Sonic Drilling with a Cavitation Hydraulic Vibrator: Theoretical Computation of the Energy Performance and Rate of Penetration

Abstract

A new approach to the numerical analysis of the dynamic interaction of rock and the drill string structure for sonic drilling of wells by using a cavitation hydraulic vibrator is proposed. The approach is formulated on the basis of the developed mathematical model of the “drill string - rock” dynamic system. The parameters of mud oscillations and mechanical vibrations of the drilling tool, as well as oscillation power at contact interaction of the bit with the rock, were determined. It is possible to establish the resonant frequencies of drilling tool mechanical vibrations and the rock, necessary for an increase in rate of penetration. The penetration rate was calculated in two ways: using the concept of hydraulic vibration power and drill vibration power. It has been established that the use of a drilling tool with a cavitation vibrator is more effective compared to other means of exciting the drilling tool vibration load. The intensification of the rock destruction process when drilling with a cavitation hydraulic vibrator is carried out due to an increase in the vibration energy transferred to the rock and a decrease in the energy intensity of the rock destruction process in resonant drilling modes.

1. Introduction

Exploration and production of hydrocarbon energy resources at great depths in difficult mining and geological conditions often leads to problems that make the construction of a well unprofitable. The construction of wells for the production and transportation of hydrocarbons requires significant capital investments required to complete the drilling of the wells. This is one of the most expensive operations in the oil and gas industry. As indicated in Ref. [1], “one of the most important parameters affecting drilling cost is the rate of penetration (ROP)”. To increase ROP, the method of imposing longitudinal vibrations on a drilling tool is often used. This is achieved by using mechanical hydraulic hammers operating at vibration frequencies up to 100 Hz and by new high-frequency cavitation pulse devices for the creation of hydraulic pressure pulses in the drilling fluid, developed and implemented in the last decade.

1.1 Mechanical hydraulic shock devices

Despite the significant disadvantages of mechanical hydraulic shock devices, such as design complexity, the presence of moving quickly wearing parts, low efficiency, etc. [2], work to improve their characteristics continues to be carried out. Thus, in the last decade, in the Russian Federation, the theory of volumetric hydraulic impact machines has been developed and original hydraulic impact and distribution devices have been created [3].

Research into submersible hydraulic shock systems with higher efficiency is being carried out in a number of other industrial countries. This is evidenced by the abundance of new works on this problem [4, 5, 6, 7, 8, 9].

So, a new drilling rig was proposed with an oscillator based on a positive displacement motor with a cam mechanism [4]. Its design is developed to implement the necessary hydraulic shock impacts on the drill at drilling wells. Mathematical modeling of longitudinal vibrations of the “drill string – oscillator – drilling tool” dynamic system was carried out by integrating a system of nonlinear differential equations given in Ref. [4], describing the dynamic processes of the oscillator structure and the fluid medium in its flow part by using the finite element method. The simulation results allowed the authors of the work to obtain the dynamic characteristics of the system (vibration displacement and vibration speed, as well as the vibration spectrum) at various positions of the drill string. As for the frequency of the dynamic action of the oscillator on the drill bit, it is clear from the spectrogram obtained by the authors using the Fourier transform that the fundamental mode of the oscillations of the vibration movement of the drill corresponds to an oscillation frequency of 6 Hz. At the same time, the authors of Ref. [4] claim that the maximum response of the dynamic system in experimental studies was observed at a frequency of 12 Hz. Data from field experiments showed “that the new design could increase ROP obviously”. Analysis of the kinematics and dynamics of such a drilling tool is described in detail in Ref. [5].

The working mechanism and dynamic characteristics of the new rotary impact tool with a disk impact system were studied in Ref. [6]. Field tests have confirmed the effectiveness of using such a tool during axial vibrations of a drill bit with the forced vibration frequency of 20 Hz. The authors claim that penetration rates have increased by more than 30% and “that the tool has a significant effect of increasing the ROP”.

To create longitudinal vibrations of a drill bit, the authors of research [7] developed a new tool with the generation of high-frequency axial impacts by a pair of mating cams. The main efforts of the authors of the work were aimed at developing a reliable and safe design in difficult drilling conditions. It is stated that “this tool has relatively good ROP improving effect”.

Based on the analysis of the mechanism of rock destruction under complex shock loads, the authors of Ref. [8] considered a new tool for combined impact drilling with a shock load of up to 8.6 kN and an impact frequency reaching 43 Hz. This paper states that “under cyclic loading, as the number of impacts increases, the degree of rock destruction increases significantly”. This makes it possible to reduce the drill impact load threshold to preserve the integrity of the drilling tool and premature wear of its working surface.

A mathematical model of vibration, as a key means of analyzing the dynamics and optimizing downhole tools, is considered in Ref. [9] using the example of a hydrogenerator to improve the efficiency of drilling various wells. The dynamic parameters of the vibration system were determined by calculation: vibration frequencies, displacement, speed and acceleration. Satisfactory agreement between the results of numerical calculations and experimental testing has been established, which confirms the adequacy of the proposed model.

Analyzing working processes in the devices proposed in Refs. [4, 5, 6, 7, 8, 9], it is necessary to note their complexity. The presence of moving parts, rubber seals, etc., significantly reduces their service life. At the same time, mathematical models that describe the dynamics of the working process of these devices in case of interacting with destroyed rock are very useful in educational terms.

1.2 High-frequency hydraulic vibrators

In parallel with the improvement of the design of impact mechanisms, a second promising scientific direction developed. According to this direction, to increase the speed of well penetration, periodically stall-type hydrodynamic cavitation is used. Such cavitation flow, realized on hydraulic resistances of various types (washer, disk, impeller blade, Venturi tube, etc.), can lead to an increase in ROP due to:

• effective removal of drilled rock from under the drilling tool by pulsating jets, excluding its repeated “grinding”;

• axial alternating vibrations of the drill bit in the sonic frequency range;

• reducing the friction of the drill string against the wellbore during its longitudinal vibrations;

• resonance phenomena that occur when the oscillation frequency of the drilling tool is close to the natural frequency of the rock being destroyed.

Examples of such studies of cavitation periodically stalled flows to increase drilling speed were presented by authors from various countries at conferences and published in works on geomechanics and well drilling. For example, laboratory studies performed by Babapour & Butt in Ref. [10] showed the possibility of increasing ROP by applying an axial vibration force to the bit using a pulsed cavitation drilling tool. Its design includes a Venturi tube that creates cavitation bubbles in the drilling mud. It is argued that the collapse of cavitation bubbles formed inside the tool generates pressure pulsations and oscillatory forces on the bit. Researchers using a Venturi tube that creates a pulsating jet to effectively clear drilled rock from the bottom came to the same conclusion [11].

Using three-dimensional (3D) numerical analysis technologies and high-speed photography, the authors of Ref. [12] obtained new data on the influence of the geometric parameters of the nozzle on the dynamic characteristics of the flow realized. The study found that “the optimal values of nozzle throat length and divergent angle are twice the throat diameter and 40°, respectively”.

An analytical model of pulsation pressure mud characteristics in drilling tool was created in Ref. [13]. The results showed that tool design and operating conditions have a significant impact on the frequency and amplitude of drill bit pressure pulsations. The use of this technology can allow optimization of frequency and amplitude, taking into account the lithology of the formations being drilled. This will lead to lower drilling costs and “lead to a larger number of oil and natural gas plays being profitable”.

A simplified model of a high-frequency harmonic vibration-impact system of a drill bit-rock was considered in Ref. [14]. It analyzes the influence of rock hardness, energy dissipation, impact frequency and other parameters on the drilling efficiency. Numerical modeling of the operation of such a system to study its dynamic characteristics showed that when the frequency of forced oscillations of the drill bit is close to the natural frequency of the rock being destroyed, the drilling process is greatly intensified.

Resonance phenomena were also studied in the technological process of geological exploration work with core selection in case of drilling with a polycrystalline diamond bit [15]. For identical drilling conditions at resonance drilling compared to traditional technology, an increase in ROP of up to 180% was established. At the same time, a high quality was achieved with extracted core. Experimental and field studies performed in Ref. [16] demonstrated that downhole vibrations in the drill string are the most effective methods of reducing friction and improving axial force transmission in high-angle, long-reach wells.

Note that the natural vibration frequencies of some studied hard rock samples (hardness is indicated on the Mohs scale), obtained from different sources, is shown in Figure 1 and are about 37 kHz.

Figure 1.

Natural vibration frequency of some studied hard rock samples: 1 is siltstone with iron additives, 2 is siltstone (hardness up to 6 units), 3 is argillite (up to 2 6 units), 4 is conglomerate of other compounds (up to 6–7 units), 5 is conglomerate of other compounds (up to 6 units), 6 is granite (up to 7 units) and 7 is metamorphic rock (up to 6–7 units).

At the same time, modern studies carried out in Ref. [17] on the experimental drilling of various rocks using the vibration method have shown that a significant increase in the ROP is observed in the frequency range from 1 to 10 kHz.

Such relatively low values of the frequencies of forced vibrations of the bit in comparison with the frequencies of vibrations of the rock (see Figure 1), obtained in the above-mentioned forced vibrations of the drill bit, which are the most effective in terms of ROP, can be explained by the presence of natural heterogeneities of the rock mass and the appearance of differently inclined cracks in the contact zone of the drill bit tool and destroyed material [18, 19]. So, numerical modeling was carried out with subsequent experimental confirmation of the destruction zone near the point of axial impact [18]. Based on the basic principles of mechanical vibrations, the mechanism and effect of rock destruction using high-frequency vibrations of a polycrystalline diamond bit was studied in Ref. [19]. The influence of oscillation frequency on the drilling speed and the magnitude of the dynamic response of the rock has been established. The authors of the above-mentioned work came to the conclusion that the application of high-frequency vibrations to a polycrystalline diamond bit can increase its service life, increase the drilling speed and, as a result, reduce the cost of drilling.

In recent years, researchers from different countries have been considering the possibility of using ultrasonic vibrations in rock destruction technologies [20, 21, 22], including the drilling operations of oil and gas wells. The review [23] summarizes the results of studies on the use of sound and ultrasonic vibrations in successful technologies implemented in related industries, and examines the problems improving high-frequency vibrations during cavitation of the flushing liquid, and opportunities were determined for increasing the magnitude of high-amplitude impacts and achieving resonance of the drilling tool with the destroyed rock. However, as the authors of the review “believed that, at present, the application of ultrasonic-assisted rock-breaking technology in the oil- and gas-drilling industry still faces some problems and challenges”. One of these problems is the propagation of ultrasonic vibrations in downhole conditions, insufficient knowledge of the frequency spectrum and amplitude characteristics.

Thus, considering various methods of initiating high-frequency vibrations on a drilling tool, we can come to the conclusion that hydrodynamic cavitation of a periodically stall type can serve as a reliable source of intensification of the technological process of well construction.

So, Chinese researchers to overcome a number of problems due to the high cost of drilling during exploration and construction of ultra-deep wells (2000–6000 m), a new downhole tool was developed. The main element of this instrument was a hydraulic generator of cavitation jet, the pulsations of which are amplified in the resonance chamber. A detailed description of this tool is given in Ref. [24]. A study on the working mechanism of the new device and its dynamic characteristics are presented in Ref. [25]. At flushing mud flow rates higher than 32 L/s, the tool realizes mud pressure fluctuations with a range of ∆Р ≈ 2.1 ÷ 2.2 MPa and with the base frequency of 10 Hz. Superimposed on the fundamental harmonic oscillations are pulsations of increased frequency (approximately 80 Hz) with a range of pressure fluctuations of ∆P ≈ 0.56 ÷ 0.60 MPa.

The penetration rate and cutting efficiency of hydraulic pulsed cavitation jet drilling have been proven in field conditions in oil fields throughout China by drilling more than 100 wells with a maximum depth of 6162 m. At the same time, the penetration rate increased by more than 25%. According to downhole tool developers, this achievement is due to jet pulsation, cavitating erosion and the effect of local negative pressure, as well as improved bottom hole cleaning efficiency.

Despite the positive results of field tests, in the opinion of the authors of this work, the new downhole tool has the following disadvantages:

• the presence in its design of a moving part (impeller) and rubber cuffs, which significantly reduce the inspection period and service life;

• strict requirements for the purity of the flushing liquid due to small gaps between the impeller blade and its structure;

• additional energy consumption of drilling fluid for the drive to rotate the impeller;

• low frequencies of pulse action in conditions of deep drilling of hard rocks lead to strong vibrations, a decrease in drilling speed and premature failure of equipment [25].

Thus, the analysis of recent studies on the influence of the frequency of axial vibration loading of a drilling tool showed that:

• during sonic drilling, the natural vibration frequency of the rock in the zone of its destruction is approximately an order of magnitude lower than the values shown in Figure 1, which is explained by the formation of a network of differently inclined microcracks in the zone of its contact with the drilling tool [18];

• at the same time, the strength of the rock decreases and the speed of penetration increases up to two times [17].

The above frequencies of excitation of drill bit oscillations (from 1000 Hz to 10 kHz) can be achieved by installing a cavitation hydraulic vibrator [26] in the drill string.

The purpose of the presented work is to elaborate on the theoretical calculation of the energy parameters in the cross-section of the drilling tool and the ROP of the well for the case of drilling using a cavitation hydraulic vibrator of the specified type.

To achieve this goal, the following tasks were solved:

• calculation of hydraulic and mechanical vibration forces on a drilling tool;

• computer modeling with numerical determination of the resonant frequencies of pressure oscillations p₂ of the flushing fluid and the drilling tool vibration acceleration Z using experimental data;

• comparative analysis of the drilling efficiency using a high-frequency mechanical vibrator and a cavitation hydraulic vibrator.

2. Layout diagram of cavitation hydraulic vibrator as part of the drill string

Using the accumulated experience in the development and industrial testing of cavitation technologies in various industries [27], including mining [28], at the Institute of Technical Mechanics of the National Academy of Sciences of Ukraine (ITM NASU) created a new direction in the development of submersible impact machines. The key element in such devices was a cavitation generator (hereinafter referred to as the generator is a Venturi tube of special geometry), which implements shock vibrations in the acoustic frequency range and above [29]. At certain operating modes of the generator, the pressure oscillations of the flushing liquid in the flow part behind the generator exceed the pressure at its inlet. Shock vibrations, interacting with the drill structure, are transformed into longitudinal vibrations of the string housing with vibration accelerations of several thousand g.

The image (in 3D projection) of cavitation hydraulic vibrator and its layout diagram as part of the drill string are shown in Figure 2.

Figure 2.

a is the image (in 3D projection) of cavitation hydraulic vibrator and b is its layout diagram as part of the drill string).

As can be seen from Figure 2a, the hydraulic vibrator consists of a housing 1, which is a hollow structure with conical threads at the ends for installation in a drill string. A cavitation generator 2 is installed at the entrance to the hydraulic vibrator housing with hydraulic input channel 5 to level the velocity field. The hydraulic vibrator body contains a flow hydraulic channel 3 with a diameter equal to the diameter of the generator diffuser at the outlet. The housing also provides space for installing a pressure pulsation sensor 4 and turnkey milled planes 6 for easy installation of the hydraulic vibrator as part of the drill string.

As can be seen from the layout diagram (Figure 2b), the cavitation hydraulic vibrator 2 is mounted by means of conical threads in the drill string 1 with a rock-cutting tool 3. When the flushing fluid passes through the generator, as a result of the periodic growth and separation of the cavern 5, it collapses in the high-pressure zone 6 in the flow channel of the hydraulic vibrator. In this case, high-frequency self-oscillations of shock-type liquid pressure are realized with a range ΔР exceeding the pressure value at the inlet to the generator [29]. The structure, consisting of elements from the entrance to the hydraulic vibrator to the rock-cutting tool, is further designated as a drilling rig.

The testing schematic for such a drilling rig, its geometric parameters and the main test results under various operating modes are given in Ref. [26]. Figure 3 shows dynamic process as an example in the cross-section of a rock-cutting tool. By the pressure p₂, one can judge the quality of flushing of the drilled rock, and the vibration acceleration of the projectile drill structure Z₂ characterizes the ROP. The same figure also shows the vibration acceleration Z₁ of the drill string at the vibrator entrance.

Figure 3.

Pressure p₂ and vibration accelerations Z₂ in the cross-section of the rock-cutting tool and at the entrance to the hydraulic vibrator Z₁ vs. time t.

Analysis of the given oscillograms indicates the main (first mode) frequency of pressure oscillations p₂, realized by the cavitation hydraulic vibrator in the operating mode under study, is 397 Hz. The vibration modes of higher frequencies are “superimposed” to this dynamic process. The manifestation of these frequencies is associated with the influence of the length of the hydraulic line of the drill string, as well as with such dynamic phenomena as repeated collapses of the detached part of the cavern in the flow channel of the hydraulic vibrator and the interaction of the fluid in it with mechanical vibrations of the drill string structure.

Frequency of the first mode of oscillation vibration acceleration Z₂ in the cross-section of the drill tool, as a rule, coincides with the frequency of cavitation pressure oscillations p₂. At the same time, high-frequency mechanical vibrations are imposed on them due to higher fashion designs. This leads to the appearance of resonance phenomena, as will be shown below, and subsequently to the development of differently inclined cracks in the rock.

Regarding the vibrations above the hydraulic vibrator (section with the parameters of p₁, Z₁), then in the work performed earlier at the ITM NASU [29], it was established that pressure oscillations p₂ upstream are completely “cut off” (suppressed) by the high compliance of the cavitation cavern located behind the throat section of the generator. Mechanical vibrations of the drill string structure in the initial section Z₁ of the cavitation hydraulic vibrator (see Figure 3) amount to no more than 7% of the vibrations of the structure in the cross-section of the rock-cutting tool.

The developed drilling rig underwent a full range of experimental studies on hydraulic (at ITM NASU) and drilling test stands (at SKB “Geotekhnika”, Podolsk) [26, 30] and confirmed its performance in field conditions [31].

3. Theoretical approach to computation of the hydraulic and mechanical oscillatory powers of the drill string on the rock-cutting tool

When creating a new type of drilling tool using a cavitation pulse device to intensify the process of drilling a well, the development and application of methods for theoretically predicting the parameters of work processes in the installation become of great importance. These methods must be reliable, but at the same time quite simple, and they must use a minimum number of specified empirical constants.

The mathematical model of the dynamic system “drilling tool with a hydraulic vibrator” proposed in Ref. [32] made it possible to obtain not only qualitative, but also quantitative agreement between the calculated and experimental parameters for the elastic fastening of the drilling tool on the stand. In order to take into account the mass forces of the drill pipe and the axial load created by the hydraulic cylinder when drilling wells, as well as the force of interaction between the rock-cutting tool and the rock, a mathematical model of the “drill tool with a cavitation hydraulic vibrator – rock” system was subsequently developed [33].

The mathematical model of the “drilling tool with a hydraulic vibrator” dynamic system proposed in Ref. [32] made it possible to obtain not only qualitative, but also quantitative agreement between the calculated and experimental parameters for the elastic fastening of the drilling tool on the stand. In order to take into account the mass forces of the drill pipe and the axial load created by the hydraulic cylinder, as well as the force of interaction between the rock-cutting tool and the rock, a mathematical model of the “drill tool with a cavitation hydraulic vibrator – rock” system was subsequently developed [33].

To better coordinate the calculated and experimental data, the dependences of dissipative losses on the amplitude of vibrations of the drill structure in the corresponding finite elements of the system were introduced into the model [33]. This made it possible to calculate the dynamic parameters of the system in accordance with experimental data and, subsequently, to perform a theoretical study of the hydraulic [34] and mechanical [35] powers of the drilling rig depending on the cavitation flow regime.

3.1 Mathematical model of the “drill bit with cavitation hydraulic vibrator: Rock” dynamic system

The energy characteristics of devices operating in the cavitation stall flow mode have been studied in a number of works. In particular, the goal of work [36] was to study the mechanism for increasing ROP. At the same time, the efficiency of converting the hydraulic energy of the pulse jet into mechanical specific energy (the amount of energy required to destroy a unit volume of rock) was determined. According to this theory, “the major mechanisms of improving ROP in pulsed-jet drilling are changing the breaking strength of rock and improvement of downhole-cuttings-cleaning efficiency”.

The mathematical model of the “drill bit with a cavitation hydraulic vibrator – rock” system was considered as an elastic-mass system consisting of finite elements. The total number L of finite elements used, which determines the longitudinal length of the i-th structural element, was selected based on the relationship [37] between the length of the element and the minimum wavelength of oscillations of the dynamic system in the frequency range under study:

$$\Delta l_i\le \frac{a_{\mathit{as}}}{10f_{\mathit{max}}},$$E1

where a_as is the sound speed in the material of the drill string structure and f_max is the maximum oscillation frequency of the frequency range under study.

The finite element schematic for computation modeling of dynamic interaction of drill string longitudinal oscillations and cavitation self-oscillations flow in hydrovibrator is shown in Figure 4. For the energy efficiency estimation and optimization of the drilling operation, the mathematical model of the “drill string with a hydraulic vibrator – rock” dynamic system must describe for the drill string various structural elements’ time changes of such parameters as vibration displacement, vibration velocity, vibration acceleration, as well as pressure and fluid flow rate oscillations in the flow path of the drill string’s corresponding elements. The mathematical modeling of the longitudinal vibrations of the drill string was carried out by integrating a system of differential equations describing the dynamic processes of the drill string structure and the fluid in its flow path.

Figure 4.

The finite element schematic for computation modeling of dynamic interaction of drill string longitudinal oscillations and cavitation self-oscillations flow in a hydrovibrator.

The mathematical model of the drill string longitudinal oscillations can be described using the finite element method [38].

$$m_i\frac{d^2{z}_i}{d{t}^2}+c_i\left(z_i-z_{i-1}\right)+b_i\left(\frac{d{z}_i}{\mathit{dt}}-\frac{d{z}_{i-1}}{\mathit{dt}}\right)+c_{i+1}\left(z_i-z_{i+1}\right)+b_{i+1}\left(\frac{d{z}_i}{\mathit{dt}}-\frac{d{z}_{i+1}}{\mathit{dt}}\right)=F_i,$$E2

where: $z_i$ is the mass center coordinate of the i-th finite element from the position of dynamic equilibrium; $m_i$ is mass of the i-th finite element of the drill string; $c_i=E_i{A}_i{l_i}^{-1}$ is stiffness coefficient of the i-th finite element; $A_i$ is cross-sectional area of the i-th structural finite element; $l_i$ is longitudinal length of the i-th structural finite element; $E_i$ is Young’s modulus of elasticity of the material of the i-th structural finite element; $b_i$ is damping coefficient of the i-th finite element at small vibrations of the structure; $F_i$ is the external force, acting on the i-th finite element of drill string structure.

Taking into account the dissipation of the vibration energy for the modeling of drill string longitudinal vibrations, in turn, is a problem, since until now there are no sufficiently accurate models for describing the energy losses during elastic vibrations of such structures, and the available experimental data on the magnitude of these forces are very scarce.

To determine the damping coefficients in the equations of i-th finite element of drill string, it is proposed to use the expression [39].

$$b_i=\left(b_i^0+b_i\left(a_i\right)\right),$$E3

where: $b_i^0$ is damping coefficient of vibrations of the i-th finite element at small vibrations of the drill string structure; $b_i\left(a_i\right)$ is component of the damping coefficient of mechanical elements of a dynamic system, taking into account the increase in structure amplitude $a_i$ at considerable substantially nonlinear vibrations of the structure at resonance modes of the dynamic system.

Thus, as can be seen from Eq. (2) mentioned above, each finite element, based on the scheme of the drill string structure, was described by mass ($m_i$), elastic ($c_i$) and dissipative ($b_i$) characteristics. The logarithmic damping ratio of the system was set, based on the values of these quantities, for a specific type of connection of drill string parts and the drill string material used in its construction, and varied from 0.06 to 0.2 [40].

The string structural element of the cavitation generator can be schematically represented as a round rod with a through hole along its longitudinal axis, consisting of a “pipe”-type element of variable cross-section (see Figure 5, where $A_1$, $A_2$ are the cross-sectional areas of the flow part of the cavitation generator in its inlet and outlet sections, and l_g is axial length of the cavitation generator). Consider the flow of an incompressible fluid in a cavitation generator (see Figure 5), oscillating with longitudinal acceleration $\ddot{z}$.

Figure 5.

Simplified image of a cavitation generator.

To determine the total force effect of the liquid medium on the cavitation generator structure (as the incompressible duct) of the drill, we will use the law of conservation of momentum (in vector form). The integral external force ${\overrightarrow{F}}_A$ acting on the closed surface A, which limits the fluid volume within the flow path of the cavitation generator rigid housing, can be defined as

$${\overrightarrow{F}}_A=\frac{d\overrightarrow{M}}{\mathit{dt}}+{\int }_A\rho {\overrightarrow{U}}_a\left({\overrightarrow{U}}_a·d\overrightarrow{A}\right),$$E4

where $\overrightarrow{M}$ is the value of the momentum for the accelerating volume of fluid in the cavitation generator; $t$ is time; $\rho$ is density of the fluid medium; ${\overrightarrow{U}}_a$ is the absolute speed of fluid movement through the area $A\left(l\right)$ at the current length l of the cavitation generator structure; and $\mathit{dA}$ is elementary area directed along the normal to the fluid surface.

For the volume of fluid contained in the cavitation generator structure (see Figure 5), the value of the momentum $\overrightarrow{M}$ can be written in the form

$$\overrightarrow{M}={\int }_0^{l_g}\mathit{\rho A}\left(l\right){\overrightarrow{U}}_a\left(l\right)\mathit{dl}.$$E5

The speed $U$ of fluid movement in the cavitation generator flow path relative to its structure will be expressed for one-dimensional (1D) fluid motion as

$$U\left(l\right)=U_a\left(l\right)-\dot{z}=\frac{\dot{W}}{\mathit{\rho A}\left(l\right)},$$E6

where $\dot{z}$ is speed of the wall of the cavitation generator structure; W is the relative flow rate of drilling fluid in the pipeline wall coordinate system (steady component $\stackrel{-}{W}$ plus dynamic component $\dot{w}$); $\text{and}\rho$ is fluid density.

Using Eqs. (5) and (6), the momentum vector can then be written as

$$M=\dot{W}{l}_g+m_g\dot{z}.$$E7

Applying the theorem on the momentum of motion to a closed volume of fluid, the ${\int }_A\rho {\overrightarrow{U}}_a\left({\overrightarrow{U}}_a·d\overrightarrow{A}\right)$ in Eq. (5) for the cavitation generator entering the input (Section 1) can be expressed as

$$\rho \left(\frac{\dot{W}}{\rho A_1}+\dot{z}\right)\left(\frac{\dot{W}}{\rho A_1}{A}_1+\dot{z}{A}_1\right),$$E8

and the momentum of motion of the fluid leaving the cavitation generator (Section 2)

$$\rho \left(\frac{\dot{W}}{\rho A_2}+\dot{z}\right)\left(\frac{\dot{W}}{\rho A_2}{A}_2+\dot{z}{A}_2\right).$$E9

The net force perturbation on the cavitation generator fluid surface$F_2$ is

$$F_2=\ddot{w}{l}_g+m_g\ddot{z}+\frac{{\dot{W}}^2}{\rho }\left(\frac{1}{A_2}-\frac{1}{A_1}\right)-\rho {\dot{z}}^2\left(A_2-A_1\right)$$E10

This force must also equal the pressure-area forces on the two end cross- sections of the fluid plus the force that the duct wall imparts to the fluid (negative of the force Fg that the fluid imparts to the cavitation generator duct). Therefore,

$$F_2=p_1{A}_1-p_2{A}_2-F_g$$E11

The total force $F_g$ ($F_i$ in Eq. (2) in the section of the cavitation generator installation), acting to the cavitation generator structure (containing the attached cavity in the self-oscillating mode) with vibration acceleration $\ddot{z}$, is written in the form:

$$F_g=p_1{A}_1-p_2{A}_2-\ddot{w}{l}_g-m_g\ddot{z}-\frac{{\dot{W}}^2}{\rho }\left(\frac{1}{A_2}-\frac{1}{A_1}\right)+\rho {\dot{z}}^2\left(A_2-A_1\right),$$E12

where $p_1$, $p_2$ are pressures and $m_g$ is the fluid mass in the cavitation generator flow channel.

The external forces F_i in Eq. (2), acting to the drill string structure from the hydraulic channels of cavitation hydrovibrator and drill pipes, are calculated by a similar way.

Thus, longitudinal vibration accelerations ${\ddot{z}}_i$ of the drilling assembly structure occur under the action of the dynamic components of the pressure forces of the drilling fluid (in the mode of cavitation self-oscillations) through the contact surfaces of the liquid medium (flushing fluid) with the drilling assembly structure. For a separate structural finite element with internal liquid filling in the mathematical model of the ‘drill string with a hydraulic vibrator – rock’ dynamic system, these forces in the projection on the longitudinal direction of drill string movement are the sum of the fluid pressure forces in the end sections of the structural element (taking into account the changes in the flow area of the flow paths) and from the forces of inertia of the liquid medium associated with the acceleration of the drilling string structure.

3.2 Mathematical model of the dynamics of cavitation formations in the cavitation generator

The equations of fluid cavitation oscillatory processes used to calculate the dynamic pressure in the flow channel of the cavitation hydrovibrator are described in accordance with Ref. [33]. The study of physical phenomena in the cavitation hydrovibrator by Ref. [29] indicated the existence in the fluid flow of the two zones with a periodic change in the volume of the cavitation cavity. There is the zone of growth of the attached cavity $V_c^{\prime }$ (attached cavern at Figure 4), located directly behind the critical section of the generator, and the zone with the collapse of the cavity $V^{″}$ (zone with the collapse of the cavity at Figure 4) after its separation from the generator diffuser. The periodic change in the length of the attached cavitation cavity in the generator diffuser is taken into account here, like in Ref. [33], by introducing the boundary condition at the entrance to the diffuser part of the cavitation generator, i.e., by dependence of fluid pressure on the volume of the attached cavity $p_2^{\prime }\left(V_c^{\prime }\right)$ .

The system of differential equations describing the process of oscillating fluid motion in the drill string flow channel with the cavitation hydrovibrator is given in Ref. [33] and here has the form:

equations of unsteady fluid motion from the entrance to the cavitation hydrovibrator to the point of contact of the drill string with the rock:

$$p_2^{\prime }\left(V_c^{\prime }\right)=p_1+a_1{\dot{W}}_1^2+J_1\frac{d{\dot{W}}_1}{\mathit{dt}}+{\rho }_f{l}_1\frac{d^2{z}_1}{d{t}^2},$$E13

$$p_{j-1}=p_j+a_j{\dot{W}}_j^2+J_j\frac{d{\dot{W}}_j}{\mathit{dt}}+{\rho }_f{l}_j\frac{d^2{z}_j}{d{t}^2}j=2,3,\dots ,N-1,$$E14

$$p_{N-1}={\stackrel{-}{p}}_3+a_N{\dot{W}}_N^2+J_N\frac{d{\dot{W}}_N}{\mathit{dt}}+{\rho }_f{l}_N\frac{d^2{z}_N}{d{t}^2},$$E15

mass conservation equation

$$-C_i\frac{d{p}_i}{\mathit{dt}}={\dot{W}}_{i+1}-{\dot{W}}_i+C_i{r}_i\left(\frac{d{\dot{W}}_{i+1}}{\mathit{dt}}-\frac{d{\dot{W}}_i}{\mathit{dt}}\right),i=1,2,\dots ,N-1,i\ne K$$E16

equation of mass conservation in the cavity collapse section

$$\rho \frac{d{V}_c^{″}}{\mathit{dt}}-C_i\frac{d{p}_i}{\mathit{dt}}={\dot{W}}_{i+1}-{\dot{W}}_i+C_i{r}_i\left(\frac{d{\dot{W}}_{i+1}}{\mathit{dt}}-\frac{d{\dot{W}}_i}{\mathit{dt}}\right),i=K$$E17

equation of the dynamics of the cavity volume

$$\frac{d{V}_c^{″}}{\mathit{dt}}=\xi ·O\left(t\right),$$E18

Rayleigh equation [41].

$$\frac{\mathit{d\xi }}{\mathit{dt}}=\frac{1}{2V_c^{″}}\left[-\frac{\left(p_K-p_c\right)}{{\rho }_f{C}^{\ast }}{V^{″}}_c^{4/3}+\frac{\chi }{C^{\ast }}+\frac{1}{3}{\xi }^2\right]·O\left(t\right),$$E19

where N is the total number of finite elements used for describing fluid oscillations during its translational motion from the critical section of the hydrovibrator to dynamic contact with the rock massive; K is the number of the “fluid” finite elements (K = 2), depicting the process of collapse of the cavitation cavity; $V_c^{″}\left(t\right)$ is the volume of the collapsing cavern; $V_c^{\prime }\left(t\right)$ is the volume of the settled cavity in the diffuser part of the cavitation generator; $O\left(t\right)$ is control cavitation process function (the $O\left(t\right)$ values up to the point in time of the periodic collapse of the cavity are set equal to zero, then to 1 until the end of the process of collapse of the cavity); $\xi$ is the time derivative of the cavitation volume, when cavity collapses; $a_i=\left({\stackrel{-}{p}}_{i-1}-{\stackrel{-}{p}}_i\right)/{\dot{W}}^2$ is the hydraulic resistance coefficient of the i-th ‘fluid’ finite element; $J_i=4l_i/\left(\pi d_{i-1}{d}_i\right)$ is coefficient of inertial resistance of the i-th ‘fluid’ finite element ($l_i$, $d_{i-1}$, $d_i$ are length and diameters of the “fluid” finite element); $C_i=0,5\left(V_i+V_{i+1}\right)/c^2$ is compliance due to fluid compressibility; $V_i$, $V_{i+1}$ are volumes of adjacent elements; c is speed of sound in the fluid medium of hydrovibrator; $r_i=\mathit{\epsilon c}/A_i^{\mathit{liq}}$ is the coupling coefficient of the flow rate [42], usually taken as a certain part of the value of the characteristic impedance $c/A_i^{\mathit{liq}}$ ($A_i^{\mathit{liq}}$ is cross-sectional area of jth element); p_c is fluid vapor pressure; $C^{\ast }=0.5\ast 3^{-1/3}{\left(4\pi \right)}^{-2/3}$ and $\chi =-2·3^{-\left(1/3\right)}·{\left(4\pi \right)}^{\left(1/3\right)}·\sigma V_c^{″}{\rho }^{-1}-4/3·\eta {\rho }_f{V_c^{″}}^{\left(1/3\right)}\xi$ are functions; $\eta$ and $\sigma$ are viscosity and surface tension of the fluid; ${\rho }_f$ is the fluid density.

The process of collapsing a cavitation cavity in the flow part of the hydrovibrator is described by using (10) Rayleigh equation [41]. A shape of the equation was changed in Ref. [33] for convenience of recording and subsequent integration by switching from the “radius of the cavity” variable to a “volume of cavity” variable.

The length $l_c$ of the attached cavity $V_c^{\prime }$ and the coordinates $l_s$ of the cavity $V^{″}$ collapse, as well as the total volume of cavities (it is accepted that $V^{″}=V_c^{\prime }$) in the flow part of the cavitation hydrovibrator and the frequency of cavitation oscillations, were determined using the experimental dependences of these quantities on flow parameters obtained earlier in Ref. [29]. The value of the control function $O\left(t\right)$ in Eq. (18) was set according to the oscillation period determined by the experimental frequencies of cavitation oscillations.

The maximum length $l_c$ and volume $V_c^{\prime }$ of the attached cavity in the diffuser of the generator were determined using the expressions given in Ref. [29].

3.3 Boundary conditions for modeling axial vibrations of a drill string

The boundary conditions on the finite element model of longitudinal oscillations at the beginning and at the end of the drill string sections were recorded taking into account the following:

• drilling fluid oscillations do not propagate upward (against the movement of fluid flow) due to the fact that the existence of a cavitation cavity in a hydrovibrator “lowers” the speed of sound in the drill pipe. This “cuts off” the passage of fluid flow disturbances through the hydrovibrator cavitation cavity section above the installation site of the hydrovibrator in drill string;

• the bottom part of the drill string is “pressed” to the rock surface with the axial force of 9.8 kN that implies taking into account the dynamic properties of the rock for calculating its dynamic interaction with the drill bit. Elastic-dissipative properties of the rock massive were taken into account by introducing into model contact interaction of the drill string finite elements (describing the drill bit movement) the rigidly fixed finite element (rock) with a large equivalent mass and damping coefficient. The modulus of elasticity, density and acoustic impedance of the rock were determined from reference data [29, 43] using the expression

$$Z\left(\mathit{j\omega }\right)=R+X\left(\mathit{j\omega }\right)$$E20

where: R is determined by the equivalent mass of rock in dynamic interaction with the drill bit and X($\mathit{j\omega }$) is determined by the elastic-inertial properties of the rock.

3.4 Results of numerical determination of the dynamic characteristics of the “drill tool with a hydraulic vibrator: Rock” system

The time dependences of the drill string displacement, vibration velocity and vibration acceleration of structural elements, fluid pressure and flow rate amplitudes, the volumes of the cavity attached and collapsing cavity in hydrovibrator flow path as a result of numerical integration of differential equations of the ‘drill string with a hydraulic vibrator – rock’ dynamic system by the Runge-Kutta method for different cases of cavitation parameter (as hydrovibrator operational mode) are computed. These dependences were obtained at the drill pump discharge pressure Р_p = 4 MPa and the flow rate of the flushing fluid 2.5 l/s, the drill axial static load F = 9.8 kN and values of the parameter of cavitation $\tau$= 0.12, 0.16, 0.184, 0.2, 0.34, 0.415, 0.475 correspond to the test studies of the drill string experimental sample for drilling a well with the diameter of 76 mm in field conditions. The test results are given in Ref. [33].

Figure 6 demonstrates the calculated time dependences of pressure p₂, volumetric flow rate Q, vibration velocity $v$ and dynamic force F at the drill bit cross-section for the value of the cavitation parameter τ = 0.16, as an example.

Figure 6.

The calculated time dependences of pressure p₂, volumetric flow rate Q, vibration velocity $\mathrm{v}$ and dynamic force F at the drill bit cross-section.

As it follows from the given dependences, the oscillatory process is impulsive in nature. At the indicated value of the cavitation parameter τ, the fundamental frequency f_cav of cavitation oscillations is 323 Hz. The peak-to-peak values of pressure oscillations ΔP do not exceed 6.19 MPa, the volumetric flow rate sweep ∆Q does not exceed 3.17 l/s, the vibration velocity sweep v is about 35 m/s and the peak-to-peak values ∆F of vibration force are less than 5 kN. From the calculation results, it can be seen that the drill structure oscillation frequency equal to 970 Hz is superimposed on the fundamental harmonic of the frequency f_cav of cavitation pressure oscillations. This fact is due to the dynamic interaction of the drill string structure and the fluid in its flow path.

The processing of the results of numerical simulation of the oscillatory process of the drill string in the section of the drilling bit by using the “oscilloscope” data analysis program allows performing a comparative analysis with experimental data at field conditions, given from Ref. [33]. Figure 6 shows the calculated and experimental dependences of the peak-to-peak values ΔP of the pressure oscillations on the cavitation parameter τ in the flow part of the hydraulic vibrator at the rock-cutting tool section. It also shows the calculated and experimental frequencies f_cav of pressure pulses in the range of variations of the cavitation parameter τ from 0.1 to 0.475. Figure 6 also shows the calculated and experimental frequencies of hydraulic oscillations f_cav and the frequencies of forced vibrations of the structure of the drill string (the first f _m1 and second f_m2 vibration modes).

In Figure 7 (the left-hand side), the calculated and experimental dependences are presented on the cavitation parameter τ the range of oscillations in fluid pressure ΔP₂, as well as the repetition rates of hydraulic pulses f_cav in the flow channel of the hydraulic vibrator and the first and second modes of mechanical vibrations f_m1, f_m2 rock-cutting tool. Here is the calculated dependence on the cavitation parameter τ volumetric flow ΔQ.

Figure 7.

Dependences on the cavitation parameter τ of the range of pressure oscillations ΔP₂, volume flow ΔQ, pulsation frequency f, vibration velocity Δv and vibration force ∆F in the cross-section of the drill bit.

The nature of the dependences ΔР₂ and ΔQ on the cavitation parameter τ is nonlinear. As the value of the parameter τ increases from 0.1 (with an increase in the backwater pressure P₂ at P₁ = const), the ranges of pressure oscillations in ΔP₂ and volume flow ΔQ increase, reaching a maximum value of 6.19 MPa and 317 L/s at τ = 0.16, and then decrease. The maximum value of the oscillatory pressure value in the cross-section of the sensor installation is approximately 1.5 times higher than the discharge pressure P₁. Despite the complexity of the processes occurring in the flow channel of the cavitation hydraulic vibrator over the entire studied range of changes in the value of τ, not only better, but also quantitative agreement was obtained between the calculated values of ΔР₂ and f_cav with experimental data, including in resonant modes. Experimental and calculated dependences of the frequency of cavitation self-oscillations f_cav on the cavitation parameter τ are close to linear in nature and with increasing value of τ the frequency increases from 200 Hz to 1400 Hz. The first mode frequency of forced oscillations of the drill structure corresponds to the hydraulic frequency of fluid oscillations in the flow channel of the hydraulic vibrator. This is clearly illustrated in Figure 7. The second frequency (mode) of mechanical vibrations drilling tool is three times or higher than the first frequency. Note that as the value of parameter τ changes from 0.12 to 0.475, the duty cycle of the impact process decreases.

The oscillatory parameters of pressure and volumetric fluid flow determine the hydraulic power of the cavitation hydraulic vibrator, which characterizes the quality of removal of drilled rock from under the rock-cutting tool.

The parameters that determine the mechanical power, and therefore the drilling speed, include vibration velocity and vibration force. Theoretical dependences of the vibration speed range Δv and vibration force ΔF on the cavitation parameter τ in the cross-section of the drill bit are shown in Figure 7.

From this figure, it is clear that the increase in the amplitudes of forced oscillations of drilling fluid pressure (see Figure 7) at cavitation numbers from 0.1 to 0.16 leads to an increase in vibration speed and vibration forces. The maximum values of the vibration velocity and vibration force on the rock-cutting tool for this design were obtained at τ = 0.16 and are approximately 4.7 m/s and 4.1 kN. Let us note that, as shown in Ref. [17], the theoretical dependences of the amplitude of vibration acceleration oscillations ΔZ and experimental data in the studied range of the cavitation parameter τ agree satisfactorily with each other, including in resonant modes with a high level of oscillations of the system under study.

Satisfactory agreement between theoretical and experimental vibrational parameters ΔР₂ and ΔZ subsequently made it possible to determine its hydraulic and mechanical power.

3.5 The drill string’s hydraulic and mechanical oscillatory power computation at the rock drill bit

Process of change in time t, hydraulic N_hi and mechanical N_mi oscillatory powers was determined by the formulas:

$$N_{\mathit{hi}}\left(t\right)=p_2\left(t\right)·Q\left(t\right),$$E21

$$N_{\mathit{mi}}\left(t\right)=v\left(t\right)·F\left(t\right),$$E22

where: p₂(t), Q(t), v(t) and F(t) are the current values of pressure, volumetric flow, vibration velocity and vibration force in the cross-section of the drill bit.

The calculation of average values of hydraulic N_hav [34] and mechanical N_mav [35] oscillatory powers was determined by the expression

$$N_{\mathit{hav}}\mathbf{=}\frac{\mathbf{1}}{{\mathit{T}}_{\mathit{im}}}{\int }_0^T\mathit{\delta P}\left(t\right)·\mathit{\delta Q}\left(t\right)·\mathit{dt}$$E23

where $\mathit{\delta P}\left(t\right)$ and $\mathit{\delta Q}\left(t\right)$ are deviations of current values of pressure and volumetric flow rate of liquid from their steady-state values, $T_{\mathit{im}}$ is pulse duration

$$N_{\mathit{mav}}=\frac{1}{T_{\mathrm{im}}}{\int }_0^T\delta F_{\Sigma }\left(t\right)·\mathit{\delta v}\left(t\right)·\mathit{dt}$$E24

where $\delta F_{\Sigma }\left(t\right)$ and $\mathit{\delta v}\left(t\right)$ are deviations of current values of drill string forces and vibration velocities at drill bit section from their stationary values. The $\delta F_{\Sigma }\left(t\right)$ is determined by dissipative, elastic and inertial forces acting between the interacting elements of drill string structure and drill bit.

As an example, the results of calculating the time processes of hydraulic N_hi and mechanical N_mi oscillatory powers determined by formulas (21) and (22) are shown in Figure 8 for the cavitation parameter τ = 0.16.

Figure 8.

The process of changing over time the hydraulic N_hi and mechanical N_mi oscillatory powers in the cross-section of the drilling tool at τ  = 0.16.

Analysis of dependences N_hi = f(t) and N_mi = f(t) presented in Figure 8 shows the time history process of power change is shock in nature. The maximum peak values of hydraulic power in the cross-section of the drilling tool follow with a frequency corresponding to the main frequency of cavitation oscillations realized in the flow channel of the hydraulic vibrator. At the same time, the main mode of forced cavitation oscillations of mechanical power is superimposed on the dynamic components of higher modes of natural oscillations of the drilling rig structure.

Processing time history dependences N_hi = f(t) and N_mi = f(t), similar to those shown in Figure 8, allowed us to establish the change in the maximum peak values (in the pulse) from the cavitation parameter τ . These dependences N_hi = f(τ) and N_mi = f(τ) are shown in Figure 9 (on the right).

Figure 9.

Dependences of the maximum peak values of hydraulic N_hi and mechanical N_mi (right) and their average values N_hav and N_mav (on the left) of the cavitation parameter τ.

The presented dependences indicate that at a fixed pressure at the inlet to the hydraulic vibrator, with an increase in the cavitation parameter τ from 0.12 to τ = 0.16, the peak values of N_hi and N_mi sharply increase to 43.5 and 22 kW. With further growth of the cavitation parameter τ (from a value of 0.16 to τ = 0.2), there is a sharp decrease in the level of vibrational energy ranges N_hi, N_mi followed by a smooth decrease to values of 11.4 and 5.8 kW, respectively.

The same Figure 9 (on the left) shows the calculated dependences of the average values of oscillatory hydraulic N_hav and mechanical N_mav powers determined in accordance with (23) and (24), as well as experimental data on hydraulic power N_hexp on the cavitation parameter τ [34]. At the same time, when determining the maximum peak and calculating the average values of mechanical oscillatory power, only its positive values were taken into account. That is the values characteristic of the contact of the drilling tool with the destroyed rock.

A comparison of average calculated values of N_hav and experimental data N_hexp on hydraulic power on the cavitation parameter τ in the cross-section of the drilling tool (see Figure 4 on the right) indicates their qualitative and quantitative agreement.

Over the entire studied range of changes in the cavitation parameter, τ the numerical average values of the hydraulic oscillatory power of the liquid flow, both theoretical and experimental, do not exceed the power of the stationary flow at the inlet to the hydraulic vibrator, which is defined as:

$$N_{h1}=P_1·Q_{\mathit{cr}},$$E25

where $P_1$ is the pressure at the inlet to the hydraulic vibrator taking into account the liquid column. $Q_{\mathit{cr}}$ is volumetric flow rate of liquid through the generator.

The maximum calculated value of the efficiency of converting a stationary fluid flow into a discrete pulse flow in the case considered was approximately 76%, with a value of the cavitation parameter τ equal to 0.16, and the minimum 20% at τ = 0.475.

3.6 Spectral analysis of the dynamic process and analysis of the results obtained

As part of the vibroacoustic analysis of the drill string with the cavitation hydraulic vibrator, spectral analysis was performed using experimental data from field tests given in Ref. [16], in the frequency range from 1 to 12 kHz. It is in this range that a drilling rig with a cavitation hydraulic vibrator realizes a fairly high performance of its dynamic process.

Spectral analysis of signals from pressure and vibration acceleration sensors was performed on the basis of their amplitude spectra using standard computational procedures based on the calculation of the complex coefficients of the Fourier series of a periodic signal using the Fast Fourier Transform (FFT) algorithm.

When processing cavitation shock vibrations of pressure amplitudes and vibration accelerations due to the significant nonlinearity of the vibration shape, with the nonlinear nature of the FFT signal determines only the harmonic components of the signal. This is especially evident from the pressure amplitude spectrum. However, the vibration frequencies in a coupled system (hydraulic system and structure) can be analyzed.

As an example, Figure 10 shows the amplitude spectra of pressure p₂ (ASP) and vibration accelerations of the structure Z(ASV) in the cross-section of the drilling tool for cavitation parameter values of τ =0.19 and 0.41.

Figure 10.

Amplitude spectra of pressure p₂ (ASP) and vibration accelerations of the structure Z (ASV) in the cross-section of the drilling tool.

For the cavitation parameter value of τ = 0.19, resonance phenomena in the system under study (dynamic pressure p₂) are observed in Ref. [17] at the first frequency of cavitation oscillations equal to 397 Hz, as well as at the second and third natural frequencies, which are 790 and 1170 Hz. When the cavitation parameter is τ = 0.41, system pressure resonances p2 are observed, as in the first case, at the fundamental frequency of cavitation oscillations, equal to 1114 Hz. The second and third natural frequencies are equal to 1980 and 3520 Hz, respectively. Increase in oscillation frequency above values of the third natural frequency (both at τ = 0.19 and τ = 0.41) leads to an intensive decrease in the resonant processes of the drill string.

Resonant processes of vibration acceleration of the drill bit for both τ = 0.19 and τ = 0.41 were observed, respectively, at frequencies of 1.65, 4.57 kHz and near frequency 9 kHz.

Thus, the analysis of the amplitude spectra of pressure p₂ and vibration acceleration Z in the cross-section of a drill bit made it possible to establish a range of values of the resonant frequencies of vibration accelerations with increased efficiency of the drilling operational process.

3.7 Theoretical computation of drilling speed

The rate of penetration of a well is the determining parameter for assessing the profitability of constructing a well. The theoretical computation of ROP in this work was performed by two ways.

The first traditional method of calculating the drilling speed is based on the calculation of the volume of rock destroyed by a drill bit per unit time. The volume of the destroyed rock V per unit time is directly related to the amount of power N (kW) supplied to the face and the energy intensity of destruction per unit volume of rock $A_V$

$$V=\frac{N}{A_V}$$E26

The volume of rock destroyed per unit time can be determined using the values of the mechanical drilling speed $v_{\mathrm{м}}$(m/h) and the borehole bottom area F (m²)

$$V=v_{\mathrm{м}}F,$$E27

From the equality of formulas for determining the volume of destroyed rock (26) and (27), we obtain

$$v_{\mathrm{м}}=\frac{N}{A_{V}F}.$$E28

From this expression, it follows that the ROP is proportional to the amount of power supplied to the face and is inversely proportional to the energy intensity of rock destruction and the area of the face.

The second method of calculation is based on the assumption that during sonic drilling, the speed is determined by high-frequency impacts of a high-energy level, and the drill bit rotation is of an auxiliary nature. In this case, the ROP is determined by the vibration displacement Δx(m) and the frequency f of mechanical impacts of the drilling tool

$$v_{\mathrm{м}}=∆x·f$$E29

For the first case (28), Figure 11 shows that the energy intensity of destruction of hard rocks $A_V$ was determined for hard rocks in accordance with Ref. [44], and the power supplied to the bottom N was equated to the peak hydraulic power in the cross-section of the drilling tool N_hi. For the second case (29), vibration displacements Δx and their frequencies were determined from calculated data (see Figure 6, for example). The indicated dependences correspond to the conditions of field experiment at the inlet pressure of the hydraulic vibrator and at the flow rate of drilling mud and are given in Ref. [33].

Figure 11.

Calculated dependences of the ROP on the operating mode of the hydraulic vibrator.

From Figure 11, it is clear that the calculated values of the drilling speed sharply increase to 90 m/h (by N_hi) and 85 m/h (by Δx), with subsequent decrease to 38 m/h (by N_hi) and 25 m/h (by Δx) at τ = 0.475 for the fixed pressure at the inlet to the hydraulic vibrator Р₁ (as the cavitation parameter τ increases from 0.12 to 0.16). It should be noted that there is both qualitative and quantitative agreement between the given dependences.

Figure 11 shows the proposed working area of the hydraulic vibrator with a range of vibration displacement values from 0.64 mm (at τ = 0.2) to 0.11 mm (at τ = 0.475). When the hydraulic vibrator operates at τ < 0.18, the calculated values of vibration displacements reach values of more than 1 mm, which can have a negative impact on the drill bit for diamond drilling of relatively deep wells. This assumption is subject to further experimental verification and theoretical clarification.

Considering the presented results, it can be argued that for rocks of approximately equal strength, the use of a drill bit with a cavitation hydraulic vibrator is more effective compared to other means of exciting the vibration load of a drill bit. For example, the maximum penetration speed using a hydraulic low-frequency oscillator (up to 100 Hz) was 22 m/h [45], and the high-frequency mechanical vibration impactor at the frequency of 2500 Hz reached the value of 34 m/h [19]. At the same time, the penetration speed with a cavitation hydraulic vibrator at a frequency of approximately 2500 Hz (see Figure 7a at τ = 0.25) is 43 m/h and it is approximately 28% higher than the speed with a mechanical high-frequency vibration hammer.

4. Conclusions

The new approach for calculating the dynamic interaction of the drill string structure with a cavitation hydraulic vibrator and rock is proposed for sonic drilling. The approach is based on the advanced mathematical model of longitudinal vibrations of the “drill string with hydraulic vibrator - rock” dynamic system, developed taking into account dissipation that depends on the drill string structure vibrations’ amplitude. For a specific drill string design, the following features are computed at the drill bit:

• parameters of hydraulic (pressure, cavitation volume, flow and frequency) and mechanical vibrations (force, displacement, speed and frequency) in case the drill bit comes into contact with the rock;

• hydraulic and mechanical components of oscillatory power in various operating modes of the cavitation hydraulic vibrator, characterizing the efficiency of cleaning well cuttings and drilling speed;

• resonant frequencies of mechanical vibrations (as a result of the dynamic interaction of the drilling tool and the rock being destroyed) based on computer calculations being 1.65 and 4.57 kHz;

• calculated dependences of ROP on hydraulic power and vibration displacements. It has been established that for rocks the use of a drilling tool with the developed cavitation hydraulic vibrator is more effective compared to other means of exciting the drill bit vibration loading. For example, the ROP with the cavitation hydraulic vibrator at a frequency of approximately 2500 Hz is 43 m/h and is approximately 28% higher than the speed with a mechanical high-frequency vibration hammer.

Thus, the intensification of the rock destruction process for sonic drilling with a cavitation hydraulic vibrator is carried out due to an increase in the energy transferred to the rock and a decrease in the energy intensity of the rock destruction process at resonant vibration modes of the drill bit and the destroyed rock.

These computations make it possible to select rational modes of vibration loading of the bit with the cavitation hydraulic vibrator to increase the ROP during well construction and avoid restrictions on drilling operating parameters.