Actual,wellbore,tortuosity,evaluation,using,a,new,quasi-threedimensional,approach

Jun Jing ,Wnyong Y ,Cong Cao ,Xiaomin Ran

a School of Mechatronic Engineering,Southwest Petroleum University,Chengdu,China

b CNPC Chuanqing Drilling Engineering Company Limited,Xi’an,China

c National Engineering Laboratory for Low-permeability Oil &Gasfield Exploration and Development,Xi’an,China

d CNPC BOHAI Drilling Engineering Company Limited,Tianjin,China

e Sichuan Vocational College of International Technology,Guangyuan,China

Keywords:Wellbore trajectory Deflection angle Peak-valley principle Evaluation method

ABSTRACT The irregular wellbore trajectory caused by the wellbore deviation and fluctuation makes a significant effect on the torque and drag in extending and direction drilling,especially for wellbore trajectory with obvious deviation in the drilling direction.As a consequence,a new quasi-three-dimensional wellbore tortuosity evaluation method is developed.The new method incorporates the effect of fluctuation frequency and amplitude of oscillating wellbore trajectory;a weight coefficient index that quantifies the effect of tortuosity of one segment trajectory to the entire trajectory;a ‘Peak-Valley’ principle that can decompose the irregular wellbore trajectory in various scale lengths.The studies show that the deflection angle between the segments of tortuous wellbore increases the torque and drag by strengthening the contact behaviors between the drillstring and borehole.Therefore,the deflection angle is introduced to quantify the effect of deviation in the drilling direction on wellbore tortuosity.The evaluation results of two field cases demonstrate the new method which is adapted to the wellbore trajectory fluctuating with various characteristics and can reflect the actual state of wellbore tortuosity with severe oscillation more effectively and accurately.

The wellbore trajectory is not a smooth curve but deviating and fluctuating around the designed path on various scale length in practice.Lubinski et al.[1]described it as a ‘crooked hole’.The factors causing the tortuosity can be divided into three categories:(1) Geological factor.It includes the stratigraphic anisotropy,dip angle and soft and hard staggered formations.(2) Downhole tool and bottom hole assembly (BHA) factor.The structure (blade and gauge section;cutter number,size and layout strategy),cutting force anisotropy,walking angle of the drill bit,steerable system and stabilizer (size,location and number) determine the drilling direction by affecting the force characteristics of BHA [2,3].(3) Drilling mode factor.The sliding and rotating drilling mode affect the dynamic characteristics of BHA which are associated with the wellbore tortuosity.

Severe wellbore tortuosity will cause troubles to the drilling and completion,such as buckling,increase of torque and drag,excessive casing wear and cuttings deposition,and so on[4-7].Especially in the direction and extending drilling,the tortuosity will lead to the loss of drilled depth of wellbore even if the measured depth is enough [8-11].Therefore,the wellbore tortuosity evaluating and controlling are becoming more and more important.But it only can be achieved under the guidance of accurate tortuosity evaluation.Based on the sizes of drill bit and collar,the maximum allowable deviation of wellbore trajectory can be calculated by the crooked hole country formula[4,12].The dogleg severity(DLS)of wellbore is used to quantify the tortuosity which is associated with the use of steerable system to maintain and/or correct the wellbore.Some improvements were made to improve its accuracy and applicability[13-17].Based on the analysis of structure characteristics of BHA,Nobbs [18]combined the directional model and torque and drag model to calculate the DLS.Abughaban [19]used the second derivative of Advanced Spline-Curve model to calculate the DLS.Although this method is more robustness,it requires that the curve equation (cubic piecewise polynomial function) of wellbore trajectory should be continuous and can be derived at any point.However,the DLS is not an effective way to quantify the micro tortuosity occurred in a much smaller scale length.The tortuosity can be derived by comparing the torque and drag simulated results with the field recording data [20].Heisig [21]and Marland [22]proposed a wellbore tortuosity evaluation method which can work in a small-scale length.However,it requires a large amount of depth-based bending moment data of BHA which is measured by a special strain measurement tool.The bended wellbore forces the drill string to bend.Therefore,the elastic energy of the drillstring can be used to evaluate the wellbore tortuosity[23-25].However,this method only focuses on the wellbore section which increases the elastic energy.In addition,the energy required to bend the various sizes of drillstring in different sizes of wellbores is different.Hence,the complicated standardization is required to achieve the comparison.Bang[26]proposed an ‘effective diameter’ method to evaluate the irregular undulating wellbore on a much smaller scale length.It is defined as the maximum diameter of a straight cylinder that can be inserted into the section without distorting.It is similar to the crooked hole country formula.

In conventional wisdom,the wellbore tortuosity can be divided into the absolute tortuosity and relative tortuosity.The absolute tortuosity is the actual tortuosity.The relative tortuosity can be calculated by the actual tortuosity subtracting the planned tortuosity.In a sense,it reflects the amount of the actual wellbore trajectory deviating from the planned one.The planned wellbore is much smoother and its tortuosity is given when design.Therefore,the tortuosity evaluation of the actual wellbore trajectory is the key factor.Peters [27]proposed a new simple method to satisfy the need.This method divides the wellbore trajectory into a number of continuous small segments.Then,the ratio of curve length of each segment to its corresponding chord length is used to quantify the wellbore tortuosity.But it will not be accurate when dealing with the wellbore oscillating with high frequency and low amplitude.Zhou,Y [28]developed it and proposed a real-time tortuosity evaluation method which is also named as the tortuosity density method.But it does not consider the influence of deviation in the drilling direction.Although,these methods mentioned above made great contributions to the wellbore tortuosity evaluation,some key factors are still underestimated or ignored.Therefore,based on the tortuosity density method,a new quasi-three-dimensional evaluation method is developed.The new method takes into account more key factors and quantifies their influences on the wellbore tortuosity in a more reasonable way.Then,the comparative analysis with existing method on wellbore tortuosity index of two field cases are made to verify the accuracy and advantages of the new method.

2.1.Characteristics

Based on the scale length of the unit period of the oscillating wellbore,the actual wellbore tortuosity can be divided into three categories:Macro,medium and micro tortuosity.Macro tortuosity means the wellbore corrects back to the designed trajectory after deviating by the steerable system.The deviation amplitude in the drilling direction is big.The medium tortuosity means the wellbore trajectory oscillates around the designed trajectory,just like the case wells in Fig.1 (a).Its scale length is smaller than the macro tortuosity.Due to the drift size caused by this deviation,the BHA components with a big outside diameter,just like the collar,may not pass through the wellbore smoothly.Additionally,the medium tortuosity can cause a loss on the effective drill depth and change the contact behaviors between the drillstring and borehole wall[29].The increase of torque and drag and drillstring buckling will be inevitable when it becomes worse.The micro tortuosity is caused by the vibration and whirl of drill bit and it occurs in most wellbores with the form of undulating and spiral wellbore,see Fig.1(b).Commonly,the micro tortuosity is always considered as a kind of wellbore roughness.The drill bit with long gauge section and antiwhirl feature can reduce the micro tortuosity greatly in practice.

2.2.The effect of wellbore tortuosity

The drag force of drillstring is greatly affected by the variations of contact surface topography,actual contact area and the contact behaviors between the drillstring and wellbore [30-32].For instance,as shown in Fig.2,the suspended drillstring is formed between two adjacent peaks of the oscillating wellbore trajectory.

The offset distance (Hω) limits the deviation range of the wellbore.The smaller the offset distance is,the higher requirements on wellbore trajectory controlling will be.The rotary steerable system and adjustable downhole tools will be used to achieve this requirement.Based on the superposition principle of elastic mechanics,the relationship between the offset distance and the span length of the suspended drillstring under the action of axial force,moment and gravity is established.

The offset distance of the suspended drillstring under the action of the moment can be calculated by Eq.(1).

where M is the moment act on the drillstring,N.m;x is the distance to an end of the suspended drillstring,m;I is the span length of the suspended drillstring;E and I are the elasticity modulus and inertia moment of drillstring.

The offset distance of the suspended drillstring under the action of the gravity and axis force can be calculated by Eq.(2) [33].

where m is the mass of the drillstring,kg;F is the axis force,N.

The effect of the span length on drag force and contact area between the drillstring and wellbore are analysed in simulation way[34].The simulation model of pushing the φ126 mm drillstring down into the horizontal section of the case well in Fig.1(a) is established.The length of the φ216 mm horizontal wellbore is 200 m.The numerical procedure applies a constant axial velocity(0.5 m/s) on the top of the drillstring.The friction coefficient between drillstring and wellbore is tuned with the aid of field data longer span length will lead an increase of the drag force and contact area between the drillstring and wellbore.The suspended drillstring begins to contact the wellbore when the span length is longer than 16 m.This phenomenon also implies that the offset distance of the drillstring is bigger than that of wellbore.The contact area between the drillstring and wellbore continues to increase to 1.432 m2when the span length is 20 m.from a horizontal well in China.From the variation of the drag force with respect to the span length(Fig.3(a)),it can be found that the

Fig.1.The irregular wellbore trajectories in practice (a) Actual wellbore trajectory oscillates around the designed trajectory (b) The 3D undulating wellbore.

Fig.2.The sketch map of suspended drillstring.

The relationship between the drag force and the span length when drilling forward is presented in Fig.3(b).The axial force and rotation speed applied on the drillstring model are 80 kN and 60 rpm,respectively.The drag force increases from 7.9 kN to 8.54 kN when the span length increases from 10 m to 16 m.Then,due to the contact area between the drillstring and wellbore keeps stable,the drag force also keeps stable even the span length increases to 20 m.This phenomenon indicates that under the effect of various loads,stable multi-point contact between the drillstring and wellbore is formed.

Fig.3.The effect of span length on drag force and contact area (a) Pushing the drillstring down (b) Drilling forward.

The process of evaluating the wellbore tortuosity using the quasi-three-dimensional evaluation method is divided into four steps generally.

(1) Project the wellbore trajectory to the inclined and the azimuth planes and two independent projection curves can be obtained.

(2) Divide the two projection curves into multiple portions with a certain length.

(3) Then,based on the fluctuating characteristic of projection curve,the portions are decomposed into a series of consecutive segments of convex or concave curves.

(4) At last,evaluate the tortuosity of the projection curve and get the tortuosity index (Q-3D index) of the entire wellbore trajectory.

For the macro-tortuosity and medium-tortuosity,the tortuosity indexes will be calculated first.Then the Q-3D index of entire wellbore trajectory can be obtained by coupling them.For the micro-tortuosity,higher-frequency logging data is required to draw the wellbore trajectory in detail.

3.1.Pre-processing

The key step using the quasi-three-dimensional evaluation method evaluating the wellbore tortuosity is decomposing the portions of projection curve into a series of segments.Commonly,the irregular wellbore trajectory can be considered as the wave curve with a certain fluctuation amplitude and frequency.The wave crest is the peak point of the convex curve while the wave trough is the valley point of the concave curve.Basing on the crests and troughs of a wave curve,the ‘Peak-Valley’ principle is proposed in this research to decompose the irregular projection curves of wellbore trajectory.It defines that the breakpoint of one segment is the geometric middle point between the adjacent peak and valley points of the projection curve,like the breakpoints（A1，A2，A3...）in Fig.4.If the geometric middle point is not on the projection curve,the point on the curve which is close to it will be the breakpoint.

The projection curve equation is S=S（x，y，z） and its first and second order equations are continuous and can be derived.Two adjacent points i（xi，yi，zi）and j（xj，yj，zj）will be the peak and valley points,if

and their second derivative

The breakpoint k（xk，yk，zk） can be obtained by Eq.(4).

According to the ‘Peak-Valley’ principle,all breakpoints of the projection curves can be obtained.The curve between two adjacent breakpoints is defined as a segment.Therefore,a portion projection curve(Lp)consists of a series of consecutive segments of convex or concave curves.The bend curve length of a segment(lpq)is lps-q.The length of the chord line connecting the beginning and end breakpoints of a segment is lpc-q.The end point of front segment is the beginning point of the subsequent segment.

A scenario may happen when a straight trajectory exists between two segments.If the straight trajectory is tangent to the curve of segment,the effect of the straight trajectory on tortuosity can be ignored because it will not aggravate the bending of drillsting.Otherwise,if there is a deflection angle between the straight trajectory and the adjacent segment,this straight trajectory should be considered as an independent segment.During the process of evaluating the wellbore tortuosity,two important factors must be considered.One is the fluctuating amplitude and frequency of the actual wellbore trajectory,the other is the deviation of the actual wellbore trajectory in the drilling direction.The deflection angle in this research is used to reflect the deviation of wellbore trajectory in the drilling direction,which is ignored by existing methods.This angle is defined as the angle between the chord lines of two adjacent segments,just like the angle θ1in Fig.4.Due to the buildup rate limitation of BHA,the deflection angle is an acute angle.The comparison simulation analysis is conducted to demonstrate the effect of deflection angle on wellbore tortuosity.As shown in Fig.5,the curve length and chord lengths of segments 1,2,3 and 4 are the same.But there is a deflection angle(θi=8。)between the segment 3 and 4.Based on them,the simulation models of φ 127 mm drillstring passing through φ 216 mm wellbore are established.The friction coefficient between drillsting and wellbore is 0.15.The crooked wellbore will bend the drillstring.The severer the wellbore tortuosity is,the severer the deformation of drillstring is.Therefore,the elastic energy generated by the drillstring elastic deformation can be used to evaluate the wellbore tortuosity.Fig.5(b)reflects the elastic energy of drillstring in the wellbore trajectory with deflection angle is much bigger.The deflection angle exacerbates the wellbore tortuosity and strengthens the contact behaviors between the drillstring and borehole,which in turn results in the increasing of the torque and drag.Therefore,the effect of the deflection angle of wellbore trajectory on wellbore tortuosity cannot be ignored.

Fig.4.A portion of the projection curve is decomposed into a series of consecutive segments.

Fig.5.The effect of deflection angle on the elastic energy of drillstring.

3.2.Wellbore tortuosity evaluation approach

Based on the fluctuating characteristics of the wellbore trajectory,the projection curve (S) of the wellbore trajectory is divided into a series of short portions(S1，S2，···Ss，···Sw-1，Sw)with certain lengths.The tortuosity indexes of these portions will be used to evaluate the actual wellbore tortuosity.

Then,one short portion (Sp) is decomposed into a series of consecutive segments of convex or concave curves.

The ratio of the curve length (lps-q) of one segment (spq) divided by its chord length (lpc-q) is used to account for the fluctuate amplitude of this segment.The curve lengths of segments are different,the contributions they made to the tortuosity of entire wellbore trajectory are different.Therefore,the ratio of the curve length of one segment to the corresponding portion length is defined as the weight coefficient index.This index quantifies the effect of tortuosity of a segment trajectory to the entire trajectory.Then the tortuosity index of the one portion can be calculated by Eq.(7).

where Lpis the length of the short portion Sp.The value of n is equal to the number of segments,which is defined as an amplification coefficient to reflect the effect of the fluctuation frequency of trajectory on wellbore tortuosity.The wellbore trajectory will be a single convex or concave curve when n is equal to 1.If the beginning and end points of a segment are Q-（xQ-，yQ-，zQ-）and Q+（xQ+，yQ+，zQ+）,the curve length lps-qand chord length lpc-qcan be calculated by Eqs.(8) and (9).

The tangent value of the deflection angle is used to quantify its influence on wellbore tortuosity.For instance,if the curve in Fig.4 is the projection curve in inclined plane,the related breakpoints of θ1are A2（x2，y2，z2）A3（x3，y3，z3）,A4（x4，y4，z4）.The quantification index of deflection angle (θ1) between the segments of A2A3and A3A4can be calculated by Eq.(10).

Therefore,the quantification index (Γ（Sp）) of the projection curve be calculated by Eq.(11).

The m is the number of deflection angles.If the Γ（Sp）=0,it means that all the chord lines of segments are on the same line and the wellbore trajectory does not deviate in the drilling direction at all.Based on the tortuosity index of one portion and the quantification indexes of deflection angles,a new index TΘΓ（Sp） used to evaluate the tortuosity of projection curve can be calculated by Eq.(12).

In this way,the tortuosity indexes of projection curves in inclined and azimuth planes can be calculated.Then the Q-3D index(TΘΓ（Sp）Q-3D) used to evaluate the actual wellbore trajectory can be calculated by Eq.(13).

The locations of breakpoints on two projection curves may be different.In order to solve this matter,the modification on breakpoint locations will be made before evaluating.The first step is determining the main projection curve which makes the greatest effect on the wellbore tortuosity.Then,according to the breakpoint locations on the main projection curve,the locations of corresponding breakpoint on the secondary projection curve are adjusted.If the breakpoint location on the secondary projection curve can not be adjusted,the standardized value of the tortuosity indexes will be used.The standardized value is based on the tortuosity indexes of the two adjacent segments and their according effective lengths.It assumes that the tortuosity indexes of the two adjacent segments of the secondary projection curve are TΘΓ（S′i）and TΘΓ（S′i+1）,and their according lengths are l′iand l′′i+1,respectively,see Fig.6.The standardized value of the segment i of the secondary projection curve can be calculated by:

Fig.6.Breakpoints on two projection curves are not at the same location.

In order to verify the accurate of various wellbore tortuosity evaluation methods,a series of 2D curves in Fig.7 are tested.The fluctuating frequency of yellow and red curves are the same,but the amplitude of yellow curve is larger.The green and the yellow curves have the same fluctuating amplitude,but the frequency of green curve is lower.The red and blue curves have the same fluctuating amplitude,but the frequency of the red curve is much higher.Although the fluctuation amplitude and frequency of black and red curves are the same,the black curve bends downward in the macro scale length.It means that the deflection angles exist among the segments of black curve.The comparison among the ls/lcindexes,TD (tortuosity density) indexes,Q-3D indexes and DLS (dogleg severity)of these curves are made to verify the disadvantages and advantages of the method proposed in this research.The ls/lcindex is the ratio of the arc length of a single segment to the chord length,which can be used to show the bending degree of a single segment.The TD index uses the ratio of wellbore trajectory fluctuation times to wellbore trajectory length in a given length of wellbore trajectory as a parameter to describe wellbore trajectory distortion [28].The ratios of tortuosity indexes of all curves to the tortuosity index of green curve are shown in Fig.8.

Fig.7.2D curves with different amplitude,frequency and deflection angle.

Fig.8.The comparison among the ls/lc index,TD index,DLS and Q-3D index.

The ls/lcindexes of all curves are similar,which is hard to offer some useful suggestions to evaluate the wellbore tortuosity.The TD indexes quantified by the tortuosity density method pays more attention to the effect of the fluctuating frequency of the wellbore trajectory on wellbore tortuosity.Hence,the TD index experiences a continuous rise at the blue curve while the curves of the Q-3D index and DLS decrease.The ls/lcindexes,TD indexes and DLS of red and back curves are same because their fluctuating amplitude and frequency are the same.However,the deflection angle of black curve leads to an increase on the Q-3D index of the of black curve,which is 1.2 times bigger than that of red curve.It can be concluded that the quasi-three-dimensional evaluation method which considers the influence of the deflection angle on the wellbore tortuosity is a more effective way to evaluate the actual wellbore trajectory in more detail.

To demonstrate the advantages of the proposed method on wellbore tortuosity evaluation,the comparison of the tortuosity indexes of two typical field cases quantified by various methods are made.The fluctuating characteristics of wellbore trajectories of the two wells are different.

4.1.Field case 1

The horizontal section of the field case 1 is located at a vertical depth of 3045 m and its corresponding measured depth is ranging from 3341 m to 4445 m.The 3D wellbore trajectory of the horizontal section and its projection curves in inclined and azimuth planes are shown in Fig.9.The fluctuating frequency of the projection curve in azimuth plane is low and its maximum amplitude is only 1.12 m.Although the fluctuating frequency of the projection curve in inclined plane is low,its fluctuating amplitude reaches 8.2 m.Additionally,for the projected curve in the inclined plane,the obvious deviation in the drilling direction can be found at 250 m-360 m,410 m-500 m and 720 m-800 m,just as the red circles marked in Fig.9(b).

Fig.9.The wellbore trajectory of the field case 1.

Fig.10(a) presents the variation of TD index and Q-3D index of the horizonal section of the field case 1 with respect to the depth.It can be found that the projection curve in inclined plane makes the greater contributions to the wellbore tortuosity.Although both curves of the TD index and Q-3D index show a similar trend,the curve of Q-3D index keeps rising at the depth of 250 m-360 m while the curve of TD index sharply decreases.That’s because the deflection angles caused by the deviation in the drilling direction of the projection curve in inclined plane are not quantified by the TD index.The same scenario also happens at the depth of 720 m-800 m.The DLS of the horizontal wellbore trajectory of field case 1 is show in Fig.10(b).Although the DLS increases at the depth of 720 m-800 m,it also decreases at the depths of 250 m-360 m and,410 m-500 m.From the analysis above,it can be concluded that the quasi-three-dimensional evaluation method proposed in this research is much more sensitive to the deflection angle of the wellbore trajectory than others.The new evaluation method is a more effective way to reflect the actual state of wellbore tortuosity with severe oscillation.

4.2.Field case 2

The field case 2 also is a horizontal well whose wellbore trajectory is shown in Fig.11.The horizontal section is located at a vertical depth of 1069 m and its corresponding measured depth is ranging from 1194 m to 2211 m.The 3D wellbore trajectory of whole well and the projection curves of horizontal wellbore in inclined and azimuth planes are shown in Fig.11.The maximum fluctuating amplitudes of projection curves in inclined and azimuth planes are 2.78 m and 0.41 m which are 34%and 36.6%of the field case 1.However,the fluctuating frequency of two projection curves of the field case 2 is much higher than that of field case 1.The projection curve in inclined plane in macro scale length appears to be concave while the projection curve in azimuth plane appears to be convex.

Fig.10.The wellbore tortuosity of the horizontal section of field case 1.

Fig.11.The wellbore trajectory of the field case 2.(a) The 3D trajectory of field case 2 (b) The projection curves of horizontal wellbore.

The TD index and Q-3D index of wellbore trajectory of field case 2 along the depth of well are shown in Fig.12(a).The tortuosity of the projection curve in the inclined plane is much higher than that in azimuth plane.Therefore,the former curve makes the greatest contribution to the wellbore tortuosity of field case 2.Although the projection curve in the inclined plane bends downward in macro scale length,no obvious deflection angle exists between the segments.For the new method proposed in this research,the number of segments is defined as an amplification coefficient to reflect the effect of the fluctuating frequency of trajectory on wellbore tortuosity.The higher fluctuating frequency of wellbore trajectory will result in a higher tortuosity index.Hence,although the Q-3D index and TD index show the same trend,the value of former is still 5.3%larger than the latter.In addition,the Q-3D index of the field case 2 is much higher than that of field case 1 and it reaches 0.5 at the depth of 200 m-400 m.The variation of the DLS of the wellbore trajectory with respect to the depth,presented in Fig.12(b),shows the same trend as the Q-3D index.The analysis above reflects that the accuracy of the new method is enough to evaluate the wellbore trajectory with various fluctuating characteristics in practice.

Fig.12.The wellbore tortuosity of the horizontal section of field case 2.(a)The TD index and Q-3D index (b) The dogleg severity.

The quasi-three-dimensional approach offers a novel way to evaluate the wellbore tortuosity.Compared with the existing methods,the new method considers more comprehensive and indepth factors,but it also has its own preconditions of use.First,in order to reflect the fluctuating characteristic of wellbore trajectory more truly,the high frequency logging data (logging every 1 m) is needed to reconstruct the wellbore trajectory.Therefore,the prerequisites for using this method are more difficult.Second,considering the influence of deflection angle of the wellbore trajectory,this method can evaluate the tortuosity of wellbore trajectory which with severe oscillation more accurately.However,to the smooth wellbore trajectory,the effect of this new method is similar to the conventional method.

This method provides a more perfect guidance for the wellbore trajectory controlling in drilling operation.The accurate evaluation of the completed well trajectory will be helpful to determine the drilling quality and whether the well trajectory structure meets the design standards.It also improves the judgment accuracy of the torque and drag when casing,and provides guidance for formulating reasonable strategy of the completion.

A new quasi-three-dimensional quantization method is proposed in this research for evaluating the wellbore tortuosity.The following conclusions can be summarized:

(1) The fluctuating characteristics of the actual wellbore trajectory are various.Based on the scale length of the unit fluctuating period of the wellbore trajectory,the wellbore tortuosity can be divided into three main categories macro,medium and micro.The macro tortuosity is used to describe the obvious deviation of the wellbore trajectory in drilling direction.The medium tortuosity means the wellbore trajectory fluctuates around the designed path,which significantly affects the torque and drag when drilling.The micro tortuosity is considered as the wellbore roughness commonly,which affects the frictional coefficient between the drillstring and wellbore wall.

(2)The quasi-three-dimensional evaluation method proposed in this research decomposes the wellbore trajectory into a series of segments.The contribution of the curve length of one segment to the length of the entire wellbore trajectory is introduced as a weight coefficient.The innovative deflection angle is also introduced as another index to quantify the effects of the trajectory’s orientation and trajectory segments’orientations.The analysis of elastic energy of the deformed drillstring and the comparison of the tortuosity indexes of 2D curves show that the deflection angle will aggravate the wellbore tortuosity and cannot be ignored for a decent and accurate tortuosity evaluation.

(3) The comparative analysis of the wellbore tortuosity of the field case 1 and 2 with different evaluation methods show that the new method has the same accuracy as the existing methods.For the wellbore trajectory which experiences obvious deviation in the drilling direction,the tortuosity index quantified by the new method is more accurate.In addition to the establishment of a better actual tortuosity evaluation method,the findings of the present research can also be used as a guidance for wellbore trajectory controlling and for well drilling and completing to be used in the oil field.

Declaration of competing interests

The authors declared that they have no conflicts of interest to this work.

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgements

This study is supported by Open Fund(PLN201921)of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation(Southwest Petroleum University),and scientific research starting project of SWPU (No.2019QHZ008).