Copyright. Alberto López Manríquez

Transcription

1 Copyright by Alberto López Manríquez 003

2 The Dissertation Committee for Alberto López Manríquez Certifies that this is the approved version of the following dissertation: FINITE ELEMENT MODELING OF THE STABILITY OF SINGLE WELLBORES AND MULTILATERAL JUNCTIONS Committee: Augusto L. Podio, Co-Supervisor Kamy Sepehrnoori, Co-Supervisor Martin E. Chenevert Eric B. Becker Eric P. Fahrenthold Carlos Torres-Verdín

3 FINITE ELEMENT MODELING OF THE STABILITY OF SINGLE WELLBORES AND MULTILATERAL JUNCTIONS by Alberto López Manríquez, B.S., M.S. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin May 003

4 Dedication I dedicate this work to my adorable children Karla Elizabeth and Carlos Alberto, hoping it inspires them to pursue great and ambitious goals in their lives. I am grateful with my wife Adriana for her understanding and support during the long period of time required to complete this project. To my parents José Jesús and Consuelo, this work is dedicated with love.

5 Acknowledgements I want to express my sincere gratitude to Professors Augusto L. Podio and Kamy Sepehrnoori for their guidance through this research project. I appreciate their support, patience and tolerance during the development of this work. I extend my appreciation to the other members of the Dissertation Committee, Dr. Eric B. Becker, Dr. Eric P. Fahrenthold, Dr. Martin E. Chenevert, and Dr. Carlos Torres Verdín for their suggestions to complete this work. I would like to acknowledge the invaluable help and knowledge provided by all my professors during my studies at the University of Texas at Austin. Their knowledge added priceless value to my academic career. My gratitude is also to all the staff in the Petroleum and Geosystems Engineering and Civil Engineering Departments who with their daily activities contributed to keeping everything running smoothly. I would also like to thank my fellow student Baris Guler who helped me to set up the software and computing system needed to develop this research. Finally, I express my sincere indebtedness to all those persons in Petroleos Mexicanos who believed in this project and authorized the financial support necessary to carry it out successfully. v

6 FINITE ELEMENT MODELING OF THE STABILITY OF SINGLE WELLBORES AND MULTILATERAL JUNCTIONS Publication No. Alberto López Manríquez, Ph.D. The University of Texas at Austin, 003 Supervisors: Augusto L. Podio and Kamy Sepehrnoori This dissertation describes investigation of the stability of single holes and multilateral junctions in order to optimize their design. The investigation is based on finite element three-dimensional modeling using the commercial software ABAQUS. The stability of single holes and multilateral junctions was analyzed at different orientations in a three-dimensional in-situ stress field. Traditional stressdisplacement analysis in steady-state was coupled with transient phenomena to compute strain and stress behaviors and changes in pore pressure due to disturbances created by drilling. This coupled analysis allowed for the inclusion of time dependent processes and the non-linear processes that influence the behavior of the system compounded by rock, fluids contained in the rock, and insitu stresses. vi

7 The three-dimensional wellbore stability modeling presented here overcomes the limitations of common assumptions in wellbore stability analysis, such as linear poroelasticity, homogeneous and isotropic formations, and isotropic in-situ stress field, because this modeling accounts for the sources of non-linearity affecting the strain and stress responses of rock. This study showed that precise knowledge of the in-situ stress field is an important geomechanical parameter needed to optimize the orientation of a single wellbore and the orientation of the lateral at the junction in a multilateral scenario regarding stability. In addition, performing stress-displacement analysis of multilateral junctions identified critical areas regarding failure in the junction area. Geometry, placement, and orientation of the junction were analyzed, and the results provided a real insight to propose strategies to optimize drilling and completion design of multilateral wells. Comparisons of the predictions of this numerical approach with experimental data recently published showed that this numerical approach is reliable for simulating the steady-state phenomena and some transient phenomena encountered in wellbore stability analysis of both single holes and multilateral junctions. vii

8 Table of Contents List of Tables...xii List of Figures...xiii Chapter 1: Introduction Importance of wellbore stability Multilateral well completion scenarios Organization of this dissertation...5 Chapter : An overview of wellbore stability modeling Wellbore stabilty: background Wellbore stability: literature review Single well stability analysis Multilateral well stability analysis Constitutive models Basic Constitutive Relationships Critical State and the Cambridge Model (Cam-Clay) Failure criterion Tensile failure criteria Compressive failure criteria Is the intermediate stress really important Wellbore closure...37 Chapter 3: Statement of the problem Elasticity Differential equations of equilibrium Stress-displacement relationships Stress-strain relationships Displacement formulation of problems in elasticity Stresses around boreholes...54 viii

9 3. Poroelasticity Background in poroelasticity Terzaghi's principle Biot's theory Stress-strain relationships Displacement formulation of problems in poroelasticity Stresses around boreholes Boundary conditions Switching from a boundary value problem to wellbore stability analysis...67 Chapter 4: Numerical approach to the solution of the wellbore stability problem Computational Modeling Analytical and Numerical solutions Constitutive models available in ABAQUS Model Definition Model's geometry for analysis in a single hole Drilling simulation in a single hole Model's geometry for analysis in a multilateral scenario Drilling simulation in a multilateral scenario Wellbore stability mathematical model General assumptions Governing equations Isothermal analysis Hydraulic diffusion analysis Phenomena in steady state Stress-displacement analysis in elasticity Stress-displacement analysis in poroelasticity Transient phenomena...89 ix

10 Rate of Deformation Coupled stress-hydraulic diffusion analysis Solution method used in ABAQUS Wellbore inclination and azimuth variation...96 Chapter 5: Discussion of results Stability of a single wellbore Phenomena in steady state Effect of assuming different constitutive models: stress-displacement analysis Effect of wellbore inclination and azimuth variation: stress-displacement analysis Effect of rock anisotropy: stress-displacement analysis Transient phenomena Rate of deformation Coupled stress-hydraulic diffusion analysis Wellbore stability in multilateral scenarios Phenomena in steady state Elastic stress-displacement analysis Effect of increasing the junction angle Effect of varying the diameter of the lateral hole Effect of varying the orientation of the lateral hole Effect of changing the depth of placement of the junction Independence between holes Complex multilateral scenarios Chapter 6: Conclusions and recommendations x

11 Appendix ABAQUS Input File...03 Nomenclature...09 References...14 Vita...19 xi

12 List of Tables Table.1 Classification of wellbore stability models (from Fonseca 1998)...40 Table. Categorization of Peak-Strength Criterion (from McLean 1990a)...41 Table 5.1 Data from a drained triaxial test (from Atkinson and Bransby 1978) Table 5. Isotropic compression test results (from Atkinson and Bransby 1978) Table 5.3 Effect of varying M value on hole closure Table 5.4 Values of parameters for various clays (from Atkinson and Bransby 1978) Table 5.5 Effect of varying λ s and κ s values on hole closure Table 5.6 Stress level imposed to analyze wellbore orientation Table 5.7 Transversely isotropic rock properties used for the sensitivity analysis Table 5.8 Orthotropic rock properties used for sensitivity analysis Table 5.9 Effect of rate of penetration on hole closure Table 5.10 Material properties for a coupled stress-diffusion (from Chen et al. 000) xii

13 List of Figures Figure 1.1 Completion levels 1 and according to the Technical Advancement of Multilateral, TAML...7 Figure 1. Completion levels 3 and 4 according to the Technical Advancement of Multilateral, TAML...8 Figure 1.3 Completion level 5 according to the Technical Advancement of Multilateral, TAML...9 Figure 1.4 Completion level 6 according to the Technical Advancement of Multilateral, TAML...10 Figure.1 Geometries at the multilateral junction (from Aadnoy and Edland 1999)...41 Figure. Definition of independence distance (from Aadnoy and Edland 1999)...4 Figure.3 Comparison between stresses for elastic and plastic solution (from Charlez 1997a)...4 Figure.4 Elastic, hardening, and perfectly plastic behaviors...43 Figure.5 Yield surface (from Atkinson and Bransby 1978)...43 Figure.6 Physical phases in plastic collapse (from Charlez 1997a)...44 Figure.7 Elastic wall in the three-dimensional p :q :v space (from Atkinson and Bransby 1978)...44 Figure.8 Elastic wall and the corresponding yield curve (from Atkinson and Bransby 1978)...45 xiii

14 Figure.9 Behavior during isotropic compression and unloading. Hardening law (from Atkinson and Bransby 1978) Figure.10 Strain increments during yield. Flow rule (from Atkinson and Bransby 1978)...46 Figure.11 A yield curve as predicted from the Cambridge model (from Atkinson and Bransby 1978)...47 Figure.1 Correlation λ s κ s (from Charlez 1997a)...47 Figure.13 Common yield surfaces (from McLean 1990b)...48 Figure 4.1 Pure compression behavior of clay (form ABAQUS/Standard User's manual, Version 6.1, 000)...99 Figure 4.. Model mesh for a single hole one step Figure 4.3 Effect of mesh refinement in the radial direction on the accuracy of radial stress calculations Figure 4.4 Effect of mesh refinement in the tangential direction on the accuracy of radial stress calculations Figure 4.5 Effect of mesh refinement in the tangential direction on the accuracy of tangential stress calculations...10 Figure 4.6 Improved accuracy obtained of radial stress calculations in the nearest region to the wellbore when using unequally spaced elements...10 Figure 4.7 Multi-layer model for multi-step drilling Figure 4.8 Mutilateral mesh scenario (open view) Figure 4.9 Mutilateral mesh scenario (close view) xiv

15 Figure 4.10 Transformation system for a deviated wall (from Fjaer et al 199) Figure 5.1 Stress distribution around a wellbore: Elastic case Figure 5. Comparison of tangential stresses Figure 5.3 Contour plot showing the extent of the plastic zone Figure 5.4 Comparison between tangential stress solutions Figure 5.5 Comparison between radial stress solutions Figure 5.6 Analysis of compressive failure for the elements in the immediate vicinity of the wellbore Figure 5.7 Effect of M variation on tangential stress response: Cam-Clay...15 Figure 5.8 Tangential stress behavior...15 Figure 5.9 Representation of the principal in-situ stresses in a shallow formation in a tectonically active stressed region (σ H >σ h >σ v ) Figure 5.10 Maximum Mises and Mean stresses vs hole deviation for three different azimuth values in a shallow formation (elastic rock) Figure 5.11 Effect of varying angle deviation on the maximum p and q values in a shallow formation (elastic rock) Figure 5.1 Maximum hole closure vs wellbore inclination in a shallow formation (elastic rock) Figure 5.13 Representation of the principal in-situ stresses in an intermediate formation in a tectonically active stressed region (σ H >σ v >σ h ) xv

16 Figure 5.14 Maximum Mises and Mean stresses vs hole deviation for three different azimuth values in an intermediate formation (elastic rock) Figure 5.15 Effect of varying angle deviation on the maximum Mean and Mises effective stresses in an intermediate formation (elastic rock) Figure 5.16 Maximum hole closure vs wellbore inclination in an intermediate formation (elastic rock) Figure 5.17 Representation of the principal in-situ stresses in a deep formation in a tectonically active stressed region (σ v >σ H >σ h ) Figure 5.18 Maximum Mises and Mean stresses vs hole deviation for three different azimuth values in a deep formation (elastic rock) Figure 5.19 Effect of varying angle deviation on the maximum Mean and Mises effective stresses in a deep formation (elastic rock) Figure 5.0 Maximum hole closure vs wellbore inclination in a deep formation (elastic rock) Figure 5.1 Maximum Mises stress vs hole deviation for three different azimuth values in a deep formation (elastic and elastoplastic cases) Figure 5. Comparison of maximum hole closures between the elastic and the non-elastic cases for three different azimuths in a deep formation xvi

17 Figure 5.3 Effect of varying inclination angle on the maximum Mises and Mean effective stresses. Deep formation (elastic and elastoplastic cases) Figure 5.4 Maximum Mises stresses vs hole deviation at three different R t values in a deep transversely isotropic formation (elastic rock) Figure 5.5 Comparison of the maximum p and q values when varying the deviation angle. Different R t. Transversely isotropic formation Figure 5.6 Maximum hole closure vs wellbore inclination. Different R t. Transversely isotropic formation Figure 5.7 Maximum Mises stresses vs hole deviation at three different R p values in a deep orthotropic formation (elastic rock) Figure 5.8 Maximum hole closure vs wellbore inclination. Different R p. Orthotropic formation Figure 5.9 Rate of deformation influence on the uniaxial stress-strain curves and failure of sandstone (from Cristescu and Hunsche 1998) Figure 5.30 Comparison of hole closure between one-step and multi-step analysis Figure 5.31 Progress of drilling with time showing hole closure behind the advancing face of the wellbore Figure 5.3 Comparison between pore pressure distribution around a wellbore for both solutions: elastic and elastoplastic...17 Figure 5.33 Contour plot showing pore pressure distribution around a wellbore after three hours (t=3) xvii

18 Figure 5.34 Pore pressure distribution as a function of time and radial distance from the wellbore wall Figure 5.35 Pore pressure distribution as a function of radial distance from the wellbore wall for different permeability conditions Figure 5.36 Effect of yield stress variation on the response of pore pressure distribution around a wellbore Figure 5.37 Effect of fluid compressibility variation on the response of pore pressure distribution around a wellbore Figure 5.38 Distribution of the radial and tangential stresses at the junction area Figure 5.39 Contour plot showing Mises stress Figure 5.40 Contour plot showing displacement in the x-direction Figure 5.41 Stresses in the p :q plane showing changes in the stress cloud Figure D representation showing the three regions A, B, and C identified at the junction area Figure 5.43 Effect of variation of the junction angle on the stress cloud Figure 5.44 Effect of variation of the diameter of the lateral well on the stress cloud Figure 5.45 Contour plot of displacements when the lateral is oriented with an azimuth (a=90 o ) Figure 5.46 Contour plot of Mises stresses when the lateral is oriented with an azimuth (a=90 o ) xviii

19 Figure 5.47 Contour plot of Mises stresses showing failure in the lateral wellbore at a higher stress level Figure 5.48 Contour plot of Mises stresses showing that the most likely region to fail after the junction when the lateral is oriented with an azimuth (a=0 o ) is the mainbore Figure 5.49 Contour plot of Mises stresses showing breakout orientation when the lateral is oriented with an azimuth a=90 o ) Figure 5.50 Contour plot of Mises stresses showing breakout orientation when the lateral is oriented with an azimuth(a=0 o ) Figure 5.51 Contour plot of Mises stresses when the lateral is oriented with an azimuth(a=0 o ) Figure 5.5 Contour plot of Mises stresses when the lateral is oriented with an azimuth (a=0 o ) and in a deep formation Figure 5.53 Contour plot of Mises stresses when the lateral is oriented with an azimuth (a=90 o ) and in a deep formation Figure 5.54 Stress distribution in the region between the two boreholes showing independence between them Figure 5.55 Contour plot of Mises stresses of a horizontal mainbore with two laterals when the mainbore is oriented with an azimuth (a=0 o ) Figure 5.56 Contour plot of Mises stresses of a horizontal mainbore with two laterals when the mainbore is oriented with an azimuth (a=45 o ) Figure 5.57 Contour plot of Mises stresses of a horizontal mainbore with two laterals when the mainbore is oriented with an azimuth (a=90 o ) xix

20 Chapter 1: Introduction This chapter has the aim to present a general overview of the importance of wellbore stability in drilling. Initially, brief comments are presented about general aspects of this topic, followed by a general overview of the scenarios encountered in multilateral technology and some remarks about the organization and the structure of the dissertation. 1.1 IMPORTANCE OF WELLBORE STABILITY Wellbore stability analysis has been the subject of study and discussion for a long time. The integrity of the wellbore plays a important role in many well operations during drilling, completion, and production. Problems involving wellbore stability occur principally through changes in the original stress state due to removal of rock, interactions between rock and drilling or completion fluids, temperature changes, or changes of differential pressures as draw down occurs. For the particular drilling case, support provided originally by the rock is replaced by hydraulic drilling fluid pressure; this creates perturbation and redistribution of stresses around the wellbore that can lead to mechanical instabilities. These instabilities can cause lost circulation or hole closure in the case of tensile or compressive failure respectively. In severe situations, hole closure can cause stuck pipe and loss of the wellbore. These events lead to an increase of drilling costs. The causes of instability have been classified into either mechanical or chemical effects. A significant amount of research has been focused on these two 1

21 aspects of instability; the last one mainly oriented to instability in shales. Although there exists a significant amount of articles related to wellbore stability, most of them address the study of stability in the vicinity of the wellbore for a single hole. When two holes interact, the interference that a lateral hole causes on the stresses around the mainbore is particularly interesting. However, information about research conducted in a multilateral scenario where two holes interact is limited. Therefore, the review of literature presented focuses on the status of a specific area of multilateral wells: the stability of the junction between the mainbore and the lateral hole. During the last years, complex well architecture has been implemented as a new technique to increase well productivity, such as drilling secondary branches from an existing well. The evolution of multilateral technology has created a wide range of completion scenarios. Hogg (1997) recognizes that although these new scenarios have brought new expectations in reservoir management, they have also created a new set of obstacles, concerns, and risks. To develop a better understanding of multilateral applications, capabilities, and required equipment, an oil industry forum on the Technical Advancement of Multilateral (TAML) was created, and a multilateral classification scheme was developed. Vullings and Dech (1999) give a complete description of the main characteristics of this multilateral classification scheme.

22 1. MULTILATERAL WELL COMPLETION SCENARIOS According to Hogg (1997), several factors must be taken into account when one considers a multilateral project. First, since the goal of the multilateral is to enhance hydrocarbon recovery, it is crucial to have a good understanding of reservoir behavior. Secondly, wellbore stability plays an important role; geological characteristics of the rock must be considered. In addition, even if the lateral junction is initially competent, the completion system should be designed for the life of the well. A final consideration for multilateral completion design should be the need for future workovers requiring re-entry into the lateral or mainbore with the purpose of periodic cleanouts, stimulations, or any other kind of workover. It is interesting to note that although drilling plays a very important role in multilateral activity, the multilateral classification scheme is based on completion rather than drilling characteristics. TAML categorizes the multilateral completion process into levels as a function of risk and complexity. The goal of multilateral completions is to achieve a junction with full mechanical and hydraulic integrity by increasing the level of complexity. According to the TAML classification, there are six different levels of multilateral completion. The simplest system is Level 1, consisting of branches drilled from a main open hole. Because little or no completion equipment is required, there is no mechanical support or hydraulic isolation. The advantage of this system is its low cost and simplicity. However, the lack of casing limits the installation of completion equipment, and as a consequence, there is no production control. 3

23 Furthermore, this kind of completion is limited to competent formations able to provide borehole stability. The next step in complexity is Level. At this level, the mainbore is cased while the lateral bore is openhole or with a simple slotted liner. The presence of casing in the mainbore helps to reduce the risk of borehole collapse, but this is only true in the case where the formation is competent in the junction area. Figure 1.1 illustrates completion levels 1 and according to TAML. The next level of completion is Level 3. This scenario requires the mainbore to be cased and cemented; the lateral well is cased with a liner, but it is not cemented. The main advantage of this completion is the mechanical support given by the casings at the junction area. Therefore, the junction is partially protected from potential collapse. It is important to remark that although mechanical support is given, there is no hydraulic isolation at the junction. Level 4 is exactly the same as level 3 from a drilling point of view. However, the main difference is that both holes are cased and cemented. For this reason, it is considered that the junction is mechanically protected from collapse. However, there is no complete hydraulic isolation at the junction since the cement may be unable to support large differential pressure, or it could fail over time as drawdawn pressure increases. Figure 1. illustrates the characteristics of the levels mentioned above. Only levels 5 and 6 provide pressure integrity at the junction, and only level 6 provides full mechanical and hydraulic integrity. As shown in Figure 1.3, level 5 completion requires a complex configuration of isolation packers to isolate 4

24 the junction and provide pressure integrity. In this case, both holes are cased and cemented, and isolation packers provide three sealing points in the well. Two of the three are at the junction area in the mainbore; the first one is above, and the second below. The third one is in the lateral, below the junction. This arrangement allows isolation of the junction, and as a result, better hydraulic isolation is achieved where completion equipment works in conjunction with the cement. Finally, it is important to remark that pressure integrity is achieved with completion equipment. The principal characteristic of level 6 completion is that mechanical and hydraulic integrity at the junction are achieved with the casing using a pre-formed metal junction, which is installed with the casing itself. Thus, mechanical and hydraulic integrity are obtained with the casing rather than using completion equipment. This condition brings some advantages over the lower levels. In addition to avoiding the risk of handling isolation packer assemblies, it helps to prevent and to reduce problems related to the quality of the cementing job and the cement material properties. Figure 1.4 illustrates level 6 completion. 1.3 ORGANIZATION OF THIS DISSERTATION This introductory Chapter 1 deals with brief comments about the importance of wellbore stability, mainly in drilling. A general overview about multilateral well completion scenarios is described. Chapter serves two purposes. First, it summarizes different approximations to the solution of the problem of wellbore instability and reviews single and multilateral well stability 5

25 analyses. With this, the reader has the opportunity to compare what is the state-ofthe-art in each area. Secondly, it points out the importance of choosing an appropriate constitutive model and an adequate failure criterion to reproduce rock mechanical behavior and rock failure. Chapter 3 presents the general theory of material mechanical behavior. It is an overview of the basic formulation of the general problem in elasticity. For the case of rock analysis, material porosity is introduced in these theories. Once the formulation of the general problem is stated, then we switch from the boundary value problem to the wellbore stability analysis problem. Chapter 4 aims to support the decision of choosing a commercial finite element program to conduct this research. It presents the general considerations for constructing the models using the commercial package. Chapter 5 presents the analysis of the results obtained by simulation of particular cases. Finally, Chapter 6 presents the conclusions and recommendations for future work. 6

26 Figure 1.1 Completion levels 1 and according to the Technical Advancement of Multilateral, TAML (web site of Baker Hughes URL 7

27 Figure 1. Completion levels 3 and 4 according to the Technical Advancement of Multilateral, TAML (web site of Baker Hughes URL 8

28 Figure 1.3 Completion level 5 according to the Technical Advancement of Multilateral, TAML (web site of Baker Hughes URL 9

29 Figure 1.4 Completion level 6 according to the Technical Advancement of Multilateral, TAML (web site of Baker Hughes URL 10

30 Chapter : An overview of wellbore stability modeling This chapter serves two purposes. First, it presents a review of the literature that is relevant to the solution of the single and multilateral well stability problems. It provides the opportunity to compare what has been done in each area. Second, it reviews the importance of choosing an appropriate constitutive model as well as an adequate failure criterion to analyze wellbore stability problems..1 WELLBORE STABILITY: BACKGROUND Simulation of wellbore stability has the purpose of predicting the redistribution of stresses around the wellbore as result of drilling, completion, or production operations. The most important elements needed to simulate geomechanical problems are the rock s constitutive behavior model and an appropriate failure criterion. Constitutive behavior models used to forecast wellbore stability range from those using the theory of elasticity to more complex models which take into account the theories of elasticity and plasticity, porosity of the materials, temperature, and time dependent effects. Comparison of the stresses obtained by using some of these constitutive models with an adequate rock failure criterion determines whether the rock around the borehole is likely to fail or not. Fonseca (1998) and McLean and Addis (1990a) include in their works a classification of wellbore stability models. Table.1 shows some special features that characterize those models for specific purposes. 11

31 . WELLBORE STABILITY: LITERATURE REVIEW..1 Single well stability analysis An attempt to analytically formulate the wellbore stability problem was done by Bradley (1979a). He used Kirsch s equations combined with the solution proposed by Fairhurst in 1968 to develop analytical expressions of stress distribution around inclined boreholes using the linear elastic theory. Charlez (1991) explains that Kirsch s equations were formulated to calculate stresses in an infinite plate subjected to an initial state of stress. Kirsch s solution states that the presence of a circular hole at the center of the plate produces a disturbance within the solid plate that modifies the initial stress condition. Because Kirsch s equations were derived from the assumption that rock was isotropic and homogeneous, Bradley s equations keep this condition. Plane strain condition is also assumed, indicating that the strain component parallel to the wellbore axis is negligible compared to the radial and tangential strain components. In addition, Bradley assumed there was no interaction between drilling mud and in-situ formation fluid. Bratli et al. (1983) initially investigated the sand problem, which occurs during production in poorly consolidated sandstones. They focused on the mechanisms that destabilize the sand behind perforation openings and extended this theoretical stress analysis to cylindrical wellbores to study stability. Because they assumed the existence of poorly consolidated material, failure was considered to be located in a zone around the wellbore, known as the plastic zone. 1

32 They analyzed the rock stress behavior in this region where high effective stress concentration occurs. Aadnoy and Chenevert (1987) and Aadnoy (1988) use Bradley s approach to make a detailed analysis about how borehole inclination can influence borehole stability. They considered two different compressive failure criteria to analyze borehole collapse: the von Mises and the Jaeger criteria. The first of these takes into account the intermediate principal stress while the second neglects this. Jaeger s criterion, which is an extension of the Mohr-Coulomb criterion, is useful for laminated sedimentary rocks because it considers the existence of a plane of weakness that may affect rock behavior. McLean and Addis (1990b) also use Bradley s solution, but they focus their analysis by selecting an appropriate failure criterion to compute safe-drilling fluid densities. They found that when using a linear elastic constitutive model, the criteria that do not consider the influence of the intermediate principal stress are likely to underestimate the strength of the rock. Earlier work was conducted by considering rock as homogeneous and isotropic. Aadnoy (1988) and Ong and Roegiers (1993) attempt to provide a better understanding of the effects of rock properties anisotropies on the stability of a wellbore. The assumptions they made are that rock behaves as linear elastic formation, a condition of plane strain prevails, and there is no interaction between in-situ formation fluids and drilling mud. To fully describe the mechanical behavior of the rock, the number of elastic constants that Ong and Roegiers suggest is five: two moduli of elasticity, two Poisson s ratios and one shear 13

33 modulus. They concluded that anisotropy strongly influences rock stability, especially when wellbore inclinations are high or horizontal. Detournay and Cheng (1988) and Cui et al. (1997) presented analytical solutions for a circular wellbore embedded in a homogeneous and isotropic formation, which behaves linearly and according to the poroelastic theory. These solutions are the first attempts to formulate the time-dependent problem originated from the diffusion process through the porous medium related to the hydraulic conductivity of the rock. These solutions are restricted to the condition where the wellbore axis coincides with the direction of the vertical principal stress. Cui et al. give the analytical solution for a circular wellbore, whose axis is inclined with respect to the principal stresses, in a linear, poroelastic, homogeneous, and isotropic formation where the in-situ stresses are anisotropic. They separate the problem into three parts: poroelastic plane-strain, elastic uniaxial, and elastic antiplane shear problem. In the first part, they assume that only in-plane displacements are different from zero. For the second part of the problem, they claim that the solution is uni-axial and is given by a constant vertical stress anywhere. For the third part, they explain that the disturbance caused by the removal of wellbore rock during drilling is introduced to the analytical problem by a sudden change of shear stress at the wellbore wall. Finally, they find the final solution by superposition. For wellbore stability analysis purposes, Drucker-Prager is used as failure criterion. Another interesting reference is Fonseca (1998). The objective of his work was to develop a chemical poroelastic model applicable to shales. He considered 14

34 the poroelastic solution proposed by Detournay and Cheng (1998) to conduct his research. To investigate the chemical aspect of the instability problem, he took a microscopic approach of the forces acting in a clay-fluid system, which is based on the Double-Layer Verwey and Overbeek (DLVO) theory and a macroscopic approach that evaluates the influence of osmotic potential between shale and fluid. He found that for a water based mud-shale one-dimensional system, the total flow of fluid into or out of the shale is driven by two mechanisms: hydraulic pressure and chemical potential. He reported that the chemical potential can be introduced into a wellbore stability model as a pore pressure alteration, and it is controlled by the ratio between the water activity of the shale and the water activity of the drilling fluid. He concluded that by controlling the water activity of the mud it is possible to produce a chemical potential that counterbalance the hydraulic pressure so that the shale behaves as an impermeable formation. A particular case where a mud with a water activity lower than the water activity of the shale will induce flow of water out of the shale. This condition is beneficial for the stability of the wellbore. Abousleiman et al. (1999) developed software, called Pore-3D, to predict stability problems during drilling. They claimed that traditional analytical solutions for wellbore stability, which are based on Bradley s (1979a) work, fail to capture the coupled-time dependent phenomenon of stress variation around the wellbore. They stated that only the analytical solution recently developed by Cui et al. (1997) considers the coupled-time dependent phenomenon of stress variation, stating: 15

35 The solutions of theories of poroelasticity, porochemoelasticity, porothermoelasticity, and poroviscoelasticity as well as their elastic, chemoelastic, thermoelastic, and viscoelastic counterparts are included in PORE-3D. Assumptions involved in this software are that rock formation behaves linearly when its stress-strain response is analyzed. Moreover, rock formation is considered homogeneous and isotropic of infinite extent following the poroelasticity theory. Their development is based on the Cui et al. (1997) poroelastic solution. Based on the solution proposed by Lomba et al. (000a, 000b) to find the solute concentration profile in the formation and the poroelastic solution proposed by Detournay and Cheng (1988), Yu et al. (001) developed a three dimensional model to investigate the stress behavior around a wellbore taking into account chemical and thermal effects in shale formations. They claimed that existing models, allowing for chemical effect, only take into account the osmotic pressure effect but do not consider the effect of diffusion of solutes. They concluded that due to differences between solute concentrations of the drilling fluid and the pore fluid, competition between water and solute fluxes occurs, altering pore pressure, which may lead to instabilities. In recent years, a new modeling approach of wellbore stability has arisen. Since finite element theory was successfully implemented in other disciplines, researchers in geomechanics focused their attention on this theory. Pan and Hudson (1988) developed a couple of nonlinear axisymmetric finite element models in -D and 3-D to study the behavior of stresses and displacements in the rock surrounding tunnel excavations. They used an elasto-viscoplastic model 16

36 proposed by Zienckiewicz and Cormeau (1974) that considers the time-dependent response of the rock associated with its plastic properties. They directed their study to find the differences between the results predicted by assuming plane strain in the -D model versus the results obtained by the 3-D model. Development of the 3-D model gave them the opportunity to compare the results of classical analysis in -D, a one-step tunnel excavation, versus multi-step analysis in 3-D. Among other conclusions, they found that modeling tunnel excavations in -D underestimates deformation compared with the results of the 3-D analysis. They concluded that this discrepancy obeys the plastic response of the rock behind the tunnel face, a response that a -D model cannot reproduce. Ewy (1993) also used commercial finite element software to study the behavior of sedimentary rocks to analyze wellbore stability in directional and horizontal wells. He assumed rock formation behaves according to the elastoplastic theory. He developed a model in three dimensions (3-D) by assuming that a thin slice of elements orthogonal to the well axis may represent the rock behavior. Similar analysis was done by Zervos et al. (1998), who modeled wellbore stability of weak sedimentary rocks for a wide range of wellbore orientations and deviations. They found that the risk of hole closure increases as wellbore inclination increases. Orientation of the wellbore becomes important only for deviations between 30 and 60 degrees. Also wellbores with inclinations of up to 15 degrees can be treated as vertical wells while for inclinations of more than 75 degrees, wellbores can be analyzed as horizontal wells. 17

37 Chen et al. (000) developed two numerical models to investigate the pore pressure diffusion effect in shales. They compared numerical predictions obtained using a linear and a nonlinear elastoplastic model against those obtained using experimental observations done with a thick-walled hollow cylinder of synthetic shale. The analyses demonstrated that for more accurate predictions of stresses and deformations around a wellbore embedded in shale, the nonlinear model should be considered because its results showed good agreement with the results of the laboratory tests... Multilateral well stability analysis There exists a considerable amount of publications related to wellbore stability in a single hole. However, this situation changes radically with respect to analyses of stability in multilateral junctions. Aadnoy and Edland (1999) investigated the effect of wellbore geometry on the stability of multilateral junctions. They assumed that the geometry around the junction takes different configurations. Above the junction, the hole geometry is circular, which becomes oval at the junction. Then it splits into two adjacent boreholes below that point, to finally separate in two independent circular holes. Figure.1 illustrates this situation. They found a relationship between the tangential stress and a stress concentration factor (K s ) at the wall of the wellbore as shown in Equation.1. They used elasticity theory to set their model. The tangential stress σ θ for an isotropic stress field is represented as follows: 18

38 σ θ = P + K ( σ P ) = K σ ( K 1) P (.1) w s H w s H s w where K s = stress concentration factor P w = borehole pressure σ Η = Maximum horizontal stress Their approach rests on the assumption that each geometry corresponds to a different stress concentration factor. First, for circular holes, K s is a constant with a value equal to two, K s =. Second, for oval holes, K s factor is not unique as is found with circular holes. Instead, there is a K s value for each of the axes of the oval geometry. These K s values are not constant, and they are a function of n and m values as Equation. shows. K s = f ( n, l) (.) where l = the vertical/horizontal hole size ratio for the ellipse n = empirical geometric parameter Values n and l are functions directly of the geometry of the oval. According to Aadnoy and Froitland (1991), for the adjacent boreholes condition, the K s factor is defined as a function of the distance between holes and the borehole radius. They found a dimensionless separation distance between holes 19

39 where the adjacent boreholes can be treated as two independent circular holes. This distance is expressed as ξ = d/r w where d is the distance between borehole centers and r w is the borehole radius. Figure. illustrates this situation. They established that the condition to treat the boreholes as independents is ξ > 3. Τhis model assumes that the two holes are of the same diameter. However, according to the multilateral completion scheme presented in the previous chapter, holes have different diameters. The only exception exists in level 6 completion, where split holes are of the same diameter. Aadnoy and Edland (1999) considered the Mohr-Coulomb failure criterion, and they also assumed the medium to be isotropic and homogeneous. Their main conclusion was that the junction is a critical region where the stress concentration increases as the hole becomes oval. They found that the oval and the two adjacent holes configurations create extreme conditions for fracturing and collapse respectively. Bayfield et al. (1999) showed a particular case of stability at the junction considering the completion level 6, which means that the junction is cased, and its integrity is achieved with the casing itself using a pre-formed metal junction. They performed finite element analysis using a commercial finite element software to predict the burst and collapse strengths of the pre-formed junction and then to evaluate the effects of internal and external pressure on the pre-formed junction, varying the angle between the mainbore and lateral, and cementing the junction. Their main conclusions are as follows: 0

40 Increasing the junction angle from.5 to 5 degrees does not significantly increase burst and collapse strengths. Steel reinforcement of the pre-formed junction can significantly increase junction strength. Cement support to the junction can improve burst strength, depending on the adequate placement of the cement and the cement properties. This work aimed at analyzing the resulting stresses along the tubular, the pre-formed junction, rather than the stress behavior of the rock itself. Fuentes et al. (1999) present an analysis based on three-dimensional finite element model, using commercial software to estimate the stress distribution at the junction. To set up their particular model, they assumed the formation to be homogeneous sand without shales with no flow between wellbore and formation. No chemical effects were considered. Other considerations in the model were that the axes of the global system coincide with the direction of the principal stresses, and the lateral well is in the direction parallel to the maximum horizontal stress. The two previous assumptions simplify the problem since no shear stresses occur when the axes of the system coincide with the direction of the principal stresses. They used an elastoplastic constitutive model to predict the mechanical behavior of a sand formation in Lake Maracaibo, Venezuela. Comparing stresses around the junction region against a compressive failure criterion, they found that the 1

41 region between the two holes is where stress concentration increases and failure is more likely to occur..3 CONSTITUTIVE MODELS Previously, it was mentioned that one of the most important elements to predict rock behavior is the constitutive behavior model. Choosing an appropriate constitutive model to simulate rock behavior deeply affects the accuracy of the results. In this respect, there is still debate over the applicability of some constitutive models to particular conditions. For instance, it is commonly thought that wellbores are presumably stronger than the linear elasticity theory predicts. McLean and Addis (1990a) pointed out that the results of laboratory tests over a variety of hollow cylinder rock samples show that failure occurs at pressures up to 8 times the failure pressure predicted by linear elasticity used in conjunction with a failure criterion that does not consider intermediate stress. In the same way, Charlez (1997a) remarked the significance of plasticity and hardening effects on stress behavior around wellbores. He mentioned, by comparing the solution based on a plastic constitutive model to a purely elastic solution, that a plastic zone surrounding a wellbore exists, which the purely elastic constitutive model is unable to predict. Figure.3 illustrates the comparison between the plastic and the elastic solutions. The zero value on the radius axis of this figure corresponds to the wellbore wall. Slight difference can be seen when comparing the radial stress of the plastic and the elastic solutions. However, considerable relaxation of the tangential stress occurs in the region nearest to the wellbore (low radius values).

42 Based on this substantial difference of the tangential stress behaviors between the elastic and the plastic solutions, Charlez (1997a) concluded that there exists a plastic zone surrounding the wellbore..3.1 Basic Constitutive Relationships Although it is beyond the scope of this work to give an explanation for each one of those constitutive models used to describe rock behavior, it is necessary to briefly mention the principal characteristics of some of the basic constitutive relations. The simplest relationship is elastic, which is the foundation for all aspects of rock mechanics. This theory is based on the concepts of stress and strain, which are related according to Hooke s law: σ = Eε (.3) The proportionality constant E between stress σ and strain ε is the elastic modulus. The other parameter required for this model is Poisson s ratio, which is a measure of lateral expansion relative to longitudinal contraction. It is defined as follows: ε y ν = (.4) ε x 3

43 However, rocks have what is called void space, which is actually occupied by fluids. Consequently, elasticity theory for solid materials does not satisfy this condition, and the poroelasticity concept arises. When we talk about poroelasticity, immediately we should think about two components: solids and fluids. Therefore, in addition to the variables involved in elasticity, new variables related to void space and fluid content appear. Thus, a complete description of rock behavior under this theory requires more than the two simple parameters considered in the elasticity theory. Wang (000) divides poroelastic constants into six different categories: (1) compressibility bulk modulus, () Poisson s ratio, (3) storage capacity, (4) poroelastic expansion coefficient, (5) pore pressure buildup coefficient, and (6) shear modulus. There are three basic material constants: bulk modulus, poroelastic expansion coefficient, and storage coefficient. However, in order to define a complete set, a fourth constant has to be considered. This last constant should include a property related to shear deformation. For instance, Biot and Willis (1957) suggested the set {G, 1/K, 1/K u, S}, where 1/K u is the compressibility coefficient obtained in an unjacketed test and S is the storage coefficient. G is the shear modulus. Detournay and Cheng (1988) selected {G,α, ν, ν u } as the complete set of poroelastic constants, where α is the Biot constant, and ν and ν u are the Poisson s ratios obtained in jacketed and unjacketed tests respectively. What happens when elasticity theory is unable to match rock behavior? Above the elastic limit, the elasticity theory is unable to predict material behavior. 4

44 Therefore, an appropriate definition of failure criterion and post-failure behavior are important. Fjaer et al. (199) mentions that the immediate option for postfailure behavior is the plasticity theory, although other options such as the bifurcation theory are available. Fjaer et al. (199), Naylor and Pande (1981), and Atkinson and Bransby (1978) agree that the main concepts supporting the plasticity theory are yield criterion, hardening rule, flow rule, and plastic strains. Simple definitions about each concept are as follows: Yield criterion is the point where irreversible changes occur in the rock. It separates states of stress, which cause only elastic strains, from those which cause plastic and elastic strains. The hardening rule describes how rocks under certain conditions might sustain an increasing load after the initial failure. Flow rule defines the direction of the vector of the plastic strain increment, δε p, related to the yield surface. Plastic strains take place when a sample is forced beyond its elastic limit. Total strain, δε, can be expressed as the summation of the vectors of elastic (δε e ) and plastic (δε p ) strain increments: δε + e p = δε δε (.5) A typical strain-stress diagram for an elastoplastic material is shown in Figure.4. Three different regions can be identified: elastic, hardening, and perfectly plastic behaviors. Atkinson and Bransby (1978) explain that yielding, hardening, and failure may be represented on a diagram with axes σ : σ : ε ' a ' c p as it is shown in Figure.5, where ' σ a and ' σ c are the axial and compressive 5

45 effective stresses respectively applied on a sample in a conventional triaxial test. This figure shows a set of yield curves such as G a G c, each curve associated with a particular plastic strain value ε p. All curves together define a particular yield surface shown in Figure.5. The yield surface is limited by the curve Y a Y c which corresponds to the yield when ε p =0 and by the failure envelope F a F c. The hardening behavior is represented by the response of curve G a G c to plastic strains, ε p. It is accepted that clays are the main cause of wellbore instability problems during drilling. Therefore, it is important to have a constitutive model equipped to handle clay behavior. The literature review showed that, in general, wellbore stability analysis is done by considering either elasticity or poroelasticity theories. However, recent numerical approaches have taken into account the plastic response of rock. Charlez (1991) and Brignoli and Sartori (1993) point out the importance of two classical elastoplastic models used in geomechanics that take into account the role of clays. These are the Cambridge model (Cam-Clay) and the Laderock model, which are both critical state models..3. Critical State and the Cambridge model (Cam-Clay) Charlez (1991, 1997a) reviews the limits of the Cam-Clay model. He remarks that there are different phases in the plastic collapse mechanism associated to rock s volumetric deformation. He establishes that, for ductile rocks under increasing loading, three phases are observed: (1) rupture of bonds, () plastic collapse, and (3) consolidation. These regions are illustrated in Figure.6. 6

46 However, not all rocks exhibit these three phases. For instance, for unconsolidated sands and shales, there is not enough cohesion between the grains. Hence, only a consolidation phase exists. It is here where soil mechanics begins playing an important role, and the Cam-Clay model can be selected to analyze shale behavior. Atkinson and Bransby (1978) discussed in depth the theory of critical state. Because it is complex, here it is presented only in a brief description of its principles. There are important definitions in soils mechanics, such as the Roscoe and Hvorslev surfaces, representing the state boundary surfaces for normal and overconsolidated materials respectively. Elastic wall, critical state line, normal consolidation line, and swelling line are also important elements of soils mechanics analysis. These elements are represented in the q :p :v space, shown in Figure.7, where q and p are known as stress invariants in terms of the effective ' ' ' stresses (, σ σ ) σ defined according to Equations.6 and.7. v is the specific 1, 3 volume of the sample defined as v=1-e, and e is the void space. For a general three dimensional state of stress, q and p become: q p eff eff 1 = q' = 1 = p' = 3 [( σ ' σ ') + ( σ ' σ ') + ( σ ' σ ') ] ( σ ' + σ ' + σ ') (.6) ' ' For a triaxial stress state, where the horizontal stresses are equal ( σ = σ 3 ) ' ' and ( σ = σ σ = ) 1 axial; σ radial, these q and p values are determined by the following equations: 7

47 q p eff eff = q' = σ 1 = p' = 3 axial σ ( σ + σ ) axial radial = τ radial (.7) Atkinson and Bransby (1978) provide the details on how to merge the yield criterion, the hardening rule, the flow rule, and plastic strains into the Cam- Clay model. First, they introduce the concept of elastic wall to show the wall s corresponding yield curve. Figure.8(a) illustrates the concept of elastic wall in the three-dimensional space p :q :v, with its corresponding projections to the p :q and p :v planes shown in Figures.8(b) and.8(c). These projections are the yield curve (L,M,N) on the p :q plane and the swelling line (L,M,N ) on the p :v plane respectively. Atkinson and Bransby (1978) say, For sample states on the elastic wall and below the state boundary surface, the strains will be purely elastic and recoverable. They also define that plastic strains only occur when the sample state touch the state boundary surface, shown in Figure.7. In this sense, the state boundary surface plays equal role to the yield surface illustrated in Figure.5 in pure plasticity. Secondly, the hardening rule is obtained by an isotropic compression and swelling laboratory test on a sample. Typical results of this kind of test are shown in Figure.9. Isotropic compression is represented along the normal consolidation line to point B, then swelling to point D, compression to point B then point C, and finally swelling to point E. It is assumed that the sample behaves elastically 8

48 everywhere except during the loading from B to C where plastic irrecoverable volumetric strain occurs. These particular plastic strains give enough information to calculate the hardening behavior of the rock. A flow rule representation is given in Figure.10. The direction of the vector plastic strain increment δε p, represented by the (QR) vector, is normal to p p the yield curve. The flow rule then relates the gradient ( d dε ) ε / of vector (QR) with the stress applied to the sample represented by (OQ) vector. The Cam-Clay model, which is valid for normally consolidated materials, offers one of the alternatives to relate the components of the elastoplasticity theory. The flow rule is expressed by Equation.8. s v dε dε p v p s = M q' p' (.8) where M is defined as the slope of the critical state line, identified in Figure.11, on the p :q plane. The yield curve associated with this flow rule is calculated by the following equation: q' + ln ' Mp p' ' p x = 1 (.9) 9

49 where p x is the value of p at the intersection of the yield curve with the projection of the critical state line at point X as it is shown in Figure.11. At this particular point, the equation of the Cam-Clay state boundary surface can be obtained and expressed as follows: Mp' q' = ( Γ + λs ks v λs ln p' ) (.10) λ k s s where λ s and κ s coefficients are the slopes for the normal consolidation line and the swelling line respectively, and Γ is defined as the value of v corresponding to p =1.0 knm on the critical state line. Charlez (1997a) published a correlation between λ s and κ s useful over a large range of values, which is shown in Figure.1. There exists a direct relationship between these two coefficients where large λ s values correspond to large κ s values. The state boundary surface intersects the v:p plane along the normal consolidation line where q =0 (see Figure.7), and Equation.10 reduces to: v = N λs ln p' (.11) where N = Γ + λ k. s s 30

50 For the critical state line, the specific volume of the sample v is defined by v = Γ λs ln p', and the Equation.10 simplifies to: q ' = Mp' (.1) The constitutive relationships discussed in this section will be used in this work to predict rock behavior in wellbore stability analysis. Chapter 3 presents the basic formulation of a general boundary value problem using the elastic and poroelastic constitutive relationships..4 FAILURE CRITERION Properly choosing the failure criterion is as important as the correct selection of the constitutive model. The simplest type of criterion is based on the assumption that the system remains mechanically stable until a certain stress or strain failure value is achieved (Charlez 1997b). For instance, in a purely elastic analysis, the stresses are compared against a peak-strength criterion, normally defined in terms of principal stresses. However, the view that the failure of the system depends on a single localized point has been debated and considered pessimistic. On the other hand, when plastic properties of the rock are taken into account, rock behavior is characterized by a yield criterion. In this case, plastic strains develop once the stress state reaches the yield criterion instead of at a peak-strength point. 31

51 .4.1 Tensile Failure Criteria According to McLean and Addis (1990b), tensile failure in a wellbore initiates when the minimum effective stress ' σ min at the wall of the wellbore is greater than the tensile strength of the formation σ t. Then failure occurs when: σ t < σ ' min (.13) He proposes that once tensile failure occurs at the wellbore wall, the criterion to evaluate whether the tensile fracture will propagate inwards the formation is given by the following relationship: P w σ min (.14).4. Compressive Failure Criteria In contrast to the simplicity of tensile failure criterion, compressive criterion requires more analysis. There are numerous failure criteria proposed to predict the failure of rock in compression. One of the well-known criteria is the Mohr-Coulomb class B criterion. This criterion can be expressed in terms of principal stresses as follows: 1+ sin f c cosf σ max po = ( σ min po ) + (.15) 1 sin f 1 sin f 3

52 where c = cohesion of the sample, p o = pore pressure, and f = angle of internal friction. On the other hand, the Drucker-Prager (extended Von Mises) category A criterion is expressed as follows: oct o ( p ) τ = τ + m σ (.16) oct o where τ o and m are Drucker-Prager parameters defined in Equation.17. τ σ oct oct 1 = 3 1 = 3 ( σ σ ) + ( σ σ ) + ( σ σ ) ( σ + σ + σ ) max max int int min int min min max (.17) There are three alternatives in using this criterion for investigating wellbore stability: the outer, the middle, and the inner Drucker-Prager circles. The values of τ o and m for each alternative are given by Equations (.18). McLean and Addis (1990b) discussed the differences in predicting mud weight values as a result of choosing these different compressive failure criteria. His conclusions are ambiguous; he says that by using any of the three alternatives given, they may be successful in one situation, but extremely unrealistic under different conditions. He presented two different cases of wellbore stability in sandstones. For the first case, he concluded that, for vertical wells, the Mohr-Coulomb criterion was in agreement with the inner and middle circle versions of the Drucker-Prager criterion. As wellbore deviation increases, the two versions of the 33

53 Drucker-Prager criterion predicted higher mud density requirements than Mohr- Coulomb. On the other hand, the outer circle version of Druker-Prager was in agreement with real data values of vertical and horizontal wells. However, for the second case, when weaker sand with lower cohesion and friction angle was used, the results between the outer circle version of Drucker- Prager and the real data field were no longer in agreement. He concluded that linear failure criteria are applicable to wellbore stability analysis. Only in the cases of very weak formations with a uniaxial strength less than 1500 psi (10 MPa), a nonlinear criterion may be justified. Figure.13(a) shows the projection of the Mohr-Coulomb criterion and one of the Drucker-Prager circles. Figure 13(b) compares all the Drucker-Prager circles with the Mohr-Coulomb criterion in the π plane (a plane perpendicular to the line defined when the three principal stresses are equal ( σ σ = σ ) max =. int min Outer circle: Middle circle: Inner circle: sin f m = 3 sin f sin f m = 3+ sin f m = 6 sin f 9 + 3sin f τ o ccosf = 3 sin f ccosf τ o = (.18) 3 + sin f τ 0 = 6c cosf 9 + 3sin f.4..1 Is the intermediate stress really important? The importance of whether the intermediate stress (σ int ) should or should not be taken into account in the failure criterion for wellbore stability purposes has been discussed for a long time, and is being debated. Several authors have 34

54 discussed at length the importance of σ int : Mogi (1967), Handin et al. (1967), McLean and Addis (1990a), Addis and Wu (1993), and numerous others. There are a variety of reasons for the disagreement related to the significance of the socalled intermediate stress. Mogi (1967) established that there are three critical disagreement points that arise from experimental uncertainties. First, an unknown degree of anisotropy in rocks, second, inhomogeneity of stress distribution, and third, lack of accuracy of the failure stress measurements. In this respect, Ong and Roegiers (1993) agreed with the influence of anisotropy as source of uncertainty. He pointed out that the usual assumption of rock strength isotropies have been deemed to be inadequate in describing rock failure under field conditions. To address this problem, he used a three-dimensional anisotropic failure criterion in conjunction with the linear elastic theory. The great limitation of this approach is that to fully describe the mechanical response of rock, the number of elastic constants required to perform analyses is five: two moduli of elasticity E 1 and E, two Poisson s ratios ν 1 and ν, and one shear modulus G, information which most of the time is unavailable. Likewise, Handin et al. (1967) established that assumptions such as: constant temperature, constant strain rate, and mechanical properties of rock depending only on the state of stress in the material, cause lack of accuracy of failure criteria. Contrarily, he believed that rock properties are functions at least of these three factors: state of stress, temperature and strain rate. His main conclusions were that in the ductile response region, the Von Mises yield 35

55 condition holds reasonably; in addition, in both the brittle region and the brittleductile transition zone, the shear strength depends on the intermediate stress. Mogi (1967) agreed with the first of Handin s conclusions by saying that In ductile states failure stresses of rocks are roughly independent of pressure, so that the Von Mises criterion seems to apply, as for ductile metals. McLean and Addis (1990a) separated the different peak-strength failure criteria in four categories (A, B, C, and D) shown in Table.. According to McLean, the main problem with many of the criteria which consider the intermediate stress is that they give great importance to the influence of that stress. He recognized its importance, but he believed the manner in which Mogi (1967) incorporated σ int into the failure criterion for competent rock as more reliable. Instead of expressing the stress invariants σ oct and τ oct in the traditional way (Equations.19), Mogi introduces a weight factor, δ, into the normal effective stress σ oct, which is function of the rock properties (Equation.0). He found that for the four brittle rocks that he tested (Westerly granite, Dunham dolomite, Darley Dale sandstone, and Solenhofen limestone), the δ value is nearly the same ( δ 0. 1). τ σ oct oct 1 = 3 1 = 3 ( σ σ ) + ( σ σ ) + ( σ σ ) ( σ + σ + σ ) max max int int min int min min max (.19) where σ oct and τ oct are octahedral normal and octahedral shear stresses. 36

56 1 σ + [ σ + δσ σ ] oct = max int min (.0) where σ max, σ int, and σ min are the maximum, intermediate, and minimum principal stresses expressed at the wellbore wall by: σ θ + σ z σ θ σ z σ max = + + τ θz σθ + σ z σθ σ z σ int = + τθz (.1) σ min = P w where σ r, σ θ, and σ z are the radial, tangential, and axial stresses, and τ θz is the shear component. Besides Mohr-Coulomb and Drucker-Prager, there are many others different compressive failure criteria based on the evaluation of stresses. However, the importance of the intermediate stress and the consideration that failure of rocks depends on a single localized point remains in controversy. Consequently, in recent years, a different class of criterion that evaluates the maximum wellbore closure allowed has emerged Wellbore Closure Wellbore closure depends on the stress-strain response of the rock and the stress field, and it can be used as a criterion to identify wellbore instability. Rather 37

57 than using an ultimate strength limit, wellbore closure is based on the evaluation of strains until certain strain value is achieved. Ewy (1993) defined this criterion based on the clearances needed around the drilling tools to allow them to work properly. The wellbore closure allowed would depend on the size of the tools being used and the kind of operations being carried out. Therefore, it is not unique. The review of the literature showed that the accepted value of wellbore closure allowed is % of the wellbore radius. Due to its simplicity and physical meaning, this is an important parameter in analyzing wellbore stability since wellbore closure directly affects drilling operations. The literature review has shown that most of the research in wellbore stability has focused on two main aspects. These are wellbore instabilities attributed to mechanical and chemical effects. In addition, most of the research has been oriented towards the study of stability in the vicinity of a single wellbore. Significant research has not been conducted in multilateral scenarios to understand the behavior of rock in the region where two or more wellbores intersect, region known as the junction. The primary objectives of this research are as follows: First, to understand rock behavior during drilling of a single wellbore and then of the junction between the mainbore and the lateral hole, and second, to propose strategies for design of multilateral wells. Selection and implementation of an appropriate constitutive model constitutes an important task of this research to understand the effects of rock anisotropy and stress anisotropy on wellbore stability. Other aspects such as geometry and placement of the 38

58 junction and orientation of the lateral wellbore are addressed to evaluate their effect on the stability of the junction. 39

59 Table.1 Classification of wellbore stability models (from Fonseca 1998). Reference Model type Special features Bradley (1979) Fuh et al. (1988) Aadnoy and Chenevert (1987) Linear elasticity Directional wells McLean and Addis (1990) Zhou et al. (1996) Santarelli (1987) Stress-dependent For laboratory analysis. elasticity Includes pre-peak yielding Wang (199) Stress-dependent Includes the chemical Mian et al. (1995) elasticity effect (water content concept) Paslay and Cheatham Linear elasticity Allowancee for fluid (1963) flow Hsiao (1998) Linear poroelasticity Stress at wellbore wall Yew and Liu (199) Mody and Hale (1993) Linear poroelasticity Stress at wellbore wall and chemical effect (osmotic potential Sherwood (1993, 1994, 1995) Wong and Heidug (1995) Detournay and Cheng (1988) Cui (1995) Ewy (1991) Mc Lean (1989) concept) Linear poroelasticity Chemical effect (chemical potential of each species). For laboratory analysis Linear poroelasticity Simulates the instantaneous drilling effect Elasto-pasticity Wetergaard (1940) Veeken et al. (1989) Elasto-plasticity Incorporate hardening and softening behavior 40

60 Table. Categorization of Peak-Strength Criterion. (from McLean 1990a). Linear Criterion Non-Linear Criterion Function of σ x, σ y, & σ z Category A e.g. Drucker-Prager Category C e.g. Pariseau Function of σ x & σ z only Category B e.g. Mohr-Coulomb Category D e.g. Hoek-Brown Single hole Oval Hole Two Adjacent holes Two Independent holes Figure.1 Geometries at the multilateral junction. (from Aadnoy and Edland 1999) 41

61 sy σx σx d= ξrw σy Figure. Definition of independence distance (from Aadnoy and Froitland 1991). Stress (MPa) Tangential stress:elastic Tangential stress:plastic Plastic zone Radial stress:elastic Radial stress:plastic Radius (m) Figure.3 Comparison between stresses for elastic and plastic solution (from Charlez 1997a). 4

62 σ Perfectly plastic Hardening Elastic Figure.4 Elastic, hardening, and perfectly plastic behaviors. ε Figure.5 Yield surface (from Atkinson and Bransby 1978). 43

63 1 3 P Initial material. Grains free after rupture of the bonds. 3. Consolidation Volumetric deformation Figure.6 Physical phases in plastic collapse (from Charlez 1997a). Figure.7 Elastic wall in the three-dimensional p :q :v space (from Atkinson and Bransby 1978). 44

64 Figure.8 Elastic wall and the corresponding yield curve (from Atkinson and Bransby 1978). 45

65 Figure.9 Behavior during isotropic compression and unloading. Hardening law (from Atkinson and Bransby 1978). Figure.10 Strain increments during yield. Flow rule (from Atkinson and Bransby 1978). 46

66 Figure.11 A yield curve as predicted from the Cambridge model (from Atkinson and Bransby 1978) ks λ s Figure.1 Correlation λ s k s (from Charlez 1997a). 47

67 Figure.13 Common yield surfaces (from McLean 1990b). 48

68 Chapter 3: Statement of the problem This chapter presents the basic formulation of a general boundary value problem in elasticity and poroelasticity. Once the formulation is stated, then we switch from a boundary value problem to a wellbore stability problem by imposing a suitable failure criterion. 3.1 ELASTICITY In practice, generally only the simplest rock properties data are available for wellbore stability investigation. Because the linear elasticity theory only requires two parameters: Young s Modulus (E) and Poisson s ratio (ν), to describe a general problem of an isotropic and homogeneous medium, its applicability becomes feasible. The solution of a given problem considering the elasticity theory consists on the determination of the stress, strain, and displacement components. Chou and Pagano (1967) shows in detail the formulation of a general boundary value problem in elasticity. After the formulation of the boundary value problem is presented, analytical equations are obtained to compute stresses around circular wellbores Differential equations of equilibrium For a three dimensional scenario, the equilibrium equations governing variation of stresses in a body from point to point can be expressed as follows: 49

69 = = = z yz xz z y zy xy y x zx xy x F y x z F z x y F z y x τ τ σ τ τ σ τ τ σ (3.1) where τ xy = τ yx, τ xz = τ zx, and τ yz = τ zy. F x, F y, and F z are body forces acting in each direction. Only six (σ x, σ y, σ z, τ xy, τ yz, and τ zx ) of the nine stress components are independent. Equations 3.1 are three equations involving six variables so that additional equations are required for a complete solution of the stress distribution in a body Strain-displacement relationships The additional equations required are in the form of strain-displacement Equations 3.. z w y v x u z y x = = = ε ε ε z u x w y w z v x v y u zx yz xy + = + = + = γ γ γ (3.) which relate the displacements to the point deformations.

70 3.1.3 Stress-strain relationships Preceding sections showed two sets of equations: the equilibrium 3.1 and the strain-displacements Equations 3.. Equations 3.1 involve only stress components while Equations 3. involve strain and displacements. This section presents the relation between these two sets of equations. Up to this point, material properties have not been mentioned. Establishing a relationship between Equations 3.1 and 3. depends on the mechanical properties of the particular material under consideration. For a particular medium considered to be elastic, the equations relating stress, strain, stress-rate, and strainrate take the form of generalized Hooke s law: σ = Eε (3.3) Hooke s law involves only stresses and strains independently of the stressrate or strain-rate and consists of the following equations. ε ε ε x y z 1 = E 1 = E 1 = E [ σ ν ( σ + σ )] x [ σ ν ( σ + σ )] [ σ ν ( σ + σ )] z y y z x z x y γ γ γ xy yz zx 1 = τ G 1 = τ G 1 = τ G xy yz zx (3.4) 51

71 Equations 3.5 are more common expressions of generalized Hooke s law, solved for stresses in terms of the strain components. σ σ σ τ τ τ x y z xy yz zx = Gε = Gε = Gε = Gγ = Gγ = Gγ xy yz zx x y z + λ + λ + λ ( ε ) x + ε y + ε z ( ε x + ε y + ε z ) ( ε + ε + ε ) x y z (3.5) follows: Values G and λ are called Lame s constants, and they are defined as E G = 1 λ = ( + ν ) νe ( 1+ ν )( 1 ν ) (3.6) Displacement formulation of problems in elasticity The 15 Equations 3.1, 3., and 3.5 involve 15 variables (six stresses, six strains, and three displacements). To handle this problem there are some reduction procedures. Here, only the solution in terms of displacement for a three dimensional problem is shown. It consists of three expressions in terms of displacement. 5

72 53 By introducing Equation 3. into 3.5, we get six stress-displacement relationships. x u G x + = λε σ + = y v y u G xy τ y v G y + = λε σ + = x v y w G yz τ (3.7) z w G z + = λε σ + = z u x w G zx τ = + + = z y x ε ε ε ε z w y v x u + + (3.8) where ε is the volumetric strain. Combining Equations 3.7 with Equations 3.1, we have nine equations and nine variables (six stresses plus three displacements). By introducing Equations 3.7 into 3.1, we get three equations in terms of displacements. ( ) 0 = x F u G x G ε λ ( ) 0 = y F v G y G ε λ (3.9) ( ) 0 = z F w G z G ε λ where z y x + + = (3.10)

73 Stresses around boreholes Bradley (1979a) and Fjaer et al. (199) among others present analytical equations to compute stresses around boreholes. They assumed a state of plane strain. This assumption simplifies the computation of stresses around boreholes because the displacement in the direction parallel to the wellbore axis is assumed zero (w = 0). Components u and v are function only of x and y: u=u(x,y), v=v(x,y). Assuming plane strain, the governing equations reduce to eight: two equilibrium equations, three stress-displacement relations, and three strain components. Equilibrium equations: 0 0 = + + = + + y xy y x xy x F x y F y x τ σ τ σ (3.11) Stress-displacement relations: 0 = = + = + + = + + = yz xz xy y x y u x v G y v G y v x u x u G y v x u τ τ τ λ σ λ σ (3.1)

74 55 Strain components: = 0 = = + = = = xz yz z xy y x x v y u y v x u γ γ ε γ ε ε (3.13) These eight equations can be reduced to two equations in terms of displacements u and v. They are the following: ( ) ( ) 0 0 = = y x F y v x u x G v G F y v x u x G u G λ λ (3.14) Bradley (1979a) and Fjaer et al. (199) show a set of equations in cylindrical coordinates r, θ, z, useful to compute stress behavior around wellbores. This set is given by Equations 3.16 that are the solution of Equations Stresses and strains in cylindrical coordinates relate to the cartesian coordinate system according to the set of equations 3.15.

75 56 ( ) ( ) θ τ θ τ τ θ τ θ τ τ θ θ τ θ θ σ σ τ σ σ θ θ τ θ σ θ σ σ θ θ τ θ σ θ σ σ θ θ θ sin cos sin cos sin cos cos sin cos sin cos sin cos sin sin cos xz yz z yz xz rz xy x y r z z xy y x xy y x r = + = + = = + = + + = z w v u r r u z r = + = = ε θ ε ε θ 1 + = + = + = z v w r z u r w r v v u r z rz r θ γ γ θ γ θ θ (3.15) The derivation of the stress solution is in Jaeger and Cook (1979), and the final equations are given by Bradley (1979a) and Fjaer et al. (199). These are as follows: sin cos r r P r r r r r r r r r r w w w w xy w w y x w y x r = θ τ θ σ σ σ σ σ sin 3 1 cos r r P r r r r r r w w w xy w y x w y x = θ τ θ σ σ σ σ σ θ ( ) + = θ τ θ σ σ ν σ σ sin 4 cos 4 4 r r r r w xy w y x v z

76 4 4 σ x σ y rw rw rw rw τ θ 1 3 sin θ + τ 1 3 r = cos θ xy + r r r r τ θ z r ( ) = τ sin + cos 1 + w xz θ τ yz θ r τ rz r ( ) = τ cos + sin 1 w xz θ τ yz θ r (3.16) where P w is the wellbore pressure and r w is the wellbore radius. At the wellbore wall, Equations 3.16 reduce to: σ σ σ τ τ τ r θ z rθ θz rz = P = σ = σ = 0 = = 0 w x v + σ ν ( σ x σ y ) cosθ 4τ xy sin [( σ σ ) cosθ + τ sin θ ] ( τ sin θ + τ cosθ ) xz y x y yz xy θ P w (3.17) Although Equations 3.16 are derived form the assumption of a state of plane strain in a linear elastic and homogeneous material, these equations are useful to understand stress behavior around boreholes. 57

77 3. POROELASTICITY 3..1 Background in poroelasticity A better mechanical properties representation for a rock formation is to consider the existence of void space in the rock. Recently, several authors have contributed to analyze poroelastic response of the rock under stress by developing analytical solutions for a circular wellbore in a homogeneous and isotropic formation, which behaves linearly and according to the poroelastic theory. Detournay and Cheng (1988), Cui et al. (1977), and Bratli et al. (1983) solutions are some of these. Because fluid now occupies the void space, the two components of this new system are solid and fluid. Wang (000) points out that two basic phenomena underlie poroelastic behavior: Solid-to-fluid coupling and fluid-to-solid coupling. The first occurs when a change in applied stresses produces change in fluid pore pressure, and the second when change in fluid pressure produces change in the volume of the porous medium Terzaghi s principle Wang (000) among other authors points out that in general geomechanical studies considering poroelasticity lead to Terzaghi s formulation. Terzaghi conducted, between 1916 and 195, laboratory experiments on rock samples to understand soils behavior. As a result, he derived the consolidation equation in one-dimension for these experiments, which is analogous to the diffusion Equation expressed in 3.18, where p is the excess water pressure, and c c is a diffusivity factor known as consolidation coefficient. 58

78 p t = c c p z (3.18) The effective stress concept (σ eff ), attributed to Terzaghi, is defined as the total stress σ total minus formation pressure P o. σ = σ P (3.19) eff total o This equation has been used extensively in rock mechanics to represent the state of stress of a given fluid saturated porous formation Biot s theory While Terzaghi s approach was derived in one-dimension, Biot introduced in 1941 his three dimensional theory for poroelasticity known as General theory of three-dimensional consolidation. He defined several different coefficients to characterize rock-fluid behavior. They are known as poroelastic material constants. By monitoring the water exchanged by flow into or out of the rock sample, Biot defined a quantity called variation of fluid content, ζ. This quantity is given by ζ= P o /R, and it is related to the proportionality constant called the specific storage coefficient at constant stress, 1/R. The second constant refers to compressibility of the system. It is 1/H, and it is known as the poroelastic expansion coefficient. The third coefficient, the drained bulk 59

79 modulus (K), is obtained by measuring the volumetric strain caused by applied stress, holding the pore pressure constant. Additional coefficients, such as Skempton s coefficient (B) and Biot- Willis coefficient (α), can be derived and expressed in terms of these three main constants. Skempton s coefficient is defined as the ratio of the induced pore pressure to the change in applied stress for undrained conditions (ζ=0). Biot- Willis coefficient is defined as the ratio of increment of fluid content with respect to the volumetric strain holding the pore pressure constant. They are expressed as follows: B = α = R H ζ ε = = K H E K = 3 1 ( + ν ) 3p ( σ + σ + σ ) x y z ζ = 0 (3.0) where ( σ σ + σ )/ 3 x + is the mean normal stress, and p is the excess water y z pressure. According to Wang (000), the three basic material constants (1/R, 1/H, 1/K) characterize the linear poroelastic behavior of a rock-fluid system. A fourth independent constant, shear modulus or drained or undrained Poisson s ratio, is required to complete the poroelastic constitutive equations when shear stresses are present. 60

80 Equation 3.1 now defines the effective stress. The range for α is φ α 1 where φ is porosity. Coefficient α is less than one when the solid is compressible (i.e., the change in volumetric strain is greater than the variation of fluid content). When the solid is incompressible, this coefficient is exactly one (α=1), and the Equation 3.1 simplifies to Terzaghi s equation. σ eff = σ αp (3.1) total o 3.. Stress-strain relationships As in the elastic case, the poroelastic problem consists of calculating stress, strain, and displacement components. Two new variables play a role in poroelasticity: pore pressure and variation of fluid content in the system. Since this procedure is analogous to elasticity analysis, the equilibrium Equations (3.1) and the strain-displacements Equations (3.) fully apply again. The relationship between these two sets of equations is now modified according to Biot s theory. Establishing the relationship between Equations (3.1) and (3.) in poroelasticity depends on the coefficients defined by Biot s theory. The generalized Hook s law is modified by terms that include the pore pressure of the medium as shown in Equations

81 1 ε x = x E 1 ε y = y E 1 ε z = z E 1 γ xy = σ G 1 γ yz = σ G 1 γ xz = σ G 1 ζ = 3H [ σ ν ( σ + σ )] [ σ ν ( σ + σ )] [ σ ν ( σ + σ )] xy yz xz ( σ + σ + σ ) x y y z x z z x y + p + 3H p + 3H p + 3H p R (3.) The basic variables in a three-dimensional problem in poroelasticity include six stress components, three displacements, pore pressure and the variation of fluid content. These eleven unknowns are solved according to eleven equations. They are as follows. First, Equations 3., which include seven equations: six stress-strain relations plus one for pore pressure. Secondly, Equations 3.1 that are the three equilibrium equations, and finally, the diffusion Equation 3.18 obtained by combining Darcy s law with the continuity equation Displacement formulation of problems in poroelasticity Analysis analogous to that done previously for elasticity is applied here to obtain the solution in terms of displacement for a three-dimensional problem. It consists of three expressions in terms of displacement, which include the 6

82 63 contribution of pore pressure. These are analogous to Equations 3.9 and expressed as follows: 0 1 = F x x p u G z x w y x v x u G α ν 0 1 = F y y p v G z y w y v x y u G α ν (3.3) 0 1 = F z z p w G z w y z v x z u G α ν The partial differential equation governing fluid flow is obtained by combining Darcy s law with the continuity equation. A particular expression of this diffusion equation for a poroelastic medium is given by Charlez (1991) and Wang (000) as follows: p k t t p 1 = + µ ε α η (3.4) where µ and k are fluid viscosity and permeability of the porous medium respectively, 1/η is the specific storage coefficient, and ε represents volumetric strain. z y x BK R ε ε ε ε α η + + = = = 1 1 (3.5)

83 The term ε α in Equation 3.4 couples the time dependence of strain t into the diffusion equation for a porous medium Stresses around boreholes To find an analytical solution for stress distribution and displacements around a circular wellbore in a linearly poroelastic formation even considering it is homogeneous and isotropic requires a complex development. Among others, particular solutions can be found in Bratli et al. (1983), Cui et al.(1997), and Detournay and Cheng (1988). Other sources such as Fjaer et al. (199) and Fonseca (1998) refer also to these solutions. The original set of equations for stress distribution in a linearly poroelastic formation where the two horizontal stresses are isotropic can be found in Bratli et al. (1983) and appear in Fjaer et al. (199); they are equations presented in Equations

84 65 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = = + + = ln ln ln ln ln ln ν ν α ν ν σ ν σ σ α ν ν σ σ σ α ν ν σ σ σ θ r r r r r r r P P r r r P r r r r r r r r r P P r r r r r P r r r r r r r r r P P r r r r r P o w w o w w o w o w w hor v z o w o o w o w w o o w o w w hor hor w o o o w o w w o o w o w w hor hor r (3.6) It is assumed that there exists a boundary at a finite radial distance r o, measured from the center of the wellbore. The following condition is assumed (r o >>r w ). 3.3 BOUNDARY CONDITIONS A complete boundary value formulation consists of the governing equations and the boundary conditions. The boundary conditions applicable to the stress-displacement Equation 3.3 and to the diffusion Equation 3.4 depend on

85 the physical phenomenon analyzed. Chapter 4 presents the assumptions to attempt simultaneous solutions for stress-displacement and pore pressure distribution in the physical problem considered in this study. The choice of boundary conditions is made among the following: 1) Essential boundary conditions: The value of the dependent variable is specified at the boundary. In the case in which the displacement components are prescribed over the entire boundary, the boundary conditions for the stress-displacement Equation 3.3 are expressed as follows: ( x, y, z, t) u : v ( x, y, z, t) = v : w ( x y, z, t) = w u =, (3.7) For the diffusion Equation 3.4, the essential boundary condition is pore pressure prescribed over the boundary. In this case, the boundary condition for the diffusion Equation 3.4 is the following: ( x y, z, t) p p, = (3.8) ) Natural boundary conditions: The value of the first derivative of the dependent variable is prescribed at the boundary. The boundary conditions for the stress-displacement Equation 3.3 are the following where the stress, σ, is prescribed over the entire boundary: 66

86 ( x, y, z t) = σ : E v( x, y, z, t) = σ : E w( x y, z, t) = σ E u,, (3.9) For the diffusion Equation 3.4, the boundary conditions are given by the flow rate, q, specified at the boundary. k p, µ ( x, y, z t) = q (3.30) In Chapter 4, it is shown that the complete boundary value formulation of the physical problem is stated in terms of a combination of the boundary conditions here presented. Essential boundary conditions are specified at the outer boundaries for both the stress-displacement and the diffusion equations. Natural boundary conditions are prescribed at the inner boundary for both equations. Becker et al. (1981) define this kind of problem as a mixed boundary-value problem. The physical meaning of imposing these particular boundary conditions is fully explained in Chapter SWITCHING FROM A BOUNDARY VALUE PROBLEM TO WELLBORE STABILITY ANALYSIS The boundary value formulations stated in this chapter allow finding the solution for stress and strain around wellbores. This solution is completely dependent on the constitutive model applied to the analysis. The connection between these boundary value formulations and wellbore stability analysis is given by a failure criterion. Different kinds of failure criteria were mentioned in 67

87 the last chapter. They range from the simplest peak-strength criterion to complex criteria based on a failure surface. In addition, wellbore closure is being recently considered as a criterion to determine wellbore instability. These failure criteria will be used to analyze and discuss wellbore stability in Chapter 5. 68

88 Chapter 4: Numerical Approach to the Solution of the Wellbore Stability Problem The review of literature in Chapter discusses the status of the study of wellbore stability around the junction in multilateral wells and concludes that the current trend towards analyzing this problem is by implementing numerical solutions in 3-D. Finite element techniques have proven to be reliable in areas such as aerospace and structure analysis. As a result of this success, researchers have turned their attention to use finite element theory in modeling geomechanical problems and recently in wellbore stability analysis. This chapter serves to support the decision of using commercial finite element software to conduct this investigation. The second major part of this chapter presents the mathematical representation of the physical phenomena studied. 4.1 COMPUTATIONAL MODELING According to Starfield and Cundall (1988), by comparing rock mechanics problems with other areas of mechanics such as aerospace or structural mechanics, rock mechanics modeling falls into the class of problems dealing with limited amount of data. This leads to the question of why mathematical or computational models are considered viable tools to forecast the behavior of rock in the absence of enough information. One of the reasons to think about computational modeling to simulate rock mechanic problems is accessibility to more versatile and powerful computer packages that have been successfully applied in other areas. As a consequence of this versatility, these computer 69

89 packages have increased their ability to handle geological detail in construction of appropriate models. Easy access to high-performance computers provides to the modeler an important tool. Although the limited amount of geological data is a concern in modeling process, it is necessary to accept and recognize that to reproduce real events; it should be necessary to construct a model with the same complexity as reality. An alternative to overcome this situation is simplifying the model by applying appropriate assumptions Analytical and Numerical solutions The main aspects that support the decision of working with numerical methods are founded on the statements of researchers who support numerical simulation. Hibbit, Karlsson, and Sorensen (000a) state that all physical phenomena behave non-linearly and the three different sources of non-linearity are due to material, boundary conditions, and geometry. The first type of non-linear problem is associated with material properties. As mentioned in Chapter, for unconsolidated sands and shales there is not enough cohesion between grains, and only a consolidation deformation mechanism exists. According to Chen et al. (000), to estimate displacements and changes in stress distribution for these materials, non-linear constitutive models should be considered. Chen agrees with Charlez (1997a) by stating that two conditions are required to obtain an analytical solution from a given initial boundary value problem in elastoplasticity. First, the shape of the plastic zone must be known in advance to couple the elastic equations with the corresponding 70

90 ones for the plastic zone. Charlez stated that although for an isotropic initial stress field the shape of the plastic zone can be known, for an anisotropic stress field this shape remains unknown, and only numerical approaches can be applied to find a solution. The second requirement is that the constitutive models representing material behavior for the elastic and the plastic zones must be expressed linearly in order to superpose their solutions. Other sources of non-linear behavior related to material properties are for materials exhibiting either strain-rate dependence or post-failure behavior. The second source of non-linearity is due to boundary conditions. When boundary conditions change during a particular analysis, non-linearity occurs. For instance, in hydraulic diffusion analysis, boundary conditions may change from no-flow condition to flow condition. This change causes variations in pore pressure, which are time dependent. This phenomenon can be properly incorporated numerically. The third source is related to changes in the geometry of a body during a given analysis. Necas and Hlavacek (1981) pointed out that the concept of small strain tensor, defined in classical mechanics, raises the question of when a given strain tensor represents the real deformation of the body. Hibbit, Karlsson, and Sorensen (000a) explain that when a problem is defined as a smalldisplacement analysis, the problem is linearized, ignoring any possible geometric non-linear response. The alternative to a small-displacement analysis is to include large-displacement effects. This alternative allows taking into account the geometric non-linear response of the body. 71

91 Because some of the sources of non-linearity mentioned before are expected in the physical phenomenon studied in this research, a numerical approach arises as the most likely tool to attempt a solution. Selection of a constitutive model is not arbitrary. As many of the researchers in wellbore stability mention, strictly there will always be a need to compare the model predictions against laboratory data and calibrate the model if possible. ABAQUS version 6., developed by Hibbit, Karlsson, and Sorensen (000a), is a finite element software developed initially to study problems related to structural analysis. Because of its success, it has become a general purpose modeling software package. ABAQUS is equipped to handle different constitutive models to represent material behavior, and as such, it was chosen as the commercial finite element software to conduct this research. 4. CONSTITUTIVE MODELS AVAILABLE IN ABAQUS In former chapters, it was mentioned that the solution to a particular boundary value problem depends on the constitutive model used in the analysis. Since ABAQUS is a general purpose finite element software, it allows considering different constitutive models. These models range from the purely elastic model, passing through models that take into account void ratio such as poroelasticity to complex models that incorporate plasticity. Elastoplastic models, particularly those based on the theory of critical state introduced by Roscoe and Burland during the 1960 s at Cambridge, are considered good tools to reproduce shales behavior. In order to analyze and compare rock behavior with respect to the 7

92 constitutive model, we discuss the following constitutive models: elastic, poroelastic, and poroelastoplastic. For the simplest case of elasticity, Young s modulus and Poisson s ratio are the required parameters. If a porous medium is considered, then in addition to the elastic parameters, the following parameters are required: bulk modulus of rock and fluid, shear modulus, average rock porosity and average rock permeability, densities of rock and fluid contained. Two different elastoplastic models are addressed: Drucker-Prager and Cam-Clay. These models are described by Hibbit, Karlsson, and Sorensen (000a). For these models, in addition to the information mentioned above, hardening and post-failure behaviors of rock are required. The results of a triaxial compression test are required to calibrate the Drucker-Prager model. Two tests are required to calibrate the Cam-Clay model: a hydrostatic compression test and a triaxial compression test. For the Cam-Clay model, laboratory test results must be expressed in terms of critical state variables. The hydrostatic compression test consists of applying equal compression forces in all directions to a rock sample. This test provides the initial shape of the yield surface, a 0. a 0 = 1 e1 exp e0 ks ln λ k s s p 0 (4.1) where e 1 is defined by the intersection of the consolidation line with the void ratio axis as shown in Figure 4.1 while e 0 is the void ratio measured at the beginning of 73

93 the test. The logarithmic bulk modulus λ s, and the swelling coefficient κ s represent the slopes of the consolidation line and the swelling line respectively. p :q were defined in Equations.7 in Chapter. Void ratio, e, is related to the measured volume change as follows: 1 + e exp( ε ) = (4.) 1 + e 0 Details of the methodology of each test can be found in Atkinson and Bransby (1978). Briefly, a compression test consists of measuring the volume of water expelled from the sample at different confining pressures, p. Specific volume values v are obtained from the relative density G s and water saturation S w of the sample v = 1+ e = 1+ S G w s. On the other hand, a triaxial test allows the calibration of the yield parameter M, which is defined as the slope of the critical state line on the p :q plane. 4.3 MODEL DEFINITION Model s geometry for analysis in a single hole A 3-D finite element model (FEM) constructed using hexahedral elements was used to predict the behavior of rock formation surrounding a single wellbore whose diameter is 8 ½ inches ( m). Because the initial state of stress is altered over a distance of 5 to 7 times the wellbore radius, the model consists of a square region of 3.0 by 3.0 meters, which is equivalent to 15 radii. It helps to 74

94 represent better the boundary conditions at infinity. Figure 4. illustrates the model. Mesh refinement calculations were done to assure accuracy of the results. Figures 4.3 and 4.4 show results of mesh refinements for the radial and tangential directions using equally spaced quadratic elements. Rock is assumed to be elastic with the following properties: E=10000 MPa and ν=0.30. The analysis assumes a plane stress condition for a vertical wellbore of radius equal to 0.1 meters in a stress field σ x =σ y =10 MPa. Wellbore pressure imposed is P w =7 MPa. Figure 4.3 shows the radial stress as a function of radial distance varying the number of equally spaced quadratic elements in the radial direction (N r ). The results are compared with the analytic elastic solution. It can be seen how the accuracy of the results increases significantly as N r increases from 7 to 8 elements. This plot shows that accuracy of the results is highly sensitive to mesh size in the radial direction. The relationship between the size of the element and the number of elements in the radial direction (N r ) is as follows: element size=0.164 m. for N r =7, element size=0.08 m. for N r =14, and element size=0.041 m. for N r =8. These results show that to obtain accuracy in results, the size of the equally spaced quadratic elements in the radial direction must be equal or smaller than m. Figures 4.4 and 4.5 illustrate the radial and the tangential stresses, respectively, as a function of radial distance varying the number of quadratic elements in the tangential direction (N θ ) from 8, 16, 3, and 64. Because all the curves in both Figures 4.4 and 4.5 converge to a single curve, it can be concluded 75

95 that mesh size in the tangential direction does not affect accuracy of the results for the radial and tangential stresses in the range of N θ values analyzed. Figure 4.3 also shows that the initial state of stress is altered over a distance of 5 to 7 times the wellbore radius. Beyond this zone, the solution tends to the initial conditions. In order to improve the accuracy of the results in the nearest region to the wellbore, a denser concentration of unequally spaced quadratic elements was applied instead of equally spaced quadratic elements. Figure 4.6 shows these results Drilling simulation in a single hole Traditional stress-displacement analyses assume the wellbore has been previously drilled. It is assumed that a cylindrical hole preexists when the analysis is performed. In contrast, this model allows simulation of the process of drilling in sequential steps. A cylindrical hole also exists in this model, which constitutes the wellbore, but in order to simulate as close as possible the process of drilling a sequence of steps is followed. The first step of a given analysis consists of trying to equilibrate geostatic forces acting on the system. Secondary steps simulate the process of drilling. There are three different strategies to find initial equilibrium and to simulate drilling. The first strategy follows Hibbitt, Karlsson, and Sorensen (000b) methodology when they simulated tunnel excavations. It consists of applying, during the first step, concentrated loads at the nodes located in the wellbore wall (inner boundary). These loads must be in equilibrium with the initial stress field, 76

96 and they are applied as reaction forces to restore as close as possible the state of stress existing before drilling. To do that, it is necessary to determine from a previous independent analysis the magnitude and location of these loads, and then add them to the model manually at the correct nodes of the wellbore wall. Once these loads are applied and equilibrium is achieved during the first step, drilling process simulation begins in a second step by reducing those loads to a value of well pressure desired, one layer at a time until the wellbore penetration is achieved. This procedure is tedious for fine meshes or large models such as those defined in Sections and and illustrated in Figures 4.7 and 4.8 in this Chapter 4. Therefore, this strategy is unpractical and a second and simpler approach may be used. The second strategy consists of applying during the initial step; instead of concentrated loads at nodes, distributed loads on the wellbore wall. These distributed loads are in equilibrium with the initial stress field and equivalent to pressure applied inside the wellbore. Again, once equilibrium is achieved, the second step consists of reducing those loads to a value of well pressure desired. It solves the problem of adding loads manually to the model, which was mentioned before for fine or large meshes. When a highly anisotropic initial stress field exists, an independent analysis has to be done in order to estimate the internal wellbore pressure required to equilibrate the model, and then apply this pressure as a distributed load on the wellbore wall. The third alternative is using what in theory is known as ghost elements. This strategy is based on filling the wellbore with additional ghost elements. In 77

97 this way, the model does not require any load at the inner boundary to find equilibrium. Deactivating those ghost elements that represent drilled rock, by reducing their stiffness to negligible values simulates the drilling process. Then, fluid pressure inside the wellbore is applied in the form of distributed loads on the wellbore wall. In this case, since the ghost elements are deactivated during the drilling step, it is necessary to be careful about boundary conditions and any kind of nodal forces applied to the surfaces that are in contact with those ghost elements. To simulate drilling in a single step, the model consists of (8x64x1) 179 hexahedral elements and 1864 nodes and uses a single thin 0.05 meters layer, as shown in Figure 4.. In order to keep the aspect ratio of the hexahedral elements, it is recommended that the size of the layer in the vertical direction be in accord with the size of the elements in the radial direction. On the other hand, to simulate drilling in a multi-step process, the model uses ten layers, and it consists of (8x16x10) 4480 hexa hedral elements. This kind of model is defined as a multilayer model. Figure 4.7 illustrates this case Model s geometry for analysis in a multilateral scenario A totally different 3-D finite element model shown in Figure 4.8 was constructed by using the pre-processor ABAQUS/CAE for simulation of the lateral junction. The dimensions and characteristics of the model are as follows. The mainbore diameter is 1 ½ inches ( m), and the lateral wellbore diameter is 8 ½ inches ( m). The lateral wellbore is constructed in the 78

98 direction of the σ x principal stress with an inclination of.5 o. Considering the junction angle equal to.5 o, the height of the window created in the mainbore is about 5.05 meters long. It is an axis-symmetric rectangular region 1. by 0.4 by 8.0 meters, consisting of hexahedral quadratic elements and nodes. Figure 4.9 shows a closer view of this model. This representation was chosen after several attempts using different model dimensions. The initial dimensions of the model were 3.0 by 1.5 by 8.0 meters. However, the computer used to carry out the analysis was unable to handle the finite element code needed to perform the numerical calculations of this model; therefore, resizing of the model was done to allow execution of the code. By doing this resizing, the computing problems were solved. Section in Chapter 5 will show how this resizing causes alteration of the stress behavior in the region near to the boundaries, but it does not affect the stress behavior in the region between the holes where instabilities are expected due to the presence of the lateral well Drilling simulation in a multilateral scenario The purpose of this research is to observe the influence of drilling a second hole from the mainbore. Simulation of drilling can also be done in a single or multiple step analysis like in the situation of a single hole. Some restrictions apply for drilling simulation process in a multilateral scenario. Drilling simulation of the lateral wellbore for the multilateral case considers that the mainbore was drilled previously, and it exists at the time the lateral wellbore is being drilled. Because of the complexity and large number of nodes and elements required in 79

99 constructing a model involving a multilateral scenario, only the second strategy previously described is applicable to simulate the drilling of the lateral well. This strategy consists of applying, during the initial step, distributed loads on both wellbore walls the mainbore wall and the lateral wall. Initially, distributed loads applied at the wellbore wall of the lateral well are in equilibrium with the initial stress field while distributed loads applied at the wellbore wall of the mainbore represent hydrostatic pressure created by the drilling fluid. The second step consists of reducing the loads applied at the lateral wellbore wall to a wellbore pressure value equal to the wellbore pressure imposed at the mainbore. 4.4 WELLBORE STABILITY MATHEMATHICAL MODEL This section presents the assumptions and equations to compute stress distribution and displacements around wellbores. Conventional stress analysis is fully coupled with fluid flow equations to attempt simultaneous solutions for stress/displacement and pore pressure distribution General assumptions General assumptions are as follows: Static equilibrium (No inertial forces acting). It is accepted that the model represents rock formation. Rock formation is homogeneous. Temperature remains constant during each particular analysis. 80

100 The axes of the global coordinate system are parallel to the in-situ principal stresses. Mass diffusion process is not taken into account Governing equations Chapter 3 presented the governing equations involved in the solution of a general stress-displacement problem in elasticity and poroelasticity. The final governing equation depends on the constitutive model considered. In general, the final governing equation can be written in vector form as follows: A U + B U + C p + F = 0 (4.3) k where A, B, and C are material constants, and F k represents body forces assuming negligible inertial effects. Diffusion processes occur in porous media. Three different diffusion processes can be identified affecting wellbore stability. They are pore pressure diffusion associated with hydraulic conductivity of rocks, thermal diffusion, and mass diffusion process related to ions exchange between formation fluids and drilling mud. This last is recognized as the chemical effect. According to Charlez (1991) and Wang (000), the first of these three processes can be mathematically represented by the diffusion Equation 4.4. In this equation, 1/η is the specific storage coefficient, ε represents volumetric strain 81

101 defined by the bulk volume variations, α is Biot s coefficient, and L is called latent heat. This diffusion equation coupled with the Equation 4.3 describes a wellbore stability problem considering hydraulic diffusivity. 1 p ε Lρ T k + α = p η t t Tη t µ (4.4) Lomba et al. (000a) developed a model to calculate the transient pressure profiles and solute diffusion through low permeability shales. The solute concentration profile is defined according to the mass diffusion equation: Cs = Deff t C s (4.5) where C s is the concentration of solute, and D eff is the diffusivity of the diffusing material. They found that both hydraulic and mass diffusion processes induce the flow of solute and water. The coupled equation to represent these phenomena is expressed by Lomba et al. (000a) as follows: K t c p I II f nrtk p = c f C s (4.6) 8

102 In this equation, K I was defined as hydraulic diffusivity and nrtk II compressibility while represents diffusivity. c f c f is fluid The mathematical relationship that describes a problem considering both hydraulic and mass diffusion processes can be obtained from the Equations 4.4 and 4.6 and expressed as follows: 1 + α η t p ε Lρ T I II t Tη t K = c f nrtk p + c f C s (4.7) Hence, to describe a wellbore stability problem in elasticity and poroelasticity, the stress-displacement formulation given by Equation 4.3 couples with Equation 4.7. Because ABAQUS version 6.1 is not equipped to couple both the hydraulic and the mass diffusion processes with the stress-displacement problem, the pore pressure alteration induced by chemical potential cannot be quantified Isothermal analysis Taking into account the general assumptions stated in Section 4.4.1, we obtain the following for the particular case of an isothermal analysis, where Lρ Tη T t = 0 (4.8) 83

103 Equation 4.7 reduces to: 1 + α η t p ε I II t K = c f nrtk p + c f C s (4.9) Hydraulic diffusion analysis Because hydraulic diffusivity is the diffusion process addressed in this research, the second term in the right hand side of the Equation 4.9 becomes zero, and the Equation 4.9 reduces to the following. 1 p ε + α η t t = K c f I p (4.10) Both Equation 4.3 and Equation 4.10 constitute the mathematical representation of the physical phenomena studied in this research Phenomena in steady state Stress-displacement analysis in Elasticity The simplest case to analyze is to consider a rock formation, which behaves according to the linear elastic theory. In addition to the general assumptions stated in Section 4.4.1, these other assumptions are required: 84

104 Rock s porosity is negligible such that the influence of fluid contents on rock behavior is not taken into account. Rock s behavior can be modeled as a perfect elastic material. Because porosity is assumed negligible, diffusive processes do not occur. No time dependent effects are involved (i.e., the rate of deformation is independent of the rate of loading). Because of these assumptions, all the terms in Equation 4.10 vanish, and the Equation 4.3 constitutes the mathematical representation this stressdisplacement problem. In this case, Equation 4.3 is rewritten as follows: ( G) U + G U + F = 0 λ (4.11) + k where the material constants A, B, and C have been substituted. A = B = G C = 0 ( λ + G) Equation 4.11 is the vector form of Equations 3.9 derived in Chapter 3, which are rewritten in Equations

105 ε ( λ G) + G u + F = 0 + x x λ + G + G v + Fy y = (4.1) λ + G + G w + Fz z = ( ) 0 ( ) 0 Boundary conditions. The first alternative to represent boundary conditions at far field is by using what in ABAQUS is defined as infinite elements. Hibbitt, Karlsson, and Sorensen (000a) suggest that these infinite elements can be used in conjunction with finite elements in boundary value problems defined in unbounded domains where the region of interest is relatively small compared to the surrounding medium. Infinite elements were applied at far field in the model; however, the computer used to perform the analysis was unable to handle the finite element code needed to perform the numerical calculations due to memory capacity. Therefore, a second alternative was assumed, which solved the problem. This second alternative consisted of specifying the magnitudes of the displacements at far field equal to zero. Equations 4.13 give these boundary conditions. u v ( xb, yb, zb ) ( xb, yb, zb ) (, y, z ) w x or b U ( x b b, y b b, z b = u = v b b = w ) = U = 0 = 0 b b = 0 = 0 (4.13) 86

106 The boundary condition at the wellbore wall is given by the first derivative of the displacements. Stress is specified at this boundary and represented by Equation E U ( x, y, z ) = P, (4.14) o o o w where E is Young s modulus representing the material properties, and P w is the wellbore pressure Stress-displacement analysis in Poroelasticity To perform this kind of analysis, in addition to the general assumptions imposed in Section 4.4.1, the following assumptions are required. A single fluid saturates the porous medium. Drilling fluid (mud) creates a membrane on the wellbore wall representing filter cake. Permeability of this membrane is low enough to be neglected. The filter cake is assumed impermeable. Mass diffusion (chemical interaction) is neglected under the assumption that the filter cake acts as a perfect barrier impeding filtrate to invade formation. Under this condition, it is assumed that in-situ formation fluids do not get into contact with the drilling fluid. This condition avoids ion exchange into or out the formation, and chemical interaction between fluids can be neglected. 87

107 88 No time dependent effects are involved (i.e., the rate of deformation is independent of the rate of loading). Because of these assumptions, all the terms in Equation 4.10 vanish, and Equation 4.3 constitutes the mathematical representation of the stressdisplacement problem in poroelasticity. In this case, Equation 4.3 is written as follows: 0 1 = + + F k p U G U G α ν, (4.15) where the material properties are expressed as follows: α ν = = = C G B G A 1 In expanded form, Equation 4.15 is written as follows: 0 1 = F x x p u G z x w y x v x u G α ν 0 1 = F y y p v G z y w y v x y u G α ν (4.16) 0 1 = F z z p w G z w y z v x z u G α ν

108 Boundary conditions. before. The boundary conditions are given by Equations 4.13 and 4.14 as stated Transient phenomena Until recently, wellbore stability had been mainly analyzed as a steadystate phenomenon. The review of literature in Chapter showed that most authors actually recognize wellbore instability as a time dependent problem. Charlez (1997a) classifies these time dependent problems in two categories. First, deformation and rupture in rocks exhibiting plastic behavior. Second, diffusion processes through porous medium Rate of Deformation Multi step drilling analysis (MSDA) has the purpose of simulating the first of these effects, material deformation as a function of time, rate of deformation. To achieve this, rather than considering that the borehole is drilled instantaneously, MSDA considers the process of drilling in sequential steps. This condition gives the model the opportunity of behaving as function of time. Each step in an analysis is divided into multiple increments. The user defines the total time of each step and suggests the first time increment. Then ABAQUS controls automatically time increments during a step to obtain a solution in the least 89

109 possible computational time. These time increments depend on the severity of the nonlinear response of each particular problem. Because this kind of analysis is based on a multi-layer model, drilling process can be simulated in several drilling steps according to the number of layers. The multi-layer model and the simulation of the drilling were described in Section In order to set mathematically this problem, these other assumptions are considered in addition to the general assumptions defined in Section 4.4.1: Rock s porosity is negligible such as the rock behaves as a solid. Rock s behavior obeys to an elastoplastic constitutive relationship. Because porosity is assumed negligible, diffusion processes do not occur. As a consequence of these assumptions, the only term remaining in Equation 4.10 is the coupled term representing the rate-dependent deformation behavior of the material, which is given according to the equation: w r = ε (4.17) t Because an elastoplastic model is being used to carry out the analysis for this particular case, stress-strain relationships are non-linear. Equation 4.3 is no longer applicable because of plastic material properties. In order to set up this new 90

110 problem, rather than defining the problem in terms of differential equations, variational principles for energy are applied. Necas and Hlavacek (1981) and Doltsinis (000) define a stress-strain problem of equilibrium in terms of the principle of virtual work for a static stress field σ as follows. The virtual work of the internal forces (left hand side of Equation 4.18) equals the virtual work of the external forces, which are body forces per unit volume f at any point within the material V plus surface tractions per unit area t on the surface S bounding this volume (right-hand side). Equation 4.18 represents a complete statement of the stress-strain problem of equilibrium in terms of displacements in three-dimensions. σ : δddv f δvdv + t δvds, (4.18) V = V S whereδd is defined as a virtual rate of deformation, and δv is a virtual velocity field. This equation is the basic equilibrium statement for the formulation of a problem in the finite element theory. Equation 4.18 coupled with Equation 4.17 represents the physical phenomenon of stress behavior and deformation in rocks exhibiting plastic behavior. Initial conditions. This is given according to the strain rate dependence of the material. 91

111 ε ( x, y, z,0) = ε (4.19) 0 Boundary conditions. The boundary conditions are given by Equations 4.13 and Coupled stress-hydraulic diffusion analysis Another cause of wellbore instabilities associated with time is the fluid diffusive process through a porous medium. In order to observe the response of a porous formation, Charlez (1997a, 1997b) proposed the simulation in two different steps, which must be carried out successively. The first step is a simulation of the drilling process, which consists in decreasing the pressure applied in the wellbore. The second step simulates the hydraulic diffusion response of the porous medium. In order to set this problem mathematically, in addition to the general assumptions defined in Section 4.4.1, these assumptions are required: A single fluid saturates the porous medium. The drilling fluid (mud) creates an impermeable membrane on the wellbore wall (filter cake). Under this condition, hydraulic diffusion is allowed within the system, but mass diffusion (chemical interaction) is neglected. Because the filter cake acts as a perfect barrier impeding filtrate to invade formation, it is 9

112 assumed that in-situ formation fluids do no get into contact with the drilling fluid. This condition avoids ion exchange into or out the formation, and chemical interaction between fluids can be neglected. Equation 4.10 fully applies to this case, and it is rewritten in Equation p ε + α η t t = K c f I p (4.0) Equation 4.0 coupled with Equation 4.18 constitutes the mathematical representation of the coupled stress-hydraulic diffusion problem. Initial conditions. p( x, y, z,0) = p ( x, y, z,0) 0 ε = ε 0 (4.1) Boundary conditions. Conditions during the drilling step at the inner and the outer boundaries are respectively specified in Equations 4. and 4.3. Due to the existence of the 93

113 filter cake, no flow condition at the wellbore wall is imposed. It is represented by Equation 4.. k p µ ( x, y, z, t) = 0 o o o (4.) Because the outer boundary is assumed at a finite distance far away from the wellbore, r o, a pressure boundary condition, Equation 4.3, is prescribed at this boundary. This equals the initial pore pressure of the porous medium. ( xb yb, zb, t) po p, = (4.3) 4.5 SOLUTION METHOD USED IN ABAQUS Hibbitt, Karlsson, and Sorensen (1998) give a complete description of the formulation of a strain-stress finite element analysis. This section only describes the basics of this formulation. Equilibrium in terms of the principle of virtual work is defined according to the following equation. σ : δddv f δvdv + t δvds (4.4) V = V S contained, f w. For a porous medium, body forces f include the weight of total liquid 94

114 f w ( S φ + φ ) ρ g = (4.5) w t w The term ( S φ ) w φ + in Equation 4.5 includes the fraction of water that is free to move through the porous medium, t S w φ, plus the volume of irreducible water per unit of total volume, φ t. S w is the water saturation that is free to move, S w = Vw Vp. φ is porosity, = V p Vb gravitational acceleration. φ. ρ w is water density, and g is the Because I N represents the internal forces and P N represents the external forces, the virtual work equation can be rewritten as Equation 4.6. I I N N = P P N N = 0 (4.6) where I N and P N are respectively. I P N N = = V σ : δddv V f δvdv + t δvds S (4.7) When Equation 4.7 is discretized in terms of the virtual velocity field δ v and the virtual rate of deformation δ D, the resultant system of equations forms the basis of a finite element analysis. It can be expressed in the following form. ( ) = 0 F N x N (4.8) 95

115 F N is the force component associated to the current approximation of x N for a system of N equations and N unknowns. For non-linear problems, ABAQUS uses Newton s method as a numerical technique for solving the non-linear equilibrium Equation 4.8. Newton s method assumes that after iteration i+1, an approximation N xi 1 + to the solution has been obtained. The difference between this solution and the solution at iteration i is expressed by the term N dxi 1 +. At this stage, the approximate solution is then. N N N xi+ 1 xi + dxi + 1 = (4.9) in N F i Convergence of Newton s method is achieved by ensuring that all entries (residual forces) and N dxi 1 + are sufficiently small. 4.6 WELLBORE INCLINATION AND AZIMUTH VARIATION To analyze the effect of wellbore inclination and azimuth variation, two different alternatives can be used. The first is Fjaer s (199) approach. The second alternative is by constructing a particular tri-dimensional grid for each desired combination of inclination and azimuth of the wellbore. This can be done using the pre-processor ABAQUS/CAE. Fjaer s approach is used in this study to analyze the effect of wellbore inclination and azimuth variation on wellbore stability. He proposes a stress transformation from a global coordinate system (x,y,z ) where the axes are parallel to the direction of the principal stresses to an 96

116 orthogonal local coordinate system (x,y,z) where the z-axis is parallel to the axis of the borehole. The global coordinate system (x,y,z ) is oriented so that the x - axis is parallel to the maximum horizontal stress σ H, y -axis is parallel to the minimum horizontal stress σ h, and z -axis is parallel to the vertical stress σ v. Figure 4.10 shows this coordinate system transformation for a deviated well. This transformation is expressed by using the direction cosines which depend on the azimuth a and the wellbore inclination i. To transform from the (x,y,z ) to the (x,y,z) coordinate system, the azimuth a is defined as the angle between the x -axis and the projection of the x-axis on the (x :y ) plane while the inclination i is defined as the angle between the z -axis and the z-axis. The direction cosines matrix [l], and the transformation of the in-situ stress tensor [σ ] from the global to the local frame [σ 0 ] are given by the following equations. [s 0 ] = [l] [s ] [l] T where cos acosi sin acosi [ l] = sin a cosa cos asin i sin asin i σ H 0 0 ' σ = 0 σ h σ v o o o σ xx τ xy τ xz o o o o σ = τ yz σ yy τ yx o o o τ zx τ zy σ zz sin i 0 cosi (4.30) 97

117 The final expressions are as follows: ' l xx ' l xy ' l xz ' ' = cos a cos i = sin a = cos a sin i l yx ' ' = sin a cosi = cosa = sin asin i (4.31) l yy ' ' = sin i l = 0 = cosi yz l zx l zy l zz Expressed in the (x, y, z) coordinate system, the in-situ stresses σ H, σ h, and σ v become: σ σ σ τ τ τ o x o y o z o xy o yz o zx = l = l = l xx yx zx xx ' σ ' σ ' σ yx H H H + l + l + l H xy yy zy ' σ ' σ ' σ h h h xy + l + l + l yy xz yz zz ' σ ' σ ' σ h v v v = l ' l ' σ + l ' l σ + l ' l ' σ (4.3) = l = l yx zx ' l ' σ + l ' l σ + l zx xx H H yy zy zy ' l ' σ + l ' l σ + l xy h h xz yz zz ' l ' l yz zz xz ' σ ' σ v v v 98

118 e 1 = locates initial consolidation lnp =0 e -k s = slope of swelling line -λ s = slope of consolidation line ln p Figure 4.1 Pure compression behavior of clay (from ABAQUS/Standard User s manual, Version 6.1, 000). 99

119 Figure 4.. Model mesh for a single hole one step. 100

120 Stress (MPa) Radius (m) Exact Nr=7 Nr=14 Nr=8 Figure 4.3 Effect of mesh refinement in the radial direction on the accuracy of radial stress calculations. Stress (MPa) Radius (m) N=8 N=16 N=3 N=64 Figure 4.4 Effect of mesh refinement in the tangential direction on the accuracy of radial stress calculations. 101

121 Stress (MPa) Radius (m) N=8 N=16 N=3 N=64 Figure 4.5 Effect of mesh refinement in the tangential direction on the accuracy of tangential stress calculations. Stress (MPa) Radius (m) Exact Numerical Figure 4.6 Improved accuracy obtained of radial stress calculations in the nearest region to the wellbore when using unequally spaced elements. 10

122 Figure 4.7 Multi-layer model for multi-step drilling. 103

123 Figure 4.8 Mutilateral mesh scenario (open view). 104

124 Figure 4.9 Mutilateral mesh scenario (close view). 105

125 σ v z z i y σ h a y x θ σ H x Figure 4.10 Transformation system for a deviated well (from Fjaer et al. 199). 106

126 Chapter 5: Discussion of Results This chapter presents the discussion of results obtained with the models described in Chapter 4. This chapter is divided into two major sections. The first contains the discussion related to wellbore stability analysis in a single hole while the second deals with wellbore stability in multilateral scenarios. In order to present the results in an orderly manner, these sections are also divided into two subsections: phenomena in steady state and transient phenomena. This chapter also presents the discussion of the effect of taking into account rock anisotropies on the stability of an inclined wellbore. A series of plots and tables are presented to analyze and discuss stress distribution around wellbores. 5.1 STABILITY OF A SINGLE WELLBORE Phenomena in Steady State Effect of assuming different constitutive models: stress-displacement analysis The objective of these analyses is to point out how constitutive models impact stress behavior around wellbores. This section presents the results of simulations carried out assuming rock properties correspond to a poorly consolidated, soft, homogeneous, and isotropic shale formation. The first analysis is performed assuming this rock can be characterized as an elastic material; the second analysis is conducted assuming this rock follows elastoplastic behavior. 107

127 Sections and in Chapter 4 defined the geometry of the model and the mathematical representation for stress-displacement analysis around a single hole respectively. The initial state of stress imposed for this case is the corresponding to a normally stressed region, σ x =10 MPa, σ y =10 MPa, and σ z =30 MPa, where the maximum principal in-situ stress is vertical and the other two principal in-situ stresses are horizontal and equal or nearly equal. Wellbore pressure is P w =0 MPa. The rock is homogeneous and isotropic which behaves as a linear elastic material with the following properties: Young s Modulus, E=10000 MPa and Poisson s ratio, ν=0.5. Figure 5.1 illustrates the behavior of the radial and tangential stress components assuming the rock is an elastic material. Zero value in the axis of radius represents the wellbore wall. Two different constitutive models were used to characterize rock formation as an elastopastic material: Drucker-Prager and Cam-Clay models. The Drucker-Prager parameters were obtained from a triaxial test for a clay soil published by Atkinson and Bransby (1978). The results of this test are presented in Table 5.1. Confining pressure and pore pressure were 40 and 80 MPa respectively. The Cam-clay parameters were obtained from the same triaxial test data presented in Table 5.1 and an isotropic compression test for a clay soil also published by Atkinson and Bransby (1978). The results from this isotropic compression test are shown in Table 5. Figure 5. illustrates the comparison of behavior of the tangential component of stress for a rock characterized by the Cam-Clay model versus the elastic solution. A substantial relaxation of the tangential stress is observed in the 108

128 region near the wellbore. Many authors, Fjaer (199) and Charlez (1997) among them, recognize this region as the plastic zone. This relaxation zone is attributed to high effective stress concentration, which causes the plastic response of the rock. For this particular case, the plastic zone extends approximately 0.05 meters into the formation, which approximately is equivalent to one half of the wellbore radius. Figure 5.3 shows a contour plot of the Mises stress for this case where Mises stresses are expressed in [MPa]. The extent of the plastic zone, where a high stress concentration occurs, is shown in red. Figure 5.4 shows the comparison of the tangential stress component for the three cases: elastic, Cam- Clay, and Drucker-Prager solutions. Both Drucker-Prager and Cam-Clay curves exhibit a maximum stress level inside the formation and a slightly lower stress level at the wellbore wall. The radial component of stress is only slightly affected. Comparison of radial stress behavior for these cases is shown in Figure 5.5. Plotting the effective mean stress versus the effective Mises stress for the elements in the immediate vicinity of the wellbore helps to visualize how the relaxation of the tangential stress in this zone increases stability. Figure 5.6 illustrates this plot for the three cases analyzed: elastic, Cam-Clay, and Drucker- Prager. It shows how the effective stresses increase with respect to the initial state of stress. The elastic solution exceeds the hypothetical failure envelope while Cam-Clay and Drucker-Prager solutions remain in the stable region. It is interesting to remark that the three different effective Mises stresses for the elastic case shown in Figure 5.6 constitute three different regions in the model, regions A, B, and C. The region A represents the group of nodes that form the wellbore 109

129 wall while regions B and C represent those nodes at a distance 0.05 and 0.05 meters within the formation respectively. Instead, for the Cam-Clay and the Drucker-Prager models, the Mises stresses for these three regions A, B, and C tend to converge to a single point. A simple sensitivity analysis was done for the parameters involved in the Cam-Clay model. Figure 5.7 shows the effect of varying M, the slope of the critical state line on the p :q plane, on the tangential stress behavior. All other parameters remained constant. It can be seen how low M values increase the extent of the plastic zone and produce additional relaxation of the tangential stress. These results are in agreement with results published by Charlez (1997) showing the physical effect of varying parameter M. He concluded that low M values relax the tangential stress at the borehole wall. Hole closure is computed from the maximum radial displacement calculated at the wellbore wall. Table 5.3 shows wellbore closure computations as a percentage of the wellbore radius for the different M values. This is called case A. The same sensitivity analysis was done for different stress levels. A tectonically active stressed region, where all the principal in-situ stresses are unequal is assumed. The initial state of stress is σ x =15 MPa, σ y =10 MPa, and σ z =0 MPa for case B, and σ x =5 MPa, σ y =0 MPa, and σ z =30 MPa for case C. Wellbore pressure is P w =0 MPa for both cases. Wellbore closure results are shown in Table 5.3. From these results, it can be seen that when low or intermediate stress levels are applied such as cases A and B, changing the value of parameter M does not modify significantly hole closure. However, at a higher 110

130 stress levels such as the applied in case C, a slight decrement of M value (e.g. M=.0 to M=1.9) affects significantly hole closure. It can be concluded that for any stress condition, cases A, B, and C, low M values lead to additional hole closure. In order to analyze the effect of bulk modulus λ s and swelling coefficient κ s parameters on stress behavior, calculations were carried out for different λ s and κ s values. Three particular cases are based on data for various clays published by Atkinson and Bransby (1978). These data are given in Table 5.4. In Section.3., Chapter, it was shown that there exists a direct relationship between these two coefficients. In general, large λ s values correspond to large κ s values. The results of tangential stress calculations for these three cases are shown in Figure 5.8. It can be seen how the tangential stress in the nearest region to the wellbore wall relaxes for London clay and Kaolin samples. The largest relaxation occurs for Kaolin, which has the highest λ s value and the intermediate κ s value even though Kaolin has the highest M value. On the other hand, there is not relaxation of the tangential stress when low λ s and κ s values are applied such as for Weald clay. Hole closures for these three cases are shown in Table 5.5 and compared with hole closure for the elastic case. Only Kaolin sample experiences an additional hole closure. Equation 4.1 states that λ s and κ s values define the initial shape of the yield surface for the Cam-Clay model. Therefore, these results demonstrate that high λ s and κ s values are associated with additional relaxation of the tangential stress and increase of the extent of the plastic zone. The shadowed zone 111

131 in Figure 5.8 shows the difference between the plastic zones computed for London clay and Kaolin samples Effect of wellbore inclination and azimuth variation: stressdisplacement analysis The model geometry and mathematical representation for stressdisplacement analysis around a single hole was defined in Sections and in Chapter 4. This section discusses the results of analyzing the effect of wellbore inclination (i) and azimuth variation (a) on wellbore stability. Because this analysis is done assuming rock is homogeneous and isotropic, this section also serves as a foundation to later discuss results for an anisotropic porous medium. The effect of wellbore inclination and azimuth variation on wellbore stability has been widely discussed in literature. Bradley (1979a), Aadnoy and Chenevert (1987), and Zervos et al. (1998) among others discussed this topic. Bradley (1979a) concluded that in normally stressed regions (σ v >σ H =σ h ), vertical wellbores are more stable to collapse and to fracture than inclined wellbores. Aadnoy and Chenevert (1987) agreed with Bradley s conclusion when they reported that isotropic formations become more sensitive towards collapse the higher the wellbore inclination. They also concluded that in a tectonically active region (σ v >σ H >σ h ), stability regarding collapse could be improved by orienting the wellbore in the same direction as the minimum principal in-situ stress. Zervos et al. (1998) conducted elastoplastic finite element analysis of inclined wellbores assuming an isotropic formation. They reported that for the particular stress 11

132 condition they imposed into their analysis (σ v >σ H >σ h ), hole closure in general increases with wellbore inclination. They also concluded that in wellbores with inclinations from 30 o to 60 o the role of the azimuth is important when analyzing wellbore stability towards collapse. Finally they stated that wellbores with inclinations up to 15 o can be treated as vertical wellbores, and wellbores with inclinations more than 75 o can be treated as horizontal wellbores. A total of 90 different runs were completed to analyze the effect of wellbore inclination and azimuth variation on the stability of a single wellbore in a homogeneous and isotropic formation. The states of stresses imposed for this parametric study are shown in Table 5.6. The minimum horizontal stress, σ h, is always assumed /3 times the maximum horizontal stress, σ H. These states of stresses correspond to a tectonically active stressed region, where all the principal in-situ stresses are unequal, and the maximum is not necessarily vertical. They were associated with depth as follows: Shallow: Intermediate: Deep: σ H >σ h >σ v σ H >σ v >σ h σ v >σ H >σ h Three different kinds of plots are used to discuss the results obtained. The first kind of plot illustrates the variation of the maximum Mean effective stress p and maximum Mises effective stress q calculated at the wellbore wall when the inclination angle varies from 0 o, 30 o, 45 o, 60 o, and 90 o. The second kind of plot 113

133 illustrates behavior of the maximum Mean effective stress p and maximum Mises effective stress q on the p :q plane. The third kind of plot shows the maximum hole closure, calculated from the maximum radial displacement at the wellbore wall expressed as a percentage of the wellbore radius. Figures 5.9 through 5.0 serve to present the results of stress-displacement analysis assuming rock is homogeneous, isotropic, and behaves as a linear elastic material with the following properties: E=10000 MPa and ν=0.68. Figure 5.9 shows the representation of the principal in-situ stresses in a formation at a shallow depth. For a deviated wellbore in a shallow formation oriented with azimuth zero degrees (a=0 o ), parallel to the direction of the maximum horizontal stress (σ H ), Figure 5.10a shows that increasing the inclination angle (i) in the range from 0 o to 60 o reduces p and q values. For inclination angles higher than 60 o, p and q values show a slight increment. This behavior is also seen in Figure 5.11 when a=0 o. The stress pair of points (p,q ) move down and shift to the left in the direction of lower effective mean stress when (i) is between 0 o and 60 o. For inclination angles higher than 60 o, (p :q ) values move up and to the right in the direction of higher effective mean stress. On the other hand, when a deviated wellbore is oriented with azimuth (a=90 o ), parallel to the direction of the minimum horizontal stress (σ h ), Figure 5.10c shows that p and q values increase as the inclination angle increases. Figure 5.11 shows how stresses increase as inclination angle increases from 0 o to 90 o when a=90 o. These behaviors indicate that increasing the inclination angle in parallel direction to the maximum 114

134 horizontal stress (σ H ), azimuth zero (a=0 o ), improves wellbore stability regarding collapse in a shallow formation. Figure 5.1 corroborates this conclusion because hole closure decreases as inclination increases when a=0 o. Hole closure values are smaller for (a=0 o ) than the other hole closure values computed for the other azimuth values. Figure 5.13 shows the representation of the principal in-situ stresses in a formation at an intermediate depth. Figures 5.14 and 5.15 show that in general, a deviated wellbore in an intermediate formation is more stable towards collapse than a vertical wellbore because p and q stresses decrease as inclination increases. A wellbore oriented with azimuth zero degrees (a=0 o ), parallel to the direction of the maximum horizontal stress (σ H ), represents the most stable condition. Figure 5.16 shows that hole closure values are smaller for (a=0 o ) than the other hole closure values computed for different azimuths in the range of inclination angles from 30 o to 90 o. For deep formations, the behavior of p and q varies with respect to the shallow an intermediate formation cases. Figure 5.17 shows the representation of the principal in-situ stresses in a formation at a deep depth. The results indicate that in general, increasing the inclination angle causes increment of p and q values as shown in Figure The behavior shown in the p :q plane, Figure 5.19, indicates that in general a wellbore becomes unstable with tendency towards borehole collapse as inclination increases. Hole closure behavior in Figure 5.0 confirms this statement. The behavior shown in these plots suggests that at deep depths, wellbore trajectories close to the vertical should be pursued. However, 115

135 many times drilling oriented wells is needed to reach hydrocarbons zones. When this happens, we can infer from Figures 5.18 through 5.0 that drilling deep deviated wells with an azimuth (a=90 o ), parallel to the minimum horizontal stress (σ h ), and inclination angles less than (i=45 o ) constitutes the least adverse wellbore stability condition. Particular statements for the limit case (i=90 o ) can be done. For instance, the results in Figures 5.19 and 5.0 indicate that drilling a horizontal well (i=90 o ) in parallel direction to the minimum horizontal stress (a=90 o ) is the most stable condition. In contrast, drilling a horizontal well in parallel direction to the maximum horizontal stress (a=0 o ) is the least stable condition. The same analysis of the effect of wellbore inclination (i) and azimuth variation (a) on wellbore stability is now done taking into account the non-elastic behavior of rock. The Drucker-Prager model is used to predict the mechanical behavior of rock whose elastic properties are assumed to be the same as the analysis previously presented: E=10000 MPa and ν=0.68. Two different yield stress values (Y o ) are imposed, and the results are compared with the elastic case. First, the yield stress value is assumed to be equal to the magnitude of the minimum horizontal stress, Y o = 67 MPa. The second analysis is done assuming an arbitrary lower yield stress, Y o = 0 MPa. The results obtained for the cases of shallow and intermediate formations taking into account the non-elastic behavior of rock are the same as the results obtained previously assuming the elasticity theory. For this reason they are not shown and no further discussion is needed. These results indicate that at the low and intermediate stress levels imposed in this 116

136 analysis, the non-elastic response of rock is negligible. However, when the analysis is done at a higher stress level, corresponding to deep formations, the results change. Figures 5.1 through 5.3 serve to present the discussion about these results. Some of the conclusions achieved previously remain valid. For instance, wellbore trajectories close to the vertical should be pursued at deep depths. In the same way, when needed, oriented wells should be drilled parallel to the minimum horizontal stress (a=90 o ). Figures 5.1 through 5.3 support these statements and serve to discuss the effect of varying yield stress on wellbore stability. Figures 5.1 show that the Mises stress curves for Y o = 67 MPa follows the same behavior as the Mises stress curves for the elastic case until certain inclination angle is reached. These curves separate at a different deviation angle. Figure 5.1a shows that when azimuth=0 o, both curves separate at a deviation angle (i=30 o ). Figure 5.1b shows that they separate at a deviation angle (i=45 o ) when azimuth=45 o, and finally, when azimuth=90 o, they separate at a deviation angle (i=60 o ) as shown in Figure 5.1c. Figure 5. shows that the magnitude of hole closure computed when Y o = 67 MPa is in all cases equal to the magnitude of hole closure computed when pure elasticity is assumed. Different results are found when a lower yield stress is imposed, Y o = 0 MPa. Figure 5.1 shows that in general Mises stresses experience a significant relaxation. Figure 5. show that although hole closure trends are the same, hole closure magnitudes increase. Comparing the magnitudes of hole closures between the cases Y o = 67 MPa and Y o = 0 MPa results in differences up to 14 %. 117

137 When plotting the results on the p :q plane (see Figure 5.3), the separation points previously described in Figure 5.1, are visualized again. These separation points (i=30 o, i=45 o, and i=60 o ) indicate that for these particular cases (a=0 o, a=45 o, and a=90 o ) the rock behaves elastically whenever wellbore inclination angles (i) remain equal or lower than these values. When a wellbore is inclined at a higher angle than these values, the rock is likely to exhibit nonelastic response. It is important to note that analyzing wellbore stability using a different constitutive model than the elasticity theory requires using both a stress failure criterion and a strain failure criterion. To demonstrate this statement, let compare the particular case when Y o = 67 MPa versus the elastic case. Figure 5.3a shows the comparison of stresses in the p :q plane for both cases: elastic and elastoplastic. p and q stresses for both cases exceed the failure envelope. This indicates that according to the stress failure criterion imposed by the failure envelope, rock formation is unstable and fails. However, according to the strain failure criterion (e.g., maximum % of hole closure allowed), hole closure remains below the maximum hole closure allowed as illustrated in Figure 5.a. For this particular case, it should be concluded that under the stress field conditions imposed, this wellbore is stable against collapse. This suggests that analyzing wellbore stability regarding collapse using a peak-strength criterion is pessimistic when rock exhibits non-elastic behavior, and a yield criterion should be taken into account. 118

138 Effect of rock anisotropy: stress-displacement analysis This section deals with the shortcomings caused by the usual assumption of isotropic rock properties. This section provides a basic understanding of the effects that laminated sedimentary rock anisotropy causes on wellbore stability when analyzing wellbore inclination and azimuth variation. This analysis assumes the rock is a linear elastic but anisotropic formation whose elastic constants relate to a bedding plane orientation. In order to define the orientation of the bedding plane, a rock property coordinate system is arbitrarily attached to the global coordinate system (x,y,z ), where the axes are parallel to the direction of the principal in-situ stresses. Figure 4.10 in Chapter 4 illustrates this global coordinate system. When imposing this arbitrary rock property coordinate system, it is assumed a 0 o angle of the bedding plane relative to the horizontal plane defined by the two principal in-situ horizontal stresses. The analysis is divided in two parts. The first part assumes the simplification of a transversely isotropic porous medium and the second part assumes an orthotropic porous medium. To describe anisotropic behavior of rock, it is assumed the rock exhibits a transversely isotropic behavior. It means that the elastic properties are assumed to be the same in the horizontal direction but different in the vertical direction (transverse plane). Table 5.7 shows the data used for this parametric study, which does not have the purpose of simulate real field conditions but analyze the effect of high rock anisotropy on the stability of an inclined wellbore. The state of stress is the same as the one associated with deep formations defined in Table 5.6. According to Ong and Roegiers (1993) and Hibbitt et al. (000), this material 119

139 definition allows setting the number of independent elastic properties to five: Two elastic moduli, one for the horizontal plane E xy and the other for the transverse plane E xz, two Poisson s ratios ν xy and ν xz, and one shear modulus, G. The degree of anisotropy R t is defined in terms of Young s moduli by the ratio R = E t xy E xz. Sensitivity analysis is done for different degrees of anisotropy R t : R t =1, R t = R t =5, and R t =10. The results are compared with the results obtained assuming the rock is isotropic, where R t =1. It is important to remark two important aspects about the results obtained in this analysis. First, the set of Figures 5.4 through 5.6 show that in general, increasing the degree of anisotropy slightly increases the Mises stresses and causes additional hole closure of a deviated well. Figures 5.4a, b, and c show that at inclination angles lower than 30 o the effect of the anisotropy of rock on Mises stress magnitudes is negligible. Figures 5.5a, b, and c show that the anisotropy of rock reduces the mean stresses moving the (p :q ) pair of points to the left in the direction of lower mean effective stress. Further inspection of Figures 5.5a, b, and c shows that wellbore stability in an anisotropic rock is improved with increasing the azimuth of the deviated wellbore in the direction of the minimum principal in-situ stress (a=90 o ). This statement reinforces the conclusion achieved before with respect to the orientation of a deviated or a horizontal wellbore in an isotropic formation. Secondly, according to the sources reviewed, Chenevert and Gatlin (1965) and Podio (1968), they reported that the degree of anisotropy R t found in laminated sedimentary rocks, sandstone and shales, is less than two (R t <). This xz 10

140 situation allows to concentrate this analysis in comparing results between R t =1 and R t = values. Figures 5.6 a, b and c show that hole closures for R t = are greater than hole closures for R t =1. The maximum differences between hole closures calculated for these two R t values are about 6.7%, and they occur when an inclined wellbore is drilled with azimuth 45 o and an inclination angle higher than 30 o. These results indicate that when a deviated wellbore is drilled into an anisotropic formation, it is slightly more unstable than one drilled into an isotropic formation. No further discussion is needed with respect to the effect of varying inclination (i) and azimuth (a) on wellbore stability because the behavior of the curves in Figures 5.4 through 5.6 follows the same trend as the isotropic case (R t =1). The conclusions achieved in the last Section with respect to wellbore orientation in an isotropic rock formation under an isotropic stress field fully apply. The second part of the analysis assumes the rock behaves as an orthotropic formation. It implies that the elastic properties are different in both the horizontal and the vertical plane. According to Hibbitt et al. (000), this material definition requires nine independent elastic properties: an elastic modulus, a Poisson s ratio, and a shear modulus for each one of the three principal directions (x,y,z ). Two degrees of anisotropy are defined in terms of Young s moduli: one in the horizontal plane, R p = Ex Ey and the other in the transverse plane R t = Ex Ez. Sensitivity analysis is done for two different degrees of anisotropy in the horizontal plane R p keeping R t constant (R t =). Table 5.8 shows the data used 11

141 to simulate the two cases for different R p values. The state of stress imposed corresponds to that associated with deep formations. Comparison of the results is done with those obtained assuming the rock behaves accordingly to the transversely isotropic theory. This allows visualizing the effect of varying anisotropy in the horizontal plane on wellbore stability of inclined wellbores. Figures 5.7 and 5.8 show these results from which the following important aspects are pointed out. The most important changes occur when the azimuth increases (see Figure 5.7). It can be noted that when azimuth is (a=0 o ), no significant changes occur between the curves from Case I (R p =1.5) to Case II (R p =). In contrast, when azimuth is (a=45 o or a=90 o ) the magnitude of Mises stresses show major changes. These results show that the two different Young s moduli assumed in the horizontal plane create an additional weakness condition in regards to wellbore collapse of a deviated wellbore. This is particularly important when the azimuth of the wellbore changes towards the direction of the minimum horizontal principal in-situ stress. Figures 5.8b and 5.8c show that this statement is true for azimuths 45 o and 90 o and inclination angles lower than 60 o. As the wellbore inclination increases above 60 o, the effect that an orthotropic rock formation causes on the stability of the wellbore is less pronounced. Finally, the results obtained in this study are limited to the effect that anisotropies of laminated sedimentary rocks cause on wellbore stability when the angle of the bedding plane is 0 o. Abaqus is capable to handle any angle of the 1

142 bedding plane; therefore, further analysis is recommended about the effect of varying the angle of the bedding plane on wellbore stability of deviated wellbores Transient phenomena Rate of deformation Section. in Chapter included the review of the research conducted by Pan and Hudson (1988) related to time-dependent response of rock associated with its non-elastic properties. They explained that modeling tunnel excavations using a two-dimensional numerical model underestimates deformation compared with the results obtained from a three-dimensional numerical model. They concluded that this discrepancy obeys the non-elastic response of the rock behind the tunnel face, a response that a two-dimensional model cannot reproduce. This section shows the discussion of the results obtained with the three-dimensional model described in Section A stress-displacement analysis is performed coupled with the time dependent response of rock associated with its rate of deformation as described in Section in Chapter 4. The initial state of stress imposed in this analysis was σ x =61 MPa, σ y =61 MPa, and σ z =68 MPa. Wellbore pressure was computed assuming water is in a vertical wellbore. Rock is assumed to be homogeneous and isotropic with the following elastic properties: Young s Modulus, E=500 MPa and Poisson s ratio, ν=0.. Rate of deformation data were obtained from a uniaxial test for sandstone published by Cristescu and Hunsche (1998). The results of this test are presented in Figure 5.9. Comparison of hole closures, calculated from the radial 13

143 displacement at the wellbore wall expressed as a percentage of the wellbore radius, were computed for simulation of drilling in two different modes. Figures 5.30 and 5.31 show the results of this analysis. The first mode of drilling was assuming the wellbore is drilled instantaneously, assumption applied in a two-dimensional elastic model and usually in a three-dimensional elastic model. This mode of simulation of the drilling is defined in this study as drilling in a single step. Hole closure computed by this mode was 0.63 %. The same computations of radial displacement at the wellbore wall were done assuming the rock behaves elasto-plastically and accordingly to the Drucker-Prager constitutive model. Simulation of the drilling in this case was done in five successive steps following the second strategy described in Section 4.3. in Chapter 4. This mode of simulation of the drilling is defined in this study as drilling in a multi-step analysis. Figure 5.30 shows that larger hole closures were found when the multi-step analysis was performed. A maximum hole closure value of % was computed. For these particular conditions, the difference between hole closures computed was 5.87 %. Assuming a constant rate of penetration equals 1 m/hr. The 0.5 m thickness model was assumed to be drilled in five sequential steps of 6 minutes each one. The total time of the simulation was 30 minutes. The initial time step suggested was t i =3 min. then ABAQUS controlled automatically the time increments during each step. Figure 5.31 shows the progress of drilling. It can be seen that after the first step (t=6 min) the maximum hole closure is 0.88 %. As 14

144 time progresses and drilling continues, hole closure increases to a maximum value of % at the end of the last step (t=30 min). The effect of rate of penetration on hole closure computations was analyzed by executing three different cases at three different rates of penetration: 1, 10 and 0 m/hr. Table 5.9 shows the results of the effect of the rate of penetration on hole closure. It can be seen that hole closure values approach to the elastic solution as the rate of penetration increases. From these results it can be concluded that a three-dimensional model in conjunction with simulation of drilling in a muti-step process is the only mode that accounts correctly for the non-elastic behavior of a formation associated with its rate of deformation, which causes deformation of the wellbore after it has been drilled. This is an effect that the elasticity theory is unable to quantify Coupled Stress-hydraulic diffusion analysis In order to show the time dependent response of pore pressure during simulation of drilling, this section discusses the results obtained when a stressdisplacement analysis is coupled with a hydraulic diffusion analysis. The governing equations and assumptions taken into account for this modeling are described in Section in Chapter 4. Analysis of this coupled phenomenon is performed using two different constitutive models: the elastic and the Drucker- Prager elastoplastic. Material properties used in this analysis are listed in Table 5.10 and were obtained from data published by Chen et al. (000) for synthetic shale. The initial state of stress applied was σ x =σ H =61 MPa, σ y =σ h =55 MPa, and 15

145 σ x =σ v =68 MPa. (σ H =0.9σ v and σ h =0.9σ H ). In order to visualize the diffusion process through a porous medium due to the hydraulic conductivity of the formation, it is followed the modeling procedure proposed by Charlez (1997b) where two different analyses must be carried out successively. First, simulation of drilling was performed by implementing the second strategy for simulation of drilling described in Section 4.3. in Chapter 4. Wellbore pressure was decreased from 55 MPa to 39 MPa, process simulated in a step time of three hours with time increments of an hour each. It was assumed that the drilling fluid created an impermeable filter cake on the wellbore wall. The initial pore pressure was 31 MPa assuming a pressure gradient of psi/ft. The second part of the analysis simulates a period time of 4 hours at constant wellbore pressure equal to 39 MPa. This part of the analysis was with the purpose of simulating propagation of pore pressure due to the hydraulic conductivity of the rock. Figure 5.3 illustrates the comparison between pore pressure behavior as a function of the radial distance for both solutions: the elastic and the Drucker Prager elastoplastic. This is the pore pressure response at three hours (t=3), immediately after the simulation of drilling has finished. It can be seen that the coupled elastic-hydraulic diffusion analysis does not detect variations in pore pressure. However, when the Drucker Prager elastoplastic model is coupled with the hydraulic diffusion analysis, pore pressure reaches its lowest value at the wellbore wall and increases rapidly to reach its initial value at a short distance within the formation. The difference between both behaviors is attributed to relaxation of rock in the region near the wellbore, a phenomenon that the 16

146 elasticity theory is unable to quantify as previously stated in Section Maximum hole closures were computed for both cases: 0.40 % and % for the elastic and the elastoplastic cases respectively. Figure 5.33 shows a contour plot of pore pressure in the region near the wellbore showing in red the underpressurized zone at (t=3) hours. Figure 5.34 shows the time dependent pore pressure response during the second phase of the modeling. Results are shown at time (t=7) hours. One can see a pore pressure wave displacing into the formation as time progresses. The rate of pore pressure propagation is controlled by the permeability conditions of the formation, defined by the hydraulic diffusivity value K I. Figure 5.35 shows a comparison of the under-pressurized zone and pore pressure profiles for three different K I values. Formations with higher K I induce less severe pore pressure reduction and propagate faster pore pressure than formations with lower K I. It was found that for an elastoplastic model such as the Drucker-Prager, the yield stress of rock affects the response of the pore pressure curve. Figure 5.36 shows how pore pressure response in the vicinity of the wellbore is affected when varying the yield stress. In general, as yield stress decreases, lower pore pressure values are computed in the region near the wellbore and the extent of the underpressurized zone extends into the formation. Otherwise, as expected, pore pressure response approaches the elastic solution as yield stress increases. The effect of fluid compressibility, c f,, on the response of pore pressure around a wellbore was analyzed for four different fluid compressibility values: a totally incompressible fluid (c f =0), water at atmospheric conditions (c f =4.79E-4 17

147 1/MPa), a slightly compressible fluid such as oil containing dissolved gas at (c f =1.0E-3 1/MPa), and a compressible gas (methane) at p=31 MPa and T=100 o C (c f =1.89E- 1/MPa). Figure 5.37 shows that for a compressible fluid, pore pressure behavior varies slightly in the region near the wellbore. For the other fluid compressibility values, pore pressure behavior does not change significantly. 5. WELLBORE STABILITY IN MULTILATERAL SCENARIOS 5..1 Phenomena in Steady State Elastic stress-displacement analysis This section describes the results of the stress-displacement analysis in a multilateral scenario. The geometry of the model and the assumptions taken into account were defined in Sections 4.3. and in Chapter 4. A normally stressed formation is assumed where σ x =10 MPa, σ y =10 MPa, and σ z =30 MPa, wellbore pressure P w =0 MPa. Rock formation is assumed to be homogeneous and isotropic which behaves as a linear elastic material with the following properties: E=10000 MPa and Poisson s ratio ν=0.5. Figure 5.38 illustrates the stress distribution around the main and lateral holes as a function of radial distance three meters below the junction in the direction of the x-axis of the model. Zero value in the x-axis in Figure 5.38 corresponds to the axis of the mainbore. The two sections where no data appear correspond to the main and lateral wellbores. The purpose of this plot is to show the interference originated at the junction area due to the presence of the lateral hole. Maximum values of tangential stress occur in the region between the two 18

148 holes (0.15<r<0.30). The maximum tangential stress occurs at a radius 0.3 m (r=0.30). This is located on the wall of the lateral wellbore at its closest point to the mainbore. A high value of tangential stress is also computed at (r=0.16), which corresponds to the point on the mainbore wall closest to the lateral well. Figure 5.39 shows a contour plot of the Mises stress. This plot confirms that maximum stress values are achieved in the region between both holes. This behavior in the region between the holes can be interpreted as an additional weakness condition affecting wellbore stability in regards to collapse. Figure 5.40 shows a contour plot of displacements in the x-direction. This contour plot illustrates two important events. First, the mainbore closes uniformly. Secondly, the lateral wellbore experiences closure at its farthest side with respect to the mainbore but enlargement at its closest side even under an isotropic state of stress. A scale factor of 300 is used in this plot in order to make these events visible. Analysis of stresses in the p :q plane for all the elements forming the lateral wellbore wall is shown to illustrate rock behavior. These p :q pair of points form a stress cloud, a concept introduced by Bradley (1979b). Figure 5.41 illustrates this plot with data from the very first stage of the analysis (initial or equilibrium conditions). This plot shows that the stress cloud tends to converge to the initial stress in the stable region, below the hypothetical failure envelope. However, once simulation of drilling of the lateral hole is done (final conditions), the stress cloud is modified. The same Figure 5.41 illustrates the stress cloud at this final stage. It can be seen how the stress cloud that originally tended to 19

149 converge now changes its shape and disperses. These particular changes in shape and position of the stress cloud on the p :q plane represent weakness of the rock that can lead to mechanical instability around the junction. Three regions were identified in this stress cloud plot and correlated to their corresponding location at the junction. Figure 5.4 shows the 3-D representation of the junction area where the three main regions are identified. Region A in Figure 5.4 corresponds to the closest elements to the mainbore forming the lateral wellbore wall. Region A in Figure 5.41 groups stresses that exceed the failure envelope. Region B in Figure 5.4 corresponds to the portion of the lateral wellbore wall farthest from the mainbore. In Figure 5.41, this region is identified in the stable zone, below the failure envelope. Finally, tensile stresses shown in Figure 5.41 correlate with those points located in the window created to initiate the lateral well, identified as Region C on Figure 5.4. Both regions A and C are mechanically unstable. Region A is unstable towards collapse while Region C is unstable towards fracture. Identification of these regions gives the opportunity to discuss with respect of the weakness of the junction area during petroleum field operations. Once the junction has been drilled, changes in wellbore pressure may create additional instabilities at the junction. When increasing wellbore pressure, a fracture may be initiated in the Region C, which can lead to circulation losses of drilling or completion fluids. On the other hand, reduction of wellbore pressure can lead to wellbore collapse in the region between the two holes. Changes in fluid density during drilling or completion operations change the wellbore pressure at the 130

150 junction. In addition, there are other drilling and completion operations that change the wellbore pressure. For instance, when making a trip or running casing into the wellbore, wellbore pressure changes at the junction. If drillstring or a casing string is lowered into the wellbore, the wellbore pressure increases. This effect is known as surge pressure. If the drillstring or casing is pulled from the wellbore, the wellbore pressure decreases, effect known as swab pressure. This exemplifies how it is important to take into account the stability of the junction not only during the drilling but also during the completion operations. Furthermore, the integrity of the junction must be designed for the entire life of the well. The completion design must take into account how the formation behaves as the wellbores produce and pressure drawdown occurs in the hydrocarbons reservoir. A junction that initially is competent may eventually fail on time as drawdown occurs. Previously, it was stated that the window created to initiate the lateral well and the region between the two holes are unstable regions. The following two sections discuss the effect of modifying the geometry of the junction area on wellbore stability Effect of increasing the junction angle The window in the junction area was identified as a critical zone regarding fracture. Increasing the junction angle reduces the height of the window. This section discusses the effect of increasing junction angle as a mean to influence wellbore stability at the junction. 131

151 In this study, the junction angle was varied from.5 o to 5 o to 10 o. Analysis of stress cloud in the p :q plane for all the elements forming the lateral wellbore wall is included to visualize this effect. Figure 5.43 illustrates the corresponding stress cloud for junction angles.5 o, 5 o, and 10 o. It can be seen how even increasing the junction angle from.5 o to 10 o, the shape of the stress cloud only changes slightly. Two results should be pointed out. First, the stress cloud for the 10 o junction angle is less disperse than the corresponding stress cloud for the.5 o junction angle. Secondly, the magnitudes of p and q values remain practically the same. From the first statement, it may be stated that the junction of a lateral wellbore drilled at a 10 o junction angle is more stable than one drilled at.5 o because of changes of the shape of the stress cloud. On the other hand, no significant changes in magnitude of p and q stresses indicate that the position of the stress cloud is not modified. Maximum radial displacements in the main and lateral wellbores are computed for the three cases of.5 o, 5 o, and 10 o junction angles. The maximum values are wellbore enlargements found in the lateral wellbore wall. For the.5 o case, the maximum lateral hole enlargement expressed as the percentage of the wellbore radius was %. For the 5 o case, the maximum hole enlargement is 0.19 %, and for the 10 o case, this value is %. The difference between the hole enlargements is negligible. Thus, for the particular conditions imposed in this analysis, it is concluded according to these results that wellbore stability benefits expected at the junction area when the junction angle is increased is limited. 13

152 Effect of varying the diameter of the lateral hole Another alternative to modifying the height of the window is varying the size of the lateral wellbore. This section is devoted to analyzing the effect of changing the diameter of the lateral well on wellbore stability in the junction area. The diameter of the lateral well was varied from to 8.5 to 6.75 inches, keeping constant the 1.5 inches diameter of the mainbore. These diameters were chosen according to conventional bit size combinations available in the oil industry when planning multilateral wells. The results were analyzed using stress cloud plots and the maximum radial displacement computed during the analysis in both the mainbore and lateral wellbore. Figure 5.44 shows the limit cases when the lateral hole is and 6.75 inches diameter. By comparing Figures 5.44a and 5.44b, it can be noted that there are not significant changes between the two stress clouds. The stress cloud (b), corresponding to the 6.75 in. diameter lateral hole, shows a slight change in shape with respect to the stress cloud (a) because stress cloud (b) is less disperse than stress cloud a. When analyzing radial displacements in both wellbores for each one of the geometries, again the maximum radial displacements are found at the lateral wellbore wall and they represent wellbore enlargements. The difference between wellbore enlargements in both cases (a) and (b) is negligible. In conclusion, varying the diameter of the lateral wellbore does not significantly affect the mechanical stability of the junction area. 133

153 Effect of varying the orientation of the lateral hole Papanastasiou et al. (00) presented the study of the stability of a multilateral junction based on experimental results and numerical modeling. They performed physical tests in a true triaxial cell on cubical blocks of weak sandstone with two holes intersecting at.5 o. Deformation of wellbore walls and development of breakouts were monitored with a video camera placed either into the lateral wellbore or into the mainbore. They compared their experimental results with numerical modeling based on a generalized plane strain formulation. Details on the experimental procedure, wellbore deformation calculations, and the numerical modeling can be consulted in Papanastasiou et al. (00). They characterized the rock using the elasticity theory and reported the following elastic parameters: Young s modulus E=500 MPa and Poisson s ratio ν=0.. They concluded that their numerical model predicts reasonably well the area around holes that is prone to failure, but it underestimates the stress level at which failure initiates. They reported that the rock tested exhibited a pronounced elastic brittle behavior. They also concluded that for the state of stresses imposed, the most stable direction for a lateral to be drilled is parallel to the maximum principal in-situ stress. Based on this last conclusion, this section has the purpose of discussing the effect of varying orientation (azimuth) of the junction as a mean to influence wellbore stability in the junction area. An elastic stress-displacement analysis using the same elastic constants as Papanastasiou et al. (00) is done with the model and assumptions defined in Sections and in Chapter 4. Although the physical dimensions of the 134

154 cubical blocks, the angle of the junction, and the diameters of the mainbore and lateral wellbores are different between the actual model and the blocks used in Papanastasiou s experiments, the results achieved in Section and in this chapter demonstrated that wellbore stability is not significantly affected when the junction angle and the diameter of the lateral hole change. Therefore, it is assumed that the results from the actual model can be satisfactorily compared with Papanastasiou s experimental results. Because this analysis is limited to a particular state of stress, it is not the purpose of this analysis comparing actual results with those obtained by Papanastasiou et al. (00) in its full extent. This comparison is limited the following state of stresses σ x =σ H =30 MPa, σ y =σ h =18 MPa, and σ z =σ v =18 MPa, same as the stress level where they reported failure in the mainbore. No wellbore pressure is applied, P w =0 MPa. Simulations were carried out at two different orientations of the lateral wellbore. The first is for the lateral in the direction of the maximum horizontal insitu stress, azimuth equals zero degrees (a=0 o ). The second is for the lateral in the direction of the minimum horizontal in-situ stress, azimuth equals ninety degrees (a=90 o ). The results using the actual model and the results found by Papanastasiou et al. from their experimental tests have some differences, but for the most part they have similarities. The first difference is the location of failure. Based on their experimental results (deformation of the mainbore), they reported the onset of failure in the mainbore when the lateral wellbore is oriented with an azimuth (a=90 o ), while the actual model, based on the maximum hole closure allowed ( %), predicts stable 135

155 mainbore and lateral holes. Secondly, they do not report failure in the lateral wellbore in any case neither in experimental results nor numerical modeling. However, the results from the actual model show that once the junction fails, the lateral wellbore will fail at a higher stress level. The results obtained with the actual model for the lateral oriented in the direction of the minimum horizontal in-situ stress (a=90 o ) are shown in a contour plot of displacements (see Figure 5.45). This Figure shows that the maximum displacements computed are wellbore closures equivalent to 0.433% found in the window and less than the maximum hole closure allowed. Figure 5.46 shows a contour plot of the Mises stresses. It can be seen that failure occurs first in the junction, where the maximum Mises stresses are computed. At a higher stress level, the most likely region to fail is the lateral wellbore wall. Figure 5.47 shows a contour plot of stresses showing failure of the lateral wellbore at a higher stress level. Despite the differences, there are more similarities between results. First, when the lateral is oriented with an azimuth (a=0 o ), both results found the onset of failure in the junction. Secondly, both results also predict that when increasing the stress level, the mainbore fails after the junction has failed. Figure 5.48 shows this situation. The orange area shows how stress concentration in the mainbore is high. In addition, although the actual model does not predict the creation of breakouts, from the high stress concentration areas seen in Figures 5.49 and 5.50 it can be inferred that the direction of the breakouts is in agreement with the physical results that Papanastasiou et al. found in their experimental work. Furthermore, 136

156 they concluded that the most stable direction for a lateral to be drilled is parallel to the maximum principal in-situ stress just as the actual model predicts. This conclusion is founded on the comparison of the maximum values of Mises stress computed by the actual model in each simulation. Figure 5.51 shows the results obtained with the actual model for the lateral wellbore oriented in the direction of the maximum horizontal in-situ stress (a=0 o ). By comparing the maximum Mises stresses computed between both simulations (see Figures 5.46 and 5.51), it can be seen that higher Mises stress values are computed when the lateral wellbore is oriented parallel to the minimum principal in-situ stress. There are some explanations to the differences found between both results. First, they reported failure based on the deformation measured during the experimental results. During these measurements, they explained the difficulties they faced to identify the instant at which failure initiated due to the lighting conditions and position of the video camera. They reported that in some of the experiments it was impossible to measure deformation of the lateral wellbore. This explains why they do not report failure in the lateral wellbore. The conclusions achieved with the actual model are based on the Mises stresses computed rather than the deformations (hole closures). Because Papanastasiou et al. reported an elastic brittle behavior of the rock, deformations were not expected to be large before brittle failure occurred. This justifies why in this particular case, hole closure allowance criterion is not useful in predicting failure in numerical simulations. 137

157 Effect of changing the depth of placement of the junction The purpose of this section is to study the effect of changing the depth of placement of the junction from a shallow or intermediate formation to a deep formation on the stability of the junction itself. The stress level imposed in the study by Papanastasiou et al. (00), σ H >σ h =σ v, is associated with depth as in Section Either a shallow formation or an intermediate formation corresponds to this stress condition according to the following classification. Shallow: Intermediate: Deep: σ H >σ h >σ v σ H >σ v >σ h σ v >σ H >σ h Two new simulations are conducted imposing the following stress condition σ x =σ H =30 MPa, σ y =σ h =18 MPa, and σ z =σ v =50 MPa. One simulation is for the lateral in the direction of the maximum horizontal in-situ stress (a=0 o ) and the other for the lateral in the direction of the minimum horizontal in-situ stress (a=90 o ). Comparison of the results obtained for both orientations of the lateral wellbore (a=0 o ) and (a=90 o ) serves to discuss about the most stable direction for a lateral to be drilled when the junction needs to be placed at a deep depth. Figures 5.5 and 5.53 show contour plots of the Mises stresses for orientations of the lateral wellbore (a=0 o ) and (a=90 o ) respectively. It can be seen that lower Mises stress values are computed when the lateral wellbore is oriented parallel to the maximum principal in-situ stress. This indicates that independently of the depth of placement of the junction, the most stable junction is with the lateral wellbore 138

158 oriented parallel to the maximum principal in-situ stress. Once interaction between the mainbore and the lateral wellbore has finished, they can be treated as single holes, and the orientation of the lateral should be designed according to the conclusions reached in Section 5.1 with respect to stability of a single wellbore. Now, comparison of the results illustrated in Figures 5.46 and 5.51 with the results shown in Figures 5.5 and 5.53 serves to further discussion about the effect of placing the junction at a different depth. Figure 5.46 can be directly compared with Figure 5.5, while Figure 5.51 can be compared with Figure From these comparisons, it can be seen that the maximum Mises stresses computed at the junction area are found when the stress level imposed corresponds to a shallow or intermediate formation. Lower Mises stress values are computed when the junction is assumed to be placed at a deep formation. These results indicate that junctions should be placed in deep formations, as close as possible to the hydrocarbons zones. This conclusion is based on the mechanical response of rock and assuming that both the shallow or intermediate formation and the deep formation have the same rock properties. Other criteria such as wellpath design, equipment, and re-entry capability of the lateral wellbore must be taken into account to decide the placement of the junction Independence between holes The junction area is defined as the region where a mainbore and a lateral well are connected. This has been identified as a region where mechanical instabilities are likely to happen. Common sense suggests that there is a 139

159 separation distance between the two holes where interaction between them no longer exists. Beyond this separation distance the two holes become independent of each other, and they can be treated as single and independent holes. This section has the aim of showing how analyzing stress response in the region between the two holes helps to find that separation distance. The initial state of stress imposed for this analysis is σ x =10 MPa, σ y =10 MPa, and σ z =30 MPa, wellbore pressure P w =0 MPa. Rock formation is assumed to be homogeneous and isotropic which behaves as a linear elastic material with the following properties: E=10000 MPa and Poisson s ratio ν=0.5. Figure 5.54 shows the response of the radial and tangential stresses around the main and lateral wells as a function of radial distance in the direction of the x- axis of the model. This Figure 5.54 shows the region between the boreholes at a distance of about 0 meters below the junction, where the separation distance between the two holes is (d=0.87 m). The axis of the mainbore is located at coordinate (r=0 m). The mainbore wall corresponds to the coordinate (r=0.16 m), and the wall of the lateral hole is at coordinate (r=1.08 m). From this plot, it can be said that because the radial and tangential stresses tend to the initial state of stress condition in the region between the two holes (0.54<r<0.76 m), both wellbores have become independent. These results are valid only for the particular conditions imposed for this analysis. Further analysis should be done to find whether the separation distance where the two holes become independent is affected by other parameters such as the state of stress level or the non-elastic behavior of rock. 140

160 Complex Multilateral Scenarios Up to the knowledge of the author, no research has been conducted in analyzing the effect of two lateral wellbores with the same starting point from the mainbore on wellbore stability of the junction. Section discussed the effect of changing the placement of the junction from a shallow or intermediate formation to a deep formation. That discussion is limited to consider the mainbore is vertical with the lateral wellbore oriented in the direction of one of the principal in-situ stresses. That analysis is applicable to a multilateral scenario where the junction is placed somewhere above the hydrocarbons zone, in the overburden. When reservoir management requires construction of a multilateral in a single producing formation, a different analysis is required to study the stability of the junction. The junction is assumed to be located into the producing formation with the mainbore and the two laterals lying on the horizontal plane. This section has the aim of providing a basic understanding about the effect of three wellbores interacting on the stability of the junction when the junction is placed in a producing formation. Three elastic stress-displacement analyses are done at three different orientations of the mainbore wellbore. The first is for the mainbore in the direction of the maximum horizontal principal in-situ stress, azimuth equals zero degrees (a=0 o ). The second is for the mainbore with an azimuth equals 45 degrees (a=45 o ), and the third for the mainbore in the direction of the minimum horizontal principal in-situ stress, azimuth equals ninety degrees (a=90 o ). The elastic 141

161 constants are the same as in Papanastasiou et al. (00) study: E=500 MPa and ν=0.. The stress level applied is the following σ x =σ H =30 MPa, σ y =σ h =18 MPa, and σ z =σ v =50 MPa. The results obtained are shown in contour plots of Mises stresses in Figures 5.55 through The first comment about these plots is that the region that is prone to failure is in any case the junction. Secondly, comparison of these figures confirms that drilling a horizontal well in the direction of the minimum horizontal principal in-situ stress (a=90 o ) constitutes the most stable condition. It can be seen in these figures that once the junction fails the next region to fail is the mainbore. Red in these contour plots indicates zones that are more likely to fail. It can be seen in Figure 5.55 (a=0 o ) a red zone in the mainbore, which indicate failure in the mainbore while in Figure 5.57 (a=90 o ) the no presence of a red zone indicates that the wellbores remain stable. The third important aspect from these plots is that the stability of the junction is slightly affected by the azimuth of the mainbore. Maximum Mises stresses are computed when the mainbore is oriented with a=90. When a=0 o, the maximum Mises stress is 10 MPa. When a=45 o, then the maximum Mises stress is 104 MPa, and the maximum Mises stress is1089 MPa when a=90 o. 14

162 Table 5.1 Data from a drained triaxial test (from Atkinson and Bransby 1978). Axial force (N) Change of length (mm) Volume of Water expelled (mm 3 x10 3 ) Volumetric strain Axial strain (Fraction) Area (m x10-3 ) q (Mpa) Table 5. Isotropic compression test results (from Atkinson and Bransby 1978). Cell pressure Volume of Volume of the Specific ln p water expelled sample volume (MPa) (cm 3 ) (cm 3 )

163 Table 5.3 Effect of varying M value on hole closure. Model Case A Low stress level Case B Intermediate stress level Hole closure (% of radius) Case C High stress level Elastic Cam-Clay M= Cam-Clay M= Cam-Clay M= Cam-Clay M= * Cam-Clay M= * Cam-Clay M= * Cam-Clay M= * Cam-Clay M= * Cam-Clay M= * Cam-Clay M= * * * Excessive deformation occurs so that the plasticity-algorithm used by Abaqus is unable to find a solution. 144

164 Table 5.4 Values of parameters for various clays (from Atkinson and Bransby 1978) London clay Weald clay Kaolin λ s κ s Γ M Table 5.5 Effect of varying λ s and κ s values on hole closure. Model Hole closure (%) Elastic 0.11 London clay 0.11 Weald clay 0.11 Kaolin 0.16 Table 5.6 Stress level imposed to analyze wellbore orientation Case Depth σ v σ H σ h *P w [ft] [MPa] [MPa] [MPa] [MPa] Shallow Intermediate Deep *P w was calculated assuming water in the wellbore. 145

165 Table 5.7 Transversely isotropic rock properties used for sensitivity analysis. Parameter Value Parameter Value E xy MPa ν xy 0.68 E xz 5000 MPa when (R t =) ν xz E xz 000 MPa when (R t =5) σ x 100 MPa E xz 1000 MPa when (R t =10) σ y 67 MPa G xz 319 MPa when (R t =) σ z 10 MPa G xz 1614 MPa when (R t =5) G xz 893 MPa when (R t =10) Table 5.8 Orthotropic rock properties used for sensitivity analysis. Parameter Value Parameter Value E x MPa ν xy 0.68 E z 5000 MPa ν xz = ν yz E y 6667 MPa when (R p =1.5) CASE I σ x 100 MPa E y 5000 MPa when (R p =) CASE II σ y 67 MPa σ z 10 MPa G yz 3036 MPa when (R p =1.5) G xy 3943 MPa G yz 77 MPa when (R p =) G yz 77 MPa 146

166 Table 5.9 Effect of rate of penetration on hole closure. Rate of penetration [m/hr] Hole closure [%] Difference respect to one-step simulation (elastic) [%] One-step (Elastic) Table 5.10 Material properties for a coupled stress-diffusion analysis (from Chen et al. 000). Properties Units Model Density of the sample Kg/m3 78 Bulk modulus sample Gpa Shear modulus sample Gpa 7.7 Friction angle sample Degrees 37 Cohesion sample MPa 6.3 Dilation angle sample Degrees 0 Tensile strength sample MPa.07 Porosity sample % 4.3 Mobility ratio sample (m/s)/(pa/m) 5.14E-0 Bulk modulus fluid GPa.0 Density fluid Kg/m

167 Stress (MPa) Corresponding to the Wellbore wall Radial Radius (m) Tangential Figure 5.1 Stress distribution around a wellbore: Elastic case. 0.0 Extent of the plastic zone 18.0 Relaxation of the tangential stress Stress (MPa) Corresponding to the wellbore wall Radius (m) Elastic Cam-Clay Figure 5. Comparison of tangential stresses 148

168 Figure 5.3 Contour plot showing the extent of the plastic zone. 149

169 Maximum tangential stress values located inside the formation Stress (MPa) Corresponding to the wellbore wall Radius (m) Elastic Cam-Clay D-Prager Figure 5.4 Comparison between tangential stress solutions Stress (MPa) Corresponing to the wellbore wall Radius (m) Elastic Cam-Clay D-Prager Figure 5.5 Comparison between radial stress solutions. 150

170 Effective Mises stress [MPa] Failure Envelope Elastic Cam-Clay Drucker-Prager Initial state Region A B C Effective mean stress [MPa] Elastic Cam-Clay D-Prager Envelope Initial State Figure 5.6 Analysis of compressive failure for the elements in the immediate vicinity of the wellbore. 151

171 Tangential stress (MPa) Corresponding to the wellbore wall Radius (m) M=1.5 M=1.1 M=1.0 M=0.9 Figure 5.7 Effect of M variation on the tangential stress response: Cam-Clay 0.0 Difference between the extent of the plastic zones 18.0 Stress (MPa) Corresponding to the wellbore wall Radius (m) Kaolin London clay Weald clay Figure 5.8 Tangential stress behavior 15

172 σ v =1 MPa z σ h =15 MPa y x Global coordinate system σ H =.5 MPa Figure 5.9 Representation of the principal in-situ stresses in a shallow formation in a tectonically active stressed region (σ H >σ h >σ v ). 153

173 50 Stresses [MPa] Inclination [degrees] a) azimuth=0 Misses Mean 50 Stresses [MPa] b) azimuth=45 Inclination [degrees] Misses Mean Stresses [MPa] c) azimuth=90 Inclination [degrees] Misses Mean Figure 5.10 Maximum Mises and Mean stresses vs hole deviation for three different azimuth values in a shallow formation (elastic rock). 154

174 50 Failure envelope 40 0 o 90 o Mises [MPa] o 90 o 90 o a=0 a=45 a=90 Mean [MPa] Figure 5.11 Effect of varying angle deviation on the maximum p and q values in a shallow formation (elastic rock). Hole closure [%] a=0 a=45 a=90 Inclination [degrees] Figure 5.1 Maximum hole closure vs wellbore inclination in a shallow formation (elastic rock). 155

175 σ v =48 MPa z σ h =40 MPa y x Global coordinate system σ H =60 MPa Figure 5.13 Representation of the principal in-situ stresses in an intermediate formation in a tectonically active stressed region (σ H >σ v >σ h ). 156

176 Inclination [degrees] a) azimuth=0 Stresses [MPa] Misses Mean Inclination [degrees] b) azimuth=45 Stresses [MPa] Misses Mean Stresses [MPa] c) azimuth=90 Misses Mean Inclination [degrees] Figure 5.14 Maximum Mises and Mean stresses vs hole deviation for three different azimuth values in an intermediate formation (elastic rock). 157

177 Mises effective [MPa] Failure envelope 90 o 90 o 90 o 0 o a=0 a=45 a=90 Mean effective [MPa] Figure 5.15 Effect of varying angle deviation on the maximum Mean and Mises effective stresses in an intermediate formation (elastic rock). Hole closure [%] Inclination [degrees] a=0 a=45 a=90 Figure 5.16 Maximum hole closure vs wellbore inclination in an intermediate formation (elastic rock). 158

178 σ v =10 MPa z σ h =67 MPa y x Global coordinate system σ H =100 MPa Figure 5.17 Representation of the principal in-situ stresses in a deep formation in a tectonically active stressed region (σ v >σ H >σ h ). 159

179 Stresses [MPa] a) azimuth=0 Misses Inclination [degrees] Mean Stresses [MPa] b) azimuth=45 Misses Inclination [degrees] Mean Stresses [MPa] c) azimuth= Misses Mean Inclination [degrees] Figure 5.18 Maximum Mises and Mean stresses vs hole deviation for three different azimuth values in a deep formation (elastic rock). 160

180 Mises effective [MPa] o o 90 o 0 o Failure envelope o a=0 a=45 a=90 Mean effective [MPa] Figure 5.19 Effect of varying angle deviation on the maximum Mean and Mises effective stresses in a deep formation (elastic rock). Hole closure [%] a=0 a=45 a=90 Inclination [degrees] Figure 5.0 Maximum hole closure vs wellbore inclination in a deep formation (elastic rock). 161

181 Mises effective [MPa] Separation point Inclination [degrees] a) azimuth=0 Elastic Yo=67 MPa Yo=0 MPa Mises effective [MPa] Separation point Inclination [degrees] b) azimuth=45 Elastic Yo=67 MPa Yo=0 MPa Mises effective [MPa] c) azimuth=90 o Separation point Inclination [degrees] Elastic Yo=67 MPa Yo=0 MPa Figure 5.1 Maximum Mises stress vs hole deviation for three different azimuth values in a deep formation (elastic and elastoplastic cases). 16

182 Hole closure [%] Maximum hole closure allowed Inclination [degrees] a) azimuth=0 Elastic Yo=67 MPa Yo=0 MPa Hole closure [%]. Maximum hole closure allowed Inclination [degrees] b) azimuth=45 Elastic Y0=67 MPa Yo=0 MPa..0 Maximum hole closure allowed Hole closure [%] Inclination [degrees] c) azimuth=90 Elastic Yo=67 MPa Yo=0 MPa Figure 5. Comparison of maximum hole closures between the elastic and the non-elastic cases for three different azimuths in a deep formation. 163

183 Mises effective [MPa] Failure envelope 90 o 0 o 30 o 90 o 90 o 0 o 0 a) azimuth= Mean effective [MPa] Elastic Yo=67 MPa Yo=0 MPa Mises effective [MPa] Failure envelope 90 o 45 o 0 o 0 o 90 o Mean effective [degrees] b) azimuth=45 Elastic Y0=67 MPa Yo=0 MPa o 90 o Mises effective [MPa] Failure envelope 90 o 0 o 0 o Mean effective [degrees] c) azimuth=90 Elastic Yo=67 MPa Yo=0 MPa Figure 5.3 Effect of varying inclination angle on the maximum Mises and Mean effective stresses. Deep formation (elastic and elastoplastic cases). 164

184 Mises effective[mpa] Rt=1 Rt= Rt=5 Rt= a) azimuth=0 Inclination [degrees] 00 Rt=1 Rt= Rt=5 Rt=10 Mises effective [MPa] b) azimuth=45 Inclination [degrees] Mises effective [MPa] c) azimuth=90 Rt=1 Rt= Rt=5 Rt=10 Inclination [degrees] Figure 5.4 Maximum Mises stresses vs hole deviation at three different R t values in a deep transversely isotropic formation (elastic rock). 165

185 Rt=1 Rt= Rt=5 Rt=10 00 Mises effective [MPa] Failure envelope 0 o 0 o 90o 90 o a) azimuth=0 Mean effective [MPa] Rt=1 Rt= Rt=5 Rt=10 00 Mises effective [MPa] Failure envelope 90o 90 o 0 o 0 o b) azimuth=45 Mean effective [MPa] 00 Rt=1 Rt= Rt=5 Rt=10 Mises effective [MPa] o 90 o 0 o 0 o Failure envelope c) azimuth=90 Mean effective [MPa] Figure 5.5 Comparison of the maximum p and q values when varying the deviation angle. Different R t. Transversely isotropic formation. 166

186 Hole closure [%] Rt=1 Rt= Rt=5 Rt= a) azimuth=0 Inclination [degrees] Rt=1 Rt= Rt=5 Rt=10 Hole closure [%] b) azimuth=45 Inclination [degrees] Hole closure [%] Rt=1 Rt= Rt=5 Rt= c) azimuth=90 Inclination [degrees] Figure 5.6 Maximum hole closure vs wellbore inclination. Different R t. Transversely isotropic formation. 167

187 Rt= Rp=1.5 Rp= 0 Mises effective [MPa] a) azimuth=0 Inclination [degrees] 0 Rt= Rp=1.5 Rp= Mises effective [MPa] b) azimuth=45 Inclination [degrees] 0 Rt= Rp=1.5 Rp= Mises effective [MPa] c) azimuth=90 Inclination [degrees] Figure 5.7 Maximum Mises stresses vs hole deviation at three different R p values in a deep orthotropic formation (elastic rock). 168

188 Hole closure [%] Rt= Rp=1.5 Rp= a) azimuth=0 Inclination [degrees] Rt= Rp=1.5 Rp= Hole closure [%] b) azimuth=45 Inclination [degrees] Hole closure [%] Rt= Rp=1.5 Rp= c) azimuth=90 Inclination [degrees] Figure 5.8 Maximum hole closure vs wellbore inclination. Different R p. Orthotropic formation. 169

189 Figure 5.9 Rate of deformation influence on the uniaxial stress-strain curves and failure of sandstone (from Cristescu and Hunsche 1998). 170

190 One-step Hole closure (%) Multi-step Figure 5.30 Comparison of hole closure between one-step and multi-step analysis. Depth [m] Hole closure (%) Advancing face of the wellbore at corresponding times t=6 t=1 t=18 t=4 t=30 Figure 5.31 Progress of drilling with time showing the hole closure behind the advancing face of the wellbore. 171

191 Pore pressure (MPa Corresponding to the wellbore wall Elastic Radius (m) Initial pore pressure Elastoplastic Figure 5.3 Comparison between pore pressure distribution around a wellbore for both solutions: elastic and elastoplastic. 17

192 Figure 5.33 Contour plot showing pore pressure distribution around a wellbore after three hours (t=3). 173

193 Pore pressure(mpa Corresponding to the wellbore wall Radius (m) t=3 hr t=7 hr Figure 5.34 Pore pressure distribution as a function of time and radial distance from the wellbore wall. Pore pressure (MPa) Radius (m) Corresponding to the wellbore wall Ki=5.14E-0 Ki=5.14E-18 Ki=5.14E-17 Figure 5.35 Pore pressure distribution as a function of radial distance from the wellbore wall for different permeability conditions. 174

194 Pore pressure (MPa Corresponding to the wellbore wall Radius (m) Yo=45 MPa Yo=55 Mpa Yo=68 MPa Elastic Figure 5.36 Effect of yield stress variation on the response of pore pressure distribution around a wellbore. Pore pressure(mpa) Corresponding to the wellbore wall Radius (m) cf=0 cf=4.79e-4 cf=1.0e-3 Cf=1.89E- Figure 5.37 Effect of fluid compressibility variation on the response of pore pressure distribution around a wellbore. 175

195 Maximum tangential stress Region between the two holes Stress (MPa) Lateral wellbore Mainbore x-axis Radius (m) Radial Tangential Figure 5.38 Distribution of the radial and tangential stresses at the junction area. 176

196 Figure 5.39 Contour plot showing Mises stress. 177

197 Closure Enlargement Figure 5.40 Contour plot showing displacement in the x-direction. 178