Research on friction characteristics of drill string in whole well section of gas drilling based on finite element method

2.1. Beam element model

A global coordinate system $O$-$XYZ$ is established in Fig. 1 to describe the trajectory of the well, with the origin $O$ located at the wellhead, the $X$-axis points due north, the $Y$-axis points due east, and the $Z$-axis points down to the center of the earth. To more accurately describe the actual state of the drill string in the borehole, a element coordinate system $o$-$xyz$ is instituted on the beam element. The $x$-axis points to the tangential direction of the borehole axis, while the $y$-axis points to thenormal direction of the borehole axis, and the $z$-axis is determined by the right-hand rule [16].

Fig. 1Drill string’s finite element model

A drill string beam element consists of two nodes and twelve degrees of freedom. The three-dimensional motion of the drill string can be represented by the displacement vector of the beam element:

where, subscript $i$ represents the $i$th node; $u$, $v$ and $w$ are linear displacements along the $x$, $y$, and $z$ directions, respectively, m; ${\theta }_x$, ${\theta }_y$ and ${\theta }_z$ is the angular displacement around the $x$, $y$, and $z$ axes, respectively, rad; ${\left[.\right]}^T$ is the transpose of the matrix; $\mathbf{N}$ is a type function matrix.

Lagrange equation is used to establish the dynamic equation of drill string in element coordinate system [17]:

where, $T_e$ is the kinetic energy, J; $U_e$ is the potential energy, J; ${\dot{\mathbf{q}}}_e$ represents the reciprocal of the generalized variable with respect to time; ${\mathbf{F}}_e$ is the node force vector of the beam element, N.

The overall kinetic energy of a beam element is comprised solely of translational and rotational kinetic energy, and can be represented as [18]:

where, $l_e$ denotes the length of beam element, m; $e$ represents the eccentricity of the beam element, m; $\rho$ represents the density of the beam element, kg/m³; $\dot{u}$, $\dot{v}$ and $\dot{w}$ represent the translation velocity of the node in the $x$, $y$ and $z$ axes, respectively, m/s; ${\dot{\theta }}_x$, ${\dot{\theta }}_y$ and ${\dot{\theta }}_z$ represent rotation speeds around the $x$, $y$, and $z$ axes, respectively, rad/s; $A$ represents the cross-sectional area of the beam element, m²; $\mathrm{\Omega }$ represents the speed of the beam unit, r/min; $\beta$ denotes the initial angle between the line connecting the center of gravity of a element to its shape center, relative to the $z$-axis, rad; $J_1$ and $J_2$ represent torsional moment of inertia and transverse moment of inertia per unit length, respectively, kg·m².

For beam elements subjected to axial load, torsional deformation and bending deformation, their potential energy is expressed as follows [19]:

where, $E$ is the elastic modulus of the drill string, GPa; $I_p$ signifies the polar moment of inertia for the cross-sectional geometry of the beam element, m⁴; $G$ is the shear modulus of the drill string, GPa; $I_{yz}$ is the moment of inertia of the beam element cross-section, m⁴.

The external force vector includes the element gravity, the unbalance force and the contact force between the drill string and the wellbore wall. The density of circulating gas in air drilling is small, and the buoyancy of the air on the drill string can be ignored. Where the unit gravity equivalent nodal force can be expressed as:

where, the gravity component of the beam element per unit length in $x$ axis and $y$ axis is expressed as $q_x$ and $q_y$, respectively.

The unbalance force of the drill string unit due to mass eccentricity can be expressed as:

where, $p_y$ and $p_z$ represent the centrifugal force components in the $y$ and $z$ direction of the beam element, respectively.

The drill string-wellbore wall contact force vector can be expressed as:

where, $F_y$ and $F_z$ represent the components of friction along the $y$-axis and $z$-axis of the beam element respectively; $T_f$ represents the friction torque between the drill string and the wellbore wall.

By substituting Eqs. (3-7) into Eq. (2) and assembling the matrix, the dynamic equation can be expressed as follows:

where, $\mathbf{K}$, $\mathbf{M}$ and $\mathbf{C}$ are stiffness, mass and damping matrices respectively; $\ddot{\mathbf{q}}$, $\dot{\mathbf{q}}$ and $\mathbf{q}$ are generalized acceleration, velocity and displacement vectors, respectively; $\mathbf{F}$ is resultant force vector.

2.2. Boundary condition

3.1. Numerical calculation

The Generalized method is employed to solve the nonlinear dynamic equation of the drill string system, ensuring high accuracy, while the Newton iterative method is utilized for rapid convergence. The solution flow of the drill string system is illustrated in Fig. 2. An approximate solution to the dynamic equation can be obtained through the generalized-$\alpha$ method [21-22]:

where, ${\alpha }_r$, ${\delta }_r$, ${\eta }_r$, ${\beta }_r$, ${\epsilon }_r$ and ${\gamma }_r$ are independent dimensionless parameters in the Generalized-$\alpha$ method; $\mathrm{\Delta }t$ represents the time step, s; Subscript $k$ represents the $k$-th time step.

Fig. 2Solution process of finite element model

Polynomial expansion of Eq. (12) is performed and higher-order terms are ignored. The displacement iteration expression of the drill string system can be expressed as:

where, ${\mathbf{R}}_{k+1}^{\left(p\right)}$ is the residual force vector; ${\mathbf{J}}_{k+1}^{\left(p\right)}$ is the Jacobian matrix and the superscript $p$ represents the number of iterations.

Repeat Eq. (15) until the displacement and accumulation tolerance is satisfied:

where, $\epsilon \text{'}$ is convergence error.

3.2. System parameter

The frictional characteristics of the drill string system were investigated using the air drilling horizontal well model of Jinqian 511-7-H1, located in Sichuan Basin with a vertical depth of 1982 m and a sounding depth of 2844 m. Three-hole size Φ152.4 mm was designed and nitrogen drilling was implemented. The node model of borehole trajectory is shown in Fig. 3. The drill string system comprises drill pipe, heavy weight drill pipe, drill collar, stabilizer and drill bit. The finite element method is used to divide the drill string into several beam elements, with an element length of 6 meters for the deflecting section and horizontal section, and 8.5 meters for the vertical section. To enhance computational efficiency while ensuring unconditional stability in numerical analysis, a time step of 0.0015s is set when solving the finite element model of drill string system dynamics.

The drill string structure parameters are: Φ152.4 mm PDC bit × 9.19 m + Φ120 mm drill collar × 9.19 m + Φ146.1 mm stabilizer ×1.05 m + Φ120 mm drill collar × 17.58 m + Φ101.6 mm heavy weight drill pipe × 283.6 m + Φ101.6 mm drill pipe × 2038.93 m.

Fig. 3Borehole trajectory model of horizontal well

4.1. Lateral motion characteristics of drill string

The influence of WOB and rotational speed on drill string lateral vibration are illustrated in Fig. 4, where node 160 is located in the vertical section, node 280 is located in the deflecting section, and nodes 320 and 395 are located in the horizontal section. The displacements of each node on the drill string in the $y$ and $z$ directions increase significantly as the rotational speed increases from 30 r/min to 60r/min, as shown in Fig. 4(a) and (b). The $y$-direction displacement of nodes 160, 280, 320, and 395 increases by 35 %, 12 %, 8 %, and 2 % respectively. Node 160 in the vertical section exhibits no contact with the wellbore wall, while the motion trajectorys of nodes 280 and 320 are concentrated on the lower left of the shaft wall due to the influence of gravity. The larger the node number, the greater the lateral motion range of the drill string. Node 320 executes circular motion around the wellbore at a rotational speed of 60 r/min, leading to an increase in the frequency of contact. Node 395 is located in the position of the stabilizer, since the stabilizer diameter is slightly smaller than the hole diameter, the maximum lateral motion range of node 395 is smaller than that of node 320. The acceleration of the drill string rotation elevates friction between the drill string and the borehole wall, which is unevenly distributed, potentially causing bending or vibration of the drill string. At high rotational speeds, such bending and vibration can expand the contact area between the drill string and the well wall, thereby affecting the trajectory of the drill string [23]. Fig. 4(c) and (d) demonstrate the impact of varying WOB on the lateral displacement of the drill string. The lateral movement range of the drill string increases in correlation with the increase in WOB, while the lateral displacement of the drill string gradually increases as the distance between the node and the bit decreases, similar to the trend of speed change. Therefore, the WOB and the rotational speed should be reduced as much as possible under the premise of meeting the rate of penetration, so as to reduce the contact frequency between the drill string and the wellbore wall, and reduce the drilling friction.

Fig. 4Lateral displacement of drill string with different parameters

a)${\mathrm{\Omega }}_{rt}\mathrm{}$= 30 r/min

b)${\mathrm{\Omega }}_{rt}\mathrm{}$= 60 r/min

c) WOB = 10 kN

d) WOB = 70 kN

4.2. The influence of borehole curvature on friction of drill string

The influence of borehole curvature on the friction between drill string and wellbore wall is shown in Fig. 5(a). The vertical section is 0-1400 m, the deflecting section is 1400-2000 m, and the horizontal section is 2000-2844 m. As can be seen from Fig. 5(a), with the increase of borehole curvature in the deflecting section, the friction on the drill string gradually increases. The borehole curvature increases from 1°/30 m to 4°/30 m, and the maximum friction force increases by 8 %, and from 7°/30 m to 10°/30 m, the maximum friction force increases by 13 %. When the borehole curvature is 1°/30 m, the friction resistance value of the whole well section is 125 kN. When the borehole curvature is increased to 10°/30 m, the friction resistance increases to 170 kN, and the friction force increases more in the deflecting section than in the horizontal section. Fig. 5(b) shows the curve of average friction with borehole curvature when the drill string is raised and lowered. As can be seen from Fig. 5(b), the average lifting friction on the drill string is less than the average lowering friction on the drill string. With the increase of borehole curvature in the deflecting section, the drag lifted on the drill string and lowered on the drill string gradually increases. When the borehole curvature is 12°/30 m, the lifting friction on the drill string in the whole well section is 225 kN and the lowering friction is 283 kN, which increases 58 kN. Therefore, reasonable drilling parameters should be selected when drill string is lowered to prevent excessive friction. In the design of well structure, the borehole curvature of the hole should be reduced as much as possible, so as to reduce friction and improve drilling efficiency.

Fig. 5Effect of borehole curvature on drill string friction

a) Friction of drill string in the whole well section

b) Friction of lifting and lowering the drill string

4.3. The influence of friction coefficient on friction of drill string

The influence of friction coefficient on the friction between drill string and wellbore wall is shown in Fig. 6(a). As can be seen from Fig. 6(a), with the increase of friction coefficient, drilling friction increases significantly, and the increase is getting larger. When the friction coefficient increases from 0.2 to 0.3 and 0.4 to 0.5, the friction resistance increases sharply. The relative increase of the former reached 17 kN, up 21 % year-on-year; The latter has a relative increase of 26 kN, an increase of 25 % year-on-year. In addition, the change of friction coefficient has a significant effect on the horizontal drill string, with an average increase of 50 kN per kilometer. Fig. 6(b) shows the curve of average friction with the friction coefficient when the drill string is lifted and lowered. As can be seen from Fig. 6(b), the average lifting friction on the drill string is smaller than that on the lowering friction, because there is less downward pressure when the drill string is lifted. When the friction coefficient is 0.15, the average friction is 259 kN; when the friction coefficient increases to 0.4, the average friction resistance reaches 302 kN, which increases 43 kN. Therefore, in the process of drilling, the cleanliness of the borehole should be increased as much as possible, so as to reduce the drill string friction.

Fig. 6Effect of friction coefficient on drill string friction

a) Friction of drill string in the whole well section

b) Friction of lifting and lowering the drill string

Fig. 7Friction torque of drill string nodes under different WOBs

a) WOB = 10 kN

b) WOB = 30 kN

c) WOB = 50 kN

d) WOB = 70 kN

4.4. The drill string friction torque under different WOBs

The drill string friction torque under different WOBs is shown in Fig. 7. Node 220 and 280 are located in the deflecting section, nodes 320 and 380 are located in the horizontal section, and node 395 is located near the bit. According to Fig. 7(a)-(d), It is evident with the increase of WOB, the friction torque between the drill string and wellbore wall significantly increases, and the torque required to drive the drill string to work normally is also larger. When the WOB is 10 kN, the range of friction torque is small, and the maximum friction torque is 90 N∙m; When the WOB is 70 kN, the friction torque fluctuates in the range of 0-2.1 N∙m. An increase in weight on bit causes the drill string to be subjected to a greater axial force, resulting in excessive bending of the drill string in the hole. Drilling string bending increases the area of contact with the wall, which increases friction and torque and increases the risk of wall collapse. Too high weight on bit or too low weight on bit may cause resonance of drill string and increase fatigue damage of drill string [24]. As the distance between node and bit decreases, the friction torque gradually increases, and the average friction torque of drill string in horizontal section is larger than that in vertical section and deflecting section. Therefore, at the deflecting section and horizontal section, it is recommended that the WOB should not exceed 50 kN to extend the service life of the drill string and increase the stability of the wellbore wall [25].

Fig. 8Friction torque of drill string nodes under different rotational speeds

a)${\mathrm{\Omega }}_{rt}$ = 30 r/min

b)${\mathrm{\Omega }}_{rt}\mathrm{}$= 40 r/min

c)${\mathrm{\Omega }}_{rt}$ = 50 r/min

d)${\mathrm{\Omega }}_{rt}\mathrm{}$= 60 r/min

4.5. The drill string friction torque under different rotational speeds

The impact of varying rotational speeds on the drill string friction torque is analyzed using a consistent WOB of 30 kN. The amplitude and fluctuation frequency of friction torque increase obviously with the increase of rotation speed, as shown in Fig. 8. When the rotation speed is 30 r/min, 40 r/min, 50 r/min and 60 r/min, the average fluctuation size is 120 N∙m, 150 N∙m, 210 N∙m and 310 N∙m, respectively. This indicates that as rotational speed increases, the likelihood of the drillstring coming into contact with the borehole wall is heightened, which is easy to cause damage to borehole wall. which is consistent with the “the greater lateral vibration displacement and the more frequent the contact between drillstring and wellbore wall” in literature [17]. In addition, the unbalance force generated by the rotation increases with higher rotation speeds, which increases the possibility of the drill string colliding with the wellbore wall. The friction torque of drill string in horizontal section increases significantly, indicating that there is a certain risk of sticking in this section, and the horizontal section should be used as the key well section for anti-wear and drag reduction. Consequently, to minimize the harm caused to drilling tools, it is recommended to choose a speed range of 30-50 r/min.