EXTENDING THE REACH OF DRILLING: BETTER WELLBORE TRAJECTORY AND TORQUE & DRAG MODELS

Transcription

1 EXTENDING THE REACH OF DRILLING: BETTER WELLBORE TRAJECTORY AND TORQUE & DRAG MODELS by Mahmoud Abughaban

2 Copyright by Mahmoud Abughaban, 2017 All Rights Reserved

3 A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Petroleum Engineering). Golden, Colorado Date Signed: Mahmoud Abughaban Signed: Dr. Alfred William Eustes III Thesis Advisor Golden, Colorado Date Signed: Dr. Erdal Ozkan Professor and Department Head Department of Petroleum Engineering ii

4 ABSTRACT Extended reach [ER] drilling refers to the practice of directionally drilling to a given geological target located at a significant horizontal distance [horizontal departure] from the drilling rig. The ability to perform complex ER operations has become increasingly important in recent years, as the average length of the wellbore horizontal section continue to be extended. Drilling these complex wells, whether it is to reach reserves far from existing facilities or to expose reservoir sections for production and reservoir management advantages, requires accurate planning to reduce the borehole friction and ensure reaching desired targets within defined accuracy. This dissertation aims to extend the horizontal section and push the drilling capabilities near their limits without taking undue risks with the drilling system. These risks include stuck pipe, drillstring wear, and fatigue. This objective will be achieved by [1] calculating accurate measurements of borehole positions [true vertical depth, northing and easting], [2] calculating truer measures of wellbore tortuosity and geometric torsion that define the shape and the undulation of the borehole, and [3] predicting accurate measurements of drillstring to borehole friction forces from torque and drag [T&D] models. The novelty of the proposed models is the ability to estimate more realistic bending effects, accurately predict the contact forces between the drillstring and the wellbore, and solve T&D parameters from surface to total depth in reasonable time using standard engineering computers. The current 3D borehole trajectory model based on the minimum curvature method [MCM] tend to mathematically smoothen the wellpath. This is due to the assumption that the borehole trajectories are composed of constant curvature arcs. This assumption creates an artificial low tortuosity expressed as dogleg severity [DLS] which leads to the miscalculation of borehole positions, and generating unreliable predictions of T&D. iii

5 Today s T&D models are either based on the assumption of continuous drillstring to wellbore contact [the soft-string model] or intermittent contact due to drillstring stiffness [the stiff-string model]. In both cases, the bending parameter, the change in the rate of curvature and the geometric torsion in the T&D equilibrium equations are nil, because the wellbore trajectory is based on the MCM. In the scope of this study, a non-constant curvature trajectory model, the Advanced Spline-Curves [ASC] borehole trajectory model, has been developed to overcome these limitations. The principal method proposed using the spline-curves does not make the potentially unrealistic assumption of a constant curvature arc between survey measurements. It provides realistic results and accurately calculates the spatial course of the wellpath. This model is straight-forward and has the robustness and flexibility to calculate complex 3D wellbore trajectories. Based on this non-constant curvature 3D borehole trajectory model, an extended stiff-string 3D T&D model [the ASC 3D T&D model] is developed. This model includes the geometric torsion, the wellbore curvature, the change in the rate of wellbore curvature and the drillstring bending stiffness in its equilibrium equations. Various applications of the ASC borehole trajectory model and the ASC 3D T&D model are presented, and their results have been validated using field cases with real-time data. The ASC borehole trajectory model has been validated using two methodologies: [a] five horizontal wells with actual survey data recorded at one survey per foot using the high resolution continuous gyroscopic [HRCG] survey tool and [b] two synthetic wellbore examples of known trajectory values. The ASC 3D T&D model has been validated using field cases with real-time forces that define T&D measured at the surface by comparing two methodologies: [a] a semi-analytical approach pseudo-high resolution [PHR] survey data generation from the ASC borehole trajectory model at one survey per foot as an input in current industryapproved T&D software, and [b] the ASC 3D T&D model compared to actual drilling data. The calculated borehole trajectory utilizing the ASC model guarantees curvature continuity along the entire wellpath with significant improvement in wellbore positioning and iv

6 tortuosity as compared to MCM. This allows the introduction of the change in the rate of curvature and the geometric torsion measurements. The calculated T&D outputs from the ASC 3D T&D model provide a more accurate view of the drilling conditions downhole, including the downhole weight and torque on bit. The use of these models yields the best-known solution for the industry to design longer horizontal section. It provides multiple advantages to drilling efficiency and borehole quality. In terms of drilling efficiency, it helps to evaluate drilling equipment and procedures while drilling ER and complex horizontal wells. This allows drilling engineers to update the driller with surface weight on bit and torque parameters to improve the drilling performance. In terms of borehole quality, it increases the calculated borehole position accuracy and calculates truer measures of tortuosity without increasing the number of surveys taken. Thus, the ASC borehole trajectory model and the ASC 3D T&D model are not only an alternative approach to accurately approximate wellbore trajectories and model downhole T&D parameters to be used in real-time operation centers [RTOC]; it can also serve as a step toward drilling automation. v

7 TABLE OF CONTENTS ABSTRACT iii LIST OF FIGURES xi LIST OF TABLES xv LIST OF SYMBOLS xvii LIST OF ABBREVIATIONS xx ACKNOWLEDGMENTS xxii DEDICATION xxiii CHAPTER 1 INTRODUCTION What Limits Extending the Reach of Drilling? Challenges and Critical Technologies Wellbore Trajectory Modeling Torque and Drag Modeling Motivation of Dissertation Objectives of Dissertation The Novelty of Dissertation Wellbore Position, Tortuosity and Geometric Torsion Calculation Drillstring to Wellbore Wall Contact Forces Calculation Organization of Dissertation CHAPTER 2 THREE DIMENSIONAL WELLBORE TRAJECTORY Reasons for Drilling Directional Wells vi

8 2.2 Coordinate Systems and Maps Map Projections Reference Directions Deflection Tools Jetting Action Whipstocks Positive Displacement Motors and Bent Subs or Housing Rotary Steerable Systems Wellbore Surveying Tools Measurement While Drilling Tools The Gyro Survey Tools Wellbore Tortuosity and Geometric Torsion Wellbore Tortuosity Geometric Torsion Prior Work of Three-Dimensional Wellbore Trajectory Calculations The Tangential Method The Average Angle Tangential Method The Balanced Tangential Method The Radius of Curvature Method The Minimum Curvature Method Three-Dimensional Wellbore Trajectory Calculation in the Current Dissertation Work CHAPTER 3 THREE DIMENSIONAL TORQUE AND DRAG MODELING vii

9 3.1 Components of Torque and Drag Buoyancy Factor Friction Coefficient Axial Load, Buckling and Horizontal Lock-up Torque and Drag General Vector Differential Form Prior Work of Torque and Drag Models The Soft-String Model The Stiff-String Model Torque and Drag in the Current Dissertation Work CHAPTER 4 TECHNICAL APPROACH - THREE DIMENSIONAL ADVANCED WELLBORE TRAJECTORY MODEL Problem Setup Process Assumptions and Boundary Conditions Initial Approaches Bézier Curves Spherical indicatrix of tangents method [SIT 2] Building the Model Data Set Reprocessing Development of the Advanced Spline-Curve Borehole Trajectory Model Wellbore Positioning Representation Tortuosity and Geometric Torsion Representation The ASC borehole Trajectory Model Sample Calculation Quantification of the Advanced Spline-Curves Borehole Trajectory Calculation Model viii

10 4.3.1 Field Data Results Field Case Field Case Field Case Field Case Field Case Synthetic Wellpath Example Synthetic Wellpath Synthetic Wellpath ASC Borehole Trajectory Model Results Summary CHAPTER 5 TECHNICAL APPROACH - THREE DIMENSIONAL TORQUE AND DRAG MODEL Problem Setup Process Assumption and Boundary Conditions Initial Approaches Building the Model Model the tangent, normal and binormal unit vectors to the approximating circle Development of the Torque and Drag Model for the Advanced Spline-Curve Trajectory Estimating Downhole Weight on Bit and Torque Adaptation of the T&D Model for a Non-Constant Curvature Trajectory Model Quantification of the ASC 3D Torque and Drag Model Field Case ix

11 5.4.2 Field Case Depth-Based Field Data from Field Case Validating the Estimated Downhole Weight on Bit and Torque on Bottom. 127 CHAPTER 6 TECHNICAL RESULT AND DISCUSSION Torque and Drag Effect Field Application Results Summary Wellbore Positioning Effect Wellbore Tortuosity Effect CHAPTER 7 CONCLUSIONS AND RECOMMENDATION The Advanced Spline-Curves Model The Torque and Drag Model Recommendations Future Work REFERENCES CITED APPENDIX A - MINIMUM CURVATURE METHOD APPENDIX B - LIST OF DEFINITIONS x

12 LIST OF FIGURES Figure 1.1 Step-out ratio example of an extended-reach well modified from Kibsgaard Figure 1.2 Extended reach drilling database plot created by Kwantis Figure 1.3 Extended reach drilling limit due to drag forces created by Fazaelizadeh.. 4 Figure 1.4 Extended reach drilling limit due to the rotary torque created by Fazaelizadeh Figure 1.5 The evolution of extended reach drilling modified from Cox Figure 1.6 Minimum curvature method represented by a circular arc wellpath Figure 1.7 Left: Slide then rotate to build 10 over 100-ft section. Right: Rotate then slide to build 10 over 100-ft section Figure 2.1 Reasons to drill directional wells represented by Greg and Stefan Figure 2.2 Geoid surface of the Earth represented by Jamieson Figure 2.3 WSG-84 geoid height represented by the NGA Figure 2.4 Geographic coordinate system represented by Jamieson Figure 2.5 Map projection examples based on various plane surfaces and the orientation of the surface represented by Jamieson Figure 2.6 Universal Transverse Mercator Projection represented by Jamieson Figure 2.7 Magnetic declination Figure 2.8 Toolface orientation of a deflection tool Figure 2.9 Jetting tool represented by Mitchell and Miska Figure 2.10 Baker Hughes Positive Displacement Motor represented by Krueger Figure 2.11 Rotary Steerable Systems represented by Ferndaddy: [a] Baker Hughes AutoTrak; [b] Schlumberger s PowerDrive Xtra xi

13 Figure 2.12 Baker Hughes Inteq MWD tool components represented by Choudhary. 31 Figure 2.13 Dynamically tuned gyroscope represented by Jamieson Figure 2.14 Tortuosity comparison: micro-tortuosity vs large scale tortuosity Figure 2.15 Concrete block tests showing a long transition between the slide and rotation zones by Stockhausen et al Figure 2.16 Lip produced at the bottom of the borehole when transiting from rotary to sliding tested by Stockhausen et al Figure 2.17 Geometric torsion from HRCG survey using the ASC borehole trajectory model, resolution 90 ft Figure 2.18 Effective diameter from HRCG survey, resolution one survey per foot (Courtesy of Adrián Ledroz) Figure 2.19 Actual wellbore shape from HRCG survey, resolution one survey per foot (Courtesy of Adrián Ledroz) Figure 2.20 Three-dimensional borehole trajectory methods represented by Jamieson Figure 3.1 Buckling phenomena in a horizontal section created by Fazaelizadeh Figure 3.2 Local [ t, n, b ] and global [x,y,z] coordinate system in 3D Figure 3.3 Force and moment balance FBD: [a] Moment looking downhole; [b] Force and moment balance equilibrium Figure 3.4 Drillstring position related to the borehole Figure 4.1 Examples of Bézier curves approximation represented by Sampaio Figure 4.2 Field Case 1: Inclination and azimuth from actual survey data Figure 4.3 Field Case 1: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.4 Field Case 1: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.5 Field Case 2: Inclination and azimuth from actual survey data xii

14 Figure 4.6 Field Case 2: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.7 Field Case 2: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.8 Field Case 3: Inclination and azimuth from actual survey data Figure 4.9 Field Case 3: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.10 Field Case 3: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.11 Field Case 4: Inclination and azimuth from actual survey data Figure 4.12 Field Case 4: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.13 Field Case 4: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.14 Field Case 5: Inclination and azimuth from actual survey data Figure 4.15 Field Case 5: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.16 Field Case 5: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution Figure 4.17 Synthetic Wellpath 1: 3D horizontal section Figure 5.1 Spline-curve and circular arc osculation Figure 5.2 Field Case 1: ER well vertical section Figure 5.3 Field Case 1: ER well contact force. DLS in units of /100 ft Figure 5.4 Field Case 1: ER well torque results while ROB Figure 5.5 Field Case 1: ER well axial load while ROB Figure 5.6 Field Case 1: ER well axial load while pickup Figure 5.7 Field Case 1: ER well axial load while slackoff xiii

15 Figure 5.8 Field Case 1: ER well torque while 120 RPM and 60 klbf WOB Figure 5.9 Field Case 1: ER well axial load while 120 RPM and 60 klbf WOB Figure 5.10 Field Case 2: Backward-nudge horizontal well vertical section Figure 5.11 Field Case 2: Backward-nudge horizontal well contact force. DLS in units of /100 ft Figure 5.12 Field Case 2: Backward-nudge horizontal well torque while ROB Figure 5.13 Field Case 2: Backward-nudge horizontal well axial loading while ROB. 120 Figure 5.14 Field Case 2: Backward-nudge horizontal well axial loading while PU Figure 5.15 Field Case 2: Backward-nudge horizontal well torque while 75 RPM and 25 klbf WOB Figure 5.16 Field Case 2: Backward-nudge horizontal well axial loading while 75 RPM and 25 klbf WOB Figure 5.17 Field Case 2 Backward-nudge horizontal well hookload while ROB Figure 5.18 Field Case 2: Backward-nudge horizontal well hookload while pickup Figure 5.19 Field Case 2: Backward-nudge horizontal well hookload while slackoff. 128 Figure 5.20 Field Case 2: Backward-nudge horizontal well torque while 75 RPM and 25 k-lbf WOB Figure 6.1 Side force prediction using multiple T&D models Figure 6.2 TVD gross error using multiple borehole trajectory models Figure 6.3 Tortuosity error comparison between MCM and ASC borehole trajectory model Figure 6.4 Liquid loading phenomena represented by Khan xiv

16 LIST OF TABLES Table 1.1 History of extended reach drilling Table 3.1 Range of friction factors Table 4.1 ASC model sample calculation example Table 4.2 Field Case 1: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Table 4.3 Field Case 1: The ASC borehole trajectory model accuracy improvement from low resolution to higher resolution Table 4.4 Field Case 2: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Table 4.5 Field Case 2: The ASC borehole trajectory model accuracy improvement from low resolution to higher resolution Table 4.6 Field Case 3: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Table 4.7 Field Case 3: The ASC borehole trajectory model accuracy improvement from low resolution to higher resolution Table 4.8 Field Case 4: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Table 4.9 Field Case 5: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Table 4.10 Synthetic Wellpath 1: Maximum error comparison at n = Table 4.11 Synthetic Wellpath 1: Maximum error comparison at n = Table 4.12 Synthetic Wellpath 2: Maximum error comparison at α = 10,ξ = 4,δ = Table 4.13 Synthetic Wellpath 2: Maximum error comparison at α = 1000,ξ = 100,δ = xv

17 Table 5.1 Field Case 1: Drillstring design Table 5.2 Field Case 1: T&D results comparison Table 5.3 Field Case 2: T&D results comparison Table 5.4 DWOB academic example with 175 klbf measured HL Table 6.1 Average of absolute value percentage of variation between the ASC 3D T&D model and Program A [PHR 1-ft and 90-ft survey intervals input] 133 Table 6.2 Accuracy of models in predicting T&D parameters Table 6.3 Accuracy of models in predicting wellbore position Table 6.4 Performance prediction matrix for gas wells experiencing liquid loading. 139 xvi

18 LIST OF SYMBOLS Cross-sectional Area A Advanced Spline Function Coefficients A,B,C,D Bending Stiffness EI Binomial Shear Force F b Sinusoidal Critical Buckling Load F cr Friction Force F f Helical Buckling Load F hel Normal Force F n Axial [Tangential] Load F t Axial Force Increment F t Stiffness Force F s Results of Spline curve s Integrations to Solve for the Vector z h, u, v Moment [Torque] M t Number of Survey-Stations n Radius of Curvature R Outer Radius of the Drillstring Contact Point r p Measured Depth on the Space Curve s Advanced Spline Function T(s) z-components of the Tangential, Normal and Binormal Vector t z,n z,b z Drillstring Effective Buoyed Weight W b xvii

19 Side Force W c Drag Force W d External Fluid Load w ef Internal Fluid Load w if A Curve in a 3D Space X(t) Vector z of the General Spline System of Equations z Coefficients of a Curve in a 3D Space α,ξ,η,δ Buoyancy Factor β Coefficient of the Arc Length s γ Maximum Error Over the Entire Wellpath ε Determinate Vector Formed by the First, Second and Third Derivatives of the Function T(s) ζ Azimuth θ Contact Force Direction ϑ Wellbore Tortuosity κ Change in the Rate of Curvature dκ/ds Normalized Tangential Vector on the Space Curve λ Normalized Tangential Vector for Easting Coordinates [x axis] λ E Normalized Tangential Vector for Northing Coordinates [y-axis] λ N Normalized Tangential Vector for true vertical depths [z-axis] λ TVD Friction Factor µ Fluid Density ρ Geometric Torsion τ xviii

20 Inclination φ The Arc Length s for t[0,1] ϕ(t) Turning Angle of a Curve ψ xix

21 LIST OF ABBREVIATIONS Average Angle Tangential Method AATM Advanced Spline-Curves ASC Bottom Hole Assembly BHA Balanced Tangential Method BTM Degree of Curvature Continuity of Order n Cn Coordinate Reference Systems CRS Dogleg Severity DLS Extended Reach ER Global Positioning Systems GPS Geodetic Reference System GRS80 Horizontal Departure HD High Resolution Continuous Gyroscopic HRCG The Industry Steering Committee on Wellbore Surveying Accuracy ISCWSA Kick off Point KOP Logging While Drilling LWD Minimum Curvature Method MCM Measured Depth MD Measurement While Drilling MWD North American Datum NAD27 North American Datum NAD83 xx

22 Polycrystalline Diamond Compact PDC Positive Displacement Motor PDM Pseudo-high resolution PHR Rotary Steerable System RSS Rate of Penetration ROP Radius of Curvature Method RCM Spherical Indicatrix of Tangent SITn Tangential Method TM Three-Dimension D Torque and Drag T&D Torque on Bit TOB True Vertical Depth TVD United States Geological Survey USGS Universal Transverse Mercator UTM Vertical Section VS The World Geodetic System WGS84 Weight on Bit WOB xxi

23 ACKNOWLEDGMENTS A special thanks to my family. Words cannot express how grateful I am to my mother, and father for all of the sacrifices that you ve made on my behalf. Your prayer for me was what sustained me thus far. I would like to express my special appreciation and thanks to my advisor Professor Alfred William Eustes III and my co-advisor John P. de Wardt. You have both been tremendous mentors to me. I would like to thank you for encouraging my research and for allowing me to grow as a research scientist and a teaching instructor. Your advice on both research as well as on my career has been priceless. I would also like to thank my committee members, Professor D.V. Griffiths, Professor Mark Miller, Professor John Steel, and Professor Azra Tutuncu for serving as my committee members even at hardship. I also want to thank you for letting my defense be an enjoyable moment, and for your brilliant comments and suggestions, thanks to you. I would also like to thank all of my friends who supported me in writing, and their encouragement to strive towards my goal. At the end I would like express appreciation to my beloved wife for her support in the moments when there was no one to answer my queries. xxii

24 In loving memory of my brother Ahmed Abughaban (September 20, April 12, 2015) xxiii

25 CHAPTER 1 INTRODUCTION Extended reach [ER] drilling refers to the process of directing the wellbore along a threedimensional [3D] trajectory to a predetermined target or several targets, located at a significant horizontal distance from the drilling rig. ER wells are typically drilled vertically to a given depth, known as the kick off point [KOP] and then built to an angle of inclination that allows sufficient horizontal distance from the drilling rig towards the target and then held constant near or at horizontal and extended into the reservoir. Drilling these complex horizontal wells is accompanied by with significant challenges, and therefore, the planning and well construction process is very important to ensure reaching desired targets within defined accuracy and reducing the induced friction between the drillstring and the wellbore wall. ER drilling is a technology that aims to improve the reservoir drainage and maximize its productivity at the least cost while reducing well-site footprints which minimizes environmental impact. These wells are often drilled because the surface location directly above the reservoir is inaccessible, either because of natural or man-made obstacles. Examples include reservoirs under populated areas, environmentally sensitive areas, or located beneath water. Relief wells, a complex design example, are also drilled directionally to control a blowout well. The relief well is deviated to intercept the uncontrolled well in the reservoir. Sidetrack is another application of directional drilling that involves drilling a secondary wellbore away from an original wellbore. This sidetracking may be done to bypass salt domes, unusable section of the original wellbore, or a lost drillstring in a hole [also known as a fish]. It is possible to have multiple sidetracks to enhance the production further by extending the producing zone through number of branches radiating from the main wellbore known through re-entry or multilateral drilling. 1

26 1.1 What Limits Extending the Reach of Drilling? The first ER well was drilled in 1975 with an absolute departure value of 10,900-ft as illustrated in Table 1.1. In the late 1980s and early 1990s the industry began drilling these types of wells more regularly. Table 1.1: History of extended reach drilling Year Maximum Departure [ft] Maximum TVD [ft] ,900 31, ,082 31, ,203 31, ,197 31, ,197 40, ,956 40, ,765 40,230 Step-out ratio is defined as the horizontal departure [HD] divided by the total vertical depth [TVD] and water depth as illustrated in Figure 1.1 and can be calculated as follow: Step-out ratio = HD TVD Figure 1.1: Step-out ratio example of an extended-reach well modified from Kibsgaard(2011). 2

27 Generally, a well is defined as an ER well when it has a step-out ratio of 2 or more. This definition also implies that as the ratio of horizontal departure to TVD increases, the well difficulty increases. Therefore, the step-out ratio also provides an effective measure of the overall drilling difficulty. Figure 1.2 illustrates the ER drilling database for wells worldwide. One interpolation from this plot is that there are four depth levels or zones of ER drilling reach: [1] low reach of less than 10,000-ft horizontal departure, [2] medium reach from 10,000-ft to 15,000-ft horizontal departure, [3] extended reach from 15,000-ft to 20,000-ft horizontal departure, and [4] very extended reach that occurs above 20,000-ft horizontal departure. Figure 1.2: Extended reach drilling database plot created by Kwantis (2015). The limit of ER drilling is reached if one of the following occurs: Borehole instability arises, due to time exposure, geomechanics issues, drilling fluids interaction or pressure differential. 3

28 Excessive drag preventing the drillstring from traveling to the bottom of the hole, i.e. the friction exceeds the force available to push the drillstring to the bottom of the hole as shown in Figure 1.3. Figure 1.3: Extended reach drilling limit due to drag forces created by Fazaelizadeh (2013). The torque required to overcome the rotational friction is greater than the thread makeup torque, i.e. rotational friction exceeds the torque available to turn the drillstring at the bottom of the hole, as shown in Figure 1.4. Figure 1.4: Extended reach drilling limit due to the rotary torque created by Fazaelizadeh (2013). A substantial growth in ER drilling was observed in the early 1990s as illustrated in Figure 1.5 showing the ER drilling evolutionary plot in metric units. To expand the ER drilling envelope even further, several challenges must be addressed. 4

29 Figure 1.5: The evolution of extended reach drilling modified from Cox (2016). 1.2 Challenges and Critical Technologies As drilling challenges continue to evolve, new projects are now expected to achieve measured depths of more than 50,000 ft. Among the technical limitations will be the capability to handle the torque required to drill on bottom and the tensile [load] requirements to complete the casing plans. Overcoming these limitations demands further advancement in technologies that have been found to be vital to the success of ER drilling. These technologies include wellbore trajectory modeling, T&D modeling, borehole stability, hole cleaning and fluid dynamics. The focus in this dissertation will be on wellbore trajectory and T&D modeling. 5

30 1.2.1 Wellbore Trajectory Modeling The standard surveying technique of today employs downhole measurement while drilling [MWD] directional sensor tools. MWD tools measure the direction of the Earth gravity by using three orthogonally mounted accelerometers as the principal measured components. Each accelerometer consists of a magnetic mass [like a pendulum] suspended in an electromagnetic field. Gravity deflects the mass from its null position. Sufficient current is applied to the sensor to return the mass to the null position. This current is directly proportional to the gravitational force acting on the mass. The use of magnetic instruments requires nonmagnetic equipment to avoid magnetic interference. Another surveying technique is to use the gyroscopic survey instrument. This tool is a north seeking/ rate gyro technology that works on the principal of spinning mass and it is operated in a continuous measurement mode. The use of the gyroscopic survey instrument is currently considered the most accurate surveying technique for the oil industry since it is immune to the magnetic interference and hence it can be run in casing rather than open hole. Wellbore surveying measurements are performed at several discrete survey-stations along the wellpath, typically at 30-ft to 90-ft intervals. These surveys are then used to calculate the wellbore trajectory. In the industry, several methods have been proposed to calculate the wellbore trajectory between survey-stations. In practice, no mathematical model can predict the exact curvature; however, a robust mathematical model presented in Chapter 4 can predict the curvature of the wellpath more accurately than current models. The industry standard 3D well trajectory model is the minimum curvature method [MCM]. It tends to mathematically smoothen the wellpath between survey-stations creating an artificially low tortuosity and miscalculation of the actual TVD, northing and easting. This leads to the underestimation of T&D magnitude. Representing the actual wellpath precisely and accurately predicting the wellbore shape and geometry are critical elements in T&D modeling, particularly in drilling ER and complex horizontal wells. 6

31 1.2.2 Torque and Drag Modeling One of the main challenges while drilling ER wells flows from the fact that surface indicators such as hookload, surface torque and standpipe pressure measurements provide only a partial picture of downhole conditions. Accurate T&D modeling in real-time and the ability to analyze downhole forces are extremely helpful tools that reduce much of the risk of a drilling program. Real-time monitoring of an operation can alert the rig crew to assist in making sound decisions using engineering principles to change the well conditions. Torque level and axial drag in ER wells are generally affected by the build or drop sections. Typically, the borehole has high and repeated tortuosity. In these conditions, the drillstring is in extensive contact with the wall. The longer the deviated section, the more accurate estimations of the downhole T&D conditions are required to optimize the delivery of the power transmitted from the surface. In vertical holes, estimated T&D are usually within tolerable levels, however, as a vertical well is drilled deeper, T&D values will increase. This is caused by the hidden tortuosity along the borehole between survey-stations. Such phenomenon will affect the downhole weight and torque on bit that are determined from surface sensors. Predicting these downhole conditions is critical to determining if a well can be drilled within equipment capabilities. The buckling of drillpipe resulting from excessive bending loads is another issue that impacts ER wells. This phenomenon causes an increase in the contact force between the drillstring and the wellbore. As the drillstring weight is concentrated at the buckled portion of the drillstring, a lock-up will occur due to axial friction build up in the extended build section with high angle (Wu 1996). This can lead to helical buckling developing, eventually preventing the pipe from moving in the hole. Today s T&D modeling proceeds from the soft-string model or the stiff-string model [explained in Chapter 3]. In both models, the wellbore trajectory is calculated using the MCM, assuming a constant curvature arc between survey-stations. With this assumption, the geometric torsion [τ] and the change in the rate of wellbore curvature [dκ ds] are nil 7

32 (Mitchell and Miska 2011). This generates unreliable estimates of T&D magnitudes, since the true wellbore curvature [κ] between survey-stations does change. Studies have shown that the change in curvature could be reasonably small, hence, the temptation to assume curvature to be constant. However, it has been proven that the effect of the bending moment in the T&D equilibrium equations cannot be effectively analyzed based on this assumption. In the scope of this dissertation, a 3D T&D model has been extended for a non-constant curvature borehole trajectory model [the ASC model] that is presented in Chapter 5. This model includes τ, κ, dκ ds and the drillstring bending stiffness [EI] to predict more accurate drillstring contact forces and a more realistic bending stiffness effect. 1.3 Motivation of Dissertation To overcome the challenges and technical limitations of drilling ER wells, further advancements in technology and modeling are required. To improve current T&D modeling, three approaches could be applied: 1. Simply including all the terms in the T&D equilibrium equations. 2. Developing a more advanced wellbore trajectory calculation method. 3. Considering different fluid loads and include dynamics in the T&D equilibrium equations. The first approach has been analyzed and tested in the industry. Including all the terms in the T&D equilibrium equations is called the stiff-string T&D model. This approach differs from the industry standard model, known as the soft-string model, which neglects the bending moment. Stiff-string model considers the drillstring bending stiffness and annulus clearance. Thus far, many stiff-string models have been developed. However, none of them has a standard formulation in the industry (Wu 1996). The main limitation of this model is the use of a constant curvature trajectory calculation [MCM], which assumes constant curvature arc and, hence, constant bending moment. 8

33 Different wellbore trajectory computation methods have been used in the past, these methods are explained in Chapter 2. The MCM is the most common and most accurate model from the defined algorithms, and is effectively the industry standard. However, there are major limitations and issues associated with this method: The MCM is based on a piecewise approximation assuming a constant curvature arc between survey-stations. This has been shown with high resolution surveys to be not true during sliding and rotate modes using positive displacement motors [PDM]. Although the MCM is continuous in the first derivative [C1], it exhibits discontinuity in the second order derivative [C2], i.e. the change in the rate of curvature [dκ ds]. Figure 1.6 shows a wellpath between three survey-stations represented by a circular arc fitted to the measured data, such that the survey-stations are tangential to a circle. The two arcs are sharing a common tangent at the middle survey-station [station 2] causing the change in the rate of wellbore curvature [dκ ds] to be nil. Figure 1.6: Minimum curvature method represented by a circular arc wellpath. The discontinuity illustrated causes an artificially low tortuosity created from the constant radius of curvature represented by each circle s radius. This leads to TVD error accumulating along the wellpath, causing the underestimation of T&D magnitude. This is due to the assumption that the bending part in the force and moment equilibrium equations is 9

34 negligible. This is one of the main weaknesses of using the MCM, as it does not represent the real drillstring configuration due to the omission of bending stresses. The potential for significant TVD errors along the wellpath is illustrated in Figure 1.7. Consider survey points A and B, with the trajectory along a 100 ft path both cases start and end with the same inclination, azimuth and MD. If it is assumed that the angle is building 10 from A to B, the well paths are different as the tangent on the right comes first, followed by a curve, and vice-versa on the left. In both cases the MD is the same, the azimuth direction has not changed and the inclinations have risen by 10 ; hence, the surveys would be identical (Jamieson 2012). This phenomenon is also known by the slide and rotate pattern drilling effect (Stockhausen et al. 2012), in which the borehole is in gauge while sliding and over gauge while in rotation, causing high DLS fluctuation. Figure 1.7: Left: Slide then rotate to build 10 over 100-ft section. Right: Rotate then slide to build 10 over 100-ft section. In this dissertation, the underestimation of T&D in the drilling process has been investigated through theoretical modeling. This has entailed investigating the mathematical models for T&D calculations in the light of a robust, more advanced and continuous 3D wellbore trajectory model, the Advanced Spline-Curves [ASC] borehole trajectory model. This approach will be shown to improve T&D modeling, and accurately predict forces, bending 10

35 moments, and contact loads along the wellbore. When the wellbore trajectory is modeled using the ASC borehole trajectory model, the wellbore positions are first computed. From this trajectory, wellbore curvature [κ], geometric torsion [τ] and the change in the rate of wellbore curvature [dκ ds] are determined. These parameters will be included in the force and moment equilibrium equations. This approach will allow the industry to represent the actual wellpath more precisely, resulting in more accurate representation of the wellbore shape and geometry and consequently better estimation of the T&D parameters. Unlike the current method of borehole trajectory calculation, the MCM, this new model does not make the potentially unrealistic assumption of a constant curvature arc between survey-stations. Accurately determining T&D parameters is a technology enabler in drilling ER and complex horizontal wells. It is also an important factor in the evaluation process in designing and running downhole completion and production equipment and optimizing the bottomhole assembly [BHA] (Liu et al. 2004; Menand et al. 2006). These factors will improve the drilling performance while minimizing the occurrence of catastrophic drillstring failure (Stockhausen and Lesso 2003). 1.4 Objectives of Dissertation The main objectives of this dissertation are: 1. Accurately determine wellbore positions to reduce wellbore positioning uncertainties. This includes the calculation of: (a) True vertical depth (b) Easting coordinates (c) Northing coordinates 11

36 2. Accurately predict the wellbore shape and geometry and present the wellpath precisely. This includes the calculation of: (a) Wellbore tortuosity (b) Geometric torsion 3. Accurately predict drillstring to wellbore wall contact forces from T&D models. This includes the calculation of: (a) Hookload (b) Rotary torque (c) Axial load With these three main objectives, the engineer can better apply drilling operational parameters, improve rate of penetration[rop] and minimize the non-productive time[npt]. This will move the industry forward to design longer horizontal sections and reduce ER drilling cost and time. 1.5 The Novelty of Dissertation This dissertation pushes the boundaries for wellbore trajectory and T&D models. In terms of wellbore trajectory, the proposed model, described in Chapter 4, has the ability to accurately measure wellbore positions, wellbore shape and geometry, and present the wellpath precisely. In terms of T&D modeling, the proposed model, described in Chapter 5, has the ability to estimate bending effects more realistically, accurately predict the contact forces between the drillstring and the wellbore, and solve T&D parameters from surface to total depth in reasonable time using standard engineering computers. Accurately estimating wellbore configurations [positions, shape and geometry] and T&D parameters will improve the drilling performance in ER and complex horizontal wells and brings operational value to the oil and gas industry. 12

37 1.5.1 Wellbore Position, Tortuosity and Geometric Torsion Calculation This dissertation describes a novel methodology for approximating the wellbore position calculation using the ASC borehole trajectory model. The calculated trajectory is based on a curve defined by a piecewise cubic polynomial function using splines with a guaranteed degree of continuity of the resulting curve up to the third order derivative [C3] along the entire wellpath. This approach possesses the robustness and flexibility to calculate complex, 3D wellbore trajectories. Cubic splines have been used routinely to solve many mathematical applications such as approximation theory, computer graphics, data fitting and other complex numerical solutions. In addition, splines are a key component of aircraft Automated Flight Manual systems, which is used in aircraft controls (Grandine 2005). Thus, using the ASC borehole trajectory model is not only an alternative approach to accurately approximate wellbore trajectories in the oil field; it can also serve as step forward to drilling automation Drillstring to Wellbore Wall Contact Forces Calculation When the wellbore trajectory is modeled using the Advanced Spline-Curves [ASC] borehole trajectory model, the wellbore positions are first computed. From this trajectory, τ, κ and dκ ds are determined. These parameters will be included in the force and moment equilibrium equations. Introducing the stiffness and bending moment terms in the T&D model will have the potential to greatly reduce the under-prediction of T&D parameters as illustrated in Chapter 5. Contact forces will be more accurately estimated under downhole conditions. This model becomes more important in ER wells with large side turns where contact side forces on the drillstring are not properly estimated using current T&D models. 1.6 Organization of Dissertation This dissertation addresses two main topics: [1] 3D wellbore trajectory models, and [2] T&D modeling. It first presents an overview of extended reach drilling and moves to an explanation of the 3D wellbore trajectory and T&D models development and validation. 13

38 Chapter 2: Three Dimensional Wellbore Trajectory This chapter includes a review on the reasons for drilling directional wells, the different coordinate systems and maps employed, and typical deflection and wellbore surveying tools. The fundamentals of the past and current 3D wellbore trajectory models employed by the oil and gas industry are presented. Chapter 3: Three Dimensional Torque and Drag Modeling This chapter includes a review on the general vector differential form of the T&D modeling. The current T&D models are discussed in this chapter. The differences between the soft-string model and the stiff-string model are explained. Finally, the historical perspective, and past and current methods and technologies of T&D are presented. Chapter 4: Technical Approach - Advanced Wellbore Trajectory Model This chapter presents the details of the 3D advanced wellbore trajectory model. It starts with a general description of constructing a curve that passes through survey-stations. Then the assumptions and boundary conditions on the model are examined. From there, the non-constant curvature trajectory model [The Advanced Spline-Curves Model] is derived to compute the wellbore positioning [TVD, northing and easting], the wellbore tortuosity and the geometric torsion. The model is validated using five field studies and two synthetic wellpath examples. Finally, a comparison of the current wellbore trajectory models with the ASC model derived in this dissertation demonstrates the favorable results of the model s accuracy. Chapter 5: Technical Approach - Three Dimensional Torque and Drag Model This chapter presents the improvement of current T&D models. It starts with general approaches to improve the accuracy of T&D models. Then the assumptions and boundary conditions on the model are shown. This sets the guidelines to derive the force and moment equilibrium equations to predict T&D outputs. The general procedure and the derivation of 14

39 the model are described. Lastly, the model is validated using two field cases with real-time forces that define T&D measurements at the surface. Chapter 6: Technical Result and Discussion This chapter presents technical results on the drilling process. These are summarized in terms of three factors, namely, T&D, wellbore positioning and wellbore tortuosity. A discussion on the results, and recommendations on both the ASC borehole trajectory model and the extended 3D T&D model are presented. Chapter 7: Conclusions and Recommendation This chapter concludes the work effort based on the analysis and field applications presented in this dissertation. Recommendations on the impact of the ASC borehole trajectory model and the extended 3D T&D model are presented. A discussion of future research plans, and the potential to successfully extend the horizontal departure in ER wells is included. 15

40 CHAPTER 2 THREE DIMENSIONAL WELLBORE TRAJECTORY This chapter includes a review on the reasons for drilling directional wells, the different coordinate systems and maps employed, and typical deflection and wellbore surveying tools. The fundamentals on the past and current 3D wellbore trajectory models in the oil and gas industry are presented. Directional drilling is the art and science of drilling a wellbore along a predetermined trajectory at a given measured depth, inclination and azimuth. The tools and methods used are determined by the complexity of the wellpath and the level of accuracy required to follow that trajectory. 2.1 Reasons for Drilling Directional Wells Strong economical and environmental pressures have increased the use of directional drilling for many reasons, as illustrated in Figure 2.1. Figure 2.1: Reasons to drill directional wells represented by Greg and Stefan (2011). 16

41 Six reasons for drilling directional wells: Inaccessible surface location: Directional wells are often drilled because the surface location directly above the reservoir is inaccessible, either because of natural or manmade obstacles. Examples include reservoirs under cities, mountains and lakes. Reducing the number of offshore structures: The conventional approach for a large subsea oilfield has been to install a fixed platform on the sea-bed, from which many directional wells may be drilled. Relief wells: Directional techniques are used to drill relief wells to control a well blowing out. The relief well is deviated to pass as close as possible to the uncontrolled well in the reservoir. Heavy mud is pumped into the reservoir to overcome the pressure and bring the wild well under control. Sidetrack from existing wells: Sidetracking out of an existing wellbore is another application of directional drilling. This sidetracking may be done to bypass an obstruction in the original wellbore or to explore the extent of the producing zone in a certain sector of a field. Re-entry and multi-lateral wells: Re-entry and multi-lateral drilling employs the full range of directional drilling tools and techniques. This includes planning, wellbore engineering, and careful consideration of the numerous aspects of drilling straight and deviated holes. Re-entry and multi-lateral well s configurations can include dual stacked, dual opposing, dual opposing stacked, spokes, lateral tie-back, and herringbone. Geological requirements: Directional drilling is also applicable in geological fault drilling. It is sometimes difficult to drill a vertical well in an inclined fault plane. Often, the bit will deflect when passing through the fault plane; and, sometimes the bit will follow the fault plane. To avoid this problem, the well can be drilled on the up dip or down dip side of the fault and deflect into the producing formation. 17

42 2.2 Coordinate Systems and Maps The primary objective of this section is to introduce and highlight the importance of geodesy as a discipline while raising awareness of the concepts surrounding Coordinate Reference Systems [CRS]. A secondary objective is to provide an appreciation of the risks and consequences of failing to understand the importance of geodesy and CRS in the oil and gas industry. Earth geodesy is the study of the shape of the Earth. The Earth has an irregular and constantly changing surface. Models of the surface of the Earth are used in navigation, surveying, and mapping. Topographic and sea-level models attempt to model the physical variations of the surface of the Earth. Gravity models and geoid are used to represent local variations in gravity that change the local definition of a level surface. The geoid is the model used to represent the surface of the Earth over both land and ocean as though the surface resulted from gravity alone [Figure 2.2]. Figure 2.2: Geoid surface of the Earth represented by Jamieson (2012) Geographic coordinate system [GCS] is a reference system used to measure horizontal and vertical distance on a planimetric map that is achieved through a datum. Datum is the basis for planar coordinate systems and the basis for all geodetic survey work. 18

43 Utilizing certain ellipsoids of the Earth, a datum represents a set of parameters and controlpointsusedtoaccuratelydefinethe3dshapeoftheearth. Thereareover100datums in existence around the world. At a minimum, a complete datum provides definition for orientation, scale, and dimensions for the reference ellipsoid. Some sample datums include: The North American Datum 1927 [NAD27]: Based on Clarke 1866 Ellipsoid. This datum is the basis for the United States Geological Survey [USGS] quad maps and is measured in units of feet. NAD27 is currently being phased out. The North American Datum 1983 [NAD83]: Based on the Geodetic Reference System 1980 [GRS80] Ellipsoid. NAD83 is measured in units of meters. The World Geodetic System 1984 [WGS84] Datum: Based on WGS84 Ellipsoid [Figure 2.3]. This datum is the basis for maps of the world and large scale maps and is the basis for Global Positioning Systems [GPS]. WSG84 defines heights for the entire Earth in units of meters. In general, the spheroid, on an ellipsoid, is the basis for the datums which are used in coordinate systems. It is constructed using the geodetic data from the geoid. Maps are created by projecting the Earth on a flat plane. In many cases these projections will cause distortions. Different projections preserve some properties at the expense of others. These properties are: Area: a quantitative measure of a 2D shape or figure. Shape: the external boundary of an object and its form. Direction: the relative position of one point with respect to another. Orientation: the direction of a particular path. Distance: numerical description of how far objects are apart on the surface. Scale: the ratio of a particular distance on the ground to its distance in a map. 19

44 Figure 2.3: WSG-84 geoid height represented by the NGA (2013) Any position on the Earth s surface can be described in term of degrees north or south of a latitude, and degrees east and west of a longitude. The latitude and longitude are network of imaginary lines that are superimposed on the Earth surface. The latitude lines run parallel from the equator plane [0 latitude] to both the north [at 90 N] and the south pole [at 90 S]. The longitude lines, also known as meridians, run to the east or west from Greenwich in England [0 longitude] and are perpendicular to the equator and meet at the poles. In effect, the Earth has 90 integer latitude lines between the equator and each pole ranging from 90 N to 90 S, and 360 integer longitudes ranging from 180 E to 180 W. For example, Marquez Hall in Golden is located at N and W. Figure 2.4 shows the GCS on a globe. 20

45 Figure 2.4: Geographic coordinate system represented by Jamieson (2012) Map Projections Whether the Earth is treated as a sphere or a spheroid, its 3D surface must be translated into 2D projections [i.e. flat map sheet] to determine relative positions on or within the Earth. This mathematical transformation is commonly referred to map projections. A map projection uses mathematical formulas to relate spherical coordinates on the globe to flat, planar coordinates. The standard Mercator projection [cylindrical] is produced by placing a cylinder over the Earth with the contact point circumference around the equator. Points on the Earth will be projected onto the inside of the cylinder, the cylinder is then unwrapped and laid flat. There are different projections available as shown in Figure 2.5. All projections cause different types of distortions. Some projections are designed to minimize the distortion of one or two of the data s characteristics. Borehole surveying in the oil and gas industry requires a map or rectangular grid system that preserves the locality area and shape of the surveyed position. The current industry acceptable projection systems are the Universal Transverse Mercator, and the Lambert conical orthomorphic [transverse conic]. 21

46 Figure 2.5: Map projection examples based on various plane surfaces and the orientation of the surface represented by Jamieson (2012). The Lambert conical projection is the most common projection used in the United States; it incorporates a greater extension of the east and west and lesser extension of north and south. The Universal Transverse Mercator [UTM] projection is the most common projection worldwide using a horizontal cylinder over the Earth with the contact point circumference around a chosen meridian as shown in Figure Reference Directions Typically, there are three azimuth reference systems reported by wellbore surveying tools [discussed in the next section]; either in reference to True Geographic North, Grid North or Magnetic North. 22

47 Figure 2.6: Universal Transverse Mercator Projection represented by Jamieson (2012). True Geographic North: This is the direction of the geographic North Pole, also known as the meridian direction, shown on maps by the meridian longitude. This direction is indicated by the rotation axis of the Earth. Grid North: Grids enable survey points to be computed and plotted in rectangular coordinates to simplify the calculations for bearing and distance. These models project the curved surface of the Earth into flat surfaces in maps by dividing the world into number of latitudes and longitudes with defined distortions. The grid north is equal to true north only at the central meridian on a given map. Away from the central meridian, a convergence angle is required to correct the grid north to true north and vice-versa. Magnetic North: This reference direction points toward the magnetic north of a compass. This is subject to time variation due to the movement of the north and south magnetic poles. The variation may cause error in the magnetic measurements, causing interpretation error in surveying, if not accounted for. In drilling, the wellbore surveys recorded using a magnetic tool are initially given an azimuth reading from the magnetic north and then converted to either true north or grid north for final reporting. Such tools require a declination correction to convert magnetic 23

48 north to true north. The divergence between the two norths is measured in degrees and referred to the declination angle. The declination is negative if the magnetic north is located to the west of the true north and is positive if the magnetic north is on the west of the true north as shown in Figure 2.7. Figure 2.7: Magnetic declination. 2.3 Deflection Tools Deflection tools are used to deflect the wellbore from its current position to a predetermined direction using the toolface orientation [TFO]. TFO can be expressed as a direction from north or highside of the wellbore. The magnetic TFO refers to the angular measurement from magnetic north to the tool orientation. The highside of the toolface indicates whether a deflection tool is facing up, down, to the left or right, relative to the wellbore [Figure 2.8]. The wellbore can be deflected using several wellbore deviation technologies, such as: jetting action; whipstocks; 24

49 Figure 2.8: Toolface orientation of a deflection tool. 25

50 positive displacement motors with a bent sub or housing; and rotary steerable systems. The two most common used for steering are positive displacement motors and rotary steerable systems Jetting Action In jetting, typically the drill bit is equipped with small diameter nozzles that produce high velocity fluids from the bit. With the advancement of bit designs, bits can be fitted with asymmetric nozzles, one large jet nozzle oriented to the toolface and two or more smaller nozzles on the opposite orientation as illustrated in Figure 2.9. If the drillstring is stationary during the jetting operation, the smaller size nozzles will cause greater erosion on the side wall causing a deviation of the well. Since this technique relies on hydraulics to deviate the wellbore, it is only effective in soft formations. One disadvantage of the jetting action and main reason it is rarely used today is the high tortuosity produced from the hydraulic deflection. Figure 2.9: Jetting tool represented by Mitchell and Miska (2011). 26

51 2.3.2 Whipstocks Whipstocks consist of a long-inverted steel wedge that is concave on the bit side to allow the deflecting drilling assembly to be oriented for the deviation process operation. Whipstocks are widely used as a deflection tool in multilateral wells and to mill casing windows during sidetrack operations in existing wells. There are two main types of whipstocks, a retrievable and a permanent type whipstock: Retrievable whipstocks: Commonly used in multilateral well design. The whipstock is set hydraulically against the casing wall below the lateral window. Once an opening in the casing wall is drilled, the whipstock is retrieved. Permanent whipstocks: Typically set permanently on the bottom of the hole or on top of a cement plug. These whipstocks consist of an inclined wedge with hard steel surface that forces the bit to penetrate through the rock rather than the whipstock itself allowing the BHA to deviate from the main axis of the wellbore Positive Displacement Motors and Bent Subs or Housing The most popular method of deflection is the Positive Displacement Motor [PDM]. In this method, a bent sub or housing is located at the end of the drillstring near the bit. Bent subs or housings enable the orienting of the bit in the desired direction at a specified tilt angle. The PDM generates its mechanical power from mud circulation to provide rotation and torque to the bit without the need for rotation of the drillstring. The design of PDM consists of four basic components [Figure 2.10]: bypass valve; power module [rotor and stator]; universal joint assembly or deflection device; and bearing assembly with a drive sub [bit box]. 27

52 Figure 2.10: Baker Hughes Positive Displacement Motor represented by Krueger (1996). The PDM configuration can be used for drilling both tangent sections, known as the rotating mode, and building sections, known as the sliding mode. When a PDM is operated in the rotating mode, the entire drillstring rotates. This redresses the bent angle and the bit goes straight. When it is time to change the direction of the well, the tool will be oriented in the direction of the bent, the PDM will allow the bit to rotate, independently from the drillstring, and the non-spinning drillstring will slide down the hole and build angle Rotary Steerable Systems Rotary Steerable systems [RSS] allow 3D control of the bit with simultaneous rotation of the drillstring by the action of three pads contained within a non-rotating sleeve. In the industry, RSS fall into two broad categories: 1. Push-the-bit: This system uses pads on the outside of the tool which press against the wellbore causing the bit to change its toolface in the opposite direction causing a direction change, as shown in Figure 2.11 [a]. 2. Point-the-bit: This system causes the bit toolface to change direction by bending the main shaft running through the entire tool with an eccentric ring, as shown in Figure 2.11 [b]. 28

53 Figure 2.11: Rotary Steerable Systems represented by Ferndaddy (2016): [a] Baker Hughes AutoTrak; [b] Schlumberger s PowerDrive Xtra. ER drilling using the RSS tool has made it possible to drill complex trajectory wells. This is due to tool s capability to overcome the high axial drag that might have prevented the PDM tool and bit from sliding. RSS also minimize the fluctuation in DLS applied to both rotation and slide sequence as the bit is controlled with simultaneous rotation of the drillstring. 2.4 Wellbore Surveying Tools Wellbore surveying measurements are performed at several discrete survey-stations along the wellpath, typically at 30 to 90-ft intervals. These survey-stations commonly use either the Earth s gravitational field, or the local magnetic field as the principal measured components. Instrument performance models convert the data to three-component measurement vectors, namely: [1] measured depth [MD], describing the distance along the wellbore from some reference point, [2] inclination [φ], describing the deviation angle from a vertical plane, and [3] azimuth [θ], describing the horizontal deviation from either magnetic or true north. The three-component measurement vectors [MD, φ and θ] are then used to calculate the wellpath of approximated curves, connecting the survey-stations to approximate the borehole positions [true vertical depth, northings and eastings] and wellbore curvature [tortuosity and geometric torsion]. 29

54 Wells are surveyed for the following reasons: While planning: to avoid well to well collisions, insure geological objective can be achieved with drilling confidence, and intercept a blowout. While drilling: to avoid well to well collisions, steer to a geological objective, monitor the progress of the well, and manage the wellbore tortuosity. Post drilling: to tie well data to geological and reservoir models and fulfill requirements by the regulatory authorities. There are two main types of surveying tools [1] magnetic survey tools, designed to measure both hole inclination and azimuth using the Earth s magnetic field, such as measurement while drilling tools, and [2] non-magnetic survey tools, designed to measure both hole inclination and azimuth not dependent upon the Earth s magnetic field, such as the principal of spinning mass or rate inertial technology, and gyro survey tools. Survey measurements are placed as close as physically possible to the bit Measurement While Drilling Tools The Measurement While Drilling [MWD] tool is a type of well logging that incorporates the measurement tools into the drillstring and provides real-time information to help with steering the drill bit. The MWD tool uses magnetometers and accelerometers to determine the borehole inclination and azimuth, using the Earth s magnetic field, during the actual drilling, as shown in Figure The tool transmits measurements from the downhole sensors to the surface. Typically, measurements include wellbore position, bit orientation, temperature, drillstring dynamics and real-time drilling information. This information is transmitted either by pressure pulsation up to the surface through the drilling mud or by a very low frequency radio wave to a surface antenna. 30

55 Figure 2.12: Baker Hughes Inteq MWD tool components represented by Choudhary (2011) 31

56 2.4.2 The Gyro Survey Tools A Gyro survey tool tends to keep its axis oriented to a fixed direction in space. Measurements are calculated by the Earth s spin rate and gravitational forces. These measurements must be corrected to the rotation of the Earth and the position on the Earth s surface. There are three types of gyro survey tools: 1. Conventional Gyro: The conventional gyro is used to sense and accurately measure components of the Earth s spin rate from which the azimuth can be calculated. The basic components of the gyro tool are [a] compass card to measure the hole azimuth fromthehorizontalspinofthegyroaxisand[b]aplumbbobassemblyoverthecompass to measure the hole inclination. 2. North Seeking Gyro: A high resolution continuous gyroscopic [HRCG] survey tool [also known as north seeking/ rate gyro tool] collects survey measurements using dual-axis dynamically tuned gyroscope [Figure 2.13]. This tool incorporates accelerometers that measure components of the specific force due to gravity. This data is used to compute borehole inclination and determine the position of all the sensors axis with respect to the high-side of the hole. The system is operated in a continuous measurement mode and is currently considered the most accurate for the oil industry to improve wellbore positioning (Stockhausen and Lesso 2003). A limitation of the HRCG survey tool is that the data is acquired on a wireline run, post drilling. 3. Rate Inertial Gyro: This instrument uses two triaxial packages, one to establish the true north using a ring laser gyro, and one to measure the accelerations in 3D using an accelerometer. The rate inertial gyro is the most expensive type of gyro, and is used mainly in cased hole surveys and wellhead orientation. In all cases, the Earth s magnetic field is not used to measure directions. Therefore, these tools can be used inside steel pipe with no interference to the survey readings. 32

57 Figure 2.13: Dynamically tuned gyroscope represented by Jamieson (2012). 2.5 Wellbore Tortuosity and Geometric Torsion Dogleg severity [DLS] is the most commonly known measure of wellbore tortuosity in the industry. It is associated with using the minimum curvature method [MCM]. It can be defined as the measure of the change in wellbore orientation over a given section of the well. The geometric torsion is defined as the degree of wellbore diameter irregularity. Both are critical elements in representing the wellbore shape and geometry and consequently better estimating the T&D parameters in the drilling of ER and complex horizontal wells Wellbore Tortuosity Wellbore tortuosity [κ], defined as the degree of wellbore deviation from the smooth trajectory, is calculated from inclination and azimuth data between two survey-stations and the measured depth interval between the stations. It is typically normalized to a section length of 100-ft and expressed in units of /100-ft [ /30-m] and it is calculated as follows: 33

58 DLS i = 180 π κ i 100 (2.1) Considered by itself, DLS is not a reliable indicator of tortuosity, (Brands and Lowdon 2012) since it is calculated using the curvature represented by a single constant arc between survey-stations. This assumption results in underestimating the degree of wellbore irregularity. This can result in a significant difference in friction factors and T&D magnitude. In extreme cases, this can lead to stuck pipe incidence, which is a factor that limits the distance horizontal wells can be drilled, given equipment capabilities. To overcome this limitation, a better representation of wellbore curvature is required. Wellbore tortuosity as a general term is separated into planned tortuosity [κ 1 ], microtortuosity [κ 2 ], and large-scale tortuosity [κ 3 ]. Planned tortuosity can be expressed as the overall shape of the planned trajectory. Micro-tortuosity is the induced deviation from the planned wellbore trajectory while drilling at a much smaller survey interval, typically every one to three feet. Micro-tortuosity is only visible when using a high resolution survey tool such as the HRCG survey tool recorded at approximately one survey per foot. Large-scale tortuosity is often considered to be synonymous with DLS, that is the type most commonly measured in industry practice. Large-scaled tortuosity is the type that is detected in scenarios which use the MCM and record surveys at the typical 90-ft survey interval (Gaynor et al. 2001). This is because MCM mathematically smoothens the wellbore creating an artificially low tortuosity. The difference between large-scale tortuosity and micro-tortuosity is illustrated in Figure The DLS using the HRCG survey tool at 90-ft survey interval representing large scale tortuosity effect, and surveys taken at one ft survey intervals, representing the microtortuosity effect, are shown in Figure The high fluctuation observed from micro-tortuosity has been explained through testing concrete samples. Tests done by Stockhausen et al. (2012) verified an initial theory that the borehole is in-gauge while sliding and over-gauge while in rotation. 34

59 Figure 2.14: Tortuosity comparison: micro-tortuosity vs large scale tortuosity. This phenomenon is shown in Figure 2.15, a concrete block cross-section drilled with PDM started with rotating mode to drill a tangent and then slide to build-up using a 1.41 bent sub. At the end of the rotation section, a lip was produced at the bottom of the block, as shown in Figure This transition causes an immediate increase in inclination at the rotation to side interface with visible hole size change. This phenomenon, referred to as the Stockhausen effect, is caused by the drilling mode changes. The Stockhausen effect confirms a non-constant curvature between survey-stations. It is also an evidence that the curvature calculated using the MCM at typical 90-ft survey intervals is miscalculated, causing significant error in borehole positions and wellbore curvature, when compared to continuous high resolution surveys (Stockhausen et al. 2012). 35

60 Figure 2.15: Concrete block tests showing a long transition between the slide and rotation zones by Stockhausen et al. (2012) Geometric Torsion Geometric torsion [τ], defined as the degree of wellbore diameter irregularity, is an important parameter to monitor and control wellbore trajectories and for estimating T&D parameters. Therefore, understanding the use of this parameter results in a realistic representation of the wellpath behavior and minimizes the occurrence of drilling problems. Wellbore diameter irregularity has a direct impact on the wellbore bending, which causes restrictions to the clearance a device can pass through or at which it may be permanently placed (Bang et al. 2015). This result can be displayed as a graph of the maximum allowed diameter as a function of measured depth or as a 3D view of the actual wellbore shape. The geometric torsion from typical drilling surveys using the HRCG survey tool at 90-ft resolution is shown in Figure As can be observed, there is a high torsional region between 3,800-ft and 3,900-ft. 36

61 Figure 2.16: Lip produced at the bottom of the borehole when transiting from rotary to sliding tested by Stockhausen et al. (2012). Further investigation of the wellpath was conducted using the effective diameter method proposed by Bang et al. (2015). The principal technique used in this method relies primarily on the basics of wellbore displacements. For a 5 inch casing field case, the maximum diameter plot [Figure 2.18] effectively presents that the wellbore diameter is less than two inches over a length of 100-ft between 3,800-ft and 3,900-ft. The actual shape of the problematic region of investigation is shown in Figure Such results bear directly on designing and running downhole completion and production equipment (Bang et al., 2015). Therefore, this region, between 3,800-ft and 3,900-ft, should be avoided for the setting of downhole equipment. This example demonstrates the operational value of a better borehole trajectory model to the industry. The operator can use such a model with typical drilling surveys at 90- ft intervals and determine the geometric torsion to identify potential downhole problems. Once the region of investigation is narrowed, a survey recorded at one survey per foot can be requested for further investigation using the effective diameter method. 37

62 Figure 2.17: Geometric torsion from HRCG survey using the ASC borehole trajectory model, resolution 90 ft. 2.6 Prior Work of Three-Dimensional Wellbore Trajectory Calculations Drilling a well entails the creation of mathematical expressions describing the wellpath in 3D, from start point to the target. The importance of describing the wellpath correctly is key to: Accurately directing horizontal wells at specific geological targets. Reducing the occurrence of drilling problems and avoiding collision. Maximizing a well s long-term production potential. Effectively mapping the subsurface of reservoirs to specify the lithology and pressure regimes. 38

63 Figure 2.18: Effective diameter from HRCG survey, resolution one survey per foot (Courtesy of Adrián Ledroz). There are several calculation models used to represent 3D wellbore trajectories. The different calculation methods have their own way of describing the wellpath, and are often used in different stages of the drilling process. The wellpath is assumed to comprise a series of straight or curved lines and circular arcs. The goal is to a find a mathematical model to best represent the wellpath. The simplest model uses straight-lines, while the more complex model uses the shape of a sphere or cylinder to describe the curve between the two points. If the wellpath is straight, as it would be in case of a tangent, even the easiest model would give accurate results. However, when the path is curved, simple models will not give accurate results, and the model used to describe the wellpath accurately perforce becomes more complex. The complexity of a model is a function of the smoothness of the model curve employed. The smoothness of this curve can be described by the degree of its derivative continuity and can be categorized as follow: 39

64 Figure 2.19: Actual wellbore shape from HRCG survey, resolution one survey per foot (Courtesy of Adrián Ledroz). zero-order continuous derivative [C0]: Same position at each side of the survey-station [i.e. straight-lines]; first-order continuous derivative [C1]: Same tangent at each side of the survey-station [i.e. circular arc]; and second-order continuous derivative [C2]: same curvature at each side of the surveystation [i.e. spline-curves]. Because of the properties of the integration, the continuity for the interpolation function for the tangent vectors on the unit sphere is always one degree higher than the degree of the computed trajectory space curve. For instance, if the space curve [curvature] for the wellbore trajectory model is required to be continuous at the second order derivative [C2], the interpolation function must have a curvature continuity at the third order derivative [C3]. 40

65 Several methods have been used for wellpath computation based on piecewise approximations of the wellpath with a constant line or circular arc. These are: the tangential method [TM], the average angle tangential method [AATM], the balanced tangential method [BTM], the radius of curvature method[rcm] and the minimum curvature method[mcm], as shown in Figure All these methods involve trajectory assumptions. The accuracy in determining the true course of a borehole is the key on selecting the appropriate method of trajectory calculations (Wolff and de Wardt 1981). Figure 2.20: Three-dimensional borehole trajectory methods represented by Jamieson(2012) The Tangential Method The TM is the first model and the simplest to implement. It assumes the segments of the trajectory to be approximated as straight-lines between the two surveys with inclination and azimuth angles along the northing and easting coordinates. The wellpath is tangent to these angles throughout the section length. Although this method has probably been the most widely used approach in the past, it is the most inaccurate of the available methods causing sizable error, as it does not account for previous inclination and azimuth angles. TM is a discontinuous vector function of a zero degree of trajectory continuity [C0] that is also known as the piece-wise constant method. 41

66 2.6.2 The Average Angle Tangential Method The drawback of neglecting the inclination and azimuth angles of the previous surveystationisaddressedintheaatm,whichisanimprovementtothetm.thismethodassumes that the inclination and azimuth angles are constant and equal to the average value of two survey points on the trajectory. AATM is also a discontinuous vector function with zero degree continuity [C0] The Balanced Tangential Method The third approach is the BTM, incorporates a further improvement whereby the inclination and azimuth at the survey are applied half-way forward into the next interval and half-way back into the previous. This method is more accurate than the previous methods. However, it also has errors when applied. All of these techniques represent the wellpath as a straight line. This assumption causes significant errors to accumulate along the wellpath The Radius of Curvature Method The radius of curvature method uses the inclination and azimuth measured at the upper and lower ends of the trajectory to generate a space curve having a circular arc shape when viewed in both the vertical and horizontal planes. This method assumes that the wellpath lies on a cylinder whose axis is vertical, with radius equal to the radius of curvature in the horizontal plane. It determines the length of the arc between the upper and lower ends of the course length in the horizontal plane. This works reasonably well for near vertical wells, however, it deviates significantly for anything not near vertical The Minimum Curvature Method With the recent computational improvements, the wellbore trajectory computational models moved toward a curvature representation that was introduced by Wolff and de Wardt (1981) and known as the MCM. This method assumes that the wellpath can fit on the surface of a sphere and have one radius in a 3D plane such that, two successive points on the 42

67 trajectory lies on a circular arc. MCM is effectively the industry standard and considered the most common and most accurate model. Through out initial analysis on simple splinecurves, the MCM calculations are derived from the basic spline, referred to as the natural spline-curve [derivations are illustrated in Appendix A]. 2.7 Three-Dimensional Wellbore Trajectory Calculation in the Current Dissertation Work In the current dissertation work, the ASC borehole trajectory model is proposed to calculate the borehole trajectory. This spline-curve denoted by T is a continuous function that consists of cubic polynomial pieces, in which the first, the second and third derivatives [T, T and T ] are continuous at all survey-stations. During the process of developing this robust 3D wellbore trajectory model, the following guidelines were considered: 1. The mathematical model must have the ability to compute the wellpath trajectory positions [TVD, northing and easting] along the entire wellpath with high accuracy. 2. The computed curvature must provide significant improvement in accuracy of the calculated borehole position, wellbore tortuosity and geometric torsion than all current models. 3. The resulting curvature must be continuous at the first order derivative [C1] and the second order derivative [C2] at any point along the entire wellpath. These guidelines will reduce borehole path uncertainty, increase the confident level of the definitive survey and more effectively represent the drillstring configuration for T&D calculations. The development process and validation of the ASC borehole trajectory model are presented in Chapter 4. 43

68 CHAPTER 3 THREE DIMENSIONAL TORQUE AND DRAG MODELING Torque is defined as the measure of a forces tendency to produce torsion and rotation about an axis. These forces can be generated around the wellbore from different sources, namely: mechanical torque, frictional torque, and bit torque. Vibrations of the drillstring could also contribute to additional torque. Drag is anaxial force which isgenerated only when thepipeis moved in an axial direction and dependent on the friction factor coefficient. The drag force always has an opposite direction to the pipe motion. Torque and drag [T&D] has been the subject of intense interest during the execution of ER drilling. In vertical holes, T&D forces are usually within tolerable levels. High values are an indicator of a more specific problem and should be investigated to determine the actual cause. In deviated holes, the drillstring is in more intimate contact with the wall. The longer the deviated section, the greater the requirement for downhole conditions data to optimize the delivery of power from the surface to the bit. T&D analysis is primarily used to determine if a well can be drilled within equipment capabilities. This includes considering drillstring, casing, completion and surface equipment. The evaluation criteria includes torque on bit, maximum overpull, axial load and buckling. The factors evaluated include the buoyancy factor, friction factor coefficient, wellbore tortuosity [κ], geometric torsion [τ], the change in the rate of the curvature [dκ/ds], the drillstring bending stiffness [EI], among others. This chapter includes a review on the general vector differential form of the T&D modeling. The current T&D models are discussed in this chapter. The differences between the soft-string model and the stiff-string model are explained. Finally, the historical perspective, and past and current methods and technologies of T&D are presented. 44

69 3.1 Components of Torque and Drag T&D models assume that the drillstring is made up of short elements joined by connections that transmit tension, compression, and torsion, but not the bending moment. The calculations start at the bottom of the string and proceed upward to the surface. Each short element consists of buoyed weight, axial load, friction force and normal force perpendicular to the contact surface of the wellbore Buoyancy Factor The effect of buoyancy forces during different operations is important. The effective drillstring weight in a fluid is equal to the drillpipe weight multiplied by the buoyancy factor. The buoyancy factor is defined as: β = 1 ρ oa o ρ i A i ρ steel (A o A i ) (3.1) In the case of an equal fluid density above equation is simplified as follow: β = 1 ρ o ρ steel (3.2) For a given drillstring, the effective buoyed weight [w b ] in lbf for an element along the drillstring can be calculated as follows: where, β = Buoyancy factor, unitless; ρ = Density, lb/in 3 ; A = Cross-sectional area, in 2 ; w b = βw L W =The weight per foot of an element along the drillstring, lbf/ft; L = The measured length of an element along the drillstring, ft. 45

70 3.1.2 Friction Coefficient The friction coefficient [µ] plays an important role in T&D modeling calculations and is frequently a source of confusion. It is defined as the roughness between the drillstring and wellbore wall. A friction coefficient can be estimated from the ratio of the shear force in the normal direction [f n ] to the friction force [F f ], defined as an acting force against the drillstring movement. The friction coefficient is a dimensionless scalar value and its value can vary along the borehole. The range of friction factors are shown in Table 3.1 (Mitchell and Samuel 2009). Table 3.1: Range of friction factors Fluid Type Friction Factors Cased hole Open hole Oil-based Water-based Brine Polymer-based Synthetic-based Foam Air There are two approaches on using the friction coefficient for T&D calculations. Either one friction factor value is used for the entire well or, if significant portions of both cased and open hole exists, two or more friction factors are used. One for the drillstring inside the casing and one for the drillstring in the open hole. By monitoring hole conditions, apparent friction coefficients are back-calculated to match the data from the field with those from the modeling data. The model s procedure requires iteration over directional survey data and numerical integration between the axial forces of the drillstring movement in the wellbore and the survey-stations. Since the type of drilling fluid used affects the friction in the the formation, ranges of friction factors must be specified for each fluid type. The higher the friction factor, the higher the chance of the occurrence of drilling problems. Problems related to friction factor could be either hole geometry issues from inclination 46

71 and azimuth changes with high tortuosity or other factors related to hole cleaning, cutting transport and hydraulics Axial Load, Buckling and Horizontal Lock-up The buckling of drillpipe resulting from excessive axial and bending loads is another issue that impacts ER wells. The amount of force required to initiate the buckling differs from one section to another, and depends on whether the section is vertical, curved, or horizontal (Wu 1996). Real-time monitoring of bending load along the drillstring is vital to identify the starting point and distance of buckling. This information are then considered in T&D analysis to determine if the well can be drilled within equipment capabilities. According to Wu (1996), buckling due to excessive loads causes an increase in the contact force between the drillstring and the wellbore that leads into sinusoidal buckling once it exceeds the critical load. The critical load can be calculated using Equation 3.3: EI w c F cr = 2 (3.3) r p where, F cr =The critical buckling force, lbf; EI = Bending moment, lbf/in 2 ; w c = Contact force magnitude, lbf; r p = The outside radius of the contact point, in. As the drillstring weight is concentrated at the buckled portion of the drillstring, a lockup due to axial friction build up will occur in the extended build section with high angle (Wu 1996). This causes helical buckling to develop. This phenomena is accompanied by lost weight on bit, causing the drillpipe to lock-up in a spiraling manner against the sides of the borehole, and eventually preventing the pipe from moving in the hole. To avoid helical buckling, the additional drag created in the post-buckled portion must be accurately calculated in the T&D models and compensated for when choosing drilling parameters (Aarrestad and Blikra 1994) using Equation 3.4: 47

72 F hel = 1.83 F cr (3.4) The onset of buckling will depend on the stiffness of the drillstring components and the outer diameter of components in relation to the wellbore. This is important for T&D modeling, since helical buckling will cause a great increase in the side force between the drillstring and the wellbore wall. Buckling is often seen in small diameter pipe sizes and coiled tubing. Additional friction forces will be generated due to the helical buckling. The recognition of this fact is vital in the T&D analysis. Wu (1996) showed that these additional friction forces increase with the rate of square of the axial load. The axial load behavior changes with sinusoidal and helical buckling as it turns into a nonlinear distribution as shown in Figure 3.1. In the event of helical buckling, it is recommended to release the weight on bit and begin surface rotation. Figure 3.1: Buckling phenomena in a horizontal section created by Fazaelizadeh (2013). 48

73 3.2 Torque and Drag General Vector Differential Form In the T&D computational process, forces and moments are summed from the bottom upwards to produce the cumulative loads on the drillstring. Initial conditions usually are specified starting from the bottom. For different drilling operations, such as rotating off bottom [ROB], pickup [PU], slackoff [SO] and rotary drilling [RD], the inputs would be weight on bit [WOB] as well as hookload and torque that are measured at the surface. During the ROB operation the drillstring rotates without any axial movement. There is no weight or torque on bit, as the bit is not engaged with the formation. PU is the action of lifting the bit off bottom by raising the drillstring out of the hole. SO is the action of lowering the drillstring into the hole. The PU weight and SO weight are used to estimate the friction between the drillstring and wellbore wall. RD is the process of rotating the drillstring, while applying WOB, to create a hole in the ground. During all four operations, the drillstring will exert forces and moments along the wellbore. To derive the force and moment equilibrium equations, the rectangular coordinate system [x,y,z], with the conventional unit vectors i,j,k and the Frenet-Serret local system of coordinates, with its unit tangent, normal and binormal vectors [ t, n, b ] are useful tools (Mitchell and Miska 2011). Figure 3.2 illustrates these coordinate systems. In the process of modeling torque [moment], it is assumed that the pipe will climb up the side of the wellbore as it rotates as shown in the free body diagram [FBD] illustrated in Figure 3.3[a]. Drag, is proportional to the hookload and dependent on the friction coefficient. Drag always acts in the opposite direction to the drillstring motion. The general balance equilibrium of forces and moments are shown in the FBD in Figure 3.3 [b]. For a drillstring, the internal forces applied consist of the tangential [axial] force [F t ] and two shear forces: one in the normal direction [F n ] and one in the binomial direction [F b ]. The external forces consist of the pipe effective weight [w b ], the side force [w c ] which is normal to the pipe contact force and, lastly, the drag force [w d ] which is force induced due to the friction. 49

74 Figure 3.2: Local [ t, n, b ] and global [x,y,z] coordinate system in 3D. Figure 3.3: Force and moment balance FBD: [a] Moment looking downhole; [b] Force and moment balance equilibrium. 50

75 All six forces are added to obtain the general vector differential form of the balance equilibrium of forces, given by Equation 3.5. df t ds tz + df n ds nz + df b bz + w b + w c + w d = 0 (3.5) ds For a drillstring with bending stiffness [EI], the general form of the moment [M t ] balance equilibrium is given by Equation 3.6: where, (EI dκ ds +F n) b z +(M t κ EIκτ F b ) n z + dm t tz + m = 0 (3.6) ds F t = Tangential [axial] force, lbf; F n = Shear force in the normal direction, lbf; F b = Shear force in the binormal direction = w b b z µr p, lbf; t z,n z,b z = z-components of the tangential, normal and binormal vector; m = Distributed external moment = rp w d, ft-lbf; M t = Torque [Moment], ft-lbf; w b = Effective weight of the drillstring, lbf; w c = Contact force magnitude, lbf/ft; w d = Drag force magnitude, lbf; ϑ = Wellbore contact angle, degrees. When the wellbore trajectory is assumed to be composed of constant curvature arcs between survey-stations from the MCM, the geometric torsion [τ] and the change in the rate of wellbore curvature [dκ ds] will be nil. Thus, Equation 3.6 becomes useful when the wellbore is assumed to be modeled as a spline using the ASC borehole trajectory model to derive the ASC 3D T&D equilibrium equations. 3.3 Prior Work of Torque and Drag Models T&D software has existed since the 1990s; however, some confusion still exists over the validity of these models that are used to characterize drilling operations, particularly in the 51

76 applications for ER wells. The mathematical models used to predict the drillstring behavior inside the wellbore, are either [1] soft-string model, that assumes a continuous drillstring to wellbore contact as it is laying at the low side of the wellbore [i.e. the bending stiffness of the drillstring is neglected], or [2] stiff-string model, that assumes an intermittent drillstring to wellbore contact that depends on the wellbore tortuosity [i.e. the change in the bending moment is assumed to be constant along the wellbore curved section] The Soft-String Model The soft-string model, often called a cable or a chain method, neglects the drillstringstiffness effects. This means that the drillstring is treated as a heavy cable, chain, or rope lying along the low side of the wellbore. The axial tension and torque forces are supported by the drillstring and the contact forces are supported by the wellbore. It also assumes that the drillstring has continuous surface contact area with the low side of the hole and it takes the shape of the wellbore. This can lead to errors in T&D predictions. Throughout the history of T&D modeling there has been one model consistently assumed to be the typical drillstring model; this is the theory developed by Johancsik et al. (1984) and then standardized by Sheppard et al. (1987). In 1984, Johancsik et al. began with a softstring T&D theory assuming a normal force was caused from tension against the curvature and the weight of the pipe section based on Coulomb s friction model. Both torque and drag are caused entirely by sliding friction forces that result from contact of the drill string with the wellbore. The force balance for an element of the pipe considers that the normal component of the tensile force was acting on the element contributing to the normal force. Applying the condition of equilibrium along the axial and normal directions, the normal force presented by Johancsik et al. (1984) is derived in Equation 3.7. F n = [(F t (θ 2 θ 1 )sin( φ 2 +φ 1 2 )) 2 +(F t (φ 2 φ 1 )+w b sin( φ 2 +φ 1 )) 2 ] 1 2 (3.7) 2 Once the drillstring and hole configuration are specified, the T&D calculation starts at the bottom of the drillstring and proceed stepwise upward as follows: 52

77 The tangential [axial] force increment of an element [ F t ] at any point in the well is calculated from Equation 3.8: F t = w b cosφ avg ±µf n (3.8) Where plus [+] means PU and minus [ ] means SO of the drillstring, depending on which direction the friction will be, such that drag always acts in the opposite direction of the drillstring motion. The axial tension on the lower end of the element in lbf is F t1. The axial tension on the upper end of the element in lbf: F t2 = F t1 + F t (3.9) In the vertical section, where the inclination angle [φ] is zero, the normal force term will diminish. While in the horizontal section, where the inclination [φ] is 90, the weight term will be zero. This also applies for the torque. In the vertical section, the axial force has no contribution to the torque parameter. The torque [M] is calculated using the radius of the drillstring (r p ) using the following equation: M 2 = M 1 +µf n r p (3.10) It should be noted that the drillstring may consist of different components with different unit weights and outside radius, such as drillpipe, heavy weight drillpipe and drill collars which should be all taken into consideration in calculating T&D parameters. Later, Sheppard et al. (1987) changed the model to a differential form that included the effects of mud pressure. In this conceptualization, effective tension is used instead of true tension. This subsequent model has been used in many different facets of the oil and gas industry for both straight and curved sections. In 1986, H-S Ho (Aston et al. 1998) created a more comprehensive model that satisfied all the force equilibrium equations and moment equilibrium equations by assuming that 53

78 the drillstring has shear forces. These models were standardized by Mitchell and Samuel (2009). This was accomplished by mathematically deconstructing previous models, where it was determined that the impact of bending moment was substantial and not effectively accounted for. One major shortcoming of this soft-string model, as stated early, is that bending stiffness is neglected The Stiff-String Model The current alternative to the soft-string model is the stiff-string model, which considers the drillstring bending stiffness and annulus clearance. Even though there have been many stiff-string models developed, none of them has a standard formulation in the industry (Wu 1996). Beside the scientific modeling differences between soft-string and stiff-string models, field cases and numerous studies have shown that the soft-string model underestimates the drillstring torque, fails to predict the buckling onset, and poorly estimates the contact forces between the drillstring and wellbore wall. It also assumes that the drillstring has continuous surface contact area with the low side of the hole that leads to errors in T&D predictions. In practice, the drillstring position relative to the wellbore may be on the high side, low side, right side or left side depending on the wellbore section and drillstring operation, as illustrated in Figure 3.4. The assumption of this model may be inappropriate leading to inaccurate T&D results. In 1962, Lubinski (Aston et al. 1998) was the first to consider the calculation of the drillstring deflection while in tension using the Euler beam theory. Later, Paslay and Cernocky (1991) extended the study for the drillstring being in compression. Both models included the bending moment and the drillstring stiffness. Furthermore, the deflection of the drillstring in the wellbore and the severity of buckling, using the same methodology adopted by Lubinski, has been studied and validated by Paslay and G. Handelman in 1964, Christman in 1976, Walker and Friedman in 1977, Chandra in 1986, Dareing in 1991, Rocheleau in 1992, J. Wu and H. Juvkam-Wold in 1993 and Li Zifeng in 1994 (Paslay and Cernocky 1991). 54

79 Figure 3.4: Drillstring position related to the borehole. In 2007, Mason and Chen(2007) analyzed numerous companies stiff-string T&D software and concluded that all yielded similar results given comparable inputs. Some of the concerns around the validity of the model s conclusions were addressed in this study. For instance, hole size does not have a significant effect on T&D results. Also, the friction factors for running casing have been estimated at twice that of drilling operations. Following this study, Mason and Chen (2007) pointed out different effects that have to be considered: [1] the hydrodynamic viscous force, which is the drag force as a result of pipe movement in opposite direction of the drilling fluid flow, [2] the wellbore tortuosity effect, a well with high tortuosity shows higher T&D values, [3] the buckling effect, the excessive axial load causes the drillstring to buckle. These factors must be taken into consideration to have a more realistic T&D result. Mitchell and Samuel (2009) extended the previous model by adding the stiffness of the drillstring. This model assumes the drillstring to act as a beam. The bending moments are calculated over the beam as a result of forces acting on them. The solution gave better insight into the friction behavior throughout the wellbore. Later, Mitchell and Samuel(2009) pointed out an important finding that T&D models use a dynamic friction and not a static friction, therefore, the standard T&D models perform poorly in ER and complex wells. 55

80 Later in 2014, a new dynamic 3D analytical approach was published (Tikhonov et al. 2014), which incorporates a new criterion and calculates accurate and more robust contact forces per joint using a stiff-string model. This model follows the same assumption of the Lubinski-Paslay-Cernocky bending-stress-magnification factor, where the deflection is not constant and is changing along the BHA. The only limitation of this model is the use of a constant curvature trajectory calculation [MCM], which assumes constant bending and, hence, constant bending moment. 3.4 Torque and Drag in the Current Dissertation Work Current models provide unreliable estimation of T&D magnitudes due to the constant curvature arc assumptions between survey-stations in the MCM, the most common model used in the industry. This assumption causes the bending parameter in the T&D equilibrium equation to be discontinuous at survey-stations. Two methodologies have been considered in this dissertation to overcome the limitations of current T&D models: 1. Using a non-constant curvature trajectory [The ASC model] on a semi-analytical approach: survey data generation at a pseudo-high resolution [PHR] of one survey per foot intervals as an input in a current industry-approved T&D software. 2. Developing an extended 3D stiff-string T&D model for non-constant curvature borehole trajectories [The ASC 3D T&D model]. In Chapter 5, the process to improve current T&D models is explained. Then the assumptions and boundary conditions on the model are shown. This sets the guidelines to derive the force and moment equilibrium equations to predict T&D outputs. The general procedure and the derivation of the model are described. The effect of bending moment on T&D magnitude are analyzed. And lastly, the validation process of the model using two field cases is presented. 56

81 CHAPTER 4 TECHNICAL APPROACH - THREE DIMENSIONAL ADVANCED WELLBORE TRAJECTORY MODEL This chapter presents the details of the 3D advanced wellbore trajectory model. It starts with a general description of constructing a curve that passes through survey-stations. Then the assumptions and boundary conditions on the model are shown. From there, the non-constant curvature trajectory model [The Advanced Spline-Curves Model] is selected and derived to compute the wellbore positioning [TVD, northing and easting], the wellbore tortuosity and the geometric torsion. The model is validated using five field studies and two synthetic wellpath examples. Finally, a comparison of the current wellbore trajectory models with the ASC model derived in this dissertation demonstrates the superior results of the model s accuracy. Accurately representing well profiles has become more challenging, given ER and complex horizontal design wells. As noted in Chapter 2, the limitations of the minimum curvature method [MCM], the current industry standard, present major problems with wellbore trajectory computations and curvature accuracy. The oil and gas industry should consider not using this model under these conditions. Through intensive theoretical modeling, quantification and validations process described below, the advanced spline-curves [ASC] borehole trajectory model was selected in this dissertation. This model is based on a cubic piecewise polynomial function using splines with a guaranteed degree of continuity of the resulting curvature up to the second order derivative [C2] along the entire wellpath. These characteristics provide robustness, flexibility and significant improvement to the accuracy of the calculated borehole position, wellbore tortuosity and geometric torsion; particularly in ER and complex horizontal wells. 57

82 The ASC borehole trajectory model validation process shows excellent precision in wellbore trajectory calculations with a higher accuracy than currently used methods. This accuracy was tested using two approaches: [a] five horizontal wells with actual survey data recorded at one survey per foot using the HRCG survey tool, and [b] two synthetic wellbore examples of known trajectory values. 4.1 Problem Setup Process A natural way to construct a curve from a set of given survey-stations is to force the curve to pass through these surveys. The simplest example of this is a straight-line from point to point. In this chapter, the following fundamental problem is considered: Given a set of survey-stations [points in space], determine a smooth curve that approximate the path between the stations with a single solution process. The algorithm for determining the curve involve a relatively small number of elementary arithmetic operations. This ensures that the model can solve borehole trajectory parameters in a reasonable time using standard engineering computers and efficiently present the wellpath in real-time drilling activities with a high level of accuracy Assumptions and Boundary Conditions During the process of developing a robust 3D wellbore trajectory model, the following guidelines were considered: 1. The mathematical model must have the ability to compute the wellpath trajectory positions [TVD, northing and easting] along the entire wellpath with high level of accuracy. 2. The computed curve must provide significant improvement in accuracy of the calculated borehole position, wellbore tortuosity and geometric torsion than all current models. 3. The resulting curvature must be continuous at the first derivative [C1] and the second derivative [C2] at any point along the entire wellpath. 58

83 These guidelines will reduce borehole path uncertainty, increase the confidence level of the definitive survey, better represent the drillstring configuration for T&D calculations and make drilling operation more efficient in cost and time Initial Approaches The process to obtain approximate coordinates of points along the trajectory usually start from a known point at the surface. Each method must make some assumptions about how the trajectory is recreated from the measurements. The simplest model to describe the curve between two points uses straight-lines, while the more complex model uses the shape of a sphere, cylinder, spline or helix. If the wellpath is straight, as it would be in case of a tangent, even the easiest methods would give accurate results. However, when the wellpath is curved, these easy approximations will fail to give accurate results, and the use of more complex models becomes a necessity. The complexity of the model is a function of the curve smoothness, that is described by the degree of continuity [discussed in Section 2.2]. In the literature, several methods have been used for wellpath computation based on piecewise approximations with a constant linear or circular arc [presented in Chapter 2]. There are other curve approximation methods such as the Bernstein polynomial basis, i.e. Bézier curves approximation and spherical indicatrix of tangents [SIT 2] method, among others Bézier Curves Bézier curves were the first approach taken during this dissertation work. Bézier curves consist of polynomial functions that can evaluate the curve at given fixed survey-stations and join the points with approximated curves. Throughout the analysis, several limitations were raised: Bézier curves require [a] a high degree function to get decent accuracy, which increases the complexity of evaluating curves at many survey points, and [b] it requires a set of control points, two defining the end points and two points [called handles, in some applications] controlling the tangents at the endpoints in a geometric way. These control 59

84 points are adjusted by the user to control the shape of the curve. Figure 4.1 illustrates some examples of Bézier curves with their control points. These limitations restricted the goal of a robust wellbore trajectory model to guarantee a single solution process. Figure 4.1: Examples of Bézier curves approximation represented by Sampaio (2016) Spherical indicatrix of tangents method [SIT 2] SIT 2 method is another form of the Bernstein polynomial basis. It is a smooth curve that ensures at least curvature-continuity [C2] at any point along the curve and can compute complex curvatures along the entire wellpath. However, the accuracy level of SIT 2 is questionable due to the numerical considerations, such as the sectional Bézier spline approximation, that is required to construct the interpolating spline (Gfrerrer and Glasser 2000). This limitation restricts the SIT 2 method from accurately representing the wellpath as illustrated in the model quantification section of this Chapter Building the Model Simple construction, ease and accuracy of evaluation, capacity to approximate complex shapes through curve fitting and interactive curve design are the main guidelines to develop and build the 3D wellbore trajectory model. The spline-curves used in the chosen model are easily computed by solving a system of linear equations without the need to resort to any 60

85 kind of successive approximation scheme, numerical considerations or control polygons. The ASC borehole trajectory model proposed has the following characteristics: 1. Based on a cubic polynomial piece-wise function that is continuous up to the third order derivative [C3]. 2. Requires solving linear system of equations with defined boundary conditions. 3. Results are based on a single solution process. With these five characteristics, the borehole position accuracy is improved. This provides the possibility for reduced borehole path uncertainty and increase the confidence level in the definitive survey Data Set Reprocessing During the validation process of the ASC borehole trajectory model, original field data set recorded from the high resolution continuous gyroscopic [HRCG] survey tool at one survey per foot is sampled to simulate typical drilling surveys at 90-ft resolution. More data points were introduced to simulate the convergence to a more accurate solution. A second approach was used during the validation process which required the generation of a set of survey-stations by using a complex synthetic well example of a known trajectory wellpath values. This is a typical mathematical approach to verify the accuracy of a given model. The higher the confidence level of the proposed model in preserving the shape properties of the original function, the higher its accuracy. This approach can generate survey-stations at any specified resolution for an effective model validation process. 4.2 Development of the Advanced Spline-Curve Borehole Trajectory Model The ASC function [T] is a continuous function that has the ability to approximate the wellbore positions, tortuosity and geometric torsion along the entire wellpath (Abughaban et al. 2016). The mathematical steps, to compute the wellbore trajectory are outlined below. 61

86 4.2.1 Wellbore Positioning Representation The advanced spline-curve, on an interval from surface location [s 0 ] to an n th point [s n ] in the wellbore, is a function that consists of polynomial pieces of degree 3 on [s i,s i+1 ], i = 0,...,n 1 and such that T, T, and T are continuous at s i, i = 1,...,n 1. Given n + 1 survey data points: Measured depth [MD i ], inclination [φ i ], and azimuth [θ i ], such that s i = MD i, i = 0,...,n. The normalized tangent vector [λ i ] for the given surveys is calculated according to Equation 4.1: λ Ei λ i = λ Ni = λ TVDi where, sinφ i sinθ i sinφ i cosθ i cosφ i λ Ei : Normalized tangent vector for easting coordinates [x-axis]; λ Ni : Normalized tangent vector for northing coordinates [y-axis]; λ TVDi : Normalized tangent vector for true vertical depths [z-axis]. The goal is to find the ASC function [T(s i )] such that:, i = 0,...,n (4.1) T(s i ) = Y (s i ) = λ i, i = 0,...,n (4.2) The spline-curves general equation (Atkinson 1978; Cheney and Kincaid 2005) for i = 0,...,n 1 is shown in Equation 4.3: where: T(s) = A i +(s s i )B i +(s s i ) 2 C i +(s s i ) 3 D i, s [s i,s i+1 ] (4.3) A i = λ i, B = λ i+1 λ i h i h i 6 z i+1 h i 3 z i, C i = z i 2, D i = z i+1 z i 6h i (4.4) Note that there are n 1 equations and n+1 unknown vectors z i and h i = s i+1 s i,i = 0,...,n 1. These conditions require a set of two boundary conditions to obtain a unique interpolating spline. For survey calculations, the first set point is selected at the start point [s 0 ] and the second set point is selected at the target [s n ] as follows: 62

87 Boundary condition at the start point: Let the third derivative T (s 0 ) be continuous at s 1 : T (s 0 ) = T (s 1 ) Using Equation 4.3 it can be shown that This condition yields: T (s 0 ) = 6D 0 = z 1 z 0 h 0 = T (s 1 ) = 6D 1 = z 2 z 1 h 1 Boundary condition at the end point: z 0 = z 1 + h 0 h 1 (z 1 z 2 ) (4.5) Let the third derivative T (s n ) be continuous at s n 1 : T (s n 1 ) = T (s n ) Using Equation 4.3 it can be shown that T (s n 1 ) = 6D n 1 = z n 1 z n 2 h n 2, T (s n ) = 6D 1 = z n z n 1 h n 1 This condition yields: z n = z n 1 + h n 1 h n 2 (z n 1 z n 2 ) (4.6) The general system of equations governing the vector z = [z 0,z 1,...,z n ] T can be written in the following matrix form (Cheney and Kincaid 2005): 63

88 1 0 h 0 u 1 h 1 h 1 u 2 h 2 h 2 u 3 h h n 3 u n 2 h n 2 h n 2 u n 1 h n z 0 z 1 z 2 z 3.. z n 2 z n 1 z n = v 0 v 1 v 2 v 3.. v n 2 v n 1 v n where the nomenclatures h i,u i and v i are results of integrations to solve for z i, and are set as follow: h i = s i+1 s i, i = 0,...,n 1 and for each component of λ i : u i = 2(h i 1 +h i ), i = 1,...,n 1 v i = 6(β i β i 1 ), i = 1,...,n 1 β i = 1 h i (λ i+1 λ i ) This matrix is symmetric, positive definite and diagonally dominant and the linear system is uniquely solvable. Substituting Equation 4.5 and Equation 4.6 in the general matrix will yield the ASC borehole trajectory general matrix form, as follows: (u 1 +h 0 + h2 0 h 1 ) (h 1 h2 0 h 1 ) h 1 u 2 h 2 h 2 u 3 h h n 3 u n 2 h n 2 (h n 2 h2 n 1 h n 2 ) (u n 1 +h n 1 + h2 n 1 h n 2 ) z 1 z 2 z 3.. z n 2 z n 1 = v 1 v 2 v 3.. v n 2 v n 1 64

89 and z 0 = z 1 + h 0 h 1 (z 1 z 2 ), z n = z n 1 + h n 1 h n 2 (z n 1 z n 2 ) This splits into three tridiagonal linear systems, with the same matrix, in the first, second and third component vectors of z i, i = 1,...,n 1. Once the system is solved, z 0 and z n are computed using Equation 4.5 and Equation 4.6. Then A i, B i, C i, D i are calculated using Equation 4.4. Hence, each component of T is now known over the entire wellpath by Equation 4.3 for each component of λ i. The approximation of the 3D space curve is determined using the values of s i = MD i and calculated tangent vectors λ i from Equation 4.1 by using the ASC borehole trajectory model. The approximated wellpath trajectory positions are obtained by integrating the approximate spline T(s) from Equation 4.3 for all components of λ i and i = 0,...,n as follows: i 1 ˆ Y(s i ) j=0 s j+1 s j T(s)ds = i 1 j=0 h i A Ei + h2 i 2 B Ei + h3 i 3 C Ei + h4 i 4 D Ei h i A Ni + h2 i 2 B Ni + h3 i 3 C Ni + h4 i 4 D Ni h i A TVDi + h2 i 2 B TVDi + h3 i 3 C TVDi + h4 i 4 D TVDi Tortuosity and Geometric Torsion Representation (4.7) From the earlier defined ASC function T = Y (s), the curvature [κ] is computed from its second derivative, Y (s), as follows: such that: κ i = Y (s i ) 2 (4.8) Y (s i ) = T (s i ) = { λi+1 λ i h i h i λ i λ i 1 h i 1 z 6 i+1 h i z 3 i, i = 0,...,n 1 h i 1 z 6 i 1 + h i 1 z 3 i, i = n (4.9) 65

90 Alternatively, geometric torsion will be used as a more rigorous approach to describe the wellbore shape. The torsion parameter [τ] can be calculated as the determinate matrix formed by the first, second and third derivatives of the ASC function divided by the curvature squared and is given by: τ i = det[y (s i )Y (s i )Y (s i )] κ 2 i (4.10) wherey (s)iscomputedfromequation4.2,y (s)usingequation4.9andy (s)iscomputed as follow: Y (s i ) = z i (4.11) The ASC borehole Trajectory Model Sample Calculation Given the following nine survey points [i = 0,...,9] in Table 4.1. Calculate the wellbore positioning, wellbore tortuosity and geometric torsion. Table 4.1: ASC model sample calculation example...i... Measured Depth [s i ] Inclination [φ i ] Azimuth [θ i ] Feet Degree Degree Step 1: Compute the tangent vectors for TVD, Nothings and Eastings: λ TVDi = [cosφ i ], λ Ni = [sinφ i cosθ i ], λ Ei = [sinφ i sinθ i ] 66

91 hence, λ TVDi = , λ Ni = , λ Ei = Step 2: Set the left hand side [A] of the ASC borehole trajectory general matrix form [Az=b]: h i = s i+1 s i, u i = 2(h i 1 +h i ) hence, yields, A = h i = (u 1 +h 0 + h2 0 h 1 ) (h 1 h2 0 h 1 ) h 1 u 2 h 2 h 2 u 3 h 3 ft, u i = h 3 u 4 h 4 h 4 u 5 h h 5 u 6 h 6 ft h 6 u 7 h 7 (h 7 h2 8 h 7 ) (u 8 +h 8 + h2 8 h 7 ) 67

92 hence, A = Step 3: Compute the right hand side [b] of the ASC borehole trajectory general matrix form for TVD, Nothing and Eastings: v TVDi = 6( λ TV D i+1 λ TVDi h i λ TVD i λ TVDi 1 h i 1 ) hence, v Ni = 6( λ N i+1 λ Ni h i v Ei = 6( λ E i+1 λ Ei h i λ N i λ Ni 1 h i 1 ) λ E i λ Ei 1 h i 1 ) v TVDi = 1.795E E E E E E E E 04 1, v ft N i = 5.490E E E E E E E E 03 1, v ft E i = 1.747E E E E E E E E 03 1 ft 68

93 Step 4: Solve the ASC borehole trajectory general matrix for [z]: z TVDi = E E E E E E E E E E 08 1 ft 2, z Ni = 3.798E E E E E E E E E E 06 1 ft 2, z Ei = 3.205E E E E E E E E E E 06 1 ft 2 Step 5: Calculate the parameters A i, B i, C i, D i using Equation 4.4. Integrate the function T given by Equation 4.3 for each component of λ i. The integration is the change in wellbore positioning as follows [refer to Equation 4.7]: TVD i = A TVDi h i 1 + B TVD i 2 h 2 i 1 + C TVD i 3 h 3 i 1 + D TVD i h 4 i 1 4 N i = A Ni h i 1 + B N i 2 h2 i 1 + C N i 3 h3 i 1 + D N i 4 h4 i 1 E i = A Ei h i 1 + B E i 2 h2 i 1 + C E i 3 h3 i 1 + D E i 4 h4 i 1 hence, TVD i = ft, N i = ft, E i = ft 69

94 Step 6: Compute the second derivative of the function T given by Equation 4.3. the curvature [κ] is the norm of the second derivative of T TVD, T N and T E : hence, κ TVDi =T TVD i = κ Ni =T N i = κ Ei =T E i = λ TVDi+1 λ TVDi h i λ Ni+1 λ Ni h i λ Ei+1 λ Ei h i h i 6 z TVD i+1 h i 3 z TVD i h i 6 z N i+1 h i 3 z N i h i 6 z E i+1 h i 3 z E i κ TVDi = E E E E E E E E E 03 1, κ ft N i = E E E E E E E E E 04 1, κ ft E i = E E E E E E E E E 04 1 ft Now compute the norm of κ TVDi,κ Ni,κ Ei : κ i = κ TVDi κ Ni κ Ei = /100f t 70

95 Step 7: Compute the determinate matrix [ζ i ] formed by the first, second and third derivatives of the function T given by Equation 4.3. The geometric torsion [τ i ] is computed by dividing ζ i over κ 2 i: ζ i = det λ TVDi+1 λ Ni+1 λ Ei+1 ( λtv Di+1 λ TV Di h i λ Ni+1 λ Ni h i λ Ei+1 λ Ei h i h i 6 z TVD i+1 h i 3 z TVD i ) z TVDi h i 6 N i+1 h i 3 N i z Ni h i 6 E i+1 h i 3 E i z Ei 1 ft 3 hence, τ i = ζ i κ 2 i = /ft 4.3 Quantification of the Advanced Spline-Curves Borehole Trajectory Calculation Model All trajectory computational models are subject to errors which limit the accuracy of wellbore positions and curvature. To test and validate the accuracy of the models, two examples have been used: 1. Five horizontal wells with actual survey data recorded at one survey per foot using the HRCG survey tool. 2. Two complex synthetic well examples of known wellbore trajectories. The ASC borehole trajectory model, MCM and average angle tangential method [AATM] were compared for accuracy. Results from both approaches, outlined below, favors the ASC model to be the most accurate when compared to all models. 71

96 4.3.1 Field Data Results Five horizontal wells were used to validate and test the accuracy of the ASC borehole trajectory model. The resulting wellbore position, wellbore tortuosity [κ] and geometric torsion [τ] from the actual survey data recorded at one survey per foot resolution, will represent the baseline ground truth of the wellpath during the comparison process. Other parameters such as vertical section, toolface, build rate and turn rate have been calculated. The quantification of error will be presented as the maximum error over the entire wellpath and the well landing point at total depth [TD]. During the validation processes, a positive error will represent a shallow well placement and a negative error will represent a deep well placement for all models. The effect of slide sheet, representing the rotating and sliding patterns of the drilling process to generate additional actual survey-stations, was tested as part of a convergent rate analysis. Since slide-sheets are not available for the five surveyed well, the actual data set recorded at one survey per foot were used to simulate the 90-ft, 45-ft and 30-ft survey resolution as additional surveys. The sampled new surveys are then reprocessed using the ASC borehole trajectory model and compared to the baseline [one foot survey resolution] Field Case 1 Field Case 1 is a complex well trajectory, drilled vertically to 9,460 ft and build 13 /100- ft reaching 65 inclination at TD [9,969 ft MD]. Field Case 1 appears to be a problematic well that required various trajectory corrections throughout the drilling process to reach the target. This is clearly observed in the fluctuation in inclination and azimuth measures as shown in Figure 4.2. The baseline of the landing points for Field Case 1 are at 9,852-ft TVD, 520-ft north and 133-ft west as illustrated in the vertical section in Figure 4.3. The ASC model preserves the baseline for the wellpath very accurately, compared to both MCM and AATM, with gross errors less than one foot in TVD, northing and easting coordinates. 72

97 Figure 4.2: Field Case 1: Inclination and azimuth from actual survey data. The accuracy of the ASC borehole trajectory model for Field Case 1 is presented in Table 4.2 by the maximum error over the entire wellpath and the landing point at TD. The ASC borehole trajectory model exhibits the highest overall accuracy when compared to MCM and AATM. The AATM shows the most accurate landing TVD, However the northing shows significant inaccuracy at approximately 100-ft shallower depth; proving that AATM fails to predict an accurate wellbore trajectory. Table 4.2: Field Case 1: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM. Wellbore Field Case 1 (9868 Surveys) Trajectory Maximum Error Well Landing Error Computational TVD N E κ τ TVD N E Model [ft] [ft] [ft] [ /100-ft] [1/ft] [ft] [ft] [ft] ASC Model MCM N/A AATM N/A N/A

98 Figure 4.3: Field Case 1: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 74

99 Tortuosity from the ASC borehole trajectory model shown in Figure 4.4 presents the highest accuracy throughout the entire wellpath with a maximum error of 8 /100-ft oppose to the MCM that presents poor estimation of tortuosity throughout the entire wellpath with a maximum error of 16 /100-ft. The MCM shows negligible measurement of tortuosity along the entire wellpath. At 4,000-ft and 8,000-ft actual tortuosity was observed to be very high using 90-ft and 1-ft survey resolution. At those two depth intervals MCM continued to show negligible results, however, the ASC borehole trajectory model offers more accurate and closer to reality estimates of tortuosity. Also, at 10,000-ft the ASC borehole trajectory model shows reasonable accuracy compared to the MCM. This behavior is expected, owning to the constant curvature assumption in the MCM. The 8 /100-ft error in the wellbore tortuosity from the ASC model is inevitable, as no mathematical model can predict the exact curvature precisely. However, a robust mathematical model such as the ASC borehole trajectory model is closer to realistic values. The sensitivity of the ASC borehole trajectory model to sliding sheet concept is significant with a noticeable improvement for all positions. These results are illustrated in Table 4.3 with significant improvement using the 45-ft survey resolution at 79% in the TVD, 86% in northings and 37% in eastings. Higher accuracy was observed in the 30-ft resolution at 96% in the TVD, 95% in northings and 92% in eastings. Table 4.3: Field Case 1: The ASC borehole trajectory model accuracy improvement from low resolution to higher resolution Survey Interval Field Case 1 TVD Error [ft] Northing Error [ft] Easting Error [ft] ASC 90 ft [111 Surveys] ASC 45 ft [221 Surveys] ASC 30 ft [330 Surveys]

100 Figure 4.4: Field Case 1: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution Field Case 2 Field Case 2 is a smooth well profile with low tortuosity over the entire wellpath with a KOP at 1,000 ft while building 2.3 /100-ft to reach 90 inclination at 5,840-ft MD. It then holds the angle until 7,500 ft MD and dropping at a rate of 1.15 /100-ft to reach 50 inclination at TD [10,155 ft MD] as shown in Figure 4.5. The accuracy of the models has been compared to the baseline for the wellpath with landing points at 5,429-ft TVD, 6,649-ft Northings and 2,916-ft Eastings. Field Case 2 is an example of a case where MCM and ASC model yield similar results. The similarity in the wellpath projection between MCM and ASC model are presented in Figure 4.6. The same was observed when comparing the overall tortuosity of ASC model with MCM as shown in Figure 4.7. The error in Field Case 2 using both the ASC borehole trajectory model and the MCM are negligible over the entire wellpath, as illustrated in Table

101 Figure 4.5: Field Case 2: Inclination and azimuth from actual survey data. Table 4.4: Field Case 2: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Wellbore Field Case 2 [8502 Surveys] Trajectory Maximum Error Well Landing Error Computational TVD N E κ τ..tvd.. N E Model [ft] [ft] [ft] [ /100-ft] [ /ft] [ft] [ft] [ft] ASC Model MCM N/A AATM N/A N/A

102 Figure 4.6: Field Case 2: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 78

103 Figure 4.7: Field Case 2: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 79

104 On the slide sheet concept for Field Case 2, results show an average improvement at all three positions, where a 60% improvement was observed using a 45-ft survey resolution interval and almost a 100% improvement using a 30-ft survey resolution as shown in Table 4.5. Table 4.5: Field Case 2: The ASC borehole trajectory model accuracy improvement from low resolution to higher resolution Survey Interval Field Case 2 TVD Error [ft] Northing Error [ft] Easting Error [ft] ASC 90 ft [129 Surveys] ASC 45 ft [223 Surveys] ASC 30 ft [317 Surveys] Field Case 3 Field Case 3 is the most relevant example of this dissertation to prove the accuracy of the ASC model when compared to survey data recorded at one survey per foot using the HRCG survey tool. This is a more complex well profile as shown in both Figure 4.8 and Figure 4.9. The wellpath builds to an inclination of 90 at 7,700-ft MD and holds the angle horizontally to TD at 16,545-ft. The accuracy of the models has been compared to the wellpath baseline values of the landing points at 7,146-ft TVD, 9,573-ft Northings and 400-ft Eastings. The accuracy of the ASC borehole trajectory model for Field Case 3 is presented in Table 4.6 by the maximum error over the entire wellpath and the landing point at TD. The ASC borehole trajectory model exhibits the highest overall accuracy. Table 4.6: Field Case 3: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Wellbore Field Case 3 [16443 Surveys] Trajectory Maximum Error Well Landing Error Computational TVD N E κ τ..tvd.. N E Model [ft] [ft] [ft] [ /100-ft] [ /ft] [ft] [ft] [ft] ASC Model MCM N/A AATM N/A N/A

105 Figure 4.8: Field Case 3: Inclination and azimuth from actual survey data. 81

106 Figure 4.9: Field Case 3: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 82

107 Tortuosity from the ASC borehole trajectory model shown in Figure 4.10 presents the highest accuracy at 90-ft survey interval throughout the entire wellpath with maximum error of 6.4 /100-ft oppose to the MCM that presents poor estimation of tortuosity throughout the entire wellpath with maximum error of 13.7 /100-ft. The MCM shows negligible measurement of tortuosity along the entire wellpath. From 6,000 to 7,000-ft a high tortuosity was observed from the one survey per foot resolution. MCM still showing negligible results while the the ASC borehole trajectory model offers better accuracy. Figure 4.10: Field Case 3: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution. On the slide sheet concept for Field Case 3, results show significant improvement for horizontal wells at all three positions, where a 70% improvement was observed using a 45-ft survey resolution interval and almost 100% improvement using a 30-ft survey resolution as shown in Table

108 Table 4.7: Field Case 3: The ASC borehole trajectory model accuracy improvement from low resolution to higher resolution Survey Interval Field Case 3 TVD Error [ft] Northing Error [ft] Easting Error [ft] ASC 90 ft ASC 45 ft ASC 30 ft Field Case 4 FieldCase4isanothercomplexwellprofile, asshowninbothfigure4.11andfigure4.12. The accuracy of the models has been compared to the baseline of the wellpath landing points at 4,116-ft TVD, 10,607 ft Northings and 9,382 ft Eastings. Figure 4.11: Field Case 4: Inclination and azimuth from actual survey data. 84

109 At closer scale on the vertical section [Figure 4.12], the ASC borehole trajectory model preserves the baseline very well with maximum TVD error of 1.38-ft over the entire wellpath, 0.08-ft in the Eastings and 0.70-ft in the Northings. Figure 4.12: Field Case 4: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution. The accuracy of the ASC borehole trajectory model is better presented by the maximum error over the entire wellpath and landing point at TD, as illustrated in Table 4.8. The ASC borehole trajectory exhibits the highest overall accuracy when compared to all methods. Tortuosity from the ASC borehole trajectory model shown in Figure 4.13 presents the highest accuracy at 90-ft survey interval throughout the entire wellpath with maximum error of 3.48 /100 ft. The MCM shows negligible measurement of tortuosity at the horizontal section [6,300-ft to 16,812-ft MD] at a maximum error of 5 /100-ft. This can be explained by the artificially low tortuosity resulted by the assumption of the constant curvature arc in the MCM. 85

110 Table 4.8: Field Case 4: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Wellbore Field Case 4 [15128 Surveys] Trajectory Maximum Error Well Landing Error [ft] Computational TVD N E κ τ..tvd..... N... E Model [ft] [ft] [ft] [ /100-ft] [ /ft] [ft] [ft] [ft] ASC Model MCM N/A AATM N/A N/A Figure 4.13: Field Case 4: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 86

111 The effect of slide sheet analysis was tested as part of a convergent rate analysis. Field Case 4 shows some noticeable improvement of 40 % at 45-ft survey resolution and 70 % at 30-ft resolution for all borehole positions Field Case 5 Field Case 5 is a complex well profile, as shown in both Figure 4.14 and Figure The wellpath builds to 90 at 5,300-ft MD and holds the angle horizontally until 14,931-ft MD then drop to 71 at TD [16,563-ft MD]. The accuracy of each model has been compared to the baseline for the wellpath with landing points at 4,417-ft TVD, 7,737-ft Northings and 11,233-ft Eastings. Figure 4.14: Field Case 5: Inclination and azimuth from actual survey data. Similarly, at closer scale on the vertical section [Figure 4.15], the ASC borehole trajectory model preserves the baseline very well with maximum TVD error of 1.34-ft over the entire wellpath, 1.39-ft in Eastings and 1.49-ft in Northings. 87

112 Figure 4.15: Field Case 5: Vertical section from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 88

113 The accuracy of the ASC borehole trajectory model is also presented by the maximum error over the entire wellpath and the landing point at TD for Field Case 5 as illustrated in Table 4.9. As all previous field cases, the ASC borehole trajectory model exhibits the highest overall accuracy when compared to MCM and AATM. Table 4.9: Field Case 5: Maximum error comparison of the ASC borehole trajectory model, MCM and AATM Wellbore Field Case 5 [14609 Surveys] Trajectory Maximum Error Well Landing Error [ft] Computational TVD N E κ τ.tvd... N.. E Model [ft] [ft] [ft] [ /100-ft] [ /ft] [ft] [ft] [ft] ASC Model MCM N/A AATM N/A N/A Tortuosity from the ASC borehole trajectory model shown in Figure 4.16 presents the highest accuracy at 90-ft survey interval throughout the entire wellpath with maximum error of 1.93 /100 ft. Once again, the MCM shows negligible measurement of tortuosity at the horizontal section [5,300-ft to 14,931-ft] at a maximum error of 3 /100-ft. On the slide sheet concept for Field Case 5, results show noticeable improvement at all three positions, where a 60% improvement was observed using a 45-ft survey resolution interval and a 90% improvement using a 30-ft survey resolution Synthetic Wellpath Example A typical mathematical approach to verify the accuracy of the ASC borehole trajectory model is the use of a complex synthetic well example of a trajectory wellpath with a predetermined set of variables. The ASC borehole trajectory model outlined preserves the shape and properties of the original function very accurately compared to the MCM, the AATM and the SIT 2 model developed by Gfrerrer and Glasser (2000). The quantification of error in this section are presented as the maximum error over the entire wellpath in comparison to the exact values of the curve for easting, northing, TVD, κ, and τ. 89

114 Figure 4.16: Field Case 5: Wellbore tortuosity from actual survey data at one survey per foot and the sampled 90-ft survey resolution. 90

115 Synthetic Wellpath 1 Consider a complex synthetic wellpath given by: x 1 (t) αt 2 X(t) = x 2 (t) = ξt sin(ηt), t [0,1], X(0) = x 3 (t) δ(2t t 2 ) where such that: α = 6000, ξ = 300, η = 80, δ = 3000 x 1 (t) = E exact, x 2 (t) = N exact, x 3 (t) = TVD exact, The arc length s for t [0,1], also expressed as MD is defined by Equation 4.12: ˆt s = ϕ(t) = 3 [x k (σ)]2 dσ (4.12) and hence 0 k=1 ϕ (t) = 3 [x k (σ)]2 dσ (4.13) k=1 It is assumed that ϕ (t) 0 for t [0,1]. The curve X in term of s is given by: y 1 (s) Y(s) = y 2 (s) = X(ψ(s)), s [0,ϕ(1)] (4.14) y 3 (s) where ψ is the inverse function of ϕ, that is t = ψ(s). If follows from Equation 4.14 that: y 1(s) Y (s) = y 2(s) = X (t), s [0,ϕ(1)] (4.15) y 3(s) ϕ (t) assume that: 0 = t 0 < t 1 < < t n 1 < t n = 1 The corresponding points in the interval [0, ϕ(1)] are: s i = ϕ(t i ), i = 0,...,n (4.16)

116 Divide the subinterval [0, 1] into n subinterval using the equally spaced points. [In this example n=225,450 will be used]: t i = i, i = 0,...,n (4.17) n and for t [0,1], the arc length s i = MD i, can be computed by approximating each interval over [t j 1,t j ] using the composite Simpson s Rule with 200 subintervals. Then we require the formulas: Y (s i ) = ϕ (t i )X (t i ) ϕ (t i )X (t i ) (4.18) [ϕ (t i )] 3 Y (s i ) = [ϕ (t i )] 2 X (t i ) 3ϕ (t i )ϕ (t i )X (t i )+3[ϕ (t i )] 2 X (t i ) ϕ (t i )ϕ (t i )X (t i ) [ϕ (t i )] 5 (4.19) for s [0,ϕ(1)] and 3 [ x k (t i)x k (t i) ] ϕ (t i ) = ϕ (t i ) 3 k=1 ϕ (t i ) = k=1 ϕ (t i ) x k (t i)x k (t i)+[x k (t i)] 2 ϕ (t i ) 3 [ϕ (t i )] 2 k=1 [x k (t i)x k (t i)] These formulas are derived using 4.15 and 4.13 respectively and will be used to calculate exact tortuosity and torsion. Y (s i ) is defined as the exact tangent vector λ i, such that: y 1(s i ) = λ Ei, y 2(s i ) = λ Ni, y 3(s i ) = λ TVDi from which, survey-stations [MD i, φ i, θ i ] are calculated as follow: MD i = s i, φ i = cos 1 (λ TVDi ), θ i = tan 1 ( λ E i λ Ni ) Following the steps listed under the wellbore positioning representation section, the approximated wellbore positions [easting, northing and TVD] are obtained using Equation 4.7, and the wellbore tortuosity and the geometric torsion using Equations 4.8 and 4.10 respectively. 92

117 The quantification of error in this section are presented in comparison to the exact values of the curve for easting, northing, TVD, κ, and τ as follows: ε E = max Eexact Ẽ, ε N = max Nexact Ñ, ε TVD = max TVD exact TVD i=0,...,n i=0,...,n i=0,...,n ε κ = max i=0,...,n κ exact κ, ε τ = max i=0,...,n τ exact τ The computation results and errors using the ASC borehole trajectory model, MCM, AATM and SIT2 are presented in Table 4.10 for n = 225 and Table 4.11 for n = 450. Table 4.10: Synthetic Wellpath 1: Maximum error comparison at n = 450 Wellbore Trajectory 450 Surveys TVD Error N Error E Error κ Error τ Error Computational Model [ft] [ft] [ft] [ /100ft] [ /ft] ASC Model MCM N/A AATM N/A N/A SIT N/A Table 4.11: Synthetic Wellpath 1: Maximum error comparison at n = 225 Wellbore Trajectory 225 Surveys TVD Error N Error E Error κ Error τ Error Computational Model [ft] [ft] [ft] [ /100ft] [ /ft] ASC Model MCM N/A AATM N/A N/A SIT N/A It is evident from the comparison tables that the ASC borehole trajectory model yields a significantly higher accuracy than the MCM, AATM, or SIT2 models. This is also noted in the 3D horizontal section of the synthetic wellpath shown in Figure Even for a reduced number of surveys down to 250, it is shown that the ASC borehole trajectory model still gives more accurate results as compared to all other methods in wellbore positioning, wellbore curvature and geometric torsion. 93

118 Figure 4.17: Synthetic Wellpath 1: 3D horizontal section. 94

119 Synthetic Wellpath 2 Consider a synthetic wellpath 2 given by: x 1 (t) δsin(ξt) X(t) = x 2 (t) = δ δcos(ξt) x 3 (t) αt, t [0,1], X(0) = Different choices of α,β,δ have been tested, below two choices were chosen to present the results: α = 10, ξ = 4, δ = such that α = 1000, ξ = 100, δ = 100 x 1 (t) = E exact, x 2 (t) = N exact, x 3 (t) = TVD exact note that X (t) = x 1(t) x 2(t) x 3(t) = δξ cos(ξt) δξ sin(ξt) α hence, the arc length s for t [0,1], also expressed as MD is defined by Equation 4.20: ˆt s = ϕ(t) = 3 ˆt [x k (σ)]2 dσ = γdt = γt (4.20) where 0 k=1 0 γ = α 2 +δ 2 β 2 Since t = s/γ, the curve X in term of s is given by y 1 (s) δsin(ξs/γ) Y(s) = y 2 (s) = X(s/γ) = δ δcos(ξs/γ) y 3 (s) αs/γ.., s [0,γ] (4.21) 95

120 It follows from Equation 4.21 that y 1(s) Y (s) = y 2(s) = y 3(s) y 1(s) Y (s) = y 2(s) = y 3(s) and (δξ/γ) cos(ξs/γ) (δξ/γ) sin(ξs/γ) α/γ (δξ 2 /γ 2 )sin(ξs/γ) (δξ 2 /γ 2 )cos(ξs/γ) 0, s [0,γ] (4.22), s [0,γ] (4.23) Y (s i ) 2 = δ ξ 2 /γ 2 Divide the subinterval [0, γ] into 100 subintervals using the equally spaced points Y (s i ) is defined as the exact tangent vector λ i, such that: s i = iγ, i = 0,...,100 (4.24) 100 y 1(s i ) = λ Ei, y 2(s i ) = λ Ni, y 3(s i ) = λ TVDi from which, survey-stations [MD i, φ i, θ i ] are calculated as follow: MD i = s i, φ i = cos 1 (λ TVDi ), θ i = tan 1 ( λ E i λ Ni ) Following the steps listed under the wellbore positioning representation section, the approximated wellbore positions [easting, northing and TVD], wellbore tortuosity and geometric torsion are obtained. This synthetic wellpath example of a known trajectory path is used to demonstrate the accuracy of the ASC borehole trajectory at different choices of α,β,δ compared to all other methods. The computation results and errors using the ASC borehole trajectory model, MCM, and AATM are presented in Table 4.12 and Table 4.13 Synthetic wellpath 2 is also strong evidence that the ASC borehole trajectory model yields a significantly higher accuracy than the MCM and the AATM as presented in the comparison tables. 96

121 Table 4.12: Synthetic Wellpath 2: Maximum error comparison at α = 10,ξ = 4,δ = 150 Wellbore Trajectory α=10, ξ=4, δ=150 TVD Northing Easting Curvature Torsion Computational Model Error [ft] Error[ft] Error [ft] Error [ /100ft] Error [ /ft] ASC Model 7.2E E E E E-08 MCM 7.46E E E E-10 N/A AATM 9.14E E N/A N/A Table 4.13: Synthetic Wellpath 2: Maximum error comparison at α = 1000,ξ = 100,δ = 100 Wellbore Trajectory α=1000, ξ=100, δ=100 TVD Northing Easting Curvature Torsion Computational Model Error [ft] Error[ft] Error [ft] Error [ /100ft] Error [ /ft] ASC Model 1.00E E E E E-04 MCM 9.14E E E E-06 N/A AATM 4.30E E E+03 N/A N/A ASC Borehole Trajectory Model Results Summary In summary, it is observed, from the study of five wells with actual survey data, that the smoother the well profile [low tortuosity] the closer the results and accuracy of the MCM and ASC model. However, moving toward a more complex and tortuous wellpath, the ASC borehole trajectory model shows up to 50% improvement in the accuracy of wellbore positions. The calculated wellbore tortuosity and the geometric torsion using the ASC borehole trajectory model also shows a high level of precision and significant improvement than the results provided by currently used methods. Hence, increased borehole position accuracy from while drilling surveys and truer estimates of wellbore tortuosity and geometric torsion is possible with the ASC borehole trajectory model. The use of the proposed model has potential to reduce borehole path uncertainty, provide better estimates and projections of the drillstring configuration for T&D calculations and improve the accuracy level of wellpath trajectory in ER and complex horizontal wells. This is shown in the next Chapter. 97

122 CHAPTER 5 TECHNICAL APPROACH - THREE DIMENSIONAL TORQUE AND DRAG MODEL T&D are critical elements in ER and complex horizontal wells with high and repeated tortuosity. Current T&D models assume constant curvature arcs between survey-stations. This assumption causes the bending parameter in the T&D equilibrium equations to be nil. This leads to inaccurate estimates of T&D magnitudes, since the curvature does change. To overcome the technical limitations of current T&D models, further advancements in the computational modeling is required. This led to the development of the ASC 3D T&D model. The proposed model is a phase II initiative of the non-constant curvature trajectory model: the ASC borehole trajectory model. The ASC 3D T&D model is a stiff-string model that includes explicit values for the curvature [κ], the geometric torsion [τ], the change in the rate of wellbore curvature [dκ/ds] and the drillstring bending stiffness [EI]. These features allow the ability to estimate more realistic bending effects and contact forces. It also allows to solve T&D parameters from surface to total depth in reasonable time using standard engineering computers. This chapter presents the improvement of current T&D models. It starts with general approaches to improve the accuracy of T&D models. Then the assumptions and boundary conditions on the model are shown. This sets the guidelines to derive the force and moment equilibrium equations to predict T&D outputs. The general procedure and the derivation of the new T&D model are described. Lastly, this model is validated using two field cases with real-time forces that define T&D measurements at the surface. The validation process consists of two methodologies: [a] a semi-analytical approach: survey data generation from the ASC borehole trajectory model at a pseudo-high resolution [PHR] of one survey per foot intervals as an input in a current industry-approved T&D software, and [b] the ASC 3D T&D model compared to actual drilling data. 98

123 5.1 Problem Setup Process To improve current T&D model, three approaches could be applied: 1. Including all the terms in the force and moment equilibrium equations. 2. Developing a more advanced wellbore trajectory calculation method. 3. Considering different fluid loads and including dynamics into the force and moment equilibrium equations. The first approach has been analyzed and tested in the industry. Including all the terms in the force and moment equilibrium equations is called the stiff-string T&D model. This approach is opposed to the industry standard model known as the soft-string model. In both cases, the bending parameter, the change in the rate of curvature and the geometric torsion in the equilibrium equations are nil because the wellbore trajectory is based on the MCM. These assumptions result in an underestimation of the drillstring torque, failure to predict the buckling onset and poor estimation of the contact forces between the drillstring and wellbore (Tikhonov et al. 2014). The second approach consist of developing a more advanced wellbore trajectory calculation method. This method has been tested and validated in the scope of this dissertation. In the literature, many wellbore trajectory models have been proposed; however, the oil and gas industry continues to rely on MCM as the standard. As noted earlier in Chapter 2, the limitations of MCM present major problems with wellbore trajectory computations and curvature accuracy. The oil and gas industry can no longer effectively use this model (Stockhausen and Lesso 2003). A non-constant curvature trajectory model [the ASC model] has been developed and used to derive the force and moment equilibrium equations to predict T&D outputs. Unlike the MCM, the ASC borehole trajectory model does not make the potentially unrealistic assumption of a constant curvature arc between survey measurements. This provides an 99

124 advantage for accurate borehole curvature calculations and drillstring to wellbore modeling prediction. The third and most complex approach is to include the fluid loads and dynamics into the force and moment equilibrium equations(mitchell and Miska 2011). This approach is beyond the scope of this dissertation work as it requires an extensive knowledge of fluid dynamics in the drillstring and the annulus, including internal fluid velocity, drillstring eccentricity and cutting transport effect Assumption and Boundary Conditions During the model development, the friction forces will be modeled using the Coulomb friction concept. This system is subject to several assumptions, boundary constraints and load conditions (Abughaban et al. 2017). The required assumptions are as follow: 1. The drillstring is a simple beam with homogenous properties and a single circular cross sectional shape [tool joints are ignored]. 2. The pipe curvature and torsion are assumed to be the wellbore curvature and torsion. 3. The drillstring diameter and material properties can be grouped depending on the bottom hole assembly [BHA] design. 4. The rotary speed [RPM], weight on bit [WOB] and pump rate [SPM] at the surface are constant [steady state]. 5. The internal and external fluid loads are assumed to be zero [ w if = w ef = 0]. 6. The drillstring internal loads are not considered [P i A i = 0] Initial Approaches Generating survey data semi-analytically using the ASC borehole trajectory model was the first approach taken during this analysis. The general spline equation from the ASC borehole trajectory model is used to calculate measured depth [MD i ], inclination [φ i ] and 100

125 azimuth [θ i ] between any given survey data at a pseudo-high resolution [PHR] interval rate. The generated PHR survey results output are then used as an input in an industry-approved T&D software [Program A]. The purpose of this analysis is to increase the number of survey-stations, using a semianalytical approach which will efficiently reduce current T&D model uncertainties. surveystations can be generated from the ASC model as follow: Step 1: Compute the general spline-curves equation using the specified boundary conditions [described in Chapter 4] For i = 0,...,n 1: T(s i ) = A i +(s s i )B i +(s s i ) 2 C i +(s s i ) 3 D i, s [s i,s i+1 ] (5.1) where: A i = λ i, B i = λ i+1 λ i h i h i 6 z i+1 h i 3 z i, C i = z i 2, D i = z i+1 z i 6h i Step 2: Solve the function T(s) for the approximation of the wellbore positions [TVD, northing and easting] at any step change in measure depth [s i ]: T i (s i ) = T TVDi T Ni T Ei Step 3: Compute the inclination [φ i ] and azimuth [θ i ] at each measured depth [MD i ] selected as follow: where MD i = s i, φ i = cos 1 (T TVDi ), θ i = tan 1 ( T E i T Ni ) T(s) = Advanced spline general function; A, B, C, D =Advanced spline function coefficients; λ i = Normalized tangential vector on the space curve. 101

126 Results from the initial approach show reasonable improvement in accuracy since the PHR survey data generated redresses the limitation of current T&D models. However, the effect of the bending moment in the T&D equilibrium equations cannot be effectively analyzed using this approach. Thus, the force and moment equilibrium equations should be extended for a non-constant curvature borehole trajectory model [shown in the next section] Building the Model In the scope of this dissertation, a non-constant curvature trajectory model [the ASC model] was developed and used to derive the force and moment equilibrium equations to predict T&D outputs. During the process of developing the extended T&D model, the following guidelines were considered: 1. The force and moment equilibrium equations are derived for a non-constant curvature trajectory [The ASC model] in which the curvature and torsion are continuous. 2. The effect of wellbore irregularities, geometry and tortuosity are considered. 3. The tangent, normal and binormal unit vectors are modeled to the approximating circle. These guidelines will allow to effectively analyze the effect of the bending moment in the T&D equilibrium equations, predicting accurate measurements of drillstring to borehole contact forces, and solve T&D parameters from surface to total depth in reasonable time Model the tangent, normal and binormal unit vectors to the approximating circle To model the tangent, normal and binormal unit vectors [points T 1 and T 2 on a local section of the spline-curve shown in Figure 5.1], a circle is drawn that closely fits the points while forcing the curve and the circle to osculate. This will satisfy the conditions where both curves will have the same tangent and curvature at the point where they meet. 102

127 Figure 5.1: Spline-curve and circular arc osculation. Thus, the radius of curvature [R] can be expressed as follows: R i = (1+ ( Y ) 2 )3 2 Y, i = 0,...,n (5.2) where Y (s) and Y (s) are computed as follow: Y (s i ) = λ i, i = 0,...,n Y (s i ) = T (s i ) = { λi+1 λ i h i h i λ i λ i 1 h i 1 z 6 i+1 h i z 3 i, i = 0,...,n 1 h i 1 z 6 i 1 + h i 1 z 3 i, i = n and, the tangent, normal and binormal unit vectors can be expressed as follow for i = 1,...,n 1 (Mitchell and Miska 2011): 103

128 ( ) cosφi+1 cosφ i cosψ i t zi = cos(φ i ) cos(κ i h i )+ sin(κ i h i ) (5.3) sinψ i ( ) cosφi+1 cosφ i cosψ i n zi = cos(φ i ) sin(κ i h i )+ cos(κ i h i ) (5.4) sinψ i b zi = cos(φ i)sin(φ i+1 )sin(θ i+1 θ i ) sin(ψ i ) (5.5) where, Y (s) = The advanced spline-curves function T(s i ); Y (s) = The first derivative of the advanced spline-curves function T(s i ); λ i = Normalized tangential vector on the space curve; ψ i = Turning angle of a curve = κ i s i, degrees. 5.2 Development of the Torque and Drag Model for the Advanced Spline-Curve Trajectory When the wellbore trajectory is modeled using the ASC borehole trajectory model, the wellbore positions are first computed. From this trajectory, τ, κ and dκ ds are determined. These parameters are now included in the force and moment equilibrium equations as follows: Tangent Component df t ds F nκ+w b t z ±µw c = 0 (5.6) dm t ds = 0 (5.7) Note that from Equation 5.6: the plus [+] means hoisting and minus [-] means lowering of the pipe, which dictates the direction of the friction, such that drag always acts in the opposite direction that the drillstring is moving. 104

129 Normal Component df n ds +F tκ+w b n z +w c cosϑ+f b τ = 0 (5.8) κm t EIκτ F b µw c r p sinϑ = 0 (5.9) Binormal Component df b ds F nτ +w b b z w c sinϑ = 0 (5.10) EI dκ ds +F n µw c r p cosϑ = 0 (5.11) From Equations [5.7], [5.9] and [5.11] the magnitude of the ASC contact force [lbf/ft]: w c = (F n +EI dκ ds )2 +(F b +EIκτ) 2 µr p (5.12) where,.and the direction [ ]: [ ] ϑ = tan 1 F b EIκτ F n +EI dκ ds (5.13) F t = Tangential [axial] force, lbf; F n = Shear force in the normal direction, lbf; F b = Shear force in the binormal direction = w b b z µr p, lbf; M t = Torque [Moment], ft-lbf; w b = Effective weight of the drillstring, lbf; w c = Contact force magnitude, lbf/ft; w d = Drag force magnitude, lbf; ϑ = Wellbore contact angle, degrees. 105

130 Introducing the stiffness and bending moment terms in this model will have the potential to greatly improve the accuracy of T&D parameters and more effectively estimate the downhole conditions. This model becomes more important in ER wells with large side turns that directly contribute to the contact side forces on the drillstring that is not properly estimated using traditional T&D models Estimating Downhole Weight on Bit and Torque In this study, the ASC 3D T&D model is used to estimate the downhole weight on bit [DWOB] and torque on bit [TOB] while considering the friction factor induced from both the axial and rotational loadings. These loads are the main contributor to the energy losses from the surface (Aarrestad and Blikra 1994). The estimated DWOB and TOB parameters will allow drilling engineers to update the driller with surface WOB and torque parameter to improve the ROP. The DWOB is reduced primarily by the axial friction corresponding to the drillstring position and the static weight of the drillstring immersed in the fluid. The TOB will be reduced by the rotational friction that represents the friction in contact between two rotating bodies. These expressions can be illustrated in Equations [5.14] and [5.15]: DWOB = w b L F t HL surface (5.14) and, TOB = Torque surface F rotational (5.15) where, L = The measured length of an element along the drillstring, ft; HL surface = Hookload measured at the surface, lbf; Torque surface = Torque measured at the surface, ft-lbf; F rotational = Forces due to the rotation of the drillstring, lbf. For this study, it is assumed that the axial and rotational friction between the drillstring and the wellbore depend primarily on the normal force induced by the drillstring weight and the tension along the drillstring. Thus, applying WOB will cause a reduction in the 106

131 tensile force since some portion of the drillstring is in compression. This phenomenon will significantly reduce the rotational forces (Aston et al. 1998). In a perfect vertical well, the surfacewobandsurfacetorquecouldbeequaltodwobandtobastheaxialandrotation friction could be nil. In contrast, ER wells, especially those with some degree of DLS, will have a significant reduction in the true DWOB and TOB from the friction force due to the tension along the wellbore. This effect requires a more accurate model to reliably predict downhole conditions. An academic example has been selected to illustrate how DWOB and TOB can be estimated. The use of downhole sensors that measures real-time DWOB is required to validate the accuracy of these estimations. 5.3 Adaptation of the T&D Model for a Non-Constant Curvature Trajectory Model The main task is to calculate the axial effective force in the drillstring. If the axial force F ti+1 and the normal force F ni+1 at the bottom are known, the axial force at the top segment can be calculated using Equation 5.16 as follows: where, and, F = w b cos(φ avg ) F ti = F ti+1 F ni+1 κ s+w b t z s±µw c s (5.16) F ni+1 = w bi+1 s i+1 sin(φ avgi+1 )+ ( F ti+1 + F ti+1 ) sin( ψ i+1 2 ) (5.17) [ 1+µsin( ψ ) ] 2 cos( ψ ) +µtan(φ avg ) +2µF t sin( ψ 2 )+µf stiffness (5.18) 2 However, Mitchell and Miska(2011) have conducted analysis stating that the z-component of the tangential and normal vector is a non-linear function between the survey-stations at a given measured depth. Consequently, a numerical approach is required to calculate the drag force, and a closed form solution will not be applicable for the ASC 3D T&D model. In this case, a first approximation of the unit contact force must be obtained using Equation 107

132 [5.12], from which the first approximation of the axial force at the top can be calculated. An average axial force is calculated to obtain a second approximation of the contact force. Thus, a second approximation of the axial force and normal force is found. This iterative process is repeated until the axial force approximation reaches a tolerance level chosen to be less than 1%. Final values are then reported. To effectively validate and evaluate the results accuracy the ASC 3D T&D model, realtime forces that define T&D measured at the surface and at the bottom of the drillstring are required. The field data includes hookload and surface torque measured at the surface for different drilling operations such as rotating off bottom [ROB], rotary drilling [RD], pickup [PU] and slackoff [SO]. Given comparable inputs for an industry-approved T&D program [Program A] and the newly developed model, the T&D outputs show good agreement of results until a high-angled, high-tortuous section of the wellbore is encountered. At that point, the ASC 3D T&D model predicted additional axial and torsional loads. These forces are not predicted in current T&D models that assumes constant curvature arcs from the MCM. 5.4 Quantification of the ASC 3D Torque and Drag Model The field data analysis for the verification and validation of the extended T&D model and the estimation of DWOB and TOB from surface hookload measurements are presented. Two field cases have been used to demonstrate the challenges during different drilling operations [ROB, RD, PU and SO]. The testing and validating procedure of the model accuracy consists of two methodologies: 1. Semi analytical approach: Data generation from the ASC borehole trajectory model at a pseudo-high resolution of one survey per foot [PHR 1 ft]. 2. Using the ASC 3D T&D model compared to actual drilling data for two field cases. The two approaches applied in this study are shown to be consistent and give a close match for accuracy against field data as compared to other models used in the industry. 108

133 5.4.1 Field Case 1 Well 1 is an extended reach well with a double build trajectory as shown in Figure 5.2. The first build section begins at 1,600-ft[490-m] and ends at 6,900-ft[2,100-m] at a maximum inclination of 8. The second build goes to 90 inclination from 6,900-ft [2,100-m] to 8,200-ft [2,500-m] with a maximum DLS of 11 /100-ft. The well remains horizontal until a total depth [TD] of 23,000-ft [7,010-m] with a maximum DLS of 5 /100-ft. Figure 5.2: Field Case 1: ER well vertical section. The well is an 8.5-in hole with 10-ppg mud weight. A friction factor of 0.20 is used in the cased-hole section from the surface until 8,200-ft[2500-m], and a friction factor of 0.25 is used in the open-hole section. These friction factor values were chosen to standardize the model validation process. The BHA consists of an 8.5-in polycrystalline diamond compact [PDC] bit, a 6.75-in mud motor, a 6.25-in DS lb/ft drillcollar, a 5-in heavy weight drillpipe with NC-44 connections, and a 5-in S lb/ft drillpipe with NC50 connections. The BHA components for Field Case 1 are summarized in Table

134 Table 5.1: Field Case 1: Drillstring design Field Case 1 Drill Component OD [in] ID [in] Length [ft] Count Weight per foot [lb/ft] PDC Bit 8.5 N/A N/A Mud Motor 6.75 N/A NC-44 DC MWD/LWD NC-44 DC HT-50 HWDP NC-50 DP The comparison of hookload, axial loading and surface torque parameters calculated using different models, including the ASC 3D T&D model shows good agreement of results. However, major differences are observed when the wellbore is experiencing high and repeated doglegs and tortuosity. The ASC 3D T&D model, shown by the blue solid line in Figure 5.3, shows a high contact force distribution between the wellbore and the drillstring throughout both the highly tortuous zone from 5,000ft [1,525 m] to 8,000ft [2,440 m] and the moderately tortuous zone from 10,000 ft [3,050 m] to 12,000 ft [3,660 m]. All other models predict no contact in the second build [10,000-ft (93,050-m) to 12,000-ft (3,660-m)] and fail to capture the additional drillstring to wellbore contact force. For the basic drilling operations [ROB, PU, SO and RD], the following findings can be highlighted upon closer examination: Rotating off Bottom [ROB]: For hookload results, the difference between the T&D programs used are within tolerable limits [2 to 3%] since the friction is in the circumferential direction with no effect from the axial direction (Tikhonov et al. 2014). The rotating torque shown in Figure 5.4 has a somewhat higher difference compared to previous models at a maximum variation of 8% [approximately 10 klbf-ft difference]. The axial load illustrated in Figure 5.5 has experienced the highest variation, among all models, of 12% [approximately 20 klbf difference]. One of the main reasons for these differences is the discrepancy in the contact forces distribution. 110

135 . Figure 5.3: Field Case 1: ER well contact force. DLS in units of /100 ft. 111

136 Figure 5.4: Field Case 1: ER well torque results while ROB. Figure 5.5: Field Case 1: ER well axial load while ROB. 112

137 Pickup [PU]: The hookload has experienced high discrepancy of 17% [approximately 120 klbf difference]. This is due to the additional loads from the drillstring to wellbore contact, the wellbore irregularity and tortuosity considerations in the force equilibrium equation. The rotating torque did not experience a major effect with a difference of less than 2%. The axial load has experienced a higher difference at 14% [approximately 40 klbf difference] as shown in Figure 5.6. This is clear evidence that the impact of bending moments in the axial direction is substantial which is not effectively accounted for in current T&D models [program A]. This is a clear evidence that the impact of bending moment is substantial in T&D calculation. Figure 5.6: Field Case 1: ER well axial load while pickup.. 113

138 Slackoff [SO]: The hookload did not experience a major effect, compared to other models, with difference less than 4%. The rotating torque showed a slight difference with a maximum discrepancy of 6%. The same trend was observed for the axial load illustrated in Figure 5.7 with good agreement of results from surface to TD. During the SO operation, the drillstring experiences compressional loading when in contact with the wellbore wall. Thus, the wellbore geometry and tortuosity plays an important role in accurately calculating T&D. Figure 5.7: Field Case 1: ER well axial load while slackoff. Rotary Drilling [RD]: Several noticeable differences are observed between the models for all three parameters. The hookload has experienced a high discrepancy of 21% [approximately 25 klbf difference]. The rotating torque also experienced a high discrepancy of 18% [approximately 9 kft-lbf maximum difference] as shown in Figure 5.8. A similar trend is observed for the axial load [Figure 5.9] with a 23% discrepancy [approximately 18 klbf difference]. 114

139 Figure 5.8: Field Case 1: ER well torque while 120 RPM and 60 klbf WOB. The observed discrepancies for all basic drilling operations are primarily due to the low predicted contact force. Of course, accurately predicting the contact forces is important to prevent drillstring failures and catastrophic events such as stuck pipe, casing and drillpipe wear and drillstring fatigue. Table 5.2 illustrates the discrepancy of results between current models for surface torque and hookload parameter. The under-estimation of results is clear when using the traditional, industry-approved T&D software with typical 90-ft survey interval [Program A - 90 ft]. Reasonable improvement in accuracy is observed when using the generated high resolution surveyresultsfromtheascboreholetrajectorymodelasaninputinprograma[programa - PHR 1 ft]. The ultimate level of accuracy is shown from the extended T&D model [ASC 3D T&D Model - 90 ft]. This model efficiently reduces current T&D model under-estimations. 115

140 Figure 5.9: Field Case 1: ER well axial load while 120 RPM and 60 klbf WOB. Table 5.2: Field Case 1: T&D results comparison ROB PU SO RD Rig Operation ST HL HL HL ST HL [kft-lbf] [klbf] [klbf] [klbf] [kft-lbf] [klbf] ASC 3D T&D Model [90 ft] Program A [PHR 1 ft] Program A [90 ft] Key: HL = hookload; ST = surface torque; kft-lbf = 1000 ft-lbf. 116

141 5.4.2 Field Case 2 Well 2 is a backward-nudge horizontal well [Figure 5.10]. The nudge begins at 1,600-ft [490-m] and ends at 3,800-ft [1,160-m]. The first build was drilled over a 1,400-ft [425-m] section starting at 2,840-ft [865-m] to 6,600-ft [2,010-m] with a maximum DLS of 2 /100-ft. The second build was drilled over a 4,000-ft section from 3 inclination until the well became horizontal [90 inclination] at 10,670-ft [3,250-m] and remained horizontal until 17,122-ft [5,220-m], with a maximum DLS of 18 /100-ft. Figure 5.10: Field Case 2: Backward-nudge horizontal well vertical section. The in casing set at 7,704-ft [2,350-m] has a friction factor of 0.2. The 8.75-in open hole below the shoe has a friction factor of The mud weight is ppg. The BHA consists of an 8.75-in PDC bit, a 6.75-in mud motor, a 6.25-in DS lb/ft drillcollar and a 5-in S lbf/ft drillpipe with NC50 connections. 117

142 A similar trend of results was observed for Field Case 2 for all three force parameters [hookload, rotary torque and axial loading] indicating good agreement among programs. However, major differences of the contact force are observed across the high and repeated tortuosity zones from 9,000-ft [2,745-m] to 17,000 ft [5,182-m]. The ASC 3D T&D model was able to capture the additional drillstring to wellbore contact force through that zone as shown in Figure Figure 5.11: Field Case 2: Backward-nudge horizontal well contact force. DLS in units of /100 ft.. 118

143 The following findings from the basic drilling operations can be highlighted: Rotating off Bottom [ROB]. The results among the T&D models used are within acceptable variation. The hookload has experienced a difference of less than 2 % [approximately 5 klbf difference], since there is no effect from the axial direction. The rotating torque estimated by using the ASC 3D T&D model indicates the lowest value in the vertical section due to the contact force distribution effect. A higher torque is estimated at the build section with high tortuosity, as expected, with a 20% maximum discrepancy [approximately 3 kft-lbf difference]. The fluctuation of torque values is illustrated in Figure Axial loading showed a higher difference with a maximum discrepancy of 8% [approximately 20 klbf difference] as shown in Figure Figure 5.12: Field Case 2: Backward-nudge horizontal well torque while ROB. 119

144 Figure 5.13: Field Case 2: Backward-nudge horizontal well axial loading while ROB. 120

145 Pickup [PU]. The hookload estimations showed a large difference between models, with a maximum discrepancy of 12% [approximately 9 klbf difference]. The rotating torque showed a smaller difference with a maximum discrepancy of 4%. A similar trend is observed for the axial load in Figure 5.14 with good agreement [2% discrepancy] across different models from surface to TD. Figure 5.14: Field Case 2: Backward-nudge horizontal well axial loading while PU. Slackoff [SO]. The surface hookload estimated by all models were similar, with a difference of 3%. Similar results are observed for the rotating torque and the axial load from surface to TD with a difference of 6% and 2% respectively. 121

146 Rotary Drilling [RD]. There were large discrepancies observed in the surface hookload estimates between models, with a maximum discrepancy of 22% [approximately 14 lbf difference]. Unlike other models, the ASC 3D T&D model predicted the lowest estimation of rotating torque at the build section. The difference ranges between 20 and 32% [approximately 4 kft-lbf difference] as shown in Figure Observations show that in high-angled wells, the overall torque tends to drop while drilling because most of the drillstring is in compression and the tension profile in the build section is reduced (Paune and Abbassian 1996). The axial load has a noticeable difference across the build section of 18% [approximately 30 klbf difference], as shown in Figure Figure 5.15: Field Case 2: Backward-nudge horizontal well torque while 75 RPM and 25 klbf WOB.. 122

147 Figure 5.16: Field Case 2: Backward-nudge horizontal well axial loading while 75 RPM and 25 klbf WOB. 123

148 Comparing the hookload, the surface torque and the axial load from different models and inputs validates the ASC 3D T&D model. The basic drilling operations agrees with the ASC 3D T&D model with acceptable variation when using PHR survey data generated at one survey per foot interval [PHR 1 ft] as an input in Program A. Higher discrepancy is observed when using the typical 90-ft survey interval, as shown in Table 5.3]. This is clearly because the traditional T&D model fails to capture the additional drillstring to wellbore contact force in the high tortuosity interval [from 9,000-ft to 17,000-ft] due to the MCM assumption that the borehole trajectory is composed of constant curvature arcs. Table 5.3: Field Case 2: T&D results comparison ROB PU SO RD Rig Operation ST HL HL HL ST HL [kft-lbf] [klbf] [klbf] [klbf] [kft-lbf] [klbf] ASC 3D T&D Model [90 ft] Program A [PHR 1 ft] Program A [90 ft] Key: HL = hookload; ST = surface torque; kft-lbf = 1000 ft-lbf Depth-Based Field Data from Field Case 2 Depth-based data for Field Case 2 were obtained from daily morning reports. These data provide an entire day s operations [24 hours of drilling] by a single data point that limits its accuracy in representing the day s operation. Nevertheless, these data are the only consistent measurements of drilling T&D acquired during the drilling operation. This data will be used to validate the ASC 3D T&D model. The available data represent nine-days of drilling operation over a 6,500-ft [1,980-m] drilled section. Rotating off Bottom [ROB]. The surface hookload should match the buoyed weight of the drillstring measured at the top-drive assembly. Any difference observed would likely be from the mud weight error, as reality assumes a single mud weight value for the entire well. Error from the entered components of the drillstring and the BHA in 124

149 the modelare comparedto theones used in the fieldcould contribute tothe differences. The results from ROB hookload [Figure 5.17] show an accurate agreement at the build section with high tortuosity from 10,000-ft [3,048-m] to 11,000-ft [3,350-m] MD with predictions within 3%, which is within the industry-acceptable variation for all data inputs. Figure 5.17: Field Case 2 Backward-nudge horizontal well hookload while ROB. Pickup [PU]. The axial loading and friction factors directly impact the hookload. This is shown by the data in Figure The MCM model significantly underestimates the hookload by 8% [approximately -80 klbf difference]. This large difference can be explained by the MCM model s failure to account for the contact force in the high tortuosity interval. 125

150 Figure 5.18: Field Case 2: Backward-nudge horizontal well hookload while pickup. 126

151 Slackoff[SO]. The hookload shows an accurate agreement at the build section with high tortuosity. However, an underestimated parameter, using the ASC 3D T&D model, was observed around 12,500-ft [-9.5% average difference]. Following this, from 12,500- ft onwards, the PHR input at one foot in Program A overestimated the hookload with an average difference of 19.2%, as illustrated in Figure Similar results are observed from the typical 90-ft input with an underestimated parameter of 13% difference at the high tortuosity zone, followed by an overestimation of 14% difference at the horizontal section. Since the ASC 3D T&D model is assuming a greater side force due to high tortuosity, a lower slack-off weight should be expected because more of the drillstring is being supported by the wellbore friction rather than the hookload. Hence, the qualitative difference between the ASC 3D T&D model vs. field data is not well understood, although the qualitative character of the model prediction appears to be correct in the author s view. Rotary Drilling [RD]. The comparison of surface torque against the field data is shown in Figure Generally, the results from the ASC 3D T&D model shows close agreement to the field data compared to both inputs of PHR survey at one one foot intervals and the typical 90-ft intervals in Program A. This variation could be attributed to the model s predicted contact forces, particularly at the high tortuous zone. Once again, it should be noted that the traditional T&D model [Program A], that relies on the MCM, significantly underestimates the torque due to the failure of capturing contact forces in high tortuosity intervals. 5.5 Validating the Estimated Downhole Weight on Bit and Torque on Bottom Once the axial and rotational effective forces against the drillstring movement and rotation are calculated, a newly developed model estimates DWOB and TOB. These parameters are estimated using surface hookload and torque measurement during the basic drilling operations [ROB, RD, PU and SO]. 127

152 Figure 5.19: Field Case 2: Backward-nudge horizontal well hookload while slackoff. 128

153 Figure 5.20: Field Case 2: Backward-nudge horizontal well torque while 75 RPM and 25 k-lbf WOB. 129

154 Step 1: Calculation of the forces acting on the bottom of a drillstring can be computed for a given drillstring specification, survey data and friction coefficient. Step2: TheestimationsofDWOBandTOBareaniterativeprocess,suchthatdifferent downhole values will be estimated until the measured and calculated hookload reaches a tolerance level less than 1%. tolerance HL measured HL calculated HL measured 1% Step 3: The final estimated DWOB and TOB values are chosen [Table 5.4] Table 5.4: DWOB academic example with 175 klbf measured HL DWOB [klbf] Calculated Hookload [klbf] Hookload Tolerance < 1% SWOB = DWOB = The academic example [10,000-ft MD s-shape synthetic well] shows losses of 8-klbf in WOB at the bottom [an estimated 32% loss], as illustrated in Table 5. This claim will require the use of downhole sensors that measure real-time DWOB to validate the accuracy of this estimation. However, this analysis was conducted using the ASC 3D T&D model. Based on the validation and evaluation process in this chapter, the ASC 3D T&D model offers significant improvement in estimating the stiffness and bending moment. It also allows a more accurate prediction of the actual downhole parameters. This will enable drilling engineers to update the driller with optimized WOB and surface torque parameters to improve the ROP without taking undue risks to the drillstring, particularly in high angle horizontal wells. 130

155 CHAPTER 6 TECHNICAL RESULT AND DISCUSSION Extended reach [ER] drilling refers to the practice of directionally drilling to a given geological target located at a significant horizontal distance from the drilling rig. The ability to perform complex ER operations has become increasingly important in recent years as the length of the wellbore horizontal section continues to be extended. The drilling challenges continue to evolve and push the drilling capabilities near their limits. To expand the ER drilling envelope even further, several challenges have been addressed in this dissertation. To overcome the challenges and technical limitations of ER drilling, the following approaches were applied: 1. The development of a non-constant curvature trajectory model [The Advanced Spline- Curves Model]: This model has been tested and validated for accuracy of borehole positions [TVD, northing and easting], wellbore tortuosity and geometric torsion. The calculated borehole trajectory favors the ASC model to be the most accurate when compared to the most utilized model in the industry, the MCM. The ASC model gives better wellbore positioning, more realistic drillstring configuration and the introduction of the change in the rate of curvature and the geometric torsion measurements. 2. The development of force and balance equilibrium equations for the non-constant curvature borehole trajectory model. This extended T&D model [The ASC 3D T&D model] was tested and validated for model accuracy of drillstring contact forces estimation and bending stiffness effect. The predicted T&D parameters are shown to be consistently accurate when compared to field data during different drilling operations such as rotating off bottom [ROB], rotary drilling [RD], pickup [PU] and slackoff [SO]. 131

156 This chapter presents the technical results on the drilling process. The effects are summarized into three effects, namely, T&D, wellbore positioning and wellbore tortuosity. A discussion on the results and recommendations on both the ASC borehole trajectory model and the extended 3D T&D model are presented. 6.1 Torque and Drag Effect T&D modeling is regarded as an invaluable process to predict and mitigate drilling problems. Excessive T&D parameter, caused by tight-hole, borehole instability, key seating, differential sticking, and poor hole cleaning, are considered the main factors that limit the directional distance a well can be drilled, especially for ER and complex horizontal wells. Traditionally, T&D models are either based on a continuous drillstring to wellbore contact [soft-string model] or intermittent contact due to drillstring stiffness [stiff-string model]. In both cases, the bending parameters, the change in the rate of curvature and the geometric torsion in the force and moment equilibrium equations are nil because the wellbore trajectory is based on the MCM which does not model these parameters. The use of an advanced T&D model such as the ASC 3D T&D model will provide engineers with a clear view on downhole conditions. This allows a better understanding on T&D effects in ER wells and complex horizontal wells. The developed ASC 3D T&D model incorporates many more features in comparison to traditional T&D models. It introduces the wellbore tortuosity [κ], the geometric torsion [τ], and the change in the rate of the curvature [dκ/ds] in the force and moment equilibrium equation. This gives the model robustness to estimate bending effects more realistically, accurately predict the contact forces and solve T&D parameters from surface to TD in reasonable time using standard engineering computers Field Application Results Summary Two field cases have been tested to compare the variation of the ASC 3D T&D model against an industry approved T&D software [Program A]. The input used in Program A are: 132

157 1. Actual field surveys with typical drilling surveys at 90-ft intervals. 2. Semi-analytical surveys generated from the ASC borehole trajectory model using a pseudo-high resolution surveys at one foot intervals [PHR 1 ft]. The absolute value relative variation between the two methods in program A with both inputs are summarized in Table 6.1. Table 6.1: Average of absolute value percentage of variation between the ASC 3D T&D model and Program A [PHR 1-ft and 90-ft survey intervals input] Program A ROB Pickup Slackoff Rotary Torque PHR 1-ft 90-ft PHR 1-ft 90-ft PHR 1-ft 90-ft PHR 1-ft 90-ft HL < 4 < ST < 6 < AL < 3 < Key: HL = hookload; ST = surface torque; AL = axial load Based on the study analysis and field applications, the use of the current industrystandard T&D model produces valid T&D predictions in low tortuosity wellbore. The higher the build angles and more extended the reach, the greater the impact of bending stiffness and the change in curvature. From Table 6.1, input data using the typical 90-ft survey interval shows the least level of accuracy in predicting T&D parameters specifically in pickup and rotary drilling. Higher level of accuracy is observed using the PHR surveys generated at one-foot. The highest level of accuracy is achieved by using the ASC 3D T&D model as it accounts for the change of the rate of curvature and more accurately predicts the drillstring contact force. The field cases demonstrate the challenges during different operations such as rotating off bottom [ROB], rotary drilling [RD], pickup [PU] and slackoff [SO]. Major differences are observed when the wellbore is experiencing high and repeated tortuosity [red box shown in the contact force figure]. The ASC 3D T&D model, shown by the blue line in Figure 6.1, shows a high contact force distribution between the wellbore and the drillstring throughout 133

158 both the highly tortuous zone from 5,000ft [1,525 m] to 8,000ft [2,440 m] and the moderately tortuous zone [the second build] from 10,000 ft [3,050 m] to 12,000 ft [3,660 m]. The ASC 3D T&D model captured the additional drillstring to wellbore contact force.. Figure 6.1: Side force prediction using multiple T&D models. The extended T&D model [the ASC 3D T&D model] provides improved capabilities to predict the contact force along the drillstring in ER and horizontal and vertical wells with high tortuosity. These capabilities enable more accurate prediction of drilling parameters including hookload, torque and axial load. Table 6.2 summarizes the accuracy of different models for specific well categories. The acceptable estimation of variable is subjective to different factors. In this analysis, the model is considered acceptable when the variation of T&D results is less than 6 to 8 % in all basic drilling operations. 134

159 Table 6.2: Accuracy of models in predicting T&D parameters Model Well Categories Low tortuosity Extended reach High tortuosity Soft String Program A - 90 ft Program A - PHR 1 ft ASC 3D T&D Model Key: = acceptable estimation of T&D parameters; = underestimation due to model assumptions Since T&D are critical elements in the drilling of ER and horizontal and vertical wells with high and repeated tortuosity, the models developed in this dissertation are recommended to better estimate the downhole drilling forces and reduce the potential for catastrophic events while drilling, such as stuck pipe, and drillstring fatigue and wear phenomena. 6.2 Wellbore Positioning Effect When drilling ER and complex horizontal wells, the ability to land the wellbore accurately into the target reservoir is a critical step in delivering a well that achieves its long-term production potential. Accurate measurement of the TVD of the well will provide good information about the reservoir geology from subsurface mapping including well-to-well correlation. From the study of the five wells with actual survey data recorded at one survey per foot, the following was observed: given a smooth well profile[low tortuosity], both MCM and ASC show similar accuracy. given a complex and tortuous wellpath, the ASC borehole trajectory model provides higher accuracy level than the MCM. The improvement in accuracy in wellbore positions is up to 50% as shown in Figure 6.2. Table 6.3 summarized the accuracy of wellbore position for different borehole trajectory models at specific well category. 135

160 Figure 6.2: TVD gross error using multiple borehole trajectory models. Table 6.3: Accuracy of models in predicting wellbore position Model Well Categories Low tortuosity Extended reach High tortuosity AATM MCM ASC Model Key: = acceptable estimation of wellbore position; = underestimation due to model assumptions 136

161 Since wellbore positioning accuracy is critical in landing ER and complex horizontal wells, conducting subsurface mapping of reservoirs and maximizing its long-term production potential; it is recommended to use the ASC borehole trajectory model. This model will provide the directional driller with accurate TVD parameters allowing better well landings than current borehole trajectory model predictions. 6.3 Wellbore Tortuosity Effect The ability to predict the actual wellbore tortuosity is a critical factor in successful ER drilling operation. Several wellbore trajectory calculation methods have been proposed to calculate the wellbore tortuosity between the survey-stations. In reality, no mathematical model can predict the exact curvature precisely, however, a robust mathematical model such as the ASC borehole trajectory model can predict the curvature of the wellpath more realistically, as shown in Figure 6.3 where the red line is the tortuosity ground truth, blue is the ASC borehole trajectory model at a typical 90-ft survey resolution and the orange is the MCM at the same survey resolution. The MCM shows negligible measured tortuosity along the entire wellpath while the ASC borehole trajectory model shows better precision. At 4,000-ft and 8,000-ft high tortuosity was recorded. The ASC model shows high tortuosity levels while the MCM was still showing negligible tortuosity. Also at 10,000-ft, the ASC borehole trajectory model shows reasonable accuracy compared to the MCM with large error [16 /100 ft underestimation]. This behavior is expected due to the constant curvature assumption in the MCM. Accurate measures of wellbore tortuosity using the ASC borehole trajectory model not only impact the drilling operation of ER and complex horizontal wells, it also has a strong impact on the well productivity and performance. An extended analysis on the productivity period of the well was conducted as part of the 2016 Externship Program at the Colorado School of Mines. 137

162 Figure 6.3: Tortuosity error comparison between MCM and ASC borehole trajectory model. The analysis involved examining 30 horizontal gas wells that experienced liquid loading. Liquid loading is a production issue caused by the accumulation of liquid in the bottom of the wellbore causing a drop of gas velocity below the critical velocity required to produce the liquid. This is illustrated in Figure 6.4 as the wells on the left are producing the liquid from the well. Once the gas rate drops, less liquid is produced and accumulates in the the bottom of the wellbore. Wellbore tortuosity, expressed in this analysis by the absolute cumulative dogleg [The normalized summation of all the doglegs in the entire wellpath], shows a strong correlation between the early occurrence of liquid loading in gas wells. Thus, wellbore trajectory has an impact on when the well experiences liquid loading. This is supported by the correlation between the higher complexity of the well, with high cumulative dogleg and the early occurrence of liquid loading. 138

163 Figure 6.4: Liquid loading phenomena represented by Khan (2010). Throughout the analysis, a study of the sensitivity for different operating variables affecting liquid loading was conducted. This includes the liquid flow rate, the tubing pressure, and the cumulative dogleg. A variable impact matrix has been prepared by Logan Salewski and Yuyang Li from the Petroleum Engineering Department under the supervision of Dr. Rosmer M. Brito during the 2016 Externship Program, as shown in Table 6.4: Table 6.4: Performance prediction matrix for gas wells experiencing liquid loading Liquid Flow Rate Median Tubing Normalized Cum DL Well Performance Case # (2 bpd Median) pressure (219 psi) (21.5 deg/1000 ft) (894 Days) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 139