A generalized mobility-based well control dynamic radius calculation method for complex well types of oil and gas reservoirs

Abstract

The application relates to a generalized mobility-based oil and gas reservoir complex well type well control dynamic radius calculation method, which is used for predicting unput wells. First, based on the generalized mobility thought, a productivity equation is established or determined through field measurement, inflow dynamic curves are calculated and drawn, outflow dynamic curves are calculated and drawn through wellbore pipe flow equation calculation, and single-well reasonable productivity is calculated and predicted; a single-well pressure unstable flow model is established, and an unstable flow well bottom pressure solution is obtained; in combination with the constraint condition of wellhead pressure in the injection-production process, the single-well reasonable productivity and the single-well pressure unstable flow model calculated well bottom pressure, reasonable injection-production flow rates at different time points are calculated and determined; and in combination with the formation pressure constraint condition and the material balance equation, the optimal well control dynamic radius is iteratively calculated. The application is more in line with the actual production working conditions.

Description

Technical Field

[0001] This invention relates to the field of well control range calculation technology for oil and gas reservoirs, specifically to a method for calculating the dynamic radius of well control for complex well types based on generalized mobility and considering formation pressure constraints and wellhead pressure constraints. Background Technology

[0002] In the design and adjustment of oil and gas reservoir development, the study of well control radius has always been an important topic. It is related to the well network deployment, the determination of reasonable well network density, the determination of well control range and affected range, and guides the well network design adjustment, the implementation of production measures, and the evaluation of well network controlled reserves.

[0003] Currently, there are several different definitions and calculation methods for well control radius research. One method is the detection radius method, which mainly calculates the propagation distance of pressure pulse waves. Li Chuanliang et al. (2002) explored the calculation formula for the detection radius, Wang Xinhai et al. (2008) gave several special calculation methods for detection radius such as non-Darcy flow and non-Newtonian fluids, and Cui Disheng et al. (2005) derived a calculation method for the detection radius of radial composite reservoirs. However, this method mainly calculates the pressure propagation distance, which differs significantly from the actual well control radius. Although the pressure has propagated, it cannot actually control the distant reserves, and the calculated value of the detection radius is usually too large. The second method is the production data analysis method, which mainly calculates the utilization range and dynamic reserves by combining production flow and pressure data with modern production decline analysis theories. Liao Wei et al. (2019) evaluated the dynamic reserves and well control radius of gas wells based on the production instability analysis method. However, this method requires rich daily production data and bottom hole pressure data, which is not suitable for unexploited wells or wells where such data is unavailable. For wells with bottom-hole pressure data, analysis is not possible; the third is the critical pressure gradient method, which is mainly for the starting pressure gradient in low-permeability oil and gas reservoirs. When the pressure gradient in the formation is less than the starting pressure gradient, the outer reservoir does not seep, thus obtaining the well control radius. Fu Chunmei et al. (2009) considered the starting pressure gradient of low-permeability gas reservoirs to study the effective gas supply radius of gas wells. This type of method has certain limitations, and it is difficult to determine the starting pressure gradient value; the fourth is the steady flow method, which back-calculates the well control radius through the Darcy equation for steady flow, but it is limited to conventional vertical well homogeneous models and is only applicable to steady flow conditions. It is not applicable to unstable flow and complex well types and reservoir models; there are also other different methods. Gou Honggang (2005) proposed a method for calculating the radius considering the influence of flow rate ratio, but its flow rate standard is difficult to determine; Tang Ligen (2018) combined the unstable seepage equation and the material balance equation to establish a set of methods for calculating the well control radius of gas storage. The method is for gas storage and the parameters such as pressure and flow rate need to be determined manually, which is difficult.

[0004] In practical applications, the well control dynamic radius is related to many parameters such as production flow rate, formation pressure, bottom hole pressure, and production time. In particular, the pressure parameter has a significant impact on the well control dynamic radius. Existing well control radius methods all focus on the influence of a single parameter and do not consider all factors. Therefore, it is necessary to further study a well control dynamic radius calculation method that considers the constraints of parameters such as formation pressure and bottom hole pressure, as well as other factors such as reasonable production capacity. Summary of the Invention

[0005] This invention aims to address the aforementioned problems by proposing a method for calculating the well control dynamic radius of complex well types in oil and gas reservoirs based on generalized mobility.

[0006] The technical solution of this invention is as follows: A method for calculating the well control radius of complex well types in oil and gas reservoirs based on generalized mobility, characterized by the following: S1: For existing development data wells, calculate the well control radius of existing development data wells by combining the initial formation pressure, current formation pressure test data, and flow data during production and injection processes with the material balance equation. S2: For prediction of production wells that have not yet been put into production, firstly, based on the concept of generalized flowability, establish and solve the production capacity equation or determine the production capacity equation through field measurement, calculate and draw the inflow dynamic curve, and at the same time establish the wellbore flow equation to calculate and draw the outflow dynamic curve. Based on the node analysis method combined with the inflow dynamic curve and the outflow dynamic curve, calculate and predict the reasonable production capacity of a single well. S3: Based on generalized mobility, a single-well pressure unsteady flow model is established, and the unsteady flow bottom hole pressure solution is obtained; S4: Combining the constraints on wellhead pressure during the injection and production process, the predicted reasonable production capacity of a single well, and the calculation of the bottom hole pressure using a single well pressure unstable flow model, the reasonable injection and production flow rate at different times is determined through iterative calculation. S5: Iteratively calculate the optimal well control radius by combining formation pressure constraints and material balance equations.

[0007] Furthermore, the specific process of step S1 is as follows: The well control radius equation is derived by inversely using the mass balance equation, and then directly calculated by substituting parameters into the well control radius equation. The well control radius equation is as follows: (1) The equation for the controlling radius of a gas reservoir well is: (2) The equation for the well control radius is: (3) In the formula: r e - Well control radius, m; N f- Cumulative oil flow rate (positive for produced oil, negative for injected oil), m 3 B o -Oil volume coefficient, m 3 / m 3 ; - Porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 ;p i - Original formation pressure, Pa; p- Pressure, Pa; G f - Cumulative gas flow rate (positive for produced gas, negative for injected gas), m 3 Z i - Natural gas deviation factor corresponding to the original formation pressure, dimensionless; Z - Natural gas deviation factor, dimensionless; W f - Cumulative water flow (positive for production, negative for injection), m 3 B w -Water volume coefficient, m 3 / m 3 B wi -Water volume coefficient under initial conditions, m 3 / m 3 .

[0008] In the equations, the cumulative flow rate subscript is uniformly marked with 'f', which can represent both produced and injected flow. Alternatively, as is commonly used in the industry, the subscript 'p' can be used to represent produced flow and the subscript 'i' to represent injected flow. In this case, the formation pressure difference 'p' in the equations... i -p must be an absolute value.

[0009] Furthermore, the specific process of step S2 is as follows: S21: Based on the concept of generalized flow to characterize the flow law of fluids, for any fluid motion equation, it can be written in the form of a generalized mobility model: v = -λ▽p; where λ is the generalized mobility, m 2 / (Pa·s); v represents the fluid velocity, m / s; ▽p represents the pressure gradient, Pa / m; the fluid flow law is characterized using the above generalized mobility model; S22: The relationship between formation pressure, bottom hole pressure, and flow rate is determined by calculating the production capacity equation, and then the inflow dynamic curve of bottom hole pressure and flow rate is plotted. S23: Convert the wellhead pressure to the bottom hole pressure using the wellbore flow equation; calculate the corresponding bottom hole pressure by giving different nozzle diameters and wellhead pressures, and give different flow rates, and plot the flow rate versus bottom hole pressure curve, i.e., the outflow dynamic curve. S24: By using the node analysis method, the inflow dynamic curve and the outflow dynamic curve are superimposed on a single graph. The flow rate at the intersection of the inflow dynamic curve and the outflow dynamic curve is the predicted reasonable production capacity of a single well.

[0010] Furthermore, the specific process of step S3 is as follows: A complete single-well pressure-unsteady flow model is constructed using the reservoir control equations, initial conditions, internal and external boundary conditions, and auxiliary equations; and the unsteady flow bottom hole pressure solution is obtained; specifically: When the fluid type is oil or water, a single-phase liquid, the solution is obtained directly using pressure. When the fluid type is gas or multiphase, the concepts of pseudo-pressure and pseudo-time are introduced to transform the equations before solving them. In addition, the unsteady flow bottom hole pressure solution is obtained by using the Laplace transform and inverse transform for the fully perforated vertical well circular boundary model; the unsteady flow bottom hole pressure solution is obtained by using the Laplace spatial point source solution method for the horizontal well model, combined with Fourier transform, superposition principle and inverse Laplace transform; the unsteady flow bottom hole pressure solution is obtained by using the Laplace spatial line source solution method for the fractured well model, combined with superposition principle; for models with finite or infinite conductivity in the wellbore or fracture, the flow distribution and pressure in the wellbore are coupled for calculation, and the unsteady flow bottom hole pressure solution is obtained by solving a system of linear equations. In addition, if the analytical method cannot solve the problem, a numerical method is used: by meshing, a numerical model is established using numerical methods such as finite difference, finite element, finite volume or boundary element to obtain the solution of the bottom pressure of the unsteady flow.

[0011] Furthermore, the specific process of step S4 is as follows: S41: Using the predicted reasonable production capacity of a single well obtained in S2 as the initial injection and production flow rate, single well production or injection is carried out. The bottom pressure at different times is calculated through the unstable flow bottom pressure solution obtained in S3. S42: During single-well production or injection, the wellhead pressure is constrained in combination with the surface engineering conditions. The surface engineering conditions are the wellhead pressure limit values ​​during operation. The wellhead pressure limit values ​​are the minimum wellhead pressure or the maximum wellhead pressure. The wellhead pressure limit values ​​are converted to the bottom of the well by combining the wellbore flow equation with the injection and production flow rates to obtain the bottom of the well pressure limit values. By comparing the unstable flow bottom hole pressure solution obtained in S3, the injection and production flow rates are iterated to ensure that the bottom hole pressure meets the constraints, thereby obtaining reasonable injection and production flow rates at different times. When the injection and production flow rates are different at different times, the unstable flow bottom hole pressure solution obtained in S3 needs to be combined with the superposition principle to calculate the bottom hole pressure values ​​at different times under variable flow injection and production conditions.

[0012] Furthermore, the specific process of step S5 is as follows: When a single well is produced or injected under initial formation pressure conditions, the reasonable injection-production flow rate at the set time is obtained through S4. The formation pressure at the set time is obtained through the mass balance equation. By giving a predicted formation pressure value, the calculated formation pressure is compared with the predicted formation pressure value, and the well control radius is iteratively modified until the calculation meets the accuracy requirements, thereby obtaining the optimal well control radius that meets the formation pressure constraint conditions.

[0013] The technical advantages of this invention are as follows: This invention is based on the concept of generalized mobility, which simplifies the application of complex problems and enables wider applicability. Generalized mobility unifies different linear and nonlinear fluid motion equations, allowing fluids stored in different regions and at different scales to use the same form of motion equations to construct unified governing equations. Therefore, it is easier to discretize the model, reduce the complexity of coupled problems, and make model solving simpler and more unified. The well control dynamic radius proposed in this invention is similar to the existing concepts of operational range and well control range, both of which represent the size of the range that a single well can control or operate. The difference is that the well control dynamic radius proposed in this invention takes into account its variation with injection and production time and formation pressure, and also takes into account the pressure constraints during the injection and production process, which is more in line with the actual field situation.

[0014] Compared with conventional methods such as detection radius and production data analysis, this invention comprehensively considers wellhead pressure constraints, formation pressure constraints, and variable flow constraints. It applies the material balance equation, the unsteady flow equation, the wellbore flow equation, and the production capacity equation for calculation, which can better obtain a more reliable well control dynamic radius that meets the constraints, and the calculation method is more in line with the actual production conditions on site. Attached Figure Description

[0015] Figure 1 This is a detailed flowchart of the well control dynamic radius calculation of the present invention.

[0016] Figure 2 This is a schematic diagram of the physical model of the homogeneous circular closed boundary model of a vertical well established in specific embodiment 1 of the present invention.

[0017] Figure 3 This is a schematic diagram of the physical model of the homogeneous rectangular closed boundary model of the horizontal well established in specific embodiment 2 of the present invention.

[0018] Figure 4 This is a schematic diagram of the physical model of the radial composite circular closed boundary model of the two zones of a vertical well established in specific embodiment 3 of the present invention. Detailed Implementation

[0019] Example 1

[0020] 1. A method for calculating the well control radius of complex well types in oil and gas reservoirs based on generalized mobility, the method is as follows:

[0021] S1: For existing development data wells, calculate the well control radius of existing development data wells by combining the initial formation pressure, current formation pressure test data, and flow data during production and injection processes with the material balance equation. The specific process is as follows: For existing development data wells, the well control radius equation is derived from the material balance equation. The well control radius is then directly calculated by substituting parameters into the well control radius equation. The material balance equation for a closed elastically driven reservoir is as follows: (4) The expression for the control radius of a reverse thrust well is: (5) The mass balance equation for a constant-volume gas reservoir is: (6) The expression for the control radius of a reverse thrust well is: (7) The mass balance equation for a water well is: (8) The expression for the control radius of a reverse thrust well is: (9) In the formula: p i - Original formation pressure, Pa; p - Current formation pressure, Pa; N f - Cumulative oil flow rate (positive for produced oil, negative for injected oil), m 3 B o -Oil volume coefficient, m 3 / m 3 ;r e - Well control radius, m; φ - Porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 Z - Natural gas deviation factor, dimensionless; Z i - Natural gas deviation factor corresponding to the original formation pressure, dimensionless; G f - Cumulative gas flow rate (positive for produced gas, negative for injected gas), m 3 G - Geological reserves of natural gas, m 3 W f- Cumulative water flow (positive for production, negative for injection), m 3 B w -Water volume coefficient, m 3 / m 3 B wi -Water volume coefficient under initial conditions, m 3 / m 3 ; The existing development data wells all have reservoir porosity, effective thickness, volume factor, comprehensive compressibility factor, and gas deviation factor. Initial formation pressure, current formation pressure, and flow rate data during production injection can be obtained through testing, thus allowing direct calculation of the well control radius r. e Currently, formation pressure can be calculated either through on-site static pressure testing or by measuring pressure.

[0022] S2: For prediction of production wells that have not yet been put into production, firstly, based on the concept of generalized flowability, establish and solve the production capacity equation or determine the production capacity equation through field measurement, calculate and draw the inflow dynamic curve, and at the same time establish the wellbore flow equation to calculate and draw the outflow dynamic curve. Based on the node analysis method combined with the inflow dynamic curve and the outflow dynamic curve, calculate and predict the reasonable production capacity of a single well. The specific process is as follows: For wells not yet in production, forecasting is required. This forecasting process considers many influencing factors and involves complex calculations. A detailed calculation procedure for the well control radius is provided here. Figure 1 As shown. Based on the concept of generalized flow to characterize the flow law of fluids, any fluid motion equation can be written in the form of a generalized mobility model: v = -λ▽p; where λ is the generalized mobility, m 2 / (Pa·s); v represents the fluid velocity, m / s; ▽p represents the pressure gradient, Pa / m; the fluid flow law is characterized using the above generalized mobility model; The system obtains reservoir physical properties such as permeability and porosity, and provides predicted production time and formation pressure parameters. The predicted production time is calculated using a decreasing analysis, and the predicted formation pressure can be calculated through field static pressure testing or bottom hole pressure. During injection, the maximum pressure limit is considered. In gas storage wells, formation pressure can be the upper and lower limits of the proven gas storage operation pressure. Assuming a well control radius value in the range of [200m, 1000m], considering injection or production, construct a production capacity equation; Regarding capacity equations, the forms of capacity equations differ depending on the type of fluid. Generally, there are binomial capacity equations and exponential capacity equations: For oil and water wells, the binomial productivity equation is usually expressed in pressure form: (10) Where, p avg The mean formation pressure is expressed in MPa; p wf q is the bottom hole pressure, MPa; q is the flow rate, m³ / d; A1 is the laminar flow coefficient in pressure form, MPa / (m³ / d). 3 / d); B1 is the pressure-form turbulence coefficient, MPa / (m 3 / d) 2 . For gas wells, the binomial productivity equation is usually expressed in pseudo-pressure form or pressure square form: (11) (12) Where m represents the pseudo-pressure, MPa 2 / (mPa·s). A2 is the laminar flow coefficient in pseudo-pressure form, in MPa. 2 / (mPa·s) / (m 3 / d); B2 is the pseudo-pressure turbulence coefficient, MPa 2 / (mPa·s) / (m 3 / d) 2 A3 is the laminar flow coefficient in the form of pressure square, in MPa. 2 / (m 3 / d); B3 is the pressure square form of the turbulence coefficient, MPa 2 / (m 3 / d) 2 . For oil and water wells, the exponential productivity equation is: (13) For gas wells, the exponential productivity equation is: (14) (15) Where C1 is the pressure square form of the capacity coefficient, (m 3 / d) / MPa 2n C2—Pseudo-pressure form capacity coefficient, (m 3 / d) / (MPa 2 / (mPa·s)) n n1 is the exponent, which is dimensionless. Different well types have different productivity equations, mainly reflected in the coefficients of the productivity equation. The basic forms are similar. The productivity equation connects the formation and the bottom of the well, establishes the relationship between formation pressure, bottom-hole pressure and flow rate, and plots the inflow dynamic curve of bottom-hole pressure and flow rate. Then, the wellbore flow equation is constructed, connecting the wellhead and the bottom of the well to establish the relationship between the wellhead pressure, the bottom of the well pressure, and the flow rate. (16) Where: p - pressure, Pa; z - height, m; ρ - density, kg / m³ 3 g - gravitational acceleration, N / kg; θ - tilt angle, °; u - acceleration, m / s² 2 f - frictional resistance, dimensionless; d ti - Inner diameter of the oil pipe, in meters; The calculation is usually performed by dividing the well into micro-segments. The entire well is divided into several micro-segments. For each micro-segment, parameters such as well diameter, flow rate, fluid density, and inclination angle are determined. The outflow pressure can then be calculated from the inflow pressure. The calculation starts from the wellhead and continues to the bottom of the well to obtain the bottom pressure. Given parameters such as wellhead pressure and tubing size, the bottom hole pressure under different flow conditions is calculated, and the curve of the relationship between flow rate and bottom hole pressure, i.e., the outflow dynamic curve, is plotted. By overlaying the inflow and outflow dynamic curves onto a single graph using node analysis, the flow rate at the intersection of the inflow and outflow dynamic curves represents the predicted reasonable production capacity of a single well.

[0023] S3: Based on generalized mobility, a single-well pressure unsteady flow model is established, and the unsteady flow bottom hole pressure solution is obtained; The specific process is as follows: The single-well pressure unsteady flow model needs to consider the unstable flow state of the formation, as well as the well type, boundary conditions, reservoir type, etc., to construct and solve the model. The governing equations of the model are constructed by the equations of motion, state, and continuity. At the same time, the initial conditions and internal and external boundary conditions are considered to form a mathematical model of unsteady flow. A commonly used mathematical model for the three-phase flow of oil, gas, and water can be written as follows: Considering the source and sink terms, the oil phase governing equations are as follows: (17) Consider the water phase governing equations for source and sink terms: (18) Consider the gas-phase governing equations for source and sink terms: (19) In the formula, φ represents porosity (%), and B represents porosity (%). w B o B g λ represents the volume coefficients of water, oil, and gas, respectively, and is dimensionless; w , λ o , λ gThe generalized mobility of water, oil, and gas, respectively, is expressed in m. 2 / (Pa·s);ρ wsc , ρ osc , ρ gsc Ground reference densities for water, oil, and gas, respectively, in kg / m³ 3 ;R s The dissolved gas-oil ratio is dimensionless; S w S o S g q represents the saturation levels of water, oil, and gas, respectively; w q o q g These are the source and sink items for water, oil, and gas, respectively, in kg / (m³). 3 ·s); The space is represented by m; t is represented by s; t max The total time is s; p w p o p g Ω represents the pressure of water, oil, and gas, respectively, in Pa; Ω represents the storage space; ▽ represents the Hamiltonian operator; ∂ represents the partial derivative symbol. Saturation and capillary force auxiliary equations: (20) (twenty one) (twenty two) In the formula: p cow p cgo These are the capillary pressures at the oil-water and gas-oil interfaces, respectively, in Pa. Boundary condition equations: (twenty three) (twenty four) (25) Where: g w g o g g ∂Ω represents the boundary functions of water, oil, and gas on the reservoir boundary, respectively, and is dimensionless; ∂Ω represents the reservoir boundary (the boundary includes the inner and outer boundaries); c w,1 c o,1 c g,1 These are the pressure coefficients for water, oil, and gas under the boundary conditions of the reservoir, respectively, in 1 / Pa; c w,2 c o,2 c g,2 Let n be the coefficients of the directional derivative terms of water, oil, and gas along the outward normal direction of the boundary conditions of the reservoir, respectively, in s / m; ∂Ω Let m be the direction of the outward normal to the boundary of the storage collective. Initial condition equations: (26) (27) In the formula: k = o, w, g; l w , l o , l g Let f be the initial saturation distribution function for water, oil, and gas, respectively; w f o f g Let be the initial pressure distribution functions for water, oil, and gas, respectively, in Pa. When considering generalized mobility, the equation of motion is written in the form of a generalized mobility model: v = -λ▽p, where λ is the generalized mobility and m 2 / (Pa·s); v represents the fluid velocity, m / s; ▽p represents the pressure gradient, Pa / m; the generalized mobility λ is a function of spatial location and time. The flow characteristics of fluids in the pipe flow regions of wells, pipes, cracks, fissures, cavities, holes, hollows, caves, and seepage regions of porous media are characterized by the above generalized mobility model. Among them, the generalized mobility can characterize the Darcy flow law of Newtonian fluids in the seepage region of porous media. Considering Newtonian fluids and Darcy's law, the Darcy (1856) seepage formula can be obtained. (28) Where λ is the generalized mobility, m 2 / (Pa·s); μ is the viscosity of the fluid, Pa·s; K is the permeability, m 2 . Generalized mobility can also characterize the laminar flow characteristics of non-Newtonian fluids in pipe flow regions such as wells, pipes, fractures, fissures, cavities, holes, caves, and karst caves. It can be obtained from the Hagen-Poiseuille (1839, 1840) formula. (29) Where λ is the generalized mobility, m 2 / (Pa·s); μ is the viscosity of the fluid, Pa·s; d is the hydraulic diameter of the pipe, m. Meanwhile, generalized mobility can characterize the turbulent flow characteristics of Newtonian fluids in pipe flow regions such as wellbores, pipes, fractures, fissures, cavities, caves, and karst caves. Based on the Darcy-Weisbach (1845) formula, Colebrook (1938) friction factor formula, Qstwald-DeWaele (1923, 1925) power-law fluid viscosity formula, and Hirasak-Pope (1974) shear rate formula, it can be obtained... (30) (31) Where d is the hydraulic diameter of the pipe, in meters; ρ represents the fluid density, in kilograms per cubic meter of water. 3 v represents fluid velocity, m / s; μ eff The effective viscosity of the fluid is expressed in Pa·s; α represents roughness in meters; W(·) represents the Lambert W function; n² represents the power-law exponent, which is dimensionless; and H represents the consistency coefficient in Pa·s. n . In practical applications, the model is simplified and processed according to the actual reservoir conditions to obtain a suitable mathematical model for solving, thereby obtaining the bottom hole pressure solution. The bottom hole pressure can then be calculated based on a given flow rate.

[0024] S4: Combining the constraints on wellhead pressure during the injection and production process, the predicted reasonable production capacity of a single well, and the calculation of the bottom hole pressure using a single well pressure unstable flow model, the reasonable injection and production flow rate at different times is determined through iterative calculation. The specific process is as follows: S41: Using the predicted reasonable production capacity of a single well obtained in S2 as the initial injection-production flow rate Q, single well production or injection is carried out. The bottom pressure at different times is calculated through the unstable flow bottom pressure solution obtained in S3. S42: While performing the unsteady flow model calculation, the bottom hole pressure is calculated based on the wellbore flow equation, combined with the wellhead pressure constraint and flow rate. The wellhead pressure constraint refers to the constraint on the wellhead pressure considering surface engineering conditions such as surface pipeline transportation and pipeline injection. There is usually a minimum or maximum wellhead pressure value. The flow rate refers to the calculated reasonable injection and production flow rate. The wellbore flow equation is the same as the wellbore flow equation used in the previous reasonable production calculation process. Finally, the maximum or minimum value of the corresponding bottom hole pressure can be calculated. The bottom hole pressure p calculated by comparing with the unsteady flow equation wf1 The bottom hole pressure constraint value p obtained from the wellbore flow equation wf2 When p is extracted wf1 Is it greater than p? wf2 If yes, proceed to the next time step for calculation; otherwise, limit the injection / production flow rate to [0, Q], take the injection / production flow rate as (0+Q) / 2, and recalculate p. wf1 If the calculated p wf1 Greater than p wf2 Then the injection / collection flow rate range is limited to [(0+Q) / 2,Q], if p wf1 Less than p wf2 The injection / collection flow rate range is then limited to [0, (0+Q) / 2]. A bisection method is used for iterative calculation until |p wf1 -p wf2|<10 -3 At this point, the flow rate is the reasonable flow rate for that moment; when injecting, p is judged. wf1 Is it less than p? wf2 If yes, proceed to the next time step for calculation; otherwise, limit the injection / production flow rate to [0, Q], take the injection / production flow rate as (0+Q) / 2, and recalculate p. wf1 If p wf1 Less than p wf2 Then the injection / collection flow rate range is limited to [(0+Q) / 2,Q], if p wf1 Greater than p wf2 The injection / collection flow rate range is then limited to [0, (0+Q) / 2]. A bisection method is used for iterative calculation until |p wf1 -p wf2 |<10 -3 At this point, the injection and production flow rate is the reasonable flow rate for that moment; repeat this process until all moments have been calculated, and finally the flow rate and bottom hole pressure at different moments are calculated.

[0025] S5: Iteratively calculate the optimal well control radius by combining formation pressure constraints and material balance equations. Finally, the mass balance equation is used to determine whether the formation pressure reaches the predicted formation pressure within the predicted time under the assumed well control radius. Material balance equation for closed elastically driven reservoirs: The mass balance equation for a constant-volume gas reservoir is: The mass balance equation for a water well is: In the formula: p i - Original formation pressure, Pa; p r1 - Formation pressure reached within the predicted timeframe, in Pa and N. f -Cumulative oil flow rate, m 3 B o -Oil volume coefficient, m 3 / m 3 ;r e - Well control radius, m; φ - Porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 Z - Natural gas deviation factor, dimensionless; Z i - Natural gas deviation factor corresponding to the original formation pressure, dimensionless; G f -Cumulative gas flow rate, m 3 G - Geological reserves of natural gas, m 3 W f-Cumulative water flow, m 3 B w -Water volume coefficient, m 3 / m 3 B wi -Water volume coefficient under initial conditions, m 3 / m 3 .

[0026] Among them, the cumulative oil flow N f The cumulative gas flow rate and cumulative water flow rate are calculated based on the flow rate at different times obtained from the previous calculations. The well control radius r e To predict the initially assumed well control radius, the final formation pressure p can be calculated using the mass balance equation. r1 Determine the relationship between the given predicted formation pressure p and the actual formation pressure. r2 Difference, when extracted, if p r1 Greater than p r2 Then the well control radius needs to be reduced, and the range should be set to [0, r]. e ], take the radius as (0+r e ) / 2, recalculate p r1 If the calculated p r1 Greater than p r2 The range is then limited to [0, (0+r)]. e If p r1 Less than p r2 The range is then limited to [(0+r)]. e ) / 2,r e The bisection method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control radius; if p r1 Less than p r2 Then the well control radius needs to be increased. First, take r. e Recalculate at 1.5 times the original value, if the calculated p... r1 Greater than p r2 The range is then limited to [r] e 1.5*r e If the calculated p r1 Less than p r2 Get the current r e Calculate using 1.5 times the value, until the calculated p... r1 Greater than p r2 After obtaining the defined range, the binary search method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control radius; when injecting, if p r1 Less than pr2 Then the well control radius needs to be reduced, and the range should be set to [0, r]. e ], take the radius as (0+r e ) / 2, recalculate p r1 If the calculated p r1 Less than p r2 The range is then limited to [0, (0+r)]. e If p r1 Greater than p r2 The range is then limited to [(0+r)]. e ) / 2,r e The bisection method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control radius; if p r1 Greater than p r2 Then the well control radius needs to be increased. First, take r. e Recalculate at 1.5 times the original value, if the calculated p... r1 Less than p r2 The range is then limited to [r] e 1.5*r e If the calculated p r1 Greater than p r2 Get the current r e Calculate using 1.5 times the value, until the calculated p... r1 Less than p r2 After obtaining the defined range, the binary search method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control dynamic radius. This completes the entire prediction process for the well control dynamic radius of wells that are not yet in production and have no data.

[0027] In practical applications, the response well control radius can be calculated by selecting the appropriate model based on the actual reservoir conditions.

[0028] Specific Experiment Example 1 - Homogeneous Circular Closed Boundary Model of Oil Production Vertical Well; This specific experimental example 1 proposes a method for calculating the well control dynamic radius of a homogeneous circular closed boundary model for oil production vertical wells; The physical model corresponding to this model assumes the following: a vertical well at the center of a circular, homogeneous, closed boundary reservoir ( Figure 2 ); The following conditions must be met: ① The oil well produces at a constant rate; ② The reservoir is horizontally of uniform thickness and isotropic, and the single-phase flow of oil satisfies Darcy's law; ③ Under the original conditions, the formation pressure distribution is uniform and constant; ④ The effects of gravity and capillary force are ignored, and the effects of wellbore reservoir effect, temperature, and other factors on flow are not considered.

[0029] S1: For existing development data wells, directly calculate the well control radius: (32) In the formula: p i - Original formation pressure, Pa; p - Current formation pressure, Pa; N f- -Cumulative oil flow rate, m 3 B o -Oil volume coefficient, m 3 / m 3 ;r e - Well control radius, m; φ - Porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 .

[0030] S2: Constructing a model for prediction based on undeveloped wells: Assume a well control dynamic radius r e Range [200m, 1000m]; Productivity equation for vertical wells in oil reservoirs: (33) In the formula: p r - Mean formation pressure, Pa; p wf - Bottom hole pressure, Pa; q - Flow rate, m³ / s 3 / s;B o -Oil volume coefficient, m 3 / m 3 λ - generalized mobility, m 2 / (Pa.s); h - effective thickness, m; t - time, s; φ - porosity, %; C t -Comprehensive compressibility, 1 / Pa; r w - Wellbore radius, m; S - Skin factor, dimensionless; γ is Euler constant, 0.5772. The flow equation in a wellbore is: (34) Where: p - pressure, Pa; z - height, m; ρ - density, kg / m³ 3 g - gravitational acceleration, N / kg; θ - tilt angle, °; u - acceleration, m / s² 2 f - frictional resistance, dimensionless; d ti - Inner diameter of the oil pipe, in meters. The inflow dynamic curve and outflow dynamic curve are calculated based on the production capacity equation and the wellbore flow equation, respectively. The reasonable production capacity Q is calculated based on the nodal analysis method.

[0031] S3: The mathematical model for pressure-unsteady flow is: (35) (36) (37) (38) Dimensionless variables are defined as follows: (39) (40) (41) (42) In the formula: p D -Dimensionless pressure, dimensionless; t D - Dimensionless time, dimensionless; r D - Dimensionless distance, dimensionless; r eD -Dimensionless outer boundary radius, dimensionless; h - effective thickness, m; t - time, s; λ - generalized mobility, m 2 / (Pa.s); p - pressure, Pa; p i - Original formation pressure, Pa; B o -Oil volume coefficient, m 3 / m 3 ;q sc -Reference flow rate, m 3 / s; φ - porosity, %; C t - Overall compressibility coefficient, 1 / Pa; l - Reference length, m; r e - Well control radius, m; r - Distance, m; ∂ is the sign of the partial derivative; Model Solving Applying a Laplace transform to the above formula yields: (43) (44) (45) Therefore, the general solution and the conditions for the definite solution can be obtained as follows: (46) (47) (48) in, . Solving the system of linear equations to obtain the general solution and undetermined coefficients (49) Laplace Unsteady Space Bottom-of-Well Pressure Solution: (50) In the formula: -Laplace space is dimensionless and has no pressure. - Laplace space dimensionless bottom-hole pressure, dimensionless; s - dimensionless time t D The corresponding Laplace variables are dimensionless; A0, B0, ε0 are intermediate variables, dimensionless; K0 is a BesselK0 function; K1 is a BesselK1 function; I0 is a BesselI0 function; I1 is a BesselI1 function.

[0032] S4: After obtaining the bottomhole pressure solution for unsteady flow, set the initial injection-production flow rate equal to the reasonable production capacity Q, and calculate the bottomhole pressure p based on the bottomhole pressure solution and the initial injection-production flow rate. wf1 While performing calculations using the single-well pressure unsteady flow model, the bottom hole pressure p is calculated based on the wellbore flow equation, combined with wellhead pressure constraints and injection / production flow rates. wf2 Determine p wf1 Is it greater than p? wf2 If yes, proceed to the next calculation; otherwise, limit the flow range to [0, Q], take the flow as (0+Q) / 2, and recalculate p. wf1 If the calculated p wf1 Greater than p wf2 The range is then limited to [(0+Q) / 2,Q], if p wf1 Less than p wf2 The range is then limited to [0, (0+Q) / 2]. A bisection method is used for iterative calculation until |p... wf1 -p wf2 |<10 -3 The flow rate at this time is the reasonable flow rate at that moment. The calculation is continuously performed according to this process until all moments are calculated. Finally, the injection and production flow rates and bottom hole pressure at different moments are calculated.

[0033] S5: The material balance equation for a circular reservoir can be: (51) In the formula: p i - Original formation pressure, Pa; p r1 - Formation pressure reached within the predicted timeframe, in Pa and N. f -Cumulative oil flow rate, m 3 B o -Oil volume coefficient, m 3 / m3 ;r e - Well control radius, m; φ - Porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 . Determine the well control radius r under the assumed well control radius. e Below, within the predicted time, the formation pressure p r1 Has the predicted formation pressure p been reached? r2 If p r1 Greater than p r2 Then the well control radius needs to be reduced, and the range should be set to [0, r]. e ], take the radius as (0+r e ) / 2, recalculate p r1 If the calculated p r1 Greater than p r2 The range is then limited to [0, (0+r)]. e If p r1 Less than p r2 The range is then limited to [(0+r)]. e ) / 2,r e The bisection method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control radius; if p r1 Less than p r2 Then the well control radius needs to be increased. First, take r. e Recalculate at 1.5 times the original value, if the calculated p... r1 Greater than p r2 The range is then limited to [r] e 1.5*r e If the calculated p r1 Less than p r2 Get the current r e Calculate using 1.5 times the value, until the calculated p... r1 Greater than p r2 After obtaining the defined range, the binary search method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control dynamic radius.

[0034] Specific Experimental Example 2 – Homogeneous Rectangular Closed Boundary Model of Gas Production Horizontal Well;

[0035] This specific experimental example 2 proposes a method for calculating the well control dynamic radius of a homogeneous rectangular closed boundary model for gas production horizontal wells; The physical model corresponding to this model assumes the following: a horizontal well at the center of a rectangular homogeneous closed boundary reservoir ( Figure 3 ); The following conditions must be met: ① The gas well produces at a constant rate; ② The reservoir is horizontally uniform and isotropic, and the gas flows in a single phase, satisfying Darcy's law; ③ Under the original conditions, the formation pressure distribution is uniform and constant; ④ The gas compressibility and viscosity are considered to vary with pressure, with the gas compressibility and deviation factor calculated using the DPR method, and the gas viscosity calculated using the Lee method; ⑤ The effects of gravity and capillary force are ignored, and the effects of wellbore reservoir effect, temperature, and other factors on the flow are not considered.

[0036] S1: For existing development data wells, directly calculate the well control radius: (52) In the formula: p i - Original formation pressure, Pa; p - Current formation pressure, Pa; r e - Well control dynamic radius, m; φ - Porosity, decimal; h - Effective thickness, m; Z - Natural gas deviation factor, dimensionless; Z i - Natural gas deviation factor corresponding to the original formation pressure, dimensionless; G f -Cumulative gas flow rate, m 3 .

[0037] S2: Constructing a model for prediction based on undeveloped wells: Assume a well control dynamic radius r e , range [200m, 1000m]. The productivity equation for horizontal wells in gas reservoirs can be adopted as follows: (53) in: , . Where: m - pseudo-pressure, Pa 2 ·m 2 / (Pa.s); p r - Mean formation pressure, Pa; p wf - Bottom hole pressure, Pa; q - flow rate, m 3 / s;p sc - Pressure under standard conditions, Pa; T f - Gas reservoir temperature, K; h - Effective thickness, m; T sc - Temperature under standard conditions, K; - Intermediate variable; r w - Wellbore radius, m; D - Non-Darcy flow coefficient, 1 / (m) 3 / s); L - Horizontal well length, m; z w-Z-axis well location, m; Z-natural gas deviation factor; λ r -Radial generalized mobility, m 2 / (Pa.s); The flow equation in a wellbore is: (54) Where: p - pressure, Pa; z - height, m; ρ - density, kg / m³ 3 g - gravitational acceleration, N / kg; θ - tilt angle, °; u - acceleration, m / s² 2 f - frictional resistance, dimensionless; d ti - Inner diameter of the oil pipe, in meters. The inflow and outflow dynamic curves are calculated based on the production capacity equation and the wellbore flow equation, respectively, and the reasonable production capacity Q is calculated based on the nodal analysis method.

[0038] S3: The mathematical model for pressure-unsteady flow is as follows: First, establish a homogeneous infinite boundary point source model as follows: (55) (56) (57) (58) (59) (60) Dimensionless variables are defined as follows: (61) (62) (63) (64) (65) (66) The pseudo-stress is defined as: (67) In the formula: C t - Composite compression coefficient, 1 / Pa; h - Effective thickness, m; l - Reference length, m; p - Pressure, Pa; p i - Original formation pressure, Pa; q sc -Reference flow rate, m 3 / s;T f -Gas reservoir temperature, K; T sc -Temperature under standard conditions, K; p sc- Pressure under standard conditions, Pa; r - Distance, m; t a -Pseudo-time, s; λ r -Radial generalized mobility, m 2 / (Pa.s);λ z -Vertical generalized mobility, m 2 / (Pa.s); z- Distance along the axis, in meters (m); z w - The center point location of the point source in the axial direction, m; φ - porosity, %; ∂ is the sign of the partial derivative; p D -Dimensionless pressure, dimensionless; t D - Dimensionless time, dimensionless; r D -Dimensionless distance, dimensionless; L D -Dimensionless thickness, dimensionless; z D - Dimensionless vertical distance, dimensionless; z wD - Vertical position of the center point of the dimensionless point source, dimensionless; δ - Height of the point source (approaching zero), m; Model Solving Applying the Laplace transform to the above formula yields... (68) (69) (70) (71) (72) Solving for the given information, we get: (73) Applying the superposition principle to the homogeneous infinite point source model yields the rectangular closed boundary point source model. Integrating the pressure solution of the homogeneous rectangular closed boundary point source model along the horizontal well, the bottom-hole pressure solution of the rectangular closed boundary of the horizontal well is obtained as follows: (74) (75) In the formula: -Laplace space is dimensionless and has no pressure. - Laplace space dimensionless bottom-hole pressure, dimensionless; s - dimensionless time t D Corresponding Laplace variables; l - reference length, m; K0 - BesselK0 function; d1 - distance from well center to left boundary, m; d2 - distance from well center to right boundary, m; d3 - distance from well center to lower boundary, m; d4 - distance from well center to upper boundary, m; x D- Dimensionless distance in the x-direction, dimensionless; y D - Dimensionless distance in the y-direction, dimensionless; x wD - Dimensionless well position in the x-direction, dimensionless; y wD - Dimensionless well position in the y-direction, dimensionless; r D - Dimensionless distance, dimensionless; z wD - The vertical position of the center point of the dimensionless point source, dimensionless; p wD - Dimensionless bottom hole pressure, dimensionless; i, m, n - summation counting parameters.

[0039] S4: After obtaining the bottomhole pressure solution for unsteady flow, set the initial injection-production flow rate equal to the reasonable production capacity Q, and calculate the bottomhole pressure p based on the bottomhole pressure solution and the initial injection-production flow rate. wf1 While performing calculations using the single-well pressure unsteady flow model, the bottom hole pressure p is calculated based on the wellbore flow equation, combined with wellhead pressure constraints and injection / production flow rates. wf2 Determine p wf1 Is it greater than p? wf2 If yes, proceed to the next calculation; otherwise, limit the flow range to [0, Q], take the flow as (0+Q) / 2, and recalculate p. wf1 If the calculated p wf1 Greater than p wf2 The range is then limited to [(0+Q) / 2,Q], if p wf1 Less than p wf2 The range is then limited to [0, (0+Q) / 2]. A bisection method is used for iterative calculation until |p... wf1 -p wf2 |<10 -3 The flow rate at this time is the reasonable flow rate at that moment. The calculation is continuously performed according to this process until all moments are calculated. Finally, the injection and production flow rates and bottom hole pressure at different moments are calculated.

[0040] S5: The material balance equation for a rectangular gas reservoir can be: (76) In the formula: p i - Original formation pressure, Pa; p r1 - The formation pressure reached within the predicted timeframe, in Pa; r e - Well control dynamic radius, m; φ - Porosity, decimal; h - Effective thickness, m; Z - Natural gas deviation factor, dimensionless; Z i - Natural gas deviation factor corresponding to the original formation pressure, dimensionless; G f -Cumulative gas flow rate, m 3 . Determine the well control radius r under the assumed well control radius. e Below, within the predicted time, the formation pressure pr1 Has the predicted formation pressure p been reached? r2 If p r1 Greater than p r2 Then the well control radius needs to be reduced, and the range should be set to [0, r]. e ], take the radius as (0+r e ) / 2, recalculate p r1 If the calculated p r1 Greater than p r2 The range is then limited to [0, (0+r)]. e If p r1 Less than p r2 The range is then limited to [(0+r)]. e ) / 2,r e The bisection method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control radius; if p r1 Less than p r2 Then the well control radius needs to be increased. First, take r. e Recalculate at 1.5 times the original value, if the calculated p... r1 Greater than p r2 The range is then limited to [r] e 1.5*r e If the calculated p r1 Less than p r2 Get the current r e Calculate using 1.5 times the value, until the calculated p... r1 Greater than p r2 After obtaining the defined range, the binary search method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control dynamic radius.

[0041] Specific experimental example 3: Radial composite circular closed boundary model of two zones in a vertical oil well;

[0042] This specific experimental example 3 proposes a method for calculating the well control dynamic radius of a two-zone radial composite circular closed boundary model of an oil production vertical well; The physical model corresponding to this model assumes the following: a vertical well at the center of a circular two-zone radially composite closed boundary reservoir ( Figure 4 ); The following conditions must be met: ① The oil well produces at a constant rate; ② The reservoir is horizontally of uniform thickness and isotropic, and the single-phase flow of oil satisfies Darcy's law; ③ Under the original conditions, the formation pressure distribution is uniform and constant; ④ The effects of gravity and capillary force are ignored, and the effects of wellbore reservoir effect, temperature, and other factors on flow are not considered.

[0043] S1: For existing development data wells, directly calculate the well control radius: (77) In the formula: p i - Original formation pressure, Pa; p - Current formation pressure, Pa; N f -Cumulative oil flow rate, m 3 B o -Oil volume coefficient, m 3 / m 3 ;r e - Well control radius, m; r1 - Inner zone radius, m; φ1 - Inner zone porosity, decimal; φ2 - Outer zone porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 .

[0044] S2: Constructing a model for prediction based on undeveloped wells: Assume a well control dynamic radius r e Range [200m, 1000m]; The productivity equation for vertical wells in radial composite reservoirs can be: (78) In the formula: B o -Oil volume coefficient, m 3 / m 3 h - effective thickness, m; p r - Mean formation pressure, Pa; p wf - Bottom hole pressure, Pa; q - Flow rate, m³ / s 3 / s;r w - Wellbore radius, m; S - Skin factor, dimensionless; λ1 - Generalized mobility in the inner zone, m 2 / (Pa.s); λ2 - generalized mobility in the outer region, m 2 / (Pa.s);r e - Well control radius, m; r1- Inner zone radius, m. The flow equation in a wellbore is: (79) Where: p - pressure, Pa; z - height, m; ρ - density, kg / m³ 3 g - gravitational acceleration, N / kg; θ - tilt angle, °; u - acceleration, m / s² 2 f - frictional resistance, dimensionless; d ti - Inner diameter of the oil pipe, in meters. The inflow and outflow dynamic curves are calculated based on the production capacity equation and the wellbore flow equation, respectively, and the reasonable production capacity Q is calculated based on the nodal analysis method.

[0045] S3: The mathematical model for pressure-unsteady flow is: (80) (81) in (82) Inner boundary conditions are (83) The interface conditions are (84) (85) in (86) The outer boundary conditions are (87) Initial conditions are (88) Dimensionless variables are defined as follows: (89) (90) (91) (92) (93) In the formula: B o -Oil volume coefficient, m 3 / m 3 C t - Composite compression coefficient, 1 / Pa; h - Effective thickness, m; l - Reference length, m; p - Pressure, Pa; p i - Original formation pressure, Pa; q sc -Reference flow rate, m 3 / s;r e - Well-controlled dynamic radius, m; r1 - Inner zone radius, m; r - Distance, m; φ1 - Inner zone porosity, decimal; φ2 - Outer zone porosity, decimal; t - Time, s; λ1 - Inner zone generalized mobility, m 2 / (Pa.s); λ2 - generalized mobility in the outer region, m 2 / (Pa.s); ∂ represents the partial derivative; p 1D -Dimensionless internal zone pressure, dimensionless; p2D -Dimensionless outer zone pressure, dimensionless; t D - Dimensionless time, dimensionless; r D - Dimensionless distance, dimensionless; r eD - Dimensionless outer boundary radius, dimensionless; r 1D -Dimensionless interior radius, dimensionless; Model Solving By performing a Laplace transform on the mathematical model, we can obtain (94) (95) (96) (97) (98) (99) The general solution is (100) (101) Based on the boundary value conditions, the bottom hole pressure solution can be obtained as follows: (102) in: In the formula: -Laplace space has dimensionless interior pressure and is dimensionless; -Laplace space is dimensionless outer region pressure and has no dimension. - Laplace space dimensionless bottom-hole pressure, dimensionless; s - dimensionless time t D The corresponding Laplace variables; b1, b2, g - intermediate variables, dimensionless; K0 - BesselK0 function; K1 - BesselK1 function; I0 - BesselI0 function; I1 - BesselI1 function.

[0046] S4: After obtaining the bottomhole pressure solution for unsteady flow, set the initial injection-production flow rate equal to the reasonable production capacity Q, and calculate the bottomhole pressure p based on the bottomhole pressure solution and the initial injection-production flow rate. wf1 While performing calculations using the single-well pressure unsteady flow model, the bottom hole pressure p is calculated based on the wellbore flow equation, combined with wellhead pressure constraints and injection / production flow rates. wf2 Determine p wf1 Is it greater than p? wf2If yes, proceed to the next calculation; otherwise, limit the flow range to [0, Q], take the flow as (0+Q) / 2, and recalculate p. wf1 If the calculated p wf1 Greater than p wf2 The range is then limited to [(0+Q) / 2,Q], if p wf1 Less than p wf2 The range is then limited to [0, (0+Q) / 2]. A bisection method is used for iterative calculation until |p... wf1 -p wf2 |<10 -3 The flow rate at this time is the reasonable flow rate at that moment. The calculation is continuously performed according to this process until all moments are calculated. Finally, the injection and production flow rates and bottom hole pressure at different moments are calculated.

[0047] S5: The material balance equation for a circular composite reservoir can be: (103) In the formula: p i - Original formation pressure, Pa; p r1 - Formation pressure reached within the predicted timeframe, in Pa and N. f -Cumulative oil flow rate, m 3 B o -Oil volume coefficient, m 3 / m 3 ;r e - Well control radius, m; r1 - Inner zone radius, m; φ1 - Inner zone porosity, decimal; φ2 - Outer zone porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 . Determine the well control radius r under the assumed well control radius. e Below, within the predicted time, the formation pressure p r1 Has the predicted formation pressure p been reached? r2 If p r1 Greater than p r2 Then the well control radius needs to be reduced, and the range should be set to [0, r]. e ], take the radius as (0+r e ) / 2, recalculate p r1 If the calculated p r1 Greater than p r2 The range is then limited to [0, (0+r)]. e If p r1 Less than p r2 The range is then limited to [(0+r)]. e) / 2,r e The bisection method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control radius; if p r1 Less than p r2 Then the well control radius needs to be increased. First, take r. e Recalculate at 1.5 times the original value, if the calculated p... r1 Greater than p r2 The range is then limited to [r] e 1.5*r e If the calculated p r1 Less than p r2 Get the current r e Calculate using 1.5 times the value, until the calculated p... r1 Greater than p r2 After obtaining the defined range, the binary search method is used for iterative calculation until |p r1 -p r2 |<10 -3 At this point, the radius is the final well control dynamic radius. Specific Implementation Example 4: Taking the closed boundary model of a circular homogeneous vertical well for gas production as an example, calculations are performed using actual data. Basic parameters: original formation pressure 30 MPa, predicted formation pressure 10 MPa, predicted production time 300 days, minimum wellhead oil pressure 5 MPa, effective thickness 11 m, porosity 0.08, permeability 3 mD, skin factor 0, well depth 2000 m, wellbore radius 0.1 m, bottom hole temperature 100℃, overall compressibility coefficient 0.0258 MPa. -1 . First, assuming a well control radius of 220m, calculate a reasonable production rate of 17.68 × 10⁻⁶ m. 4 m 3 / d, let the initial injection-production flow rate equal to the reasonable production rate, and calculate the bottom-hole pressure p on the first day according to the unsteady flow equation. wf1 The pressure is 23.07 MPa. The bottom hole pressure p is calculated based on the wellbore flow equation. wf2 It is 5.94 MPa, p wf1 > p wf2 The calculation continues at the same injection / production flow rate for the next time step. The calculated bottom hole pressures at different times are shown in the table below. When the calculation reaches day 71, the bottom hole pressure p is calculated using the unsteady flow equation. wf1 The wellbore pressure p is calculated using the wellbore flow equation, which is 5.64 MPa. wf2 It is 5.94 MPa, p wf1 < pwf2 The flow rate is limited to [0, 17.68×10]. 4 First, change the flow rate to 8.84 × 10. 4 m 3 / d, calculate p wf1 Given a pressure of 12.41 MPa, calculate p. wf2 It is 5.83 MPa, |p wf1 -p wf2 |=6.5781, which does not meet the accuracy requirement. Proceed to the next iteration calculation. See the table below for the specific iteration process. After 12 iterations, the flow rate was 17.428 × 10⁻⁶. 4 m 3 After / d, |p wf1 -p wf2 |=0.000542<10 -3 To meet the accuracy requirements, the injection / import flow rate at that moment is set to 17.428 × 10⁻⁶. 4 m 3 / d, with the injection / production flow rate, proceed to the next time step for calculation, and repeat this iterative process until the calculation time reaches 300d. After calculating the flow rate over all time periods, based on the assumed well control radius of 220m and the mass balance equation, the formation pressure p at 300 days is calculated. r1 It is 6.87 MPa, which is less than the predicted formation pressure p. r2 The well control radius needs to be increased to 330m for recalculation, based on a pressure of 10MPa. The formation pressure p at 300 days needs to be calculated. r1 The pressure is 11.604 MPa, which is 10 MPa greater than the predicted formation pressure. r1 -p r2 The value is 1.604, which does not meet the accuracy requirements. The range is limited to [220, 330] for iterative calculation. The specific iterative process is shown in the table below. After 13 iterations, when the well control radius is 297.1826m, the formation pressure p at 300 days is calculated. r1 It is 10.00016 MPa, |p r1 -p r2 |=0.000164<10 -3 If the accuracy requirements are met, the size of the well control dynamic radius at this point is the final calculated well control dynamic radius.

Claims

1. A method for calculating the well control radius of complex well types in oil and gas reservoirs based on generalized mobility, characterized in that: The method is as follows: S1: For existing development data wells, calculate the well control radius of existing development data wells by combining the initial formation pressure, current formation pressure test data, and flow data during production and injection processes with the material balance equation. S2: For prediction of production wells that have not yet been put into production, firstly, based on the concept of generalized flowability, establish and solve the production capacity equation or determine the production capacity equation through field measurement, calculate and draw the inflow dynamic curve, and at the same time establish the wellbore flow equation to calculate and draw the outflow dynamic curve. Based on the node analysis method combined with the inflow dynamic curve and the outflow dynamic curve, calculate and predict the reasonable production capacity of a single well. The specific process is as follows: S21: Based on the concept of generalized flow to characterize the flow law of fluids, for any fluid motion equation, it can be written in the form of a generalized mobility model: v = -λ▽p; where λ is the generalized mobility, m 2 / (Pa·s); v represents the fluid velocity, m / s; ▽p represents the pressure gradient, Pa / m; the fluid flow law is characterized using the above generalized mobility model; S22: The relationship between formation pressure, bottom hole pressure, and flow rate is determined by calculating the production capacity equation, and then the inflow dynamic curve of bottom hole pressure and flow rate is plotted. S23: Convert the wellhead pressure to the bottom hole pressure using the wellbore flow equation; calculate the corresponding bottom hole pressure by giving different nozzle diameters and wellhead pressures, and give different flow rates, and plot the flow rate versus bottom hole pressure curve, i.e., the outflow dynamic curve. S24: By using the node analysis method, the inflow dynamic curve and the outflow dynamic curve are superimposed on a single graph. The flow rate at the intersection of the inflow dynamic curve and the outflow dynamic curve is the predicted reasonable production capacity of a single well. S3: Based on generalized mobility, a single-well pressure unsteady flow model is established, and the unsteady flow bottom hole pressure solution is obtained; The specific process is as follows: A complete single-well pressure-unsteady flow model is constructed using the reservoir control equations, initial conditions, internal and external boundary conditions, and auxiliary equations; and the unsteady flow bottom hole pressure solution is obtained; specifically: When the fluid type is oil or water, a single-phase liquid, the solution is obtained directly using pressure. When the fluid type is gas or multiphase, the concepts of pseudo-pressure and pseudo-time are introduced to transform the equations before solving them. In addition, the unsteady flow bottom hole pressure solution is obtained by using the Laplace transform and inverse transform for the fully perforated vertical well circular boundary model; the unsteady flow bottom hole pressure solution is obtained by using the Laplace spatial point source solution method for the horizontal well model, combined with Fourier transform, superposition principle and inverse Laplace transform; the unsteady flow bottom hole pressure solution is obtained by using the Laplace spatial line source solution method for the fractured well model, combined with superposition principle; for models with finite or infinite conductivity in the wellbore or fracture, the flow distribution and pressure in the wellbore are coupled for calculation, and the unsteady flow bottom hole pressure solution is obtained by solving a system of linear equations. In addition, if the analytical method cannot solve the problem, a numerical method is used: by meshing, a numerical model is established using numerical methods such as finite difference, finite element, finite volume or boundary element to obtain the solution of the bottom pressure of the unsteady flow. S4: Combining the constraints on wellhead pressure during the injection and production process, the predicted reasonable production capacity of a single well, and the calculation of the bottom hole pressure using a single well pressure unstable flow model, the reasonable injection and production flow rate at different times is determined through iterative calculation. S5: Iteratively calculate the optimal well control radius by combining formation pressure constraints and material balance equations.

2. The method for calculating the well control radius of complex well types in oil and gas reservoirs based on generalized mobility according to claim 1, characterized in that: The specific process of S1 is as follows: The well control radius equation is derived by inversely using the mass balance equation, and then directly calculated by substituting parameters into the well control radius equation. The well control radius equation for an oil well is as follows: (1) The equation for the well control radius of a gas well is: (2) The equation for the well control radius is: (3) In the formula: r e - Well control radius, m; N f -Cumulative oil flow rate, m 3 B o -Oil volume coefficient, m 3 / m 3 ; - Porosity, decimal; h - Effective thickness, m; C t -Comprehensive compressibility, 1 / Pa; B oi -Crude oil volume factor under initial conditions, m 3 / m 3 ; p i - Original formation pressure, Pa; p- Pressure, Pa; G f -Cumulative gas flow rate, m 3 Z i - The natural gas deviation factor corresponding to the original formation pressure, dimensionless; Z - Natural gas deviation factor, dimensionless; W f -Cumulative water flow, m 3 B w -Water volume coefficient, m 3 / m 3 B wi -Water volume coefficient under initial conditions, m 3 / m 3 .

3. The method for calculating the well control radius of complex well types in oil and gas reservoirs based on generalized mobility according to claim 2, characterized in that: The specific process of S4 is as follows: S41: Using the predicted reasonable production capacity of a single well obtained in S2 as the initial injection and production flow rate, single well production or injection is carried out. The bottom pressure at different times is calculated through the unstable flow bottom pressure solution obtained in S3. S42: During single-well production or injection, wellhead pressure is constrained by surface engineering conditions. These surface engineering conditions are the wellhead pressure limit values ​​during operation, which are either the minimum or maximum wellhead pressure. The wellhead pressure limit values ​​are converted to the bottom hole pressure limit values ​​using the wellbore flow equation and the injection / production flow rates to obtain the bottom hole pressure limit values. By comparing these with the unsteady flow bottom hole pressure solution obtained in S3, the injection / production flow rates are iterated to ensure that the bottom hole pressure meets the constraints, thereby obtaining reasonable injection / production flow rates at different times. When the injection / production flow rates are different at different times, the unsteady flow bottom hole pressure solution obtained in S3 needs to be calculated using the superposition principle to calculate the bottom hole pressure values ​​at different times under variable flow injection / production conditions.

4. The method for calculating the well control radius of complex well types in oil and gas reservoirs based on generalized mobility according to claim 3, characterized in that: The specific process of S5 is as follows: single-well production or injection is carried out under the initial formation pressure conditions. When the set time is reached, the reasonable injection and production flow rate at the set time is obtained through S4. The formation pressure at the set time is obtained through the material balance equation. By giving the predicted formation pressure value, the calculated formation pressure is compared with the predicted formation pressure value, and the well control radius is iteratively modified until the calculation meets the accuracy requirements, thereby obtaining the optimal well control radius that meets the formation pressure constraint conditions.

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