Coupling and Damping Effects on the Dynamics of Submerged Expanded Tubes in Borehole

Hydraulic expansion of submerged tubes is accomplished by propelling a mandrel through it using differential pressure. This process deforms the tube beyond its elastic limit. Toward the end of the expansion process, the mandrel pops out of the tube resulting in displacement, stress, and pressure waves propagating through the system. A mathematical model has been developed to describe the dynamics of the tube-fluid system due to the pop-out phenomenon. The model takes into consideration the coupling effect between fluids and the structure, as well as the inherent system damping of its response. An analytical solution describing the wave propagation in the tube-fluid system was obtained. The model identified the potential failure locations and showed that the inherent system damping reduced the chances of failure but could not eliminate it completely. In addition, it showed that the coupling effect was more prominent in the tube as compared to the outer and inner fluids. Furthermore, a sensitivity analysis was conducted in order to investigate the effect of the geometrical and material properties on the response. The sensitivity analysis showed that the coupling effect vanished with the increase in tube stiffness and reached an asymptotic value with an increase in formation stiffness.


Introduction
Solid expandable tubular (SET) technology is a down-hole process consisting of expanding the diameter of a tube by pushing or pulling a mandrel through it.Recent interest and further research in SET will be able to provide solutions for many unsolvable problems having to do with well drilling and completion, such as extended reach applications, mono-diameter wells, cost-effective repair of damaged zones and the conservation of hole-sizes.
The principle behind SET is simple.A mechanical expansion device, known as a mandrel, is pushed through the tube hydraulically by back pressure across the mandrel.The forward movement of the mandrel expands the tube to the required diameter.
The first SET test was performed at Royal Dutch Shell in The Hague (Fillipov et al. 1999).The test results were encouraging and paved the way for significant research work in this field.Recently, there has been an increase in applications in well engineering due to their many successes in deep water applications as well as its cost effectiveness in solving many long standing problems faced during the construction and operation of wells (Owoeye et al. 2000).Most of the researchers have focused their efforts on determining the power required for down-hole tube expansion and its effect on mechanical properties including burst and collapse strengths (Fillipov et al. 1999;Mark et al. 2000).Ruggier et al. (2001) attributed the decrease in collapse pressure to the length and thickness variation, the Bauschinger effect, and residual stresses.Stewart et al. (1999) conducted a laboratory test on a 3.5-inch outer diameter tube and showed that the elongation at fracture and uniform strain decreased, while the yield and ultimate tensile strength increased.Limited successful applications of SET in well construction and remediation were reported by Daigle et al. (2000).Due to the complexity of the process, the researchers have focused their effort only on determining the power required for down-hole and simulated expansion, and post-expansion behavior using the finite element method.Pervez et al. (2005) used a non-linear explicit finite element analysis to study the effects of expansion ratio, friction coefficient, and mandrel angle on tube expansion.It was found that for small conical angles in mandrels and in large expansion ratios, the failure could not be avoided even at low values of friction coefficient.In a subsequent study, Pervez et al. (2005) concluded that the tube wall thickness decreased with an increase in mandrel angle, expansion ratio, and friction coefficient.The tube length often shortened for expansion under tension for most of the loading mechanisms, but it elongated at high drawing force.
During the hydraulic expansion process, elastic energy is stored in the tube, and in inner and outer fluids.When the mandrel pops out of the tube, the stored energy is suddenly released resulting in an excitation of the whole assembly.This dynamic condition is called the pop-out phenomenon.Nothing has been found in the literature that specifically addresses the pop-out phenomenon.The authors have developed a mathematical model without damping effects to predict the system response as well as the propagation of displacement, pressure, and stress waves in the tube, and in the inner and outer fluids (Karrech et al. 2004;Pervez et al. 2006).It was found that the three mediums interact with each other immediately after the pop-out phenomenon takes place, where a small disturbance in one medium influences the others.When the tube is purely elastic, the effect of the outer fluid on the inner fluid is substantially higher (Seibi et al. 2004).This reduces significantly if the tube material is elastic perfectly plastic.Aarrestad et al. (1986) studied a similar problem regarding the dynamics of submerged drill strings.In this study, the damping effects were included but the coupling between the drill string and the fluid were ignored.It was found that frequency-dependent damping more accurately responded than constant damping when compared with experimental results.For the same problem, Lea (1996) carried out a modal analysis including the coupling effect between the drill string and fluid.This work neglected damping effects.Wang and Bloom (1999) investigated the effect of geometry on the natural frequencies and damping ratios of an inclined submerged tube.
In general, an oscillating system dissipates energy while vibrating.Therefore, damping is considered one of the most important characteristics of the system unless it is not desirable by design.It is certainly useful in SET, where it may limit the resonant amplitudes of the structures and drive system, stabilize instruments and sub-systems, or reduce the danger of tube failure.Virtually all phases of the response of tubefluid system are affected by the damping, yet no efforts have been made for their proper characterization.
The primary goal of this work was to predict the system response and determine the coupling effects between the propagating waves in the tube and its inner and outer fluids while considering the inherent damping present in the system.Due to the difficulties in mathematically representing a real damping mechanism, a proportional damping approximation has been employed here.The influence of formation stiffness, tube material properties, and fluid characteristics on coupling effects is also investigated here.
During the expansion process, the tube, with inner and outer post-expansion radii of r i and r o , respectively, is subjected to an inner pressure, p i and an outer pressure, p o .When the mandrel leaves the tube, the inner pressure and the axial stress in the tube suddenly drop to zero.During the expansion process, elastic energy is being stored in the fluid and the tube depending on its dimensions and mechanical properties.

Coupled Equations of Motion
Using the above-mentioned assumptions, the continuity of the mass and momentum of the fluids as well as the equations of motion for the tube-fluid system, the wave equations associated with the inner fluid, elastic perfectly plastic tube, and the outer fluid, respectively, can be expressed as The derivation of these governing differential equations is shown in the Appendix.The coupling effect is mainly dependent on the cross sectional dynamic change and the material properties such as densities, fluid's bulk modulus, the formation shear modulus, and the tube's elastic constants.The above-mentioned equations of motion can be written in a matrix form where the off-diagonal terms describe the coupled behavior of wave propagation in the three mediums.where w = {w i w p w o } T represents the displacement vector in the three mediums.A is the matrix of known coefficients and I is the 3 x 3 identity matrix.Using the transformation = we -i( t+KZ) , Eqn. 4 can be expressed as a linearized eigenvalue problem as follows: (5) (6) where the subscripts 'i', 'p' and 'o' refer to the inner fluid, the tube, and the outer fluid, respectively.The terms n and S n denote the angular frequencies and wave propagation speeds for the tube, and the inner and outer fluids.

Damping Approximation
The damping effects have been neglected while deriving the equation of motion (Eqn.4) and the eigenvalue problem (Eqn.5) of the tube-fluid system.Although neglecting damping simplifies the derivation, it jeopardizes complete understanding of the dynamics of the system.In order to obtain relevant results, damping must be taken into consideration in determining the system response in terms of displacement, pressure, and stress waves through the three mediums.In general, the damping force in a dynamic equilibrium equation is proportional to velocity.Hence, augmenting Eqn. 5 with a damping force term gives (7) Where D represents the damping matrix.The ways in which the tube-fluid system may be damped are diverse and can incorporate many different kinds of mechanisms.For instance, the tube vibration can be damped due to the viscous effects of the surrounding fluids.In addition, the visco-elastic effect of the solid mediums further enhances damping.According to Apostal et al. (1990), the damping matrix of a submerged drill-string in a borehole contains terms based on the effect of proportional, structural, and viscous damping.The effect of Rayleigh or proportional damping on the bottom-hole assembly (BHA) vibrational response is described by D r = o A + 1 B, where o and 1 are proportionality coefficients.The structural damping, which is assumed to be proportional to the displacement but in phase with the velocity of a harmonically oscillating BHA, is decomposed into two separate terms: D s = 2 B + 3 B c .The first term characterizes the behavior of the BHA alone while the second term represents the possible contact between the structure and the borehole, and B c is the equivalent contact (or formation) stiffness matrix.Also, the viscous damping describes the energy dissipation by laminar fluid friction.Aarrestad et al. (1986) showed that the fluid viscous damping was negligible as compared to the breathing effects (ie. the effect of the propagating waves on the environment due to the radial contraction of the tube); hence, it can be ignored in the present system.In addition, the tube did not come in direct contact with the formation, as the structure did not experience bending due to popout.Therefore, the damping of the system under consideration can be described by a damping matrix, which is proportional to the equivalent mass and stiffness matrices: The constants o and 1 satisfy the relation where is the damping ratio and is the natural frequency of the system.Using Eqn. 9 for the first two modes of vibration gives (10) The dynamic response is essentially governed by the lower modes.Therefore, in the above equation, the constants o and 1 can be determined once the frequencies of the first two modes and their corresponding damping ratios are known.However, the values that bracket the likely range of frequencies of interest can also be used for certain applications.
2 2 Fossen and Johansen (2002) performed an experiment on a similar submerged cylinder in a frame to determine the natural frequencies and damping ratios.
The tests for different fluids were carried out at Institutt for Teknisk Kybernetikk at Norges Teknisk-Naturvitenskapelige Universitet (NTNU).These tests experimentally determined the damping ratios used in Eqn. 10 along with the first two fundamental frequencies of the system to calculate unknown constants o and 1 .

Solution of Governing Equations
The orthogonality properties of the undamped mode shapes are used to decouple the equations of motion (Eqn.7).Using the transformation = y , Eqn. 7 reduces to (11) where is the modal matrix consisting of independent mode shapes of the associated undamped system.Use of the orthogonality condition reduces the coefficients in the above equation to Hence, the equation of motion corresponding to the nth mode of vibration can be written as follows: (12) The solution requires initial and boundary conditions associated with mandrel pop-out at the end of expansion process.Figure 1 shows the tube-fluid system where the tube is fixed at the bottom end and is free at the top end.Hence, the boundary conditions can be expressed as: During the expansion process, the tube is subjected to uniform axial stress, p, inner pressure, q i , and outer pressure, q o , which is thereby independent of z.With the sudden release of the mandrel, the stress and pressure at the free end of the tube along with the inner and outer fluids drop to zero, leading to the following initial conditions: where n = i, p, o and k = 1, 2, 3, …., oo, and

Results and Discussions
Once the mandrel pops out, the tube-fluid system is excited by the sudden release of energy stored in the system.This initial excitation at the free end of the tube results in stress and pressure waves propagating in the three mediums.To investigate the dynamic effects of the pop-out phenomenon, a typical field case with an expansion ratio of 20% and a friction coefficient of 0.1 was considered.The geometrical and material parameters were the inner radius of the tube- A conical shaped mandrel was used for expansion with an optimal cone angle of 20° (Pervez et al. 2005).In order to quantify the post-expansion system response, the drawing force required to expand the tube was determined using a finite element analysis (Pervez et al. 2005).At the end of the expansion process, the inner fluid pressure was 57 MPa and the axial stress in the tube was 149 MPa.These values then became the initial conditions for the tube-fluid system.The boundary conditions were governed by the fixed end condition at the bottom end of the tube.

Damping and Coupling Effects (15)
It is important to note that the superposition of the axial stress and displacement of the tube was allowed because the deformation in the z-direction was purely elastic [Appendix 6].
Figure 2 shows the waves propagating in the three mediums with respect to time and position along the zaxis of the tube.It can be seen that the tube experienced a high apparent frequency of 64 Hz as compared to the inner and outer fluid frequencies, which were 18.43 and 16.41 Hz, respectively.In terms of displacements, it can be seen in Figs. 2 (a), (c), and (e) that the inner and outer fluid tubes reached 0.6 m, 1.4 cm and 2.3 cm, respectively, when the mandrel popped out.In addition, Fig. 2 shows that the displacements satisfied the initial conditions defined by Eqn. 13.The displacements were maximized at the free end (z = L) and equal to zero at the fixed end (z = 0).It can also be noticed that the initial displacement of the inner fluid at the tube's free end was quite high as compared to the tube length (5%) but this result was expected with the high inner fluid pressure.Initially, the response corresponded to the excitation imparted to the system.Figures 2(b), (d), and (f) show that the inner and outer pressure and the axial stress were equal to 57 MPa, zero Pa and 149 MPa, respectively, when the pop-out took place.After pop-out, the displacements, stress, and pressures started decaying and interacting with each other as shown analytically by Eqn.14.Also of note is that the amplitudes of the fluid pressures and tube axial stress were maximal at the fixed end (z = 0) of the system and equal to zero at the free end (z = L).
Note that the inner fluid mode was slightly affected by the radial flexibility of the tube.The wave propagation speed increased by 0.5% relative to the sonic speed.The effect on the tube was almost negligible.However, the outer fluid mode was substantially affected and the wave propagation speed was lowered by 11% as compared to the corresponding uncoupled wave propagation speed.This was due to the low shear modulus of the formation and the radial flexibility of the tube.For each eigenvalue, , there was a corresponding eigenvector describing the contribution in terms of amplitude of each medium on the propagation modes.The three eigenvectors, as columns, are given in the following matrix: The eigenvectors were normalized in order to study the contribution of each medium to the others.It can be observed that the off-diagonal terms representing the coupling effect were, in general, small in magnitude.For example, in the second mode, a vibration amplitude of 1 mm in the tube induced a vibration amplitude of 0.0542 mm and 0.0137 mm in the inner and outer fluids, respectively.This means that each mode was strongly located within its own medium and the disturbances in other mediums were very small.
Interestingly, the inner fluid mode was contained within itself and did not produce any disturbances to the tube or outer fluid (two zeros in first column).This is due to the elastic perfectly plastic tube material.The inner fluid was in contact with the plastic region of the tube, while the outer fluid was in contact with the elastic region.
The changes in the fluid properties of the outer fluid due to the presence of mud will enhance the inherent system damping.

Effects of Geometrical and Material Properties
The response of the formation-fluids-tube system is highly influenced by the material properties and geometry.The objective of this section is to study the effects of the formation shear modulus and tube stiffness on the response of the structure in terms of natural frequency and coupling between the propagating waves.
Figures 5 (a) and (b) show that the natural frequencies and wave propagation speeds increase with tube stiffness.It is important to note that according to Eqns. 15 and 16, the predominant mode of system vibration corresponds to the tube.Therefore, the components of this mode are reported in Fig. 6, which shows that the coupling effects vanish with an increase in tube rigidity.This result was expected as the coupling effects are mainly related to the stiffness of the tube and formation as can be seen through Eqns.9 through 11.When the stiffness of the tube increases, the cross-sectional change of the inner and outer fluid medium reduces.This results in less interaction between the three mediums.
On the other hand, it can be seen in Figs.7 (a) and 8 (a) that the frequency of the tube and the interaction between the tube and inner fluid do not change significantly with the formation shear modulus.However, the frequency of the outer fluid and the amplitude of the coupling term between the outer fluid and tube increase with the formation shear modulus, as shown in Figs.7 (b) and 8 (b).This means that in the predominant mode of vibration, the component associated with the annulus is much more affected by the formation shear modulus variation than the others.This phenomenon is equivalent to the garden hose effect where larger hoses yield lower fluid speeds.Moreover, when the stiffness of the formation increases, the amount of

Conclusions
A mathematical model for tubes submerged in inner and outer fluids inside boreholes was developed to investigate the dynamics of fluids and expanded tubes due to the input energy imparted to the system at the end of the expansion process.This model took into consideration the damping as well as the coupling effects.An analytical solution was obtained for the displacement, and for the stress and pressure wave propagation.It was found that the expanded tube has a localized critical region in the vicinity of the fixed end of the tube, which, under the right conditions, may fail.The inherent system damping helps to reduce the failure, but it is not enough to completely avoid it.An alternative solution is to provide external corrective damping in the system, which would be costly and difficult to implement in field conditions.Furthermore, a sensitivity analysis was conducted in order to study the effects of the geometry and the formation material properties on the response of the system in terms of frequency, wave propagation speeds.and coupling effects.The sensitivity analysis showed that the fundamental frequency of the outer fluid varied significantly with the formation shear modulus.But the tube fundamental frequency was insensitive to variation in the formation shear modulus.The reverse was true when tube stiffness varied.More studies would be required to study the use of solid expandable tube technology in drilling mono-diameter wells efficiently with less cost and time.Future work may also include other aspects such as the formation porosity or the non-conservativity of the fluid mediums at the end of the expansion process.

Figure 2 .Figure 3 .
Figure 2. Response of tube-fluid system in terms of axial displacement, stress and pressures

Figure 6 .Figure 7 .Figure 8 .
Figure 6.Variation of the first (a) and third (b) components of the predominant mode of vibration with respect to the tube stiffness