# Extension Limit of a Straight-Swirling Mixed Jet Bit and Its Influential Factors in Radial Jet Drilling

Authors: Peng Du; Weikang Xia; Chaoxiong Yu
Journal: Geofluids
 (2022)

---

## Abstract

Radial jet drilling (RJD) is applied to low-permeability and unconventional natural gas. A self-propelled bit chiefly affects the borehole length and drilling efficiency. Herein, for a straight-swirling mixed jet (SSMJ) bit, a prediction model of the drilling extension limitLEL was established by analysing the forces and pressure loss of RJD system. LEL of a specific project was calculated and the pressure loss and forces distribution were obtained. Additionally, the influence laws of the main factors on LEL were studied using the established model. The results indicated that the high-pressure hose contributes the main system pressure loss, and the recoil force of backward nozzles is the sole driving force of RJD. The recoil force of forward nozzle and friction between the hose and borehole constitute the main resistances. Increasing the pump pressure pb, backward nozzle diameter db, and jet diffusion angle θx and reducing forward nozzle diameter df, impeller central hole diameter dm, and backward nozzle inclination angle θb are conducive to improving LEL. However, larger pb and db increase the hydraulic consumption, besides larger θx or smaller df, and dm will reduce the rock-breaking capability. From a sensitivity analysis, pb and dm have the maximum and minimum influence on LEL, respectively. Finally, prediction value LEL of an oil well in Eastern Sichuan can be up to 63.7 m. The results provide a guidance for the hydraulic parameters and key component selection, additionally bit structure optimization of RJD.


---

## Body

## 1. Introduction

China is rich in unconventional natural gas resources of coalbed methane (CBM) and shale gas, etc. [1]. Of these, the total CBM resources in China that are within 2000 m from the ground are estimated to be 36.81 trillion cubic meters [2, 3]. Most of the production wells are located in the southern Qinshui Basin and eastern Ordos Basin [4, 5]. Radial jet drilling (RJD) technology was imposed in the 1980s [6]. The use of RJD to exploit low-permeability oil-gas reservoirs, CBM, and shale gas is a hot topic and an important direction for the development of oil-gas resource drilling [7–10]. A RJD technique utilizes hydraulic energy to create several lateral holes in different directions and levels with several different lengths. These lateral holes are made by milling the casing with a small bit and then extending these holes laterally using high-pressure hydraulic jetting, which can improve the recovery of oil and gas with lower cost [11]. In 2011, Petrobel Company drilled five 50 m radial wells and one 90 m well at two stratums at No.1 well in Belayin Oil Field [12]. The successful application of RJD in the world reflects its promising future.Predicting the extension limit of the RJD can optimize the layout of boreholes, and it is significant to improve their output. To date, some authors researched the main factors influencing the extension limit, such as the self-propelled force of the bit and system pressure loss. Buset et al. researched the self-propelled force of multiple nozzle jet bits of a RJD system [13]; Ma et al. established a calculation model of pressure loss for RJD system by analysing the pressure loss of a coiled tube and high-pressure hose [14]; Zhang et al. designed a jet radial horizontal drilling simulation experiment system of the casing windowing and evaluated the self-propelled force of a multiorifice nozzle [15].A self-propelled bit, the “core” of RJD, chiefly affects the borehole length and drilling efficiency. RJD technology changes the direction from a vertical to horizontal direction using a diverter [15]. The turning radius of the interior trail within the diverter is small, which limits the dimension of the bit. Therefore, only a simple structure can be selected in the design of bit. Research [16–19] indicated that the jets used for drilling can be divided into three main types: straight, swirling, and a combination thereof. Li et al. researched sandstone-breaking characteristics by experiment and studied the influence of working conditions of lithology, axis length, jet pressure, and standoff distance [20]. Liu et al. adopted the FLUENT combined with RNG turbulence model to analyze the multinozzle jet flow field and study the effects of inclination angle of forward nozzles and standoff distance [21]. Du et al. researched the damage characteristics of straight jet, swirling jet, and SSMJ and then obtained the velocity fields of the three jets by 3DPIV [22]. Overall, a straight jet can drill a deep hole, yet the diameter is smaller because the energy of the jet is more centralized. A swirling jet can be used to drill a larger diameter hole, but it has a boss at the centre owing to the central low-velocity zone of the jet. A combination of a dual-jet nozzle or an SSMJ nozzle can combine the advantages of both a straight jet and swirling jet [23–25]. Few articles have been published on the research on predicting the extension limit of an SSMJ and its influential factors.With an SSMJ as the research objective, this study established a model to predict the extension limit of a RJD system by analysing the distributions of force and pressure loss. Thereafter, the drilling extension limit of a specific project was calculated and the pressure loss and forces were obtained. Finally, the influence laws of the main parameters (pump pressure, diameter of the forward nozzle and backward nozzles, diffusion angle of the jet, inclination angle of the backward nozzle, etc.) on the extension limit were studied using the established prediction model; additionally, the influence degree of parameters was investigated. We conducted these studies in an attempt to provide a guidance for the construction parameter selection of the RJD.


## 2. Mathematical Model

During drilling, the horizontal section of the RJD system has a self-propelled force of the bit (Fp), the pressure of an external fluid on the bit (Fo), friction between the diverter and high-pressure hose (Frz), friction between the borehole and high-pressure hose (Frk), and friction between the borehole and bit (Fxk), as shown in Figure 1. Owing to the lower flow rate of the system and the use of a pure water jet during operation, the viscosity is low, and the viscous resistance of the fluid in the borehole to the hose is also small, which can be ignored. The resultant force F on the RJD system is expressed as
(1)F=Fp−Fo−Frz−Frk−Fxk.Figure 1
Force model of the RJD system.


### 2.1. Self-Propelled Force of the Bit

An SSMJ bit composes of a body and an impeller with a central hole and several slots (Figure1). Its basic operating principle is as follows: the fluid flows into the mixing chamber through the central hole and slots of the impeller, generating a low-speed, straight-swirling flow. An SSMJ with a high axial velocity and peripheral rotational intensity is then created successively under the pressurization of the nozzle outlet, and the extended section ensures a specific target distance when breaking rock. In addition, the jets created by the backward nozzles can push the bit and high-pressure hose forward. A straight jet has only a one-dimensional velocity; however, for a swirling jet, the trajectory is approximately a helix, and the velocity of any particle is a space vector, its axial velocity u is parallel to the jet axis, and its radial and tangential velocities v and w are perpendicular to the jet axis [26, 27]. When the system is operating, the forces of the bit include the recoil force of the backward nozzles (Fb) and recoil force of the forward nozzle (Ff). Naturally, the self-propelled force of the bit Fp is Fb minus Ff.


#### 2.1.1. Recoil Force of the Forward Nozzle

The recoil force of the SSMJ forward nozzle was produced from two parts: straight jet and swirling jet. If the forward nozzle and the impeller central hole diameter aredf and dm, respectively, the flow area of the swirling jet is the difference between the forward nozzle area and the central hole area, and the equivalent diameter of the swirling jet can be obtained:
(2)dx=df2−dm2.(a)
Recoil Force of the Straight Jet. The equation to calculate the recoil force of the straight jet acting on the bit is [27]
(3)Fm=1.56μfdm2pi,where pi is the nozzle inlet pressure (MPa) and μf is the flow coefficient of the forward nozzle(b)
Recoil Force of the Swirling Jet. Swirling jet is a three-dimensional flow different from the straight one. The axial velocityu of any particle of the swirling jet is perpendicular to v and w; thus, v and w do not produce an axial recoil force. According to the law of mass conservation, the total flow rate qx is the sum of the axial flow rate qu, radial flow rate qv, and tangential flow rate qw. The recoil force of the swirling jet acting on the jet bit is only related to qu and can be expressed as
(4)Fx=0.745qupiThe axial velocity distribution along the radial of the swirling jet is approximately shaped as “M”; therefore, for ease of calculation, we can assume that all fluid particles of the swirling jet are distributed on the cone with a diffusion angle ofθx. Moreover, the approximate relationship between the axial velocity flow rate (qu) and total flow rate (qx) can be obtained:
(5)qu=qxcosθx2.Additionally,qx is
(6)qx=2.1μfdx2pi.Combining Equations (4), (5), and (6), we obtain
(7)Fx=1.56μfdf2−dm2picosθx2.Finally, the forward nozzle recoil force is(8)Ff=Fm+Fx=1.56μfpidm2+df2−dm2cosθx2.


#### 2.1.2. Recoil Force of Backward Nozzles

Referring to the analysis method in the previous section, the equation for the backward nozzle recoil force can be written as(9)Fb=∑b=1nb1.56μbdb2picosθb,where nb is the number of backward nozzles and db, θb, and μb are the diameter, inclination angle, and flow coefficient of a backward nozzle, separately.Combining Equations (8) and (9), the self-propelled force of the SSMJ bit can be obtained:
(10)Fp=1.56pi∑b=1nbμbdb2cosθb−μfdm2−μfdf2−dm2cosθx2.


### 2.2. Pressure of the External Fluid on the Bit

In the axial direction, the drill bit is subjected to the pressure of the fluid in the hole on the front end face. The following equation of this pressureFo is obtained:
(11)Fo=π4podo2−df2,where do is the outer diameter of the bit (mm) and po is the pressure of the external fluid on the front end face of the bit (MPa). To be specific, po is the sum of the annulus pressure loss and ground pressure, and the ground pressure is 0; thus, the value of po is the annulus pressure loss, which can be calculated according to the relevant research [14].


### 2.3. Friction Resistance of the System

The friction resistance (F’) of the RJD system primarily includes the friction between the diverter and hose (Frz), the friction between the borehole and the hose (Frk), and the friction between the borehole and the bit (Fxk). Of these, Frz can be obtained through a laboratory test, while
(12)Frk=μrkGrlrk,Fxk=μxkGx,where μrk is the friction coefficient between the hose and borehole, Gr is the submerged weight of a unit length of the hose (N/m), lrk is the friction section length of the hose (m), μxk is the friction coefficient between the bit and borehole, and Gx is the submerged weight of the jet bit (N).Substituting Equations (10), (11), and (12) into (1), the force model of the RJD system can be obtained:
(13)F=1.56pi∑b=1nbμbdb2cosθb−μfdm2−μfdf2−dm2cosθx2−π4podo2−df2−Frz−μrkGrlrk−μxkGx.


### 2.4. Pressure Loss of the System

We need to calculate the pressure loss of the RJD system to obtain the inlet pressure of the bit (pi). Ignoring the leakage at the joint and the local pressure loss such as hose diameter variation and bending, the system pressure loss is primarily generated by the coiled tubing (Δpcti) and the high-pressure hose (Δpr):
(14)pi=pb−∆pcti−∆pr,where pb is the pump pressure (MPa).


#### 2.4.1. Theoretical Pressure Loss

The pressure loss along the coiled tubingΔpcti includes the spiral section loss Δpcti1 and straight section loss Δpcti2, i.e., Δpcti=Δpcti1+Δpcti2, which can be obtained from the previous studies [14].The theoretical pressure loss of the hose can be obtained using the following equation [28]:
(15)∆pr=Kr×59.7q2lrdri5Re0.25,where Kr=1, dri is the hose inner diameter (mm), q is the volume flow rate (L/min), lr is the total length of the hose (m), and Re is the Reynolds number, which should be written as λq/dri, and the coefficient λ is 1.12×104 under turbulent condition. According to the previous studies [14], the theoretical value of hose pressure loss is quite deviation from the actual, and then, it will greatly affect the judgment of the inlet pressure value of the bit. The correction of Equation (15) is necessary to be performed through an experimental test.


#### 2.4.2. Experimental Test

In this study, several groups of tests were executed under different factors of the flow rate, hose inner diameter, and length; then, the measured pressure loss values under different factors were obtained; moreover, the parameters under each factor were substituted into Equation (15) to obtain the calculated value. By comparing the measured and calculated values, the correction factor Kr is introduced to Equation (15), and the deviation of Kr is analyzed to evaluate the effectiveness of the test. Eventually, Kr is taken as an average value.The following apparatuses were used (Figure2): a high-pressure pump, an SSMJ bit, high-pressure hoses of 1/4” (inner diameter of 6.35 mm) and 3/8” (inner diameter of 8 mm), a flowmeter, and two pressure gauges. The two gauges were, respectively, allocated at the inlet and outlet of the tested hose; therefore, the difference reading between pressure gauges No.1 and No.2 was the pressure loss value of the hose when the system operated steadily. The test results are listed in Table 1. Kr is introduced to correct Equation (15). The values of Kr under different conditions range from 0.23 to 0.27, with the relative standard deviation of 1.3%. A small deviation rate proves the effectiveness of the test; then, the final correction factor is taken as an average value, i.e., Kr=0.25.Figure 2
Schematic of the test system.Table 1
Test data of the high-pressure hoses.
Hose length (m)Hose inner diameter (mm)Flow rate (L/min)TestedΔpr (MPa)CalculatedΔpr (MPa)Correction factorKr106.3511710.841.680.26106.35552.811.180.25206.3511720.083.350.24206.35555.622.350.251081173.413.130.26108550.83.520.242081177.126.260.27208551.67.040.23


### 2.5. Mathematical Model of the Extension Limit

When the resultant forceF>0, the RJD system can continuously drill forward. As the drilled radial borehole length increases, the friction increases and F decreases gradually until F=0. Herein, the system is in an equilibrium condition of forces and the bit will not continue to drill forward with the hose; eventually, the value lrk under this condition is exactly the extension limit of the RJD system. Substituting F=0 into Equation (13), we can obtain the prediction model of the extension limit LEL:
(16)LEL=1.56pi∑b=1nbμbdb2cosθb−μfdm2+df2−dm2cosθx/2−π/4podo2−df2−Frz−μxkGxμrkGr.


## 2.1. Self-Propelled Force of the Bit

An SSMJ bit composes of a body and an impeller with a central hole and several slots (Figure1). Its basic operating principle is as follows: the fluid flows into the mixing chamber through the central hole and slots of the impeller, generating a low-speed, straight-swirling flow. An SSMJ with a high axial velocity and peripheral rotational intensity is then created successively under the pressurization of the nozzle outlet, and the extended section ensures a specific target distance when breaking rock. In addition, the jets created by the backward nozzles can push the bit and high-pressure hose forward. A straight jet has only a one-dimensional velocity; however, for a swirling jet, the trajectory is approximately a helix, and the velocity of any particle is a space vector, its axial velocity u is parallel to the jet axis, and its radial and tangential velocities v and w are perpendicular to the jet axis [26, 27]. When the system is operating, the forces of the bit include the recoil force of the backward nozzles (Fb) and recoil force of the forward nozzle (Ff). Naturally, the self-propelled force of the bit Fp is Fb minus Ff.


### 2.1.1. Recoil Force of the Forward Nozzle

The recoil force of the SSMJ forward nozzle was produced from two parts: straight jet and swirling jet. If the forward nozzle and the impeller central hole diameter aredf and dm, respectively, the flow area of the swirling jet is the difference between the forward nozzle area and the central hole area, and the equivalent diameter of the swirling jet can be obtained:
(2)dx=df2−dm2.(a)
Recoil Force of the Straight Jet. The equation to calculate the recoil force of the straight jet acting on the bit is [27]
(3)Fm=1.56μfdm2pi,where pi is the nozzle inlet pressure (MPa) and μf is the flow coefficient of the forward nozzle(b)
Recoil Force of the Swirling Jet. Swirling jet is a three-dimensional flow different from the straight one. The axial velocityu of any particle of the swirling jet is perpendicular to v and w; thus, v and w do not produce an axial recoil force. According to the law of mass conservation, the total flow rate qx is the sum of the axial flow rate qu, radial flow rate qv, and tangential flow rate qw. The recoil force of the swirling jet acting on the jet bit is only related to qu and can be expressed as
(4)Fx=0.745qupiThe axial velocity distribution along the radial of the swirling jet is approximately shaped as “M”; therefore, for ease of calculation, we can assume that all fluid particles of the swirling jet are distributed on the cone with a diffusion angle ofθx. Moreover, the approximate relationship between the axial velocity flow rate (qu) and total flow rate (qx) can be obtained:
(5)qu=qxcosθx2.Additionally,qx is
(6)qx=2.1μfdx2pi.Combining Equations (4), (5), and (6), we obtain
(7)Fx=1.56μfdf2−dm2picosθx2.Finally, the forward nozzle recoil force is(8)Ff=Fm+Fx=1.56μfpidm2+df2−dm2cosθx2.


### 2.1.2. Recoil Force of Backward Nozzles

Referring to the analysis method in the previous section, the equation for the backward nozzle recoil force can be written as(9)Fb=∑b=1nb1.56μbdb2picosθb,where nb is the number of backward nozzles and db, θb, and μb are the diameter, inclination angle, and flow coefficient of a backward nozzle, separately.Combining Equations (8) and (9), the self-propelled force of the SSMJ bit can be obtained:
(10)Fp=1.56pi∑b=1nbμbdb2cosθb−μfdm2−μfdf2−dm2cosθx2.


## 2.1.1. Recoil Force of the Forward Nozzle

The recoil force of the SSMJ forward nozzle was produced from two parts: straight jet and swirling jet. If the forward nozzle and the impeller central hole diameter aredf and dm, respectively, the flow area of the swirling jet is the difference between the forward nozzle area and the central hole area, and the equivalent diameter of the swirling jet can be obtained:
(2)dx=df2−dm2.(a)
Recoil Force of the Straight Jet. The equation to calculate the recoil force of the straight jet acting on the bit is [27]
(3)Fm=1.56μfdm2pi,where pi is the nozzle inlet pressure (MPa) and μf is the flow coefficient of the forward nozzle(b)
Recoil Force of the Swirling Jet. Swirling jet is a three-dimensional flow different from the straight one. The axial velocityu of any particle of the swirling jet is perpendicular to v and w; thus, v and w do not produce an axial recoil force. According to the law of mass conservation, the total flow rate qx is the sum of the axial flow rate qu, radial flow rate qv, and tangential flow rate qw. The recoil force of the swirling jet acting on the jet bit is only related to qu and can be expressed as
(4)Fx=0.745qupiThe axial velocity distribution along the radial of the swirling jet is approximately shaped as “M”; therefore, for ease of calculation, we can assume that all fluid particles of the swirling jet are distributed on the cone with a diffusion angle ofθx. Moreover, the approximate relationship between the axial velocity flow rate (qu) and total flow rate (qx) can be obtained:
(5)qu=qxcosθx2.Additionally,qx is
(6)qx=2.1μfdx2pi.Combining Equations (4), (5), and (6), we obtain
(7)Fx=1.56μfdf2−dm2picosθx2.Finally, the forward nozzle recoil force is(8)Ff=Fm+Fx=1.56μfpidm2+df2−dm2cosθx2.


## 2.1.2. Recoil Force of Backward Nozzles

Referring to the analysis method in the previous section, the equation for the backward nozzle recoil force can be written as(9)Fb=∑b=1nb1.56μbdb2picosθb,where nb is the number of backward nozzles and db, θb, and μb are the diameter, inclination angle, and flow coefficient of a backward nozzle, separately.Combining Equations (8) and (9), the self-propelled force of the SSMJ bit can be obtained:
(10)Fp=1.56pi∑b=1nbμbdb2cosθb−μfdm2−μfdf2−dm2cosθx2.


## 2.2. Pressure of the External Fluid on the Bit

In the axial direction, the drill bit is subjected to the pressure of the fluid in the hole on the front end face. The following equation of this pressureFo is obtained:
(11)Fo=π4podo2−df2,where do is the outer diameter of the bit (mm) and po is the pressure of the external fluid on the front end face of the bit (MPa). To be specific, po is the sum of the annulus pressure loss and ground pressure, and the ground pressure is 0; thus, the value of po is the annulus pressure loss, which can be calculated according to the relevant research [14].


## 2.3. Friction Resistance of the System

The friction resistance (F’) of the RJD system primarily includes the friction between the diverter and hose (Frz), the friction between the borehole and the hose (Frk), and the friction between the borehole and the bit (Fxk). Of these, Frz can be obtained through a laboratory test, while
(12)Frk=μrkGrlrk,Fxk=μxkGx,where μrk is the friction coefficient between the hose and borehole, Gr is the submerged weight of a unit length of the hose (N/m), lrk is the friction section length of the hose (m), μxk is the friction coefficient between the bit and borehole, and Gx is the submerged weight of the jet bit (N).Substituting Equations (10), (11), and (12) into (1), the force model of the RJD system can be obtained:
(13)F=1.56pi∑b=1nbμbdb2cosθb−μfdm2−μfdf2−dm2cosθx2−π4podo2−df2−Frz−μrkGrlrk−μxkGx.


## 2.4. Pressure Loss of the System

We need to calculate the pressure loss of the RJD system to obtain the inlet pressure of the bit (pi). Ignoring the leakage at the joint and the local pressure loss such as hose diameter variation and bending, the system pressure loss is primarily generated by the coiled tubing (Δpcti) and the high-pressure hose (Δpr):
(14)pi=pb−∆pcti−∆pr,where pb is the pump pressure (MPa).


### 2.4.1. Theoretical Pressure Loss

The pressure loss along the coiled tubingΔpcti includes the spiral section loss Δpcti1 and straight section loss Δpcti2, i.e., Δpcti=Δpcti1+Δpcti2, which can be obtained from the previous studies [14].The theoretical pressure loss of the hose can be obtained using the following equation [28]:
(15)∆pr=Kr×59.7q2lrdri5Re0.25,where Kr=1, dri is the hose inner diameter (mm), q is the volume flow rate (L/min), lr is the total length of the hose (m), and Re is the Reynolds number, which should be written as λq/dri, and the coefficient λ is 1.12×104 under turbulent condition. According to the previous studies [14], the theoretical value of hose pressure loss is quite deviation from the actual, and then, it will greatly affect the judgment of the inlet pressure value of the bit. The correction of Equation (15) is necessary to be performed through an experimental test.


### 2.4.2. Experimental Test

In this study, several groups of tests were executed under different factors of the flow rate, hose inner diameter, and length; then, the measured pressure loss values under different factors were obtained; moreover, the parameters under each factor were substituted into Equation (15) to obtain the calculated value. By comparing the measured and calculated values, the correction factor Kr is introduced to Equation (15), and the deviation of Kr is analyzed to evaluate the effectiveness of the test. Eventually, Kr is taken as an average value.The following apparatuses were used (Figure2): a high-pressure pump, an SSMJ bit, high-pressure hoses of 1/4” (inner diameter of 6.35 mm) and 3/8” (inner diameter of 8 mm), a flowmeter, and two pressure gauges. The two gauges were, respectively, allocated at the inlet and outlet of the tested hose; therefore, the difference reading between pressure gauges No.1 and No.2 was the pressure loss value of the hose when the system operated steadily. The test results are listed in Table 1. Kr is introduced to correct Equation (15). The values of Kr under different conditions range from 0.23 to 0.27, with the relative standard deviation of 1.3%. A small deviation rate proves the effectiveness of the test; then, the final correction factor is taken as an average value, i.e., Kr=0.25.Figure 2
Schematic of the test system.Table 1
Test data of the high-pressure hoses.
Hose length (m)Hose inner diameter (mm)Flow rate (L/min)TestedΔpr (MPa)CalculatedΔpr (MPa)Correction factorKr106.3511710.841.680.26106.35552.811.180.25206.3511720.083.350.24206.35555.622.350.251081173.413.130.26108550.83.520.242081177.126.260.27208551.67.040.23


## 2.4.1. Theoretical Pressure Loss

The pressure loss along the coiled tubingΔpcti includes the spiral section loss Δpcti1 and straight section loss Δpcti2, i.e., Δpcti=Δpcti1+Δpcti2, which can be obtained from the previous studies [14].The theoretical pressure loss of the hose can be obtained using the following equation [28]:
(15)∆pr=Kr×59.7q2lrdri5Re0.25,where Kr=1, dri is the hose inner diameter (mm), q is the volume flow rate (L/min), lr is the total length of the hose (m), and Re is the Reynolds number, which should be written as λq/dri, and the coefficient λ is 1.12×104 under turbulent condition. According to the previous studies [14], the theoretical value of hose pressure loss is quite deviation from the actual, and then, it will greatly affect the judgment of the inlet pressure value of the bit. The correction of Equation (15) is necessary to be performed through an experimental test.


## 2.4.2. Experimental Test

In this study, several groups of tests were executed under different factors of the flow rate, hose inner diameter, and length; then, the measured pressure loss values under different factors were obtained; moreover, the parameters under each factor were substituted into Equation (15) to obtain the calculated value. By comparing the measured and calculated values, the correction factor Kr is introduced to Equation (15), and the deviation of Kr is analyzed to evaluate the effectiveness of the test. Eventually, Kr is taken as an average value.The following apparatuses were used (Figure2): a high-pressure pump, an SSMJ bit, high-pressure hoses of 1/4” (inner diameter of 6.35 mm) and 3/8” (inner diameter of 8 mm), a flowmeter, and two pressure gauges. The two gauges were, respectively, allocated at the inlet and outlet of the tested hose; therefore, the difference reading between pressure gauges No.1 and No.2 was the pressure loss value of the hose when the system operated steadily. The test results are listed in Table 1. Kr is introduced to correct Equation (15). The values of Kr under different conditions range from 0.23 to 0.27, with the relative standard deviation of 1.3%. A small deviation rate proves the effectiveness of the test; then, the final correction factor is taken as an average value, i.e., Kr=0.25.Figure 2
Schematic of the test system.Table 1
Test data of the high-pressure hoses.
Hose length (m)Hose inner diameter (mm)Flow rate (L/min)TestedΔpr (MPa)CalculatedΔpr (MPa)Correction factorKr106.3511710.841.680.26106.35552.811.180.25206.3511720.083.350.24206.35555.622.350.251081173.413.130.26108550.83.520.242081177.126.260.27208551.67.040.23


## 2.5. Mathematical Model of the Extension Limit

When the resultant forceF>0, the RJD system can continuously drill forward. As the drilled radial borehole length increases, the friction increases and F decreases gradually until F=0. Herein, the system is in an equilibrium condition of forces and the bit will not continue to drill forward with the hose; eventually, the value lrk under this condition is exactly the extension limit of the RJD system. Substituting F=0 into Equation (13), we can obtain the prediction model of the extension limit LEL:
(16)LEL=1.56pi∑b=1nbμbdb2cosθb−μfdm2+df2−dm2cosθx/2−π/4podo2−df2−Frz−μxkGxμrkGr.


## 3. Results and Discussion

### 3.1. Project Setting

A field construction condition of an oil well in Eastern Sichuan, China, with a radial well depth of 1500 m and casing model of 7” (inner diameter of 168.6 mm) was selected as the analyzed case of this study, and the working pressure of the field pump is 60 MPa. A pure water jet, with a temperature of approximately 90°C under this operating condition, a density of 965.4 kg/m3, and a viscosity of 0.315×10−3 Pa·s, was used. A drilling experiment using an SSMJ bit has been executed previously; then, a borehole with a diameter of 25 mm was obtained, and μrk and μzk are both 0.3. According to experience and the previous laboratory tests, the flow coefficients of the forward and backward nozzles μf and μb are both set as 0.83; furthermore, the friction between the hose and diverter Frz was obtained as 20 N through a ground test.A coiled tubing with a model of 1.5” was used, and its main parameters of lengthlct and outer and inner diameter dcto and dcti are shown in Table 2; additionally, Table 2 shows the length lr and outer and inner diameter dro and dri and submerged weight Gr of high-pressure hose with a model of 5/16”; the self-propelled bit was made by cemented carbide, and its specific dimensions are also shown in Table 2, such as density ρz, length lz, outer diameter do, inner diameter di, forward nozzle diameter df, impeller central hole diameter dm, jet diffusion angle θx, number of backward nozzles nb, diameter of backward nozzle db, and inclination angle of backward nozzle θb.Table 2
Main parameters of the project.
lct (m)dcto (mm)dcti (mm)lr (m)dro (mm)dri (mm)Gr (N/m)ρz (g/cm3)lz (mm)do (mm)di (mm)df (mm)dm (mm)θx (°)nbdb (mm)θb (°)200038.131.7512014.382.315301881.816031.330


### 3.2. Pressure Loss Analysis

The construction parameters were substituted into Ma et al.’s study [14] and Equation (15) to obtain the pressure loss distribution along the coiled tubing and high-pressure hose, separately. The main pressure loss of the system was observed to emanate from the high-pressure hose, which was 20.1 MPa, accounting for 92.6% of the total.The hose parameters (inner diameterdri and length lr) have a significant impact on the pressure loss (Figure 3). The pressure loss increases as a quadratic function with the increase in lr, which will reduce the extension limit. Therefore, the length of the hose should be as short as possible. During engineering construction, lr must be chosen according to the designed radial borehole length, and a little longer than the designed borehole length should be most suitable. The pressure loss is highly sensitive to the variation in dri; therefore, dri should be as large as possible, provided that the high-pressure hose can smoothly pass through the diverter.Figure 3
Hose pressure loss influenced by the hose length.


### 3.3. Extension Limit Results and Analysis

The construction parameters of an oil well in Eastern Sichuan were placed into Equation (16), and the extension limit of the RJD system LEL was calculated as 63.7 m. The force distribution in the horizontal section of RJD is shown in Figure 4. We observe that the recoil force (Fb) of the backward nozzle is the sole driving force for the system to drill forward; thus, increasing the equivalent diameter of the backward nozzles (any nozzle diameter or number) is the most effective method of increasing the drilling depth. The recoil force (Ff) of the forward nozzle is the main resistance of the system, accounting for 66.96% of the total resistance. Therefore, the diameter of the forward nozzle should be appropriately reduced, provided that its rock-breaking performance satisfies the design requirements. The friction between the hose and borehole (Frk) increases with the increase in drilling depth. When the axial force of the system is balanced, Frk also reaches the maximum. Herein, Frk accounts for 20.42% of the total resistance. A hose with a light weight and smooth outer surface should be selected to reduce Frk. In the bit structure design, the roundness of the drilled hole is used as an evaluation index. The friction between the hose and diverter (Frz) accounts for 9.3% of the total resistance. A hose with a small turning radius should be selected in a RJD system to reduce Frz. In the design of diverter, the hose and bit should be investigated on whether they can pass through the diverter smoothly.Figure 4
Force distribution of the RJD system.


### 3.4. Influential Factor Analysis for Extension Limit

#### 3.4.1. Pump Pressure

We changed the pump pressurepb and maintained other parameters in the engineering example to obtain the variation law of extension limit LEL with pb under different inner diameters of the hose (Figure 5). We observed that the extension limit increases approximately linearly with the increase in pb. Besides, the inner diameter of the high-pressure hose has a significant influence on LEL. When the inner diameter is 6.4 mm, the drill bit can drag the high-pressure hose forward after pb reaches 45 MPa, whereas when the diameter is 8 mm, the drill bit can drag the high-pressure hose forward when pb reaches 25 MPa. The larger the inner diameter, the greater the slope, additionally the higher the rise rate of LEL, which indicates that the increase in the inner diameter of the hose can effectively reduce the pressure loss and improve the inlet pressure of the bit.Figure 5
Extension limit influenced by the pump pressure.


#### 3.4.2. Forward Nozzle Diameter

We changed the forward nozzle diameter (df) and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 6). We observed that when the pump pressure is 60 MPa, the decline rate of LEL is greater than that when 40 MPa. This is because when the pump pressure is relatively high, the flow rate of the system is relatively large. As df increases, the increase rate of the system pressure loss is also faster, which increases the decline rate of the nozzle inlet pressure compared with when the pump pressure is relatively low, and finally increases the decline rate of the drilling extension limit. Under the same pump pressure, LEL decreases approximately linearly with the increase in df; therefore, reducing the diameter of the forward nozzle is conducive to increasing LEL. However, the main function of the forward nozzle is to generate an SSMJ for rock-breaking to enable the system to continuously drill forward. The smaller diameter of the forward nozzle weakens the energy of the jet and the rock-breaking performance of the bit becomes worse. Therefore, when selecting df, we should also combine with rock-breaking research and appropriately reduce df under the premise that the rock-breaking efficiency satisfies the operating requirements.Figure 6
Extension limit influenced by the forward nozzle diameter.


#### 3.4.3. Jet Diffusion Angle

We changed the jet diffusion angle (θx) by adjusting the slot dip angle of the impeller and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 7). We observed that under the same pump pressure, LEL increases as a cosine curve with the increase in θx. By increasing θx, the axial velocity of the swirling jet decreases, and then, the recoil force of the forward nozzle decreases. The decrease in the system resistance can improve LEL of the system, but an excessively large diffusion angle will also weaken the rock-breaking efficiency of the SSMJ. Therefore, the flow field and erosion performance analysis must be combined to obtain the optimal θx.Figure 7
Extension limit influenced by the diffusion angle.


#### 3.4.4. Impeller Central Hole Diameter

The variation law ofLEL was obtained by changing the diameter of the impeller central hole dm (Figure 8). Under the same pump pressure, LEL decreases in a quadratic function with the increase in dm. Therefore, the diameter can be appropriately reduced to increase LEL. However, according to the previous research, when dm is small, the central boss easily appears in the borehole bottom, which hinders the advancement of the bit and reduces the drilling speed. Therefore, the size of the central hole should satisfy the larger LEL and drilling speed. When dm is greater than 1.5 mm, LEL decreases to a lower value, and the decline rate reaches 40% under the pump pressure of 60 MPa. Thus, dm less than 1.5 mm is more appropriate; additionally, the specific results can be further investigated combining the drilling efficiency test.Figure 8
Extension limit influenced by the impeller central hole.


#### 3.4.5. Backward Nozzle Diameter

We changed the backward nozzle diameter (db) and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 9). Under the same pump pressure, LEL increases approximately linearly with the increase in db. According to the data, the increase rate of LEL decreases with the increase in db. For example, under the condition of a pump pressure of 60 MPa, when db changes from 1.22 to 1.24 mm, LEL increases by 8.55 m, whereas for a db changes from 1.36 to 1.38 mm, LEL increases by 7.85 m. Increasing db is the most effective method of increasing the drilling depth. However, at the same pump pressure, the increase in db increases the flow rate, additionally increases the water consumption and pressure loss of the system, and reduces the increase rate of LEL. Therefore, considering the system loss and economy, db cannot increase indefinitely and should be determined according to the drilling extension limit required by the construction design.Figure 9
Extension limit influenced by the backward nozzle diameter.


#### 3.4.6. Backward Nozzle Inclination Angle

We changed the inclination angle of the backward nozzle (θb) to obtain the variation law of LEL (Figure 10). Under the same pump pressure, LEL decreases as a cosine function with an increase in θb. Except for providing the SSMJ a driving force, the backward nozzle also has the function of reaming. Reducing θb is conducive to increasing LEL, but its reaming capacity decreases, and too small θb will cause the fluid to erode the high-pressure hose. According to the prediction model of LEL, when the pump pressure is 60 MPa and θb is 0, LEL is 111.91 m. When θb increases to 20°, 30°, and 40°, LEL decreases to 90.20, 63.68, and 27.69 m, respectively. When θb reaches 48.4°, LEL is 0. Overall, LEL decreases relatively rapidly when θb exceeds 30°. Therefore, it is reasonable to select θb of 30° to ensure both a large LEL and borehole diameter.Figure 10
Extension limit influenced by the backward nozzle inclination angle.


### 3.5. Parameter Sensitivity Analysis

In this study, the method of grey relational analysis (GRA) was used to investigate the influence degree of six factors above on the extension limit, such as pump pressurepb, forward nozzle diameter df, jet diffusion angle θx, impeller central hole diameter dm, backward nozzle diameter db, and backward nozzle inclination angle θb. GRA refers to the uncertain correlation between objects or between subsystems and between factors and main behavior. Its basic assignment is to analyze and determine the influence degree between factors or the contribution degree of factors to main behavior, based on the micro or macro geometric proximity of behavior factor sequences. The detailed calculation process is as follows:
(1)
Determine the reference and comparison sequences: taking the reference sequence reflecting the characteristics of system behavior asy, and the comparison sequence affecting system behavior as xi, the corresponding equation is as follows:
(17)y=ykk=1,2,⋯,n,xi=xikk=1,2,⋯,n,where k and i are the group k and column i of a specific quantity value in the matrix composed of sequences, yk is the reference sequence value of the data group k and its unit is related to the chosen physical quantity, and xik is the value of the i’th influential factor of data group k and its unit is also related to the selected physical quantity(2)
Dimensional normalization processing of the data: it is necessary to unify the unit for different parameters, usually the dimensional normalization methods including mean value, initial value, maximization, and minimization. The initial value method was adopted, since it is suitable for the data with a tendency and regularity and is often used in comprehensive evaluation, such as GRA(3)
Calculation of relation coefficient: the equation is as follows:(18)ξik=miniminkx0k−xik+ρ∙maximaxkx0k−xikx0k−xik+ρ∙maximaxkx0k−xik,where ρ is the resolution coefficient and has a value range between 0 and 1, usually as 0.5. The smaller ρ, the greater the difference between relation coefficients, and the stronger the discrimination ability. After calculating the relation coefficient, take the average value of each coefficient as the grey relation coefficient of the factor, and the calculation equation is
(19)ri=1n∑k=1nξik,where ri is the relation degree of the i’th influential factorTake the extension limitLEL of the jet bit under different parameters as the reference sequence, which is yo, and take pb, df, θx, dm, db, and θb as the comparison sequence, which are x1 to x6, respectively. It is found that taking LEL as the evaluation index, the order of parameter sensitivity from large to small is pb, θx, df, db, θb, and dm (Table 3).Table 3
Calculation results.
Comparison sequenceRelation coefficientSensitivity orderpb0.529631θx0.518742df0.518673db0.518564θb0.518525dm0.518426


## 3.1. Project Setting

A field construction condition of an oil well in Eastern Sichuan, China, with a radial well depth of 1500 m and casing model of 7” (inner diameter of 168.6 mm) was selected as the analyzed case of this study, and the working pressure of the field pump is 60 MPa. A pure water jet, with a temperature of approximately 90°C under this operating condition, a density of 965.4 kg/m3, and a viscosity of 0.315×10−3 Pa·s, was used. A drilling experiment using an SSMJ bit has been executed previously; then, a borehole with a diameter of 25 mm was obtained, and μrk and μzk are both 0.3. According to experience and the previous laboratory tests, the flow coefficients of the forward and backward nozzles μf and μb are both set as 0.83; furthermore, the friction between the hose and diverter Frz was obtained as 20 N through a ground test.A coiled tubing with a model of 1.5” was used, and its main parameters of lengthlct and outer and inner diameter dcto and dcti are shown in Table 2; additionally, Table 2 shows the length lr and outer and inner diameter dro and dri and submerged weight Gr of high-pressure hose with a model of 5/16”; the self-propelled bit was made by cemented carbide, and its specific dimensions are also shown in Table 2, such as density ρz, length lz, outer diameter do, inner diameter di, forward nozzle diameter df, impeller central hole diameter dm, jet diffusion angle θx, number of backward nozzles nb, diameter of backward nozzle db, and inclination angle of backward nozzle θb.Table 2
Main parameters of the project.
lct (m)dcto (mm)dcti (mm)lr (m)dro (mm)dri (mm)Gr (N/m)ρz (g/cm3)lz (mm)do (mm)di (mm)df (mm)dm (mm)θx (°)nbdb (mm)θb (°)200038.131.7512014.382.315301881.816031.330


## 3.2. Pressure Loss Analysis

The construction parameters were substituted into Ma et al.’s study [14] and Equation (15) to obtain the pressure loss distribution along the coiled tubing and high-pressure hose, separately. The main pressure loss of the system was observed to emanate from the high-pressure hose, which was 20.1 MPa, accounting for 92.6% of the total.The hose parameters (inner diameterdri and length lr) have a significant impact on the pressure loss (Figure 3). The pressure loss increases as a quadratic function with the increase in lr, which will reduce the extension limit. Therefore, the length of the hose should be as short as possible. During engineering construction, lr must be chosen according to the designed radial borehole length, and a little longer than the designed borehole length should be most suitable. The pressure loss is highly sensitive to the variation in dri; therefore, dri should be as large as possible, provided that the high-pressure hose can smoothly pass through the diverter.Figure 3
Hose pressure loss influenced by the hose length.


## 3.3. Extension Limit Results and Analysis

The construction parameters of an oil well in Eastern Sichuan were placed into Equation (16), and the extension limit of the RJD system LEL was calculated as 63.7 m. The force distribution in the horizontal section of RJD is shown in Figure 4. We observe that the recoil force (Fb) of the backward nozzle is the sole driving force for the system to drill forward; thus, increasing the equivalent diameter of the backward nozzles (any nozzle diameter or number) is the most effective method of increasing the drilling depth. The recoil force (Ff) of the forward nozzle is the main resistance of the system, accounting for 66.96% of the total resistance. Therefore, the diameter of the forward nozzle should be appropriately reduced, provided that its rock-breaking performance satisfies the design requirements. The friction between the hose and borehole (Frk) increases with the increase in drilling depth. When the axial force of the system is balanced, Frk also reaches the maximum. Herein, Frk accounts for 20.42% of the total resistance. A hose with a light weight and smooth outer surface should be selected to reduce Frk. In the bit structure design, the roundness of the drilled hole is used as an evaluation index. The friction between the hose and diverter (Frz) accounts for 9.3% of the total resistance. A hose with a small turning radius should be selected in a RJD system to reduce Frz. In the design of diverter, the hose and bit should be investigated on whether they can pass through the diverter smoothly.Figure 4
Force distribution of the RJD system.


## 3.4. Influential Factor Analysis for Extension Limit

### 3.4.1. Pump Pressure

We changed the pump pressurepb and maintained other parameters in the engineering example to obtain the variation law of extension limit LEL with pb under different inner diameters of the hose (Figure 5). We observed that the extension limit increases approximately linearly with the increase in pb. Besides, the inner diameter of the high-pressure hose has a significant influence on LEL. When the inner diameter is 6.4 mm, the drill bit can drag the high-pressure hose forward after pb reaches 45 MPa, whereas when the diameter is 8 mm, the drill bit can drag the high-pressure hose forward when pb reaches 25 MPa. The larger the inner diameter, the greater the slope, additionally the higher the rise rate of LEL, which indicates that the increase in the inner diameter of the hose can effectively reduce the pressure loss and improve the inlet pressure of the bit.Figure 5
Extension limit influenced by the pump pressure.


### 3.4.2. Forward Nozzle Diameter

We changed the forward nozzle diameter (df) and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 6). We observed that when the pump pressure is 60 MPa, the decline rate of LEL is greater than that when 40 MPa. This is because when the pump pressure is relatively high, the flow rate of the system is relatively large. As df increases, the increase rate of the system pressure loss is also faster, which increases the decline rate of the nozzle inlet pressure compared with when the pump pressure is relatively low, and finally increases the decline rate of the drilling extension limit. Under the same pump pressure, LEL decreases approximately linearly with the increase in df; therefore, reducing the diameter of the forward nozzle is conducive to increasing LEL. However, the main function of the forward nozzle is to generate an SSMJ for rock-breaking to enable the system to continuously drill forward. The smaller diameter of the forward nozzle weakens the energy of the jet and the rock-breaking performance of the bit becomes worse. Therefore, when selecting df, we should also combine with rock-breaking research and appropriately reduce df under the premise that the rock-breaking efficiency satisfies the operating requirements.Figure 6
Extension limit influenced by the forward nozzle diameter.


### 3.4.3. Jet Diffusion Angle

We changed the jet diffusion angle (θx) by adjusting the slot dip angle of the impeller and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 7). We observed that under the same pump pressure, LEL increases as a cosine curve with the increase in θx. By increasing θx, the axial velocity of the swirling jet decreases, and then, the recoil force of the forward nozzle decreases. The decrease in the system resistance can improve LEL of the system, but an excessively large diffusion angle will also weaken the rock-breaking efficiency of the SSMJ. Therefore, the flow field and erosion performance analysis must be combined to obtain the optimal θx.Figure 7
Extension limit influenced by the diffusion angle.


### 3.4.4. Impeller Central Hole Diameter

The variation law ofLEL was obtained by changing the diameter of the impeller central hole dm (Figure 8). Under the same pump pressure, LEL decreases in a quadratic function with the increase in dm. Therefore, the diameter can be appropriately reduced to increase LEL. However, according to the previous research, when dm is small, the central boss easily appears in the borehole bottom, which hinders the advancement of the bit and reduces the drilling speed. Therefore, the size of the central hole should satisfy the larger LEL and drilling speed. When dm is greater than 1.5 mm, LEL decreases to a lower value, and the decline rate reaches 40% under the pump pressure of 60 MPa. Thus, dm less than 1.5 mm is more appropriate; additionally, the specific results can be further investigated combining the drilling efficiency test.Figure 8
Extension limit influenced by the impeller central hole.


### 3.4.5. Backward Nozzle Diameter

We changed the backward nozzle diameter (db) and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 9). Under the same pump pressure, LEL increases approximately linearly with the increase in db. According to the data, the increase rate of LEL decreases with the increase in db. For example, under the condition of a pump pressure of 60 MPa, when db changes from 1.22 to 1.24 mm, LEL increases by 8.55 m, whereas for a db changes from 1.36 to 1.38 mm, LEL increases by 7.85 m. Increasing db is the most effective method of increasing the drilling depth. However, at the same pump pressure, the increase in db increases the flow rate, additionally increases the water consumption and pressure loss of the system, and reduces the increase rate of LEL. Therefore, considering the system loss and economy, db cannot increase indefinitely and should be determined according to the drilling extension limit required by the construction design.Figure 9
Extension limit influenced by the backward nozzle diameter.


### 3.4.6. Backward Nozzle Inclination Angle

We changed the inclination angle of the backward nozzle (θb) to obtain the variation law of LEL (Figure 10). Under the same pump pressure, LEL decreases as a cosine function with an increase in θb. Except for providing the SSMJ a driving force, the backward nozzle also has the function of reaming. Reducing θb is conducive to increasing LEL, but its reaming capacity decreases, and too small θb will cause the fluid to erode the high-pressure hose. According to the prediction model of LEL, when the pump pressure is 60 MPa and θb is 0, LEL is 111.91 m. When θb increases to 20°, 30°, and 40°, LEL decreases to 90.20, 63.68, and 27.69 m, respectively. When θb reaches 48.4°, LEL is 0. Overall, LEL decreases relatively rapidly when θb exceeds 30°. Therefore, it is reasonable to select θb of 30° to ensure both a large LEL and borehole diameter.Figure 10
Extension limit influenced by the backward nozzle inclination angle.


## 3.4.1. Pump Pressure

We changed the pump pressurepb and maintained other parameters in the engineering example to obtain the variation law of extension limit LEL with pb under different inner diameters of the hose (Figure 5). We observed that the extension limit increases approximately linearly with the increase in pb. Besides, the inner diameter of the high-pressure hose has a significant influence on LEL. When the inner diameter is 6.4 mm, the drill bit can drag the high-pressure hose forward after pb reaches 45 MPa, whereas when the diameter is 8 mm, the drill bit can drag the high-pressure hose forward when pb reaches 25 MPa. The larger the inner diameter, the greater the slope, additionally the higher the rise rate of LEL, which indicates that the increase in the inner diameter of the hose can effectively reduce the pressure loss and improve the inlet pressure of the bit.Figure 5
Extension limit influenced by the pump pressure.


## 3.4.2. Forward Nozzle Diameter

We changed the forward nozzle diameter (df) and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 6). We observed that when the pump pressure is 60 MPa, the decline rate of LEL is greater than that when 40 MPa. This is because when the pump pressure is relatively high, the flow rate of the system is relatively large. As df increases, the increase rate of the system pressure loss is also faster, which increases the decline rate of the nozzle inlet pressure compared with when the pump pressure is relatively low, and finally increases the decline rate of the drilling extension limit. Under the same pump pressure, LEL decreases approximately linearly with the increase in df; therefore, reducing the diameter of the forward nozzle is conducive to increasing LEL. However, the main function of the forward nozzle is to generate an SSMJ for rock-breaking to enable the system to continuously drill forward. The smaller diameter of the forward nozzle weakens the energy of the jet and the rock-breaking performance of the bit becomes worse. Therefore, when selecting df, we should also combine with rock-breaking research and appropriately reduce df under the premise that the rock-breaking efficiency satisfies the operating requirements.Figure 6
Extension limit influenced by the forward nozzle diameter.


## 3.4.3. Jet Diffusion Angle

We changed the jet diffusion angle (θx) by adjusting the slot dip angle of the impeller and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 7). We observed that under the same pump pressure, LEL increases as a cosine curve with the increase in θx. By increasing θx, the axial velocity of the swirling jet decreases, and then, the recoil force of the forward nozzle decreases. The decrease in the system resistance can improve LEL of the system, but an excessively large diffusion angle will also weaken the rock-breaking efficiency of the SSMJ. Therefore, the flow field and erosion performance analysis must be combined to obtain the optimal θx.Figure 7
Extension limit influenced by the diffusion angle.


## 3.4.4. Impeller Central Hole Diameter

The variation law ofLEL was obtained by changing the diameter of the impeller central hole dm (Figure 8). Under the same pump pressure, LEL decreases in a quadratic function with the increase in dm. Therefore, the diameter can be appropriately reduced to increase LEL. However, according to the previous research, when dm is small, the central boss easily appears in the borehole bottom, which hinders the advancement of the bit and reduces the drilling speed. Therefore, the size of the central hole should satisfy the larger LEL and drilling speed. When dm is greater than 1.5 mm, LEL decreases to a lower value, and the decline rate reaches 40% under the pump pressure of 60 MPa. Thus, dm less than 1.5 mm is more appropriate; additionally, the specific results can be further investigated combining the drilling efficiency test.Figure 8
Extension limit influenced by the impeller central hole.


## 3.4.5. Backward Nozzle Diameter

We changed the backward nozzle diameter (db) and maintained other parameters in the engineering example to obtain the variation law of LEL (Figure 9). Under the same pump pressure, LEL increases approximately linearly with the increase in db. According to the data, the increase rate of LEL decreases with the increase in db. For example, under the condition of a pump pressure of 60 MPa, when db changes from 1.22 to 1.24 mm, LEL increases by 8.55 m, whereas for a db changes from 1.36 to 1.38 mm, LEL increases by 7.85 m. Increasing db is the most effective method of increasing the drilling depth. However, at the same pump pressure, the increase in db increases the flow rate, additionally increases the water consumption and pressure loss of the system, and reduces the increase rate of LEL. Therefore, considering the system loss and economy, db cannot increase indefinitely and should be determined according to the drilling extension limit required by the construction design.Figure 9
Extension limit influenced by the backward nozzle diameter.


## 3.4.6. Backward Nozzle Inclination Angle

We changed the inclination angle of the backward nozzle (θb) to obtain the variation law of LEL (Figure 10). Under the same pump pressure, LEL decreases as a cosine function with an increase in θb. Except for providing the SSMJ a driving force, the backward nozzle also has the function of reaming. Reducing θb is conducive to increasing LEL, but its reaming capacity decreases, and too small θb will cause the fluid to erode the high-pressure hose. According to the prediction model of LEL, when the pump pressure is 60 MPa and θb is 0, LEL is 111.91 m. When θb increases to 20°, 30°, and 40°, LEL decreases to 90.20, 63.68, and 27.69 m, respectively. When θb reaches 48.4°, LEL is 0. Overall, LEL decreases relatively rapidly when θb exceeds 30°. Therefore, it is reasonable to select θb of 30° to ensure both a large LEL and borehole diameter.Figure 10
Extension limit influenced by the backward nozzle inclination angle.


## 3.5. Parameter Sensitivity Analysis

In this study, the method of grey relational analysis (GRA) was used to investigate the influence degree of six factors above on the extension limit, such as pump pressurepb, forward nozzle diameter df, jet diffusion angle θx, impeller central hole diameter dm, backward nozzle diameter db, and backward nozzle inclination angle θb. GRA refers to the uncertain correlation between objects or between subsystems and between factors and main behavior. Its basic assignment is to analyze and determine the influence degree between factors or the contribution degree of factors to main behavior, based on the micro or macro geometric proximity of behavior factor sequences. The detailed calculation process is as follows:
(1)
Determine the reference and comparison sequences: taking the reference sequence reflecting the characteristics of system behavior asy, and the comparison sequence affecting system behavior as xi, the corresponding equation is as follows:
(17)y=ykk=1,2,⋯,n,xi=xikk=1,2,⋯,n,where k and i are the group k and column i of a specific quantity value in the matrix composed of sequences, yk is the reference sequence value of the data group k and its unit is related to the chosen physical quantity, and xik is the value of the i’th influential factor of data group k and its unit is also related to the selected physical quantity(2)
Dimensional normalization processing of the data: it is necessary to unify the unit for different parameters, usually the dimensional normalization methods including mean value, initial value, maximization, and minimization. The initial value method was adopted, since it is suitable for the data with a tendency and regularity and is often used in comprehensive evaluation, such as GRA(3)
Calculation of relation coefficient: the equation is as follows:(18)ξik=miniminkx0k−xik+ρ∙maximaxkx0k−xikx0k−xik+ρ∙maximaxkx0k−xik,where ρ is the resolution coefficient and has a value range between 0 and 1, usually as 0.5. The smaller ρ, the greater the difference between relation coefficients, and the stronger the discrimination ability. After calculating the relation coefficient, take the average value of each coefficient as the grey relation coefficient of the factor, and the calculation equation is
(19)ri=1n∑k=1nξik,where ri is the relation degree of the i’th influential factorTake the extension limitLEL of the jet bit under different parameters as the reference sequence, which is yo, and take pb, df, θx, dm, db, and θb as the comparison sequence, which are x1 to x6, respectively. It is found that taking LEL as the evaluation index, the order of parameter sensitivity from large to small is pb, θx, df, db, θb, and dm (Table 3).Table 3
Calculation results.
Comparison sequenceRelation coefficientSensitivity orderpb0.529631θx0.518742df0.518673db0.518564θb0.518525dm0.518426


## 4. Conclusions

Through analysing the force and pressure loss of the RJD system, a model to predict the drilling extension limitLEL was established. The LEL value of a specific project was calculated and the pressure loss and forces were obtained. The influence laws of the main parameters on LEL were studied using the established prediction model. The conclusions are summarized as follows:
(1)
The sensitivity of pressure loss to the variation in the inner diameter is higher, because the diameter of high-pressure hose is significantly lower than that of the coiled tubing, and the main pressure loss of the system results from the high-pressure hose. The backward nozzle recoil force is the only driving force of the system, and the forward nozzle recoil force and friction between the hose and borehole contain the main resistance of the system. The drilling extension limit prediction value of the project in this article can reach up to 63.7 m(2)
Under the same pump pressure,LEL of the system increases approximately linearly with the increase in backward nozzle diameter, yet this will significantly increase hydraulic consumption and pressure loss; it decreases linearly with the increase in the forward nozzle diameter. Therefore, the design of the rear nozzle should not blindly pursue the large LEL but also consider that the hydraulic consumption and pressure loss are within a reasonable range; the forward nozzle diameter can be as small as possible on the premise of ensuring rock-breaking efficiency. LEL increases with the increase in the swirling jet diffusion angle; the diffusion angle can also be reduced to ensure the capability of rock-breaking rock. LEL decreases as a quadratic function with the increase in the backward nozzle installation angle; when the angle is larger than 30°, the decrease in LEL rate is high. From parameter sensitivity analysis, the factors that have the greatest and least influence on LEL are pb and dm, respectively. Hence, increasing pb is the most direct and effective way to raise LEL, while the design of dm does not depend on the requirements of LEL, but on its rock-breaking capacity(3)
This research provides a theoretical guideline for predicting the extension limit of RJD under a specific working condition, optimizing the bit structure, and choosing high-pressure hose and hydraulic parameters. The specific bit structure parameters should be further obtained through rock-breaking performance tests combined with this study. In addition, laboratory simulation of RJD experiment and field test will be performed in the future



---

*Source: 3078233-2022-05-23.xml*