# Flow structure and rock-breaking feature of the self-rotating nozzle for radial jet drilling

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## Abstract

For improving the hole-enlarging capability, roundness and rock-breaking efficiency of the nozzle in radial jet drilling, a new structure of self-rotating nozzle was put forward. The flow structure and rock-breaking features of the self-rotating nozzle were investigated with sliding mesh model and labortary tests and also compared with the straight and the swirling integrated nozzle and multi-orifice nozzle which have been applied in radial jet drilling. The results show that the self-rotating jet is energy concentrated, has longer effective distance, better hole-enlarging capability and roundness and impacts larger circular area at the bottom of the drilling hole, compared with the other two nozzles. Forward jet flow generated from the nozzle is peak shaped, and the jet velocity attenuates slowly at the outer edge. Due to periodic rotary percussion, the pressure fluctuates periodically on rock surface, improving shear and tensile failures on the rock matrix and thereby enhancing rock-breaking efficiency. The numerical simulation results of the flow structure of the nozzle are consistent with the experiments. This study provides an innovative approach for radial jet drilling technology in the petroleum industry.

## Keywords

Self-rotating nozzle Flow field characteristic Numerical simulation Rock-breaking## 1 Introduction

## 2 Self-rotating nozzle structure and its working principle

## 3 Numerical model and boundary conditions

### 3.1 Simulation model

The outer diameter of the nozzle rotator is 18 mm, with a length of 30 mm. The leading end surface of the rotator is round, and the main inner flow diameter is 3 mm. There are 3 forward orifices and 4 backward orifices on the rotator. The forward orifices have the same diameter of 0.8 mm, with the intersection angles *α*, *β*, *γ* of their axes to the nozzle axis being 7°, 13° and 20°, respectively. Backward orifices also have the same diameter of 1.2 mm, with the same 30° intersection angles of their axes to the nozzle axis, and each has an eccentricity of 0.8 mm. The leading end of the nozzle is 10 mm away from the bottom of the outer flow field. The horizontal cross section of the outer flow field is circular, 14 mm in radius and 50 mm in height.

### 3.2 Mesh of computational domain

### 3.3 Control equations

In numerical simulations, the RNG *k*–*ε* turbulence model was used to obtain a detailed visual flow field structure of the rotating jet. The set of equations of the model is similar to that of the standard *k*–*ε* model, but with an additional *R* in the transmission equation of *ε*. Thus, the model can more accurately describe swirling and impinging jet flows (Chen et al. 2003; Anderson et al. 2009). Buoyancy forces are not taken into account for incompressible fluids. The control equations in the RNG *k*–*ε* model mainly include continuity and momentum equations as well as transmission equations of the turbulent kinetic energy *k* and the dissipation rate ε.

- (1)Continuity equation$$\frac{\partial \rho }{\partial t} + \frac{{\partial \rho u_{i} }}{{\partial x_{i} }} = 0$$(1)
- (2)Momentum equation$$\rho \frac{{\partial u_{i} }}{\partial t} + \rho u_{j} \frac{{\partial u_{i} }}{{\partial x_{j} }} = - \frac{{\partial \overline{p} }}{{\partial x_{i} }} + \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu_{0} + \mu_{\text{t}} } \right)\left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right) - \frac{2}{3}\left( {\rho k + \mu_{\text{t}} \frac{{\partial u_{i} }}{{\partial x_{i} }}} \right)\delta_{ij} } \right]$$(2)
- (3)Transmission equations$$\frac{\partial (\rho k)}{\partial t} + \frac{{\partial (\rho u_{i} k)}}{{\partial x_{i} }} = \frac{\partial }{{\partial x_{i} }}\left( {\alpha_{k} \left( {\mu_{0} + \mu_{\text{t}} } \right)\frac{\partial k}{{\partial x_{i} }}} \right) + \mu_{\text{t}} S^{2} - \rho \varepsilon$$(3)$$\frac{\partial (\rho \varepsilon )}{\partial t} + \frac{{\partial (\rho u_{i} \varepsilon )}}{{\partial x_{i} }} = \frac{\partial }{{\partial x_{i} }}\left( {\alpha_{\varepsilon } \left( {\mu_{0} + \mu_{\text{t}} } \right)\frac{\partial \varepsilon }{{\partial x_{i} }}} \right) + C_{1\varepsilon } \frac{\varepsilon }{k}\mu_{\text{t}} S^{2} - C_{2\varepsilon } \rho \frac{{\varepsilon^{2} }}{k} - R$$(4)$$S = \sqrt {2S_{ij} S_{ij} } ,S_{ij} = \frac{1}{2}\left( {\frac{{\partial u_{j} }}{{\partial x_{i} }} + \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right)$$where \(\rho\) is the fluid density;$$R = \frac{{C_{\mu } \rho \eta^{3} \left( {1 - \eta /\eta_{0} } \right)}}{{1 + \beta \eta^{3} }}\frac{{\varepsilon^{2} }}{k}$$
*t*is time; \(\overline{p}\) is pressure; \(u_{i}\), \(u_{j}\) (\(i,\;j = 1,\;2,\;3\)) represent the velocity; \(x_{i}\), \(x_{j}\) represent coordinate axis; \(\delta_{ij}\) is Kronecker symbol; \(\mu_{0}\) is the fluid viscosity; \(\mu_{\text{t}}\) represents the turbulence viscosity, \(\mu_{\text{t}} = \rho C_{\mu } k^{2} /\varepsilon\); \(C_{\mu }\), \(C_{1\varepsilon }\), \(C_{2\varepsilon }\) are empirical coefficients set to 0.09, 1.42, 1.68, respectively, in the model; \(\alpha_{k} = \alpha_{\varepsilon } \approx 1.393\); \(\eta \equiv {{Sk} \mathord{\left/ {\vphantom {{Sk} \varepsilon }} \right. \kern-0pt} \varepsilon }\), \(\beta = 0.012.\)

*R* is the difference between the RNG *k*–*ε* model and the standard *k*–*ε* model. The existence of *R* enables the RNG *k*–*ε* model to simulate flow with high strain and allows the flow lines to be more accurately depicted.

Solutions to the discretized equation were reached using the tridiagonal matrix algorithm (TDMA), in which the low relaxation iteration was done line by line. The coupled and iterative solution for continuity and momentum equations were worked out using the SIMPLEC algorithm.

### 3.4 Parameters and boundary conditions

As the nozzle rotator has a high rotational velocity, the RNG *k*–*ε* model was adopted, which has a higher accuracy for flow fields with high flow curvature and strain rate (Song et al. 2004, 2017). At the same time, a SIMPLE algorithm was used to solve the continuity equations and momentum equations with coupling iteration. The rotational speed of the rotator is set to 0, 25, 50 and 75 rps to be discussed later and the test results are shown in Fig. 4.

- (1)
*Inlet boundary*The inlet velocity depends on inlet conditions in which*k*and*ε*are computed from the equations below:$$I = 0.16(Re)^{1/8}$$(5)$$k = 1.5\left( {v_{\text{in}} I} \right)^{2}$$(6)where$$\varepsilon = {{C_{\mu }^{0.75} k^{1.5} } \mathord{\left/ {\vphantom {{C_{\mu }^{0.75} k^{1.5} } {(0.07L_{\text{in}} )}}} \right. \kern-0pt} {(0.07L_{\text{in}} )}}$$(7)*Re*is the Reynolds number;*v*_{in}is the inlet velocity;*I*is the turbulent intensity;*L*_{in}is the length of the mixing section at the inlet. - (2)
*Outlet boundary*The static pressure for the outlet is set to 0.1 MPa. - (3)
*Wall boundary conditions*The relative angular velocity at the rotator wall boundary is set to zero. Non-slip boundary condition is applied to other walls.$$U_{ + p} = \frac{1}{\kappa }\ln \left( {Ey_{ + p} } \right)$$(8)where \(U_{ + p}\) is the dimensionless velocity of point$$y_{ + p} = \frac{{C_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} y_{p} }}{\upsilon }$$(9)*p*near the wall; \(\kappa\) is the Karman constant, \(\kappa = 0.4\);*E*is the wall roughness for the hydraulically smooth surface \(E = 9.0\); \(y_{p}\) is the distance computed from point*p*to the wall; \(\upsilon\) is the kinematic viscosity. To guarantee the applicability of the law of logarithmic distribution for velocity, the value of \(y_{ + p}\) is set in the range as:$$11.5\sim30 \le y_{ + p} \le 200\sim400$$

## 4 Results and discussion

### 4.1 Distribution and variation trend of the jet impact zone

### 4.2 Flow characteristics of forward jets

#### 4.2.1 Axial velocity attenuation

*Y*-axis is the normalized axial velocity

*v*/

*v*

_{max}(

*v*is the axial velocity of the jet;

*v*

_{max}is the maximum axial velocity of the jet), and

*X*-axis is the normalized jetting distance

*D*/

*d*(

*D*is the jetting distance;

*d*is the equivalent diameter of forward orifices). A larger jet angle of forward orifices forms a larger rotational radius at the bottom of the hole; as a result, the jetting distance decreases. It can be seen that when forward jets eject from orifices, the axial velocities of jets rapidly attenuate as the normalized jetting distance increases, and the velocities reduce to 0 at the boundary wall. Furthermore, the axial velocity of the 20° jet is lower than that of the other two jets, which means the larger the jet angle, the more the jet velocity attenuates. Near the boundary wall, three jetting velocities gradually become similar. It is concluded that the larger the jet angle is, the greater the influence of the wall effect on the jet, and the faster the attenuation of the axial velocity is.

#### 4.2.2 Axial velocity distribution

### 4.3 Impact of rotational speed of rotor on flow characteristics

*α*,

*β*,

*γ*. The axial velocity attenuation of jet with 20° intersection angle is obviously affected by the rotational speeds set from 0 to 75 rps. The rock-breaking depth is mainly influenced by jet velocity and its erosion duration. Since the orifice with 20° intersection angle has less acting time on the rock and the velocity attenuates faster, it yields shallower breaking depth compared with multi-orifice and straight and swirling integrated nozzles illustrated in later rock test, and a T-type crater is formed at the bottom of the hole.

- (1)
The axial velocity of the forward jet at different rotational speeds presents a peak distribution at the bottom of the hole. The maximum values of the three peaks are basically the same, and the size of the velocity distribution area is basically unchanged. The axial velocity distribution is basically independent on the rotational speed.

- (2)
Under different rotational speeds, the radial velocity of the three forward jets corresponds to the bottom hole position with three peaks, and the peak value is basically the same. There are “peaks” with smaller peaks among the three peaks. The analysis shows that the distance between the three jets is small, and the diffuse flow produced by the two forward jets is mutually contradictory and strengthened, thus forming a radial velocity concentrated zone.

- (3)
The tangential velocity of forward jets with different intersection angles corresponds to three peaks in the figure. The tangential velocity of jet with 20° intersection angle is the highest, followed by the 13° angle, and the lowest is the 7° angle. With an increase in rotational speed, the peak of the tangential velocity of forward jet increases. From the mechanism of rock fragmentation, it can be seen that the high tangential velocity of jet is helpful to the tensile and shear failure of rock and to reduce the threshold pressure of rock breakage (Shen 1998).

Through numerical simulation of the flow structure of self-rotating jets, flow structure, it is found that forward orifices including angles *α*, *β*, *γ* have an important influence on jet flow characteristics. Four groups of nozzles were manufactured with the configurations of forward orifice with intersection angles *α*–*β*–*γ* set as 7°–13°–20°, 7°–12°–20°, 7°–11°–18° and 7°–11°–20°. In order to make the rotor rotate smoothly, the centerlines of the forward orifices and the rotor are kept in the same planes, whereas the axes of two backward orifices offset from the axes of the rotor by 0.4 mm, and the equivalent diameter of all forward orifices is 2 mm.

Combining numerical simulations and the rock-breaking tests, the self-rotating nozzle shows the similar axial velocity attenuation in comparison with the multi-orifice nozzle, and thus it can achieve desirable rock-breaking depth. At the same time, because of the rotating impact mode of the self-rotating nozzle, a circular impact surface is formed at the bottom of the hole, so the impact breakage makes the drilling hole more smooth and round which is a benefit for reducing friction during hose movement and reaching longer radial distance. Furthermore, compared with the straight and swirling integrated nozzle, the self-rotating nozzle has an better energy concentrated jet flow core, longer effective jetting distance, and faster radial and tangential velocities; thus, rock can be effectively eroded, and the hole profile is much clearer. Therefore, different types of nozzles generate different types of jets and form various impact patterns on a rock. The multi-orifice jet is energy concentrated, longer in effective jetting distance but limited in impact area; the straight and swirling integrated jet has three-dimensional velocity, more diffusivity, but faster velocity attenuation and shorter effective operating range, resulting in a shallow hole in the rock. The self-rotating jet is energy concentrated and longer in effective jetting range; compared with the other two nozzles, it has stronger reaming capability as well as good drilling roundness, which are exactly what RJD requires.

## 5 Conclusions

In this paper, a new type of self-rotating nozzle was put forward in order to achieve a longer and rounder hole in radial jet drilling. From the numerical simulation and the rock-breaking test, the nozzle is energy concentrated and has a long effective jetting distance like a multi-orifice nozzle, whereas its three-dimensional velocities are like the straight and swirling integrated nozzle. Therefore, it can drill a rounder and larger hole as compared with the other two nozzles. Simulation reveals that the forward jet flow feature of the self-rotating nozzle has three peak points, among which the outer velocity attenuates slower. Compared with the straight and swirling integrated jet, it can guarantee more effective central erosion and breakage of the outer edge and generate periodic pressure fluctuation on the rock surface, which has the characteristic of pulse jets and assists to improve rock penetration. The tangential velocity of forward jets of the self-rotating nozzle increases as the rotational speed goes up, which enhances shear and tensile failure and achieves a higher rock-breaking efficiency.

The self-rotating nozzle has a potential application, and a number of factors like the nozzle structure and working parameters affect the rock-breaking efficiency. Therefore, more rock-breaking tests need to be conducted for optimizing the nozzle design and work parameters.

## Notes

### Acknowledgements

The authors express their appreciation to the supports from Natural Science Foundation of China (Grant No. 51274235), Shandong Provincial Natural Science Foundation (Grant No. ZR2019MEE120) and the Major project of CNPC (Grant No. ZD2019-183-005).

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