Load transfer mechanism of flexible drill string with hinges based on dynamic relaxation method

3.1. Universal joint connection element

The universal joint connection element has two coincident nodes and two relative rotational degrees of freedom, enabling “universal rotation” connection. Fig. 5 depicts the geometry and node positions of the element. Nodes $I$ and $J$ coincide in location, and at both nodes, local Cartesian coordinate systems need to be defined. The directions of the local coordinate systems follow the right-hand rule, and the local coordinate axis $y$ aligns with the main axis of the universal joint.

Fig. 5Coordinates of the universal joint connection element

Based on the coordinate system at the nodes of this element, it is relatively easy to describe the motion constraints. The displacement constraint of the universal joint element at any given time is represented by ${\delta }^I={\delta }^J$, and the rotational constraint is represented by ${e_x}^I\cdot {e_z}^J=0$.

The first Cardan angle (rotation around the $x$-axis) and the third Cardan angle (rotation around the $z$-axis) of the relative position of node $I$ with respect to node $J$ in the local coordinate system [24] are respectively given by Eq. (1):

The changes of the relative rotation angle of the two local coordinate systems $\mathrm{\Delta }\phi$ and $\mathrm{\Delta }\psi$ can be represented as Eq. (2):

By combining the beam elements with the universal joint connection elements, and going through the process of coordinate transformation and assembly, we can derive the overall equilibrium equation for the flexible drill string with hinges as Eq. (3):

where, ${\mathbf{F}}_0$ is the node force vector of flexible drill string, N; ${\mathbf{K}}_0$ is the global stiffness matrix of flexible drill string, N/m; ${\mathbf{\delta }}_0$ is the node displacement vector of flexible drill string, m.

3.2. Gap element method for nonlinear analysis of beam-to-beam contact

3.2.1. The basic idea of the gap element

The basic idea of the gap element method is to treat the gap between contacting bodies as a virtual element, merging the gap and the contacting bodies into a single entity [25-26]. In three-dimensional beam-to-beam contact problems, the geometric shape of the gap element is circular, with its thickness approaching zero. The stiffness of this virtual element needs to be adjusted based on the specific contact conditions.

The gap element possesses the following physical characteristics:

(1) When the flexible drill string is not in contact with the outer tube, it does not affect the free movement of the flexible drill string, and the compression stiffness approaches zero.

(2) When the flexible drill string is in contact with the outer tube, its compression stiffness is either a given value or becomes sufficiently large. It can resist mutual intrusion of objects while allowing the flexible drill string to move under the conditions of frictional resistance.

The inner edge of the gap element is in contact with the outer surface of the flexible drill string, meaning that the inner diameter of the gap element is equal to the outer diameter d of the flexible drill string. The outer edge of the gap element is connected to the inner surface of the outer tube, which means that the outer diameter of the gap element is equal to the inner diameter $D$ of the outer tube. As shown in Fig. 6, the orthogonal coordinate system $yOz$ is aligned with the local coordinate system of the beam element. The initial gap $E_{oi}$ of the gap element $i$ in any direction $n$ can be expressed as Eq. (4):

4

$E_{oi}=\frac{1}{2}\left(D_i-d_i\right).$

The deformation of gap element in the $n$-direction is described $U_{Gi}$ as Eq. (5):

5

$U_{Gi}={\left(v_i^2+w_i^2\right)}^{\frac{1}{2}},$

where, $v_i$ is the displacement of the $i$ node of the flexible drill string along the local axis $y$, m; $w_i$ is the displacement of the $i$ node of the flexible drill string along the local axis $z$, m.

The strain of the gap element can be represented as Eq. (6):

6

${\epsilon }_{Gi}=\frac{U_{Gi}}{E_{oi}}.$

Fig. 6Schematic diagram of different contact states of the gap element

a) The initial state

b) The compressed state

c) The contact state

5.1. Theoretical analysis of the contact plane model between hinged beam and outer beam

In order to validate the correctness of the model for flexible drill string inside an outer tube and the dynamic relaxation method, a study object was selected consisting of a controllable hinged inner beam and an outer beam. The inner beam has an elastic modulus of $E$, a diameter of $D_1$, a moment of inertia of $I$, and a length of 2$l$. The controllable hinge has a rotational limit of$\beta$. There is a gap of $\mathrm{\Delta }$ between the inner beam and the outer beam. The left end of the inner beam is fully fixed, while the right end is supported by a movable hinge support. The outer beam is fully fixed, as shown in Fig. 7.

Fig. 7Plane model for the contact of hinged beam and outer beam

Let’s assume that when the hinge point of the inner beam reaches its rotational limit, the critical concentrated force at that point is denoted as $F_r$. The internal forces acting on the inner beam are shown in Fig. 8. $F_{Ax}$, $F_{Ay}$ and $M_A$ represent the support reactions and the reaction couple at point $A$, while $F_{By}$ represents the support reaction at point $B$.

Fig. 8Inner beam force diagram when hinge reaches rotation limit

The deflection of the $AC$ segment and the $BC$ segment is represented $y_{F_{r}1}$ and $y_{F_{r}2}$ respectively. Based on the equilibrium equation and boundary conditions, the equations representing the angular rotation and deflection of the $AC$ segment and $BC$ segment of the internal beam are as follows Eqs. (23-24).

For the AC segment:

where, $E$ is the Young modulus.

For the $BC$ segment:

The angles at point $C$ of the beams in section $AC$ and section $BC$ are respectively ${\theta }_{CA}$ and ${\theta }_{CB}$, and the relative angles $\beta$ at the hinge can be expressed as Eq. (25):

Let $E=$2.1×10¹¹Pa, $D_1=$ 10mm, $I=$7.84×10^-9m⁴, $l=$ 1 m, and the rotational limit of the hinge is $\beta =$ 1°. According to Eq. (25), when $F_r=$ 34.48 N, the deflection at the hinge point under the action of $F_r$ is denoted as ${\delta }_r$:

To ensure that the hinge point of the internal beam reaches the rotational limit before making contact with the outer beam, the following conditions should be met: $F>F_r$ and $\mathrm{\Delta }$ >${\delta }_r$. Therefore, taking $F=$ 80 N and $\mathrm{\Delta }$ = 8 mm. According to the principle of superposition, we can split $F$ into two parts, $F_r$ and $F\text{'}$, such that $F=F_r+F\text{'}$. The deflection or angular rotation of a specific section of the internal beam under the action of $F$ is equal to the superposition of the deflection or angular rotation under the individual actions of $F_r$ and$F^{\text{'}}$. When $F_r$ reaches the rotational limit, the hinge joint becomes a rigid connection at the internal beam joint. The internal beam transforms into a hyperstatic structure. Under the influence of $F^{\text{'}}$, the deflection at point $C$ increases, coming into contact with the outer beam. Additionally, the hinge joint is capable of transmitting bending moments. Remove the restraint of the inner beam of the $B$ end support and replace it with $F_{By}$ The contact boundary constraint of the outer beam and the internal beam is replaced by $F_C$. The internal beam is subjected to the forces and constraints as shown in Fig. 9.

Fig. 9Inner beam force diagram

The deflection of the $AC$ segment of the internal beam under the separate actions of $F^{\text{'}}$ and $F_C$ is represented by $y_{F^{\text{'}}1}$ and $y_{F_{C}1}$ respectively. The deflection of the $BC$ segment of the internal beam under the separate actions of $F^{\text{'}}$ and $F_C$ is represented by $y_{F^{\text{'}}2}$ and $y_{F_{C}2}$ respectively. The deflection of the internal beam under the sole action of $F_{By}$ is denoted by $y_{F_{By}}$. Based on the continuity conditions at point $C$ and the conditions of zero angular rotation and deflection at fixed end $A$, the expressions for the deflection equations can be derived. The total deflection of the internal beam, denoted as $y_Z$, is equal to the algebraic sum of $F_r$, $y_{F^{\text{'}}}$, $y_{F_C}$ and$y_{F_{By}}$. The inner beam and the outer beam are in contact at $x=l$, then Eq. (27):

Due to the constraint of the internal beam at point $B$ in the $y$-direction, the total deflection at point $B$ should be zero. Therefore, the compatibility equation can be expressed as follows Eq. (28):

From this, we can obtain the following Eqs. (29-30):

After determining $F_C$ and $F_{By}$, the internal beam transforms into a cantilever beam. Based on the equilibrium conditions, we can determine the support reactions at the $A$ end of the internal beam, denoted as $F_{Ay}$ and the moment reaction, denoted as $M_A$, as well as the bending moment at point $C$, denoted as $M_C$ Eqs. (31-33):

5.2. Model verification

The flexible drill string model and dynamic relaxation method were used to analyze the contact plane model between the hinged inner beam and the outer beam. The function expression of the lateral force with respect to time is as follows Eq. (34):

It should be noted that the controllable universal hinge is capable of rotating in two directions. In order to achieve planar hinge rotation that occurs only within the $XY$ plane, in numerical analysis, the first Cardan angle can be constrained to prevent its rotation. Through analysis, the constraint reactions at the $A$ end of the internal beam, denoted as $F_{Ay}$ (constraint force) and $M_A$ (constraint moment), can be determined as shown in Fig. 10(a) and Fig. 10(b) respectively. The constraint reaction at the $B$ end of the internal beam, denoted as $F_{By}$, is shown in Fig. 10(c). The deflection at the hinge point $C$ as a function of time is depicted in Fig. 10(d).

From Fig. 10, it can be observed that during the time interval of 0 to 0.45 s, the support reactions $F_{Ay}$ and $M_A$ at the fixed end $A$ of the internal beam, as well as the support reaction $F_{By}$ at point $B$, exhibit significant fluctuations. The deflection ${\delta }_C$ at the hinge point $C$ shows relatively small fluctuations, while $F_{Ay}$, $M_A$, $F_{By}$, and ${\delta }_C$ all tend to reach a balanced and stable state when $t$ > 0.45 s. Compared the theoretical analysis and numerical analysis results of the support reactions, moment reactions, contact forces, deflections, and bending moments at the constraint locations of contact plane model of outer beam with internal controllable hinged inner beam. The comparison can be seen in Table 1.

Fig. 10Variation curves of different parameters with time

a) Curve of restraining force $F_{Ay}$ at point $A$ with time

b) Curve of the restraining moment $M_A$ at point $A$ with time

c) Curve of restraining force $F_{By}$ at point $B$ with time

d) Curve of deflection ${\delta }_C$ at point $C$ with time

The numerical solution compared to the results obtained from theoretical derivation shows a maximum error of only 0.14 %, confirming the feasibility of flexible drill string with hinged in outer tube model and the dynamic relaxation method (Table 1).

6.1. Analysis of results for different hinge rotation limit results

The dynamic relaxation method is employed to analyze the contact nonlinearity of the flexible drill string with hinges. The functional expressions of the applied torque and axial force at the top of the flexible drill string as a function of time are given as follows Eqs. (35-36):

By conducting a contact nonlinearity analysis on the flexible drill string with hinges, we obtained the torque and axial force variations at the bottom of the flexible drill string for different hinge rotation limits, as shown in Fig. 11.

Fig. 11Variation curve of internal force at the bottom of flexible drill string with time

a) The torque at the bottom

b) The axial force at the bottom

According to Fig. 11(a), it can be seen that when t is in the range of 0 s to 0.4 s, the torque applied at the top end of the flexible drill string gradually increases, and the torque at the bottom end shows an increasing trend. After 0.4 s, the torque applied at the top end of the flexible drill string remains constant. The torque at the bottom end reaches an equilibrium and stable state after 0.6 s. The torque at the bottom end of the flexible drill string with different hinge rotation limits degree does not vary significantly. According to Fig. 11(b), it can be observed that when $t$ is in the range of 0 s to 0.4 s, the axial force applied at the top end of the flexible drill string gradually increases, the bottom axial force of the flexible drilling string is showing an overall increasing trend. After 0.4 s, the bottom axial force of the flexible drill string, with a hinge rotation limit of less than 4.5°, reaches a balanced and stable state after more than 0.7 seconds. The bottom axial force of a flexible drill string with a hinge rotation limit greater than or equal to 4.5° reaches a balanced and stable state after more than 0.5 seconds. As the hinge rotation limit increases, the axial force transmitted to the bottom of the flexible drill string gradually decreases. When $t=$ 1 s, both the torque and axial force at the bottom of the flexible drill string have reached a balanced and stable state.

When $t=$1 s, the calculation results for the different hinge rotational limits degree of the flexible drill string are shown in Table 2. The variations of torque and axial force along the axis of the flexible drill string are illustrated in Fig. 12.

Based on Table 2 and Fig. 12(a), it can be observed that the torque of the flexible drill string decreases gradually as the length along the axis increases. As the hinge rotation limit increases, the torque values of the flexible drill string exhibit larger fluctuations. The total resistance torque between the flexible drill string and the outer tube also increases. However, the variation of torque at the bottom end of the flexible drill string is irregular and does not differ significantly. This is due to the presence of hinges in the flexible drill string, some of which increase torque while others decrease it. As a result, the maximum torque loss rate is only 6.77 %. When the hinge rotation limit is 3.0°, the torque transmitted to the bottom of the flexible drill pipe is 1907.6 N·m, and the torque loss rate is 4.6 %. Compared with the calculation results in reference [30], the torque loss rate is 3.4 % higher. This is because the paper considers the relative rotation characteristics of the flexible drill pipe articulated, resulting in increased deformation of the flexible drill pipe and increased frictional torque between the flexible drill pipe and the outer pipe, so the torque transmitted to the bottom is relatively smaller and the torque loss rate is greater.

Based on Table 2 and Fig. 12(b), it can be observed that the axial force of the flexible drill string decreases gradually as the length along the axis increases. The decreasing trend of axial force in the flexible drill string is more pronounced within the length range of 0 m to 2.2 m. The decreasing trend of axial force in the flexible drill string is relatively gentle within the length range of 2.2 m to 5.0 m. As the hinge rotational limit increases, the total frictional resistance between the flexible drill string and the outer tube becomes greater. This results in a smaller axial force transmitted to the bottom end of the flexible drill string. When the hinge rotational limits degree are 5.0° and 5.5°, the axial forces transmitted to the bottom end of the flexible drill string are 8.1 kN and 1.5 kN, respectively. The axial force loss rates are 59.5 % and 92.5 %, respectively.

Fig. 12Torque distribution and axial force distribution of flexible drill string

a) Torque distribution

b) Axial force distribution

The third Cardan angle at the hinge point of the flexible drill string is illustrated in Fig. 13. The deformation of the flexible drill string in the $XY$ plane is depicted in Fig. 14. Based on Fig. 13 and Fig. 14, it can be observed that as the hinged rotational limit increases, the relative angle of the third Cardan angle at the hinge point of the flexible drill string becomes larger. This results in a more pronounced “sawtooth-like” deformation of the flexible drill string in the $XY$ plane. That is because the flexible drill string utilizes hinges to meet the bending requirements for creating short-radius horizontal wellbore sections. When the wellbore curvature radius is constant, for different hinge rotational limits degree, the initial third Cardan angle at the same hinge point of the flexible drill string is equal and approximately 2.83°. Currently, the upper limit of the third Cardan angle has reached the hinged rotational limit, and the current third Cardan angle is equal to the sum of the initial third Cardan angle and the relative angle of rotation. Therefore, as the hinged rotational limit increases, the relative angle of rotation of the third Cardan angle at the hinge point of the flexible drill string becomes larger.

Fig. 13Third Cardan angle at the hinge of the flexible drill string

a) 3.0°

b) 3.5°

c) 4.0°

d) 4.5°

e) 5.0°

f) 5.5°

Fig. 14Deformation diagram of flexible drill string in XY plane

a) 3.0°

b) 3.5°

c) 4.0°

d) 4.5°

e) 5.0°

f) 5.5°

The contact force between the flexible drill string and the outer tube is shown in Fig. 15. According to Fig. 15, it can be observed that the contact force between the flexible drill string and the outer tube is randomly distributed along the axial direction. The contact force is discontinuous and intermittent. This is because the flexible drill string is treated as a variable cross-section beam. Therefore, only the flexible drill string at the maximum cross-section comes into contact with the outer tube. As the hinged rotational limit increases, the maximum contact force between the flexible drill string and the outer tube also increases.

Fig. 15Contact forces between flexible drill string and the outer tube

a) 3.0°

b) 3.5°

c) 4.0°

d) 4.5°

e) 5.0°

f) 5.5°

6.2. Analysis of results for different single section lengths

Through a nonlinear contact mechanics analysis of the flexible drill string with hinges of different single section lengths inside the outer tube, the curves of torque and axial force variations over time at the bottom end of the flexible drill string are obtained. These curves are illustrated in Fig. 16.

Fig. 16Variation curve of internal force at the bottom of flexible drill string with time

a) The torque at the bottom end

b) The axial force at the bottom

According to Fig. 16(a), it can be observed that from $t=$0 s to 0.4 s, as the torque applied at the top end of the flexible drill string gradually increases, the torque at the bottom end shows an increasing trend. After 0.4 s, the torque applied at the top end of the flexible drill string remains constant, and the torque at the bottom end reaches a steady state after 0.5 s. The torque at the bottom end of the flexible drill string does not differ significantly for different single section lengths. According to Fig. 16(b), it can be observed that from $t=$0 s to 0.4 s, as the axial force applied at the top end of the flexible drill string gradually increases, the axial force at the bottom end of the flexible drill string shows an overall increasing trend. After 0.6 s, the axial force at the bottom end of the flexible drill string for different single section lengths reaches a steady state. The flexible drill string with a single section length of 200 mm experiences the highest transmitted axial force at the bottom end, while the flexible drill string with a single section length of 150 mm experiences the lowest transmitted axial force at the bottom end.

Overall, the transmitted axial forces at the bottom end of the flexible drill string do not differ significantly for different single section lengths.

When $t=$1 s and the hinged rotational limit is 4.0°, the calculated results for different single section lengths of the flexible drill string are presented in Table 3. The variations of torque and axial force along the axis of the flexible drill string are illustrated in Fig. 17.

Fig. 17Distribution of internal forces along the axis of flexible drill string

a) Torque distribution

b) Axial force distribution

Combining Table 3 and Fig. 17(a), it can be observed that the torque of the flexible drill string exhibits a generally decreasing trend along the length of the axis. When the single section length varies, the total frictional resistance torque between the flexible drill string and the outer tube exhibits irregular changes. However, the torque values transmitted to the bottom end of the flexible drill string do not differ significantly. The maximum torque loss rate is only 5.84 %.

Based on Table 3 and Fig. 17(b), it can be observed that the axial force of the flexible drill string exhibits an overall decreasing trend along the length of the axis. The decreasing trend of the axial force in the flexible drill string is more pronounced in the length range of 0 m to 3.2 m. The decreasing trend of the axial force in the flexible drill string is relatively gradual in the length range of 3.2 m to 5.0 m. There is not a significant difference in the axial force at the bottom end of the flexible drill string for single section lengths greater than 150 mm.

The deformation of the flexible drill string in the $XY$ plane is illustrated in Fig. 18. According to Fig. 19, it can be observed that as the single section length decreases, the “sawtooth-like” deformation of the flexible drill string in the $XY$ plane becomes more pronounced. This is because with smaller single section lengths, the relative angular displacement at the joints of the flexible drill string becomes larger.

Fig. 18Deformation diagram of flexible drill string in XY plane

a) 150 mm

b) 160 mm

c) 170 mm

d) 180 mm

e) 190 mm

f) 200 mm

Fig. 19Contact force between flexible drill string and outer tube

a) 150 mm

b) 160 mm

c) 170 mm

d) 180 mm

e) 190 mm

f) 200 mm

The contact forces between the flexible drill string and the outer tube are depicted in Fig. 19. According to Fig. 19, it can be observed that the contact forces between the flexible drill string and the outer tube are randomly distributed along the axial direction. The contact forces exhibit intermittent and discontinuous behavior. As the length of the flexible drill string increases, the maximum contact force between the flexible drill string and the outer tube initially increases and then decreases.