Research on the dynamic behavior of flexible drilling tools in ultrashort-radius radial horizontal wells

A flexible drilling tool is a special drilling tool for ultrashort-radius radial horizontal wells. This tool is composed of many parts and has the characteristics of a multibody system. In this paper, a numerical method for the dynamic analysis of flexible drilling tools is proposed. The flexible drill tool is discretized into spatial beam elements, while the multilayer contact of the flexible drilling tool is represented by the multilayer dynamic gap element, and the dynamic model of the multibody system for the flexible drilling tool’s multilayer contact is established, considering the interaction force between the drill bit and the rock. The nonlinear dynamic equation is solved using the Newmark method and Newton–Raphson method. An analysis of the dynamic behavior of a flexible drilling tool is conducted. The results indicate that the flexible drilling tool experiences vortex formation due to the interaction between the flexible drilling pipe and the guide pipe, leading to increased friction and wear. This situation hinders safe drilling operations with flexible drilling tools. The collision force of the flexible drilling tool near the bottom of the hole is more severe than that of the other tool types, which may lead to failure of the connection.


Mechanical model
The dynamic model of flexible drilling tools is illustrated in Fig. 2. At the top, a constant rotational speed is maintained on the flexible drill pipe.At the bottom, a bit-rock interaction model is added.The displacement boundary conditions for flexible drilling tools are presented in Table 1.
A simplified model of drill bit rock interaction is applied to the bottom of the flexible drill pipe 18 .The calculation formula is shown in Eq. (1): where T OB is the total torque generated by the friction of the drill bit, η is the dynamic friction factor between the bit and the rock, D bit is the bit diameter, and W OB is the weight of the bit.

Dynamic finite element model
The flexible drilling tool is a complex structure that can be effectively analyzed using the finite element method 19 .The flexible drilling tool is a slender structure, and a beam element is used for dispersion.Figure 3 shows a spatial beam element.The spatial beam element has 12 nodal coordinates that describe the translations and slopes of the two nodes 20 .The displacement vector of the spatial beam element nodes is as follows: The displacement of any point in the spatial beam element is: (2) where N is the shape function matrix.
where ξ = x l .The stiffness matrix and mass matrix of the space beam element are where H is the matrix of differential operators and D is the elastic matrix.
where A is the section area of the space beam element; I z and I y are the second moments of the area about the neutral axes z and y, respectively; and I x is the polar moment of inertia of the area.
The damping matrix is damped proportionally.
The relationship between the components of a spatial beam element and the components of the global coordinate system is described in terms of the local coordinate system.
where the coordinate transformation matrix Ŵ.

Ŵ=
� where (X,x) represents the angle between the global coordinate axis and the local coordinate axis.The finite element model of the flexible drill pipe, as depicted in Fig. 4, consists of five discrete space beam elements representing each section of the pipe based on its cross-section.The hinge point is located between elements e r i+1 and e r i+2 .Similarly, the guide pipe is discretized into equal sections using beam elements with an equivalent number of nodes as the flexible drill pipe.
The stiffness matrix, mass matrix and damping matrix of the flexible drill pipe and guide pipe are assembled according to the calculation formula of the spatial beam element.
where K r , M r and C r are the global stiffness matrix, the mass matrix and the damping matrix of the flexible drill pipe, respectively.K d , M d and C d are the global stiffness matrix, the mass matrix and the damping matrix of the guide pipe, respectively.
(3) d e =Nq e (4) www.nature.com/scientificreports/By disregarding the flexible pipe, interpipe contact, and interaction between the pipe run and borehole wall, we can independently formulate the dynamic equation of the flexible tool.
where qr (t) , qr (t) and q r (t) are the acceleration array, the velocity array and the displacement array of the flexible drill pipe, respectively.qd (t) , qd (t) and q d (t) are the acceleration array, the velocity array and the displacement array of the guide pipe, respectively.F r and F d are the equivalent nodal force arrays of the flexible drill pipe and guide pipe, respectively.
The kinematic constraints imposed by joint points have not been taken into account in the previously mentioned dynamic equations of flexible drill pipes.The coordinate systems n for the joint points of the flexible drill pipe should be established at nodes n with respect to the origin O in the global coordinate system are determined as follows:  www.nature.com/scientificreports/ The vector diameters of the hinge point coordinate systems n and n r1 , with respect to the origin O of the global coordinate system, are denoted by r (i+1) r and r (i+2) r , respectively.is the cosine matrix of the deformed hinge coordinate system relative to the element coordinate system.q (i+1) r and q −i+2) r represent the displacement vectors of nodes n (i+1) r2 and n (i+2) r1 , respectively.The node-relative translational constraints at the locations where elements e (i+1) r and e (i+2) r coincide can be determined by constraining both the magnitude and direction of the vector h The rotational constraints for elements e r i+1 and e r i+2 are as follows: If the flexible drill pipe is connected by k hinges, then the overall articulated constraint equation of the flexible drill pipe can be derived.
Equation (14) takes the derivative of time, and the velocity constraint equation is where The Lagrange multiplier method is employed to incorporate the constraint equation into the overall motion equation of the flexible drill pipe, which can be derived as follows: The symbol r (q r , t) denotes the set of constraint equations governing the flexible pipe within the universal joint, while T rq represents the constraint equations associated with the Jacobian matrix.Additionally, joint r signifies the Lagrange multiplier array pertaining to hinged constraints.The Lagrange multiplier matrix associated with it is derived: Equation ( 17) is abbreviated as follows: where The interaction between the flexible drill pipe and guide pipe, as well as between the guide pipe and the well bore, is characterized by employing a dynamic gap element.A schematic representation of this dynamic gap element and the corresponding collision contact force can be found in Fig. 6.
The local coordinate system of the gap element should align with both the local coordinate systems of the flexible drill string beam element and guide pipe beam element.The displacement of any node I within the dynamic gap element can be represented as follows: where q i represents the displacement array of node I in the flexible drilling tool beam element and N G denotes the shape function matrix of the gap element.
The stiffness matrix of the dynamic gap element is determined: where the matrix K e G represents the stiffness of the dynamic gap element and G k denotes its compressive stiffness during contact.
During the operation of the flexible pipe and pipe run, no contact is observed between the pipe run and borehole wall.The dynamic gap elements adhere to kinematic principles, with their velocity determined accordingly: The acceleration of the dynamic gap element is: where θ represents the torsional angular velocity of the flexible drill tool and θ represents the angular accelera- tion of the flexible drill tool.
The collision between the flexible drill pipe and the guide pipe, as well as between the guide pipe and the shaft wall, generates reaction forces, resistance distances, and bending moments: where F Gn is the normal contact force, F Gt is the tangential friction resistance, F GA is the axial friction resistance, µ 1 and µ 2 are friction coefficients, M Gt is the bending moment caused by the tangential friction resistance, and M GA is the bending moment caused by the axial friction resistance.
The additional forces mentioned above are transformed into the corresponding nodal forces of the gap element: where F Gty and F Gtz are the components of F Gt and M GAy and M GAz are the components of M GA .
The internal dynamic gap element between the flexible drill pipe and the guide pipe was established, and the external dynamic gap element between the guide pipe and the wall bore was constructed (Fig. 7).
The initial gap of the inner dynamic gap element in any direction is: The inner dynamic stiffness matrix of the gap elements is:

Model solving algorithm
Due to the presence of a nonlinear term in the equation governing the dynamics of an articulated flexible drill, both the Newmark method and Newton Raphson method are employed for solving this nonlinear differential equation 21 .
Applying Newmark to Eq. (34) gives the equilibrium equation at the time: Table 2. Contact state conditions.

Free state
Inner-layer contact Outer-layer contact Two-layer contact state Vol.:(0123456789) www.nature.com/scientificreports/where where �t ξ is the time step and κ and γ are parameters in the Newmark method.Equation ( 37) is a nonlinear system of equations that is iteratively calculated by the Newton-Raphson method where where ψ is the number of iterations and K T is the tangential stiffness matrix.The above iterative calculation formula applies the expression for the displacement increment iteration. Moreover,

Verification of the numerical method
The numerical calculation method in this paper is verified by a numerical example in reference 12 .There is a controllable hinge connection beam inside the beam (Fig. 8).The outer beam is completely fixed.The left end of the inner beam is fully fixed, and the right end is movable hinge support.Point C is subjected to a concentrated load The analytical solution of the displacement in the y direction at point C is: Fr is the critical concentration force at the hinge required to reach the rotation limit; F1 is the critical concentration force of contact between the inner beam and the outer beam.
(37) www.nature.com/scientificreports/A comparison between the analytical solution and the numerical solution is shown in Fig. 1.As shown in Fig. 9, the maximum error is 0.9%, and the calculation model satisfies the accuracy conditions.

Results and discussion
The calculation parameters of the numerical simulation are shown in Table 3.The variation curves of the rotational speed of the flexible drill pipe at well inclination angles of 0°, 22.5°, 45°, 67.5° and 90° with time are shown in Fig. 12. Figure 12b,c show that the rotational speed of the flexible drill pipe is gradually amplified and fluctuates widely with increasing well inclination angle.

△
The rotational speeds of the guide pipes at positions D 1 and D 2 are shown in Fig. 13.Under the influence of the rotation of the flexible drill pipe, the guide pipe has a revolution velocity around the hole, but the revolution velocity is small.The rotation period of the guide tube is the same as that of the flexible drill pipe.Under the influence of the flexible drill pipe and bit weight, the rotational velocity of the guide pipe fluctuates between − 0.74 and 0.97 r/min.

Collision contact analysis
Considering the randomness of the collision force and acceleration caused by the sampling frequency, their effective values will be used as the basis for analysis in the following text.The collision force between the flexible drill pipe and the guide pipe is depicted in Fig. 14. Figure 14a shows that the intense collisions between the flexible drill pipe and guide pipe are concentrated at the bottom of the hole, reaching a maximum value of 19.97 kN at an inclination angle of 88°, corresponding to the lowermost section of the flexible drill pipe.The temporal evolution of this collision force is depicted in Fig. 14b, revealing periodic characteristics in terms of the collision force between these two pipes at this position.The collision period lasts for 1.5 s, with each individual collision lasting for approximately 1.27 s and a maximum recorded collision force measuring up to 44.42 kN.It can be seen that the impact of the flexible drill pipe is mainly concentrated in the position near the drill bit, which is the vulnerable part.
The collision force curve between the guide pipe and the shaft wall is shown in Fig. 15. Figure 15a shows that severe collisions between the guide pipe and the borehole wall are concentrated at the bottom of the hole, with a maximum value of 4.82 kN and a borehole inclination of 81°.The change curve of the collision force at this position over time is shown in Fig. 15b. Figure 15b shows that the maximum collision force is 12.81 kN, the collision period between the guide pipe and the shaft wall at this position is the same as that between the flexible drill pipe and the guide pipe at the same position, which is 1.5 s, and the duration of each collision is 0.56 s.
The variation curves of the effective acceleration in the three directions of the flexible drill pipe with respect to the well inclination angle are shown in Fig. 16.The vibration fluctuation in the y direction (transverse direction) of the flexible drill pipe is large, taking the positions A ry1 , A ry2 and A ry3 of the acceleration curve in the y direction, as shown in Fig. 16b-d, respectively.The figure shows that the vibration of the flexible drill pipe has a certain periodicity, and the vibration of the flexible drill pipe is close to the collision period.Combined with the collision force curve, it can be concluded that the violent vibration of the flexible drill pipe is caused by the collision.The acceleration curve of the guide pipe is shown in Fig. 17.Although the guide pipe is subjected to greater impact, its vibration is smaller than that of the flexible drill pipe.As can be seen from Fig. 17b, the vibration of the guide pipe is relatively gentle.The reason for this difference is that the guide pipe is a continuous body in structure, while the flexible drill pipe is a multi-stage hinged multi-body system, so the vibration of the flexible drill pipe is more intense.

Figure 2 .
Figure 2. Flexible drilling tool dynamics model. https://doi.org/10.1038/s41598-024-57742-3 r1 ), and the vector h (i+1,i+2) r connects the joint points.The kinematic constraint equation of articulation can be defined by the kinematic relation between the vectors shown in Fig.5.The radius vectors of nodes n

Figure 4 .
Figure 4. Finite element model of flexible drill pipe.

Figure 6 .
Figure 6.A diagrammatic drawing of the dynamic gap element.(a) Schematic representation of the dynamic gap element, (b) Schematic representation of the contact force during collision.

Figure 7 .
Figure 7. Schematic diagram of the two-layer contact dynamic gap elements.(a) initial state, (b) free state, (c) contact state.

Figure 8 .
Figure 8. Schematic diagram of the inner and outer beams.

Figure 9 .
Figure 9.Comparison between the analytical and numerical solutions for the hinge point displacement.

Figure 10 .
Figure 10.Displacement curve of the flexible drill pipe.(a)Overall deformation of flexible drill pipe, (b) D r1 Motion trajectory, (c) D r1 Displacement curve with time, (d) D r2 motion trajectory, (e) D r2 motion trajectory.

Figure 11 .Figure 12 .
Figure 11.Displacement curve of the guide pipe.(a) Overall deformation of guide pipe, (b) D d1 Motion trajectory, (c) D d1 Displacement curve with time, (d) D d2 Motion trajectory, (e) D d2 Displacement curve with time.

Figure 13 .Figure 14 .
Figure 13.The change in the rotational speed of the guide pipe with time.

Figure 15 .Figure 16 .Figure 17 .
Figure 15.Collision force curve between the guide pipe and shaft wall.(a) Effective values of collision forces at various well angles, (b) Time history curve at point C d .

Table 1 .
The displacement boundary conditions of the flexible drilling tool.

Table 3 .
Dynamic analysis and calculation parameters of flexible drilling tools.