 Open Access
 Total Downloads : 168
 Authors : Abdennour C. Seibi, Fathi Boukadi, Mohammed AlKhozaimi, Tasneem Pervez, Racha Trabelsi
 Paper ID : IJERTV6IS030345
 Volume & Issue : Volume 06, Issue 03 (March 2017)
 Published (First Online): 29032017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Development of Operating Envelopes for Drillpipes Running through Medium to High Curvature Wells
A. C. Seibi1*, R. Trabelsi1, F. Boukadi1 1University of Louisiana,
Department of Petroleum Engineering Lafayette, Louisiana, USA
T. Pervez2,
2Sultan Qaboos University,
College of Engineering Mechanical Engineering Department, P.O. Box 33
AlKhod 123, Oman
M. AlKhozaimi3,
3Petroleum Development Oman
P.O. Box 33, Mina ElFahl 123, Oman
Abstract – The present paper aims at establishing elastic/plastic operating envelopes of drillpipes running through medium to high curvature wellbores. The operating envelopes will serve as guidelines for drilling engineers by identifying the operating limits of various API drillpipes used in curved wells before undergoing any activities. A two dimensional mathematical model, which incorporates various field parameters that affect the running operation in medium to high curvature boreholes, has been developed. The developed model is capable of studying the effect of the vertical force at the kick off point (k.o.p), the back push (horizontal) force at the end of curve (e.o.c), radius of curvature, drillpipe bending stiffness, and formation roughness on the running operation. Two boundary conditions (fixedfixed and pinnedpinned) at both ends of the drillpipes (k.o.p and e.o.c) were treated. The developed model estimates the required running force as well as the induced stresses of potential API drillpipes in curved well bores. The stress state, in drillpipes running through curved sections, was used to develop operating envelopes for drillpipes running through typical well configurations. The model also enables field engineers to select appropriate drillpipes according to operating field conditions in order to avoid any unexpected failure.
INTRODUCTION
Directional drilling is commonly used to reach lateral targets within oil and gas reservoirs. Current directional drilling technology provides the capability to drill, navigate and control hole paths along a predetermined well path. Mechanical friction between drillpipes and wellbore presents a major concern in directional drilling because it i) increases the surface power required to rotate the drillpipe,
ii) may cause drillpipes to get stuck to the formation making it difficult or impossible to pull out of the borehole; and iii) makes it difficult to establish and estimate a given weight on drillbits while drilling.
In horizontal drilling, the buildup section represents the most critical part of the drilling operation. Available literature published in the past presented different models to predict the behavior of drillpipes running through curved holes [1 11]. Of particular importance to this subject is
the work done by Seibi et al. [9 12] and Martinez et al.
[13] who conducted experimental and numerical studies on pipes running through curved holes and concluded that the end forces are affected by the contact between the pipe and formation due to lateral pipe deformation. These studies were further investigated experimentally by Kuru et al. [14] to study the effect of buckling on the axial force and concluded that i) the end support conditions of the tubular have a significant effect on tubular buckling, and ii) tubular buckling controls the contact force, and hence, the axial force. Most of the existing models do not consider the effect of various parameters affecting the stress state in drillpipes while running through curved sections. These parameters such as drag force, drillpipe weight, drillpipe bending stiffness, and borehole curvature may limit the extension of horizontal wells. Prediction of drag forces as well as bending moments play a great role in moving from vertical to horizontal drilling because it assists drilling engineers in designing well paths with appropriate radii of curvatures and selecting appropriate weights on bits to avoid any unexpected failure. Although extensive lab, field, and simulation work dealing with this problem has been performed, none has considered the operating limits of drillpipes running through curved sections. Therefore, the present paper focuses on the development of a mathematical model capable of establishing elastic/plastic working envelopes through careful examination of the induced stress state in drillpipes running through deviated wells.MATHEMATICAL MODEL
The problem of drillpipes running through curved sections was modeled as a multispan simply supported beam where two cases of boundary conditions at both ends of the curved sections were treated. Fixedfixed and pinnedpinned boundary conditions at the k.o.p and e.o.c of the curved sections were considered. Figure A.1 shows the free body diagram of an infinitesimal element of a running drillpipe through a curved section from which equilibrium equations
were obtained. The governing equation of an infinitesimal
M n1 11n11 11n1
element of a drillpipe running through curved boreholes is 1
given by:
12 n1
d2 M
ds2

dV
ds
d 2 w
P ds2 0
M n1 22 n1 M1 21n1
(1)
where M, V, P, and w denote respectively the bending moment, shear force, compressive normal force, and drillpipe radial deflection. In this model, the normal force was assumed to be constant throughout each span. Using the definition of the total curvature of the beam defined by
1
21 n1
Mn1 21n11 22 n1 M1 21n1
(4)

1 w d 2 w
n1 11n11 12 n1 M1 11n1
R R 2
ds 2
and assuming Eulers beam
(5)
bending yields to the following fourth order differential equation:
Equations (4) and (5) represent respectively the bending moment and slope at the beginning ( 0 , k.o.p) and
d4w 1 P d2w 1 P
ds4


R2 EI ds2
q
EI R
end of drillpipe ( 90
, e.o.c). The developed
(2)
mathematical model is capable of predicting the reaction
force at the contact points, compressive forces at the k.o.p,
Solution of Equation (2) is given by:
w( s ) As B C cosh s D sinh s 1 s2
2
(3)
Since the curved drillpipe was divided into many small elements, solution of Equation (2) was performed in multi steps using the transfer matrix method. The constants of integration were obtained from the boundary conditions at the free ends (k.o.p and e.o.c) of the curved drillpipe. Using the assigned values of the drillpipe deflection and slope, Equation (3) and its derivative lead to the following general form (see Appendix):
and stress distribution in each drillpipe span. The magnitude of compression in the top span (k.o.p) represents the force required to push the tubular through 90deg curved borehole sections.
SOLUTION METHOD
The transfer matrix method was used to solve the problem of multispan simply supported beams representing a typical drillpipe running through 90 degrees curved sections. A computer program was developed to estimate the compressive force, radial displacement, shear force, and induced stresses along the multispan curved drillpipe. Table 1 shows the main steps involved in solving this problem. It is worth noting that results of the preceding sections are used to solve for the adjacent sections.
Table 1: Flowchart of the solution procedure
1
2
3
4
5
6
7
8
Step
Process
, , , , , , a , b , , and
i i i i i i i i 1 1
calculate i1 and i1
Q1 V1 at first span
Calculate P1 using (A.39) for a given weight on bit P0
For i =2, 3, , n 1

Input desired load on the bit (P0), radius of curvature (R), size and modulus of elasticity of drillpipe, coefficient of Coulomb friction between drillpipe and borehole (Âµ).

Calculate body forces on each section.

Initiate an iterative loop to calculate the following parameters (Appendix):

For a given end condition (fixedfixed or simply supported boundary conditions at the k.o.p and e.o.c) use (4) to estimate bending moment and slope at first support of the first span.

Calculate all i and M i .

Compute all constants Ci , Di , Bi , and Ai

Calculate deflection, slope, moment, and shear force on any section.

Using known internal shear forces at the beginning and the end of each section, update normal reactions Qi at each support. For i =1
Qi Vi Vi1
Vi : shear force at the beginning of the span
Vi 1 : shear force at the end of the span

Calculate all new values of Pi using Eq. (A.39)

Compare new values of Pi obtained in step 9 with values of Pi estimated in steps 4 to 8 until convergence is achieved with 1 kN accuracy.
9
10
RESULTS AND DISCUSSION:
The effect of bending stiffness (EI ) , formation roughness, radius of curvature, and required back push force caused by drag force in the horizontal section on the running forces,
are studied. Different case studies were used to predict the running force and induced stresses of drillpipes running through medium to high curvature borehole sections (see Table 2). The mechanical properties of selected API grade
drillpipes are summarized in Table 3. In the model, the drillpipe was divided into multisections (spans) of 5 arc angle. The number of sections depends on the radius of curvature of curved sections. Two different boundary conditions at both ends of the drillpipe consisting of: 1) fixedfixed drillpipe ends and 2) pinnedpinned drillpipe ends were considered.
Table 2: Drillpipes dimensions used in this study
Drill drillpipes size
inches (mm)
Drill drillpipes Weight lbf/ft (N/m)
Grade
Radius of curvature (m)
Friction Coefficient
2 3/8 (60.3 )
4.85 (70.8)
X
10, 25, 50, 75, 100, 200
0.1, 0.2, 0.3, 0.4
3 Â½ (88.9 )
13.3 (194)
E & X
10, 25, 50, 75, 100, 200
0.1, 0.2, 0.3, 0.4
4 Â½ (114.3 )
16.6 (242)
E & X
10, 25, 50, 75, 100, 200
0.1, 0.2, 0.3, 0.4
5 (127.0 )
19.5 (285)
E
10, 25, 50, 75, 100, 200
0.1, 0.2, 0.3, 0.4
Table 3: Mechanical properties of API grade drillpipes
Grade
Ultimate tensile strength (MPa)
Yield strength (MPa)
Modulus of elasticity (GPa)
E75
689
517
207
X95
724
655
207
G105
793
724
207
S135
1000
931
207
The effect of formation roughness on the running force was studied for four different values of coefficient of friction, 0.1, 0.2, 0.3, and 0.4, using a drillpipe (bending stiffness EI of 3.87 x 105 Nm2 ) under fixedfixed boundary condition. A radius of curvature of 200 m and a load on bit of 20 kN are arbitrarily selected. The relationship between the running force and the inclination angle for different coefficients of friction is shown in Figure 1. It is obvious that as the coefficient of friction increases, the running force required to pushing the drillpipe increases. Variation in magnitude of the running force increases as the inclination angle increases and exhibit a nonlinear behavior. This means that the formation roughness or drag force is one of the key factors in affecting the running process and should be carefully considered in any drilling operation. Figure 2 shows the effect of formation roughness on drillpipe bending stress. It can be seen that the bending stress exhibits a reversed sign ranging from positive to negative peaks as the drillpipe moves forward.
This stress variation is observed for all coefficients of friction. It is worth mentioning that for low friction coefficient (Âµ < 0.3), the peak values of the bending stress is below the yield stress of the selected API drillpipes (see Table 2). However, the stress value exceeds the yield stress of the drillpipes for higher coefficients friction (Âµ > 0.3) and becomes severe starting from an inclination angle of 65 degrees. This variation of stress and increase in its magnitude can be the result of a combination of factors related to the dogleg severity along the well path and tubular buckling in curved sections where the stress variation takes sinusoidal shapes due to high friction force as indicated by Wu and Wold [15]. Thereby, when the drillpipe buckles in a sinusoidal shape the bending stress becomes compressive and tensile along the curved section. This may lead to the possibility of drillpipes fatigue failure in all cases.
100
90
80
70
60
50
40
30
20
10
0
Âµ = 0.1
0
10
20
30
40
50
60
70
80
90
Inclination angle, degrees
Âµ = 0.3
Âµ = 0.4
Âµ = 0.2
Bending stress, MPa
Running force, KN
Fig. 1: Effect of formation roughness on running force
2480
1980
1480
980
480
20
520
1020
1520
2020
2520
3020
Âµ = 0.1
Âµ = 0.2
Âµ = 0.3
Âµ = 0.4
0 10 20 30 40 50 60 70 80 90
Inclination angle, degrees
Fig. 2: Effect of formation roughness on bending stress
Another important parameter that has significant effects on the drillpipe stress state is the drillpipe weight/size (bending stiffness, EI). Four different drillpipes specific weights of 70.8, 194, 242, and 285 N/m under fixedfixed boundary conditions were considered. These drillpipe sizes were arbitrarily chosen giving a range of drillpipes bending stiffnesses of 6.71 x 104, 3.87 x 105, 8.26 x 105, and 1.23 x 106 Nm2. A radius of curvature of 50 m and a friction coefficient of 0.2 which provides low stress level as compared to the yield stress of the drillpipes were selected to study the effect of drillpipe bending stiffness on running forces. A load of 30 kN was applied at the bit. Figure 3 shows the variation of the running force as a function of the inclination angle. It can be seen that the vertical (running) force increases as the weight of the drillpipe increases implying that the higher the drillpipe bending stiffness, the higher the running force. The figure also shows that the running force is very low for all drillpipe sizes up to an inclination angle of 40 degrees beyond which the force
starts to level off to much higher values. This increase in the running force becomes more apparent as the drillpipe bending stiffness gets higher. However, for low drillpipe bending stiffness, the running force is very low and does not exhibit any variation with respect to the inclination angle. This bhavior was observed by Seibi [11]. This suggests the use of the cable model for this particular case, which neglects the effect of drillpipe bending stiffness. Further advancement of drillpipes along the curved section results in excessive induced bending stresses. Figure 4 shows the variation of the bending stress versus the inclination angle for different drillpipe sizes. It can be observed that drillpipes of high specific weights experience high alternating bending stresses, which exceed the yield stress of the drillpipes, for inclination angles higher than 30 degrees. This increase in stress level may be due to excessive compressive force at the end of curve which is much higher than the critical buckling force in curved [15] which may lead to sinusoidal or helical buckling modes;
Running force, KN
thereby, leading to cyclic stress variation. This indicates that there is a very high chance for fatigue failure to take
place; thereby, careful selection of drillpipes before undergoing any activity must be performed.
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
W = 70.8 N/m W = 194 N/m W = 242 N/m
W = 285 N/m
0 10 20 30 40 50 60 70 80 90
Inclination angle, degrees
Bending stress, MPa
Fig. 3: Effect of bending stiffness on running force
800000
600000
400000
200000
0
200000
400000
600000
800000
W = 70.8 N/m W = 194 N/m W = 242 N/m W = 285 N/m
0 10 20 30 40 50 60 70 80 90
Inclination angle, degrees
Fig. 4: Effect of bending stiffness on bending stress
The effect of the radius of curvature of curved hole sections on the running force is studied by considering three radii of curvatures i.e. 50, 100, and 200 m. The developed model was used to calculate the running forces for a drillpipe running through the buildup section. The drillpipes used in this case has respectively an outer and inner diameter of 0.0603 and 0.0507 m and a specific weight of 70.8 N/m under fixedfixed boundary conditions. The other field parameters such as friction coefficient, mud density, and load on bit were given constant values of 0.3, 198 Kg/m3, and 10 kN, respectively. Figure 5 shows the instantaneous vertical (running) force required to push the drillpipes
through high to medium curvature wellbores (50, 100, and 200 m). It can be observed that the running force at the
k.o.p increases as the radius of curvature decreases. A substantial increase in the vertical force is observed for a radius of curvature of 50 m as compared to the other two radii of curvatures. The running force is almost the same for all radii of curvatures up to an inclination angle of 40 degrees. For instance, for the case of a radius of curvature of 50 m, the running force starts to increase gradually between 40 50 degrees and exhibits a sharp increase beyond the 50 degrees inclination angle. This increase in magnitude is mainly related to the increase in drag forces at
150
200
250
R = 50 m
R = 100 m R = 200 m
Running force, KN
contact points indicating that the total force required to push the drillpipe (OD = 0.0603 m) through curved wells is highest for holes with shorter radii of curvatures (R < 50 m) and lowest for higher radii of curvatures (R > 200 m). This phenomenon is mainly attributed to wellbore curvature and drillpipe bending stiffness effects which become prominent for short curvature wellbores (severe doglegs). However, the small increase in the running force for both radii of
curvatures of 100 and 200 m within the first 60 degrees is mainly attributed to the drillpipe weight which is acting in the same direction as the running force. Therefore, the weight in these two cases helps the drillpipe advance further downward along the curved hole. However, for inclination angles exceeding 60 degrees, a slight increase in the running force is observed for both radii of curvatures as a result of further resistance caused by drag forces.
300
50
0
0
10
20
30
40
50
60
70
80
90
Inclination angle, degrees
100
Fig. 5: Effect of radius of curvature on running forces
Bending stress, MPa
The induced bending stress in the drillpipe running through the three radii of curvatures is shown in Figure 6. The figure shows that the drillpipe experiences an alternating high bending stress as it advances through short radii of curvatures (R < 50 m). This phenomenon is attributed to the high compressive force at the end of curve cuasing pipe buckling as well as pipe bending stiffness which requires a much high stress level to bend the drillpipe and follow the curved path. This observation was made by Seibi [12] where pipe bending becomes more apparent for high
curvature wellbores. Similar observations can be made for the radius of curvature of 200 m but with less severity. The high stress values may lead to unexpected drillpipe failure. It is worth mentioning that the compressive force in the drillpipe was observed to increase with respect to the inclination angle as it advances from the build up point to the end of curve. This phenomenon was observed by Dareing [4] for a pullout operation where the pullout force increases with respect to the inclination angle.
100000
80000
60000
40000
20000
0
20000 0
40000
60000
80000
R = 50 m R = 100 m
R = 200 m
Inclination angle, degrees
90
80
70
60
50
40
30
20
10
Fig. 6: Effect of radius of curvature on bending stress
Working Envelope for Drillpipes under Running Operations
This section aims at providing drilling engineers with proper guidelines that predetermine the possibility of running drillpipes through curved sections without undergoing any plastic deformation. These guidelines are presented in the form of a working envelope that was developed based on the induced stress levels in drillpipes. In other words, the induced normal stress caused by drillpipes bending and axial loads was compared against the yield stress of various API drillpipes. The criterion adopted in this study was based on stress values higher than the yield stress indicating that drillpipes undergo plastic deformation which may result in drillpipes buckling
or sticking to the formation. Based on this hypothesis, a working envelope was developed for various field cases to aid drilling engineers in i) selecting suitable drillpipes as per field operating conditions in order to avoid any unexpected failure and ii) making sound decisions while developing well paths before undergoing any drilling activities.
The aforementioned results related to the effect of drillpipes bending stiffness, radius of curvature, wellbore curvature, coefficient of friction, and boundary conditions were used to develop the failure/operating envelop. The parameters used in this study are summarized in Table 4.
Table 4 Parametric matrix used in this study
Wob kN
Radius of curvature (m)
Drillpipe weight (N/m)
Friction coefficient
10
10
78.8
0.1
15
25
194
0.2
20
50
242
0.3
25
75
285
0.4
30
100
200
Table 5 shows the obtained results for various field scenarios where shaded areas indicate that drillpipes can safely run through the curved holes or particular field cases without undergoing any plastic deformation. Whereas, the unshaded areas indicate that drillpipe failure may take place since the induced stresses on the drillpipes exceed the yield stress. For instance, a drillpipe with a specific weight of 70.8 N/m can be ran without any problem into a well having a radius of curvature of 100 m and a load on bit of 10 kN with a coefficient of friction of 0.4; while a drillpipe with a specific gravity of 194 N/m cannot run safely. The table also shows that drillpipes of 242 N//m and 285 N/m cannot be safely run into curved wells at all selected field conditions when a 10 kN load on bit is applied. Similar observations can be made in the case of pinnedpinned boundary conditions.
CONCLUSIONS
The major steps in developing a twodimensional mathematical model based on Dareing and Ahlers model were described in details. Various field parameters that affect the drillpipes in buildup sections, such as coefficient of friction, radius of curvature, and bending stiffness, were incorporated in the developed model. Calculations show that the soft string model used to determine the running force is an approximation method provided that there are no severe local doglegs within the buildup section. It was also found that the running force increases as the inclination angle, coefficient of friction, drillpipe bending stiffness, and radius of curvature increase along the build up section. Moreover, a useful operating envelope for various drillpipe sizes and field conditions was developed. This envelope is a useful tool to field engineers during well planning phase.
REFERENCES

Birades M., 1988, Static and Dynamic ThreeDimensional BHA Computer Models, SPE Paper No. 15466, SPE Drilling Engineering, Vol. 3, 2, pp: 160 166.

Dareing D.W. and Ghodwani S., 1970, Directional Drilling and Circular Arc Deflection of Long Beam, SPE Paper No. 3097, Meeting of the Society of Petroleum Engineers of AIME, 4 7 Oct., Houston, Texas.

Dareing D.W., 1971, Drilling Directional Holes Having Constant Curvature, SPE Paper No. 3508, Meeting of the Society of Petroleum Engineers of AIME, 4 7 Oct., Houston, Texas.

Dareing D.W. and Ahlers C.A, 1991, "Tubular Bending and PullOut Forces in HighCurvature Wellbores, Journal of Energy Resources Technology, Vol.113, pp.133139.

Haduch G.A, Procter R.L. and Samuels D.A., 1994, "Solution of Common Stuck Drillpipes Problems through the Adaptation of Torque/Drag Calculations, SPE Paper No. 27490, SPE/IADC Drilling Conference 15 18 February, Dallas, Texas.

Ho HS., 1988, An Improved Modeling Program for Computing the Torque and Drag in Directional and Deep Wells, SPE Paper No. 18047, SPE Annual Technical Conference & Exhibition, 2 5 October, Houston, Texas.

Johancsick C.A., Friesen D. B. and Dawson R., 1984, Torque and Drag in Directional WellsPredictions and Measurement, SPE Paper No. 11380, J. of Petroleum Technology, Vol.36, 6, pp: 987 992.

Paslay P.R. and Cernocky E.P., 1991, Bending Stress Magnification in Constant Curvature Doglegs With Impact on Drillpipes and Drillpipes, SPE Paper No. 22547, SPE Annual Technical Conference & Exhibition, 6 9 October, Dallas, Texas.

Seibi A.C., 2000, Running Force Measurement in High Curvature Wellbores," Journal of Experimental Techniques, March/April, pp. 3135.

Seibi A.C. and AlHashmi M.A., 1998, Effects of Drillpipes/Formation Interaction on The Running Force in HighCurvature Wellbores, ASME/JSME Joint Pressure Vessels and Piping Conference, San Diego, CA, July 26 30, PVP Vol. 375, pp. 8994.

Seibi, A.C. and AlShabibi, A.M., 1998, Drillpipes Bending and Running Forces in Medium to High Curvature Wells Using Finite Element Analysis," Journal of Energy Resources Technology, Vol.120, pp. 263267.

Seibi, A., 2001, Running Force in Medium to High Curvature Wellbores: An Experimental Study and Numerical Simulation of Laboratory and Field Cases, Journal of Energy Resources Technology, Vol. 123, pp: 133 137.

Martinez, A., Miska, S., Kuru, E., Sorem, J., 2000, Experimental Evaluation of the Lateral Contact Force in
APPENDIX
Governing Equations of Drillpipes Running Through Curved Sections
The governing equations are derived by considering the equilibrium of an infinitesimal element of a drilldrillpipes running through curved boreholes. Figure A.1 shows a free body diagram of a differential element under compression. The drilldrillpipes was modeled as a multispan simply supported beam with multiple supports representing contact between the drilldrillpipes and borehole walls starting from the kickoff point till the end of curve.
R
P + dP
Horizontal Wells, Journal of Energy Resources Technology, Vol. 122, pp: 123 128.

Kuru, E., Martinez, A., Miska, S., Qiu, W., 2000, The Buckling Behavior of Pipes and Its Influence on the Axial Force Transfer in Directional Wells, Journal of Energy Resources Technology, Vol. 122, pp: 129 135.

Wu, J., Juvkam Wold, H. C., The Effect of Wellbore Curvature on Buckling and Lockup, J. of Energy Resources Technology, Vol. 117, pp: 214 218.
d
qds
V
M P
M + dM
a
V + dV
w + dw
w
n t
Fig. A.1: Free body diagram of differential element under compression in terms of polar coordinates
Using the equilibrium equation in the normal direction and neglecting higher order terms leads to the following first order of differential equation:
dV q P
(A.1)
R
ds
where, q is a distributed force per unit length.
Using the equilibrium equation of the moment about point a and neglecting the higher order terms, a first order differential equation is given by:
dM V P dw 0
ds ds
(A.2)
Differentiating equation (A.2) with respect to s leads to the following equation:
d2M ds2


dV
ds
d 2w
P ds2 0
(A.3)
Note that the dependent variable, w , is the radial displacement of the beam where positive displacement is taken in the inward
direction. Considering the total curvature of the beam in terms of polar coordinates defined by assuming Euler Bending, the bending moment takes the form:
1 1 w
R R2

d 2w ds2
and
1 w d 2w
M EI R R2
ds2
(A.4)
Differentiating Equation (A.4) with respect to s twice becomes,
d 2M ds2
EI 1
R2
d 2w ds2
d 4w

ds4
(A.5)
Combining of Equations (A.1), (A.3), and (A.5) results in a fourth order differential equation in terms of radial displacement given by:
d 4w 2 d 2w
where,
ds4 ds2
(A.6)
2 P
1
and
1
q P Rq P
EI
R2
EI
REI
R
The compressive force is assumed to be constant over a given section.
Solution Method
Solution to Equation (A.6) gives the deviation of the radial displacement of a drillpipes from the centerline of the target well path of a typical borehole and takes the following form:
ws 1 s2 A s B C cos( s) D sin( s)
(A.7)
where,
Rq P
RP EI

i i i i i i i
R
Using the boundary conditions at the beginning of the ith section (where ith support is located, s 0 and w(0) wi ) in (A.7) gives:
w(0) Bi Ci wi
(A.8)
Similarly, at the other end of the ith section (where the ( i 1)th support is located), s li , and these values in equation (A.7) gives:
w(li ) wi 1 . Substituting
w(l ) 1 l 2 A l B C cos( l ) D sin( l ) w
(A.9)
i 2 i i i i i i
i
i
Substituting Bi from (A.8) into (A.9) and solving for Ai yields:
i i i
i i i 1
A (
1 l ) C
(cos(ili ) 1) D sin(ili )
(A.10)
l
where, i is defined as
i
i
wi 1 wi
li
i 2 i i
i li
The rate of change of w with respect to ' s ' is given by:
dw
ds is Ai Cii sin(is) Dii cos(is)
(A.11)
Using equation (A.11), the slope at the beginning of ith section (at s 0 ) can be calculated as:
i Ai Dii
(A.12)
The bending moment at the beginning of the ith section is given by:
1 wi
d 2wi
Mi EI R R2
Differentiating Equation (A.7) twice with respect to ' s ' and substituting s = 0 for ith point gives:
ds2
(A.13)
d w
2
2
i i Ci i ds2
Substituting Equation (A.8) and Equation (A.14) into Equation (A.13) results in:
(A.14)
M EI 1 Bi Ci C 2
(A.15)
i R R2 i i i
Substituting
Ai and Bi
into (A.12) and (A.15), respectively, results in a simplified form for the slope and bending moment at
the beginning of ith section:
1 1 cos( l ) sin( l )
( i l ) i i Ci i i i Di
(A.16)
i 2 i i li
li
R
M EI 1 wi
C 2EI
(A.17)
Solving equations (A.16) and (A.17) for the constants Ci and Di
i
gives:
R2
i i i
Ci i
i a i
(A.18)
where,
(1 cos( l ))
Di Mi
sin( l )
i i
i
1 l
i
i
i i
li
(A.19)
2EI 0
i
and
1
i 2
i l i
a i
1 w
(A.20)
EI i i
R R2
Similarly, at the end of the ith section when s li , the slope at the ( i 1)th support is:
i1 ili Ai Cii sin(ili ) Dii cos(ili )
(A.21)
Substituting constant
Ai into (A.21) and simplifying, Equation (A.21) becomes:
1 l
(1 cos(ili )) sin(
i1
2 i i
i i
li
ili )Ci
(A.22)

cos( l ) sin(ili ) D
i i i
i
li
Equation (A.22) represents the slope at the end of ith section. Similarly, the bending moment at the end of ith section, when s li (at ( i 1)th support) can be expressed as:
M EI 1 wi1 R C 2EI cos( l
) D 2EI sin( l )
(A.23)
R
i 1
R
i i i
i i i i i i
Expressing equations (A.22) and (A.23) in a matrix form leads to:
i 1
Ci b
(A.24)
M i D i
where,
i 1
i
(1 cos(i li )) sin( l )
cos( l ) sin(i li )
l i i i
i i i l
(A.25)
i i i
EI 2 cos( l ) EI 2 sin( l )
and
i i i
i i i
2
1 l
b i
i i i
(A.26)
EI 1 wi 1 R
R R i
Ci
Substituting constants D from Equation (A.18) into Equation (A.24) gives:
i
i 1
i
i i
a
b i
(A.27)
Mi 1
Mi i
As can be seen from equation (A.27) , the slope and bending moment at the end of a span ( i 1) can be expressed in terms of the slope and bending moment at the beginning of the span.
The generalized form of equation (A.27) can be written as:
i 1
1
a b
(A.28)
1
where,
M
i 1
i i i M
i i i i i
i
1
,
i 1,2,…., n 1
(A.29)
M i M i
i 1
and where i
and i
are the matrices and vectors of constants of order 2×2 and 1×2, respectively, which are updated during
each load increment where the initial values are given by
1 0
and
0
. Rewriting Equation (A.28) as:
1 0 1
1
i 1
1
0
(A.30)
where,
M
i 1
i1 M
1
i1
i1
i i i
(A.31)
and
i1 i i

a b
i i i
(A.32)
If the multispan is composed of n sections, the number of supports along the multispan is n 1. Hence equation (A.29) will extend from 1 to n 1 and Equation (A.30) becomes:
n1
1
= 11
12
1
11
(A.33)
M
n1 M
n1
M
n1
1
21 22 n1
1 21n1
Equation (A.33) relates the slope and bending moment at the two ends of the multispan.
If 1 and n1
are given as end
constraints, the bending moments at the two ends of the multi span can be obtained using following equations:
n1 11 n11 11 n1
M1
12
n1
(A.34)
Mn1 21 n11 22 n1 M1 21 n1
(A.35)
Equation (A.34) represents the bending moment at the beginning of the tubular ( 0 ) and Equation (A.35) represents the
bending moment at the end of the tubular ( 90 ) and vice versa. If
M1 and M n1 are given as end constraints, the slopes at the two ends of the tubular are given by:
M n1 22 n1 M1 21 n1
1
21 n1
(A.36)
n1 11 n11 12 n1 M1 11 n1
(A.37)
Equations (A.36) and (A.37) represent the slope at the beginning of tubular ( 0 ) and at the end of tubular ( 90 ), respectively.
Compression over a given span
The assumption of a constant compressive force throughout a given section simplifies the undertaken problem while obtaining realistic engineering results. Figure A.2 shows the forces on a given span. The reaction force, Qi , and the compressive force, Pi , at the ith support were determined from the equilibrium conditions.
i
i
Qi ÂµQi
Pi t
i+1
n
ith span
BFWi li
i ith support
Pi1
Fig. A.2: Force diagram of a given span
Summation of forces in the normal direction gives the reaction and compressive forces, respectively:
Q Pi 1 sin(i ) BFWili cos(i i )
i cos( ) sin( )
(A.38)
i i
Summation forces in the tangential direction gives:
Pi (Pi 1 Qi ) cosi Qi sini BFWili sini i
where, P0 is the load applied at the bit. The body force is estimated by the following expression:
(A.39)
q BFW
cos
i
(A.40)
i i i
2
Table 5 Parameters for safe running operation for drillpipes with different specific weights and at various field conditions
Weight (N/m) lluP kcBB N 
70.8 
194 
242 
285 

10000 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 

0.1 
0.1 
0.1 
0.1 

0.2 
0.2 
0.2 
0.2 

0.3 
0.3 
0.3 
0.3 

0.4 
0.4 
0.4 
0.4 

15000 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 

0.1 
0.1 
0.1 
0.1 

0.2 
0.2 
0.2 
0.2 

0.3 
0.3 
0.3 
0.3 

0.4 
0.4 
0.4 
0.4 

20000 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 

0.1 
0.1 
0.1 
0.1 

0.2 
0.2 
0.2 
0.2 

0.3 
0.3 
0.3 
0.3 

0.4 
0.4 
0.4 
0.4 

25000 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 

0.1 
0.1 
0.1 
0.1 

0.2 
0.2 
0.2 
0.2 

0.3 
0.3 
0.3 
0.3 

0.4 
0.4 
0.4 
0.4 

30000 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 
R Âµ 
10 
25 
50 
75 
100 
200 

0.1 
0.1 
0.1 
0.1 

0.2 
0.2 
0.2 
0.2 

0.3 
0.3 
0.3 
0.3 

0.4 
0.4 
0.4 
0.4 
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