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A stress-strain model for the undrained response of oil sand Cheung, Ka Fai Henry 1985

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A STRESS-STRAIN MODEL FOR THE UNDRAINED RESPONSE OF OIL SAND  by  KA FAI HENRY/CHEUNG B.Sc,  University of Manchester, 1983  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering  We accept this thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1985 © KA FAI HENRY CHEUNG » ?985  In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements f o r an advanced degree at THE UNIVERSITY OF BRITISH COLUMBIA, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study.  I  further agree that permission for extensive copying of this thesis f o r scholarly purposes may be granted by the Head of my Department or by his or her representative. I t i s understood that copying or publication of this thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of C i v i l Engineering THE UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, B.C. CANADA, V6T 1W5  Date:  A p r i l , 1985  ABSTRACT  An e f f i c i e n t undrained model for the deformations analyses of o i l sand masses upon undrained loading i s presented i n t h i s thesis. An analysis which couples the s o i l skeleton and pore f l u i d s i s used.  The s o i l skeleton i s modelled as a non-linear e l a s t i c - p l a s t i c  i s o t r o p i c material.  In undrained conditions, the constitutive  r e l a t i o n s h i p s f o r the pore f l u i d s are formulated based on the i d e a l gas laws.  The coupling between the s o i l skeleton and the pore f l u i d s i s  based upon volume compatibility. The undrained model was v e r i f i e d with the experimental r e s u l t s and one dimensional expansion of s o i l sand cores.  Comparisons between  computed and measured responses are i n good agreement and suggest that this model may  prove useful as a tool i n evaluating undrained response  of o i l sand. The response of a wellbore i n o i l sand upon unloading was analysed using the developed model.  Such analyses are important i n the  r a t i o n a l design of o i l recovery systems i n o i l sand.  iii  TABLE OF CONTENTS  Page Abstract  i i  Tabele of Contents  i i i  L i s t of Tables  vii  L i s t of Figures  viii  L i s t of Symbols  x  Acknowledgements CHAPTER 1  xiii  Introduction  1.1  Introduction  1  1.2  Behaviour of O i l Sand  2  1.3  The Scope  5  1.4  The Organization of Thesis  6  CHAPTER 2  Review of Previous Work  2.1  Introduction  2.2  Mathematical Model for Undrained Behaviour  8  of O i l Sand  CHAPTER 3  2.2.1  Harris and Sobkowicz  8  2.2.2  Dusseault  12  2.2.3  Byrne and Janzen  14  Stress-Strain Model  3.1  Introduction  17  3.2  Development of the Undrained Model  17  3.3  Constitutive Relations 3.3.1  Incremental Non-linear E l a s t i c S o i l s Model  22  3.3.2  Incremental Non-linear E l a s t i c Fluid Model  25  iv  Page  3.3.2.1  General  25  3.3.2.2  Partly M i s c i b l e Gas/Liquid Mixture  27  a.  Air/Water Mixture  28  b.  Carbon Dioxide, Air/Water  30  Mixture c. CHAPTER 4  Gas/Bitumen and Water Mixture  F i n i t e Element Formulations  4.1  Introduction  4.2  The Plane Strain Formulation  4.3  4.4 CHAPTER 5  36  4.2.1  The Constitutive Matrix [D]  36  4.2.2  The Strain Displacement Matrix [B]  40  4.2.3  The S t i f f n e s s Matrix [K]  44  The Axisymmetric  Formulation  4.3.1  The Constitutive Matrix [D]  46  4.3.2  The Strain Displacement Matrix [B]  50  4.3.3  The S t i f f n e s s Matrix [K]  53  Load Shedding Formulation  55  Comparisons with E x i s t i n g Solutions  5.1  Introduction  5.2  Comparisons with Theoretical Solutions  5.3  33  60  5.2.1  E l a s t i c Closed Form Solutions  60  5.2.2  E l a s t i c - P l a s t i c Closed Form Solutions  61  Comparisons with Observed 5.3.1  Data  One Dimensional Unloading of O i l Sand  68  Page 5.3.2 CHAPTER 6  T r i a x i a l Tests on Gassy Soils  72  Stresses Around a Wellbore or Shaft i n O i l Sand  6.1  Introduction  79  6.2  General Model Description  80  6.3  Theoretical Solutions for Stresses Around a Wellbore 6.3.1  6.4  80  Stresses 6.3.1.1  Stresses i n E l a s t i c Zone  83  6.3.1.2  Stresses i n P l a s t i c Zone  85  6.3.1.3  Radius of P l a s t i c Zone  87  6.3.2  Stability  89  6.3.3  Pore Pressure P r o f i l e  90  Comparisons of Predicted Response and Closed Form Solution  6.5  93  Analysis of Response of Borehole i n O i l Sand on Unloading 6.5.1  Undrained Response  6.5.2  Drained Response  6.5.3  Implications of Undrained and Drained Analysis  6.6 CHAPTER 7  93 100  104  Application of o i l recovery  104  Summary and Conclusions  108  Bibliography  110  Appendix A  114  Appendix B  116  Appendix C  119  VI  Page Appendix D Appendix E Appendix F Appendix G Appendix H Appendix J  . _,  136  vii LIST OF TABLES  Table  Page  A comparison of computed and measured r e s u l t s (Test 11, Sobkowicz)  74  Vlll  LIST OF FIGURES  Figure  _ __  1.1  Schematic spring analogy for o i l sand  3.1  S t r e s s - s t r a i n curves f o r drained t r i a x i a l  Page  3  tests on loose sand  24  3.2  Phase diagram f o r gassy s o i l  32  3.3  Phase diagram for o i l sand  34  4.1  Stresses associated with load shedding  56  5.1  thick wall cylinder  62  5.2  Stresses and displacements around c i r c u l a r opening in an e l a s t i c material  5.3  63  Stresses and displacements around c i r c u l a r opening in an e l a s t i c - p l a s t i c material  64  5.4a  Model f o r one dimensional unloading of o i l sand  70  5.4b  Response of o i l sand to one dimensional unloading  5.5  Comparisons of predicted and observed pore pressure  5.6  Comparisons of predicted and observed strains  77  6.1  Outline of the wellbore problem  81  6.2a  F i n i t e element mesh for wellbore problem  82  6.3b  Mohr Coulomb f a i l u r e envelope  84  6.3  Idealised flow to a wellbore  92  .... ..  71 76  LIST OF FIGURES  -  Continued  Figure 6.4  Page Stresses around a wellbore i n an e l a s t i c - p l a s t i c material  6.5  Undrained  94 response of wellbore i n o i l sand on  unloading  96  6.6  Pore pressure p r o f i l e around a borehole  101  6.7  Drained response of a wellbore i n o i l sand on unloading  6.8  Comparisons of undrained and drained  102 response  of a wellbore i n o i l sand at a support pressure of 3500 kPa 6.9  Comparisons of undrained and drained  105 response  of a wellbore i n o i l sand at a support of 3200 kPa  pressure 106  X LIST OF SYMBOLS  The following i s a l i s t of the commonly used symbols i n t h i s thesis.  Multiple use of several symbols i s unavoidable because of the  complexity of the formulations. The symbol w i l l be defined  immediately  i n the text where the use of symbols d i f f e r s from those l i s t e d below.  SYMBOL  MEANING  3  compressibility  A  change  a  stress  e  strain  v  Poisson's r a t i o  (j)  f r i c t i o n angle  a  temperature s o l u b i l i t y constants  a  r  r a d i a l stress  o"g  tangential stress  a z  v e r t i c a l stress  B  bulk modulus  e  . void ratio  E  Young's modulus  G  shear modulus  H  Henry's constant  k K  permeability apparent bulk f l u i d modulus  xi  bulk f l u i d modulus n  porosity  P  pressure  q  flow rate  r  radius (variable)  R  radius (constant)  R r  c  w  radius of p l a s t i c zone radius of wellbore  S  saturation  T  temperature  u  pore f l u i d pressure  V  volume  xii  Subscripts 1  final  a  air  f  pore f l u i d  g  gas  i  i n i t i a l ( i n t e r n a l i n Chapter 6)  0  o i l (outer i n Chapter 6)  S  s o i l skeleton  v  volumetric  w  water  cb  combined  dg  dissolved gas  fg  free gas  Tg  t o t a l gas  Superscripts e  elastic  f  final  1  initial  p  plastic  xiii  ACKNOWLEDGEMENTS I am greatly indebted to my supervisor, Professor P.M. Byrne, for his guidance, encouragement end enthusiastic interest throughout t h i s research.  I would also l i k e to thank Professor Y.P.Vaid f o r  reviewing the manuscript  and making valuable suggestions.  My  colleagues, U. Atukorala, F. Salgado, J . She, H. V a z i r i and e s p e c i a l l y , C. Lum, shared a common active interest i n s o i l mechanics.  I thank  them a l l f o r t h e i r h e l p f u l discussions and constructive c r i t i c i s m s . Appreciation i s extended to Ms. S.N. Krunic for typing the manuscript  and her patience during the preparation of t h i s t h e s i s .  Support and assistance provided by the Natural Science and Engineering Research Council of Canada i s acknowledged with deep appreciation. A s p e c i a l thanks i s extended to my family and P r i s c i l l a , f o r their constant support and encouragement.  1  1.1  INTRODUCTION Many schemes for o i l recovery require open excavations, tunnels  or wellbore i n o i l sand.  As a r e s u l t , an accurate and e f f i c i e n t  analysis of stresses and deformations around these openings i s becoming increasingly important.  Mathematical models have been developed by  Byrne et a l (1980), Dusseault (1979), and Harris and Sobkowicz (1977) to investigate these problems.  An e f f i c i e n t undrained model f o r  analysing the stresses and deformations around open excavations, tunnels and wellbores i n o i l sand i s presented i n t h i s t h e s i s . O i l sand i s comprised of a dense sand skeleton with i t s pore spaces f i l l e d with bitumen, water and free or dissolved gas.  The  presence of bitumen reduces the e f f e c t i v e permeability of o i l sand, hence undrained conditions occur on rapid unloading.  Gas evolves from  pore f l u i d s during unloading when the pore f l u i d pressure i s below gas saturation pressure.  Because gas exsolution takes some time to occur,  two undrained conditions a r i s e , (1) an immediate or short term condition i n which there i s no time f o r gas exsolution, and (2) a long term or equilibrium condition i n which complete gas exsolution has occured.  Both of these conditions are considered i n this thesis.  rate of gas exsolution i s not considered herein.  The  The drained analysis  with pore pressures under steady-state conditions i s also addressed. The sand skeleton i s modelled as a non-linear e l a s t i c - p l a s t i c porous material.  The f l u i d s s t r e s s - s t r a i n relationships f o r undrained  condition are formulated on the basis of ideal gas law. For the undrained condition, the pore pressure changes ;.re computed from the constant of volume compatibility between the sand skeleton and the pore fluids.  It i s assumed that the pore f l u i d pressures are known i n the  drained analysis.  2 V a l i d a t i o n of the stress s t r a i n model i s made by comparing responses with t h e o r e t i c a l solutions and observed data.  The observed  data are the experimental results i n Sobkowicz's doctorate thesis. The stresses and deformations around a wellbore i n o i l sand upon unloading i s investigated using the new s t r e s s - s t r a i n model.  1.2  BEHAVIOUR OF OIL SAND O i l sand i s comprised of a dense, highly incompressible,  uncemented, interlocked skeleton with pore spaces f i l l e d by water, bitumen, and dissolved or free gas.  The interpenetrative structure  leads to the low i n - s i t u void r a t i o and high shear strength. The response of o i l sand i s mainly governed by the rate of loading.  Undrained conditions occur as a r e s u l t of:  1)  low e f f e c t i v e permeability to pore f l u i d s  2)  large amount of dissolved gas i n pore f l u i d s  3)  rapid unloading  The response of o i l sand may be physically modelled by a set of springs shown i n Figure 1.1. The o i l sand i s s p l i t into two load carrying components - s o i l skeleton and pore f l u i d s .  The s o i l skeleton  compressibility, 3 , characterizes the deformation of s o i l skeleton, g  which r e s u l t s i n a change i n e f f e c t i v e s t r e s s .  The pore f l u i d  compressibility, 8^, characterizes the deformation of pore f l u i d s (because of free gas) which results i n a change i n pore pressure.  3  a  +  Aa  X X X X 4 4 X 4 K=>  O  6  § § f o  § Water Soil Skeleton  5 \ § Stress = a'+ Aa § )  «o § Oil (bitumen)  Pore Fluid Pressure = u + Au  o § Gas Rigid Frame, No Lateral Yield  a = a' + u Ao = Aa'+ A u A V = /(Aa, Au)  Fig.1.1 - Schematic spring analogy for o i l sand ( After Dusseault , 1979 )  4  Strain compatibility between the pore f l u i d s and s o i l skeleton controls the r e l a t i v e magnitudes of the changes i n pore pressures and e f f e c t i v e stress which together equal the change i n t o t a l s t r e s s .  Stress changes  (unloading) are shared between the sand skeleton and pore f l u i d s according to t h e i r c o m p r e s s i b i l i t i e s .  When the pore pressure i s above  the gas saturation pressure and the o i l sand i s 100% saturated, the stress changes w i l l be accommodated by the pore f l u i d s because t h e i r compressibilities are lower.  Once the pore pressure drops below the  gas saturation pressure, gas s t a r t s to evolve which increases the fluids' compressibilities.  Therefore the sand skeleton becomes the  less compressible phase and takes up the stresses rather than the pore fluids.  When the e f f e c t i v e stress drops to zero, the s o i l skeleton  i s very compressible r e l a t i v e to the pore f l u i d s ;  hence any further  decrease i n t o t a l stress i s e n t i r e l y accommodated by the pore pressure. The gas exsolution takes some time to occur, hence two undrained conditions (Sobkowicz,  1982) a r i s e .  The expressions 'short  term' and 'long term' w i l l be applied to these processes exclusively.  1)  'Short term' undrained i n which there i s no time for gas exsolution  2)  'Long term' undrained i n which equilibrium state has been reached, i . e . completion of gas exsolution.  In the f i e l d , the unusal behaviour of o i l sand manifests i n a number of ways.  They include:  5  1)  Volumetric expansion of 5 to 15% occurred when core samples were l e f t i n an unconfined state, i . e . not retained by p l a s t i c core sleeves (Dusseault 1980;  2)  Core samples spontaneously  Byrne et a l 1980).  s p l i t l o n g i t u d i n a l l y and  perpendicularly to the core axis, effervescence was observed on several freshly recovered cores (Hardy and Hemstock 1963). 3)  Retrogression of slopes i n o i l sand on rapid excavation.  4)  O i l sand at the base of excavation subjects to softening and heaving, followed by settlement on reloading.  When the decrease i n external stress occurs over a period of time, the evolved gas has time to drain o f f and e f f e c t i v e stress does not go to zero which results i n an undisturbed sand skeleton. i n - s i t u density and hence high shear strength i s retained.  Its high  Such  situations can be seen on the exposures of o i l sand deposits along the Valley of Athabasca River where erosion (unloading) by the r i v e r has occurred over thousands of years.  These o i l sand deposits are standing  on steep stable slopes with slope angles i n excess of 60° and heights up to 60 metres, exhibiting high strength (Harris and Sobkowicz 1977).  1.3  THE SCOPE The purpose of this study i s to present a s t r e s s - s t r a i n model  ( V a z i r i , 1985) f o r the deformation analyses of o i l sand masses, i . e . to explain the behaviour of o i l sand as described i n Section 1.2. Modelling the undrained response of o i l sand requires the pore f l u i d s pressure to be numerically evaluated.  For the undrained  condition, pore f l u i d pressures are computed from the constraint of  volumetric compatibility between the sand skeleton and pore f l u i d phases. Compressibilities  of sand skeleton and pore f l u i d s have to be  evaluated i n the new s t r e s s - s t r a i n model.  A non-linear s t r e s s - s t r a i n  r e l a t i o n s h i p (Duncan et a l 1970) i s adopted for the sand skeleton. compressiblities  The  of pore f l u i d s are formulated on ths basis of ideal  gas laws. The new s t r e s s - s t r a i n model i s incorporated into f i n i t e programmes (INCOIL, MHANS).  element  For the v a l i d a t i o n of the s t r e s s - s t r a i n  r e l a t i o n s h i p , the computed results are compared with the t h e o r e t i c a l solutions and observed data. The validated programme was used to study the unloading response of stress and deformations around a deep wellbore i n o i l sand.  1.4  ORGANIZATION OF THE THESIS This thesis consists of seven chapters. A review of previous  work on undrained models f o r o i l sand i s given i n Chapter 2.  This  chapter concentrates on the examinations of their c a p a b i l i t i e s and shortcomings. A s t r e s s - s t r a i n model which was developed by V a z i r i i s presented i n Chapter 3.  Appropriate s o i l skeleton and pore f l u i d s  constitutive relationships are also recommended i n t h i s chapter. Chapter 4 summarizes the f i n i t e element formulations used i n the development of the programme. V a l i d a t i o n of the developed model by comparing predicted response with existing t h e o r e t i c a l solutions, f i e l d and laboratory  observations i s mentioned i n Chapter 5.  The response of a wellbore  o i l sand upon unloading i s investigated i n Chapter 6 using the validated f i n i t e element programme. A summary of work, and major conclusions are presented i n Chapter 7.  8  CHAPTER 2  2.1  : REVIEW OF PREVIOUS WORK  INTRODUCTION Theoretical solutions f o r the undrained response of o i l sand  were not a v a i l a b l e u n t i l 1977 because of i t s unusual behaviour.  Due to  the increase i n demand for construction i n o i l sand formation, such as open p i t mining, tunnels and deep shafts, a considerable amount of research has been done on this topic since 1977. These developed t h e o r e t i c a l r e l a t i o n s share the same basic approach of coupling the s o i l skeleton and pore f l u i d s together.  Pore  pressure changes are computed from the constraint of volumetric compatibility between the s o i l skeleton and pore f l u i d s .  Harris and  Sobkowicz's model i s capable of evaluating pore pressure change upon a stress or/and temperature change.  This model was then extended by  Byrne et a l and incorporated i n a f i n i t e element programme. Dusseault's approach i s r e s t r i c t e d to one dimensional problems. A c a r e f u l examination of these t h e o r e t i c a l models and t h e i r applications i s made and several shortcomings are also reviewed.  2.2  MATHEMATICAL MODELS FOR OIL SAND  2.2.1  Harris and Sobkowicz (1977) Harris and Sobkowicz presented a mathematical model to analyse  the undrained response of o i l sand subjected to changes of stress and/or temperature.  This i s the f i r s t a n a l y t i c a l model developed to  explain the behaviour of o i l sand such as:  1)  Movement and s t a b i l i t y of slopes and tunnels formed i n o i l  sand. 2)  Settlements or heave of structures placed i n o i l sand (e.g. hot o i l tank).  3)  Heave at the_base of excavations i n o i l sand.  The model can be explained by the same spring analogy as shown i n Figure 1.1. The response of o i l sand to changes i n stresses or/and to changes i n temperature may be computed by the following equations:  a)  One-Dimensional Analysis  Au can be obtained from a quadratic equation  L * Au + M * Au + N = 0  2.2  2  where  L = 3 s Tl M = 3 (P. - Aoi) + n + Pi — (n H s i g T w w a 1 /  N = P  ±  [n (1 -  1  v  - A 3 ] 0 l  s  + n H ) oo'  " P i AT ^ -  ( n ^+  n^)  10  b)  Two-Dimensional Analysis  from s t r a i n compatibility again  Aa  3  + A  Au =  (Ao ! - Ao ) -  3  n6 1+  *  2.3  f  s  and  P Au  z  + Q Au + R = 0  2.4  where  P = B s  T1 Q = 8 ( P . - X) + n + P i (n H s i g T w w v  R = P. [n l g L  1  (1 - — ) T  + n H ) oo'  —  - X 0 ] - P, AT — s i T J  X = A02 + Ap (Aai - Ao"2)  and = volumetric B  g  n  strain  = s o i l skeleton  compressibility  = porosity of s o i l  n  Q  = porosity of o i l  n  w  = porosity of water  n  = porosity of gas  Aa  = change i n t o t a l stress  ^  v  (n a w w  + n a ) 0 0  11  P  = i n i t i a l pore f l u i d pressure  Au  = change i n pore f l u i d pressure  Ao"i = change i n major p r i n c i p a l stress Ao"3 = change i n minor p r i n c i p a l stress T  a  = standard temperature (288 °K) = i n i t i a l pore f l u i d temperature °K  Tl  = f i n a l pore f l u i d temperature °K  AT  = change i n temperature = temperature s o l u b i l i t y constant for gas i n water = temperature s o l u b i l i t y constant for gas i n o i l  H  W  = Pressure s o l u b i l i t y constants f o r gas i n water  H  q  = Pressure s o l u b i l i t y constants f o r gas i n o i l  Ap  = s o i l skeleton dilatancy factor  Harris and Sobkowicz examined the response of o i l sand during excavation and reloading of a square footing, and the behaviour of a tunnel excavation i n the same material.  Their results w i l l allow an  assessment of the a p p l i c a b i l i t y and shortcoming of the theory.  1)  This solution..incorporated a linear constitutive relationship f o r the s o i l .  As o i l sand behaves l i k e an  e l a s t i c - p l a s t i c material, there w i l l be a p l a s t i c  zone  developed adjacent to the tunnel wall on unloading when the effective.stresses are such that f a i l u r e (Mohr Coulomb C r i t e r i a ) occurs. 2)  The extent of the p l a s t i c zone i s only an approximation because r e d i s t r i b u t i o n of stresses has not been taken into  account during the formation of p l a s t i c zone. 3)  2.2.2  An i t e r a t i v e procedure i s required  to obtain  Au  DUSSEAULT Dusseault (1979) extended the one dimensional Skempton's B  equation to analyse the behaviour of cohesionless amount of free or dissolved gas i n pore f l u i d s .  materials with large He presented a more  rigorous derivation for the compressibility of pore f l u i d s , and coupled the compressibility with that of s o i l skeleton  ( e - a' relationship)  i n Skempton's B equation. This model shares the same basic idea as the previous (Harris and  Sobkowicz), which has two  skeleton and pore f l u i d s .  one  load carrying components - s o i l  The r e l a t i v e c o m p r e s s i b i l i t i e s of these  components control the magnitude of changes in pore pressure and effective  stress which together balance the change i n t o t a l stress i n  an undrained state. The  t r a d i t i o n a l Skempton's B equation i s  2.5  Aa 1 + n-  8^ = compressibility of pore f l u i d (water) 3  The  = compressibility of s o i l  extended one i s  skeleton  ^  = l / [ l + f(u,o)]  2.6  a+bAn(a-u)-e -e +H e -Hi e f(u,a) =  .  D  [eg + e 3 + o o w w  J J  U  a, b = void r a t i o - e f f e c t i v e stress relationship parameters e o', ew', e g = void r a t i o s : o i l», water,» e>gas H , H = Henry's constants : o i l , water o' w 3,  3  Q  u, a  w  = compressibility  : o i l , water  = current values of pore pressure and  t o t a l stress  Dusseault applied this model to investigate the response of element of o i l sand upon unloading. and a deeper one  (500 m).  He examined a shallow case (15  His results w i l l allow an assessment of  an m) the  a p p l i c a b i l i t y and shortcoming of the theory:  1)  In order for the solution to be numerically stable, reduction  further  i n t o t a l stress i s assumed to be e n t i r e l y taken  by the pore pressure once e f f e c t i v e stress drops to zero. 2)  This solution incorporated relationship for the  3)  a non-linear constitutive  soil.  Accurate e - a' relationship or compressibility of o i l sand i s extremely d i f f i c u l t to obtain since they are very sensitive to sample disturbances. samples have been cored so f a r .  No undisturbed o i l sand  4)  2.2.3.  An i t e r a t i v e procedure i s required to get Au.  Byrne and Janzen Byrne et a l (INCOIL, 1983) developed an incremental a n a l y t i c a l  f i n i t e element method for predicting stresses and deformations i n excavations and around tunnels i n o i l sand using a nonlinear e l a s t i c sand skeleton with shear d i l a t i o n .  An extension to Harris and  Sobkowicz's model was used to evaluate the pore pressure change, Au. The f i n i t e element program, INCOIL, can handle both undrained and drained analyses.  For the undrained condition, pore f l u i d  pressures are computed from the constraint of volumetric compatibility between the sand skeleton and the pore f l u i d s . For the f u l l y drained condition, i t i s assumed that the pore f l u i d pressures are known. The general framework of the f i n i t e element model for o i l sand are  as follows:  [K]  (fi) = (Af) - (K ) (Au)  2.7  w  where  u Ti | — * AT * (n a +n a ) - u fn * ( 1 - — ) T w w o o ' «oL g T ' v  Au =  u  a  L  v  a  Ti L  (n H +n H ) + n - Ae w w o o g v  Derivation of equation 2.7 i s presented i n Appendix A  Ae  1 v J  2.8  15  The e f f e c t i v e stress may  (Ae) = [c]  be evaluated from  (A6)  2.9  (Aa') = [D»] (Ae)  2.10  where  [c]  = the matrix which depends on element geometry  [D'J  = the matrix property matrix ( a f f e c t i v e stress)  [Aa']  = the change i n e f f e c t i v e stress vector  [Ae]  = the change i n s t r a i n vector  [K]  =» the element s t i f f n e s s matrix  (A6)  = the incremental element nodal deflections vector  (Af)  = the incremental element nodal forces vector  (K )  = the pore pressure load vector  n , n , n^  = porosity : water, o i l and gas phase  a  = temperature s o l u b i l i t y constant : water, o i l  w  w >  H , W  u  Q  a  Q  H  Q  = pressure s o l u b i l i t y constant : water, o i l = reference (atmospheric)  pressure  T  a  = reference temperature (288°K)  T  Q  = i n i t i a l temperature (°K)  Tl  - f i n a l temperature (°K)  AT  = change i n temperature ( T i - T ) (°K) o = i n i t i a l absolute pore pressure  U  Q  Ae  = volumetric s t r a i n (compression p o s i t i v e )  They examined the response of c y l i n d r i c a l shaft i n o i l sand on reduction of support pressure and the response of an element of o i l sand to one dimensional unloading.  The l a t t e r case i s to simulate core  samples of o i l sand l e f t i n an unconfined state.  Their results  will  allow an assessment of the a p p l i c a b i l i t y and shortcoming of the theory:  1)  The predicted expansion of the core on unloading i s small compared with those measured i n the f i e l d which i s 5-15% when core samples of o i l sand are l e f t i n an unconfined state.  2)  An i t e r a t i v e procedure i s required to obtain Au.  CHAPTER 3  3.1  : STRESS-STRAIN MODEL  INTRODUCTION It i s noted that the mathematical models described i n Chapter  2 have quite a few shortcomings.  Therefore, a more sophisticated and  e f f i c i e n t undrained model was developed (Naylor 1973, V a z i r i 1985) and i s incorporated i n a f i n i t e element programme. A t o t a l stress approach coupling the s o i l skeleton and pore f l u i d s i s used .  Pore pressure changes are computed from the  constraint of volume compatibility between the s o i l skeleton and pore f l u i d s under undrained conditions. Separate c o n s t i t u t i v e relationships f o r s o i l skeleton and pore f l u i d s are required i n the new undrained model.  An  incremental  non-linear e l a s t i c and i s o t r o p i c s t r e s s - s t r a i n model as described by Duncan et a l i s adopted for the s o i l skeleton.  Depending on the  component of the pore f l u i d s , d i f f e r e n t formulations f o r the non-linear e l a s t i c and i s o t r o p i c s t r e s s - s t r a i n relationships of pore f l u i d are derived.  3.2  DEVELOPMENT OF UNDRAINED MODEL The e f f e c t i v e stress concepts (Terzaghi) i n conventional  soil  mechanics seem to be applicable to determine the shear strength of o i l sand (Hardy and Hemstock).  However, the pore pressures of f l u i d s i n  the o i l sand have to be numerically evaluated i n the e f f e c t i v e stress approach.  Hence, a quantitative r e l a t i o n s h i p between the magnitudes of  stress release and pore pressure i s required. When a saturated s o i l mass i s subjected to undrained loading,  18 stress change must be shared between the s o i l skeleton and pore f l u i d (Bishop and E l d i n 1950).  A theoretical expression for the relationship  between t o t a l stress change and r e s u l t i n g pore f l u i d pressure change was derived by Skempton (1954) which i s the Skempton's B equation. Using the f i n i t e element method, C h r i s t i a n (1968) introduced an e f f e c t i v e stress approach for s o i l s subjected to undrained loading, enabling r e s u l t i n g pore f l u i d pressure to be evaluated. Programmes incorporating this alternative are r e l a t i v e l y i n e f f i c i e n t .  Naylor  (1973) developed a more elegant approach which allows excess pore pressure to be computed e x p l i c i t l y i n terms of material skeleton s t i f f n e s s parameters and an independently s p e c i f i e d pore f l u i d stiffness.  However, Naylor only considers s o i l s that are two-phase  system - solids and water. Due to the presence of bitumen, free and dissolved gas i n o i l sand, the above mentioned approaches are inadequate to describe the undrained behaviour of o i l sand.  Not u n t i l 1977, Harris and Sobkowicz  developed an a n a l y t i c a l expression incorporating a linear constitutive r e l a t i o n s h i p f o r the o i l sand, to r e l a t e the change i n pore pressure to the change i n total stress.  Dusseault (1979) extended Skempton's one  dimensional B equation to model the equilibrium behaviour of o i l sand. Byrne et a l (1980) studied the behaviour of o i l sand by using f i n i t e element method.  They extended Harris and Sobkowicz's model and  incorporated i t into Christian's f i n i t e element formulation. As there are shortcomings of the approaches developed by Harris and Sobkowicz, Dusseault and Byrne et a l , a more sophisticated numerical model i s required.  V a z i r i adopted Naylor's approach and  extended i t to model the undrained behaviour of o i l sand using f i n i t e  elements. The s t r e s s - s t r a i n model f o r o i l sand behaviour i s based on the following assumptions.  1)  Volumetric change of s o i l skeleton i s governed by the e f f e c t i v e stress.  2)  Liquids and gas i n the voids are at the same pressure, i . e . effect of surface tension between the pore f l u i d phases are neglected.  3)  Free gas i n the pores behaves i n accordance with c l a s s i c gas laws with respect to pressure.  Gas comes out from  solution i n accordance with Henry's law. Henry  solubility  constants are constant. 4)  The compressibility of s o i l grains i s n e g l i g i b l e , and has no contribution to the volume changes.  5)  The gas Is i n the form of occluded bubbles inside the pore fluid.  The t o t a l stress constitutive law may be written as:  (Aa) = [D] (Ae)  3.1  where (Aa), (Ae) are the incremental t o t a l stress vectors and s t r a i n vectors respectively, and [D] i s the material property matrix ( t o t a l stress).  Computation of element s t i f f n e s s matrix, assembly and  solutions f o r displacements proceeds along the standard l i n e s . analysis yields a t o t a l stress f i e l d .  The  Since the s o i l skeleton and the pore f l u i d s deform together when conditions are undrained, strains - i n a macroscopic sense - are the  same i n each phase.  Thus i n addition to Equation 3.1  (Aa') = [D»] (Ae)  (Au)  = K  = K  (Ae  a  v  n  3.2  + Ae 2 + A e ) 2  3 3  (Ae ) v'  3.3a  3.3b  i n which prime means e f f e c t i v e , [D'] i s the material property matrix (affective stress), K  i s the apparent pore f l u i d bulk modulus and (Au  i s the pore pressure change vector. The actual pore f l u i d modulus, K^, i s related to the apparent one as  f = — a n K  K  where n i s the s o i l  3.4  porosity.  Derivations of equations 3.3 and 3.4 are presented i n Appendix B. Equation 3.3 may be expressed i n a form compatible with equation 3.1 and 3.2 as:  (A° ) = [D ] (Ae) f  where (Aa ) = [Au f  3.5  f  AU  AU  0  0  0]  T  Since the pore f l u i d cannot transmit shear, [D^] can be expressed i n terms of apparent bulk modulus,  r  [D ] = K f a F  3  0  3  °3 0  3.6  3  where I 3 i s as 3 x 3 matrix with a l l the elements equal to 1, whereas O3 are 3 x 3  n u l l matrices.  The p r i n c i p l e of e f f e c t i v e stress may be used to r e l a t e the changes i n e f f e c t i v e stress and pore presusre caused by the applied loads to the corresponding change i n t o t a l stress  (Ao) = (Aa') + (Aa )  3.7  f  Substituting from Equations 3.1, 3.2, and 3.3 into 3.7,  yields:  [D] = [D«] + [D ]  3.8  f  The e l a s t i c material i s now considered to be two phase, with the s t i f f n e s s defined by e f f e c t i v e stress moduli, E, B and pore f l u i d apparent builk modulus K^. The material properties of [D'] and  are read i n separately.  They are combined i n the programme automatically using equations and 3.8.  3.6  Thereafter, computation of element s t i f f n e s s , assembly and  solution of displacements and hence strains proceed along the standard lines.  The e f f e c t i v e stress and pore pressure are obtained by  Equations 3.2 and 3.5  respectively.  This approach allows stresses, porepressure and deformation i n s o i l mass, with nearly incompressible to highly compressible pore f l u i d s , to be evaluated by f i n i t e elements f o r undrained conditions. Naylor studied the response of undrained t r i a x i a l test on clay using t h i s model.  The end platens were r i g i d and rough.  But the computed  excess pore pressures near the centre of the sample were i n good agreement with the theoretical solutions (Cam-Clay).  In the case of  drained analysis, Equations 2.7 to 2.10 are used instead, assuming a l l the pore f l u i d pressures are known.  3.3  CONSTITUTIVE RELATIONS  3.3.1  Incremental Non-linear Elastic Soil Skeleton An incremental non-linear e l a s t i c - p l a s t i c and  isotropic  s t r e s s - s t r a i n s o i l model as described by Duncan et a l (1970) i s employed i n this thesis.  In this approach, two independent e l a s t i c  parameters are required to represent non-linear s t r e s s - s t r a i n and volume change behaviour. the Poisson's r a t i o , v.  These are usually the Young's modulus, E, and The shear modulus, G, and the bulk modulus B,  are the more fundamental parameters because they separate shear or d i s t o r t i o n and volume components of s t r a i n and would be the most desirable ones to use.  However, the shear modulus i s d i f f i c u l t to  obtain d i r e c t l y i n laboratory testings and f o r t h i s reason Duncan et a l (1980) used the Young's modulus and the bulk modulus as their parameters.  two  The Young's modulus i s very s i m i l a r i n character to the  shear modulus as both are a measure of d i s t o r t i o n a l response. Therefore, E and B are used i n t h i s thesis.  The stress-dependent E and B are usually obtained from laboratory tests.  A t y p i c a l example for sand i s shown i n Figure 3.1.  The d i s t o r t i o n a l response can be reasonably approximated by modified hyperbolas (Konder) and the volumetric response i n exponential form. They are expressed as follows:  The tangent Young's modulus  (l-sin4.) ( a i - 0 3 )  R  E  where  t  E  ±  = [l-  =-j — 2oj sin<j>  —  L  = Kg P  a  (p-) a  2  1 J  E, i  3.9  3.10  n  and  <J. = <(>!- A* log (—)  3.11  The tangent bulk modulus  B  where  E  i  t  = K  B a<F-> a P  m  = i n i t i a l Young's modulus = Young's modulus number  n  = Young's modulus exponent  3  '  1 2  24  16  Fig.3.1  - S t r e s s - S t r a i n c u r v e s f o r d r a i n e d t r i a x i a l t e s t s on l o o s e sand ( A f t e r Byrne and E l d r i d g e , 1982 )  \  = bulk modulus number  m  = bulk modulus exponent  P  = atmospheric pressure a  °{> R  f  = major and minor p r i n c i p a l e f f e c t i v e stress = failure ratio = f r i c t i o n angle at confining stress of 1 atm  •l A<{>.  = decrease i n f r i c t i o n angle f o r a tenfold increase i n confining stress  The procedures for evaluating these parameters from laboratory tests are described i n d e t a i l by Duncan et a l (1980) and Byrne and Eldridge (1982).  3.3.2  Incremental Non-linear Fluid Modulus  3.3.2.1 General An incremental non-linear e l a s t i c and i s o t r o p i c s t r e s s - s t r a i n f l u i d model i s employed here.  Since the f l u i d cannot transmit shear,  the s t r e s s - s t r a i n relations are defined only by one e l a s t i c  parameter,  the bulk f l u i d modulus, K^, which i s a measure of volumetric response. Before the derivation of K^, Henry's law and Boyle's law must be mentioned  because these laws govern the derivations.  Pore f l u i d s may be immiscible, miscible, or a combination of the two.  Examples of both miscible and immiscible f l u i d s w i l l be  considered i n this thesis.  That i s , water undersaturated with a i r and  carbon dioxide, water and bitumen saturated with gas (methane, (X^)The f i r s t combination i s to describe the pore f l u i d s of the gassy s o i l s which Sobkowicz (1982) used i n his laboratory testings. combination i s to similate o i l sand pore f l u i d s .  The second  If both free gas and l i q u i d s are present i n the pore f l u i d s , and the gas i s soluble i n the pore l i q u i d i n a certain extent, the pore f l u i d compressibility w i l l be both pressure dependent and influenced by the s o l u b i l i t y relationship.  Hence, Boyle's and Henry's laws are  appropriate f o r describing these volume and pressure relationships.  1)  Boyle's law (Laidler et a l ) : The volume of a free gas i s inversely proportional to the pressure applied to i t when the temperature  i s kept constant.  Mathematically,  V a j  3.13  where V i s the volume, P i s the absolute pressure 2)  Henry's law ( L a i d l e r et a l ) :  The mass of gas m dissolved  by a given volume of solvent at constant temperature, i s proportional to the pressure of the gas i n equilibrium with the solution.  Mathematically,  m (dissolved gas) = H * P  3.14  where H i s Henry's s o l u b i l i t y constant, P i s the absolute pressure.  In other words, the volume of dissolved gas i s constant i n a f i x e d volume of solvent at constant temperature when the volume i s measured at P  27  (dissolved gas) = H * V (solvent)  3.15  Most gases obey Henry's law when the temperature i s not too low and the pressure i s moderate.  If several gases from a mixture of gases  dissolve i n a solution, Henry's law applied to each gas independently, regardless of the pressure of the other gases present i n the mixture. H i s both temperature and pressure dependent, p a r t i c u l a r l y for natural gases i n hydrocarbons (Burcik, 1956).  Since the v a r i a t i o n of H on  pressure i s not very s i g n i f i c a n t , i t i s assumed that H i s independent of pressure i n t h i s thesis.  3.3.2.2 Partly Mlsclble Gas/Liquid Mixture Definitions:  1)  Pore f l u i d s compressibility, 3  3.16a  where V  i s the volume of pore f l u i d s , P i s the absolute  pressure, assuming surface tension e f f e c t s are neglected,  3.16b  28  2)  Compressibility of a gas g  1  d  v  8  where V  a)  i s the volume of gas.  Air/water Mixture Let the i n i t i a l volume of free gas and water i n a s o i l element be:  V* and V fg w  Thus the t o t a l volume of free and dissolved gas  vi = Tg  fg  + H V w  3.18  For a change i n pressure, the volume of water i s assumed constant as the compressibility of water i s i n s i g n i f i c a n t compared with the pore gas.  Applying Boyle's law to the .,1  t o t a l volume of gas (Fredlung 1973, Sobkowicz 1982) i n the element, f i V = V Tg Tg V  V  i * — P P  3.19  f  Then the new volume of free gas after change i n pressure  f f v = V fg Tg V  V  f - V dg V  29 P Tg  ( V  P.  fg  +  V  V  dg  dg  ) * Pf  " dg  3.20  V  Change of free gas i n the s o i l element  fg  fg  fg p  fg  - ( fg L V  By  +  V  dg'  P  ,!) dg' *  P  dg  f  +AP  v  fg  ra»  3.21  definition  g  i fg  AP  (v* + fg i fg  V, ) dg' 1  #  v  ±+  Also, by d e f i n i t i o n  1 V  * f  1 P AP  d  f dP V  3.22  30  ,  dV.  V  V  L  dV  dP  dP  P  L f  1 - S + SH P, + AP  J  + AP  +  S  e  VwJ  - w  3.23a  As AP approaches zero,  3  f  P  +  Sg  w  3.23B  and  b)  Carbon Dioxide, Air/water Mixture  By d e f i n i t i o n , the compressibility of pore f l u i d s , 3 , f  3  f  1_ " V  , = - — V  ^ *  f*f  dP  dV dV f—& + — dP dP r  L  J  i n which there may have a i r and carbon dioxide as free gas, V* . g  Consider the phase diagram i n Figure 3.2, the volume of solids i s assumed to be 1 u n i t ,  the volume of void i s e  units according to d e f i n i t i o n of void ratios  Therefore  V* e - e fg w V " e  3.25a  V* H e + H „ e dg _ a w co2 w V  3.25a  =  f  f  Substituting equations 3.25 into 3.24a, y i e l d s  , , _ 1 r !, = — I f e L  ( e - e ) + H e + H _ e w a w co2 w -i „ , r g e P + AP w w  As AP approaches zero,  L  Q  J  „  ,  3.24b 0 /  32  •+ •w  Air & C02  v  9  Water  Solids  F i g . 3 . 2 - Phase diagram for gassy s o i l  33  . _ 1 r 3 = _ ^ f e a  L  ( e - e ) + H e + H _ e w a w co2 w , . -1 + 8 e J P w w  -  . .  3.24c  J  and  K  1  -  2s £  c)  Gas (Methane)/Bitumen and Water Mixture Again, by d e f i n i t i o n the compressibility of pore f l u i d s  1 8 = - — f V  d  f  *  dP  P  f  i V  V  d v  L f  rr ^  ^  dv  dP  x  dP  J  (V* + V* ) p , ° P^ + AP  [  g  v  8  - V 0 ww  - V. B. ] b b  3.26a  J  Consider the phase diagrams i n Figure 3.3, the volume of solids i s assumed to be 1 u n i t , the volume of voids i s e units according to d e f i n i t i o n of void r a t i o s  Therefore  (e - e  w  - e ) b  3.27a  F i g . 3 . 3 - Phase diagram for o i l sand  where Hg/ > ^g/b W  a  r  e  Henry's s o l u b i l i t y constants of gas  i n water and bitumen respectively.  Substituting Equations 3.27 into 3.26a, and AP approaches zero, y i e l d s ,  B  = I f e  [ L  (e - e - e, ) + H , e 2 * -^!_w P  + H  e, g/b_J>  ,  +  w w  b b  J  3.26b  and  CHAPTER 4 4.1  : FINITE ELEMENT FORMULATIONS  INTRODUCTION Two types of formulations are presented herein which are  suitable for modelling a variety of problems encountered i n practice. Depending on the nature of the problems, they w i l l f a l l into one of the following categories: 1)  Plane s t r a i n - 2 Dimensional, e.g. tunnels, shaft, e t c .  2)  Axisymmetric - 3 Dimensional, e.g. t r i a x i a l  test,  wellbore, e t c . The s o i l i s modelled by isoparametric q u a d r i l a t e r a l or triangular elements. Stress d i s t r i b u t i o n formulations are added to cope with problems where p l a s t i c zones are developed during loading or unloading.  4.2.  THE PLANE SRAIN FORMULATIONS  4.2.1  The Constitutive Matrix [P] Plane s t r a i n problems are characterized by the following two  properties: 1)  no d e f l e c t i o n i n z d i r e c t i o n  2)  f i r s t derivative of the x and y deflections with respect to z are zero.  Therefore  where u, v, w are the displacements i n x, y and z d i r e c t i o n s respectively with corresponding subscripts, 1, 2 and 3 r e s p e c t i v e l y .  From the generalized  Hooke's law for incremental e l a s t i c i t y  Aei j = [Aon - v ( A 0 2 2 Ae 2 2  =  ~  [AO"22  v  ^  a  i l  +  Ac )]/E  4.2a  Ao" )]/E  4.2b  3 3  +  33  - v ( A o ^ + Aa 2)]/E  Ae  3 3  = [Aa  Ae  1 2  = Ao /G  4.2d  Ae  2 3  = Aa /G  4.2e  Ae  3 1  = Aa /G  4.2f  2  33  12  23  31  After substituting the conditions 4.2, i t follows  4.2c  from equation 4.1b into equations  that:  Aoi  3  Ao2  3  = Ao  = 0  4.3a  = Aa 2 = 0  4.3b  3 1  3  where Aa^.. i s the incremental shear stress with the d i r e c t i o n indicated by the subscripts. With the above eliminations,  the incremental stress and s t r a i n  vectors become  (Aa) = [ A a u  Ao"22  Ao-12]  4.4a  (Ae) = [ A e  n  Ae  4.4b  Ae ]  2 2  12  The constitutive relations for a plane s t r a i n problem i n t o t a l stress analysis are written (Naylor, 1973) as:  Ao  r l-v  n  Ao" 2 1 2  + K  a  0  l-v  0  0  d-2v) 2  (l+v)(l-2v)  2  Aa  V  1  1  1  0  1  1  0  ll Ae 22  0  0  0  Ae 12  A  e  4.5a  or  (Aa) = ([D'l + [D ]) f  (Ae)  4.5b  where  E  = tangent Young's modulus  v  = tangent Poisson's r a t i o  K cL  = apparent tangent bulk f l u i d modulus  The constitutive relations can also be written i n another form which i s adopted i n INCOIL.  equivalent  *  Aa  2 2  Ao  1 2  <  B' + G'  B' - G'  0  B  B* - G'  0  1  - G 0  0  G'  <  + K  1  1  0  1  1  0  0  0  0  Ae •  A £  n  22  Ae  1 2  or  (Aa) - ([ ']) + [D ]) (Ae) n  f  Where  •a  G'  i  _  3B 2(l+v)  =  E  2(l+v)  B  =  tangent bulk modulus  E  =  tangent Young's modulus  v  =  tangent Poisson's r a t i o  K  =  apparent tangent bulk f l u i d modulus  Equation 4.6 can also be written as  (Aa) = [D]  (AS)  40  where  [D] i s the constitutive matrix  4.2.2  The Strain Displacement Matrix [B] For isoparametric elements, the geometry (x,y) and  displacement (u,v) are both expressed by te same shape functions and are approximated as:  (*) = [N] («)  4.10  O  4.11  and  - [M] («')  where  Nj  0  N  Ni  0  (6)  - [x! y!  x  2  y  2  x  3  y  3  (6')  =  u  2  v  3  u  3  v  3  W  to  [uj  V!  2  0 N  N 2  3  0  0 N  U  k  3  0  X l +  uit  0  NJ y j  -  v ] 4  i n which ( 6 ) i s the nodal coordinate vector and ( 6 ' ) i s the incremental  nodal displacement vector  and  N  x  =  (l-s) (l-t)/4  N  2  =  ( l - s ) (l+t)/4  N  3  =  (1+s) (l+t)/4  -  (1+s) ( l - t ) / 4  x y = nodal coordinates i n x and y d i r e c t i o n s 1, i respectively u. v. = incremental nodal displacement i n x and y 2» 3 directions respectively. s,t are l o c a l coordinates  The incremental s t r a i n vector can be expressed i n terms of displacement as follows: *  3u/3x Ae Ae  1 2  13  =  3v/8y 3u/3y + 3v/3x  Substitution of u and v from equation 4.11 into equation 4.12 y i e l d  3N  3Ni Ae  11  3N  2  3x~  3x~ 3N  Ae 22  Ae  0  W 3Ni  3N  3y  3x  W  u  l  v  l  u  2  v  2  0 3N  2  2  3N\  3  3No  3Nq  3N.  3N  3x  3y  3x  3y~  3N1+:  4  37" I  4.10  3~  u  v  0  3x~  3y~~  3Nj  13  3N^  3  3  Equation 4.10 can also be written i n matrix notation as  (Ae)  - [B] (6)  4.13  where  [B] i s the s t r a i n displacement matrix  However, the shape functions  for isoparametric elements are  defined with respect to the l o c a l coordinates s and t and therefore cannot be d i f f e r e n t i a t e d  d i r e c t l y with respect to the global x, y  axes. In order to overcome this d i f f i c u l t y i t i s necessary to obtain  the derivatives of the two sets of coordinates and t h i s can be achieved through the chain rule of p a r t i a l d i f f e r e n t i a t i o n . For p l a i n s t r a i n problems, the derivatives are related as  3_ 3s  3_ 3x  3_ 3t  3_ 3y  4.14  where  3x 3s  3y 3s  M-  i s c a l l e d the Jacobian matrix 3y 3t  3x 3t  )  Hence the derivatives w*r»t» x and y can be expressed as derivatives w»r«t s and t as follows  3_ 3x 3_ 3y  -1  3s 4.15  [j]  3t  The s t r a i n displacement matrix i n Equation 4.13 are evaluated numerically, using Gaussian quadrature over q u a d r i l a t e r a l regions. quadrature rules are a l l of the form  // f (s,t) ds dt -  n E i-1  r. I K j-1 1  K. f ( s , t.) 2  1  J  The  where K^, K\ are weighting functions and s^, t^ are coordinate position within the element. A 2 x 2 Gauss quadrature i s used to evaluate the s t r a i n displacement.  4.2.3  The Stiffness Matrix [K] The s t i f f n e s s matrix for the force displacement relationship  i s obtained by the p r i n c i p l e of v i r t u a l work.  For a v i r t u a l nodal  displacement vector ( 6 ) , the external work done, W(ext), by the g  external force vector ( f ) caused by v i r t u a l displacements i s written as  W(ext) - {6)1 ( f )  4.16  The v i r t u a l s t r a i n vector caused by the v i r t u a l displacements vector i s written as  (^) = [B] ( 6 )  4.17  e  Hence the internal work done, W(ext), caused by the v i r t u a l strain i s written as  W(int) = J  where  ( A e ) (Ao) t dA 1  4.18a  t = thickness of the element A = area of the element  Substituting equation 4.17 into 4.18a y i e l d s  W(int) = J  [ « ] * [ B ] [D] [B] [6] t dA T  A  4.18b  Applying p r i n c i p l e of v i r t u a l work  W(ext) = W(int)  4.19  (f) = /  4.20  Therefore  [ B ] [D] [B] [6] t dA T  A  (f) = [K] (6)  ' 4.21  T where [K] = /  [B] [D] [B] t dA i s c a l l e d the s t i f f n e s s matrix f o r  the element. The global s t i f f n e s s matrix, [K ], i s obtained by assembling a l l the element s t i f f n e s s matrices together.  The procedure of  assembling the element matrix i s based on the requirement of 'compatibility' at the element nodes.  This means that at the nodes  where elements are connected, the values of the unknown nodal degrees of freedom are the same for a l l the elements joining at that node. global force  The  vector, [ F ] , i s assembled by adding nodal loads of each of the elements sharing the node. procedure (e.g.  Displacements are calculated using  standard  Gaussian elimination) to solve the simultaneous  equations,[K ] [s]  = [F],represented by the global s t i f f n e s s matrix and  the force vector.  Strains can be computed from equation 4.13 or 4.36  a f t e r knowing the elements nodal displacements.  After solving f o r  displacements and s t r a i n s , the e f f e c t i v e stress and pore pressure can be computed from  (Aa') = [D'j (Ae)  and  (Au) = [D ] (Ae) f  4.3  THE AXISYMMETRIC FORMULATIONS  4.3.1  The Constitutive Matrix [D| Axisymmetric problems are characterized by the following  properties:  1)  Symmetry of both geometry and loading  2)  Stress components are independent of*the angular (0) coordinates.  Hence  v = 0  4.22a  A e  13  Ae  2 3  "  A e  31  = Ae  =  4.22b  0  = 0  3 2  4.22c  where u, v, w are displacements i n the r , z and 0 directions with corresponding cubscripts, 1, 2 and 3 r e s p e c t i v e l y . From the generalized Hooke's law for incremental e l a s t i c i t y  = [ l o - v (Ao 2  + Ao )]/E 33  4.23a  2 2  =  [ACT  + Aa )]/E n  4.23b  33  "  [ "33  + Aa )]/E  4.23c  Ae  n  Ae A e  u  2 2  Ao  2  - v (A033 ~  v  (Aa  22  n  Aei3  = Ao /G  4.23d  Ae  1 2  = Aa /G  4.23e  Ae  2 3  = Aa /G  4.23f  13  12  23  After substituting the conditions from Equations 4.22b and 4.22c i n t o Equations 4.23, i t follows that  Aai3 = A a i = 0  4.14a  Aa  4.15a  3  where  A  o i  j  2 3  = Aa  = 0  3 2  i s the incremental shear stress with the d i r e c t i o n indicated  by the subscripts. With the above eliminations, the incremental stress and s t r a i n vectors become  (Ao) = [ A a  n  Aa  2 2  Aa  3 3  Aa ] 1 2  4.25  48  (Ae) = [ A e n  Ae 2 2  Ae  Ae  3 3  1 2  4.26  ]  The constitutive relations for an axisymmetric problem i n t o t a l stress analysis are written (Naylor, 1973) as  Aa Aa  2 2  Aa  3 3  Aa  1-v  11  (l+v)(l-2v)  1 2  V  V  0  1-v  V  0  V  1-v  0  0  0  1  1  1  0  Ae  n  1  1  1  0  Ae  2 2  1  1  1  0  0  0  0  0  [D-]  +  + K  l-2v  4.27a Ae  1 2  Also  (Air) =  [ D J  (Ae)  where  E  =  tangent Young's modulus  v  =  tangent Poisson's r a t i o  K  =  apparent tangent bulk modulus  4.27b  The constitutive r e l a t i o n s can also be written i n another equivalent form which i s adopted i n INCOIL.  Ao  ] B'+G'  B'-G'  B'-G'  0  A022  ] B'-G'  B'+G'  B'-G  0  ^33  ] B'-G'  B'-G'  B'+G'  0  Ao  n  1  1  1 2  + K  1  1  1  1  0  1  1  1  1  0  1  1  1  1  0  0  0  0  0  0  A  e  ll I  Ae  2 2  33 Ae  j  I  4.28a  1 2  or  [Ao]  - ([D«] +  B'  =  [ D ] ) [Ae] f  where  3  B  2(l+v) .  _ J 2(l+v)  B  =  tangent bulk modulus  E  =  tangent Young's modulus  v  =  tangent Poisson's r a t i o  G  ,  4.28b  K  =  a  apparent tangent bulk f l u i d modulus  Equation 4.28 may be written i n matrix notation as  (Aa) = [D] (Ae)  4.29  where [D] i s the constitutive matrix.  4.3.2  The Strain Displacement Matrix [B] The isoparametric elements, the geometry ( r , z) and  displacements (u, v) are both expressed by the same shape functions and are approximated  as:  c •>  r  -  W  =  [ N ]  4.33  ( « ' )  0  N  0  N ,  0  N  [N]  4.32  (6)  x  0  2  N  3  0  N  N q  0  z  3  r  v  3  4  0  =  (6)  =  (>!  z  l  r  (6')  =  [u!  v  x  u  0  N ,  2  2  z  2  r  v  2  u  3  3  k  Ul+  N  Zlf  b  ]  v„]  i n which (fi) i s the nodal coordinate vector, ( 6 ' ) i s the incremental  nodal displacement vector and the shape functions  are the same as  given i n Section 4.2.2.  and  r  nodal coordinates i n x and y directions  i» i Z  respectively, incremental nodal displacements i n x and y  u., v.  J  J  directions respectively.  The incremental s t r a i n vector can be expressed i n terms of displacements as follows:  Ae  s  11  i Ae  3v 3r  2 2  4.34  u r  I Ae 33 Ae  3u 3r  1 2  )  3v 3r  3u 3z  Substitution of u and v from Equation 4.33 into Equation 4.30 y i e l d s  3N  3N  X  Ae  11  3r  3r~  3Ni Ae 2 2  A e  33  Ae  1 2  0  3N  2  0  3z~  Nl — r 3Nj  3Nj  3z~  IF  0  TT~  3N N — r 3N  0 N  0 2  3z~"  3N  3  3r~  3N  3  3z~ 0  r 3  0 3N  2  0 2  0  3  3N 3  37"  w  3  52  r 0  W  vi  j  u | 2  3z~  0  v u  3  v  3  0 r  31^  4.35  2  UI+  3r~  Equation 4.35  can also be written i n matrix notation as  (Ae)  =  [B]  (6)  4.36  where  [B] i s the s t r a i n displacement matrix  However, the shape functions  for isoparametric elements are  defined with respect to the l o c a l coordinates s and t therefore cannot be d i f f e r e n t i a t e d d i r e c t l y with respect to the global x, y axes. In order to overcome this d i f f i c u l t y i t i s necessary to obtain a r e l a t i o n s h i p between the derivatives of the two sets of and  this can be achieved  coordinates  through the chain rule of p a r t i a l  differentiation.  ds  3_ 3r  a_ at  3_ 9z  _8_  4.37  where  [J]  -  3r 3s  3z 3s  3r 3t  3z_ St  Hence the derivatives wvt  i s called the Jacobian matrix  x and y can be expressed as derivatives  w»r«t s and t as follows  3_ 3r 3_ 3z  [j]  3_ 3s  -1  4.38  3_ 3t  A similar Gauss quadrature mentioned i n Section 4.2.2. i s employed to evaluate the above [B] matrix numerically.  4.3.3  The S t i f f n e s s M a t r i x [K] The s t i f f n e s s matrix for the force displacement relatinship i  obtained by the p r i n c i p l e of v i r t u a l work. displacement vector (5) > e  For a v i r t u a l nodal  the external work done, W(ext), by the  external force vector ( F ) caused by the v i r t u a l displacements i s written as  W(ext) = [s)  T e  (f)  4.39  The v i r t u a l s t r a i n vector caused by the v i r t u a l displacements vector i  written as  (Ai) = [B] ( 6 )  4.40  e  Hence the internal work done, W(int), caused by the v i r t u a l s t r a i n i s written as  W(int) = /  where  (Ae)  T  (Aa) dV  4.41a  V = volume of the element  Substituting Equation 4.40 into 4.41a y i e l d s  W(int) = J  Y  [&f  [ B ] [D] [B] [6] dV T  e  4.41b  Applying the p r i n c i p l e of v i r t u a l work  Wext = Wint  4.42  Therefore  (0  " /  [ B ] [D] [B] T  v  (f) = [K] (6)  [6] dV  4.43  4.44  where  /  [B]  [ D ] [ B ] dV i s c a l l e d the s t i f f n e s s matrix for the  element. The procedure of obtaining element stresses and s t r a i n s i s the same as described i n Section 6.2.3.  4.4  LOAD SHEDDING ( P L A I N S T R A I N )  (Byrne 1983)  Problems arise when any element within the solution domain v i o l a t e s the f a i l u r e c r i t e r i a (Mohr Coulomb).  That i s , for  unloading  of a shaft or tunnel, a p l a s t i c zone usually developes adjacent shaft.  to the  The extent of p l a s t i c zone and hence volume changes are only  approximated since the stress r e d i s t r i b u t i o n has not been considered during the formation of the zone. The analysis predicts the stress path ABC unloading.  But the stress state at C (Figure 4.1)  c r i t e r i o n (Mohr Coulomb).  instead of ABD  on  v i o l a t e s the f a i l u r e  If load shedding technique i s used, the  overstress can be distributed to the adjacent  elements by applying  an  appropriate set of nodal forces described herein and brings the stress path BC back to the correct The overstress, Ax,  BD. i n the element can be removed by  subtracting the computed stresses by Aa^i, A a  2 2  and A a  i 2  amount as  shown i n Figure 4.21b.  where Aa. . have the same notation as those i n Section 4.2  Aa (Ao)  =  Aa  12  4.3  Aa]  11  Aa 22  and  =  [T]  Aa-  Ax  4.45 13  56  Fig.4.1 - Stresses associated with load shedding  where  cos26  L_ 2  cos29 2 sin29  1 4 . cos29 2 2 sin26  (T) =  cos26 2  i s called the 0  transformation  0  matrix  A 0 3 = minor p r i n c i p a l stress = 0 Ao"i = major p r i n c i p a l stress = 0 1 - 0 3 tan A 13 T  =  2  (45 + -|-)  p r i n c i p a l shear stress = 0  The derivation of these stresses changes and the transformation matrix i s i n Appendix E, The removal of these overstresses can be achieved by applying a set of nodal forces which i s obtained by the principle of v i r t u a l work. vector.  The incremental nodal force vector causes a v i r t u a l  displacement  Hence the external work done, W(ext), can be written as  W(ext) = ( s ) ^ ( A f )  e  4.46  The incremental v i r t u a l s t r a i n vector caused by the v i r t u a l displacement vector i s  Ui)  e  =  [B]  ( « )  4.47 e  58  Therefore,  the i n t e r n a l work, done, W(int), i s  W(int) = /.  (Ae)  A.  T  (Ac)  6  t dA  4.48a  6  Substitution of Equation 4.43 to 4.48 y i e l d s  W(int) - /  Applying  ( 6 ) [B] T  ( A C ) t dA  4.48b  the p r i n c i p l e of v i r t u a l work  W(ext) = W(int)  Hence  ,  [Af ]  g  = t/  4.49  [ B ] [ A a ] dA  4.50  T  A  e  where [B] i s the s t r a i n displacement matrix i n Section 4.2.2. and  [ A o ] i s the stress vector shown i n Equation 4.41. e  The f a i l e d element w i l l have a stress change of Ao"n> A6 2  a  n  d  2  Ao"i2-  However, the computed stresses may not l i e on the f a i l u r e  envelope due to the a p p l i c a t i o n of the nodal forces.  Therefore  i t e r a t i o n s may be required to bring this to the assigned  tolerance.  The loading shedding technique presented herein gives the same results compared with INCOIL (a' = constant). m  However, the number of  i t e r a t i o n s required to bring the computed stresses back to the f a i l u r e envelope i s l e s s . With the incorporation of load shedding technique, the sand  skeleton i s modelled as a non-linear e l a s t i c - p l a s t i c porous material. The s o i l skeleton i s coupled with the pore f l u i d s i n the undrained model.  For undrained conditions, t h i s model allows stresses, pore  pressure and deformations of o i l sand masses to be evaluated by f i n i t e elements.  60  CHAPTER 5  5.1  : COMPARISONS WITH EXISTING SOLUTIONS  INTRODUCTION It i s important to check the v a l i d i t y of the developed model  before any major application.  Two types of comparisons are presented  herein which are suitable for checking the s t r e s s - s t r a i n model of the s o i l skeleton and also the newly developed gas law model. For the v a l i d a t i o n of the a n a l y t i c a l procedure, the computed r e s u l t s are compared with e l a s t i c and e l a s t i c - p l a s t i c closed form solutions.  The gas law model i s validated by comparing  computed  solutions with observed data, such as expansion of o i l sand cores and t r i a x i a l tests on gassy s o i l s .  5.2  COMPARISONS WITH THEORETICAL RESULTS  5.2.1  Elastic Closed Form Solutions The theory for e l a s t i c closed form solution was f i r s t  developed by Timoshenko (1941).  The plane s t r a i n solutions of stresses  and displacements i n a thick wall cylinder are presented herein:  Stresses  a P. - b P i o 2  a  r  — —r~>  Q  (P - P ) a b i o 2  .  — + —TTV  ?  b^ - a  a  °  2  -T\—9  (b^ - a ) r  z  z  (P - P ) a b  2  (b  2  b  - a  2  2  2  - a ) r 2  2  5.1a  z  P. - b P  2  2  2  5  '  l b  Displacement  n N (1+v)N . _ /i (l-2v)  (1+v) E  i  P  P  and  a2p  -  0  where  ( - 1 ~ b P o')r ^TZTTI a'  =  o  P  r  e  s  s  u  (  i~ o a b (b - a ) r  P  P  z  r  e  2  v  o n  )  2  5.1c  2  z  t n e  inner surface of cylinder  = Pressure on outer surface of cylinder J  E  = Young's modulus  v  = Poisson's r a t i o  a, b, r are defined i n Figure The  5.1  response of unloading a thick wall cylinder i s being  investigated. The stresses and displacements predicted by the programme are i n remarkably good agreement with the closed form solution as shown i n Figure 5.2.  5.2.2  Hence the a n a l y t i c a l procedure i s validated  Elastic-Plastic Closed Form Solution For a tunnel or shaft problem, the i n i t i a l state of stress  w i l l be the same throughout the domain.  As the support pressure of the  tunnel drops, yielding w i l l occur i f the strength of the s o i l i s exceeded.  Y i e l d i n g develops on the i n s i d e face f i r s t as a p l a s t i c  annular zone and extends r a d i a l l y outward i f the support pressure i s reduced further.  Hence theie e x i s t s a p l a s t i c and an e l a s t i c zone i n  the domain concentrically.  The  solutions of stresses and displacements  Fig.5.1 - Thick wall cylinder  63  o o O  Radii (r/r ) 0  E = 3000 MPa v = 1/3 i n i t i a l stress : a = a, = 6000 kPa final stress : o = 2500 kPa inside radius : r„ = 1 m r  r  Fig.5.2 - Stresses and displacements around c i r c u l a r in an e l a s t i c  material  opening  o o o  64  -  Tl  -  J  -  J  2o  •Q—  00 CO  0  0  0  j_> o  >  closed  p  o  o  o  form  program  0) Ed 1  1  1  _l  4  v-  1  6  Radii (r/r )  1  i  8  10  0  = 3000 kPa  E  1/3 40° i n i t i a l stress final stress inside radius V  0  1  L  B  e  o a r  r r 0  •= o = 6000 kPa = 1500 kPa = 1m e  F i g . 5 . 3 - S t r e s s e s and d i s p l a c e m e n t s around c i r c u l a r elastic-plastic  material  opening i n an  for the p l a s t i c and e l a s t i c zone w i l l be quite d i f f e r e n t . The stresses and displacements for the e l a s t i c zone are just an extension of Equation 5.1-by setting b to i n f i n i t y .  They may  be  written as  Stresses r  =  a P o + (P, -^V i - Po ) r"  5.2a  9  =  a P - (P, - P ) —jr o i o' r  5.2b  2  a  v  /  1  2  a  a  v  z  Displacement E  where  r  v  i  o  P^  =  Pressure on inner surface of cylinder  P  =  Pressure on outer surface of cylinder  E  =  Young's Modulus  v  =  Poison's Ratio  o  J  Different investigators, Gibson and Anderson (1951), Ladanyi (1963), Vesic (1972) and Hughes et a l developed closed form solutions of stresses and displacements for the p l a s t i c zone.  Hughes et a l  presented the more elegant solutions which are presented herein: A l l the sand i s assumed to f a i l with a constant ratio of p r i n c i p a l stresses, so that  N = tan  2  (45 + |)  5.3  The equilibrium equation that must be s a t i s f i e d i s  do a* - al -^+^ i-0 dr r  5.4  Substituting for a' from Equation 5.3, boundary conditions of a^  =  integrating and using outer  at r = R.  a'  in  = (1-N) Jin (|)  5.5  R  where  a' r  = r a d i a l stress at r within the p l a s t i c zone  o  = r a d i a l stress at the outer boundary, R, of the  R  r  p l a s t i c zone  = P  (1 - sin<j>)  p  R  - a [-2. ( l - s i n * ) ] I  5.6  l-sinfr 2 s i n  *  5.7  Equation 5.5 governs the d i s t r i b u t i o n of the r a d i a l effective  stress  within the p l a s t i c zone. Continuity of stresses and displacements between the p l a s t i c and e l a s t i c zone must be maintained. into Equation 5.2c, contact i s  Hence Equation 5.6 i s substituted  the displacement, U  at the e l a s t i c - p l a s t i c zone  U  R  E" o R  P  S  i  n  5.8  *  Hughes et a l also show that the displacements, u, within the p l a s t i c zone  ^=(|)  where  j  n + 1  n = tan  2  (^|)  5.9  (45 +  u = d i l a t i o n angle  The response of unloading a tunnel i s investigated. comparisons  The  i n Figure 5.3 show that the analysis of stresses and  displacements i n e l a s t i c - p l a s t i c materials predicted by the programme i s i n good agreement with the closed form solutions.  The minor  discrepancies are due to the l i m i t of Poisson's r a t i o and the coarseness of the mesh.  An upper l i m i t of 0.499 i s adopted i n the  programme (MHANS) to maintain numerical s t a b i l i t y , whereas the actual value should be 0.5.  The agreement i n displacements and the extent of  the p l a s t i c zone also confirm that the load shedding technique i n the programme (MHANS) i s working properly. This i s a s a t i s f a c t o r y check o the programme i n drained analysis. Unfortunately, the load shedding technique i n INCOIL cannot b successfully tested.  The element type i n this programme i s QM-6.  Non-equilibrium of stresses a r i s e i n QM-6 elements at high Poisson's r a t i o (v > 0.4) i f the geometry of the elements i s non-rectangular.  Since the elements i n a f i n i t e element mesh f o r modelling plane s t r a i n shaft problems are not rectangular, non-equilibrium of stresses arise before load shedding i s required.  5.3  COMPARISONS WITH OBSERVED DATA  5.3.1  One Dimenional Unloading of Oilsand This i s an opportunity f o r the new  with some f i e l d data.  gas law model to be checked  Since the unloading i s 1-D,  the v a l i d i t y of the  schematic spring analogy model (Figure 1.1) can also be demonstrated. Unconfined  oilsand core taken from d r i l l e d holes swells by 5  to 15% of the o r i g i n a l volume (Dusseault 1980,  Byrne et a l 1980).  Maximum expansion potential generally cannot be reached because expansion stops when there i s adequate intercommunication to permit flow of gas out of the sample. the s o i l f a b r i c .  of gas voids  This leads to disruption of  This i s the equilibrium saturation point i n which gas  becomes mobile and the maximum gas saturation value i s 15% (Amyx et a l 1960).  However, the t o t a l amount of expansion i s impossible to predict  and can only be measured for i n d i v i d u a l cores. The core l i n e r s are s p e c i f i c a l l y designed oversized to prevent the jamming of the core within the b a r r e l .  Radial expansion i s assumed  to be completed when the o i l sand core i s brought up from the d r i l l e d hole.  The l i n e r s and steel containers are assumed to be r i g i d and  f r i c t i o n l e s s so that only a x i a l expansion of the core i s allowed. Therefore, the oilsand core can be modelled as 1-D  unloading after  recovery. The i n i t i a l l y high stressed core specimen i s modelled as shown i n Figure 5.4a.  V e r t i c a l stress i s reduced  from 750 kPa to zero.  stresses, pore pressure and displacements are shown i n Figure  5.4b.  The  It can be seen that the change i n t o t a l stress i s apportioned between the e f f e c t i v e stess and the pore pressure as suggested by the spring analogy model.  Case A:  Gas saturation pressure = 100 kPa. I n i t i a l l y pore pressure i s above gas saturation pressure and  saturation remains 100%. Therefore loads come off from the pore f l u i d s while e f f e c t i v e stress remains f a i r l y constant because pore f l u i d i s the s t i f f e r phase as shown i n Figure 5.4b.  As pore pressure drops  below the gas saturation pressure, gas s t a r t s to evolve which causes the pore f l u i d to become f l e x i b l e .  Stress change w i l l be taken up by  the s o i l skeleton on further unloading u n t i l zero e f f e c t i v e s t r e s s . Further reduction on boundary load at zero e f f e c t i v e stress i s e n t i r e l y accommodated by the pore f l u i d s . Case Bt  Gas Saturation pressure = 300 kPa. Because of gas exsolution the pore f l u i d s s t a r t as a f l e x i b l e  phase so e f f e c t i v e stress drops to zero with no appreciable change of pore pressure on unloading.  Reduction of boundary stress beyond this  stage i s e n t i r e l y taken by the f l u i d phases as the s o i l skeleton e s s e n t i a l l y has no s t i f f n e s s at zero e f f e c t i v e stress as shown i n Figure 5.4b.  It may be seen from figure 5.4b that there are no appreciable displacements when the e f f e c t i v e stress i s p o s i t i v e .  The displacements  e s s e n t i a l l y come from unloading at zero e f f e c t i v e stress.  The t o t a l  displacements upon t o t a l removal of v e r t i c a l stress l i e within the range of 5 to 15% of the o r i g i n a l length.  70  unload to  450 kPa 300 kPa  •Im  s •  = 1565  n  -= 1000  m  - 65° ' 0.9 =• 0.243 =  =  A<f> =  0.5 0.25 o 22  f n o n == 0.045 w = 0.2 H o = 0.02 H w = 100% S R  5.4  -  Model f o r one dimensional unloading of o i l sand  - u = 100 kPa u = 300 kPa  o  bb  w  bb  CO  c  '5o u  _  \ j 150  g.5.4  300  L 450  i  i  600  i  External Load (kPa)  i  750  900  - Response of o i l sand to one dimensional unloading  72 The r e s u l t s show that the new stress s t r a i n model has the excellent c a p a b i l i t y of predicting undrained response of o i l sand.  5.3.2  T r i a x i a l Tests on Gassy S o i l s (Sobkowicz 1982) Performance of gassy s o i l on laboratory t r i a x i a l tests are  reported by Sobkowicz (1982).  A review of his work shows that the gas  law model i s generally correct.  An appraisal of the predictive  c a p a b i l i t i e s of the gas law model i s made by comparing predicted and laboratory observed response of the immediate pore pressure, the immediate (short term) B value, the equilibrium pore presusre, the equilibrium (long term) B value, and value of saturation and displacements. Since only test 11 i n Sobkowicz's thesis i s documented i n d e t a i l , a comparison between predicted and observed response for t h i s test i s made to evaluate the v a l i d i t y of the undrained model.  Analysis  i s performed using the same i n i t i a l condition as test 11: S = 99.75%, n = 32.28%, a = 1403.3 kPa, u = 652.3 kPa and 3 = 9E-6 k P a . s - 1  The  compressibilkity of s o l i d i s comparable with that of water (3 = 5 x 10  -7  k P a ) so that the B (short term) value w i l l not equal to one even -1  for f u l l saturation.  These components are converted into parameters to  be read i n by the programme.  The conversion and parameters are shown  i n Appendix F. The unloading  sequence i s shown i n Table 5.1.  During any phase of the i s o t r o p i c unloading  test, pore  pressure responses are predicted from the knowledge of s o i l  skeleton  and f l u i d compressibilities which are a function or e f f e c t i v e stress,  pore pressure, saturation and porosity. set  For short term reponse, H i s  to zero i n Equation 3.26b since there i s no time for gas  exsolution.  H . , ^ = 0.02 and H „ , ^ = 0.86 are used for the air/water co2/water  equilibrium response when gas exsolution i s complete. The comparisons are summarized i n Table 5.1 and are presented graphically i n Figures 5.5 and 5.6.  1)  They include:  predicted undrained response by the present undrained model  2)  measured undrained response (Test 11, Sobkowicz  1982)  A careful examination of Table 5.1 and Figures 5.5 and 5.6 show that the predictive c a p a b i l i t y of the gas law model i s remarkably good, especially for the long term undrained response.  The minor  discrepancies are due to the loss of gas from the sample as the r e s u l t of gas d i f f u s i o n and leakage through the membrane. The observed immediate pore pressure are higher than the predicted values because of the time elapse (15 to 30 seconds) between reducing the t o t a l stress and taking the f i r s t reading. Thus, the predicted short term B i s always higher than the observed ones. It can be seen that the stress reduction i s apportioned between s o i l skeleton and pore f l u i d s , depending on their compressibilities.  The sample i n Test 11 was i n i t i a l l y  saturated with  respect to a i r i n water and undersaturated with respect to carbon dioxide i n water. P  On unloading, during the f i r s t few phases, as  . < P < P . , _ , a small amount of gas exsolves for H co2/water air/water' 0 /  (air/water) = 0.02.  &  The change of e f f e c t i v e stress and pore pressure  TABLE 5.1a A Comparison of Computed and Measured Results (Test 11, Sobkowicz)  Total  Short Term B  S a l w r a "t.» ori ( %,)  Long Term B  Phase  Stress (kPa)  Predicted  A  1322.4  0.897  0.694  0.523  0.606  99.65  99.67  B  1220.5  9.843  0.69  0.477  0.433  99.50  99^54  C  1112.1  0.781  0.68  0.378  0.403  99.30  99.38  D  978.2  0.705  0.64  0.021  0.011  98.90  99.11  E  883.6  0.628  0.548  0.022  0.016  98.61  98.93  F  766.4  0.584  0.508  0.024  0.034  98.22  98.93  G  654.9  0.548  0.482  0.026  0.07  97.80  98.27  H  559.0  0.536  0.50  0.031  0.155  97.38  97.90  Jl  457.3  0.569  0.590  0.129  0.21  96.40  J2  1  1  1  1  95.15  J3  1  1  1  1  94.66  Measured  Predicted  Predicted:  Results predicted by Programme  Measured:  Results measured i n Test 11 by Sobkowicz  Measured  Predicted  Measured  TABLE 5.1b  A Comparison of Computed and Measured Results (Test 11, Sobkowicz)  Measured Strains (%) Phase  Horizontal  Vertical  Porosity (%) Volumeteric  Predicted  Measured  A  0.112 E - l  0.112 E - l  0.336 E - l  32.30  32.30  B  0.275 E - l  0.275 E - l  0.825 E - l  32.33  32.33  C  0.490 E - l  0.490 E - l  0.147  32.38  32.36  D  0.929 E - l  0.919 E - l  0.2757  32.46  32.42  E  0.124  0.124  0.372  32.53  32.46  F  0.167  0.167  0.501  32.62  32.53  G  0.214  0.214  0.624  32.72  32.61  H  0.261  0.261  0.783  32.80  32.69  Jl  0.376  0.376  1.128  J2 J3  .5.5 - Comparisons of predicted and observed pore pressure  77  are  roughly the same because the c o m p r e s s i b i l i t i e s of both s o i l  skeleton and pore f l u i d s are comparable. s i m i l a r to those of unsaturated s o i l s . P  This c h a r a c t e r i s t i c i s On further unloading, as P <  „ , _ , a large amount of gas exsolves because the high s o l u b i l i t y co2/water' ° ° a J  (H  „ = 0.86) coz  of carbon dioxide i n water.  This causes a sudden  increase i n f l e x i b i l i t y of the f l u i d phase and hence most of the load i s transferred to the s o i l skeleton.  When the e f f e c t i v e stress i n the  skeleton approaches zero, the f l u i d once again becomes the s t i f f e r phase, hence the B value rises to one.  This i s the t y p i c a l behaviour  of gassy s o i l on unloading. The predicted and measured displacements are i n remarkably good agreement.  This indicates that the input parameters (Appendix f )  and the r a t i o of the parameters, and Cheung) are generally correct.  = 0.6 K^, and n = 2m = 0.5  (Byrne  CHAPTER 6 - STRESSES AROUND A WELLBORE OR SHAFT IN OIL SAND  6.1  INTRODUCTION The response of a wellbore i n o i l sand upon unloading i s  considered because i t i s an important problem i n o i l recovery i n o i l sand.  In general, knoweldge of the stress solutions around a borehole  i s of great importance i n several s i t u a t i o n s :  1)  borehole s t a b i l i t y  2)  hydraulic f r a c t u r i n g  3)  production or i n j e c t i o n  A theoretical solution for stresses around a wellbore was developed by Risnes et a l (1982) and the equations are presented herein.  Validation of the programme (MHANS) for drained analysis i s  made by comparing computed response with the closed form solutions developed by Risnes et a l . A linear e l a s t i c - p l a s t i c constitutive r e l a t i o n s h i p i s used f o r the above v a l i d a t i o n . Upon v a l i d a t i o n of the programme, i t was used to study the behaviour of a wellbore i n o i l sand upon unloading.  Undrained and  drained analyses were performed to obtain the short term and long term response respectively.  In the undrained analysis, the gas exsolution  i s assumed to be very fast r e l a t i v e to the construction of wellbore. In the drained analysis, the pore pressure p r o f i l e i s estimated by using Dupuit's theory (Section 6.3.3).  6.2  GENERAL MODEL DESCRIPTION The wellbore under consideration i s supported by f l u i d  pressure.  As a model, a v e r t i c a l c y l i n d r i c a l hole through a horizontal  layer of o i l sand i s considered.  The geometry, f i n i t e element mesh,  and i n i t i a l conditions of the problem are shown i n Figure 6.1 and 6.2. Loading and geometry are assumed to be symmetrical around the well axis.  Only r a d i a l displacement a f t e r the i n i t i a l overburden loading  are considered.  These correspond to the assumption of axisymmetric and  plane s t r a i n conditions. The sand formation homogeneous and i n i t i a l l y  i s assumed permeable, i s o t r o p i c ,  f u l l y saturated.  The material i s assumed  e l a s t i c - p e r f e c t l y p l a s t i c and obeys Mohr Coulomb f a i l u r e c r i t e r i o n . Only stress solutions for  >  at the e l a s t i c - p l a s t i c  boundary w i l l be investigated.  6.3  THEORETICAL SOLUTIONS FOR STRESSES AROUND A BOREHOLE A closed form solution for streses around a well, using a  l i n e a r e l a s t i c - p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p , can be obtained  from  Risnes et a l (1982). The derivations of the stress solutions follow that of Risnes et a l (1982) with two additional assumptions.  1)  Insitu state of stress i s considered initially,  2)  to be i s o t r o p i c  i . e . a = o. = a . r o z  The Mohr Coulomb f a i l u r e c r i t e r i o n i n a porous material  is  f = a{ - 2S tana - 0 3 tan a. = 0 2  6.1  Fig.6.1 - Outline of the problem  76 elements , 78 nodes Fig.  6.2  F i n i t e element mesh for wellbore problem  where S i s the cohesion intercept (or apparent cohesion) a i s the f a i l u r e angle, i . e .  + -|—  <f>' i s the internal f r i c t i o n angle These symbols are also explained i n Figure  6.3.1  6.2b  Stresses  6.3.1.1 Stresses In Elastic Zone The stresses around a hole i n an e l a s t i c thick wall cylinder, with porous material saturated with f l u i d , may  be written as follows:  R2,  'r  =  °ro  < ro " r i > F ^ R T o  +  0  o  0  [l i  £n(R . -°>  » "i  '  *n(R /R.)o  R^*  ( P  ^ o  2  i ' 2(l-v)  R?  °9 * °ro  (^) ]  +  (  °ro " °ri> R ^ R T o  _  P  o  [*n (^)  ) i  ;  R i  t  1  +  1-2* 2(l-v)  i  v  - 1])  o i '  6.3  Fig.6.2 b - Mohr Coulomb f a i l u r e envelope  6.4  The  procedure for obtaining  these stress solutions i s given i n Appendix  G.  6.3*1.2 Stresses i n Plastic Zone As long as f = 0 (Equation 6.1),  a p l a s t i c zone w i l l start to  develop at the borehole wall, and then expanding i n size as support pressure i s decreased.  Equation 6.1 w i l l apply within the p l a s t i c  zone. If the stress state at the boundary between the e l a s t i c and p l a s t i c zone i s considered, the e l a s t i c stress solutions at this boundary are given by Equation 6.2 to 6.3, with R = r = R., a =a o i rc r and P = P.. With the assumption of no f l u i d flow ( i . e . P = P ) and c i c o' , r  R  Q  »  R^, the stress solutions from Equations 6.2, 6.3 and 6.4 may be  written as:  a  rc  = a  = 2a  a  The  zc  = a  6.4  rc  ro  zo  - a  rc  6.5  6.6  e l a s t i c solutions (Figure 5.2) show c l e a r l y that the r a d i a l stress  w i l l be the smallest at the boundary between e l a s t i c and p l a s t i c zones, and following the assumption (1) of i n i t i a l i s o t r o p i c stress, the state of stress at the e l a s t i c - p l a s t i c boundary w i l l be a < a < a . rc zc 9c n  The stress solutions within the p l a s t i c zone may be derived by combining Coulomb f a i l u r e c r i t e r i o n (6.1) and the equation of equilibrium. do -* dr  a  -a  + JL-± r  =0  6  .7  The stress solutions for the p l a s t i c zone may be written as:  For R. < r < R i c  a  r  = P. + ir^r; Zn | - + ^- (2S tana - -^r-) i  2irhk  R^  t  2Trhk  [(I-)* - 1]  6.8  For R. < r < R i c  °9 -  p  i  +  2¥ht c * ir> F < 1+  n  [(t + 1) (I-) - l] 11  R  For R, < r < R.  i  +  2S t a n a  - iSk> 6.9  87  °z " i 2^hk < * R7> I P +  +X  N  +  (  ° - zSfc t  2S tan  ( t + L)  <!/ 6.10  For R, < r < R b c  a  - (P +  z  «a-  i  £ n  f-) + v R '  2Trhk  l i l - + Cl-y)(l-2v) 2irhk  1-v  6.11  <|)1 where t = tan a - 1, a = 45° + -j2  u = fluid viscocity The procedure to derive Equations 6.7 to 6.11 i s given i n Appendix H.  6.3.1.3 Radius of the P l a s t i c Zone Radius of the inner P l a s t i c Zone R, b At the boundary of inner and outer p l a s t i c zones, the tangential stress and v e r t i c a l stress given by equations 6.9 and 6.11 are equal.  Setting Equations 6.9 equal to 6.11 and r = R^, y i e l d s .  a  x  (^V + a i  2  = 0  6.12  88  where  - £ (2S tana -  a i  a 2  *  =  ( 1  7  .  v  [(t + 1) - v (t + 2)]  JiJL - (1^) 2irhk l - v  )  ( 2 S  t a n a  ' 2$!k>  ( 1  "  zo  (ff v  P  ) cr  2 v )  Radius of the E n t i r e P l a s t i c Zone There are two requirements that must be s a t i s f i e d at the boundary of e l a s t i c and p l a s t i c zones 1)  Mohr Coulomb c r i t e r i o n must hold  2)  Continuity  of r a d i a l stress  Inserting r a d i a l stress from Equation 6.8 and tangential stress from 6.3 into Mohr Coulomb f a i l u r e c r i t e r i o n 6.1, the r e s u l t i n g equation f o r the radius of p l a s t i c zone R^ i s  b  R  x  + b  6  C c  2  R  £n ^  R  +  b  2  R R in Y~ • in jp- + b c i  £  7  n  -2. +  R in ^  R  b  + i  3  R  b Q  +  2  £  n  b l +  R  2  R _° + c  £  b g  n  =  |_  R  + ,. 2 b  R  £  n  _ |  0  6.13  89  where  bx = (2S tan a -  b •  B  - - ^r  1  2  Ci R o  t  "  1  2  2irhk  b  - - b^ R  6.3.2  9  = - C  <  +  P + Q  V  +  -  2  p  i  1  a 2 5 tan  uq  t+2  i 2TThk  t  b  2  v  =  V  2uhk  b5  8  o -  ( P  v  2 (1-v)  b  2  1  3 * " 2(1=*)  =  R ^  J  3  R  2  Stability It i s noted that the r a d i a l stress component i n Equation 6.8  consists of two r-dependent terms, one logarithmic and one to the power  90  of t . The l a s t term w i l l become dominant when the exponent t has a value greater than about two.  Setting  °  C = 7-t  (2S  tan a -  ^rtjr) Ri~ 2irhk  6.14  t  If C i s positive, r a d i a l stress i n the p l a s t i c zone w i l l increase with r, and combined p l a s t i c - e l a s t i c solutions are possible.  But when the  flow rate q i s large enough to cause C to become negative, r a d i a l stress w i l l decrease with increasing distance r , and combined solutions are not possible. Hence, there exists a s t a b i l i t y c r i t e r i o n .  C>0  6.15  with the l i m i t  •X—T-  2irhk  =  2S  tan  a  6.16  This study concentrates on o i l sand which has S = 0.  If the wellbore  i s supprted only by f l u i d pressure, equation 6.16 indicates that i n s t a b i l i t y arises when flow into the wellbore occurs.  6.3.3  Pore Pressure P r o f i l e When steady-state conditions around the wellbore have been  reached, the pore pressure i n the s o i l elements may be estimated i f the piezometric surface i s known.  Dupuit developed a theory which enables  the quantity of steady-state seepage and the piezometric surface around a well to be evaluated.  His theory i s based on three assumptions:  91  1)  the hydraulic gradient i s equal to the slope of the free surface and i s constant with depth,  2)  for small i n c l i n a t i o n s of the l i n e of seepage the streamlines may  3)  be taken as h o r i z o n t a l .  The permeability of the s o i l Is constant  With the terminology  i n Figure 6/2a. the flow when steady state  conditions e x i s t i s given by:  Q - w k (h  J^J^  - h )  2  2  2  L  6.17  w  and the location of the free surface i s  h - h! + . ,—rJin (—) £n(R/r ) r ' w w 2  2  2  h  2  = hi 1  2  / T i  v  6.18  There Is a controversy about the location of the piezometric surface predicted by Dupuit's theory, especially i n the v i c i n i t y of the well.  This i s because the surface of seepage i s omitted i n Dupuit's  prediction.  But the problem i s modelled as a disk of sand below the  seepage surface (Figure 6.3b). disk.  Hence equation 6.16  pressure  profile.  Only r a d i a l flow i s assumed i n the sand  i s accurate enough to estimate the pore  h,  ^^—  f Flow  surface of seepage  7"~ F.F. M e s h  Fig.6. 3 - Idealised flow to a wellbore  93  6.4.  Comparisons of Predicted Response and Closed Form Solution The response of unloading a borehole, with l i n e a r  e l a s t i c - p l a s t i c porous material, i s investigated. A drained analysis was performed.  The i n i t i a l and f i n a l conditions of the problem are  shown i n Figure 6.4(a).  When the f l u i d support pressure i s higher than  the i n i t i a l pore pressure, no flow from the borehole into the sand formation i s assumed.  Since the f i n a l f l u i d support (4100 kPa) i s  higher than the i n i t i a l pore pressure, flow into the borehole i s not considered herein. The stress solutions computed by the programme are i n good agreement with the closed form solutions mentioned i n Section 6.3, and shown i n Figure 6.4.  The radius of the entire p l a s t i c zone i s small  which shows that the borehole i s stable at the f i n a l f l u i d  support  pressure of 4100 kPa.  6.5.1  Undrained Response  The undrained non-linear e l a s t i c - p l a s t i c model w i l l now be used to study the response of a wellbore on unloading.  The f i n i t e element  mesh and i n i t i a l conditions are shown i n Figure 6.1b.  Only the long  term undrained response w i l l be investigated i n t h i s thesis because this condition i s f e l t to be more r e a l i s t i c  (t * 0) and i t was also  shown that the long term undrained condition i s more c r i t i c a l than the immediate response  (Sobkowicz, 1982) i n terms of s t a b i l i t y .  The wellbore i s unloaded by decreasing the t o t a l stress at the wellbore wall, and the stress solutions and displacements are shown graphically i n Figures 6.5.  A careful examination of these figures  indicates some interesting r e s u l t s :  94  W <=>  10  20  30  40  50  Radii (r/r ) 0  E = 60 MPa v = 0.45 + = 35° i n i t i a l stress final stress inside radius outside radius  Or O  = 0 = Oz = = 600 k P a  lb  =  R  =  r  8  4000  kPa  0.125 m 6.7 m  (a)  Fig.6.4--  Stresses around  a wellbore  in  an e l a s t i c - p l a s t i c  material  95  o o  O  00  ><  o o CD  CL  CO CO CD  o o  u CO  o o  (D CVJ  > o  Cd  °  o  o closed form program  -oCO >  o o  CM  O  0) V—I  w °  J  L 10  J  20  30  Radii (r/r )  40  I  50  0  (b) Fig.6.4 - Stresses around a wellbore  in an e l a s t i c - p l a s t i c material  Fig.6.5 - Undrained response of a wellbore in o i l sand on unloading  97  Fig.6.5  - Undrained  r e s p o n s e of a w e l l b o r e i n o i l sand  on u n l o a d i n g  98  Fig.6.5 - Undrained response of a wellbore in o i l sand on unloading  99  1)  The support pressure can be reduced below the i n i t i a l  pore  f l u i d pressure and the wellbore Is s t i l l stable; i n s t a b i l i t y i s defined when large displacements start to occur at the wellbore wall. 2)  I n s t a b i l i t y occurs at a support pressure of approximately 2500 kPa.  3)  The size of the p l a s t i c zone remains small as long as the support pressure i s higher than 2500 kPa (Figure 6.5  b).  Once the support pressure drops below 2500 kPa, the size of the p l a s t i c zone increases rapidly (Figure 6.5 c ) . 4)  Pore f l u i d pressure changes only occur i n the p l a s t i c zone.  The evaluation of the f l u i d pressure response  depends on volumetric s t r a i n Ae e l a s t i c zone, Ae  e  r  v  = Ae  e P + Ae . v v  But i n the  = - A e^ and Ae = 0 so that Ae = 0, 6 z v e  e  and hence no change i n pore f l u i d pressure i s predicted.  5)  Once i n s t a b i l i t y has been reached, a l i q u i d zone with zero e f f e c t i v e stress associated with a large p l a s t i c zone w i l l form adjacent to the wellbore.  These zones w i l l extend  into the sand formation rapidly upon further reduction of support pressure, leading to large displacements.  100 6.5.2  Drained Response For the drained condition, the pore f l u i d pressure i s assumed  to be known.  Dupuit's theory described i n Section 6.4.3  i s adopted to  estimate the pore pressure p r o f i l e around the borehole i n this study. Two  t y p i c a l pore pressure p r o f i l e s are shown i n Figure 6.7 with R = 100  and 150 m with the f l u i d support pressure f l u i d at 3200 kPa.  There are  some intermediate pore pressure p r o f i l e s between support pressure of 3500 kPa to 3200 kPa, depending on the number of increments on unloading, but they are not shown here.  When the f l u i d support  pressure i s above 3500 kPa, no flow from the wellbore into the sand formation i s assumed. The wellbore i s unloaded i n the same manner as f o r the undrained analysis. Figure 6.7.  The results of the stress solutons are shown i n  A careful examination of these r e s u l t s indicate some  interesting points:  1)  To maintain borehole s t a b i l i t y , the support pressure cannot be reduced to less than the i n i t i a l pore pressure; i n s t a b i l i t y i s defined as large displacements start to occur at the wellbore w a l l .  2)  The stress solutions only d i f f e r by a few percent when the input pore pressure p r o f i l e s are generated by using R = 100 and 150 m.  Therefore, only one set of stress solutons  i s presented here i n Figure 3)  6.7.  Once s t a b i l i t y has been reached, a l i q u i d zone with zero e f f e c t i v e stress associated with a large p l a s t i c zone w i l l extend into the sand formation rapidly upon further  F i g . 6 . 6 - Pore pressure p r o f i l e around a borehole  102  200  300  400  500  600  700  Support Pressure (kPa) (X10 ) 1  (a) Fig.6.7 - Drained response of a wellbore in o i l sand on unloading  104  reduction of support pressure, leading to large displacements.  6.5.3  Implications of Undrained and Drained Analyses The analyses show that there are l i m i t s on the f l u i d  support  pressure reduction i n order to maintain borehole s t a b i l i t y . Comparisons of both analysis are made at support pressure of 3500 kPa and 3200 kPa.  3500 kPa i s the c r i t i c a l pressure below which  i n s t a b i l i t y occurs i n drained condition.  It i s noted that i n Figure  6.9 the p l a s t i c zone i n the undrained analysis i s much smaller than the one i n drained analysis.  At a support pressure of 3200 kPa, the  borehole i s obviously unstable under drained conditions (Figure 6.9), showing a large p l a s t i c zone and consequently large displacements.  But  the borehole only exhibits a small p l a s t i c zone at t h i s support pressure (3200 kPa) under undrained conditions (Figure 6.9). This i s because the pore pressure around the wellbore i s lower i n the undrained case, which results i n a higher e f f e c t i v e stress. Consolidation i s the process which bridges the f u l l y and drained conditions.  undrained  An interesting point i s that the pore pressure  w i l l increase around the borehole during consolidation, leading to lower e f f e c t i v e stress.  Hence, the long term drained condition i s less  stable than the undrained condition.  6.6  Application to O i l Recovery For o i l production, the f i n a l support pressure must be reduced  below the i n - s i t u pore f l u i d pressure.  Based on the undrained  with the wellbore supported only by f l u i d pressure, the support  analyses  g.6.8  - Comparisons of undrained and drained  response  of a wellbore in o i l sand at a support pressure of 3500kPa  106  Fig.6.9  - Comparisons of undrained and drained response of a wellbore in o i l sand at a support pressure of 3200kPa  107 pressure can be reduced below the i n - s i t u pore f l u i d pressure and the wellbore i s s t i l l stable.  This allows the construction of the wellbore  and i n i t i a l reduction of f l u i d support pressure below i n - s i t u pore f l u i d pressure.  However, for the drained condition, the wellbore  becomes unstable which causes collapse of the well and hence no o i l production. I n s t a b i l i t y r e s u l t s i n the formation of large l i q u i d and p l a s t i c zones (Figure 6.9)  around the wellbore.  Since the permeability  i n the l i q u i d and p l a s t i c zones are higher due to the expansion of sand skeleton, i t i s desirable to have these zones around the o i l production well.  This e f f e c t i v e l y increases the diameter of the w e l l . To enhance o i l production and maintain wellbore s t a b i l i t y , a  screen may be i n s t a l l e d to provide e f f e c t i v e support pressure a f t e r the l i q u i d and p l a s t i c zones have formed. The three dimensional and viscous e f f e c t s have not been considered i n the analysis, however they may wellbore.  help i n s t a b i l i z i n g the  108 CHAPTER 7  : SUMMARY AND CONCLUSIONS  A new stress-stress r e l a t i o n s h i p f o r modelling the undrained response of o i l sand has been presented. s o i l skeleton and pore f l u i d s i s used.  An analysis which couples the The pore pressure changes are  computed from the constraint of volume compatibility. Separate s t r e s s - s t r a i n models are required f o r both s o i l skeleton and pore f l u i d s i n this analysis.  The conventional hyperbolic  s t r e s s - s t r a i n model described by Duncan et a l i s adopted f o r the s o i l skeleton.  The pore f l u i d s s t r e s s - s t r a i n relationships are formulated  on the basis of i d e a l gas laws. The developed model i s incorporated into a f i n i t e element programme for analysing the deformation behaviour of gassy s o i l s (e.g. o i l sand).  Upon v a l i d a t i o n i n Chapter 5, i t i s shown that the  undrained model i s capable of predicting the response of unsaturated to gassy s o i l s .  Naylor has shown that t h i s model can be used to predict  the response of saturated s o i l s . For non-rectangular QM-6  elements, equilibrium cannot be  achieved for Poisson's r a t i o values greater than 0.4, but higher order element can remedy t h i s . In the study of the response of wellbore i n o i l sand upon unloading, the f l u i d support pressure can be reduced below i n - s i t u pore f l u i d pressure under undrained condition and the wellbore i s s t i l l stable.  However, for the drained conditin, the f l u i d support pressure  cannot be reduced below the i n - s i t u pore f l u i d pressure i n order to maintain wellbore s t a b i l i t y . For o i l production, the f l u i d support must be reduced below i n - s i t u pore pressure which results i n formation of large l i q u i d and  109  p l a s t i c zones.  It i s desirable to have these zones around the wellbore  because the diameter of the well i s e f f e c t i v e l y  larger.  To enhance o i l production and maintain wellbore s t a b i l i t y , a screen may be i n s t a l l e d to provide e f f e c t i v e l i q u i d and p l a s t i c zones have formed.  support pressure after  the  BIBLIOGRAPHY  Atukorala, U.D.  F i n i t e Element Analysis of Fluid Induced Fracture  Behaviour i n Oilsand. Biot, M.A.  M.A.Sc. Thesis, U.B.C. 1983.  General Theory of Three-Dimensional Consodilation.  Journal  of Applied Physics, V o l . 12, February 1941. Bishop, A.W.  The Influence of an Undrained Change i n Stress on the  Pore Pressure i n Porous media of Low Compressibility. Geotechnique, V.23, N.3. 1973. Bishop, A.W.  The Influence of System Compressibility on the Observed  Pore Pressure Response to an Undrained Change i n Stress i n Saturated Rock. Brooker, E.W.  Geotechnique, V.26, 1976.  Tar Sand Mechanics and Slopes Evaluation.  10th Canadian  Rock Mechanics Symposium, Department of Mining Engineering, Queen's University, 1975. Byrne, P.M. and Cheung, H. Sand Masses.  S o i l Parameters f o r Deformation Analysis of  S o i l Mechanics Series No. 81, Department of C i v i l  Engineering, U.B.C. 1984. Byrne, P.M. and Eldridge, T.  A Three parameter Dilatant E l a s t i c  Stress-Strain Model f o r Sand.  S o i l Mechanics Series No. 57,  Department of C i v i l Engineering, U.B.C. 1982. Byrne, P.M. and Janzen. W.  Soilstress.  S o i l Mechanics Series No. 52,  Department of C i v i l Engineering, U.B.C. 1981. Byrne, P.M. and Janzen, W.  INCOIL.  S o i l Mechanics Series No. 80,  Department of C i v i l Engineering, U.B.C. 1984. C h a t t e r j i , P.K., Smith, L.B., Insley, A.E. and Sharma, L. of Saline Creek Tunnel i n Athabasca O i l Sand. Geotechnical Journal, 1, 197 9.  Construction  Canadian  Christian, J.T.  Undrained Stress D i s t r i b u t i o n by Numerical Methods.  J.S.M.F.D., A.S.C.E., S.M.6, 1968. Cook, R.D.  Concepts and Applications of F i n i t e Element Analysis. 2nd  Edition, John Wiley & Sons, 1981. Duncan, J.M. and Chang, C.Y. Nonlinear Analysis of Stress and S t r a i n in S o i l s .  J.S.M.F.D., A.S.C.E., S.M.5, 1970.  Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P.  Strength, Stress-  Strain and Bulk Modulus Parameters for F i n i t e Element Analysis of STresses and Movements i n S o i l Masses.  Report No. UCB/GT/80-01,  August 1980. Dusseault, M.B.  Sample Disturbance i n Athabasca O i l Sand.  Journal of  Canadian Petroleum Technology, V.19, N.2, Dusseault, M.B.  Undrained Volume and Stress Change Behaviour of  Unsaturated Very Dense Sands.  Canadian Geotechnical Journal,  Volume 16, 1979. Dusseault, M.B. and Morgenstern, N.R. Sands.  Shear Strength of Athabasca O i l  Canadian Geotechnical Journal, V.15, N.2, 1978.  Dusseault, M.B. and Morgenstern, N.R.  Characteristics of Natural  Slopes i n the Athabasca O i l Sands, V.15, N.2, 1978. Florence, A.L. and Schwer, L.E.  Axisymmetric Compression of a  Mohr-Coulomb Medium Around or Circular Hole.  International  Journal f o r Numerical and A n a l y t i c a l Methods i n Geomechanics, Vol. 2, 1978. Fredlund, D.G.  Density and Compressibility of Air-water Mixture.  Canadian Geotechnical Journal, V.3, 1976.  Geertsma, J . Some Rock-Mechanical Aspects of O i l and Gas Well Completions.  Paper EUR 38 presented at the 1978 SPE European  Offshore Petroleum Conference and Exhibition, London, October 24-26. Hardy, R.M. and Hemstock, R.A. Athabasca O i l Sands.  Shear Strength Characteristics of  K.A. Clark Volume, Alberta Research Council,  Information Series No. 45, 1963. Harr,  Groundwater and Seepage.  McGraw-Hill  Harris, M.C., Poppen, S. and Morgenstern N.R.  Tunnels i n O i l Sand.  Journal of Canadian Petroleum Technology, 1979. Harris, M.C. and Sobkowicz, J.C. Engineering Behaviour of O i l Sand. The O i l Sands of Canada-Venezuela,  1977. CIM Special Volume 17.  Hughes, J.M.O., Wroth, C P . and Windle, D. Sands.  Pressuremeter Tests i n  Geotechnique 27, 1977.  Naylor, D.J. Discussion.  Proceedings of the Symposium on the Role of  P l a s t i c i t y i n S o i l Mechanics, Cambridge,  13-15, p. 291-294.  September 1973. Pasley, P.R. and Cheatham, J.B. Rock Stresses Induced by Flow of Fluids Into Boreholes.  Soc. Pet. Eng., J . , p. 85-94, March 1963.  Risnes, R., B r a t l i , R.K. and Horsrud, P. Wellbore.  Sand Stresses Around a  Society of Petroleum Engineering Journal, Col. 22, No.  6, 1982. Smith, L.B. and Burn, P.M.  Convergence-Confinement  for Shafts and Tunnels i n Oilsands.  Method of Design  Conference on Applied  Oilsands Geoscience, Edmonton, Alberta, 1980.  113  Sobkowicz, J.C.  The Mechanics of Gassy Sediments.  Ph.D. Thesis,  University of Alberta, C i v i l Engineering Department, 1982. Timoshenki, S.  Strength of Materials.  V o l . 2, 1941, Van Nostrand, New  Yord. Todd,  .  V a z i r i , H.  Ground Water Hydrology. John Wiley and Sons, Inc.  Forthcoming Ph.D. Thesis, Department of C i v i l Engineering,  U.B.C. 1985.  APPENDIX A The constitutive relationship may be written:  (Aa') = [ D ' ] (Ae)  i n which  A.l  (Ao ) i s the incremental stress vector, (Ae) i s the 1  incremental s t r a i n vector and [D*] i s the incrmental e f f e c t i v e stress s t r a i n matrix. The incremental s t r a i n vector i s related to the nodal displacements  by:  (Ae) =  [B]  (S)  A.2  i n which [B] i s a matrix that depends on element geometry. By the p r i n c i p l e of v i r t u a l work, the external work done by the v i r t u a l displacement  i s equal to the i n t e r n a l work done by the  increment of v i r t u a l s t r a i n s :  (5)  i n which  T  (f)  ( f ) - /A U e )  =  T  (Ao') dA + J .  (Ae") (}) Au dA T  A.3  the element force vector  (Aa') =  the element incremental e f f e c t i v e stress vector  (Au)  the element incremental pore pressure vector  =  Substituting for (Ae) and (Aa') from Equations A . l and  A.2,  (6) (f) = (6) [ B ] [D'] [B] (6) A + (8) [ ] (*) T  T  T  o  T  B  Au A  A.4 e  i n which A  e  i s the area of the element with unit thickness,  Rearrangement of Equation A.4 y i e l d s  (6) = ( f ) - ( K j (Au)  [K]  i n which  [K] = [ B ]  T  [K  T  w  ]  =  L  [B] 1  A.5  [ D ] [ B ] A^  ( 1 )e  ^o'  A  where [K] i s the element s t i f f n e s s matrix and (& ) i s a load vector w  associated with the pore pressure.  APPENDIX B  Assumptions: 1)  the volume^ of solids i s 1 unit, then the volume of voids i s e units by the d e f i n i t i o n of void r a t i o ,  2)  the solids are incompressible.  The undrained response of the element under a change on external pressure w i l l be  Skeleton  Aa ' (Ae ) v'SK B m  S K  Fluid  i n which  (Ae )_ = v f  B.l  B.2  (Ae ) , (Ae ) , = volumetric s t r a i n : Skeleton, F l u i d v SK v f ' Bg^,  =  bulk modulus: Skeleton, f l u i d  Aa ' m  =  mean e f f e c t i v e stress change  Au  =  pore pressure change  Since the s o i l skeleton and the pore f l u i d s deform together when the conditions are undrained, i n additions to Equations B . l and  117  Compatibility  e (A£ ) y  or  -  f  (1 + e) t^)^  B.3a  (Ae ) = - (Ae ) v f n v SK  B.3b  i n which e i s the void ratio and n i s the porosity. Substituting B.3b into B . l , the pore pressure change, Au, may be written as:  A  u  =  K  < v>SK  a  A e  f = —, a n  B  K  i n which  K  the apparent bulk f l u i d modulus  In f i n i t e element analysis  Au = K  = K  a  (Ae ) v  a  (Ae  n  + Ae  2 2  + Ae ) 33  '  4  118 APENDIX C  Poisson's r a t i o i s equal to 0.5 i n undrained analysis theoretically.  This means that bulk modulus i s i n f i n i t e and numerical  i n s t a b i l i t y w i l l arise (this sentence does not sound r i g h t ) . Therefore the default of Poisson's r a t i o , v, i s always l e s s than 0.5 (e.g. 0.495) to maiontain s t a b i l i t y and accuracy.  In the t o t a l stress  model developed i n Section 3.2, i t i s the combined Poisson's r a t i o , for matrix [D] that controls the o v e r a l l numerical s t a b i l i t y , not just v. The e l a s t i c moduli i n matrix [D'] are related as  G =  E 2(l+v)  C.I  B =  E 3(l-2v)  C.2  Assuming the e l a s t i c moduli i n matrix [D] = [ D ' ] + [ D ] are related as F  G cb  C.3  B  C.4  cb  119  i n which s u f f i c cb means combined, G = G , as pore f l u i d does not cb transmit shear. Just consider the d i r e c t stress terms i n the constitutive relationship Aa Aa  L +K  n  1 2  a L*M+K a L*M+K  ss  Ao-33  L  a  L*M+K a L+K a L*M+K a  L*M+K a L*M+K a L +K a  Ae  n  Ae  2 2  A  E  33  C.5  4 i  M n  w  h  i  U. c  T  h  E(l-V)  (l+v)(l-2v)  =  1-v  Substituting  C.2 into C.5, y i e l d s ,  Aff Ao  2 2  0-  3 3  A  P+K  H  =  a Q+K a Q+K a  Q+K a P+K a Q+K a  Q+K a Q+K a P+K a  Ae  u  Ae  2 2  A  E  33  C.6  120  v i n which  „ 3B(l-v) P = /  Q =  3  (  1  B  V  +  v  1+v  Adding the d i r e c t stress i n Equation C.6 and rearranging  T  (  A  q  "  +  Ag  22  (Ae^i + Ae 2 2  +  A g  +  33)  =  B(l-v)  ^£33)  +  2BV  l-v  +K  a  1+v  = B cb  Eliminating E  cb  from Equations C.3 and C.4, y i e l d s  3B V  cb  C.7  =  c b  - 2G  6B°. + 2G cb  Substituting C.7 into C.8, rearranging  0  ,  8  121  APPENDIX D  In the incremental e l a s t i c method, two i t e r a t i o n s are performed to obtain the tangent moduli of the s o i l skeleton.  Hence two  i t e r a t i o n s are also employed to evaluate the tangent bulk f l u i d modulus i n order to make the procedure compatible.  The parameters at the end  of previous increment w i l l be used i n the analysis of the present increment, and then updated at the end of t h i s increment. The tangent bulk f l u i d modulus and the procedure to update the parameters are shown herein: Equation 3.24e and 3.26b are programmed as:  e-n (1+e) + H n w ww L  l  0  r  3 i - f  + 3 N ww  J  D.l  e-n (1+e) - n (1+e) + H,n, (1+e) + H n (1+e) o w b b ww L  f  +  (1+e)]  r  „ _ 1 82 - e  +  (1+e) + H „n (1+e) co2 w p  W  1  +  e  >  ]  Vw  The parameters i n Equation D.l and D.2 may be updated according to the following f o a u l a e based on the assumptions:  D.2  volume of solids i s 1 unit bitumen, water and s o i l solids are incompressible  Ae  =  e^  =  S e f f c  £  (1+e) Ae v  e  =  i +  A  e  S.e, 1i  D.3  ^"^  D.5  n (1+e) + Ae  -81+e + Ae  D.6  n (1+e)  <  - rzzn  b  D. 7  1+e+Ae  n'  "w  =  n (1+e) w r£—n 1+e+Ae  P  =  u + Au  D.8  D.9  123  APPENDIX E  Refer to Figure  4.1b  °1  9  =  an  °12  cosec 29  E.l  + a2 2  03 =  x  +  ^  ~ °12  cosec 28  E.2  = 0  1 3  E.3  The overstress i n the element i s removed by reducing AOj, A a , 3  AT  1 2  as follows  AO!  =  Oi - a  Ao-c,  =  0  E.5  =  0  E.6  Ax  1 2  tan  3  2  (45° + |-)  E.4  These change i n p r i n c i p a l stresses can be expressed  i n terms o f  stresses i n x-y space:  A  o  ll  AO} + A a = 0  3  AO} - A a + *  3  cos 29  E.7  124  Aoi +  Aa-i  Ao"i - Aero cos 29  Aa 2 = 2  -  AOi  Aa  1 2  AO:  cos 29  =  E.8  E.9  Equations E.7, E.8 and E.9 may be expressed i n matrix form  Ac  r Aoi  n  Ao 2  [T]  2  Aa  Aa  E.10  3  Ati3  1 2  [ T ] i s the transformation matrix  [T]  =  1_ 2  cos 29 2  1_ 2  cos 29 2  i 1_  cos 29 2  1_ 2  cos 29 2  2  s i n 20  s i n 20  APPENDIX F  Back Calculation of S o i l Parameters Given the compressibility of s o i l 0  g  = 9 E-b k P a  e f f e c t i v e stress o§ = 751 kPa, the bulk modulus K  - 1  and the  number can be B  backcalculated  by assuming a value of m.  From Equation 3.12 B  =  *B a p P  (  *'  ) m  a  s  1  a  Substituting 3 ,  and P  S  and assuming m = 0.25 (Byrne and Cheung)  cl  F.2,  = 665  Adopting the relationship K B K  E  = 1108  Other parameters are depicted Cheung reports. written as:  = 0.6 K (Byrne and Cheung) E  from Byrne and Eldridge on Byrne and  A complete set of s o i l parameters f o r Test 11 may b  \  =  1108  n  =  0.5  h  "  665  m  =  0.25  =  0.8  R  f  <t>'  = 42°  A<f>'  =  8°  n  =  0.3228 (e = 0.4767)  S  =  0.9975  APPENDIX G  E l a s t i c Stress Solution If the f l u i d pressure i s included, the displacement u of an e l a s t i c material may be written as  (X + 2G) ^dr (dr T + ~) r ' + 3 dr i  where  A  =  G  =  3  C - 1 - -p-— b  0  G.l  (1+v) (l-2v)  E  2 (1+v)  which i s assumed equal to 1  C  = sand matrix compressibility  C, .= sand bulk compressibility b  The pressure may be expressed by Darcy's law i n r a d i a l form  iP _ dr  "q 2irhKr  ,  The stresses are written as  o  r  =  X e  e  v  + 2G e + P r 6  G.3  a.  =  X e  e  v  6  a  =  X e  Z  e e Where e , e r 9 + e  V  and e  0  e  + 2G  + P  G.4  + 2G e + P z  G.5  0  e  e 6 6 are the e l a s t i c s t r a i n components and e = e + e z v r v  G.6  e  z  Assuming the i n i t i a l loading cause a deformation only i n the v e r t r i c a l d i r e c t i o n , but no displacement i n the horizontal directions (e  r  = e  n  9  = 0), the i n i t i a l v e r t i c a l s t r a i n e , i s given by G.5 as zo a  :  zo  -P zo o X + 2G  =  G , /  Assuming only r a d i a l displacement a f t e r i n i t i a l loading, the strains are  e  r  e e  =  —  dr  u  9 = 7  e  G.8  e = e z zo  G  '  y  By solving Equation G.l with the boundary conditions  a  a  and combining  r  r  = a . when r = R. ri i  = a  ro  when r = R, i  the results with Equations G.8, G.9 and G.10, the stres  solutions (Equations 6.2, 6.3, 6.4) can be found by i n s e r t i n g the result i n Equations G.3 through G.5  APPENDIX H  P l a s t i c Stress Solutions (a < a < a„) r z 0 The conditions must be s a t i s f i e d within the p l a s t i c zone  1)  Equilibrium  da  2)  a  - o„  Mohr Coulomb f a i l u r e  f = a. - a t a n 0 r  The  2  criteria  a + (tan  2  a-1) P - 2S tan a = 0  H.2  flow rule associated with y i e l d condition i s  e  P  , 9f ^ 2 = X = - X tan a r da r z  el  0  e  - X ~ - - X 9a  H.3a  H.3b  Q  P-\||--0 z da  H.3c  From Equations H.3a and H.3b, i t follows that  e£ +  tan  2  a = 0  H.4  The t o t a l s t r a i n components may be written as  e  r  e  e  =  e  r  +  r  E  H  '  = Eg + e^  g  e  = e  z  H.5b  p  z  + e  5 a  = e zo  z  H.5c  = 0 because i t i s assumed that there i s only r a d i a l displacement a f t e r i n i t i a l loading. Combining Equations H.5a and H.5b, i n s e r t i n g into H.4, gives  e  o  + e. tan^ a = 0  r  E  e r  +  e o tan^ a 0  E  q  H.6  Applying Hooke's law of e l a s t i c i t y for porous material, y i e l d s  Ee® = a - v (a + a ) - (l-2v)P r r o z  H.7a  E E Q = a . - v ( o - + a ) - (l-2v)P 0 9 r z  H.7b  fl  v  Ee  e  z  = a  z  - v (a + a ) - (l-2v)P r 9 Q  H.7c  Substituting Equation H.5c  a  = Ee  z  into H.7c  + v (a  zo  r  + a.) + (l-2v)P 8  Combining Equations H.7 c r i t e r i o n Equation H.2  [tan * a + 1 - v ( t a n 1  yields  and H.8,  i n s e r t i n g into the y i e l d  with the s t r a i n r e l a t i o n i n H.6,  a + 1) 1  2  2  a  J  a -  = 2G £ r  + 2G e„ t a n r 9  + [tan  2  a (tan  + (tan  2  a + 1) (1 - 2 v)]p - [ t a n  + v (tan  2  a  H.8  '  2  1) - v (tan * a -  a  1)  1  2  2  i t gives  a ( l - v ) - v] 2 Stan a  1) 2G E  +  H.9 zo  and  [tan * a + 1 - v ( t a n 1  a + l) ] o  2  2  Q  = 2Ge  o  + [v (tan * a - i ) - ( t a n 1  + tan  2  a (tan  + v tan  2  2  2  tan  2  a + 2Ge  a -  2  a + 1) 2Ge  tan* a  1)  a + 1) (l-2v)]p - [v ( t a n  a (tan  n  6  r  2  a + 1) - l ] 2Stan a  H.10 zo  Substituting Equations H.9 and H.10 i n t o equilibrium Equation H.l, together with the strain-displacement relations G.8 and G.9, the displacement equation may be written as  r 2  2 — j+ r dr  - utan  + (tan  2  a = 7^7 (- [ t a n  2  a + 1) (l-2v)]  + v (tan  2  j^hk  +  a - 1) 2G e  a (tan  2  ( t a f l 2  a  +  1  2  )  a - 1) - v(tan  <  1 _ 2 v  )  2  S  1  t  a  n  l+  - 1)  a  H.ll  zo  J  The displacement solution of Equation H . l l i s  2Gu =  ,  A.^  tan a , . - tan a , _ + A r + Br 2  „ .. H.12  2  2  and the corresponding strains are  2Ge  r  = tan  2  .0 ^ r  1  ^ -  1  - tan  2  a A  2  r  _  t  a  n  2  a  + B H.13  00 A tan a 2Ge = A ^ 2  Q  Where Aj and A conditions.  2  . , + A  2  -tan a r 2  ._ + B  „ ,, H.14  are constants of integrations which depend on i n i t i a l  134  a v ,-i [f2? tan'•'a+l  ^ l-2v -i tan^a-l  z  B =  L  - v 2G e  Substituting  + - — 2 — r JJ  uq o ui 2irhk  -  l-2v „ tan ct-l  0  7—2 — r 2 Stana z  H.15  zo  Equations H.13  and H.14  into Equations H.9  and  H.10  together with Darcy's law f o r r a d i a l flow,  P = P *  + i  -Hi- £n — 2TThk  X  R  n  H.16  i  gives  a  r  = P, + -^rr In-Ii 2irhk R,  oz  + 2tan  A l  p  u  y  t a ^— r  a = - + o^T * 9 I 2irhk  a  - -\- (2Stana - ^ 3 - ) t 2TThk  A l  + 2tan*a — T  n  t r  I  H.17  7" (2Stana - t a n t  2  a -^tr") 2trhk  H.18  a  = P. + z  0  i  .* Jin ZTrhk  ^S_] + 2irhk  (  1  J  A  2tan  where  —  2  R  7  )  (  1  1-v  4Stan a - ( t a n  t  "  2  V  (a  )  z  a + 1),  L  -P ) - + v ( t a n a + 1), zo o 2  l  a - r '  t = tan  2  a - 1  T = tan  4  a + 1 - v (tan  H.19  2  a +l )  2  The constant of Integration Aj can be found by i n s e r t i n g the boundary condition, o  r  = P. when r = R., into H.17. i i '  136  APPENDIX J  The quantity of steady-state seepage at any distance r from the centre of the w e l l i s  Q = k 2irrh ^ dr  J«l  x  on setting the l i m i t of integration,  _  R r  ,  h  yields  2  hi  w  1  and  Q  =  IT  k (h2 - h')  A n  I  J.2 w  Equating J . l and J.2,  2rh§  yields  =  (4  -h?)  j^sbr-'C  J.3  w  integrating  J.3  h  2  = C Jin r + D  J.4  137 where  D i s the constant of integration  Substituting the boundary conditions r =  and h = h^ i n J.4,  yields,  2  D = hi - C Jin r  J.5 XJ  *  Substituting J.5 into J.4, y i e l d s  2 h  l ~ ? = hi + - — 5 - 7 — In R/r w 2  1  h  h  r  An — r  J.6 w  

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