Underground gas storage is an effective method to cope with the seasonal imbalance of natural gas consumption and alleviate the contradiction between supply and demand^[1]. With working gas volume accounting for about 75% of the total working gas volume of various types of gas storages worldwide^[2], the gas storage rebuilt from a gas reservoir is the most important type of underground gas storage in the world. The injection-production capacity of a gas well is one of the most important parameters in design and evaluation of gas storage. Xu et al. and Tan et al. studied the dynamic operational status of several in-service gas storages owned by CNPC and the results showed that the injection-production capacity of gas wells could affect the dynamic operation status of gas storages and restrict their capacities for peak shaving^[3,4]. In addition, Zhou et al. studied the parameter configurations of ground facilities for gas storage in the Bohai Sea and many other areas in China, showing that the injection-production capacity of gas wells has a greater influence on the operating parameters, equipment safety, and reliability of ground facilities of gas storage^[5,6,7]. The high-intensity multi-cycle injection and production of gas storage leads to periodic changes in the net stress of reservoir rocks in the gas storage. The physical properties of rocks in the reservoir can be severely damaged under alternating stress, which can further affect the injection-production capacity of gas wells and restricts the safe and long-term operation of gas storage^[8]. It is therefore urgent to determine the permeability-stress sensitivity of rocks in gas storage reservoirs and the injection-production capacity of gas wells in these reservoirs to optimize the selection of gas storage reservoir locations and accurately predict multi-cycle operational dynamics.

Many studies previously focused on the single-cycle stress sensitivity of reservoir rock, so the relationship between the net stress and overburden permeability of the reservoir rock has been well understood^[9,10], and some models between net stress and overburden permeability have been established^[11,12] and widely used in the engineering and numerical simulation of oil and gas reservoirs^[13,14]. However, few studies focused on the multiple-cycle stress sensitivity of reservoir rocks. Several researchers carried out experiments on rock samples from gas storage rebuilt from sandstone gas reservoirs, including Dazhangtuo in the Dagang Oilfield NE China and Wen 96 in Zhongyuan Oilfield, Central China and artificial sandstone samples, and reached the conclusion that under multi-cycle intensive injection and production in underground gas storage reservoirs, the rock permeability would decrease with the increase of net stress and number of injection-production cycles; and the decrease in permeability would become slow with the increase of number of injection-production cycles^{[15,16,17,18,19]}. Preliminary results have been obtained on the effect of net stress and injection-production cycles on the overburden permeability of reservoirs, but two questions are still waiting to be answered: (1) The internal mechanisms of the law of overburden permeability of rock changing with net stress and injection-production cycle have not been fully clarified. (2) The current research findings are still in a quailtative research stage, and models between rock’s overburden permeability with rock permeability, net stress and the net stress variation period have not yet been established. This means that the multi-cycle stress sensitivity of reservoir rocks cannot be considered in researches on gas reservoir engineering and numerical simulation.

The permeability-stress sensitivity tests were carried out on standard rock samples with different physical properties from the Wen 23 gas storage reservoir in the Zhongyuan Oilfield under multi-cycle injection-production conditions. The overburden permeability of rock samples under different net stresses were measured, and the internal mechanism and change laws of overburden permeability of the rock sample with net stress on the rock sample and injection-production cycle were analyzed. The relationships between the stress sensitivity index of permeability and the permeability of the reservoir rock and operation cycle of gas storage, as well as the relationship between initial overburden permeability and the rock permeability were established. A method for calculating the overburden permeability of gas storage reservoirs under multi-cycle injection production was proposed to study the influence of the stress sensitivity of reservoir on the injection and production capacity of the gas storage.

In light of the physical property distribution range of reservoir rocks of the Wen 23 gas storage, 6 standard cores were selected as experimental rock samples. The samples had a porosity range from 10.24% to 19.92%, and the permeability of (0.037 3-28.438 0)×10^-3 μm² (Table 1). In this paper, permeability refers to the absolute permeability of rock. The gas used in the experiment was nitrogen with a purity of 99.99%.

The Wen 23 gas storage is a gas storage transformed from a gas reservoir, which has an overburden pressure of 66.33 MPa, the upper operating pressure limit of 38.62 MPa (consistent with the original formation pressure of the gas reservoir), which corresponds to the minimum net stress of 27.71 MPa; and the lower limit operating pressure of 19.06 MPa which corresponds to the maximum net stress of 47.27 MPa. Based on the upper limit of operating pressure of the gas storage, the initial net stress in the experiment was set at 27.71 MPa, the interval of the net stress measuring points was designed at 2.5 MPa, and the maximum net stress in the experiment was 47.27 MPa. The experimental temperature was 120 °C, which is the formation temperature of the Wen 23 gas storage reservoir.

In the experiment, the overburden permeability tests were designed in seven variation processes (Table 2). The overburden permeabilities under different net stresses were measured in the net stress variation mode of keeping confining pressure constant and changing pore pressure, and the experimental setup is shown in Fig. 1. The experimental steps are as follows: (1) The rock sample was placed into the core holder, the confining pressure was set at 66.33 MPa and kept constant; nitrogen was injected to increase the pressure to 38.62 MPa. The net stress was successively increased from the initial net stress to the maximum net stress according to the predetermined interval, and the overburden permeability of the rock sample at each net stress was measured. (2) The overburden permeability of the rock sample at each net stress point was measured in the process of net stress drop from the maximum net stress to the initial net stress according to the predetermined interval. (3) Repeat steps (1) and (2) until all samples were tested.

The test results of the overburden permeability experiment (Fig. 2) show that the overburden permeability decreases with the increase of net stress. This is mainly because with the increase of net stress exerted on the rock sample, the rock sample is compressed and the pore throat radius of the rock sample decreases, in turn, the overburden permeability of the rock sample decreases. It can also be seen from Fig. 2 that the relationship curves between overburden permeability and net stress in all the net stress variation processes successively move downward with the increase in the number of net stress changing process (the overburden permeability decreases). This is because with the increase of net stress acting on the rock sample, the rock sample is compressed and deforms (elastic deformation and plastic deformation), and the overburden permeability decreases with the increase of the number of the net stress changing process, indicating that the rock sample has some plastic deformation when the net stress increases and the rock sample is compressed. The phenomenon that the overburden permeability of reservoir rock decreases with the increase of net stress is the “stress sensitivity” of the rock permeability, and the phenomenon that the overburden permeability of the reservoir rock decreases with the increase of net stress changing cycle is called the “cyclic stress sensitivity” of rock permeability.

Currently, power function, exponential function, binomial function or logarithmic function are often used to describe the relationship between the overburden permeability and net stress^[20]. According to the experimental results (Fig. 1), the overburden permeability and net stress in this area show a good power function relationship, so the power function can be used to describe the relationship between the overburden permeability and net stress in a single net stress changing process, i.e.:

Transform Eq. (1) into the dimensionless form as follows:

where a denotes the dimensionless overburden permeability when the dimensionless net stress (ratio of net stress to initial net stress) is 1 in each net stress changing process, which reflects the retention rate of the overburden permeability under the initial net stress in each net stress variation process. In the first net stress changing process, the overburden permeability (K) under the initial net stress is the initial overburden permeability (K_i). Therefore, the value for a in the first net stress variation process is 1. The expression of 1-a reflects the rate of reduced overburden permeability of the rock caused by the close of pore throats due to plastic deformation when the net stress rises and the rock is compressed, which is usually called irreversible permeability loss rate. b reflects the strength of permeability sensitivity to stress; the larger the value of b, the faster the dimensionless overburden permeability decrease with the increase of dimensionless net stress, and the higher the permeability sensitivity to the stress will be, and vice versa.

According to the experimental results and Eq. (2), the dimensionless overburden permeability and dimensionless net stress were obtained (Fig. 3). Through regression analysis, the permeability retention rate and the stress sensitivity index of permeability of each rock sample in each net stress changing process were determined (Table 3).

Fig. 4 shows the relationship curves between the permeability retention rate and the operation cycles of gas storage of the different rock samples. It can be seen with the increase of operation cycles number, the increase of net stress compresses the rock sample and causes plastic deformation, so the number of closed throats in the rock sample increases, accordingly, decreasing amplitude of overburden permeability goes up, the permeability retention rate goes down, and the permeability retention rate decrease with the increase in the operational cycles’ number of gas storage. With the increase in the operational cycles number of gas storage, the rock sample becomes tighter, its cyclic plastic deformation degree decreases, and the number of closed throats decreases. Accordingly, the decreasing amplitude of cyclic overburden permeability of rock sample decreases, and the decreasing speed of permeability retention rate with the increase of operation cycles number of gas storage goes down.

It can be seen from the relationship curves between the stress sensitivity index of permeability and net stress variation process of different rock samples (Fig. 5): (1) In net stress rising processes, the stress sensitivity index of permeability decreases with the increase in the number of net stress changing processes. As with the increase of number of net stress changing processes, the rock samples become tighter, and the number of throats that can be closed under compression decreases continuously. Consequently, the decreasing amplitude of permeability of the rock samples reduces and the stress sensitivity index of permeability decreases. (2) In the processes of net stress decrease, the stress sensitivity index of permeability hardly changes with the increase in the number of net stress changing processes. Since only the elastic deformation can recover when the net stress decreases, and the stress changes in all rising processes of net stress are the same, so the elastic deformation laws in the rising processes of net stress are the same, and the elastic deformation recovery laws in the net stress decrease processes are the same. Therefore, the stress sensitivity index of permeability does not change with the increase of the net stress decrease processes number.

It can also be seen from Fig. 5 that with the increase in the number of net stress changing processes, the stress sensitivity indexes of permeability in the rising process and decreasing process of net stress of some rock samples keep approaching each other. During the increase of net stress, the rock has both elastic and plastic deformations; while during the decrease of net stress, only elastic deformation can recover. For a specific rock sample, the sum of the elastic and plastic deformations during a net stress increase process is greater than the elastic deformation during a net stress decrease process, so the stress sensitivity index of permeability during the increase of net stress is greater than that during the decrease of net stress. When the plastic deformation in a rising process of net stress approaches 0 with the increase in the number of net stress changing processes, the deformation during the increase of net stress of the rock sample is equal to that during the decrease of net stress. In this case, the stress sensitivity index of permeability in the rising process of net stress is equal to that in the decreasing process of net stress.

It can be seen from the relationship curve between the irreversible loss rate of permeability and permeability of different rock samples in the same operation cycle of gas storage (Fig. 6) that the lower the permeability of the rock sample, the greater the irreversible loss rate of permeability, and the stronger the cyclic stress sensitivity of the rock sample will be. This is mainly because the lower the permeability of the rock sample, the more the number of throats will be closed due to plastic deformation when the rock sample is compressed during the increase of net stress.

It can be seen from the relationship curve between the stress sensitivity index of permeability and permeability of different rock samples in the same stress changing process (Fig. 7) that the lower the permeability of the rock sample, the higher the stress sensitivity index of the permeability and the stronger the stress sensitivity of the rock sample will be. This is because the lower the permeability of the rock sample, the smaller the pore throat radius of the rock sample, the more throats will be closed when the net stress rises and the rock sample is compressed, and the greater the decreasing amplitude of the permeability of the rock sample will be.

According to the above analysis, Eq. (2) can be used to describe the relationship between the overburden permeability and net stress of a rock sample in a single process of net stress change, and in Eq. (2), both the permeability retention rate and stress sensitivity index of permeability are related to rock permeability and the operation cycle of gas storage.

Relationship between the stress sensitivity index of permeability and permeability of the core samples in Table 3 are shown in Fig. 8. Regression analysis of these data shows that in the same net stress changing process, the stress sensitivity index of permeability and permeability of the rock sample show a better power function relationship (with relevant regression parameters shown in Table 4), and can be expressed by a generalized function relationship as follows:

In the processes of net stress decrease, as the stress sensitivity index of permeability of a specific rock sample hardly changes with the increase in the number of net stress changing processes (Fig. 9). Therefore, e and f remain almost unchanged, and the relationship curves between e and f with the number of net stress changing processes are basically horizontal lines. The values of e and f in each decreasing process of net stress were averaged, and then substituted into Eq. (3), to obtain the calculation model for stress sensitivity index of permeability in the net stress decreasing process of the gas storage as follows:

During an increasing process of net stress, with the increase in the number of net stress changing processes, the stress sensitivity index of permeability of a specific rock sample decreases and approaches that of the decreasing process of net stress. The lower the permeability of the rock sample, the fast the decreasing and approaching speed of the rock sample is (Fig. 8). Therefore, with the increase of the number of net stress changing processes, e and f in the rising process of net stress gradually approach to e and f in the decreasing process of net stress (Fig. 9).

Δe and Δf between the increasing and decreasing processes of net stress show good power function relationships with the number of net stress changing processes. Through power function regression analysis, the relationship models between Δe and Δf with rising processes of net stress can be established (Fig. 10). During the decreasing processes of net stress, e and f are constant values, 0.006 383 and 0.344 7 respectively. Then, the relationship models between e and f with the operation cycles of gas storage in the rising process of net stress are as follows:

By substituting Eqs. (5) and (6) into Eq. (3), the calculation equation for the stress sensitivity index of permeability during the increasing process of net stress is as follows:

If the rock permeability is given, Eq. (4) can be used to calculate the stress sensitivity index of permeability in the decreasing process of net stress, and Eq. (7) can be used to calculate the stress sensitivity index of permeability in the rising process of net stress during any operation cycle of gas storage.

The relationships between the dimensionless overburden permeability and dimensionless net stress in different net stress changing processes are continuous. The dimensionless net stress and dimensionless overburden permeability at the end of each net stress changing process are the starting points of the following net stress

changing process (Fig. 3). The permeability retention rate in the first net stress changing process is always 1. According to the continuity of the relationship curve between the dimensionless overburden permeability and dimensionless net stress, when the permeability of the reservoir rock is given, the permeability retention rate for any net stress changing process can be calculated recursively through combination of Eqs. (2), (4), and (7).

With rock permeability and operation cycle of gas storage given, the stress sensitivity index of permeability and permeability retention rate can be calculated. The dimensionless overburden permeability under the given dimensionless net stress can be calculated by using Eq. (2). The permeability of core sample measured in laboratory and the reservoir permeability derived from well logging interpretation are generally rock permeability under smaller net stress. Therefore, after the dimensionless overburden permeability under the given dimensionless net stress is calculated, to calculate the rock’s overburden permeability, it is necessary to establish the conversion relation between the initial overburden permeability (the overburden permeability under the initial net stress in the first process of stress changing) and permeability of the rock (the permeability of rock under a lower net stress).

Based on analysis of test results of the six core samples (Fig. 11), the relationship between the initial overburden permeability and permeability of the gas storage is as follows:

The overburden permeability is

After the dimensionless overburden permeability is calculated, the overburden permeability can be calculated with Eq. (9). With given permeability of the reservoir rock, by combining Eqs. (2), (4), (7), and (9), the overburden permeability of the rock at any operational cycle of gas storage and any net stress can be calculated with the above-mentioned method.

The reservoirs of Wen 23 gas storage have an average permeability of 1.001×10^-3 μm², and average thickness of 69 m. The wells in the gas storage have an average well spacing of 300 m, and a wellbore radius of 0.078 55 m. Based on the average reservoir permeability, the overburden permeability at different gas storage operational cycles under different net stresses can be calculated by using the method proposed in this paper. Then, the equation for radial quasi-steady flow production in the gas plane^[21] can be used to calculate the open flow capacity of a gas well as follows:

${{\psi }_{\text{R}}}-{{\psi }_{\text{wf}}}=\left[ \frac{{{p}_{\text{sc}}}T}{\text{1}{{\text{0}}^{\text{3}}}\pi {{K}_{\text{g}}}h{{Z}_{\text{sc}}}{{T}_{\text{sc}}}}\left( \ln \,\frac{{{r}_{\text{e}}}}{{{r}_{\text{we}}}}-\frac{3}{4} \right) \right]{{q}_{\text{gsc}}}+$

From the relationship curves between overburden permeability, open flow capacity of gas well, and operation cycle of gas storage (Figs. 12 and 13), it can be seen that the overburden permeability decreases rapidly in the initial stage and slowly in the later stage with the increase of operation cycles of gas storage. The open flow capacity of gas well (production capacity or injection capacity of gas well) also decreases rapidly in the initial stage and slowly in the later stage.

At the initial overburden permeability of reservoir of 0.772 9×10^-3 μm², the calculated overburden permeability under the initial net stress in the 30^th operation cycle of gas storage is 0.693 5×10^-3 μm²; the irreversible loss rate of permeability is 10.27%; the initial open flow capacity of the gas well is 62.19×10⁴ m³/d, and the open flow capacity of the gas well under the initial net stress in the 30^th operation cycle of gas storage is 57.41×10⁴ m³/d, with a loss rate of 7.68%.

The calculated overburden permeability under the maximum net stress in the 30^th operation cycle of gas storage is 0.669 7×10^-3 μm², which is 0.103 2 ×10^-3 μm² lower than the initial overburden permeability, with a loss rate of 13.35%. The calculated open flow capacity of the gas well under the maximum net stress in the 30th operation cycle of the gas storage is 55.94×10⁴ m³/d, which is 6.25×10⁴ m³/d lower than the initial open flow capacity, with a loss rate of 10.05%.

The production-injection capacity of the gas well decreases most significantly during the first five operational cycles of the gas storage, and then it decreases slowly. The loss rates of open flow capacity under the initial net stress and the maximum net stress during the first five operational cycles are 6.21% and 8.78%, respectively, accounting for 80.86% and 87.36% of the total loss of open flow capacity of gas well in all 30 operational cycles.

Based on the variation law of overburden permeability during multi-cycle injection and production from the above study, the lower the rock permeability, the stronger the stress sensitivity and the cyclic stress sensitivity will be (Figs. 6 and 7), and the more significantly the injection-production capacity of gas well will decrease. Reservoirs with too low permeabilities are therefore not suitable for gas storage.

The reservoirs in the Wen 23 gas storage have a permeability range from 0.01×10^-3 μm² to 100.00×10^-3 μm². The total permeability loss rate was calculated by the method for calculating overburden permeability during multi-cycle injection production. The results show that when the reservoir permeability is lower than 0.30×10^-3 μm², the total permeability loss rate increases sharply with the decrease of permeability. Therefore, the minimum permeability of selected reservoirs in the Wen 23 gas storage is set at 0.30×10^-3 μm². The reservoirs with permeability lower than the lower limit have stronger cyclic stress sensitivity and stress sensitivity, and therefore can’t meet the demand of intensive injection and production of gas storage.

The retention rate of rock permeability decreases rapidly initially and then more slowly with the increase of operating cycles of gas storage. During the increase of net stress, the stress sensitivity index of permeability decreases with the increase in the number of net stress changing processes; while during the decrease of net stress, the stress sensitivity index of permeability hardly changes with the increase in the number of net stress changing processes. With the increase of the number of net stress changing processes, the stress sensitivity index of permeability of a rock sample during the increase of net stress tends to approach that of the rock sample during the decrease of net stress.

The lower the permeability of the rock, the higher the irreversible loss rate of permeability, and the higher the cyclic stress sensitivity of the rock will be. The lower the permeability of the rock, the higher the stress sensitivity index of permeability, and the higher the stress sensitivity of the rock will be.

The stress sensitivity of reservoir permeability has a significant influence on the injection-production capability of the gas well, and its effect is most obvious during the first few cycles.

a—permeability retention rate;

b—stress sensitivity index of permeability;

C_UGS—number of operational cycles of gas storage;

e—coefficient of stress sensitivity index equation;

f—exponent of stress sensitivity index equation;

h—effective thickness of reservoir, m;

K—overburden permeability, 10^-3 μm²;

K₀—permeability of the rock sample, 10^-3 μm²;

K_D—dimensionless overburden permeability;

K_i—initial overburden permeability (the overburden permeability under the initial net stress during the first net stress changing process), 10^-3 μm²;

K_g—effective permeability of the gas phase, 10^-3 μm²;

m—coefficient of the relation equation of overburden permeability and net stress;

n—index of relation equation of overburden permeability and net stress;

p_eob—net stress, MPa;

p_eobi—initial net stress, MPa;

p_sc—pressure under standard conditions, MPa;

q_gsc—gas production of gas well under standard conditions, m³/ks;

r_e—discharge radius of gas well, m;

r_we—equivalent radius of wellbore, m;

T—formation temperature, K;

T_sc—temperature under standard conditions, K;

Z_sc—gas compression factor under standard conditions;

β—non-Darcy seepage coefficient, pm^-1;

Δe—coefficient difference of the stress sensitivity index equations during an increase and decrease processes of net stress;

Δf—exponent difference of the stress sensitivity index equations during an increase and decrease processes of net stress;

$\mu_{gi}$—gas viscosity under initial formation conditions, mPa·s;

$\rho_{gsc}$—gas density under standard conditions, kg/m³;

$\psi_{wf}$—pseudo-pressure of gas under bottomhole flowing pressure, MPa²/(mPa·s);

$\psi_{R}$—pseudo-pressure of gas under average formation pressure, MPa²/(mPa·s).