A method and device for analyzing lateral vibration of a variable parameter drill string

Abstract

The application discloses a kind of variable parameter drill string transverse vibration analysis method and device, it is related to variable parameter structure dynamics field, method includes: obtaining drill string along well depth's drilling tool combination information, to construct along well depth change's structure and mechanics parameter function accordingly, establish transverse vibration control equation and boundary condition;To control equation space discrete and construct relevant matrix, solve generalized eigenvalue and obtain vibration characteristic parameter, and then construct generalized integral transformation relationship;Combined with external excitation, establish and solve modal dynamics equation, restore vibration response to physical space after result, and output engineering decision basis after analysis.This application solves the technical problems that existing method is difficult to adapt to the change of structure and mechanical properties of drill string and the influence of complex environment, leading to inaccurate vibration analysis results and insufficient calculation efficiency, achieves the technical effect of accurately obtaining vibration-related response and key indicators, and provides reliable support for engineering optimization and adjustment.

Description

Technical Field

[0001] This invention relates to the field of variable parameter structural dynamics technology, and in particular to a method and apparatus for analyzing the lateral vibration of a variable parameter drill string. Background Technology

[0002] Lateral vibration analysis of the drill string is crucial for drilling safety, drill string fatigue life assessment, and drilling parameter optimization. Existing techniques often employ constant-section or piecewise constant-parameter beam models, relying on orthogonal eigenfunctions of self-adjoint operators, and solving through modal superposition or traditional generalized integral transformations. However, actual drill strings consist of multiple drill strings connected in series, including drill pipe and weighted drill pipe. Parameters such as bending stiffness vary significantly with well depth, and non-conservative effects such as drilling fluid damping exist, leading to non-self-adjoint control operators and no longer orthogonal eigenfunctions. Traditional methods suffer from modeling disconnect from reality, high computational costs, lack of modal interpretability, and inability to accurately obtain vibration responses, failing to meet the needs of drill string assembly optimization and drilling safety management. Summary of the Invention

[0003] This application provides a method and apparatus for analyzing the lateral vibration of a drill string with variable parameters, which solves the technical problem that existing methods are difficult to adapt to changes in the structure and mechanical properties of the drill string and the influence of complex environments, resulting in inaccurate vibration analysis results and insufficient computational efficiency.

[0004] The first aspect of this application provides a method for analyzing the lateral vibration of a drill string with varying parameters. The method includes: acquiring information about the drill string assembly arranged along the well depth direction; constructing structural and mechanical parameter functions describing the changes in the drill string along the well depth direction based on the drill string assembly information; establishing a control equation for the lateral vibration of the drill string and corresponding boundary conditions based on the structural and mechanical parameter functions; spatially discretizing the control equation for the lateral vibration of the drill string to construct a stiffness matrix, an additional matrix, and a mass matrix, and solving the corresponding generalized eigenvalue problem to obtain characteristic parameters of the lateral vibration of the drill string; constructing a generalized integral transformation relationship for the analysis of the lateral vibration of the drill string based on the characteristic parameters of the lateral vibration of the drill string; establishing a modal dynamics equation based on the generalized integral transformation relationship under given external excitation conditions, and solving the modal dynamics equation to obtain the solution result; restoring the solution result to physical space based on the solution result of the modal dynamics equation to obtain the lateral vibration response of the drill string, and analyzing the lateral vibration response of the drill string to obtain analysis results for engineering decision-making.

[0005] A second aspect of this application provides an analysis device for the lateral vibration of a variable-parameter drill string. The device includes: a drill string assembly information acquisition module for acquiring information about the drill string assembly arranged along the well depth direction; a structural and mechanical parameter function construction module for constructing structural and mechanical parameter functions describing the changes in the drill string along the well depth direction based on the drill string assembly information; a drill string lateral vibration control equation construction module for establishing the drill string lateral vibration control equation and corresponding boundary conditions based on the structural and mechanical parameter functions; and a drill string lateral vibration characteristic parameter acquisition module for spatially discretizing the drill string lateral vibration control equation, constructing a stiffness matrix, an additional matrix, and a mass matrix, and... The system employs a multi-module approach: solving the corresponding generalized eigenvalue problem to obtain the characteristic parameters of the drill string's lateral vibration; a generalized integral transformation relationship construction module to construct a generalized integral transformation relationship for drill string lateral vibration analysis based on the drill string's lateral vibration characteristic parameters; a solution result acquisition module to establish modal dynamics equations based on the generalized integral transformation relationship under given external excitation conditions, and to solve the modal dynamics equations to obtain the solution results; and an analysis result acquisition module to restore the solution results to the physical space based on the solution results of the modal dynamics equations, obtain the drill string's lateral vibration response, and analyze the drill string's lateral vibration response to obtain analysis results for engineering decision-making.

[0006] One or more technical solutions provided in this application have at least the following technical effects or advantages:

[0007] This application obtains drill string assembly information along the well depth direction, constructs structural and mechanical parameter functions that vary along the well depth, establishes lateral vibration control equations considering variable parameter characteristics, axial tension, and non-conservative effects, obtains vibration characteristic data through spatial discretization of characteristic parameters, and constructs biorthogonal generalized integral transformation relationships, solves modal dynamic equations and recovers physical space response, calculates stress and fatigue life indices at key locations, and adapts them to external lateral excitation conditions and boundary constraints. This allows for accurate acquisition of the drill string lateral vibration response along the well depth direction, making the drill string lateral vibration analysis results more consistent with engineering practice. It provides reliable support for drill string assembly optimization, stabilizer placement, and drilling parameter adjustment, achieving the technical effect of accurately acquiring vibration-related responses and key indicators, and providing reliable support for engineering optimization and adjustment. Attached Figure Description

[0008] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly explained below. As is obvious, the accompanying drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without any creative effort.

[0009] Figure 1 This is a flowchart illustrating a method for analyzing the lateral vibration of a drill string with variable parameters, provided in an embodiment of this application.

[0010] Figure 2 This is a schematic diagram of the structure of an analysis device for the lateral vibration of a drill string with variable parameters provided in an embodiment of this application.

[0011] Figure labeling: Module 1 for obtaining drill string assembly information, Module 2 for constructing structural and mechanical parameter functions, Module 3 for constructing control equations for lateral vibration of drill string, Module 4 for obtaining characteristic parameters of lateral vibration of drill string, Module 5 for constructing generalized integral transformation relationship, Module 6 for obtaining solution results, and Module 7 for obtaining analysis results. Detailed Implementation

[0012] This application provides a method and apparatus for analyzing the lateral vibration of a drill string with variable parameters, which solves the technical problem that existing methods are difficult to adapt to changes in the structure and mechanical properties of the drill string and the influence of complex environments, resulting in inaccurate vibration analysis results and insufficient computational efficiency.

[0013] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Description. Obviously, the described embodiments are merely some, not all, of the embodiments described in this application. All other embodiments obtained by those skilled in the art based on the embodiments described in this application without inventive effort are within the scope of protection of this application.

[0014] It should be noted that the terms "first," "second," etc., in the specification and accompanying drawings of this application are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion; for example, a process, method, system, product, or server that includes a series of steps or units is not necessarily limited to those explicitly listed, but may include those not explicitly listed or inherent to such processes, methods, products, or devices. Steps or modules.

[0015] Example 1, as Figure 1 As shown, an analysis method for lateral vibration of a drill string with variable parameters is provided, wherein the method includes:

[0016] Obtain information on the drill string arrangement along the well depth direction.

[0017] Specifically, the process begins by collecting information on the assembly of at least one drill string, consisting of drill pipe, weighted drill pipe, drill collars, and a bottom-hole assembly (BHA). This information includes the outer diameter, inner diameter, length, and material parameters of each drill string segment, as well as the location and type of stabilizers. Next, the drill string is divided into several segments along the well depth, and the starting and ending points of each segment are determined, thus defining its depth range and length. Finally, based on the outer and inner diameters of each segment, key geometric parameters such as the cross-sectional area and moment of inertia are calculated, laying the foundation for subsequent structural and mechanical parameter functions.

[0018] Based on the drill string assembly information, a function describing the structural and mechanical parameters of the drill string as it changes along the well depth is constructed.

[0019] Optionally, based on the geometric parameters and material properties of each drill string section, and combined with the well depth interval division results, structural mechanical parameter functions such as bending stiffness and linear density that continuously vary along the well depth are constructed, while simultaneously considering non-conservative effects such as axial tensile force distribution, drilling fluid damping, and wellbore friction. Then, a control equation for the lateral vibration of the drill string adapted to these variable parameter characteristics is established. The space is discretized using the finite element method or higher-order finite difference method, and the generalized eigenvalue problem is solved to obtain the characteristic frequencies and left and right eigenfunctions. Based on the linear density, a biorthogonal relationship is constructed. The high-dimensional control equation is transformed into a low-dimensional modal dynamics equation through a generalized integral transformation. A solution method adapted to the working conditions is used to obtain the generalized coordinates. The lateral vibration response along the well depth is recovered through inverse transformation, and then the stress and fatigue life related indicators at key locations are calculated.

[0020] Based on the aforementioned structural and mechanical parameter functions, the control equations for the transverse vibration of the drill string and the corresponding boundary conditions are established.

[0021] In one embodiment of this application, based on beam theory, a control equation for the lateral vibration of the drill string is established by comprehensively considering linear density, bending stiffness, axial tension, and non-conservative effects along the well depth, such as the gyroscopic effect and Coriolis force caused by drill string rotation, wellbore contact friction, and drilling fluid action. The corresponding boundary conditions are determined according to the actual constraints at the wellhead and drill bit ends, and equivalent local springs and damping terms are introduced at the stabilizer location to reflect the stabilizer's constraint effect on the lateral vibration of the drill string.

[0022] The control equations for the lateral vibration of the drill string are spatially discretized to construct the stiffness matrix, additional matrix, and mass matrix. The corresponding generalized eigenvalue problem is then solved to obtain the characteristic parameters of the lateral vibration of the drill string.

[0023] Specifically, under free vibration conditions, neglecting external excitation and explicit damping terms, the governing equations are separated into variables, yielding a spatial eigenvalue problem. This problem is discretized along the well depth direction, transforming it into a generalized eigenvalue problem in matrix form. A correlation matrix is ​​constructed, including bending stiffness, axial tension, and non-conservative effects terms. Solving this matrix yields multi-order right and left eigenvectors and their corresponding characteristic frequencies. Finally, through interpolation or shape functions, the discrete eigenvectors are restored to left and right eigenfunctions in continuous space, serving as characteristic parameters of the drill string's lateral vibration.

[0024] Based on the lateral vibration characteristic parameters of the drill string, a generalized integral transformation relationship for lateral vibration analysis of the drill string is constructed.

[0025] Specifically, using linear density as the weighting function, a biorthogonal relationship is established between the left and right characteristic functions, and the biorthogonal matrix is ​​calculated. Based on this biorthogonal relationship, the generalized integral forward and inverse transformation relationships of the drill string's lateral vibration are constructed respectively, and based on these two transformation relationships, the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix for vibration analysis are constructed.

[0026] Under given external excitation conditions, modal dynamic equations are established based on the generalized integral transform relationship, and the modal dynamic equations are solved to obtain the solution results.

[0027] Specifically, under given external lateral excitation conditions, a generalized integral transformation is performed on the control equations for the lateral vibration of the drill string to obtain the dynamic equations in modal space. Then, a numerical integration method is used to solve these modal dynamic equations, obtaining the response results of each modal's generalized coordinates as a function of time, i.e., the solution results.

[0028] Based on the solution results of the modal dynamics equations, the solution results are restored to the physical space to obtain the lateral vibration response of the drill string. The lateral vibration response of the drill string is then analyzed to obtain analytical results for engineering decision-making.

[0029] Specifically, after solving the modal dynamics equations and obtaining the results, the results are first organized into a sequence of modal generalized coordinates changing over time. These modal generalized coordinates can be stored sequentially as time histories of each truncated modal order, and correlated with the corresponding right eigenfunction. Establish a one-to-one correspondence. Here, n represents the modal order, t represents time, and z represents the well depth coordinate. This indicates the truncation mode order used in the calculation. This represents the generalized coordinates of the nth modal after correction by a bioorthogonal matrix.

[0030] Subsequently, the solution results in modal space are restored to physical space based on the inverse generalized integral transform. Following the modal superposition method, at any well depth z, the spatial function values ​​of all truncated modes are multiplied term by term with their corresponding modal time functions and summed to obtain the lateral displacement response. The formula used is: .in This represents the lateral displacement of the drill string at depth z and time t. It is the nth right characteristic function. To and The accompanying corrected modal generalized coordinates. During engineering implementation, at each time step... With each well depth node Discrete displacement field data can be obtained by performing a single summation.

[0031] After obtaining the displacement response, to obtain the velocity and acceleration responses, a conventional method of time differentiation of the modal generalized coordinates while keeping the spatial basis functions unchanged is adopted. The time history of the modal generalized coordinates is used to simultaneously obtain its first and second time derivatives through difference or integration algorithms. Then, the physical space response is recovered in the same modal superposition form. The formula used is:

[0032] , ,in, and These represent lateral velocity and lateral acceleration, respectively. and These represent the first and second time derivatives of the modified modal generalized coordinates, respectively. In engineering implementation, these derivatives can be directly output during the numerical integration process of solving the modal dynamics equations. The derivatives of the velocity and acceleration fields can be obtained by calculating the derivatives using the difference between adjacent time steps in post-processing and substituting them into the above equation.

[0033] After completing the displacement-velocity-acceleration recovery, in order to obtain the bending moment and stress parameters required for engineering decisions, the spatial second derivative of the displacement field is calculated using the post-processing method of existing beam theory. First, at each time step... The second-order difference or the derivative of the shape function is obtained by performing a second-order finite difference along the well depth direction. Then, using the bending stiffness function that varies along the well depth direction... The formula used to calculate the bending moment response and further calculate the bending normal stress response is as follows: , .in Indicates bending moment, Represents the bending stiffness function. Indicates bending stress. This represents the distribution of the moment of inertia of the cross section with depth. This represents the distance from the outer fiber of the cross section to the neutral axis. In engineering implementation, the second spatial derivative of the displacement field is first calculated using a well-depth mesh, and then multiplied point-by-point by... The bending moment field is obtained, and finally combined with and The stress time history of key well sections and key cross sections is obtained by converting each point into a stress field.

[0034] In obtaining After a period of time, fatigue assessment procedures are followed to perform cycle counting and life calculations at key locations. Specifically, stress sequences at locations of engineering interest are selected and mean-reduced or adjusted as needed. Then, rainflow counting methods are used to... The number of cycles for each stress amplitude range is obtained by statistical analysis of stress cycles, and then calculated in combination with the fatigue curve parameters of the material. It damages and provides fatigue life indicators, outputting the peak stress at each key location, the distribution of dangerous well sections, and the corresponding fatigue life or damage results, which can be used for drill string assembly optimization, stabilizer layout optimization, and drilling parameter adjustment.

[0035] By performing a generalized inverse integral transformation on the solution of the modal dynamics equations and further calculating engineering indicators such as bending moment and bending stress, the vibration and fatigue evaluation results that can be directly used for drill bit assembly and working condition optimization decisions were obtained.

[0036] Furthermore, the method provided in this application embodiment includes:

[0037] Obtain drill string assembly information consisting of multiple drill string segments connected sequentially along the well depth direction. This assembly information includes the outer diameter, inner diameter, length, and material parameters of each drill string segment, as well as the location and type of the stabilizer. The drill string assembly includes one of the following: drill pipe, weighted drill pipe, drill collar, and bottom hole assembly (BHA). Divide the drill string along the well depth direction into several drill string segments, and determine the corresponding well depth interval for the i-th drill string segment, as follows: Where i is the drill string segment number, and Let represent the starting and ending positions of the i-th segment of the drill string in the depth direction, respectively. This represents the length of the i-th drill string segment; the cross-sectional geometric parameters of the i-th drill string segment are obtained based on the outer and inner diameters, as follows: , ;in, and Let these represent the outer diameter and inner diameter of the i-th segment of the drill string, respectively. Let represent the cross-sectional area of ​​the i-th segment of the drill string. Let represent the moment of inertia of the i-th segment of the drill string.

[0038] Specifically, in drilling operations, the drill string is typically composed of multiple drill strings connected in series, each with different functions. To conduct lateral vibration analysis of the drill string, it is first necessary to obtain complete and accurate information on the drill string assembly. Those skilled in the art can collect relevant information on each drill string segment connected sequentially along the well depth by consulting the drill string configuration list and on-site measurement records. The types of drill strings involved include common types such as drill pipe, weighted drill pipe, drill collars, and bottom hole assembly (BHA). The specific parameters collected must cover the outer diameter, inner diameter, length, and material parameters of each drill string segment. Simultaneously, the location and type of stabilizers must be clearly identified, as these parameters form the basis for subsequent analysis.

[0039] After obtaining the basic parameters of the drill string assembly, the drill string is divided into sections according to the well depth direction, and the well depth interval of each section is determined. When performing mechanical analysis on the segmented structure, the interval division method is used. First, a well depth coordinate system is established, with the wellhead as the starting reference point and the bottom of the well as the ending point. According to the actual connection sequence of the drill string, each section is assigned a unique serial number i. Then, based on the actual installation positions of each section recorded on-site, the starting position of the i-th section in the well depth direction is determined. and termination position Then through the formula Calculate the length of this section of the drill string. This clarifies the spatial distribution range of each drill string segment, allowing subsequent parameter calculations to accurately correspond to specific well depth ranges and avoiding confusion between parameters from different drill string segments.

[0040] Then, the cross-sectional geometric parameters of each section of the drill string are calculated. Since the cross-section of the drill string is circular, the formula for calculating the geometric parameters of a circular hollow cross-section is directly used:

[0041] For cross-sectional area , in the formula This represents the outer diameter of the i-th segment of the drill string, that is, the diameter of the outer contour of the drill string. π represents the inner diameter of the i-th segment of the drill string, i.e. the diameter of the hollow part inside the drill string. π is the constant of pi. When calculating, first calculate the square of the outer diameter and the square of the inner diameter respectively, then subtract the square of the inner diameter from the square of the outer diameter, multiply the result by π and then divide by 4 to finally obtain the cross-sectional area of ​​the i-th segment of the drill string. This parameter is used to calculate the linear density of the drill string in the subsequent calculation.

[0042] For the moment of inertia Ii of the cross section, the same method is used. and First, calculate the fourth power of the outer diameter and the fourth power of the inner diameter for these two parameters. Then, subtract the two, multiply by π, and divide by 64. The result is the moment of inertia of the i-th segment of the drill string. This parameter reflects the drill string's ability to resist bending deformation and is the core basis for subsequent calculations of bending stiffness.

[0043] By collecting actual drill string parameters, dividing the well depth range, and calculating the cross-sectional parameters using geometric formulas, the spatial range and key geometric characteristics of the drill string segments were accurately obtained, providing reliable basic data for subsequent lateral vibration analysis of the drill string.

[0044] Furthermore, the method provided in this application embodiment includes:

[0045] Based on the cross-sectional area of ​​each drill string section Moment of inertia of cross section Material elastic modulus and density The bending stiffness and linear density parameters for each drill string section are determined as follows: , ;in, This represents the bending stiffness of the i-th segment of the drill string. Let represent the linear density of the i-th drill string segment; based on the start and end positions of each drill string segment in the well depth direction, the bending stiffness and linear density parameters are mapped to piecewise functions that vary along the well depth direction, as follows: ; Where z represents the well depth direction coordinate; Indicates the total number of drill string sections; and These represent the starting and ending positions of the i-th segment of the drill string in the direction of well depth, respectively. The heaviside function is used; based on factors such as drill string weight, well inclination, friction, and drilling hydraulic pressure difference, an axial tensile force function N(z) distributed along the well depth direction is established. Wherein, the axial tensile force function N(z) This function is used to characterize the axial tensile force borne by the drill string at different well depths, and the axial tensile force function is either a continuously distributed function or a piecewise linear distributed function along the well depth direction.

[0046] Optionally, based on the i-th segment of the drilling tool depth range determined in the aforementioned steps When constructing structural and mechanical parameter functions that vary along well depth z, based on basic data such as the outer diameter, inner diameter, length, and material parameters of each section of the drill string, a piecewise constant function construction method from the field of drill string dynamics analysis is adopted. For example, when solving the drill string vibration equation using the piecewise constant method (PT method), it can accurately reflect the parameter mutation characteristics of the segmented drill string, and the calculation results are closer to the theoretical solution than the Runge-Kutta method of the same order, which is fully adapted to the actual engineering application scenario.

[0047] Constructing line density function At the same time, the calculation logic of mass distribution in existing materials mechanics is strictly followed, and the parameter values ​​are entirely based on the calculation results of the aforementioned steps and existing standard data. For any well depth coordinate z, the drill string segment to which it belongs is first determined: if z falls within the well depth range of the i-th drill string segment... Within this range, the linear density function takes the value of a constant for that segment. ,Right now .in, The material density of the i-th segment of the drill string can be obtained directly from the API RP 7G standard document or the technical manual provided by the drill string manufacturer. For example, the drill pipe density of AISI4145HM material and the density of chromium-manganese austenitic alloy of non-magnetic drill collar are both publicly available constants in the industry. Let be the cross-sectional area of ​​the i-th segment of the drill string, derived from the aforementioned steps. The formula is used to calculate that, and These are the outer and inner diameters of the i-th drill string segment, respectively, collected in the preceding steps; z is a continuous coordinate variable along the well depth direction, with its value range covering the entire drill string length from the wellhead to the bottom of the well. The core function of this function is to quantify the mass distribution per unit length of the drill string, providing accurate load input parameters for the calculation of the inertial force term in the subsequent lateral vibration equation.

[0048] Constructing the bending stiffness function At this time, the classic calculation model of bending stiffness in drill string mechanics analysis is adopted to ensure the industry universality of parameter correlation and calculation logic. When the well depth z is within the range of the i-th drill string segment... When, the bending stiffness function takes the value ,Right now .in, The elastic modulus of the drill bit material in the i-th segment is an inherent mechanical property of the material and can be determined by consulting the steel mechanical properties manual or API standards. For example, the elastic modulus of E75 steel grade drill pipe is constant at 206 GPa. The elastic modulus values ​​of different steel grades of drill bits have formed a unified standard in the industry. Let be the moment of inertia of the i-th segment of the drill string, derived from the aforementioned steps. The parameters are obtained by calculation using the formula. and The values ​​are completely consistent with those in the linear density function, ensuring the continuity of parameter transfer. This function directly characterizes the drill string's ability to resist bending deformation and is a core parameter that determines key characteristics such as the natural frequency and mode shape of the drill string's lateral vibration. Its construction logic is completely consistent with the stiffness parameter handling method when analyzing drill string buckling and vibration problems using the finite element method.

[0049] Constructing the axial force function N(z) At that time, referring to the segmented calculation method of drill string axial force in drilling engineering, and combining the actual working condition parameters, the constant was completed. The determination of the well depth range for the i-th segment of the drill string. The axial force function takes the value of the constant axial force in that segment. ,Right now . The calculation employs a piecewise cumulative summation method, specifically based on key operating parameters including the wellhead suspended weight, the buoyant weight of the drill string in the i-th segment and above, drilling fluid buoyancy, and drilling pressure. The buoyant weight is calculated using the buoyancy coefficient method, through the formula... calculate, and These are the outer and inner cross-sectional areas of the drill bit, respectively. Here, g represents the drilling fluid density, and g is the acceleration due to gravity. The hanging weight at the wellhead can be directly read from the hook load sensor at the drilling site, while the drilling pressure is set according to drilling process requirements and determined through adjustments to the drill string's force balance. Meanwhile, The calculations can also refer to known theories such as the Johansick flexible rod model and the three-dimensional drill string force model, taking into account the influence of well inclination angle and azimuth angle on axial force transmission to ensure... It can accurately reflect the actual axial load borne by the i-th segment of the drill string. The function's role is to reflect the modulation effect of axial pressure or tension on the lateral vibration characteristics of the drill string, providing load boundary conditions that conform to actual working conditions for the vibration control equation.

[0050] By employing a piecewise constant function construction method, relying on basic parameters and industry standard data, and following a coherent process of interval judgment, parameter substitution, and constant determination, the linear density, bending stiffness, and axial force functions that continuously vary along the well depth were accurately constructed. This achieved a continuous characterization of the drill string's piecewise parameters, providing complete and reliable parameter support for the subsequent establishment and semi-analytical solution of the drill string's lateral vibration equation.

[0051] Furthermore, the method provided in this application embodiment includes:

[0052] Based on beam theory, a linear density function considering variations along the well depth direction is established. Bending stiffness function Axial tensile force function N(z) The non-conservative effect of the drill string lateral vibration control equation is as follows:

[0053] ; This represents the lateral displacement of the drill string at depth z and time t in the well. This represents the equivalent damping coefficient distributed along the well depth direction. This represents the gyroscopic effect and Coriolis force term caused by the rotation of the drill string. This represents the non-conservative effects caused by wellbore contact, friction, and drilling fluid action. The external lateral excitation is represented; based on the constraint conditions at the wellhead end and the drill bit end, boundary conditions corresponding to the control equation for the lateral vibration of the drill string are established; at the location where the stabilizer is set, equivalent local springs and damping terms are introduced to characterize the constraint effect of the stabilizer on the lateral vibration of the drill string.

[0054] Specifically, based on the Euler-Bernoulli beam vibration theory in drill string dynamics analysis, the core idea of ​​establishing the control equation for the lateral vibration of a variable-parameter drill string is derived through the force balance of a micro-element. First, a length of [missing information] is selected from the entire drill string determined in the preceding steps. The infinitesimal element is defined with the well depth z as its axial coordinate, where z ranges from the wellhead (0) to the bottom of the well (L) along the entire drill string length. The two end faces of the infinitesimal element correspond to the well depths z and z+, respectively. Place.

[0055] Next, a lateral force analysis was performed on the micro-element. The core forces causing the drill string's lateral vibration include inertial force, bending internal force, and additional bending force caused by axial force. The inertial force originates from the lateral acceleration of the micro-element; its magnitude is proportional to the mass and lateral acceleration of the micro-element, and its direction is opposite to the vibration direction. The bending internal force is the internal force generated by the drill string resisting bending deformation, and it exhibits a gradient change along the well depth direction. The axial force N(z) When the micro-element undergoes lateral deformation, it generates an additional bending moment, which is then converted into an additional lateral force. This effect cannot be ignored in the analysis of drill strings in deep wells and horizontal wells, and is a necessary consideration in the vibration modeling of drill strings with variable parameters.

[0056] Then, based on Newton's second law, the lateral force equilibrium equations for the infinitesimal element are established, and the force terms are transformed into mathematical expressions and simplified. The final lateral vibration control equations for the drill string are as follows: .in, ρA(z) represents the lateral displacement of the drill string at depth z and time t, in meters, and is the core variable describing the lateral vibration state of the drill string; t represents time, in seconds, covering the entire time-domain process of vibration; ρA(z) is the constructed linear density function, in kg / m, reflecting the mass distribution per unit length of the drill string; EI(z) is the constructed bending stiffness function, in units of... N(z) characterizes the drill string's ability to resist bending deformation; is the constructed axial force function, in Newtons, representing the axial load distributed along the well depth; z represents the well depth coordinate, in meters, and is the position parameter along the drill string axis. The first term on the left side of the equation is the inertial force term, the second term is the bending force term, and the third term is the additional bending force term caused by the axial force. The balance of these three terms constitutes the core relationship of vibration control, ensuring the coherence and feasibility of the theory.

[0057] After establishing the governing equations, corresponding boundary conditions need to be set based on the actual drilling conditions. The boundary conditions of the drill string must match the actual constraint states at the wellhead and bottom. At the wellhead boundary (z=0), the drill string is suspended by the hook. In actual operation, only a small lateral swing is allowed, but there is no significant lateral displacement, and the bending moment is approximately zero. Therefore, the boundary conditions are set as: w(0,t)=0 and ∂²w(0,t) / ∂z²=0. Here, w(0,t)=0 indicates that the lateral displacement of the drill string at the wellhead is always zero, and ∂²w(0,t) / ∂z²=0 corresponds to the constraint state where the bending moment at the wellhead is zero, which conforms to the elastic support characteristics of the hook suspension.

[0058] At the bottom boundary (z=L), the drill string, drill bit, and bottom hole assembly are connected. When the drill bit contacts the rock, lateral displacement is restricted, and the bottom hole bending moment tends to be balanced. Therefore, the boundary conditions are set as: w(L,t)=0 and ∂²w(L,t) / ∂z²=0. w(L,t)=0 indicates that the drill string at the bottom hole has no lateral displacement due to the contact between the drill bit and the rock, while ∂²w(L,t) / ∂z²=0 corresponds to the actual working condition of bottom hole bending moment balance, ensuring a close fit between the model and the actual working condition.

[0059] By adopting the infinitesimal force balance method in beam vibration theory, combined with the constructed variable parameter function and actual drilling constraints, a complete control equation for the transverse vibration of the drill string and corresponding boundary conditions were established. The physical meaning and value basis of each parameter were clearly defined, providing an accurate and feasible mathematical model support for the subsequent semi-analytical solution of the equation.

[0060] Furthermore, the method provided in this application embodiment includes:

[0061] Under free vibration conditions, neglecting external excitation and explicit damping terms, the variable separation of the control equation for the transverse vibration of the drill string yields the following expression: ;in, Represents spatial characteristic functions, Let represent the circular frequency; substituting the expression into the control equation for the transverse vibration of the drill string, we obtain the spatial eigenvalue problem, as follows: ;in, Represents eigenvalues. Represents the line density function. Describes a spatial differential operator; and the spatial differential operator The following conditions are met:

[0062] ;in, and The operator term contains non-conservative terms; the spatial eigenvalue problem is discretized along the well depth direction, transforming it into a generalized eigenvalue problem in matrix form, as follows: Where K is the stiffness matrix composed of bending stiffness and axial tensile force terms, G is the additional matrix introduced by non-conservative effects, and M is the mass matrix. Let n be the right eigenvector of order n. Let be the nth order eigenvalue; and simultaneously solve the corresponding left eigenvalue problem, as follows: ;in, It is the left eigenvector of order n; Indicates conjugate transpose; Representing complex conjugation; by interpolation or shape functions, the discrete right and left eigenvectors are respectively restored to right eigenfunctions in continuous space. With left characteristic function The lateral vibration characteristic parameters of the drill string are obtained.

[0063] Specifically, firstly, when obtaining characteristic parameters under free vibration conditions, variable separation is performed to transform the spatiotemporal coupling problem into a spatial eigenvalue problem. Specifically, external excitation and explicit damping terms are ignored, and the lateral displacement is treated using a separation method. ,in For spatial characteristic functions, Given the angular frequency, substituting this expression into the control equation for the lateral vibration of the drill string yields the spatial eigenvalue problem. and ,in As a weighting function, it appears on the right-hand side of the eigenvalue problem, and the output is about... The expression for the spatial eigenvalue problem.

[0064] Next, it will be clear The composition, ,in The term corresponds to the contribution of bending stiffness. The item corresponds to the contribution of axial tensile force. and This corresponds to the operator contribution that includes non-conservative effect terms. In engineering implementation, this can be... and As an equivalent linearization result of linear operators written into matrix addendum terms or velocity-related terms in a predetermined discrete manner, it can be compared with... and The discrete terms are entered together into the same eigenvalue solution process, and the output is the discretely implementable operator decomposition and the correspondence between each term.

[0065] Subsequently, the spatial eigenvalue problem is discretized along the well depth direction using structural dynamics eigenvalue solving techniques, transforming the continuous operator problem into a generalized eigenvalue problem in matrix form. Specific methods may include finite element method or higher-order finite difference method. In practice, nodes are arranged within the well depth interval, and values ​​are taken on each element or difference template. , , The numerical values ​​are used to assemble the mass matrix M, the stiffness matrix K formed by the bending stiffness and axial tensile force terms, and the discrete contributions of the non-conservative operators are assembled into an additional matrix G, thus forming... ,in Let n be the right eigenvector of order n. Given the nth order eigenvalues, the output is a K, G, M and generalized eigenvalue problem that can be solved directly by calling numerical eigenvalue algorithms.

[0066] Because the system contains non-conservative effects that cause the operator to be non-self-adjoint, it is necessary to simultaneously solve the corresponding left eigenvalue problem to obtain the left eigenvector for subsequent bioorthogonal processing. Specifically, after obtaining K, G, and M, according to... Solve the left characteristic problem in the form of , where For the nth-order left eigenvector, the above conjugate transpose With complex conjugate treatment To adapt to the case of complex eigenvalues, the output is the same as that of each eigenvalue. The corresponding set of right eigenvectors With left eigenvector .

[0067] After obtaining the discrete left and right eigenvectors, in order to obtain the lateral vibration characteristic parameters of the drill string and meet the engineering requirements of continuous well depth representation, interpolation or shape function recovery methods are used to map the discrete vectors back to the continuous well depth space. Specifically, this is done by using the discrete nodes... and To control the values, a continuous right-hand characteristic function is reconstructed within each cell using a cell shape function or piecewise interpolation consistent with the discretization method. With continuous left characteristic function This ensures that the result is strictly equal to the solution at the nodes and continuous in the well depth direction, thus outputting a continuous set of eigenfunctions and their corresponding eigenvalue sequences that can be used for subsequent bioorthogonal integration and mode transformation. That is, characteristic frequency information.

[0068] By obtaining the generalized eigenvalue problem and simultaneously solving the left and right features to restore the continuous eigenfunction, we can obtain the lateral vibration characteristic parameters in the variable parameter drill string model containing non-conservative effects, which can be directly used for subsequent generalized integral transformation and response calculation, thereby improving the feasibility and accuracy of engineering analysis.

[0069] Furthermore, the method provided in this application embodiment includes:

[0070] With line density function As a weighting function, a bioorthogonality relationship is established between the right and left eigenfunctions, as follows: ;in, Denotes the m-th left characteristic function. Denotes the right-hand characteristic function of order n. The elements of the bioorthogonal matrix are represented; based on the bioorthogonality relation, the generalized integral positive transform of the drill string lateral vibration is constructed as follows: And construct the inverse transform corresponding to the generalized integral forward transform, as follows: ;in, The modal generalized coordinates are corrected by the aforementioned biorthogonal matrix; based on the aforementioned generalized integral transformation relationship, the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix are constructed for the lateral vibration analysis of the drill string.

[0071] Specifically, firstly, using the line density function Establishing a bioorthogonal relationship as a weight function involves, specifically, based on the existing modal orthogonal integral, assigning each pair of left eigenfunctions... With right characteristic function Multiply together Numerical integration was then performed over the well depth interval from 0 to L to obtain the elements of the bioorthogonal matrix. ,Right now In engineering implementation, trapezoidal integrals or Gaussian integrals can be directly used to integrate the function values ​​at discrete well depth nodes, and the output is an invertible bioorthogonal matrix S and its elements. .

[0072] After obtaining the bioorthogonal matrix, the positive transformation of the generalized integral transformation is constructed, specifically the transformation of the given lateral displacement in physical space. By projecting step by step, we obtain:

[0073] When the project is implemented, Instantaneous values ​​on the well depth grid and and The generalized coordinates of each modality can be obtained by multiplying each point and then performing numerical integration. The output is a modal space coordinate vector. .

[0074] Since bioorthogonal matrices are generally off-diagonal, to ensure that the constructed generalized inverse integral transform can directly reconstruct the transverse displacement field in physical space, it is necessary to perform matrix correction on the modal coordinates according to the existing bioorthogonal modal expansion method. Specifically, this involves using the already calculated bioorthogonal matrix... , For the modal generalized coordinate vector obtained by the positive transformation A linear transformation is performed to obtain the modal generalized coordinate vector after correction by a bioorthogonal matrix. The two satisfy the relationship. In engineering implementation, this can be achieved by numerically inverting matrix S, or equivalently by solving the aforementioned system of linear equations. Thus, the modified modal generalized coordinates of each order are obtained. This coordinate sequence serves as the direct input for subsequent physical quantity recovery.

[0075] After obtaining the corrected modal generalized coordinates, an inverse transform corresponding to the generalized integral forward transform is constructed to recover the lateral vibration response in physical space. Lateral displacement in physical space. The expression is .in It is the nth right characteristic function. This refers to the selected truncation mode order. This will be determined during engineering implementation. Calculate each order at each well depth node z. The value is then compared with the corresponding time. By multiplying each term and summing them, the lateral displacement field along the well depth at that moment can be obtained. The displacement response can be continuously updated over time.

[0076] After determining the forward and inverse transformation relationships, the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix are constructed based on this generalized integral transformation relationship. Specifically, the modal projection assembly method is used, where the mass, damping, and stiffness terms in the governing equations are respectively... Multiply by the corresponding term and multiply by Integrate over the interval from 0 to L, or in discrete form, project the left and right eigenvectors onto the original mass matrix M, the equivalent matrix of the damping term, the stiffness matrix K, and the additional matrix G to obtain Mg, Cg, and Kg for the modal dynamics equations. The output is the matrix coefficients that can be directly used for numerical integration in modal space.

[0077] pass By using a biorthogonal matrix and defining forward and inverse transformations accordingly, and assembling Mg, Cg, and Kg, the physical space control equations are stably transformed into modal space under non-self-adjoint variable parameter conditions, thus facilitating efficient solution and accurate reconstruction of vibration response.

[0078] Furthermore, the method provided in this application embodiment includes:

[0079] Under a given external lateral excitation condition, the generalized integral transformation of the control equation for the lateral vibration of the drill string yields the modal dynamics equation, as follows:

[0080] ;in, For modal generalized coordinate vectors, , and These represent the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix, respectively, constructed using the generalized integral transformation relation. The equivalent modal excitation vector is represented; the modal dynamics equation is solved using a numerical integration method to obtain the solution results of the generalized coordinates of each mode changing with time.

[0081] In one embodiment, after obtaining the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix, under given external lateral excitation conditions, a generalized integral transformation is performed on the control equations for the lateral vibration of the drill string. This transforms the control equations, originally distributed along the well depth direction, into dynamic equations in modal space, with the following form: ,in, Let Mg, Cg, and Kg be the modal generalized coordinate vectors, representing the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix, respectively, constructed through the generalized integral transformation relation. This represents the equivalent modal excitation vector corresponding to the external transverse excitation.

[0082] After obtaining the above modal dynamics equations, the equations are numerically solved according to the solution process for multi-degree-of-freedom structural dynamics systems. Specifically, the initial values ​​and initial states of the modal generalized coordinate vectors are first set based on the drilling conditions, and an appropriate time step is selected as the calculation step parameter. Then, within each time step, the values ​​of the modal generalized coordinate vectors are gradually updated based on the balance relationships between the terms in the modal dynamics equations, allowing them to evolve continuously over time.

[0083] During the numerical solution process, the generalized mass matrix Mg, the generalized damping matrix Cg, and the generalized stiffness matrix Kg remain unchanged as known system parameters. The equivalent modal excitation vector is updated over time according to the external excitation conditions. The calculation process proceeds sequentially until the entire analysis period is completed, thereby obtaining the complete solution results of the generalized coordinates of each modality changing over time.

[0084] By converting the control equations for the lateral vibration of the drill string into modal dynamics equations and solving them using step-by-step numerical integration, the modal response time history of the lateral vibration of the drill string can be obtained under the premise that the calculation process is clear and the steps are feasible.

[0085] In summary, the method for analyzing the lateral vibration of a drill string with variable parameters provided in this application has the following technical advantages:

[0086] This application obtains drill string assembly information along the well depth, constructs structural and mechanical parameter functions that vary along the well depth, establishes lateral vibration control equations with variable parameters and non-conservative effects, solves characteristic parameters by spatial discretization, constructs generalized integral transformation relationships, solves modal equations and recovers physical space response, calculates stress and fatigue life indices at key locations, making the lateral vibration analysis of the drill string more realistic, achieving the technical effect of accurately obtaining vibration-related responses and key indices, and providing reliable support for engineering optimization and adjustment.

[0087] Example 2, as Figure 2 As shown, based on the same inventive concept as in Embodiment 1 above, this application provides an analysis device for the lateral vibration of a variable parameter drill string, the device comprising:

[0088] Drill string assembly information acquisition module 1, which is used to acquire drill string assembly information arranged along the well depth direction.

[0089] The structural and mechanical parameter function construction module 2 constructs structural and mechanical parameter functions describing the changes of the drill string along the well depth direction based on the drill string assembly information.

[0090] The drill string lateral vibration control equation construction module 3 is used to establish the drill string lateral vibration control equation and corresponding boundary conditions based on the structural and mechanical parameter functions.

[0091] The drill string lateral vibration characteristic parameter acquisition module 4 is used to spatially discretize the control equation of the drill string lateral vibration, construct the stiffness matrix, the additional matrix and the mass matrix, and solve the corresponding generalized eigenvalue problem to obtain the drill string lateral vibration characteristic parameters.

[0092] The generalized integral transformation relationship construction module 5 constructs a generalized integral transformation relationship for the analysis of the drill string's lateral vibration based on the drill string's lateral vibration characteristic parameters.

[0093] The solution result acquisition module 6 is used to establish the modal dynamics equation based on the generalized integral transformation relationship under given external excitation conditions, and solve the modal dynamics equation to obtain the solution result.

[0094] The analysis result acquisition module 7, based on the solution results of the modal dynamics equation, restores the solution results to the physical space, obtains the lateral vibration response of the drill string, and analyzes the lateral vibration response of the drill string to obtain analysis results for engineering decision-making.

[0095] Furthermore, the drill string assembly information acquisition module 1 is used to perform the following steps:

[0096] Obtain drill string assembly information consisting of multiple drill string segments connected sequentially along the well depth direction. This assembly information includes the outer diameter, inner diameter, length, and material parameters of each drill string segment, as well as the location and type of the stabilizer. The drill string assembly includes one of the following: drill pipe, weighted drill pipe, drill collar, and bottom hole assembly (BHA). Divide the drill string along the well depth direction into several drill string segments, and determine the corresponding well depth interval for the i-th drill string segment, as follows: Where i is the drill string segment number, and Let represent the starting and ending positions of the i-th segment of the drill string in the depth direction, respectively. This represents the length of the i-th drill string segment; the cross-sectional geometric parameters of the i-th drill string segment are obtained based on the outer and inner diameters, as follows: , ;in, and Let these represent the outer diameter and inner diameter of the i-th segment of the drill string, respectively. Let represent the cross-sectional area of ​​the i-th segment of the drill string. Let represent the moment of inertia of the i-th segment of the drill string.

[0097] Furthermore, the structure and mechanical parameter function construction module 2 is used to perform the following steps:

[0098] Based on the cross-sectional area of ​​each drill string section Moment of inertia of cross section Material elastic modulus and density The bending stiffness and linear density parameters for each drill string section are determined as follows: , ;in, This represents the bending stiffness of the i-th segment of the drill string. Let represent the linear density of the i-th drill string segment; based on the start and end positions of each drill string segment in the well depth direction, the bending stiffness and linear density parameters are mapped to piecewise functions that vary along the well depth direction, as follows:

[0099] ; Where z represents the well depth direction coordinate; Indicates the total number of drill string sections; and These represent the starting and ending positions of the i-th segment of the drill string in the direction of well depth, respectively. The heaviside function is used; based on factors such as drill string weight, well inclination, friction, and drilling hydraulic pressure difference, an axial tensile force function N(z) distributed along the well depth direction is established. Wherein, the axial tensile force function N(z) This function is used to characterize the axial tensile force borne by the drill string at different well depths, and the axial tensile force function is either a continuously distributed function or a piecewise linear distributed function along the well depth direction.

[0100] Furthermore, the drill string lateral vibration control equation construction module 3 is used to perform the following steps:

[0101] Based on beam theory, a linear density function considering variations along the well depth direction is established. Bending stiffness function Axial tensile force function N(z) The non-conservative effect of the drill string lateral vibration control equation is as follows:

[0102] ; This represents the lateral displacement of the drill string at depth z and time t in the well. This represents the equivalent damping coefficient distributed along the well depth direction. This represents the gyroscopic effect and Coriolis force term caused by the rotation of the drill string. This represents the non-conservative effects caused by wellbore contact, friction, and drilling fluid action. The external lateral excitation is represented; based on the constraint conditions at the wellhead end and the drill bit end, boundary conditions corresponding to the control equation for the lateral vibration of the drill string are established; at the location where the stabilizer is set, equivalent local springs and damping terms are introduced to characterize the constraint effect of the stabilizer on the lateral vibration of the drill string.

[0103] Furthermore, the drill string lateral vibration characteristic parameter acquisition module 4 is used to perform the following steps:

[0104] Under free vibration conditions, neglecting external excitation and explicit damping terms, the variable separation of the control equation for the transverse vibration of the drill string yields the following expression: ;in, Represents spatial characteristic functions, Let represent the circular frequency; substituting the expression into the control equation for the transverse vibration of the drill string, we obtain the spatial eigenvalue problem, as follows: ;in, Represents eigenvalues. Represents the line density function. Describes a spatial differential operator; and the spatial differential operator The following conditions are met:

[0105] ;in, and The operator term contains non-conservative terms; the spatial eigenvalue problem is discretized along the well depth direction, transforming it into a generalized eigenvalue problem in matrix form, as follows: Where K is the stiffness matrix composed of bending stiffness and axial tensile force terms, G is the additional matrix introduced by non-conservative effects, and M is the mass matrix. Let n be the right eigenvector of order n. Let be the nth order eigenvalue; and simultaneously solve the corresponding left eigenvalue problem, as follows:

[0106] ;in, It is the left eigenvector of order n; Indicates conjugate transpose; Representing complex conjugation; by interpolation or shape functions, the discrete right and left eigenvectors are respectively restored to right eigenfunctions in continuous space. With left characteristic function The lateral vibration characteristic parameters of the drill string are obtained.

[0107] Furthermore, the generalized integral transform relation construction module 5 is used to perform the following steps:

[0108] With line density function As a weighting function, a bioorthogonality relationship is established between the right and left eigenfunctions, as follows: ;in, Denotes the m-th left characteristic function. Denotes the right-hand characteristic function of order n. The elements of the bioorthogonal matrix are represented; based on the bioorthogonality relation, the generalized integral positive transform of the drill string lateral vibration is constructed as follows: And construct the inverse transform corresponding to the generalized integral forward transform, as follows: ;in, The modal generalized coordinates are corrected by the aforementioned biorthogonal matrix; based on the aforementioned generalized integral transformation relationship, the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix are constructed for the lateral vibration analysis of the drill string.

[0109] Furthermore, the solution result acquisition module 6 is used to perform the following steps:

[0110] Under a given external lateral excitation condition, the generalized integral transformation of the control equation for the lateral vibration of the drill string yields the modal dynamics equation, as follows:

[0111] ;in, For modal generalized coordinate vectors, , and These represent the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix, respectively, constructed using the generalized integral transformation relation. The equivalent modal excitation vector is represented; the modal dynamics equation is solved using a numerical integration method to obtain the solution results of the generalized coordinates of each mode changing with time.

[0112] The variable parameter drill string lateral vibration analysis device provided in this embodiment of the invention can execute the variable parameter drill string lateral vibration analysis method provided in any embodiment of the invention, and has the corresponding functional modules and beneficial effects of the method.

[0113] Although this application makes various references to certain modules in the apparatus according to the embodiments of this application, any number of different modules can be used and run on user terminals and / or servers. The various units and modules included are only divided according to functional logic, but are not limited to the above division, as long as the corresponding functions can be achieved; in addition, the specific names of each functional unit are only for easy distinction between each other and are not intended to limit the scope of protection of this invention.

[0114] The specific embodiments described above do not constitute a limitation on the scope of protection of this application. Those skilled in the art should understand that various modifications, combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application should be included within the scope of protection of this application. In some cases, the actions or steps described in this application can be performed in a different order than that shown in the embodiments and still achieve the desired results. Furthermore, the processes depicted in the accompanying drawings do not necessarily require a specific or sequential order to achieve the desired results. In some embodiments, multitasking and parallel processing are also possible or may be advantageous.

Claims

1. A method for analyzing the lateral vibration of a drill string with variable parameters, characterized in that, The method includes: Obtain information on the drill string arrangement along the well depth direction; Based on the drill string assembly information, construct a function describing the structural and mechanical parameters of the drill string as it changes along the well depth direction; Based on the structural and mechanical parameter functions, establish the control equation for the transverse vibration of the drill string and the corresponding boundary conditions; The control equations for the lateral vibration of the drill string are spatially discretized to construct the stiffness matrix, additional matrix, and mass matrix. The corresponding generalized eigenvalue problem is then solved to obtain the characteristic parameters of the lateral vibration of the drill string. Based on the lateral vibration characteristic parameters of the drill string, a generalized integral transformation relationship for lateral vibration analysis of the drill string is constructed. Under given external excitation conditions, modal dynamic equations are established based on the generalized integral transform relationship, and the modal dynamic equations are solved to obtain the solution results; Based on the solution results of the modal dynamics equations, the solution results are restored to the physical space to obtain the lateral vibration response of the drill string. The lateral vibration response of the drill string is then analyzed to obtain analytical results for engineering decision-making. Construct functions describing the structural and mechanical parameters of the drill string as it changes along the well depth, including: Based on the cross-sectional area of ​​each drill string section Moment of inertia of cross section Material elastic modulus and density The bending stiffness and linear density parameters for each drill string section are determined as follows: in, This represents the bending stiffness of the i-th segment of the drill string. This represents the linear density of the i-th segment of the drill string; Based on the start and end positions of each drill string section in the well depth direction, the bending stiffness and linear density parameters are mapped to piecewise functions that vary along the well depth direction, as follows: Where z represents the well depth direction coordinate; Indicates the total number of drill string sections; and These represent the starting and ending positions of the i-th segment of the drill string in the direction of well depth, respectively. For the Heaviside function; Based on the factors of drill string self-weight, well inclination, friction and drilling hydraulic pressure difference, an axial tensile force function N(z) distributed along the well depth direction is established. The axial tensile force function N(z) is used to characterize the axial tensile force borne by the drill string at different well depth positions, and the axial tensile force function is a continuous distribution function or a piecewise linear distribution function along the well depth direction. Establish the governing equations for the lateral vibration of the drill string and the corresponding boundary conditions, including: Based on beam theory, a linear density function considering variations along the well depth direction is established. Bending stiffness function The axial tensile force function N(z) and the non-conservative effect of the control equation for the transverse vibration of the drill string are as follows: This represents the lateral displacement of the drill string at depth z and time t in the well. This represents the equivalent damping coefficient distributed along the well depth direction. This represents the gyroscopic effect and Coriolis force term caused by the rotation of the drill string. This represents the non-conservative effects caused by wellbore contact, friction, and drilling fluid action. Indicates external lateral stimulus; Based on the constraints at the wellhead and drill bit ends, boundary conditions corresponding to the control equation for the transverse vibration of the drill string are established. At the stabilizer setting location, an equivalent local spring and damping term are introduced to characterize the stabilizer's restraining effect on the lateral vibration of the drill string. Obtain the characteristic parameters of the drill string's lateral vibration, including: Under free vibration conditions, neglecting external excitation and explicit damping terms, the variable separation of the control equation for the transverse vibration of the drill string yields the following expression: in, Represents spatial characteristic functions, Indicates angular frequency; Substituting the expression into the control equation for the transverse vibration of the drill string, the spatial eigenvalue problem is obtained as follows: in, Represents eigenvalues. Represents the line density function. Represents the spatial differential operator; And the spatial differential operator The following conditions are met: ; in, and This indicates an operator term that contains non-conservative terms; Discretizing the spatial eigenvalue problem along the well depth direction transforms it into a generalized eigenvalue problem in matrix form, as follows: Where K is the stiffness matrix composed of bending stiffness and axial tensile force terms, G is the additional matrix introduced by non-conservative effects, and M is the mass matrix. Let n be the right eigenvector of order n. It is the nth order eigenvalue; Simultaneously solve the corresponding left eigenvalue problem, as follows: in, It is the left eigenvector of order n; Indicates conjugate transpose; Indicates complex conjugation; By using interpolation or shape functions, the discrete right and left eigenvectors can be restored to right eigenfunctions in continuous space. With left characteristic function The lateral vibration characteristic parameters of the drill string are obtained. The generalized integral transformation relation for the analysis of lateral vibration of the drill string is constructed, including: With line density function As a weighting function, a bioorthogonality relationship is established between the right and left eigenfunctions, as follows: in, Denotes the m-th left characteristic function. Denotes the right-hand characteristic function of order n. Represents the elements of a bioorthogonal matrix; Based on the aforementioned bioorthogonality relationship, a generalized integral positive transform of the drill string's lateral vibration is constructed as follows: And construct the inverse transform corresponding to the generalized integral forward transform, as follows: ; in, These are the modal generalized coordinates corrected by the aforementioned bioorthogonal matrix; Based on the generalized integral transformation relationship, the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix are constructed for the analysis of drill string lateral vibration.

2. The method for analyzing the lateral vibration of a variable-parameter drill string as described in claim 1, characterized in that, Obtain information on the drill string arrangement along the well depth direction, including: Obtain information on the drill string assembly formed by connecting multiple drill string segments sequentially along the well depth direction. The drill string assembly information includes the outer diameter, inner diameter, length, and material parameters of each drill string segment, as well as the location and type of the stabilizer. The drill string assembly includes one of the following: drill pipe, weighted drill pipe, drill collar, and bottom hole assembly (BHA). The drill string is divided into several drill string segments along the well depth direction. The corresponding well depth interval for the i-th drill string segment is determined as follows: Where i is the drill string segment number. and Let represent the starting and ending positions of the i-th segment of the drill string in the depth direction, respectively. Indicates the length of the i-th segment of the drill string; The cross-sectional geometric parameters of the i-th segment of the drill string are obtained based on the outer and inner diameters, as follows: , ; in, and Let these represent the outer diameter and inner diameter of the i-th segment of the drill string, respectively. Let represent the cross-sectional area of ​​the i-th segment of the drill string. Let represent the moment of inertia of the i-th segment of the drill string.

3. The method for analyzing the lateral vibration of a variable-parameter drill string as described in claim 1, characterized in that, The modal dynamics equations are solved to obtain the solution results, including: Under a given external lateral excitation condition, the generalized integral transformation of the control equation for the lateral vibration of the drill string yields the modal dynamics equation, as follows: ; in, For modal generalized coordinate vectors, , and These represent the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix, respectively, constructed using the generalized integral transformation relation. Represents the equivalent modal excitation vector; The modal dynamics equations are solved using a numerical integration method to obtain the solution results of the generalized coordinates of each mode changing with time.

4. An analysis device for the lateral vibration of a drill string with variable parameters, characterized in that, An apparatus for implementing the analytical method for lateral vibration of a variable-parameter drill string according to any one of claims 1-3, the apparatus comprising: The drill string assembly information acquisition module is used to acquire information on the drill string assembly along the well depth direction; The structural and mechanical parameter function construction module constructs structural and mechanical parameter functions that describe the changes of the drill string along the well depth direction based on the drill string assembly information. The drill string lateral vibration control equation construction module is used to establish the drill string lateral vibration control equation and corresponding boundary conditions based on the structure and mechanical parameter functions. The drill string lateral vibration characteristic parameter acquisition module is used to spatially discretize the control equation of the drill string lateral vibration, construct the stiffness matrix, additional matrix and mass matrix, and solve the corresponding generalized eigenvalue problem to obtain the drill string lateral vibration characteristic parameters. The generalized integral transform relationship construction module constructs a generalized integral transform relationship for the analysis of the drill string's lateral vibration based on the drill string's lateral vibration characteristic parameters. The solution result acquisition module is used to establish modal dynamics equations based on the generalized integral transform relationship under given external excitation conditions, and to solve the modal dynamics equations to obtain the solution results; The analysis result acquisition module, based on the solution results of the modal dynamics equations, restores the solution results to the physical space, obtains the lateral vibration response of the drill string, and analyzes the lateral vibration response of the drill string to obtain analysis results for engineering decision-making.

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