Casing Design Methodology for Casing While Drilling

Transcription

1 Casing Design Methodology for Casing While Drilling Karunakar Charan Nooney Karunakar Charan Nooney Casing Design Methodology for Casing While Drilling

2 Casing Design Methodology for Casing While Drilling By Karunakar Charan Nooney in partial fulfilment of the requirements for the degree of Master of Science in Applied Earth Sciences at the Delft University of Technology, to be defended publicly on Wednesday December 16, 015 at 04:00 PM. Supervisor: Prof. Dr. Ir. J.D. Jansen Thesis committee: Prof. Dr. Ir. J.D. Jansen Prof. Dr. W.R. Rossen, Prof. Dr. A.V. Metrikine E.G.D. Barros An electronic version of this thesis is available at

3 Title : Casing Design Methodology for Casing While Drilling Author(s) : Karunakar Charan Nooney Date : December 16, 015 Professor(s) : Prof. Dr. Ir. Jan-Dirk Jansen Supervisor(s) : Prof. Dr. Ir. Jan-Dirk Jansen Postal Address : Section for Petroleum Engineering Department of Geoscience & Engineering Delft University of Technology P.O. Box 508 The Netherlands Telephone : (31) (secretary) Telefax : (31) Copyright 015 Section for Petroleum Engineering All rights reserved. No parts of this publication may be reproduced, Stored in a retrieval system, or transmitted, In any form or by any means, electronic, Mechanical, photocopying, recording, or otherwise, Without the prior written permission of the Section for Petroleum Engineering

4 Abstract In the current plans of Delft Aardwarmte Project (DAP), it is considered to perform the drilling operation by using pipes which remain in the well after drilling thus acting as a casing, the so-called Casing While Drilling (CwD) technique. Due to the absence of drill pipe tripping prior to casing the well, this technique results in reduced drilling time compared to conventional drilling. Additionally, potential downhole problems due to drill pipe tripping are precluded. This thesis presents a simulation based approach to selecting casing steel of suitable grade capable of withstanding typical loads encountered while drilling of the well and during its producing life. The developed algorithm is then used in the design of the casing string for the proposed DAP geothermal producer well. The algorithm first considers the effect of uni-axial stresses on casing due to defined burst and collapse pressure loads encountered due to loss of well control while drilling or in the production phase to make a preliminary selection. The effect of axial stress due to buoyed weight of casing and the bending stress due to wellbore curvature is then used to re-evaluate the design against the same collapse and burst loads. This is performed by using a bi-axial approach for the former and the Von- Mises triaxial stress criteria for the latter. A Johancsik torque and drag model developed in MATLAB is used to predict drag values during tripping. The bi-axial and Von-Mises stress analysis approach is repeated to include the effect of the computed pull-out drag forces. The associated torque values are used to compute torsional stresses and to identify casing connections of appropriate torque capacity. The final step in the algorithm is to simulate the loads occurring during drilling and calculate the equivalent Von Mises stress values throughout the casing string. Typical drilling loads considered include torque and casing lateral vibration which induce torsional and bending stresses respectively. It was identified that rather than bending stress due to whirling or buckling, torsional stress was more likely to cause casing string failure. This is due to its relatively higher magnitude and the weaker maximum torque capacities of conventional casing connections. Additionally, the MATLAB tools developed for analysing buckling and whirling are used to compute the critical load for inducing sinusoidal buckling as a function of wellbore inclination and the critical rotary speed at surface to induce lateral vibration for varying weight on bit (WOB) respectively. From the generated mode shapes, the bending stress magnitude at each node and therefore the points of maximum stress occurrence for bucking and whirling are also identified. ii

5 Acknowledgements First and foremost, I would like to thank my parents without whom none of this would have been possible. Their unconditional love and support is the foundation on which I have built my career thus far. Thanks are also due to my elder brother whose keen insight from having completed a Master s degree himself benefited me tremendously in the planning and execution of various activities throughout my MSc studies. I am extremely grateful to my thesis supervisor, Professor Jan-Dirk Jansen for accepting me as his student. I am constantly amazed by how he finds time to fulfil his responsibilities as thesis supervisor inspite of the numerous demands on his time. Without his understanding and guidance, this project would never have reached completion. I would also like to thank Professor W.R Rossen, Professor A.V Metrikine & Eduardo Barros for consenting to form part of the thesis assessment committee and evaluate this project. I would be seriously remiss not to acknowledge the moral support of my friends here in Delft, Anand Sundaresan, Akshey Krishna, Bharadwaj Rangarajan, Jeyakrishna Sridhar & Saashwath Swaminathan. Thanks for everything guys, I couldn t have done it without you! Last but not the least, I would like to thank the management at Cost Engineering Consultancy, Zwijndrecht for their cooperation in agreeing to defer the starting date of my employment so that I could complete this thesis. iii

6 Contents Abstract... ii Acknowledgements... iii Contents... iv 1 Introduction Casing While Drilling Non-Retrievable BHA Retrievable BHA Liner While Drilling Casing Pipe Connections for CwD Thesis Outline... 4 Design Algorithm Torque and Drag Analysis Implementation of Torque and Drag Model Model Validation Static Deflection Derivation of Governing Equation Results Deflection Validation of Model with Analytical Solution Slope & Bending Moment for various Weight-on-Bit Conditions Buckling Analysis Governing Equation Results Natural Frequency of Lateral Vibration for Rotating Casing Derivation of Governing Equation Results Comparison with Analytical Solution Variation of Natural Frequency with Applied Weight on Bit Application of Developed Concepts to DAP Producer Initial Well Data Stresses Considered Axial Stress Bending Stress Torsional Stress iv

7 7..4 Hoop Stress Radial Stress Von Mises Stress Power Law Fluid Rheology Model Casing Design Load Cases Surface Casing Intermediate Casing Production Liner Casing Selection Based on Uniaxial Loading Criteria Surface Casing Production Casing Axial Loading Wellbore Trajectory Surface Casing Intermediate Casing Liner Axial Load due to Well Bore Trajectory Combined Loading Collapse with Axial Loading Von Mises Analysis for Burst Loading Torque & Drag Analysis Axial Loading due to Drag Forces Torque Analysis Wellbore Pressure Distribution Drilling Loads Liner Bending Stress Due to Whirl /8 Casing Bending Stress Due to Buckling Von Mises Analysis of Drilling Loads during 7 Liner Section Conclusions Recommendations... 5 Nomenclature List of Abbreviations Bibliography Appendix A. Implementation of FDM for Beam Deflection Appendix B. Implementation of FDM for Buckling Analysis... 6 Appendix C. Implementation of FDM for Whirling Analysis v

8 Appendix D. Supporting Tabular Data Appendix E. Torque Analysis & Corresponding Selection of Casing Connection Appendix F. Fluid Hydraulics Frictional Pressure Losses vi

9 List of Figures Figure 1 Non Retrievable BHA [1]... Figure Retrievable BHA. Combination image generated from [] & [3]... Figure 3 Retrievable BHA for drilling with Liner [5]... 3 Figure 4 BTC with Torque Shoulder [36]... 4 Figure 5 Typical Buttress Threaded Connection [10]... 4 Figure 6 Overview of the Casing Design Process... 5 Figure 7 Downhole Forces on Casing [11]... 7 Figure 8 Force Balance on Drill String Element [13]... 8 Figure 9 Discretization of Drill String into Nodes [7]... 9 Figure 10 Snapshot of EXCEL spread sheet used for accepting Input String Data for T&D Model Figure 11 Sample Output Plots for Surface Hook Load & Cumulative Surface Torque Figure 1 1 1/4" OH Section Hook Load Measurements for Well South Sangu Figure /" OH Section Hook Load Measurements for SS Figure 14 Forces acting in radial direction [19] Figure 15 Variation of axial load in drill string [19] Figure 16 Plot of Deflections at Inclination of 10 Degrees Figure 17 Clamped beam with Pinned End on which is exerted a constant lateral load and axial compressional force. [0] Figure 18 Verification of Numerical Model Figure 19 Variation in Bending Moment for Hold Inclination of 10 Degrees Figure 0 Casing initially resting on lower side of wellbore Figure 1 Mode shapes at 0 degrees Inclination... 1 Figure Plot of numerical & analytical critical loads as a function of wellbore inclination... 1 Figure 3 BHA Whirl [5]... Figure 4 Section of BHA rotating about axis [19]... 3 Figure 5 Mode shapes for the first natural frequency of Lateral Vibration... 4 Figure 6 Dependence of natural frequency on the applied Weight on Bit... 5 Figure 7 Drilling Window for Wellbore Fluid Gradient... 8 Figure 8 Planned Trajectory of Producer... 8 Figure 9 Graphical representation of Design Loads vs. Rated Strength Figure 30 Measurement of Key Wellbore Trajectory Parameters Figure 31 Plot of Design Load vs. Rated Strength for Collapse Figure 3 Plot of Pick Up Drag Forces When Tripping Out of String... 4 Figure 33 Design Collapse Load vs. Strength for DAP Producer Figure 34 Increased FOS for Burst Loading Due to Effect of Drag Forces Figure 35 Cumulative Torque Observed at Surface for Various Drilled Depths Figure 36 Flow Path for Drilling Fluid in 7" Liner Drilling Phase Figure 37 Fluid Pressure Distribution for 7" Liner Drilling Phase Figure 38 Whirling Mode Shapes Figure 39 Natural Frequency of Lateral Vibration vs. WOB Figure 40 Bending Stress Due to Whirl Figure 41 Buckling Mode Shapes and Dependency of Critical Load on WOB Figure 4 Bending Stress due to Buckling in 9 5/8" Intermediate CSG Figure 43 7" Liner Von-Mises Stress Analysis for Drilling Operation Figure 44 Discretization of Drill String [19] vii

10 List of Tables Table 1 Error Percentages for Simulated Hook Load Results... 1 Table Flow Rate for Minimum Annular Velocity [30] Table 3 Factor of Safety [31] Table 4 Production Casing Specifications Table 5 Summary of Required Material Properties based on Design Loads Table 6 Surface Casing Properties Table 7 Intermediate CSG Properties Table 8 Liner Hold Section Properties Table 9 Build Up Section Properties Table 10 Power Law Input Viscometer Measurements Table 11 Casing String Design for DAP Producer Geothermal Well... 5 Table 1 Von Mises Stress Analysis at Survey Points for Burst Loading Scenario Table 13 Simulated Pick-Up Drag Values at Survey Points for 7 Liner Drilling Table 14 Von Mises Analysis at Various Survey Points for 7'' Deviated N80 Liner Section Table 15 Simulated Bending Stress due to Whirling for 7 Liner Stand (30 M) Table 16 Von Mises Tri Axisal Stress Analysis of 7" Liner for Drilling Conditions at Survey Points Table 17 Bending Stress due to Buckling for 9 5/8" Intermediate CSG Stand (30 metres)... 7 Table 18 Torque Analysis at Survey Points for 7" N80 Liner Section Table 19 Connection Make-Up Torque Capacities from Manufacturer Catalogue Table 0 Pressure Loss Computed By Power Law Rheology viii

11 1 Introduction 1.1 Casing While Drilling The primary motivation for this thesis is the plan of the Delft Aardwarmte Project (DAP) to drill the producer geothermal well by employing the CwD technique. Due to the absence of drill pipe tripping, this technique results in reduced drilling time compared to conventional drilling whilst also precluding numerous downhole problems. The simulation based approach presented in this thesis is designed to select a suitable grade of casing steel for the DAP geothermal producer well but this algorithm can also be extended towards composite materials as well. The reduced weight of composite materials in comparison to steel would allow DAP to use rigs of smaller draw-works capacities without reduction in target depths thereby resulting in significant cost savings. The CwD has the following advantages in comparison to conventional drilling: Eliminates the need for tripping of drill string prior to casing the well. The primary benefit of doing so is the reduction in rig operating time. Additionally, potential downhole problems such as loss of well control, surge and swab pressures when the casing is being run in or hole collapse due to presence of unstable zones in the wellbore can be avoided. Problematic thief zones which absorb drilling fluid completely, thereby causing a loss of primary well control can be bypassed in this technique due to the plastering effect of the casing. Due to the low annular clearances between casing and borehole wall, the casing effectively smears the drilling fluid particles into the borehole wall creating a superior filter cake thereby bypassing these zones. Low annular volumes enables higher flow velocities which facilitates hole cleaning. It is not without its disadvantages however and these are: Low annular clearances lead to higher frictional flow losses which necessitate the use of higher capacity pumps for same drilled depths in comparison to conventional drilling In order to execute a CwD operation, the rig hoisting equipment (usually the top drive) has to be modified to accommodate the casing. Additionally, other special equipment such as torque rings, modified elevators and tongs have to be employed which result in a further increase in costs There are two variations in bottom hole assemblies (BHAs) used for this technique which are applied based on the well trajectory and target depth requirement Non-Retrievable BHA This technique is used for shallow depth vertical sections in formations with soft to medium hardness levels only. Typically, the top hole section is drilled using this BHA. No drill collars are used however stabilizers may be used in regular intervals of one stand. Other cementing equipment such as float shoe and collar may also be used. Typically, special polycrystalline diamond compact (PDC) bits which are soft enough to be drilled through by conventional bits used in the next drilling phase are preferred. The flow path for circulating of drilling fluid is identical to conventional drilling operations. 1

12 Figure 1 Non Retrievable BHA [1] 1.1. Retrievable BHA Figure Retrievable BHA. Combination image generated from [] & [3] The retrievable BHA as the name suggests can be retrieved by wireline after the section has been drilled prior to cementing. The primary components are the pilot bit for drilling the initial hole, an under reamer for subsequently enlarging it, rotary steerable or mud motors, Measurement While Drilling (MWD)/ Logging While Drilling (LWD) tools and the drill lock/latch assembly. The DLA is locked in such a way to prevent relative axial and torsional motion between the BHA and the casing [4]. Seals used in the DLA as shown in Figure prevent the flow of drilling fluid into the casing and instead divert into the BHA [4]. Any type of formation can be drilled at any depth without placing any restrictions on wellbore inclination & depth or formation hardness.

13 1.1.3 Liner While Drilling In Liner while Drilling operations, the drill pipe used to run in the liner includes the liner hanger for supporting the weight of the liner and the BHA for drilling. Consequently, the circulation path for drilling fluid is entirely through the inner drill string as a result of which, the entire liner is not exposed to internal fluid pressure as shown in Figure Casing Pipe Connections for CwD Figure 3 Retrievable BHA for drilling with Liner [5] Typical casing connections are designed for one time use only where the casing connection is made up at surface and run into the hole. They are not designed for handling torsional and alternating stresses induced by rotation of the string. Therefore special casing connections have to be designed with high torque transmission capability. As per [1], [6], [7], [8]& [9] the preferred connections used are the API Buttress Threaded Coupled (BTC) connections with metal to metal seal. Depending upon the specific well trajectory, these connections may also provide insufficient torque capacity. Consequently, special metal torque rings are inserted between casing threads to further increase the maximum torque capacity by providing a positive torque shoulder which prevents thread crushing by over-torqueing and acting as a spacer between the two threads. 3

14 Figure 5 Typical Buttress Threaded Connection [10] Figure 4 BTC with Torque Shoulder [36] 1. Thesis Outline Chapter presents the design algorithm to be followed for the selection of casing steel grade with a brief discussion of each step in the sequence. Chapter 3 introduces the torque and drag model starting with a brief discussion of the need for calculating wellbore torque and drag forces and the subsequent implementation of the Johancsik model in this thesis by using a combination of an MS EXCEL spread sheet for accepting input data and MATLAB for computing actual forces and plotting the results. Steps taken for checking the accuracy of the model are also presented. Chapters 4, 5, and 6 are concerned with the theoretical derivation of the governing equations or Eigen value formulations for buckling and whirling analysis. The starting point is the derivation for the deflection of the string under the influence of the external load applied at surface. In addition to the theoretical derivation, the governing equations are solved using the finite difference method in MATLAB for which the details are contained in Appendix A, Appendix B and Appendix C. Each model is validated by comparing with analytical solutions from literature. Chapter 7 focuses on the application of the algorithm and supporting MATLAB tools developed, to the DAP producer well casing design. The implementation of the previously defined algorithm along with the analysis of results obtained is described comprehensively. Supporting data for bending stress due to Buckling & Whirling, Torque, Drag & Von Mises stress obtained from the MATLAB tools is tabulated in Appendix D & E for reference. Appendix E also presents a simple example of selecting a suitable buttress casing connection from a manufacturer catalogue on the basis of rated make-up torque capacity. Implementation of the hydraulic model based on the Power law rheology model for calculating the wellbore pressure distribution is reported. Supporting hydraulic calculations are summarized in tabular format in Appendix E. Lastly, in Chapter 8 the conclusions derived from the thesis work are presented and suggestions are made towards extending specific aspects of this thesis. 4

15 Design Algorithm Figure 6 Overview of the Casing Design Process 5

16 In this section, the overall process of selecting the casing steel for the CwD application is conveyed in the form of an algorithm displayed in Figure 6 above. The starting point of the algorithm is the formation strength and fluid pressure gradient data (section 7.1) needed for calculating bottom hole pressure to maintain hydrostatic balance with formation fluid. The fluid properties of plastic viscosity, yield point and shear rate are used to calculate the wellbore dynamic fluid pressures on the basis of the power law rheology model as shown in Appendix F. In the next step, the collapse and burst loads are estimated for making the preliminary selection of casing on the basis of the uniaxial loading criteria (section 7.4). The selected grade of casing steel is also verified for axial loading due to buoyed weight and bending due to wellbore curvature (section 7.6). As axial loads reduce collapse resistance of the casing string, a combined load analysis for hoop and axial loads is performed (section 7.7.1) along with a tri-axial stress analysis for the burst scenario (section 7.7.). If in each of these cases, the ratio of material yield strength to the equivalent stress is found to be greater than the design factor of safety, than the design is considered to be effective. The next step is the simulation of torque and drag forces induced in the casing string when drilling and tripping out of the wellbore respectively (section 7.8 ). The drag forces are used to compute an increased axial load for which the combined loading analysis is repeated to ensure that the casing string does not fail to this increased axial load (section ). If failure occurs, then the next grade of casing steel is selected and the process is repeated again starting with the revised torque and drag analysis. If however the design does not fail, then the computed torque values (Appendix E ) are used to calculate the torsional stress. In the final step, torsion together with axial stress due to whirling are included in the von Mises triaxial stress analysis (section 7.10) to simulate the effect of combined loads on the casing string during the drilling process. The torque values are also used to select the casing buttress connection of required torque capacity (Appendix E ). The radial and hoop stress are determined by the dynamic fluid pressures inside the casing and in the casing open hole annulus and are obtained from the result of the hydraulic model. While the buckling and torque & drag analysis assume that the casing string is always in contact with the wellbore, the whirling analysis considers the casing to be perfectly centered such that the radial clearance is uniform throughout. These assumptions are needed to overcome the difficulty associated with assessing the contact points in a realistic scenario where there are many uncertainties such as wellbore tortuosity, washed-out holes, and varying diameters of casing centralizers, stabilizers and collars. In short, the nature of the analysis contained in this thesis is more qualitative in nature rather than quantitative. 6

17 3 Torque and Drag Analysis Figure 7 Downhole Forces on Casing [11] The analysis of torque and drag force encountered by the casing string during the process of tripping in and out of the wellbore is critical to the design process. It is often the limiting factor in drilling deviated and extended reach wells as a result of which trajectories are designed to minimize normal contact forces and consequently the torque and drag as much as feasible. Torque and drag forces are generally caused by poor downhole conditions such as key-seating, sloughing shales etc. or due to sliding friction associated with the wellbore trajectory. Sliding friction acting between the wellbore and the casing string gives rise to an increase in the tensile force acting along the longitudinal axis of the casing string which is known as drag. Because friction acts in the opposite direction of string movement, drag causes an increase in the hook load measured at surface above the free rotating weight of the string during pull out and a reduction in hook load while running in. The increased axial force component during pull out significantly reduces the collapse resistance of the casing and the maximum along- hole depth capable of being reached by a rig with a fixed hook load capacity. The reduction in the axial force when running in can limit the free movement of the string under its own weight into the hole at higher inclinations and can lead to buckling if the driller exceeds the critical weight on bit applied at the surface in order to push the string deeper. When the casing string is rotating in the wellbore with a weight on bit during drilling, the torque available at the bit is much less than the external torque applied at surface by the rotary or top drive. This is due to losses occurring throughout the casing string due to the frictional forces acting at the point of contact between the wellbore and the casing string creating a resisting moment with vector direction opposite to that of casing rotation. As torque is directly proportional to the frictional force, increase in drag forces is usually accompanied by increasing torque loss as well. Consequently, the limiting factor for rotating the casing while reciprocating (for better conditioning of drilling fluid) or for drilling deeper and with greater inclination is often the maximum torque deliverable at surface by the rig and the torque carrying capacity of the casing connections. 7

18 For designing the casing string, it is therefore necessary to use a torque and drag model which can simulate the various loads during drilling and tripping operations so that the requisite grade of casing steel with yield strength greater than design axial and torque loads by a planned factor of safety can be selected. As per [1]the most widely used model in the industry is the soft string model first developed by Johancsik et al. [13]. The Johancsik equations were later rewritten in differential form by Sheppard [14] who also included the effect of buoyancy due to mud pressure in the model. The Johancsik model assumes that Torque and drag are caused primarily by sliding friction force with other smaller contributors neglected The drill string is in continuos contact with the borehole wall Normal (side) forces due to pipe bending stiffness are neglected. ie, the drill string is modelled as a flexible chain or soft string model with no bending moments Figure 8 Force Balance on Drill String Element [13] Johancsik states that sliding friction force is the product of coefficient of dynamic friction between drill string and borehole wall and the normal force acting at the point of contact of the two surfaces. He then calculates the normal force or net side load, by performing a force balance as shown in Figure 8. Net side load is equated to the normal components of the axial force due to bending of drillstring in curved (build up or drop off) section of wellbore and gravity force due to its weight. Johancsik discretizes the drill string into elements of finite length which transmit incremental axial force and torque to the next section. The analysis starts from the bit and proceeds towards the top. As shown in Figure 9 at node point i, the axial force transmitted by node point i 1 is used as the input for calculating the normal force acting uniformly on the element Si Si 1 and the axial force F. Thus, axial force and torque values are cumulatively summed up to obtain the actual loads. i 1 8

19 Figure 9 Discretization of Drill String into Nodes [7] The incremental form of the Johancsik equations for axial force and torque as presented in [7] are: n i 1 i Fn Fo ( si si 1) wi cos( ) ini i1, (3.1.1) n o i i i i1 n r N, (3.1.) where, F i w s Axial load at node i, N Buoyed specific weight of casing, N Distance coordinate along the measured depth of wellbore trajectory, m Friction coefficient Inclination angle, radians Azimuth angles, radians Torque, Nm The contact force by: Ni computed from the force balance as used in(3.1.1) and (3.1.) above is given i1 i i i1 i1 i i i1. (3.1.3) Ni ( si1 si ) wi sin Fi sin Fi si1 s i si1s i From(3.1.3), it can be seen that the normal force vanishes for vertical wellbores as inclination reduces to zero and azimuth remains constant and hence the model returns zero drag and torque values for vertical sections. This is however untrue in practice because the wellbore cannot be drilled strictly vertical and will always possess low inclination values upto 3~4 degrees. 9

20 3.1 Implementation of Torque and Drag Model The Johansik model is implemented in MATLAB using an excel spread sheet to accept the input parameters of drill string specifications and the well survey data (Measured Depth, Inclination & Azimuth). Other input parameters include Drilling Fluid Density Friction Factor WOB for Drilling TOB in case of downhole motor Figure 10 Snapshot of EXCEL spread sheet used for accepting Input String Data for T&D Model The output of the model is the plot of drag and torque forces acting on the casing string at a particular bit depth. It also computes the plot of hook load measured at surface versus the drilled depth which is used for estimating the required rig draw works capacity. It is also used for validating the model by comparing simulated hook loads with real time data obtained from field measurements. Figure 11 Sample Output Plots for Surface Hook Load & Cumulative Surface Torque 10

21 3. Model Validation For the purpose of verifying the accuracy of the torque and drag model, real time hook load measurement data from offshore well South Sangu-4 drilled by Santos Ltd. is used to compare against the simulated hook loads. This data is obtained from MSc thesis report of T.Chakraborty [15]. The author back calculates the friction factor for this well from the real time measurements using the Halliburton Well Plan simulator. These same friction factors along with drill string specification and other input parameters are fed into the developed torque and drag model to compare with the Santos real time hook load data set for an 1 ¼ and 8 ½ open hole sections. Figure 1 1 1/4" OH Section Hook Load Measurements for Well South Sangu-4 Figure /" OH Section Hook Load Measurements for SS-4 11

22 Table 1 Error Percentages for Simulated Hook Load Results Well South Sangu 4 1 ¼ OH 8 ½ OH Friction Factor, Maximum POOH 6.87 % 11.11% Error Maximum RIH Error 8.18% 7% As can be seen from the table above, there is an average error of approx. 8.3% between the simulation results and real time data. This can possibly be attributed to sophistication of the WellPlan simulator. Although it is also based on the soft string model [16], it also considers effect of contact surface area [17] and Hydrodynamic Viscous Drag forces [18]. The latter is induced on the string when tripping by the drilling fluid in the wellbore. Implementation of the same would require a more realistic hydraulics model and is beyond the scope of this thesis work. 1

23 4 Static Deflection 4.1 Derivation of Governing Equation The static deflection analysis begins with the derivation of the governing differential equation for the lateral displacement of the casing string by considering a section of the casing of length dz shown below. The z axis is chosen as the longitudinal axis of the borehole with the origin at the bit. The analysis will be carried out for one single of casing pipe of standard length. Figure 14 Forces acting in radial direction [19] Let the radial component of the buoyant weight of steel per unit length be defined as: wy Agh sin. (4.1.1) Similarly, the longitudinal component of buoyant weight of steel per unit length is: wz Agh cos. (4.1.) Additionally, let = Density of drilling fluid, kg / m m 3 = Density of tubular, kg / m m h 1, buoyancy factor N = axial force acting through the casing F 0 = applied weight on bit y = lateral deflection in the y-z plane s y =shear force in the drill string in the y-z plane = Inclination from the vertical 3 13

24 The axial load varies linearly throughout the casing string as shown in Figure 15. At the drill bit, the load is compressive in nature owing to the weight of the overlying portion of the string slacked off by the driller in order to exert the weight on bit. This is why drill collars possess a greater internal cross-section to reduce the likelihood of buckling. Towards the top, the drill pipes are in tension due to the weight of the string underneath. The point at which the tensile and compressive forces negate each other thereby reducing the axial force to zero is known as the Neutral point indicated by the pointl. In general, the drill string is designed in such a way that the neutral point falls within the ' 0 BHA. The applied weight on bit is used to determine the position of the neutral point by using the relation: F Agh cosl. (4.1.3) Thus the axial load is given by the relation: 0 0 N F0 wz z. (4.1.4) Figure 15 Variation of axial load in drill string [19] From [0], we know that the angle of deflection of the beam with respect to the longitudinal axis: dy, (4.1.5) dz and the shear force: S y 3 d y EI. (4.1.6) dz 3 From Figure 14, we can form the equilibrium equation of forces in the radial direction as: dsy dn sy dz wydz sy N sin N dz)(sin d) 0. (4.1.7) dz dz For small deflections, sin( d) d. (4.1.8) Substituting (4.1.8) in(4.1.7), we get: 14

25 dsy dn d dz wydz dz N dz 0. (4.1.9) dz dz dz Substituting(4.1.4),(4.1.5) & (4.1.6) in(4.1.9), and dividing throughout by z : 4 d y dy d( F0 wz z) d y EI w ( 4 y F0 wzz) 0, (4.1.10) dz dz dz dz d 4 y ( 4 0 z ) d y dy z y EI F w z w w. (4.1.11) dz dz dz Substituting (4.1.3) in(4.1.11), and dividing throughout by contained within the casing string, we finally obtain: ACM to account for the drilling fluid 4 EI d y gh cos d y dy ghsin ( l 4 0 z) ACM dz C M dz dz. (4.1.1) CM Equation (4.1.1) is the static equilibrium equation for forces acting on a section of the casing string in the y-z plane. A similar method can be applied to obtain the equilibrium equation for the x-z plane which is similar to (4.1.11) except for the presence of the lateral gravity component w. The two equations can be combined using polar coordinates to obtain the complete deflection equation in the wellbore. Equation (4.1.1) can be expressed in a dimensionless form as follows: y EI, (4.1.13) ' y 0 4 ACmL t, (4.1.14) 0 z w, (4.1.15) L w l y w, (4.1.16) L l L ' 0 0. (4.1.17) Making use of these terms, the following dimensionless equation is obtained: 4 ' ' ' d y ghcos ' d y dy ghsin 4 l0 w. (4.1.18) dw L0C m dw dw L0C m 15

26 Equation (4.1.18) can be rewritten in the form of a series of a series of algebraic equations by using the finite difference method [19] and solved at various node points throughout the drill string in order to find the nodal displacements as shown in Appendix A. 4. Results 4..1 Deflection Figure 16 Plot of Deflections at Inclination of 10 Degrees As a test case, deflection of a drill collar with outer and inner diameters 7.5 inches (190.5 mm) and.81 inches (71.37 mm) respectively at an inclination of 10 degrees to the vertical was plotted in Figure 16 and as expected, with increasing weight on bit, the observed deflection is found to increase. 4.. Validation of Model with Analytical Solution The analytical solution considered here is of a beam which is clamped on one end and pinned on the other end. It is subjected to an axial compressive force and a constant lateral force (gravity). 16

27 Figure 17 Clamped beam with Pinned End on which is exerted a constant lateral load and axial compressional force. [0] As per [0], the analytical expression for deflection in this situation is given by: where, X = Deflection, m ux 4 ql cos u ql x l x Mb sin kx x x l 1 ( ). (4..1) 4 164EIu cosu 8EIu P sin kl l P = Longitudinal compressive force, N l = Length of beam, m q, Buoyed uniform lateral load, N m kl l P u EI M b, Bending moment = ql ( u), Nm 8 ( u) ( u) Trigonometric factor representing influence of P on X ( u) 3(tan uu). u 3 Trigonometric factor representing influence of P on deflection angle at the beam ends = u u tan u. Numerical values of the functions ( u) and ( u) for varying u are tabulated in [0]. The numerical deflection for the test drill collar was then computed with an inclination of 90 degrees on the well trajectory. Both curves were then plotted on the same graph as shown in Figure

28 Figure 18 Verification of Numerical Model From Figure 18, it can be seen that the results of the numerical model are in close agreement with the analytical solution for deflection. 4.3 Slope & Bending Moment for various Weight-on-Bit Conditions d y As per [0], bending moment of a beam is EI which is then computed at each node point for dw the drill collar test case in section 4. using the Finite Difference method. The result is plotted for the test drill collar case in Figure 19. We see that as the Weight on Bit increases, the Bending Moment also increases. Figure 19 Variation in Bending Moment for Hold Inclination of 10 Degrees 18

29 5 Buckling Analysis 5.1 Governing Equation When the weight on bit causes the axial load to increase beyond a certain critical value, the drill string becomes unstable and buckles transversely to come in contact with the wellbore. When drilling inclined holes, the drill string in spite of resting on the lower side of the borehole will first buckle into a sinusoidal shape. Upon further increase of the weight on bit, the drill string will eventually transition into a helical shape when the axial load approaches the second critical load. Determining the limiting value of the allowable weight on bit is an important part of the well design process due to the numerous disadvantages associated with buckling such as failure of down hole tools, BHA being subjected to fluctuating fatigue loads and the reduced drilling progress. Figure 0 Casing initially resting on lower side of wellbore In the following analysis, the tendency for drill strings to buckle in inclined well bores will be studied by determining the critical load for sinusoidal buckling when the string is in contact with the borehole wall. This is done by using the finite difference method to discretize the governing differential equation and reshape it into an Eigen value problem. The governing differential equation is derived in [1]by combining the equations for deflection in the y-z (4.1.1)& the x-z plane using complex coordinates to form a fourth order non-linear differential equation with variable coefficients. This equation is linearized by a Jacobian transformation [1]to obtain the equation for the deflection in the y-z plane as: 4 d y d y dy wy EI ( F 4 0 wzz) w 0 z y. (5.1.1) dz dz dz c where c D D, radial clearance between casing and wellbore wall in m. Wellbore Casin g In equation(5.1.1), the first term on the left hand side corresponds to the flexural rigidity of the tubular. The second term on the left represents the effect of the uniformly varying axial load along the longitudinal axis of the casing on the lateral deflection. The third term quantifies the gravity effect on the casing string which is the buoyed weight per uniform length. Finally, the fourth term represents a restoring force. From Figure 0, it can be seen that the natural tendency of the casing is to rest on the lower side of the wellbore. When the casing is displaced laterally or sideways from this 19

30 position during buckling, the resistance of the wellbore wall to this motion induces a restoring force in the casing proportional to the magnitude of its displacement from the equilibrium position. The radial clearance term c, can be dimensionalized as: c D D ' Wellbore Casin g, (5.1.) L where L = length of casing section between drill bit and stabilizer, m. The same procedure used to obtain the dimensionless form of the governing equation for static deflection in section 4.1 is repeated here to obtain the scaled governing equation for the stability analysis in the y-z plane. By substituting the dimensionless parameters (4.1.13) to (4.1.17) & equations (4.1.1),(4.1.), and (5.1.) in (5.1.1), we obtain d y ghcos d y dy ghsin dw L C dw dw L C c 4 ' ' ' ' ' 4 l0 w 0 y ' 0 m 0 m. (5.1.3) The homogenous linear differential equation (5.1.3) is discretized using the finite difference method with similar clamped & pinned boundary conditions described in Appendix A and then rearranged into an eigen value problem as: A y B y, (5.1.4) A By 0 Where = weight on bit multiplier for determining critical load.. (5.1.5) The Eigen value problem defined in (5.1.5) is solved by implementing the finite difference technique in Appendix B. 5. Results The following curves were plotted for the previously defined drill collar test case using MATLAB to represent the buckling of the casing string in the wellbore. The deflection is scaled to the annular clearance between the drill collar outer diameter and wellbore diameter specified by the chosen bit diameter. In Figure 1, the eigen vector solution of equation (5.1.5) is plotted as the buckled mode shape of the casing string in a vertical wellbore. The numerically computed solution is then compared with the standard analytic formula for the critical load to induce sinusoidal buckling in inclined or vertical wellbores as presented in []: F crit EI Agh sin. (5..1) r 0

31 Figure 1 Mode shapes at 0 degrees Inclination Figure Plot of numerical & analytical critical loads as a function of wellbore inclination From Figure, we see that the numerical solution computed corresponds reasonably to the analytical solution for increasing weight on bit and angle of inclination. It is also observed that the critical load increases at a much faster rate at lower wellbore inclinations. 1

32 6 Natural Frequency of Lateral Vibration for Rotating Casing In order to gain an insight into the dynamic loading conditions of the BHA, the lateral or bending vibrations will be studied in this section with the aim of determining the natural frequencies and their dependence on the applied weight on bit. Historically, of the three different vibration modes, lateral vibrations were the last to be studied, as they typically went undetected at the surface owing to their relatively higher frequency which caused them to attenuate much faster. They have been identified as the leading cause of drill string failure by [3]. They can be caused by amongst other things, improperly balanced, bent or off-centre BHA components [4]. Interaction of the drill bit with the formation or of the drill string with the wellbore may also lead to lateral vibrations. Lateral vibrations are a complex phenomenon, an important subset of which is the forward synchronous whirl. When the instantaneous centre of rotation of the BHA is eccentric or away from its centre of gravity, a whirling motion occurs and when the BHA rotates about its instantaneous centre in the same direction as its revolution around the borehole axis due to the torque applied by the rotary table, it is known as forward synchronous whirl. By performing an Eigen value analysis of the natural frequency of lateral vibration, the critical rotary speed at which forward synchronous whirl will occur can be theoretically determined and can then be used by the driller as a rough guide on which rotary speed range for drilling ahead without causing resonant vibrations. Figure 3 BHA Whirl [5] Another significant subcategory is the backward whirl which occurs when the directions of rotation of the BHA about its instantaneous centre and the rotary table are different. One of the causes of backward whirl as described by [6] and [3]is the friction between the stabilizers and the wellbore which can induce the backward whirl of the entire BHA.

33 6.1 Derivation of Governing Equation Figure 4 Section of BHA rotating about axis [19] For determining the governing differential equation of lateral vibration for a rotating BHA, the inertia force must be defined so that it can be included in the subsequent force balance equation as: ( d y sin ) i M F A C b t. (6.1.1) dt Where F i = inertia force = rotary speed b = offset of the centre of gravity of the section from the geometric centre along the Y-axis C M of BHA M M m 1 ' s, added mass coefficient to account for drilling fluid contained per unit length Recalling the previously defined equation (4.1.11) for the static force equilibrium in the Y-Z plane, the aspect of rotation of the BHA is included in it by substituting the inertia force thereby obtaining: d 4 y ( 4 0 ) d y dy z ( ) sin z d y M y M EI F w z w A C w A C b t. (6.1.) dz dz dz dt This is the governing equation for lateral vibration from which the resonant frequencies are to be determined. To execute the Eigen value analysis, the excitation force on the RHS in equation (6.1.) is considered to be zero. It is also assumed that the lateral force due to gravity w will not influence the natural frequency and hence can be ignored. Therefore, equation (6.1.) reduces to: y 3

34 d 4 y d y dy 4 0 z z d y M EI ( F w z) w A C ( ) 0. (6.1.3) dz dz dz dt Let standard harmonic function described below be the solution to (6.1.3) y.. y i Substituting (6.1.4)&(6.1.5) in(6.1.3), and dividing throughout by y z e t, (6.1.4) i t e. (6.1.5) A CM : 4 EI d y gh cos d y dy ( l 4 0 z) y 0 M M A C dz C dz dz. (6.1.6) Equation (6.1.6) is now the equation to be discretized and solved in similar fashion to previous examples as an Eigen problem in the form of: A y B y, (6.1.7) A By 0. (6.1.8) Where is the Eigen value which when multiplied with the rotary speed is yields the natural frequency. Implementation of Finite Difference method to solve this Eigen value problem is detailed in Appendix C. 6. Results 6..1 Comparison with Analytical Solution Figure 5 Mode shapes for the first natural frequency of Lateral Vibration 4

35 Using MATLAB, the Eigen value statement was solved for the example drill collar of outer and inner diameters 7.5 inches (190.5 mm) and.81 inches (71.37 mm) respectively rotating at a speed of 10 RPM with no applied weight on bit at 0 degrees inclination from the vertical. The mode shapes shown in Figure 5 above are plotted for a normalized deflection where the maximum deflection value of 1 corresponds to the maximum annular clearance between the drill collar outer diameter and the borehole diameter defined by the specified bit diameter. The computed natural frequency was found to be 0.73 Hz (43.67 RPM). To confirm the accuracy of the tool, the solution was compared with the analytical solution for the bending natural frequency of a beam with similar boundary conditions of simply supported or pinned at one end and clamped or fixed at the other end [7] given by: EI Hz (6..1) 4 AL From the above, it is confirmed that the tool developed is in close agreement with the analytical solution. The reason for the computed natural frequency being higher than its analytical counterpart is because although the weight on bit is zero, the linearly varying tension (equal to the weight of the buoyant drill string beneath that point) throughout the drill string increases the resistance to bending and hence results in a higher natural frequency. 6.. Variation of Natural Frequency with Applied Weight on Bit Figure 6 Dependence of natural frequency on the applied Weight on Bit From Figure 6, It can be seen that as the applied weight on bit increases, the natural frequency reduces. This phenomenon will appear intuitive after examining the terms of the Eigen value 5

36 statement for the governing equation (6.1.6) and comparing it with the analytical expression for buckling under excitation [8]reproduced below: k m GkG0 v 0 (6..) k - Stiffness which corresponds to the flexural rigidity 4 d y EI dz 4 kgo - Geometric stiffness term capturing the effect of axial loading. The minus sign indicates compressive load which is consistent with the typical load distribution in the BHA. This term d y dy corresponds with ( F0 wzz) w z dz dz - Natural frequency corresponding to rotary speed Thus it can be clearly seen from(6..), when k GO increases as shown in Figure 6, the natural frequency decreases until eventually it reduces to zero. The corresponding weight on bit is the buckling load and hence a further check on the accuracy of the buckling result computed earlier in section 5. is obtained because of the agreement in the computed buckling load of 306 kn in both cases for similar input conditions of wellbore inclination (vertical), tubular material properties and dimensions. 6

37 7 Application of Developed Concepts to DAP Producer The concepts discussed thus far can now be applied towards selecting the appropriate steel grade for the DAP geothermal producer well. This will be done by following the previously proposed algorithm for casing string design. 7.1 Initial Well Data The following assumptions are made regarding the initial well data: o Overburden gradient = 6.89 kpa / m o Entire well is drilled with same mud weight o Surface temperature = 5 C o Assume ideal gas behaviour for Methane 3 Formation fluid density is taken to be 144 kg / m which is obtained from the study of total dissolved solids observed in the water samples contained in the report of IF technologies Netherlands This means that in any forthcoming calculations, mud gradient g ( ) kpa / m. (7.1.1) To maintain sufficient overbalance, the gradient of the drilling fluid is taken as 1 kpa / m It was not possible to obtain any data pertaining to the formation fracture gradient from the DAP. Therefore the Hubbert and Willis equation [9] for wellbore injection pressure required to initiate fracture propagation was used as an approximation of the formation strength gradient as follows: 1 P F (1 ) 3 D Where, F Fracture Gradient in psi/ft. P Pore pressure gradient in psi/ft. D 1 {1 ( )} psi / ft 15.0 kpa / m. (7.1.) Note that this formula assumes the overburden gradient to be 1 psi/ft. which is a reasonable assumption in the absence of specific data as per Hubert and Willis The producer has a planned TVD of 300m. Using the geothermal gradient, the bottom hole temperature is found to be: 7

38 T 0.03 d( TVD) 8.1 ( ) C. (7.1.3) This is below the boiling point of water so the occurrence of steam as a potential well bore influx can be ruled out. The only gaseous influx which can occur is typically Methane [9] with or without H S in small quantities. The design for burst will then be carried out considering the case of methane influx into the casing. Figure 7 Drilling Window for Wellbore Fluid Gradient Figure 8 Planned Trajectory of Producer 8

39 7. Stresses Considered The following section consists of a brief recap of some basic mechanical engineering concepts pertaining to stress.. During the drilling process, the tubular used will be subjected to any or all of the following stresses simultaneously. The fundamental formulas used to calculate these stresses are obtained from [7] Axial Stress The axial stress at any point on a tubular is the tension caused by the gravity component of the buoyed weight of the section below it, exerting a loading force vertically downwards. 4F, ( D D ) 1 (7..1) where, D 1 D Outer Diameter, m Inner Diameter, m 7.. Bending Stress Bending stress in the tubular acts along its longitudinal axis and can be caused when the tubular undergoes bending due to curvature of the wellbore trajectory, tubular buckling and due to lateral vibration. For the former, it is calculated using the formula: r b E, (7..) R where, b E r R Bending Stress, Pa Young s Modulus of Elasticity, Pa Radius of tubular, m Well bore radius of curvature, m Bending stresses due to whirling and buckling are calculated as the product of the flexural rigidity with the second order derivative of the nodal displacements obtained from simulation results as will be discussed in sections and respectively. 9

40 7..3 Torsional Stress The maximum torsional stress is induced in casing or drill pipe due to the torque or twisting force applied at the surface by the top drive or rotary table system as: 1rT T, (7..3) J where, T J T Torsional Stress, Pa Applied Torque, Nm Polar Moment of Inertia, 4 m 7..4 Hoop Stress Hoop stress acts in the circumferential direction of the tubular and is caused by a difference in internal and external pressures. This form of stress becomes especially relevant in well control situations such as collapse and burst loading and is given by the formula: r ( r r / r ) r ( r r / r ) i i o o i o h p i p o ro ri ro ri, (7..4) where, Hoop Stress, Pa h r io, Internal or external radius, m p io, Internal or external Pressure, Pa 7..5 Radial Stress Radial stress is also caused by internal & external pressure differences but acts orthogonal to the hoop stress. It is calculated as: r ( r r / r ) r ( r r / r ) i i o o i o r p i p o ro ri ro ri. (7..5) 7..6 Von Mises Stress The Von Mises stress is a theoretical construct which allows for the conversion of a triaxial stress state into an equivalent uniaxial value for comparing with a material s uniaxial yielding criterion. The various stress components calculated in the preceding section are combined to obtain the resultant Von Mises stress equivalent as follows: VM 6 r h a r h a t. (7..6) 30

41 7.3 Power Law Fluid Rheology Model In order to determine the circulating fluid pressures in the wellbore, the pressure drop due to frictional losses needs to be estimated. The frictional pressure drop when added to the hydrostatic pressure at a particular depth gives a simple estimation of the flowing wellbore pressures. This pressure drop is a function of the fluid viscosity. Most drilling fluids are typically Non-Newtonian and therefore, fluid viscosity cannot be determined by usual means. Consequently, various fluid rheology models were developed to calculate fluid viscosity. The power law is one such rheology model for Non-Newtonian fluids which relates the shear stress to the shear rate as follows: n K, (7.3.1) where, K n Shear stress, Pa Shear rate, s 1 Consistency index, PaS Flow behaviour index n Given the laboratory measurements of the drilling fluid shear rate, the values of constants K and n in equation (7.3.1) can be determined as: n log, 300 (7.3.) 5.11( 600), K (7.3.3) 10 n where, XXX Measurement of shear rate made in a fluid viscometer at XXX rpm, s 1 The minimum fluid flow velocity required for lifting of cuttings in the annulus is determined from Table by looking for flow rate corresponding to the annulus hole size. Table Flow Rate for Minimum Annular Velocity [30] For a given fluid velocity, the effective viscosity of the fluid in the wellbore under ideal conditions can then be determined in field units as: n1 96V 3n1 e 100 K, D 4 n n (7.3.4) 31

42 where e V Effective viscosity, cp Flow velocity, m s D D for annular flow, m. D o i D i for pipe flow, m. The frictional pressure losses are dependent on the friction factor between fluid flow and the wellbore. The friction factor is in turn affected by the nature of the flow, i.e. laminar or turbulent. This is quantified by the Reynolds number as: where, kg Fluid density,. 3 m N VD (7.3.5) Re, The friction factor is then calculated for laminar flow as: f 16, (7.3.6) N Re Additionally, the friction factor for turbulent flow is: 1 n log Re, f n n (7.3.7) where, f Friction factor Finally, the frictional pressure drop can be calculated as follows: dp f V. (7.3.8) dz D where, z dp dz Wellbore longitudinal axis, m Frictional pressure drop,. Pa m 3

43 7.4 Casing Design Load Cases The operational pressures which can be expected in the worst case scenarios are first computed for the different casing sections in order to select casing steel grades of suitable yield strength. These pressures are then multiplied with specifc safety factors to arrive at the design loads against which the casing must be rated. Table 3 Factor of Safety [31] Surface Casing The analysis of design load is first carried out for the 13 3/8 surface casing with planned shoe at 350M as shown in Figure Collapse Scenario External pressure is due to the column of mud outside the casing which is present when the casing was run Internal pressure is taken as complete evacuation due to fluid loss when drilling subsequent section Collapse pressure at casing shoe located at 350m depth: mud gradient Depth kpa. (7.4.1) Corresponding design load including the factor of safety(fos) for collapse: F. O. S Collapse Load 1.400kPa 5040 kpa. (7.4.) 33

44 Burst Scenario In the event of a poor cement job, the fluid in the spaces where cement is absent or has channelled has density similar to fresh water. Therefore external pressure is calculated as per fresh water gradient of 9.81kPa/m Internal pressure is due to Gas influx. At the casing shoe, the influx pressure is given by the hydrostatic pressure equivalent of the formation strength at that point: ( Fracture Grad. Depth) kpa. (7.4.3) The external pressure at casing shoe is: kpa. (7.4.4) Burst pressure at casing shoe is calculated as the difference between external and internal pressure which is 184.kPa Corresponding design load including the FOS for burst at shoe is given as: F. O. S Burst Load kpa. (7.4.5) Temperature at the casing shoe is calculated by linearly interpolating between previously calculated bottom hole temperature of 77. C and surface temperature of 5 C: (77.1 5) T 5 [ 350] 33 C. (7.4.6) 300 The average temperature is: (5 33) K. (7.4.7) By using ideal gas law with appropriate compressibility factor for methane, internal pressure at surface is calculated as: gm ( hh1) ZRTavg Pe 1 P e (0 350) Pa. (7.4.8) Where M, Molecular mass of gas = 16 g/mole for Methane Z, Compressibility factor = 1 R, Ideal gas constant = gm/(mol.kelvin )for Methane Since there is no external pressure due to fresh water gradient at surface, the burst pressure is the internal pressure itself 34

45 Corresponding design load including the FOS for burst at surface is given as: F. O. S Burst Load kpa. (7.4.9) Tabulating the results of computed design loads for surface casing, Design Collapse (kpa) Design Burst(kPa) Surface Shoe(300M) Intermediate Casing The intermediate casing comprises of the 9 5/8 liner casing for which the loads encountered will be computed in a similar manner. Repeating the procedure followed previously, the results of the design load computation are summarized as: Design Collapse (kpa) Design Burst(kPa) Surface Shoe(700M) Production Liner For the production liner, the worst case loading scenario for burst loading is defined as entire casing filled with a column of gas during production. Consequently, it is not necessary to consider the formation strength in this condition and the formation fluid pressure will be sufficient as stated in [7]. As the liner will be tied back to the intermediate casing, the intermediate casing will also be exposed to the production loads. Consequently, the burst pressure will be computed up to surface to compare with the previously computed design loads for the intermediate casing. The higher of the two loads will then be chosen as the effective design load for the intermediate casing. Design Collapse (kpa) Design Burst(kPa) Surface Shoe(00M) Clearly, the burst gradient of the liner plotted upto surface is higher and so the gradient of the production liner will be adopted for the intermediate casing as well. 7.5 Casing Selection Based on Uniaxial Loading Criteria Typical choice of steels from the API-5CT standards for use in Geothermal wells as per [9] are K55, H40, J55, C75 & L80 which are chosen for relatively low tensile strength which reduces the tendency for hydrogen embrittlement. Subsequent selections will therefore use these recommendations as a starting point. 35

46 7.5.1 Surface Casing Collapse Loading As per [3], the yield strength of 13 3/8 diameter K55 surface casing casing, Y p N This yield strength is de-rated by a factor of because of exposure to temperature of production fluid up to80 C degrees as a conservative estimate as per [6] Thermally de-rated yield strength is then: Y N N 6 6 p (7.5.1) On the basis of calculated D/t ratio for this steel, theapi-5c3 Transition Collapse [3] formula was to be applicable as follows: F PT Yp[ G] D/ t MPa ( ) [ ] 35.0 (7.5.) Where P T Rated Collapse Pressure Burst Loading Applying the API-5C3 [3] formula for calculating the minimum internal yield pressure or burst strength: Yt P p 0.875[ ] D [ ] ( ) MPa. (7.5.3) The selected material is thus found to withstand the expected design loads 7.5. Production Casing Collapse Loading In the DAP proposal, the production casing comprises of the intermediate casing run upto 700m TVD and the liner hung off from the intermediate casing shoe upto the TD. By following a similar procedure to the preceding section, the results are tabulated below. 36

47 Table 4 Production Casing Specifications Casing 9 5/8 Intermediate 7 Liner Material 53.5 ppf, L80 9 ppf, N80 Applicable Collapse Formula Plastic Plastic Thermally Derated Yield Strength, Y N N p Derated Collapse Rating, P 4.91MPa 46.66MPa T Burst Loading Yt P For 9 5/8, 53.5ppf, L80 grade steel: p 0.875[ ] 5.13 MPa. D (7.5.4) Yt P For 7, 9ppf N80 grade steel: p 0.875[ ] 55.8 MPa. D (7.5.5) Tabulating the results, we obtain: Table 5 Summary of Required Material Properties based on Design Loads O.D(Inches) GRADE Density(PPF) Derated Yield (10 6 N) Collapse (10 6 MPa) Burst(Mpa) (10 6 MPa) Surface 13 3/8 K Intermediate 9 5/8 L Liner 7 N Figure 9 Graphical representation of Design Loads vs. Rated Strength 37

48 7.6 Axial Loading Wellbore Trajectory Figure 30 Measurement of Key Wellbore Trajectory Parameters The well trajectory shown in Figure 7 must be converted into survey data points containing Measured Depth, Inclination & Azimuth in order to be usable for the developed Torque & Drag model. This is done by using trigonometric relations to calculate the position of key points in the trajectory such as the Kick-off point etc. as shown in Figure 30. From Figure 30, the radius of curvature, r for the build-up section is calculated as 443.3Metres and the maximum wellbore inclination to the vertical, is found to be 43 or 0.75 radians. Consequently, arc length BC is: r 33.7m. (7.6.1) The total measured depth is then found to be: m. 43 Lastly, the Dog Leg Severity (DLS) is calculated to be: per 30m Surface Casing Once the wellbore survey data is obtained, the various axial load properties of the different casing grades selected are tabulated below: Table 6 Surface Casing Properties Unit Weight 795 N/M Density Kg/m3 Buoyancy Factor Buoyed Unit Weight N Axial Load at Top of Hold Section (TVD-1000m).36 X 10 5 N F.O.S 1.8 Design Axial Load 4.4 X 10 5 N Yield Strength 36.3 X 10 5 N 38

49 7.6.3 Intermediate Casing Table 7 Intermediate CSG Properties Unit Weight 781 N/M Density Kg/m3 Buoyancy Factor Buoyed Unit Weight N Axial Load at Top Joint 4.6 X 10 5 N F.O.S 1.8 Design Axial Load 8.31 X 10 5 N Yield Strength 5.8 X 10 5 N Liner Hold Section at 43 Deg. Inclination For section CD in Figure 30, Table 8 Liner Hold Section Properties Unit Weight 43 N/M Density Kg/m3 Buoyancy Factor Buoyed Unit Weight N Buoyed & Inclined Weight N Axial Load at Top of Hold Section (TVD-1000m) 4.3 X 10 5 N F.O.S 1.8 Design Axial Load 7.77 X 10 5 N Build Up Section Table 9 Build Up Section Properties Measured Depth[m] Inclination [deg] Inclination [rad] Buoyed & Inclined Weight E E E E E E E E E E E X 10 5 N 39

50 7.6.5 Axial Load due to Well Bore Trajectory Axial load due to well bore trajectory arises from the bending stress acting along the longitudinal axis of the casing string in the wellbore due to the curvature of the wellbore trajectory. Bending stress is computed as per (7..) for which the variables are: E Young s Modulus of Elasticity = Pa r Outer radius of liner at which maximum bending occurs = m R Well bore radius of curvature = m b Pa (7.6.) Axial Load due to Bending: b Area(7" N80) N. (7.6.3) The total axial load at the uppermost joint of 7 liner is then given by summing the unit inclined buoyed weights in the build-up and hold sections with the axial load due to bending: =4.3 X X N (7.6.4) 6 Multiply axial load computed in (7.6.4) with the FOS of 1.8 gives design axial load of N 6 In comparison to the thermally de-rated yield strength N, the design is found to be within the rated limit. 7.7 Combined Loading Collapse with Axial Loading To determine reduced collapse strength for 7 liner casing, we substitute the maximum axial load obtained at the topmost joint of the 7 casing string in(7.6.4) in the Reduced Von-Mises D collapse with axial loading formula defined as: Y pa S a a ( ) 0.75( ) Yp. Yp Yp S (7.7.1) Where, S a Y p Axial Load, Pa Yield Strength 40

51 Y pa ( ) 0.75( ) Pa (7.7.) Using the reduced yield strength due to axial loady collapse load for the 7 N80 liner was found to be: pa, for the Plastic collapse formula [3], the rated PT [.0667] MPa. (7.7.3) 17.5 Similarly, for the top joint of the 9 5/8 intermediate L80 casing string which forms the top section of the production casing, the procedure above is repeated to obtain: Maximum axial load at surface due to combined effect of 7 and 9 5/8 combination string: N N, (7.7.4) This gives the reduced yield strength and rated collapse load of: Ypa ( ) 0.75( ) Pa, (7.7.5) PT [.0667] MPa. (7.7.6) 17.5 Figure 31 Plot of Design Load vs. Rated Strength for Collapse 41

52 7.7. Von Mises Analysis for Burst Loading For the tri-axial Von Mises stress analysis, the critical point where failure is likely to occur is the inner diameter. Therefore, the Bending stress calculations are repeated for the inner diameter as follows for the 7 N80 liner casing using the same data as above: b Pa (7.7.7) Axial load due to bending is now: b Area(7" N80) N. (7.7.8) Consequently, total axial load at the topmost joint of the 7 liner will be: =4.3 X X N. (7.7.9) The results of the triaxial stress analysis computed at the survey points which are at 30metres interval from each other are recorded in Appendix D,Table Torque & Drag Analysis Figure 3 Plot of Pick Up Drag Forces When Tripping Out of String 4

53 5 Maximum axial loading including the effects of pick up drag forces is now N which is obviously higher than the previous maximum calculated at the same position including only the effects of buoyant forces and well bore inclination previously. The increased friction observed in build-up section is due to implicit relationship between string axial tension and the normal force as can be seen in(3.1.3). This analysis was carried out with friction factor for Delft sand of 0.46 corresponding to the maximum observed dynamic friction value for steel in contact with dry sand obtained from the experimental investigation performed in [33]. Analysis carried out only for 7 N80 liner section as the 9 5/8 and 13 3/8 sections are vertical sections and would therefore not experience any normal (side) force component. Tabulated drag values are published in Appendix D, Table Axial Loading due to Drag Forces Total Axial load at the uppermost joint of 7 liner will now include the effects of drag in addition to the axial stress caused by bending of the liner in the build-up section. Therefore the design calculation for the liner casing subjected to axial loading alone will be: 5 5 =(8.7.4)X N. (7.8.1) 6 This gives a design axial load using FOS 1.8 of N. Therefore the design is found to be within the rated limit when comparing against thermally de-rated yield strength value obtained from 6 Table 5, i.e N. Additionally, the combined loading scenarios for collapse and burst are now repeated to include the effect of the increased axial loading due to drag forces as follows: Collapse with Axial Loads Including Drag Forces To determine reduced collapse strength for 7 liner casing, we substitute the new maximum axial load obtained at the topmost joint of the 7 casing string including the effect of drag forces in equation (7.7.1) to obtain: Ypa ( ) 0.75( ) Pa (7.8.) Using the reduced yield strength due to axial loady rated collapse load for the 7 N80 liner was found to be: pa, from the API Plastic collapse formula [3], the PT [.0667] MPa. (7.8.3) 17.5 This is only just above the design collapse load observed in section and Figure 9 so depending upon the company specific policy of the operator, it may be decided to consider the next grade of casing for the liner to remove any chance of failure due to collapse loading. Similarly, for the top joint of the 9 5/8 intermediate L80 casing string which forms the top section of the production casing, the procedure above is repeated to obtain the maximum axial load at surface due to combined effect of 7 and 9 5/8 combination string: 43

54 5 5 De-rated collapse strength due to axial loading is then: N N. (7.8.4) Ypa ( ) 0.75( ) Pa (7.8.5) This gives reduced collapse strength of: PT [.0667] MPa. (7.8.6) 17.5 Figure 33 Design Collapse Load vs. Strength for DAP Producer Von Mises Analysis for Burst Loading Including Effect of Drag Forces Figure 34 Increased FOS for Burst Loading Due to Effect of Drag Forces 44

55 As expected, when the Von Mises analysis is repeated at the survey points including the effect of drag forces, the FOS is observed to increase as shown in Figure 34 due to the supporting effect of the axial loading on the collapse strength of the casing pipe. Tabulated Von Mises stress analysis values at survey points are published in Appendix D, Table Torque Analysis Figure 35 Cumulative Torque Observed at Surface for Various Drilled Depths As can be seen from Figure 35, the maximum surface torque required to overcome the cumulative torque losses while drilling the inclined section of the well is of the order of approx. 1 kn. Due to constant wellbore inclination after 1000m TVD, the torque increases linearly whereas it increases at relatively higher rate in the build-up section. The simulated torque and corresponding torsional stress at survey points are tabulated in Appendix E for reference. As previously stated, the torque analysis is integral for selecting casing buttress connections with suitable torque capacity required for drilling operations. Sample manufacturer catalogue presenting connections with improved torque capacity is also present in Appendix E. Computed torsional loads which used as input parameters for simulating the combined Von-Mises stresses in the liner section during the process of drilling the inclined section are also tabulated. 7.9 Wellbore Pressure Distribution Based on drift diameter of 9 5/8 L80 intermediate casing, the bit size used for drilling next open hole section was chosen to be 8 ½. The following assumptions were then made for implementing the power law rheology model to calculate pressure distribution of drilling fluid in wellbore during drilling: 45

56 Fluid Pressure [MPa] From [30], the discharge for obtaining minimum annular velocity required for lifting the cuttings is m s 3 Typical viscometer readings required as input for power law model obtained from [34] and shown in Table 10 Pressure drop across the bit for 8 ½ PDC is typically approx. 5MPa [34] Table 10 Power Law Input Viscometer Measurements Viscometer Reading RPM Shear Rate (1/Sec) Figure 36 Flow Path for Drilling Fluid in 7" Liner Drilling Phase Wellbore Fluid Pressure Distribution Distance from Stand Pipe [M] Wellbore Hydrostatic Circulating Wellbore Pressure Figure 37 Fluid Pressure Distribution for 7" Liner Drilling Phase 46

57 As shown in Figure 36, The drilling fluid from the mud pumps flows into the wellbore first through the 5 drill pipe used to hang the liner from the surface from the liner hanger and then through the retrievable BHA necessary for drilling inclined sections. Consequently, the interior of the 7 Liner is not exposed to fluid pressures at all. Only the external surface is in contact with fluid flow in the annulus between open hole and Intermediate casing. The frictional pressure losses due to fluid flow as calculated by the power law model are tabulated in Appendix F. The total wellbore pressure at any depth as shown in Figure 37 is than obtained by summing the hydrostatic pressure at that point with the annular pressure losses as previously stated Drilling Loads Liner Bending Stress Due to Whirl Figure 38 Whirling Mode Shapes Figure 39 Natural Frequency of Lateral Vibration vs. WOB 47

58 During Drilling, the casing string is likely to be deflected laterally when the rotary speed applied at surface by the rotary table approaches the critical value as predicted by the MATLAB tool developed in chapter 6. Using the material properties of N80 Liner casing, the range of rotary speed values for a particular WOB which cause resonance or whirling can be seen in Figure 39. This tool may be used as a quick estimate for determining the RPM range to drill at so that the alternating bending stresses associated with whirling which can cause string failure due to fatigue can be avoided. For drilling the deviated sections, the retrievable BHA needs to be used as discussed in Chapter 1. Consequently, the casing stand (30metres) is modelled with the fixed or clamped boundary condition at both ends. This is because the loads from the retrievable BHA are transferred onto the lowermost stand of the liner casing at the bottom through internal stabilizers and through the Drill- Lock assembly at the top of the stand (section 1.1.). For simplicity, it is assumed that the weight on bit transfers directly to the bottom of the liner stand. The corresponding mode shapes for this boundary condition are shown in Figure 38. From these mode shapes, the corresponding bending stress is calculated and tabulated results for bending stress across one liner stand are published in Appendix D, Table 15. Bending stress is also plotted in Figure 40 from which it can be observed that the critical points occur at 1.4metres from both end points. As expected, a maximum bending stress of 15MPa is experienced at the midpoint of the casing stand which equates to approximately 10% of the total axial load at the topmost joint of the 7 liner. Knowledge of where maximum bending stresses are likely to occur can be applied to the design of casing pipe with composite materials which can then be selectively strengthened at the appropriate points. The advantage of doing so is that overall pipe weight can be reduced and cost of hiring drilling rigs can also be brought down as the hook load capacities needed for drilling to the target depth will be comparatively lesser. Figure 40 Bending Stress Due to Whirl 48

59 /8 Casing Bending Stress Due to Buckling Figure 41 Buckling Mode Shapes and Dependency of Critical Load on WOB It is advisable to investigate the magnitude of the axial stress induced by buckling of casing in static conditions to ensure that the selected casing steel grade will not fail under buckling. As an example, the driller will sometimes apply WOBs that exceed the critical load threshold in order to clear a downhole block. However for identical boundary conditions as seen from Figure 1 & Figure 5, the mode shapes and hence the bending stresses induced are the same. In the case of the 9 5/8 Intermediate casing which is being used in drilling the completely vertical top section of the wellbore, the drillable non-retrievable BHA specified is used. This can be modelled using a different end condition of pinned or hinged at the bit. Although the magnitude of maximum bending stress occurring will be determined by the casing-wellbore annular clearance, the location of the points of maximum bending stress may be different and it is therefore of interest in order to specify additional strengthening points when using composite materials for the casing. The results obtained from the MATLAB buckling analysis tool developed in chapter 5 for the 9 5/8 intermediate casing with the pinned at bit and fixed at stabilizer boundary end condition are shown in Figure 41. Also of interest is the plot of critical frequency as a function of the wellbore inclination which provides a limit on the maximum WOB which should be used by the driller in normal conditions. 49

60 Figure 4 Bending Stress due to Buckling in 9 5/8" Intermediate CSG From Figure 4, the point of maximum bending stress observed is observed at approximately 10metres from the bit or at the beginning of the nd single of the casing stand which is not the case in the mode shapes observed for the Pinned at both ends boundary condition examined so far. Bending stress is also significantly high and only slightly lesser at the stabilizer placed at the ending of the casing stand (30 metres from bit). The magnitude of this additional bending stress (1.74 MPa) tabulated in Appendix D is clearly not significant to alter the casing selection. Note that the value of bending stress is determined by the annular clearance which generally tends to be low when drilling with casing. This observation corresponds with the low values of bending stress encountered in [1] when drilling the vertical hole section with the non-retrievable BHA. 50

61 TVD[M] 7.11 Von Mises Analysis of Drilling Loads during 7 Liner Section Liner Stress Distribution at TVD Axial[Mpa] Torsional[Mpa] Radial[Mpa] Hoop[Mpa] Von Mises[Mpa] Yield [Mpa] Stress [MPa] Figure 43 7" Liner Von-Mises Stress Analysis for Drilling Operation The final test of the selected design is to verify the resultant Von Mises stress due to triaxial loading and compare it with the yield stress throughout the casing string for the 7 Liner section for typical drilling conditions. The hoop and radial stresses are calculated based on the fluid pressure gradient computed [section 7.9]. The axial stress includes the effect of buoyed string weight, bending stress due to wellbore curvature in the build-up section [section 7.6.5] and bending stress due to whirling [section ]. Since the string is rotating in the wellbore, there will be no drag forces. However, induced torsional stress [section 7.8.] is included in the analysis. A graphical representation of the result is shown in Figure 43 where the resultant von Mises stress is clearly lower than the yield stress by a FOS ranging between &.3, thereby validating the design for drilling the final section. Tabulated values of the stress analysis at survey points are also include in Appendix D Table 16 for reference. It is observed that the maximum torsional stress is 38% of the total axial stress which indicates the importance of the torque analysis and the selection of appropriate casing connection with enhanced torque capacity in order to prevent design failure. 51

62 8 Conclusions An algorithm was developed to select the required API steel grade of casing pipe for use in the Casing While Drilling technique which evaluates the effect of conventional casing Burst & Collapse pressure loads and Drilling loads i.e. Torque, Drag, Buckling and Whirling. Supporting tools were developed in MATLAB to calculate the Drilling loads. The algorithm was applied towards the DAP producer Geothermal Well with the following recommendations being made: Casing Table 11 Casing String Design for DAP Producer Geothermal Well Shoe TVD (m) Outer Diameter (Inches) Grade Nominal Weight (dan/m) Surface /8" K Intermediate /8" L Liner 00 7" N The effect of induced bending stresses due to buckling and whirling of the Casing String was not found to be significant due to the relatively small annular clearances available. Maximum Bending Stress due to whirling was calculated to be 10.6% of the axial stress. For a casing stand of 30m length, the maximum bending stress due to whirling when modelled with the boundary conditions of fixed or clamped at both ends was observed at the midpoint and at a distance of 1 m from both ends. Similar points of maximum stress due to buckling with the boundary conditions clamped at the stabilizer and pinned or hinged at the bit were located at 10.5 metres from the bit (close to casing joint between 1 st & nd single) and at the stabilizer. The critical rotary RPM as a function of the WOB for inducing whirling of the casing was calculated by the developed MATLAB tool to help the driller avoid string failure due to fatigue loads caused by alternating bending stresses. The critical weight on bit for inducing sinusoidal buckling of casing string was also determined for varying wellbore inclination from the vertical axis The effect of torque on casing string was found to be significant with a maximum torsional stress calculated as 38% of the total axial stress which is close to four times the bending stress due to whirling. Buttress threaded connections with torque rings to boost the makeup torque capacity must be used to avoid string failure 8.1 Recommendations The hydraulic model used to model the wellbore pressure distribution during drilling has to be expanded to more realistically simulate fluid pressure losses and to include effects such as surge and swab pressure loads The torque and drag model must be altered to include hydrodynamic effects of viscous drag and the effect of bending stiffness of the casing The boundary conditions used to simulate drilling with the retrievable BHA can be expanded to more accurately represent downhole conditions The algorithm used in the thesis can be adapted to include the analysis of composite materials for the CwD application. 5

63 Nomenclature A Cross sectional area of casing, m, Added mass coefficient Radial clearance in wellbore, m E F 0 Young s Modulus of Elasticity, Pa Applied weight on bit, N m g Acceleration due to gravity, s Buoyancy factor i l Area Moment of Inertia, 4 m Length of segment of interest between bit and first stabilizer, m Dimensionless effective length of compression Mass of volume of mud displaced by solid cylinder of same outer diameter (per metre), Kg Mass of steel per m, Kg / m Number of segments into which casing is divided into S y w Shear stress, Pa Dimensionless distance along Z axis Angle subtended by deflected beam with respect to its longitudinal axis Inclination from vertical, radians kg Density of Mud, 3 m kg Density of Steel, 3 m Dimensionless time Dimensionless angular velocity 53

64 List of Abbreviations BHA CwD DAP FOS LWD MWD PDC WOB Bottom Hole Assembly Casing while Drilling Delft Aardwarmte Project Factor of Safety Logging while Drilling Measurement while Drilling Polycrystalline Diamond Compact Weight on Bit 54

65 Bibliography [1] F. Sanchez, M. Turki, and M Cruz, "Casing While Drilling: A New Approach to Drilling Fiqa Formation in Oman," Journal of Petroleum Technology, pp. 3-33, June 01. [] Tommy Warren, Bruce Houtchens, and Garret Maddell, "Directional Drilling with Casing," Journal of Petroleum Technology, pp. 17-4, March 005. [3] Schlumberger, "Using Casing to Drill Directional Wells," Oil Field Review, pp , 005. [4] Hendry Shen, "Feasibility Study on Combining Casing with Drilling with Explandable Casing," NTNU, Trondheim, MSc Thesis 007. [5] Abubakar Mohammed, Chika Judith Okeke, and Ikebudu Abolle-Okoyeagu, "Current Trends and Future Development in Casing Drilling," International Journal of Science and Technology, vol., no. 8, pp , August 01. [6] British Gas, "Well Engineering and Production Operations Management System," BG, Casing Design Manual 001. [7] Ted G. Byrom, Casing and Liners for Drilling and Completion, 1st ed.: Gulf Professional Publishing, 01. [8] M.Enamul Hossain, Fundamentals of Sustainable Drilling Engineering.: Scrivener-Wiley. [9] Geoelec, "Report on Geothermal Drilling," European Union, 013. [10] Xie Jueren and Gang Tao, "Analysis of Casing Connections Subjected to Thermal Cycle Loading," in SIMULIA Customer Conference, Alberta, 010. [11] B. Bennetzen, J. Fuller, and E. Isevcan. Schlumberger. [Online]. [1] Robert F. Mitchell and Robello Samuel, "How Good is the Torque & Drag Model?," SPE Drilling Engineering, pp. 6-7, March 009. [13] C. A. Johancsik, D. B. Friesen, and Rapier Dawson, "Torque and Drag in Directional Wells- Prediction and Measurement," Journal of Petroleum Technology, pp , June [14] M. C. Sheppard, C. Wick, and T. Burgess, "Designing -Well Paths To Reduce Drag and Torque," SPE Drilling Engineering, pp , Decemeber [15] Tanmoy Chakraborty, "Performing simulation study on drill string mechanics, Torque and Drag," NTNU, Msc Thesis 01. [16] Halliburton, "Landmark WellPlan User Manual," 003. [17] E E Maidla and A. K. Wojtanowicz, "Field method of assessing borehole friction for directional well casing," in SPE Middle East Oil Show, Manama, [18] M. Fazaelizadeh, G Hareland, and B S Aadnoy, "Application of New 3-D Analytical Model for 55

66 Directional Wellbore Friction," Modern applied science, pp. -, 010. [19] Olivier F. Rey, "Dynamics of Unbalanced Drill Collars in a Slanted Hole," Massachusetts Institute of Technology, Cambridge, MSc Thesis, [0] Gere Timoshenko, Theory of Elastic Stability.: Mc-Graw Hill, [1] D.A Boddeke, "Buckling & Post Buckling Behaviour of a Drill String in an Inclined Borehole During Drilling for Oil & Gas," TU Delft, Delft, MSc Thesis [] Rapier Dawson, "Drillpipe Buckling in Inclined Holes," Journal of Petroleum Technology, [3] Jan Dirk Jansen, "Nonlinear Dynamics of Oilwell Drillstring," TU Delft, Delft, PhD Thesis [4] Kim Vandiver and Rong-Juin Shyu, "Case Studies of the Bending Vibration and whirling Motion of Drill Collars," SPE Drilling Engineering, [5] Schlumberger. [Online]. [6] Rong-Juin Shyu, "Bending Vibrations of Rotating Drill strings," Massachusetts Institute of Technology, Cambridge, PhD Thesis [7] S. Singiresu Rao, Mechanical Vibrations.: Pearson, 004. [8] Ray Clough and Joseph Penzien, Dynamics of Structures.: McGraw Hill, [9] Hubbert and Willis, "Mechanics Of Hydraulic Fracturing," Petroleum Transactions, AIME, [30] TU Delft, "Drill String, Drill Bit & Hydraulics Lecture Notes," TU Delft, Delft, Well Engineering Manual. [31] NZ Code of practice for geothermal wells., 001. [3] American Petroleum Institute, "Bulletin on Formulas and Calculations for Casing, Tubing, Drill Pipe, and Line Pipe Properties," API, [33] Steven Leijnse, "Friction Coefficient Measurements for Casing While Drilling with Steel and Composite Tubulars," TU Delft, Delft, MSc Thesis AES/PE/10-10, 010. [34] Texas A&M. (00, October) PETE 411 Well Drilling - Pressure Drop Calculations. [Online]. MM4N/PETE_411_Well_Drilling_powerpoint_ppt_presentation [35] TESCO Corp., "Casing & Tubing Torque Tables for API Buttress with TESCO MLT Rings," TESCO Corp., Field Make-Up Handbook Table,. [Online]. [36] Tesco Corp. Tesco Corporation. [Online]. [37] P D Spanos, A M Chevallier, N P Politis, and M L Payne, "Oil & Gas Well Drilling : A Vibrations 56

67 Perspective," The Shock and Vibration Digest, vol. 35, no., pp , March

68 Appendix A. Implementation of FDM for Beam Deflection Figure 44 Discretization of Drill String [19] To simplify the implementation of FDM, the constant terms of equation(4.1.18) are grouped into the following coefficients: The equation to be discretized now reads: ghcos a, (A.1) a 1 L0 Cm ghsin. (A.) L0 Cm d y d y dy dw dw dw 4 ' ' ' ' a 4 1 l0 w a. (A.3) The length of the BHA being investigated here is the first casing single of standard length, 10 metres. It is bounded by the bit on one end and the first stabilizer on the other end. This single is considered to be divided into N segments of length 1 N each or N 1 node points. The Taylor series is then used to approximate the differential terms in equation (4.1.18) into their algebraic equivalent by using the central difference method as follows: d y dw 4 ' 4 N ( y 4 ( j) 4y( j1) 6y j 4 y( j1) y( j) ), (A.4) ' d y dw ( ), (A.5) N y( j1) y j y( j1) ' dy N y j1 y j1. (A.6) dw 58

69 The term w in equation (A.3) represents the longitudinal distance along the casing single from the origin at the bit. To facilitate the discretization process, it is represented by the formula: j w. (A.7) N Where the value of j varies from 0 at the bit to N at the stabilizer as depicted in Figure 44. By examining the terms carefully in(a.4),(a.5)&(a.6), it can be seen that extra points which fall outside the casing single will be required at the end points of the Bit and the first stabilizer. Imaginary points can be taken which will be related to known points within the domain of the casing single by making use of the boundary conditions at the bit and the stabilizer. Since the casing single is essentially a beam element, the stabilizer owing to the fact that it restricts movement in all directions is modelled as a clamped end. The dimensionless boundary conditions are: ' y L 0, (A.8) ' dy dw L 0. (A.9) Substituting (A.6) & (A.8) in the boundary condition defined by (A.9), we can obtain the imaginary point at the stabilizer as: N yj 1 yj 1 y N1 N1 0, (A.10) y. (A.11) The bit however is chosen to be a pinned end. Therefore the boundary conditions are: y, (A.1) Z 0 0 d y ' ( Z 0) 0 dw. (A.13) Making use of (A.5) by substituting in (A.13), boundary condition simplifies to: Where y a y, (A.14) a4 1. (A.15) Now that the boundary conditions have been defined and applied to determine the imaginary points, the next step is to substitute the finite difference approximations defined in(a.4) (A.5),& (A.6) and also the relation (A.7) in the governing differential equation(a.3) to obtain: 59

70 y j N y j1 4N a1l 1N a1 j N y 6N a1l 1N a1 jn 4 an 1 4 y j1 4N a1l 1N a1 jn y j N a.(a.16) Equation (A.16) is the discretized general equation applicable for node points 3 j N 3 which can be further simplified by grouping constant coefficients as shown below: Thereby obtaining C1 4 N, (A.17) 4 ( an 1 ) C 4N a1l 1N, (A.18) C 6N a l N, (A.19) ( an 1 ) C4 4N a1l 1N. (A.0) c j c c y y c a N y a jn y a jn y c a. (A.1) j 1 j j1 4 1 j 1 Similarly, making use of the relations defined in (A.11) &(A.14), the finite difference formulation at the node points j 1,, N & N 3are evaluated as: At j 1, At j, y [ C a C a N] y [ C a N] y [ C] a. (A.) y [ C a N] y [ C 4a N] y [ C 4 a N] y [ C ] a. (A.3) At jn, At J N 1, yn 4 C1 yn 3[ C ( N ) a1n ] yn [ C3 a1n ( N )]. (A.4) y [ C ( N 1) a N] a N y C y [ C ( N 1) a N] y [ C a N( N 1) C ] a. (A.5) N3 1 N 1 N Finally, the N 1equations can be arranged in the form of matrices according to the expression [ A] y [ B]. (A.6) This can then be solved by using standard functions in MATLAB. The diagonal matrix A containing the constant coefficients is: 60

71 A1 C3 a4c1 a1n C4 a1n C C a1n C3 4a1N C4 4a1N C C1 C a13n C3 6a1N C4 a13n C C1 C C1 C a1 jn C3 ja1n C4 a1 jn C C1 C C C ( N ) a N C a N( N ) C ( N 1) a N C1 C ( N 1) a1n C3 a1n( N 1) C1 (A.7) Y Is a vector containing the displacements at each node: And lastly, [ B] is given by: y y y y y y ' 1 ' ' j ' N ' N 1, (A.8) a [ B]. (A.9) a a 61

72 Appendix B. Implementation of FDM for Buckling Analysis The matrix A defined in (5.1.5) is similar to the earlier coefficient matrix defined earlier in Appendix A except for the parameter denoting the weight on bit which is contained in the B matrix. Recalling equation(5.1.3): 4 ' ' ' ' ' d y ghcos d y dy ghsin ' ghcosl 0 d y 4 w y. (B.1) ' dw L0C m dw dw L0C mc L0 Cm dw Substituting finite difference approximations(a.4) to (A.6) and the terms (A.1) & (A.) in (B.1), we obtain the general equation for nodes3 j N 3 as: a y j N y j1 4N a1 j N y j 6N a1 jn ' c 4 an 1 4 y j1 4N a1 jn y j N. (B.) ' ' ' j1 1 0 j 1 0 j1 1 0 y a l N y a l N y a l N Equation (B.) can be simplified by making use of: C, (B.3) ( an) N, (B.4) 4 a C6 6N, ' c (B.5) 4 ( an 1 ) C7 4N, (B.6) C C1 N 4 a ln, (B.7) ' ' C9 al 10N. (B.8) Thus obtaining: C1 j jn y C1 y C y C y C y y C a N y C a jn y C a j j1 1 j 3 1 j1 4 1 j j1 5 j 6 j1 5. (B.9) 6

73 Equation (B.9) is the general equation which is discretized at all N 1 node points to obtain a set of N 1 linear equations which can be solved in MATLAB by using standard Eigen functions. MATLAB returns two matrices as the output of the Eigen value analysis. The first is a diagonal matrix containing the Eigen values or the critical loads. The lowest Eigen value thus observed is the first critical load to induce sinusoidal buckling. The second matrix contains the Eigen vectors or the mode shapes in each column. The first mode shape corresponding to the lowest critical load is found in the first column and so on. The matrix A defined in(5.1.5) is thus given by: A1 C6 a4c1 a1n C7 a1n C C5 a1n C6 4a1N C7 4a1N C C1 C5 a13n C6 6a1N C7 a13n C C1 C C1 C5 a1 jn C6 ja1n C7 a1 jn C C1 C C1 C5 ( N ) a1n C6 a1n ( N ) C7 ( N 1) a1n C1 C5 ( N 1) a1n C6 a1n( N 1) C1 (B.10) And the diagonal matrix matrix Bis given by: B C9 C C8 C9 C (B.11) C8 C9 C C8 C9 63

74 Appendix C. Implementation of FDM for Whirling Analysis The implementation of the finite difference method for identifying natural frequencies is essentially the same as that described previously in Appendix A & Appendix B. Here, the following terms can be defined to simplify equation(6.1.6): Resulting in: Let EI a 1 AC, (C.1) a M ghcos. (C.) C a a l z y dz dz dz 4 d y d y dy 1 4 ( 0 ) 0 M. (C.3) H LN L, (C.4) N where L = length of casing single N =Number of segments the BHA single is divided into Then the discretized general equation based on (C.3) is obtained by substituting the finite difference approximations (A.4) to (A.6) yielding: a1 a1 al0 a j a a1 al0 a j ( ) y 4 j ( 4 ) y 4 j1 (6 ) y 4 j H LN H LN H LN H LN H LN H LN H LN H LN. (C.5) a al a a j a ( 4 ) y ( ) y y H H H H H j1 4 j j LN LN LN LN LN The constant coefficients in (C.5) are grouped together as: C a C, (C.6) H LN a a l a a a l a 4 4, (C.7) HLN HLN HLN HLN HLN HLN C a al 6, (C.8) HLN HLN a al a C 4. (C.9) HLN HLN HLN 64

75 Equations (C.6) to (C.9)are then substituted in (C.5) to yield the following general equation: a j a j a j ( C ) y ( C ) y ( C ) y ( C ) y ( C ) y y. (C.10) 1 j j1 3 j 4 j1 1 j j HLN HLN HLN Thus, Afrom(6.1.8) is given by: A1 a a C3 a4c1 C4 C HLN HLN a a a C C3 4 C4 C H LN H LN H LN a j a j a j C1 C C3 C4 C H LN H LN H LN 0 C1 C C1 C C1 C1 a( N ) a( N ) a( N ) C1 C C3 C4 H LN H LN H LN a ( N 1) a ( N 1) C C C (C.11) 1 3 C3 C1 H LN H LN The same boundary conditions as described in Appendix A are used to fill in the first and last two rows of the matrix. Similarly, B is a diagonal matrix of same dimensions as A : B (C.1)

76 Appendix D. Supporting Tabular Data Table 1 Von Mises Stress Analysis at Survey Points for Burst Loading Scenario TVD Axial Stress Radial Stress Hoop Stress Von Mises Stress FOS [M] [Mpa] [Mpa] [Mpa] [Mpa]

77

78 Table 13 Simulated Pick-Up Drag Values at Survey Points for 7 Liner Drilling Measured Depth (m) Axial Pick-Up Load ( x 10^5 N) Measured Depth (m) Axial Pick-Up Load ( x 10^

79 Table 14 Von Mises Analysis at Various Survey Points for 7'' Deviated N80 Liner Section TVD Axial Stress Including Von Mises FOS TVD Axial Stress Including Von Mises FOS [M] Drag Forces [Mpa] Stress [Mpa] [M] Drag Forces [Mpa] Stress [Mpa]

80 Distance From Bit(M) Table 15 Simulated Bending Stress due to Whirling for 7 Liner Stand (30 M) Bending Stress (MPa) Distance From Bit(M) Bending Stress (MPa) Distance From Bit(M) Bending Stress (MPa)

81 Table 16 Von Mises Tri Axisal Stress Analysis of 7" Liner for Drilling Conditions at Survey Points TVD Axial Torsional Radial Hoop Von Mises FOS TVD Axial Torsional Radial Hoop Von Mises FOS [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa] [Mpa]

82 Table 17 Bending Stress due to Buckling for 9 5/8" Intermediate CSG Stand (30 metres) Distance From Bit(M) Bending Stress (MPa) Distance From Bit(M) Bending Stress (MPa) Distance From Bit(M) Bending Stress (MPa)

83 Appendix E. Torque Analysis & Corresponding Selection of Casing Connection Table 18 Torque Analysis at Survey Points for 7" N80 Liner Section Measured Depth[m] Torque [Nm] Torsional Stress [Mpa] Measured Depth[m] Torque [Nm] Torsional Stress [Mpa]

84 Table 19 Connection Make-Up Torque Capacities from Manufacturer Catalogue Data presented in Table 19 is obtained from Tesco Corp. manufacturer catalogue [35]. As previously stated, the torque capacities of typical buttress connections used for casing in geothermal wells is insufficient to withstand high torque loads during drilling and even while rotating the casing string when running in to clear blockages. Thus special torque rings have been designed and tested by manufacturers to increase the make-up torque capacities of buttress connections substantially.the ideal connection based on the outer diameter of the liner (177.80mm) and the inner diameter (157.07mm) of the same grade (N-80) as the pipe body for the 7 Liner casing string is highlighted in the table above. With the addition of the torque rings, the maximum rated make up torque for this connection will clearly contain the maximum simulated torque of approx.0,000 Nm from Table 18 and is thus the recommended choice. 74