# Critical Length and Collapse of Interlayer in Rock Salt Natural Gas Storage

Authors: Yingjie Wang; Jianjun Liu
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The existence of an interlayer has a significant effect on the stability of a rock salt gas storage cavity; therefore, an uncontrollable collapse of the interlayer would cause a series of issues. In this study, three types of mechanical instability criteria are comprehensively calculated. The limit radius of the interlayer is computed under different criteria, and the collapse radius of the interlayer is obtained by comparison. The calculation results of the mathematical model are highly accurate with respect to actual engineering logging data, in general with over 90% of accuracy. It is demonstrated that, besides the physical and mechanical characteristics of the sandwich, the location of the interlayer in the cavity and concentration of the brine also have an important effect on the collapse of the interlayer. The brine at the bottom of the cavity is nearly saturated. Therefore, an interlayer at this location does not easily collapse. The mathematical model established in this study is used in the seismic design and prediction of interlayer collapse during the construction of salt-cavern gas storage facilities in China.


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## Body

## 1. Introduction

The solution mining technology of injecting water to the salt bed, dissolving the rock salt, and pumping brine is the most common approach to create underground gas storage. In the process of solution mining of a salt cavern, the soft interlayer would rapidly collapse with the dissolution of the surrounding rock salt, forming a mound at the bottom. In comparison, the hard interlayer would acquire the status of a cantilever owing to its high insoluble content and hardness. Thus, preventing the continuity of the cavity walls and reducing the rate of solution mining have a tremendous effect on the shape and volume of the salt cavity [1–4]. The basic construction situation of gas storage cavern in interlayers salt rock is shown in Figure 1. In recent years, instability of the interlayer in rock salt-cavern gas storage has occurred frequently at home and abroad. There are frequent accidents such as leakage of the gas storage, destruction of the gas storage, and subsidence or collapse of the surface, which have caused great economic loss and environmental damage. In addition, the instability and collapse of the hard interlayer can also cause damage to the tube and loss of the cavity volume [5, 6].Figure 1
Schematic diagram of salt rock underground natural gas storage.Recently, the most common method to model an interlayer is the slab model, and numerous mechanical mechanisms for the interlayer collapse have been proposed [7–11]. Li et al. analyzed the mechanical mechanism of interlayer instability based on the critical radial stress and circular plate radial buckling and summarized the form of the interlayer collapse into two categories: insufficient strength and stability failure [12]. Jiang et al. researched the interlayer stability of the single rock salt-cavity roof using a clamped circular plate mutation model and determined the necessary condition for the instability of the rock salt-cavity roof strata [13]. Bekendam and Paar analyzed the deflection and radial stress of the roof interlayer in a single cavity and predicted the collapse of the interlayer based on the tensile strength [14]. Shi et al. solved the equilibrium differential equation corresponding to the interlayer level of radial pressure and vertical pressure using a circular plate model and obtained the mezzanine deflection and curvature of the solution [15]. Liang et al. proposed the principle for determining the limit operating pressure of the bedded rock salt gas storage after an analysis of the characteristics of the rock salt mechanics principle, primarily including the cavity roof stability and creep deformation control of the surrounding rock [16, 17].According to the actual construction of gas storage caverns in China, the problem of interlayer stability in the entire process of solution mining and in the operation period involves complex mechanical and technical issues, among which the mechanical problem needs to be resolved urgently [18]. Therefore, it has a very positive relevance in the study of the interlayer in solution mining and in the analysis of the effect of the interlayer on the stability of a gas storage cavern.In this study, a mathematical model of the critical radius of an interlayer is proposed and compared with an actual rock salt interlayer collapse. The results show that the mathematical model is in strong agreement with the actual scenario and can be used as the basis for determining the interlayer collapse in a salt-cavity gas storage.


## 2. Mathematical Model of Interlayer

### 2.1. Stress Analysis of Interlayer

Owing to the strong probability of rock salt deformation, the initial stress state of a deep rock salt mine is generally under hydrostatic pressure [19, 20]. The initial geostress of an interlayer can be calculated by the following formula:(1)σh=σv=−γ0H=−∑i=1nγihi,where σh is the horizontal geostress, σv is the vertical crustal stress, γ0 is the average unit weight, H is the depth of the interlayer, n is the number of interlayers above the depth of H, and γi and hi are the unit weight and thickness of the interlayer on layer i; it is specified that the compressive stress is negative.The interlayer in a cavity can be simplified as a fixed circular thin plate, and thus, the salt cavity can be treated as an axial-symmetric sphere. The stress on the interlayer is shown in Figure2, where d is the diameter and t is the thickness of the interlayer. In addition, the borehole diameter is usually 224.5 mm, which can be ignored if the exposure length of the sandwich is approximately 40–60 m.Figure 2
Force analysis of the interlayer. (a) Vertical load. (b) Horizontal load.
(a)
(b)The difference between the vertical load on both sides of the sandwich is(2)q=γmt+Pt−Pb=γm−γbt,where γm is the unit weight of the interlayer; t is the thickness of the interlayer; γb is the unit weight of brine; Pt is the brine pressure on the top surface of the interlayer, and the value of Pt is γbH−t/2; and Pb is the brine pressure on the bottom surface of the interlayer, and the value of Pb is γbH+t/2.The value of the radial pressure on the edge of the interlayer can be described by the following formula:(3)Pr=Mγ0H,where M, which is called the radial pressure coefficient, is the ratio of radial pressure on the edge of the interlayer to initial stress.The factors of the radial pressure coefficientM are sandwich depth H, interlayer length d, cavity height h, height position of the interlayer in the salt chamber η (the ratio of the distance from the sandwich to the bottom of the cavity to the height of the salt chamber), thickness of the interlayer t, elastic modulus E of the interlayer, elastic modulus E′ of the interlayer socked in brine, and elastic modulus of the rock salt Es. E′ is equal to E at the start and less than E after the interlayer is softened. In addition, ν is not considered when the change interval is small.The effect of each parameter on the radial pressure coefficient is set for the numerical experiment. The contents of the experimental scheme are listed in Table1.Table 1
Values of each parameter.
Number
Factor (unit)
Value
1
H (m)
1000
1250
1500
1750
2000
2
d (m)
20
30
40
50
60
3
H (m)
40
60
80
100
120
4
Η (m)
0.250
0.375
0.500
0.625
0.750
5
t (m)
1.5
2.5
3.5
4.5
5.5
6
E (GPa)
6.0
7.5
9.0
10.5
12.0
E′ (GPa)
6.0
7.5
9.0
10.5
12.0
E
s (GPa)
4.0
5.0
6.0
7.0
8.0
7
E (GPa)
3.750
5.625
7.500
9.375
11.250
E′ (GPa)
3.750
5.625
7.500
9.375
11.250
E
s (GPa)
5.000
5.000
5.000
5.000
5.000
8
E (GPa)
7.5
7.5
7.5
7.5
7.5
E′ (GPa)
7.5
6.0
4.5
3.0
1.5
E
s (GPa)
5.0
5.0
5.0
5.0
5.0The radial pressure coefficient obtained from the test is listed in Table2, where the last column indicates the coefficient of variation of the radial pressure coefficient in each set of experiments. A large coefficient of variation implies a high degree of dispersion of the data [21–23] and, consequently, a strong effect of the factors on the radial pressure coefficient.Table 2
Experimental values of the radial pressure coefficient.
Number
Radial pressure coefficientM
Variation coefficient (%)
1
1.461
1.463
1.464
1.465
1.465
0.118
2
1.461
1.492
1.496
1.485
1.466
1.059
3
1.423
1.461
1.472
1.474
1.473
1.491
4
1.425
1.453
1.461
1.455
1.429
1.139
5
1.510
1.431
1.374
1.330
1.294
6.144
6
1.496
1.496
1.496
1.496
1.496
0.006
7
1.254
1.385
1.496
1.592
1.676
11.254
8
1.496
1.409
1.309
1.194
1.058
13.388It can be seen from Tables1 and 2 that the radial pressure coefficient M is mainly related to the thickness of the sandwich layer t and the ratios of the elastic modulus E/Es and E′/E.The radial pressure coefficient is obtained from the three main factors derived from the test results of the numerical calculation experiment as(4)M=1−E′E+E′E0.7333e3.523/t+4.743e0.1250E/Es.The fitting effect based on Equation (4) is shown in Figure 3.Figure 3
Fitting effect of the radial pressure coefficient: (a) elastic modulus ratioE′/E= 1 m; (b) interlayer thickness t =2 m.
(a)
(b)It can be found from the numerical experiments that a large interlayer thickness implies a small radial pressure coefficient, a large elastic modulus of the interlayer corresponds to a large radial pressure coefficient, and the radial pressure coefficient decreases linearly with the decrease in the elastic modulus of the interlayer and finally converges to 1.


### 2.2. Deformation and Stress Distribution of Interlayer

There are two types of loading in the salt chamber: radial pressurePr and brine pressure γbH resulting in the sandwich pressure and vertical load q resulting in sandwich bending. Therefore, the stress of the interlayer is expressed as follows, based on the principle of superposition:(5)σ=σi+σii,where σ is the stress of the interlayer, σi is the compressive stress caused by the radial pressure and brine at the edge of the interlayer, and σii is the bending stress caused by the vertical uniform load q.In a cylindrical coordinate system,σi is(6)σri=σθi=−Mγ0H,σzi=−γbH.The interlayer can be considered as a small deflection of the circular thin bending plate when the thickness and deflection of the interlayer are small, i.e.,(7)t≤a5,(8)ωmax≤t5,where a is the radius of the interlayer and ωmax is the maximum deflection.Under the action of the vertical uniformly distributed loadq, the deflection of the circular thin plate can be calculated as(9)ω=qa464D1−r2a22,where D is the bending stiffness given as D=Et3/121−μ2, in which μ is Poisson’s ratio.Equation (8) can be written in the form of Equation (10) based on Equations (2) and (9):(10)31−μ2γm−γba416Et3≤15.Therefore, it is feasible to use the superposition method if Equations (7) and (10) are satisfied, the mezzanine has a small deflection curve, and the neutral plane is considered as a plane.In a circular thin plate, the relationship between the stress component and internal force is(11)σr=12Mrt3z,σθ=12Mθt3z,τrθ=12Mrθt3z,τθz=6Qθt3t24−z2,τrz=6Qrt3t24−z2.For the axisymmetric bending, the relationship between the internal force and deflection is(12)Mr=−Dd2ωdr2+μ1rdωdr,Mθ=−D1rdωdr+μd2ωdr2,Mrθ=Qθ=0,Qr=−qr2.Substituting Equations (2) and (9) into Equations (11) and (12), we get(13)σrii=3γm−γb4t21+μa2−3+μr2z,σθii=3γm−γb4t21+μa2−1+3μr2z,τrzii=3γm−γbt2z2−t24r,τrθii=τθzii=0.The equilibrium equation of the axisymmetric problem is(14)∂σz∂z+∂τrz∂r+τrzr=0.Substitutingτrzii given in Equation (13) into the above formula yields(15)σzii=−2γm−γb12−zt2t+z.Therefore, it can be ignored because the values ofτrzii and σzii are small.Substituting Equations (6) and (13) into Equation (5), the interlayer stress can be given as follows:(16)σr=−Mγ0H+3γm−γb4t21+μa2−3+μr2z,σθ=−Mγ0H+3γm−γb4t21+μa2−1+3μr2z,σz=−γbH.where τrz, τrθ, and τzθ are all equal to zero so that σr, σθ, and σz are the three principal stresses of the interlayer. Furthermore, σz is equal throughout the interlayer.The minimum and maximum values ofσr and σθ are obtained from the center and edge of the top and bottom surfaces of the interlayer, whose positions are represented as (r = 0, z = −t/2), (r = 0, z = t/2), (r = a, z = −t/2), and (r = a, z = t/2).Comparing the maximum values in the latter and previous parts yields(17)3γm−γba24tMγ0H<0.05.When the value of the ratio is small, it implies that the bending effect is not significant, and then the stress on the interlayer is calculated directly according to Equation (6). In addition, if Equation (7) is not satisfied, which implies that the interlayer is not a thin plate, then it can also be calculated according to Equation (6) when the vertical uniformly distributed load q is small and mainly the stress on the interlayer is considered.In conclusion, the radial pressure coefficient has a significant impact on the interlayer stress. Moreover, the effects of a large radius ratio on the deformation and stress of the interlayer are initially considered in Equations (7), (10), and (17).


### 2.3. Critical Length of Interlayer

The critical radius of the interlayer is the maximum radius when it begins to collapse [24]. The interlayer collapse can be classified into two categories according to the mechanics: stability conditions and strength criteria. Therefore, the failure modes are also correspondingly classified into these two categories and are further divided into four types: buckling instability, bending instability, tensile failure, and shear failure. The critical radius of the interlayer, a1–a4, can be calculated from the stability condition or strength criterion at the beginning of the failure mode. First, the relevant mechanisms and patterns come into effect while the value of a is small.Therefore, the critical length of the interlayer is(18)dcr=2mina1,a2,a3,a4.Equation (18) considers numerous mechanisms of interlayer collapse and compares the possibility of each mechanism with the radius.The process ofa1–a4 calculation in the failure mode of the interlayer is as follows:(1)
Buckling instability of the interlayerBased on the radial buckling instability of the circular plate, the radial pressure of the sandwich edge is at least(19)Mγ0H=1.224E1−μ2ta2.The radius of the interlayer is(20)a1=t1.224E1−μ2Mγ0H1/2.(2)
Bending instability of the interlayerWhen both sides of Equation (10) are equal, the deflection is large and the radius is(21)a2=16Et3151−μ2γm−γb1/4.(3)
Tensile failure of the interlayerThe maximum tensile stress criterion is(22)σ3−σt=0,where σt is the tensile strength.It is found that the tensile failure is most likely to occur along the radial direction of the top edge of the interlayer, and the radius of the interlayer is(23)a3=4tMγ0H+σt3γm−γb.(4)
Shear failure of the interlayerThe Mohr–Coulomb criterion expressed by the principal stress is(24)σ1−σ31+sinφ1−sinφ+c2cosφ1−sinφ=0,where c is the cohesion and φ is the internal friction angle.Similarly,σ1 and σ3 of the four types of locations are calculated, and it is found that the shear failure occurs first at the edge of the bottom surface of the interlayer. The principal stress at the bottom edge of the interlayer is(25)σ1=−Mγ0H−3γm−γb4ta2,σ3=−γbH.Substituting the above equation into Equation (24) yields the radius of the interlayer as(26)a4=4t3γm−γb1/2⋅γbH1+sinφ1−sinφ+c2cosφ1−sinφ−Mγ0H1/2.In addition, the rock salt interlayer is softened by the brine, and accordingly, its various parameters would change. It has been analyzed that both the elastic modulusE and radial pressure coefficient M decrease. The strength of the interlayer is reduced. In reference to the strength reduction method,(27)c′=cλ,tanφ′=tanφλ,where λ is the strength reduction factor and c′ and φ′ are the cohesion and internal friction angles after the interlayer is softened; the tensile strength σt is also reduced to σt′.Simultaneously, Poisson’s ratio of the interlayer is increased, which can be considered as follows:(28)AE′En=0.5−μ′0.5−μ,where μ is Poisson’s ratio after the interlayer is softened; parameters A and n can be fitted by the experimental data, and the values of the parameters are all set to 1.Therefore, the calculation model of the critical radius of the interlayer involves the above formulas.


## 2.1. Stress Analysis of Interlayer

Owing to the strong probability of rock salt deformation, the initial stress state of a deep rock salt mine is generally under hydrostatic pressure [19, 20]. The initial geostress of an interlayer can be calculated by the following formula:(1)σh=σv=−γ0H=−∑i=1nγihi,where σh is the horizontal geostress, σv is the vertical crustal stress, γ0 is the average unit weight, H is the depth of the interlayer, n is the number of interlayers above the depth of H, and γi and hi are the unit weight and thickness of the interlayer on layer i; it is specified that the compressive stress is negative.The interlayer in a cavity can be simplified as a fixed circular thin plate, and thus, the salt cavity can be treated as an axial-symmetric sphere. The stress on the interlayer is shown in Figure2, where d is the diameter and t is the thickness of the interlayer. In addition, the borehole diameter is usually 224.5 mm, which can be ignored if the exposure length of the sandwich is approximately 40–60 m.Figure 2
Force analysis of the interlayer. (a) Vertical load. (b) Horizontal load.
(a)
(b)The difference between the vertical load on both sides of the sandwich is(2)q=γmt+Pt−Pb=γm−γbt,where γm is the unit weight of the interlayer; t is the thickness of the interlayer; γb is the unit weight of brine; Pt is the brine pressure on the top surface of the interlayer, and the value of Pt is γbH−t/2; and Pb is the brine pressure on the bottom surface of the interlayer, and the value of Pb is γbH+t/2.The value of the radial pressure on the edge of the interlayer can be described by the following formula:(3)Pr=Mγ0H,where M, which is called the radial pressure coefficient, is the ratio of radial pressure on the edge of the interlayer to initial stress.The factors of the radial pressure coefficientM are sandwich depth H, interlayer length d, cavity height h, height position of the interlayer in the salt chamber η (the ratio of the distance from the sandwich to the bottom of the cavity to the height of the salt chamber), thickness of the interlayer t, elastic modulus E of the interlayer, elastic modulus E′ of the interlayer socked in brine, and elastic modulus of the rock salt Es. E′ is equal to E at the start and less than E after the interlayer is softened. In addition, ν is not considered when the change interval is small.The effect of each parameter on the radial pressure coefficient is set for the numerical experiment. The contents of the experimental scheme are listed in Table1.Table 1
Values of each parameter.
Number
Factor (unit)
Value
1
H (m)
1000
1250
1500
1750
2000
2
d (m)
20
30
40
50
60
3
H (m)
40
60
80
100
120
4
Η (m)
0.250
0.375
0.500
0.625
0.750
5
t (m)
1.5
2.5
3.5
4.5
5.5
6
E (GPa)
6.0
7.5
9.0
10.5
12.0
E′ (GPa)
6.0
7.5
9.0
10.5
12.0
E
s (GPa)
4.0
5.0
6.0
7.0
8.0
7
E (GPa)
3.750
5.625
7.500
9.375
11.250
E′ (GPa)
3.750
5.625
7.500
9.375
11.250
E
s (GPa)
5.000
5.000
5.000
5.000
5.000
8
E (GPa)
7.5
7.5
7.5
7.5
7.5
E′ (GPa)
7.5
6.0
4.5
3.0
1.5
E
s (GPa)
5.0
5.0
5.0
5.0
5.0The radial pressure coefficient obtained from the test is listed in Table2, where the last column indicates the coefficient of variation of the radial pressure coefficient in each set of experiments. A large coefficient of variation implies a high degree of dispersion of the data [21–23] and, consequently, a strong effect of the factors on the radial pressure coefficient.Table 2
Experimental values of the radial pressure coefficient.
Number
Radial pressure coefficientM
Variation coefficient (%)
1
1.461
1.463
1.464
1.465
1.465
0.118
2
1.461
1.492
1.496
1.485
1.466
1.059
3
1.423
1.461
1.472
1.474
1.473
1.491
4
1.425
1.453
1.461
1.455
1.429
1.139
5
1.510
1.431
1.374
1.330
1.294
6.144
6
1.496
1.496
1.496
1.496
1.496
0.006
7
1.254
1.385
1.496
1.592
1.676
11.254
8
1.496
1.409
1.309
1.194
1.058
13.388It can be seen from Tables1 and 2 that the radial pressure coefficient M is mainly related to the thickness of the sandwich layer t and the ratios of the elastic modulus E/Es and E′/E.The radial pressure coefficient is obtained from the three main factors derived from the test results of the numerical calculation experiment as(4)M=1−E′E+E′E0.7333e3.523/t+4.743e0.1250E/Es.The fitting effect based on Equation (4) is shown in Figure 3.Figure 3
Fitting effect of the radial pressure coefficient: (a) elastic modulus ratioE′/E= 1 m; (b) interlayer thickness t =2 m.
(a)
(b)It can be found from the numerical experiments that a large interlayer thickness implies a small radial pressure coefficient, a large elastic modulus of the interlayer corresponds to a large radial pressure coefficient, and the radial pressure coefficient decreases linearly with the decrease in the elastic modulus of the interlayer and finally converges to 1.


## 2.2. Deformation and Stress Distribution of Interlayer

There are two types of loading in the salt chamber: radial pressurePr and brine pressure γbH resulting in the sandwich pressure and vertical load q resulting in sandwich bending. Therefore, the stress of the interlayer is expressed as follows, based on the principle of superposition:(5)σ=σi+σii,where σ is the stress of the interlayer, σi is the compressive stress caused by the radial pressure and brine at the edge of the interlayer, and σii is the bending stress caused by the vertical uniform load q.In a cylindrical coordinate system,σi is(6)σri=σθi=−Mγ0H,σzi=−γbH.The interlayer can be considered as a small deflection of the circular thin bending plate when the thickness and deflection of the interlayer are small, i.e.,(7)t≤a5,(8)ωmax≤t5,where a is the radius of the interlayer and ωmax is the maximum deflection.Under the action of the vertical uniformly distributed loadq, the deflection of the circular thin plate can be calculated as(9)ω=qa464D1−r2a22,where D is the bending stiffness given as D=Et3/121−μ2, in which μ is Poisson’s ratio.Equation (8) can be written in the form of Equation (10) based on Equations (2) and (9):(10)31−μ2γm−γba416Et3≤15.Therefore, it is feasible to use the superposition method if Equations (7) and (10) are satisfied, the mezzanine has a small deflection curve, and the neutral plane is considered as a plane.In a circular thin plate, the relationship between the stress component and internal force is(11)σr=12Mrt3z,σθ=12Mθt3z,τrθ=12Mrθt3z,τθz=6Qθt3t24−z2,τrz=6Qrt3t24−z2.For the axisymmetric bending, the relationship between the internal force and deflection is(12)Mr=−Dd2ωdr2+μ1rdωdr,Mθ=−D1rdωdr+μd2ωdr2,Mrθ=Qθ=0,Qr=−qr2.Substituting Equations (2) and (9) into Equations (11) and (12), we get(13)σrii=3γm−γb4t21+μa2−3+μr2z,σθii=3γm−γb4t21+μa2−1+3μr2z,τrzii=3γm−γbt2z2−t24r,τrθii=τθzii=0.The equilibrium equation of the axisymmetric problem is(14)∂σz∂z+∂τrz∂r+τrzr=0.Substitutingτrzii given in Equation (13) into the above formula yields(15)σzii=−2γm−γb12−zt2t+z.Therefore, it can be ignored because the values ofτrzii and σzii are small.Substituting Equations (6) and (13) into Equation (5), the interlayer stress can be given as follows:(16)σr=−Mγ0H+3γm−γb4t21+μa2−3+μr2z,σθ=−Mγ0H+3γm−γb4t21+μa2−1+3μr2z,σz=−γbH.where τrz, τrθ, and τzθ are all equal to zero so that σr, σθ, and σz are the three principal stresses of the interlayer. Furthermore, σz is equal throughout the interlayer.The minimum and maximum values ofσr and σθ are obtained from the center and edge of the top and bottom surfaces of the interlayer, whose positions are represented as (r = 0, z = −t/2), (r = 0, z = t/2), (r = a, z = −t/2), and (r = a, z = t/2).Comparing the maximum values in the latter and previous parts yields(17)3γm−γba24tMγ0H<0.05.When the value of the ratio is small, it implies that the bending effect is not significant, and then the stress on the interlayer is calculated directly according to Equation (6). In addition, if Equation (7) is not satisfied, which implies that the interlayer is not a thin plate, then it can also be calculated according to Equation (6) when the vertical uniformly distributed load q is small and mainly the stress on the interlayer is considered.In conclusion, the radial pressure coefficient has a significant impact on the interlayer stress. Moreover, the effects of a large radius ratio on the deformation and stress of the interlayer are initially considered in Equations (7), (10), and (17).


## 2.3. Critical Length of Interlayer

The critical radius of the interlayer is the maximum radius when it begins to collapse [24]. The interlayer collapse can be classified into two categories according to the mechanics: stability conditions and strength criteria. Therefore, the failure modes are also correspondingly classified into these two categories and are further divided into four types: buckling instability, bending instability, tensile failure, and shear failure. The critical radius of the interlayer, a1–a4, can be calculated from the stability condition or strength criterion at the beginning of the failure mode. First, the relevant mechanisms and patterns come into effect while the value of a is small.Therefore, the critical length of the interlayer is(18)dcr=2mina1,a2,a3,a4.Equation (18) considers numerous mechanisms of interlayer collapse and compares the possibility of each mechanism with the radius.The process ofa1–a4 calculation in the failure mode of the interlayer is as follows:(1)
Buckling instability of the interlayerBased on the radial buckling instability of the circular plate, the radial pressure of the sandwich edge is at least(19)Mγ0H=1.224E1−μ2ta2.The radius of the interlayer is(20)a1=t1.224E1−μ2Mγ0H1/2.(2)
Bending instability of the interlayerWhen both sides of Equation (10) are equal, the deflection is large and the radius is(21)a2=16Et3151−μ2γm−γb1/4.(3)
Tensile failure of the interlayerThe maximum tensile stress criterion is(22)σ3−σt=0,where σt is the tensile strength.It is found that the tensile failure is most likely to occur along the radial direction of the top edge of the interlayer, and the radius of the interlayer is(23)a3=4tMγ0H+σt3γm−γb.(4)
Shear failure of the interlayerThe Mohr–Coulomb criterion expressed by the principal stress is(24)σ1−σ31+sinφ1−sinφ+c2cosφ1−sinφ=0,where c is the cohesion and φ is the internal friction angle.Similarly,σ1 and σ3 of the four types of locations are calculated, and it is found that the shear failure occurs first at the edge of the bottom surface of the interlayer. The principal stress at the bottom edge of the interlayer is(25)σ1=−Mγ0H−3γm−γb4ta2,σ3=−γbH.Substituting the above equation into Equation (24) yields the radius of the interlayer as(26)a4=4t3γm−γb1/2⋅γbH1+sinφ1−sinφ+c2cosφ1−sinφ−Mγ0H1/2.In addition, the rock salt interlayer is softened by the brine, and accordingly, its various parameters would change. It has been analyzed that both the elastic modulusE and radial pressure coefficient M decrease. The strength of the interlayer is reduced. In reference to the strength reduction method,(27)c′=cλ,tanφ′=tanφλ,where λ is the strength reduction factor and c′ and φ′ are the cohesion and internal friction angles after the interlayer is softened; the tensile strength σt is also reduced to σt′.Simultaneously, Poisson’s ratio of the interlayer is increased, which can be considered as follows:(28)AE′En=0.5−μ′0.5−μ,where μ is Poisson’s ratio after the interlayer is softened; parameters A and n can be fitted by the experimental data, and the values of the parameters are all set to 1.Therefore, the calculation model of the critical radius of the interlayer involves the above formulas.


## 3. Application

To verify the mathematical model, the rock salt gas storage in Jintan, China, is simulated as an example. The Jintan storage project should be able to supply approximately 15 million standard cubic meters to the marketplace per day. To achieve this objective, a total of 57 rock salt caverns will be solution mined and six existing caverns will be recreated [25]. Technically, to ensure the overall stability of the gas storage, the average thickness of the rock salt available is approximately 140 m, with the depth ranging from approximately 900 to 1240 m underground.According to the four critical radii of the above interlayer, we analyzed the interlayer collapse of the rock salt gas storage cavities and compared the calculated results with the field measurement results. The interlayers analyzed in this study originate from the three rock salt gas storage cavities in the Jintan area of China, which are labelled as A, B, and C. Among these, two interlayers are present in cavity A, one interlayer is in cavity B, and three interlayers are in cavity C.The physical and mechanical characteristics of the interlayer and rock salt, as listed in Table3, are based on the results of the previous experiments of rock mechanics conducted on the strata in this area.Table 3
Physical and mechanical characteristics of rock.
Type of rock
ρ (kg·m−3)
E (GPa)
μ
c (MPa)
φ (°)
σ
t (MPa)
Interlayer
2500
4.28
0.205
1.2
32
0.6
Rock salt
2200
3.60
0.282
1.0
40
0.4The critical radius of each interlayer is calculated according to the established mathematical model. The basic characteristics of the interlayer and the results of the critical radius are listed in Tables4 and 5, and the contrast between the calculation results and the actual measured data is shown in Figure 4.Table 4
Critical length of each interlayer under different conditions.
Number
H (m)
γ
b (N/m3)
M
t (m)
E′/E
E/Es
a
1 (m)
a
2 (m)
a
3 (m)
a
4 (m)
A1
950
9800
1.29
3.8
1.0
1.2
54.61
33.36
25.49
14.06
A2
965
10780
1.27
2.2
0.9
1.2
31.61
22.17
19.44
12.82
B
970
10780
1.27
4.0
1.0
1.2
55.60
70.00
110.00
74.00
C1
1020
9800
1.51
1.4
1.0
1.2
17.94
15.78
17.31
5.68
C2
1030
10780
1.32
2.8
0.9
1.2
38.20
26.56
23.07
14.10
C3
1050
11760
1.43
2.0
1.0
1.2
25.00
41.60
86.20
44.90Table 5
Critical length and collapse condition of the interlayer (m).
Number
Design radius (L1)
Critical radius (L2)
Rest part length (L3)
Unstable part length (L4)
Logging length (L5)
Accuracy (|L5 − L3|)/L3 (%)
A1
20
14.06
5.94
14.06
6.5
90.6
A2
25
12.82
12.18
12.82
13.2
91.6
B
25
55.60
25
0
20
80
C1
20
5.68
2.96
17.04
3.2
91.9
C2
25
14.10
10.9
14.1
11.5
94.5
C3
20
25.00
20
0
20
100Figure 4
Collapse length of each interlayer.It can be seen from Table5 that the calculation results of the mathematical model established in this study have a high degree of accuracy with respect to the actual engineering results, and it is generally above 90%. From Figure 3, it is understood that the main factors affecting the interlayer collapse are the thickness and position of the interlayer, while different positions imply different concentrations of the brine. For example, interlayer A1 is thicker than interlayer A2; however, because of the low concentration of the brine at the position of interlayer A1, it is obvious that the interlayer is eroded by the brine. If there are weak parts or cracks in the interlayer, it would be easier for it to collapse and the logging length would be small.For interlayer C3, although its thickness is small, the interlayer is at the bottom of the cavity where the brine concentration is nearly saturated, and the solubility of the interlayer is 0. So the destruction of the interlayer is very low, and it is difficult for the interlayer to collapse in this case.Concurrently, because the thickness of interlayer B is large, it is not easy for it to collapse. However, in the process of cavity creation, the middle part of the interlayer is highly affected by the middle pipe and brine circulation, and a hollow is formed, of approximately 10 m in diameter. Thus, the accuracy of the collapse of interlayer B is low.


## 4. Conclusions

The main conclusions drawn from this study are as follows:(1)
This study defined the ratio of radial pressure to initial stress as the radial pressure coefficient, which makes the calculation of the interlayer stress simple and with satisfactory precision. This study considered the mechanical mechanism of the interlayer collapse in the proposed critical-length calculation model. This mathematical model comprehensively considered the strength and position of the interlayer and the concentration of brine, which is more suitable for an actual engineering problem.(2)
In general, the value ofa4 is smallest for most interlayers, which means the main form is shear failure of the interlayer collapse. The critical length of the interlayer is related to the position of the interlayer in the cavern. Because the concentration of brine is close to saturation, which has a little effect on the mechanical properties of interlayer characteristics, the instability of the interlayer is not easy.(3)
The critical-length calculation model has been used to analyze the collapse of numerous salt cavities in Jintan, China. It is demonstrated that the model has a high accuracy and can provide a good reference for the collapse of the interlayer. Therefore, it has significant relevance and practical application value for modelling the engineering method of rock salt gas storage.



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*Source: 8658501-2018-09-02.xml*