Optimal design of a global force-balanced polycrystalline diamond compact bit considering wear condition

Abstract

Polycrystalline diamond compact bits are one of the most widely used oil and gas drilling tools in the world. With wear, a large unbalanced lateral force and bending moment exist. These force and moment contribute not only to bit lateral vibration and whirl but also to wellbore tilt and enlargement, which will then cause early bit failure and low drilling efficiency. In this article, considering wear condition, a single cutter force model is proposed. Lateral force and bending moment models are constructed based on space-force theory. An optimal cutter layout model considering cutter wear is established. The matching approach for the optimal model is discussed based on Kriging surrogate model and genetic algorithm. Then, an optimization case is presented. The results show that the bit force models are in line with the actual drilling condition. The optimal approach is efficient. After optimization, the lateral force to weight on bit ratio is reduced by 10.99%, and the bending moment to torque on bit ratio is reduced by 30.43%. This result is a significant improvement in the force condition and stability of the polycrystalline diamond compact bit; ultimately, the whirl and tilt motion can be reduced, and the drilling efficiency can be improved.

Introduction

Polycrystalline diamond compact (PDC) bits are widely used to drill wells for the exploration and exploitation of hydrocarbons because of their high rock-breaking efficiency. The PDC bit is composed of a matrix body, flute, blades, individual cutting teeth and so on. To date, PDC bits have comprised 75%–80% of the global market. ¹

The PDC bit is mainly affected by two forces and two moments during the drilling process: weight on bit (WOB) F_w, lateral force F_l, torque on bit (TOB) M_t and bending moment M_b, as shown in Figure 1. F_w and M_t drive the PDC cutters to break rocks. Meanwhile, a PDC bit is subjected to F_l and M_b caused by the bit–rock interaction. F_l and M_b are perpendicular to the drilling direction, and they only impede rock breaking. A large unbalanced F_l is the main cause of bit whirl. In addition, M_b contributes not only to bit lateral vibration or whirl but also to wellbore enlargement and tilt. Whirl and tilt motion will cause early bit failure and low drilling efficiency.

Figure 1.

The forces on a PDC bit.

To improve the PDC bit life and drilling efficiency, the studies of reducing the lateral force and bending moment have been carried out in recent decades. In the 1990s, the method of reducing the lateral force by changing the cutter layout was put forward to enhance the bit stability, and these bits are called ‘force-balanced’ PDC bits.^2,3 The concept of a ‘global force-balanced’ bit was proposed in 2005. These bits have balanced or minimal lateral forces and bending moments. ⁴ Global optimization algorithms, such as particle swarm optimization (PSO) and genetic algorithms (GAs), have been used to design the cutter layout structure of ‘force-balanced’ PDC bits.^5,6 A grouping cutter layout method was proposed to keep the lateral force balanced under transit drilling conditions. ⁷ Based on the aforementioned research, multilevel force-balanced drilling tools and methods are researched. The PDC bit will be in a good force state during the drilling process when the ratio of M_b to M_t is less than 4%, and the ratio of F_l to F_w is less than 4%. ⁸ A general approach of backup cutter layout to extend bit life without sacrificing rate of penetration was discussed. ⁹

The aforementioned studies have greatly improved the drilling performance of PDC bits. However, in those studies, the wear of a PDC bit is not considered in the process of cutter layout design, and the specific computing methods for reducing M_b are not mentioned. In fact, most PDC bits cannot maintain the global force-balanced state with cutter wear, which is one of the important factors restricting the rock-breaking efficiency and stability of PDC bits. In addition, a bit wearing is a normal state during drilling, and it is not reasonable to design a ‘global force-balanced’ bit without considering wearing conditions.

In this article, an optimal design of the global force-balanced PDC bit considering wear condition is carried out to improve drilling performance of a wear PDC bit and to achieve the balance state of lateral force and bending moment at the same time.

Development of bit force models

Single cutter force model

Existing single cutter force model

Single cutter force models can be divided into two-dimensional (2D) cutter force models (Figure 2(a)) and three-dimensional (3D) cutter force models (Figure 2(b)). The 2D cutter force model usually considers the effect of the back rake angle α, which is defined as the acute angle between the cutter surface and axis OZ (same as the drilling direction). The cutter is subjected to cutting force f_c and penetration force f_p during rock breaking. f_c and f_p are parallel to axis OY and axis OZ, respectively. Axis OY is the direction of the cutter movement.

Figure 2.

Single cutter force models: (a) 2D single cutter force model and (b) 3D single cutter force model.

Most single cutter force experiments are based on the 2D single cutter force model. Cheatham and Daniels first performed a series of tests on the drillability of PDC cutters. They concluded that the cutting force f_c is proportional to the cutting area. ¹⁰ Detournay and Defourny developed the cutter force model, which divides the cutter/rock interaction into two processes: rock cutting and frictional contact. By combining these two processes, a relationship between specific energy (E) and drilling strength (S) is obtained. This model is called the E-S model. ¹¹ The E-S model is complemented by introducing characteristic contact length and contact strength. Contact length is an objective measure of bit bluntness, and contact strength is the maximum stress that can be transmitted at the cutter wear flat–rock interface. ¹² Considering the wearing condition, the effects of the wear height, angle α, cutting area, and cutting arc length on the cutter forces were researched by systematic experiments, ¹³ and the cutter force model was determined to be

$$\{\begin{array}{c}{f}_c={\mathrm{e}}^{\left(\frac{h_w}{d\cos \alpha }\right)}{p}_c{K}^{q_c}{A}^{0.7564}L/L_{\mathrm{e}}\hfill \\ f_p={\mathrm{e}}^{\left(\frac{h_w}{d\cos \alpha }\right)}{p}_p{K}^{q_p}{A}^{0.5472}L/L_{\mathrm{e}}\hfill \end{array}$$

(1)

$$\{\begin{array}{c}{p}_c=0.481{\mathrm{e}}^{6.9291\sin \alpha };q_c=2.4154{\left(\cos \alpha \right)}^{4.8154}\hfill \\ \multicolumn{1}{c}{p_p=4.832{\mathrm{e}}^{3.1724\sin \alpha };q_p=1.8196{\left(\cos \alpha \right)}^{1.6419}}\end{array}$$

(2)

In equations (1) and (2), d is the diameter of the cutting teeth. K represents the rock drillability. A represents the cutting area which is the contact surface of the rock and cutter (the sand texture in Figure 3). Besides, the yellow zones in Figure 3 represent the cutting surface of cutter with no contact rock. L is the cutting arc length, which represents the part of the cutting tooth that is in contact with the rock during drilling (Figure 3(a)). L_e represents the equivalent arc length when the area of the ordinary cutting surface is equal to that of the standard cutting surface (Figure 3(b)). h_w represents the wear height of the cutting teeth (Figure 3(c)).

Figure 3.

Cutting parameters of cutters: (a) ordinary cutting surface, (b) standard cutting surface and (c) wear height.

However, the aforementioned experimental studies lack the parameter of side rake angle β, which is defined as the acute angle between the cutting tooth surface and axis OX, as shown in Figure 2(b). This omission will result in the inaccuracy of the force model of the PDC bit.

When considering side rake angle β, the cutter is subjected to three forces, f_c, f_p and f_l, whose directions are parallel to OY, OZ and OX, respectively (Figure 2(b)). At this point, the single cutter force model is 3D, and it more closely represents actual drilling conditions. The influence of the angle β, cutting depth, and rock properties on the cutter forces is studied by a series of experiments, and the test results show that angle β has no significant effect on forces f_c and f_p. Force f_l approximately linearly increases with angle β. ¹⁴ Chen et al. ¹⁵ believe that the cutting force model is related to the cutting area shape which can be divided into three types. Shi ¹⁶ concluded that there is a linear relationship between the three forces f_c, f_p and f_l as expressed in equation (3). However, the existing 3D single cutter force models are relatively rough since it does not consider the wear factor of cutters

$$\{\begin{array}{c}{f}_c=C_1{f}_p\hfill \\ \multicolumn{1}{c}{f_{\mathrm{l}}=C_2{f}_p}\end{array}$$

(3)

where C₁ and C₂ are constants determined by the rock properties and angles α and β.

A new single cutter force model

According to equations (1) and (2) and some conclusions of the existing 3D cutter force model, a new 3D single cutter force model is proposed, as shown in equation (4)

$$\{\begin{array}{c}{f}_p=e^{\left(\frac{h_w}{d\cos \alpha }\right)}{p}_p{K}^{q_p}{A}^{0.5472}\frac{L}{L_{\mathrm{e}}}\hfill \\ f_c=\frac{p_c}{p_p}{K}^{(q_{\mathrm{c}}-q_{\mathrm{p}})}{A}^{0.2092}{f}_p\hfill \\ f_l=\frac{\beta }{300}{f}_p\hfill \end{array}$$

(4)

In equation (4), force f_l increases with angle β linearly, and it is 10% of force f_p when angle β is equal to 30°. ¹⁶ The forces f_p and f_c describe the deformation in equation (1).

This 3D single cutter force model (equation (4)) is more comprehensive and accurate than the 2D model because the parameters of back rake angle, side rake angle, cutting arc length, cutting area, rock property, and wear degree are all considered.

Lateral force model of PDC bit

Based on the new single cutter force model, the lateral force model is established in this section. The cylindrical coordinate system ORΘH, whose axis OH is the centre axis of the PDC bit and parallel to the drilling direction, is established as shown in Figure 4(a). The axial component force f_a (parallel to axis OH), cutting component force f′_c (circumferential force) and radial component force f_r (pointing towards axis OH along radial direction form cutter centre) in the coordinate system ORΘH can be obtained by decomposition and synthesis of the forces f_p, f_c and f_l on a single cutter.

Figure 4.

Force model of a single cutter on a PDC bit: (a) cutter force model in the cylindrical coordinate system ORΘH, (b) force analysis in plane M and (c) force analysis in plane N.

First, make plane M through the cutter centre and axis OH. Make plane N perpendicular to the axis OH and through the cutter centre, as shown in Figure 4(a). Then, only f_p and f_l exist, as shown in Figure 4(b), because f′_c is perpendicular to plane M. Create a Cartesian coordinate system OR′H′ in the plane M. Axes OR′ and OH′ are the reverse directions of axes OR and OH, respectively. Forces f_p and f_l can be decomposed into axes OR′ and OH′ with equation (5) for both the inner cone cutter and outer cone cutter

$$\{\begin{array}{c}{f}_{a1}=f_p\cos \gamma \hfill \\ f_{r1}=f_p\sin \gamma \hfill \\ f_{a2}=-f_l\sin \gamma \hfill \\ f_{r2}=f_l\cos \gamma \hfill \end{array}$$

(5)

In Figure 4, the axial component force f_a can be obtained by adding f_a1 and f_a2. The radial component force f_r can be obtained by the superposition of forces f_r1 and f_r2. The cutting component force f′_c is the same as the cutting force f_c because both of them are aligned with the cutting direction. At this point, the forces f_a, f′_c and f_r in the coordinate system ORΘH can be expressed as

$$\{\begin{array}{c}{f}_a=f_{a1}+f_{a2}=f_p\cos \gamma -f_l\sin \gamma \hfill \\ f_r=f_{r1}+f_{r2}=f_p\sin \gamma +f_l\cos \gamma \hfill \\ f{\prime }_c=f_c\hfill \end{array}$$

(6)

where γ is the normal angle of the cutter on a PDC bit.

Then, the WOB F_w and TOB M_t can be expressed as

$$\{\begin{array}{c}{F}_w=\sum _{i=1}^{N_c}{f}_a^{(i)}\hfill \\ \multicolumn{1}{c}{M_t=\sum _{i=1}^{N_c}\left(f{\prime }_c^{(i)}{R}_c^{(i)}\right)}\end{array}$$

(7)

where N_c is the amount of cutting teeth on the bit, and i is the serial number of cutters from the bit centre to the edge. $f_a^{(i)}$ , $f{\prime }_c^{(i)}$ and $R_c^{(i)}$ are f_a, f′_c and R_c of the ith cutter, respectively; R_c represents the radial distance from cutter centre to axis OH.

To obtain the lateral force F_l of the PDC bit, the forces f_a, f′_c and f_r need to be decomposed and synthesized in plane N, as shown in Figure 4(c). Among them, f_a can be ignored because it is perpendicular to plane N. Establish a Cartesian coordinate system O′XY, whose O′X direction is the same as axis OR. f′_c and f_r can be decomposed into axes O′X and O′Y

$$\{\begin{array}{c}{f}_{x1}={f\prime }_c\sin {\theta }_c\hfill \\ \multicolumn{1}{c}{f_{y1}=-{f\prime }_c\cos {\theta }_c}\\ \multicolumn{1}{c}{f_{x2}=-f_r\cos {\theta }_c}\\ \multicolumn{1}{c}{f_{y2}=-f_r\sin {\theta }_c}\end{array}$$

(8)

where θ_c represents the angle between plane M and plane ORH and is named the circumferential angle.

Then, the lateral force F_l can be represented by equation (9)

$$\{\begin{array}{c}{F}_{\mathrm{l}}=\sqrt{f_x^2+f_y^2}\hfill \\ \multicolumn{1}{c}{f_x=\sum _{i=1}^{N_c}[f_{x1}^{(i)}+f_{x2}^{(i)}]=\sum _{i=1}^{N_c}[f{\prime }_c^{(i)}\sin {{\theta }_c}^{(i)}-f{\prime }_r^{(i)}\cos {{\theta }_c}^{(i)}]}\\ \multicolumn{1}{c}{f_y=\sum _{i=1}^{N_c}[f_{y1}^{(i)}+f_{y2}^{(i)}]=\sum _{i=1}^{N_c}[-f{\prime }_c^{(i)}\cos {{\theta }_c}^{(i)}-{f_r}^{(i)}\sin {{\theta }_c}^{(i)}]}\end{array}$$

(9)

The circumferential angle θ_s of F_l is slightly different in different quadrants, as shown in equation (10)

$$\{\begin{array}{c}{\theta }_s=\arctan \left(\frac{F_x}{F_y}\right);f_x\ge 0\mathrm{and}{f}_y\ge 0\hfill \\ {\theta }_s=180+\arctan \left(\frac{F_x}{F_y}\right);f_x<0\hfill \\ {\theta }_s=360+\arctan \left(\frac{F_x}{F_y}\right);f_x\ge 0\mathrm{and}{f}_y<0\hfill \end{array}$$

(10)

Bending moment model of a PDC bit

To date, there have been no public reports on the calculation of the bending moment for PDC bits. To optimize the cutter layout design for a global force-balanced PDC bit smoothly, the bending moment model of a PDC bit is discussed in this section based on space-moment theory.

Moment of force about a point

In the space-force system, the orientation of the rotation axes, which is perpendicular to the moment plane and cross the centroid, is different because the lines of action of all the forces are not in a plane. As shown in Figure 5(a), the resultant moment of forces F and F ₁ about point O is difficult to obtain, since the orientation of the rotation axes of F and F ₁ are different.

Figure 5.

Moment of force about a point in space force system: (a) The moment of forces F and F ₁ about point O and (b) the projection of vectors r _AO and F .

Take force F as the research object. The moment of force F can be expressed by the vector product, as shown in equation (11)

$${\mathit{M}}_O\left(\mathit{F}\right)={\mathit{r}}_{\mathrm{AO}}\times \mathit{F}$$

(11)

Figure 5(b) shows the unit vectors i , j and k along the direction of the coordinate axes OX, OY and OZ, respectively. The projections of vectors ${\mathit{r}}_{\mathrm{AO}}$ and F on the corresponding coordinate axis are represented by (x, y, z) and (F_x, F_y, F_z). Then, F and ${\mathit{r}}_{\mathrm{AO}}$ can be expressed by equation (12)

$$\{\begin{array}{c}{\mathit{r}}_{\mathrm{AO}}=x\mathit{i}+y\mathit{j}+z\mathit{k}\hfill \\ \multicolumn{1}{c}{\mathit{F}=F_x\mathit{i}+F_y\mathit{j}+F_z\mathit{k}}\end{array}$$

(12)

Combining equations (11) and (12), M _O ( F ) can be expressed by equation (13)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{M}}_O\left(\mathit{F}\right)={\mathit{r}}_{\mathrm{AO}}\times \mathit{F}=\left|\begin{array}{ccc}\mathit{i}\hfill & \mathit{j}\hfill & \mathit{k}\hfill \\ \multicolumn{1}{c}{x}& y\hfill & z\hfill \\ \multicolumn{1}{c}{F_x}& F_y\hfill & F_z\hfill \end{array}\right|}\\ \multicolumn{1}{c}{=(y{F}_z-z{F}_y)\mathit{i}+(z{F}_x-x{F}_z)\mathit{j}+(x{F}_y-y{F}_x)\mathit{k}}\end{array}$$

(13)

Set n forces in the space-force system: F ⁽¹⁾,…, F ^(m),…, F ⁽ⁿ⁾. The moment of force F^(m) about point O can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{M}}_O\left({\mathit{F}}^{(m)}\right)=}& (y^{(m)}{F}_z^{(m)}-z^{(m)}{F_y}^{(m)})\mathit{i}\hfill \\ \multicolumn{1}{c}{}& +(z^{(m)}{F_x}^{(m)}-x^{(m)}{F_z}^{(m)})\mathit{j}\hfill \\ \multicolumn{1}{c}{}& +(x^{(m)}{F}_y^{(m)}-y^{(m)}{F}_x^{(m)})\mathit{k}\hfill \end{array}$$

(14)

where (x^(m), y^(m), z^(m)) and (F_x^(m), F_y^(m), F_z^(m)) are the projections of the mth ${\mathit{r}}_{\mathrm{AO}}$ and F , respectively.

Then, the resultant moment M _O of forces F ⁽¹⁾,…, F ^(m),…, F ⁽ⁿ⁾ about point O can be expressed as equation (15)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{M}}_O=}& \sum _{m=1}^n(y^{(m)}{F}_z^{(m)}-z^{(m)}{F_y}^{(m)})\mathit{i}\hfill \\ \multicolumn{1}{c}{}& +\sum _{m=1}^n(z^{(m)}{F_x}^{(m)}-x^{(m)}{F_z}^{(m)})\mathit{j}\hfill \\ \multicolumn{1}{c}{}& +\sum _{m=1}^n(x^{(m)}{F}_y^{(m)}-y^{(m)}{F}_x^{(m)})\mathit{k}\hfill \end{array}$$

(15)

Development of the bending moment model

For a PDC bit with N_c cutters, the force state F and position r _AO of the ith cutter can be expressed by equation (16)

$$\{\begin{array}{c}{\mathit{r}}_{\mathrm{AO}}^{(i)}=R_c^{(i)}\cos {\theta }_c^{(i)}\mathit{i}+R_c^{(i)}\sin {\theta }_c^{(i)}\mathit{j}+H_c^{(i)}\mathit{k}\hfill \\ {\mathit{F}}^{(i)}=(f_c^{(i)}\sin {\theta }_c^{(i)}-f_r^{(i)}\cos {\theta }_c^{(i)})\mathit{i}+(-f_c^{(i)}\cos {\theta }_c^{(i)}-f_r^{(i)}\sin {\theta }_c^{(i)})\mathit{j}+f_a^{(i)}\mathit{k}\hfill \end{array}$$

(16)

where H_c is the axial position of the centre point of the cutter and X⁽ⁱ⁾ represents the parameter X of the ith cutter (X can be r _AO , F , R_c, θ_c, etc.) as shown in Figure 6.

Figure 6.

Moment of a PDC bit.

According to equation (15), the resultant force moment $M_O$ about point O for all the cutters can be expressed by equation (17)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{M}}_O=}& \sum _{i=1}^{N_c}(y^{(i)}{F}_z^{(i)}-z^{(i)}{F_y}^{(i)})\mathit{i}+\sum _{i=1}^{N_c}(z^{(i)}{F_x}^{(i)}-x^{(i)}{F_z}^{(i)})\mathit{j}\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=1}^{N_c}(x^{(i)}{F}_y^{(i)}-y^{(i)}{F}_x^{(i)})\mathit{k}\hfill \\ \multicolumn{1}{c}{}& \{\begin{array}{c}{x}^{(i)}={R_c}^{(i)}\cos {{\theta }_c}^{(i)}\hfill \\ \multicolumn{1}{c}{y^{(i)}={R_c}^{(i)}\sin {{\theta }_c}^{(i)}}\\ \multicolumn{1}{c}{z^{(i)}={H_c}^{(i)}}\\ \multicolumn{1}{c}{{F_x}^{(i)}={f_c}^{(i)}\sin {{\theta }_c}^{(i)}-{f_r}^{(i)}\cos {{\theta }_c}^{(i)}}\\ \multicolumn{1}{c}{{F_y}^{(i)}=-{f_c}^{(i)}\cos {{\theta }_c}^{(i)}-{f_r}^{(i)}\sin {{\theta }_c}^{(i)}}\\ \multicolumn{1}{c}{{F_z}^{(i)}={f_a}^{(i)}}\\ \multicolumn{1}{c}{i=1,\dots ,N_c}\end{array}\hfill \end{array}$$

(17)

In equation (17), $\sum _{i=1}^{N_c}(y^{(i)}{F}_z^{(i)}-z^{(i)}{F_y}^{(i)})$ is the moment M_OX about axis OX, $\sum _{i=1}^{N_c}(z^{(i)}{F_x}^{(i)}-x^{(i)}{F_z}^{(i)})$ is the moment M_OY about axis OY, and $\sum _{i=1}^{N_c}(x^{(i)}{F}_y^{(i)}-y^{(i)}{F}_x^{(i)})$ is the moment M_OZ about axis OZ. M_OZ is equal to the TOB.

The bending moment M_b of the PDC bit is related to only M_OX and M_OY, since the direction of the bending moment B is perpendicular to M_OZ. M_b can be expressed by equation (18)

$$\begin{array}{c}{M}_b=\sqrt{M_{\mathrm{OX}}^2+M_{\mathrm{OY}}^2}\hfill \\ \{\begin{array}{c}{M}_{\mathrm{OX}}=\sum _{i=1}^{N_c}(y^{(i)}{F}_z^{(i)}-z^{(i)}{F_y}^{(i)})\hfill \\ \multicolumn{1}{c}{M_{\mathrm{OY}}=\sum _{i=1}^{N_c}(z^{(i)}{F_x}^{(i)}-x^{(i)}{F_z}^{(i)})}\end{array}\hfill \end{array}$$

(18)

There are unknown parameters f_c, f_r and f_a in the lateral force model (equation (9)) and bending moment model (equation (18)). According to equations (4)–(6), f_c, f_r and f_a are functions of the wear height h_w, cutting area A and cutting arc length L.

Given the cutter layout parameters and drilling parameters of a PDC bit, cutting area A and cutting arc length L can be obtained by the zero-point traversal (ZPT) method. ¹⁷ Wear height h_w can be expressed by equation (19), which can be solved by iterative computations based on abrasive wear theory ¹⁸

$${\mathrm{dh}}_{\mathrm{w}}^{(i)}=\frac{f_p^i{R}_c^{(i)}\cos {\alpha }^{(i)}{L}_w^{(1)}}{f_p^{(1)}{R}_c^{(i)}\cos {\alpha }^{(1)}{L}_w^{(i)}}{\mathrm{dh}}_w^{(1)}$$

(19)

where $L_{\mathrm{w}}^{(i)}$ is the cutting weight of the ith cutter.

Experimental verification

To verify the correctness of the new single cutter force model (equation (4)), a rock-breaking test for PDC bits A and B is carried out as shown in Figure 7. The two bits have the same structure, except for the blade angle and back rake angle. The diameter of each bit is 100 mm and the number of cutter is 6.

Figure 7.

Experimental PDC bits.

The WOB F_W and lateral force F_l of the bits during rock breaking are mainly tested. The drilling rate is set 30 r/min, and the rock is sandstone.

Test equipment and principle

The test equipment mainly includes micro-bit rock-breaking machine, resistance strain gage, signal line, DH5923N dynamic signal analysis instrument, pressure sensor, and so on, as shown in Figure 8.

Figure 8.

The main experimental equipment.

Figure 9 is the test schematic of bits forces during rock breaking. The test is completed on a micro-bit rock-breaking machine (No. 1 in Figure 9). The power source (No. 2) of the micro-bit rock-breaking machine provides a certain speed and WOB to drive the bit (No. 4) breaking rock (No. 5), and the rock is fixed by a holding device (No. 6). The test component (No. 3) attached to the bit converts the stress signal into an electrical signal. Then, the electrical signal is communicated to the DH5923N dynamic signal analysis instrument (No. 10) through the signal line (No. 8) and modulator (No. 9). The electrical signal is further processed and converted by DH5923N dynamic signal analysis instrument. At last, the personal computer (PC) (No. 12) picks up the signal by network cable (No. 11) from DH5923N dynamic signal analysis instrument. The forces conditions of PDC bits during drilling can be obtained by PC analysis.

Figure 9.

Test schematic of bits forces.

Test results

The lateral forces and WOB of bits A and B are obtained from the test. There are eight groups of data: lateral forces of unworn PDC bits A and B; lateral forces of wear PDC bits A and B; WOB of unworn PDC bits A and B and WOB of wear PDC bits A and B, as shown in Figure 10. The lateral forces of both bits increase after bits wear. However, the WOB of both bits almost keeps unchanged.

Figure 10.

The lateral forces and WOB of bits A and B: (a) lateral forces of bits A and B under different wear condition and (b) WOB of bits A and B under different wear condition.

The calculated force values of the bits A and B can be obtained based on the established force model (equations (4)–(9)). Compared with the mean values of the test data, the maximum calculation error of forces on PDC bit is 11.2%, and the mean error is 6.45% as shown in Table 1. Therefore, the bit force models established in this section have a small error and can be used as the basis for future research.

Table 1.

The error of the calculated value compared with measured value.

Different bit	FW (kg)			Fl (kg)
	Measured value	Calculated value	Calculation error (%)	Measured value	Calculated value	Calculation error (%)
New bit A	318	300.1	5.6	1.7	1.5	11.2
Wear bit A	365	345.5	5.3	18.4	17.2	6.7
New bit B	302	288.3	4.6	2.8	2.6	5.8
Wear bit B	314	300.1	4.4	9.5	8.8	8.0

Optimal global force-balanced PDC bit design considering wearing condition

Construction of the optimization model

Objective function

To make the PDC bit reach the global force-balanced state in all drilling moments, the optimal objective function is set to minimize the mean value of the ratio of the lateral force to the WOB and the ratio of the bending moment to the TOB throughout the drilling process, as shown in equation (20)

$$\mathrm{Min}:\{\begin{array}{c}\mathrm{Mean}\left(R_F\right)=\frac{1}{T}\sum _{t=1}^T\frac{F_l\left(t\right)}{F_w\left(t\right)}\hfill \\ \multicolumn{1}{c}{\mathrm{Mean}\left(R_M\right)=\frac{1}{T}\sum _{t=1}^T\frac{M_b\left(t\right)}{M_t\left(t\right)}}\end{array}$$

(20)

where R_F is the ratio of the lateral force F_l to the WOB F_w; R_M is the ratio of the bending moment M_b to the TOB M_t; t represents a moment in the drilling process; and T is the total drilling time.

In equation (20), the wear condition of PDC cutters at each drilling time t is updated by numerical iteration of the ZPT method; then, the PDC forces and moments can be obtained according to equation (9), equation (18) and the wear condition at each drilling time t.

Design variables

When the formation properties and drilling parameters remain unchanged, the lateral forces and bending moments are determined by the position of the cutting teeth on the bit. There are six position parameters, namely, the circumferential angle θ_c, radial position R_c and axial position H_c for determining the position of the cutter centre, and the back rake angle α, side rake angle β and normal angle γ for determining the direction of the cutter surface (Figure 11). These parameters (R_c, H_c, θ_c, α, β and γ) are also known as cutter layout parameters.

Figure 11.

The cutter layout parameters.

The parameters (R_c, H_c and γ) at the cutter centres are difficult to adjust because they are determined by many factors, such as cutter number, cutter diameters, bit profile and cover coefficient at bottom hole. In addition, the back rake angle α is a fixed value for a particular formation. At this point, the parameters (R_c, H_c, γ and α) of all the cutters are not set as design variables in this article.

The effect of side rake angle β is to assist in chip removal. The side rake angle β can be adjusted on a large scale when the hydraulic cleaning effect is good. In addition, the circumferential angle θ_c has been shown to be closely related to the lateral force of the PDC bit. Therefore, the parameters (β and θ_c) are design variables.

The optimal model

Combined with the analysis results of the objective function and design variables, the optimal model is constructed in equation (21)

$$\begin{array}{c}\mathrm{Min}:\{\begin{array}{c}\mathrm{Mean}\left(R_F\right)=\frac{1}{T}\sum _{t=1}^T\frac{F_l\left(t\right)}{F_w\left(t\right)}\hfill \\ \multicolumn{1}{c}{\mathrm{Mean}\left(R_M\right)=\frac{1}{T}\sum _{t=1}^T\frac{M_b\left(t\right)}{M_t\left(t\right)}}\end{array}\hfill \\ \mathrm{subject}\mathrm{to}:\{\begin{array}{c}{\beta }^{\mathrm{lb}}\le {\beta }^{\left(i\right)}\le {\beta }^{\mathrm{ub}}\hfill \\ \multicolumn{1}{c}{{\theta }_c^{\mathrm{lb}}\le {{\theta }_c}^{\left(i\right)}\le {\theta }_c^{\mathrm{ub}}}\\ \multicolumn{1}{c}{i=1,2,3,\dots ,N_c}\end{array}\hfill \end{array}$$

(21)

where ${\beta }^{\mathrm{lb}},{\theta }_c^{\mathrm{lb}},{\beta }^{\mathrm{ub}}\mathrm{and}{\theta }_c^{\mathrm{ub}}$ are the lower and upper boundaries of parameters $\beta \mathrm{and}{\theta }_c$ .

Optimization approach

The optimal model for a global force-balanced PDC bit has two distinct features. First, it is a multi-objective optimization (MOP) because there are two objective functions. Second, the optimization model has many independent variables. For a PDC bit with 29 cutters, the model has 58 design variables. In terms of these two features, the MOP algorithm and surrogate model technique are applied to solve the optimization model.

MOP algorithm

MOP is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. In the feasible region, a solution is a Pareto optimal solution if none of the solutions can dominate it. All the Pareto optimal solutions together are called the Pareto optimal solution set. The objective function values corresponding to the Pareto optimal solution set are known as the Pareto front or Pareto frontier.

There are two kinds of methods used to solve the MOP problem: the normalized method and the non-normalized method. The normalized optimal methods usually convert multiple objectives into a single objective and include the weighting, goal programming and ε-constraint methods. The normalized method is mature because it inherits other algorithms to solve the single-objective optimization problem. However, the normalized method has some drawbacks. (1) It is difficult to compare the objectives when the dimension is different. (2) It cannot address the optimal problem whose Pareto frontier is non-convex. (3) The solving efficiency is low.

The non-normalized method is a direct optimization technology based on Pareto theory. It does not require converting multiple objectives into a single objective. At present, the non-normalized algorithms are mainly based on improved intelligent algorithms such as GA, PSO, and neural network. A GA, named the elitist non-dominated sorting genetic algorithm (NSGA-II), was proposed by Kalyanmoy Deb in 2000. It is less complicated than the non-dominated sorting genetic algorithm (NSGA) because a fast non-dominated sorting approach, crowding distance and crowded comparison operator are applied in the calculating process. The total complexity of NSGA-II is mN², where m is the number of objectives, and N is the size of the population. With properties of a fast sorting procedure, an elitist strategy and a parameterless approach, NSGA-II is one of the best MOP algorithms thus far. In this article, NSGA-II is used to find the optimal solution with the optimal model (equation (21)).

Surrogate model technology

For the optimal model (equation (21)), it takes hundreds or thousands of calculations to find the optimal solution based on NSGA-II. Each calculation requires approximately 2.5 h because the model needs to solve a series of calculations, such as the single cutter force, lateral force and bending moment of the new PDC bit, and force status of the PDC bit with different degrees of wear. Therefore, it is almost impossible to solve the optimal model using only NSGA-II because the calculation time cost is too high.

The surrogate model is an approximate model whose computation requirement is small, but its calculation result is similar to that of the original model. The computing efficiency of the surrogate model is usually hundreds times higher than that of the original model. Construction of the surrogate model requires three steps: (1) generate initial design points by a design of experiment (DOE) method, such as orthogonal array (OA), Latin hypercube (LH) and optimal Latin hypercube (OLH). ¹⁹ (2) Obtain the function value of the initial design points by experiment or emulation. (3) Construct the surrogate model based on the initial design points and the function value. There are many methods with which to construct a surrogate model; for example, the response surface model, Kriging model, radial basis functions (RBFs) and back propagation (BP) neural network approaches.

In this article, the Kriging surrogate model is used to approximate the optimization model because it has strong applicability to linear and highly nonlinear functions.

The Kriging surrogate model can be expressed as a regression model $f(x{)}^T\beta$ and a random function (stochastic process) $z(x)$ as equation (22). ²⁰ Set the order of $f(x{)}^T\beta$ to 0, because a low-order function has higher smoothness

$$y\left(x\right)=f{\left(x\right)}^T\beta +z\left(x\right)$$

(22)

The Kriging predictor can be set to

$$\hat{y}\left(x\right)=f{\left(x\right)}^T{\beta }^{*}+r{\left(x\right)}^T{\gamma }^{*}$$

(23)

where

$$\begin{array}{c}{\beta }^{*}={\left(F^T{R}^{-1}F\right)}^{-1}{F}^T{R}^{-1}Y\hfill \\ \gamma *=R^{-1}(Y-F{\beta }^{*})\hfill \end{array}$$

(24)

where f is the basis function for the regression model; β is a matrix of regression parameters; r represents a vector of correlation; F is the coefficient matrix of the design sites; R is the matrix of stochastic process correlations and Y is the response of the initial design points.

Then, the mean squared error (MSE) $\phi (x)$ of the predictor $\hat{y}(x)$ is

$$\begin{array}{cc}\multicolumn{1}{c}{\phi \left(x\right)}& =E\left[{(\hat{y}\left(x\right)-y\left(x\right))}^2\right]\hfill \\ \multicolumn{1}{c}{}& ={\sigma }^2(1+u^T{\left(F^T{R}^{-1}F\right)}^{-1}u-r^T{R}^{-1}r)\hfill \end{array}$$

(25)

where $u=F^T{R}^{-1}r-f$ and ${\sigma }^2$ can be obtained by a generalized least squares fit

$${\sigma }^2=\frac{1}{m}{(Y-F{\beta }^{*})}^T{R}^{-1}(Y-F{\beta }^{*})$$

(26)

where m represents the number of initial design points.

The unworn PDC bit B in Figure 7 is taken as an example to illustrate the establishment of Kriging surrogate model. The main parameters of the bit B are shown in Table 2. Two design variables are set. One is the back rake angle α, and all cutters are set at the same α. The other is circumferential angle θ_c of cutters 2, 4 and 6. Keep the remaining parameters of bit B unchanged.

Table 2.

The main parameters of PDC bit B.

Cutter number	Rc (mm)	Hc (mm)	θc (°)	α (°)	β (°)	Diameter (mm)
1	7.11	36.31	360	15	0	13.44
2	14.20	38.89	188.4	18.5	0
3	22.52	41.91	360	15.3	0
4	28.57	44.09	188.4	19.4	0
5	38.51	45.31	360	18	0
6	45.68	43.75	188.4	15.1	0

Set the variation range of parameters θ_c and α to (170–200) and (10–25), respectively. Then, OLH is used to generate the initial design points, which are uniformly distributed in the variables interval, as shown by the red points in Figure 12(a). Calculate the corresponding lateral force F_l of each initial design points based on the constructed force models (equations (4)–(10)). Then, the Kriging surrogate model of lateral force (Figure 12(b)) can be built using equations (22)–(26) according to the mapping relationship from (θ_c, α) to F_l.

Figure 12.

The construction of the surrogate model.

The other nine design points (the black points in Figure 12(a)) are generated randomly to verify the accuracy of the Kriging model. Calculate the lateral forces F_l of nine design points using equations (4)–(10). According to the parameters (θ_c, α, Fl_l) of nine points, plot the spatial points, as shown by the black dot in Figure 12(b). All the black dots are very close to the Kriging surface model. This indicates that the Kriging model is constructed correctly with a high precision.

The aforementionedis the construction process of a simple Kriging model only with two variables. When the number of design variables increases, the construction process of the surrogate model will become more complicated, which will be elaborated in the following section ‘Optimization case’.

MOP process for a global force-balanced PDC bit

In the optimal design for a global force-balanced PDC bit, the ZPT method is employed to calculate the cutting area A and cutting arc length L of each cutter. Combined with the single cutter force model equation (4), WOB and TOB model equation (7), lateral force model equation (9), bending moment model equation (18) and cutter wear model equation (19), R_F and R_M can be calculated. The Kriging surrogate model is used to build the function relationship between the ratio (R_F, R_M) and the design variables (β, θ_c). NSGA-II is used to find the Pareto optimal solutions and Pareto frontier. The MOP process for a global force-balanced PDC bit is shown in Figure 13.

Figure 13.

The MOP process for a global force-balanced PDC bit.

The first step (block 1) is to construct the optimal model which includes determination of the objective function, design variables and constraint condition. Then, generate a design variable set S₀ (block 2). The distribution of samples in set S₀ directly affects the construction and accuracy of the Kriging surrogate model. Therefore, the samples are often generated by the DOE.^21–24 In this article, the optimal Latin hypercube design method (Opt LHD), which is more uniform and has higher flexibility, is used to generate the samples of set S₀ because the MOP problem of the global force-balanced PDC bit has many design variables.

In the following steps, R_F and R_M of each sample of set S₀ are obtained by the ZPT algorithm and bit force models (block 3). Then, the Kriging models M_rf and M_rm are created. M_rf represents the function relationship between S₀ and R_F, and M_rm represents the relationship between S₀ and R_M (block 4). The Pareto optimal solution set S₁ and Pareto frontier Y_f can be calculated by the NSGA-II algorithm based on the models M_rf and M_rm (block 5).

To verify the accuracy of the models M_rf and M_rm, the ZPT algorithm and bit force models are used to obtain R′ _F and R′ _M of the Pareto optimal solution S₁ and the exact Pareto frontier Y′_f (block 6). To determine whether the accuracy of the models M_rf and M_rm meet the requirement (block 7); the judging criteria can be given by

$$E_{or}\le \epsilon$$

(27)

where

$$\{\begin{array}{c}{E}_{\mathrm{or}}=\sum _{i=1}^n\sqrt{G_{i1}^2+G_{i2}^2}\hfill \\ \multicolumn{1}{c}{G=Y_f-{Y\prime }_f}\end{array}$$

(28)

where ε represents the upper limit of error of the models M_rf and M_rm; G is an n × 2 matrix; n is the number of Pareto optimal solution set S₁; and G_ij represents the element in the ith row and the jth column of the matrix G.

If E_or is not smaller than ε, insert the Pareto optimal solution set S₁ into set S₀, and add R′_F and R′_M to R_F and R_M, respectively (block 8). Then, repeat the construction of the Kriging models and optimal process until E_or is smaller than ε. Finally, export the Pareto optimal solution set S₁ and its functional values R′ _F and R′ _M (block 9).

Optimization case

Cutter layout parameters for a PDC bit

The cutter layout structure of the PDC bit in Figure 14 is discussed in this section. The PDC bit has 6 spiral blades and 29 cutters. The diameters of the PDC bit and cutter are 215.9 and 19 mm, respectively. The recommended revolutions per minute (RPM) is 120 r/min, and the rate of penetration (ROP) is 28.8 m/h. The design parameters of the cutter layout structure of the PDC bit are shown in Table 3.

Figure 14.

Cutter layout structure of the PDC bit: (a) front view of the cutter layout structure and (b) upwards view of the cutter layout structure.

Table 3.

The design parameters of the cutter layout structure.

Number	θc (°)	β (°)	Number	θc (°)	β (°)
1	−2.5	15	16	159.44	15
2	174.75	15	17	218.94	15
3	−8	15	18	278.51	15
4	169	15	19	−21.89	15
5	228.17	15	20	37.71	15
6	287.23	15	21	97.32	15
7	−13.7	15	22	156.95	15
8	45.38	15	23	216.58	15
9	104.49	15	24	276.23	15
10	163.64	15	25	−24.12	15
11	222.82	15	26	35.54	15
12	282.04	15	27	95.2	15
13	−18.7	15	28	154.88	15
14	40.62	15	29	214.56	15
15	100	15

Combined with drilling parameters (RPM and ROP) and the ZPT method, the cutting parameters (cutting arc length, cutting area, cutting volume etc.) can be solved. The result shows that the cutter layout of the PDC bit is in line with the law of equal cutting volume and has good cutting performance when the PDC bit is unworn. However, with continuous drilling, F_l, M_b, R_F and R_M all increase with the wear height of the cutters; this is detrimental to keeping the drilling stable and efficient and avoiding early failure of the PDC bit. This PDC bit is used as the research object in the next section.

Creation of the optimal model

There are 58 design variables for the PDC bit with 29 cutters because each cutter has two variables (β and θ_c). The number of design variables can be reduced according to the cutter layout characteristics of the PDC bit.

Number the spiral blades of the PDC bit from 1 to 6 counterclockwise; the blade with the first cutter is No. 1, as shown in Figure 15(a).

Figure 15.

The blades of the PDC bit: (a) blade number and (b) structure parameters for the blade.

The spiral of the blade is an Archimedes spiral. It is expressed as

$$R_c=\frac{R_p}{{\theta }_{\mathrm{span}}}({\theta }_{\mathrm{star}}-{\theta }_c)$$

(29)

where R_p is the radius of the PDC bit; θ_span is the blade span angle; θ_star is the starting angle of the blade (Figure 15(b)); (R_c, θ_c) represents a point on the blade spiral in terms of the radial position and circumferential angle of the cutter.

Then, the design variable θ_c of the cutters can be replaced by the variables θ_span and θ_star of the blades because R_p and R_c are constants, as shown in equation (30)

$${\theta }_c={\theta }_{\mathrm{star}}-\frac{R_c}{R_p}{\theta }_{\mathrm{span}}$$

(30)

Fix θ_star of blade 1 (the red curve in Figure 15(a)), take θ_star of the rest of the blades as variables, and make θ_span of all the blades equal. Then, the number of design variables related to θ_c is reduced from 29 to 6, that is, θ_star2, θ_star3, θ_star4, θ_star5, θ_star6 and θ_span (Table 4). θ_star and θ_span fluctuate around their original values, and the fluctuation amplitude is 15.

Table 4.

Parameters of blades.

Blade no.	Cutter no.	Starting angle (°)		Span angle (°)
		Design variables	Original value	Design variables	Original value
1	1,3,7,13,19,25		0	θ span	25.44
2	8,14,20,26	θ star2	60
3	9,15,21,27	θ star3	120
4	2,4,10,16,22,28	θ star4	180
5	5,11,17,23,29	θ star5	240
6	6,12,18,24	θ star6	300

To further reduce the variables of the optimal model, set β of the 1st and 29th cutters as design variables. The rest of the cutter β values are obtained by the linear interpolation of the two points $H_d^{(i)}\le 3.5$ and $(R_c^{(29)},{\beta }^{(29)})$ . The variation range of β is set to 0°–20°.

Finally, the optimization model can be expressed as

$$\begin{array}{c}\mathrm{Minimize}:\{\begin{array}{c}\mathrm{Mean}\left(R_F\right)=\frac{1}{T}\sum _{t=1}^T\frac{F_l\left(t\right)}{F_w\left(t\right)}\hfill \\ \multicolumn{1}{c}{\mathrm{Mean}\left(R_M\right)=\frac{1}{T}\sum _{t=1}^T\frac{M_b\left(t\right)}{M_t\left(t\right)}}\end{array}\hfill \\ \mathrm{subject}\mathrm{to}:\{\begin{array}{c}45\le {\theta }_{\mathrm{star}2}\le 75\hfill \\ \multicolumn{1}{c}{105\le {\theta }_{\mathrm{star}3}\le 135}\\ \multicolumn{1}{c}{165\le {\theta }_{\mathrm{star}4}\le 195}\\ \multicolumn{1}{c}{225\le {\theta }_{\mathrm{star}5}\le 255}\\ \multicolumn{1}{c}{285\le {\theta }_{\mathrm{star}6}\le 315}\\ \multicolumn{1}{c}{10\le {\theta }_{\mathrm{span}}\le 40}\\ \multicolumn{1}{c}{0\le {\beta }^{(1)}\le 20}\\ \multicolumn{1}{c}{0\le {\beta }^{(29)}\le 20}\\ \multicolumn{1}{c}{H_d^{(i)}\le 3.5}\\ \multicolumn{1}{c}{i=1,\dots ,29}\end{array}\hfill \end{array}$$

(31)

$H_d^{(i)}\le 3.5$ means that the PDC bit will be considered to have failed when the wear height of any bit cutter reaches 3.5 mm.

Generation of the design variables set

Opt LHD is used to create 100 initial design samples with eight design variables. The initial design points evenly fill the entire design space since the scatter matrix of any two variables is uniformly distributed as shown in Figure 16.

Figure 16.

The distribution for the initial design samples.

Analysis and optimal results

The optimization model (equation (31)) can be solved according to the optimal approach described in Figure 13. ε, which represents the accuracy requirement of the Kriging models (block 7 in Figure 13), is set to 0.05. The population size of NSGA-II is set to 16, and the number of generations is set to 40. The process of finding Pareto solutions is shown in Figure 17. Abscissas and ordinates are the values of R_M and R_F, respectively.

1. During the evolving process of the first 10 generations, the distribution of the objective values (R_M, R_F) is scattered, and it is not convergent, as shown by the green points in Figure 17(a).

2. In the next 10 generations, the objective values, which are marked with the blue points in Figure 17(b), gradually converge in the lower left corner of the coordinate system. However, there is an obvious discontinuous zone between the blue points, which indicates that further optimization is needed to find a better Pareto solution set.

3. The distribution of the solutions from 21st generation to the 30th generation (the red points in Figure 17(c)) shows that many solutions (the red points) are improved compared with the previous 20 generations. Most of the red points concentrate on the edge of the points set. It has noticeable convergence and forms a Pareto frontier.

4. The solutions (the cyan points in Figure 17(d)) from 31st generation to 40th generation are still basically at the Pareto frontier generated by the 21–30 generations. The only difference is the case that the points at the Pareto frontier are denser. There is no further Pareto improvement that can be made without damaging the constraints and the optimization model, in which case, the optimization process can be terminated and the Pareto optimality is reached.

5. Exact the Pareto optimal set according to the Pareto theory. Forty-nine Pareto optimal solutions are obtained as shown in the red points in Figure 17(e).

6. Take nine points from the 49 Pareto optimal solutions. Calculate the exact values R_F and R_M of the nine Pareto optimal solutions according to ZPT method, the single cutter force model equation (4), WOB and TOB model equation (7), lateral force model equation (9), bending moment model equation (18) and cutter wear model equation (19). There are nine calculating results, also called non-dominated solutions (NDSs), represented by pentagrams in Figure 17(f). It is noted that the exact values R_F and R_M have a slight deviation from the Pareto optimal solutions, which is due to the error of the Kriging model.

Figure 17.

The optimization process: (a) the solutions of the first 10 generations, (b) the solution distribution of the first 20 generations, (c) the solutions distribution of the first 30 generations, (d) the solutions distribution of the 40 generations, (e) the Pareto optimal set and (f) the non-dominated solutions.

For a global force-balanced PDC bit, R_F should be less than 5% and R_M less than 15%. There are five NDS (the pentagrams in the dotted rectangle in Figure 17(f)) whose R_F is smaller than 5% and R_M is smaller than 15%. The corresponding cutter layout parameters are displayed in Table 5. In contrast to traditional PDC bits, the optimal PDC bit has a side rake angle β that decreases from inside to outside the bit. The span angle of the blade θ_span increases by 24.69% compared with the original PDC bit shown in Figure 14.

Table 5.

Cutter layout parameters of the five Pareto frontier.

Non-dominated solutions			Cutter layout parameters
No.	RF (%)	RM (%)	β (1)	β(29)	θ span	θ star2	θ star3	θ star4	θ star5	θ star6
1	3.32	13.56	14.2	6.0	31.5	72.5	127.2	191.9	250.3	306.4
2	3.52	12.72	14.2	5.9	31.8	72.5	127.2	193.7	248.5	303.4
3	3.65	11.56	14.2	6.1	32.3	73.0	129.9	192.5	250.3	300.0
4	4.10	10.43	14.2	6.0	31.5	72.5	127.8	191.9	250.3	292.2
5	4.43	9.94	14.4	6.0	31.5	72.4	127.2	193.8	250.3	290.5

Figure 18 shows the layout structure of the blades of NDSs 1, 3 and 5 (green curves). Compared with the original blades (blue curves), the optimal starting angles of blades 2, 3, 4 and 5 are all greater, while the optimal starting angle of blade 6 is smaller.

Figure 18.

Layout structure of blades: (a) blades layout structure of Pareto frontier 1, (b) blades layout structure of Pareto frontier 3 and (c) blades layout structure of Pareto frontier 5.

As shown in Figure 19, the PDC bit of NDS 1 (green curve) has a maximal R_F when the PDC bit is not worn (drilling time is equal to zero). With the gradual wear of the PDC bit, R_F decreases slightly. When the drilling time is greater than 0.63, the PDC bit of NDS 1 has the smallest R_F. Compared with the original PDC bit described in Figure 14, the mean R_F of the PDC bit of NDS 1 is reduced by 10.99%, and the maximum R_F is smaller than 5%.

Figure 19.

RF of different PDC bits.

The mean R_F of the PDC bit of NDS 3 (blue curve) is similar to that of the original PDC bit (red curve). The maximum R_F is greater than 5%.

Although the PDC bit of NDS 5 (brown curve) has the smallest R_F when PDC bit is not worn, R_F continues to increase with the wear of the PDC bit. The maximum R_F is greater than 10%.

Therefore, regarding the balance of lateral force, the cutter layout structure of NDS 1 fully meets the design requirement. The cutter layout structure of NDS 3 basically meets the requirements.

As shown in Figure 20, the PDC bits of NDSs 1, 3 and 5 all have smaller R_M compared with the original PDC bit, and the R_M values are smaller than 15%. Therefore, the cutter layout structure of NDSs 1, 3 and 5 are all in line with the bending moment balanced design.

Figure 20.

RM of different PDC bits.

Combined with the aforementioned analysis, the cutter layout structure of NDS 1 is selected as the final optimal result. Compared with the original PDC bit, R_F of the optimal PDC bit is reduced by 10.99% and R_M is reduced by 30.43%. The maximal R_F is lower than 5%, and the maximal R_M is less than 15%. The optimal results satisfy the global force-balanced PDC bit design.

A set of cutter layout structures corresponding to the minimum R_F and R_M can be obtained by the optimization approach proposed in this article. A PDC bit with this cutter layout structure is capable of maintaining a balance of the lateral force and bending moment during the drilling process even with continuous wear of the cutting teeth, which means that the lateral vibration degree and the probability of whirl of the PDC bit is reduced. Furthermore, trip times can be reduced, and drilling efficiency can be improved.

Discussion

To date, all the research of cutter layout design for force-balanced PDC bits has focused on the new PDC bit, without considering cutter wear. In fact, a bit wearing is the normal state during drilling, and lateral force and bending moment are no longer balanced after bit wears. It is not reasonable to design a ‘lateral force-balanced’ bit or ‘global force-balanced’ bit without considering wearing conditions.

To solve the aforementioned problems, we have performed a series of studies. An algorithm ZPT ¹⁷ was created to solve the cutting parameters of the worn teeth, and ZPT makes it possible to calculate force status of wear PDC bit. Then, mechanical properties study ²⁵ of wear PDC bit was analysed based on ZPT method. Cutter layout optimal design of a PDC bit with wear was carried out ¹⁸ to reduce the lateral force.

In this article, the optimal design for cutter layout is further studied. A single 3D cutter force model is proposed, and it is more comprehensive and accurate because the parameters of side rake angle, wear degree, back rake angle, rock property, and so on are considered. The lateral force model is recreated based on the 3D single cutter force model. In addition, the bending moment model is constructed based on space-force theory, since there is no public report on the method of calculating the bending moment of PDC bit. The established single cutter force model, lateral force model and bending moment model make it possible to analyse the global force-balanced state of wear PDC bits. Then, cutter layout optimal design of a PDC bit to achieve global force-balanced state is carried out with NSGA-II and Kriging surrogate model. After optimization, the lateral force to WOB ratio is reduced by 10.99%, and the bending moment to TOB ratio is reduced by 30.43%. This result is a significant improvement in the force condition and stability of the PDC bit; ultimately, the whirl and tilt motion can be reduced, and the drilling efficiency can be improved. This study perfects the theory and approach to design a global force-balanced PDC bit by cutter layout optimization and reduces the whirl and tilt motion of PDC bit and further improves drilling efficiency.

Conclusion

A new optimization approach for global force-balanced PDC bits considering wear condition is proposed in this article; this approach can improve the drilling properties of the bit. The detailed conclusions are as follows:

1. The lateral force model and bending moment model are obtained by the synthesis of the forces and moments of all the cutters based on the new single cutter force model, which considers cutter layout parameters and wear condition. The use of the lateral force model and bending moment model allow the design of a global force-balanced PDC bit.

2. The optimal design model for a global force-balanced PDC bit is proposed. The side rake angle β and circumferential angle θ_c are set as the design variables. The optimal objective function is set to minimize the mean values of R_F and R_M for worn cutters at all drilling times. The optimal model is more similar to actual drilling requirements because it considers lateral force, bending moment and cutter wear.

3. The optimal approach is proposed to solve the optimal model. The Kriging surrogate model is used to build the functions between the ratios (R_F and R_M) and the design variables (β and θ_c). NSGA-II is used to find the Pareto optimal solutions and Pareto frontier. The optimal approach makes the optimization process more efficient.

4. An optimization case for a PDC bit with 29 cutters is presented. After optimization, R_F is reduced by 10.99% and R_M is reduced by 30.43%. The PDC bit is capable of maintaining the balance of lateral force and bending moment with wear. The lateral vibration degree and the probability of whirl of the PDC bit can be reduced, and the drilling efficiency can be improved.