Drillstring vibration analysis using differential quadrature method

Journal of Petroleum Science and Engineering 70 (2010) 235–242

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Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / p e t r o l

Research paper

Drillstring vibration analysis using differential quadrature method H. Hakimi, S. Moradi ⁎ Mechanical Engineering Department, Shahid Chamran University, Ahvaz, 61355, Iran

a r t i c l e

i n f o

Article history: Received 8 January 2009 Accepted 20 November 2009 Keywords: Differential quadrature method Drillstring dynamics Nonlinear geometry Vibrations

a b s t r a c t In this study, the differential quadrature method (DQM) is applied to analyze the drillstring vibrations in a near vertical hole. First, a nonlinear static analysis is performed to find the effective length of the string where it rests on the borehole wall. The exact form of the beam curvature is used to formulate the string. To model the contact between different parts of the drillstring and the borehole wall, the formation is modeled by a series of springs placed through its length. Then the DQM is applied to the nonlinear differential equations of drillstring sections and those defining the edge and interface boundary conditions. The Newton–Raphson algorithm is used to solve the system of nonlinear equations. Next a free vibration analysis is carried out to determine the natural frequencies of the drillstring. Using effective length derived from static analysis, free vibration analysis is performed to find the lateral natural frequencies of the drillstring, while the full length of the string is used to compute its axial and torsional natural frequencies. The numerical results obtained from a series of case studies confirm the efficiency and accuracy of the method in dealing with drillstring vibration problems. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Drilling industry has a common problem of severe drillstring vibrations. The drillstring consists of several drill pipes, drill collars, stabilizers and connections (crossover sub). They are under some heavy and complex dynamic loadings, caused by different sources such as bit and drillstring interactions with the formations, torque exerted by rotary table or top drive, buckling and misalignment. By producing different state of stresses, these loads may result in excess vibrations and lead to failure of the drilling tools. Moreover, rotation of the rotary table or top drive on the surface may transform into a turbulent movement in downhole. Three forms of vibrations have been identified for drillstring are axial, torsional and lateral vibrations (Spanos et al., 2003) (Fig. 1). If drillstring excites near one of its natural frequencies, it starts to absorb the energy. This energy boosts the amplitude of string vibration, increases bending and impacts with the borehole, and leads to early fatigue of tools and reduction of bit life. Moreover, impacts with the borehole wall tend to form the over gauge hole or produce problems with directional control of the well as well as increasing the surface torque (Aldred and Sheppard, 1992). To solve the problem, one should analyze the drillstring to find its natural frequencies and modal shapes, then by changing the drilling parameters, avoids the resonance of the drillstring.

⁎ Corresponding author. Tel./fax: +98 611 3330408. E-mail address: [email protected] (S. Moradi). 0920-4105/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2009.11.016

Investigation into the problem of drillstring vibrations dates back to the early sixties by Baily and Finnie (1960a,b). They used a graphical approach to compute the natural frequencies of the string. Since then, there have been increasing interests in modeling the dynamic behavior of drillstring by using both analytical and numerical techniques. The analytical approach has been the basis for the early analyses (Aarrestand et al., 1986; Jansen, 1991). Yigit and Christoforou (1996, 1998) modeled the drillstring based on the assumed mode method. Their models account for the coupling between axial and transverse vibrations (Yigit and Christoforou, 1996), and between torsional and transverse vibrations (Yigit and Christoforou, 1998). Navarro-Lopez and Cortes (2007) used a lumped parameter model to analyze torsional vibration and stick–slip phenomenon in a vertical drillstring. The complexity of the problem and the development of fast new computers have opened the door to versatile numerical techniques. Several researchers have employed effective numerical methods such as finite difference method (FDM) and finite element method (FEM) to treat the problem. Khan (1986) used the FDM to solve the axial and torsional vibrations of drillstring, while neglecting the added mass and damping effects. Rey (1983) derived the differential equation of motion for lateral vibration of bottom hole assembly (BHA) between the bit and stabilizers, and solved it by FDM. He examined the effects of torque and weight on the bit and stabilizers, damping of mud and deviation of the BHA from vertical direction. Shyu (1989) studied the coupling between axial and lateral vibrations and the whirling of drillstring using FDM. Baird et al. (1985) used FEM to find transient response of a rotating BHA under the interaction of the formation. Burgess et al. (1987) used FEM to model the lateral vibration of

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intermediate point xi can be approximated by the weighted linear sum of the function values as N dn f ðxi Þ ðnÞ = ∑ cik f ðxk Þ n dx k=1

i = 1; …; N

ð1Þ (n)

In Eq. (1), the domain is divided into N discrete points and cik are the weighting coefficients of the nth derivative where n ≤ N − 1. To insure the integrity of the method, two factors deserve particular attention. These are the values of the weighting coefficients, and the position of discrete variables. For proper weighting coefficients, f(x) must be approximated by a test function. The weighting coefficients may be determined explicitly once and for all discrete points, irrespective of the position and number of sampling points as proposed by Shu and Richards (1992). They used Lagrange interpolating function as the test function and derived the following recurrence formulae for the weighting coefficients ð1Þ

cij = ðkÞ cij

=

Fig. 1. Three forms of drillstring vibration. ðmÞ

cii drillstring. They performed a static nonlinear analysis to find the location where the string above the last stabilizer touch the well bore wall. Then, using the drillstring length from bit to this point, they evaluated the response of the string by a harmonic analysis. Apostal et al. (1990) developed a three dimensional finite element model to investigate the harmonic response of BHA. Damping in the form of proportional, structural and viscous were included in their model. Khulief and Al-Naser (2005) used the Lagrangian approach to formulate the finite element model of a rotating vertical drillstring. Bellman and Casti (1971) and Bellman et al. (1972) introduced the differential quadrature method (DQM) in the early seventies. Since then, several researchers have applied the method to solve varieties of problems in different areas of science and technology. The method has been shown to be a powerful contender in solving systems of differential equations. Therefore, it has become an alternative to the existing numerical methods such as FDM and FEM. One can mention two advantages when comparing the DQM with FEM: (1) simpler algorithms and therefore easier implementation; (2) more efficient solution by requiring less discretizing points. Due to these advantages, DQM has gained increasing popularity in solving different kinds of engineering problems in recent years (Bert and Malik, 1996; Moradi and Taheri, 1999). To the authors' best knowledge, the validity of DQM in treating the drillstring vibrations has not yet been explored. This work is performed to validate the applicability of the DQM in the analysis of drillstring vibrations. In order to model the lateral vibrations, a static nonlinear problem is solved to find the effective drillstring length. The model adopted here consists of the coupled large deformation equations of a beam. Then, axial, torsional and lateral natural frequencies and their corresponding mode shapes are computed by modal analysis. 2. Differential quadrature method The DQM was introduced by Bellman and Casti (1971) for solving linear or nonlinear differential equations. It states that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of function values at some intermediate points in the domain of that variable. Consider a function f = f(x) on the domain a ≤ x ≤ b. Based on the preceding definition, the nth-order differential of the function f at an

∏ðxi Þ ðxi −xj Þ⋅∏ðxj Þ 2

i; j = 1; …; N and j≠i 3

ðk−1Þ cij ðk−1Þ ð1Þ k4cii ⋅cij − x −x 5 i j N

ðmÞ

= − ∑ cij

2 ≤ k ≤ N−1

ð2Þ

m = 1; …; N−1

j=1 j≠i

where N

∏ðxi Þ = ∏

ðxi −xj Þ

ð3Þ

j=1 j≠i

The above relations are not affected by the number of sampling points and thus, significantly reduce the computational effort. Furthermore, the spacing of sampling points can significantly affect the accuracy of the solution. An effective method is to select the zeros of shifted orthogonal polynomials. A simple and yet effective choice is the roots of shifted Chebyshev polynomials in the [0,1] domain, presented as xi =

  1 2i−1 1− cos π 2 2N

ð4Þ

3. Governing equations The geometry of the drillstring is shown in Fig. 2. The string consists of several tubular beams where each represents a drill pipe, drill collar, stabilizer, crossover sub or a bit. The well is deviated from vertical direction by a constant but small angle ϕ. In order to find the axial and torsional natural frequencies, the corresponding wave equations of the beam are solved using the entire length of the string. However, for lateral natural frequencies, the effective length should be used in the modal analysis. In a deviated hole the string tends to contact the borehole wall. Therefore, to perform modal analysis in transverse direction, it is required to do a static analysis to find the location where the string above the last stabilizer rests against the well bore wall. The effective length of the string starts from the bit and ends at this point. 3.1. Nonlinear static analysis Part of the drillstring has large deformation when it rests on the borehole wall. To formulate the static problem, each of the drillstring regions is considered as a separate beam. A region is part of the string

H. Hakimi, S. Moradi / Journal of Petroleum Science and Engineering 70 (2010) 235–242

237

Using the kinematical expressions and equilibrium equations developed before, one can write the nonlinear differential equations describing each region in terms of the displacement components, as Ek Ak

d2 uk dwk d2 wk + 2 dz ⋅ dz2 dz

! = −pk

k = 1; ⋯; m

ð9aÞ

0

!3   2 1 2 4 2 3 d wk dwk d wk d wk 3 d wk ⋅ 1−4 dwk B C 9 B C dz dz2 B C dz ⋅ dz2 ⋅ dz3 dz4 Ek Ik B − − C 5 7 3 B C  2   2     2  @ A dwk dwk dwk 2 2 2 1+ 1+ 1+ dz dz dz " !    2 #  2 2 d uk dwk d wk dwk duk 1 dwk 2 d wk = fk + + ⋅ + −Ek Ak 2 dz dz ⋅ dz2 dz dz dz2 dz2 k = 1; ⋯; m

ð9bÞ The boundary conditions consist of the axial and transverse, as well as the continuity conditions. The axial and transverse boundary conditions at the ends indicate that

Fig. 2. Typical drillstring configuration.

that has the same geometric and material properties. Based on the Von Karman kinematic approach, the kinematical relationship for each beam can be expressed by k

ε = εk0 =

εk0 −xκk ðzÞ

k = 1; …; m

  duk 1 dwk 2 + 2 dz dz

ð5Þ

where u and w are the axial displacement and lateral deflection of the beam, εk is the strain at any depth z in region k, εk0 and κ are axial strain and flexural curvature of the middle surface of regions, respectively. The exact nonlinear curvature of the beam can be written as 2

d wk k dz2 κ =   2 3 2 k 1 + dw dz

ð6Þ

ð10Þ

at the top ðz = LÞ P = Trig ⋅ cos ϕ; w = 0; wz = 0

ð11Þ

where L is the length of drillstring and Trig is the hook load. The continuity conditions along the interfaces of the regions must be satisfied for the axial forces and displacements, as well as transverse deflections, slopes, moments and shear forces ui = ui + 1 ; Pi = Pi + 1 wi = wi + 1 ; wi;z = wi + 1;z ; Qi = Qi

Pk = Ek Ak εk0

ð7Þ

Mk = Ek Ik κk

P and M are the axial force and bending moment, and E, A and I are the modulus of elasticity, cross-sectional area and area moment of inertia for each region, respectively. Using the classical beam theory in the presence of axial load, the equilibrium equations for each region can be written as

ð8a; bÞ k = 1; ⋯; m

where pk and fk are axial and transverse loads per unit length, respectively.

+ 1;

Mi = Mi +

1

ð12Þ

where Q is the shear force. Moreover, the transverse deflection is zero at stabilizers. In order to model the contact between different parts of drillstring and formation, the borehole wall is modeled by a series of springs (Spanos and Chevallier, 2000) as shown in Fig. 3. The annular space between the outer diameter of the string and the borehole wall is represented by gap = w−½v−tolk 

ð13Þ

tolk is the radial difference between the string and the borehole wall for kth region, and v is the deformation of wall defined by v=−

The load/moment–strain/curvature relations are defined by

dPk = −pk dz   dwk d Pk ⋅ 2 d Mk dz = fk − dz dz2

at the bit ðz = 0Þ u = 0; w = 0; M = 0

Fn K

ð14Þ

Fn is the normal contact force generated as the contact happens. The formation is modeled by a series of springs, which its stiffness is K (Spanos and Chevallier, 2000). Although the stiffness of springs can be varied with respect to the stiffness and depth of formations, in this study their values consider to be constant. If gapN 0, the two surfaces do not contact each other and Fn = 0. However, if gap b 0, the string penetrates into the borehole wall and the normal contact force is not zero. As the two surfaces contact each other, the gap becomes zero. Thus, the following relation can be defined to represent these two conditions ð15Þ

gap⋅Fn = 0

In order to satisfy the foregoing relation, a normal constraint function is defined using the penalty function approach (Bhatti, 2000) by ψðgapk ; Fnk Þ =

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   gapk + Fnk gapk −Fnk 2 − +μ=0 2 2

k = 1; …; m ð16Þ

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H. Hakimi, S. Moradi / Journal of Petroleum Science and Engineering 70 (2010) 235–242

The boundary conditions can be expressed in differential quadrature form by N1

ð2Þ

u11 = 0; w11 = 0; ∑ C1j1 wj1 = 0

at z = 0 :

ð19Þ

j=1

1 Nm ð1Þ 1 at z = L : ∑C u + 2 lm j = 1 Nm jm jm 2lm Nm

∑

wNm m = 0;

j=1

Nm

!2

ð1Þ

∑ CN

j=1

ð1Þ CNm jm wjm

m jm

wjm

=

Trig cosϕ E m Am

ð20Þ

=0

The axial continuity boundary conditions become uNi i = u1ði + 1Þ

!2    Ni Eði + 1Þ Aði + 1Þ 1 Ni ð1Þ 1 ð1Þ ∑ CNi ji uji + 2 ∑ CNi ji wji − li j = 1 Ei Ai 2li j = 1 2 Nði + 1Þ Nði + 1Þ 1 1 ð1Þ ð1Þ ×4 ∑ C u + 2 ∑ C wjði lði + 1Þ j = 1 1jði + 1Þ jði + 1Þ 2lði + 1Þ j = 1 1jði + 1Þ

!2 3 5=0 + 1Þ

ð21Þ and the transverse boundary conditions are

Fig. 3. Well borehole contact with drillstring.

wNi i = w1ði

+ 1Þ N

where μ is a small positive number, which its value depends on the requested numerical accuracies.

ði + 1Þ 1 Ni ð1Þ 1 ð1Þ ∑C w = ∑ C li j = 1 Ni ji ji lði + 1Þ j = 1 1jði

" 1 Ni ð2Þ ∑ CNi ji wji : 1 + li j = 1

3.2. Differential quadrature application

f

DQM can be applied to the system of nonlinear differential equations, and their corresponding boundary and contact equations, using the nondimensionalized variable Z =z /L. Applying DQM to Eqs. (9a) and (9b) results in

1 Nk ð2Þ 1 ∑C u + 2 lk j = 1 ijk jk lk

!

Nk

∑

j=1

!

Nk

ð2Þ Cijk wjk

ð1Þ

∑ Cijk wjk

j=1

p l =− kk E k Ak

ð17aÞ

⋅

∑

j=1

ð4Þ Cijk wjk Nk

3I − k7 Ak lk 1 − 2 lk −

3 2l3k

∑

j=1 Nk

Nk 9I ð3Þ V − k5 ∑ Cijk wjk Ak lk j = 1 !3 "

V

!

ð2Þ

∑ Cijk wjk

j=1

ð1Þ

∑ Cijk wjk

j=1

!

Nk

ð1Þ

!

Nk

∑

j=1 ð1Þ

ð2Þ Cijk wjk

Nk

! ð1Þ

∑ Cijk wjk V

j=1

j=1

1 − 2 lk !2

∑ Cijk wjk

j=1

=

Nk

! ð1Þ

∑ Cijk ujk

j=1

Nk

1 ð1Þ ∑C w lk j = 1 ijk jk

f

Nði

1 lði

ð2Þ

∑ Cijk wjk

j=1

+ 1Þ

ð3Þ

j=1

+ 1Þ

 Iði

ð18Þ

lði

l2ði

Nði

−

Nði

+ 1Þ

∑

+ 1Þ j = 1

ð1Þ C1jði + 1Þ wjði + 1Þ

!2 3−3=2 5

g

=0

+ 1Þ

li lði +

j=1 Nði

1 + 1Þ

j=1

 Eði + − Ei

+ 1Þ

ð1Þ

1 lði

Nði

+ 1Þ

1Þ

 Iði + Ii

ð1Þ

∑ C1jði

j=1

+ 1Þ

1Þ

 lði

+ 1Þ wjði +

!2

li + 1Þ

!2 3−3=2 5 1Þ

g

1Þ

+ 1Þ wjði

∑ C1jði

j=1

ð1Þ

∑ CNi ji wji

!2

! ð1Þ

!

Ni

3 l3i

!2 #−5=2

41 +

∑ C1jði

+ 1Þ

lði

!2 #−3=2

1Þ wjði + 1Þ :



Ii 3

⋅41 +

fk lk Ek Ak k

+ 1Þ

Ei

2

!2 #−1

Ei

!   Iði + 1Þ li Ii lði + 1Þ

2

∑ C1jði +

+ 1Þ

× −

(r)

+ 1Þ

1

+

1 Ni ð1Þ ∑C w li j = 1 Ni ji ji

1+

j=1

(

!

where Cijk are the weighting coefficients for the rth order derivative along the non-dimensionalized Z axis, ujk and wjk are the axial and transverse displacements of the jth point in the kth region, respectively. V is defined as

V = 1+

1 Ni ð1Þ ∑C w li j = 1 Ni ji ji

!2 "

ð2Þ

 Eði −

!2 #

ð17bÞ

Nk

⋅

5 2

i = 3; …; Nk −2; k = 1; …; m

"

ð2Þ C1jði + 1Þ wjði + 1Þ :41

" 1 Ni ð3Þ ∑ CNi ji wji : 1 + li j = 1

1−4⋅ ∑ Cijk wjk !

Nk

ð2Þ

∑ Cijk ujk

j=1

Nk

!

Nk

7 2

∑

Ni

3 2

ð2Þ Cijk wjk

+ 1Þ

j=1

+ 1Þ

 Eði −

2

⋅ ∑ CNi ji wji

!

Nk

lði

!2 #−3=2

ð22aÞ

i = 2; …; Nk −1; k = 1; …; m

Ik Ak l3k

Nði

1

1 Ni ð1Þ ∑C w li j = 1 Ni ji ji

+ 1Þ wjði + 1Þ

+ 1Þ

+ 1Þ wjði +

Nði

!2

+ 1Þ

ð2Þ

∑ C1jði +

j=1

g

1Þ wjði

+ 1Þ

!2 3−5=2 Fniði + 1Þ l2i 5 − =0 1Þ Ei Ii

ð22bÞ

where the index i varies from 1 to m − 1. The contact equations become 1 2

ffi  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   1 1 1 + 1 Fnik + wik + tolk − −1 Fnik + wik + tolk + μ = 0 K 4 K



i = 1; …; Nk ;

k = 1; …; m

ð23Þ The Newton–Raphson method was used to solve the nonlinear system of equations resulting from applying DQM to the governing system of differential equations, their corresponding boundary and contact conditions.

H. Hakimi, S. Moradi / Journal of Petroleum Science and Engineering 70 (2010) 235–242

3.3. Free vibration analysis

Table 1 Drillstring configuration used by Burgess et al. (1987).

Axial, torsional and transverse natural frequencies of the drillstring can be evaluated using the corresponding wave equation in each direction. The axial and torsional differential equations of motion for the drillstring can be written as (Thomson, 1998) ∂2 uk 1 ∂2 u − 2 2k = 0 2 ∂z ca ∂t ∂2 θk 1 ∂2 θ − 2 2k = 0 2 ∂z ct ∂t

ca =

sffiffiffi E ρ

k = 1; …; m

ð24Þ

4

k = 1; …; m

ð25Þ

"

2

#

∂ wk EI ∂ wk ∂ wk ∂wk + k + g ⋅ cos ϕ ðl′ek −zk Þ − = 0 k = 1; …; r ρAk ∂z4 ∂z ∂t 2 ∂z2 ð26Þ where r is the region number which is contacted the borehole wall. l′ek is defined by l′ek =

ð27Þ

Using the method of separation of variables, and applying the DQM, one obtains

 2 Nk L ð2Þ ∑ C Θ = −λt ⋅Θik lk j = 1 ijk jk

Length (m)

O.D. in (mm)

I.D. in (mm)

Drill pipe Drill collar Stabilizera Drill collar Stabilizera Bit

37.2 161.3 2 9.4 1.95 1

3.5 (88.9) 4.75 (120.7) 4.75 (120.7) 4.75 (120.7) 4.75 (120.7) 6.25 (158.8)

2.06 (52.3) 2.25 (57.2) 2.25 (57.2) 2.25 (57.2) 2.25 (57.2) 0

Blade O.D. is 158.8 mm.

Axial and torsional vibrations at the bit ðz = 0Þ : u = 0; T = 0

ð34aÞ

at the top ðz = LÞ : P = 0; θ = 0

ð34bÞ

at the interfaces of the regions: ui = ui

+ 1;

Pi −Pi

+ 1

= 0; θi = θi

+ 1;

Ti −Ti

+ 1

ð34cÞ

=0

Transverse vibration at the bit ðz = 0Þ : w = 0; M = 0

ð35aÞ

at the contact point with borehole wall : w = 0; wz = 0

ð35bÞ

at the interfaces of the regions:

Pk ρAk g ⋅ cosϕ

 2 Nk L ð2Þ ∑ C U = −λa ⋅Uik lk j = 1 ijk jk

Tools

a

sffiffiffiffi G ct = ρ

where θ is the torsional rotation, ρ and G are mass density and shear modulus, ca and ct are the axial and torsional wave propagation velocities, respectively. The lateral differential equation of motion for a drillstring deviated from the vertical direction by a constant angle ϕ, while taking into account the effect of axial force, can be expressed by Rey (1983) 2

239

k = 1; …; m

ð28Þ

wi = wi + 1 ; wi;z = wi +

1;z ;

Mi −Mi

ð29Þ

  4 Nk A Ik L ð4Þ ∑C W I Ak lk j = 1 ijk jk " # Nk Nk 4 ð2Þ ð1Þ ρAgL cosϕ + ðlek −Zik Þ ∑ Cijk Wjk − ∑ Cijk Wjk = λl Wik k = 1; …; r j=1 j=1 EIlk



ð30Þ

= 0; Qi −Qi

+ 1

ð1Þ

at the bit ðz = 0Þ : U11 = 0; ∑ C1j1 Θj1 = 0

ð36aÞ

Nm

ð1Þ

at the top ðz = LÞ : ∑ CNm jm Ujm = 0 ; ΘNm m = 0

ð36bÞ

j=1

at the interfaces of the regions: UNi i = U1ði

+ 1Þ ;

 Ni Aði ð1Þ ∑ CNi ji Uji −

j=1

+ 1Þ



Ai

!N

ði + 1Þ

li lði

+ 1Þ

ð1Þ

∑ C1jði

j=1

+ 1Þ Ujði + 1Þ

4 ω2 L2 ω2 L2 2 ρAL ; λt = ; λl = ω 2 2 EI ca ct

l′ek

u θ w ; Θ = ; W = ; lek = L L L L

ð31Þ

ΘNi i = Θ1ði

+ 1Þ ;

Ni

ð1Þ

∑ CNi ji Θji −

j=1

 Jði

+ 1Þ

Ji



!N

ði + 1Þ

li lði +

1Þ

ð1Þ

∑ C1jði +

j=1

1Þ Θjði + 1Þ

=0

ð36dÞ where J is the polar moment of inertia of the cross-sectional area of the regions.

ð32Þ Table 2 Material properties of drillstring and formation.

L, A, and I are L = ∑lk ; A = ∑Ak ; I = ∑Ik

=0

ð36cÞ

ω is the natural frequency. The following non-dimensionalized variables have been used U=

ð35cÞ

j=1

where λa =

=0

The differential quadrature representation of the boundary conditions of Eqs. (34a), (34b), (34c), (35a), (35b) and (35c) can be written as Axial and torsional vibrations N1

k = 1; …; m

+ 1

ð33Þ

The boundary conditions are the same as Eqs. (10)–(12) except that in axial direction the string is free at the rotary table. Moreover, the drillstring is fixed at the rotary table and free at the bit in torsional vibrations. These conditions can be expressed as

Drillstring Modulus of elasticity (E) Mass density (ρ) Shear modulus (G)

214 × 109 N m− 2 7850 kg m− 3 82 × 109 N m− 2

Formation Stiffness (K)

100 × 106 N m− 1

240

H. Hakimi, S. Moradi / Journal of Petroleum Science and Engineering 70 (2010) 235–242

Fig. 5. Lateral modal shapes of drillstring.

Fig. 4. Drillstring deflection under static load.

where the subscripts b and i are boundary and interior points used for writing the differential quadrature, respectively. Transforming Eq. (38) into a general eigenvalue form in terms of interior points, results in

Transverse vibration N1

ð2Þ

at the bit ðz = 0Þ : W11 = 0; ∑ C1j1 Wj1 = 0

ð37aÞ

j=1

Nr

ð1Þ

at the contact point with borehole wall : WNr r = 0; ∑ CNr jr Wjr = 0 j=1

ð37bÞ

½AT 

8 8 9 9
ð39Þ

The solution of this eigenvalue problem by a standard eigensolver provides the natural frequencies and corresponding modal shapes. 4. Results and discussion

at the interfaces of the regions: WNi i = W1ði

+ 1Þ N

1 Ni ð1Þ 1 ði + 1Þ ð1Þ ∑ CNi ji Wji = ∑ C li j = 1 li + 1 j = 1 1jði Ni

∑

j=1

 Iði

ð2Þ CNi ji Wji −

 Ni Iði ð3Þ ∑ CNi ji Wji −

j=1

+ 1Þ



Ii + 1Þ

Ii

li lði



!2 N

ði + 1Þ

ð2Þ

∑ C1jði

j=1

+ 1Þ

li lði

+ 1Þ Wjði + 1Þ

!3 N

ði + 1Þ

ð3Þ

∑ C1jði

j=1

+ 1Þ

ð37cÞ + 1Þ Wjði + 1Þ

=0

+ 1Þ Wjði + 1Þ

=0

By using these boundary conditions, some of the terms of Eqs. (28)– (30) become redundant. Therefore, the combination of Eqs. (28)– (30), (36a), (36b), (36c), (36d), (37a), (37b) and (37c) give the following system of linear equations



Abb Aib

9 9 88 > > > > > > > Θ > > >  = <:W; > Abi 8 9b = λ⋅ 0 Bib Aii > > > > > > > Θ > > > > ; :: ; W i

9 9 88 > > > > > > > Θ > > > = <:W; > 0 8 9b Bii > > > > > > > Θ > > > ; :: ; > W i

To verify the accuracy and reliability of the DQM, the formulation developed in the preceding section is used to investigate several drillstring vibration case studies. Two drillstrings, which their natural frequencies have already been measured experimentally, are investigated. A drillstring used by Burgess et al. (1987) is considered first. The drillstring configuration is listed in Table 1. They used FEM to find the natural frequencies of the drillstring. The drillstring is divided into five regions. The first region includes the bit, while half of the first stabilizer is included in the second region. The third region includes half of the first stabilizer, the first drill collar and half of the second stabilizer. The next region consists of half of the second stabilizer and the rest of the drill collars and the last region has the whole length of the drill pipes. Both centers of the stabilizers are pivoted. Material properties of drillstring and formation are tabulated in Table 2. The

ð38Þ

Table 3 Results for lateral natural frequencies.

DQM Burgess et al. (1987)

First frequency (Hz)

Second frequency (Hz)

1.24 1.25

3.16 3.15

Fig. 6. Effect of number of sampling points on the first lateral resonant frequency.

H. Hakimi, S. Moradi / Journal of Petroleum Science and Engineering 70 (2010) 235–242 Table 4 Comparison of measured and model-derived axial resonant frequencies.

241

Table 6 Comparison of measured and model-derived lateral resonant frequencies.

Mode

Measured (Hz)

DQM (Hz)

BHASYS (Hz)

NATFREQ (Hz)

WHIRL (Hz)

Mode

Measured (Hz)

DQM (Hz)

BHASYS (Hz)

NADRID (Hz)

WHIRL (Hz)

1 2 3

9.7 26.3 40.3

9.78 27.67 41.29

9.06 26.01 40.74

8.7 25.4 40.1

8.53 24.86 39.43

1 2 3

1.3 2.01 2.57

1.28 1.92 3.44

1.3 2.05 2.51

1.67 – 2.49

1.21 2.03 2.54

weight on bit was recorded at 15 klb (66.7 kN) and the well deviated from vertical direction by 1°. In order to find the effective length, a nonlinear static analysis was performed using Newton–Raphson algorithm. Fig. 4 shows the deflected drillstring where most of its length rests on the borehole wall. The points correspond to rotary table, bit and the two stabilizers have zero deflection. The point at which the string contacts the formation found to be at 109 ft (33.2 m) from the bit. Next, free vibration analysis was performed to find the natural frequencies of the drillstring in axial, torsional and lateral directions. The results for lateral natural frequencies are compared with those of the Burgess et al. (1987) in Table 3, and their corresponding modal shapes are shown in Fig. 5. As can be seen from the table, the results obtained by DQM agree well with those of the finite element method. As mentioned earlier, the accuracy of the DQM results is governed by the number of sampling points that is used to discretize the domain. The influence of the number of sampling points (for each region) on the convergence of first natural frequency of drillstring is shown in Fig. 6. In our investigation, we used the same number of sampling points for all of the drillstring regions. This figure shows the good stability and rapid convergence of the DQM. To further validate the proposed method, the case study that was considered by Jogi et al. (2002) reevaluated. The drillstring length is 526.8 ft (160.6 m), the well deviated from vertical direction by 1°, and the weight on bit was recorded at 25 klb (115.7 kN). Jogi et al. (2002) used experimental methods to measure the natural frequencies of the drillstring, and then compared them with those evaluated by the available commercial software. These software packages include NATFREQ, BHASYS, NADRID and WHIRL. The software NATFREQ utilizes analytical models and performs iterative methods to obtain the axial and torsional natural frequencies of drillstring (Baily and Finnie, 1960a,b). BHASYS applies linear finite element algorithms (Paslay et al., 1992), while NADRID (Heisig, 1993) and WHIRL (Dykstra, 1996) use nonlinear FEM to compute all the natural frequencies of the drillstring. Performing a static analysis using DQM, the effective length of the string was found to be 181 ft (55.2 m). Tables 4–6 show the comparison of the measured natural frequencies with the results evaluated by DQM, NATFREQ, BHASYS, NADRID and WHIRL. As can be seen from the tables, there is a good agreement between the results obtained by DQM and those of experiment. Indeed, the DQM results are closer to the measured data than those calculated based on the aforementioned models.

5. Conclusions The DQM was applied to analyze the problem of drillstring vibrations. A nonlinear static analysis carried out to determine the

Table 5 Comparison of measured and model-derived torsional resonant frequencies. Mode

Measured (Hz)

DQM (Hz)

BHASYS (Hz)

NATFREQ (Hz)

WHIRL (Hz)

1 2 3

5.00 14.4 23.5

3.61 12.57 23.58

1.3 9.44 20.71

3.8 13.46 24.53

3.83 13.61 24.82

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