Finite Element Analysis for Control of Lateral and Torsional Vibrations in Drilling Directional and Multi-Lateral Wells

DOI : 10.17577/IJERTV5IS010339

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Finite Element Analysis for Control of Lateral and Torsional Vibrations in Drilling Directional and Multi-Lateral Wells

Odesa David Emudiaga,

Nigerian Petroleum Development Company Limited; Adewale Dosunmu and Ossia C. Victor,

University of Port Harcourt, Nigeria

Abstract – Vibration during drilling operations has a large effect on both the bottomhole assembly which in turn has significant effect on the drilling efficiency. Drillstring vibration causes damage to the drillstring and BHA components, premature bit failure, wear on tool joints and stabilizers, twist-offs where the drillstring breaks down hole due to fatigue or excessive torque and poor control of the deviation due to inadequate understanding of the tool face position and stresses especially in deviated and multilateral wells.

The primary causes of drillstring vibrations are bit/formation and drillstring/borehole interactions. Large vibration levels cause reduced rates of penetration and catastrophic failures while lower levels may lead to a reduced operating life. The benefits of addressing this problem are obvious and include reduced drilling time and costs, reduced maintenance, and lower equipment turnover.

  1. Interferences on the measurements performed during the drilling process and damage of the measurement equipment and even failure to acquire evaluation data

  2. Significant waste of energy due to increased Tripping Times and inability to run & set casing, torque and drag

  3. BHA instability, reducing the directional control.

  4. Wellbore instability occasioned by the fracturing effect on the wellbore due to BHA whirl.

Damage to the drillstring and BHA components

This research was carried out to analyse the effect of torsional and lateral vibrations on drilling efficiency when drilling directional wells. Harmonic and modal finite element analyses were adopted for the obvious advantage that it is able to approximate the real structure with a finite number of degrees of freedom. In the cause of the analysis, different critical load, critical speed for the finite element of the drillstring and the BHA and saturation points were developed for the different components of the drillstring from the Euler

Premature bit failure

Effect of Vibration on directional and multilateral drilling

Damage to the wellbore

model to determine the crippling loads that will cause each component to buckle and propagate lateral and torsional vibration.

Wear on tool joints and stabilizers

Poor control of the deviation

The results shows that it is safer to drill below the critical speed of the finite elements of the drillstring and BHA to avoid resonance and to ensure the axial forces acting on the BHA is below the critical load.


The drill-string vibrations are induced by the characteristics of the bit-rock interaction and by the impacts that might occur between the column and the borehole. If not controlled, vibrations are harmful to the drilling process causing:

  1. Premature wear and consequent damage of the drilling equipment and BHA components – bit, motor / RSS, MWD etc. resulting many times in failures, especially due to fatigue.

  2. Decrease of the rate of penetration (ROP), increasing the well cost

Fig. 1. Effect of vibration on directional and multilateral drilling

Factors that affect torsional and lateral drillstring vibrations

  1. Material of the drillstring (shear modulus). For the steel with increasing torque, shear stress increases linearly with shear strain until plastic region is reached.

  2. Drilling fluid

  3. Well geometry (Hole angle)

  4. Difference in friction between the static and dynamic friction of the drill bit and bit face

  5. The nature of vibration in drillstrings depends on the type of bit, among other factors. PDC bits work by shearing the rock rather than crush the rock. This results in a bit-rock interaction mechanism

characterized by cutting forces and frictional forces. The torque on bit and the weight on bit have both the cutting component and the frictional component when resolved in horizontal and vertical direction.

A common mode of failure of PDC bits in hard rock

In this analytical model, we would be able to control lateral vibration by controlling some factors such as under-gauge stabilizer, initial phase angle, initial deformation, WOB and ROP.

The rotation of the cross section as measured by is

drilling is that of catastrophic breakage caused by the various modes of drillstring dynamics. Typically this mode

less than 1.0, one radian

of failure takes place in advance of any appreciable wear that may dictate cutter replacement.

At any section XX distant X from the fixed point B, for the curved beam, bending moment is expressed as given as

To obtain required drilling performances, it is necessary to adjust features such as profile shape, gage and mainly cutter characteristics (shape, type and orientation). To


= 2 (Leonhard Euler equation)


optimize the drilling parameters in order to increase the ROP, you need to know the real drill bit response which is a direct function of the cutter rock interaction. Cutter rock interaction model is a critical feature in the design process. But previously used models considered only three forces

M = Force x distance

2 = ( )


2 + =


on a cutter based on the cutter-rock contact area: drag


force, normal force and side force. Such models are no longer valid with the introduction of PDC cutters with

2 + =

chamfer and special shape (Gerbaud et al, 2006).Chamfer was introduced to avoid diamond chipping when drilling hard formations. Two different mechanisms take place at

Solving the differential equation,

1 2

1 2

= cos( ) + (


the chamfer with respect to the depth of cut. If the depth of

cut is greater than the chamfer height, crushed rock is trapped between the cutting face and the rock and


+ (1)

additional forces are generated in the same way than for the cutting face crushed material.

Now, if depth of cut is lower than chamfer height, the chamfer becomes the cutting face with higher back rake angle and the chamfer forces are the cutting face forces.


example, at 45° chamfer angle and 15° back rake angle, the real back rake angle for small depth of cut becomes 60°.


Lateral vibration will cause fatigue and failure of drillstring, broaden hole and change bending angle of bit. The major difficulty however encountered in controlling lateral vibration amplitude and impact intensity is the fact that lateral vibration and its consequent drillstring impact

E = Modulus of elasticity for steel expressed in pa (from 200,000 to 220,000MPa)

= Moment of inertia of the straight section of the drillstring expressed in m4

= Free buckling length which depends on the actual length of the pipe and the way the end s are fixed, expressed in m

C1C2 = Arbitrary constants of integration

At the fixed point, B, the deflection is zero Boundary conditions:

= 0 @ = 0

= @ =


0 = 1 + 1 =

The slope at any section is given by

cannot be observed at the surface without the expensive

MWD apparatus.



) +

cos( )

Bit whirl is predominant with PDC bits because tricone bits



penetrate the bottom of the borehole more and do not allow sideways movement of the bit. Whirl generation is caused by two factors: a centrifugal force generated as a result of the high rotary speed. The added force creates more friction which further reinforces whirl. The second is the center of rotation which is no longer the centre of the bit. This fact however contradicts the basic bit design assumption that the geometric center of the bit is the center of rotation. The impact loads associated with this motion cause PDC cutters to chip, which in turn, accelerates wear.

At the fixed point, B, slope is zero

= 0, = 0


0 =


2 = 0

At A, the deflection is

= , =


= 1 cos( ) + 2 ( ) +

= cos( ) +

cos( ) =

Mean position Position after time, t sx x

= , 3 , 5

2 2 2

Where K = 1, 3, 5,

Considering the first critical value,




Fig 2. Natural frequency of free lateral vibration

= 2

k = stiffness of the drillstring (N/m)



p = 2EIa



(Euler formula)

(4 4)

m = mass (Kg)

W = weight of drillstring = mg



= static deflection due to weight of the body

Re: outside radius of the pipe expressed in m Ri: inside radius of the pipe expressed in m

= (4 4)

x= displacement of the body from the equilibrium position

wn = circular natural frequency (rad/s)


De: outside diameter of the pipe expressed in m Di: inside diameter of the pipe expressed in m

P, the crippling load is the maximum limiting load at which the column tends to have lateral displacement. Buckling occurs about the axis having least moment of inertia.

From above, the factors affecting the vibration/deflection are critical load, length of drillstring, young modulus and moment of inertia.



To determine the natural frequency of free lateral vibrations, consider a drillstring whose end is fixed and the other end carries a body of weight, W as seen in the figure below

fn = natural frequency, Hz f= frequency of mass body A = Cross-sectional area

In equilibrium position, the gravitational pull, W = mg and is balanced by force of spring, such that W = k.

Since the mass is displaced from its equilibrium position by a distance of x, as shown in Fig 2. Above and is then released, therefore after time t,

Restoring force = ( + ) =

Recall =

= ( ) 2

Accelerating force = mass x acceleration


m2 ( ) 3


Equating equation 2 and 3, the equation of motion becomes

2 =


2 + = 0


2 + = 0 4


Recall that the fundamental equation of simple harmonic equation is

2 + = 0 5


Comparing equation 4 and 5, we have,

2 =


The periodic time of the vibration is

Similarly, using shock subs are not a universal solution as they are designed for one set of conditions. When the drillstring environment changes as it often does, shock subs become ineffective and often result in increased drilling vibrations, making the situation more complex.

Fundamentally, the rotary table is driven by means of a sprocket and chain by the drawworks. It can however also be driven by an electric motor independent of the drawworks transmission on heavyweight rigs.

haft ax

haft ax


s is



= 2 G

And the natural frequency = 1 = = 1 = 1

2 2 2

Since the static deflection due to gravity, =

Taking the value of g as 9.81m/s2 and in metres Therefore, the Natural Frequency,

Fig 3. Rotary table when shaft is stationary





= 1 9.81 = 0.4985



The value of static deflection can be obtained from the y


= = G


Fig 4. Rotary table when shaft is rotating

The external forces causing vibration to the drillstring should not operate at this natural frequency of the drillstring to avoid resonance.



      Drillstrings develop vibrations, excessive wear and fatigue when run at critical rotary speeds. Drilling deeper in hard rock induces severe vibrations in drillstring which causes resonance, wear, fatigue and reduced rates of penetration and consequent premature failure of the equipment. The current means of controlling vibration by varying conditions such as reducing the rotary speed or WOB often ends up reducing the drilling efficiency.

      Lets consider a shaft of negligible mass carrying a rotor as seen in Fig 3 where G is the centre of gravity of the shaft.

      When shaft is stationery, the centre line of the bearing and the axis coincides.

      When shaft is rotating with a uniform speed, /, the centrifugal force acting radially outwards through G causing the shaft to deflect is given as

      = 2( + )


      m= mass of the rotor

      e = initial distance of centre of gravity of the rotor from the centre line of the bearing or shaft axis, when the shaft is stationery

      y = additional deflection of centre of gravity of the rotor when the shaft starts rotating at /

      The shaft behaves like a spring, therefore the resisting force to deflection f =

      For the equilibrium position,

      2( + ) =

      2 + 2 =

      ( 2) = 2

      Note: when the centre of gravity of the rotor lies between the centre line of the shaft and the centre line of the bearing, e is taken to be negative. On the other hand, if the centre of gravity of the rotor does not lie between the centre line of the shaft and the centre line of the bearing, the value of e is taken to be positive.


      The critical speed vary with length and size of drillstring



      and hole size. Two types of vibrations may occur: nodal

      = 2 = 2

      We know the circular frequency

      vibration as the pipe vibrates in the nodes as a violin string or longitudinal vibration as the pipe may vibrate as a pendulum.

      = ( )

      Nodal (lateral or transverse) vibration

      330562 + 2

      Substituting this


      = 5


      2 2

      Application: if the drilling RPM is equal to the calculated critical , there may be drill pipe, drill collar and

      As shown in Fig 4 with the dotted lines, when > , the value of y will be negative and the shaft deflects in the opposite direction.

      HWDP failure. So, the drilling RPM should be less than

      the calculated

      Axial (Longitudinal) Vibration

      In order to have the value of y always positive, both plus and minus signs are taken

      = 258000 6

      = ±


      2 2 =






      2 =

      Application: if the drilling RPM is equl to the calculated

      critical , and/or equal to the harmonic vibration, there



      Substituting =



      2 =

      may be drill pipe, drill collar and HWDP failure.

      For tri-cone bits, drill bit displacement frequencies are consistently three cycles for every bit revolution for three-







      cone bits, in other words, tri-cone bit impacts an excitation

      frequency of three times the rotary speed

      From the above expression, when = , the volume of y becomes infinite


      is the critical or whirling speed of the rotary table


      Natural frequency of longitudinal vibration, 1


      1 = / sec 7

      = = =

      Natural torsional frequency of drill collars, 2

      If Nc is the critical or whirling speed of the rotary in revolutions per second, then


      = 2662 / sec 8







      = 201 202 9

      **To prevent vibration with tri-cone bits, bit rotation must be tolerated at a speed less than 201 202. The


      2 =


      best situation for a drillstring is to operate below its lowest

      critical speed. In this region, the bit is able to continously

      **The critical or whirling speed of the rotary is the same as the natural frequency of lateral vibration but its unit will be revolutions per second

      make contact with the formation.


Frequency of vibration at the bit (excitation frequency)

= 3. / sec 10


5.0. CONCLUSION The results show that to drill safely,

  1. The rotation of the drillstring and BHA should be within the region of speed below the critical speed of the finite elements of the drillstring and BHA to avoid resonance and to ensure the axial forces acting on the BHA is below the critical load

  2. Using the modal and harmonic finite element analysis, we were able to set critical limits for different components of the drillstring and BHA, beyond which the drillstring will be subjected to lateral and torsional vibrations.

  3. Using the Euler equation, the crippling load that causes each drillstring to begin to buckle and the effect of the mud weight and length of drillstring were analyzed and incorporated in the mathematical model.

  4. One of the issues to keep close tap on is the manipulation of the WOB during torsional and lateral vibration control to ensure that the founders point is not exceeded.

  5. The greater the moment of inertia of the drillstring and BHA when considered in finite terms, the higher the tolerance and therefore the higher the allowable critical speed of the drillstring


Further studies should be carried out on microscopic analysis of transient drillstring vibration and effect of geomechanical stress on vibration.


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T1 = torque on bit T2 = surface torque

J1J2 = equivalent moment of inertia at bottom end and top end respectively

C1C2 = mud damping CS = structural damping 1 = bit speed

2 = top drive speed (surface RPM)

1 2=rotational displacement (angle) of the bit starting and top drive respectively with zero at time t = 0

k = equivalent stiffness of the drillpipe The equivalent mass moment of inertia J1 = BHAIBHALBHA

= the damping factor T =vector with efforts

q = state-space coordinate m = Mass

k = stiffness

x = displacement of m from the equilibrium position

E = Modulus of elasticity for steel expressed in pa (from 200,000 to 220,000MPa)

= Moment of inertia of the straight section of the drillstring expressed in m4

= Free buckling length which depends on the actual length of the pipe and the way the end s are fixed, expressed in m

C1C2 = constants of integration L = length of drillstring in inches

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