Gas storage well productivity calculation method in complex seepage environment

Abstract

The invention discloses a gas storage well productivity calculation method in a complex seepage environment, and the method comprises the steps: carrying out a rock core stress sensitivity experiment under multi-cycle injection and production: carrying out the multi-cycle injection and production rock core experiment, and building a stress sensitivity quantitative characterization model considering the effective covering pressure and the cycle round influence; implementing a high-speed injection-production non-darcy flow experiment: carrying out high-speed non-darcy flow experiment on cores with different permeability and porosity, and establishing a quantitative characterization model of a non-darcy flow coefficient; and construction of a gas storage capacity equation: based on the stress sensitivity quantitative characterization model and the quantitative characterization model of the non-darcy flow coefficient, forming the gas storage injection-production capacity equation through seepage field simplification and conformal transformation modeling, simultaneous operation of a gas motion equation and a state equation and construction of an unsteady seepage differential equation. According to the method, the functions of accurately predicting the productivity of the vertical well and the horizontal well of the gas storage and effectively evaluating the injection-production capacity under the multi-period injection-production condition are achieved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of oil and gas development engineering, and particularly relates to a gas storage well productivity calculation method under a complex seepage environment. BACKGROUND

[0002] With the continuous growth of global natural gas consumption, underground gas storage plays an increasingly important role in natural gas reserves and peak shaving. Since the gas storage is operated in an annual cycle, it presents the characteristics of strong injection and strong production, which leads to the rock in the reservoir being in a multi-cycle alternating stress state, and the gas seepage having a significant high-speed non-Darcy characteristic. These complex factors seriously affect the accuracy of the productivity evaluation of the gas storage well. At present, the gas storage productivity calculation method is mostly borrowed from the gas reservoir development field. However, the production system of the gas reservoir is generally stable, and the production is far lower than the requirement of strong injection and strong production during the peak shaving period of the gas storage, and it lacks the influence of the repeated stress changes caused by the periodic injection and production. Therefore, the traditional gas well productivity equation has poor applicability to the gas storage. For the high-speed non-Darcy seepage problem, some researches have proposed several correction methods, for example, Li Xiaoping et al. introduced a high-speed non-Darcy correction factor to improve the binomial productivity equation; Hu Jiangtao et al. proposed a trinomial productivity equation by experimental data regression, which comprehensively considers the high-speed non-Darcy, low-speed non-Darcy effect and stress sensitivity. However, these methods are based on the inverse problem solving of the field test data, and do not fundamentally solve the productivity prediction problem from the seepage mechanism level, nor do they reflect the rock response and gas seepage characteristics under the condition of periodic injection and production. In the prior art, although it has been recognized that the factors such as reservoir permeability, wellbore damage and well type affect the injection and production capacity of a single well, the research objects are mainly aimed at conventional gas reservoirs. In addition, although some researches have proposed a single well productivity rapid prediction method for the injection and production characteristics of the gas storage, this method also relies on the regression of the field data and lacks theoretical support.

[0003] The existing methods generally do not systematically consider the stress cycle changes and dynamic seepage effects caused by the periodic injection and production and the strong injection and production in the calculation of the productivity of the horizontal well, and are mostly limited to the qualitative analysis of the physical parameters of the reservoir, which leads to a large deviation between the productivity prediction results and the actual situation, and makes it difficult to meet the needs of the efficient operation and fine management of the gas storage. In addition, under the large pressure fluctuation of the gas storage, the productivity test results have poor regularity, which further increases the difficulty of accurately evaluating the injection and production capacity. SUMMARY

[0004] The purpose of the present application is to provide a gas storage well productivity calculation method under a complex seepage environment, to solve the problems that the traditional productivity equation ignores the periodic injection and production stress sensitivity, high-speed non-Darcy seepage and multi-field coupling effect, relies on empirical regression, has limited applicability, and is difficult to depict the dynamic of strong injection and production, so as to realize the accurate prediction of the productivity of the vertical well and the horizontal well of the gas storage and the efficient evaluation of the injection and production capacity under the condition of multi-cycle injection and production.

[0005] The technical scheme adopted by the present application is a gas storage well productivity calculation method under complex seepage environment, comprising the following steps:

[0006] Step S1: core stress sensitivity experiment under multi-cycle injection and production is implemented: multi-cycle injection and production core experiment is carried out, and a stress sensitivity quantitative characterization model considering the influence of effective overburden pressure and cycle round is established;

[0007] Step S2: high-speed injection and production non-Darcy flow experiment is implemented: high-speed non-Darcy seepage experiment of cores with different permeability and porosity is carried out, and a quantitative characterization model of non-Darcy flow coefficient is established;

[0008] Step S3: construction of gas storage productivity equation: based on the stress sensitivity quantitative characterization model and the quantitative characterization model of non-Darcy flow coefficient, the seepage field is simplified and the modeling is transformed, the gas motion equation and the state equation are associated, and the non-steady-state seepage differential equation is constructed to form the injection and production productivity equation of the gas storage.

[0009] Further, in the step S1, the stress sensitivity quantitative characterization model is:

[0010] K / K i =m((p e -p) / P i ) -n

[0011] Wherein, K is the effective permeability under the current pressure; K i is the initial permeability; m is the stress sensitivity coefficient; n is the stress sensitivity index; p e is the net overburden pressure; p is the initial net overburden pressure; P i is the initial pressure of rock.

[0012] Further, in the stress sensitivity quantitative characterization model, the coefficients m and n are respectively represented by the following formulas:

[0013] m=2.404e -2.106τ +1.045e -0.026τ

[0014] n=1.44×10 5 ·e -14.28τ +0.182e -0.031τ

[0015] Wherein, τ is the number of pressure changes.

[0016] Further, in the step S2, the quantitative characterization model of non-Darcy flow coefficient β is:

[0017]

[0018] Wherein, k is the absolute permeability of rock; Porosity.

[0019] Furthermore, in step S3, the simplification and conformal modeling of the seepage field includes: simplifying the three-dimensional seepage problem of a horizontal well into a two-dimensional seepage problem in the horizontal and vertical planes; and using the Rukowski transformation to convert the rectangular coordinate system into an elliptical coordinate system.

[0020]

[0021] Where z is a complex variable in a rectangular coordinate system; L is the length of the horizontal well section; and ω is a complex variable in an elliptical coordinate system.

[0022] Furthermore, in step S3, the seepage field simplification and conformal modeling also includes: transforming an ellipse with a major semi-axis of a and a minor semi-axis of b into a circle with a radius of (a+b) / (L / 2) through conformal modeling; transforming a horizontal well of length L into a unit circle with a radius of 1; wherein the relationship between the major semi-axis of the ellipse and the equivalent radius is:

[0023]

[0024] Where, r e The radius of a circular seepage field that is equivalent to the area of ​​an elliptical seepage field.

[0025] Furthermore, in step S3, the simultaneous equations of motion and state for the gas include:

[0026] Establish the gas motion equations that take stress sensitivity into account:

[0027]

[0028] in, denoted as p, which is the derivative of pressure p with respect to the seepage radius r; Δp is the pressure difference; μ is the viscosity; k0 is the initial permeability of the formation; v is the seepage velocity; m is the stress sensitivity coefficient; n is the stress sensitivity index.

[0029] Introducing the high-speed non-Darcy flow effect:

[0030]

[0031] Where k is the absolute permeability of the rock; Porosity is ρ; gas density is ρ.

[0032] Simultaneously considering stress sensitivity and high-speed non-Darcy effect:

[0033]

[0034] Combining the real gas law: pV=NZRT; where p is pressure; V is gas volume; N is amount of substance; Z is compressibility factor; R is the universal gas constant; and T is temperature.

[0035] Furthermore, in step S3, the construction of the unsteady seepage differential equation includes:

[0036] Establish the unsteady flow differential equation of the gas continuity equation in radial coordinates:

[0037]

[0038] Where t is time; r is the seepage radius; p(r,t) is the formation pressure at time t and location r; The pressure conductivity of a gas is expressed as:

[0039]

[0040] Where k is the absolute permeability of the rock; Porosity; μ is viscosity; C g The gas compressibility coefficient;

[0041] After considering stress sensitivity, the result is corrected to:

[0042]

[0043] Where m is the stress sensitivity coefficient; n is the stress sensitivity index; k0 is the initial permeability of the formation; and Δp is the pressure difference.

[0044] Furthermore, in step S3, the gas storage injection-production capacity equation simultaneously considers stress sensitivity and high-speed non-Darcy effect.

[0045] The extraction process has the following form:

[0046]

[0047] Where, p wf The bottom hole flowing pressure; p e ρ is the formation pressure; n is the stress sensitivity index;

[0048] The injection process has the following form:

[0049]

[0050] Where A and B are the specific expressions for Darcy's flow term and non-Darcy's flow term, respectively.

[0051] Furthermore, when constructing the gas storage injection and production capacity equation, the pressure term is processed by the integral by parts method to obtain the unsteady-state production capacity equations for the injection and production processes, respectively.

[0052] The beneficial effects of this invention are as follows: Firstly, through multi-cycle injection-production core stress sensitivity experiments, this invention establishes a universal quantitative characterization model that considers the effects of effective overburden pressure and cycle number, accurately describing the rock stress-sensitive behavior of clastic gas reservoirs during cyclic injection-production processes. Secondly, based on high-speed non-Darcy flow experiments under different permeability, porosity, and liquid saturation conditions, a quantitative characterization model for the non-Darcy flow coefficient is proposed, significantly improving the prediction accuracy of high-speed non-Darcy flow of gas during intensive injection and production. Furthermore, a calculation model for methane gas physical property parameters, encompassing the effects of temperature and pressure, is constructed, further enhancing the descriptive ability of the seepage process under multi-field coupling environments. Thirdly, by systematically establishing production capacity prediction models for horizontal and vertical wells applicable to steady-state and unsteady-state flows, accurate assessment of pressure dynamics and injection-production capacity of gas reservoirs in different injection-production cycles is achieved, providing a reliable theoretical basis for the optimized design and efficient operation of gas reservoirs. Attached Figure Description

[0053] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0054] Figure 1 This is a flowchart of the stress sensitivity experiment of the present invention.

[0055] Figure 2 This is a diagram showing the results of a multi-cycle stress sensitivity experiment on core B4.

[0056] Figure 3 This is a flowchart of a high-speed non-Darcy injection and production experiment.

[0057] Figure 4 This is a graph showing the relationship between flow rate and pressure square difference in core sample 100-2.

[0058] Figure 5 This is a comparison chart of the relationship between flow rate and pressure squared difference for different core samples.

[0059] Figure 6 This is a schematic diagram of conformal transformation in the horizontal direction.

[0060] Figure 7 This is a schematic diagram of conformal transformation in the vertical direction. Detailed Implementation

[0061] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0062] An embodiment provides a method for calculating the production capacity of a gas storage well under complex seepage conditions, comprising the following steps:

[0063] Step S1: Implementation of core stress sensitivity experiment under multi-cycle injection and production:

[0064] Under cyclic injection and production conditions in gas storage facilities, rocks exhibit significant stress sensitivity. Stress sensitivity refers to the characteristic of rock permeability changes caused by variations in net overburden pressure, resulting from deformation of pore throats and alterations in fracture status (closure or opening). Furthermore, the stress sensitivity of rocks is related to the number of cycles. During multiple overburden pressure cycles, the reservoir rocks undergo irreversible plastic deformation, leading to pore structure damage and a continuous decline in permeability. Their permeability recovery capacity weakens with increasing cycle duration.

[0065] Experiments were conducted using a series of core samples with varying permeability and porosity. The selected core samples basically cover all the physical property parameters in the gas storage reservoir, as shown in Table 1.

[0066] Table 1 List of core samples for stress sensitivity analysis

[0067]

[0068] To illustrate the experimental procedure in detail, a permeability of 94 × 10⁻⁶ is used. -3 μm 2 Taking core B4 as an example, the experimental procedure is as follows: Figure 1 As shown, the specific experimental steps are as follows:

[0069] (1) Dry the core sample;

[0070] (2) The dried core is loaded into the core holder and 0.5 MPa of axial pressure is applied to fix the core;

[0071] (3) Apply ring pressure to 2 MPa using a hand pump; this value is set as the initial net overburden pressure.

[0072] (4) Starting from the initial net overburden pressure, core permeability is tested using the steady-state method. The net overburden pressure is slowly increased at predetermined intervals (2 MPa, 4 MPa, 6 MPa, 8 MPa, 10 MPa) until the maximum set value of 10 MPa is reached (this pressure range covers the operating pressure range of most domestic gas storage facilities). Each pressure point must be maintained for at least 30 minutes until the permeability stabilizes, after which the data is recorded.

[0073] (5) After reaching the maximum net overburden pressure, the net overburden pressure is gradually reduced at the same pressure intervals until it returns to the initial pressure point. Each pressure reduction point must be maintained for more than 1 hour, and the data is recorded after the permeability stabilizes.

[0074] (6) Repeat the pressurization and depressurization process of steps (4) and (5) a total of 5 times to simulate the construction of the gas storage facility and the operation of 4 complete injection and production cycles.

[0075] The B4 core was tested according to the above procedure, and the experimental results are shown below. Figure 2 .Depend on Figure 2 It is evident that the initial change in net overburden pressure has the greatest impact on core permeability, indicating that significant deformation of the core pore structure occurs during the initial loading, leading to a substantial decrease in permeability. During the first round of depressurization, permeability recovers somewhat but cannot return to its initial level, indicating irreversible plastic deformation. With increasing injection-production cycles, the degree of deformation under repeated pressure changes gradually decreases, and the degree of permeability recovery after each depressurization also gradually decreases, with irreversible deformation showing a cumulative increasing trend. Under the same net overburden pressure, core permeability decreases with increasing injection-production cycles and gradually approaches a certain stable value.

[0076] Since the stress sensitivity parameters of core samples with different rock pore structures vary considerably, to improve the applicability of this method, quantitative expressions are used to describe the two indices affecting stress sensitivity, namely the stress sensitivity coefficient m and the stress sensitivity index n. Through fitting, the relationship between the permeability and net overburden pressure of the core samples used in this experiment is found to be more in line with a power function, as shown in equation (1):

[0077] K = mK i (Δp / P i ) -n (1)

[0078] Where K is the effective permeability under the current pressure (10) -3 μm 2 ); K i Initial permeability (10) -3 μm 2 ), which is the core permeability under the initial net overburden pressure; Δp is the core net overburden pressure (MPa); P idenoted as the initial rock pressure (MPa), typically 1 MPa; m is the stress sensitivity coefficient, and n is the stress sensitivity index, both dimensionless.

[0079] To further improve the applicability of the method and reflect the influence of periodic changes, it is proposed to express m and n as a double exponential function with the number of pressure changes τ as the variable:

[0080] m = 2.404e -2.106τ +1.045e -0.026τ (2)

[0081] n = 1.44 × 10 5 ·e -14.28τ +0.182e -0.031τ (3)

[0082] In formulas (2) and (3), τ is the number of pressure changes, which is dimensionless.

[0083] Therefore, the quantitative characterization model of stress sensitivity considering the cycle is given by formula (4):

[0084]

[0085] Where, p e ρ is the net overburden pressure (MPa); p is the initial net overburden pressure (MPa).

[0086] This model is the first to achieve a general quantitative characterization of stress-sensitive behavior under different cycle cycles.

[0087] Step S2: High-speed injection and production non-Darcy flow experiment implementation:

[0088] The high-speed non-Darcy effect refers to the phenomenon in gas flow where, when the flow velocity exceeds a certain critical value, the flow regime changes from laminar to turbulent, leading to a significant increase in flow resistance. Determining the non-Darcy flow coefficient is crucial for studying the high-speed non-Darcy effect. Conducting indoor physical simulation experiments to study the non-Darcy seepage law and establishing a quantitative relationship for the non-Darcy coefficient under high-speed flow conditions has significant engineering application value for calculating the production capacity of gas storage facilities.

[0089] In 1901, Forchheimer discovered that Darcy's equation could not accurately describe high-speed gas flow. He introduced an inertial drag term to modify Darcy's equation, resulting in the more commonly used Forchheimer equation:

[0090]

[0091] Where dp / dx represents the pressure gradient (10 -1 MPa·cm -1μ is the gas viscosity (mPa·s); k is the absolute permeability of the rock (10). - 3μm 2 ); ρ is the gas density (g / cm³) 3 ); β is the high-velocity non-Darcy flow coefficient; v is the seepage velocity (m / s).

[0092] Under low-velocity seepage conditions, fluid motion is dominated by viscous forces, and inertial effects are negligible. In this case, the seepage field conforms to the linear relationship described by Darcy's law. However, as the flow velocity increases to a certain level, inertial resistance continuously increases. At this point, inertial resistance cannot be ignored relative to viscous resistance, and seepage behavior enters the high-speed nonlinear flow region, where the Forchheimer equation applies. Compared to the oil and gas reservoir development process, the gas velocity is higher during the injection and production dynamics of underground gas storage facilities, exhibiting a significant high-speed non-Darcy seepage effect.

[0093] To ensure the universality of the proposed model, multiple sets of core samples were selected during the non-Darcy flow experiments, covering low, medium, and high permeability ranges. For each permeability range, physical simulation gas-driven water experiments were conducted to study the high-velocity non-Darcy flow characteristics and analyze the pressure difference versus flow rate (pQ) curves of non-Darcy flow, aiming to establish a nonlinear productivity equation considering high-velocity non-Darcy flow. The high-velocity non-Darcy injection-production experiment flowchart is shown below. Figure 3 As shown.

[0094] Taking a single rock core as an example, the specific experimental steps are explained below:

[0095] (1) Drying the core: Place the core in an oven and dry it at 60℃ for 12 hours. Weigh the dried core using an electronic balance and record the core mass.

[0096] (2) Vacuum saturation of the core: After weighing, the core was placed in an intermediate container and evacuated for 8 hours. Deionized water was then injected into the vacuum container containing the core. After the core was completely submerged, deionized water was continued to be injected into the intermediate container using a high-precision displacement pump until the pressure inside the container reached 14 MPa. The core was then saturated under pressure for 6 hours. The saturated core was then removed, and the deionized water on the surface of the saturated core was removed on a damp paper towel. The core was weighed and the weight was recorded. The saturated core was then immersed in deionized water for storage.

[0097] (3) Gas-driven water: Place the saturated core into the core holder, turn the hand pump, and set the confining pressure to 7 MPa. Control the nitrogen injection rate, set the displacement pressure difference to 0.1 MPa, displace the target core, and observe the liquid production at the outlet until no water is produced at the outlet (simulating the formation bound water situation), then close the gas cylinder valve.

[0098] (4) After simulating the operation of the gas storage tank through the above steps, the gas in the pipeline is vented, the gas cylinder valve is opened, and a gas drive experiment is carried out. The displacement pressure difference is controlled by controlling the nitrogen injection rate, and the gas flow rate at the outlet end is measured by using a gas flow meter under different pressure differences.

[0099] (5) Clean and organize the experimental instruments and equipment, and the experiment is over.

[0100] The above is the complete process of a high-speed injection-production non-Darcy coefficient experiment. The experiment was carried out on 8 core samples in sequence according to the designed experimental plan.

[0101] Taking core 100-2 as an example, the processed experimental results are as follows: Figure 4 As shown. By Figure 4 It can be seen that at low pressure square difference and low flow rate, the two basically show a linear relationship, which is Darcy flow; however, as the pressure square difference increases, the flow rate change gradually increases, and the pressure square difference and flow rate no longer show a simple linear relationship, but enter the nonlinear relationship stage. At this time, the turbulence effect between fluid molecules intensifies, no longer conforms to Darcy flow, and manifests as non-Darcy flow.

[0102] The experimental results of the eight sets of core samples were compared in the same semi-logarithmic coordinate system, such as... Figure 5 As shown. By Figure 5 It can be seen that, due to the different physical properties of the cores, the flow rate at the core outlet increases significantly with the increase of the production pressure differential, and the two are positively correlated; under the same production pressure differential, the greater the permeability of the core, the greater its flow rate. The flow rate and pressure differential of eight cores with different permeabilities were obtained through high-speed injection-production experiments, and regression derivation yielded a nonlinear seepage production capacity equation that better conforms to the binomial form.

[0103] Based on formula (5), a linear flow binomial capacity equation applicable to the experimental testing process was derived:

[0104]

[0105] Where p2 is the core inlet pressure (MPa); p1 is the core outlet pressure (MPa); p sc ρ is the pressure of the gas under standard conditions (MPa), typically taken as 0.101 MPa; μ is the gas viscosity (mPa·s); Z is the compressibility factor of the gas under experimental conditions, dimensionless; T is the temperature under experimental conditions (K); L1 and L2 represent the scale positions corresponding to the starting and ending points of the core sample during the experimental measurement, and the effective length of the core sample is determined by the absolute value of the difference between the two; d is the core diameter (cm); T sc Z represents the temperature under standard conditions (K), typically taken as 293.15K. scQ is the gas compressibility factor under standard conditions, a dimensionless quantity; Q is the flow rate (cm³). 3 ·s -1 M is the molar mass of the gas (g·mol⁻¹). -1 );γ g Let be the relative density of the gas, which is dimensionless; R is the universal gas constant, usually taken as 8.314 MPa·cm. 3 / (mol·K), and the gas used in the experiment was nitrogen.

[0106] according to Figure 5 The experimental results show that the specific values ​​of the non-Darcy flow coefficient β are calculated as shown in Table 2.

[0107] Table 2 Non-Darcy Flow Coefficients

[0108]

[0109] Based on the non-Darcy coefficients of the eight core samples calculated in Table 2, and by fitting the data with the core permeability and porosity, and considering the wide range of permeability in this test, the obtained data has a comprehensive applicability. Therefore, a general expression for the non-Darcy flow coefficient β can be obtained as follows:

[0110]

[0111] Where k is the absolute permeability of the rock (10 -3 μm 2 ); Porosity is a dimensionless quantity.

[0112] Step S3: Construction of the gas storage capacity equation:

[0113] S31: Simplification and conformal modeling of seepage field:

[0114] Based on the production capacity equation of horizontal wells in oil reservoirs, combined with the differences in gas-liquid seepage laws, and considering the nonlinear characteristics of gas seepage during gas storage operation (such as high-speed non-Darcy effect) and periodic injection and production characteristics, the three-dimensional seepage problem of horizontal wells is simplified into a two-dimensional seepage problem in the horizontal and vertical planes, and then the production capacity equation of horizontal wells during injection and production is derived.

[0115] The assumptions include: single-phase gas flow; horizontal wells located in the center of the gas reservoir; formation permeability being sensitive to periodic stress; the flow process being nonlinear and neglecting gravity and capillary forces; and pressure and temperature varying with the injection and production process and affecting gas physical properties.

[0116] There is a horizontal well with a horizontal section length of L in the center of the gas reservoir. Using the Jukowski transformation:

[0117]

[0118] Where z is a complex variable in a rectangular coordinate system (z-plane), z = x + iy, where x and y are rectangular coordinates in the horizontal plane, in meters (m), and i is an imaginary unit; L is the length of the horizontal well section (m); ω is a complex variable in an elliptical coordinate system (ω-plane), ω = ξ + iη, where ξ and η are the transformed elliptical coordinates (ξ, η), dimensionless.

[0119] By conformal transformation, the (x, y) coordinates in the rectangular coordinate system z-plane are converted to (ξ, η) coordinates in the elliptical coordinate system ω-plane, that is, z = x + iy is mapped to ω = ξ + iη. This transforms the ellipse with major semi-axis a and minor semi-axis b into an ellipse with radius a. The circle transforms the elliptical seepage field within the horizontal plane of the horizontal well into a circular seepage field, and converts the horizontal well of length L into a unit circle with a radius of 1, such as... Figure 6 As shown.

[0120] The area of ​​an ellipse πab and the area of ​​a circle πr e 2 Equivalent, and The relationship between the semi-major axis of the ellipse and the focal length is obtained as follows:

[0121]

[0122] Where, r e The radius (m) of a circular seepage field that is equivalent to the area of ​​an elliptical seepage field is the equivalent radius.

[0123] For horizontal wells where seepage occurs within a closed boundary in the vertical direction, conformal mapping can be used:

[0124] ω=(1-e -πz / h (1+e) -πz / h (10)

[0125] In the formula, z is the complex coordinate in the physical plane, z = x + iy. In the vertical transformation, y represents the vertical coordinate (across the thickness direction), and x is the horizontal coordinate (along the well or within the layer). The unit is meters (m). h is the formation thickness (vertical distance between the upper and lower closed boundaries), and the unit is meters.

[0126] The upper and lower closed boundaries of the z-plane are transformed into circular isopaltic boundaries with radius 1 in the ω-plane. The sink point (0,0) in the z-plane is transformed into the center (0,0) of the circle in the ω-plane, as follows: Figure 7 As shown.

[0127] Well radius r in the z-plane w In the ω-plane, it becomes R w The relation is:

[0128]

[0129] Where, r w R is the actual wellbore radius (m) of the horizontal well in the z-plane; w r is the radius of the well in the z-plane. w After conformal transformation, the equivalent radius in the ω-plane is dimensionless.

[0130] By using conformal mapping, the three-dimensional seepage problem is decomposed into the superposition of two two-dimensional problems, resulting in the equation for the horizontal seepage velocity v1 in a horizontal well:

[0131]

[0132] Where q is the actual gas well production (m³) 3 / d); p is the formation pressure (MPa); h is the effective formation thickness (m); r is the radius of the horizontal well in the horizontal plane (m); q sc Gas well production under standard conditions (m³) 3 / d);p sc Z is the pressure of the gas under standard conditions (MPa), typically taken as 0.101 MPa; Z is the compressibility factor of the gas under experimental conditions, dimensionless; T is the temperature under experimental conditions (K); T sc Z represents the temperature under standard conditions (K), typically taken as 293.15K. sc R is the gas compressibility factor under standard conditions, a dimensionless quantity; R is the universal gas constant, typically taken as 8.314 MPa·cm. 3 / (mol·K), and the gas used in the experiment was nitrogen.

[0133] The equation for the vertical seepage velocity v2 in a horizontal well is:

[0134]

[0135] Where q is the actual gas well production (m³) 3 / d); p is the formation pressure (MPa); L is the length of the horizontal section of the horizontal well (m); r is the radius of the horizontal well in the horizontal plane (m); q sc Gas well production under standard conditions (m³) 3 / d);p sc Z is the pressure of the gas under standard conditions (MPa), typically taken as 0.101 MPa; Z is the compressibility factor of the gas under experimental conditions, dimensionless; T is the temperature under experimental conditions (K); T sc Z represents the temperature under standard conditions (K), typically taken as 293.15K. sc R is the gas compressibility factor under standard conditions, a dimensionless quantity; R is the universal gas constant, typically taken as 8.314 MPa·cm. 3 / (mol·K), and the gas used in the experiment was nitrogen.

[0136] S32: Simultaneous equations of motion and equations of state for gases

[0137] The gas motion equations, combined with the stress sensitivity experimental model, are expressed as follows:

[0138]

[0139] in, Δp is the derivative of pressure p with respect to the seepage radius r; Δp is the pressure difference (MPa); r is the seepage radius (m); μ is the viscosity (mPa·s); k0 is the initial permeability of the formation (10). -3 μm 2 v is the seepage velocity (m / s); m is the stress sensitivity coefficient; n is the stress sensitivity index.

[0140] Further introducing the high-velocity non-Darcy flow effect, and substituting the relationship between the non-Darcy coefficient β and core porosity and permeability into the Forchheimer equation, we obtain:

[0141]

[0142] Where k is the absolute permeability of the rock (10 - 3μm 2 ); Porosity is ρ; gas density (g / cm³) 3 );

[0143] When considering both stress sensitivity and high-speed non-Darcy effect, the gas motion equation is:

[0144]

[0145] The equation of state for a real gas is:

[0146] pV = NZRT(17)

[0147] Where p is the pressure (MPa); V is the gas volume (cm³). 3 N is the amount of substance (mol); Z is the compressibility factor; R is the gas constant (R = 8.314 MPa·cm⁻¹). 3 ·mol⁻¹·K - 1); T is the temperature (K). The expression for gas density is derived from this equation:

[0148]

[0149] Where M is the molar mass of the gas, in g·mol⁻¹ -1 ;γ gThe relative density of the gas is dimensionless.

[0150] S33: Construction of Unsteady Seepage Differential Equations:

[0151] The gas continuity equation in radial coordinates is expressed as an unsteady flow differential equation:

[0152]

[0153] Where t is time; r is the seepage radius; p(r,t) is the formation pressure at time t and location r; The pressure permeability of the gas (cm) 2 / s), the pressure permeability coefficient refers to the formation area over which pressure changes propagate per unit time, characterizing the pressure propagation speed; the larger the pressure permeability coefficient, the lower the fluid compressibility and the faster the pressure propagation; the expression is:

[0154]

[0155] Among them, C g The gas compressibility coefficient (0.1 Pa) -1 ).

[0156] After considering stress sensitivity, the compressibility coefficient is corrected as follows:

[0157]

[0158] The initial condition is that when t = 0, p(r,t) = p e When the boundary condition is r→∞, p(r,t)=p e Based on the principle of pressure drop superposition, and considering both stress sensitivity and high-velocity non-Darcy flow, the differential equation for unsteady seepage in the horizontal direction is:

[0159]

[0160] Where, p e This represents formation pressure, expressed in MPa.

[0161] The differential equation for unsteady seepage in the vertical direction is:

[0162]

[0163] S34: Solving the injection-procurement differentiated capacity equation:

[0164] To accurately characterize the effect of pressure on stress sensitivity and fluid parameters, a partial integration method is used to process the pressure term, avoiding the direct use of the average pressure value. Taking the extraction process as an example, the pressure term is integrated as follows:

[0165]

[0166] Where, p wf The bottom hole flowing pressure is (MPa).

[0167] Using the equivalent seepage resistance method, the unsteady-state productivity equation considering stress sensitivity and high-speed non-Darcy effect in the extraction process is derived:

[0168]

[0169] The construction of the gas storage capacity equation differs significantly between injection and production processes. During injection, the wellhead pressure is generally higher than the injection pressure, and the formation pressure increases continuously with injection. Conversely, during production, the wellhead pressure is lower than the production pressure, and the formation pressure decreases continuously with production. The method of handling pressure directly affects stress sensitivity. Through integration by parts, a calculation term relating to pressure, i.e., the left-hand side of the equation, can be obtained. Compared to previous methods, this calculation term is not simply a pressure difference, but rather relates to the formation pressure p. e and bottom hole flowing pressure p wf The combination of these forms more accurately reflects the impact of internal and external boundary conditions on production capacity.

[0170] In formulas (25) and (26), the right side consists of the sum of two terms: the first term on the right side, i.e. The specific expression for the Darcy flow term (A) represents the pressure loss generated by gas conforming to the Darcy flow law in a porous medium; the second term on the right, namely... The specific expression for the non-Darcy flow term (B) characterizes the additional pressure loss caused by the high-speed flow of gas in the near-wellbore zone deviating from Darcy's law.

[0171] Therefore, the unsteady-state productivity equation for the injection process of a horizontal well in a gas storage facility, considering stress sensitivity and high-speed non-Darcy effect, has the following new form:

[0172]

[0173] This invention, based on the fundamental differences in pressure distribution range and seepage mechanism during injection and production, is the first to employ a separate modeling approach, resulting in two distinct capacity equations for the injection and production stages. This departs from the traditional practice of using a single equation to describe both processes simultaneously, thus more accurately reflecting the dynamic characteristics of the periodic operation of gas storage facilities. By establishing a periodic stress-sensitive quantitative characterization model, a non-Darcy flow coefficient quantitative characterization model, and a high-precision gas property parameter calculation model, the core challenge of accurately describing the periodic injection and production dynamics of gas storage facilities using traditional methods is systematically solved. Differentiated capacity equations applicable to both injection and production processes are systematically derived using seepage mechanics theory, completely overcoming the limitations of traditional single capacity equations. This significantly improves the prediction accuracy of pressure dynamics and injection / production capacity of gas storage facilities in different injection and production cycles, providing a reliable theoretical foundation and technical support for the optimized design and efficient operation of gas storage facilities.

[0174] The various embodiments in this specification are described in a related manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the system embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0175] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for calculating the productivity of a gas storage well in a complex seepage environment, characterized in that, The method comprises the following steps: Step S1: Core stress sensitivity experiment under multi-cycle injection and production is implemented: multi-cycle injection and production core experiment is carried out, and a stress sensitivity quantitative characterization model considering the influence of effective overburden pressure and cycle round is established; Step S2: High-speed injection and production non-Darcy flow experiment is implemented: high-speed non-Darcy seepage experiments of cores with different permeabilities and porosities are carried out, and a quantitative characterization model of non-Darcy flow coefficient is established; Step S3: Construction of gas storage productivity equation: based on the stress sensitivity quantitative characterization model and the quantitative characterization model of non-Darcy flow coefficient, the productivity equation of the gas storage is formed through seepage field simplification and conformal transformation modeling, simultaneous establishment of gas motion equation and state equation, and non-steady-state seepage differential equation construction.

2. The method of claim 1, wherein, In the step S1, the stress sensitivity quantitative characterization model is: K / K i = m((p e -p) / P i ) -n where K is the effective permeability at the current stress; K i is the initial permeability; m is the stress-sensitivity coefficient; n is the stress-sensitivity exponent; p e is the net overburden pressure; p is the initial net overburden pressure; P i is the initial stress of the rock.

3. The method of claim 2, wherein, In the stress sensitivity quantitative characterization model, the coefficients m and n are respectively represented by the following formulas: m = 2.404e -2.106τ + 1.045e -0.026τ n = 1.44 x 10 5 ·e -14.28τ + 0.182e -0.031τ Wherein, τ is the number of pressure changes.

4. The method of claim 1, wherein, In the step S2, the quantitative characterization model of non-Darcy flow coefficient β is: where k is the absolute permeability of the rock; is the porosity.

5. The method of claim 1, wherein, In the step S3, the seepage field simplification and conformal transformation modeling comprises: simplifying the three-dimensional seepage problem of the horizontal well into a two-dimensional seepage problem in the horizontal plane and the vertical plane; and converting the rectangular coordinate system into an elliptical coordinate system by using the Joukowski transformation: Wherein, z is a complex variable in the rectangular coordinate system; L is the length of the horizontal well section; and ω is a complex variable in the elliptical coordinate system.

6. The method of calculating gas storage well productivity in complex percolation environment according to claim 5, characterized in that, In the step S3, the seepage field simplification and conformal transformation modeling further comprises: converting the ellipse with a long semi-axis a and a short semi-axis b into a circle with a radius of (a+b) / (L / 2) by the conformal transformation; and converting the horizontal well with a length L into a unit circle with a radius of 1; wherein the relationship between the long semi-axis of the ellipse and the equivalent radius is: where r e is the radius of a circular flow field equivalent to the area of the elliptical flow field.

7. The method of calculating gas storage well deliverability in complex flow environment according to claim 1, characterized in that, In the step S3, the simultaneous establishment of the gas motion equation and the state equation comprises: Establishing a gas motion equation considering stress sensitivity: wherein, is the derivative of pressure p with respect to seepage radius r; Δp is the pressure difference; μ is the viscosity; k0 is the initial permeability of the formation; v is the seepage velocity; m is the stress sensitivity coefficient; and n is the stress sensitivity index. Introducing high-speed non-Darcy flow effect: where k is the absolute permeability of the rock; is the porosity; p is the gas density; Simultaneously considering stress sensitivity and high-speed non-Darcy effect: Combining the real gas state equation: pV=NZRT; wherein, p is pressure; V is gas volume; N is the amount of substance; Z is the compression factor, R is the gas universal constant; and T is temperature.

8. The method of calculating gas storage well deliverability in complex flow environment according to claim 1, characterized in that, In the step S3, the non-steady-state seepage differential equation construction comprises: Establishing an unsteady seepage differential equation of the gas continuity equation under the radial coordinate: where t is time; r is the seepage radius; p(r, t) is the formation pressure at time t and position r; is the compressibility factor of the gas and is given by: where k is the absolute permeability of the rock; is the porosity; μ is the viscosity; C g is the gas compressibility factor; After considering stress sensitivity, it is revised as: Wherein, m is the stress sensitivity coefficient; n is the stress sensitivity index; k0 is the initial permeability of the formation; and Δp is the pressure difference.

9. The method of calculating gas storage well deliverability in complex flow environment according to claim 1, characterized in that, In the step S3, the gas storage injection and production productivity equation simultaneously considers stress sensitivity and high-speed non-Darcy effect, For the production process, it has the following form: where p wf is the bottom hole flowing pressure; p e is the formation pressure; n is the stress sensitivity index; For the injection process, it has the following form: Wherein, A and B are specific expressions of the Darcy flow term and the non-Darcy flow term respectively.

10. The method of calculating gas storage well productivity in complex percolation environment according to claim 9, characterized in that, In the construction of the gas storage injection and production productivity equation, the pressure term is processed by the method of integration by parts to obtain the non-steady-state productivity equation of the injection and production processes respectively.

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