(1) Construction of hierarchy model

Set the target level as the top level of the evaluation system. The criterion level and sub-criteria level constitute the middle level of the evaluation system. The evaluation system allows the criterion and sub-criteria layers to be composed of multiple layers; the criteria of the previous level will roughly determine the requirements of the next level. The evaluation object is the lowest level of the hierarchical structure. The sub-criteria level, the criterion level, and the decision-making goal are the bridges to establish the connection with the evaluation object.

(2) Construction of judgment matrix

The importance of any two evaluation indicators at each hierarchy level needs to be compared for a specific criterion level of evaluation indicators. The comparison results of the pairwise evaluation index are described by the improved three-scale method. Then the pairwise evaluation judgment matrix A corresponding to the index level is constructed, and the judgment matrix element is the scale value aij. The three-scale process is improved from the traditional nine-scale way. The method of judging the scale value is shown in Table 1.

(3) Hierarchical Single Sorting and Consistency Check

For the eigenvector of the largest eigenroot λmax of the judgment matrix A, it is normalized and denoted as W, which is called the weight vector. The element of W is the sorting weight of the relative importance of the element at the same level as a particular element at the previous level. This process is called single-level sorting. A consistency test is required as to whether the order of the level list can be confirmed. The consistency index is calculated by CI. The smaller the CI, the greater the consistency. The degree of inconsistency of A can be measured by the value of λmax−n (n is the order of the judgment matrix A). Define the consistency index as:

(1)CI=λmax−nn−1

CI = 0, there is complete consistency; the closer the CI is to 0, there is satisfactory consistency; the more significant the CI, the more serious the inconsistency.

To facilitate the measurement of CI, the random consistency index RI is introduced:

(2)RI=CI1+CI2+CI3+⋯+CInnwhere the random consistency index RI is related to the order of the judgment matrix, and the corresponding relationship is shown in Table 2.

Considering that random reasons may cause the deviation of consistency when testing whether the judgment matrix has satisfactory consistency, it is also necessary to compare CI and RI to obtain the test coefficient CR. The formula is as follows:

(3)CR=CIRI

Generally, when the test coefficient CR < 0.10, the judgment matrix A is considered to pass the consistency test. Otherwise, it fails.

(4) Improved judgment matrix

Based on the judgment matrix A, it is stipulated that:

(4)ri=∑j=1naij (i=1,2,3,…;,n) 

Set rmax=max{ri}, rmin=min{ri}, the element kij of the judgment matrix K is:

(5)kij={ri−rjrmax−rmin(rmaxrmin−1)+1=ri−rj+rminrminri≥rj[rj−rirmax−rmin(rmaxrmin−1)+1]−1=rminrj−ri+rminri<rj

Based on the judgment matrix constructed above, to obtain a judgment matrix that satisfies the consistency of judgment, it is necessary to transform the judgment matrix K to a certain extent with the help of the concept of an optimized transfer matrix. Therefore, the transformed judgment matrix can naturally meet the consistency test index CR requirements, and the ranking weight value of the comprehensive evaluation index can be obtained conveniently and directly.

The formula for constructing the transfer matrix is as follows:

(6)pij=lg⁡kij

The element qij of the optimal transfer matrix Q is:

(7)qij=∑k=1n(pik−pjk)/n

The matrix element yij of the quasi-optimal consistent matrix Y is:

(8)yij=10qij

Solve the eigenvector w¯ corresponding to the maximum eigenvalue λmax of the to-be-optimized consistent matrix Y, and then normalize the vector w¯=[w1¯,w2¯,⋯,wn¯]T, namely:

(9)ui=wi¯∑i=1nwi¯

U=[u1,u2,u3,…,un]T, each component in U is the ranking weight of n elements in the same layer to a specific evaluation index in the previous layer. The weight is the subjective weight calculated based on the improved analytic hierarchy process.

The entropy weight method EWM is an objective weighting method used to measure the amount of information. Based on the degree of variation of each indicator, entropy weight is obtained through information entropy. Information entropy can describe the degree of disorder and the quantitative degree of effectiveness of a natural system. The gas storage operation stability evaluation system is open, and changes due to different places and times. Therefore, the traditional objective weighting method cannot clearly describe the law of influencing indicators. In using the entropy method to construct the judgment matrix in order to determine the index weight, the interference of human factors should be avoided as much as possible, so that the evaluation index assignment result can be more consistent with the actual situation, detached as it is from human bias.

A gas storage with m operating conditions is drawn up, and each operating condition has n evaluation indicators, judgment matrix is constructed:

(10)X=(xij)m×n

References the Eq. (24) be normalized:

(11)x~ij=xij∑j=1nxij

Quoting Eq. (25) to calculate the entropy value of the evaluation index:

(12)ei=−1ln⁡n∑j=1nxij~ln⁡(xij~)

Use Eq. (26) to determine the entropy weight of the evaluation index:

(13)vi=1−ei∑i=1m(1−ei)

V={v1,v2,⋯,vm}, The weight is the objective weight of the evaluation index calculated based on the entropy weight method.

The evaluation index weight calculated by the improved AHP is called the subjective weight [10]. The evaluation index weight estimated by the EWM is called the objective weight, and the combined weight is composed of both the AHP and EWM. The variance maximization method can effectively solve the problem of the distribution of subjective and objective weights. The smaller the evaluation index value difference, the smaller the effect of the index, the smaller the weight is assigned, and the greater the weight is given to it. The combined weight wi~=αui+βvi is expressed linearly by the subjective weight ui and objective weight vi, where α, β≥0, and α2+β2=1.

Normalize the judgment matrix A=(aij)m×n to get R=(rij)m×n, where rij¯=1n∑j=1nrij(i=1,2,3,⋯,m).

Refer to Eqs. (27) and (28) to calculate α,β, which are expressed as follows:

(14)α=11+∑i=1m∑j=1n(rij−rij¯)2ui∑i=1m∑j=1n(rij−rij¯)2vi

(15)β=11+∑i=1m∑j=1n(rij−rij¯)2vi∑i=1m∑j=1n(rij−rij¯)2uiwhere α represents the subjective weight distribution coefficient, β represents the objective weight distribution coefficient.

Through calculating subjective weight ui, objective weight vi and combined weight wi~, the constant weight of comprehensive index evaluation can be finally determined.

The Norton Power-Law model (MODEL cpower) is selected as the constitutive model for creep calculation of the interbedded salt rock underground gas storage for this work. Based on the corresponding geological survey data of the studied interbedded salt rock underground gas storage, the static parameters and rheological parameters of each rock layer of the established calculation model of the salt rock underground gas storage with interlayer were obtained, as are shown in Tables 4 and 5.

At present [14], the research results show that when the width of the calculation model is more than five times the diameter of the caverns of the salt rock underground gas storage group, the influence of the boundary effect on the deformation of the surrounding rock of the salt rock underground gas storage group can be reasonably ignored. The research object of this paper is a karst cavern of underground gas storage containing an interbedded salt rock with a burial depth of 1000 m. The length and width of the entire calculation area are set to 500 m, and the height is set to 800 m. The short radius is 50 m, the long radius is 80 m, and the interlayer passes through the cavity. The corresponding physical model and numerical calculation model are shown in Fig. 5.

The model separately designed three gas storage cavity shapes, namely: ellipsoid, egg-shaped, and spherical, to study the influence of different cavity shapes on the operation stability of the gas storage. Fig. 6 shows the calculation model of the three cavity shapes.

For specific underground gas storages containing interbedded salt rock, the scoring model of the cavity shape index is shown in Eq. (17).

(17)M1=(1−Δ/d)×100

From Eq. (17), the cavity shape index score decreases linearly with the increase of the cavity creep rate. According to the results of cavern creep displacement, it can be concluded that the three gas storage cavity shapes are: ellipsoidal (82.91), egg-shaped (87), and spherical (86.94).

Operating pressure index scoring model, as shown in Eqs. (18) and (19).

(18)M2={00<Rp<1−50cos⁡(2πRP)+501≤RP<20RP≥2

(19)RP=PPminwhere M2 is the score of the operating pressure index U2. RP is the operating pressure ratio. P is the internal operating pressure, MPa. Pmin is the minimum allowable working pressure, MPa.

Fig. 7 presents the creep displacement result of the cavity under the different operating pressure.s time goes by, the maximum displacement continues to increase, and the volume of the cavern keeps shrinking; under the same operating time, the maximum displacement will decrease with the increase of operating pressure; when the operating pressure is greater than or equal to 8 MPa, the maximum displacement changes approximately linearly with the time; when the operating pressure is lower than 4 MPa, the maximum displacement increases significantly in the first 15 years, and grows slowly in the next 15 years. Therefore, it is reasonable to set 8 MPa as the minimum operating pressure of the underground gas storage with interbedded salt rock. Meanwhile, to avoid leakage of the gas storage due to excessive operating internal pressure during the operation of the gas storage, the operating pressure should also be less than 16 MPa. From Eq. (18), It can be seen that the pressure ratio should be maintained between 1 and 2, and the best value is 1.5.

This section starts with four aspects of thickness, number, inclination, and stiffness, using the simulation software FLAC 3D to research its stability, and obtain the indexes score model.

The scoring model of the interlayer index is shown in Eqs. (20)~(23).

(20)M31={50cos⁡(πh6)+500≤h<60h≥6where M31 is the sub-item score of the interlayer thickness index U31. h is the thickness of the interlayer, m.

(21)M32={50cos⁡(πm3)+500≤m<30m≥3where M32 is the sub-item score of the interlayer number index U32. m is the number of the interlayer, piece.

(22)M33={00≤θ<7.550cos⁡(0.2πθ−π)+507.5≤θ≤17.50θ>17.5where M33 is the sub-item score of the interlayer inclination index U33. θ is the inclination of the interlayer, °.

(23)M34={−1500RQ2+600RQ0≤RQ<0.2−250RQ2+1500RQ+2650490.2≤RQ<3−10RQ2+60RQ+103≤RQ<520RQ2−320RQ+128035≤RQ<80RQ≥8

(24)RQ=QMQSwhere M34 is the sub-item score of the interlayer stiffness ratio index U34. RQ is the stiffness ratio of the interbedded salt rock. Qi is the stiffness of the interlayer, N/m. Qs is the stiffness of the salt rock, N/m.

The above formula presents the score model of the interlayer thickness, number, inclination, and stiffness ratio, respectively. As the thickness and quantity of the interlayer increase, it can be seen that corresponding index scores gradually decrease. In the storage construction process, the area with less interlayer thickness and quantity should be selected as much as possible. For the interlayer inclination, 7.5°~17.5° should be chosen as far as possible to maintain stability better. The stiffness ratio is defined as the ratio of the stiffness of the mudstone interlayer to the salt rock. When the stiffness ratio is minimal, the score is particularly susceptible to it in fact, and it increases rapidly with the increase of stiffness ratio, and then slowly increases to the peak value from there. When the stiffness ratio increases, on the other hand, the index scoring gradually begins to decrease. When the stiffness ratio takes the maximum and minimum values, the corresponding index scores are lower. In actual, practical engineering applications, we are trying to keep the stiffness ratio between 2 and 4, where 3 is the best.

For the long-term safe operation of the interbedded salt rock underground gas storage, setting a certain thickness of the roof salt rock layer can offset or weaken the influence of the cavern deformation caused by the salt rock creep on the upper rock layer and the ground surface. The actual field experience shows that the low permeability of the roof and floor of the salt cavern and the surrounding salt rock layer guarantee the tightness and stability of the salt rock UGS. The salt rock layer index includes three sub-indices: the roof thickness, the floor thickness, and the thickness of the surrounding salt rock.

(25)M41={50cos⁡[π(2Rr+1)]+500≤Rr<0.51000.5≤Rr<150cos⁡[π(Rr+1)]+501≤Rr<20Rr≥2

(26)Rr=hrdwhere M41 is the score of the roof thickness index U41. Rr is the ratio of the thickness of the roof salt rock to the maximum diameter of the cavern. hr is the thickness of the roof salt rock, m.

d is the maximum diameter of the salt cavern, m.

From the above score model, As the thickness of the roof salt rock increases, the stability score gradually rises; when the value is 0.5 to 1, the score is the highest and continues to increase, the stability is quickly destroyed until the score tends to 0. As floor thickness and surrounding thickness index are similar to the score of roof thickness, not the same. In practical engineering applications, the site should be selected for construction where the roof, floor, and surrounding thickness are close to the diameter of the cavern.

The mechanical properties of salt rock include elastic modulus, cohesion, internal friction angle, and steady-state creep rate. The scoring model of mechanical properties is shown in Eqs. (27)~(30), respectively.(27)M51={12E0≤E≤5−8E2+320E+1300455<E≤20100E>20where M51 is the score of the elastic modulus index U51. E is the elastic modulus of the salt rock, Gpa.

(28)M52={60C0≤C≤1−2.5C2+25C+37.51<C≤5100C>5where M52 is the score of the cohesion index U52. C is the cohesion of the salt rock, Mpa.

(29)M53={φ2/150≤φ≤30−0.1φ2+10φ−15030<φ≤50100φ>50where M53 is the internal friction angle index U53 score. φ is the angle of internal friction, °.

(30)M54={−40ε2/9+1000≤ε≤320ε2/3−80ε+2403<ε≤60ε>6where M54 is the steady-state creep rate index U54 score. ε is the steady-state creep rate, 10⁻⁴/h.

For the elastic modulus, it is to determine the elastic modulus stress required to plastically deform under the applied force of rock salt. The greater the elastic modulus, the stronger the ability of the salt rock to resist external deformation, and, accordingly, the less likely it is to deform during actual operation. The higher the elastic modulus, the better the stability; in real engineering applications, attempting to choose high elastic modulus to build the UGS with remains critical. Salt rock cohesion is defined as the mutual attraction between adjacent parts of the rock, which reflects the strength of the salt rock to a certain extent; the greater the cohesion, the greater the mechanical strength of the salt rock. The internal friction angle reflects the shear strength of salt rock. The higher the internal friction angle, the better the salt rock gas storage stability. Therefore, the high internal friction angle should be selected as far as possible to ensure the operational stability of the salt rock gas storage. The creep rate refers to the derivative of elongation to the time when the salt rock undergoes creep deformation; the lower the steady-state creep rate, the less prone to deformation of the salt rock, the stronger the ability to resist external stress interference, and the better the stability of salt rock gas storage [17].