Research on the characteristics of earthworm-like vibration drilling

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Contents lists available at ScienceDirect

Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

Research on the characteristics of earthworm-like vibration drilling Peng Wang a, *, Hongjian Ni a, **, Xueying Wang b, Ruihe Wang b, Shuangfang Lu a a b

Research Institute of Unconventional Oil & Gas and Renewable Energy, China University of Petroleum (East China), Qingdao 266580, China School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Earthworm-like drilling Friction reduction Load transfer Position optimization

The load transfer difficulty caused by borehole wall friction limits the rate of penetration (ROP) and extendedreach limit of complex structural well enormously. A new friction reduction technology called “earthworm-like drilling”, which can improve the load transfer and extended-reach limit, is proposed in this paper. A mathematical model based on “soft-string” model is developed to analysis the characteristics of this technology. Simulation results indicate that more stick-slip and load transfer issue are caused by higher friction. The earthworm-like drilling is more effective in reducing friction than single-point vibration drilling and less effective than multipoint vibration drilling because of the pulse pressure attenuation. However, this disadvantage can be offset by adding the number of axial oscillators. Amplitude and frequency of pulse pressure and the installation position of axial oscillators have great impact on the friction reduction and load transfer. An optimizing model based on projection gradient method is developed and used to optimize the position of three axial oscillators in a threedimensional horizontal well. The weight on bit (WOB) increases significantly after the optimized position and the new position of axial oscillators move towards bottom of well and close to each other. Results verify the feasibility and advantages of earthworm-like drilling, and lay a solid theoretical foundation for its application in oil field drilling.

1. Introduction Complex structural well is a series of well types with the characteristic of horizontal well and can be classified as directional well, horizontal well, extended reach well and so on (see Fig. 1). Drilling a complex structural well can ensure hitting the pay zone successfully and achieve good formation protection. Besides, it's compatible with the future stimulation to help boost the well production efficiently and economically and improve the final recovery rate. There is great difficulty in controlling the well trace of complex structural well, which make directional drilling be a key technology to build the complex structural well. At present, two well track controlling modes have formed, i.e. slide steering controlling mode and rotatory steering controlling mode. The slide steering controlling mode has wider application than rotatory steering controlling mode because of its better cost-performance. However, the drill-string do not rotate under slide steering mode, which results in huge friction between drill-string and borehole wall and decreases the axial load transfer efficiency. The weight component of upper drill-string can't be transmitted to the drill bit, and the rate of penetration (ROP) and extended-reach limit are decreased. Therefore,

decreasing the friction between drill-string and borehole wall during directional drilling of complex structural well has great significance, and is an important issue of petroleum drilling engineering for many years (Gao et al., 2009). Many scholars have carried out research on friction reduction, which in general can be divided into decreasing normal contact force or frictional coefficient. The methods of decreasing normal contact force include optimizing well track, using light drill pipe. The methods of decreasing frictional coefficient include developing high performance lubricant, using cylindrical roller sub and non-rotating protective joint (Wang et al., 2017a). However, all these methods belong to passive friction reduction methods, which achieve limited application effect. As early as 1983, Roper proposed the idea of decreasing the friction between drill-string and borehole wall by adding vibrators in drill-string (Roper and Dellinger, 1983). Until recent years, several petroleum technology service companies began to carry out application research of this idea, and focused on development of vibrator (Maidla et al., 2005; Steve et al., 2016). In addition to the development of vibrators, several scholars research the friction reduction mechanism and load transfer rule of drill-string under vibration conditions through simulation and modelling

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (P. Wang), [email protected] (H. Ni). https://doi.org/10.1016/j.petrol.2017.10.027 Received 29 June 2017; Received in revised form 8 September 2017; Accepted 9 October 2017 Available online 14 October 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

assembly of multi-point vibration is exactly similar to the segmented body of earthworm. Every vibrator and adjacent drill-string is equivalent to one somite of earthworm. Earthworm can generate backward wave passing along its body by controlling the shrink and relaxation of muscle of somite, and can adjust the friction force between its somite and wall of hole by controlling the bristle stretching out and retracting into the somite. The earthworm moves forward, combined with squeezing and devouring soil, a cave is formed (Mezoff et al., 2004; Ren, 2009). In this process, earthworm decomposes the required huge friction of whole body moving towards into smaller friction of every somite. Meanwhile, there is always bristle of a somite penetrating into soil as fulcrum to drive neighbouring somite moving towards in the process of movement, which achieves making full use of friction. Refer to the decomposition and using of friction, the authors propose a new friction reduction idea called earthworm-like drilling scheme (see Fig. 2). A hydraulic pulse generator and more than one axial oscillators are mounted in the drill-string from bottom to top. The hydraulic pulse generator can generate successive pulse pressure through a mechanical column valve. The axial oscillators can elongate axially with a maximum of 10 mm-15 mm (equals to the moving clearance 7) after receiving the positive pulse pressure generated by hydraulic pulse generator, while shorten to original condition after the pulse pressure disappearing. The upper sub 1 and sleeve 2 are connected to a whole by thread, and the center shaft 4 and lower sub 3 are connected to another whole by thread. All the parts of axial oscillator are revolution solid except the match position of bottom of center shaft 4 and sleeve 2 (see section view A-A in Fig. 2) which is hexagon. This specific hexagon structure makes the center shaft can only slide axially relative to sleeve 2 to transmit axial load and torsion torque of drill-string. The upper sub 1, center shaft 4 and sleeve 2 constitute a high-pressure chamber 9. The center shaft 4 and sleeve 2 constitute a low-pressure chamber 10. In the process of drilling, the high-pressure chamber fills with high-pressure drilling fluid and the low-pressure chamber fills with low-pressure drilling fluid of annulus. There are four forces acting on the center shaft 4 in axial direction, including: ①pressure difference of high-pressure chamber and lowpressure chamber; ②spring force difference of disc springs in highpressure chamber and low-pressure chamber; ③force from lower sub 3; ④weight component of center shaft 4. The displacement and density of drilling fluid, bit nozzle pressure-drop and circulating pressure loss when drilling a well are in a specific range and can be calculated. Then the force balance of ①, ② and ④ can be realized through choose suitable disc spring stiffness of axial oscillator. Therefore, small force from lower sub 3 can make the moving clearance 7 equal to zero. As the drill-string segment where axial oscillators installed are compressed, all the axial oscillators can keep “closed” during the process of drilling if there is no positive pulse pressure. When drilling, the positive pulse pressure excited

Fig. 1. Classification of complex structural well (λ ¼ horizontal depth=vertical depth).

(Parbon et al., 2010; Wicks et al., 2012; Ritto et al., 2013; Shor et al., 2015; Gee et al., 2015; Wilson and Noynaert, 2017; Wang et al., 2017b). The mechanism of friction reduction by vibrating drill-string can be summarized as follows: (a) the axial and torsional vibrations change the static friction to dynamic friction (Skyles et al., 2012); (b) the axial and torsional vibrations change the direction of dynamic friction, which decreases the average friction force during a vibrating cycle (Gutowski and Leus, 2012, 2015); (c) the lateral vibration decreases the normal contact force periodically (Tolstoi et al., 1973). In these models, the exciting force of vibrator is usually treated as sinusoidal or cosinoidal acting on disperse nodes. In fact, the exciting force generated by disc valve structure of vibrators is bidirectional (such as axial oscillation tool (Alali et al., 2012): upward 25% and downward 75%), and the stiffness of the vibrators equals to the stiffness of disc spring used in axial oscillator sub. Inattention to the bidirectional vibrating and disc spring stiffness is an important reason for the deviation between simulation results of models and practical application effect. Meanwhile, the deficiency of limited action distance by adding only one vibrator in drill-string is increasingly apparent as designed horizontal displacement of complex structural wells increasing. The “multi-point vibration technology” realized by adding several vibrators in drill-string has become the development tendency of friction reduction technology by vibrating drill-string, also is a reserved technique meeting the increasingly urgent friction reduction demand in the future (Gee et al., 2015). Thought and method of bionics have permeated into many subjects and industries. For example, many drilling and pipe protection problems have been resolved by using bionics theory in petroleum industry, and bionics increasingly influences the technical innovation concept and thought of petroleum exploration and development. Based on the thought of “multi-point vibration technology”, the authors find that the

Fig. 2. Schematic diagram of earthworm-like drilling. Parts of axial oscillator: 1-upper sub; 2-sleeve; 21- low-pressure hole; 3-lower sub; 4- center shaft; 41-high-pressure hole; 5,6-disc spring; 7- moving clearance; 8-seal; 9-high-pressure chamber; 10-low-pressure chamber. 61

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Gao, 1995). In this paper, the effect of axial earthworm-like excitation on the load transfer characteristics in directional drilling of complex structural well is focused and the following assumptions are adopted

by hydraulic pulse generator with certain frequency and amplitude spreads up inside the drill-string. The axial oscillators elongate orderly when the pulse pressure passing through, and shrink under the axial compressive force of drill-string after the pulse pressure disappearing. Under the combined action of positive pulse pressure and axial compressive force, the axial oscillators vibrate periodically. The drillstring will move forward smoothly under the action of weight component in axial direction. Besides, the pressure consumption is much less than the manner of multi-point vibration technology realized by mounting several hydroscillators (Gee et al., 2015) as there is only one hydraulic pulse generator. The drill-string in Fig. 2 has similar structure and propagation mode of backward wave with earthworm. However, the drill-string assembly cannot adjust friction of every “body segment” proactively as earthworm. The drill-string only can adjust the weight component acting on the bottom hole assembly through controlling the hook load, which means that the drill-string can control the axial force acting on its backmost “body segment”. Before the development of earthworm-like friction reduction directional drilling, several questions should be clarified. Is earthworm-like drilling more effective in friction reduction than the present single-point vibrating technology or even multi-point vibrating technology? If the answer is yes, what are the load transfer characteristics and influence factors of earthworm-like drilling? How the load transfer effect of earthworm-like drilling in the condition of low frequencies as the frequency of backward wave during earthworm crawling is low? All these questions confuse us and determine the feasibility of earthwormlike vibration drilling technology. In this paper, the load transfer characteristics, advantage and feasibility of earthworm-like drilling are studied. Firstly, a model used to analysis the forces and motion of drill-string with one or several axial oscillators is established, and a finite difference method with secondorder accuracy is adopted to solve this numerical model. Secondly, the load transfer effect under different parameters (such as amplitude and frequency of pulse pressure, installation distance and phase difference of axial oscillators) of 3-point earthworm-like drilling are analyzed to find the relationship of weight on bit (WOB) and optimum parameters. The load transfer effect under low frequency (<10 Hz), which can be called “wriggle drilling”, is studied by changing amplitude of pulse pressure. Thirdly, an optimizing model based on projection gradient method is developed and used to optimize the position of three axial oscillators in the drill-string of a horizontal well. The purpose of this paper is to prove the feasibility and advantages of earthworm-like drilling and lay a solid theoretical foundation for its application in petroleum and natural gas drilling.

(a) The cross section of drill-string is annular. Ignoring the clearance between drill-string and borehole. The center line of drill-string coincides exactly with the center line of borehole and the drillstring keeps uniform contact with borehole wall. (b) Every axial oscillator is seem as a length of drill-string, and sinusoidal exciting force of same amplitude and frequency and inverse direction are applied on both ends of the axial oscillator, respectively. (c) The damping forces acting on drill-string include friction and viscous damping of drilling fluid. The mechanical resistances resulting from the drill-string pressing into the borehole wall and sticking are not considered. (d) Only the axial dynamic effect of drill-string is considered, and ignoring the shear force and bending moment of the cross section of drill-string. 2.2. Equilibrium equation A micro-element ds is extracted from the drill-string to analysis the forces acting on drill-string. In natural curvilinear coordinates (! e t, ! e n, ! e b ), the forces acting on micro-element include internal tension force ! ! T , normal contact force F , friction force Ff , damping force and buoyant weight of drilling, as shown in Fig. 3. According to equilibrium condition of forces, the following kinematic equation can be obtained

 
ds ! ! ! T ðs þ ds; tÞ  T ðs; tÞ þ F s þ ; t ds  Ff ðs; tÞ 2    

∂u s þ ds2 ; t ds ds ds! eg ds! e t þ gs s þ þ c s þ ;t 2 2 ∂t   

∂2 u s þ ds2 ; t ! ds ds A sþ ds ¼ρ sþ et 2 2 ∂t 2

(1)

where Ftop is the hook load; WOB is weight on bit; ! e t, ! e n and ! e b are unit base vectors in natural curvilinear system; ! e g is vector of submerged drillstirng weight; T is internal tension force; F is normal contact force; Ff is axial friction force; c is drilling fluid drag; ρ is density of drill-string; gs is linear buoyant weight of drill-string; u is axial displacement of drillstring; s is well depth; A is cross-section area of drill-string; t is computing time. The following simpler equilibrium equation can be derived from the

2. Model A mathematical model based on “soft-string” model (Ho, 1988) is developed to analysis the load transfer characteristic under axial vibration conditions in this section. Considering the complexity of borehole system, several assumptions are adopted before modelling. Governing differential equation, boundary and initial conditions are respectively derived in section 2.2 and 2.3. A finite difference method with second-order accuracy is derived in section 2.4 to solve the model. 2.1. Assumptions The forces applied on the drill-string are actually very complicated during the rotary drilling process, which make drill-string present complex state of motion (rotation, revolution and swirling) and deformation (bending, sinusoidal and helical buckling). The drill-string does not rotate in directional drilling process and huge friction and drag are applied on drill-string by borehole wall, which make the rotate and lateral motions can be ignored and the axial motion become the main motion of drill-string. Besides, the existence of friction and drag make the buckling less likely to happen, especially in horizontal section (Gao and

Fig. 3. Forces acting on micro-element of drill-string.

62

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Taylor expansion (Canuto and Tabacco, 2008) in (s, t) of variables by omitting higher order terms:

has the following form

  ! ∂ T ðs; tÞ ! ∂u ! þ F ðs; tÞ  Ff ðs; tÞ þ cðs; tÞ e t þ gs ðsÞ! eg ∂s ∂t

Ff ¼ μF

∂2 uðs; tÞ! ¼ ρðsÞAðsÞ et ∂t 2

! Separating the internal force T and distributed lateral contact force ! e n, ! e b) F into components in natural curvilinear coordinates (! e t, !

(3)

! e n þ Fb ðs; tÞ! eb F ðs; tÞ ¼ Fn ðs; tÞ!

(4)

2.3. Boundary and initial conditions In order to solve the equilibrium equation, boundary conditions and initial conditions are essential. Section 2.3.1 and 2.3.2 give the boundary conditions and initial conditions, respectively. 2.3.1. Boundary conditions In the actual drilling process, drillers try to adjust the weight on bit (WOB) to desired value by controlling hook load on wellhead and keep it constant, then the hook load equals to the difference between buoyant weight of drill-string, friction force and nominal WOB in axial direction. The sum of friction force and nominal WOB can be considered as constant value, while the friction force and real WOB mutual transformation because of the heterogeneity of rocks and stick-slip motion of drill-string. The hook load is calculated by static analysis of drill-string in the condition of frictional coefficient equal to static friction coefficient and nominal WOB equal to zero and keeps constant in the later calculation.

where, Tt ðs; tÞ, Tn ðs; tÞ and Tb ðs; tÞ are internal tension force acting on e n and ! e b directions, respectively; Fn , Fb are the cross section in ! e t, ! lateral contact force acting on drill-string in principal normal direction ! e n and binormal direction ! e b , respectively. Using the Frenet-Serret formulas (Ho, 1988) for the centerline of the borehole

! e g ⋅! e t ¼ cosα kα ! e n ¼ sinα e g ⋅! kb kφ ! e b ¼  ðsinαÞ2 e g ⋅! kb

∂u Tt js¼0 ¼ Gt  WOB  Ff ¼ EA ∂s s¼0

(5)

kα kφ ! e g ¼ cosα! e t þ sinα! e n þ sin2 α! eb kb kb

ujs¼L ¼ ROP⋅t (6)

kα sinα ¼ 0 kb

(7)

In ! e b direction

kφ Fb þ gs ðsinαÞ2 ¼ 0 kb

(8) Pðx; tÞ ¼ P' sinð2πftÞ

According to engineering mechanics theory, the internal tension force Tt in axial direction for slender rod can be written in the following form

2 P ðxÞ ¼ P0 ⋅exp  aD '

Tt ¼ EA

∂u ∂s

(9)

2 #12 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "
kα kφ F 2n þ F 2b ¼ Tkb þ gs sinα þ gs ðsinαÞ2 kb kb

(14) sffiffiffiffiffiffiffiffiffiffiffi ! πμpv f x ρ

(15)

where Pðx; tÞ is pulse pressure applied on each axial oscillator; P0 is the amplitude of pulse pressure generated by hydraulic pulse generator; x is the distance between axial oscillator and hydraulic pulse generator; P' is amplitude of pulse pressure after propagating x; a is the wave velocity of pulse pressure inside the drill-string; D is inner diameter of drill-string; μpv is viscocity of drilling fluid; f is frequency of pulse pressure. The exciting force of each axial oscillator can be obtained

where E is the elastic(Young's) modulus of drill-string. The distributed lateral contact force between drill-string and borehole wall can be obtained by combining equations (7) and (8)

F¼

(13)

where ROP is rate of penetration; L is the whole length of drill-string; t is computing time. The hydraulic pulse generator begins to work after the drill-string comes to a steady state. The pulse pressure is sine shaped with amplitude of P0 and frequency of f . The pulse pressure propagates upward and decreases with the increase of propagation distance, while the frequency keeps constant. The pulse pressure applied on all axial oscillators can be calculated by the following formulas (Wu et al., 2015; Desbrandes, 1988)

In ! e n direction

Fn þ Tt kb þ gs

(12)

where Tt js¼0 represent the hook weight; Gt is the axial gravity component of the whole drill-string. In most cases, axial oscillation tool is necessary only when the rate of penetration (ROP) is low because of its expensive rent. The motion of drill-string ineluctably appears stick-slip phenomenon during drilling process either the ROP is high or low. The stick-slip motion of drill-string will result in varying WOB and ROP, which will in turn affect the stickslip. Therefore, the ROP is set to a constant value to observe the effect of vibration on friction reduction and load transfer more clearly.

where α, φ and α are deviation angle, azimuth angle and mean deviation angle, respectively; kα , kφ and kb are rate of change of deviation angle, rate of change of azimuth angle and total bending curvature, respectively. The following scalar equations can be obtained by substituting equations (3)–(5) into equation (2): In ! e t direction

∂Tt ∂u ∂2 u  Ff ðs; tÞ  c þ gs cosα ¼ ρA 2 ∂t ∂t ∂s

(11)

where μ is the friction coefficient in ! e t direction; μs is static friction coefficient; μd is dynamic friction coefficient; v is the velocity of drill-string.

(2)

! T ðs; tÞ ¼ Tt ðs; tÞ! e t þ Tn ðs; tÞ! e n þ Tb ðs; tÞ! eb

if v  0; μ ¼ μs if v > 0; μ ¼ μd

(10)

Fe ¼ Ae P' sinð2πftÞ

The friction force Ff is calculated by Coulmob friction model, which

63

(16)

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

where Fe is the exciting force; Ae is the carrying area of pulse pressure of each axial oscillator.

 g ¼ ∫ p

2.3.2. Initial conditions Assuming the initial velocity of drill-string equals to zero. The initial displacement of drill bit was set to zero, and the initial displacement of the other part of drill-string are calculated by static force balance.

∂u ¼0 ∂t t¼0



pþ12 h

 gs cosαds

Substitute equations (19)-(24) into (18), we can get the recurrence algorithm of drill-string axial displacement

uðp; k þ 1Þ ¼ (17)

ujt¼0 ¼ ΦðTt Þ; uðLÞjt¼0 ¼ 0

 2τ2 2Ap 1 1 T pþ2  T p2  Ff ðp; kÞ þ gp þ 2 uðp; kÞ 2Ap þ C p τ τ  
p p C τ  2A uðp; k  1Þ þ 2τ2

where Φ is the initial displacement distribution of drill-string.

(25) Difference solution format (25) is explicit. If knowing the displacements of foregoing two time horizons, the displacement of drill-string at any moment can be calculated. Then the drilling parameters, such as axial tension force and weight on bit, can be obtained. To ensure the solution is convergent, the time step τ and space step h should satisfy a certain relationship. It can be proved that the difference scheme of equation (25) is conditionally stable by using Fourier error analysis method (Anderson, 1995). Convergence condition can be expressed as follows

2.4. Solution method The above model is elastic wave propagation problems in essence. In this section, finite difference method is chosen to solve the model. Dispersing the equilibrium equation by central difference scheme. The equilibrium equation then are transformed into algebraic equations and solved using MATLAB programming. Firstly, the whole drill-string is divided into N units from top of the well to the bottom. Setting time step τ, space step h and forming difference grids. Then the node displacement of drill-string can be expressed as uðp⋅h; k⋅τÞ, abbreviated as uðp; kÞ. There ∂ðEA ∂uÞ is nonlinear term ∂s ∂s in the equilibrium equation. The equilibrium 



  equation is integrated for random node p in interval p  12 h; p þ 12 h

h τ  qffiffi E ρ

Earthworm-like vibration drilling is a new idea for solving the stick slip and load transfer difficulty caused by higher friction. Section 3.1 verifies its advantages and effectiveness through comparing the friction reduction effect of earthworm-like vibration drilling with single-point vibration drilling and multi-point vibration drilling based on the model developed in section 2. Further more, section 3.2 researches the effect of main influence factors on friction reduction efficiency, which can provide reference for the parameter determination of earthworm-like vibration drilling in the future application.

  pþ12
p12 1 ∂u ∂u uðp; k þ 1Þ  uðp; k  1Þ pþ2 h ∫   cds EA  EA  ∂s ∂s 2τ p1 h 2

 þ∫





pþ12 h

 gs cosαds  sgnðuðp; kÞ  uðp; k  1ÞÞ∫ 

p12 h



pþ12 h

 Fμds

p12 h

  1 uðp; k þ 1Þ  2uðp; kÞ þ uðp; k  1Þ pþ2 h ∫ ¼   ρAds τ2 p1 h

(26)

3. Results and discussion

to deal with the nonlinear preferably. The following equation can be obtained




(24)

p12 h

(18)

3.1. Comparison of three vibration manners

2

The purpose of earthworm-like drilling is to improve the friction reduction and load transfer effects. It has application prospect and research necessity only when it is more effective than conventional single-point vibration and multi-point vibration drilling. The friction reduction effect of above three vibration manners are compared in this section. A one-dimensional horizontal segment with length of 3000 m is adopted to analyze the characteristics of earthworm-like drilling,as shown in Fig. 4. Applying axial force Ftop on the top of the drill-string. The value of Ftop equals to the static friction of the whole drill-string. This means that the WOB increment after applying vibration equals to the friction released. The other calculation parameters are shown in Table 1. The positions of axial oscillation tools are x1 ¼ 750m, x2 ¼ 1125m, x3 ¼ 1500m, x4 ¼ 1875m, x5 ¼ 2250m, respectively. For single-point vibration drilling, only one axial oscillator and one hydraulic pulse generator is mounted at x5 (2250 m from drill bit). Three axial oscillators and three hydraulic pulse generators are respectively mounted at x1 , x3 and x5 for 3-point vibration drilling. For 3-point earthworm-like drilling, only one axial oscillators and one hydraulic pulse generator are mounted at x5 and another two axial oscillators are mounted at x1 and x3 , respectively. For 5-point earthworm-like drilling, one axial oscillator and one hydraulic pulse generator are mounted at x5 and another four axial oscillators are mounted at x1 , x2 , x3 and x4 , respectively. All hydraulic pulse generators can generate positive pulse pressure, meanwhile, axial oscillators can only receive these pulse pressure which will gradually

where sgnðÞ is the sign function; h is space size; τ is time step. If denote 1

¼pþ12
∂u uðp þ 1; kÞ  uðp; kÞ EA ¼ EA ∂s h

(19)

1

p12
∂u uðp; kÞ  uðp  1; kÞ EA ¼ EA ∂s h

(20)

T pþ2 ¼

T p2 ¼

 Ff ðp; kÞ ¼ sgnðuðp; kÞ  uðp; k  1ÞÞ∫   C ¼ ∫ p

 A ¼ ∫ p



pþ12 h

 Fμds

(21)

p12 h



pþ12 h

 cds

(22)

p12 h



pþ12 h

 ρAds

(23)

p12 h

64

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Fig. 4. One-dimensional horizontal segment with length of 3000 m.

has the highest maximum velocity. The 5-point earthworm-like drilling has the lowest maximum and average velocity. It can be concluded that the motion of drill-string in the manner of 5-point earthworm-like drilling relatively gentle and homogeneous. Fig. 7 shows the maximum and mean of acceleration along the drillstring during vibration process. From Fig. 7, the drill-string between the first and third vibrators in the manner of 3-point vibration drilling has higher maximum acceleration. The maximum acceleration of other vibration manners appear on xexcite 5 ¼ 2250m, and gradually decay to both sides. The maximum acceleration of 3-point earthworm-like drilling is relatively low. The average acceleration of all vibration manners almost the same and take the maximum value at axial oscillators positions. From the above analysis, we can see that the friction reduction and load transfer effect of earthworm-like drilling manner is better than single-point vibration drilling and multi-point vibration drilling due to its full use of pressure energy. Meanwhile, the earthworm-like drilling manner has relatively low velocity and acceleration which is good for the life of drill-string and downhole safety.

Table 1 Parameters used in the calculations. drill-string

drilling fluid

boundary parameters

solving parameters

outer diameter(mm) inner diameter(mm) density(kg/cm3) elasticity modulus E(GPa) density (kg/m3) drag c (N⋅s/m2) plastic viscosity μpv (mPa⋅s) amplitude of pulse pressure P0 (MPa) frequency of pulse pressure f (Hz) wave velocity a (m/s) static frictional coefficient μs dynamic frictional coefficient μd bottom hole ROP(m/s) initial WOB (N) time size τ (s) space size h (m) computing time t (s)

127 108.6 7850 210 1.25 12 30 4 24 1200 0.35 0.25 0.001 0 0.0005 5 50

damp as the propagation distance increasing. The hydraulic pulse generators begin to work when the computing time equals to 0.5s. Fig. 5 shows the change of WOB after applying different manners of vibrations. As can be seen from Fig. 5, the single-point vibration drilling has the lowest WOB, which indicates that single-point vibration drilling has the worst friction reduction and load transfer effect. The order of the other three tested manners from bad to good is 3-point earthworm-like drilling, 3-point vibration drilling and 5-point earthworm-like drilling. It can be concluded that multi-point vibration drilling is more effective than single-point vibration drilling under the same conditions. The earthworm-like drilling is less effective than multi-point vibration drilling because of the pulse pressure attenuation. However, this disadvantage can be offset by adding the number of axial oscillators. Fig. 6 shows the maximum and mean of velocity along the drill-string during the process of vibration. From Fig. 6, the single-point vibration drilling has the highest average velocity and higher maximum velocity in most positions than earthworm-like drilling. The drill-string between the first and third axial oscillators in the manner of 3-point vibration drilling

3.2. Influence factors of earthworm-like drilling The friction reduction and load transfer effects of earthworm-like drilling are affected by geology and engineering parameters. In this section, 3-point earthworm-like drilling is taken for example to research the main influence factors of earthworm-like drilling, including amplitude and frequency of pulse pressure, installation distance of axial oscillators and phase difference of exciting force of different axial oscillators. 3.2.1. Amplitude of pulse pressure Fig. 8 shows the relationship of WOB with amplitude of pulse pressure. The frequency of pulse pressure f equals to 24 Hz. From Fig. 8, the

0.4

Velocity (m/s)

0.3

0.2

0.1

0.0

Fig. 5. Weight on bit (WOB) under different types of vibration.

0

1000

2000

Fig. 6. Velocity along the drill-string. 65

3000

Depth (m) maximum: 5-point earthworm-like drilling 3-point earthworm-like drilling 3-point vibration drilling single-point vibration drilling mean: 5-point earthworm-like drilling 3-point earthworm-like drilling 3-point vibration drilling single-point vibration drilling

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Acceleration (g)

9

6

3

0

0

1000

Depth (m)

maximum: 5-point earthworm-like drilling 3-point vibration drilling mean: 5-point earthworm-like drilling 3-point vibration drilling

2000

Fig. 9. Effect of frequency on WOB.

3000

3-point earthworm-like drilling single-point vibration drilling

vibrating drill-string axially including: (a) changing static friction to dynamic friction; (b) changing the direction of friction. However, the reversed friction is not conductive to the downward movement of drillstring, while only the exciting force together with the buoyant weight of the drill-string below neutral point can push the drill-string move downward. Based on these mechanism, the maximum value of the increased WOB after applying vibrations equals to the friction released by the whole drill-string changing from static frictional coefficient to dynamic frictional coefficient plus the amplitude of exciting force. The increased WOB will increase the extended reach limit to some extent. Therefore, if we can decrease the frequency appropriately and improve the amplitude of pulse pressure at the same time, the maximum value of WOB can further increase. As the frequency is low, we can call it earthworm-like wriggle drilling for the moment. Fig. 10 shows the relationship of WOB with amplitude of pulse pressure under low frequency (<10 Hz). From Fig. 10(a), the WOB increases with the increase of amplitude of pulse pressure under the condition of f ¼ 2Hz. Compared with Fig. 9, the WOB has been improved obviously by increasing amplitude of pulse pressure as the other parameters keep the same. However, the increment of WOB is lower than the conditions of high frequencies. Fig. 10(b) and (c) show the relationship of WOB with amplitude of pulse pressure under f ¼ 6Hz and f ¼ 10Hz, respectively. From Fig. 10(b) and (c), the WOB increases markedly with the increase of frequency. Therefore, better friction reduction and load transfer effect can be achieved by improving amplitude of pulse pressure in the case of medium frequencies.

3-point earthworm-like drilling single-point vibration drilling

Fig. 7. Acceleration along the drill-string.

mean value of WOB increases gradually with the increase of amplitude of pulse pressure. The fluctuation amplitude of WOB and stick-slip interval decrease with the increase of amplitude of pulse pressure. The reason for this is that larger proportional of drill-string can keep dynamic friction state under higher amplitude of pulse pressure. All these change of WOB is beneficial for drilling. Therefore, the amplitude of pulse pressure of hydraulic pulse generator should be improved in the application process on the premise of maintaining safety. 3.2.2. Frequency of pulse pressure Frequency of pulse pressure is another important parameter for earthworm-like drilling because it influences the attenuation and maximum propagation distance of pulse pressure. Fig. 9 shows the relationship of WOB with frequency of pulse pressure when amplitude of pulse pressure equals to 2 MPa. From Fig. 9, the WOB increases first and then decreases with the increase of frequency, and gets maximum value when f ¼ 96Hz, which means 96Hz is the optimum frequency for this case. The reason for this relationship is that more proportional of drillstring can keep dynamic friction state as frequency increases to 96 Hz. Meanwhile, the amplitude of pulse pressure also decay gradually with the increase of frequency, which will reduce the friction reduction effect. The increase of frequency will increase the vibration energy acting on drillstring on premise of maintaining amplitude of pulse pressure constant, while also increase the attenuation of pulse pressure and decrease the propagation distance. Therefore, there is optimal frequency value for every case. From the present knowledge, the mechanism of friction reduction by

3.2.3. Installation distance of axial oscillators Earthworm-like drilling needs to mount several axial oscillators in drill-string to guarantee friction reduction effect. The vibrations of drillstring in overlap region will be weaken or strengthen if the vibrations generated by neighbouring axial oscillators overlap with each other, which will results in poor friction reduction effect or drill-string fatigue damage. Therefore, the distance between axial oscillators is another important parameter for earthworm-like drilling. Fig. 11 shows the effect of distance between axial oscillators on WOB. From Fig. 11, the maximum and fluctuation amplitude of WOB increase as the distance between axial oscillators decreases. The reason for this phenomenon can be explained by the velocity and acceleration distribution of drill-string in Fig. 12 and Fig. 13. The maximum velocity and acceleration improves significantly as the distance between axial oscillators decreases, which results in the severe fluctuation of WOB. 3.2.4. Phase difference of exciting force The distance between axial oscillators has noticeable impact on the fluctuation of WOB described in section 3.2.3. We can infer that one of the reasons for this phenomenon is the distance influences the phase difference of exciting force acting on different axial oscillators, in other words, the working time difference of axial oscillators, and influences the interaction of axial oscillators at last. The velocity of pulse pressure is

Fig. 8. Effect of amplitude of pulse on WOB. 66

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Fig. 10. Effect of pulse pressure amplitude on WOB. (a) f ¼ 2Hz; (b) f ¼ 6Hz; (c) f ¼ 10Hz.

Fig. 11. Effect of installation distance of vibrators on WOB.

another influence factor, and influenced by composition and properties of drilling fluid, size and material property of drill-string. In order to exclude the influence of working time difference of different axial oscillators on WOB, Fig. 14 shows the influence of working time difference of neighbouring axial oscillators. From Fig. 14, we can see that the working time difference has no influence on the maximum and minimum of WOB. It can be deduced indirectly that the maximum or minimum of WOB is not affected by the velocity of pulse pressure.

Fig. 12. Velocity along the drill-string.

earthworm-like drilling. 4.1. Optimization model Aiming at the increment of weight on bit. In order to ensure the strength of drill-string, the axial force Tt and variation amplitude of axial ~ of drill-string should less than the strength Tmax and fatigue force ΔT ~ max , respectively. They usually have the following relationship strength T ~ T max ¼ ð0:2  0:5ÞTmax . Besides, the minimum interval of vibrators Dmin should exceed a certain value in case vibrators interfere with each other. Considering the length of drill pipe in site is between 9 and 10 m, we set

4. Optimal design of earthworm-like drilling The above calculation are carried out in a horizontal hole section. The actual well profile usually includes straight section, deviated section and curved portion. Therefore, optimal design based on the mathematical model developed in section 2 is necessary for the field application of 67

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

gradient method. The nonlinear conditions in the constraint conditions need to be linearized before solving. The partial derivative of gi ðxÞ with respect to x can be calculated using interpolation method:

    ∂gi gi … xj1 xj þ Δxj xjþ1 …  gi … xj1 xj xjþ1 … ¼ Δxj ∂xj ¼ 1; …; m; j ¼ 1; …; n

i (30)

∂gi Calculating all ∂x every time, n þ 1 points are need for approximate j calculation. Assuming the original value of the k step iteration k1 k1 xik1 xiþ1 , and taking the first-order approximations of xk1 ¼ ½xi1 the Taylor series expansion of gi ðxÞ near xk1 , the following equation can be obtained

n

   X ∂gk1 i xj  xk1 gi ðxÞ ¼ gi xk1 þ j ∂xj j¼1

gi ðxk1 Þ and

∂gik1 ∂xj

(31)

are constant for every iteration. Therefore, constraint

conditions are converted into first-order form, which means that the constraint can be expressed as the sum of product of matrix and unknown quantity and a constant. Then the nonlinear constraint problem (equation (29)) is translated into the following linear constraint problem

Fig. 13. Acceleration along the drill-string.

minf ðxÞ ¼ minðΔWOBÞ x 2 Rn s:t: Ax þ b  0

(32)

P ∂g k1 ∂g k1 where Aij ¼ ∂xi j , bi ¼ gi ðxk1 Þ  nj¼1 ∂xi j xjk1 . This linearly constrained problem can be solved by projection gradient method after linearization. Calculation steps are as follows (a) Taking the initial feasible point xð0Þ , i.e. Axð0Þ þ b  0, permissible error ε > 0, let k ¼ 0. P (b) Calculating IðxðkÞ Þ ¼ fij nj¼1 Aij xj þ bi ¼ 0; i ¼ 1; …; mg;
 b1 , make A1 xðkÞ ¼ b1 , dividing A and b into a ¼ ðA1 ; A2 Þ, b ¼ b2 A2 xðkÞ > b2 . 1 (c) If IðxðkÞ Þ ¼ Φ, then P ¼ I, else P ¼ I  A1 ðAT1 A1 Þ AT1 . ðkÞ ðkÞ ðkÞ (d) Let d ¼ P∇f ðx Þ; if d  ε, go to step (f), else go to Step e. min (e) Find αk > 0, make f ðxðkÞ þ αk dðkÞ Þ ¼ f ðxðkÞ þ αk dðkÞ Þ , 0  α  α0 where

Fig. 14. Effect of phase difference of vibrators on WOB.

Dmin ¼ 9m. Combining the objective function and constraint conditions as follows

max 8 f ðxÞ ¼ ΔWOB T  Tmax < s:t: ΔT~  ΔT~ max : xiþ1  xi  Dmin

(27)

α0 ¼

~ is the variation where ΔWOB is the increment of weight on bit; ΔT ~ max is the amplitude of axial force; Tmax is the strength of drill-string; ΔT fatigue strength of drill-string; xi is the position of the ith axial oscillator. We rewrite equation (27) as standard form of nonlinear constrained optimization problem

minf ðxÞ ¼ minðΔWOBÞ x 2 Rn ; y 2 Rs p ðx; yÞ  0 i ¼ 1; 2; …; m s:t: i y ¼ hðxÞ

8 > > <

(

> > : min

AT2 d  0 )  AT2 x  b2 i  T    A d <0  AT2 d i 2 i þ∞;

else let xðkþ1Þ ¼ xðkÞ þ αk dðkÞ , k ¼ k þ 1, go to Step (b). 1

(f) Calculating λ ¼ ðAT1 A1 Þ AT1 ∇f ðxðkÞ Þ, if λ  0, then xðkÞ is the optimal solution, else let λir ¼ minfλij g < 0; A1 ¼ ðai1 ; …; air1 ; airþ1 ; …; aik Þ, dðkÞ ¼ P∇f ðxðkÞ Þ, go to Step (e).

(28)

1

P ¼ I  A1 ðAT1 A1 Þ AT1 ,

4.2. Example where f ðxÞ is objective function; x is parameter waiting for optimizing, which need to be solved by model developed in section 2; y ¼ hðxÞ is the procedure parameter. Substituting y ¼ hðxÞ into constraint condition pi ðx; yÞ  0. Equation (28) can be further simplified

minf ðxÞ ¼ minðΔWOBÞ x 2 Rn s:t: gi ðxÞ  0 i ¼ 1; 2; …; m

Take a three-dimensional horizontal well for example. The well profile is shown in Fig. 15. The depth of the well is 4200 m. The kickoff point and landing point are 1290 m and 2190 m, respectively, and the build-up rate is 0.0524 rad/30 m. The drill-string is make up of ∅127mm drill pipe. Other parameters used in this example are listed in Table 1. The positions of axial oscillators are optimized to get maximum WOB by applying the optimization method described in section 4.1. The initial positions of the three axial oscillators are 1740 m, 2945 m and 3700 m, respectively. The hydraulic pulse generator is mounted at the same position with the axial oscillator nearest drill bit. The amplitude and frequency

(29)

where gi ðxÞ  0 is the constraint condition. The above nonlinear optimization problem is solved by the projection 68

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Fig. 15. Well profile of a horizontal well.

vibration drilling is more effective than single-point vibration drilling under the same conditions. Compared with multi-point simultaneous vibration, the drop off of friction reduction effect of earthworm-like drilling due to pulse pressure attenuation can be offset by adding number of axial oscillators. Amplitude and frequency of pulse pressure and the installation positions of axial oscillators have great impact on the effect of friction reduction and load transfer. Frequency also affects the attenuation and propagation distance of pulse pressure aroused by hydraulic pulse generator. Besides, an optimizing model based on projection gradient method is established and used to optimize the positions of three axial oscillators in a three-dimensional horizontal well. The WOB increases significantly after the optimized position. The new positions of axial oscillators move towards bottom of well and close to each other. This paper discussed the feasibility, advantages and influence factors of earthworm-like drilling, which make us realize the earthworm-like drilling is a technology with a wide range of applications. Some simplification and issues should be considered in future research, such as

Fig. 16. Comparison of WOB before and after optimized position.

of pulse pressure generated by hydraulic pulse generator are 4 MPa and 30 Hz, respectively. Fig. 16 shows the change of WOB with time before and after optimized position of axial oscillators. From Fig. 16, the WOB increases about 30 kN after optimized position of axial oscillators. The new positions of axial oscillators are 2684 m, 3300 m and 3921 m, respectively. Compared with the initial positions of axial oscillators, the axial oscillators move towards to bottom of the well. The reason for this is that the friction between horizontal segment of drill-string and borehole is larger than the building up section on the premise of ignoring the effect of bending moment on normal contact force. Setting the positions of axial oscillators in horizontal segment is conductive to decrease friction. Besides, the axial oscillators move close to each other after optimized position. The reason for this can refer to section 3.2.3.

(a) The soft-string model is adopted in the mathematical model. For wells with smaller curvature, it can meet the accuracy requirement in site. Large deviation will be caused in high curvature interval. (b) In the actual drilling process, the drill-string includes joints and centralizers, and borehole wall is not smooth or isodiametric, and the clearance between drill-string and borehole wall can not be ignored. These conditions are bound to influence the friction reduction and load transfer of earthworm-like drilling. Therefore, the actual conditions should be considered in the future research. (c) Vibration is seen as arch-criminal of drill-string fatigue damage all the time, especially axial vibration. However, everything has two sides. Rational utilization of drill-string vibration can solve many problems in drilling engineering. Therefore, the safety analysis of drill-string, as a crucial component of earthworm-like drilling technology, should be fully considered in the future research.

5. Conclusions A new friction reduction technology, called “earthworm-like drilling”, is proposed in this paper. Compared with single-point vibration drilling, earthworm-like drilling can improve friction reduction and load transfer effect to a greater extent. Compared with multi-point simultaneous vibration drilling, earthworm-like drilling just needs to consume energy of one hydraulic pulse generator, which makes it more feasibility on site. In order to verify its feasibility and advantages, a mathematical model based on “soft-string” model is established, and a finite difference method with second-order precision is used to solve the model. The comparison results of three vibration manners indicate that more stickslip and load transfer issue are caused by higher friction, which can be alleviated obviously under vibrating conditions. The multi-point

Acknowledgments The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 51374234, 51704323, 41604103, 51404287), and China Postdoctoral Science Foundation (Grant No. 2016M602224), and Natural Science Foundation of Shandong Province (Grant No. ZR2017BEE053). The authors express their appreciation for the comments of the anonymous reviewers and the editors.

69

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71

Nomenclature

Roman symbols a propagation velocity of pulse pressure inside the drill-string, m/s A cross-section area of drill-string, m2 Ae carrying area of pulse pressure of each axial oscillator, m2 c drilling fluid drag, N⋅s=m2 D inner diameter of drill-string, m Dmin minimum interval of axial oscillators, m E elastic(Young's) modulus of drill-string, Pa ! vector of submerged drillstirng weight eg ! e , ! e e , ! t

n

b

unit base vectors in natural curvilinear system f frequency of pulse pressure, Hz f ðxÞ objective function F, Fn , Fb normal contact force and its components in ! e n and ! e b direction, N Fe exciting force, N Ff axial friction force, N hook load, N Ftop linear buoyant weight of drill-string, N/m gs gðxÞ constraint condition Gt axial component of gravity of the whole drill-string, N kα rate of change of deviation angle, rad/m kφ rate of change of azimuth angle, rad/m total bending curvature, rad/m kb L length of drill-string, m P pulse pressure applied on each axial oscillator, Pa P0 amplitude of pulse pressure excited by hydraulic pulse generator, Pa P' amplitude of pulse pressure after propagating x, Pa q propagation distance of pulse pressure, m ROP rate of penetration, m/s s well depth, m t computing time, s Tmax strength of drill-string, N T, Tt ; Tn , Tb internal tension force and its components in ! e t, ! e n, ! e b direction, N u axial displacement of drill-string, m v axial velocity of drill-string, m/s WOB weight on bit, N x parameter waiting for optimizing xi position of the ith axial oscillator, m ~ ΔT variation amplitude of axial force, N ~ max fatigue strength of drill-string, N ΔT ΔWOB increment of weight on bit, N Greek symbols α deviation angle, rad φ azimuth angle, rad Φ initial displacement distribution of drill-string, m α mean deviation angle, rad ρ density of drill-string, kg/m3 μ instantaneous friction coefficient in ! e t direction μs static friction coefficient μd dynamic friction coefficient μpv plastic viscosity of drilling fluid, Pa⋅s

Anderson, J.D., 1995. Computational Fluid Dynamics-The Basics with Applications. McGraw-Hill Education, New York, pp. 105–113. Canuto, C., Tabacco, A., 2008. Taylor expansions and applications. In: Mathematical Analysis I. Universitext. Springer, Milano, pp. 223–255. Desbrandes, R., 1988. Status report: MWD technology. II: data transmission. Petrol. Eng. Int. 60, 48–54.

References Alali, A., Akubue, V.A., Barton, S.P., Gee, R., Burnett, T.G., 2012. Agitation tools enables significant reduction in mechanical specific energy. In: SPE Asia Pacific Oil and Gas Conference and Exhibition Held in Perth, Australia, 22–24 October, SPE 158240.

70

P. Wang et al.

Journal of Petroleum Science and Engineering 160 (2018) 60–71 Shor, R.J., Dykstra, M.W., Hoffmann, O.J., Coming, M., 2015. For better or worse: applications of the transfer matrix approach for analyzing axial and torsional vibration. In: SPE/IADC Drilling Conference and Exhibition Held in London, United Kingdom, 17–19 March, SPE/IADC 173121. Skyles, L., Amiraslani, Y., Wilhoit, J., 2012. Converting static friction to kinetic friction to drill further and faster in directional holes. In: IADC/SPE Drilling Conference and Exhibition Held in San Diego, California, USA, 6–8 March, SPE 151221. Steve, J., Chad, F., Junichi, S., Jeff, L., 2016. A new friction reduction tool with axial oscillation increases drilling performance: field-testing with multiple vibration sensors in one drill string. In: IADC/SPE Drilling Conference and Exhibition Held in Fort Worth, Texas, USA, 1–3 March, SPE 178792. Tolstoi, D.M., Borisova, G.A., Grigorova, S.R., 1973. Friction regulation by perpendicular oscillation. Sov. Physics-Doklady 17, 80–86. Wang, X.Y., Ni, H.J., Wang, R.H., Liu, T.K., Xu, J.G., 2017a. Modeling and analyzing the movement of drill string while being rocked on the ground. J. Nat. Gas Sci. Eng. 39, 28–43. Wang, X.M., Chen, P., Ma, T.S., Liu, Y., 2017b. Modeling and experimental investigations on the drag reduction performance of an axial oscillation tool. J. Nat. Gas Sci. Eng. 39, 118–132. Wicks, N., Pabon, J., Auzerais, F., Kats, R., Godfrey, M., Chang, Y., Zheng, A., 2012. Modeling of axial vibrations to allow intervention in extended reach wells. In: SPE Deepwater Drilling and Completions Conference Held in Galveston, Texas, USA, 20–21 June, SPE 156017. Wilson, J.K., Noynaert, S.F., 2017. Inducing axial vibrations in unconventional wells: new insights through comprehensive modeling. In: SPE/IADC Drilling Conference and Exhibition Held in Hague, The Netherlands, 14–16 March, SPE/IADC 184635. Wu, J.F., Wang, R.H., Zhang, R., Sun, F., 2015. Propagation model with multi-boundary conditions for periodic mud pressure wave in long wellbore. Appl. Math. Model. 39, 7643–7656.

Gao, B.K., Gao, D.L., 1995. Possibility of drill stem buckling in inclined hole. Oil Drill. Prod. Technol. 17, 6–11. Gao, D.L., Tan, C., Tang, H., 2009. Limit analysis of extended reach drilling in South China Sea. Pet. Sci. 6, 166–171. Gee, R., Hanley, C., Hussain, R., Canuel, L., Martinez, J., 2015. Axial oscillation tools vs. Lateral vibration tools for friction reduction-what's the best way to shake the pipe?. In: SPE/IADC Drilling Conference and Exhibition Held in London, United Kingdom, 17–19 March, SPE/IADC 173024. Gutowski, P., Leus, M., 2012. The effect of longitudinal tangential vibrations on friction and driving forces in sliding motion. Tribol. Int. 55, 108–118. Gutowski, P., Leus, M., 2015. Computational model for friction force estimation in sliding motion at transverse tangential vibrations of elastic contact support. Tribol. Int. 90, 455–462. Ho, H.S., 1988. An improved modeling program for computing the torque and drag in directional and deep wells. In: SPE Annual Technical Conference and Exhibition Held in Houston, Texas, 2–5 October, SPE 18047. Maidla, E., Haci, M., Jones, S., Cluchey, M., Alexander, M., Warren, T., 2005. Field proof of the new sliding technology for directional drilling. In: SPE/IADC Drilling Conference Held in Amsterdam, Netherlands, 23–25 February, SPE 92558. Mezoff, S., Papastathis, N., Takesian, A., 2004. The biomechanical and neural control of hydrostatic limb movements in Manduca sexta. J. Exp. Biol. 207, 3043–3053. Parbon, J., Wicks, N., Chang, Y., Dow, B., Harmer, R.J., 2010. Modeling transient vibrations while drilling using a finite rigid body approach. In: SPE Deepwater Drilling and Completions Conference Held in Galveston, Texas, USA, 5–6 October, SPE 137754. Ren, L.Q., 2009. Progress in the bionic study on anti-adhesion and resistance reduction of terrain machines. Sci. China Ser. E Technol. Sci. 52, 273–284. Ritto, T.G., Escalante, M.R., Sampaio, R., Rosales, M.B., 2013. Drill-string horizontal dynamics with uncertainty on the frictional force. J. Sound Vib. 332, 145–153. Roper, W. F., Dellinger, T. B., 1983. Reduction of the frictional coefficient in a borehole by the use of vibration. Patent Publication No.US 4384625.

71