I hereby declare that, except where specifically indicated, the work submitted herein is my own original work.

Transcription

1 Finite Element Studies on the Mechanical Stability of Arbitrarily Oriented and Inclined Wellbores Using a Vertical 3D Wellbore Model by Di Zien Low (F) Fourth-Year Undergraduate Project in Group D, 2010/2011 I hereby declare that, except where specifically indicated, the work submitted herein is my own original work. Date: Signature: 1

2 LIST OF CONTENTS PART 1: TECHNICAL ABSTRACT... 5 PART 2: INTRODUCTION Definition of Wellbore Stability Factors Affecting Wellbore Stability Past Works on Developing Wellbore Stability Models Computational Finite-Element Analysis Aims of Project PART 3: THEORY AND DESIGN OF EXPERIMENT Elastic Stress Transformation for and Arbitrarily Oriented Borehole Stresses and Strains in Cylindrical Coordinate General Elastic Solution for a Borehole in Cylindrical Coordinate Wellbore Inclination in Planes Perpendicular to and Finite-Element Wellbore Models Considered Dimensions, Meshes and Elements of the Wellbore Model PART 4: EXPERIMENTAL TECHNIQUES AND PROCEDURES Experimental Data and Local In-Situ Stress Calculations The Geostatic Stage Important ABAQUS Keywords The Drilling Stage Data Extraction and Analysis PART 5: RESULTS AND DISCUSSION Comparing Mathematical and ABAQUS Models under Isotropic Loads Comparison of Results with Data Published by Zhou et al (1996) PART 6: CONCLUSION PART 7: FUTURE WORKS PART 8: REFERENCES PART 9: APPENDIX

3 LIST OF FIGURES 2.1 Borehole Breakout Schematic In-Situ Coordinate System Coordinate System for Deviated Borehole Borehole Orientation and Coordinates Local Borehole Stresses and Strains in Cylindrical Coordinate System Plane Stress Transformation Equations Mohr s Circle of Stress for Plane Stress Transformation Symmetrical Quarter-Vertical Wellbore Model Symmetrical Half-Vertical Wellbore Model Full-Inclined Wellbore Model Dimensions of the Symmetrical Half-Vertical Wellbore Model Mesh and Elements of the Symmetrical Half-Vertical Wellbore Model Partition around the Wellbore Mesh around the Wellbore Node C3D20RP Element Type Inclination Plane for Case #A and Case #B Boundary Conditions at the Geostatic Stage Loads at the Geostatic Stage Cylindrical Coordinate System Drilling Fluid Pressure Supports the Borehole Wall in the Drilling Stage Pore Pressure Applied to Back and Side Surfaces, Others Assumed Impermeable Circumferential Stress Contours of Case 4B (=60 ) at the end of Drilling Stage Case 1A Mathematical Model Case 1A ABAQUS Model Case 2A ABAQUS Model Case 2B ABAQUS Model Case 3A ABAQUS Model Case 3B ABAQUS Model Case 4A ABAQUS Model Case 4B ABAQUS Model Chart Produced by Zhou et al (1996)

4 LIST OF TABLES 4.1 Well Data for All Cases (1A, 2A, 2B, 3A, 3B, 4A and 4B) Far Field Total and Effective Stresses Rock Type and Properties Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 1A Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 2A Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 2B Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 3A Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 3B Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 4A Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 4B Result Figures and Loading Cases

5 PART 1: TECHNICAL ABSTRACT This study aims to define a comprehensive method that can transform a vertical wellbore model into any arbitrarily oriented wellbore model, which can then be used repeatedly to analyse the stability and stress distribution around wellbores at any orientation. The finite element software package ABAQUS will be used to create the symmetrical half-vertical wellbore finite-element model, which will then be used to analyse the stability and stress distribution around wellbores inclined at in planes perpendicular to the far field minimum and maximum principal stresses where = 0 and = 90. The staged approach will be implemented by first bringing the normal and shear stresses as well as pore pressure acting on the block of soil into equilibrium at the Geostatic Stage; a cylindrical borehole will then be removed from the block of soil using model change at the Drilling Stage. Soil properties such as density, elasticity, friction angle, cohesion, dilation, void ratio and pore pressure will be applied to the 3D finite-element wellbore model. The wellbore model will then be transformed and inclined in two different azimuth planes, one perpendicular to the far-field maximum horizontal stress, and the other perpendicular to the far-field minimum horizontal stress. The effects of isotropic and increasing anisotropic far-field horizontal stresses on the wellbore stability and stress distribution around wellbore will also be examined at constant far-field overburden stress. The mathematical model published by Jaeger and Cook (1979) for an arbitrarily oriented wellbore in an isotropic linear elastic material and the research findings of Zhou et al (1996), which suggested that inclined wellbores can be more stable than a vertical well in an extensional stress regime, will be used to verify that the methods proposed in this study to transform a vertical finite-element model into any arbitrarily oriented and inclined wellbore by changing the normal and shear stress applied to the model is acceptable and justifiable. If this research proves to be successful, it may translate into major time savings in future works of wellbore stability analysis as vertical wellbore models are easier to build, partition, mesh, debug and analyse compared to inclined wellbore models. Besides that, there will be no need to create countless finite-element models with wellbores set at different angles to analyse the stress distribution around wellbores at a various inclination and orientation, only one comprehensive vertical wellbore model will be required to test everything. 5

6 This study intends to prove the following hypotheses: 1. Appropriate sets of normal and shear stresses can be applied to the surfaces of a 3D finite-element vertical wellbore model to transform it into an inclined wellbore model; 2. By applying different sets of normal and shear stresses, the same vertical wellbore model can be reused and transformed into wellbores at different inclination and orientation; 3. The wellbore stability and stress distribution around an arbitrarily oriented and inclined wellbore can be analysed using a single 3D finite-element vertical wellbore model. The steps below highlight the experimental procedures taken in this study: Step 1: Identify the magnitudes of the far-field principal stresses Step 2: Determine the azimuth (α) and inclination () angles required for the test Step 3: Calculate local in-situ normal and shear stresses base on required α and Step 4: Build the 3D finite-element vertical wellbore model using ABAQUS Step 5: Apply material properties, boundary conditions to the wellbore model Step 6: Referring to the stresses calculated in Step 3, apply loads to the model Step 7: Run Geostatic Stage, check that stresses are in equilibrium and displacement is zero - If failed, return to Steps 4, 5, 6 and 7 to debug the problem, repeat until OK Step 8: Run Drilling Stage, remove borehole using Model Change and apply mud pressure - If failed, return to Steps 4, 5, 6, 7 and 8 to debug the problem, repeat until OK Step 9: Extract Data from the nodes around the wellbore using Field Output Requests Step 10: Repeat Steps 2 9 to analyse wellbores at different azimuth (α) and inclination () The results show that the mathematical theory published by Jaeger and Cook (1979) and finite-element result agrees that a vertical wellbore is most stable under isotropic horizontal stress. Under the same loading, both analysis also agree with each other that maximum stress anisotropy occurs at = 90, i.e. when the wellbore is horizontal. Besides that, the result published by Zhou et al (1996) agrees quite well with the results obtained in this study. For starters, vertical wellbores will always be the most stable under isotropic conditions. Then, as the stress horizontal stress anisotropy increases, the value of increases as well, which then increases the required inclination angle to give the most stable wellbore. Overall, the hypotheses were proven to be successful, but more work can still be done by creating a full vertical wellbore model to test the wellbore transformation in other azimuth planes. 6

7 PART 2: INTRODUCTION The stability of a wellbore is one of the major issues surrounding drilling operations. It is the main cause of non-productive time during drilling operations and costs the oil and gas industry worldwide more than USD 6 billion annually. 1 Before a well is drilled, subsurface rocks are under well-balanced stress conditions with the three in-situ principal stresses: vertical, horizontal maximum and horizontal maximum, which also produce shear stresses on planes within the rock mass. 2 However, this equilibrium state will change when cylindrical sections of rocks are removed at the drilling stages and replaced by drilling fluids at specific mud weights to provide temporary support. If the wellbore is not supported, the rock formation around the wellbore might become unstable and collapse. According to Kang et al (2009), drilling fluid can only partially support the normal stresses on the wellbore wall but not the shear stresses, as the original rock does, and this result in stress concentration and redistribution around the wellbore. In such cases, rock failures or breakouts will occur at the fractured zones of the borehole wall where the tangent of the circular borehole is parallel to the direction of the maximum horizontal stress 3, as shown in Figure 2.1. The greater the stress anisotropy between the maximum and minimum horizontal stresses, the more serious the breakouts will be. These rock deformations redistribute the stress around the borehole until a new equilibrium is achieved. Figure 2.1: Borehole Breakout Schematic 3 Wellbores can be drilled vertically, inclined, or horizontally (i.e. 90 inclination) and it has been widely recognised that highly deviated, extended-reach and horizontal wells can offer economic benefits through lower development costs, faster production rates, and higher recovery factors. 4 Since the mid-20 th century, the use of controlled directional drilling technology has increased dramatically to reach otherwise inaccessible reserves. 5 However, inclined and horizontal wells may be prone to mechanical instability in high in-situ stresses. 6 7

8 2.1 Definition of Wellbore Stability According to Kang et al (2009), a wellbore is considered to be stable when its diameter equals to the drill-bit diameter and remains the same shape for an extended period of time. Besides that, Kang et al (2009) also stated that wellbore instability can be considered as a function of how rocks respond to the induced stress concentration around the wellbore during various drilling activities. However, the general understanding is that as long as the borehole surface remains intact and does not collapse during and after drilling operations, the wellbore can be considered as stable. On the other hand, Zhou et al (1996) suggested the concept of minimum stress anisotropy around a wellbore when the research group tries to identify the optimum drilling direction and inclination angle under various magnitudes of far-field principal stresses. This concept appears to be more appropriate and will be adopted in this study to assess the stability of wellbores across various orientations. Hence, the definition of a stable wellbore will be one that has the minimum stress anisotropy around its borehole surface. 2.2 Factors Affecting Wellbore Stability 2 Wellbore instability is a very complicated phenomenon as various factors can affect the stress distribution around the wellbore. Many wellbore stability models were presented in the past to address the wellbore stability issue. Most of the existing models examine one or a combination of two or more of the following effects on wellbore stability: mechanical, hydraulic, chemical, and thermal. A comprehensive study regarding the major factors affecting wellbore stability has been done by Kang et al (2009) and is summarised as follow: (a) Rock Mechanical Properties Rock mechanical properties such as Young s modulus, Poisson s ratio, Biot s constant, rock porosity, permeability, bulk density, cohesive strength, tensile strength, and internal friction etc. are among the parameters that can affect borehole behaviour. Although rock properties cannot be controlled by drilling engineers, a good knowledge of the rock mechanical properties can help well planners predict the stability of a wellbore and minimise risk by choosing a suitable well path before drilling. 8

9 (b) Far-field Principal Stresses The in-situ principal stresses are calculated from the three far-field principal stresses: overburden stress, maximum horizontal stress and minimum horizontal stress. For wellbore stability analysis, the overburden stress is relatively easy to get from density logs, but a lot of work needs to be done to determine the magnitude and direction of the maximum and minimum horizontal stresses in the field. It has been observed in the field that wellbores deform differently under different horizontal stress conditions. This clearly shows that good understanding of the magnitude and direction of far-field principal stresses are important to analyse the stability of a wellbore. (c) Wellbore Trajectory In order to transform the far-field principal stresses into the in-situ stresses around a wellbore, the azimuth and inclination angle of a wellbore are required. Normal and shear stresses acting on the rock formation in the near-wellbore region are functions of the azimuth and inclination, and this will be discussed in detail in Part 3 of this report. Unlike rock properties and far-field principal stresses, wellbore trajectory is something that can be determined by the drilling engineer. Careful design of wellbore trajectory during well planning may help to avoid any borehole failure in the drilling operations. 7 (d) Pore Pressure Based on the effective stress concept, stresses acting on the rock formation in the nearwellbore region are partially supported by the pore pressure within the rock formation. Any changes in the pore pressure can affect the stress state around the wellbore. Hence, accurate pore pressure prediction is important in wellbore stability analysis as well. (e) Mud Weight Drilling fluids within specific mud weights are used to provide temporary support to the borehole walls in drilling operations. Drilling fluids need to be heavy enough to support the borehole and prevent it from collapsing. A common practice in the field to maintain wellbore stability is by increasing the mud weight. However, the mud weight must not be too high as heavy drilling fluid may exert too much hydrostatic pressure on the borehole wall and cause the rock formation to fracture. 9

10 (f) Drilling Fluid and Pore Fluid Chemicals Due to the different composition and concentration of chemicals in the drilling fluid and pore fluid, a chemical potential difference can be generated between these two fluids while drilling through rock formation. Such chemical potential difference can cause fluid to flow in or out of the pores and result in pore pressure redistribution. The effects due to chemical potential difference can be neglected for highly permeable formations such as sandstone, but is a major concern for low permeability formations such as shale. This is because the induced fluid flow can generate significant pore pressure propagation and redistribution in low permeability formations. This is also one of the reasons why the drilling fluid composition and concentration are closely monitored and controlled in the field to prevent shale formations from failure. (g) Temperature As drilling operations extend to deep wells, temperature differences of about 60 C to 70 C between the circulated drilling fluid and the rock formation can result in tens of MPa of thermally induced stresses in the rock formation. 8 In addition to the thermally induced stresses, temperature can also change the chemical potential in the drilling fluid and pore fluid, resulting in transient fluid flow in the near-wellbore region. (h) Time Chemical and thermal effects can cause pore pressure propagation in the rock formation and thus redistributes the stresses around the wellbore over a period of time. Hence, wellbore stability is also time-dependent phenomenon. This study intends to incorporate wellbore stability factors (a) to (e) into a 3D finite-element wellbore model subjected to isotropic and anisotropic far-field principal stresses to examine local in-situ stress distribution around the wellbore. Vertical overburdened stress and mud weight will remain constant and the in-situ stresses around the wellbore will be observed at various inclination angles in two azimuth planes. The computational result will then be compared to the mathematical model published by Jaeger and Cook (1979) for an arbitrarily oriented borehole, which assumes plane strain condition normal to the borehole axis, as well as by Zhou et al (2009), to gain better understanding about wellbores stability issues. 10

11 2.3 Past Works on Developing Wellbore Stability Models A lot of work has been done since Bradley (1979a, 1979b) towards the determination of the magnitude and orientation of in-situ stress in around a wellbore. Bradley presented a mathematical model which calculates the elastic stress distribution around a wellbore under plane strain condition. 9 This simplifying assumption allowed Bradley to use Drucker-Prager failure criterion to evaluate the shear failure of the formation at the borehole wall. 10 The elastic wellbore stability models only takes into account the far field mechanical principal stresses around a wellbore and the hydrostatic pressure exerted by the drilling fluid onto the wellbore surface, but not the pore pressure in the rock formation, to minimise input variables and to simplify calculations. However, lab experimental results and field experience indicate that pore pressure does affect stress distribution around the wellbore and instability could also occur in the near-wellbore region, not only on the wellbore surface. Hsiao (1988), Yew and Li (1988), Zhou et al (1996) are among the researchers who have extended Bradley s elastic model to study the wellbore stability of deviated boreholes under different stress conditions. Hsiao (1988) used the theory to investigate the stability of a horizontal well. 11 Yew and Li (1988) developed a 3D elastic model to study the fracture failure of a deviated well. 12 Zhou et al (1996) pointed out that in an extensional stress regime, i.e. when the maximum principal stress is the overburden stress, a deviated wellbore can be more stable than a vertical well. The study also suggested that wellbores parallel to the minimum horizontal principal stress direction have the least possible failure, and at this direction, the most stable wellbore inclination can be determined by the ratio of the maximum horizontal principal stresses to the vertical stresses. The higher the ratio, the greater the inclination angle from the vertical will be to achieve minimum stress anisotropy around the wellbore, hence a more stable wellbore. The extended mathematical models of Bradley (1979a, 1979b) that were published by Jaeger and Cook (1979) and Zhou et al (2009) will be used to verify that the method used in this study to transform a 3D vertical finite-element wellbore model into any arbitrarily oriented wellbore is correct, and to justify that the stress distribution around the wellbore generated by this transformed finite-element wellbore model is acceptable. 11

12 2.4 Computational Finite-Elements Analysis With the increase in computational power, development of sophisticated software and the reduction in costs, the extensive use of computational techniques such as finite-element methods to model wellbore stability has increased in recent years. The latest finite-element software packages allow one to combine the mathematical theories of solid mechanics such as elasticity, plasticity, poroelasticity and viscoelasticity, with different failure theories and wellbore components to create more realistic and sophisticated models. 13 The finite-element model may exhibit linear or non-linear behaviour and can still be tested and calibrated using experimental data to give realistic visual representation and analysis about what is actually happening around the wellbore. The ability of finite-element software to generate stress contours and displacement plots at incremental time frames throughout the drilling operation allows the computational analysis to be pieced together to create a continuous video, which can then be used to provide better feel and insights into how and why the wellbore actually deforms, which may otherwise be impossible. Over the past ten years, with the rapid development of robust non-linear finite-element method techniques that are suitable to analyse complex geomechanics, soil mechanics and rock mechanics, the finite-element software package ABAQUS (SIMULIA) 14 has successfully been used to perform finite-element analysis to analyse horizontal-casing integrity 15, cement sheath integrity 16, sand production 17, hydraulic fracturing 18 and many more. While some researchers prefer to concentrate on analysing the stress state at a particular stage of life-of well without considering the previous loading and deformation history, Gray et al (2007) suggested that the staged approach should be used and stated that, The staged approach imitates construction of the well, following all or some of the development stages such as drilling, casing, cementing, completion, hydraulic fracturing, and production. By use of the finite-element method, the stress state at each stage is modelled, and stage variables such as the amount of damage and plasticity along with loading and boundary conditions are transmitted to the model for the next stage. This technique eliminates the need to guess the initial state of stress, amount of plasticity and damage. This study will use and adapt the staged approach proposed by Gray et al (2009) to analyse the stability of a wellbore. 12

13 2.5 Aims of Project This study will focus on creating a vertical wellbore model using the ABAQUS finite-element software package, and then establishing a comprehensive method to transform this vertical model into any arbitrary oriented and inclined wellbore model. This allows the same vertical wellbore model to be used repeatedly to examine the stability and stress distribution around wellbores at any orientation by changing the magnitude and direction of local in-situ normal and shear stress acting on the vertical wellbore model only. The staged approach will be implemented by first bringing the normal and shear stresses as well as pore pressure acting on the block of soil into equilibrium at the Geostatic Stage; a cylindrical borehole will then be removed from the block of soil using model change at the Drilling Stage. Soil properties such as density, elasticity, friction angle, cohesion, dilation, void ratio and pore pressure will be applied to the 3D finite-element wellbore model. The wellbore model will be transformed and inclined in two different azimuth planes, one perpendicular to the far-field maximum horizontal stress, and the other perpendicular to the far-field minimum horizontal stress. The effects of isotropic and increasing anisotropic far-field horizontal stresses on the wellbore stability and stress distribution around wellbore will also be examined at constant far-field overburden stress. The mathematical model published by Jaeger and Cook (1979) for an arbitrarily oriented wellbore in an isotropic linear elastic material and the research findings of Zhou et al (1996), which suggested that inclined wellbores can be more stable than a vertical well in an extensional stress regime, will be used to verify that the methods proposed in this study to transform a vertical finite-element model into any arbitrarily oriented and inclined wellbore by changing the normal and shear stress applied to the model is acceptable and justifiable. If this research proves to be successful, it may translate into major time savings in future works of wellbore stability analysis as vertical wellbore models are easier to build, partition, mesh, debug and analyse compared to inclined wellbore models. Besides that, there will be no need to create countless finite-element models with wellbores set at different angles to analyse the stress distribution around wellbores at a various inclination and orientation, only one comprehensive vertical wellbore model will be required to test everything. 13

14 PART 3: THEORY AND DESIGN OF EXPERIMENT 3.1 Elastic Stress Transformation for an Arbitrarily Oriented Borehole 19,20 The in-situ principal stresses define the coordiante system (x, y, z ) as shown in Figure 3.1, where is taken to be parallel to z, to be parallel to x and to be parallel to y. Then, a second coordinate system (x, y, z) is introduced to transform the in-situ coordinate system. The z-axis points along the axis of the hole, the x-axis points towards the lowermost radial direction direction of the hole, and the y-axis horizontal is horizontal (see Figure 3.2). Figure 3.1 In-situ coordinate system Figure 3.2 Coordinate system for deviated borehole A transformation from (x, y, z ) to (x, y, z) can be obtained in two operations, as shown in Figure 3.3: 1. A rotation α around the z -axis, 2. A rotation around the y-axis. Figure 3.3: Borehole orientation and coordinates This enables transformation from insitu principal stresses (σ v, σ H, σ h ) into local borehole coordinate system stresses (σ o x, σ o y, σ o z, τ o yz, τ o xz, τ o xy ) mathematically by using Equation (1) 14

15 According to Jaeger and Cook (1979), for an arbitrarily oriented borehole shown in Figure3.3, the rotation of the stress tensor from the global in-situ coordinate system (σ v, σ H, σ h ) to a local borehole Cartesian coordinate system (σ x o, σ y o, σ z o, τ yz o, τ xz o, τ xy o ) is given by { }...(1) { } { } Note: Superscript o on the stresses denote that these are virgin formation stresses. 3.2 Stresses and Strains in Cylindrical Coordinate To examine the stresses and strains in the rock surrounding a borehole, it will be convenient to express the local borehole Cartesian coordinates system derived from the in-situ principal stresses in Equation (1) into local cylindrical coordinates (r, θ, z). The cylindrical coordinate stresses and strains at a point in a plane perpendicular to the z-axis are shown in Figure 3.4 Figure 3.4: Local Borehole Stresses and Strains in Cylindrical Coordinate System r 3.3 General Elastic Solution for a Borehole in Cylindrical Coordinate (Fjaer et al, 2008) Assuming plane strain normal to the borehole axis in the local cylindrical coordinates (r, θ, z) where r represents the distance from the borehole axis, θ the azimuth angle relative to the x-axis, and z is the position along the borehole axis, for an arbitrary borehole with radius R w, excess fluid pressure p w acting on the surface of the borehole wall, and formation Poisson s ratio v fr, the general elastic solution for (σ r, σ θ, σ z, τ rθ, τ θz, τ rz ) can be written as follow 4 : ( ) ( ) ( )...(2) 15

16 ( ) ( ) ( )...(3) * ( ) +...(4) ( ) ( )...(5) ( ) ( )...(6) ( ) ( )...(7) The borehole influence is given by the terms in and, which vanish rapidly with increasing radial distance from the borehole axis. The general elastic solutions depend on angle, indicating that the stresses vary with position around the wellbore. Generally, the shear stresses are non-zero. Thus, and are not principal stresses for arbitrary orientations of the well. At the borehole surface ( = ), the equations can be simplified to:...(8) ( )...(9) ( )... (10)... (11) ( )... (12)... (13) where = in-situ effective vertical principal stress = in-situ effective major horizontal principal stress = in-situ effective minor horizontal principal stress = Poisson s ratio of the formation = angle between and the projection of the borehole axis onto the horizontal plane = angle between the borehole axis and the vertical direction = polar angle in the borehole cylindrical coordinate system = excess fluid pressure in the borehole = mud pressure less pore pressure in formation = stress tensor in the local borehole Cartesian coordinate system = stress tensor in the local borehole cylindrical coordinate system 16

17 3.4 Wellbore Inclination in Planes Perpendicular to and Special conditions exist when the wellbore rotates in planes perpendicular to the far-field minimum principal axis at = 0 and in planes perpendicular to the far-field maximum principal axis at = 90. The following equations are derived from Equation (1). At = 0 and for any inclination angle { }... (14) Quick Check: σ y τ yz σ { } { } τ xy At = 90 and for any inclination angle Quick Check: { }... (15) σ y τ yz σ H { } { } τ xy This analysis shows that when the wellbore is inclined in the planes perpendicular to the farfield minimum and maximum principal axis at = 0 and = 90, the local in-situ shear stress that acts on the wellbore model only exist in the X-Z plane, i.e. within the same plane where the wellbore is inclined. Shear stresses in the X-Y and Y-Z plane will be zero. This may be useful when it comes to identifying a suitable finite-element model for this study. Figure 3.5: Plane Stress Transformation Equations 17

18 If the wellbore is inclined in the plane perpendicular to the far-field minimum principal axis ( = 0 ), it can be assumed that =, = and = 0. Then, from Figure 3.5 and using the equilibrium of an elementary triangle, the following equations can be derived: Similar equations can be derived if the wellbore is inclined in the plane perpendicular to the far field maximum principal axis ( = 90 ). If the 0, the equations will look like this:... (16)... (17)... (18) According to Crandall et al (1972), these equations can be further simplified into graphical representation, i.e. the Mohr s circle of stress. 21 By first applying double-angle trigonometric relations to Equations (16), (17) and (18), the Mohr s circle of stress can be then be drawn.... (19)... (20)... (21) Figure 3.6: Mohr s Circle of Stress for Plane Stress Transformations τ σ aa τ ab 2 σ xx τ xy σ σ yy - τ xy σ bb - τ ab 18

19 3.5 Finite-Element Wellbore Models Considered Figure 3.7: Symmetrical Quarter-Vertical Wellbore Model In this study, shear stress needs to be applied to the surfaces of the model to create an inclined wellbore. However, the symmetrical quarter-vertical model faces complications as shear stress do not form complete loops around the model. This model is only suitable to analyse vertical or horizontal wellbores that is perpendicular to, and. Figure 3.8: Symmetrical Half-Vertical Wellbore Model The half-vertical wellbore model allows shear stress to a make complete loop around the X-Z plane, i.e. the vertical wellbore can be rotated in the X-Z plane. However, the model is limited to rotate perpendicularly to or as these are the only directions where shear stress around X-Y and Y-Z plane are zero. Otherwise, a full model will be required. Full-Vertical Wellbore Model This combines two symmetrical half-vertical wellbore model to create a block of soil with a cylindrical vertical wellbore drilled through the middle. The full vertical wellbore model, allows the wellbore model to be transformed into any orientation and inclination as shear stresses can form complete loops around any of the X-Y, X-Z and Y-Z planes, without the shear stress symmetrical issues that exist for the quarter and half vertical wellbore models. However, a full vertical wellbore model contains twice as many element as the half model. This means that a lot more resources, time and computing power is required to run the finite-element analysis, which are the main things that this research study seriously lack of. 19

20 Figure 3.9: Full-Inclined Wellbore Model The full-inclined wellbore model is an alternative to analyse the wellbore stability of an arbitrarily oriented and inclined wellbore. No stress transformation calculation will be required for this model as the far-field stress that acts normally to the block surfaces and the cylindrical wellbore that has been carefully created at a specific angle will do the job. It must be recognised that an inclined wellbore can only analyse wellbore stability and stress distribution around a wellbore at one inclination only. A lot of inclined wellbore models may need to be created if various analyses at different orientation and inclination are required. Hence, it is fair to say that inclined wellbore models are not as reusable as vertical wellbore models that make use stress transformations. Also, because local in-situ normal and shear stresses can be calculated from the far-field principal stresses base on the desired azimuth and inclination angles, and these local in-situ normal and shear stresses can then be applied easily to the surfaces of the same vertical wellbore model by simple change of numbers to create any desired orientation and inclination, only one comprehensive vertical wellbore model that has been properly built and thoroughly checked will be required. As one of the aims of this study is to establish a comprehensive method that can transform a vertical wellbore model into any arbitrary oriented and inclined wellbore model, as well as considering the limited resources, time and computing power, the symmetrical half-vertical wellbore finite-element model will be used to analyse the stability and stress distribution around wellbores inclined at in rotational planes perpendicular to and. 20

21 3.6 Dimensions, Meshes and Elements of the Wellbore Model Figure 3.10: Dimensions of the Symmetrical Half-Vertical Wellbore Model 0.5m 0.5m 0.5m 0.5m 0.5m 0.5m 1m R1 R2 R3 1m 2m Figure 3.11: Mesh and Elements of the Symmetrical Half-Vertical Wellbore Model 21

22 Figure 3.12: Partition around the Wellbore # Figure 3.13: Mesh around the Wellbore R3 = 0.35m Borehole Boundary Borehole Boundary R2 = m R1 = 0.06m Two additional concentric cylindrical partitions will be made to facilitate the convergence of mesh lines towards the centreline of the borehole, as well as to ensure that the number and size the elements can be controlled more effectively throughout the wellbore model. Element type C3D20RP will be assigned to and used in the entire wellbore model. According to the ABAQUS Analysis User s Manual, each C3D20RP element is a 20-node brick that analyse triquadratic displacements, trilinear pore pressures, with reduced integration. This will do the job of analysing pore pressures, stresses and displacements in the model. The model will be 10 element-layers thick and have finer elements around the vicinity of the borehole. Figure 3.14: 20-Node C3D20RP Element Type There will be 20 elements per 180 arc of the borehole and the overall layout of the mesh and relative size of the elements can be found in Figures 3.11 and A rough calculation suggests that the wellbore model will have a total of 6280 elements and nodes. The soil parameters for the model, the displacement and pore pressure boundary conditions, and the direction and magnitude of the stresses that needs to be applied on which surface of the wellbore model and at what stage will be discussed in detail in the next section. 22

23 PART 4: EXPERIMENTAL TECHNIQUES AND PROCEDURES As discussed earlier, this study aims to define a comprehensive method that can transform a vertical wellbore model into any arbitrarily oriented wellbore model, which can then be used repeatedly to analyse the stability and stress distribution around wellbores at any orientation. With the considerations of limited resources, time and computing power, the symmetrical half-vertical wellbore finite-element model will be used to analyse the stability and stress distribution around wellbores inclined at in planes perpendicular to the far field minimum and maximum principal stresses where = 0 and = 90. This study intends to prove the following hypotheses: 4. Appropriate sets of normal and shear stresses can be applied to the surfaces of a 3D finite-element vertical wellbore model to transform it into an inclined wellbore model; 5. By applying different sets of normal and shear stresses, the same vertical wellbore model can be reused and transformed into wellbores at different inclination and orientation; 6. The wellbore stability and stress distribution around an arbitrarily oriented and inclined wellbore can be analysed using a single 3D finite-element vertical wellbore model. The steps below highlight the experimental procedures taken in this study: Step 1: Identify the magnitudes of the far-field principal stresses Step 2: Determine the azimuth (α) and inclination () angles required for the test Step 3: Calculate local in-situ normal and shear stresses base on required α and Step 4: Build the 3D finite-element vertical wellbore model using ABAQUS Step 5: Apply material properties, boundary conditions to the wellbore model Step 6: Referring to the stresses calculated in Step 3, apply loads to the model Step 7: Run Geostatic Stage, check that stresses are in equilibrium and displacement is zero - If failed, return to Steps 4, 5, 6 and 7 to debug the problem, repeat until OK Step 8: Run Drilling Stage, remove borehole using Model Change and apply mud pressure - If failed, return to Steps 4, 5, 6, 7 and 8 to debug the problem, repeat until OK Step 9: Extract Data from the nodes around the wellbore using Field Output Requests Step 10: Repeat Steps 2 9 to analyse wellbores at different azimuth (α) and inclination () 23

24 4.1 Experimental Data and Local In-Situ Stress Calculations The example problem published by Gray et al (2007) will be used and adapted in this study Table Well Data for All Cases (1A, 2A, 2B, 3A, 3B, 4A and 4B) True Vertical Depth of the Well ft m Estimated Hole Diameter Being Drilled 9.5 inch m Overburden Pressure Gradient 1 psi/ft kpa/m Pore Pressure Gradient 0.62 psi/ft kpa/m Table Far-Field Total and Effective Stresses Case 1A 2A, 2B 3A, 3B 4A, 4B Unit Total Vertical Stress, kpa Total Horizontal Maximum Stress, kpa Total Horizontal Minimum Stress, kpa Pore Pressure, kpa Effective Vertical Stress, kpa Effective Horizontal Maximum Stress, kpa Effective Horizontal Minimum Stress, kpa Difference Between Effective Horz Stress, kpa K 1 = / = K 2 = / K 3 = / = Average Effective Horz Stress, ( ) / kpa All cases have the same vertical and average horizontal stresses Case 1A: Isotropic Horizontal Stresses, well inclination in α = 0 azimuth plane only Case 2A: Anisotropic Horizontal Stresses, well inclination in α = 0 azimuth plane Case 2B: Anisotropic Horizontal Stresses, well inclination in α = 90 azimuth plane Case 3A: Larger Anisotropic Horizontal Stresses, well inclination in α = 0 azimuth plane Case 3B: Larger Anisotropic Horizontal Stresses, well inclination in α = 90 azimuth plane Case 4A: Largest Anisotropic Horizontal Stresses, well inclination in α = 0 azimuth plane Case 4B: Largest Anisotropic Horizontal Stresses, well inclination in α = 90 azimuth plane 24

25 Figure 4.1: Inclination Plane for Case #A and Case #B σ v Inclination Plane for Ca e #A Inclination Plane for Ca e #B σ H σ α ) α ) Table Rock Type and Properties Rock Type Young's Modulus, E (kpa) Poisson's Ratio, v Friction Angle, φ ( ) Cohesive Strength, c (kpa) Dilation Angle, ψ ( ) Density, ρ (kg/m 3 ) Permeability, k (m/s) Void Ratio R3c3 2.70E E E Table 4.4 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 1A σ σ = σ total τ kpa σ τ R = 5492 τ σ H = σ v = σ 2 C = σ kpa σ - τ τ Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa 25

26 Table 4.5 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 2A τ kpa σ σ = σ total σ H = R = 4323 C = σ τ σ v = σ kpa τ σ σ - τ τ Azimuth Plane, α degree Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa Table 4.6 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 2B τ kpa σ σ = σ H total σ = R = 6661 C = σ τ σ v = σ kpa τ σ σ - τ τ Azimuth Plane, α degree Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa 26

27 Table 4.7 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 3A τ kpa σ σ = σ total σ H = R = 3768 C = σ τ σ v = σ kpa τ σ σ - τ τ Azimuth Plane, α degree Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa Table 4.8 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 3B τ kpa σ σ = σ H total σ = R = C = σ τ σ v = σ kpa τ σ σ - τ τ Azimuth Plane, α degree Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa 27

28 Table 4.9 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 4A τ kpa σ σ = σ total σ H = σ τ R = 2045 σ v = C = σ kpa τ σ σ - τ τ Azimuth Plane, α degree Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa Table 4.10 Stresses Applied to ABAQUS Model at Various Well Inclinations for Case 4B τ kpa σ σ = σ H total σ = R = 8939 C = σ τ σ v = σ kpa τ σ σ - τ τ Azimuth Plane, α degree Well inclination, degree = C R cos(2) kpa = kpa = C + R cos(2) kpa = R sin(2) kpa Pore pressure, u kpa Effective kpa Effective kpa Effective kpa 28

29 4.2 The Geostatic Stage This is the stage where the external loads acting on the finite-element soil block model are being brought into equilibrium with internal loads, pore pressures and boundary conditions. This is very important because it creates a finite-element soil block model that has similar characteristics to the soil that is pre-stressed underground before a wellbore is being drilled. Before the loads are being applied, boundary conditions such as symmetrical faces, base rollers and pore pressures need to be applied to the finite-element soil block model. The boundary conditions applied to the model have been indicated in Figure 4.2 below. The external loads will be the in-situ normal and shear stresses calculated in Step 3 of the experimental procedures, i.e. the total stresses, and in Tables 4.4 to These external loads will then need to have equal magnitude but opposite internal loads and pore pressures to keep the forces in equilibrium. The internal loads are Effective, Effective and Effective in the same tables. Assuming undrained conditions, internal and external shear stresses will be but in opposite directions. The surfaces and direction in which these normal and shear stresses act upon are shown in Figure 4.3 below. Figure 4.2: Boundary Conditions at the Geostatic Stage Pore Pressure Assigned to the Whole Model Top Free Surface Back Free Surface Right Side Free Surface Left Side Free Surface Y Z X Front Symmetrical Surface (YSYMM) Bottom Rollers ( U3 = 0 ) Centreline Anti-Slide Pins ( U1 = 0 ) Note: U1 = Displacement in x-direction, U3 = Displacement in z-direction 29

30 Figure 4.3: Loads at the Geostatic Stage τ 13 σ 33 Top, Right τ 13 σ 11 σ 11 Z X τ 13 Bottom, Left τ 13 Note: is being applied perpendicularly onto the back surface of the model 4.3 Important ABAQUS Keywords Figure 4.4: Cylindrical Coordinate System The origin (0, 0, 0) will be positioned at the base of the model with the z-axis placed along the centreline of the cylindrical borehole. ABAQUS is capable of transforming Cartesian coordinates automatically into cylindrical coordinates. For SYSTEM=CYLINDRICAL, points a and b lie on the polar axis of the cylindrical system. The local axes are: 1 = radial, 2 = circumferential, 3 = axial. To assign the cylindrical coordinate system to the model, the following lines into the Parts section of the ABAQUS input file: * ORIENTATION, NAME=R-THETA, SYSTEM=CYLINDRICAL 0, 0, 0, 0, 0, 1 3, 0 1 st Data Line: X-coordinate of point a, Y-coordinate of point a, Z-coordinate of point a, X-coordinate of point b, Y-coordinate of point b, Z-coordinate of point b 2 nd Data Line: Local direction about which additional rotation is given (3 = axial), Additional rotation defined by a single scalar value. Default = 0. 30

31 After the coordinate system has been set, initial pore pressure, void, internal normal and shear stresses at the beginning of the Geostatic step can be assigned to the model using the ABAQUS input file as well. The following lines can be added after Assembly, before Materials: * INITIAL CONDITIONS, TYPE=PORE PRESSURE NODESET, * INITIAL CONDITIONS, TYPE=RATIO NODESET, 0.33 Note: Initial Pore Pressure and Void Ratio are assumed to be uniform throughout the 1m thick soil block model * INITIAL CONDITIONS, TYPE=STRESS ELEMENTSET, , , , 0, , 0 Data Line: Element Set Name, - Effective, - Effective, - Effective, -, -, - (The initial internal stress example uses Case 3A data for α = 0 and = 30 from Table 4.7) Once the 3D finite-element soil block model has been created, with the cylindrical borehole partitioned, the local in-situ normal and shear stresses applied, the whole model properly meshed with the assigned element types, and the important ABAQUS keywords typed into the input file, the model is ready to be submitted, run and analysed by the finite-element program. If the model and its input parameters are correct, this Geostatic Stage will take only a few minutes to be analysed as it is just a one-step time increment analysis. If the finite-element program finds a problem with the soil block model, the analysis will be aborted. This means that Steps 4, 5, 6 and 7 of the experimental procedures need to be revisited to find out what actually went wrong with the finite-element soil block model, and whether all elements, parameters and loads have been defined correctly. After a few trials and corrections, the analysis may eventually be completed. At this point, the analysis result will have to be thoroughly checked to ensure that everything is correct. All the stresses must be in equilibrium and there should be zero displacement or distortion throughout the whole model. The maximum tolerance to the displacement will be m. If the external stresses are not in equilibrium with the pore pressure and internal stresses, the soil block model will be distorted in some way. Steps 4, 5, 6 and 7 of the experimental procedures will need to be revisited again until everything meets the requirement. 31

32 4.4 The Drilling Stage In the staged approach analysis, it is extremely important that results from each stage are being thoroughly checked and verified before feeding them into subsequent stages for further analysis. This means that if minor problems are not dealt with in the Geostatic Stage, it can expected that the problems will snowball into larger issues and the final result at the end of the Drilling Stage will not be accurate at all. Therefore, pore pressure, external and internal stresses must be in equilibrium at the end of Geostatic Stage and the soil block model must have zero distortion before the Drilling Stage can commence. All the displacement boundary conditions and in-situ normal and shear stresses applied on the wellbore model surfaces are propagated from the Geostatic Stage to the Drilling Stage. At the end of the Geostatic Stage and before the start of the Drilling Stage, three important changes will need to take place to change the soil block model into a wellbore model: (i) Removing the cylindrical borehole section from the soil block model to represent drilling operations using the Model Change Interaction function available in ABAQUS; (ii) Replacing the removed cylindrical borehole section with drilling fluid which exerts hydrostatic pressure onto the borehole surface by applying a mechanical pressure load onto the borehole surface, where the magnitude will be the same as pore pressure within the formation to simplify wellbore stability analysis; (iii) Reapplying the same magnitude of pore pressure to the left, right and back surfaces only to make them permeable, which also make the top, bottom, front symmetrical and borehole wall surfaces impermeable, to further simplify the analysis by assuming that fluid does not flow into the borehole and affect its stability in the drilling stage. Once these three changes have taken place at the end of the Geostatic Stage and before the start of the Drilling Stage, the soil block model now becomes a wellbore model. The Drilling Stage analysis can then be started as the borehole deforms under the existing in-situ normal and shear stresses propagated from the Geostatic Stage. The Drilling Stage analysis will take a longer time, ranging from 20 minutes to a few hours, depending on the far-field stress anisotropy and the elastic or plastic distortion of the borehole elements. 32

33 Figure 4.5: Drilling Fluid Pressure Supports the Borehole Wall in the Drilling Stage Figure 4.6: Pore Pressure Applied to Back and Side Surfaces, Others Assumed Impermeable Similar to the Geostatic Stage, ABAQUS might find problems with the wellbore model and decides to abort the analysis. Steps 4 to 8 of the experimental procedures may need to be revisited to debug the problem and rerun the analysis until it can be completed successfully. 33

34 4.5 Data Extraction and Analysis The following Field Output Requests will be used across the Geostatic and Drilling Stages: E - Total Strain Components EE - Elastic Strain Components PE - Plastic Strain Components PEEQ - Equivalent Plastic Strain POR S U VOIDR - Pore Pressure - Stress Components and Invariants - Translations and Rotations - Void Ratio The advantages of using ABAQUS is that any of the Field Outputs Requests information can be extracted from any node point or any element in the finite-element wellbore model at any time frame across the Geostatic and Drilling Stages. For example, the effective hoop stress σ θ around the wellbore surface halfway between the top and bottom surfaces can be easily extracted using the Path function in ABAQUS, as shown in Figure 4.6. The middle layer was chosen because it is the furthest layer away from any edge effects induced by the shear stresses acting on the top and bottom surfaces. All the information collected from the nodes along the Path can then be saved, plotted as graphs, compared and analysed. Besides that, a more general representation of the result can be displayed as coloured contours as well. Figure 4.7: Circumferential Stress Contours of Case 4B ( = 60 ) at the end of Drilling Stage Edge Effects Path (Node List) 34

35 PART 5: RESULTS AND DISCUSSION The aim of this study is to define a comprehensive method that can transform a vertical wellbore model into any arbitrarily oriented wellbore model, which can then be reused to analyse the stability and stress distribution around wellbores at any orientation by changing the stresses acting on the model. Hence, this study examined the following hypotheses: Hypothesis 1: Appropriate sets of normal and shear stresses can be applied to the surfaces of a 3D finiteelement vertical wellbore model to transform it into an inclined wellbore model; Hypothesis 2: By applying different sets of normal and shear stresses, the same vertical wellbore model can be reused and transformed into wellbores at different inclination and orientation; Hypothesis 3: The wellbore stability and stress distribution around an arbitrarily oriented and inclined wellbore can be analysed using a single 3D finite-element vertical wellbore model. The first part of this study focuses on understanding the mathematical models published by Jaeger and Cook (1979). Then using the Case 1A isotropic horizontal far-field stresses, effective hoop stress is plotted against θ and in Figure 5.1 a-c. This set up a framework in which the first set of finite-element analysis result may have an established mathematical model to be compared to and verified with. Once the method of transforming a 3D finiteelement vertical wellbore model into an inclined wellbore is proven to be credible, the effects of horizontal far-field stress anisotropy on inclined wellbores were further examined. Table 5.1: Result Figures and Loading Cases Figures 5.1 a-c 5.2 a-c 5.3 a-c 5.4 a-c 5.5 a-c 5.6 a-c 5.7 a-c 5.8 a-c Cases 1A 1A 2A 2B 3A 3B 4A 4B α Analysis Jaeger & Cook ABAQUS ABAQUS ABAQUS ABAQUS ABAQUS ABAQUS ABAQUS σ H - σ h 0 kpa 4376 kpa 6894 kpa kpa Horz Stress Isotropic Anisotropic Larger Anisotropic Largest Anisotropic 35

36 Case 1A: Mathematical model with isotropic loads (σ H = σ h ) Figures 5.1 a-c plots the effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of the drilling stage, at various well inclinations ( 0 90 ) with azimuth α = 0. Equations (1) and (9) were used to generate the plots below. Figure 5.1a Figure 5.1b -4.00E E E E E E E E E E E E+07 θ E E E E E E+07 Figure 5.1c θ -3.00E E E E E E E+07 36

37 Case 1A: ABAQUS finite-element analysis with isotropic loads (σ H = σ h ) Figures 5.2 a-c plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling stage, at various well inclinations ( 0 90 ) with azimuth α = 0. For isotropic loading, rotating the well in azimuth planes α = 0 and α = 90 give the same stress distribution. The local in-situ stresses calculated in Table 4.4 were applied to the finite-element vertical wellbore model to generate these results. Figure 5.2a Figure 5.2b -4.00E E E E E E E E E E E E+07 θ E E E E E E+07 Figure 5.2c θ -3.50E E E E E E E+07 37

38 5.1 Comparing Mathematical and ABAQUS Models under Isotropic Loads Figures 5.1 a-c and Figures 5.2 a-c are almost identical with each other despite the fact that one is produced using theoretical mathematical equations, and the other using a vertical finite-element wellbore model that is transformed into various inclined wellbores models. This was achieved by applying a set of pre-calculated normal and shear stresses onto the vertical wellbore surfaces, which effectively created the conditions as if the analyses were done using full inclined wellbore model, as shown in Figure 3.9. If several inclined wellbore models with different borehole inclinations were used to analyse the same problem, then the conventional method is expected to generate effective hoop stress plots similar to Figures 5.1 a-c. However, this may take a lot of time as it is not easy than to build a finite-element model, let alone a few of them with different specifications. Hence, if there is a simpler and more effective way of creating an inclined borehole, then the method should seriously be considered as it can save a lot of time and resources. Both Figures 5.1b and 5.2b shows that stress anisotropy around the borehole surface is minimum a when =0. This means that both the mathematical theory and finite element result agrees that a vertical wellbore is most stable under isotropic horizontal stress. Besides that, both analysis also agree with each other that maximum stress anisotropy occurs when =90, i.e. when the wellbore is horizontal. The magnitudes of stress are almost identical and the nodes on Figures 5.1a and 5.2a occur around the same value of θ as well. With so many similarities, it can be said that a 3D finite-element vertical wellbore model can be transformed into an inclined wellbore model by applying appropriate sets of normal and shear stresses to the vertical wellbore model surfaces, under the condition that the normal and shear stresses are calculated using the mathematical equations put forward in Part 3 and 4 if this report. Hence, Hypothesis 1 has been proven to be correct. At the same time, as different sets of normal and shear stresses calculated in Table 4.4 has been applied to the same vertical wellbore model and it has been reused many times to analyse the stress distribution around the boreholes at different inclination and orientation. Hence, Hypothesis 2 can be proven correct as well. 38

39 Case 2A: ABAQUS analysis with anisotropic loads (σ H - σ h = 4676 kpa) and α = 0 Plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling step, at various well inclinations ( 0 90 ) rotating in the α = 0 azimuth plane, perpendicular to the minimum horizontal stress axis. Figure 5.3a Figure 5.3b -3.00E E E E+07 θ E E E E E E E E E E E E+08 Figure 5.3c θ -2.00E E E E E E E E E+08 39

40 Case 2B: ABAQUS analysis with anisotropic loads (σ H - σ h = 4676 kpa) and α = 90 Plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling step, at various well inclinations ( 0 90 ) rotating in the α = 90 azimuth plane, perpendicular to the maximum horizontal stress axis. Figure 5.4a Figure 5.4b -4.00E E+07 θ E E E E E E E E E E+07 Figure 5.4c θ -4.00E E E E E E+07 40

41 Case 3A: ABAQUS analysis with larger anisotropic loads (σ H - σ h = 6895 kpa) and α = 0 Plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling step, at various well inclinations ( 0 90 ) rotating in the α = 0 azimuth plane, perpendicular to the minimum horizontal stress axis. Figure 5.5a Figure 5.5b -3.00E E E E E E E+07 θ E E E E E E E E E E+08 Figure 5.5c θ -2.00E E E E E E E E E+08 41

42 Case 3B: ABAQUS analysis with larger anisotropic loads (σ H - σ h = 6895 kpa) and α = 90 Plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling step, at various well inclinations ( 0 90 ) rotating in the α = 90 azimuth plane, perpendicular to the maximum horizontal stress axis. Figure 5.6a Figure 5.6b -4.00E E+07 θ E E E E E E E E E E+07 Figure 5.6c θ -4.00E E E E E E+07 42

43 Case 4A: ABAQUS analysis with largest anisotropic loads (σ H - σ h = kpa) and α = 0 Plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling step, at various well inclinations ( 0 90 ) rotating in the α = 0 azimuth plane, perpendicular to the minimum horizontal stress axis. Figure 5.7a Figure 5.7b -2.00E E+07 θ E E E E E E E E E E E E E E E E+08 Figure 5.7c θ -2.00E E E E E E E E E+08 43

44 Case 4B: ABAQUS analysis with largest anisotropic loads (σ H - σ h = kpa) and α = 90 Plots of effective hoop stress σ θ around the wellbore surface ( 0 θ 180 ) at the end of drilling step, at various well inclinations ( 0 90 ) rotating in the α = 90 azimuth plane, perpendicular to the maximum horizontal stress axis. Figure 5.8a Figure 5.8b -2.00E E+07 θ E E E E E E E E E E E E E E+07 Figure 5.8c θ -2.00E E E E E E E E+07 44

45 5.2 Comparison of results with data published by Zhou et al (1996) Case 2B: Anisotropy Case 3B: Larger Anisotropy Case 4B: Largest Anisotropy = 0.2, = 0.78, = 0.66 = 0.2, = 0.81, = 0.63 = 0.2, = 0.90, = E E E E E E E E E E E E E E E E E E E E+07 Figure 5.9: The drilling direction (α) in the chart below by Zhou et al (1996) is equal to 90, which is comparable to Cases 2B, 3B and 4B in this study. It plots the deviation angle from vertical () at which stress anisotropy around the well wall is minimised in extensional stress regimes with = 0 and = 0.25, slightly higher than the = 0.2 used in Cases 2B, 3B and 4B. 4B* 3B* 2B* n H σ H / σ v 1A n σ / σ v 45