Modelling casing wear at doglegs by incorporating alternate accumulative wear

Accepted Manuscript Modelling casing wear at doglegs by incorporating alternate accumulative wear Hao Yu, Arash Dahi Taleghani, Zhanghua Lian PII:

S0920-4105(18)30398-X

DOI:

10.1016/j.petrol.2018.05.009

Reference:

PETROL 4932

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 14 February 2018 Revised Date:

20 April 2018

Accepted Date: 1 May 2018

Please cite this article as: Yu, H., Taleghani, A.D., Lian, Z., Modelling casing wear at doglegs by incorporating alternate accumulative wear, Journal of Petroleum Science and Engineering (2018), doi: 10.1016/j.petrol.2018.05.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Modelling Casing Wear at Doglegs by Incorporating Alternate Accumulative Wear Hao Yua,b, Arash Dahi Taleghania, Zhanghua Lianb a

John and Willie Leone Family Department of Energy and Mineral Engineering, The Pennsylvania State University,

University Park, PA 16802, USA State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu,

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b

610500, China

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Abstract. In some situations while drilling through severe doglegs, drill strings are inevitably touching the inner wall of casing, which may wear casing harshly. For instance, this is a frequent

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problem in wells drilled in Bohai Bay basin in East China. In large dogleg sections of the well, casing is subjected to a dual-wear accumulation contributed by the tool joint and drill pipe (alternate wearing). Nonplanar geometry of the hole and complex geometry of the rotating drill pipe may add to the complexity of the wear process. Here, we propose a mathematical model to calculate the loss

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of wall thickness of the casing due to the dual wear. A finite element model is developed to incorporate wearing of the casing alternating between the tool joint and the drill pipe. Arbitrary

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Lagrange-Eulerian (ALE) adaptive meshing and remapping technique is utilized to specify the ablation velocity vectors at the inner casing surface. Our numerical results show that the dual wear

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contributions and ever changing contact geometry in alternate wearing intensifies the rate of the casing wear 150% faster than that of the single wear models. Additionally, the curvature at point with maximum wear depth has been found to significantly impact the maximum stress of worn casing. This paper provides an accumulative wear model for accurate prediction of the casing wear and its residual strength before running the casing through the dogleg section. Key words: Directional drilling; doglegs; casing wear; dual-wear contribution

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ACCEPTED MANUSCRIPT 1. Introduction In recent years, drilling directional and horizontal wells with complicated trajectories and large doglegs have become a common practice especially for exploration and development of unconventional hydrocarbon reservoirs. However, directional drilling has also been used for various

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purposes such as drilling relief wells, drilling multiple wells from a single pad, or sidetracking from the wells with fishing problems. Therefore, the increasing complexities of directional wells escalate

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the potential risks for casing integrity, which may lead to well leakage or even well plugging and abandonment before starting production. Therefore, accurate prediction of the casing wear in large

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doglegs can be critical to save the well and avoid unnecessary costly aftermaths. Bradley et al. (1975) was one of the early references that documented the severe casing wear problem in directional wells, and conducted casing wear experiments under different operational conditions. Through the laboratory and field studies, the rotation of drill string is identified as the

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major reason for casing wear, and wear volume may obviously increase due to the concentrated contact loads in the doglegs. Best (1986) classified casing wear caused by drill string rotation as

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typical adhesive wear and abrasive wear. Through lab testing in field conditions, Best proposed that smooth, round and uniform surface of tool joint hardfacing may help to achieve a maximum contact

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area at minimum contact stress. Additionally, soft solid particles of drill mud may form a layer in the tool joint/casing contact region to avoid the metal-to-metal contact. White and Dawson (1987) used the full-scale wear-test machine to conduct the casing wear experiment for different steel grades, drilling fluid properties and contact loads. He proposed a linear casing wear-efficiency model that is based on the frictional work loss of the casing metal during the wear process. After 475 times eight-hour casing wear tests, Hall et al. (1994, 2005) proposed the concept of contact pressure threshold, which means that the casing wear may be sufficiently self-limiting unless the 2

ACCEPTED MANUSCRIPT contact pressure threshold (CPH) is reached. He concludes that higher CPH values may imply shallow ultimate wear groove depth. However, CPH is hard to be predicted and controlled due to the ever changing complicated contact geometry. Han et al. (2003) simulated the casing stress re-distribution by considering simplifying

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assumptions like uniform-thickness wear model, eccentric cylinder wear model and crescent-shaped wear model. Gao et al. (2010) used analytical methods to demonstrate the impact of different-sized

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drill strings on the casing wear and discussed three possible forms of the casing wear groove, i.e. single wear groove, sharp crescent-shaped and blunt crescent-shaped wear grooves. Later, on the

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basis of the minimum energy of the wellbore path, Kumar et al. (2013) proposed a combined “wear-energy” model to estimate the casing wear in curved sections of the wellbore to determine the wear volumes of casing by the various curvature and torsion model. They showed that the wellbore torsion was found to have a significant impact on the casing-wear downhole, while the

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effect of drill pipe on casing wear was not taken into account. Then, Kumar and Samuel (2014, 2015) argued that casing wear model should be developed not only for static wear conditions, but

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also for the operation parameters such as rotary speed, vibration and buckling of drill strings on the casing wear. They showed that using the eccentric cylinder model would lead to more extensive

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wear in comparison to the crescent-shaped model. Lin et al. (2015) presented a new algorithm for calculating the residual collapse strength of the worn casing by incorporating the ovality of inner wall and non-uniform wear due to the casing defects. More recently, Yu et al. (2016a, 2016b) proposed a numerical model of dynamic casing wear in a directional well accounting for the anisotropic tectonic stresses. Vavasseur et al. (2016) found the torque, drag and buckling of drill string could make the pipe body and tool joint contact with the casing inner wall respectively, both of which contributed to the casing wear and caused the complex wear in the casing inner wall. Later, 3

ACCEPTED MANUSCRIPT Aichinger et al. (2016) verified Vavasseur’s argument through field data using multiple-arm caliper Log. Despite fundamental studies of the casing wear conducted in the past 40 years, drilling multilateral, ultra-deep as well as extended-reach wells, which significantly complicates casing

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wear require a more sophisticated wear model. In the large dogleg section of these wells, stiff straight drill string should deform to fit the borehole trajectory, which leads to its complicated

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contact geometry with the casing. In this case, not only tool joint but also drill pipe can be rubbed against the casing surface due to the bending and/or buckling of the drill string by axial compressive

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loads. In that case, using simplified wear models like eccentric-cylinders (Dou et al., 2010; Chen et al., 2015; Lin et al., 2016) may overestimate the residual strength of the casing. In addition, the anisotropic tectonic stresses effect makes it more complex to predict the wear condition and its residual strength.

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2. Alternate Complex Casing Wear

The common approach to analyze the casing wear is simply the uniform loss of the casing thickness,

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as shown in Fig. 1a. Then, the remaining collapse and burst strength of casing could be calculated following the API 5C3 standard based on the remaining thickness (American Petroleum Institute,

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1994). However, this uniform-wear assumption may severely underestimate the remaining strength of eccentrically worn casing. Hence, many efforts have been focused on improving the casing wear model by primarily two basic models: first, the single crescent-shaped wear model (Fig. 1b), which considers only the interaction of the casing inner wall with the tool joint (Song, 1992; Gao et al., 2010; Lin et al., 2015;); and second, the eccentric cylinder wear model, which simplifies the crescent shape to a pipe with geometric eccentricity (Fig. 1c). (Dou et al., 2010; Chen et al., 2016; Zhang et al., 2016). However, it is known that the pipe body of drill string may partly or completely 4

ACCEPTED MANUSCRIPT touch the inner wall of the casing as it is under axial compressional loading especially at the dogleg section as shown in Fig. 2 (see for example Paslay et al., 1991; Vavasseur et al., 2016; Huang et al.,

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2017).

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Fig. 1. Diagrams of conventional casing wear geometries

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Fig. 2. Schematics of a drill string touching the casing through a dogleg section.

The contact pattern for the drill string is assumed to occur at a specific point based on the assumption of small deflection of drill string. Therefore, we assume the deflection is so small that the buckled drill string cannot touch the casing elsewhere except the lower side of wellbore. Based on the assumption of drilling string buckling mentioned above, drill pipe comes into contact with specific wear location where the tool joint has already contacted with casing and caused prior wear. Then tool joint comes into contact with the worn place again and continues to 5

ACCEPTED MANUSCRIPT repeat the interacting process with the existing wear. Hence, the casing wear in the large dogleg section is actually a round-and-round losing process of alternate accumulating wear contributed by the tool joint and the drill pipe, instead of the conventional assumption of tool joint-contributed

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single wear, as shown in Fig. 3.

Fig. 3. Diagrams of revolving accumulative wear geometry in a large dogleg section.

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As demonstrated in Fig. 3, the alternate accumulative wear process can be decomposed into three basic cycles: (1) Assuming the tool joint initially contacts with the casing inner wall, casing is

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worn into a single crescent-shaped groove, as shown in the yellow area, (2) Drill pipe contacts with the casing inner wall, and a smaller crescent-shaped groove is generated by drill pipe based on the

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initial wear, as marked in Fig. 3. (3) Next tool joint comes into contacting with the worn area, and casing will be first worn on the borders of smaller crescent and the overall maximum casing wear depth will not increase unless the remaining area is totally worn off, as marked in Fig. 3b. For the first wearing cycle, the depth and width of initial casing wear can be calculated as , 4

(1) (2)

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ACCEPTED MANUSCRIPT where

is the maximum depth of initial casing wear; !

represents the outside radius of tool joint;

1

is the width of initial casing wear;

represents the inside radius of casing;

1

stands for

the eccentricity of tool joint and casing center. Hence, the crescent-shaped area of casing wear contributed by tool joint can be calculated as &

2 $+ %

arcsin

'

' ( )'

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*

"

arcsin

.

(3)

2

2

2

3

2

4

5

4

5

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4

2

is the maximum depth of drill pipe and casing wear;

casing wear; 1

casing,

)

represents the outside radius of drill pipe,

!

(4)

.

(5)

is the width of drill pipe and

represents the inside radius of

stands for the eccentricity of tool joint and casing center, m. Hence, the smaller

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where

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For the second wearing cycle, the depth and width of casing wear can be calculated as

crescent-shaped area contributed by drill pipe can be calculated as *

2 $+ % 4

5

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"

'

2

&

arcsin

2

' 3

( )'

arcsin

(6)

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For the third wearing cycle, the overall maximum casing wear depth will not increase unless the remaining area is totally worn off. The critical wear area can be calculated as

"6

"8

6 6

where

2

6

"

arcsin

7

6

4

arcsin

is critical wear eccentricity,

6

, 7

,

7

7

;

9

4

(7)

5

(8)

2

arcsin

3

arcsin

:

is critical width of casing wear, "3 is the whole crescent

shaped area of casing wear by second-time tool joint, "; is critical wear area. 7

(9)

ACCEPTED MANUSCRIPT As the next drill pipe reaches the wear zone, a new round of wear cycle starts and continues to repeat these three basic cycles until wear process is over or the casing is worn out. Because the contact area of the drill string and the casing evolves nonlinearly with the drilling process, the effective contact pressure on the casing inner wall will also change continuously; hence, the

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practical wear loss and depth of each cyclic process are different with each other. Considering the anisotropic tectonic stresses interaction makes the alternate wear process very complicated to be

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solved analytically. 3. Force Analysis of Drill String

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In the curved section of a directional well, due to complex well trajectory and drill string bending and/or buckling, the force status of drill string appears more complicated than that in a vertical well. Some basic assumptions were made in the derivation: (1) Drill string is considered as an elastic beam, (2) The distance between two consecutive tool joints is assumed to be much less than the

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radius of curvature of the curved sections, hence, the relatively small deflection assumption is satisfied, (3) The curved borehole axis is assumed to have a fixed curvature, (4) The magnitude of

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the axial force is assumed to be constant that is always compressive. The direction of the axial force is perpendicular to the cross-section of the drill string and (5) The direction of the contact force on

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the casing applied by drill string is assumed to be always perpendicular to the lower side of wellbore.

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Fig. 4. Projected components of a 3D well trajectory on inclination and normal planes

Based on the above assumptions, equivalent beam-column method is developed to calculate drilling string forces. The deviation angle α, azimuth angle φ and well depth s are used to describe

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the well trajectory. β in the Fig. 4 is the angle between the deviation and inclination planes. The relationship of the angle β, the deviation angle α and azimuth angle φ is defined as 2= 2>

?@ ABC D ABC E

,

(10)

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Projected components of the well trajectory on inclination and normal planes near reference

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point-P are shown in Fig. 4. The well curvature near the reference point-P can be calculated as FG

2E

2=

9 2> :

9 2> sin H: ,

(11)

Where the projected components of well curvature on the plane O-yz and the plane O-xz can be calculated as

FIJ

2E 2>

, FKJ

2= 2>

sin H

(12)

Using Eqs. (11-12), the 3D well problem can be decomposed into two 2D problems on well inclination and normal planes respectively.

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ACCEPTED MANUSCRIPT On the plane O-yz, the governing equation of drill string deformation can be written as the fourth-order linear differential equation LM

2N I

O

2J N

2 I 2J

P sin H

Q ,

(13)

where E is the elasticity modulus of the casing, I is its second moment of area, EI represents the

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flexural rigidity of drill string, F represents the axial compressive load, q stands for the gravity load of the drill string per unit length, α is the well inclination angle at point-P, rad; m1 indicates the

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component of distributed contact force between drill string and casing on the plane O-yz; y1 indicates the lateral displacement of drill string axis with respect to the wellbore axis tangent on

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inclination plane.

Fig. 5. Equivalent additional tubular string weight based on beam-column model

As shown in Fig. 5, to a curved section, the differential between borehole axis P-s and borehole tangent P-z can be approximated as

R

R S FIJ T

(14)

Substitute R in Eq. (14) into the Eq. (13), we can obtain LM

2N I

2J N

O

2 I 2J

P sin H 10

FIJ O

Q

P

Q

(15)

ACCEPTED MANUSCRIPT where y2 is the lateral displacement of the drill string axis to the borehole axis. According to Eq (15), FIJ O is the additional load caused by wellbore curvature on the plane

O-yz, and P

P sin H

FIJ O is equivalent lateral loads in the plane O-yz. Hence, the

deformation problem of drill string with lateral string weight P sin H in curved sections is FIJ O in horizontal wellbores.

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equivalent to that with lateral string weight P sin H

On the plane O-xz, the governing equation of drill string deformation can be written as the

2N K

O

2J N

2 K

Q ,

2J

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LM

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fourth order linear differential equation

(16)

where m2 is the component of distributed contact force between drill string and casing on the plane O-yz. Similar to Eq. (14), the deviation of the borehole axis and its tangent on the plane O-xz is '

' S FKJ T

(17)

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Using Eq. (17), the Eq. (16) can be simplified as LM

2N K

O

2J N

2 K 2J

FKJ O

P

(18)

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where P is the equivalent string weight on the plane O-xz.

Q ,

Combining components of the total equivalent string weight of the drill string from Eqs. (15)

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and (18) can be written as PU

&P

P

9P sin H

2E 2>

∙ O:

where PU is the equivalent string weight in the curved section.

2∅

9 2> sin H ∙ O: ,

(19)

According to Eq. (19), the contact force of tool joint and drill pipe through point-P are XY

Xa

Z[ ∙\

9Z ABC E

3^ ∙`: 3_

9

Z[ ∙\

9Z ABC E

3^ ∙`: 3_

9

\7]

\7b

11

\7] \7b

3∅ ABC E∙`: 3_

∙\

(20)

3∅ ABC E∙`: 3_

∙\

(21)

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where XY and Xa stand for the contact force of tool joint and drill pipe on casing; c is the distance between two consecutive tool joints; c6Y and c6a represent the effective contact length of

tool joint and drill pipe on casing. 4. Numerical Simulation

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Based on what explained in previous sections, a case analysis is conducted for the casing wear in well X-3 located in Eastern China. According to caliper measurements conducted after drilling, the

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maximum wear occurred at 2562m, where the maximum principal stress is 46MPa in the E-W direction, and the minimum principal stress is 32MPa in the N-S direction (as shown in Fig. 6) The

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well deviation angle, at this point is 35.86°, with the well deviation rate of 8.77°/25m, and the azimuth deviation rate is 2.43°/25m. The length of each pipe string is 10.36m and, and the length of the tool joint is 0.4318m. The average rotary speed of the drill string was 80r/min. The elastic modulus of the formation rock is 7.2GPa and Poisson’s ratio is 0.30. Mechanical properties of the

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drill string, casing and cement are shown in Table 1. Table 1. Material parameters used in numerical simulations Outer diameter

-

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Cement sheath

Thickness

Elastic modulus

Steel grade

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Material

Yield strength Poisson's ratio

(mm)

(mm)

(GPa)

(MPa)

311.1

33.3

9.8

0.25

-

Casing

P110

244.5

11.05

210.0

0.30

758.0

Drill pipe

S135

127

9.19

210.0

0.30

931.0

Tool Joint

S135

168

20.5

210.0

0.30

931.0

A two dimensional finite element model is developed to consider elastic and plastic deformations of casing, drilling pipe and formation rock (see Fig. 6). The formation rock, cement sheath, and casing are presented with linear elements (about 20,000 nodes and 20,000 elements). 12

ACCEPTED MANUSCRIPT The drill pipe and tool joint are assumed to be rigid bodies because their hardness is far larger than the hardness of the casing material. The well pressure of 23.5 MPa is applied in the interior face of

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the casing.

Fig. 6. Finite element model under anisotropic tectonic stresses

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The model edges are constrained and tectonic stresses are assigned as initial stresses into the model. According to previous studies (Yu et al., 2016b), the most severe condition of casing wear occurs when the normal of initial contact surface becomes parallel to the direction of the minimum

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strength.

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principal stress. Therefore, we adopt this severe condition as the benchmark to study casing

We considered 5500 increments (equivalent time) here to simulate dynamic alternate accumulation of casing wear during drilling, and each increment (equivalent 1-minute rotation) includes one-time drill pipe or tool joint wearing casing process. Because a drill pipe is much longer than a tool joint, contact time of pipe with casing is longer than the contact time of tool joint with casing. Hence, in order to reasonably compare different wear processes, we set the rotatory speed of drill pipe per increment as 10 times as the initial value to shorten the contact time to one tenth of the 13

ACCEPTED MANUSCRIPT initial time period. The resultant alternate loading history of the drill pipe and the tool joint, and their corresponding rotary speeds experienced by an arbitrary point-P of the casing can be seen in

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Fig. 7.

Fig. 7. Alternate loading history and boundary conditions. The red curve represents the periodically alternate side contact force through point-P from drill pipe or tool joint in the finite element model. The blue curve is the periodically alternate rotary speed of drill pipe or tool joint

It can be seen that the curves can be divided into several cyclic a-b-c-d stages. The section ‘ab’ is the stage of casing wear contributed by drill pipe, and the load XK is the contact force from drill Xa ). The rotary speed !K is that of drill pipe (!K

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pipe (XK

!a ). The section ‘bc’ is the stage of

casing wear contributed by the tool joint, and the load XK through point-P is the side contact force XY ). The rotary speed !K is that of tool joint (!K

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from tool joint (XK

!Y ). The section ‘cd’ is

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the stage of casing wear contributed by drill pipe in the next round. Substitute well deviation angle, well deviation rate, azimuth rate and dimension of drill string of X-3 well into Eq. (20) - (21), assuming the value of axial compression F = 30 t (as shown in Fig. 5), the side contact force of drill pipe and tool joint to casing can be calculated as Np = 2.01 kN/m and Nj = 50.9 kN/m respectively. The rotary speed of drill pipe and tool joint are np = 800 r/increment and nj = 80 r/increment respectively.

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ACCEPTED MANUSCRIPT 4.1 Implementation of Dynamic Wear Process Based on the wear-efficiency model proposed by White and Dawson (1987), a portion of friction work is transformed into heat generation, and the other portion leads to wear. The wear volume can

de

f ∑ jOk c>

fgh c>

!l

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be calculated as (22) (23)

where de is wear volume; f is the wear coefficient; gh is the friction work; j is the friction

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coefficient. c> is the sliding distance; ! is rotary speed; l is the wear time. Casing wear is a

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time-dependent process due to the ever changing contact geometry and time-varying contact pressure. Hence, Eq (22) can be considered as

de

4m5

$n 4m5 )o

f $\ jp4m, r5 )r s

where o is cross-sectional area of contact geometry;

(24) (25)

4m5 is the wear depth at an arbitrary

pressure at this point.

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pointM within )r; r is the sliding distance along the traveled path ct ; p4m, r5 is the contact

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Based on the above equations, the casing wear process can be regarded as the gradual abrasion accumulation of its surface material. We used a user–defined subroutine in ABAQUS to simulate

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the wear, which is used to specify the ablation velocity vectors at the nodes that are on the casing inner wall. Arbitrary Lagrange-Euler (ALE) adaptive mesh smoothing is applied to adjust nodes in the interior side of the casing to maintain a good mesh quality in the process of simulation. The dynamic accumulative process of casing wear can be described as periodically cyclic procedures, and every cycle includes: ⅰ) solution of the contact problem using the finite element model to calculate the contact pressure and the sliding distance of each node of contact regions, ⅰ) computation of wearing rates to calculate the wear depth of each node, and ⅰ) modification of 15

ACCEPTED MANUSCRIPT casing surface geometry by a user-defined subroutine. Each wear increment is performed by

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repeating the cycles listed in Fig. 8.

Fig. 8. Flowchart for implementation of dynamic accumulative process of the casing wear

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Based on the Eq. (25), the incremental depth of wear u can be expressed as

u

Y

fjp 4r Y

Y

r

Y

Y

5,

in the ith increment at the jth node

(26)

where r is the total tangential displacement of ith cycle at the jth point; p is the contact pressure Y

Y

of ith cycle at the jth point. f is the wear coefficient, 7.0537×10-13 Pa-1 for S135 pipe and P110 casing in this paper. µ is the friction coefficient, 0.3 for S135 pipe and P110 casing in this paper. Thus the cumulative wear depth of casing at the jth node is 16

∑v u

ACCEPTEDY MANUSCRIPT Y The vector sum for all wear surfaces is

wx y

∑v u

Y

(27)

xz , y

(28)

where j is an arbitrary point of the contact surface and xz is the unit vector. y

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4.2 Simulation Results Fig. 9 shows the stress and deformation contours of an intact casing under initial anisotropic tectonic stresses. It can be observed that the displacement and stress distribution of casing presents

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symmetric distribution.

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Fig. 9. Stress and displacement contours of unworn casing

The maximum deformation displacement is only 0.269mm, and is in the direction of the

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maximum principal stress. The maximum Mises stress is only 301.064 MPa, and is in the direction with the minimum principal stress. The magnitude of stress is far from P110 casing yield strength (758 MPa).

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Fig. 10. Contours of alternate accumulative wear model when casing has reached its yield strength

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Fig. 10 demonstrates stress and displacement contours based on the proposed alternate accumulative wear model proposed here. It takes 108.18 hours of rotation (at 80 rpm) to wear the casing down to its yield strength for the alternate complex wear. It can be seen that due to the dual wear contributions by drill pipe and tool joint, eventually the casing is worn to be a similar

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double-crescent shape, interacting with the anisotropic tectonic stresses. The maximum wear depth occurs at point-o, as show in Fig. 10b. Where the tool joint contributes most to casing wear is m-o-n segment, and the drill pipe contributes most to casing wear is p-o-q segment. When the maximum

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depth of alternate accumulative wear model reaches 7.908mm, the maximum Mises stress of casing

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occurs at the location of maximum wear depth and the Mises stress decreases progressively from the maximum depth to both sides. Compared with the contours of unworn casing in Fig. 9b, the maximum Mises stress of casing increases from 301 MPa to 759 MPa, and the casing entered the yield stage.

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Fig. 11. Contours of Drill pipe-Casing single wear model when casing has reached its yield strength

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Fig. 12. Contours of Tool Joint-Casing single wear model when casing has reached its yield strength

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In order to compare the influence of working conditions on the casing ultimate strength, numerical simulations of drill pipe-contributed single wear and tool joint-contributed single wear were conducted separately. We may compare the impact of each method based on equal drilling time. It takes 185.76 and 211.98 hours of rotation (at 80 rpm) to wear the casing down to its yield strength for drill pipe-contributed and tool joint-contributed single wear models, respectively. Stress and displacement contours of two single wears when the casing just reached the yield strength were obtained, as shown in Fig. 11 and Fig. 12. It can be seen that the casing is worn like a 19

ACCEPTED MANUSCRIPT single-crescent shape, which remains consistent with outer diameter of drill pipe or tool joint, and the maximum wear position also occurs at point-o. When the casing reaches the yield strength, the corresponding maximum wear depth of casing is 7.885mm and 8.492mm respectively for these two situations. In the drill pipe-contributed single wear model, the maximum wear depth for yield

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strength is almost the same as that of alternate accumulative wear model. However, the area of alternate accumulative model prone to damage is 1.26 times bigger than that of drill

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pipe-contributed single model. In the tool joint-contributed single wear model, the maximum wear depth for yield strength is 0.59 mm greater that of alternate accumulative wear model. It is thus

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clear that different working conditions have noticeable influence on the ultimate strength of the

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casing.

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Fig. 13. Curves of maximum wear depth with increment /time under three working conditions a) wear depth over contact time; b) wear depth over drilling time

Fig. 13a and 13b demonstrate maximum wear depths of casing (at point-o) over continuous contact time and versus drilling time, respectively. It can be seen from Fig. 13a that the wear rate of

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tool joint-induced single wear model is higher due to the larger contact force within the same “contact time”. However, in the field, tool joint or drill pipe cannot continuously be in contact with the casing. If we make comparison based on the contact time (Fig. 13a), we will see less induced

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wear by the alternate wear model, however, we need to keep in mind that in the alternate wear

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model, a longer contact time is anticipated, so when comparison is based on drilling time, the alternate wear model is more destructive. The an-bn-cn-dn stage in Fig. 13b corresponds with a-b-c-d stage of loading history curve in Fig. 7. It is observed that all the three curves in Fig. 13b manifest ladder-like rising trend, which means the maximum wear depth of casing remains a periodically steady phase instead of continuous growth in the process of casing wear. In the drill pipe induced single wear model, the steady phase is due to the “blank period” of wear induced by the tool joint. In the tool joint-contributed single wear model, the steady phase is due to the “blank period” of 21

ACCEPTED MANUSCRIPT wear induced by the drill pipe. For the alternate accumulative wear model, the steady phase remains all the same due to the existing critical wear area, as discussed in Eq. (7) - (9). By comparing the three curves (Fig. 13b) within the same drilling time period, it is observed that the growth rate of alternate accumulative wear is more than 150% faster than that of the other

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two curves. One reason is due to the dual impacts of the drill pipe and the tool joint. Besides, another very important reason is that the variable contact geometry between the drill string and

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casing intensifies the contact pressure. In the early stage of the wear, the drill pipe or tool joint has mainly a point contact with the casing. The small contact area then results in high contact pressure

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and consequently fast wear according to Eq. (26). Then the contact pressure decreases non-linearly due to the continuously increasing contact area along with the wear process, hence, the rate of casing wear depth at point-o reduces accordingly over time. However, for the alternate accumulative wear model, the cyclic alternating between drill pipe and tool joint leads to the contact

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area between drill string and casing always very small. After one-time wear contribution by tool joint, the borders of previously worn casing would be worn partly or completely, which made the

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subsequent contact area of drill pipe and casing decrease greatly, even the point contact could again occur. It will result in increasing the contact pressure dramatically, and thus casing wear rate always

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keep fast at point-o in the whole alternate process.

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Fig. 14. Curves of maximum Mises stress with the wear depth under three working conditions

In order to study the influence of wear depth on the maximum Mises stress under different

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wear conditions, curves of the maximum Mises stress versus the wear depth were obtained, as shown in Fig. 14. It can be seen that the maximum Mises stress increases continuously with the increase of wear depth in different scenarios. When the wear depth is less than half the thickness of casing wall (before A-A’ line), the maximum Mises stress approximately presents linear with the

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wear depth. While the wear depth is more than half the thickness of casing wall (after A-A’ line), the maximum Mises stress increases sharply with the wear depth. It indicates that under any wear

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condition, the maximum permissible wear depth should not exceed half the thickness of the casing. In addition, it is observed that with the same wear depth, the magnitude of maximum stress of the

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tool joint-contributed single wear model is always the least one, but the magnitude of maximum stress of drill pipe-contributed single wear model tends to be equal with that of alternate accumulative wear model. In order to make a more direct and clear comparison of the stress distribution at the same wear depth, the Mises stress distribution curves of 7.908mm wear depth casing (at B-B’ line) under different working conditions are calculated in Fig. 15.

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Fig. 15. Mises stress curves of 7.908mm wear depth casing under three working conditions

According to Fig. 15, the maximum Mises stress of worn casing is not only related to the wear depth, but also closely related to the curvature at point with maximum wear depth. By comparing the drill pipe-contributed single wear and alternate accumulative wear, it can be seen that the

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maximum Mises stresses remain almost the same because of similar wear curvature. However, comparing the tool joint-contributed single wear and alternate accumulative wear, the maximum Mises stress of tool joint-contributed single wear is less than that of the alternate accumulative wear

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due to its smaller curvature (i.e. radius of the tool-joint). It also verifies that in presence of small

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curvatures, simplified wear models like eccentric-cylinders can overestimate the residual strength of the casing. Additionally, the results show that the residual strength prediction model in a directional well should be based on the size of the drill pipe rather than that of the tool joint to achieve more conservative design for burst and collapse of casing strings in the large dogleg section. 5. Conclusions This paper proposed a mechanical model for the casing wear in the dogleg section of directional wells that incorporates dual wear contributions. The proposed model is successfully implemented in 24

ACCEPTED MANUSCRIPT a commercial finite element package by developing required user-defined subroutines. The outcomes of these simulations show that due to the complicated contact geometry in the dogleg, the casing wear process is mainly an alternate accumulative phenomenon that cannot be simulated correctly without considering the contributing factors of both drill pipe and tool-joints.

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In conclusion, the dual wear factor makes the worn casing to be similar to the double-crescent shape instead of the single-crescent shape. Mises stress distribution of double-crescent worn casing

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is more concentrated near the wearing regions. Meanwhile, the dual wear factor and the revolving contact area make the wear depth at some points far greater than what single wear models show for

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the same rotation time interval. The difference of the depth of wear between complex model and single models is increasing over running time. Hence, the prediction for casing wear in the large dogleg section should be based on the proposed accumulative models to achieve a safe design. The results show that the maximum Mises stress of the worn casing has different sensibilities to the

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residual wall thickness. While the wear depth is more than half the thickness of casing wall, the maximum Mises stress increases dramatically with the maximum wear depth.

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The results also indicate that maximum stress of worn casing increases with the increasing worn surface curvature. Therefore, the residual strength prediction model of casing wear in the large

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dogleg section should be based on the size of drill pipe rather than the size of the tool joint. Acknowledgements

The authors are grateful to the support from the China Scholarship Council (CSC, File No. 201708510125) for its contribution to this paper. References 1. 2.

Aichinger, F., Dao, N., Nobbs, B., Delapierre, A., Pinault, C., Rossi, X., 2016. Systematic Field Validation of New Casing Wear Quantification Process. SPE 183386 MS. American Petroleum Institute (API). Bulletin on formulas and calculations for casing, tubing, 25

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Drill strings are inevitably touching the inner wall of casing, which may wear casing harshly. Here, we propose a mathematical model to calculate the loss of wall thickness of casing due to the dual wear. A finite element model is developed to incorporate tribology of the casing alternating between the tool joint and the drill pipe. Our numerical results show that the dual wear factor and the revolving contact area make the wear depth at some points far greater than what single wear models show for the same rotation time interval.

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