CN113076649A - Method for analyzing transverse vibration of drill column of well with complex structure - Google Patents

Method for analyzing transverse vibration of drill column of well with complex structure Download PDF

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CN113076649A
CN113076649A CN202110379349.0A CN202110379349A CN113076649A CN 113076649 A CN113076649 A CN 113076649A CN 202110379349 A CN202110379349 A CN 202110379349A CN 113076649 A CN113076649 A CN 113076649A
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drill string
coordinate system
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drill
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毛良杰
马茂原
刘清友
蔡明杰
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Southwest Petroleum University
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Abstract

The invention discloses a method for analyzing the transverse vibration of a drilling column of a well with a complex structure, which relates to the technical field of oil and gas development and comprises the following steps of S1: obtaining a wellbore trajectory for the target well; s2: establishing a satellite coordinate system to obtain a displacement vector equation; s3: carrying out coordinate system transformation to obtain a conversion relation between a satellite coordinate system and a geodetic coordinate system of each beam unit; s4: according to a Lagrange equation, a nonlinear transverse vibration model of each beam unit of the drill string under a satellite coordinate system is established, and a dynamic model of the drill string under a geodetic coordinate system is obtained through coordinate system conversion and superposition; s5: setting boundary conditions, and discretely solving a dynamic model to obtain a transverse vibration curve of the drill string during load running. According to the method, the nonlinear transverse vibration model of the drill string in the load operation process is established, the dynamic vibration characteristic prediction model of the drill string is obtained, and the vibration characteristic prediction of the drill string under the dynamic load condition is more accurate.

Description

Method for analyzing transverse vibration of drill column of well with complex structure
Technical Field
The invention relates to the technical field of petroleum and natural gas development, in particular to a method for analyzing transverse vibration of a drilling column of a complex-structure well.
Background
In the drilling construction of oil and gas wells, a drill string is the part which is most prone to faults and damages, and the fatigue life and the service strength of the drill string limit the process and the construction progress of the drilling construction. Under the conditions of severe working conditions and complex stress, the drill string is accelerated to break, puncture and lose efficacy when being worn, corroded and abnormally damaged, and the fatigue life of the drill string is difficult to predict under the influences of the working conditions, geological conditions and other factors of the drilling tool, so that the fatigue life of the drill string is difficult to effectively control in the design of a drilling process and a construction process. In the underground of wells with complex structures, such as deep wells, ultra-deep wells, directional wells, horizontal wells and the like, the stress borne by a drill string is more complex, and the problem of drill string failure is particularly prominent.
The failure modes of the drill stem comprise abrasion, cracks, puncture, deformation and the like, so that the drilling cost is greatly increased, the tripping operation time is prolonged, the normal drilling operation is greatly hindered, and the mechanical drilling speed is reduced. Generally speaking, the most important reason for causing the drill string to fail is drill string vibration, but the existing drill string vibration prediction is focused on a static drill string or under a balanced state, so that the vibration in the actual load operation process of the drill string is difficult to accurately reflect, and therefore the significance for the drill string failure prediction and safe construction guidance is small. Therefore, how to predict the dynamic vibration of the drill string in the loading process has important significance for the prediction of the drill string failure and the safe drilling construction.
Disclosure of Invention
The method aims to solve the problem that the result effectiveness of a method for predicting the fatigue life of a drill string in the prior art is not high, and provides a method for analyzing the transverse vibration of the drill string of a well with a complex structure.
In order to achieve the above object, the present application provides the following technical solutions: a method for analyzing the transverse vibration of a drill string of a well with a complex structure comprises the following steps:
s1: obtaining a borehole track of a target well according to well depth data and azimuth data of the target well;
s2: dividing a drill string of a target well into a plurality of beam units along a well track, and establishing a random coordinate system by taking the well track direction of the target well as an X axis to obtain a displacement vector equation of each beam unit;
s3: carrying out coordinate system transformation on the satellite coordinate system established by each beam unit to obtain a conversion relation between the satellite coordinate system of each beam unit and a geodetic coordinate system;
s4: according to a Lagrange equation, a nonlinear transverse vibration model of each beam unit of the drill string under a satellite coordinate system is established, and is substituted into a conversion relation between the satellite coordinate system and a geodetic coordinate system, and a dynamic model of the drill string under the geodetic coordinate system is obtained through superposition;
s5: setting the boundary condition of the drill string, substituting the drill string displacement vector equation into the dynamic model, and discretely solving the dynamic model to obtain a transverse vibration curve of the drill string during load operation.
Further, in step S1, the borehole trajectory is obtained by a cubic spline interpolation method.
Further, the wellbore trajectory is obtained by:
s11: selecting a plurality of measuring points along the target well, and logging to obtain well depth data, azimuth angle data and inclination angle data of the target wells;
s12: calculating the coordinate of each measuring point according to a curvature radius method;
s13: and obtaining the well track of the target well according to a cubic spline interpolation method.
Further, the displacement vector equation of each beam unit is as follows:
{Ui}e＝[xi,yi,zi,θxi,θyi,θzi,xj,yj,zj,θxj,θyj,θzj]；
wherein x isi、yi、ziCoordinates of a node of each beam element; xj、yj、zjRespectively the coordinates of the other node of each beam unit; thetaxi、θyi、θzi、θxj、θyj、θzjThe turning angles of the two nodes of each beam unit rotating around the coordinate axis are respectively.
Further, the conversion relation between the satellite coordinate system and the geodetic coordinate system is as follows:
Figure BDA0003012370580000021
wherein, Trans is a transformation matrix;
the transformation matrix is:
Figure BDA0003012370580000022
wherein,
Figure BDA0003012370580000023
the rotation angle of the X axis is obtained when the satellite coordinate system is converted into a geodetic coordinate system;
Figure BDA0003012370580000024
the rotation angle of the Y axis is obtained when the satellite coordinate system is converted into a geodetic coordinate system;
Figure BDA0003012370580000025
the rotation angle of the Z axis when the satellite coordinate system is converted into the geodetic coordinate system.
Further, the nonlinear lateral vibration model of the beam unit is as follows:
Figure BDA0003012370580000026
wherein,
Figure BDA0003012370580000027
{U}e,{F}egeneralized acceleration, generalized velocity, generalized displacement and external force vector under a satellite coordinate system are respectively provided; [ M ] A]e，[C]eAnd [ K ]]eRespectively representing a mass matrix, a damping matrix and a rigidity matrix under a random coordinate system.
Further, the dynamic model of the drill string in the geodetic coordinate system is:
Figure BDA0003012370580000031
wherein,
Figure BDA0003012370580000032
{ U '} { F' } is generalized acceleration, generalized speed, generalized displacement and external force vector of each node on the drill column under a geodetic coordinate system respectively; [ M']、[C′]And [ K']The drill string represents a mass matrix, a damping matrix and a stiffness matrix in the geodetic coordinate system respectively.
Further, the drill string boundary conditions include a borehole wall constraint and a drill bit constraint, and the borehole wall constraint is as follows:
xi＝0,yi＝0,zi＝0,θx＝Ωt,θy＝0,θz＝0；
wherein, thetax、θy、θzRespectively the rotation angles of the drill column around each coordinate axis; omega is the rotation speed of the drill string, rad/s; t represents time, s.
Further, in step S5, the dynamic model is discretely solved by a generalized- α method.
Further, the prediction method is also based on the following assumptions:
(1) the shaft is a circle with a uniform cross section, and the track of the well hole is continuous and smooth;
(2) considering the drill string as a three-dimensional elastic beam with uniform material and collective properties, whose deformation is within the elastic range;
(3) the connecting thread and the partial orifice plate on the drill string and at the connecting part of other equipment and the drill string are not considered;
(4) centralizers are considered large, short length drill strings.
Compared with the prior art, the invention has the following beneficial effects: the invention discloses a method for analyzing the transverse vibration of a drilling column of a complex-structure well, which is characterized in that a drill column dynamic model is established according to a Lagrange equation, the transverse displacement condition of the drill column under the actual load condition is fully considered, and the dynamic model is discretely solved by utilizing a generalized-alpha method to obtain a transverse vibration prediction model of the drill column during load operation. Meanwhile, the invention also carries out coordinate system conversion according to the complex well track of the complex structure well, converts the nonlinear transverse vibration model into a dynamic model which can be superposed, and is convenient for discrete solution of the dynamic model. In addition, the invention also adopts a cubic spline interpolation method to obtain the borehole trajectory, so that the obtained borehole trajectory is more continuous and smoother.
The method for analyzing the transverse vibration of the drill string of the well with the complex structure considers the displacement of the drill string during the load work and the collision with the well wall, obtains a dynamic model of the drill string during the load work, has few assumed conditions, better accords with the actual condition of the drill string during the load work, and has better reference and guidance functions on the drill string construction and application process by the predicted transverse vibration curve of the drill string.
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FIG. 1 is a schematic flow chart of a method for analyzing lateral vibration of a drill string of a well with a complex structure, which is disclosed by the invention;
FIG. 2 is a schematic diagram of coordinates of two nodes of a beam unit in the method for analyzing transverse vibration of a well drilling column with a complex structure disclosed by the invention;
FIG. 3 is a gravity model of each beam unit in the method for analyzing the lateral vibration of the well drilling column with the complex structure disclosed by the invention;
FIG. 4 is a schematic diagram of coordinate system transformation in the method for analyzing transverse vibration of a drill string of a well with a complex structure disclosed by the present invention;
FIG. 5 is a schematic diagram of the collision stress between the drill string and the well wall in the method for analyzing the transverse vibration of the drill string of the well with the complex structure disclosed by the invention;
FIG. 6 is a schematic diagram of a dynamic model solving flow in the method for analyzing the transverse vibration of the drilling string of the well with the complex structure, which is disclosed by the invention;
FIG. 7 is a schematic representation of a wellbore trajectory obtained in accordance with a disclosed embodiment of the invention;
FIG. 8 is a graph comparing lateral vibration characteristics of a drill string simulated in accordance with an embodiment of the present disclosure with measured data from a field site;
FIGS. 9 to 15 show a mass matrix, a gyro damping matrix, a stiffness matrix and a non-stiffness matrix in the method for analyzing the transverse vibration of the drilling column of the well with the complex structure disclosed by the invention.
Detailed Description
The present invention will be described in further detail with reference to test examples and specific embodiments. It should be understood that the scope of the above-described subject matter is not limited to the following examples, and any techniques implemented based on the disclosure of the present invention are within the scope of the present invention.
In the prior art, the vibration prediction of the drill string is mainly concentrated on a static drill string or in a balanced state, and the vibration in the actual load operation process of the drill string is difficult to accurately reflect, so that the significance of the failure prediction and safe construction guidance of the drill string is small. Therefore, how to predict the dynamic vibration of the drill string in the loading process has important significance for the prediction of the drill string failure and the safe drilling construction.
In view of the above technical problems, referring to fig. 1, the present application discloses a method for analyzing lateral vibration of a drilling string of a complex structure well, comprising the following steps:
s1: obtaining a borehole track of a target well according to well depth data and azimuth data of the target well;
s2: dividing a drill string of a target well into a plurality of beam units along a well track, and establishing a random coordinate system by taking the well track direction of the target well as an X axis to obtain a displacement vector equation of each beam unit;
s3: carrying out coordinate system transformation on the satellite coordinate system established by each beam unit to obtain a conversion relation between the satellite coordinate system of each beam unit and a geodetic coordinate system;
s4: according to a Lagrange equation, a nonlinear transverse vibration model of each beam unit of the drill string under a satellite coordinate system is established, and is substituted into a conversion relation between the satellite coordinate system and a geodetic coordinate system, and a dynamic model of the drill string under the geodetic coordinate system is obtained through superposition;
s5: setting the boundary condition of the drill string, substituting the drill string displacement vector equation into the dynamic model, and discretely solving the dynamic model to obtain a transverse vibration curve of the drill string during load operation.
It should be noted that the present invention is based on the following assumptions:
(1) the shaft is a circle with a uniform cross section, and the track of the well hole is continuous and smooth;
(2) considering the drill string as a three-dimensional elastic beam with uniform material and collective properties, whose deformation is within the elastic range;
(3) the connecting thread and the partial orifice plate on the drill string and at the connecting part of other equipment and the drill string are not considered;
(4) centralizers are considered large, short length drill strings.
The well track is not absolutely vertical but is a curved space curve during the drilling process of the complex structure well, and the well inclination angle and the azimuth angle of the complex structure well change along with the increase of the well depth. Therefore, the borehole trajectory construction of the target well needs to be performed before the drill string dynamics model is established.
In some embodiments, in step S1, the wellbore trajectory is obtained using the following steps:
s11: selecting a plurality of measuring points along the target well, and logging to obtain well depth data, azimuth angle data and inclination angle data of the target wells;
s12: calculating the coordinate of each measuring point according to a curvature radius method;
s13: and obtaining the well track of the target well according to a cubic spline interpolation method.
It should be noted that in actual drilling engineering, the basic data of a complex structure borehole include the depth, the angle of inclination and the azimuth, which can be obtained by Measurement While Drilling (MWD) tools.
In some embodiments, in step S12, the coordinates of each measurement point are obtained by:
s121, obtaining the coordinate increment between two adjacent measuring points according to a curvature radius method:
Figure BDA0003012370580000051
in the formula, delta X, delta Y, delta Z and delta S are respectively coordinate increments of two adjacent measuring points; rHAnd RsThe radius of curvature of the vertical and horizontal projections of the borehole trajectory, respectively, and m, subscripts "1" and "2" indicate the adjacent upper and lower two survey points, respectively.
Wherein R isHAnd RsCalculated by the following formula:
Figure BDA0003012370580000052
in the formula, the alpha is,
Figure BDA0003012370580000053
respectively the well inclination angle and azimuth angle, rad; s is the horizontal projection length, m; delta alpha,
Figure BDA0003012370580000054
The increment of the well inclination angle and the azimuth angle of two adjacent measuring points are respectively.
When Δ α is,
Figure BDA0003012370580000061
When one of the values is zero, then R isHAnd RsIt will not be able to calculate, and this can be handled as four cases:
a) when the value of Δ α is 0,
Figure BDA0003012370580000062
during the process, the projection of the well track in the vertical section is a straight line, and the method comprises the following steps:
Figure BDA0003012370580000063
wherein
Figure BDA0003012370580000064
b) When the delta alpha is not equal to 0,
Figure BDA0003012370580000065
in time, the horizontal projection of the wellbore trajectory is a straight line, having:
Figure BDA0003012370580000066
wherein
Figure BDA0003012370580000067
c) When the value of Δ α is 0,
Figure BDA0003012370580000068
in time, the wellbore trajectory is a straight line, with:
Figure BDA0003012370580000069
d) when Δ α ≠ 0, but α1·α2When the azimuth angle at which the inclination angle is zero is equal to the azimuth angle at the other end, that is, equivalent to the corresponding case in b), the processing is performed according to the corresponding method.
And S122, according to the coordinate increment between two adjacent measuring points and the measurement data of the first measuring point, sequentially calculating and obtaining the coordinate value of each measuring point:
Figure BDA00030123705800000610
due to the fact that the coordinates calculated by the measured data are discontinuous and not smooth, and the measuring distance is large, the description of the well track and the dynamic calculation are inaccurate. Therefore, the empirical trajectory should be obtained by processing the measured data. There are generally two approaches: interpolation, approximation. In view of the fact that the approximation method is not as operationally as the interpolation method, the cubic spline interpolation method is used in the present application to reconstruct the real wellbore trajectory, and preferably, the interpolation interval should be equal to the space step of the dynamic simulation.
In step S2, the drill string of the target well is divided into beam elements along the borehole trajectory, each beam element having two nodes, each node having 6 degrees of freedom, including 3 translations (x, y, z), two lateral rotation angles (θ)y,θz) And a torsion angle thetax. Therefore, referring to fig. 2, a coordinate system with the borehole trajectory direction of the target well as the X-axis may be established at each node, and thus the equation of the drill string displacement vector of each beam unit may be obtained as follows:
{Ui}e＝[xi,yi,zi,θxi,θyi,θzi,xj,yj,zj,θxj,θyj,θzj]；
wherein x isi、yi、ziCoordinates of a node of each beam element; xj、yj、zjRespectively the coordinates of the other node of each beam unit; thetaxi、θyi、θzi、θxj、θyj、θzjThe turning angles of the two nodes of each beam unit rotating around the coordinate axis are respectively.
Preferably, each beam element has a length of 3 m.
As in a complex configuration well bore, each drill string unit is located at a different orientation and inclination. Therefore, in order to combine the motion equations of the units, a unified geodetic coordinate system must be established, so that the node force, the node displacement and the coefficient matrix of the units can be converted into the geodetic coordinate system, and a nonlinear transverse vibration model of the whole drill string can be obtained from the nonlinear transverse vibration model of each beam unit. Therefore, the above-mentioned established random coordinate system needs to be transformed into a coordinate system first, and the specific transformation process is as follows:
as can be seen from fig. 3, the satellite coordinate system is converted into a geodetic coordinate system by rotating the satellite coordinate system, and the specific rotation angles are:
firstly rotating the angle around the X axis
Figure BDA0003012370580000071
Obtaining a coordinate system x1(X)y1z1(ii) a Second winding around y1Angle of rotation of the shaft
Figure BDA0003012370580000072
Obtaining a coordinate system x2y2(y1)z2(ii) a Finally winding around z2Angle of rotation
Figure BDA0003012370580000073
Obtaining the earth coordinate system xyz (z)2)。
Therefore, the following relation between the satellite coordinate system and the geodetic coordinate system is as follows:
Figure BDA0003012370580000074
from equation (8), the conversion relationship between the satellite coordinate system and the geodetic coordinate system can be obtained as follows:
Figure BDA0003012370580000081
setting the transformation matrix as Trans, that is, the transformation matrix Trans is:
Figure BDA0003012370580000082
meanwhile, each beam unit comprises two nodes, and each node comprises three translational displacements and three rotational displacements; therefore, the displacement of each beam element is a 12-dimensional matrix. And substituting a transformation matrix Trans to obtain a transformation relation between the satellite coordinate system and the geodetic coordinate system of each beam unit, which specifically comprises the following steps:
Figure BDA0003012370580000083
let Tr be the coordinate transformation matrix:
Figure BDA0003012370580000084
therefore, the conversion relation between the generalized displacement and the generalized external force vector of each beam unit in the satellite coordinate system and the earth coordinate system is as follows:
{U′}e＝[Tr]{U}e,{F′}e＝[Tr]{F}e (13)
in the formula, { U' }eAnd { U }eRespectively representing generalized displacement under a satellite coordinate system and a geodetic coordinate system; { F' }eAnd { F }eAnd respectively representing generalized external force vectors under a random coordinate system and a geodetic coordinate system.
In step S4, a nonlinear lateral vibration model of the beam unit may be obtained according to the lagrangian equation.
Wherein the Lagrangian equation is:
Figure BDA0003012370580000085
in the formula, T represents kinetic energy; v represents potential energy;
Figure BDA0003012370580000086
representing a velocity vector; fiRepresents an external force vector; u shapeiIndicating the amount of displacement.
The nonlinear lateral vibration model of the beam element obtained by deriving the expression of T, V and carrying in (14) a simplification is:
Figure BDA0003012370580000091
in the formula,
Figure BDA0003012370580000092
{U}e,{F}egeneralized acceleration, generalized velocity, generalized displacement, and external force vector, respectively. [ M ] A]e,[C]e,[K]eRespectively a mass matrix, a damping matrix and a stiffness matrix.
Note that the stiffness matrix [ K ]]eComprising a non-linear stiffness matrix and a linear stiffness matrix [ KL](ii) a The nonlinear stiffness matrix comprises a nonlinear stiffness matrix [ K ] corresponding to the bending deformation and the axial deformation of the drill stringNA1]+[KNA2]Nonlinear stiffness matrix [ K ] corresponding to bending deformation and torsional deformation of drill stringNT](ii) a The method comprises the following specific steps:
[K]e＝[KL]+[KNA1]+[KNA2]+[KNT] (16)
it should be noted that the quality matrix can be represented as:
[M]＝[M1]+[M2] (17)
wherein [ M ]1]Is a translational mass matrix, [ M ]2]Is a rotational mass matrix, see fig. 13, 14.
Damping matrix [ C ]]eCan be expressed as:
[C]＝[CD]+[CN] (18)
wherein [ C ]D]Showing Rayleigh damping, [ CN]Gyro damping is shown with reference to fig. 15.
[CD]＝αD[M]+βD[KL] (19)
In the formula, alphaD、βDIs a constant.
Note that the generalized external force vector { F }eIncluding the centrifugal force and gravity experienced by the two nodes of each beam element, the matrix of the generalized external force vectors is a 12-dimensional matrix.
According to fig. 4, the gravity components of each beam unit of the drill string in the three directions x, y and z are:
Figure BDA0003012370580000093
wherein q is the equivalent gravity of a drill string of unit length, and the unit is N/m; alpha is the angle between the axis of the beam element and the vertical.
Thus, the equivalent nodal force of the gravity vector is:
Figure BDA0003012370580000094
for a cross section of the drill string, the centroid and the centroid are not completely consistent, an unbalanced force (centrifugal force) is generated when the drill string rotates, and therefore, the centrifugal force generated by the rotation of the beam unit in the x, y and z directions can be expressed as:
Figure BDA0003012370580000101
wherein β is the phase angle of the center of gravity, rad; similarly, the equivalent nodal force of the centrifugal force vector can be expressed as:
Figure BDA0003012370580000102
in the formula: f. ofy＝qeΩ2cosβ，fz＝qeΩ2sin beta, e is the eccentricity of the drill string, m; β is the centroid phase angle, rad.
Furthermore, during the operation of the drill string under load, the vibration of the drill string mainly comprises transverse translational vibration and rotation, and therefore the kinetic energy of each beam unit comprises translational kinetic energy and rotational kinetic energy, and therefore the expression for the kinetic energy of each beam unit can be:
Figure BDA0003012370580000103
in the formula,
Figure BDA0003012370580000104
and
Figure BDA0003012370580000105
is the translational velocity of the node along the axis of the borehole coordinate system X, Y, Z, m/s; thetaxDenotes the torsion angle, rad, of the beam element with respect to its rigid body; thetayAnd thetazRespectively representing the rotation angles, rad, of the beam unit sections around the y axis and the z axis;
Figure BDA0003012370580000106
representing the torsional deformation speed, rad/s, of the drill string after torsional vibration;
Figure BDA0003012370580000107
and
Figure BDA0003012370580000108
respectively representing the angular velocity, rad/s, of the beam unit section around the y axis and the z axis; e represents the eccentricity of the center of gravity of the unit cross section relative to the centroid, m; omega is the actual rotating speed of the underground drill string, rad/s; ρ is the density of the drill string, kg/m3；leIs the length of the beam element, m; a is the cross-sectional area of the beam element, m2；IxIs the polar moment of inertia, m, of the beam unit cross section4；IyzIs the moment of inertia, m, of the cross section of the beam unit4(ii) a In equation 24, the first three terms represent translational kinetic energy of the beam unit, and the second two terms represent rotational kinetic energy of the beam unit.
Wherein, the potential energy expression of each beam unit is as follows:
Figure BDA0003012370580000109
wherein E is the modulus of elasticity of the drill string, Pa; g is the shear modulus of the drill string, Pa.
In the formula (15), the stiffness matrix [ K ]]Linear stiffness matrix [ K ] of (1)L]Can be derived from the first four terms of equation (25). In relation to the bending deformation and axial direction of the drill stringDeformation-corresponding nonlinear stiffness matrix [ K ]NA1]+[KNA2]Can be derived from the terms 5-9 of equation (25); nonlinear stiffness matrix [ K ] corresponding to bending deformation and torsional deformation of drill stringNT]Can be derived from the last three terms of equation (25), the linear stiffness matrix [ KL]Nonlinear stiffness matrix [ K ]NA1]、[KNA2]Nonlinear stiffness matrix [ K ]NT]As shown in FIGS. 9 to 12.
Therefore, the above equations (24), (25) are simplified using a shape function, and the format is:
Figure BDA0003012370580000111
Figure BDA0003012370580000112
equations (26) and (27) are expressed in a matrix, then the equations (14) are carried in, and meanwhile the equations (21) and (23) are carried in, and the nonlinear transverse vibration model of the beam unit under the body coordinate system is obtained through integration.
Due to the obtained conversion relationship between the geodetic coordinate system and the satellite coordinate system, it can be obtained that the nonlinear lateral vibration model of the beam unit under the satellite coordinate system can be converted into the nonlinear lateral vibration model of the beam unit under the geodetic coordinate system, and the specific conversion relationship is as follows:
Figure BDA0003012370580000113
therefore, the nonlinear lateral vibration model of each beam unit in the geodetic coordinate system is:
Figure BDA0003012370580000114
and superposing the nonlinear transverse vibration model of each beam unit to obtain a dynamic model of the whole drill string, which comprises the following specific steps:
Figure BDA0003012370580000115
in the formula,
Figure BDA0003012370580000116
{ U '} and { F' } respectively represent a generalized acceleration matrix, a speed matrix, a displacement matrix and an external force matrix of each node on the whole well drill string; [ M']，[C′]And [ K']The total mass matrix, total damping matrix, and total stiffness matrix of the full well drill string are represented, respectively.
During actual drilling, the movement of the drill string is constrained by the rotary table, so that the upper end of the drill string is hinged in the center of the borehole and is subjected to tension and torque. While the lower end of the drill string is subjected to weight on bit and torque and its lateral displacement is constrained. Furthermore, when the drill string is radially displaced more than the gap between the drill string and the borehole, the drill string will be constrained by the borehole. Thus, the drill string boundary conditions include borehole wall constraints and drill bit constraints, the borehole wall constraints being:
xi＝0,yi＝0,zi＝0,θx＝Ωt,θy＝0,θz＝0；
wherein, thetax、θy、θzRespectively the rotation angles of the drill column unit around each coordinate axis; omega is the rotation speed of the drill string, rad/s; t represents time, s.
The drill string is discretized into units and only the contact between the nodes and the borehole needs to be considered. The drill string will therefore be subjected to positive forces, tangential friction and additional friction torque when it is in contact with the borehole. When the elastic deformation of the wellbore recovers, the drill string will bounce back and the nodes will recover free, see fig. 5. Thus, the drill string is subjected to a positive force FNCan be expressed as:
Figure BDA0003012370580000121
in the formula (d)oIs a shaft straightDiameter, m; diIs the drill string diameter, m; v. ofrIs the radial velocity of the drill string, m/s; u. ofrIs the radial displacement of the drill string, m; k is a radical ofhIs the stiffness of the wellbore, N/m; v. of1And v2Respectively, the velocities before and after the node collision, m/s.
Tangential frictional force FfAnd friction torque FtorqCan be expressed as:
Figure BDA0003012370580000122
in the formula,. mu.vs) Is the friction coefficient between the drill string and the borehole, and can be calculated by a static-dynamic exponential decay model:
Figure BDA0003012370580000123
in the formula uk,usRespectively representing a dynamic friction coefficient and a static friction coefficient; deRepresents the attenuation coefficient; v. ofsThe slip ratio is shown.
The characteristics of the bit itself colliding with the bit and the rock downhole, weight on bit FwobOver time t. Thus, the bit constraint is weight on bit FwobAs time t changes, it is specifically:
Fwob(t)＝W0+Wfsin(ωft) (34)
in the formula, W0Static weight-on-bit, i.e., the weight-on-bit value applied by the upper drill string; wfIs a dynamic weight-on-bit value; omegafIs the weight on bit fluctuation coefficient.
It should be noted that the weight-on-bit coefficient of fluctuation is related to the rotational speed of the drill string and the type of drill bit:
ωf＝nbΩ
in the formula, nbFor PDC bits nbFor roller bit n 1b3; omega is the drill string speed.
In step S4, in order to obtain better convergence, the dynamic model is discretely solved by using a generalized- α method, which is shown in fig. 6, and specifically includes the following steps:
s41: obtaining a general calculation formula of the generalized-alpha method according to the basic form of the generalized-alpha method;
s42: establishing a null matrix of a mass matrix [ M ' ], a damping matrix [ C ' ] and a rigidity matrix [ K ' ], and substituting into a general calculation formula of a generalized-alpha method;
s43: assigned initial displacement d0And an initial velocity v0And obtaining an initial acceleration a according to the basic form of the generalized-alpha method0A value;
s44: setting a time step length and a limit spectrum radius, and calculating a required integral constant;
s45: calculating to obtain an effective stiffness matrix, an effective load vector, displacement at the moment t + delta t, acceleration at the moment t + delta t and speed at the moment t + delta t;
s46: and substituting the values obtained in the S45 into the dynamic model, performing iterative calculation, and solving the dynamic model.
It should be noted that the generalized- α method has the basic form:
Figure BDA0003012370580000131
Figure BDA0003012370580000132
Figure BDA0003012370580000133
in the formula (d)n、vnAnd anRespectively represent
Figure BDA0003012370580000134
And U', i.e. the displacement, velocity and acceleration of the drill string; Δ t represents the time step, s; a subscript N ∈ {0,1,2.. said., N-1 }; n represents the number of time steps.
Wherein alpha isf2,αm2,γ2And beta2The relationship and calculation between them is as follows:
Figure BDA0003012370580000141
in the formula, ρ∞2Denotes the limiting spectral radius, ρ∞2∈[0,1]。
Taking formula (37) and formula (35) into formula (36) to obtain a general calculation formula of the generalized-alpha method, wherein the general calculation formula of the generalized-alpha method is specifically as follows:
Figure BDA0003012370580000142
in addition, d is determined according to the upper boundary condition0And v0The value is zero and the acceleration in the initial case is obtained by taking it into the equation (35): a is0。
In step S44, the integral parameter and its specific calculation formula are:
Figure BDA0003012370580000143
while the effective stiffness matrix of the drill string
Figure BDA0003012370580000144
Payload vector
Figure BDA0003012370580000145
The calculation formula of (a) is respectively:
Figure BDA0003012370580000146
Figure BDA0003012370580000151
therefore, according to the expressions (17) to (19) and (25) to (27), the displacement, acceleration and speed of the drill string at the time t + Δ t can be obtained:
Figure BDA0003012370580000152
and (3) bringing the formula (43) into the formula (30), and carrying out iterative solution to obtain a solution of the dynamic model.
According to the obtained dynamic model, the transverse vibration condition of the drill string under the load condition can be simulated accurately, and the vibration condition of the drill string under the load condition, the service life of the drill string and the construction safety can be predicted accurately.
Experimental data for a1 wells are obtained using wired drill pipe technology and downhole tools, using the a1 well in an oil field as an example. The measurement tool measures weight on bit, bit torque, axial acceleration and lateral acceleration at a distance of 30 meters from the bit and delivers them to the surface. The basic parameters of the well are shown in table 1:
Figure BDA0003012370580000153
TABLE 1
From the measurement data, an A1 well bore trajectory can be obtained, which is shown in FIG. 7.
It is noted that the drill string is made up of a series of drill pipes, drill collars and weighted drill pipes (HWDP). As shown in fig. 7, the beam unit is divided using a length of 3m for the drill strings of the deflecting section and the horizontal section, and the beam unit is divided using a length of 6m for the drill string of the vertical section. To get the drill string dynamics more accurately, time Δ t is set to 0.001 s.
The dynamic vibration characteristics of the drill string under load are simulated by substituting the data into a drill string dynamic model, which is specifically shown in fig. 8. As can be seen from FIG. 8, as the static coefficient of friction increases, the lateral acceleration and torque of the bit will also increase when us0.05 (coefficient of static friction) and (coefficient of dynamic friction) uk＝0.8usThe predicted bit torque (fig. 8b), weight on bit (fig. 8a), lateral acceleration (fig. 8d) and axial acceleration (fig. 8c) are substantially in agreement with the measured data. The bit interacts strongly with the formation, so that the weight on bit fluctuates within 100% of the initial value, and the rotational speed fluctuates around 1.5 kn.m. And the transverse acceleration is far greater than the axial acceleration, which indicates that the transverse vibration is stronger than the axial vibration. The conclusion is high in coincidence degree with field data of a drill string, and the fact that the dynamic model disclosed by the application can simulate and obtain more accurate vibration characteristics under the condition of the load of the drill string is proved, and the transverse vibration of the drill string can enable a bottom drilling assembly to generate larger high-frequency bending moment fluctuation, so that the early fatigue, the failure of BHA components, the erosion of a shaft, the abrasion of a stabilizer and the like are caused and are main reasons of the failure of the drill string, so that the accurate prediction of the transverse vibration characteristics of the drill string is beneficial to the regulation and control of all-directional parameters in the load process of the drill string, and reliable data reference is provided for the prediction of the failure of.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (10)

1. A method for analyzing the transverse vibration of a drill string of a well with a complex structure is characterized by comprising the following steps:
s1: obtaining a borehole track of a target well according to well depth data and azimuth data of the target well;
s2: dividing a drill string of a target well into a plurality of beam units along a well track, and establishing a random coordinate system by taking the well track direction of the target well as an X axis to obtain a displacement vector equation of each beam unit;
s3: carrying out coordinate system transformation on the satellite coordinate system established by each beam unit to obtain a conversion relation between the satellite coordinate system of each beam unit and a geodetic coordinate system;
s4: according to a Lagrange equation, a nonlinear transverse vibration model of each beam unit of the drill string under a satellite coordinate system is established, and is substituted into a conversion relation between the satellite coordinate system and a geodetic coordinate system, and a dynamic model of the drill string under the geodetic coordinate system is obtained through superposition;
s5: setting the boundary condition of the drill string, substituting the drill string displacement vector equation into the dynamic model, and discretely solving the dynamic model to obtain a transverse vibration curve of the drill string during load operation.
2. The method for analyzing the lateral vibration of a drill string in a complex structure well according to claim 1, wherein the borehole trajectory is obtained by a cubic spline interpolation method in step S1.
3. The method of analyzing lateral vibration of a complex-structured well string according to claim 2, wherein the well trajectory is obtained by the steps of:
s11: selecting a plurality of measuring points along the target well, and logging to obtain well depth data, azimuth angle data and inclination angle data of the target wells;
s12: calculating the coordinate of each measuring point according to a curvature radius method;
s13: and obtaining the well track of the target well according to a cubic spline interpolation method.
4. The method of analyzing lateral vibration of a drill string for a complex structure well according to claim 1, wherein the displacement vector equation of each beam unit is:
{Ui}e＝[xi,yi,zi,θxi,θyi,θzi,xj,yj,zj,θxj,θyj,θzj]；
wherein x isi、yi、ziCoordinates of a node of each beam element; xj、yj、zjRespectively the coordinates of the other node of each beam unit; thetaxi、θyi、θzi、θxj、θyj、θzjTwo for each beam unitAnd (4) the rotation angle of the node around the coordinate axis.
5. The method for analyzing the lateral vibration of a drill string in a complex structure well according to claim 1, wherein the transformation relation between the satellite coordinate system and the earth coordinate system is as follows:
Figure FDA0003012370570000021
wherein, Trans is a transformation matrix;
the transformation matrix is:
Figure FDA0003012370570000022
wherein,
Figure FDA0003012370570000023
the rotation angle of the X axis is obtained when the satellite coordinate system is converted into a geodetic coordinate system;
Figure FDA0003012370570000024
the rotation angle of the Y axis is obtained when the satellite coordinate system is converted into a geodetic coordinate system;
Figure FDA0003012370570000025
the rotation angle of the Z axis when the satellite coordinate system is converted into the geodetic coordinate system.
6. The method for analyzing the lateral vibration of a drill string for a complex structure well according to claim 1, wherein the nonlinear lateral vibration model of the beam unit is as follows:
Figure FDA0003012370570000026
wherein,
Figure FDA0003012370570000027
{U}e,{F}egeneralized acceleration, generalized velocity, generalized displacement and external force vector under a satellite coordinate system are respectively provided; [ M ] A]e，[C]eAnd [ K ]]eRespectively representing a mass matrix, a damping matrix and a rigidity matrix under a random coordinate system.
7. The method of analyzing lateral vibration of a drill string for complex-structure wells according to claim 1, wherein the dynamic model of the drill string in the geodetic coordinate system is:
Figure FDA0003012370570000028
wherein,
Figure FDA0003012370570000029
{ U '} { F' } is generalized acceleration, generalized speed, generalized displacement and external force vector of each node on the drill column under a geodetic coordinate system respectively; [ M']、[C′]And [ K']The drill string represents a mass matrix, a damping matrix and a stiffness matrix in the geodetic coordinate system respectively.
8. The method of analyzing lateral vibration of a drill string for a complex formation well according to claim 1, wherein the drill string boundary conditions include a borehole wall constraint and a drill bit constraint, the borehole wall constraint being:
xi＝0,yi＝0,zi＝0,θx＝Ωt,θy＝0,θz＝0；
wherein, thetax、θy、θzRespectively the rotation angles of the drill column unit around each coordinate axis; omega is the rotation speed of the drill string, rad/s; t represents time, s.
9. The method for analyzing the lateral vibration of a drill string in a complex structure well according to claim 1, wherein in the step S5, the dynamic model is discretely solved by a generalized-alpha method.
10. The method for analyzing lateral vibration of a drill string for a complex-structure well according to any one of claims 1 to 9, wherein the prediction method is further based on the following assumptions:
the shaft is a circle with a uniform cross section, and the track of the well hole is continuous and smooth;
considering the drill string as a three-dimensional elastic beam with uniform material and collective properties, whose deformation is within the elastic range;
the connecting thread and the partial orifice plate on the drill string and at the connecting part of other equipment and the drill string are not considered;
centralizers are considered large, short length drill strings.
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