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Buckling of a drill-string in two-sectional bore-holes
International Journal of Mechanical Sciences 172 (2020) 105427
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Buckling of a drill-string in two-sectional bore-holes N.V. Shlyun∗, V.I. Gulyayev Department of Mathematics, National Transport University, M. Omelianovycha-Pavlenka Str., 1, 01010, Kiev, Ukraine
a r t i c l e
i n f o
Keywords: Two-sectional bore-holes Stability loss Eigenfunctions Buckling wavelets
a b s t r a c t The nature of the phenomena of drill-strings buckling in channels of deep curvilinear bore-holes is essentially complicated by complex combination of acting distributed forces (gravity, contact and friction), large length of the string (leading to a virtual loss of its bending stiffness), and its intricate geometry. Therefore, theoretical modelling of these effects turns out to be difficult to implement. In this article, the touched problem is stated with the use of the curvilinear flexible rods theory, differential geometry concepts, basic foundations of channel surfaces theory, and methods of numerical mathematics. The elaborated software is applied to the global analysis of the incipient stage of the drill-string buckling inside a deep bore-hole consisting of two nearly rectilinear sections bridged with curvilinear interjacent part. To solve the problem, firstly, the critical stress-strain state of the drill-string (eigenfunction) is established, and then, the shape of the buckled drill-string (eigenmode) is built. It is shown that the stated problem refers to the singularly perturbed type and so, depending on the knee angle and friction coefficient, the mode of stability loss represents a short wavelet localized at the lower end of the lower section or it is shifted to the knee zone of the bore-hole. The features of critical behavior of the drill-strings associated with the values of the knee angle of the bore-hole, system clearance, and friction coefficient, as well as critical loads and buckling mode wavelets are discussed.
1. Introduction At the present time, to extract petroleum resources, deep vertical and inclined, off-shore and ground-based bore-holes are drilled. The techniques and methods of their drilling are developing so fast that scientific understanding and substantiation have no time to refine them. Therefore, not infrequently, different unpredictable emergency effects and overall failures accompany these procedures. The gravest of these are intensive vibrations and resonances of drill-strings [1,25,41], the self-triggering of torsional relaxation vibration [14], the self-excitation of forward, backward, and superfast bit whirlings [15,19,20] the drill-string sticking during dragging and rotation [17], and its buckling. The studies related to the use of the Cosserat mechanics for analysis of drill-string dynamics were performed by [37] and [39]. The process of a drill-string buckling in a curvilinear bore-hole is connected with one of the emergency situations stimulating appearing of a series of negative frictional effects. Among these are deterioration of conductivity of the torque (TOB) and axial force (FOB) to the bit, enlargement of the process energy consumption, increase of the drill-string wear, and in ultimate cases, they can lead to the drill-string sticking. These effects become apparent in the most perceptible manner under conditions of drilling deep curvilinear bore-holes. Effective means to avoid these emergencies are to model them theoretically at the stages of the hole design and drivage. ∗
Before proceeding to solving the problem of elastic string buckling in a deep curvilinear bore-hole, indispensable preconditions should be satisfied. On the one hand, the bore-hole axis line should be prescribed in analytic form, on the other, the stress state of the drill-string should be determined. Usually, the bore-hole trajectory geometry can be described analytically only at the stage of its design. During its drilling, the bore-hole line acquires additional geometric imperfections and its axis can be represented only in a tabular form. To transfer from the tabular representation of this geometry to the analytical one, the methods of numerical approximation and interpolation can be used. Then, it becomes possible to transfer to the procedure of the drill-string stress strain state determination. It is hardly probable that the drill-string would lose its stability when it quietly lies in the bore-hole and is immovable. It will rather buckle during tripping in and out operations under compressive action of additional resistance forces appearing due to friction effects. In the general case, these forces can be calculated with the use of a quasistatic drag and torque mathematic model without taking into account inertia forces. There exist many varieties of string drag and torque models based on disregarding bending stiffness of the drill-string and influence of internal bending moments, but they, very likely, are suitable for the bore-holes with simple and smooth trajectories [4,26,31,34,35]. Since, the bending stiffness is ignored, these models are usually called the cable, thread, chain, or soft string models. However, in recent years, more and
Corresponding author. E-mail addresses: [email protected] (N.V. Shlyun), [email protected] (V.I. Gulyayev).
https://doi.org/10.1016/j.ijmecsci.2020.105427 Received 25 June 2019; Received in revised form 24 December 2019; Accepted 7 January 2020 Available online 9 January 2020 0020-7403/© 2020 Elsevier Ltd. All rights reserved.
N.V. Shlyun and V.I. Gulyayev
Nomenclature a f fcont ffr fgr i,j,k
clearance value; vector of external distributed forces vector of distributed contact forces vector of distributed friction forces vector of distributed gravity forces unit vectors of reference frame oxyz, moving on the Σ surface k1 ,k2 principal curvatures of the Σ surface kR ,kT curvature and torsion of bore-hole axis line T kx ,ky ,kz components of the drill-string axis line curvature vector l natural parameter n, b, t unit vectors of Frenet’s trihedron moving along the T curve r1 ,r2 external and internal radii of the string tube, correspondingly u, v curvilinear coordinates in the Σ surface w lateral displacements of a beam A, B, C, D points on axial line of the bore-hole E module of elasticity of the drill-string material Fx , Fy , Fz components of the internal elastic force FOB force on bit H metric multiplier I inertia moment of the drill-string cross section K curvature vector of the drill-string section L length of the drill-string Mx , My , Mz components of the internal elastic moment OXYZ global fixed coordinate system S axis line of the drill-string T axis line of the bore-hole TOB torque on bit w transverse displacement of a beam X∗ , Y∗ , Z ∗ coordinates of the bore-hole axis line 𝛼 inclination angle of the T line tangent 𝛽 inclination angle of the hyperbola asymptote tangent 2𝛽 knee angle of the bore-hole 𝛾 coefficient at the hyperbola equation 𝛿 symbol of small increment of a varying value 𝜂 coefficient at the hyperbola equation 𝜃 dimensionless parameter 𝜃 0 ,𝜃 f initial and final values of the 𝜃 parameter 𝜆 wavelet pitch 𝜇 friction coefficient 𝜉 intermediate variable 𝜌t , 𝜌m densities of tube material and mud, accordingly 𝜓 angle between the directions of curve S and u coordinate line 𝝎 analog of the Darboux vector Σ channel surface of the bore-hole wall Φ1 (u,v), Φ2 (u,v) the first and second quadratic forms of the Σ surface 𝛀 vector of total curvature
more attention is focused on elaboration of stiff string drag and torque 3D models, which are more universal and precise. One of such models was presented by Gulyayev and Andrusenko [11] and Gulyayev et al. [13]. It is founded on application of the theory of curvilinear flexible rods (Cosserat theory) taking into account bending stiffness of the drillstring tube and supplementary resistance forces. Survey of drag and torque models and their analysis are performed in the reviewing paper by Mitchell and Samuel [31]. Concerning the soft string drag and torque
International Journal of Mechanical Sciences 172 (2020) 105427
models, they noted “that, there are several commonly held assumptions about torque/drag modeling that are not valid. For instance, many believe that the torque/drag model is a single model with no bending resistance. Surprisingly, the torque/drag model cannot be a soft-string model and satisfy the equilibrium equations. Nonzero shear forces must exist, otherwise, friction loads must vanish…The lack of bending resistance is also illusory because the wellbore trajectory commonly used reduces the bending effects in the equilibrium equations to zero. Therefore, to predict correctly the buckling effects in the curvilinear bore-holes, the stiff string drag and torque models should be used for calculation of the axial compressing and stretching forces. Thereafter, it becomes possible to undertake the second stage of the analysis and to initiate the process of the buckling forces calculation. Commonly, the problems of the drill-string buckling in a curvilinear bore-hole channel are nonlinear. But their linearized statements can be used if the bore-hole trajectory geometry is not very complicated and only initial stage of the buckling process is analyzed. Investigation of the drill-string buckling forces and modes is performed with elastic drill-string instability models. There are many computer models differing by the drill-string fragments separated out for calculations, their geometries, assumptions concerning distribution of the internal axial force, and neglect or taking into account frictional phenomena, torques, and moving mud influence. Lubinsky et al. [28] and Lubinsky [27] were the first ones who investigated stability of drilling tubes in vertical bore-holes. In their analysis, they neglected the effects of the drill-string rotation and torque loading and considered the axial force to be constant. Thereupon, the problem statement became more complicated. Mitchell [29] took into account friction and gravity forces and torque. Kwon [23] built a solution in the spiral mode with variable pitch for a ponderous tube. Gulyayev et al. [12] studied stability of long drill-string with allowance made for its stretched-compressed-twisted stress-strain state, and inertia forces of rotation and internal mud flow. They showed that the stated problem is singularly perturbed [5,38] and therefore, its solutions represent variable pitch spirals concentrated in the lower segments of their length. The problems of drill-string buckling in the channels of long curvilinear bore-holes are far harder. Their additional difficulties are connected with necessity to take into account frictional effects [3,29,32] Analysis of this direction publications testifies that as a rule, the approaches used in their solutions are based on approximations of eigenmode by regular sinusoids and spirals and eigenvalues are rather guessed [7,18,22,24, 36,40]. Supposedly, this circumstance is the reason why the solutions of the problems on the drill-string buckling in curvilinear bore-holes, gained by different authors, are inconsistent with each other [30]. In papers [16,17], the question of the drill-string buckling in a curvilinear channel is considered on an approach basis when in the beginning the drill-string stress strain state is calculated throughout its length, thereafter the found axial force function is involved into the constitutive differential equation of an appropriate global 3D eigenvalue problem (the Sturm-Liouville problem) for the whole drill-string, As a result of its solution, the eigenvalue (the critical load) and eigenmode (the shape of the buckling) are calculated. Based on the elaborated software, the drill-string stability in channels of inclined rectilinear, circular, and other long curvilinear boreholes was simulated. It was established that at the initial stage of the buckling onset, it appeared in the form of a wavelet distributed inside comparatively narrow range located inside the bore-hole length or shifted to its lower edge. This phenomenon was shown to be in accordance with the kinds of the constitutive equations which were singularly perturbed [5]. So, these problems statement and solving are associated with additional scientific and numerical difficulties. Principal peculiarities of the drill-string buckling phenomena and achieved results in their understanding are given in the review papers by Cunha [6], Gao and Huang [10], and Mitchell [30]. As a rule, the buckling phenomenon depends essentially on the borehole geometry. In drilling practice, the two˗sectional trajectories with
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
AB (Fig. 1b), which is approximately similar to the initial one. The simplest approximating curve for the considered two-sectional plot proves to be the hyperbola with branches trending to rectilinear asymptotes and having smoothly changing curvature in its apex zone (Fig. 2a). Equations of this curve have the form [21]: 𝑋 ∗ (𝜃) = ±𝛾𝑐ℎ𝜃, 𝑌 ∗ (𝜃) = 0, 𝑍 ∗ (𝜃) = 𝜂𝑠ℎ𝜃,
(1)
where 𝜃 is the dimensionless parameter; 𝛾and𝜂 are the real coefficients; signs “+” and “–” relate to the right and left branches, correspondingly. The dashed straight lines in this scheme correspond to the hyperbola asymptotes. They make angle 𝛽 with axisOX∗ . It is determined by the equality Fig. 1. Schematics of two-sectional bore-holes: a) Composite trajectory with two rectilinear sections including curvature discontinuities, causing additional frictional resistance forces; b) Smoothed trajectory approximated by the hyperbola curve, contributing to frictional forces reduction.
two rectilinear parts connected with a curvilinear segment are of frequent occurrence. Inasmuch as, their critical behavior is poorly investigated, below, the global stability of the DSs in the bore-holes of this type is analyzed. The modes of critical buckling in the limits of their total lengths are constructed. 2. Geometric models of two-sectional bore-holes In bore-hole trajectory planning, it is usual to configure it as a twosectional curve with two rectilinear (nearly rectilinear) parts connected with a circular bridge (Fig. 1a). The trajectories of this type have come to be known as minimal curvature bore-holes [35], though, their curvatures are not minimal and are discontinuous at points C and D.Therefore, as indicated by Samuel [33] and Gulyayev et al. [17], at that places, the functions of contact interaction and friction forces assume large values. To exclude these singularities, it is possible to introduce smoothing insertions (clothoids or cubic parabolas) in the vicinities of these points or, generally, to replace this combined curve ABCD (Fig. 1a) by one smooth differentiable curve
tg𝛽 = lim (𝑍 ∗ ∕𝑋 ∗ ) = 𝜂∕𝛾. 𝜃→∞
(2)
Then, 𝛽 = arctg(𝜂∕𝛾).
(3)
Therefore, if the OX∗ Y∗ Z∗ system is turned by angle 𝜋/2 − 𝛽 relative to the OY∗ axis, the left-hand branch of the hyperbola will take the position shown in Fig. 2b, corresponding to the bore-hole outline presented in Fig. 1b. In fixed coordinate system OXYZ, its equations acquire the form: 𝑋 = 𝑋 ∗ sin 𝛽 + 𝑍 ∗ cos 𝛽 = −𝛾 sin 𝛽ch𝜃 + 𝜂 cos 𝛽sh𝜃, 𝑌 = 0, 𝑍 = 𝑍 ∗ sin 𝛽 + 𝑋 ∗ cos 𝛽 = 𝜂 sin 𝛽sh𝜃 + 𝛾 cos 𝛽ch𝜃.
(4)
In this case, the bore-hole axis line T is completely determined by the initial (𝜃 0 ) and terminal (𝜃 t ) magnitudes of the 𝜃 parameter and values 𝛾 and 𝜂 which determine also the 𝛽 angle. 3. Analog of the Eulerian formulation of the problem of a drill string buckling in a curvilinear bore-hole The peculiarity of the considered problem of a drill-string stability in a curvilinear bore-hole lies in the fact that only initial stage of the buckling process is analyzed when the elastic displacements are small and the string shape change is not essential. As this takes place, the critical (ultimate) values of the acting loads and buckling shape are discovered. In
Fig. 2. Hyperbolic curves, used for simulation of the two-sectional borehole trajectory a) at the initial state of mathematic modelling in the fixed Cartesian coordinates; b) in the turned Cartesian coordinate system adapted for the design simulation.
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
the applied mathematics and mechanics such problems are collectively called two-point boundary eigenvalue problem (or the Sturm-Liouville problem). The simplest statement of this problem was formulated by L. Euler in analysis of rectilinear elastic rod stability under the action of axial compressive force F. It is described by linear differential equation 𝐸𝐼
𝑑4𝑤 𝑑2𝑤 −𝐹 =0 4 𝑑𝑠 𝑑 𝑠2
(5)
with the corresponding boundary equations. Here, E is the elasticity modulus, I is the inertia moment of the rod cross section, w is the transverse displacement of the rod, s is the natural axial parameter. It is apparent that Eq. (1) has zeroth (trivial) solution w(s) = 0 for any value of F, but its value corresponding to the nontrivial solution w(t) ≠ 0 is named the eigenvalue and appropriate solution is referred to as eigenmode. The problem of global stability loss of a drill-string in the channel of a curvilinear bore-hole under action of forces of gravity, contact, friction and force on the bit is stated in the analogous form. It is necessary to find the identical internal axial force F(s) in the drill-string caused by a combination of external perturbations which generates some additional buckled mode of equilibrium along with the initial one. In this case, the found F(s) force is the eigenfunction and the constructed buckled curve is the eigenmode. Inasmuch as only initial stage of the buckling is considered (as for the case of rectilinear rod), it is possible to conceive that the buckling displacements are small and to restrict our considerations to a linearized statement of the problem. It is also described by a fourth order linear ordinary differential equation of form (5) which has however more complicated structure. To construct it, firstly, it is necessary to formulate a general nonlinear system of a curvilinear rod deforming and then, to linearize it in the vicinity of the initial undeformed state. Nontrivial solutions of this system determine the critical states of the drill-string and the modes of its buckling. It is usual to formulate the equations of the theory of curvilinear flexible rods through the parametrization of bore-hole axis line T by natural parameter l measured by the length of its trajectory. Introduce also the Frenet movable trihedron with unit vectors of tangent t, normal n, binormal b and vector of total curvature 𝛀 = 𝑘𝑅 𝐛 + 𝑘𝑇 𝐭.
(6)
Here, kR is the curvature of bore-hole axis line T, kT is the its torsion. Assume that at the initial state and during buckling, the drill-string continues to contact with bore-hole surface Σ throughout its length under action of gravity forces. Then, to express the link of the drillstring axis S with this surface, introduce movable coordinate system oxyz (Fig. 3) with unit vectors i, j, k, where vector i is directed along the internal normal to the Σ surface, vector k is collinear with the t, and vector j supplements this system to the right-hand triad. With their use, it is possible to construct an analog of the Darboux vector 𝝎 = 𝑘𝑥 𝐢 + 𝑘𝑦 𝐣 + 𝑘𝑧 𝐤.
(7)
Here, kx , ky are the corresponding components of the curvature vector K = b/R, R is the curvature radius, kz determines the system oxyz twisting. With the use of these values, the equations of equilibrium of all external and internal forces are represented as follows [2,17]: 𝑑 𝐹𝑥 ∕𝑑𝑙 = −𝑘𝑦 𝐹𝑧 + 𝑘𝑧 𝐹𝑦 − 𝑓𝑥 , 𝑑 𝐹𝑦 ∕𝑑𝑙 = −𝑘𝑧 𝐹𝑥 + 𝑘𝑥 𝐹𝑧 − 𝑓𝑦 ,
(8)
𝑑 𝐹𝑧 ∕𝑑𝑙 = −𝑘𝑥 𝐹𝑦 + 𝑘𝑦 𝐹𝑥 − 𝑓𝑧 . The equations of the moments equilibrium have an analogous form:
𝑑 𝑀𝑧 ∕𝑑𝑙 = −𝑘𝑥 𝑀𝑦 + 𝑘𝑦 𝑀𝑥 + 𝑚𝑧 .
In these equations, Fx , Fy , Fz are the components of the internal elastic forces vector 𝐅 = 𝐹𝑥 𝐢 + 𝐹𝑦 𝐣 + 𝐹𝑧 𝐤,
(9)
(10)
Mx , My , Mz are the components of the internal elastic moments vector 𝐌 = 𝑀𝑥 𝐢 + 𝑀𝑦 𝐣 + 𝑀𝑧 𝐤,
(11)
fx , fy , fz are the components of the external distributed forces vector 𝐟 (𝑙) = 𝑓𝑥 (𝑙)𝐢 + 𝑓𝑦 (𝑙)𝐣 + 𝑓𝑧 (𝑙)𝐤.
(12)
mz is the external distributed twisting moment. The f force includes the forces of gravity (fgr ), contact interaction (fcont ) and friction (ffr ), then, 𝐟 (𝑙) = 𝐟 𝑔𝑟 (𝑙) + 𝐟 𝑐𝑜𝑛𝑡 (𝑙) + 𝐟 𝑓 𝑟 (𝑙).
(13)
The system of Eqs. (8), (9) determines the elastic equilibrium of the string under action of distributed forces fgr (l), fcont (l), ffr (l) in the borehole channel. If to supplement this system with the equalities, expressing internal elastic forces and moments through the strains and the last ones through the displacements, then, it will be possible to investigate nonlinear deforming, stability, and buckling of the drill-string under different combinations of loads and during tripping in/out and drilling regimes. However, these tasks have a feature conditioned by contact and frictional interaction of the string with the hole wall. Therefore, if the DS is held in the hole in the immovable state, then infinite variety of different friction forces combinations and appropriate functions of internal axial forces may correspond to this state. So, the stated problem is statically indeterminate. Then, only when the drill-string begins to move (in lowering or lifting), all the noted forces can be computed. For this reason, the stated problem is solved in two stages. Firstly, through the use of Eqs. (8), (9), the internal quasi-static forces Fx (l), Fy (l), Fz (l) are calculated and next, with the help of the linearized equations, the task of stability is stated and solved. So far as, besides the unknown internal forces, the unknown external (contact) (fcont ) and frictional (ffr ) forces are present in Eqs. (8) as well. They are supplemented with equalities 𝐹𝑥 = −𝑑 𝑀𝑦 ∕𝑑 𝑙 − 𝑘𝑧 𝑀𝑥 + 𝑘𝑥 𝑀𝑧 , 𝐹𝑦 = 𝑑 𝑀𝑥 ∕𝑑 𝑙 + 𝑘𝑦 𝑀𝑧 − 𝑘𝑧 𝑀𝑦 ,
𝑑 𝑀𝑥 ∕𝑑𝑙 = −𝑘𝑦 𝑀𝑧 + 𝑘𝑧 𝑀𝑦 + 𝐹𝑦 , 𝑑 𝑀𝑦 ∕𝑑𝑙 = −𝑘𝑧 𝑀𝑥 + 𝑘𝑥 𝑀𝑧 − 𝐹𝑥 ,
Fig. 3. Geometric interpretation of the DS axis S contact with the bore-hole wall surface Σ during buckling deforming.
(14)
issuing from Eqs. (9), and correlations 𝑀𝑥 = 𝐸𝐼 𝑘𝑥 , 𝑀𝑦 = 𝐸𝐼 𝑘𝑦
(15)
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
Since, at the initial state, the drill-string geometry coincides with the geometry of the bore-hole, then, at this stage, kx = 0, kz = 0, Mx = 0, Mz = 0, Fy = 0, and curvature ky is determined by correlation [8] [ ] 2 1∕2 (𝑋̇ 2 + 𝑍̇ 2 )(𝑋̈ 2 + 𝑍̈ 2 ) − (𝑋̇ 𝑋̈ + 𝑍̇ 𝑍̈ ) 𝑘𝑦 (𝜃) = , (16) 3 (𝑋̇ 2 + 𝑍̇ 2 )
We will point out one more feature of this process. Since, function ky has already been known at all points lm (1 ≤ m ≤ n), derivatives dky /dl and d2 ky /dl2 defined by complicated expressions are calculated during the investigation process through the use of the finite difference method. To check stability of the drill-string prestressed by the Fz (l) force, linearize system (8, 9) at kx (l) = 0, kz (l) = 0, Mx (l) = 0, Mz (l) = 0, Fy (l) = 0
where the dot over the symbol denotes differentiation with respect to 𝜃 and variables X(𝜃) and Z(𝜃) are specified according to (1). After appropriate transformations, one gains 𝛾𝜂 𝑘𝑦 (𝜃) = . (17) 3∕2 (𝛾 2 𝑠ℎ2 𝜃 + 𝜆2 𝑐 ℎ2 𝜃)
𝑑 𝛿𝐹𝑥 ∕𝑑 𝑙 = −𝛿𝑘𝑦 𝐹𝑧 − 𝑘𝑦 𝛿𝐹𝑧 − 𝛿𝑓𝑥𝑔𝑟 − 𝛿𝑓𝑥𝑐𝑜𝑛𝑡 − 𝛿𝑓𝑥𝑓 𝑟 ,
At these states, gravity force projections 𝑓𝑥𝑔𝑟 (𝜃), 𝑓𝑧𝑔𝑟 (𝜃) are equal to ( ) ( ) 𝑓𝑥𝑔𝑟 (𝜃) = −𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 cos 𝛼, (18) ( ) ( ) 𝑓𝑦𝑔𝑟 (𝜃) = −𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 sin 𝛼.
𝑑 𝛿𝑀𝑦 ∕𝑑 𝑙 = 0,
Here, 𝑟1 , 𝑟2 are the external and internal radii of the string tube, correspondingly; 𝜌𝑡 , 𝜌𝑚 are densities of the tube material and mud, correspondingly; g is the gravity acceleration. The quasistatic friction force is expressed through the friction coefficient 𝜇 and contact force: 𝑓𝑧𝑓 𝑟 = ±𝜇 ||𝑓 𝑐𝑜𝑛𝑡 ||,
(19)
𝑓 𝑐𝑜𝑛𝑡
where, the force is assumed to be along the normal direction and to be determined by its components 𝑓𝑥𝑐𝑜𝑛𝑡 and 𝑓𝑦𝑐𝑜𝑛𝑡 : √ )2 ( )2 ( |𝑓 𝑐𝑜𝑛𝑡 | = 𝑓𝑥𝑐𝑜𝑛𝑡 + 𝑓𝑦𝑐𝑜𝑛𝑡 . | | Then, after some transforms, one can derive an ordinary differential equation for the axial force Fz (l) calculation 𝑑 𝑘𝑦 ( ) ( ) 𝑑𝐹𝑧 𝛾𝜂 = −𝐸 𝐼 + 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 sin 𝛼 𝑑𝑙 (𝛾 2 𝑠ℎ2 𝜃 + 𝜂 2 𝑐ℎ2 𝜃)3∕2 𝑑 𝑙 | | 𝑑 2 𝑘𝑦 ( ) ( ) 𝛾𝜂 | | − 𝐹𝑧 + 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 cos 𝛼 |. + 𝜇 |𝐸 𝐼 | | 𝑑 𝑙2 (𝛾 2 𝑠ℎ2 𝜃 + 𝜂 2 𝑐ℎ2 𝜃)3∕2 | | (20) It should be stressed that in this equation the differentiation is performed with respect to independent variable l while its coefficients determined by geometric Eqs. (1) are expressed by parameter 𝜃. Therefore, at first, it is necessary to transfer from 𝜃 to l at the coefficients of Eq. (20) and then, to start its integration. This transition is performed with the use of differential correlation 𝑑 𝑙 = 𝐻𝑑 𝜃,
(21)
where H is the metric multiplier determined by correlation √ √ 2 2 𝐻 = (𝑋̇ ∗ ) + (𝑍̇ ∗ ) = (𝛾sh𝜃)2 + (𝜂𝑐ℎ𝜃)2 .
𝑙(𝜃) =
∫
√ (𝛾𝑠ℎ𝜉)2 + (𝜂𝑐 ℎ𝜉)2 𝑑𝜉,
(22)
𝜃(𝑙) = 𝜃0 +
∫ 0
𝑑𝜉 . √ (𝛾𝑠ℎ𝜉)2 + (𝜂𝑐 ℎ𝜉)2
(23)
Eqs. (17)–19, (22) permit one to perform integration of Eq. (20) by the Runge-Kutta method. In its realization, the drill-string length L is divided into n small steps Δl = L/n and at the every m-th point lm = Δlm (1 ≤ m ≤ n) parameter 𝜃 values are found 𝜃𝑚 = 𝜃0 +
𝑚 −1 ∑ 𝑠=1
𝑑 𝛿𝑀𝑥 ∕𝑑 𝑙 = −𝑘𝑦 𝛿𝑀𝑧 + 𝛿𝑘𝑧 𝑀𝑦 + 𝛿𝐹𝑦 ,
1 Δ𝑙. √ 2 (𝛾𝑠ℎ𝜃𝑠 ) + (𝜂𝑐ℎ𝜃𝑠 )2
Thereupon, all other functions, depending on 𝜃, are computed.
(24)
(25)
𝑑 𝛿𝑀𝑧 ∕𝑑 𝑙 = 0. Here, symbol 𝛿 denotes the small variation of the considered displacement. In Eqs. (25) derivation, it is taken into account that variations 𝛿ky , 𝛿Fx , 𝛿Fz and 𝛿Mz are equal to zero [17]. To reduce them to the conclusive form, take into account that in the buckling process, the drill-string remains in the contact with the bore-hole wall. Then, its geometry can be represented through the geometry parameters of channel surface Σ. To perform this, introduce a family of two coordinate lines u = const and v = const on it where v (0 ≤ v ≤ 2𝜋) is the parameter determining angular position of a point on a generating circle gained by the Σ surface intersection with a normal plane, therewith points v = 0 and v = 𝜋 are at the bottom and top positions on Σ; u is the parameter coinciding with l. In buckling, every point of the drill-string axis line shifts by small value a𝛿v to new position 𝛿v on its generating circle. In doing so, the appropriate value of parameter u remains unchanged. For this reason, the 𝛿v variable is a function of variable l (or u) that entirely determines the buckled state of the drill-string. Under these assumption, orientation and curvature of the drill-string axis line S, as well as, components 𝑓𝑥𝑔𝑟 (𝑙), 𝑓𝑦𝑔𝑟 (𝑙), 𝑓𝑧𝑔𝑟 (𝑙) of the gravity force in the deformed state can be expressed through the 𝛿v(l) function that determines the curve S position in the Σ surface. In this case, curvature kx coincides with geodesic curvature kgeod of the Σ surface and curvature ky does this with its normal curvature knorm [8]. In its turn curvatures kgeod and knorm are expressed through parameters of the first (Φ1 (u,v)) and second (Φ2 (u,v)) quadratic forms of surface Σ. Then, one has 𝛼̇ sin 𝑣 . 𝑘𝑥 = 𝑘𝑔𝑒𝑜𝑑 = 𝑎𝑣′′ − √ 2 𝛾 sh2 𝜃 + 𝜂 2 ch2 𝜃
(26)
At the undeformed state, kx = 0 and its small variation in the vicinity of this state is determined as follows: (27)
Curvature ky can be represented in a general form through the normal curvature knorm of surface Σ in the curve S direction, which in its turn, is expressed in the form of the Euler theorem as follows: 𝑘𝑦 = 𝑘𝑛𝑜𝑟𝑚 = 𝑘1 cos2 𝜓 + 𝑘2 sin2 𝜓,
𝜃0 𝑙
𝑑 𝛿𝐹𝑧 ∕𝑑 𝑙 = 𝛿𝑘𝑦 𝐹𝑥 − 𝛿𝑓𝑧𝑔𝑟 − 𝛿𝑓𝑧𝑐𝑜𝑛𝑡 − 𝛿𝑓𝑧𝑓 𝑟 ,
𝛼̇ 𝛿𝑘𝑥 = 𝑎𝛿𝑣′′ − √ 𝛿𝑣. 2 2 𝛾 sh 𝜃 + 𝜂 2 ch2 𝜃
In this case, 𝜃
𝑑 𝛿𝐹𝑦 ∕𝑑 𝑙 = −𝛿𝑘𝑧 𝐹𝑥 + 𝛿𝑘𝑥 𝐹𝑧 − 𝛿𝑓𝑦𝑔𝑟 − 𝛿𝑓𝑦𝑐𝑜𝑛𝑡 − 𝛿𝑓𝑦𝑓 𝑟 ,
(28)
where k1 and k2 are the principal curvatures of the Σ surface along curves v = const and u = const, 𝜓 = adv/dl is the angle between the direction of the S line and coordinate line u. In the undeformed state, ky is expressed by Eq. (17) but its overall expression for an arbitrary case is very complicated. So, here, its small variation in the vicinity v = 0 is calculated, which turns to be equal to 𝛿𝑘𝑦 = 2(−𝑘1 + 𝑘2 ) sin 𝜓 cos 𝜓 = 0
(29)
because sin 𝜓 = 0 for v = 0. It is shown also that 𝛿𝑘𝑧 = 𝑑 𝛿𝑣∕𝑑 𝑙.
(30)
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
In a similar manner, variations 𝛿𝑓𝑥𝑔𝑟 , 𝛿𝑓𝑦𝑔𝑟 ,𝛿𝑓𝑧𝑔𝑟 can be represented as follows: ) ( ) ( 𝛿𝑓𝑥𝑔𝑟 = 0, 𝛿𝑓𝑦𝑔𝑟 = 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 (−𝑎 sin 𝛼𝑑𝛿𝑣∕𝑑𝑙 + cos 𝛼𝛿𝑣), 𝛿𝑓𝑧𝑔𝑟 = 0. (31) After appropriate transformations of system (25) with allowance made for correlations (26–31), it is reduced to one ordinary differential equation of fourth order [ ] 2 𝑑 4 𝛿𝑣 sh𝜃𝑐ℎ𝜃 𝑑 𝛿𝑣 2 2 𝑑 𝐸𝐼 − 𝐸 𝐼(𝛾 + 𝜂 ) 𝑑 𝑙4 𝑑 𝑙2 (𝛾 2 sh2 𝜃 + 𝜂 2 ch2 𝜃)3∕2 𝑑 𝑙 ⎛ ⎞ ⎟ 𝐸𝐼 𝑑 2 ⎜ 𝛼̇ − 𝛿𝑣⎟ √ 𝑎 𝑑 𝑙2 ⎜⎜ ⎟ 2 2 2 2 ⎝ 𝛾 sh 𝜃 + 𝜂 ch 𝜃 ⎠ ⎡ ⎤ ⎢ 𝑑 2 𝛿𝑣 ⎥ (𝛾 2 + 𝜂 2 )sh𝜃𝑐ℎ𝜃 𝑑𝛿𝑣 𝛼̇ 1 𝛿𝑣⎥ + 𝐹 𝑧 ⎢− + + √ 2 3∕2 𝑑𝑙 𝑎 2 2 𝑑 𝑙 2 2 ⎢ (𝛾 sh 𝜃 + 𝜂 ch 𝜃) 𝛾 2 sh2 𝜃 + 𝜂 2 ch2 𝜃 ⎥⎦ ⎣ ) ( ( ) ( ) 𝑑𝛿𝑣 cos 𝛼 + 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 sin 𝛼 − 𝛿𝑣 = 0, (0 ≤ 𝑙 ≤ 𝐿). (32) 𝑑𝑙 𝑎 In it, the operations of differentiation with respect to independent variables 𝜃 (designated by a dot) and l are used. The derivative with respect to 𝜃 is used only at the coefficients. Therefore, as noted above, in numerical integrating of functions depending on 𝜃, the transition from lm to 𝜃 m is performed with the help of procedure (24). Then, through the use of found values 𝜃 m at points lm , the coefficients of Eq. (32) are calculated and it is integrated with the method of finite differences. The second peculiarity of this equation consists in its homogeneity and so it always has trivial solution 𝛿v(l) = 0(0 ≤ l ≤ L) at any values of axial force Fz (l) presenting at coefficients of Eq. (32). But, under some values of this function which changes with the change of the force on bit and action of friction forces, Eq. (32) becomes degenerated and it acquires also a nontrivial solution alike as the trivial one. In this case, force 𝐹𝑧𝑐𝑟 (𝑙) is named eigen or critical one and the corresponding solution 𝛿vcr (l) is eigen (or buckling) mode. Solution of Eq. (32) with pinned-pinned boundary conditions is constructed by the finite difference method. Together with Eq. (20) solution, it determines entirely the system stress-strain states in the buckling process and geometrical shape of the buckled drill-string. The elaborated approach advantage consists in its suitability for the problem solving in the domain of the drill-string total length 0 ≤ l ≤ L. Then, as in a general case, function 𝐹𝑧𝑐𝑟 (𝑙) is alternating-sign and has parts of stretching and compressing, the drill-string buckling (though based on the global statement) occurs mainly in the zone of its maximal negative values. So, the problem solution permits one to find its localization, to calculate function 𝐹𝑧𝑐𝑟 (𝑙)values, character of the function 𝛿vcr (l) change throughout the drill-string length, and values of external critical forces acting on the drill-string. 4. Computer modelling results It is necessary to underline once more that Eqs. (20) and (32) has variable coefficients, therefore, their eigenvalues and eigenmodes
Fig. 4. Geometric schemes of the bore-hole outlines with the declined (I), horizontal (II), and ascending (III) lower sections selected for analysis.
cannot be found via the use of analytical methods. So, the finite difference method was used for their analysis. Besides, the stated problem is multivariate. Indeed, the buckling process depends on the value of the angle 𝛽 between the hyperbola (4) branches and the curvature of rounding between them, the string tube stiffness, the clearance value, presence of frictional effects, the value of compressive force at the lower end, kinds of boundary conditions, etc. As the comprehensive analysis of such buckling process is impossible, below, some computational results are presented for the particular values of the characteristic parameters. They make up: E = 2.1 • 1011 Pa, I = 2.7 • 10−4 m4 ,𝜌t = 7.8 • 103 kg/m3 , 𝜌m = 1.3 • 103 kg/m3 , r1 = 0.1m, r2 = 0.09m, 𝜇 = 0, 0.2, 0.3. It was assumed that the drill-string was pinned at its both ends, therefore 𝛿𝑣(0) = 𝛿𝑣(𝐿) = 0, 𝛿𝑣′′ (0) = 𝛿𝑣′′ (𝐿) = 0.
(33)
Three basic structural types of the bore-holes differing by angles 𝛽 were analyzed (Fig. 4). Their properties were determined by the parameters given in Table 1. They are united into three groups differing by the hyperbola angle 2𝛽. The first group (type I) is defined by angle 2𝛽 = 100° and positive inclination of the lower section at the angle 𝛼 0 = 10° to the horizontal (Fig. 4). The lower section of the bore-hole of the second group (type II) is horizontal (2𝛽 = 90°, 𝛼 0 = 0) and it is inclined at the negative angle 𝛼 0 = −10° for the third group (type III) of bore-holes. Their measurements are regulated through the change of parameters 𝛾 and 𝜂 and prescription of the initial (𝜃 0 ) and final (𝜃 f ) values of dimensionless parameter 𝜃. They specify also drill-string length L, depth H0 of the drill-string initial point l = 0 and maximal curvature |ky |max at the middle point of the hole. In every group, additional variation of the values of clearance (a = 0.05, 0.08m) and friction coefficient (𝜇 = 0, 0.2, 0.3) was performed.
Table 1 Geometric parameters of the considered bore-holes. Type
2𝛽
Case
𝜃0
𝜃f
𝛾 (m)
𝜂 (m)
𝛼 0 (deg)
L (m)
H0 (m)
|ky |max (m−1 )
I
100°
II
90°
III
80°
1 2 1 2 1 2
−10𝜋/9 −4𝜋/9 −10𝜋/9 −4𝜋/9 −10𝜋/9 −4𝜋/9
10𝜋/9 4𝜋/9 10𝜋/9 4𝜋/9 10𝜋/9 4𝜋/9
131 1146 170 1478 220 1934
156 1366 170 1478 185 1623
10 10 0 0 −10 −10
6631 5925 7785 6705 9258 7793
4000 3995 4033 4000 4000 4000
5.36 6.14 5.87 6.76 6.43 7.34
• 10−3 • 10−4 • 10−3 • 10−4 • 10−3 • 10−4
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
Fig. 5. Trajectory outlines (a) and corresponding curvature functions ky (l) (b) of the two-sectional bore-holes of the first group with larger (curve 1) and smaller (curve 2) curvatures in the knee zones (cases 1 and 2 in Table 1).
Initially, the analysis results for group I (2𝛽 = 100°) of the lower section inclination were considered. In Fig. 5a, trajectories 1 and 2 (cases 1 and 2) of this group are presented according to their sequence in Table 1. The curvature functions, corresponding to these curves, are demonstrated in Fig. 5b, correspondingly. As can be seen, their curvatures differ essentially in their knee zones. The computer simulation showed that the critical state appearance and modes of buckling depend primarily on the fact that the formulated problem is singularly perturbed [5] as a consequence of large length of the drill-strings and their comparatively small bending stiffness. Because of this, the drill-strings buckling modes have the shapes of harmonic wavelets localized at the vicinity of a zone with enlarged compressive axial force Fz (l) which, in its turn, depends on the bore-hole inclination angle, curvature, tripping (in or out) operation, and friction coefficient value. As a rule, maximal compressive axial force Fz (l) inside the drill-string occurs during tripping in operation at the largest values of friction coefficient 𝜇. Functions 𝐹𝑧𝑐𝑟 (𝑙) of critical axial force and modes 𝛿v(l) of the drillstring buckling for the bore-holes of type I are demonstrated in Table 2. The data fit to cases 1 and 2 shown in Fig. 5 with curvature functions 1 and 2 given in Fig. 5b. Their extremal values |ky |max , as well as total length L of the drill-strings and depths H0 of their lower ends are tabulated in Table 1. They are determined by initial geometric parameters 𝜃 0 , 𝜃 f and 𝛾, 𝜂. But here, particular attention should be played to parameter 𝛼 0 which specifies the angle of the lower section inclination to the horizontal. The point is that friction effects are associated with this angle. In fact, three values of friction coefficients 𝜇 = 0, 0.2, and 0.3 were chosen in analysis and different appearances of the buckling processes took places for them. First of all, situation 𝜇 = 0 is worthy of notice. For example, it can occur when the drill-string is lowering slowly and its arbitrary vibrations and mud flows neutralize the friction effects. Then, the drill-string is prestressed by gravity forces 𝑓𝑧𝑔𝑟 (𝑙), stretching it throughout its length, and WOB (weight on bit) force Fz (0), compressing the drill-string at its lower end l = 0. Summarized axial forces 𝐹𝑧𝑐𝑟 (𝑙) at the critical states are shown in column 𝜇 = 0 of Table 2 for cases 1 and 2 and the clearance values a = 0.05 and 0.08 m. It can be seen that axial force 𝐹𝑧𝑐𝑟 (𝑙) also consists of two nearly rectilinear sections, linearly distributed in the limits of every bore-hole section, because the bore-hole sections are nearly rectilinear. The drill-string is stretched by gravity forces inside the upper section, but stretch force 𝐹𝑧𝑐𝑟 (𝑙) diminishes in the lower section, as angle 𝛼 decreases and compressive bit force 𝐹𝑧𝑐𝑟 (0) acts on its lower end l = 0. Therefore, the lower section of the
drill-string is partially stretched and partially compressed and it can buckle in its part where the compressive force 𝐹𝑧𝑐𝑟 (𝑙) has maximal values, id est, in the vicinity of the lower boundary. Built global eigenmodes 𝛿v(l) in Table 2 confirm this conclusion. But situation changes when the friction forces are included in the mechanical process. At that time, axial force Fz (l) is determined by a disparity between tensile gravity forces 𝑓𝑧𝑔𝑟 (𝑙), compressive WOB Fz (0), and distributed friction force 𝑓𝑧𝑓 𝑟 (𝑙). In Fig. 6, they are shown for the critical state, corresponding to case I, a = 0.08m, 𝜇 = 0.2 in Table 2. The friction force (Fig. 6c) is determined by the contact force modulus (Fig. 6b) having the peak form in the knee zone where it sets against the hole wall. Therefore, the friction function has broken outline (Fig. 6c). Integral combination of these forces together with critical value of WOB force 𝐹𝑧𝑐𝑟 (0) = 274.119𝑁 generates axial force 𝐹𝑧𝑐𝑟 (𝑙) outlined in Table 2. It also consists of two near rectilinear sections with transitional curvi𝑐𝑟 = 289.076𝑁 is located linear part but this time, the extremal value 𝐹𝑧,𝑒𝑥𝑡 inside its length and in this location, the buckling zone is situated. It is significant that this effect became possible owing to the fact that friction angle 𝛼𝑓 𝑟 = arctg𝜇 = arctg0.2 = 11.31◦ = 0.197 exceeded inclination angle 𝛼 0 = 10° and so the compressive frictional effect suppressed the tensile gravity action and shifted the extremum compressive forces inside the drill-string. In the considered cases, the properties of singularly perturbed systems become distinctly apparent. As far as the models of the long flexible curvilinear rods pertain to this type of systems [5,16,17], their buckling modes have the shapes of harmonic wavelets localized at the places of maximal values of their internal axial force Fz (l). Their typical 2D shapes are presented in Fig. 7. As a rule, they are concentrated in a narrow zone of the boundary l = 0 vicinity (Fig. 7a, for case 1, a = 0.08m, 𝜇 = 0 in Table 2) or in the drill-string maximal curvature zone (Fig. 7b, for case 1, a = 0.08m, 𝜇 = 0.2). In doing so, wavelet pitch 𝜆 diminished from value 35.3 m to 24.1 m. Similar effects are discussed also by Elishakoff et al. [9]. The noted peculiarities can serve as explanation of the fact that the buckling effect and modes do not practically depend on the boundary conditions, since the buckling wavelets are located far from the drill-string ends. In case 2, the bore–hole has smaller curvature (see Table 1), the friction forces are less influential, the drill-string becomes more stable, and the buckling wavelets shift to its boundary l = 0. The results of computer modelling the drill-string stability in twosectional bore-hole of type II (Fig. 4) are shown in Table 3. Inasmuch as here, the lower section is horizontal, the influence of frictional
N.V. Shlyun and V.I. Gulyayev
Table 2 Functions of critical axial forces and modes of buckling for trajectories of type I.
International Journal of Mechanical Sciences 172 (2020) 105427
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
Fig. 6. Functions of longitudinal distributed gravity (a), contact (b), and friction (c) forces contributing into formation of internal critical axial force Fz (l), causing the localized buckling of the drill-string (case 1 in Table 2).
Fig. 7. Separated fragments of the global buckling modes in the horizontal section of the bore-hole trajectory (Table 2, case I, a = 0.08 m): a) frictionless model at μ = 0; buckling shape is the boundary wavelet; b) frictional model at μ = 0.2; buckling wavelets are shifted to the knee zone.
effects shows itself more distinctly. In the bore-holes with larger value of curvature (case 1), the buckling wavelets widen and shift to the lower zone of the vertical section. In doing so, axial force 𝐹𝑧𝑐𝑟 (0) at point l = 0 of the drill-string becomes tensile for case 1, under condition of friction presence. What this means is that it is necessary to draw the drill-string at its bottom edge in order to realize its dragging. It is also significant that critical forces 𝐹𝑧𝑐𝑟 (𝑙) decrease with clearance aincrease. The bifurcation process acquires additional peculiarities when the knee point of the bore-hole trajectory is below its both ends (Fig. 4, type III). Then, even if the drill-string is stretched at its both ends, it is compressed in the vicinity of the knee zone. In Table 4 the critical states are shown. As can be seen, here, the buckling wavelets are approaching to the knee zone. If to enlarge tensile forces Fz (0) and Fz (L) at the drillstring edges, the system will change to its precritical state, their lower values correspond to the postcritical states. Inferring these results, it can be noted that if the lower section of the hole is inclined (type I) or horizontal (type II) and friction effects are removed, maximal values of the compressive axial forces occur at the bottom zone of the drill-string. In these cases, the buckling mode represents a boundary effect in the shape of a short wavelet localized in the bottom zone of the drill-string where the hole trajectory is near rectilinear. The appropriate 3D mode is schematically shown in Fig. 8. This case is not very difficult for analysis. If during tripping in operation the friction forces are generated, they act opposite the motion direction and the maximal axial force shifts upward. Therefore, buckling wavelets also shift upward. But the situation becomes more serious if the bore-hole of the type III is studied. In this case, maximal compressive force Fz (l) is generated in the knee zone (see Table 4) and therefore, the buckling wavelets are born here (Fig. 9). The contact interaction of the drill-string with the bore-hole channel wall surface is characterized by essential obstacles conditioned by the geometrical complexity of the interface surface. In this case, the drillstring stability analysis should be especially performed.
Fig. 8. Typical 3D schematic fragment of the drill string (type I) buckling wavelet in the inclined low boundary zone (frictionless model).
Fig. 9. Typical 3D schematic fragment of the drill string (type I) buckling wavelet in the knee zone (frictional model).
N.V. Shlyun and V.I. Gulyayev
Table 3 Functions of critical axial forces and modes of buckling for trajectories of type II.
International Journal of Mechanical Sciences 172 (2020) 105427
N.V. Shlyun and V.I. Gulyayev
Table 4 Functions of critical axial forces and modes of buckling for trajectories of type III.
International Journal of Mechanical Sciences 172 (2020) 105427
N.V. Shlyun and V.I. Gulyayev
5. Conclusions 1 The problem of drill-string buckling in a deep curvilinear twosectional bore-hole is stated. The initial stage of the critical elastic bifurcation of the string under action of a force on bit and distributed gravity, contact, and friction forces is considered. Geometric properties of the drill-string trajectories are analyzed, constitutive linearized differential equations of its behavior at critical states are deduced on the basis of the theory of curvilinear flexible rods, theory of channel surfaces, and differential geometry. 2 It is shown that large length of the drill-string leads to a virtual loss of its bending stiffness and near degeneration of some summands in the constitutive equations. Therefore, they belong to the singularly perturbed type equations and their solutions look like harmonic boundary effects or localized harmonic perturbations (wavelets) shifted to the internal zone. The techniques for numerical solving these equations in the domain of the whole length of the drill-string are elaborated and the appropriate software is created. 3 To analyze the two-sectional drill-string buckling, firstly, the critical stress-strain state of the drill-string (eigenfunction) is established and then, the shape of the buckled string (eigenmode) is constructed. It is shown that depending on the knee angle, and values of the knee curvature, clearance, and friction coefficient, the mode of stability loss can represent a short harmonic wavelet localized at the lower end of the lower section or shifted to the knee zone of the bore-hole. 4 The elaborated techniques and created software can be used for prognostication and prevention of emergency situations due to the twosectional bore-holes buckling. Declaration of Competing Interest None. References [1] Ajibose OK, Wiercigroch M, Akisanya AR. Experimental studies of the resultant contact forces in drillbit–rock interaction. Int J Mech Sci 2015;91:3–11 February. [2] Antman SS. Nonlinear problems of elasticity. Springer; 2005. [3] Berger EJ. Friction modeling for dynamic system simulation. Appl Mech Rev 2002;55(6):535–76. [4] Brett JF, Beckett AD, Holt CA, Smith DL. Uses and limitations of drillstring tension and torque models for monitoring hole conditions. SPE Drill Eng 1989;4:223–9. [5] Chang KW, Howes FA. Nonlinear singular perturbation phenomena. Springer −Verlag; 1984. [6] Cunha JC. Buckling of tubulars inside wellbore: a review on recent theoretical and experimental works. SPE Drill Complet 2004;19(1):13–19. [7] Dawson R, Paslay RR. Drill pipe buckling in inclined holes. J Pet Technol 1984;36(10):1734–8. [8] Dubrovin BA, Novikov SP, Fomenko AT. Modern geometry-methods and applications. Springer - Verlag; 1992. [9] Elishakoff, I., Li, Y., Starnes, J.H., 2001. Non-Classical problems in the theory of elastic stability. Cambridge University Press, Cambridge, UK. [10] Gao DL, Huang WJ. A review of down-hole tubular string buckling in well engineering. Petr Sci 2015;12(3):443–57. [11] Gulyayev VI, Andrusenko EN. Theoretical simulation of geometrial imperfections influence on drilling operations at drivage of curvilinear bore-holes. J Petr Sci Eng 2013;112:170–7.
International Journal of Mechanical Sciences 172 (2020) 105427 [12] Gulyayev VI, Gaidaichuk VV, Solovjov IL, Gorbunovich IV. The buckling of elongated rotating drill-strings. J Petr Sci Eng 2009;67:140–8. [13] Gulyayev V, Glazunov S, Glushakova O, Vashchilina E, Shevchuk L, Shlyun N, Andrusenko E. Modeling emergency situations in the drilling of deep boreholes. Cambridge Scholars Publishing; 2019. [14] Gulyayev VI, Glushakova OV. Large-scale and small-scale self-excited torsional vibrations of homogeneous and sectional drill strings. Interact Multiscale Mech 2011;4(4):291–311. [15] Gulyayev VI, Shevchuk LV. Drill-string bit whirl simulation with the use of frictional and nonholonomic models. J Vib Acoust 2015;138(1):011021–011021-9. [16] Gulyayev VI, Shlyun NV. Influence of friction on buckling of a drill string in the circular channel of a bore hole. Pet Sci 2016;13:698–711. [17] Gulyayev VI, Shlyun NV. Global analysis of drill-string buckling in the channel of a curvilinear bore-hole. J Nat Gas Sci Eng 2017;40:168–78. [18] Huang WJ, Gao DL. Helical buckling of a thin rod with connectors constrained in a cylinder. Int J Mech Sci 2014;84:189–98. [19] Kapitaniak M, Vaziri V, Páez Chávez J, Wiercigroch M. Numerical study of forward and backward whirling of drill-string. J Comput Nonlinear Dynam 2017;12(6) November, 061009 (7 pages). [20] Kapitaniak M, Vaziri V, Páez Chávez J, Wiercigroch M. Experimental studies of forward and backward whirls of drill-string. Mech Syst Signal Process 2018;100:454–65 February. [21] Korn GA, Korn TM. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and reviews. Dover Publications; 2002. [22] Krauberger N, Saje M, Planinc I, Bratina S. Exact buckling load of a restrained RC column. Struct Eng Mech 2007;27(3):293–310. [23] Kwon YW. Analysis of helical buckling. SPE Drill Eng 1988;3(June):211–16. [24] Kyllingstad A. Buckling of tubular strings in curved wells. J Petr Sci Eng 1995;12(3):209–18. doi:10.1016/0920-4105(94)00046-7. [25] Liao M, Liu Y, Páez Chávez J, Chong ASE, Wiercigroch M. Dynamics of vibro-impact drilling with linear and nonlinear rock models. Int J Mech Sci 2018;146–147:200–10 October. [26] Liu X, Vlajic N, Long X, et al. Nonlinear motions of a flexible rotor with a drill bit: stick-slip and delay effects. Nonlinear Dyn 2013;72:61–77. [27] Lubinski A. Developments in petroleum engineering. Gulf Publishing Company; 1987. p. 438. [28] Lubinski A, Althouse WS, Logan JL. Helical buckling of tubing sealed in packers. JPT 1962;14(6):655–70. [29] Mitchell RF. Comprehensive analysis of buckling with friction. SPE Drill Complet 1996;11(3):178–84. [30] Mitchell RF. Tubing buckling the state of art. SPE Drill Complet 2008;23(4):361–70. [31] Mitchell RF, Samuel R. How good is the torque/drag model? SPE Drill Complet 2009;24(1):62–71. [32] Samuel R. Friction factors: what are they for torque, drag, vibration, bottom hole assembly, and transient surge / swab analysis. J Petr Sci Eng 2010;73(3–4):258–66. [33] Samuel, R., 2008. Systems and Methods for Modeling Wellbore Trajectories. US 886.2436 B2 Patent number US 12/145, 376. Publication date 14 Oct. 2014. [34] Sawaryn SJ, Sanstrom B, McColpin G. The management of drilling-engineering and well-services software as safety-critical systems.. SPE Drill Complet 2006;21(2):141–7 (June). [35] Sawaryn SI, Thorogood JL. A compendium on directional calculations based on the minimum curvature method. SPE Drill Complet 2005:24–36 March. [36] Sun C, Lukasiewicz S. A new model on the buckling of a rod in tubing. J Petr Sci Eng 2006;50:78–82. [37] Tucker RW, Wang C. Torsional vibration control and cosserat dynamics of a drill-rig assembly. Meccanica 2003;38:145–61. [38] Vasilyeva AB, Butuzov VF. Singularly perturbed equations at critical states. Moscow State University; 1978. (in Russian). [39] Vlajic N, Fitzgerald T, Nguen V, Balachandran B. Geometrically exact planar beams with initial pre-stress and large curvature: static configurations, natural frequencies, and mode shapes. Int J Solids Struct 2014;51:3361–71. [40] Wang X, Yuan Z. Investigation of frictional effects on the nonlinear buckling behavior of a circular rod laterally constrained in a horizontal rigid cylinder. J Petr Sci Eng 2012;91:70–8 90W22. [41] Yan Y, Wiercigroch M. Dynamics of rotary drilling with non-uniformly distributed blades. Int J Mech Sci 2019;160:270–81 September.
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Buckling of a drill-string in two-sectional bore-holes N.V. Shlyun∗, V.I. Gulyayev Department of Mathematics, National Transport University, M. Omelianovycha-Pavlenka Str., 1, 01010, Kiev, Ukraine
a r t i c l e
i n f o
Keywords: Two-sectional bore-holes Stability loss Eigenfunctions Buckling wavelets
a b s t r a c t The nature of the phenomena of drill-strings buckling in channels of deep curvilinear bore-holes is essentially complicated by complex combination of acting distributed forces (gravity, contact and friction), large length of the string (leading to a virtual loss of its bending stiffness), and its intricate geometry. Therefore, theoretical modelling of these effects turns out to be difficult to implement. In this article, the touched problem is stated with the use of the curvilinear flexible rods theory, differential geometry concepts, basic foundations of channel surfaces theory, and methods of numerical mathematics. The elaborated software is applied to the global analysis of the incipient stage of the drill-string buckling inside a deep bore-hole consisting of two nearly rectilinear sections bridged with curvilinear interjacent part. To solve the problem, firstly, the critical stress-strain state of the drill-string (eigenfunction) is established, and then, the shape of the buckled drill-string (eigenmode) is built. It is shown that the stated problem refers to the singularly perturbed type and so, depending on the knee angle and friction coefficient, the mode of stability loss represents a short wavelet localized at the lower end of the lower section or it is shifted to the knee zone of the bore-hole. The features of critical behavior of the drill-strings associated with the values of the knee angle of the bore-hole, system clearance, and friction coefficient, as well as critical loads and buckling mode wavelets are discussed.
1. Introduction At the present time, to extract petroleum resources, deep vertical and inclined, off-shore and ground-based bore-holes are drilled. The techniques and methods of their drilling are developing so fast that scientific understanding and substantiation have no time to refine them. Therefore, not infrequently, different unpredictable emergency effects and overall failures accompany these procedures. The gravest of these are intensive vibrations and resonances of drill-strings [1,25,41], the self-triggering of torsional relaxation vibration [14], the self-excitation of forward, backward, and superfast bit whirlings [15,19,20] the drill-string sticking during dragging and rotation [17], and its buckling. The studies related to the use of the Cosserat mechanics for analysis of drill-string dynamics were performed by [37] and [39]. The process of a drill-string buckling in a curvilinear bore-hole is connected with one of the emergency situations stimulating appearing of a series of negative frictional effects. Among these are deterioration of conductivity of the torque (TOB) and axial force (FOB) to the bit, enlargement of the process energy consumption, increase of the drill-string wear, and in ultimate cases, they can lead to the drill-string sticking. These effects become apparent in the most perceptible manner under conditions of drilling deep curvilinear bore-holes. Effective means to avoid these emergencies are to model them theoretically at the stages of the hole design and drivage. ∗
Before proceeding to solving the problem of elastic string buckling in a deep curvilinear bore-hole, indispensable preconditions should be satisfied. On the one hand, the bore-hole axis line should be prescribed in analytic form, on the other, the stress state of the drill-string should be determined. Usually, the bore-hole trajectory geometry can be described analytically only at the stage of its design. During its drilling, the bore-hole line acquires additional geometric imperfections and its axis can be represented only in a tabular form. To transfer from the tabular representation of this geometry to the analytical one, the methods of numerical approximation and interpolation can be used. Then, it becomes possible to transfer to the procedure of the drill-string stress strain state determination. It is hardly probable that the drill-string would lose its stability when it quietly lies in the bore-hole and is immovable. It will rather buckle during tripping in and out operations under compressive action of additional resistance forces appearing due to friction effects. In the general case, these forces can be calculated with the use of a quasistatic drag and torque mathematic model without taking into account inertia forces. There exist many varieties of string drag and torque models based on disregarding bending stiffness of the drill-string and influence of internal bending moments, but they, very likely, are suitable for the bore-holes with simple and smooth trajectories [4,26,31,34,35]. Since, the bending stiffness is ignored, these models are usually called the cable, thread, chain, or soft string models. However, in recent years, more and
Corresponding author. E-mail addresses: [email protected] (N.V. Shlyun), [email protected] (V.I. Gulyayev).
https://doi.org/10.1016/j.ijmecsci.2020.105427 Received 25 June 2019; Received in revised form 24 December 2019; Accepted 7 January 2020 Available online 9 January 2020 0020-7403/© 2020 Elsevier Ltd. All rights reserved.
N.V. Shlyun and V.I. Gulyayev
Nomenclature a f fcont ffr fgr i,j,k
clearance value; vector of external distributed forces vector of distributed contact forces vector of distributed friction forces vector of distributed gravity forces unit vectors of reference frame oxyz, moving on the Σ surface k1 ,k2 principal curvatures of the Σ surface kR ,kT curvature and torsion of bore-hole axis line T kx ,ky ,kz components of the drill-string axis line curvature vector l natural parameter n, b, t unit vectors of Frenet’s trihedron moving along the T curve r1 ,r2 external and internal radii of the string tube, correspondingly u, v curvilinear coordinates in the Σ surface w lateral displacements of a beam A, B, C, D points on axial line of the bore-hole E module of elasticity of the drill-string material Fx , Fy , Fz components of the internal elastic force FOB force on bit H metric multiplier I inertia moment of the drill-string cross section K curvature vector of the drill-string section L length of the drill-string Mx , My , Mz components of the internal elastic moment OXYZ global fixed coordinate system S axis line of the drill-string T axis line of the bore-hole TOB torque on bit w transverse displacement of a beam X∗ , Y∗ , Z ∗ coordinates of the bore-hole axis line 𝛼 inclination angle of the T line tangent 𝛽 inclination angle of the hyperbola asymptote tangent 2𝛽 knee angle of the bore-hole 𝛾 coefficient at the hyperbola equation 𝛿 symbol of small increment of a varying value 𝜂 coefficient at the hyperbola equation 𝜃 dimensionless parameter 𝜃 0 ,𝜃 f initial and final values of the 𝜃 parameter 𝜆 wavelet pitch 𝜇 friction coefficient 𝜉 intermediate variable 𝜌t , 𝜌m densities of tube material and mud, accordingly 𝜓 angle between the directions of curve S and u coordinate line 𝝎 analog of the Darboux vector Σ channel surface of the bore-hole wall Φ1 (u,v), Φ2 (u,v) the first and second quadratic forms of the Σ surface 𝛀 vector of total curvature
more attention is focused on elaboration of stiff string drag and torque 3D models, which are more universal and precise. One of such models was presented by Gulyayev and Andrusenko [11] and Gulyayev et al. [13]. It is founded on application of the theory of curvilinear flexible rods (Cosserat theory) taking into account bending stiffness of the drillstring tube and supplementary resistance forces. Survey of drag and torque models and their analysis are performed in the reviewing paper by Mitchell and Samuel [31]. Concerning the soft string drag and torque
International Journal of Mechanical Sciences 172 (2020) 105427
models, they noted “that, there are several commonly held assumptions about torque/drag modeling that are not valid. For instance, many believe that the torque/drag model is a single model with no bending resistance. Surprisingly, the torque/drag model cannot be a soft-string model and satisfy the equilibrium equations. Nonzero shear forces must exist, otherwise, friction loads must vanish…The lack of bending resistance is also illusory because the wellbore trajectory commonly used reduces the bending effects in the equilibrium equations to zero. Therefore, to predict correctly the buckling effects in the curvilinear bore-holes, the stiff string drag and torque models should be used for calculation of the axial compressing and stretching forces. Thereafter, it becomes possible to undertake the second stage of the analysis and to initiate the process of the buckling forces calculation. Commonly, the problems of the drill-string buckling in a curvilinear bore-hole channel are nonlinear. But their linearized statements can be used if the bore-hole trajectory geometry is not very complicated and only initial stage of the buckling process is analyzed. Investigation of the drill-string buckling forces and modes is performed with elastic drill-string instability models. There are many computer models differing by the drill-string fragments separated out for calculations, their geometries, assumptions concerning distribution of the internal axial force, and neglect or taking into account frictional phenomena, torques, and moving mud influence. Lubinsky et al. [28] and Lubinsky [27] were the first ones who investigated stability of drilling tubes in vertical bore-holes. In their analysis, they neglected the effects of the drill-string rotation and torque loading and considered the axial force to be constant. Thereupon, the problem statement became more complicated. Mitchell [29] took into account friction and gravity forces and torque. Kwon [23] built a solution in the spiral mode with variable pitch for a ponderous tube. Gulyayev et al. [12] studied stability of long drill-string with allowance made for its stretched-compressed-twisted stress-strain state, and inertia forces of rotation and internal mud flow. They showed that the stated problem is singularly perturbed [5,38] and therefore, its solutions represent variable pitch spirals concentrated in the lower segments of their length. The problems of drill-string buckling in the channels of long curvilinear bore-holes are far harder. Their additional difficulties are connected with necessity to take into account frictional effects [3,29,32] Analysis of this direction publications testifies that as a rule, the approaches used in their solutions are based on approximations of eigenmode by regular sinusoids and spirals and eigenvalues are rather guessed [7,18,22,24, 36,40]. Supposedly, this circumstance is the reason why the solutions of the problems on the drill-string buckling in curvilinear bore-holes, gained by different authors, are inconsistent with each other [30]. In papers [16,17], the question of the drill-string buckling in a curvilinear channel is considered on an approach basis when in the beginning the drill-string stress strain state is calculated throughout its length, thereafter the found axial force function is involved into the constitutive differential equation of an appropriate global 3D eigenvalue problem (the Sturm-Liouville problem) for the whole drill-string, As a result of its solution, the eigenvalue (the critical load) and eigenmode (the shape of the buckling) are calculated. Based on the elaborated software, the drill-string stability in channels of inclined rectilinear, circular, and other long curvilinear boreholes was simulated. It was established that at the initial stage of the buckling onset, it appeared in the form of a wavelet distributed inside comparatively narrow range located inside the bore-hole length or shifted to its lower edge. This phenomenon was shown to be in accordance with the kinds of the constitutive equations which were singularly perturbed [5]. So, these problems statement and solving are associated with additional scientific and numerical difficulties. Principal peculiarities of the drill-string buckling phenomena and achieved results in their understanding are given in the review papers by Cunha [6], Gao and Huang [10], and Mitchell [30]. As a rule, the buckling phenomenon depends essentially on the borehole geometry. In drilling practice, the two˗sectional trajectories with
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
AB (Fig. 1b), which is approximately similar to the initial one. The simplest approximating curve for the considered two-sectional plot proves to be the hyperbola with branches trending to rectilinear asymptotes and having smoothly changing curvature in its apex zone (Fig. 2a). Equations of this curve have the form [21]: 𝑋 ∗ (𝜃) = ±𝛾𝑐ℎ𝜃, 𝑌 ∗ (𝜃) = 0, 𝑍 ∗ (𝜃) = 𝜂𝑠ℎ𝜃,
(1)
where 𝜃 is the dimensionless parameter; 𝛾and𝜂 are the real coefficients; signs “+” and “–” relate to the right and left branches, correspondingly. The dashed straight lines in this scheme correspond to the hyperbola asymptotes. They make angle 𝛽 with axisOX∗ . It is determined by the equality Fig. 1. Schematics of two-sectional bore-holes: a) Composite trajectory with two rectilinear sections including curvature discontinuities, causing additional frictional resistance forces; b) Smoothed trajectory approximated by the hyperbola curve, contributing to frictional forces reduction.
two rectilinear parts connected with a curvilinear segment are of frequent occurrence. Inasmuch as, their critical behavior is poorly investigated, below, the global stability of the DSs in the bore-holes of this type is analyzed. The modes of critical buckling in the limits of their total lengths are constructed. 2. Geometric models of two-sectional bore-holes In bore-hole trajectory planning, it is usual to configure it as a twosectional curve with two rectilinear (nearly rectilinear) parts connected with a circular bridge (Fig. 1a). The trajectories of this type have come to be known as minimal curvature bore-holes [35], though, their curvatures are not minimal and are discontinuous at points C and D.Therefore, as indicated by Samuel [33] and Gulyayev et al. [17], at that places, the functions of contact interaction and friction forces assume large values. To exclude these singularities, it is possible to introduce smoothing insertions (clothoids or cubic parabolas) in the vicinities of these points or, generally, to replace this combined curve ABCD (Fig. 1a) by one smooth differentiable curve
tg𝛽 = lim (𝑍 ∗ ∕𝑋 ∗ ) = 𝜂∕𝛾. 𝜃→∞
(2)
Then, 𝛽 = arctg(𝜂∕𝛾).
(3)
Therefore, if the OX∗ Y∗ Z∗ system is turned by angle 𝜋/2 − 𝛽 relative to the OY∗ axis, the left-hand branch of the hyperbola will take the position shown in Fig. 2b, corresponding to the bore-hole outline presented in Fig. 1b. In fixed coordinate system OXYZ, its equations acquire the form: 𝑋 = 𝑋 ∗ sin 𝛽 + 𝑍 ∗ cos 𝛽 = −𝛾 sin 𝛽ch𝜃 + 𝜂 cos 𝛽sh𝜃, 𝑌 = 0, 𝑍 = 𝑍 ∗ sin 𝛽 + 𝑋 ∗ cos 𝛽 = 𝜂 sin 𝛽sh𝜃 + 𝛾 cos 𝛽ch𝜃.
(4)
In this case, the bore-hole axis line T is completely determined by the initial (𝜃 0 ) and terminal (𝜃 t ) magnitudes of the 𝜃 parameter and values 𝛾 and 𝜂 which determine also the 𝛽 angle. 3. Analog of the Eulerian formulation of the problem of a drill string buckling in a curvilinear bore-hole The peculiarity of the considered problem of a drill-string stability in a curvilinear bore-hole lies in the fact that only initial stage of the buckling process is analyzed when the elastic displacements are small and the string shape change is not essential. As this takes place, the critical (ultimate) values of the acting loads and buckling shape are discovered. In
Fig. 2. Hyperbolic curves, used for simulation of the two-sectional borehole trajectory a) at the initial state of mathematic modelling in the fixed Cartesian coordinates; b) in the turned Cartesian coordinate system adapted for the design simulation.
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International Journal of Mechanical Sciences 172 (2020) 105427
the applied mathematics and mechanics such problems are collectively called two-point boundary eigenvalue problem (or the Sturm-Liouville problem). The simplest statement of this problem was formulated by L. Euler in analysis of rectilinear elastic rod stability under the action of axial compressive force F. It is described by linear differential equation 𝐸𝐼
𝑑4𝑤 𝑑2𝑤 −𝐹 =0 4 𝑑𝑠 𝑑 𝑠2
(5)
with the corresponding boundary equations. Here, E is the elasticity modulus, I is the inertia moment of the rod cross section, w is the transverse displacement of the rod, s is the natural axial parameter. It is apparent that Eq. (1) has zeroth (trivial) solution w(s) = 0 for any value of F, but its value corresponding to the nontrivial solution w(t) ≠ 0 is named the eigenvalue and appropriate solution is referred to as eigenmode. The problem of global stability loss of a drill-string in the channel of a curvilinear bore-hole under action of forces of gravity, contact, friction and force on the bit is stated in the analogous form. It is necessary to find the identical internal axial force F(s) in the drill-string caused by a combination of external perturbations which generates some additional buckled mode of equilibrium along with the initial one. In this case, the found F(s) force is the eigenfunction and the constructed buckled curve is the eigenmode. Inasmuch as only initial stage of the buckling is considered (as for the case of rectilinear rod), it is possible to conceive that the buckling displacements are small and to restrict our considerations to a linearized statement of the problem. It is also described by a fourth order linear ordinary differential equation of form (5) which has however more complicated structure. To construct it, firstly, it is necessary to formulate a general nonlinear system of a curvilinear rod deforming and then, to linearize it in the vicinity of the initial undeformed state. Nontrivial solutions of this system determine the critical states of the drill-string and the modes of its buckling. It is usual to formulate the equations of the theory of curvilinear flexible rods through the parametrization of bore-hole axis line T by natural parameter l measured by the length of its trajectory. Introduce also the Frenet movable trihedron with unit vectors of tangent t, normal n, binormal b and vector of total curvature 𝛀 = 𝑘𝑅 𝐛 + 𝑘𝑇 𝐭.
(6)
Here, kR is the curvature of bore-hole axis line T, kT is the its torsion. Assume that at the initial state and during buckling, the drill-string continues to contact with bore-hole surface Σ throughout its length under action of gravity forces. Then, to express the link of the drillstring axis S with this surface, introduce movable coordinate system oxyz (Fig. 3) with unit vectors i, j, k, where vector i is directed along the internal normal to the Σ surface, vector k is collinear with the t, and vector j supplements this system to the right-hand triad. With their use, it is possible to construct an analog of the Darboux vector 𝝎 = 𝑘𝑥 𝐢 + 𝑘𝑦 𝐣 + 𝑘𝑧 𝐤.
(7)
Here, kx , ky are the corresponding components of the curvature vector K = b/R, R is the curvature radius, kz determines the system oxyz twisting. With the use of these values, the equations of equilibrium of all external and internal forces are represented as follows [2,17]: 𝑑 𝐹𝑥 ∕𝑑𝑙 = −𝑘𝑦 𝐹𝑧 + 𝑘𝑧 𝐹𝑦 − 𝑓𝑥 , 𝑑 𝐹𝑦 ∕𝑑𝑙 = −𝑘𝑧 𝐹𝑥 + 𝑘𝑥 𝐹𝑧 − 𝑓𝑦 ,
(8)
𝑑 𝐹𝑧 ∕𝑑𝑙 = −𝑘𝑥 𝐹𝑦 + 𝑘𝑦 𝐹𝑥 − 𝑓𝑧 . The equations of the moments equilibrium have an analogous form:
𝑑 𝑀𝑧 ∕𝑑𝑙 = −𝑘𝑥 𝑀𝑦 + 𝑘𝑦 𝑀𝑥 + 𝑚𝑧 .
In these equations, Fx , Fy , Fz are the components of the internal elastic forces vector 𝐅 = 𝐹𝑥 𝐢 + 𝐹𝑦 𝐣 + 𝐹𝑧 𝐤,
(9)
(10)
Mx , My , Mz are the components of the internal elastic moments vector 𝐌 = 𝑀𝑥 𝐢 + 𝑀𝑦 𝐣 + 𝑀𝑧 𝐤,
(11)
fx , fy , fz are the components of the external distributed forces vector 𝐟 (𝑙) = 𝑓𝑥 (𝑙)𝐢 + 𝑓𝑦 (𝑙)𝐣 + 𝑓𝑧 (𝑙)𝐤.
(12)
mz is the external distributed twisting moment. The f force includes the forces of gravity (fgr ), contact interaction (fcont ) and friction (ffr ), then, 𝐟 (𝑙) = 𝐟 𝑔𝑟 (𝑙) + 𝐟 𝑐𝑜𝑛𝑡 (𝑙) + 𝐟 𝑓 𝑟 (𝑙).
(13)
The system of Eqs. (8), (9) determines the elastic equilibrium of the string under action of distributed forces fgr (l), fcont (l), ffr (l) in the borehole channel. If to supplement this system with the equalities, expressing internal elastic forces and moments through the strains and the last ones through the displacements, then, it will be possible to investigate nonlinear deforming, stability, and buckling of the drill-string under different combinations of loads and during tripping in/out and drilling regimes. However, these tasks have a feature conditioned by contact and frictional interaction of the string with the hole wall. Therefore, if the DS is held in the hole in the immovable state, then infinite variety of different friction forces combinations and appropriate functions of internal axial forces may correspond to this state. So, the stated problem is statically indeterminate. Then, only when the drill-string begins to move (in lowering or lifting), all the noted forces can be computed. For this reason, the stated problem is solved in two stages. Firstly, through the use of Eqs. (8), (9), the internal quasi-static forces Fx (l), Fy (l), Fz (l) are calculated and next, with the help of the linearized equations, the task of stability is stated and solved. So far as, besides the unknown internal forces, the unknown external (contact) (fcont ) and frictional (ffr ) forces are present in Eqs. (8) as well. They are supplemented with equalities 𝐹𝑥 = −𝑑 𝑀𝑦 ∕𝑑 𝑙 − 𝑘𝑧 𝑀𝑥 + 𝑘𝑥 𝑀𝑧 , 𝐹𝑦 = 𝑑 𝑀𝑥 ∕𝑑 𝑙 + 𝑘𝑦 𝑀𝑧 − 𝑘𝑧 𝑀𝑦 ,
𝑑 𝑀𝑥 ∕𝑑𝑙 = −𝑘𝑦 𝑀𝑧 + 𝑘𝑧 𝑀𝑦 + 𝐹𝑦 , 𝑑 𝑀𝑦 ∕𝑑𝑙 = −𝑘𝑧 𝑀𝑥 + 𝑘𝑥 𝑀𝑧 − 𝐹𝑥 ,
Fig. 3. Geometric interpretation of the DS axis S contact with the bore-hole wall surface Σ during buckling deforming.
(14)
issuing from Eqs. (9), and correlations 𝑀𝑥 = 𝐸𝐼 𝑘𝑥 , 𝑀𝑦 = 𝐸𝐼 𝑘𝑦
(15)
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
Since, at the initial state, the drill-string geometry coincides with the geometry of the bore-hole, then, at this stage, kx = 0, kz = 0, Mx = 0, Mz = 0, Fy = 0, and curvature ky is determined by correlation [8] [ ] 2 1∕2 (𝑋̇ 2 + 𝑍̇ 2 )(𝑋̈ 2 + 𝑍̈ 2 ) − (𝑋̇ 𝑋̈ + 𝑍̇ 𝑍̈ ) 𝑘𝑦 (𝜃) = , (16) 3 (𝑋̇ 2 + 𝑍̇ 2 )
We will point out one more feature of this process. Since, function ky has already been known at all points lm (1 ≤ m ≤ n), derivatives dky /dl and d2 ky /dl2 defined by complicated expressions are calculated during the investigation process through the use of the finite difference method. To check stability of the drill-string prestressed by the Fz (l) force, linearize system (8, 9) at kx (l) = 0, kz (l) = 0, Mx (l) = 0, Mz (l) = 0, Fy (l) = 0
where the dot over the symbol denotes differentiation with respect to 𝜃 and variables X(𝜃) and Z(𝜃) are specified according to (1). After appropriate transformations, one gains 𝛾𝜂 𝑘𝑦 (𝜃) = . (17) 3∕2 (𝛾 2 𝑠ℎ2 𝜃 + 𝜆2 𝑐 ℎ2 𝜃)
𝑑 𝛿𝐹𝑥 ∕𝑑 𝑙 = −𝛿𝑘𝑦 𝐹𝑧 − 𝑘𝑦 𝛿𝐹𝑧 − 𝛿𝑓𝑥𝑔𝑟 − 𝛿𝑓𝑥𝑐𝑜𝑛𝑡 − 𝛿𝑓𝑥𝑓 𝑟 ,
At these states, gravity force projections 𝑓𝑥𝑔𝑟 (𝜃), 𝑓𝑧𝑔𝑟 (𝜃) are equal to ( ) ( ) 𝑓𝑥𝑔𝑟 (𝜃) = −𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 cos 𝛼, (18) ( ) ( ) 𝑓𝑦𝑔𝑟 (𝜃) = −𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 sin 𝛼.
𝑑 𝛿𝑀𝑦 ∕𝑑 𝑙 = 0,
Here, 𝑟1 , 𝑟2 are the external and internal radii of the string tube, correspondingly; 𝜌𝑡 , 𝜌𝑚 are densities of the tube material and mud, correspondingly; g is the gravity acceleration. The quasistatic friction force is expressed through the friction coefficient 𝜇 and contact force: 𝑓𝑧𝑓 𝑟 = ±𝜇 ||𝑓 𝑐𝑜𝑛𝑡 ||,
(19)
𝑓 𝑐𝑜𝑛𝑡
where, the force is assumed to be along the normal direction and to be determined by its components 𝑓𝑥𝑐𝑜𝑛𝑡 and 𝑓𝑦𝑐𝑜𝑛𝑡 : √ )2 ( )2 ( |𝑓 𝑐𝑜𝑛𝑡 | = 𝑓𝑥𝑐𝑜𝑛𝑡 + 𝑓𝑦𝑐𝑜𝑛𝑡 . | | Then, after some transforms, one can derive an ordinary differential equation for the axial force Fz (l) calculation 𝑑 𝑘𝑦 ( ) ( ) 𝑑𝐹𝑧 𝛾𝜂 = −𝐸 𝐼 + 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 sin 𝛼 𝑑𝑙 (𝛾 2 𝑠ℎ2 𝜃 + 𝜂 2 𝑐ℎ2 𝜃)3∕2 𝑑 𝑙 | | 𝑑 2 𝑘𝑦 ( ) ( ) 𝛾𝜂 | | − 𝐹𝑧 + 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 cos 𝛼 |. + 𝜇 |𝐸 𝐼 | | 𝑑 𝑙2 (𝛾 2 𝑠ℎ2 𝜃 + 𝜂 2 𝑐ℎ2 𝜃)3∕2 | | (20) It should be stressed that in this equation the differentiation is performed with respect to independent variable l while its coefficients determined by geometric Eqs. (1) are expressed by parameter 𝜃. Therefore, at first, it is necessary to transfer from 𝜃 to l at the coefficients of Eq. (20) and then, to start its integration. This transition is performed with the use of differential correlation 𝑑 𝑙 = 𝐻𝑑 𝜃,
(21)
where H is the metric multiplier determined by correlation √ √ 2 2 𝐻 = (𝑋̇ ∗ ) + (𝑍̇ ∗ ) = (𝛾sh𝜃)2 + (𝜂𝑐ℎ𝜃)2 .
𝑙(𝜃) =
∫
√ (𝛾𝑠ℎ𝜉)2 + (𝜂𝑐 ℎ𝜉)2 𝑑𝜉,
(22)
𝜃(𝑙) = 𝜃0 +
∫ 0
𝑑𝜉 . √ (𝛾𝑠ℎ𝜉)2 + (𝜂𝑐 ℎ𝜉)2
(23)
Eqs. (17)–19, (22) permit one to perform integration of Eq. (20) by the Runge-Kutta method. In its realization, the drill-string length L is divided into n small steps Δl = L/n and at the every m-th point lm = Δlm (1 ≤ m ≤ n) parameter 𝜃 values are found 𝜃𝑚 = 𝜃0 +
𝑚 −1 ∑ 𝑠=1
𝑑 𝛿𝑀𝑥 ∕𝑑 𝑙 = −𝑘𝑦 𝛿𝑀𝑧 + 𝛿𝑘𝑧 𝑀𝑦 + 𝛿𝐹𝑦 ,
1 Δ𝑙. √ 2 (𝛾𝑠ℎ𝜃𝑠 ) + (𝜂𝑐ℎ𝜃𝑠 )2
Thereupon, all other functions, depending on 𝜃, are computed.
(24)
(25)
𝑑 𝛿𝑀𝑧 ∕𝑑 𝑙 = 0. Here, symbol 𝛿 denotes the small variation of the considered displacement. In Eqs. (25) derivation, it is taken into account that variations 𝛿ky , 𝛿Fx , 𝛿Fz and 𝛿Mz are equal to zero [17]. To reduce them to the conclusive form, take into account that in the buckling process, the drill-string remains in the contact with the bore-hole wall. Then, its geometry can be represented through the geometry parameters of channel surface Σ. To perform this, introduce a family of two coordinate lines u = const and v = const on it where v (0 ≤ v ≤ 2𝜋) is the parameter determining angular position of a point on a generating circle gained by the Σ surface intersection with a normal plane, therewith points v = 0 and v = 𝜋 are at the bottom and top positions on Σ; u is the parameter coinciding with l. In buckling, every point of the drill-string axis line shifts by small value a𝛿v to new position 𝛿v on its generating circle. In doing so, the appropriate value of parameter u remains unchanged. For this reason, the 𝛿v variable is a function of variable l (or u) that entirely determines the buckled state of the drill-string. Under these assumption, orientation and curvature of the drill-string axis line S, as well as, components 𝑓𝑥𝑔𝑟 (𝑙), 𝑓𝑦𝑔𝑟 (𝑙), 𝑓𝑧𝑔𝑟 (𝑙) of the gravity force in the deformed state can be expressed through the 𝛿v(l) function that determines the curve S position in the Σ surface. In this case, curvature kx coincides with geodesic curvature kgeod of the Σ surface and curvature ky does this with its normal curvature knorm [8]. In its turn curvatures kgeod and knorm are expressed through parameters of the first (Φ1 (u,v)) and second (Φ2 (u,v)) quadratic forms of surface Σ. Then, one has 𝛼̇ sin 𝑣 . 𝑘𝑥 = 𝑘𝑔𝑒𝑜𝑑 = 𝑎𝑣′′ − √ 2 𝛾 sh2 𝜃 + 𝜂 2 ch2 𝜃
(26)
At the undeformed state, kx = 0 and its small variation in the vicinity of this state is determined as follows: (27)
Curvature ky can be represented in a general form through the normal curvature knorm of surface Σ in the curve S direction, which in its turn, is expressed in the form of the Euler theorem as follows: 𝑘𝑦 = 𝑘𝑛𝑜𝑟𝑚 = 𝑘1 cos2 𝜓 + 𝑘2 sin2 𝜓,
𝜃0 𝑙
𝑑 𝛿𝐹𝑧 ∕𝑑 𝑙 = 𝛿𝑘𝑦 𝐹𝑥 − 𝛿𝑓𝑧𝑔𝑟 − 𝛿𝑓𝑧𝑐𝑜𝑛𝑡 − 𝛿𝑓𝑧𝑓 𝑟 ,
𝛼̇ 𝛿𝑘𝑥 = 𝑎𝛿𝑣′′ − √ 𝛿𝑣. 2 2 𝛾 sh 𝜃 + 𝜂 2 ch2 𝜃
In this case, 𝜃
𝑑 𝛿𝐹𝑦 ∕𝑑 𝑙 = −𝛿𝑘𝑧 𝐹𝑥 + 𝛿𝑘𝑥 𝐹𝑧 − 𝛿𝑓𝑦𝑔𝑟 − 𝛿𝑓𝑦𝑐𝑜𝑛𝑡 − 𝛿𝑓𝑦𝑓 𝑟 ,
(28)
where k1 and k2 are the principal curvatures of the Σ surface along curves v = const and u = const, 𝜓 = adv/dl is the angle between the direction of the S line and coordinate line u. In the undeformed state, ky is expressed by Eq. (17) but its overall expression for an arbitrary case is very complicated. So, here, its small variation in the vicinity v = 0 is calculated, which turns to be equal to 𝛿𝑘𝑦 = 2(−𝑘1 + 𝑘2 ) sin 𝜓 cos 𝜓 = 0
(29)
because sin 𝜓 = 0 for v = 0. It is shown also that 𝛿𝑘𝑧 = 𝑑 𝛿𝑣∕𝑑 𝑙.
(30)
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
In a similar manner, variations 𝛿𝑓𝑥𝑔𝑟 , 𝛿𝑓𝑦𝑔𝑟 ,𝛿𝑓𝑧𝑔𝑟 can be represented as follows: ) ( ) ( 𝛿𝑓𝑥𝑔𝑟 = 0, 𝛿𝑓𝑦𝑔𝑟 = 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 (−𝑎 sin 𝛼𝑑𝛿𝑣∕𝑑𝑙 + cos 𝛼𝛿𝑣), 𝛿𝑓𝑧𝑔𝑟 = 0. (31) After appropriate transformations of system (25) with allowance made for correlations (26–31), it is reduced to one ordinary differential equation of fourth order [ ] 2 𝑑 4 𝛿𝑣 sh𝜃𝑐ℎ𝜃 𝑑 𝛿𝑣 2 2 𝑑 𝐸𝐼 − 𝐸 𝐼(𝛾 + 𝜂 ) 𝑑 𝑙4 𝑑 𝑙2 (𝛾 2 sh2 𝜃 + 𝜂 2 ch2 𝜃)3∕2 𝑑 𝑙 ⎛ ⎞ ⎟ 𝐸𝐼 𝑑 2 ⎜ 𝛼̇ − 𝛿𝑣⎟ √ 𝑎 𝑑 𝑙2 ⎜⎜ ⎟ 2 2 2 2 ⎝ 𝛾 sh 𝜃 + 𝜂 ch 𝜃 ⎠ ⎡ ⎤ ⎢ 𝑑 2 𝛿𝑣 ⎥ (𝛾 2 + 𝜂 2 )sh𝜃𝑐ℎ𝜃 𝑑𝛿𝑣 𝛼̇ 1 𝛿𝑣⎥ + 𝐹 𝑧 ⎢− + + √ 2 3∕2 𝑑𝑙 𝑎 2 2 𝑑 𝑙 2 2 ⎢ (𝛾 sh 𝜃 + 𝜂 ch 𝜃) 𝛾 2 sh2 𝜃 + 𝜂 2 ch2 𝜃 ⎥⎦ ⎣ ) ( ( ) ( ) 𝑑𝛿𝑣 cos 𝛼 + 𝜋 𝑟21 − 𝑟22 𝑔 𝜌𝑡 − 𝜌𝑚 sin 𝛼 − 𝛿𝑣 = 0, (0 ≤ 𝑙 ≤ 𝐿). (32) 𝑑𝑙 𝑎 In it, the operations of differentiation with respect to independent variables 𝜃 (designated by a dot) and l are used. The derivative with respect to 𝜃 is used only at the coefficients. Therefore, as noted above, in numerical integrating of functions depending on 𝜃, the transition from lm to 𝜃 m is performed with the help of procedure (24). Then, through the use of found values 𝜃 m at points lm , the coefficients of Eq. (32) are calculated and it is integrated with the method of finite differences. The second peculiarity of this equation consists in its homogeneity and so it always has trivial solution 𝛿v(l) = 0(0 ≤ l ≤ L) at any values of axial force Fz (l) presenting at coefficients of Eq. (32). But, under some values of this function which changes with the change of the force on bit and action of friction forces, Eq. (32) becomes degenerated and it acquires also a nontrivial solution alike as the trivial one. In this case, force 𝐹𝑧𝑐𝑟 (𝑙) is named eigen or critical one and the corresponding solution 𝛿vcr (l) is eigen (or buckling) mode. Solution of Eq. (32) with pinned-pinned boundary conditions is constructed by the finite difference method. Together with Eq. (20) solution, it determines entirely the system stress-strain states in the buckling process and geometrical shape of the buckled drill-string. The elaborated approach advantage consists in its suitability for the problem solving in the domain of the drill-string total length 0 ≤ l ≤ L. Then, as in a general case, function 𝐹𝑧𝑐𝑟 (𝑙) is alternating-sign and has parts of stretching and compressing, the drill-string buckling (though based on the global statement) occurs mainly in the zone of its maximal negative values. So, the problem solution permits one to find its localization, to calculate function 𝐹𝑧𝑐𝑟 (𝑙)values, character of the function 𝛿vcr (l) change throughout the drill-string length, and values of external critical forces acting on the drill-string. 4. Computer modelling results It is necessary to underline once more that Eqs. (20) and (32) has variable coefficients, therefore, their eigenvalues and eigenmodes
Fig. 4. Geometric schemes of the bore-hole outlines with the declined (I), horizontal (II), and ascending (III) lower sections selected for analysis.
cannot be found via the use of analytical methods. So, the finite difference method was used for their analysis. Besides, the stated problem is multivariate. Indeed, the buckling process depends on the value of the angle 𝛽 between the hyperbola (4) branches and the curvature of rounding between them, the string tube stiffness, the clearance value, presence of frictional effects, the value of compressive force at the lower end, kinds of boundary conditions, etc. As the comprehensive analysis of such buckling process is impossible, below, some computational results are presented for the particular values of the characteristic parameters. They make up: E = 2.1 • 1011 Pa, I = 2.7 • 10−4 m4 ,𝜌t = 7.8 • 103 kg/m3 , 𝜌m = 1.3 • 103 kg/m3 , r1 = 0.1m, r2 = 0.09m, 𝜇 = 0, 0.2, 0.3. It was assumed that the drill-string was pinned at its both ends, therefore 𝛿𝑣(0) = 𝛿𝑣(𝐿) = 0, 𝛿𝑣′′ (0) = 𝛿𝑣′′ (𝐿) = 0.
(33)
Three basic structural types of the bore-holes differing by angles 𝛽 were analyzed (Fig. 4). Their properties were determined by the parameters given in Table 1. They are united into three groups differing by the hyperbola angle 2𝛽. The first group (type I) is defined by angle 2𝛽 = 100° and positive inclination of the lower section at the angle 𝛼 0 = 10° to the horizontal (Fig. 4). The lower section of the bore-hole of the second group (type II) is horizontal (2𝛽 = 90°, 𝛼 0 = 0) and it is inclined at the negative angle 𝛼 0 = −10° for the third group (type III) of bore-holes. Their measurements are regulated through the change of parameters 𝛾 and 𝜂 and prescription of the initial (𝜃 0 ) and final (𝜃 f ) values of dimensionless parameter 𝜃. They specify also drill-string length L, depth H0 of the drill-string initial point l = 0 and maximal curvature |ky |max at the middle point of the hole. In every group, additional variation of the values of clearance (a = 0.05, 0.08m) and friction coefficient (𝜇 = 0, 0.2, 0.3) was performed.
Table 1 Geometric parameters of the considered bore-holes. Type
2𝛽
Case
𝜃0
𝜃f
𝛾 (m)
𝜂 (m)
𝛼 0 (deg)
L (m)
H0 (m)
|ky |max (m−1 )
I
100°
II
90°
III
80°
1 2 1 2 1 2
−10𝜋/9 −4𝜋/9 −10𝜋/9 −4𝜋/9 −10𝜋/9 −4𝜋/9
10𝜋/9 4𝜋/9 10𝜋/9 4𝜋/9 10𝜋/9 4𝜋/9
131 1146 170 1478 220 1934
156 1366 170 1478 185 1623
10 10 0 0 −10 −10
6631 5925 7785 6705 9258 7793
4000 3995 4033 4000 4000 4000
5.36 6.14 5.87 6.76 6.43 7.34
• 10−3 • 10−4 • 10−3 • 10−4 • 10−3 • 10−4
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
Fig. 5. Trajectory outlines (a) and corresponding curvature functions ky (l) (b) of the two-sectional bore-holes of the first group with larger (curve 1) and smaller (curve 2) curvatures in the knee zones (cases 1 and 2 in Table 1).
Initially, the analysis results for group I (2𝛽 = 100°) of the lower section inclination were considered. In Fig. 5a, trajectories 1 and 2 (cases 1 and 2) of this group are presented according to their sequence in Table 1. The curvature functions, corresponding to these curves, are demonstrated in Fig. 5b, correspondingly. As can be seen, their curvatures differ essentially in their knee zones. The computer simulation showed that the critical state appearance and modes of buckling depend primarily on the fact that the formulated problem is singularly perturbed [5] as a consequence of large length of the drill-strings and their comparatively small bending stiffness. Because of this, the drill-strings buckling modes have the shapes of harmonic wavelets localized at the vicinity of a zone with enlarged compressive axial force Fz (l) which, in its turn, depends on the bore-hole inclination angle, curvature, tripping (in or out) operation, and friction coefficient value. As a rule, maximal compressive axial force Fz (l) inside the drill-string occurs during tripping in operation at the largest values of friction coefficient 𝜇. Functions 𝐹𝑧𝑐𝑟 (𝑙) of critical axial force and modes 𝛿v(l) of the drillstring buckling for the bore-holes of type I are demonstrated in Table 2. The data fit to cases 1 and 2 shown in Fig. 5 with curvature functions 1 and 2 given in Fig. 5b. Their extremal values |ky |max , as well as total length L of the drill-strings and depths H0 of their lower ends are tabulated in Table 1. They are determined by initial geometric parameters 𝜃 0 , 𝜃 f and 𝛾, 𝜂. But here, particular attention should be played to parameter 𝛼 0 which specifies the angle of the lower section inclination to the horizontal. The point is that friction effects are associated with this angle. In fact, three values of friction coefficients 𝜇 = 0, 0.2, and 0.3 were chosen in analysis and different appearances of the buckling processes took places for them. First of all, situation 𝜇 = 0 is worthy of notice. For example, it can occur when the drill-string is lowering slowly and its arbitrary vibrations and mud flows neutralize the friction effects. Then, the drill-string is prestressed by gravity forces 𝑓𝑧𝑔𝑟 (𝑙), stretching it throughout its length, and WOB (weight on bit) force Fz (0), compressing the drill-string at its lower end l = 0. Summarized axial forces 𝐹𝑧𝑐𝑟 (𝑙) at the critical states are shown in column 𝜇 = 0 of Table 2 for cases 1 and 2 and the clearance values a = 0.05 and 0.08 m. It can be seen that axial force 𝐹𝑧𝑐𝑟 (𝑙) also consists of two nearly rectilinear sections, linearly distributed in the limits of every bore-hole section, because the bore-hole sections are nearly rectilinear. The drill-string is stretched by gravity forces inside the upper section, but stretch force 𝐹𝑧𝑐𝑟 (𝑙) diminishes in the lower section, as angle 𝛼 decreases and compressive bit force 𝐹𝑧𝑐𝑟 (0) acts on its lower end l = 0. Therefore, the lower section of the
drill-string is partially stretched and partially compressed and it can buckle in its part where the compressive force 𝐹𝑧𝑐𝑟 (𝑙) has maximal values, id est, in the vicinity of the lower boundary. Built global eigenmodes 𝛿v(l) in Table 2 confirm this conclusion. But situation changes when the friction forces are included in the mechanical process. At that time, axial force Fz (l) is determined by a disparity between tensile gravity forces 𝑓𝑧𝑔𝑟 (𝑙), compressive WOB Fz (0), and distributed friction force 𝑓𝑧𝑓 𝑟 (𝑙). In Fig. 6, they are shown for the critical state, corresponding to case I, a = 0.08m, 𝜇 = 0.2 in Table 2. The friction force (Fig. 6c) is determined by the contact force modulus (Fig. 6b) having the peak form in the knee zone where it sets against the hole wall. Therefore, the friction function has broken outline (Fig. 6c). Integral combination of these forces together with critical value of WOB force 𝐹𝑧𝑐𝑟 (0) = 274.119𝑁 generates axial force 𝐹𝑧𝑐𝑟 (𝑙) outlined in Table 2. It also consists of two near rectilinear sections with transitional curvi𝑐𝑟 = 289.076𝑁 is located linear part but this time, the extremal value 𝐹𝑧,𝑒𝑥𝑡 inside its length and in this location, the buckling zone is situated. It is significant that this effect became possible owing to the fact that friction angle 𝛼𝑓 𝑟 = arctg𝜇 = arctg0.2 = 11.31◦ = 0.197 exceeded inclination angle 𝛼 0 = 10° and so the compressive frictional effect suppressed the tensile gravity action and shifted the extremum compressive forces inside the drill-string. In the considered cases, the properties of singularly perturbed systems become distinctly apparent. As far as the models of the long flexible curvilinear rods pertain to this type of systems [5,16,17], their buckling modes have the shapes of harmonic wavelets localized at the places of maximal values of their internal axial force Fz (l). Their typical 2D shapes are presented in Fig. 7. As a rule, they are concentrated in a narrow zone of the boundary l = 0 vicinity (Fig. 7a, for case 1, a = 0.08m, 𝜇 = 0 in Table 2) or in the drill-string maximal curvature zone (Fig. 7b, for case 1, a = 0.08m, 𝜇 = 0.2). In doing so, wavelet pitch 𝜆 diminished from value 35.3 m to 24.1 m. Similar effects are discussed also by Elishakoff et al. [9]. The noted peculiarities can serve as explanation of the fact that the buckling effect and modes do not practically depend on the boundary conditions, since the buckling wavelets are located far from the drill-string ends. In case 2, the bore–hole has smaller curvature (see Table 1), the friction forces are less influential, the drill-string becomes more stable, and the buckling wavelets shift to its boundary l = 0. The results of computer modelling the drill-string stability in twosectional bore-hole of type II (Fig. 4) are shown in Table 3. Inasmuch as here, the lower section is horizontal, the influence of frictional
N.V. Shlyun and V.I. Gulyayev
Table 2 Functions of critical axial forces and modes of buckling for trajectories of type I.
International Journal of Mechanical Sciences 172 (2020) 105427
N.V. Shlyun and V.I. Gulyayev
International Journal of Mechanical Sciences 172 (2020) 105427
Fig. 6. Functions of longitudinal distributed gravity (a), contact (b), and friction (c) forces contributing into formation of internal critical axial force Fz (l), causing the localized buckling of the drill-string (case 1 in Table 2).
Fig. 7. Separated fragments of the global buckling modes in the horizontal section of the bore-hole trajectory (Table 2, case I, a = 0.08 m): a) frictionless model at μ = 0; buckling shape is the boundary wavelet; b) frictional model at μ = 0.2; buckling wavelets are shifted to the knee zone.
effects shows itself more distinctly. In the bore-holes with larger value of curvature (case 1), the buckling wavelets widen and shift to the lower zone of the vertical section. In doing so, axial force 𝐹𝑧𝑐𝑟 (0) at point l = 0 of the drill-string becomes tensile for case 1, under condition of friction presence. What this means is that it is necessary to draw the drill-string at its bottom edge in order to realize its dragging. It is also significant that critical forces 𝐹𝑧𝑐𝑟 (𝑙) decrease with clearance aincrease. The bifurcation process acquires additional peculiarities when the knee point of the bore-hole trajectory is below its both ends (Fig. 4, type III). Then, even if the drill-string is stretched at its both ends, it is compressed in the vicinity of the knee zone. In Table 4 the critical states are shown. As can be seen, here, the buckling wavelets are approaching to the knee zone. If to enlarge tensile forces Fz (0) and Fz (L) at the drillstring edges, the system will change to its precritical state, their lower values correspond to the postcritical states. Inferring these results, it can be noted that if the lower section of the hole is inclined (type I) or horizontal (type II) and friction effects are removed, maximal values of the compressive axial forces occur at the bottom zone of the drill-string. In these cases, the buckling mode represents a boundary effect in the shape of a short wavelet localized in the bottom zone of the drill-string where the hole trajectory is near rectilinear. The appropriate 3D mode is schematically shown in Fig. 8. This case is not very difficult for analysis. If during tripping in operation the friction forces are generated, they act opposite the motion direction and the maximal axial force shifts upward. Therefore, buckling wavelets also shift upward. But the situation becomes more serious if the bore-hole of the type III is studied. In this case, maximal compressive force Fz (l) is generated in the knee zone (see Table 4) and therefore, the buckling wavelets are born here (Fig. 9). The contact interaction of the drill-string with the bore-hole channel wall surface is characterized by essential obstacles conditioned by the geometrical complexity of the interface surface. In this case, the drillstring stability analysis should be especially performed.
Fig. 8. Typical 3D schematic fragment of the drill string (type I) buckling wavelet in the inclined low boundary zone (frictionless model).
Fig. 9. Typical 3D schematic fragment of the drill string (type I) buckling wavelet in the knee zone (frictional model).
N.V. Shlyun and V.I. Gulyayev
Table 3 Functions of critical axial forces and modes of buckling for trajectories of type II.
International Journal of Mechanical Sciences 172 (2020) 105427
N.V. Shlyun and V.I. Gulyayev
Table 4 Functions of critical axial forces and modes of buckling for trajectories of type III.
International Journal of Mechanical Sciences 172 (2020) 105427
N.V. Shlyun and V.I. Gulyayev
5. Conclusions 1 The problem of drill-string buckling in a deep curvilinear twosectional bore-hole is stated. The initial stage of the critical elastic bifurcation of the string under action of a force on bit and distributed gravity, contact, and friction forces is considered. Geometric properties of the drill-string trajectories are analyzed, constitutive linearized differential equations of its behavior at critical states are deduced on the basis of the theory of curvilinear flexible rods, theory of channel surfaces, and differential geometry. 2 It is shown that large length of the drill-string leads to a virtual loss of its bending stiffness and near degeneration of some summands in the constitutive equations. Therefore, they belong to the singularly perturbed type equations and their solutions look like harmonic boundary effects or localized harmonic perturbations (wavelets) shifted to the internal zone. The techniques for numerical solving these equations in the domain of the whole length of the drill-string are elaborated and the appropriate software is created. 3 To analyze the two-sectional drill-string buckling, firstly, the critical stress-strain state of the drill-string (eigenfunction) is established and then, the shape of the buckled string (eigenmode) is constructed. It is shown that depending on the knee angle, and values of the knee curvature, clearance, and friction coefficient, the mode of stability loss can represent a short harmonic wavelet localized at the lower end of the lower section or shifted to the knee zone of the bore-hole. 4 The elaborated techniques and created software can be used for prognostication and prevention of emergency situations due to the twosectional bore-holes buckling. Declaration of Competing Interest None. References [1] Ajibose OK, Wiercigroch M, Akisanya AR. Experimental studies of the resultant contact forces in drillbit–rock interaction. Int J Mech Sci 2015;91:3–11 February. [2] Antman SS. Nonlinear problems of elasticity. Springer; 2005. [3] Berger EJ. Friction modeling for dynamic system simulation. Appl Mech Rev 2002;55(6):535–76. [4] Brett JF, Beckett AD, Holt CA, Smith DL. Uses and limitations of drillstring tension and torque models for monitoring hole conditions. SPE Drill Eng 1989;4:223–9. [5] Chang KW, Howes FA. Nonlinear singular perturbation phenomena. Springer −Verlag; 1984. [6] Cunha JC. Buckling of tubulars inside wellbore: a review on recent theoretical and experimental works. SPE Drill Complet 2004;19(1):13–19. [7] Dawson R, Paslay RR. Drill pipe buckling in inclined holes. J Pet Technol 1984;36(10):1734–8. [8] Dubrovin BA, Novikov SP, Fomenko AT. Modern geometry-methods and applications. Springer - Verlag; 1992. [9] Elishakoff, I., Li, Y., Starnes, J.H., 2001. Non-Classical problems in the theory of elastic stability. Cambridge University Press, Cambridge, UK. [10] Gao DL, Huang WJ. A review of down-hole tubular string buckling in well engineering. Petr Sci 2015;12(3):443–57. [11] Gulyayev VI, Andrusenko EN. Theoretical simulation of geometrial imperfections influence on drilling operations at drivage of curvilinear bore-holes. J Petr Sci Eng 2013;112:170–7.
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