Pattern and Analysis of Corrugation-Sand Retaining Seals for Tooth Wheel Drill Bits

Abstract

The existing seal is not designed with a special sand retaining structure, and it is difficult to discharge the mud particles after entering, resulting in serious wear of the seal and premature failure. Therefore, a special-shaped sealing structure for sand blocking of roller cone bits is proposed to solve the problem of sealing wear. Starting from the structural principle, the structural parameters of the special-shaped seal of the sand-retaining flap are designed based on the O-ring, and the sealing principle is analyzed, the formula of grease leakage is derived, and the simulation and optimization analysis are completed with the aid of APDL. This proved the feasibility of the design. The influence law of structural parameters on contact pressure and leakage is obtained, and the formula of leakage is verified; The influence of structural parameters on the contact gap of the sand flap is obtained. Under the condition of optimal combination of structural parameters, the grease leakage is 43 mg/h, a decrease of 85%. Theoretically, it takes 580 h to exhaust the grease, and the contact gap of the sand flap is 0.00019479 mm, a decrease of 95%, which can greatly reduce the wear of the main seal. This study opens up a new idea for seal design.

1. Introduction

The cone bit is a commonly used rock breaking tool in petroleum drilling engineering, and its working life affects drilling efficiency and cost. Field use data shows that the main reason for the failure of the cone bit is the early failure of the sealing system; therefore, the seal is the key to determining the life of the high-speed cone bit, and the entry of abrasive media is the main reason for the failure of the seal. Improving the wear resistance and working life of the sealing system has become an important issue that needs to be solved urgently in current oil drilling projects.

At present, the most common applications for roller cone bit bearing seals are radial seals and metal seals [1,2]. For the radial seal ring, the structural design, improvement, and failure research are mainly carried out through theory, simulation, and experimental research on the sealing performance, and the purpose is to improve the life and reliability of the seal [3,4]. O-ring is the most widely used. Based on O-ring, researchers have improved the structure [5,6], the sealing performance [7], and life to a certain extent [8,9]. The failure mechanism, sealing performance, and wear capacity of the sealing ring were studied by means of theoretical calculation and other methods [10,11], and the corresponding improvement methods were proposed [12,13]. Different sealing rings have different sealing characteristics, but the problem of high temperature failure cannot be solved [14]. Therefore, bimetallic seals are designed, and its heat resistance is better than rubber seal. The researchers improve its sealing performance through structural optimization and material improvement [15,16]. However, because it takes up a large space, it will have a certain impact on the strength of the bearing. In response to this problem, the researchers designed a single metal seal [17,18]. The single metal seal has a simple structure, small volume, and few components [19,20]. Therefore, many scholars use theory, simulation, and experimental research methods [21,22] to study the effects of factors on the metal sealing performance, such as high rotation speed, rubber compression, contact pressure, structural form, structural parameters, materials, grease, assembly form, etc. [23,24], then optimize and improve the structure to improve sealing performance, service life, and drilling efficiency [25].

Although the existing sealing research results are more significant, the design concept of dynamic sand retaining structure has never been applied in the field of cone bit sealing, and the sealing mechanism of sand retaining structure in other fields is completely different from this paper. Therefore, the sand retaining structure proposed in this paper is distinctively innovative. The traditional seal has no sand-retaining structure specially designed. When mud particles enter the gap and cannot be discharged, they will cause wear and premature failure. Therefore, it is necessary to make breakthroughs and changes in the seal design concept and type, and increase the sand retaining structure in order to effectively solve the wear problem. This paper proposes a corrugated special-shaped sand retaining sealing structure for roller cone bits based on the idea of “sand retaining”, as shown in Figure 1a. The main functional structure of the new type of sealing ring is a corrugated main seal and a sand retaining flap auxiliary seal. The two parts are an integral structure, which can save space and ensure the strength and safety of the bearing. The main seal is responsible for preventing the mud medium from entering the bearing system. The cross-section of the contact with the bearing is corrugated in the circumferential direction, reducing the contact area, and can introduce grease into the seal gap to form a lubricating oil film, reducing wear and heat generation (as shown in Figure 1b). The tip of the sand retaining flap is in contact with the bearing, and the angle is acute. When the sealing ring is subjected to radial load, the sand retaining flap is deformed and the angle becomes smaller. To prevent mud particles from entering the sealed cavity, the main seal must be protected and the impact of wear must be reduced. In addition, the tip of the sand retaining flap has a wedge-shaped structure. When the sealing ring and the bearing rotate relative to each other, the grease between the sand retaining flap and the bearing can be thrown into the sealing cavity, filling the sealing cavity, and performing a lubrication effect. The contact stress is expected to be much smaller than the contact stress of the traditional O-ring seal, which is also conducive to reducing the wear of the seal ring and prolonging its life.

In this paper, the theory and simulation methods are used to design and study the ripple sand retaining performance of the cone bit, and the influence of structural parameters on the sealing performance is obtained. The parameters are optimized through the control variable method to obtain the parameter combination with the best sealing performance.

2. New Structure Sealing Theory and Model

2.1. Theoretical Basis

When analyzing the leakage problem of the sealing ring, the inverse problem theory of hydrodynamic lubrication and the P. W. Wernecke theory [26] are often used as the research basis. P. W. Wernecke theory avoids the use of complex hydrodynamic calculations. Considering that the internal friction generated by pressure and fluid viscosity is far greater than the influence of all other factors, it is a greatly simplified theoretical method compared to the inverse problem of hydrodynamic lubrication. Through analysis, the corrugated main seal structure involved in this paper can use this theory as the theoretical basis. P. W. Wernecke theory is as follows.

• Velocity distribution.

The velocity distribution in a tiny gap can be expressed by the following formula:

$$x=\frac{vy}{h}+\frac{\partial p}{\partial x}\cdot \frac{h^2}{2\eta }\left[{\left(\frac{y}{h}\right)}^2-\frac{y}{h}\right]$$

(1)

where $v$ is the moving speed, $h$ is the gap height, $\eta$ is the dynamic viscosity, and $\frac{\partial p}{\partial x}$ is the pressure gradient.

As shown in Figure 2, there is a pressure difference $\mathsf{\Delta }P$ inside and outside the seal cavity, $P_1$ represents the pressure in the seal cavity, $P_2$ represents the pressure outside the seal cavity, when the seal ring does not slide, the flow speed of the lubricating medium in the seal gap is shown in Figure 2a, the flow velocity of the lubricating medium near the wall is close to zero due to its own viscosity and wall friction, the flow velocity of the area away from the wall is determined by the pressure difference and the internal friction of the fluid. When there is no pressure difference between the inside and outside of the seal cavity, and the seal ring is sliding, the flow speed of the lubricating medium is shown in Figure 2b. The lubricating medium close to the wall of the sliding seal ring moves with the seal ring due to the friction of the wall, and the velocity distribution process close to the wall decreases to zero with the viscosity gradient. When there is a pressure difference between the inside and outside of the seal cavity and the seal ring slides in the direction of $p_2$, the flow velocity of the lubricating medium is shown in Figure 2c, in this case, the flow situation of the fluid in the gap is the positive superposition of Figure 2a,b. When there is a pressure difference between the inside and outside of the seal cavity, and the seal ring slides in the direction of $p_1$, the flow velocity of the lubricating medium in the seal gap is shown in Figure 2d; in this case, the flow situation of the fluid in the gap is the reverse superposition of Figure 2a,b.

2.

Volume flow.

$$Q=\frac{1}{2}vh-\frac{1}{12}\cdot \frac{\partial P}{\partial x}\cdot \frac{h^3}{\eta }$$

(2)

where the first term is the flow velocity generated by the wall motion, and the second term is the influence of the pressure difference. It can be seen that the sealing gap $h$ has a great influence on the leakage.

3.

Fluid friction.

The fluid shear stress is:

$$\tau =\frac{v\eta }{h}+\frac{\partial p}{\partial x}y-\frac{\partial p}{\partial x}\frac{h}{2}$$

(3)

The sealing friction is:

$$F=\pi dL\frac{v\eta }{h}-\pi dL\frac{\partial p}{\partial x}\frac{h}{2}$$

(4)

where d is the shaft diameter and L is the contact length.

When the friction generated by the fluid flow dominates:

$$F=\pi dL\frac{v\eta }{h}$$

(5)

Oil film thickness:

$$h_1=\pi dL\frac{V\eta }{F}$$

(6)

2.2. Sealing Model

In Figure 1b, the a process represents the lubricating medium leakage process of the corrugated seal ring on the right half when it passes through the middle area, and b represents the process of the lubricating medium returning to the gap between the sealing ring and the shaft hole when the corrugated sealing ring of the left half passes through the middle area. Assume that the sealing ring in Figure 1b slides to the right at the speed of $v$, and $v_1$ and $v_2$ can be obtained by orthogonal decomposition of $v$, where $v_1$ is the speed perpendicular to the edge of the corrugated belt, which represents the axial sliding speed of the reciprocating shaft seal. Additionally, $v_2$ is tangent to the edge of the corrugated belt, representing the linear velocity of the rotary shaft seal rotating in the circumferential direction.

According to P. W. Wernecke theory, the leakage is calculated as follows:

$$q=\frac{1}{2}vh+\frac{h^3\left(P_1-P_2\right)}{12\eta L}$$

(7)

where $\frac{1}{2}vh$ is the grease leakage flow caused by the sliding of the seal ring relative to the shaft hole, and $\frac{h^3\left(P_1-P_2\right)}{12\eta L}$ is the influence of fluid pressure on the leakage.

When the pressure inside the cavity is greater than the external pressure, the second term is positive, and leakage will be caused in this case. Under normal circumstances, changing the pre-compression of the seal ring can change the leakage. The fundamental reason is that the gap height $h$ is changed.

Then, the leakage $q_a$ in a process is shown in Equation (8), where $\theta$ is the angle between the velocity component perpendicular to the corrugated surface and the moving speed of the corrugated surface:

$$q_a=\frac{1}{2}vh\cos \theta +\frac{h^3\left(P_1-P_2\right)}{12\eta L}$$

(8)

The process $b$ can be regarded as the height of the sealing gap $h$ becomes $k$ times of the original gap, and the retention coefficient $k$ represents the percentage of the total leakage that the lubricating medium enters the sealing gap again after entering the external mud environment. The leakage calculation formula is as follows; in this case, the angle $\theta$ is greater than $\frac{\pi }{2}$.

$$q_b=\frac{1}{2}khv\cos \theta -\frac{k^3{h}^3\left(P_1-P_2\right)}{12\eta L}$$

(9)

Integrate the leakage at all points on the edge of the corrugation to obtain the leakage. Assuming that the corrugation is in the shape of a cosine, the number of cosine periods is $N$, the length of the amplitude is $A$, and the diameter of the sealing oil film is $D$, then:

$$\begin{gathered} q={\displaystyle \sum }_{i=0}^{N-1}\left\{{{\displaystyle \int }}_{2i\pi }^{\left(2i+1\right)\pi }\left(\frac{h^3\left(P_1-P_2\right)}{12\eta L}+\frac{1}{2}hv\sin \left(\arctan \left(A\sin \left(\frac{2Nx}{D}\right)\right)\right)\right)dx \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_{\left(2i+1\right)\pi }^{2\left(i+1\right)\pi }\left(\frac{k^3{h}^3\left(P_1-P_2\right)}{12\eta L}+\frac{1}{2}khv\sin \left(\arctan \left(A\sin \left(\frac{2Nx}{D}\right)\right)\right)\right)dx\right\} \end{gathered}$$

(10)

Assuming that the leakage of each cosine period is consistent, the above equation is simplified to obtain:

$$q=\frac{1-k}{2}NAhv{{\displaystyle \int }}_0^{\pi }\frac{\sin \left(\frac{2Nx}{D}\right)}{\sqrt{A^2\sin \left(\frac{2Nx}{D}\right)+1}}dx+\frac{\left(1+k^3\right)h^3\pi D\left(P_1-P_2\right)}{12\eta L}$$

(11)

The above formula is the calculation formula for the leakage amount of the main seal. The factors that affect the leakage include the height of the sealing gap $h$, the axial length of the sealing gap $L$, the pressure difference between inside and outside the sealing cavity $\mathsf{\Delta }P=P_1-P_2$, the number of sine (or cosine) cycles $N$, the amplitude length $A$ of the corrugated shape, and the sealing oil film diameter $D$. The calculation formula for the height of the sealing gap $h$ is:

$$h=\sqrt{0.1602\frac{\eta v{l}^2}{W}}$$

(12)

where $\eta$ is the dynamic viscosity of lubricating medium $\left(\mathrm{N}\cdot \mathrm{s}/{\mathrm{m}}^2\right)$, $v$ is the relative sliding speed of the seal ring and the tooth claw bearing surface $\left(\mathrm{m}/\mathrm{s}\right)$, $W$ is radial load on the seal ring in the axial unit length $\left(\mathrm{N}/\mathrm{m}\right)$, and $l$ is the width of the sealing gap along the axis $\left(\mathrm{m}\right)$.

3. Structural Design

3.1. Overall Structure Plan

As shown in Figure 3, the overall structural plan includes integral internal installation, integral external installation, and main and auxiliary sealed separate installations. As shown in Figure 3a, the first solution has a compact structure, but, because the large shaft diameter of the bearing is slotted, the height of the large shaft diameter is reduced. The large shaft diameter is the main load-bearing part, and ensuring the shaft diameter height of the large bearing is an important factor to ensure the strength and safety of the bearing. Therefore, slotting in the shaft diameter will have a certain impact on the strength of the bearing. As shown in Figure 3c, the third option is to slot the bearing and increase the number of parts. As shown in Figure 3b, the second scheme has a compact structure, which avoids slotting on a large shaft diameter and can ensure the original structural strength of the drill. In addition, the seal ring is embedded in the cone, and when the cone rotates, it will drive the seal ring to rotate. The cone and the seal ring remain relatively static, which reduces friction. Therefore, select option two.

3.2. Key Structural Parameters and Dimensions

The research object is an 8 1/2 inch (215.9 mm) drill bit. The cross section of the corrugated main seal is arc-shaped, which is based on the improvement of the O-ring seal. Therefore, refer to the traditional O-ring compression rate selection rule, as shown in

Table 1. Selection range of compression rate and elongation rate of O-ring.

Seal Form	Sealing Medium	Stretching Rate/%	Compression Rate/%
Static seal	Hydraulic oil	1.03~1.04	15~25
	Air	<1.01	15~25
Reciprocating seal	Hydraulic oil	1.02	12~17
	Air	<1.01	12~17
Rotating sealed	Hydraulic oil	1	5~10

.

The new type of seal ring is a rotary seal with a compression rate of 5–10%. The initial diameter of the main seal arc is 5.7 mm, and the initial thickness of the seal base is 4 mm. The key structural parameters that affect the sealing performance of the special-shaped sealing ring are shown in Figure 4.

In order to meet the compression rate and elongation rate of the seal ring, the approximate value ranges of each structural size are set as shown in

Table 2. Values of structural parameters of the new sealing ring.

Structure	Size	Name	Initial Value	Value Ranges
Main seal	Internal diameter margin	C	0.15 mm	0.1–1 mm
Main seal	Diameter	D	5.7 mm	4.5–7 mm
Main seal	Highlighting matrix height	CE	1.5 mm	0.5–2.8 mm
Matrix	Thickness	T	4 mm	3–6 mm
Sand flap	Inner diameter margin	C2	0.15 mm	0.1–1 mm
Sand flap	Tip and bearing angle	F1	25°	20–30°
Sand flap	Tip angle	F2	35°	30–45°
Cone groove	Inner diameter interference	C3	2.5 mm	1.7–3.4 mm
Cone groove	Upper and lower margin	C4	0.1 mm	0–0.5 mm
Cone groove	Lower end reserved gap	C5	2 mm	2–4 mm

. The inner diameter of the cone groove directly determines the compression rate of the seal ring, and other structural parameters are determined on the basis of satisfying the basic structural characteristics of the sealing ring.

4. Simulation Research

4.1. Simulation Model

The scan path of the corrugated main seal is defined. The scan path equation in the Cartesian coordinate system is shown in Equation (13):

$$\{\begin{array}{c}x\left(t\right)=r\sin \left(t\right)\\ y\left(t\right)=A\sin \left(Nt\right),\\ z\left(t\right)=r\cos \left(t\right)\end{array}t\in \left[{\theta }_1,{\theta }_2\right]$$

(13)

In Equation (13), $t$ is the angle of rotation along the axis ($°$), $t_1=25°$ and $t_2=35°$, $r$ is the vertical distance between each point on the scan path and the Y axis of the rotation axis ($m$), $A$ is the amplitude of the up and down fluctuations of the scanning path, and $N$ is number of sine periods ($m$).

The established simulation model and main structural parameters are shown in Figure 5.

4.2. Material Settings

The seal ring material uses nitrile rubber, and the five-parameter Mooney-Rivlin [27] constitutive model is used to define the rubber material. The five parameters are: C10 = 7.2668 MPa, C01 = −4.587 MPa, C20 = 3.353 MPa, C11 = −2.187 MPa, C02 = 0.6986 MPa, Poisson’s ratio is 0.499, density $\rho$ = 1200 kg/m³ [28].

4.3. Contact Settings

The simulation objects in this paper are rubber and metal materials. Since the deformation of rubber materials is much greater than that of metals, and the main research object is rubber rings, metal materials can be regarded as rigid bodies, and rigid-flexible contact behavior can be selected. The contact type is set to frictional contact. Contact coordination chooses augmented Lagrangian method, and contact behavior chooses asymmetric contact, which is conducive to convergence and shortens the calculation time.

4.4. Load Application

Assembly process and fluid load application.

• Assembly process.

As shown in Figure 6a, when building geometric models, separate each model and keep a certain distance. During the assembly process, the groove on the inner hole of the cone and the bearing surface form a cavity, which makes the sealing ring compressed and produces sealing contact pressure. In the simulation setting, the bearing is fully constrained, and the tooth wheel presses the seal ring close to the bearing and produces compression. As the seal ring only takes part of it for simulation, the friction-free constraint is used for two sections, which is usually used to constrain the boundary of the symmetrical model in the 3D model.

2.

Fluid load application.

Take the well depth of 1000 m, the external mud pressure is about 11.76 MPa because the pressure difference between the inside and outside of the sealed cavity is 0.3~0.7 MPa [29], the safety factor is 2, and the internal pressure of the bearing is 13.16 MPa. Apply fluid load through APDL (ANSYS Parametric Design Language), the schematic diagram of the fluid pressure boundary is shown in Figure 6b.

Figure 6. Load application: (a) Precompression loading, and (b) Schematic diagram of fluid pressure boundary.

Figure 6. Load application: (a) Precompression loading, and (b) Schematic diagram of fluid pressure boundary.

4.5. Results and Analysis

The condition for the sealing ring to have reliable sealing performance is that the contact pressure is greater than the fluid pressure. When the contact pressure is less than the fluid pressure, the iterative process will be terminated by the software detecting that penetration has occurred. Therefore, if the iterative solution can be completed, it means that no penetration has occurred, and effective sealing can be ensured.

4.5.1. Main Seal

The contact pressure cloud diagram is shown in Figure 7a. The contact pressure at the main seal protrusion is the largest, the contact state is stable, the pressure gradually decreases to both sides; there is no fault, sudden change, and no leakage. The maximum pressure value is 18 MPa, greater than the fluid pressure, and has a sealing effect. As shown in Figure 7b, because the force of the rubber is not easy to analyze, the reaction force of the bearing support is used instead of the load on the sealing ring to analyze, and solve the contact area by command to get the seal gap width.

As shown in Figure 8, when the change value of the calculation result is less than the set allowable deviation value, the number of grid cells will no longer have a significant impact on the calculation result. When the number of grid cells reaches 6000, and the number of nodes reaches 27,000, the upper and lower deviation of the calculation results shall not exceed 5%, which meets the calculation requirements. At this time, the average size of grid cells is 0.375 mm.

• set,last! selects the final step of the computation iteration
• cmsel,s,pressure! Selects all nodes in the lower half of pressure by name
• cmsel,a,pressure2! Co−select all nodes in the lower half of pressure2 by name
• esln,s,1! Select the element attached to the node
• esel,r,type,,cid1! Reselect only contact elements of type “cid_1”
• etable,estat,cont,stat! Save the list of selected cells
• esel,s,etab,estat,3! select contact section
• etable,c_area,VOLU! Select the contact element area
• ssum! Add area
• get,t_area,ssum,0,item,c_area! Store the result in t_area
• my_area=t_area/4.84! Find the seal gap width and output the result to my_area

Calculation of leakage is carried out through the MATLAB platform. The following is the code of the leak calculation function:

function [s] = LLJS(L,W); %L is the sealing gap width M, W is the radial load N/M

L = L/1000; % conversion unit is M

P1 = 11 × 10⁶;% internal pressure

P2 = 10 × 10⁶;% external pressure

k = 0.5; % exposure loss proportion

A = 0.2 × 10⁻³; % amplitude M

v = 220 × 10⁻⁶; % kinematic viscosity M2/S

rho = 930;% density KG/M3

eta = v * rho; % dynamic viscosity N*S/M2

nRotation = 80; % rotation speed/min

D = 55 × 10⁻³; % diameter M

N = 36; % number of cycles

V = nRotation * pi() * D/60; % linear velocity M/S

W = W * N/D/pi();

h = sqrt(0.1602 * eta * V * L.^2/W);

f = @(x) sin(2 * N * x/D)./(A.^2 * (sin(2 * N * x/D)).^2 + 1).^0.5;

s = (1 − k)/2 * N * A * h * V * integral(@(x)f(x),0,pi());

g = (1 + k^3) * h^3 * D * pi() * (P1 − P2)/12/eta/L;

s = (s + g) * rho * 10 × 10⁶; % conversion unit is mg

s = s * 3600; % one hour leakage mg/h

end

When used, the result can be calculated by calling the LLJS (L,W) function in the command pane. For example, in the example shown above, the width L of the sealing gap in the axial direction is 2.14 mm, the radial load of the sealing ring W is 374.21 N, the leakage is calculated through the function, and the calculation result is 5.04 mg/h.

Figure 8. Grid independence verification: (a) Influence of mesh number on the result of maximum contact pressure, and (b) Influence of mesh number on maximum Mises stress results.

Figure 8. Grid independence verification: (a) Influence of mesh number on the result of maximum contact pressure, and (b) Influence of mesh number on maximum Mises stress results.

4.5.2. Sand Retaining Flap

The gap between the tip of the sand retaining flap and the bearing surface can reflect the rationality of the structural size of the sand retaining flap, ensuring that the sand retaining flap can fit well with the bearing surface during assembly and operation. In order to measure this value, it is necessary to extract the value of the contact gap at the tip. As shown in Figure 9, the gap determines the size and the degree of wear of particles that can enter the working area of the sand flap. The pressure in the middle is the smallest, increasing along both sides, the maximum contact gap is 0.0039343 mm, the minimum contact gap is 0.0039248 mm, the gap is evenly distributed, the spacing is very small, and it has good sand retaining performance.

5. Structural Optimization

Optimization is a necessary link in mechanical design. Optimizing the structure to the best shape and size is conducive to the operation and life extension of the equipment. It is far from enough to just design the structure and reach a usable state.

This paper gives the key structural parameters of the special-shaped seal ring. Under the condition that the overall structural shape of the special-shaped sealing ring is not changed significantly, the single variable method (control variable method) is used to carry out the key structural parameters of the special-shaped sealing ring. Optimization is conducive to improving the performance and service life of the special-shaped sealing ring.

5.1. Parametric Modeling

Using the 3D model that has been established before, the key structural parameters of the new type of sealing ring are numbered through adjustment and modification of constraints, as shown in

Table 3. Model parameter table.

Sketch	Name	Meaning	Call
MainSeal	C	Main seal inner diameter margin	ANS_C@MainSeal
MainSeal	D	Main seal diameter	ANS_D@MainSeal
SealBody	CE	Height of main seal protruding matrix	ANS_CE@SealBody
SealBody	T	Matrix thickness	ANS_Thickness@SealBody
SealBody	C2	Inner diameter margin of sand flap	ANS_C2@SealBody
SealBody	F1	Angle between tip and bearing	ANS_F1@SealBody
SealBody	F2	Angle of sand retaining flap tip	ANS_F2@SealBody
Cone	C3	Interference amount of groove inner diameter	ANS_C3@Cone
Cone	C4	Upper and lower margin of the groove	ANS_C4@Cone
Cone	C5	Extension length of groove bottom	ANS_C5@Cone

.

5.2. Optimization Parameters and Goals

Use the single-variable method (control-variable method) to number the dimensions and set them according to the following steps:

• Set the structural parameters to be optimized and mark each parameter as an input parameter according to Table 1.
• Set the optimization target, the optimization target is the leakage $q$ of the main seal, the contact gap $i$ of the sand flap tip, and the contact pressure $P$ of the main seal.
• Set the value range of structural parameters to be optimized, and define the value range of each parameter according to Table 2.

5.3. Optimization Results

5.3.1. Comparison of the Influence of Various Structural Parameters

Figure 10a shows the influence range of each structural parameter on the leakage of the main seal. The length of the bar graph in the figure represents the difference between the maximum value and the minimum value of the leakage of a single structural parameter. The longer the bar, the greater the influence, and the location represents the change range of the leakage. It can be seen from the figure that the main structural parameters affecting the leakage of the main seal are the inner diameter allowance of the main seal, followed by the thickness of the matrix, and the height of the main seal arc protruding from the matrix, and the other structural parameters have a small impact on the leakage of the main seal, which can be ignored. The inner diameter allowance of the main seal has a great influence on the contact width of the seal and the bearing load of the seal ring, which can determine the deformation degree and state of the main seal arc when the seal ring is compressed. When the inner diameter margin of the main seal is too large, the contact width of the seal ring is small, and the contact pressure of the seal is too low, which may easily lead to a large amount of leakage of the seal ring under pressure fluctuation. Therefore, it is necessary to select the inner diameter margin of the main seal appropriately according to the contact pressure of the seal, rather than the maximum value. The greater the height of the main sealing arc protruding from the base, the smaller the leakage, and the more stable the pressure value. Thus, the maximum value is taken under the condition that the value does not exceed the radius of the main sealing arc, that is, the height of the arc protruding from the base is the radius of the main sealing arc. The thickness of the matrix affects the overall compression state of the special-shaped seal ring. Therefore, in the design, it is also necessary to consider its impact on the contact clearance of the sand flap.

Figure 10b shows the influence range of various structural parameters on the contact clearance of the tip of the sand baffle. From the figure, it can be seen that the main structural parameters affecting the contact clearance of the tip of the sand baffle are the inner diameter allowance of the sand baffle, followed by the thickness of the seal ring base, the angle of the tip of the sand baffle, and the angle between the tip and the bearing. Consistent with the inner diameter allowance of the main seal, the inner diameter allowance of the sand flap also directly affects the compression state of the sand flap with the compression of the seal ring, and the tip angle of the sand flap directly affects the ability of the sand flap tip to resist deformation and warpage. It can be seen from Figure 10a that the influence of this parameter on the leakage of the main seal is relatively small and can be ignored. Therefore, in the design, it is only necessary to consider the influence of the two parameters on the tip contact clearance, and match the best structural parameters. As the support of the two main functional structures, the thickness of the sealing ring matrix also has a certain impact on the deformation of the two functional structures. Therefore, the thickness of the matrix also has a certain impact on the performance of the sand flap. The angle of the tip of the sand-retaining flap and the angle between the tip and the bearing are the main parameters that determine the structural shape of the sand-retaining flap, so they can also affect the performance of the sand-retaining flap.

5.3.2. The Impact of Important Parameters on Performance

Figure 11 shows the influence of different structural parameters on sealing performance. As shown in Figure 11a, as the inner diameter margin increases, the leakage of the main seal gradually decreases, and the sealing performance is enhanced, while the contact gap at the tip of the sand retaining flap is almost unaffected. When the inner diameter margin is close to 1 mm, the contact pressure drops to 7 MPa, in this case, the main seal contact gap width also drops to 0.5 mm. Due to the complex bottom hole conditions, if a certain pressure fluctuation occurs, the sealing fluid will flow through. Therefore, the inner diameter margin of the main seal should be as large as possible, but at the same time, it is necessary to ensure that the contact pressure of the contact part is higher than the pressure on both sides of the sealing ring by a certain value to ensure that the sealing ring will not leak under pressure fluctuations.

As shown in Figure 11b, the leakage of the main seal increases with the increase in the inner diameter of the sand retaining flap. The leakage of the main seal shows a gradual increase trend, but its change value is not large. When the inner diameter margin is 0.05–0.4 mm, the contact gap of the sand retaining flap tip decreases with the increase in the inner diameter margin. When the inner diameter margin is 0.4 mm, the contact gap reaches the minimum value, which is close to zero. When the amount is 0.4–0.8 mm, the contact gap has an obvious increasing trend with the increase in the inner diameter margin. Since the tip of the sand retaining valve is relatively thin, if the contact gap between the tip of the sand retaining valve is too large, the compression will also occur in the circumferential direction during the compression process, which will cause the local tip to be deformed and lifted, and will affect the sand retaining performance. In the case of ensuring that the sand retaining flap will not be stretched in the circumferential direction, its value should be kept as small as possible. The sand retaining flap is not responsible for maintaining the pressure difference between the inside and outside of the sealing system, so it is not necessary to ensure that the contact pressure at the sand retaining flap is greater than the internal and external pressure.

As shown in Figure 11c, as the thickness of the substrate increases, the leakage of the main seal first slowly increases. When the thickness is 4.4 mm, the leakage is the largest, then gradually decreases. The tip contact gap of the sand retaining flap gradually decreases before the thickness of the substrate is 5.5 mm. When the thickness of the substrate exceeds 5.5 mm, the tip contact gap of the sand retaining flap begins to increase sharply. The thickness of the sealing matrix affects the compression state of the sealing ring as a whole, so it has a certain effect on the main seal and the sand retaining flap.

As shown in Figure 11d, as the angle between the tip and the bearing increases, the leakage of the main seal is basically unchanged, and the contact gap between the tip of the sand flap gradually increases.

As shown in Figure 11e, as the tip angle of the sand retaining flap increases, the leakage of the main seal gradually increases, but the change value is small, about 80 mg/h, and the tip contact gap gradually decreases. When the tip angle is 45°, the contact gap is close to zero.

As shown in Figure 11f, the rule of influence of the height of the protruding substrate of the main seal on various performance parameters. When the height of the protruding substrate increases and does not exceed the radius of the arc of the main seal, the leakage of the main seal gradually decreases, and the sand retaining flap tip contact gap remains basically unchanged. The height of the protruding base of the main seal can affect the axial deformation of the main seal. The smaller the value, the more the main seal can maintain its appearance. The larger the value is, the original arc shape of the main seal will be greatly deformed due to the pressure inside the sealing system. The more obvious change is that the contact width of the seal gap will decrease with the increase in this value, resulting in a decrease in leakage. When the height of the main seal protruding base is about 2.7 mm, the leakage reaches the lowest value; in this case, the contact pressure of the main seal is 16 MPa, which is much greater than the fluid pressure on both sides of the seal ring, and the seal ring will not leak a lot due to pressure fluctuations.

5.3.3. Comparison of Results before and after Optimization

• Comparison of structural parameters.

The thickness of the base body of the new type of sealing ring has an impact on the leakage of the main seal and the sand retaining performance of the sand retaining flap, but it is not the main influencing factor. Therefore, the leakage and sand retaining performance need to be comprehensively considered in the design to achieve the best performance of the special-shaped seal ring. Here, the thickness of the sealing substrate is 5.5 mm. In order to achieve the minimum leakage of the main seal, the inner diameter margin of the main seal and the height of the main seal arc protruding from the base should be the maximum value, and the groove inner diameter interference should be the minimum value,; however, in order to ensure that the seal ring can withstand pressure fluctuations, take the inner diameter margin of the main seal as 0.4 mm. In order to make the contact gap between the tip of the sand flap as small as possible, the inner diameter margin of the sand flap should be 0.4 mm, the angle of the tip of the sand flap should be the maximum 45°, and the angle between the tip and the bearing should be the minimum 20°. Therefore, the key structural parameters of the special-shaped sealing ring and their corresponding values are shown in

Table 4. Key structural parameters and their values before and after optimization.

Parameter	T	C	CE	C3	C2	F2	F1
Before optimization	4.0 mm	0.1 mm	1.5 mm	1.2 mm	0.15 mm	35°	25°
Optimized	5.5 mm	0.4 mm	2.7 mm	1.3 mm	0.4 mm	45°	20°

.

2.

Stress comparison.

Figure 12 is the stress comparison before and after the optimization of the special-shaped seal ring. As shown in Figure 12a, the maximum local stress before optimization is 7.7738 MPa at the main seal. As shown in Figure 12b, the optimized stress distribution is relatively uniform, and there is no large, concentrated stress in the main seal part, which is beneficial to prolong the service life of the main seal. There is a local stress peak at the wheel contact position, the value is 4.526 MPa, but the area where the maximum stress value is located is lower than the stress value before optimization, so it does not increase the probability of failure of the sealing matrix. In general, the stress distribution of the optimized sealing ring is better than before, and the theoretical life of the sealing ring is longer.

3.

Contact pressure comparison.

Figure 13 is the comparison of the contact pressure before and after the optimization of the special-shaped sealing ring. As shown in Figure 13, the peak of the contact pressure of the seal is basically located in the main contact area of the main seal. The contact pressure before optimization is 19.213 MPa, and after optimization is 14.742 MPa, the contact pressure before optimization is greater than the contact pressure after optimization, but in the case that the sealing can be achieved, the smaller the contact pressure, the less prone to failure of the sealing ring, and the easier it is to form a lubricating liquid film to improve the lubrication state. The contact position of the seal base and the cone also has a certain amount of contact pressure, which is beneficial to fix the relative position of the sealing ring and the cone, prevent the sealing ring and the cone from sliding with each other, and avoid the second dynamic seal pair. In general, the optimized contact pressure distribution of the main seal of the special-shaped seal ring can meet the sealing requirements and is more conducive to the formation of the lubricating liquid film.

4.

Comparison of the contact gap between the tip of the sand retaining flap.

Figure 14 is a comparison of the contact gap between the tip of the sand retaining flap before and after the optimization of the special-shaped sealing ring. As shown in Figure 14, the contact gap is 0.00019479 mm, a decrease of 95%, the abrasive medium is separated from the main seal components, which can greatly reduce the wear of the main seal. The contact gap of the sand retaining flap has been greatly reduced, and the theoretical performance of the sand retaining flap has been greatly improved.

5.

Performance comparison.

The performance indicators of the new seal ring before and after optimization are shown in

Table 5. Performance changes of new sealing ring before and after optimization.

Performance	Maximum Stress of Sealing Ring	Contact Pressure of Main Seal	Contact Width of Main Seal	Main Seal Bearing	Main Seal Leakage	Sand Retaining Flap Tip Contact Gap
Not optimized	7.774 MPa	19.213 MPa	2.45 mm	550.3 N	282.79 mg/h	0.0039343 mm
Optimized	4.526 MPa	14.742 MPa	0.86 mm	499.2 N	43.10 mg/h	0.00019479 mm

.

After the parameter optimization of the seal ring structure, the leakage of the main seal of the seal ring is reduced by 85%, and the effect is obvious. From a numerical point of view, the leakage amount is 43 mg/h, and the oil storage bag of the cone bit can be used to supplement the reserve of about 25 g. In theory, it takes 580 h for the grease to run out. If there are complicated conditions, such as pressure fluctuations, the working hours will be reduced to a certain extent. The contact gap is 0.00019479 mm, which is a decrease of 95%. Since there is currently no related auxiliary sand retaining structure for cone bit sealing, it does not explain the value of reducing the contact gap to 0.00019479 mm. However, in theory, the grinding medium of the external mud medium is separated from the main seal components, which can greatly reduce the wear of the main seal.

6. Conclusions

• Based on the P. W. Wernecke theory, the relevant formula for calculating the leakage of the corrugated main seal is deduced. The main factors affecting the leakage of the special-shaped seal ring are the seal compression state, the pressure difference $\Delta P$ inside and outside the seal cavity, and the relevant parameters of the main seal sine wave, and the actual model is obtained.
• Determine the compression state of the main seal and the contact gap of the sand retaining flap, and prove that the corrugated special-shaped sand retaining seal meets the design requirements. Calculate the leakage rate and draw the conclusion that the greater the contact width, the greater the leakage rate under the premise of the constant load.
• The main structural parameter that affects the leakage of the main seal and the contact gap between the sand flap tip are obtained. Through the single-factor optimization method, the influence law of each structural parameter on the leakage and the contact gap is obtained, and a set of optimal structural parameter combinations suitable for 8 1/2 inch (215.9 mm) drill bits are obtained, the main seal contact pressure is reduced and the leakage is reduced by 85%; it takes 580 h for the grease to run out; and the contact gap is 0.00019479 mm, a decrease of 95%, the abrasive medium is separated from the main seal components, which can greatly reduce the wear of the main seal.