Proceedings of the ASME th Biennial Conference On Engineering Systems Design And Analysis ESDA2012 July 2-4, 2012, Nantes, France

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1 Proceedings of the ASME th Biennial Conference On Engineering Systems Design And Analysis ESDA2012 July Nantes France ESDA A MULTIBODY SYSTEM APPROACH TO DRILL STRING DYNAMICS MODELING Dmitry Pogorelov Gennady Mikheev and Nikolay Lysikov 1 Laboratory of Computational Mechanics Bryansk State Technical University Bryansk Bryansk region Russia 1 lysikov@umlab.ru Lev Ring Raju Gandikota and Nader Abedrabbo 2 Weatherford International Houston Texas USA 2 nader.abedrabbo@weatherford.com KEYWORDS Drilling Modeling Multibody System Dynamics Transient Analyses Constrained Buckling Whirl ABSTRACT The selection of optimal operational parameters for drilling oil and gas wells is a complex dynamic problem that depends on multiple parameters. Numerous physical and mechanical processes such as rock cutting friction hydraulics and different modes of vibrations occur during drilling which should be accounted for in numerical models. It is widely accepted that bottom hole assembly (BHA) vibrations are the primary source of drilling equipment premature failure. Over the last 30 years progress of computational sciences has enabled the use of numerical simulations of drillstring dynamics as a useful tool to understand and mitigate sources of harmful vibrations. The majority of these models have been based on nonlinear finite elements. There are several significant limitations with this approach including an extremely high number of degree of freedom (DOF) required to represent geometries with 10 5 ratio of axial to lateral dimensions and also the complexity of modeling variable contacts in bifurcating systems. While it is relatively new for simulating drilling dynamics the advantage of the proposed rigid-flexible multibody system approach has been proven for modeling complex dynamic systems in other industries. Using a rigid-flexible multibody system approach to analyze dynamic effects both in frequency and time domains dynamic modeling of BHA and drillstring is proposed. Drillstring is simulated as a set of uniform flexible beams connected via linear viscous-elastic force elements. Each beam can undergo arbitrary large displacements as absolutely rigid body but its flexible displacements due to elastic deformations are small. A method of floating frame of reference for flexible bodies and component mode synthesis is used for modeling beams dynamics. Parameters of the coupling force elements are calculated automatically based on stiffness and inertia characteristics of the connected beams. This paper discusses the development of the rigid-flexible multibody system for modeling drillstring dynamics and the influence of model parameters on simulation accuracy and calculation time. A close match is shown between theoretical and numerical results for static and buckling problems as well as resonant frequency values. Several transient drillstring dynamics problems are analyzed for wellbores with uniform diameter. Examples of the analysis of resonant conditions during drilling planning stage are also presented. It is also shown how transient time domain analysis can provide further insights into lateral and torsional vibrations whirl behavior and effect of local wellbore curvatures on the drillstring performance. 1. INTRODUCTION Computer-aided modeling of drillstring dynamics is a powerful and widely used instrument for selection of optimal operational parameters for drilling oil and gas wells. Estimation of drillstring dynamics is based on the analysis of the BHA behavior as the main source of harmful drillstring vibrations. This part of the drillstring generally works in compressed state and most intensively interacts with the wellbore. Hard operational conditions of BHA cause constrained buckling effects. Contact interaction of deformed

2 pipes with wellbore along with rock cutting processes on the bit generates multiple excitations which can result in critical drillstring vibrations for some rotation rates. Computer-aided modeling of nonlinear dynamical processes that occur during BHA rotation in the wellbore is more time-consuming and is generally applied for scientific research. Currently two types of analysis are used for practical estimation of operational parameters on the stage of drilling planning. Study of the deformed shape of non-rotating BHA placed in the wellbore under action of expected forces referred to as Static Analysis is used for prediction of points of BHA-wellbore contact interaction. Forced frequency analysis of BHA placed in the wellbore the Linear Vibration Analysis is carried out to detect drillstring rotation rates that can result in BHA resonance vibrations and hits of BHA on wellbore walls. Simplified models of real excitations acting on an assembly are used to catch critical effects of BHA motion. Increased complexity of the problem occurs due to the necessity to account motion of drilling fluid in the annulus between an assembly and wellbore walls. Multiple approaches are used for modeling BHA dynamics. The greatest distribution to the industry was received by software products based on Finite Eelement Modeling (FEM) of an assembly [1]. The great ratio of axial to lateral dimensions [10 5 ratio] of BHA model and gross motion displacements of BHA parts cause the necessity of usage of large numbers of geometrically nonlinear finite elements (FE) in the model. The complexity of modeling variable contacts in bifurcating system increases CPU costs. The computing time is also critical for industry applications developers have to use rough meshing of BHA model being reconciled with an inevitable loss of accuracy of results. Approaches based on representation of BHA by a set of rigid bodies connected with viscous-elastic force elements known as the lumped parameter method [2] or finite rigid body approach [3] are also used for drilling dynamics study. Time domain and frequency domain simulations are carried out with methods of multibody system dynamics. Essential lack of the given approach is the discrete description of elastic and inertial properties of BHA model that causes to the necessity of usage of a large number of rigid bodies for the correct evaluation of deflected mode as well as natural frequencies and modes of an assembly. Flexibility of bodies is simulated efficiently by the Component Mode Synthesis (CMS) method implemented in up-to-date multibody system dynamics software packages (MSC.ADAMS SIMPACK Universal Mechanism etc.). The CMS method enables the correct description of spatial motion of a flexible body as a superposition of motion of local frame connected to a body and small flexible displacements relative to the local frame known as floating frame of reference. Equations of flexible displacements of a body are formulated with modal approach on the base of solution of geometrically linear FEM problem. Flexible bodies can undergo arbitrary large displacements as absolutely rigid bodies but their flexible displacements due to deformations must be small. The multibody system dynamics approach based on the usage of the CMS method for BHA model description is considered in this paper. The BHA model consisted of flexible sections connected with viscous-elastic force elements of high stiffness is proposed for time domain and frequency domain BHA dynamics study. Despite large absolute displacements of BHA components during the motion relative deformations of short BHA sections are small and can be correctly described by the CMS method. Custom software was developed on the basis of the Universal Mechanism software [4] to describe the BHA assembly and its dynamics. Several test problems related to evaluation of BHA deformed shape buckling prediction forced frequency analysis and modeling of transient dynamics of rotating BHA are considered. The applicability of the approach to BHA Static and Linear Vibration analyses is estimated by the comparison of simulation results with classical finite element solutions. 2. MATHEMATICAL MODEL The multibody system approach is used for time and frequency domain drilling analyses issues. The BHA model consists of flexible bodies interconnected with the help of viscous-elastic force elements of high stiffness (coupling elements). Flexible beams are used for modeling short uniform sections of BHA components. Equations of motion of flexible bodies are generated in accordance with the CMS method in the form of Craig- Bampton method. Drilling fluid effects are accounted for by adding frequency-dependent external damping and additional masses to the flexible beam equations [5]. BHA interactions with the wellbore are modeled with the help of discrete contact forces taking into account wellbore geometry EQUATIONS OF MOTION OF A FLEXIBLE BEAM The equation of motion of a flexible beam is as follows: is the matrix-column of generalized coordinates k is the matrix-column of generalized inertia forces are the generalized gravity applied and reaction forces and are the mass stiffness and damping matrices of the beam. The mass matrix and inertia forces are derived using the lumped-mass method. The CMS method in the form of the Craig-Bampton method [6] [7] is used to describe flexible displacements of the beams. For the main assumptions used for derivation of equation of motion of flexible beam see Appendix A.

3 2.2. BOTTOM-HOLE ASSEMBLY MODEL Flexible beams are used for modeling of BHA sections of uniform geometry and density (uniform sections). In accordance to CMS approach methodology equations of motions of beams are generated on the base of analytical solutions PIPE-WELLBORE CONTACT MODEL Contact interaction between BHA components and the wellbore is modeled discretely with the help of Circle- Cylinder contact force elements. The element models a compliant contact of a circle with a cylinder whose axis is set by a smooth curve (Figure 2). To describe the dependency of cross section of a hole from its depth the diameter of the cylinder can vary. Normal force vector Circle The uniform sections are connected in the assembly with viscous-elastic force elements of high stiffness (Figure 1). Viscous-elastic force element Bea Figure 1: Scheme of flexible beams connection. Bea As long as the flexibility and damping of the pipes are considered with flexible beam models the values of coupling element parameters (stiffness constants for three linear and angular directions and corresponded damping parameters) must theoretically be infinite. Finite values of the parameters are optimized to provide acceptable results with minimal calculation time. The stiffness constants are evaluated automatically on the basis of the connecting flexible beams characteristics. The stiffness constants are proportional to the maximal beam with factor Cylinder Figure 2: Circle-Cylinder contact model. Friction force vector The normal force ( ) depends on the penetration depth and rate. Friction force model can be described as follows: is the dynamic friction coefficient is the normal force is the sliding velocity is the empirical (small enough) value of sliding velocity. If the sliding velocity is not small the classical model of friction is used otherwise the viscous damping is considered. In the simulation the value is equal to where is the characteristic size (radius) of contacting bodies. The values of damping parameters are calculated from the stiffness inertia characteristics of the beams and the prescribed damping ratio ( ). The contact force element is added to each of the points of beam connections. The diameter of contact circle is equal to the maximal outer diameter of sibling beams. In fact the contact force is implemented between the wellbore and the end point of the flexible subsystem modeled the uniform section of greater diameter. The approach indicates that lengthy contact of a beam lying in the hole is simulated with two contact forces placed at the beam ends. The accuracy of the boundary conditions can be improved by some modification of contact force model.

4 2.4. SOLUTION PROCEDURES Time domain simulation is used for evaluation of equilibrium state of BHA in the wellbore. Initially the assembly model is placed along the wellbore centerline and completion of transient processes is fixed by attenuation of the mechanical system s kinetic energy. High efficiency of the numerical simulation of model motion is reached by using up-to-date numerical methods implemented in the Universal Mechanism program package [9] [10]. Parallel processing on multi-core computers is used for increasing the solver efficiency. Linearization of equations of the BHA model motion near the equilibrium state is used for evaluation of natural frequencies and modes as well as forced frequency analysis. The dependence of external damping and inertial characteristics of system from vibration frequency due to drilling fluid motion is taken into account. 3. TEST CASES Several tests were conducted to verify the feasibilities of the proposed approach. The accuracy of the evaluation of the deformed shape of an assembly its natural frequencies and modes and Euler and constrained buckling conditions were estimated by comparison of calculated results with theoretical and FEM solutions UNIFORM BEAM The uniform beam model was used for comparison of simulation results obtained by rigid-flexible models with analytical solutions. The 200-in length beam of 3-in diameter was modeled with four flexible beams. Each beam was described with finite element mesh including 10 geometrically linear elements. Twelve flexible modes were considered for each of the beams. The total number of coordinates of the test model was 72. The maximal error in evaluation of static deflection calculated for different types of loading was lower than 0.13% (Figure 3). (Figure 4). The amplitude-frequency characteristic obtained by the time domain simulation of harmonic-excited motion has shown agreement with analytical results [2.143 Hz] 2 [ Hz] 3 [ Hz] 4 [ Hz] Figure 4: Error in evaluation of first natural frequencies %. The values of critical axial loads resulted in Euler buckling of the beam have matched closely to the theoretical ones. In Figure 5 the critical force value is estimated from the plot of the first frequencies. First buckling shape Second buckling shape Figure 5: Evaluation of Euler buckling critical forces. Figure 3: Simulation of static deflection of uniform beam. The difference between the numerical and theoretical values of the first four natural frequencies was less than 0.25%

5 3.2. CONSTRAINED BUCKLING MODELING Test description Computer simulations was used in the prediction the BHA buckling which is considered a dangerous phenomenon. In this section we compare simulation theoretical and experimental results for helical buckling of the brass rod inside a plastic pipe. Theoretical and experimental are presented in the J. Wu paper [11]. The experimental setup for buckling of the rod in the 1-in pipe is shown in Figure 6. Measuring outfits are hidden. plastic pipe is the effective weight of the pipe per unit length in the wellbore is the radial clearance between the pipe and the wellbore is the inclination angle of the wellbore for a horizontal wellbore degrees. This expression is used for calculating the critical force of the brass rod in the test. F F hel F hel Top view brass rod F cr plastic pipe load cell fixed blocker Figure 6: J. Wu test. Experimental apparatus for buckling of rod in plastic pipe. The axial load is applied to the right end of the rod via hand-driven screw. Table 1 displays the rod and pipe parameters. Table 1. Rod and pipe parameters in the Wu test. Parameter Value Rod length m Rod diameter mm 3.2 Rod modulus of elasticity kpa Linear rod weight N/mm Pipe inside diameter mm 8.3 In accordance with theory and experimental data a rod lying on the lower side of a horizontal pipe will buckle if the axial load reaches the value called the critical buckling load. Initially the sinusoidal buckling occurs that is the rod buckles into a sinusoidal shape along the lower side of the pipe. If the axial load continues to increase after sinusoidal buckling occurs a helical buckling develops with a helix around the pipe wall. Axial force corresponding to the fully developed helical buckling is called true helical buckling load. Therefore three values of the axial load can be defined for helical buckling process (Figure 7) [11]. The critical load can be obtained from the expression for buckling a pipe in a wellbore derived by Dawson and Paslay [12] is the Young s modulus is the moment of inertia of the pipe brass rod load cell fixed blocker screw (1) Figure 7: Linear approximation of axial load during helical buckling process. The true helical buckling load equation is as follows: The average value of axial force during the helical buckling process can be written as follows: Another expression for critical buckling load was proposed [13] This value corresponds to if is computed by Eq (1). Table 2 displays the theoretical values of the forces for the test. Simulation Table 2.Theoretical force values for Wu test Force Value N Computer model of the rod includes twenty flexible beams. Ten flexible modes are used for each beam. In Figure 8 the comparison of simulation results and J. Wu experimental data is presented. The plot corresponding to the experimental data is digitized from the picture of the paper [11]. x (2)

6 The sinusoidal buckling occurs when the axial force reaches N (Figure 9). When the load increases full helical buckling with three coils develops. The value N of Fhel force is obtained by simulation. When the axial load is about N buckling form of the rod transforms from three coils to four coils. simulation between theoretical and experimental results in the beginning stage of the loading. The same effect is seen in Figure STATIC BHA ANALYSIS Validation of rigid-body multibody approach to solution of BHA Static Analysis was carried out on the basis of comparison of the results with FEM solutions. For example Figure 10 shows computed deformed shapes BHA-wellbore contact forces internal force factors and stresses experiment Figure 8: Axial load at the loading end of the rod in horizontal pipe. Simulation results and experimental data. Sinusoidal buckling Wall of the plastic pipe Sinusoidal buckling Right end view Helical buckling Right end view Helical buckling with three coils Helical buckling with four coils Figure 10: BHA Static Analysis results Figure 9: Rod buckling process in a horizontal pipe. According to simulation results the suggested simulation method provides acceptable accuracy. The force F hel obtained by simulation is close to the value in Eq. (2) and to the experimental data. In their paper [11] the authors note that the full helical buckling is observed when the axial load reaches 44 N. At the same time the authors point out the difference The comparison shows that the result of Static BHA Analysis calculated on the base of the proposed approach matches well with the results obtained in ABAQUS with geometrically nonlinear finite element model of the assembly but the corresponding computational effort to describe the model and simulation is much smaller. Static Analysis of a single scenario of BHA case loading takes about two minutes.

7 3.4. LINEAR VIBRATION ANALYSIS OF BHA Linear BHA vibration analysis corresponds to kinematic and forced harmonic excitation. The approach based on linearization of the equations of motion is widely used in the industry with relatively small computational expense. Vibration sources such as formation cutting processes on the bit contact interaction of pipes with borehole rotation of eccentric parts and operation of active BHA components (mud motors RSS units and so on) are taken into account. The approach features used for the analysis are described as follows: Any contact point found during evaluation of BHA equilibrium position is assumed to stay in contact during vibration analysis. Free parts of BHA can penetrate the wellbore without impact forces. Transient processes are not taken into account. Additional damping and added masses depending on excitation frequency according to Chen equations [5] are taken into account. Figure 11 displays simulation results for the maximal lateral displacements of BHA in a wellbore and the maximal values of forces and stresses for single excitation. imperfections of the shape of real drill collars eccentric mass is located in the beam s middle cross-section (Figure 11). Figure 12: Scheme of the test case. Simulation of drill collar rotation in the hole under constant torque and longitudinal compressive force F results in buckling effects. Figure 13 displays rotation of contact force vectors in the direction opposite to collar rotation fixed during simulation. Figure 11: Results of Linear Vibration BHA Analysis Future research will focus on estimation of vibration responses under the action of several monochromatic excitations applied at different points along the BHA BHA ROTATION MODELING Several tests were conducted to investigate feasibilities of the approach for transient analysis of specific dynamical effects of drill collar rotation. In the test case the collar pipe is represented by a jointly supported beam rotating in the straight hole. To perform Figure 13: Dynamic effects of buckled BHA rotation. 4. CONCLUSIONS A new approach for drillstring dynamics analysis has been developed on the basis of the CMS method of modeling rigid-flexible multibody mechanical systems. Specialized software has been developed for simulating the problems of BHA Static Linear Vibration and

8 Transient dynamics analyses based on the Universal Mechanism software [9]. The study has proved the feasibilities of rigid-flexible multibody system approach to CPU effective solution of various problems of computer-aided simulation of drillstring dynamics. [13] Lubinski A Developments in Petroleum Engineering Vol. One Gulf Publishing Company Houston TX pp ACKNOWLEDGMENTS The authors of this paper would like to acknowledge the support of Weatherford Ltd. and the Russian Foundation for Basic Researches grant а. 6. REFERENCES [1] Payne M. L Drilling Bottom-Hole Assembly Dynamics Ph.D. Thesis Rice University. [2] Bailey J. R. Biediger E. A. O. Sundararaman S. Carson A. D. Elks W. C. Dupriest F.E Development and Application of a BHA Vibrations Model. [3] J. Pabon N.Wicks Y.Chang B.Dow SPE and R.Harmer SPE Schlumberger 2010 Modeling Transient Vibrations While Drilling Using a Finite Rigid Body Approach SPE Deepwater Drilling and Completions Conference 5-6 October 2010 Galveston Texas USA. [4] Universal Mechanism [5] Chen S. S. Wambsganss M. W. W. and Jendrejczyk J. A Added Mass and Dumping of a Vibrating Rod in a Confined Viscous Fluid J. Applied Mechanics 43 pp [6] Craig R. R. Jr. Bampton M. C. C Coupling of substructures for dynamic analysis AIAA Journal Vol.6 No.7 pp [7] Craig R. R. Jr Coupling of substructures for dynamic analysis: an overview AIAA Paper No AiAA Dynamics Specialists Conference Atlanta GA April 5. [8] Mikheev G. V Computer-aided modeling of dynamics of systems of rigid and flexible bodies subject to small deformations Ph.D. Thesis Bryansk State Technical University. [9] Pogorelov D.Y On numerical methods of modeling large multibody systems Mechanism and machine theory Volume 34 pp [10] Pogorelov D. Yu Jacobian matrices of the motion equations of a system of bodies Journal of Computer and Systems Sciences International Volume 46 Number 4 pp [11] Wu J. Juvkam-Wold H.C. Lu R Helical Buckling of Pipes in Extended Reach and Horizontal Wells Part 1: Preventing Helical Buckling Transactions of the ASME. Journal of Energy Resources Technology September vol. 115 pp [12] Dawson R. and Paslay P.R Drillpipe Buckling in Inclined Holes Journal of Petroleum Technology Oct. pp

9 APPENDIX A DERIVATION OF EQUATIONS OF MOTION OF FLEXIBLE BEAM The main assumptions used for derivation of equations of motion of flexible beam are considered below. Positions of flexible body points are described using the floating frame of reference method. The local coordinate system (CS1) is linked to the body (Figure 14). The position of an arbitrary point K in the global coordinate system (CS0) can be presented as the sum of the radiusvector of the origin CS1 in CS0 and the vector of position of point K in CS1. z 1 z 0 x 1 1 r 01 y 1 r k1 rk0 K k d k H is the number of used modes is the N H modal matrix. The set of generalized coordinates of a deformable body includes six coordinates for the description of motion of the local frame and H modal coordinates related to flexible modal displacements. The number H depends on the type of modes used and the required accuracy of the solution. Small flexible displacements of a beam are considered as a superposition of fixed-interface normal modes and constraint modes. The description of the flexible beam model in accordance to the CMS method is carried out in several steps. Creation of a finite element model of a beam. Geometrically linear beam finite elements with 6 degrees of freedom at node are used for the description of model of uniform flexible beam (Figure 15). x 0 0 y 0 Figure 14: Position of flexible body. Let the vector be a sum of the constant vector of the point coordinates of the undeformed body and the vector of the flexible displacement. Position K in CS0 can then be presented as follows: (3) the superscript is the index of coordinate systems in which vectors are presented is the rotation matrix. Flexible displacements of the body are presented with modal approach. Equations of motion are created in the local frame by finite element method. These linear equations are written in terms of nodal DOF and their number can be extremely high. To reduce the number of equations the following approximate representation of nodal degrees of freedom is suggested: is the N 1 matrix-column of nodal DOF N is the number of DOF is the modes of flexible body is the modal coordinate (4) Selection of interface nodes of the finite element model (Figure 16). Internal nodes Interface nodes Figure 16: Interface and internal nodes of the flexible beam. As a rule the interface nodes are selected in the points that interact with other bodies of mechanical system via joints or force elements. Degrees of freedom in the interface nodes are called boundary or interface DOF. Other nodes of the finite element model of the beam are called internal. Two interface nodes located at the ends were used for the description of flexible beams implemented to the BHA parts modeling. The matrix-column can be written as follows: subscript i corresponds to internal DOF subscript b denotes boundary DOF. Figure 15: Scheme of beam finite element.

10 Calculation of fixed-interface normal modes. These modes are obtained as a solution of the eigen problem (5) when all boundary DOFs are fixed. is the number of internal DOFs is the stiffness matrix is the mass matrix is the th cycle frequency is the th fixed interface normal modes The low frequency modes are used for flexible beam motion description. The number of the modes is usually optimized with respect to the problem statement and required accuracy of the solution. Figure 17 shows a fixed interface mode example. Figure 17: Second fixed-interface normal mode. Calculation of constraint modes. A constraint mode is defined as the static shape of a beam when a unit displacement is applied to one of interface degrees of freedom while the remaining ones are fixed. Number of the constraint modes is equal to where is the number of interface nodes; is the number of degrees of freedom in the node. The stiffness matrix of the flexible beam can be written as follows: (5) Generalized mass and stiffness matrices of the beam are calculated according to the following expressions: Transformation of flexible shapes: ortho-normalization of the modes and deletion of six rigid body modes from the resulting set. To exclude the rigid body motion of the beam reference to local frame the eigenvalue problem (6) with generalized matrices of the beam is solved and six eigen pair corresponding to zero values are ignored. (6) Transformed modes are calculated according to the following expression: where is the matrix whose columns are. The resulting number of beam coordinates is equal to where U is the matrix. The equation of motion of a flexible beam is as follows: (7) is the matrix-column of generalized coordinates k is the matrix-column of generalized inertia forces are the generalized gravity applied and reaction forces and are the mass stiffness and damping matrices of the beam. The mass matrix and inertia forces are derived using the lumped-mass method. The matrix column looks like the following: The modes is the columns of the matrix Figure 18 shows examples of the constraint modes. The size of equation system (7) is equal to. x z y z x Figure 18: Constraint modes of the beam: the mode for displacement along Z (upper); the mode for rotation about X axis (lower). Calculation of generalized matrices of the flexible body. The modal matrix can be presented as the following: where is the matrix with columns.