Different from the one-way low rate production in conventional gas fields which have been developed for 10-30 years (a gradual decline process of gas reservoir pressure), the multi-cycle operation of gas field storage facility (hereinafter referred to as “gas storage”) has the characteristics of short-term intensive injection and production, high individual well production (injection) proration and frequent injection-production conversion, so as to meet the need of market peak shaving and accident emergency in freezing conditions. The continuous intensive injection and production operation of gas storage requires more strictly on the interpretation and prediction accuracy of reservoir permeability, well controlled inventory and gas injection and production capacity of gas well, so as to guide the reasonable individual well production (injection) proration and the adjustment and optimization of working system in real time, give full play to the peak shaving capacity of gas storage and realize safe and efficient operation ^[1,2,3].

The transient analysis is commonly used in oil and gas field production, which is represented by Blasingame method. In this method, the mathematical model for characterizing the transient features of individual well production rate is established by quantitatively analyzing the daily production dynamic data of oil and gas wells. Then the reservoir permeability, skin factor and well controlled reserve and other parameters are obtained by means of fitting. Finally, the production dynamic prediction of individual well and well group is conducted according to related theoretical model. This method has the advantages of reliability in theoretical method, high interpretation accuracy, low field application cost and strong popularization practicability ^[4,5,6,7]. Combined with well test interpretation method, Blasingame method has become important technical means for dynamic description of oil and gas reservoir. To overcome its limitation, domestic and foreign scholars successively put forward different typical curve fitting analysis methods, such as Agarwal-Gardner, normalized pressure integral and flow material balance ^[8,9,10,11], and carry out plenty of studies on the transient analysis methods which consider the influential factors (e.g. stress sensitivity and variable fracture conductivity) under the conditions of vertical well, vertical fracture well and horizontal well in homogeneous, radially combined, dual and triple porous media. The application in the oil and gas field production has become mature increasingly ^{[12,13,14,15,16,17,18,19,20,21]}.

The gas storage is rebuilt from the gas field at the middle and late production stages and even the depleted stage. The working condition is characterized by real-time variable rate of alternating injection and production. The requirement is higher for acquisition density and accuracy of key data (individual well flow rate and pressure) in the cyclic operation than at the stage of gas field production. Based on abundant high-frequency dynamic data acquired by means of continuous dense monitoring ^[22], the existing transient analysis methods have been used to the dynamic description of gas storage. Previously, the individual well injection and production dynamic transient analysis method was studied by referring to classic theories and methods, such as Blasingame. Wang et al. ^[23] put forward a well-control dynamic evaluation method of gas injection into gas storage, which introduced the normalized pseudopressure function and material balance pseudotime function to establish the dimensionless gas injection rate and material balance pseudotime theory chart, and derived the calculation formulas of well controlled inventory and other parameters of gas injection well. Chen et al. ^[24] established the transient analysis chart of gas injection based on the flow model of gas production, by taking the gas injection process as the reversal flow of gas production, performed fitting analysis on different gas injection cycles and predicted the gas injection capacity of gas storage. These studies have proved that introducing the classic transient theories and methods into the gas injection analysis model can basically provide transient analysis on the individual well injection and production flow state in different cycles and obtain the reservoir parameters in the state of pseudo steady flow. However, the interpretation reliability and accuracy of the analysis model is relatively low, so the interpretation results of reservoir parameters of neighboring cycles in some wells are different with weak correlation. It is indicated that the complex and special working condition of frequent injection-production conversion in gas storages cannot be reflected by the assumed conditions in the mathematical model of existing transient analysis methods, resulting in imprecise theoretical analysis and limited method application. Therefore, it is necessary to further innovate and improve the transient analysis model of injection and production and establish the theoretical chart of gas injection and production in line with the multi-cycle working characteristics of gas storage, in order to improve the reliability and accuracy of cyclic transient analysis and parameter interpretation of injection and production wells.

After analyzing the injection and production operation characteristics of gas storage rebuilt from strong-heterogeneity gas reservoir deposited in the continental environment in China and the limitations of existing transient analysis methods, in this study, a new injection and production dynamic transient analysis model of “three points and two stages” has been proposed, which is suitable for the alternating working condition of gas storage well based on the characteristics of unbalanced and transient reservoir pressure under the conditions of short-term high rate injection and production and frequent working condition conversion in gas storages. In addition, the mathematical model of gas storage injection and production dynamic transient analysis has been established, and the distribution of reservoir pressure in the spatial domain has been solved by means of numerical method, so that the mismatching problem of the initial value conditions in existing analysis models in the time domain is settled down effectively. What’s more, the atypical characteristic chart of the new model is established. The simulation and field case analysis verify the reliability and accuracy of the transient analysis and interpretation results of the new model.

Compared with the one-way production of conventional gas fields, the injection and production of gas storage has the following characteristics: (1) Frequent injection-production conversion and complex working conditions. A complete injection and production operation shall be completed within one operation cycle of gas storage (generally 1 year). Gas injection period is generally about 200 d, gas production period is about 120 d and shut-in equilibrium period at the end of gas production is about 15-20 d, when surface equipment and wells are maintained and reservoir pressure and temperature are tested. In exceptional circumstances, however, such as abrupt temperature change or pipeline accident, the gas storage shall turn into gas production (injection) period instantly from gas injection (production) period or starts injection or production operation by shortening the balance period. The working conditions are complex and convert frequently, which lead to abrupt increase or decrease of reservoir pressure and temperature in short time. Therefore, the difficulty in testing will increase. (2) The individual well injection and production rate is high, and gas production speed is fast. Generally, the average daily production (injection) proration of each gas storage well is 3-5 times that in the stage of gas field production and even higher. According to the scheme design, the gas produced within 120 d shall be about 50% of the dynamic reserves of a gas field of the same scale, and the converted gas production speed is up to 150%. The individual well high rate injection and production in short time further will enhance the influence of reservoir heterogeneity, so the flow state of reservoir fluid will be greatly different from that in the stage of gas field production. (3) The high-rate injection and production and frequent working condition conversion in a gas storage cause the frequent continuous fluctuation of reservoir pressure. What’s more, the intensive injection and production in short time will force natural gas to accumulate preferentially in the large pores with lower flow resistance and then be produced. Macroscopically, the effective pore space that can be produced under the high-rate injection and production condition of gas storage is smaller than that during the “low-rate” production of gas field. The pseudopressure curve shows that when the volume of reserves is equal, the pseudopressure of gas storage is higher than that in the stage of gas field production during gas injection, but the pseudopressure in the period of gas production is lower than that in the stage of gas field production. The pseudopressure curve of multi-cycle injection and production of gas storage presents hysteresis characteristics of different degrees, which are the most obvious at the early stage of capacity expansion and production. The frequent fluctuation of reservoir pressure and the hysteresis effect during the process of injection and production make the dynamic transient analysis more difficult.

The core basic variables in existing transient analysis of gas field production are material balance pseudotime and the normalized production rate of pseudopressure, whose mathematical definitions are shown in Eqs. (1) and (2), which assumes the initial reservoir pressure (p_i) as a constant with uniform distribution.

During actual injection and production operation of gas storages rebuilt from strong-heterogeneity gas reservoirs in China, particularly, the high-rate injection and production of each well further enhances the influence of reservoir heterogeneity and the reservoir pressure is in the obvious unbalanced and transient state at the end of high-rate intensive injection (production) in short time. In addition, the balance period of injection and production operation is generally 15-20 d, and the reservoir pressure cannot recover to the stable and balanced state within short time, so the gas storage has to step into the next production (injection) period before the reservoir pressure reaches the stable and balanced state. As a result, the reservoir pressure is in the transient state of continuous dynamic change. The stronger the reservoir heterogeneity is, the faster and more frequent the injection-production conversion of gas storage is and the larger the change range of injection and production rate is. These phenomena make transient characteristics of reservoir pressure more obvious, which is not accordant with the assumed initial conditions in the mathematical model of existing transient analysis methods, so larger parameter fitting prediction error is generated. It is indicated by applying existing methods to analyze the transient flow in a single gas production period that tiny change of initial reservoir pressure in the analysis model can result in obvious pattern change of gas well transient curve and even the phenomenon of fitting data points separating from the typical chart (Fig. 1), which will increase the interpretation errors of key reservoir parameters (e.g. permeability). The interpreted permeability at the initial reservoir pressure of 35.5 MPa and 35.0 MPa are 4.3×10^-3 and 1.5×10^-3 μm², respectively.

In order to establish the injection and production dynamic transient analysis method suitable for the alternating working conditions of gas storage, the complex working conditions of gas storage are simplified according to the principle that the injection and production process of each cycle shall be analyzed independently. In order to accurately obtain the average reservoir pressure at the start point of each injection and production analysis stage, the initial conditions of the analysis model are reconstructed, and the analysis concept of “three points and two stages” suitable for the operation characteristics of gas storage is put forward. Three points refer to three time points of cyclic gas injection and production analysis, i.e., start point of gas injection for gas storage construction (t₀), start point of injection and production analysis stage (t_s) and end point of injection and production analysis stage (t_e). Two stages refer to historical flow stage and injection and production analysis stage of cyclic gas injection and production analysis (Fig. 2).

As for the gas storages rebuilt from the gas reservoirs at the middle and late production stages, the gas reservoirs have generally been developed in the model of one-way production for 10-30 years before they are rebuilt into gas storages, and most of them have been shut in for a longer time. Therefore, it is deemed that the reservoir pressure has reached the balanced and stable state before a gas reservoir is rebuilt into a gas storage, which is in line with the theoretical assumption of existing transient analysis methods that reservoir pressure is constant at the initial moment. In this paper, this moment is defined as the initial moment of historical flow stage (t₀). The reservoir pressure at this moment can be obtained more accurately by means of field test and it is a constant. The change of flow rate and pressure during the depletion production of gas reservoir is not taken into consideration.

t_s is the start point of injection and production analysis stage, as well as the end point of one or multiple injection and production cycles after gas injection starts in a gas storage at t₀. The stage from t₀ to t_s is defined as the historical flow stage of gas storage injection and production dynamic analysis. Under the working conditions of high-rate injection and production in the gas storages rebuilt from strong-heterogeneity gas reservoirs, the reservoir pressure at the moment of t_s cannot be obtained accurately by means of field test, but can be only calculated by using the transient mathematical model of historical flow stage.

t_e is the end point of injection and production analysis stage. The stage from t_s to t_e is defined as the analysis and prediction stage of gas storage injection and production dynamic analysis, which is a single gas production or injection period that needs analyzing. Pressure or flow rate is predicted by using the transient mathematical model of analysis stage. The initial conditions used for model solution and prediction are obtained by solving the mathematical model of historical flow stage, while the preparation of related theoretical charts and the fitting and interpretation of gas storage injection and production dynamic data are carried out in the stage of injection and production analysis.

2.2.1. Basic assumptions

Taking a vertical well in the center of the reservoir with circle, homogeneous and closed boundary as the example, the basic assumptions to establish the mathematical model are as follows: (1) The reservoir is isotropic with the same thickness laterally, and the gas is single-phase flow and follows Darcy flow laws; (2) The reservoir pressure is a constant with uniform distribution before gas storage construction, but its distribution is not uniform after gas storage construction, which is in close relation to injection and production history; (3) Gas compressibility coefficient and viscosity vary with pressure, so gas compressibility coefficient and deviation factor are calculated by using DPR (Dranchuk-Purvis-Robinson) method, and gas viscosity is calculated by using Lee method ^[25]; (4) The influences of gravity and capillary pressure are ignored, and the influences of other factors(such as wellbore storage effect and temperature) on the flow are not considered.

2.2.2. Mathematical model

Under these assumed conditions, based on the gas transient flow theory, material balance pseudotime and pseudopressure are introduced to deal with the problem that real-time bottom hole flowing pressure, variable production rate and PVT property of gas storage vary with pressure. The dimensionless control equation of injection and production dynamic transient analysis under the alternating working condition of injection and production in gas storages is as follows:

where the dimensionless variables are defined as:

${{\psi }_{\text{D}}}=\frac{Kh\left( {{\psi }_{\text{i}}}-\psi \right)}{1.842\times {{10}^{-3}}q{{\mu }_{g}}{{B}_{\text{g}}}}$ ${{t}_{\text{caDd}}}=\xi \frac{3.6K{{t}_{\text{ca}}}}{\phi {{\mu }_{g}}{{C}_{\text{t}}}r_{\text{wa}}^{2}}$

${{t}_{\text{ca}}}=\frac{{{\mu }_{\text{gi}}}{{C}_{\text{ti}}}}{q}\int\limits_{0}^{t}{\frac{q}{{{\mu }_{\text{g}}}{{C}_{\text{t}}}}\operatorname{d}t}$ ${{r}_{\text{D}}}=\frac{r}{{{r}_{\text{wa}}}}$ ${{r}_{\text{eD}}}=\frac{{{r}_{\text{e}}}}{{{r}_{\text{wa}}}}$

${{r}_{\text{wa}}}={{r}_{\text{w}}}{{\text{e}}^{-S}}$ $\psi (p)=\frac{{{\mu }_{\text{gi}}}{{Z}_{\text{i}}}}{{{p}_{\text{i}}}}\int\limits_{0}^{p}{\frac{p}{{{\mu }_{\text{g}}}Z}\operatorname{d}p}$

The evaluation in the injection and production analysis stage (t_s<t<t_e) mainly focuses on the change laws of gas well pressure and production rate during the process of operation. Based on the production rate transient analysis method of normalized pressure integral, $\xi $in Equation (3) is defined as$\xi =1/\pi \left( r_{\text{eD}}^{2}-1 \right)$. The initial reservoir pressure in the stage of injection and production analysis does not distribute uniformly (t=t_s), which is in close relation to the injection and production history in the historical flow stage, and its distribution is the function of reservoir space, with mathematical model as follows:

To obtain the distribution of initial reservoir pressure in the injection and production analysis stage based on the data in the injection and production historical flow stage (0≤t≤t_s), it is necessary to conduct the calculation by using the general flow equation, based on the data of historical flow pressure, and set the $\xi $ in Equation (3) at 1. The initial reservoir pressure in the injection and production historical flow stage is a constant with uniform distribution and its initial condition is shown in Equation (5). In the historical flow stage, the balance pressure point of gas storage construction is taken as the condition at the moment of t=0. By virtue of this treatment method, the actual production situations and the model initial conditions can be satisfied simultaneously, and compared with the initial reservoir pressure of conventional gas field production, the influence of historical performance of gas field production doesn’t have to be taken into consideration, so that the model calculation load and difficulty are decreased greatly.

Inner boundary condition of gas storage injection and production dynamic transient analysis:

where$~{{q}_{\text{D}}}=\frac{q\left( t \right)}{{{q}_{\text{sc}}}}$

Outer boundary condition:

The distribution of initial reservoir pressure in the injection and production analysis stage needs solving in the model, and the analytic solution method can obtain the pressure value of all points in the reservoir by means of the superposition principle, but to calculate the pressure value of each point, each superposition has to be conducted from the start point, so the calculation load is heavy. In this case, the numerical method is adopted, which can calculate the pressure value of all points in the reservoir through one iteration. The model is solved by means of the finite difference method:

The difference equation set of reservoir control equation is obtained by ignoring the second-order derivative term, introducing r_D=e^x into Equation (3) and transforming it into difference form:

The difference equation of well grid can be obtained by introducing the inner boundary condition into the difference equation set of control equation:

Similarly, the difference equation of boundary grid can be obtained by introducing the outer boundary condition:

Above mentioned equations constitute the linear equation set of the model of injection and production analysis stage, and they are iteratively solved based on the initial conditions. The distribution function of initial pressure f(r_D) shall be calculated by using the model of injection and production historical flow stage. The difference discrete process in the model of historical flow stage is similar to that in the model of injection and production analysis stage, and its initial reservoir pressure distributes uniformly. Therefore, the pressure distribution at different moments can be calculated by using the linear equation set of this flow stage, and then the dimensionless pseudopressure solution of the model of injection and production analysis stage is obtained.

The dimensionless pseudopressure of injection and production analysis stage (ψ_D) is transformed from the balance pseudopressure of gas storage construction (ψ_i). The complex alternating working conditions of injection and production in gas storages make the selected pseudopressure of injection and production analysis stage (ψ) higher or lower than the balance pseudopressure of gas storage construction (ψ_i). As a result, the calculated ψ_D may be positive or negative, which will influence the preparation of log-log chart. In this case, the dimensionless average reservoir pseudopressure at this moment is calculated by using the area integral method, based on the pressure distribution f (r_D) at the initial moment of injection and production analysis stage:

On this basis, ${{\psi }_{\operatorname{D}}}$of injection and production analysis stage is transformed into positive value by calculating the difference between the calculated dimensionless pseudopressure of injection and production analysis stage and the dimensionless average reservoir pseudopressure, according to the injection and production situations, and thus the new dimensionless pseudopressure is obtained. For injection process:

For production process:

What’s more, the measured data are also treated similarly to match the theoretical chart. The average reservoir pressure at the start point of injection and production analysis stage is calculated based on historical flow rate and material balance equation, and the corresponding average reservoir pseudopressure is calculated according to the definition of pseudopressure. Then, the material balance pseudotime and normalized pseudopressure of measured pressure data is calculated. For injection process:

For production process:

The influence of data scattering is eliminated by using the normalized pressure integral form of production rate^[8], and modified dimensionless pseudopressure integral and dimensionless pseudopressure integral derivative are calculated respectively.

Based on chart fitting and measured data calculation, the calculation formulas of interpretation parameters (e.g. permeability, well control radius, skin factor and well controlled reserve) in the model can be obtained by means of drag fitting of measured data, combined with pressure fitting point and time fitting point.

By virtue of the proposed gas storage injection and production dynamic transient analysis model of “three points and two stages” and numerical solution method, typical multi-cycle injection and production dynamic curves based on normalized pressure of production rate under different conditions of historical flow were plotted, including dimensionless pseudopressure, dimensionless pseudopressure integral and dimensionless pseudopressure integral derivative. Fig. 3 shows the muti-cycle injection and production dynamic curve of gas storage, including two complete injection and production cycles.

The typical curves of the gas production period 2 (Fig. 4) are plotted based on the dimensionless pseudopressure which is solved by using the new injection and production dynamic transient analysis model. It is shown that the typical curves are similar to the typical curves of existing transient analysis methods (Fig. 1) in the overall shape, but they are different in the specific curve characteristics. The problem that log-log graph cannot be plotted because of the calculated negative value can be eliminated to a certain degree based on the difference from the average reservoir pseudopressure. Compared with the curves of existing transient analysis methods (Fig. 5), the dimensionless pseudopressure curve and dimensionless pseudopressure integral curve of the new injection and production dynamic transient analysis model drop at the front ends, and the dimensionless pseudopressure integral derivative curve is convex upward at the middle stage.

The theoretical curves of different injection and production flow periods (Fig. 6) are compared further. It is shown that curves of different flow periods are accordant in the overall trend, but dimensionless pseudopressure curve and dimensionless pseudopressure integral curve deviate upward and downward at the front ends in different injection and production cycles, and the middle part of pseudopressure integral derivative curve is convex upward in two gas production periods and concave downward in two gas injection periods. The phenomenon of convex upward and concave downward in the middle part of integral derivative curve is caused mainly by the nonuniform distribution of initial reservoir pressure in the injection and production analysis stage. However, the distribution and typical curves of dimensionless reservoir pseudopressure at the initial moment of gas production period 1 are plotted respectively without considering the variation of flow rate in this period (Figs. 7 and 8).

The distribution curve of dimensionless reservoir pseudopressure shows that the pressure distribution of gas injection period 1 at different flow rates is in unbalanced state. The pressure varies greatly with flow rate. And combined with the comparison diagram of typical curves, it is indicated that the more unbalanced the distribution of reservoir pressure is, the higher the convex degree in the middle part of the curve is. The change trend of pressure distribution with the distance from r_D in gas production period is contrary to that in gas injection period, so its curves are concave downward.

To sum up, due to the influence of injection and production flow history, the typical curves of different injection and production analysis stages are different, which means different injection and production stages have different typical charts. Therefore, the injection and production dynamic analysis chart under the alternating working conditions of gas storage is atypical, which makes the interpretation and analysis results more accordant with the special working conditions of gas storage.

The analysis process of new injection and production dynamic transient analysis model is shown in Fig. 9. Firstly, take the average reservoir pressure, gas well permeability and skin factor tested at the moment of t₀ as initial parameter values, and calculate the reservoir pressure distribution at the moment of t_s based on the injection and production dynamic data of historical flow. Then, plot theoretical charts and measured data curves of injection and production analysis stage, and interpret reservoir parameters by means of drag fitting of measurement curves and theoretical charts. Afterwards, compare interpretation parameters with initial parameters. If the accuracy requirement is not satisfied, the fitted reservoir parameters will be taken as the iterative values to repeat theoretical chart plotting and measurement curve fitting and interpretation until the required accuracy is reached, which means the ultimate interpretation result is obtained. The interpretation parameters in the new model include reservoir permeability, skin factor, well control radius and well controlled inventory, which can be used for productivity evaluation, inventory analysis and injection and production dynamic prediction during the following optimized operation of gas storage.

The pressure was calculated by using the natural gas transient flow equation, based on the given flow rate, permeability and well control radius. The typical curves were plotted based on bottom hole pressure and flow rate and then were interpreted by fitting with the newly proposed theoretical curves. The accuracy of interpretation results was compared with that of given parameters to verify the new model. For two flow periods with injection first and then production, the gas production period was selected to be interpreted and analyzed by using the new model and the existing transient analysis model. The fitting results of two methods are shown in Fig. 10 and Fig. 11, respectively.

The fitting result of integral derivative curve of the existing transient analysis model is worse, while the fitting result of theoretical curve and measured data of the new model is better and the convex characteristics of data can be best fitted. The drop of measured data at the front end is mainly caused by integral and integral derivative difference solution method. The fitting results of two methods are shown in Table 1. It is shown that the fitting result of the new model is basically in complete accordance with the given permeability and well controlled inventory, but there is a certain error between the fitting result of the existing transient analysis method and the given parameters. Chart fitting results and interpretation parameters preliminarily verify the accuracy of the new model.

The newly proposed injection and production transient analysis model applicable to the alternating working conditions of gas storage has been used to carry out injection and production dynamic analysis on Well H14 of Hutubi Gas Storage. The analysis result was compared with the fitting result of the existing model to prove the reliability and practicability of the new model. The reservoir in Well H14 has mid depth of 3553.8 m, effective thickness of 35 m, porosity of 15.5% and temperature of 84.78 °C. The static pressure before gas storage construction is 24 MPa, tubing ID is 0.0762 m and relative natural gas density is 0.6. The initial reservoir pressure of the principle production layer is 34.0 MPa, and before it is rebuilt into gas storage, its tested average reservoir pressure is 14.4 MPa. The daily gas injection and production and tubing pressure in five injection and production cycles are shown in Fig. 12. The flow rate is negative during gas injection and positive during gas production. The last gas production period is taken as the analysis stage, and the initial reservoir pressure is set at 32.4 MPa in the existing transient analysis method. The fitting results of existing transient analysis model and new model are shown in Fig. 13 and Fig. 14, respectively. The fitting results of multi-cycle whole history flow rate curve by two methods are shown in Fig. 15.

During the fitting process of the existing transient analysis method, only the chart and the pressure history in the selected analysis stage were fitted, while the whole pressure history in the whole injection and production cycle of gas storage was not fitted. The analysis results show that the chart can be fitted to a certain degree, but the fitting result of pressure history in the whole injection and production cycle is worse and the pressure curve drifts upward overall, which worsens the reliability of the interpretation results. It is indicated by comparing the fitting diagram of the existing model with that of the new model that the dimensionless pseudopressure integral derivative curve was upwarp earlier (in the middle period), which is completely accordant with the theoretical curve in morphology (Fig. 5). By precisely fitting the dimensionless distance of the upwarping segment, the whole curve cluster was better fitted, the fitting degree of the pressure in the whole injection and production cycle has been improved and the initial reservoir pressure doesn’t need adjusting.

The fitting results of two methods are shown in Table 2. They are different to some extent in fitted well controlled inventory, permeability and skin factor.

Fitting and interpretation of Well H14 in different injection and production cycles were conducted by using the new model (Table 3). The analysis results show that with the advance of injection and production cycle, permeability, well control radius and well controlled inventory present are in increasing trend overall, indicating that the intensive injection and production of this well in the injection and production process has improved the reservoir permeability and ramp up effect to some extent, so as to enlarge the well control range. In the meantime, skin factor presents decreasing trend overall, indicating that the near wellbore reservoir has been improved to some extent during the injection and production process of this well. It is demonstrated that the advantage of injection and production dynamic analysis is the significant improvement of the correlation of segmental analysis.

The newly proposed mathematical model of injection and production dynamic transient analysis at historical flow stage and injection and production analysis stage of gas storage numerically has solved the reservoir pressure distribution in the spatial domain, which effectively eliminated the mismatching of initial conditions of existing analysis model in the time domain. In this new model, we consider the influence of multi-cycle injection and production dynamic history on the spatial distribution of initial reservoir pressure at the injection and production analysis stage, which is more accordant with the special alternating conditions of gas storages.

As for the alternating working conditions of gas storages, the injection and production dynamic transient chart is significantly different from the existing transient chart. Specifically, the dimensionless pseudopressure curve and the dimensionless pseudopressure integral curve drop at the front end, and the intergradational zone of the dimensionless pseudopressure integral derivative curve is convex upward during the gas production period and concave downward during the gas injection period. The more obvious the unbalanced distribution of initial reservoir pressure field in the injection and production analysis stage is, the more obvious the phenomenon of convex upward and concave downward in the curve is, which means the curve under different flow histories is atypical.

The simulation case and the field case of a typical well in Hutubi has proved that the permeability, skin factor and well control factor calculated by the new method are more accordant with the actual situations, indicating that the interpretation parameters are of strong correlation and the results are reliable. It is of important theoretical and application significance to further develop the existing transient analysis methods, accurately predict the reservoir dynamic characteristics under the alternating working conditions of gas storages and guide the safe and efficient operation of gas storages.

B_g—gas volume factor, m³/m³;

B_gi—initial gas volume factor, m³/m³;

C_t—total compressibility, MPa^-1;

C_ti—initial total compressibility, MPa^-1;

G—inventory, m³;

h—effective reservoir thickness, m;

i—number of injection and production cycle;

j—number of grid node;

K—reservoir permeability, 10^-3 μm²;

m—total number of grid nodes;

M—fitting point of theoretical curve and actual curve;

n—time node;

p—pressure, MPa;

p_i—initial reservoir pressure, MPa;

p_wf—bottom hole flowing pressure, MPa;

q—gas flow rate, m³/d;

q_D—dimensionless flow rate;

q_sc—reference flow rate, m³/d;

r—radial distance, m;

r_D—dimensionless radial distance;

r_e—radial outer boundary distance, m;

r_eD—dimensionless radial outer boundary distance;

r_w—wellbore radius, m;

r_wa—effective wellbore radius, m;

S—skin factor, dimensionless;

S_g—gas saturation, %;

t—production time, d;

t₀—balance moment of gas storage construction, d;

t_ca—material balance pseudotime, d;

t_caDd—dimensionless material balance pseudotime;

t_e—end moment of analysis stage, d;

t_s—start moment of analysis stage, d;

t_sD—dimensionless start moment of analysis stage;

u—arbitrary function, pseudopressure in this paper;

x—spatial distance, m;

Z—gas deviation factor, dimensionless;

Z_i—initial gas deviation factor, dimensionless;

θ—angular coordinate axis in cylindrical coordinate system, (°);

μ_g—gas viscosity, mPa•s;

μ_gi—initial gas viscosity, mPa•s;

ξ—equation coefficient, dimensionless;

ψ—pseudopressure, MPa;

$\bar{\psi }$—average reservoir pseudopressure, MPa;

${{\overline{\psi }}_{\text{D}}}$—average reservoir pseudopressure, MPa;

ψ_D—dimensionless pseudopressure;

ψ_D,new—corrected dimensionless pseudopressure;

ψ_Di—dimensionless pseudopressure integral;

ψ_Di,new—corrected dimensionless pseudopressure integral;

ψ_Did—dimensionless pseudopressure integral derivative;

ψ_Did,new—corrected dimensionless pseudopressure integral derivative;

ψ_i—initial pseudopressure of gas storage construction, MPa;

ψ_p—pseudopressure function when the arbitrary pressure is p, MPa;

${{\psi }_{{{p}_{i}}}}$—pseudopressure function when the reservoir pressure is p_i, MPa;

${{\psi }_{{{p}_{wf}}}}$—pseudopressure function when the bottom hole pressure is p_wf, MPa;

ϕ—porosity, %.