Influence of Nonstationary Processes in Drill Rigs on the Durability of Structural Elements

Abstract

Assessing the effects that nonstationary dynamic processes have on the durability of structural elements belongs to an important trend in modern dynamics and technical diagnostics of machines. Normally, fatigue strength calculations are performed taking into account only periodically variable stresses, as steady operating modes of machines are much longer in comparison with transient modes. However, a significant role in fatigue failure in machines and engineering structures is also played by nonstationary loads. This is explained by emerging intensive oscillations in the mechanical system during accelerating, braking, or changing the operation mode of a machine unit, which often lead to the accumulation of fatigue damages in the materials of parts in heavy loaded assemblies. The combination of stationary and nonstationary dynamic loads manifests itself, particularly in drilling rigs, where technological cycles include steady motion modes, starts, and stops. This paper represents a generalized mathematical model describing nonstationary processes in the lift system of a drill rig, which considers the relationship between electromagnetic processes in asynchronous motors and mechanical oscillatory phenomena, with the purpose of determining dynamic loads and stresses in structural elements of the rigs. Nonlinear physical systems include mechanical members with both concentrated and clearly expressed distributed parameters. The durability of structural elements is evaluated by means of a computer algorithm for analysis of crack growth rates using the NASGRO equation obtained with the presence of plastic deformation zones. An example of the crown block axis illustrates the influence of nonstationary dynamic processes in drill rigs on the durability of structural elements.

1. Introduction

Drilling wells and the possibility of their further use play a key role in various branches of the world industry, including mining, oil and gas exploration, geothermal energy, water supply, and geological research [1,2,3]. These processes provide valuable information about the Earth’s interior, provide insight into geological formations, and also allow the successful development and use of a huge amount of accumulated energy resources. Ensuring the needs of humanity with raw materials, fuel, and energy resources is connected with the sustained development of drilling operations, which causes the need for further improvement of drilling technology and the condition of equipment in geological exploration and oil and gas production enterprises. Nowadays, the character of these works, since their appearance in the middle of the 19th century, has undergone significant changes and modernization. Therefore, the problem of intensifying the technological processes of drilling wells with a simultaneous increase in the technical level of drilling rigs, increasing their productivity, reliability, and durability, is urgent and requires additional research.

Note that the installation of drilling equipment for oil, gas, and geothermal wells is expensive and requires significant capital investments and ongoing cost analysis. In particular, it was investigated [4] that the cost trends for different depths differ significantly, and this is explained by variations in oil and gas prices, costs, and the availability of the main components of the well and services in certain locations. However, thanks to the development of modern technologies and scientific methodologies [5,6], drilling has become an important tool for both scientific research and industrial applications.

Modern drilling rigs are complex electromechanical systems, whole complexes of structures, machines, and mechanisms, which differ significantly both in terms of design and technical characteristics. Individual elements of the drilling complex often work in harsh, extreme operating conditions, which are accompanied by intense mechanical oscillations caused by frequent starts and stops of drive systems, changes in the forces of resistance against the movement of a drill string, uneven flow of flushing fluid in the well, imbalance of rotating parts and assemblies, variability of elastic–dissipative and inertial properties of individual segments, and installation errors of equipment and supporting structures. As a result, a comprehensive study of the static and dynamic strength of drilling equipment is a necessary condition for its rational design and effective use.

The most complex element of any drilling system is the drill string model, which has many degrees of freedom. That is why a significant number of scientific works are devoted to the study of issues of strength, stability, vibrations, and spatial oscillations of drill pipe strings [7,8,9,10], the length of which sometimes reaches 10,000 m. As a rule, various mathematical models are used to study the oscillations of a drill string, which often resort to using the finite element method. Therefore, in [11,12], for the purpose of dynamic analysis of various drilling systems in the oil and gas industry, physical and mathematical modeling of vertical, horizontal, inclined rotary, and rotary-percussion drilling systems was carried out, where torsional, axial, transverse, and combined oscillations of drill strings were considered. The problems in which the peculiarities of the static and dynamic interaction of the string with the well are studied [13,14], as well as the interaction of the drill bit with the hole [15,16] are no less relevant. In parallel, the relationship between rotational and translational oscillations of the weighted part of the string is studied, as well as the damping of oscillations due to the interaction of the string with the washing liquid and the use of an active vibration isolation system [15]. The impact of string curvature on the stress state of drill and casing pipes, as well as on the friction forces of the column against the well wall, is analyzed [13,14]. In works [17,18], the strength and reliability of fastening threaded or pin connections of drill and casing pipes are investigated. Considerable attention is paid to the automation of drilling processes and the improvement of drive systems of drilling rigs, in particular to the solution of the actual scientific and applied problem of the dynamic compatibility of the frequency characteristics of modern high-speed drive systems with the dynamic characteristics of drilling rigs [19,20,21].

In the work [22], problems related to unwanted oscillations during drilling deep oil and gas production wells using an upper drive were considered. An example of a mathematical model for demonstrating the quantitative characteristics of pulsations in a drilling tool was presented. In the article [23], resonant oscillations of an elastic rod subjected to a harmonic load were studied, taking into account the relaxation properties of the material. Features of longitudinal oscillations of a drill string with a compensator during drilling a well on the shelf were considered in the work [24]. In the proposed mathematical model, viscous friction in the pipe material is taken into account, and the significant influence of the compensator on the frequency of oscillations in the string during operation is also noted. In the work [25], the process of transferring the load from the drill string to the bit by exciting the longitudinal and torsional oscillations of the string was studied. Through the application of the finite difference method, a mathematical model was implemented. As a result, it was shown that load transfer by means of vibration processes makes it possible to avoid the tool becoming stuck and ensures smooth drilling of the well. A mathematical model and computer software for the analysis of dynamic processes occurring in the drilling pipes in the borehole under stuck drill string release by means of an impact mechanism (a jerking device) or a pulse-wave installation, equipped with an electric linear pulse motor, were presented in [26]. The influence of friction forces on the propagation of longitudinal waves in the drill pipe string was investigated.

A mathematical model of transverse fluctuations in the pipeline straight section was considered in the article [27]. Such fluctuations occur during the movement of a diagnostic piston in the pipeline. The analysis was based on the method of generalized displacements. The equations of mechanical system motion were derived from the Lagrange scheme equations of the second kind. The article [28] developed recommendations for ensuring the stability of the movement of drill strings, which can be taken into account in practice both at the design stage and during the operation of the drilling equipment.

It should also be noted that the drilling equipment can be exposed to cyclic loads (for example, due to the effect of periodic percussion loads, the influence of significant vibration processes, or the peculiarities of the operation of drilling pumps, which are generators of pulsations) [29]. Fatigue damage and microcracks most often occur in places of stress concentration or corrosion localization zones and, under favorable conditions, later grow into macrocracks that propagate inward into the material bulk. The character of crack growth depends on a number of operational factors [30,31,32], such as the stress intensity factor (SIF) at the macrocrack tip, asymmetry, frequency and shape of the loading cycle, temperature, environment, etc.). Long-term operation of drilling system equipment leads to the accumulation of various types of damage, the initiation of macrocracks, and their subsequent propagation in the material. This negatively affects the durability of individual elements, reduces their efficiency, and increases the risk of unexpected breakdowns.

Today, the problems of predicting the operational lifetime of drilling rigs are also relevant and are of crucial importance for their safe and reliable operation, as well as ensuring the structural integrity of the mechanical elements of such structures. In the world literature, various methods of durability assessment and determination of the resource of elements of mechanisms and machines subjected to cyclic loads are presented [33]. Experimental approaches are considered more reliable, but they are expensive and long term. Therefore, researchers usually rely on computer algorithms for predicting the durability of cyclically loaded structural elements. In particular, articles [34,35,36] calculated the number of cycles to failure using the AFGROW computing system, which is widely used both in engineering calculations and scientific research.

Therefore, the problems of dynamics, strength, stability, and resource prediction of drilling equipment elements constitute a sufficiently large field of scientific research and attract the attention of many scientists and scientific teams. A significant amount of research work has been carried out in this field. However, certain problems require further study, and the mathematical modeling methods should be improved and refined.

2. Mathematical Modeling of Dynamic Processes

The mathematical model is designed to describe dynamic processes that emerge in the mechanical system of a drill rig during the lifting of a pipe string. The calculation scheme of the lifting system is represented in Figure 1, where J₁, J₂—moments of inertia for an engine rotor with mechanical transmissions and drawworks drum with attached drive parts considered about the axis of the lifting shaft; m₁—the mast reduced mass evaluated with respect to the crown block mass; m₂—the mass of block with a hook and suspended equipment; m₃—the mass of the lower part of a drill string weighted with a tool; c_d and c_r—stiffness coefficients of mast and wireline; ν_d, ν_r—factors that characterize energy dissipation in correspondent segments, therewith ν_r is physically the force of linear resistance to wireline deformation, which corresponds to unit relative strain rate. The drill string is considered as a homogeneous straight rod with a step change in the cross-section. Lengths and cross-sectional areas of string segments, within which the elastic and dissipative characteristics of the rod are constant, are denoted by l_i, A_i (i = 1, 2, …, n). Motion coordinates of discrete inertial elements in the calculation model are denoted by γ₁, γ₂, y₁, y₂, y₃. Translational movements of string sections are denoted by functions u_i dependent on time and longitudinal coordinates x_i with origins in upper extreme cross-sections in each string segment. During driving system operation, the body with the moment of inertia J₁ is subjected to the torque M_E = uM_e, where M_e—electromagnetic torque of the motor, u—ratio of mechanical gears connecting the engine rotor to the drawworks drum; M_c—is the moment of interacting forces of the friction coupling halves. In the presence of slippage in the coupling, M_c = M_f∙sign(ω₁−ω₂), wherein M_f∙ is the moment of friction forces in the coupling; G₁ = m₁g, G₂ = m₂g, and G₃ = m₃g are the weight forces of the crown block, the block with a hook and suspended equipment, and the weighted lower part of the drill string with the tool (g is the acceleration of free fall); q₁, q₂, …, q_n are the weight forces of drill pipes distributed along the length of the string.

The equation of motion for discrete inertial elements of a mechanical system is written by the scheme of Lagrange’s equations of the second kind. In the initial start-up period for the drive system, the equation of rotational motion for elements with moments of inertia J₁ and J₂ is written as

$$J_1\frac{d{\omega }_1}{dt}=u[M_e-M_f\mathrm{sign}\left({\omega }_1-{\omega }_2\right)];$$

$$J_2\frac{d{\omega }_2}{dt}+\frac{\alpha }{2}{\omega }_2^2+r_p{c}_r\delta \left(1+\frac{\delta }{2l_r}\right)+r_p{\nu }_r\epsilon =u{M}_f\mathrm{sign}\left({\omega }_1-{\omega }_2\right);$$

$$\frac{d{\gamma }_1}{dt}={\omega }_1; \frac{d{\gamma }_2}{dt}={\omega }_2,$$

(1)

where drawworks drum moments of inertia J₂, length l_r, and stiffness c_r of wireline vary by the following laws:

$$J_2=J_{20}+\alpha {\gamma }_2; l_r=l_{r0}-r_p{\gamma }_2, c_r=E_r{A}_r/l_r,$$

(2)

and absolute strain δ and relative strain rate ε of the wireline are defined as

$$\delta =r_p{\gamma }_2-\left(k+2\right)y_1+k{y}_2;\xi =(\eta l_r+r_p \delta {\omega }_2) l_r^2,$$

(3)

therewith, J₂₀ and r_p—initial moment of inertia and radius of the drawworks drum; E_r, A_r, l_r0—elasticity modulus, cross-sectional area and initial length of the wireline; α = A_rρ_rr_p³—constant factor (ρ_r—average wireline density); k—pulley multiplicity; η—absolute strain rate of the wireline, and

$$\eta =r_t\omega -\left(k+2\right)v_1+k{v}_2.$$

(4)

Symbols v₁, v₂ denote the speed of movement for elements with masses m₁ and m₂.

When the angular speeds of the driving and driven parts of the drive mechanism are equalized, the torque M_f is sufficient for a clutch between friction device elements, which means that, if the following conditions are fulfilled:

$${\omega }_1={\omega }_2; abs\left(\frac{J_1}{u}\frac{d{\omega }_1}{dt}-M_E\right)\le M_f,$$

(5)

then segments with inertia moments J₁ and J₂ proceed with joint movement as described by the following equations:

$$J_{\sum }\frac{d\omega }{dt}+\frac{\alpha }{2}{\omega }^2+r_p{c}_r\delta \left(1+\frac{\delta }{2l_r}\right)+r_p{\nu }_r\xi =u{M}_E; \frac{d\gamma }{dt}=\omega ,$$

(6)

where J_Σ—total moment of inertia of rotating drive parts; γ and ω—coordinate and speed of rotation for a segment with inertia moment J_Σ.

Varying loads on elements in the lift system may lead to inequality in condition (5) being violated, which results in repeat slippages in the friction device.

The equations of motion for upper and lower blocks of pulleys are written as

$$\begin{array}{c}{m}_1\frac{d{v}_1}{dt}-\left(k+2\right)c_r\delta +c_d{y}_1-\left(k+2\right){\nu }_r\xi +{\nu }_d{v}_1=m_{1}g;\\ m_2\frac{d{v}_2}{dt}+k{c}_r\delta +k{\nu }_r\xi -N_1\left(0,t\right)=m_{2}g;\\ \frac{d{y}_1}{dt}=v_1; \frac{d{y}_2}{dt}=v_2,\end{array}$$

(7)

where g—acceleration of gravity; N₁(0,t)—longitudinal force in the initial cross-section in the first segment of the string.

The equation of longitudinal motion for string segments is written as

$$a_i^2\frac{{\partial }^2{u}_i}{\partial x_i^2}-2b_i\frac{\partial u_i}{\partial t}-\frac{{\partial }^2{u}_i}{\partial t^2}=-g_i (i=1,2,\dots ,n),$$

(8)

where a_i—propagation speed of elastic deformations wave; b_i—factor of linear resistance to string movement in the well; g_i—function that considers distributed load.

Quantities of a_i², 2b_i, g_i are defined by the following relations:

$$\begin{array}{c}{a}_i^2=\frac{E_i}{{\rho }_i}; 2b_i=\frac{{\kappa }_i}{A_i{\rho }_i};\\ g_i=g\left(1-\frac{\rho }{{\rho }_i}{\sin }^2{\alpha }_0\right)\cos {\alpha }_0-\frac{a_i^{2}f}{{\rho }_0}\mathrm{abs}\left(\frac{\partial u_i}{\partial x_i}\right)\mathrm{sign}\frac{\partial u_i}{\partial t},\end{array}$$

(9)

therewith E_i, ρ_i—elasticity modulus of the first kind and drill pipe material density; ρ—density of flushing fluid; κ_i—factor equal to the force acting on unit length in the segment during its unit speed movement; α₀—average vertical tilt angle of the string axis; f—dry friction factor of the string against walls in the well; ρ₀—average curvature radius of the well.

Boundary conditions are given for the integration of Equation (9). For the upper end of the string (if x₁ = 0), second and fourth relations (8) are to be fulfilled, therewith:

$$y_2=u_1(0,t); v_2=\frac{\partial u_1\left(0,t\right)}{\partial t}.$$

(10)

On the borders of adjacent segments and in the lower end of drill pipe string boundary conditions are

$$E_i{A}_i\frac{\partial u_i}{\partial x_i}-E_{i+1}{A}_{i+1}\frac{\partial u_{i+1}}{\partial x_{i+1}}=F_i, u_i=u_{i+1},$$

(11)

$$\left(i=1, 2, \dots , n-1\right), \mathrm{if} x_i=l_i, x_{i+1}=0;$$

$$E_n{A}_n\frac{\partial u_n}{\partial x_n}+m_3\frac{{\partial }^2{u}_n}{\partial t^2}=F_n+m_{3}g, \mathrm{if} x_n=l_n,$$

(12)

where F_i—concentrated forces of interaction between string and flushing fluid.

Thus, dynamic processes in the mechanical system of a drilling rig during lifting the drill pipe string are described by partial differential Equation (8), the boundary conditions for which are the set of Equations (1), (6)–(7), and (9)–(12), and also relations (2)–(5) and (9). The values of displacement and movement speed of continuous–discrete system parts at the initial moment of time are considered initial conditions. Therewith, the angular displacements of rotating parts in the rig are assumed to be equal to zero, and the initial displacements of the elements proceeding in translational motion are found based on deformed state analysis for an elastic system in a condition of static equilibrium. The initial speed of all elements in the rig, except for the engine rotor and rigidly joined parts, is assumed to be equal to zero. The rotor is assumed to be proceeding in steady motion in idle mode. For moments when slippage in the friction device stops or reappears, the initial conditions for integrating Equations (1) and (6) are found as correspondent kinematic characteristics of the segments at the end of the previous stage of the movement. The electromagnetic torque of the asynchronous motor, which appears in the first Equations (1) and (6), is determined by the values of currents in the stator and rotor windings, which are, in turn, obtained from differential equations of the electromagnetic state of the electric machine.

3. Mathematical Model of Electromagnetic Processes in an Asynchronous Motor

Providing proper accuracy for the analysis of dynamic processes in a drill rig requires detailed consideration of not only the elastic and inertial properties of a mechanical system but also the driving forces and useful resistance forces time changes pattern. In particular, peculiarities for determining the electromagnetic torque of the motor based on mathematical modeling of electromagnetic phenomena in an electric machine are considered.

The differential equations of electromagnetic processes in an asynchronous motor have the following form:

$$\frac{d{i}_s}{dt}=A_s(u+{\Omega }_s{\Psi }_s-R_s{i}_s)+B_s({\Omega }_r{\Psi }_r-R_r{i}_r),$$

$$\frac{d{i}_r}{dt}=A_r({\Omega }_r{\Psi }_r-R_r{i}_r)+B_r(u_s+{\Omega }_s{\Psi }_s-R_s{i}_s),$$

(13)

where i_s, i_r, u_s—matrix-columns of currents and voltages; A_s, B_s, A_r, B_r—relationship matrices of electromagnetic parameters; Ω_s, Ω_r—matrices of rotation frequencies; Ψ_s, Ψ_r—matrix-columns of flux linkages; R_s, R_r—active resistances of windings. Index s identifies stator winging, and r—rotor winding.

Matrix-columns i_s, i_r and u_s are defined as

$$i_j(j=s,r)=\mathrm{col}(i_{jx},i_{jy}); u_s=\mathrm{col}(U_m,0),$$

where i_jx, i_jy—projections of currents on the coordinate axes x, y; U_m—power supply voltage amplitude.

Square matrices A_s, B_s, A_r, and B_r are defined as

$$A_s={\alpha }_s(1-{\alpha }_{s}G), B_s=-{\alpha }_s{\alpha }_{r}G, A_r={\alpha }_r(1-{\alpha }_{r}G), B_r=B_s,$$

where

$$G=\frac{1}{i_m^2}\left[\begin{array}{cc}R{i}_x^2+T{i}_y^2& \left(R-T\right)i_x{i}_y\\ \left(R-T\right)i_x{i}_y& T{i}_x^2+R{i}_y^2\end{array}\right],$$

therewith:

$$R=\frac{1}{\rho +{\alpha }_s+{\alpha }_r}, T=\frac{1}{\tau +{\alpha }_s+{\alpha }_r}.$$

In the given expressions, i_m, i_x, i_y—magnetizing current and its components along the axes x, y; τ, ρ—values determined from the magnetization curve, which represents the dependence of the working flux linkage Ψ_m on magnetizing current; α_s, α_r—quantities reciprocal to the inductance of the stator and rotor.

Rotation frequency matrices are

$${\Omega }_s=\left[\begin{array}{cc}0& {\omega }_0\\ -{\omega }_0& 0\end{array}\right], {\Omega }_r=\left[\begin{array}{cc}0& {\omega }_0-{\omega }_r\\ {\omega }_r-{\omega }_0& 0\end{array}\right],$$

where ω₀ and ω_r—synchronous angular speed of the motor and the angular speed of the rotor, expressed in radians per second.

Matrix columns of complete flux linkages for stator and rotor windings are defined as

$${\Psi }_s=\frac{1}{{\alpha }_s}{i}_s+\frac{1}{\tau }i, {\Psi }_r=\frac{1}{{\alpha }_r}{i}_r+\frac{1}{\tau }i,$$

where

$$i=col(i_x,i_y).$$

The following values are defined as

$$i_x=i_{sx}+i_{rx}, i_y=i_{sy}+i_{ry}, i_m=\sqrt{i_x^2+i_y^2}.$$

Values τ and ρ are defined by the following expressions:

$$\tau =\frac{i_m}{{\Psi }_m}, \rho =\frac{d{i}_m}{d{\Psi }_m}.$$

(14)

Electromagnetic torque is calculated by the following formula:

$$M_e=\frac{3}{2}{p}_0\frac{1}{\tau }\left(i_{rx}{i}_{sy}-i_{ry}{i}_{sx}\right),$$

(15)

where p₀—magnetic pole pairs number.

The magnetization curve is given in the form

$${\Psi }_m=a_1{i}_m+a_2{i}_m^3+a_3{i}_m^5, \mathrm{if} i_m>i_{mk};$$

$${\Psi }_m={\alpha }_m^{-1}{i}_m, \mathrm{if} i_m\le i_{mk}.$$

(16)

where i_mk—critical magnetizing current, above which the dependence Ψ_m(i_m) is nonlinear. Therefore, τ and ρ, in accordance with expressions (7) and (9), are defined as

$$\tau ={\left(a_1+a_2{i}_m^2+a_3{i}_m^4\right)}^{-1}, \mathrm{if} i_m>i_{mk};$$

$$\tau ={\alpha }_m, \mathrm{if} i_m\le i_{mk};$$

(17)

$$\rho ={\left(a_1+3a_2{i}_m^2+5a_3{i}_m^4\right)}^{-1}, \mathrm{if} i_m>i_{mk};$$

$$\rho ={\alpha }_m, \mathrm{if} i_m\le i_{mk}.$$

(18)

Solving differential Equation (13) does not require the input of magnetization curve information (16) to computer electronic memory, since expressions (17) and (18) are used directly in calculations.

Current vector projections on the coordinate axes in the initial moment of time are equal to zero:

$$i_{sx}=0, i_{sy}=0, i_{rx}=0, i_{ry}=0.$$

(19)

The equations of motion of mechanical systems given in the previous chapter, together with the equations describing electromagnetic phenomena in the engine, form a single system of differential equations.

4. Finite-Difference Approximation of the Equations of Motion for a Stepped Drill String

The explicit finite-difference scheme of Equation (8) is written with consideration of (9) in the following form:

$$u_i\left(x_i, t+\Delta t\right)=c_{1i}{u}_i\left(x_i,t-\Delta t\right)+c_{2i}{u}_i\left(x_i-\Delta l, t\right)+c_{3i}{u}_i\left(x_i, t\right)+c_{4i}{u}_i\left(x_i+\Delta l, t\right)+c_{0i}{g}_{1i} \left(i=1, 2, \dots , n\right),$$

(20)

where Δl and Δt—values of integration steps in space and time coordinates, respectively; c_0i, c_1i, …, c_4i—approximation coefficients,

$$\begin{array}{c}{c}_{0i}=\frac{{\left(\Delta t\right)}^2}{b_i\Delta t+1},{ c}_{1i}=\frac{b_i\Delta t-1}{b_i\Delta t+1},{ c}_{2i}=\frac{{\left(\Delta t\right)}^2\left(a_i^2+g_{2i}\Delta l\right)}{{\left(\Delta l\right)}^2\left(b_i\Delta t+1\right)},\\ c_{3i}=\frac{2\left(a_i^2{\left(\Delta t\right)}^2+{\left(\Delta l\right)}^2\right)}{{\left(\Delta l\right)}^2\left(b_i\Delta t+1\right)},{ c}_{4i}=\frac{{\left(\Delta t\right)}^2\left(a_i^2-g_{2i}\Delta l\right)}{{\left(\Delta l\right)}^2\left(b_i\Delta t+1\right)};\end{array}$$

g_1i, g_2i—functions characterizing the load,

$$g_{1i}=g\left(1-\frac{\rho }{{\rho }_i}{\sin }^2{\alpha }_0\right)\cos {\alpha }_0,{ g}_{2i}=\frac{a_i^{2}f}{2{\rho }_0}\mathrm{sign}\frac{\partial u_i}{\partial x_i}\mathrm{sign}\frac{\partial u_i}{\partial t}.$$

In the process of the numerical realization of expression (20), the Courant condition is to be fulfilled:

$$\Delta t<\frac{\Delta l}{a},$$

otherwise the more complex implicit finite-difference scheme, which has absolute stability, is to be used.

A finite-difference approximation of the second Equation (7), taking into account relations (10) and (20), results in a discrete expression of the boundary condition for the upper end of the string:

$$u_1(0,t+\Delta t)=q_{10}{u}_1(0,t-\Delta t)+q_{20}{u}_1(0,t)+q_{30}{u}_1(\Delta l,t)+q_{40},$$

(21)

where

$$q_{10}=\frac{s_{10}}{s_{00}}; q_{20}=\frac{s_{20}}{s_{00}}; q_{30}=\frac{s_{30}}{s_{00}}; q_{40}=\frac{s_{40}}{s_{00}},$$

therewith:

$$\begin{array}{c}{s}_{00}=E_1{A}_1{\left(\Delta t\right)}^2+2m_2{c}_{21}\Delta l; s_{10}=E_1{A}_1{c}_{11}{\left(\Delta t\right)}^2-2m_2{c}_{21}\Delta l;\\ s_{20}=E_1{A}_1{c}_{31}{\left(\Delta t\right)}^2+4m_2{c}_{21}\Delta l;s_{30}=E_1{A}_1\left(c_{21}+c_{41}\right){\left(\Delta t\right)}^2;\\ s_{40}=E_1{A}_1{c}_{01}{g}_{11}{\left(\Delta t\right)}^2+2\left[m_{2}g-F_2\right]c_{21}{\left(\Delta t\right)}^2\Delta l.\end{array}$$

Analogically, the boundary condition (12) for the lower end of the drill string is transformed as

$$u_n\left(l_n, t+\Delta t\right)=q_{1n}{u}_n\left(l_n, t-\Delta t\right)+q_{2n}{u}_n\left(l_n, t\right)+q_{3n}{u}_n\left(l_n-\Delta l, t\right)+q_{4n},$$

(22)

where

$$q_{1n}=\frac{s_{1n}}{s_{0n}}; q_{2n}=\frac{s_{2n}}{s_{0n}}; q_{3n}=\frac{s_{3n}}{s_{0n}}; q_{4n}=\frac{s_{4n}}{s_{0n}}.$$

Here,

$$\begin{array}{c}{s}_{0n}=E_n{A}_n{\left(\Delta t\right)}^2+2m_3{c}_{4n}\Delta l;{ s}_{1n}=E_n{A}_n{c}_{1n}{\left(\Delta t\right)}^2-2m_3{c}_{4n}\Delta l;\\ s_{2n}=E_n{A}_n{c}_{3n}{\left(\Delta t\right)}^2+4m_3{c}_{4n}\Delta l; s_{3n}=E_n{A}_n\left(c_{2n}+c_{4n}\right){\left(\Delta t\right)}^2;\\ s_{4n}=E_n{A}_n{c}_{0n}{g}_{1n}{\left(\Delta t\right)}^2+2\left(m_{3}g+F_n-P_3\right)c_{4n}{\left(\Delta t\right)}^2\Delta l.\end{array}$$

Having completed the finite-difference approximation for the first of boundary conditions (11) and taking into account the second relation (11) and also Equation (20), the following dependencies are obtained for the joints of string segments:

$$u_i\left(l_i, t+\Delta t\right)=u_{i+1}\left(0, t+\Delta t\right)=q_{1i}{u}_i\left(l_i,t-\Delta t\right)+q_{2i}{u}_i\left(l_i-\Delta l, t\right)+q_{3i}{u}_i\left(l_i,t\right)+q_{4i}{u}_{i+1}\left(\Delta l, t\right)+q_{5i} \left(i=1, 2, \dots , n-1\right),$$

(23)

where

$$q_{1i}=\frac{s_{1i}}{s_{0i}}; q_{2i}=\frac{s_{2i}}{s_{0i}}; q_{3i}=\frac{s_{3i}}{s_{0i}}; q_{4i}=\frac{s_{4i}}{s_{0i}}; q_{5i}=\frac{s_{5i}}{s_{0i}},$$

there with:

$$\begin{array}{c}{s}_{0i}=E_i{A}_i{c}_{2, i+1}+E_{i+1}{A}_{i+1}{c}_{4i};\\ s_{1i}=E_i{A}_i{c}_{1i}{c}_{2, i+1}+E_{i+1}{A}_{i+1}{c}_{1, i+1}{c}_{4i}; { s}_{2i}=E_i{A}_i{c}_{2,i+1}\left(c_{2i}+c_{4i}\right);\\ s_{3i}=E_i{A}_i{c}_{2, i+1}{c}_{3i}+E_{i+1}{A}_{i+1}{c}_{3, i+1}{c}_{4i}; s_{4i}=E_{i+1}{A}_{i+1}{c}_{4i}\left(c_{2, i+1}+c_{4, i+1}\right);\\ s_{5i}=E_i{A}_i{c}_{0i}{c}_{2, i+1}{g}_{1i}+E_{i+1}{A}_{i+1}{c}_{0, i+1}{c}_{4i}{g}_{1, i+1}+2c_{2, i+1}{c}_{4i}{F}_{i}h.\end{array}$$

Performing each subsequent step in integrating Equation (8) using formulas (20)–(23) requires the values of unknown functions on two previous layers. In the moment of time $t=0$, these functions are given on one layer only. Therefore, let the finite-difference scheme (20) be transformed, taking into account the initial conditions, into such a form that allows the first step in the integration of Equation (8) to be performed:

$$\begin{array}{c}{u}_i\left(x_i, \Delta t\right)=d_{2i}{f}_{1i}\left(x_i-\Delta l\right)+d_{3i}{f}_{1i}\left(x_i\right)+d_{4i}{f}_{1i}\left(x_i+\Delta l\right)+d_{5i}{f}_{2i}\left(x_i\right)+d_{0i}{g}_{1i}\\ \left(i=1, 2, \dots , n\right),\end{array}$$

(24)

where

$$\begin{array}{c}{d}_{0i}=\frac{{\left(\Delta t\right)}^2}{2}; d_{2i}=\frac{{\left(\Delta t\right)}^2\left(a_i^2+g_{2i}\Delta l\right)}{2{\left(\Delta l\right)}^2}; d_{3i}=\frac{a_i{\left(\Delta t\right)}^2+{\left(\Delta l\right)}^2}{{\left(\Delta l\right)}^2};\\ d_{4i}=\frac{{\left(\Delta t\right)}^2\left(a_i^2-g_{2i}\Delta l\right)}{2{\left(\Delta l\right)}^2}; d_{5i}=-\Delta t\left(b_i\Delta t+1\right).\end{array}$$

Taking into account boundary conditions and Equations (21)–(23), formulas for determining unknown functions at the first step of integration in extreme sections of the stepped rod are written as

$$u_1\left(0, \Delta t\right)=p_{20}{f}_{11}\left(0\right)+p_{30}{f}_{11}\left(\Delta l\right)+p_{40};$$

(25)

$$u_n\left(l_n, \Delta t\right)=p_{2n}{f}_{1n}\left(l_n\right)+p_{3n}{f}_{1n}\left(l_n-\Delta l\right)+p_{4n};$$

(26)

$$u_i\left(l_i, \Delta t\right)=u_{i+1}\left(0, \Delta t\right)=p_{2i}{f}_{1i}\left(l_i-\Delta l\right)+p_{3i}{f}_{1i}\left(l_i\right)+p_{4i}{f}_{1, i+1}\left(h\right)+p_{5i} \left(i=1, 2, \dots , n-1\right),$$

(27)

where

$$\begin{array}{c}{p}_{20}=r_0{s}_{20}, p_{30}=r_0{s}_{30}, p_{40}=r_0\left[s_{40}-2s_{00}\Delta t{f}_{21}\left(0\right)\right];\\ p_{2n}=r_n{s}_{2n}, p_{3n}=r_n{s}_{3n}, p_{4n}=r_n\left[s_{4n}-2s_{0n}\Delta t{f}_{2n}\left(l_n\right)\right];\\ p_{2i}=r_i{s}_{2i}, p_{3i}=r_i{s}_{3i}, p_{4i}=r_i{s}_{4i}, p_{5i}=r_i\left[s_{5i}-2s_{0i}\Delta t{f}_{2i}\left(l_i\right)\right],\end{array}$$

here:

$$\begin{array}{c}{r}_1=\left(b_1\Delta t+1\right)\Delta l/\left[2E_1{A}_1{\left(\Delta t\right)}^2\Delta l+4\left(m_0+m_2\right)\left(a_1^2+g_{21}\Delta l\right){\left(\Delta t\right)}^2\right];\\ r_2=\left(b_n\Delta t+1\right)\Delta l/\left[2E_n{A}_n{\left(\Delta t\right)}^2\Delta l+4m_3\left(a_n^2-g_{2n}\Delta l\right){\left(\Delta t\right)}^2\right];\end{array}$$

$$r_i=\left(b_i\Delta t+1\right)\left(b_{i+1}\Delta t+1\right){\left(\Delta l\right)}^2/\left[2\right.E_i{A}_i\left(a_{i+1}^2+g_{2, i+1}\Delta l\right){\left(\Delta t\right)}^2+2E_{i+1}{A}_{i+1}\left(a_i^2-g_{2, i}\Delta l\right)\left.{\left(\Delta t\right)}^2\right].$$

The obtained Formulas (20)–(27) provide a comprehensive analysis of stepped drill string unsteady vibrations by solving a system of partial differential equations of hyperbolic type (8) on given initial and boundary conditions.

Therefore, the drill rig start-up operation modes are described by the set of Equations (1), (5), (9)–(11), (14) and (15), which are integrated taking into account relations (2)–(4), (6)–(8), (12), (13) and (16). To solve the problem, initial conditions are determined based on deformed state analysis for the elastic system being in equilibrium. For moments of time when slippage in the friction device stops or reappears, the initial conditions for integrating Equations (5) and (9) are found as correspondent kinematic characteristics of the segments at the end of the previous stage of the movement.

The represented mathematical model is based on the joint integration of the equations of motion for an elastic mechanical system, which has members with concentrated and distributed parameters, and the equations of the electromagnetic state for an asynchronous motor. Detailed representation of elastic and inertial properties for drill rig elements in the calculation scheme and a refined account of dynamic properties of motors provide a high degree of adequate correspondence for mathematical modeling results in the real physical process. This opens up possibilities for in-depth study of extreme operating modes of technological equipment, performing refined calculations of drive system elements and bearing structures for fatigue strength and durability.

5. Calculation Results of Nonstationary Processes in the Electromechanical System of the Drill Rig

Calculation results of nonstationary processes occurring in the lifting system of a UKB-4P drill rig, which is equipped with an SKB-4 drill machine and an AO2-71-4 asynchronous motor of 22 kW power, are considered. The height of the drill rig mast is 13 m, which provides drill pipe stands of 9–9.5 m long. The diameter of drill pipes can be 42 mm, 50 mm, or 54 mm. The well is assumed to be drilled with one tool, which eliminates the need to replace the bit during the drilling process and, accordingly, while lowering the string into the well, followed by heavy braking conditions. In this case, technological operations for the lifting drill string are the heaviest operation modes in the drill rig.

For studying start-up processes, the following values of parameters of mechanical systems were taken: u = 13.07; J₁ = 0.99 kg∙m²; J₂ = 1.55 kg∙m²; m₁ = 206.5 kg; m₂ = 39.6 kg; m₃ = 23.9 kg; c_d = 1200.0 MN/m; c_r = 1.9 MN/m; ν_d = 482.0 N∙s/m; ν_r = 2.0 N∙s/m; α = 0; r_p = 0.13 m; k = 2; n = 1; A₁ = 590 mm² (pipe diameter is 42 mm); E = 2.1∙10⁵ MPa; ρ₁ = 9066.0 kg/m³; ρ = 1200 kg/m³; b₁ = 0.4 s⁻¹; ρ₀ = 500 m; α₀ = 0.1 rad; f = 0.1.

Figure 2, Figure 3 and Figure 4 represent calculation results of the lifting process of drill pipe string with a diameter of 42 mm and a length of 455.0 m. The excess torque of the friction device, through which the lifting system is accelerated, was 60 N∙m.

Graphs of angular velocities and moments (Figure 2 and Figure 3) demonstrate acceleration of the driven part in the lifting mechanism, as well as the occurrence of repeated slippage in the friction device due to the excitation of intensive oscillations in a mechanical system. Graphic dependencies shown in Figure 4 indicate that the total forces in the string, wireline, and mast construction carry significant dynamic components, which can have an essential effect on the strength and durability of structural elements. As the curves shown in Figure 5 testify, the maximum force in the wireline, and, correspondently, maximum forces in the mast and drill string, significantly depend on the length of transported string l_max, transmitting ratio of the drive mechanism, and the excess torque of the friction device.

Dynamic phenomena must be considered both in the design process of drilling complexes and during their operation in order to justify the selection of strength reserves and increase the reliability of lifting system elements.

6. Endurance Research of the Drill Rig Crown Block Axis

To illustrate the influence of nonstationary loads on the durability of structural elements, endurance research on the crown block axis in the UKB-4P drilling rig was conducted. Prediction of the crack growth rate in the crown block axis was accomplished by NASGRO equation developed by NASA:

$$\frac{\mathit{da}}{\mathit{dN}}=C{\left[\frac{1-f}{1-R}\Delta K\right]}^n\frac{{\left(1-\frac{\Delta K_{\mathit{th}}}{\Delta K}\right)}^p}{{\left(1-\frac{K_{\mathit{max}}}{K_c}\right)}^q},$$

where a—crack length; N—number of load cycles; C, n, p, q—experimentally determined coefficients; f—Newman function; R—cycle asymmetry coefficient; ΔK—diapason of stress intensity factor (SIF) for a specific load, ΔK = K_max—K_min; ΔK_th—minimal limit value in SIF diapason; K_max—maximal value of SIF; K_c—critical value of SIF.

The minimal limit value of SIF diapason for an individual cycle that ensures crack growth is determined by the question:

$$\Delta K_{\mathit{th}}=\Delta K_0{\left(\frac{a}{{a+a}_0}\right)}^{\frac{1}{2}}/{\left(\frac{1-f}{\left(1-A_0\right)\left(1-R\right)}\right)}^{1+C_{\mathit{th}}R},$$

where ΔK₀—value of ΔK at R = 0; a₀—constructive crack length determined by the grain size in material (for steel a₀ = 3.81 ∙ 10⁻⁶); C_th—coefficient required to correct the crack growth curve at different values R: for negative R, this coefficient makes 0.1; for R ≥ 0, this coefficient can be found in NASGRO materials database; A₀—constant.

The critical value of SIF can be defined from the following equation:

$$\frac{K_c}{K_{\mathit{lc}}}=1+B_K{e}^{-{\left(A_K\frac{t}{t_0}\right)}^2},$$

where K_lc—SIF of destruction; A_k, B_k—correction factors; t—sample thickness; t₀—nominal thickness of the sample for a plane stress condition.

Nominal thickness of sample t₀ is defined as

$$t_0=2.5{\left(\frac{K_{\mathit{lc}}}{{\sigma }_T}\right)}^2,$$

where σ_T—material yield strength.

The Newman function describing the closing part of the crack has the following form:

$$f=\mathit{max}({R, A}_0+A_1{R+A}_2{R}^2{+A}_3{R}^3), \mathrm{if} R\ge 0;$$

(28)

$${f=A}_0+A_{1}R, \mathrm{if} -2\le R<0; f=A_0-2A_1 \mathrm{if} R<-2.$$

(29)

Coefficients in Equations (28) and (29) are defined by the following formulas:

$$\begin{array}{c}{A}_0=\left(0.825-0.3\alpha +0.05{\alpha }^2\right){\mathit{cos}}^{1/\alpha }\left(\frac{\pi S_{\mathit{max}}}{2{\sigma }_T}\right); A_1=\left(0.415-0.07\alpha \right)\frac{S_{\mathit{max}}}{{\sigma }_T};\\ A_2=1-A_0-A_1-A_3; A_3=2A_0+A_1-1,\end{array}$$

where α—empirical coefficient.

The lifecycle duration for the drill rig crown block axis is determined by using geometric parameters of the axis, stress spectrum, and crack growth prediction algorithm inbuilt in AFGROW.

Since the greatest dynamic loads on the crown block occur during the technological operation of lifting the drill pipe string from a borehole, the effect of these loads on crown block axis endurance is considered. For this purpose, time dependences of force in the wireline are determined for all cases of lifting pipe string with a number of stands that ranges from maximal to one. The obtained results were used to attain the stress spectrum in the material of the crown block axis and assess its endurance using AFGROW software, version 5.04.05.25, finding the final number of lifts for the string until the destruction of the axis. The axis is assumed to be made of X46 steel.

The crack grows from the perimeter of the circular cross-section towards the center of the axis; the initial length of the crack is set to 1.5 mm. Figure 6 shows the graphic dependence of the crack growth rate on the diapason of the stress intensity factor for the full load cycles. The values of the stress cycle asymmetry coefficient were set equal to 0, 0.2, 0.4, 0.6, and 0.75.

Durability research of the axis with consideration of load mode represents certain practical interest. Graphs in Figure 7 and Figure 8 illustrate the growth of a crack in the crown block axis in dependence on the number of accomplished technological processes on lifting the drill pipe strings, the length of the string, and the diameter of the drill pipes. The maximal length of a string with a diameter of 42 mm or 50 mm was taken equal to 910 m, and the minimal length was 364 m. Since the weight of a drill pipe string with a diameter of 50 mm is greater than that of a string with a diameter of 42 mm, its durability is less.

Comparative characteristics of crown block axis endurance during lifting drill pipe strings of different lengths are shown in Figure 9.

7. Conclusions

• The transient modes of a drilling rig should be analyzed using a continuum–discrete model of a mechanical system of variable structure, based on the joint integration of equations of motion and nonlinear differential equations describing electromagnetic phenomena in electric motors. This makes it possible to ensure the required accuracy of calculations for the strength of structural elements and to predict the durability of heavily loaded parts and assemblies. The efficiency of the analysis is enhanced by the developed finite-difference algorithm for calculating wave phenomena in a stepped drill pipe string.
• Nonstationary modes of operation of a drilling rig are accompanied by intense mechanical vibrations which significantly affect the operating conditions of the friction device, shaft lines, hoisting rope, drill pipe string, mast, etc. Dynamic phenomena must be taken into account both at the design stage of drilling rigs for a reasonable selection of safety margins and during their operation to justify rational operating modes and increase the reliability of drive systems and support structures.
• Based on the time dependences of the force in the rope, stress spectra in the material of the crown block axis were constructed for all cases of lifting a pipe column with the number of candles from the maximum to one. Using the AFGROW program, the endurance of the axle made of X46 steel was calculated. It is shown that the number of cycles of complete lifting of the drill string from the well before the destruction of the crown block axis is significantly affected by the maximum length of the string and its running weight. The permissible crack size decreases significantly with an increase in the maximum length and diameter of the string. The number of cycles of complete lifting of the column from the well before the crown block axis is destroyed is quite sufficient, which is explained by the relatively low load of the lifting system when lifting parts of the column of short length.