For example, the Tersanne and
Eminence salt cavern gas storages lost about 35% of its volume in
10 years, and 40% in just 2 years, respectively [13], which significantly reduced their peak capacities and availabilities. Meanwhile,
several pillars of salt cavern in Hengelo area took place failures
[14–19], causing excessive ground subsidences and collapses. It became one of the most intimidatory factors to the safety of salt caverns in that area.
Field engineering experiences and available literatures [20–28]
show that rational design of the shape and dimension of salt cavern
gas storage can effectively reduce negative effects and improve
safety. However, how to design the shape and dimension of salt
cavern is a difficult engineering problem because it is greatly influenced by many factors, such as the strata characteristics, strengths
of rock salts, rock salt creep characteristics, running parameters,
and failure criteria. Fig. 1 shows the shapes and dimensions of typical salt cavern gas storages in the world [13]. It is found that the
shapes and dimensions of salt caverns are different from each

0

Salies de Bean

500m

Manosue
Huntorf

Depth / m

West Hackbury

1500m

Melville
Tersanne

Kiel
Hauterives

2000m

Regina South Cavern 5

2500m

Eminence

Jan. 1970
Fig. 1. Shapes of typical salt cavern gas storages.

June. 1972

52

T. Wang et al. / Applied Energy 104 (2013) 50–61

Fig. 2. Schematic diagram of the cavern lower structure based on the concept of slope instability.

dx

(a)

(b)

Fig. 3. Shape calculating model of cavern lower structure.

design method was not formed. Therefore, a systematical study
on this problem is very necessary. The main motive of the paper
is to propose a new model with clear mechanism and criterion
to obtain the shape and dimension of salt cavern gas storage.
In the new model, slope instability and pressure arch concepts
are introduced. Based on the new model, field salt cavern gas
storage in China is simulated as an example, and its shape and
dimension are proposed accordingly. Moreover, volume convergence, displacement, plastic volume, safety factor, and effective
strain of the newly proposed salt cavern are validated by
comparing with that of salt caverns with existing shapes. The
results of the paper can afford some references to the shape

and dimension designs of salt cavern gas storage and other
underground spaces.
2. Mathematic model
In the design processes of salt cavern gas storage, three assumptions are used, described as follows:
(i) Thickness of rock salt formation is assumed to satisfy the
demands of actual engineering. Meanwhile, its width and
length are recognized as infinite comparing to the salt cavern diameter.

53

T. Wang et al. / Applied Energy 104 (2013) 50–61

(ii) Rock salt is assumed as an isotropic visco-elastic material
[1,2,7], because it is composed by relatively homogeneous
materials (mainly NaCl).
(iii) Salt cavern gas storage is considered as an axisymmetric
structure, which can be obtained by the circumrotation of
a cross-section.
Due to the failure mechanisms of cavern upper and lower structures are different from each other greatly, the mathematic models
of the two parts are built up separately in the sections.

where K is the SF of slope; MR is the anti-sliding moment, N m; MT is
the sliding moment, N m; L is the total length of the critical sliding
surface, m; Gi is the weight of the ith rock mass column, including
the component of gas pressure along the vertical direction, N; ai
is the dip of the ith rock mass column, degree; / is the friction angle
of rock salt, degree; c is the cohesion of rock salt, Pa.
Without loss of generality, it is hypothesized the profile of the
slope as a straight line with a equation of y = mx + n, then the formula of critical sliding surface can be written as

ys ¼ y0

2.1. Cavern lower structure

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2 ðx x0 Þ2

ð2Þ

A flat disc with a width of dx is calculated as
After the construction of salt cavern, the loads subjected to
rock salt at the original cavern location transfer to the surrounding rock masses, which causes stress concentration and may induce the instability of salt cavern. In order to reduce the
possibility of instability, cavern lower structure has to be designed into a slope to release the excessive stresses. Therefore,
the shape design of cavern lower structure is changed into a slope
instability problem.
Stability factor (SF) is usually used to evaluate whether the
slope occurs instability [29]. If SF > 1, the slope is stable; whereas
if SF 6 1, the slope may take place instability, and a critical sliding
surface will exist. Thus rock masses above the critical sliding surface are in instability state, while that below it are in stability state.
Therefore here the shape of cavern lower structure could be designed into the shape of the critical sliding surface to avoid instability. Fig. 2 presents the calculating schematic diagram to obtain
the shape and dimension of cavern lower structure according to
the concept of slope instability.
Fig. 3 shows the mechanical model of cavern lower structure
based on the concept of slope instability (Fig. 2), in which the
Cartesian coordinate system locates at the bottom center of
cavern lower structure. Right of horizontal and vertical
directions are defined as the positive directions of x-axis and
y-axis respectively.
For an isotropic rock salt slope, its SF can be expressed as [30–33]



P
MR tan / ni¼1 Gi cos ai þ cL
¼ Pn
MT
i¼1 Gi sin ai þ cL

ð1Þ

dGix ¼ ½cðyi ys Þ þ p cos ai dx

ð3Þ

Substituting Eqs. (2) and (3) into Eq. (1), SF of the slope can be
written as

P R xi
tan / ni¼1 xi 1
½rðyi ys Þ þ p cos a cos adx þ cL

Pn R xi
i¼1 xi 1 ½rðyi ys Þ þ p cos a sin adx
R tan / þ cL=c
T

¼

ð4Þ

where



Z

a




qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
mx þ n y0 þ r 2 ðx x0 Þ2 þ

c

a d

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2 ðx x0 Þ2
r

dx
ð5Þ





qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p x x0
mx þ n y0 þ r 2 ðx x0 Þ2 þ
dx
c r
a d

Z

a

ð6Þ

where t is the half length of bottom horizontal part, m; x0 and y0 are
the center coordinates of the critical sliding surface, m; r is the radius of critical sliding surface, m; p is the gas pressure, Pa; a is the
angle between the critical sliding surface and horizontal plane, degree; yi and ys are the upper and lower y-coordinates of the infinitesimal dx respectively, m; c is the rock salt weight, N/m3; a is
the half span of salt cavern, m.
According to the geometric relations in Fig. 3, the center coordinates of critical sliding surface are
2

x0 ¼

z

y0 ¼ h þ

σz = γ ⋅z
G

x

Rx

H
p

a

T
E
N

ð8Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx0 xc Þ2 þ y20

ð9Þ

where xc = a d.
The length of critical sliding surface is

F

a

ðx0 aÞða tÞ
h

ð7Þ

where a d < t < a.
The radius of critical sliding surface is



S

ða tÞ2 =2 þ h =2 ða tÞa at þ a2
a t d

y

Fig. 4. Shape calculating model of cavern upper structure.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
d þh
L ¼ 2r arcsin
2r

ð10Þ

Based on Eqs. (5)–(10), the function of SF(K) expressed by t can
be achieved. When a d < t < a, the min. value of K can be obtained
by search method [30,33]. Then, the formula of critical sliding surface and the min. SF of slope are achieved. Ultimately, all the
parameters of the shape and dimension of cavern lower structure
can be obtained.

54

T. Wang et al. / Applied Energy 104 (2013) 50–61

Table 1
Physical and mechanical property parameters of strata.
Formation type

Density
(kg/m3)

Young’s
modulus
(GPa)

Poisson
ratio

Cohesion
(MPa)

Friction angle
(degree)

Tensile
strength
(MPa)

Uniaxial compressive
limit strength (MPa)

Mudstone
Rock salt

2800
2200

10
18

0.27
0.3

1.0
1.0

35
45

1
1

/
14.72

2.2. Cavern upper structure

3. Application

As the increase of cavern volume by leaching, overlying strata
loses the support afforded by rock salt at the location of cavern,
which may cause excessive displacement and collapse of the upper
strata. However, the collapse of overlying strata will stop when it
reaches a certain extent. Then, a new self-stable structure is
formed, viz., pressure arch structure. Available engineering experiences and theories [34–37] show that the pressure arch structure
has a good self-stability characteristic. Therefore here the cavern
upper structure is designed with a shape of pressure arch to improve its stability. In order to obtain the dimensions of pressure
arch structure, the coordinate system is established (see Fig. 4).
Arc of EFG is taken as the study objective, and the moment subjected to any location of the arc is written as

3.1. Calculating parameters

x
MðxÞ ¼ Rx f ðxÞ rz x þ p
2

Z
0

x

x þ f ðxÞ f 0 ðxÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx
1 þ f 0 ðxÞ2

ð11Þ
3.2. Shape design of cavern lower structure

where Rx is the reaction force, N; rz is the vertical in situ stress, Pa;
f(x) is the shape function of pressure arch structure.
According to the pressure arch theory [38], horizontal thrust
force (T) subjected to the foot of pressure arch (point E) has to satisfy Eq. (12) in the critical stability state, expressed as

T ¼ N fk ¼ rz a fk

ð12Þ

where fk is the max. internal friction coefficient of rock salt, and its
value is Rc/10 [38]; Rc is the uniaxial compressive limit strength of
rock salt, MPa; N is the overburden pressure, and its value is rza,
N. According to Eq. (12), the loads subjected to the rock salt locating
at the foot of pressure arch structure is proportional to the arch
span, indicating a big span means a high possibility of rock salt with
failure.
As shown in Fig. 4, gas pressure can balance part of the overburden pressure, which is beneficial to the stability of cavern upper
structure. Taking into account the possibility of losing pressure
or suction in actual operations, gas pressure is assigned as 0 in
the design for the safety consideration. Then, the following equations of pressure arch in critical state can be obtained according
to the static equilibrium conditions, written as

MðxÞ ¼ Rx f ðxÞ rz x
X
X

x
¼0
2

Fy ¼ 0

ð15Þ

According to Eqs. (13)–(15), we can obtain



a
fk



a
0:4h
¼
¼ 0:272h
1:472 1:472

ð18Þ

Due to the constraint of the rock salt formation thickness, the
total design height of salt cavern gas storage should be less than
120 m, then we can obtain

h þ H 6 120
ð14Þ

x2
a fk

In order to obtain the shape and dimension of cavern lower
structure, the ratio of half span and slope height (a/h) needs to
be determined firstly. Based on the mechanical model in Fig. 3,
relations between the SF of slope and a/h obtained by Eqs. (1)–
(10) are shown in Fig. 5 when gas pressures are assigned as 0, 5,
8, 11, 14, and 17 MPa respectively. It is shown SF is decreased with
the increase of gas pressure, which indicates the max. gas pressure
determines the shape and dimension of cavern lower structure.
Therefore gas pressure with a value of 17 MPa is taken as the primary standard design condition in the example. Meanwhile, SF is
decreased firstly and then increased with increasing a/h, which
reaches the minimum at a/h = 0.4.
According to Table 1, the max. uniaxial compressive limit
strength of rock salt is 14.72 MPa, thus fk = Rc/10 = 1.472. Based
on Eq. (17), the height of cavern upper structure can be obtained as

ð13Þ

Fx ¼ 0

f ðxÞ ¼

In order to verify the proposed model, field salt cavern gas storage in China is simulated as an example. The target rock salt formation where the salt cavern gas storage locates is with depth
arranging from about 1000 to 1140 m. Its average thickness is
about 140 m. The dip angle of rock salt formation along the horizontal direction is almost 0. The max. and min. design gas pressures are 17 MPa and 5 MPa, respectively. In order to satisfy the
sealing requirements, the thicknesses of rock salts above and
below salt cavern are no less than 10 m respectively. Therefore,
the whole height of the proposed salt cavern is controlled at less
than 120 m. Physical and mechanical property parameters of rock
salt obtained by experiments are shown in Table 1.

ð16Þ

ð17Þ

When the formula of pressure arch is achieved, all dimensions
of the cavern upper structure can be gotten.

ð19Þ

According to Eqs. (18) and (19), we can get the slope height
h 6 94.34 m. Taking the sealing safety into consideration, the slope
height is valued as 90 m. Then, the half roof span (a) and height of
cavern upper structure (H) are obtained as 36 m and 24.48 m
respectively.
Fig. 6 shows the effects of slope bottom width (d) on the SF of
slope when p = 0, 5, 11, 17 MPa, a = 36 m, h = 90 m respectively.
It is presented that SF reaches its minimum when gas pressure
and slope bottom width are with values of 17 MPa and 32 m
respectively. Then, the half length of bottom horizontal part is calculated as t = a d = 4 m. Although the height and bottom width of
slope are given, there are still many possible sliding surfaces. In
this situation, the sliding surface with the min. SF is the design

55

T. Wang et al. / Applied Energy 104 (2013) 50–61
3.2

1.30

0 MPa
5 MPa
8 MPa
11 MPa
14 MPa
17 MPa

2.8

1.20
1.15

SF

SF

2.4

17 MPa
14 MPa
11 MPa
8 MPa
5 MPa

1.25

2.0

1.10
1.05

1.6

1.00
1.2

0.95

0.8
0.0

0.5

1.0

1.5

2.0

0.90
34

2.5

36

38

a/h

40

42

44

46

Friction angle / degree

Fig. 5. Effects of a/h on the SF of slope under different gas pressures.

12

Fig. 8. Effects of friction angle on the SF of slope when a = 36 m, h = 90 m, d = 32 m,
r = 142.6 m and p = 5, 8, 11, 14, 17 MPa.

0 MPa
5 MPa
11 MPa
17 MPa

10

2.0

17 MPa
14 MPa
11 MPa
8 MPa
5 MPa

1.8

8

6

SF

SF

1.6

1.4

4
1.2
2
1.0
0

0

5

10

15

20

25

30

35

40

0.8
0.8

Bottom width / m
Fig. 6. Effects of slope bottom width (d) on the SF of slope when p = 0, 5, 11, 17 MPa,
a = 36 m, h = 90 m.

1.0

1.2

1.4

1.6

1.8

Cohesion / MPa
Fig. 9. Effects of cohesion on the slope SF when a = 36 m, h = 90 m, d = 32 m,
r = 142.6 m and p = 5, 8, 11, 14, 17 MPa.

shape of salt cavern lower structure. Therefore we have to find the
position of the sliding surface with the min. SF.
Radius of sliding surface (r) plays an important role on the SF of
slope, which is discussed here. Keeping a = 36 m, h = 90 m,

0
5

2.0

Height / m

1.8

1.6

SF

0
5MPa
10MPa
15MPa

10
15
20
25

1.4

30
1.2
35
-60

1.0

0.8

-45

-30

-15

0

15

30

45

60

Roof span / m
Fig. 10. Shapes of salt cavern upper structure under different gas pressures.
0

600

1200

1800

2400

3000

Radius / m
Fig. 7. Effects of sliding surface radius on the SF of slope when a = 36 m, h = 90 m,
d = 32 m, and p = 17 MPa.

d = 32 m, and p = 17 MPa, the radius is changed to discuss its effect.
Fig. 7 shows the effects of sliding surface radius on the SF of slope.
It is shown the SF is increased with the increase of the radius. In the

56

T. Wang et al. / Applied Energy 104 (2013) 50–61

24.48m

72m

90m

142.6m

142.6m

8m

32m

(a) Section dimension

(b) 3D shape

Fig. 11. Section dimension and 3D shape of the proposed salt cavern gas storage.

Fig. 12. Salt cavern gas storages with different shapes proposed by Sobolik and Ehgartner [22]. Ratios of the max. and min. diameter are 2 in the three numerical models.

18

Enlarged bottom cavern
Enlarged middle cavern
Enlarged top cavern
Proposed cavern

18
15

Convergence / %

Gas pressure / MPa

16
14
12
10

12

9

6

8
3
6
0
4

0

2

4

6

8

10

0

2

4

6

8

10

Time / year

Time / year
Fig. 13. Gas pressure of underground salt cavern gas storage.

Fig. 14. Relations between volume convergences of four salt cavern gas storages
and time.

57

T. Wang et al. / Applied Energy 104 (2013) 50–61

-900

-900

-950

-950

-1000

Depth / m

Depth / m

-1000

-1050

-1100

-1100

-1150

-1150

-1200

-1200

-120

-80

-40

0

40

80

120

-120

-80

-40

0

40

80

X/m

X/m

(a) Enlarged bottom cavern

(b) Enlarged middle cavern

-900

-900

-950

-950

-1000

-1000

Depth / m

Depth / m

-1050

-1050

120

-1050

-1100

-1100

-1150

-1150

-1200

-1200
-120

-80

-40

0

40

80

120

-120

-80

-40

0

40

80

X/m

X/m

(c) Enlarged top cavern

(d) Proposed cavern

120

Fig. 15. Vertical displacement contours of four salt cavern gas storages after running 10 years.

same time, the min. radius of sliding surface can be obtained with a
value of 142.6 m according to Eqs. (7)–(9) and above parameters.
Therefore, when the radius of sliding surface is with the min. value

Plastic volume / cavern volume

1.6

Enlarged bottom cavern
Enlarged middle cavern
Enlarged top cavern
Proposed cavern

1.2

0.8

3.3. Shape design of cavern upper structure
According to the results of Section 3.2 and Eqs. (16) and (17),
the shape function of cavern upper structure can be gotten
under p = 0 condition, expressed as y = x2/52.99. When p – 0,
the shape function is obtained by Eqs. (11), (12), (14), and
(15), written as

0.4

0.0

of 142.6 m, the SF achieves its minimum, and the most dangerous
sliding surface is determined. According to above calculating results, the shape and dimension parameters of salt cavern lower
structure are obtained.
Figs. 8 and 9 present the effects of friction angle and cohesion of
rock salt on the SF of slope when a = 36 m, h = 90 m, d = 32 m,
r = 142.6 m, p = 5, 8, 11, 14, 17 MPa respectively. As shown in Figs.
8 and 9, the SF has nearly linear relationships with friction angle
and cohesion, indicating high friction angle and cohesion are beneficial to the stability of salt cavern lower structure.

0

2

4

6

8

10

Time / year
Fig. 16. Relations between ratios of plastic volumes/salt cavern volume of four salt
cavern gas storages and time.

0
1 Brz x2
f ðxÞ ¼ @
p
Rx
2

Z
0

1
x

x þ f ðxÞ f 0 ðxÞ C
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxA
1 þ f 0 ðxÞ2

ð20Þ

T. Wang et al. / Applied Energy 104 (2013) 50–61

-900

-900

-950

-950

-1000

-1000

Depth / m

Depth / m

58

-1050

-1100

-1100

-1150

-1150

-1200

-1200
-120

-80

-40

0

40

80

120

-120

-80

-40

0

40

80

X/m

X/m

(a) Enlarged bottom cavern

(b) Enlarged middle cavern

-900

-900

-950

-950

-1000

-1000

Depth / m

Depth / m

-1050

-1050

-1050

-1100

-1100

-1150

-1150

-1200

120

-1200
-120

-80

-40

0

40

80

120

-120

-80

-40

0

40

80

X/m

X/m

(c) Enlarged top cavern

(d) Proposed cavern

120

Fig. 17. Safety factor contours of four salt cavern gas storages after running 10 years.

0
1 Brz a
p
@
Rx
2
2



Z
0

1
a

0

x þ f ðxÞ f ðxÞ C
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxA
1 þ f 0 ðxÞ2

ð21Þ

Due to the nonlinear characteristics of Eqs. (20) and (12), Matlab 6.0 software is used in the paper to obtain the numerical solutions of the shapes of cavern upper structure. The calculating
results are presented in Fig. 10. It is shown small gas pressure
means small roof span, and small loads are subjected to the rock
salt at the pressure arch foots. Therefore, smaller roof span is better
for the roof stability. It indicates the shape and dimension of salt
cavern upper structure are determined by the min. design gas pressure in the design. In the example, the min. design gas pressure
with a value of 0 is taken as the critical condition. Then, the shape
function of cavern upper structure is y = x2/52.99.

3.4. Shape of whole salt cavern
According to the results of Sections 3.2 and 3.3, the shape and
dimension of the proposed salt cavern gas storage are obtained.
The detailed dimensions of salt cavern cross-section along depth
direction are given in Fig. 11a, and the three-dimensional shape

is shown in Fig. 11b. The entire volumes of the proposed salt cavern gas storage are 26,2310 m3.
4. Validations and discussion
In order to validate the performance of the proposed salt cavern
gas storage, numerical simulations of the proposed cavern (Fig. 11)
and the caverns (Fig. 12) used by Sobolik and Ehgartner [22] are
carried out by a commercial finite difference code FLAC3D developed by Itasca company. Then, the calculating data files are imported into the Tecplot software for post-processing. In the
numerical simulations, four salt caverns have equal volumes of
26,2310 m3. The creep law of rock salt is e_ ¼ Arn , and the corresponding parameters are obtained by the creep experiment results,
namely A = 12.0 106 MPa 3.5/a, n = 3.5. Total numerical simulation time is 10 years. Physical and mechanical property parameters
of rock salt and mudstone used in numerical simulations are
shown in Table 1. Due to the dimensions of salt cavern are much
smaller than the depth of strata, parts of overlying strata are simplified as an equivalent overburden pressure subjected to the
upper boundary to improve the calculation efficiency. The equivalent overburden pressure is about 16.1 MPa calculated by the
parameters from Table 1 and depth. The distances between the

59

-900

-900

-950

-950

-1000

-1000

Depth / m

Depth / m

T. Wang et al. / Applied Energy 104 (2013) 50–61

-1050

-1100

-1100

-1150

-1150

-1200

-1200
-120

-80

-40

0

40

80

120

-120

-80

-40

0

40

80

X/m

X/m

(a) Enlarged bottom cavern

(b) Enlarged middle cavern

-900

-900

-950

-950

-1000

-1000

Depth / m

Depth / m

-1050

-1050

-1050

-1100

-1100

-1150

-1150

-1200

120

-1200
-120

-80

-40

0

40

80

120

-120

-80

-40

0

40

80

X/m

X/m

(c) Enlarged top cavern

(d) Proposed cavern

120

Fig. 18. Effective strain contours of four salt cavern gas storages after running 10 years.

boundaries and salt cavern center take five times of the salt cavern
diameter to eliminate the boundary effects. According to the field
geological data, the maximum, minimum horizontal and vertical
in situ stresses are 29.6 MPa, 20.6 MPa, and 27.4 at the depth of
1000 m respectively. The temperature of ground is valued as
20 °C according to the average temperature of salt cavern location.
The geothermal gradient is 2.8 °C per 100 m. Then, we can calculate the corresponding temperatures at different boundaries of
the numerical models. The following boundary conditions are
applied: the bottom has zero displacement boundaries; both sides
have horizontal zero displacement boundaries in the vertical
direction. Gas pressure is simulated as cycle changes according to
the actual operations as shown in Fig. 13. In order to evaluate
the performance of the proposed salt cavern gas storage, volume
convergence, displacement, plastic volume, safety factor, and
effective strain are selected as the comparing indexes in the paper.
The calculating results are shown in Figs. 14–18.
4.1. Volume convergence
Volume convergence is one of the key parameters to evaluate
the availability of underground salt cavern gas storage, which is

defined as the ratio of volume shrinkage and salt cavern volume.
Fig. 14 presents the relation between volume convergence and
time. It is shown the volume convergences of four salt caverns all
increase with time, and that of the enlarged bottom cavern is the
largest, and then followed by the enlarged middle cavern, enlarged
top cavern, and the proposed cavern. For example, when the
running time lasts for 10 years, the volume convergences of the
caverns with enlarged bottom, enlarged middle, and enlarged
top, and the proposed cavern are 16.3%, 13.9%, 12.7%, and 9.68%
respectively. It shows the proposed salt cavern has the best performance in resisting volume convergence under the same condition.
4.2. Displacement
Fig. 15 shows the vertical displacement contours of rock
masses surrounding the salt cavern gas storages after running
for 10 years, where upward displacement is defined as positive
and downward displacement is defined as negative. It is shown
that the vertical displacement of the cavern upper structure is
mainly downward, while that of the cavern lower structure is
mainly upward. Moreover, the max. displacement of cavern roof
is much bigger than that of cavern bottom. By compassions, the

60

T. Wang et al. / Applied Energy 104 (2013) 50–61

displacement of the proposed cavern after running 10 years is
much smaller than that of three other caverns, showing the proposed salt cavern gas storage has obvious advantages in the resistance to displacement. It is mainly because the pressure arch
structure can improve the force state of cavern upper structure
and resist the vertical displacement. In the same time, Fig. 15 also
shows the salt cavern with enlarged bottom has the worst performance of them, which agrees with the research results of Sobolik
and Ehgartner [22].
4.3. Plastic volume
Fig. 16 shows the relation between ratios of plastic volume/cavern volume and time. The plastic volumes are mainly formed at the
initial stage of cavern operation, and are quickly reduced as the increase of time. Ultimately, it remains stationary with increasing
time. For example, the ratios of plastic volume/cavern volume of
enlarged bottom cavern, enlarged middle cavern, enlarged top cavern, and the proposed cavern are 1.577, 0.9868, 0.9152, and 0.8605
respectively at the beginning, while that of the four caverns are
changed to 1.086, 0.8517, 0.8393, and 0.6731 after running 1 year,
and subsequently they keep stable. It is mainly because the creep
of rock salt makes the release and redistribution of stresses in
the rock masses surrounding caverns, causing the decrease of plastic volume. The numerical results show the plastic volumes of the
proposed salt cavern are the smallest of the four caverns, indicating the proposed salt cavern has a good force structure, and can
effectively reduce the stresses and stress concentrations of the rock
masses surrounding salt cavern gas storage.
4.4. Safety factor
According to available experimental results [39–42], the rock
salt damage criteria can be expressed by volumetric strain and
principal stress. The damage potential (DP) criterion is one of the
criteria, which has been used since the mid-1990s to assess the
safety of salt cavern gas storages, written as

pffiffiffiffi
J2 ¼ b I1

ð22Þ

where I1 is the first invariant of the stress tensor, I1 = r1 + r2 + r3; J2
is the second invariant of the deviatoric stress tensor,
J2 = [(r1 r2)2 + (r2 r3)2 + (r3 r1)2]; r1, r2, r3 are the min.,
intermediate, and max. principal stress, respectively. b is a material
constant with a value of 0.27 (Tension is defined as positive) [43].
The safety factor for dilatancy can be defined in terms of the
ratio of stress invariants according to Eq. (22), written as

b I1
SF DP ¼ pffiffiffiffi
J2

ð23Þ

where SFDP is the safety factor for dilatancy. When SFDP < 1, the
shear stress is bigger than the average stress, and rock salt takes
place dilatancy damage.
DeVries et al. [43–46] found the dilatancy safety factor predicted
by Eq. (23) was with some risks. They proposed a safety factor of
about 1.5 was applied to the dilatancy safety factor. Moreover,
some earlier researchers [46,47] also defined that the DP safety factor (SFDP) indicated damage when SFDP < 1.5. Therefore, we consider
the rock salt begins to damage when SFDP with value in 1.0–1.5, and
these damage thresholds will be used in the paper.
Fig. 17 shows the safety factors of the four salt caverns after
running 10 years obtained by Eq. (23). It is presented the safety
factors of the rock masses surrounding caverns are small due to
the stress concentration, and are increased with the increase of distance to the caverns. By comparisons, the safety factors of caverns
with enlarged bottom, enlarged middle and enlarged top are all

smaller than that of the proposed cavern, and most of them are
smaller than 1.5. It shows the rock masses surrounding salt caverns
have taken place dilatancy failure. However, most safety factors of
rock masses surrounding the proposed salt cavern are larger than
1.5. The comprehensive results address that the safety of the proposed salt cavern is better than any other of the three salt caverns
under the long-term rheological loads conditions.
Fig. 17 also shows the red areas of the salt caverns with enlarged bottom, enlarged middle and enlarged top are much bigger
than that of the proposed cavern. It indicates the whole safety
redundancies and abilities of withstanding earthquake, loss of
pressure, etc., of the three salt cavern gas storages are lower than
that of the proposed salt cavern gas storage.
4.5. Effective strain
Taking into account rock salt subjecting to three dimensional
in situ stresses and its damage accompanying plastic deformation,
the effective strain is introduced in the paper to assess the damage
of rock salt surrounding caverns. The effective strain is mainly used
to define and measure the material damage by the change of
modulus before and after injury, which has been widely applied
in the damage evaluation of rock, coal, limestone, etc., for its simply definition, easy to calculate. According to available literatures
[48], the max. effective strain of rock mass surrounding cavern
should be no more than 3% in the whole design service lifetime
(usually 30 years) to avoid the creep damage of rock salt.
Fig. 18 shows the effective strain of the four salt caverns after running 10 years. It is presented the effective strains are varied severely
in the adjacent areas of caverns for stress concentration. As the increase of distance to the caverns, effective strains are decreased
and varied slightly. By comparisons, the max. effective strain of cavern with enlarged middle is the largest, which can reach about 1.08%
after running 10 years and is widely distributed at the cavern middle
parts. Then, it is followed by the max. effective strain of the caverns
with enlarged bottom, enlarged top, and the proposed cavern. The
max. effective strain of the proposed cavern is about 0.9%, which is
only taken place at small local areas. As shown in Fig. 18, we can also
find the values of effective strains of rock masses surrounding the
proposed cavern are distributed much smoother than that of the
others. Moreover, the area of effective strain contour map of the proposed cavern is much smaller than that of the others under the same
conditions. The comprehensive results address that the proposed
cavern has advantages in resisting the creep damage of surrounding
rock masses under long time running.
Meanwhile, we can see from Fig. 18 the effective strains of the
four caverns are all much smaller than 3%. It is because the
numerical simulation time is 10 years rather than the design
service lifetime (usually 30 years).
5. Conclusion
(1) A new model to design the shape and dimension of underground salt cavern gas storage is proposed in the paper.
Slope instability and pressure arch concepts are introduced
into the shape and dimension designs of cavern lower and
upper structures respectively. Moreover, field salt cavern
gas storage located in China is simulated as an example,
and its shape and dimension are proposed. The newly proposed salt cavern gas storage is validated by comparing with
the caverns with existing shapes.
(2) The shape and dimension of salt cavern lower structure are
determined by the max. design gas pressure, while that of
salt cavern upper structure are determined by the min.
design gas pressure. High cohesion and friction angle mean
good stability of salt cavern gas storage.

T. Wang et al. / Applied Energy 104 (2013) 50–61

(3) By comparing with salt cavern gas storages with existing
shapes, the proposed salt cavern has great advantages in volume convergence, displacement, plastic volume, safety factor, and effective strain, indicating the salt cavern
proposed in the paper has good engineering values.
(4) The new model proposed in the paper can be used in the
designs of some other underground space structures, such
as nuclear waste storage cavern, underground laneway,
potassium mining cavern, and compressed air storage
cavern.

Acknowledgments
The authors wish to acknowledge the financial supports of China Postdoctoral Science Foundation Funded Project (Grant No.
2012M511557) and Post-doctor Innovation Research Program of
Shandong Province (Grant No. 201102033) and National Natural
Science Foundation of China (Grant No. 51234007).
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