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Numerical modeling of horizontal drain drainage in an open pit slope Ge, Shemin 1985

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NUMERICAL  MODELING  OF HORIZONTAL PIT  DRAIN  DRAINAGE  I N AN OPEN  SLOPE  by SHEMIN GE B.  A.  S c . , WUHAN I N S T I T U T E  OF B U I L D I N G  MATERIALS,  CHINA,  1982  A  THESIS THE  SUBMITTED  I N PARTIAL  REQUIREMENTS MASTER  F U L F I L M E N T OF  FOR THE DEGREE OF  OF A P P L I E D  SCIENCE  in THE The  Department  We  of  FACULTY Mining  accept to  THE  OF GRADUATE and Mineral  this  the  thesis  required  UNIVERSITY  as  STUDIES Processing  conforming  standard  O F BRITI^tKcOLUMBIA  April,1985 ©  Shemin  Ge,  Engineering  April,1985  ^  In  presenting  requirements of  British  it  freely  agree for  this for  an  available  that  in  partial  advanced degree  Columbia,  I  agree  for  understood  that  financial  by  his  or  the  shall  reference  and  study.  I  extensive  her  shall  copying of  granted  by  the  not  be  allowed  of  make  further this  thesis  head of  representatives. publication  the  University  Library  copying or  gain  at  of  the  s c h o l a r l y p u r p o s e s may b e or  fulfilment  that  permission for  department  for  thesis  It  this  my  is thesis  without  my  written  permission.  Department  W '^'^W ho^^  of  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l Vancouver, Canada V6T 1Y3  Date  DE-6  (3/81)  £  ^  Columbia  >JK~  ft  ^f^J^J  ABSTRACT A  study  horizontal open  pit  drainage A  has  drain  system,  to  slopes.  Water  computer  model  length  INTRODUCTION 1982).  flow was  TO  the were  between  hydraulic  length. to  in  computer  used  that  their  Therefore, was  and  height the  effects  fanned  drain  element  computer  zone  was  field  were  as  show  vs.  length were  of  different  of  of  such  was  of  mining  the  from  a  design  drain  made  to  rock rock  of  and as  an  system. study  the  slopes.  The  conditions  mining  slopes  on is  progresses. drain  rows  A comparison of  patterns, made.  series  spacing  operation  simulation.  also  from  relationship  s p a c i n g between  drain  The  Anderson,  a  drain  also  feature  computer  rock  taken  &  drain  anisotropic  by  layouts,  (Wang  horizontal  vertical  in  was  assumed.  data  simulations,  the  model  obtained.  graphs  in  in  drain  obtained  the  MODELING  influence  varies  data  Columbia and  Another  suitable  investigated  drainage  the  effect.  studied.  saturated the  of  a horizontal  drains  simulations  indicated  of  of  drawdown  into  s p a c i n g and  results  table  effects  were  distribution  characteristics  drainage  by  the  flow  These  drainage  the  water  computer  head  the  water  spacing,  agreements of  evaluate  parameters  the  British  c o u l d be  determine  The  the  plotted.  They  the  GROUNDWATER  result  graphs  aid  in  to  finite  tested  Satisfactory  As  and  simulate  LORNEX M i n e  on  Two m a j o r  dimensional  constructed  the  made  drainage  slopes.  two  been  parallel  the  drain  TABLE  OF C O N T E N T S Page ii  ABSTRACT TABLE  OF C O N T E N T S  LIST  OF F I G U R E S  LIST  OF T A B L E S  iii v viii  ACKNOWLEDGEMENT  ix  Chapter  1.  INTRODUCTION  1  Chapter  2.  MATHEMATICAL MODELING  6  2.1  The  Physical  2.2  The  Mathematicl  2.3  Finite  2.4 Chapter  3.  Chapter  Element  9  Solution  13  Computer  20  Program  and V e r i f i c a t i o n  COMPUTER A N A L Y S I S Horizontal  3.2  Vertical  3.3  Anisotropy  3.4  The  Model  Model  28 Analysis  Comparision  28  Analysis  Effects of  ..47  on D r a i n a g e Different  56 Drain  Layouts  ...58  Applications  4.1  Hypothetical  4.2  The  5.  Problem  13 15 18  3.1  4.  7  Basic Theory F i n i t e Element Formulation Boundary C o n d i t i o n s  Pattern Chapter  Problem  65 Example  Application  for  65 LORNEX M i n e  SUMMARY AND C O N C L U S I O N S  72 77  5.1  Conclusions  77  5.2  Summary  78  iii  T A B L E OF  CONTENTS  (continued) Page BIBLIOGRAPHY  80  Appendix  A.  Finite  Element  Appendix  B.  Computer  Formulation  Program  Organization  iv  83 89  LIST  OF F I G U R E S  Figure  Page  1.  A Horizontal  Drain  Drainage  2.  Hypothetical  Slope  with  3.  The  4.  A s s o c i a t e d Nodes  5.  Boundary  Condition  6.  The  View  LORNEX Open  Pit  7.  M o d e l S e c t i o n o f LORNEX M i n e Conductivity Distribution  and  8.  The Data  9.  Water  Plan  Table  of  Comparison of of  Finite  2  Drains  Position and  System  in  Two  . Sections  8 of  a  Elements  Applied  Computer  Mesh  .10 16 19 24  the  Output  Hydraulic 25  and  the  Field  LORNEX M i n e Element  Slope  26 for  Horizontal  Model  29  10.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 90m  32  11.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 80m  33  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 70m  34  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 60m  35  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 50m  36  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 100m  37  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 90m  38  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 80m  39  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 70m  40  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 60m  41  12.  13.  14.  15.  16.  17.  18.  19.  v  L I S T OF F I G U R E S ("continued) Figure 20.  21.  22.  23.  24.  Page D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , D r a i n L e n g t h 100m  42  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , D r a i n L e n g t h 90m  43  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , D r a i n L e n g t h 80m  44  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , D r a i n L e n g t h 70m  45  Dimensionless Hydraulic the  Slope  Height  60m  46  Water  Table  Position  in  a  Slope  100m H e i g h t  50  27.  Water  Table  Position  in  a  Slope  120m  Height  51  28.  Water  Table  Position  in  a  Slope  140m  Height  ........52  29.  Water  Table  Position  in  a  Slope  160m  Height  53  30.  Water  Table  Position  in  a  Slope  180m  Height  54  31.  Water  Table  Position  in  a  Slope  200m H e i g h t  55  32.  The  Effects  of  Anisotropic Materials  33.  Parallel  34.  Finite  35.  The Comparison of Fanned and P a t t e r n s , S l o p e H e i g h t 150m  Parallel  The C o m p a r i s o n of Fanned and P a t t e r n s , S l o p e H e i g h t 175m  Parallel  The Comparison of Fanned and P a t t e r n s , S l o p e H e i g h t 200m  Parallel  37.  38.  The  Fanned Drain  Element  Slope  Vertical  Length  26.  and  for  Drain  for  Finite  Drainage  Mesh  200m,  Distribution  25.  36.  Element  of  Head  Mesh  Section  of  for  the  vi  a  Model  48  Patterns Fanned Drain  57 59  Pattern  60  Drain 62 Drain 63  Example  Drain 64 66  L I S T OF F I G U R E S ("continued) Figure 39.  40. A . 1.  Page Estimated Slope  Water  Estimated  Water  The  Table  Position  in  the  Sample 70  Global  Table  Matrix  Position  Assembly  in  from  LORNEX M i n e Individual  Elements B . 1.  Flow  Chart  B.2.  An E x a m p l e  75  88 for to  the Show  Computer the  Program  Computer  vii  FESHDMS  Input  Data  91 98  LIST  Tables • I. The Comparison II. III. IV.  Calculation Example  of  Two  TABLES  Sets  of  Result  Table  Table  Page 27 68  Calculation  Calculation  OF  Table  for  LORNEX M i n e  v i i i  69 74  ACKNOWLEDGEMENT I  would  Brawner this  for  like his  research I  to  continuous guidance  appreciate  W.  Poling  A.  J .  Reed and D r . would  Rulon  and  proven  to  also  faculty  Professor  and encouragement  C.  0.  during  comments  H.  J .  L.  like D.  D.  S.  my c o m m i t t e e  Miller.  Smith  to  of  for  thank  am g r a t e f u l  helpful  Dr.  Goldman. The  I  members;  A.  Dr.  to  Dr.  advice.  R.  Freeze,  discussions  with  Dr.  J .  them  have  encouragement  from  valuable.  Discussions the  the  and D r .  Dr. be  to  program.  G.  I  e x p r e s s my t h a n k s  with  members  and  graduate staff  and  Mineral  Processing  from  Xu Changyu and Lu N i n g  students,  in  the  Engineering, in  Department and  China are  the  all  of  Mining  encouragement  appreciated.  1 Chapter  1  INTRODUCTION  Groundwater open  pit  greatly that of  mine  water  rock,  stress, other  in  To  the  systems  in  conditions, cost  trenches Among  and  to  easy  long  systems. with  a  slopes  install,  A recent  vacuum has with  horizontal  drain  The  in  a  to  develops  shear  blasting  seepage  to  is  strength  increases  due  in  slope  mechanism  shear  cracks  movement  ,  is  or  forces  reduce  slope  many  are  development shown  under  its  equipment are  low in  vacuum.  cost  system  is  geologic  availability  (CANMET,  They  need  1980)  has are  little  compared  some quick  upkeep,  to  other  horizontal  drains  advantages material. shown  drain drainage  drainage  combining  great  drainage  galleries,  gravity,  lower  horizontal  drain  conductivity  drainage  several  circumstances.  by  to  different  drainage  horizontal  work  are  to  They  drains  in  necessary  There  according  and  lower  it  slope.  wells,  others  life  slope.  induced  concerns  table  effective  pressure  factors.  methods,  over  major  water  requirements,  horizontal  advantages  be  of  the  practice  pumping  these  provide  in  effective  and  the  stability  drainage  drainage,  the  the  1983)  slope  table  used  water  of  tension  may  direction  improve  water  in  shock  (Brawner,  of  reduces  pressure  and  one  A high  stability  pressure  vibrations,  stability.  and  the  water  are  stability.  hydrodynamic  acting  the  slope  reduces  the  conditions  in  in  draining A  Figure  typical 1.  2  Figure  1.-  A Horizontal Drain  Drainage  System  3 (Seegmiller,  In  the  design  been  done  or  a  drain  the  which  thesis  2.  3.  the  slopes  with  To  estimate  of  drains  as  to  work  the  is  drainage  system.  less  emphasis  During  design  for  embankments of  these  the  the  design  in  are  the  only  to  use  the  for  horizontal  primarily  with  this  presented  study  open p i t  in  on  respectively  drain  methods  of  drain  commonly b a s e d  attempting  dealt  research of  testing.  knowledge,  their  on  rigorous  even  horizontal  author's  the  no  and T e s a r i k ( 1 9 8 4 )  difference  some  mining ways  slopes  such  slope materials  the  as  the  and  the  slopes.  the  objectives hydraulic  the  the the  the  in  this  head  horizontal  different  investigate on  situ  finite  four  after  spacing  in  are  determination  and  of  level  materials  and  and a n i s o t r o p y  To e s t i m a t e  To  The  studies  from  are  drain  published  was  height  There  length  embankments,  heterogeneity  1.  To  differ  varying  aspect.  related  there  particularly,  and e r r o r  d e s i g n . While of  horizontal  Williams(1982)  element  design in  as  trial  previously  finite  literature,  mining,  this  study  embankments. two  in  years  published  a  In  such  experience two  for  1982)  parameters  past  geotechnical  guides  (Williams, has  1979)  lengths  suitable pit the  on  drains  installed  and  are  the  drain in  spacings. distance  between  rows  deeper.  influence  drainage  distribution  vertical  extends  study.  of  results.  anisotropy  of  slope  4 4.  To  compare  layouts,  To  two  fanned  achieve  followed. a  the  v a l i d i t y  LORNEX  by  Mine  the of  and  is  finite  the  performances  for  a  different  drain  The  design  pattern  is  In  the  r e l i a b i l i t y  study  practice  of  is  parameters develop  a  experiences  the  computer presented  a  selection of  provide  for  design.  shows  great  thesis model  f i r s t ,  drain  basic  More  the  potential  organization,  in  Chapter  and 2.  in  this  the  not  The  form.  a  simple  depends  of in  on  the  good  of  this  the  drain  order  to  drain  design.  and  field  research  description  model  with  heights.  intention  technique  third  spacings,  and  horizontal  simulation  development  is  needed  the  drainage  slope  evaluation is  The  graphical  system  The  from  and  a  verify  system  parameters  research  for  in  the  INTRODUCTION  drainage  varying  in  to  the  information  engineers. a  computer  and  flow  1982).  lengths  drain  geologic  to  book,  were  simulates  data  simulate  results  of  field  the  drain  layouts  horizontal  experienced  the  to  drain  interpret  methodology  Combining  In  to  of  task.  judgement  of  water  & Anderson,  horizontal  combinations  step  (Wang  which  Second,  with  from  program  different  final  model  drain  procedures  model  method.  taken  different  four  saturated  element  MODELING  of  patterns.  computer  state,  example  use  The  a  computer  to  drain  objectives,  steady  an  results  p a r a l l e l  develop  the  GROUNDWATER  step  and  to  dimensional  slope  drainage  these  F i r s t ,  rock  TO  the  area.  of  the  verification  are  computer  simulations  5  which  involve  the  simulation  are  presented  the  computer  analysis  interpretation in  Chapter  4 to  study.  Chapter  5 contains  details  of  the  finite  program  are  presented  in  are show  the  on  in  different  Chapter  the  summary  3.  models Two  application and  formulation  Appendix  A and  B,  examples of  conclusions.  element  and  and  this The  computer  respectively.  6  Chapter 2 MATHEMATICAL MODEL  Groundwater  in mining slopes i s represented by the flow  of f l u i d through porous media. E a r l i e r research in this area of  fluid  flow through porous media was based on a n a l y t i c a l  methods.  These  developed  mathematical  for  methods,  treatment  of  originally,  problems  of  were  heat  flow,  e l e c t r i c i t y and magnetic f i e l d s . With the development of the digital  computer  and  i t s widespread a v a i l a b i l i t y , many of  the important recent advances in the analysis of groundwater systems  have  been  based  on  much  d i f f e r e n t mathematical  approaches generally known as numerical methods. The element  method  is  finite  one of the numerical methods. It i s the  most widely used in research work and i s  employed  in  this  study. (Freeze & Cherry, 1979) To solve involves  groundwater  four  steps.  The  problems first  by step  numerical is  methods  to examine the  physical problem encountered in practice. The second replace  the  physical problem by an equivalent mathematical  problem, which, in more s p e c i f i c terms, i s refered to boundary  i s to  value  as  a  problem. The t h i r d i s to solve the boundary  value problem for unknowns by computer techniques. The f i n a l step  i s to interpret the numerical solution in terms of the  physical problem. The emphasis of this study was the  final  step.  placed  on  The d e t a i l s of that part, therefore, were  arranged in a separate chapter while the other  three  parts  7  are  included  2.1  The  in  Physical  The  2,  in  open  section  To  and  simulate  water  different  open  on  drain  level  water  BF,  pressure  most  significance  EG  is  from  determined  assumed  not  the  of  toe  flow  to the  by  be  on  were  the  would  is be  same  drain  the  plane, the  slope  toe  the  groundwater  by  zero  drains(CD  and  water  flows  out  region  along  these  drains.  the  drainage.  AH  in not  pressure  between  two  compared  along 2)  this for  considered  left  along  under  horizontal  hydraulic  The  Figure only  level  conditions  pressure  obtain  information  the  that  to  therefore,  EG was  horizontal  comprise  highest  boundary  has  in  on  midway  right  and  problem  the  The  slope  a  so as  is  providing  regional  the  and  likely  drains,  which  the  in  affected  of  into  interesting  lines  slope  in  materials.  line(BF)  has  enough  will  performed  parallel  far  show  problem  illustrated  pit  analyses  other  analysis.  As  on  the the  common  distribution  with  be  slope  Particularly  Along  stability  the  a  drainage. an  anisotropic  along  is  of  conditions,  distribution drains.  slope  The  the  study  section  pressure  sections.  under  pit  dimensions.  heterogeneous  the  problem  a vertical  horizontal three  chapter.  Problem  physical  encountered Figure  this  are AC,  to  head  on  and  is  boundary it.  a  As  is the  installed, but  also  8  Free Surface after Drainage Free Surface before Drainage  H drain  (a). Vertical section  (b). Horizontal section  Figure 2.-Hypothetical Slope with Drains  9  To  estimate  the  varying  heights  of  case  it  model  2.2  is  more  which  is  dimensional element  shown  However,  which  could  drains  in  for  a  a  computer  program  large  storage  result  in  a  slope  three  mesh  section  is  construction  due  often  to  very  the  cost the  and  three results  for  Therefore, section,  computer  phreatic  two  two  another  on  subject that  surface  dimensional  of  computer  codes  a vertical  much  number  of  indicated  dimensional  finite requires  large  high  embankments  three  requires  for  same  a  dimensional  design  of two  produced locations  models,  section,  one were  study.  horizontal in  the  model  evaluated  analytically  (Murray,1973). are  phreatic  validity  In  does  vertical  The  the  this  (a).  element  assumption.  of  In  vertical  drain  this  assumption  the  on  on a h o r i z o n t a l  component  on  rows,  considered.  (Tesarik,1984)  drains.  The  2  be  drain  study  the  in  The  and  Tesarik's  dimensional  used  the  time.  unknowns  must  work  Figure  problem.  time  between  to  into  computer  nearly  slope  between  Problem  model,  horizontal  in  distance  flowing  considerable  run.  the  convenient  Mathematical  Water  vertical  include  direction. of  this  by  general, achieved  surface  not  is  the for  the This  the  results  situations  relatively  is  assumption  Murray best  water  flat.  and using where  flow  the  Dupuit  has  been  Monkmeyer the the  Dupuit slope  drain  »  t w 1  IT i  1  i 1  A —  (a). Plan View  H-w-f-  (b). S e c t i o n A - A  h-"—H  (c). Section B - B  Figure 3 . - W a t e r Table Position along S l o p e  11  In a  vertical  blanket  mining  (b),  used  in  the  level,  the  a  3  a  intersection  of  the  ignoring  the  results  the flow  of  of  any  on  Using  w  steady  both state  flow  . two  reached  materials  can  be  smaller w could  with in to  the  are  water  Figure  the  3  drains, into  the  slope the  toe drain  length.  It  than  actual  be  plane  the  drains  back  and  the  in  see  the  as  real  the  reaches  and  the  by  finding  the  the  vertical  model  is  estimated  on  vertical  dimensional  drain  model  and  section,  the  is  slightly  1982)  and  no d r a i n a g e after  modelled.  Heterogeneitic  from  model  estimated  equation  flow  zone.  case,  table  along  surface  always  of  further  spacing  is  treated  not  drains,  distance  water  is  height  is  is  individual  two  surface the  drain  section  horizontal  horizontal  were  the  specific  w  the  at  completed  The  perpendicular  conditions are  In  the  water  3,  between  ,  w  conservative.(willams,  In  Figure  drain  that  In  in  water  which  simulation  level.  which  midway  (b),  at  see  length.  dimensions,  maximum.  function  to  the  drainage.  the  Figure  is  obvious drain  slope  point  simulation,  A s shown  (c),  In  three  section  be  3  slope.  in  the  would  Figure  to  in  practice.  usually table  drain  section  and  modelled.  vertical  was and  used. at  This  the  was  when  in  the  that  water  steady  state  drain  assumed  anisotropic  modelling,  means  stage  horizontal Flow  section  installation the  properties  saturated of  slope  12  The medium  steady  is  derived  state  governed from  by  the  Darcy's  Law  (Freeze  & Cherry,  1979;  Where:  x  are  and  y  flow  and y  h  is  and  the  Ky a r e  Q(x,y)  is  the  the  dimensional equation  continuity  Supplement  each  saturated which  of  was  water  4-1,  in  other,  hydraulic  directions, hydraulic  two  coordinates to  the  a  following  CANMET,  perpendicular Kx  in  flow.  1976)  two  directions  [L]  conductivities  in  x  and  [L/T]  head flow  as  unknown,  influx  into  [L] or  out  of  the  region,  [L/T]  The  analytical  hydraulic and  head  Q(x,y).  impossible  as  In  due  to  region.  To  obtain  method,  boundary  the  governing  problem  is  on  a  equation  presented.  uniquely, The  next  x,  y,  flow  i.e., step  Ky,  solution  be of  to  a  (1)  high  boundary the is  The of  in  the  by  known. to  is  set  points  be  the  Kx,  conditions.  problem and  be  region,  equation  from (1)  would  would  necessary  different  problem  of  number  of  (1)  analytical  hand,  certain  are  of  boundary  solution  usually  an  shape  other  conditions  The  the  the  values  are  describe  cases,  irregular  on  conditions flow  the  equation  function  complicated  solution, head  of  the  and  hydraulic  of  complicated  most  heterogeneity, numerical  a  solution  any These  problem. conditions  mathematical to  solve  the  13  mathematical heads  2.3  in  problem  the  Finite  by  the  are  are  having  medium.  the  simple  more  powerful  and  the  commonly finite  shapes  and  addition, a  the  the  of  hydraulic  water  the  in  The  method  are  finite  flow  method without  varying  be in  used a  to  solve  homogeneuous  other  they  two  hand,  are  becoming  methods  with  the  has  more  method  having  complicated with  solve  element  a  numerical  Compared  great  With  method  materials can  are  element  element  regions  heterogeneous  table  methods  finite  modeling.  element  Analytical  restrictions.  computer,  area.  the  obtained  the  above  solve  be  can  can  to  the  difficulty shape  ease.  In  problem because  during  the  solution.  Theory  The solution types  with  flexibility  processing  Basic  handle  finite  moving  this  only  on  digital  method,  can  deal  for  used  solution  conditions  the  groundwater  difference  and  boundary  difference  in  It  in  being  some c a s e s .  without  the  tool  used  advantages.  with  of  finite  in  exact  methods,  solutions  development  The  complicated  Numerical  approximate  of  method  mathematically  problems  solution  approaches  problems.  analytical  the  Solution  several  value  obtain  region.  Element  There boundary  study  and  of  variation of  a  principle  partial  boundary  in  mathematics  differential  conditions  is  equation  identical  to  states with  that  the  certain  minimizing  the  14  functional  which  is  found  by  a  certain  mathematical  procedure.  For  the  boundary this  p a r t i a l  conditions  section,  the  differential  associated  function  is  equation  which found  are as  discussed  I.  and  the  later  in  (1)  (Remson,1971)  I=/X[^Kx(||)2+lKy(||) +Qh]dxdy  (2)  2  Where:  R  is  the  According for  (1) head i.e.  is  function h(x,y)  discrete  the  on  the  the  areas  triangular The  region  study.  principle,  equivalent  which such  could  method  is  satisfies  continous  region  elements.  or  The  rectangular heads  divided  find  minimize  solve the  the  equation hydraulic  functional  I,  R.  which  or  to  to  that  region  hydraulic is  is  element h  the  be  under  variation  head  h(x,y)  finite  find  nodes.  the  should  satisfied  procedure,  be  to  hydraulic  The to  region  at  into  an  approximate equation  is  divided  shapes  of  associated these  nodes  N elements  nodes,  3hi  method  (3). into  a  In  this  number  the  elements  with  three  are  unknowns.  with  used  or  M unknown  of can  four If head  15  over  the  whole  region  requires  i = 1 ,M  on  every  each  unknown  node,  therefore,  procedures. solving  Finite  Element  divide  and  in  step  region  into  the  formulate  following  the  illustrated  number  of  unknowns  solution  the  small  be  one  equation  formed  could  be  into  the  equation  more  in  accurate equations  Figure  elements,  finite  by  for  certain  obtained  by  specified  process.  each  element  subregions.  31 -0 3h, As  could  be  corporating  in  subregions, the  the  would  Procedure  first  the  There  M equations  equations  conditions  The  node.  Consequently,  the  boundary  head  the for  i=1,M  4,  a  of  these  method  The  smaller  results.  each  is  to the  Then,  to  node.  on R  node  is  associated  elements  all  with  a  contribute  to  31 the  value  number  of of  •g^r*  nodes  contribution  to  element  the  obtained.  in The  Similarily,  all  (usually of  region,  derivation  its the  an  element  3  or  4),  surrounding following  procedure  is  is  s u r r o u n d e d by the  element  nodes. equations  detailed  in  For  a has  each  could  be  Appendix  16  Figure 4. —Associated Nodes and Elements  17  (1).  for  triangular  elements,  rai ^  (2).  for  e  K1 1  K12  K1 a  31 ' 3h,  K2  K  2  K2  31 ^3h,J  K ,  2  ^3 3  rectangular  ah, 31 3h,  1  2  K3  3  elements, K  1  K  2 1  i  K  3  <hi  c  (4)  1  C 1  f K  1 2  1 3  K, „  3  K2 o  K2  K 2 2  r  -\ h,  Ct  h,  C 1  \  <  31 3h, 31  L U  Kg31i  K32  K„ ,  K«  9h  Where:  superscript [k]  is  the  e and  f  element  function  of  K:3«  ^33  ft  h,  ft  C 1 v- J  R«-3  2  represent  the  stiffness material  (5)  element  number,  matrix,  which  propertities  and  is  a  element  geometry {h}  is  the  hydraulic  {c}  is  the  force  boundary  As step vector  (4)  is  and  to  {C}.  (5)  conditions  which  each element,  global  stiffness  labeling  and  (Wang  & Anderson,  strictly  element  1982).  Appendix  The A.  function  element  assemble a done  a  and  for  was  unknown is  formed  This  in  vector,  vector,  were  node  •i l l u s t r a t e d  head  in  labeling process  of  of  geometry  matrix  the [K]  and  terms  of  of  element  the  assembly  next the  geometry,  is  mesh. also  18 The  result  resulting  from  of  the assembly  t h e number  of  i s a set of  sets  of  global  equations.  [K]{H}+{C}=0  Where:  [K]  i s the global  {h}  is  the  region, {C}  The obtain  h  obtain  force  is  the solution  given  to  be i n c o r p o r a t e d  the  boundary  boundary  on  the  whole  vector  to  solve  Known H e a d Figure  in solving  types  equation  (6)  to  applied  of  (6)  a l l boundary  and t o  conditions  the equations.  Figure  in the simulation.  boundary  5,  pressure  conditions  toe  of  BG, HI,  ML, are at  drain  This  the s l o p e .  At these  positions,hydraulic  to  elevation  hydraulic regional  shows  There  their  to a selected toe head  of  the  values  groundwater  also  positions  is zero.  the  5  have  are  involved.  head  according  satisfy  Boundary:  the  equal  of equation  conditions,  conditions  different  from  vector  unknown  remaining  the  In  head  Conditions  To  1.  hydraulic  matrix  values.  Boundary  three  (6)  stiffness  i s the global  problem  equations  heads  datum. slope,  which  which  f o r HM, t h e  are  heads a r e determined  D E , JK are considered therefore,  a r e assumably conditions  applies  at  the  determined  and keep  far  specified by  constant.  the  19  Free Surface after Drainage  (a). Vertical section  (b). Horizontal section  Figure 5.—Boundary Conditions Applied  20  2.  Known F l o w AF,  IJ,  and  having flow 3.  the  Rate  are  flow  pattern.  BD i s  free  are  zero.  ease.  in  difficult a  the  and  be  A.  An  was  was  applied  to  solution,  were  elevation tolerance.  If  not,  estimated  from  the  new  solution  p r o c e s s was  are  obtained  (Desai,  2.4  Computer  Programming  The  equations The  Horizontal  solve the  Drain  head  repeated  the  surface  equations  {c}  the  using  inflow formula  condition  is  more  was  used  then  flow  boundary  if  satisfied of  crosses  approach  hydraulic  position  on  flow  adding  of  global  see  No  a  no  equations.  heads  along  they  the  until  a  free  calculated  After  the  equal  within  in  free their  specified  surface previously  satisfied  was and  results  1972).  were  computer  type  the  to  by  the  boundary:  solving  vector  estimated,  hydraulic  the  the  iterative  being a  in  of  boundaries.  surface  heads  boundaries  symmetry  boundary.  third  investigated  heads  of  the  these  handled  to  The  surface  the  free  pressure  was  of  cross  incorporated  free  obtaining  Method.  the  2  with.  2,  type  element  Appendix deal  flows  third  1 can  of  to  condition  surface  the  impermeable  because  1 and  Condition  contribution (A-10)  rate  of  surface  Condition  imaginary  No w a t e r  Combination  the  which  LK  zero  Curve  with  Boundary:  and  Program  s o l v e d by  program,  Drainage  Finite in  Verification  the  Gaussian  Element  Mining  Elimination  Simulation  Slopes,  FESHDMS,  of was  21  written  by  Terminal  System)  B r i t i s h  Columbia.  described  the  in  horizontal  author at  The The  this  and  presented  Appendix  verify  simulation. of is a 1.  In  The  slopes  sufficient  a l l  the  Unversity  the  vertical  computer  data  of  computations  the  and  numerical result  or  program  output  are  there  evaluate  are  solution  be  the of  to  physics  a  methods  checked  is  computer  the  whether  several  could  modeling  of  understanding to  are  practical  Laboratory models which  heterogeneity. construct Field  if  The  (Michigan  solution to  verify  by  not  easily  interest.  Its  available  for  a p p l i c a b i l i t y  is  limited.  Physical  The  any  the  our  solutions of  Physical  3.  MTS  in  either of  on  Solution  Analytical  2.  do  input  in  of  cases,  numerical  problems  of  run  B.  Normally,  Analytical  could  details  procedure  is  IV  Center  for  example  many  reasonable.  very  program  v a l i d i t y  groundwater  model.  an  important the  Computing  The  organization  One  FORTRAN  chapter  model.  in  in  a  the  method  data  are  d i f f i c u l t  usually Also  are much  laboratory  Measured model  Model  can are  employed  to  build  to  simulate  charaterized equipment  may  by be  rock their  needed  to  model.  Data be  tested  available in  this  against and study.  the  f i e l d  reliable.  measured This  is  data the  22  The  f i e l d  obtained  from  Columbia.  data  the  In  used  LORNEX this  investigations  were  done  and  Most  groundwater  the  Report  C.  producer  the  section  major  west  wall  as  a  was 6 which  slope  is  intensive  LORNEX  staff  providing was  was  British  hydraulical with  Golder  specialist  summarized  section,  check  the  can  a  advice. in  Golder  seen has  been  of  competent, with  section  the  and  the  computer  due  to  data  The some  The  faults.  "*"  slightly  Figure  the were  in  the  data  were  used  difference made  8  output.  relatively  assumptions  to  Figure  computer the  rock  moderately  simulation.  and  that  and  data  in  conductivity  marked  These  8  p i t .  line  major  #35.  Figure  AB  two  as  was  vertical  and  Report  measured  A  the  open  measured  the  pressure  weathered  points  obtained.  understandable  modeling.  view  along  were  from  recorded  been  i n s t a b i l i t y .  heads  of  has  the  plan  of  Columbia  Groundwater  of  molybdenum  B r i t i s h  p i t .  wall  from  comparison  mine  west  number  Golder  the  copper  of  the  slope At  in  Granodiorite  results  consistency  on  hydraulic  from  be  cause  of  the  the  the  comprised  the  available  of  pit  central  instablity  major  Bethsaida  open  the  the  shows  easily  by  verification  Highland Vally,  some  large  shows  distributions.  It  model  the  information  a  selected  Figure  shows  in  Brawner  in  The  considered  altered  is  located  province.  7  0.  the  #35.  LORNEX M i n e  on  Mine mine  Associates of  in  in  good is the  23  In  addition,  example (Wang  taken  from  & Anderson,  book,  Chapter  computer  output  as  in  shown  It verified  can by  the  computer  model  INTRODUCTION  1982).  The  6  was  is  compared  Table  be  the  data  put  into with  was TO  in  tested  by  an  GROUNDWATER M O D E L I N G the  the the  also  example  computer  result  of  in model,  the  this the  example  I.  concluded field  that  data  the  with  computer  satisfactory  model  has  results.  been  Golder  Figure  6.  The  Plan  View  of  LORNEX  Mine  Associates  K12345-  C o n d u c t i v i t y [m/s] - K = 3 E - 8 6 - -- K = 3 E - 8 -K=2E-8 7 -- K = 2 E - 8 -K=9E-9 8 -- K = 2 E - 8 -K=2E-8 9 --K=2E-6 - K = 7 E - 6 1 0 -- - K = 7 E - 6  Figure 7.-The Model Section on the West Wall of Lornex Mine and the Conductivity Distribution  "1 0  I  I  I  I  I  I  I  I  50  100  150  200  250  300  350  400  Depth f r o m 4752ft  •  I  450  1 500  1 550  Bench(ft)  Legend O  Measured Values  X  C o m p u t e d Values  Figure 8.—The C o m p a r i s o n of C o m p u t e d and M e a s u r e d Hydraulic Head Values in LORNEX Mine  27  TABLE  I.  The Comparison of  Node Number  Two S e t s  of  Computer  Outputs  H y d r a ii l i c Head V a l u e s ( m ) Wang & A n d e r s o n FESHDMS  1  4.000  4.00  2  4.000  4.00  3  4.000  4.00  4  4.000  4.00  5  3.720  3.72  6  3.695  3.69  7  3.688  3.68  8  3.686  3.68  9  3.392  3.40  10  3.366  3.37  1 1  3.357  3.36  12  3.354  3.35  13  3.000  3.08  14  3.000  3.00  15  3.000  3.00  16  3.000  3.00  28  Chapter COMPUTER  After  the  verified, Chapter  it  model  was  drains  is  2.  horizontal  and  between  drain  The  and  length  simulations  sections  for  anisotropic  in  the  has the  the  the  also  included  drainage  materials  in  have  effects  of The  vertical  in  distances it  the  should  horizontal  into  on  the  the  different  both  horizontal  2.2,  taken  modeling  for  on  section  vertical  modeling,  performance and  in  performed  section  been  the  specified  lengths.  section  drains  and  horizontal  suitable  As d i s c u s s e d vertical  were  The  and  established  problem  drainage  spacings  of  been  simulations  estimate  rows.  that  model  sections.  analyze  used to  SIMULATIONS  solve  vertical  different  was  noted  to  computer  used to  with  spacing  used  The  section  be  computer  3  account. horizontal  slope  drain  with  pattern  layouts.  3.1  Simulations  The was  to  purpose analyze  system with head was  on  of the  the  the  different  The  applied  chapter,  Figure  element  in mesh  for  on  the  this  effects  level  section  have 2.  on  s p a c i n g s and  model  conditions  Section  simulation  drainage  distributions studied.  Horizontal  (b) model  been and is  of  horizontal the  horizontal  lengths. where  and  presented Figure shown  in  The  drains  geometry  5  sections  in  hydraulic  are  installed  the  boundary  the  (b).  Figure  drain  The 9.  previous finite  Drain  50 40 30 20 10 I 0  1 1 1  1  I  1  40  1  1  — I —  1  — — 80 1  1  — I — ' — ' — ' — I — • — — 120 160 1  1  — I —  1  — • — 200  1  — I —  1  — — 24 0 1  1  — I —  1  — • — ' — I — ' — ' — ' 280 320  I  X(m)  Figure 9 . - F i n i t e Element Mesh for Horizontal Model  • ' 360  '  I  ' ' 400  ' I  440  30  In  Figure  represented  by  coordinates be  in  zero  range  lengths  y  The  drain  made  can  variable  from  under  seen  y.  In  spacings  head  for  be  all  three  upper  100m.  the  the  by  80m  by  varying  and  lower  A total  slope  of of  is  varying  the  to  can  10m.  the  The  length  boundaries.  120  The  computer  drain  heights,  spacing  spacings  from  combinations  different  drain  different  modelled  the  50m t o  the  simulation  range  were  on  that  the  direction,  lengths  pressure  lengths were  it  the  modelled.  different of  9,  runs  spacings  150m,  175m,  and and  200m.  The values  computer at  section.  a  The  simulation  number  of  hydraulic  outputs  separated  heads  on  the  drains  are  most  important  They  interpreted  into  a  Figure for  10  a  slope  slope  is  of of  The a  14  175m,  15  Figure  on  the  midway  between  to 20  the  Figure to  two  analysis.  dimensionless of  head  horizontal  stability  results  Figure  and  of  hydraulic  plots.  simulations 19  Figure  for 24  the  for  the  simulations  for  200m.  slope  to be  results  significant  impractical  heights.  slopes  can  the  nodes  for  series  show  150m,  obviously  150m  simulation  of  of of  possible  number  is  height  height  It  range  Figure  height  slope  all  to  the  line  parallel were  are  which 200m. done  of  In need  For  any  to  do  mining to  be  the  practice, drained  particular  a  fall  into  problem,  great the the  specifically.  these  reduction  simulations of  indicated  hydraulic  head  in  that a  slope  there from  31  no  drains  variations  to in  drain  results.  Within  from  slope  the  significant. less  influence  even  very  spacing  a distance toe,  the  Beyond on  the  sparsely induced  of  that  the  1.0-1.5  differences distance,  hydraulic  head  spaced  different  times in  drain  drains.  the  drainage  drain  hydraulic  length  head  s p a c i n g s have  distribution.  The  is much  32  X/L  h-Woter Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=90m ot depth 150m  Figure  10.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 5 0 m , d r a i n L e n g t h 90m  h-Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length l=80m at depth 150m  Figure  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 80m  34  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  X/L  h-Water Table Height at Distance x H—Slope Height L—Distance from Toe to Headwater Drain Length l=70m qt depth 150m  Figure  12.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 70m  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  X/L  h—Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length l=60m at depth 150m  Figure  13.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 5 0 m , d r a i n L e n g t h 60m  36  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  X/L  h-Water Table Height at Distance x H-Slppe Height L-Distance from Toe to Headwater Drain Length l=50m at depth 150m  Figure  14.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 5 0 m , d r a i n L e n g t h 50m  1  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  X/L  h—Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=100m at depth 175m  Figure  15.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 7 5 m , d r a i n L e n g t h 100m  1  h—Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length l=90m at depth 175m  Figure  16.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 7 5 m , d r a i n L e n g t h 90m  39  \  X/L  h-Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length 1=8Om at depth 175m  Figure  17.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 7 5 m , d r a i n L e n g t h 80m  40  X/L  h—Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=70m at depth 175m  Figure  18.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 7 5 m , d r a i n L e n g t h 70m  41  0  0.1  0.2  0.3  0.4  0.5 X/L  0.6  0.7  0.8  0.9  1  h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=60m at depth 175m  Figure  19.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 1 7 5 m , d r a i n L e n g t h 60m  h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length 1=10Om at depth 200m  Figure  20. Dimensionless H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , d r a i n L e n g t h 100m  43  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  X/L  h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=90m at depth 200m  Figure  21.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , d r a i n L e n g t h 90m  1  44  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  X/L  h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=80m at depth 200m  Figure  22.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , d r a i n L e n g t h 80m  1  45  h-Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length l=70m at depth 200m  Figure  23.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , d r a i n L e n g t h 70m  46  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  X/L  h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=60m at depth 200m  Figure  24.  D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n for t h e S l o p e H e i g h t o f 2 0 0 m , d r a i n L e n g t h 60m  47  3.2  Simulations  The  purpose  model  was  drain  levels.  applied finite  to  of  The  element 25.  installed  the  the  section  shown  mesh  for  level  in  with  pit  goes deeper,  the  slope  the  second  of  drains  row  distances from  20m  i.e.,  the  at  section  at  drains.  was  of  1 and  about  0.6  it  Similar  section  sections the  drains  which  are  slope  face.  The  illustrated  in  assumed to of  100m.  toe  and  The  modelled  row  of to  200m.  the  at  the  drain  installed.  from  horizontal  higher,  ranging  drains  which  of  were  The  model  are  with  slope drain  length w  the  vertical  were  120m  As  be  to  the  level, 1  is  in  the  smaller  simulations  and  length.  no  with  of  real  assumption  having  is  5.  was  are  Dupuit  was  Figure  installed.  surface  the  drains  assumption  section  or  of  is  sections  the  table  drain  the  modeling the  two  estimated  the  to  the  water  where  was  of  and  between  conditions  becomes h i g h e r  second  vertical  distance  height  levels  height  from  instead  times  modeling,  middle  the  section  than  slope  distance  used  vertical  case,  the  The  intersection w,  the  slope  were  100m,  (b)  drains  height  drain  installed is  the  two  3  the  boundary  section of  between to  the  Figure  row  on  vertical  and  this  first  simulation  suitable  been  the  Section  computer  model  The  at  Vertical  the  estimate  have  Figure  on  a  been  flow  across  symmetric drains.  made  The  Therefore,  in for  the  the the  section.  section water there  horizontal  must is,  vertical In  this  between  two  flow  toward  actually,  300-1  250-  /  200-  A  / v§,  150100-  00  50  100  150  i  200  i  250  ' — r 300  h — ' '—i—' — i — — i — 11  350  400  450  500  550  X(m)  Figure 25.-Finite Element Mesh for Vertical Model  1  600  49  some  water  flowing  understandable would  be  lower  simulations result  in  would  A systems  that  away the  be  that  was  having  a  section  head  water  modelled.  values  calculated  outgoing  slightly  w=40m  the  hydraulic  than  which  from  in  from  flow  is  It  the the  is  field  computer  ignored.  The  conservative.  used,  which  represented  the  drainage  two  parallel  the  vertical  of  computer  |=0.8-0.85  Where:  S  is  the  horizontal  drains, 1  is  The  the  were  drain  The  each  slope  height  the  water  table  results of than  drains can  range,  and  maintain  distance becomes  higher  seepage  force  unfavourable  table  and 40m  to  the  stable the  and  pit  saturated  becoming for  as  the  larger slope  would  zone  floor  26  drains  such  condition.  higher  factors  60m  saturated  results  Figure  for  on  positions  after  distance  range,  the  plotted.  that  above a  distributions  water  the  [L]  as  before  vertical  that  head  were  suggested  the  length,  obtained  calculations.  between  [L]  hydraulic  section  spacing  a  be  Greater  pit and  a  slope  not  zone  to  the  Figure  were  under  is  in  31  for show  installed. suitable  The range  condition.  the  Less  first  row  of  relatively  high  and  than  that  under  the  extends larger.  stability.  slope  vertical first  row  deeper, These  the  are  From F i g u r e  very  29,  Figure 26.—Water Table Position in the Slope  51  Figure 27.-Water Table Position in the Slope  52  Figure 28.-Water Table Position in the Slope  53  Figure 29.—Water Table Position in the Slope  •9  Figure 30.-Water Table Position in the Slope  Figure 31.-Water Table Position in the Slope  56  30,  and  the  31,  it  slope  can  face.  sufficient  to  Therefore,  they  Otherwise,  Slope  stability  3.3  the height  Because anisotropy  to  meters  100  Material  the  water  at  the  made  certain usually  of  ratios  of  very are  stage.  or  reducing  good d r a i n a g e  is  of  results.  drained  about  an  not  zone.  earlier  length  near  half  zone of  the  maximum.  Analysis  feature study  rock  an  width it  is  saturated  at  drain insure  Slope  to  of  installed  to  table  second l e v e l  height  discontinuities  special  characteristics the  a  or  were  be  necessary  face,  a  a  the  original  slope  is  simulations  of  such  requires  of  that  drains  need  are  Anisotropic  range  The  increasing  spacing  slope  seen  drain  drain  behind  be  of  in  rock  slopes,  such s l o p e s .  the  influence  materials  on  conductivities  the  in  The  of  computer  anisotropic  drainage.  two  the  A  directions  wide were  assumed:  ROC=0.001,  Where:  ROC i s  Although with  a  70m,  spaced at  the  anisotropic in  Figure  60m  slope  and  materials  can has  be a  Of  10,  100,  made  on  one  system having of  150m,  the  drainage  computer  simulation  seen  1000  Conductivities,  was  height  indicate  It  1,  drainage a  The  0.1,  Ratio  drain  slope.  32.  the  simulation  horizontal  quantitatively  0.01,  that  significant  the  and  slope the  the  on  model  results  result  Ky  length  nature  anisotropy  influence  Kx  is of  of  of can the  plotted the  rock  57  h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=70m at depth 150m Drain Spacing S=60m  Figure 32.—The Drainage Effect of Anisotropic Materials  58  drainage  effects.  parallel  to  drains ROC  the  is  further  a  The  value  a  moving  time,  under  freezing in  effects  of  was  to  made  drain  As  of  Figure  slope.  The  element  mesh  Figure  34.  the  mesh  is  drain  For  shown  levels  in  same  for  in  drains order  to  the  When  the  greater,  with  hydraulic  head  cross  parallel 9,  to  more  a  often  save  are  equipment  particularly discontinuities the  computer  performance  drainage simulation  of  a  fanned  layout.  two  boundary  the  very  To e s t i m a t e  number  are  Layouts  easier,  patterns,  33,  2.3,  Figure  were  in  and  drain  same  section  finite  the  drainage  Figure the  or  Pattern  collection  drain  parallel  have  Drain  directions.  the  direction  drainage.  2  change  pattern  water  in  of  the  conditions,  and  in  order  the  smaller.  fanned  compare  5,  an  in  perpendicular  horizontal  practice,  different  shown  to  Different  different  simulation  i.e.,  ROC,  of  make  layout  length  in  drainage  in  oriented  face,  become much  slope  installed  conductivity  with  Comparison  In  larger  favorable  change  distribution  3.4  slope  more  reaches  The  sections  used  in  the  of  over  the  same  conditions  applied  fanned drain  drains  to  drain layout,  Chapter  2.  two  specified  in  sections.  The  pattern the  is  finite  Assuming  at  1.  150m  with  a  drain  length  60m  and  spacing  60m,  2.  175m  with  a  drain  length  80m  and  spacing  80m,  3.  200m w i t h  a  drain  length  100m  and  spacing  shown  100m.  in  element  that  the  (a). Parallel Drain Pattern  (b). Fanned Drain Pattern  Figure 33.-Parallel and Fanned Drain Patterns  £  / /  'V  > > >  i0 ) 0 2 4 0 8 0 12 0 16 0 2( X(m)  28,0 . 3:20 3(50  Drain  Figure 34.-Finite Element Mesh for Fanned Drain Pattern  )0  4(  4*  61  The - c o m p u t e r drain  models  material  most  symmetric and  is  the  as  At the  These  slope,  a  if  results the  difference  hydraulic  distance  of  favourable  local  it  pattern,  is  the  ends  achieve  the  drainage  local  drains.  geology  layout  can  make  the  more  cost  be  is  as  of  drains  effective.  a  yield  in  a  of  the  pattern drain  becomes  length  from  are  nearly  the  length  of  decided,  slope  induce within  toe. for  If  a  a  the a the no  fanned  discontinuities,  i.e.,  smaller  s h o u l d be  better  that  parallel  layouts  used that  in  spacing order  to  achieved  by  c i r c u m s t a n c e s where  the  drain  to  35,  drain  oriented  fanned  toe  fanned  been  the  many  the  especially,  equivalent  Figure known  a certain  pattern  from  in  central  difference  the  has  drains,  However,  system  over  different  more  for  in  conditions exist  results  good  a  models  reduction,  fanned  used to  drainage  drain  geological  between  two  that  in  the  of  drains  length  recommended t h a t  parallel  of  head  such  parallel  the  is  from  value  that  in  it  value  slope,  twice  fanned  drain  head  the  indicate  and  and  on  plotted  length  highest  heads  number  head  results,  drain  into  distance  parallel  drain  these  the  hydraulic  fanned  assumed c o n d i t i o n s . Homogeneous  s e c t i o n s were  hydraulic  back  on  hydraulic  of  as  made  considered.  both  twice  Further  toe,  same.  for  highest  about  smaller.  only  distance  the  pattern.  three  By a n a l y z i n g  the  slope,  the  were  important  line  37.  within  under  s l o p e was  The  36,  simulations  pattern,  drainage  installation  more  the  fanned  performance  and  efficient  and  62  1-  O.h  1 O.Oh  7  Legend  :  Fanned _ parallel O.OOI0  0.1  0.2  0.3  0. 4  I  0.5  X/L  0.6  0.7  0.8  1— 0.9  h-Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length l=60m, Spacing s=60m, at depth 150m  Figure  35.  Comparison For  the  of  Fanned  Slope  and  Height  Parallel of  150m  Drain  Patterns  63  0.1  X t t  —>  /  /  T f Tanned  ,/  parallel J  02  0.1  0.4  03  05  06  X/L  07  i 0.8  i 0.9  1  h—Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length l=T0m, Spacing s=60m, at depth 175m  Figure  36.  Comparison For  the  of  Fanned  Slope  and  Height  Parallel of  175m  Drain  Patterns  *  0.1 }  >  J  $  t 1 /  i  *  0.01  —' —  t  /  _  1  -t  1  r  / 3  anned  Parallel 0.1  02  03  04  05  06  07  l  i  0.8  0.9  1  X/L  h—Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length 1=80rri. Spacing s=60m, at depth 200m  Figure  37.  Comparison For  the  of  Fanned  Slope  and  Height  Parallel of  200m  Drain  Patterns  65  Chapter  4  APPLICATIONS  The  practical  aid  for  horizontal  this  chapter,  LORNEX  Mine  estimating drainage  4.1  sample  is  the  having  this study  The  water  example, in  cross  and to  the  38.  ideal  Two  table  the  is in  to  provide  mining  slopes.  application  illustrate spacing for  head  on  right  the  slope  the  procedures drain  the a  to  horizontal  an In the  procedures  geometry  is  local  the  were  determined  boundary  height.  of  climatic  Second,  are the  for  using  system design  c o n d i t i o n s are  assumptions  position and  the  horizontal  section  conditions  to  study  for  drain  Example  relatively  Figure  case  and  this  system design  presented  length  A Hypothetic  this  drain  of  system.  For of  a  purpose  slope  is  First,  regional  environment.  kept  for  results a  slope  outlined.  made. by  the  constant  shown the  initial  groundwater The  hydraulic  and  slope material  in  is  equal  to  considered  be h o m o g e n e o u s .  If  the  slope  heterogeneous, FESHDMS hydraulic following  it  is  simulation head the  material  in  suggested to to  obtain  distribution  procedures.  the  a use  actual the  detail  instead  of  mine  computer  is  highly program  and more  realistic  going  through  I  H -x  500m  F i g u r e 3 8 . - T h e S l o p e S e c t i o n of t h e E x a m p l e  67  (1) .  To Assume a D r a i n Assuming  l=70m,  equipment, (2) .  the  the  use  section  Table  line  from  on  the  geology  drain  and  installation  the  slope  height.  slope  height  Values  selected  to  the  two  10m t o  l=70m  18  of  between  ranging  local  Figure  intersection  (3) .  h/H  length  h=l75m,  depending  specified  To O b t a i n For  Length  and  obtain  drain  the h/H  level  drains  for  values  and each  along  the  the  vertical  drain  spacing  80m.  Calculation  Insert  the  data  Calculation  obtained  Table  II.  completing  data  underlined  are  others  calculated  were  from  step  Table  III  for  this  inserted obtained  in  by  is  very  into  Table  example.  Table  the  (2)  III  II  after  The  from  simple  the  data  step  (2),  equation  h=(h/H)*H  Where: (4) .  To  Plot  and h a r e  Water  Plot  the  data  pairs,  which (5) .  h/H  To  estimated  the  (x,h)  water  Select  Selecting  a a  extent  known  water in  table  Table  table  Because this  previous  III.  in  the  as  Check  steps.  study,  it  the  stability is  not  in  by  using  Figure  39  the in  s=80m  Stability  satisfies  the  slope  shown  corresponds to  spacing which  of  from  Positions  S p a c i n g and  requirements. the  Table  all  Requirement slope  analysis  discussed  stability is  beyond  here.  TABLE I. Calculation Table  X/L=0 05 x= h/H h (m) s  10 20 30 40 50 60 70 80  x/L=0. 075 x= h/H h  x/L=0 1 x= h/H h  x/L=0. 15 x= h/H h  x/L=0. 2 x= h/H h  x/L=0. 3 x= h/H h  TABLE I I . Example C a l c u l a t i o n  Table  X/L=0. 05 x=25m (m) h/H h  x/L=0. 075 x=37.5m h/H h  x/L=0. 1 x=50m h/H h  x/L=0. 15 x=75m h/H h  10  0.00  0.00  0.00  0.00  0.024 4.20  0.087  15.23 0.15  26.25 0.28  20  0.00  0.00  0.00  0.00  0.03  5.25  0.09  15.75 0.15  26.50 0.283 49.53  30  0.00  0.00  0.005 0.78  0.037 6.48  0.10  17.50 0. 155 27. 13 0.286 50.05  40  0.002 0.26  0.01  0.045 7.88  0.11  19.25 0.16  50  0.004 0.68  0.016 2.80  0.05  0.115 20. 13 0. 165 28.88 0.292 51.10  60  0.007  1 .23  0.022 3.85  0.058 10.15 0.12  70  0.01  1 .75  0.028 4.90  0.065  1 1 .38 0. 125 21 .88 0. 175 30.63 0.298 52. 15  80  0.015 2.63  0.035 6. 13  0.071  12.43 0.13  s  1 .66  8.75  L=500m, H=175m  21 .0  x/L=0.2 x=100m h/H h  0.17  22.75 0.18  28.0  x/L=0.3 x=150m h/H h 49.00  0.289 50.58  29.75 0.295 51 .63  31.5  0.30  52.50  70  Figure 39.—Estimated Wctre Table Position in the Exampls?  71  In  this  case,  corresponding  to  a  design  for  the  equal  to  70m a n d  l=70m  and  satisfactory  the  analysis  or  to  Other Other  the  table  slope  safety  not  options  spacing, increase  and  If  the  then be  a  the  water  satisfy  the  are  either  back  drain  position  primarily length  table  the and  is  at  stability  keeping  to  length  analysis  drain  the  go  the  at  stability  system would  can  is  factor,  s p a c i n g 80m.  other  to  water  l=70m,  stability repeat  step  (5). Considerations considerations  anisotropy account  drainage  s=80m  reducing  (2)  the  s=80m,  shows  requirement,  .  if  to  properties, make  the  such etc.  final  as are  local needed to  decision.  heterogeneity, be  taken  into  72  4.2  The  Application  The It  is  real  for  appliable  without  problem  the  condition  The  the 8  this  wall  range,  it  depths  The in  effective  The The  to  slope  deep  drains  region.  to  not  graphs  the  which  ensure  this  are be  in  or  would  is of  to  could by  is  at  other  would  ideal  has  been of  I0" m/s 7  In  to  adit shallow  would  be  1983)  the  level. 4752ft  moving.  The  448ft(135m). boundary  is  estimated  be  far  enough  reach  such  not  to  effectively  5200ft from  vertical  to  with  adit  slope  is  considered  slope  drainage  the  a  The  9  the  upper  constant  be  I0~ m/s. be  a  installed  level  drainage  of  depressurize  the  kept  range  Associates,  wall  will  conductivity  Whereas  (Golder  the  the  ideal.  section,  mine  6  150m.  west  to  the  for  drains  4752ft  could  produced.  wall  liklihood  heads  were  the  system  toe  consistent  10" m/s  to  charts this  mass h y d r a u l i c  that  from  discussed.  of  the  the  In  range  100m  on  is  far  these  is  drainge.  slope  hydraulic  is  drains  drain  at  from  1300ft(400m) the  crest  reduce  Mine  horizontal  for  are  pit  likely  slope,  that  Pit  The  horizontal  height  distance the  by  horizontal  bench  2. the  the  pit  the  extreme is  usually  information  of  an  depressurized system.  which  Mine  modification.  mine  Chapter  with  mines  Pit  guidance  some  this  under  in  west  I0" m/s,  design  general  described  in  LORNEX O p e n  in  conditions  LORNEX O p e n  situation  unlikely  in  to  The where to  be  from a  far  73  The  same p r o c e d u r e s a s  followed.  The  considered (1) .  in  specific  each  l=90m  installation (2) .  To O b t a i n For  the  h=135m, are  values and drain (3) .  Table  (5) .  to the  vertical  mine  feasible  mine.  l=90m  in  for  (Golder  and  which  real  mine  section  spacing ranging  are  drain  Report  #35)  from  the  the  slope  situations  line  of  between 10m t o  slope  two  conditions  to  the  height  obtain  drain  drains  h/H  level  for  each  80m.  Calculation  Table  Plot  Water the  the  pairs,  data which  the  Select  Selecting  analysis  II.  from  step  Results  are  shown  water  table  estimated (x,h)  water  in  Table  table  S p a c i n g and  a  spacing which In  this  Check  reducing for  the the  (2) in  the  into  Table  to  for  the  s=80m,  spacing to  60m,  the  VI.  system  and  using  Figure  40  Requirement slope  water the  in  by  s=20m  the  safety  40m,  slope  shown  Stability  unsatisfactory  drainage  as  satisfies  case,  corresponding shows an  VI  in  corresponds to  a  requirements. position  obtained  Table  Plotting  design  is  this  intersection  Calculation  To  in  10  the  data  in  which  selected  the  To  this  were  Values  using Figure  Insert  (4) .  h/H  length  the  of  4.1  Length  (300ft)  the  along  section  conditions  equipment  closest  in  step.  To Assume a D r a i n Assuming  outlined  table slope  stability at  the  stability  factor.  Further  20m, ' t h e  primary  suggests that  the  drains  TABLE  s (m)  X / L = 0 05 x = 20m h h/H  10  0.000  20  III.  Calculation  Table  for  LORNEX  Mine  x / L = 0 . 075 x=30m h/H h  x/L=0. 1 x = 40m h/H h  x / L = 0 . 15 x=60m h/H h  x/L=0. 2 x = 80m h h/H  x/L=0.3 x=120m h/H  0 . 00  0.00  0.00  0.00  0 .00  0.00  0.00  0.002  0 .27  0.13  17.55  0.00  0 . 00  0.00  0.00  0.00  0 .00  0.00  0.00  0.01  1.35  0.13  17.55  30  0.00  0 . 00  0.00  0.00  0.00  0 .00  0.002  0.27  0.018  2 .43  0.14  18.9  40  0.00  0 . 00  0.00  0.00  0.000  0 .000  0.005  0.675  0.025  3 .375  0.14  18.9  50  0.00  0 . 00  0.001  0 . 135 0 . 0 0 2  0 .27  0.009  1.215  0.03  4 .05  0.15  20.25  60  0.00  0 . 00  0.001  0 . 135 0 . 0 0 2  0 .27  0.014  1 .89  0.04  5 .40  0.15  20.25  70  0.002  0 . 27  0.004  0.54  0.007  0 .945  0.02  2.70  0.048  6 .48  0.16  21 . 6 0  80  0.004  0 . 54  0.007  0.945  0.01  1.35  0.025  3.375  0.052  7 .02  0.16  21 . 6 0  L=400m,  H=135m  h  F i g u r e 4 0 . — E s t i m a t e d W a t r e Table P o s i t i o n in L O R N E X  Mine  76  have (6).  l=90m,  Other The  Considerations  geologic  shows  that  parallel  The range  of  information the  to  favourable the  s=20m.  hydraulic  the  factor  slope for  s p a c i n g c o u l d be  vertical  reported  face a  is  of  in  drain  the  in  Report the  greater  drainage  increased  40m-60m a c c o r d i n g t o  Golder  conductivity  such  spacing  in  some  is  Therefore,  areas.  levels  study  direction which  system.  #35  c o u l d be  results  in  in  the  section  3.2.  In for  conclusion,  LORNEX M i n e drain  in  the the  recommended h o r i z o n t a l upper  west  wall  length=90m,  horizontal vertical  spacing  spacing  20m-30m,  40m-60m.  is:  drain  system  77 Chapter  5  C O N C L U S I O N S AND SUMMARY  5.1  Conclusions  From program, (1) .  the the  There drain a  (2) .  computer  following  is  a  a  slope  hydraulic with  spaced drain  Different within  in  distance  in  this  research  made. head  no d r a i n s  reduction compared  at  to  the  having  system.  s p a c i n g s have  the  done  c o n c l u s i o n s are  significant  level  sparsely  simulations  different  of  1.5  to  drainage 2.0  effects  times  of  the  drain  length.  Beyond influence (3) .  (4) .  on  A suitable of  drains  The in  that  reduction vertical  is  slope  in  rock  spacing  hydraulic  distance  to  the  has  little  head.  install  range  slopes  conductivity  face  system.  If  order  2 or  of  of  drain  of  plays  the  40m  an  second  to  row  60m.  important  role  performance.  Higher to  in  suggested  anisotropy  drainage  distance,  is  more  the  distribution  values  in  favourable  ratio  of  Kx  a  direction  for to  such  Ky h a s  a  a  greater,  the  change  in  becomes  smaller  with  further  parallel drainage  value  in  hydraulic change  the head  in  the  ratio. (5) .  Fanned d r a i n s produce  and  different  parallel results  in  drains the  in  slope  region  near  drainage the  slope  78 face The drain  h y d r a u l i c head a f t e r the i n s t a l l a t i o n of fanned  drainage  drainage length  i s higher  within from  benefits  the  the  than  for parallel  d i s t a n c e about twice  slope  t o e . However  drain  of the d r a i n  the  practical  and c o s t b e n e f i t s of fanned d r a i n s a r e u s u a l l y  substantial.  5.2 Summary T h i s study comprises a s e r i e s of to  drains,  f o r h o r i z o n t a l and v e r t i c a l spacing of  drainage c h a r a c t e r i s t i c s i n a n i s o t r o p i c rock  the drainage e f f e c t s of d i f f e r e n t d r a i n p a t t e r n The  two  dimensional  finite  element  developed and used i n the s i m u l a t i o n . The was  simulations  model h o r i z o n t a l d r a i n drainage i n an open p i t s l o p e . I t  i n c l u d e s the analyses  and  computer  verified  by  the  f i e l d data  slopes  layouts.  program  computer  was  program  from the LORNEX Mine. The  procedures f o r using the r e s u l t s of the study  are  outlined  in examples i n Chapter 4. S e v e r a l assumptions were made i n the study. (1) . Darcy's law i s v a l i d , (2) . Flows are i n two dimensional (3) . The graphic  saturated  fields.  r e s u l t s a r e based on s i m u l a t i o n s i n  homogeneous m a t e r i a l s .  i  79  In the area  of  mining  slopes,  there  studied  i n the f u t u r e  horizontal are as  a an  drain few  system  aspects  extension  of  design  for  that should this  be  research  program. 1.  To  take  the  the t r a n s i e n t 2.  drainage time  i n t o account  by m o d e l i n g  flow models,  To m o d e l t h e d i r e c t i o n a l d i s c o n t i n u i t i e s o f i n more  on  detail.  rock  slope  80  BIBLIOGRAPHY  Abramowitz,  M.  and  Stegum,  Functions Applied  with  Bise,  C . J . and  Drainage  C O . ,  General  Symposium,  Byrne,  Denver,  Mining,  P.M.  and W a l t e r  for  Nonlinear Soil  R.F.  Soil  and  Freeze,  Janzen,  of  SME o f  of  Mine AIME,  Cherry,  Mine  Drainage  1979 Stability  in  1982  of  Stresses  Series the  52,  Deformations Department  University  Canada, 2nd  No.  and  December  Edition,  Program  of  of  British  1981  1978,  Van  Nostrand  Ltd. J.T.,  J.A.,  Cliffs,  Mathematical  Hillslope  May  Mine  SOILSTRESS-A Computer  Engineering,  Englewood  Sons,  Analysis  Drainage,  Slopes,  3,  Analysis  Christian,  Inc.  and  Aided  International  Rock  Volume  Co.  and  in  of  Mechanics  R.A.  A.R.,  Bureau  Engineering,  Colorado,  Mechanics,  Geotechnical Freeze,  First  Vancouver,  Reinhold(l5) C.S.  Tables,  1964  on M i n e  Engineering,  Columbia,  Desai,  the  Surface  Civil  Craig,  of  Stability  Soil,  C ,  Mining  and Math  National  Computer  Report  C O . ,  in  Graphs,  55, D.  Mathematical  1984  Proceedings  Brawner,  R.L.,  Systems,  September, Brawner,  Series  Washington,  Scyoc,  Handbook  Formulas  Math.  Standard,  I.A.,  P177-125,  Methods  McGraw-Hill,  Groundwater, N.  Models  Hydrology  Numerical  J . , of  (M.J.  1978  in  1979  Prentice-Hall,  1979 Hillslope Kirkby,  Hydrology,  Ed)  John  Wiley,  81  Ge,  Shemin,  Lecture  CIVL  573,  British Golder  Notes  1984  of  and GEOL  Columbia,  Holland,  I.  Report  and  Tapir,  August  Kolbein,  The  K.H.,  The and  T.C.,  Pazin,  Drains Div., Murray,  W.A.  Son,  for  Soil  ASCE.  and  Slopes,  November  Marlon-Lambert, of  Cir. J . ,  CANMET R e p o r t Remson,  I.,  Methods Sons. Rulon,  J . ,  the  in  P.L.,  Mine  B.L.,  of  Norway,  Engineers,  John  of  of  the  Horizontal  Geotech.  Eng.  Hydraul.  Div.,  G.M.  of  Jour.  Sept.  1973 Computer  Analysis  Supplement  December  1977  and M o l z ,  F . J . ,  Hydrology,  4-1,  CANMET,  Numerical John  Wiley  and  1971  Heterogeneous  Seegmiller,  for  Validity  Seepage,  Development  Univresity  (LORNEX  Method,  Design  Equation,  Subsurface  Inc.,  of  1977  77-30,  Hornberger,  W.S.,  Slope Manual,  Groundwater  Wall,  Element  Method  Jour,  E n g . , V91,  Pit  West  University  Choi,  and Monkmeyer,  Soc.  University  ,1945  Dupuit-Forchheimer Am.  1983;  1983  Finite  Element  Inc.  M.  the  342,  1969  Finite  Wiley  GEOL  Canada  the  Technical  Trondheim-Norway, Huebner,  of  1983; 1984,  Vancouver,  #35,  Bell,  562,  564,  Associates, Drainability Mine),  Kenney,  GEOL  of  of  Multiple  Hillslopes, British  Horizontal  Dewatering,  Seepage Ph.D  Columbia,  Mine  Drainage,  along  Thesis,  June  Drains-Their  Faces  Use  the  1984 in  Open  Pit  P r o c e e d i n g s of  the  82  First  International  Mine  Draingae  Symposium  ,  D e n v e r , C o l o r a d o , May 1979, P268 S h a r p , J . C , L e y , G.M.M. a n d S a g e , R., P i t S l o p e M a n u a l C h a p t e r 4 G r o u n d w a t e r , CANMET, CANMET R e p o r t  77-13,  November 1977, P240 S m i t h , I.M. P r o g r a m m i n g Application Inc., Sterrett,  to  t h e F i n i t e Element Method Geomechanics,  With  John W i l e y and Sons.  1982  R . J . and E d i l ,  T.B., G r o u n d - W a t e r  and S t a b i l i t y o f a S l o p e , G o r u n d  Flow  System  Water,  2 0 , 1982,  P5-11 T e s a r i k , D.R. a n d K e a l y , C D . , E s t i m a t i n g H o r i z o n t a l Design  by  Methods,  t h e F i n i t e Element and F i n i t e Bureau  Investigations:8875,  of  Mines  Drain  Difference  Report  of  1984, U. S. D e p a r t m e n t  of the  Interior W i l l i a m s , R., B l o o m s b u r g , G. a n d W i n t e r , G., I n f l o w o f H o r i z o n t a l D r a i n s i n T a i l i n g s Embankments, A Research  Contract  Reoport,  M i n e s , U. S. D e p a r t m e n t  August  Mining  1982, B u r e a u o f  of I n t e r i o r  Wang, H.F. a n d A n d e r s o n , M.P., I n t r o d u c t i o n t o G r o u n d w a t e r Modeling-Finite Methods,  Difference  and  Finite  Element  W. H. Freeman a n d Co. San F r a n c i s c o , 1982,  P288 Z i e n k i e w i c z , O.C,  The F i n i t e E l e m e n t M e t h o d ,  M c G r a w - H i l l , New Y o r k , 1977, P787  3rd Edition,  83  APPENDIX A  FINITE  The  principle  discussed  of  the  in Chapter  2.  formulation  process of  know  the  that  region  The  is  ELEMENT FORMULATION  finite  The c o n t e n t  equation  water  flow  of  (A-1)  is  (4)  in  method this  a n d (5)  in a  g o v e r n e d by e q u a t i o n  functional  element  was  appendix  (Freeze  following  so as t o  condition  obtain  must  be  |£=0  or  if  the  unknown  region  head  nodes,  the  ^ =;; T  [ K  for  (A-1),  comprised  (A-3)  x(||)|  K i  the  satisfied:  (A-3)  of  N  i s equivalent  9hf i.e.  solution  on R  R is  1979)  (A-2)  2  +  I,  2 . We  I:  2  minimize  the  saturated  & Cherry,  I=/;t5Kx(||) 4Ky(||) Qh]dxdy  To  is  in Chapter  two d i m e n s i o n a l  (1):  briefly  subregions  with  M  to:  u  (||) Ky(||)| (||) +  K  i = 1 ,M  + Q  f ]axay=0 7  (A-4)  84  Therefore, equations The  M equations  a r e M h y d r a u l i c head formulation  subregions. and  c a n be o b t a i n e d ,  solution  of  M  values.  o f (A-4) d e p e n d s  Two t y p e s  the  of element  on t h e  shape  shape  of  the  were u s e d , t r i a n g u l a r  rectangular.  Formulation  f o r T r i a n g u l a r Elements  Y (I. 3(x ,y ) 3  3  2(x  within  each element, t h e h y d r a u l i c heads  function the  of t h e i r  position,  interpolation  interpolation  function.  =(1  a r e assumed t o be a  The  generalized  is  called  form  of  x,y a r e p o s i t i o n heads  (A-5)  3  x y ){a}  {a} i s g e n e r a l i z e d  hydraulic  2  f u n c t i o n f o r the t r i a n g l e i s :  2  The  ,y )  i . e . h=h(x,y), which  h=a,+a x+a y  where:  2  coordinates  coordinates at three  nodes  of the t r i a n g l e  should  85  satisfy  (A-5),  i.e.  h  The  matrix  3  1 x,  y,  1 x  2  y  2  1 x  3  y  3  a. (A-6)  <  form i s :  (A-7)  {h}=[B]{a}  {a}  i s found  to be:  {a} = [ B ] - 4 h } =[A]{h}  Substituting  (A-8) into  h-(1  (A-8)  (A-5):  x  y)[A]{h}  =[N]{h}  In  (A-9),  _ In  _ . • equation  needed  to  [N]  is a function  „\ (A-4),  9h ^ ,  be known.  (A-9)  of x,  y,  x  1  f  y  9h ' 9 / 9 h \ 9 /9h\ ^ , ^ ( ^ ) , ^ ^ ) , A l l of the terms  1  x  f  9h a  could  h  2  ,  . .  y  2  ,  9x  911^9^ 9y 9y  9x  9hf9x'"9x  ,  y  3  are  be d e r i v e d  from  9x  9 ^ 9 ^ 9y 9y  9_,9hx 9N_  3  i=1 , 3  (A-9):  9x  x  (A-10)  86  9hT" Substituting e  t o node  ^  into  i s obtained  Li  Where: The  i  (A-10)  Nl  (A-4),  from  the contribution  the following  ]  Ae i s t h e a r e a  simplified  form  of  +K  element  ]{h}  Q N l } d x d y  e.  becomes:  (|^T)=[k,i  k  ]  l  element  formation:  ay 97^ "oT^  *r Tr dr ? dT  of  k ;  k i]{h}+{Cj}  2  (A-11)  3  H=Kx(fi)(|^)Ae Ky(fi)(Mi)Ae +  Ci=//QiNi Ae  dxdy  i = 1 ,3 and j=1,3  kfj three  and  Ci could  corner  Formulation  The  be computed  by k n o w i n g  the coordinates  of  points.  for Rectangular  interpolation  Elements  function  for rectangular  elements  h=a, + a x + a. y+a xy 2  3  t h e same  procedures  elements,  equation  (A-13)  (|^-) = [ k , i  (A-12)  ( |  Following  is  i s :  d e s c r i b e d above  for  triangular  obtained:  k 21  k ; 3  k , i ] { h } + {Ci}  (A-13)  87  Ci=;/QfNj Ae i=1,4  Assembly  of  finished, global has  and j=1,4  the Global  After  the the  the form  Matrix  formulation next  equation  dxdy  step  for  individual  elements  i s t o a s s e m b l e them  f o r the whole  s y s t e m . The  into  global  of  way t o  form  each element  node  numbering  process.(Wang  a set of equations  of:  [K]{h}+[C]=0  The  is  the global i s done  (A-14)  equation  strictly  and element  & Anderson,  (A-14)  i n terms  numbering.  1982)  from of  Figure  the  the A-1  equation geometry, shows t h e  Element contributions are computed, dispersed, and summed to form the global matrix  Nodes  Element No. e  X X X X X X X X  X  e = 5  Figure  e = 6  A.1.-The  Global  Individual  Global matrix  Matrix  i  j  m  1  1  i  3  2  2  5  3  3  T  4  5  4  4  6  5  5  5  6  7  6  6  8  7  Assembly  Elements  from  89 APPENDIX B COMPUTER B.1  Introduction The  computer  Simulation  in this  hydraulic  dimensional  capability flow  (Finite  Element  Drainage  i n Mining  Slopes),  to  water p r e s s u r e steady  state  determine  the  as w e l l as flow flow  in  a  two  field.  of the program p e r m i t s t h e  f o r the following  t h e l o c a t i o n of t h e f r e e water open p i t  2.  d i r e c t i o n of  and s a t u r a t e d  groundwater  FESHDMS  a p p e n d i x was d e v e l o p e d  head d i s t r i b u t i o n ,  r a t e s and flow  The  program,  f o rHorizontal Drain  described  1.  PROGRAM  analysis  of  problems: surface  f o r water  flow i n  slopes;  hydraulic  head d i s t r i b u t i o n  on a v e r t i c a l  or  horizontal  section; 3.  flow The  following  r a t e s and d i r e c t i o n s i n t h e flow  region.  computer p r o g r a m p e r f o r m a n c e s a r e s u b j e c t e d  t o the  conditions:  1.  Darcy's  2.  material  law i s v a l i d  f o r flows;  conductivities  can  be  heterogeneous  and  anoisotropic; 3.  flow The  region  i s confined  finite  a p p r o a c h was u s e d  or  unconfined.  e l e m e n t method by t h e v a r i a t i o n a l i n the simulation.  The r e g i o n  of  principle analysis  90  is  divided  of  the  the  equations form  into  of  a number  finite  with  of d i s c r e t e  element  the hydraulic  the equations  elements.  formulation heads  The  results  i s a s e t of  a s unknowns.  The  The  (B-1)  [K]  i s the global  {h}  i s t h e unknown  hydraulic  {C}  i s the column  vector  flow  matrix  i s given by:  [K]{h}+{C}=0  where:  linear  stiffness  matrix head  vector  associated  with  the  flux  derivation  of equation  (B-1)  has  been  shown  in  presented  in  Appendix A .  B.2  Computer  Program  Organization  Flowchart  A  flowchart  Figure  B.1.  A  program  listing  f o r the computer certain c o u l d make  amount the  program  is  o f comment program  contained  details  in the  relatively  straightforward.  Subroutines  MAIN the  main  routine  intitiates  storage  global  geometry  and g l o b a l  storage  and a c t s  as a general  control flow  space,  reads  variables, routing  the  allocates  routine.  91 Start  — r  Read heading and geometry Set dimension Allocate storage i  1 Call  NODEIN  Call  ELEMTIN  Read more d a t a PA,WT, time period  Call  CONDUC  Call  ECHO  Call  BUILD  Call  ASSEMB  Call  SOLVER  Call  COMPA  T  Call  TRIAN  u  Call  QUADS  1 Call  TRIAN  *1 C A L L  INITI  Call  QUADS  Call  BUILD  Call  ASSEMB  Call  SOLVER  t =0  *  yes  IF  H=• y  Call  i Call  IF  No  PHOUT *  COMPA  H=y  yes  ?  Call  ?  PLOUT  NO  |  Call  PHOUT  Call  PLOUT  Stop  Figure  B.1  Flow  Chart  for  the  Computer  Program  FESHDMS  92 NODEIN This routine part  i scomprised of  reads  individual  two  node  i n t e r p o l a t e s m i s s i n g nodes and data  input.  The  second  portions. data  The  from  checks  errors  first  unit in  5, node  p a r t c a l c u l a t e s t h e number o f  unknowns. ELEMTIN T h i s s u b r o u t i n e reads i n d i v i d u a l element 5,  checks  f o r errors  and  d a t a from  calculates  unit  t h e maximum  bandwidth. CONDUC S u b r o u t i n e t o r e a d c o n d u c t i v i t i e s of m a t e r i a l s and specified  known h y d r a u l i c  read  h e a d s f r o m u n i t 5.  ECHO Subroutine  to  print  to  unit  6 a l l node, element and  m a t e r i a l data f o r v e r i f i c a t i o n . INITI For t r a n s i e n t problems, condition  this  subroutine  does  initial  computation.  BUILD The  main  element  subroutine  to  form  element  m a t r i x f o r each  a n d s t o r e them f o r a s s e m b l i n g t h e g l o b a l  matrix.  T R I A N a n d QUADS C a l l e d by BUILD, t h e y c a l c u l a t e t h e e l e m e n t t r i a n g u l a r and r e c t a n g u l a r  matrix f o r  elements.  ASSEMB Assembly  individual  element  matrix  into  the  global  93  equation. COMPA For  free surface  computed  problems, t h i s  h y d r a u l i c heads  elevation  on  subroutine  free  compares t h e  surface  with  their  heads.  PHOUT Subroutine  to print  out t h e h y d r a u l i c head and p r e s s u r e  head a t t h e e n d o f c o m p u t a t i o n . PLOUT Subroutine  to plot  mesh a n d water An  example  presented  the  slope  section,  finite  element  table p o s i t i o n . of the  computer  a t t h e end o f t h i s  input  Appendix.  and  output  is  94  Variable (according  to  their  Name  appearing  Number  of  Listing sequence  in  the  program)  NNODE  total  NU  toatal  NELT  total  Number  of  ELemenTs  NMAT  total  Number  of  different  NTRNGS  total  Number  of  TRiaNGular  NQUADS  total  Number  of  QUADilateral  NB  Bandwidth  NHK  Number  PA  Atomspheric  Pressure  WT  unit  of  P1  the  first  P2  the  second Period  P3  the  third  DELT1  the  time  interval  in  first  DELT2  the  time  interval  in  second drainage  DELT3  the  time  interval  in  third  ITIME  Index  MAXIT  MAXimum I T e r a t i o n  XELT(1,I)  X coordinate  of  Ith  node  in  a ELemenT  XELT(2,I)  Y coordinate  of  Ith  node  in  a ELemenT  ESM(4,4)  Element  Stiffness  EVE(4)  Element  VEctor  BM(2,4)  inter  Number  of  of  Unknow  Known H e a d  WeighT  of  NODEs nodes  MATerial  types  elements elements  nodes  water  Period  of  Period  TIME  heads  drainage  of of  drainage drainage drainage  drainage  period period period  step  variables  times  Matrix  95  CONX  CONductivity  in  X  direction  CONY  CONductivity  in  Y  direction  STOR  STORage  HIN  INitial  H  Hydraulic  MB  Maximum  BX(MB)  X coordinate  of  corner  points  on  boundary  BY(MB)  Y coordinate  of  corner  points  on  boundary  XMAX  MAXimum X  coordinate  YMAX  MAXimum Y  coordinate  TOL  specified  TOLerance  KC  Known  Head Head  Boundary  head  4-known; MC  coefficient  Movable  corner  Condition  5-free  points  of  each  node,0-unknown;  surface  Condition  of  each  node,  0-free  to  move;  1-fixed MESH  0-no  KCE  Known  mesh p l o t flux  required;  Condition  with  no  known  flow  boundary  NSECT  O-vertical  NSTADY  O-transient;  flow  model;  of  1-mesh each  boundary;  Element, 1-element  1-horizontal  1-steady  state  plot  model  0-element with  known  Computer  Storage  Array Variable  Storage  Allocation  Diagram  A(20000) Position  in  Array  Size N1 = 1 X  NNODE N2=N1+NNODE NNODE N3=N2+NNODE  CONX  NMAT N4=N3+NMAT  CONY  NMAT N5=N4+NMAT  STOR  NMAT N6=N+NMAT  HIN  NNODE N7=N6+NNODE  H  NNODE N8=N7+NNODE NNODE N9=N8+NNODE  GVE  NNODE N10=N9+NNODE  GSM  (NNODE) N11=N10+(NNODE)  HK  NHK  Array Varable  Storage  IA(2000) Position  in  Array  Size IN1 KC  NNODE IN2=IN1+NNODE  MC  NNODE IN3=IN2+NNODE  I TYPE  NELT IN4=IN3+NELT  I MAT  NELT IN5=IN4+NELT  JELT  4(NELT) IN6=IN5+4(NELT)  NC  NELT IN7=IN6+NELT  KCE  NELT  Free node •  Figure  Fixed nixie  B.2.  An E x a m p l e  to  Show  the  Computer  Input  1St1ng of WAND a t 21:47:31 on JAN 30. 1985 f o r CC1d=BRAW Page 1 2 3 4 5  e  7 8 g 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 45.5 45.7 46 47 48 49 50 51 52 53  HEAD,EXAM, 16,18.1,8,0,0, 1 , 4.60..60.. 0.,0., 6.,0.. 6.,3.. 0. .4. , 0, 1,0.,4..4,1, 2,0.,2..4,1, 3,0.,1.,4, 1, 5,2..3.7,5,0, 6,2.,2.,0,1, 8,2.,0.,0.1, 9.4. ,3.36,5,0, 10,4.,2.,0,1, 11.4.,1.,0,1, 13,6..3..4,1, 14,6.,2.,4,1, 15,6.,1.,4,1, 16,6.,0.,4,1,,1.6,5, 2,1,1,0,2,6,1, 3,1,1.0,2,7,6, 4,1,1,0.3,7,2, 5,1,1,0,3.8,7, 6,1,1.0.4,8,3, 7. 1,1,0,5,10.9, 8,1,1.0,6,10,5, 9,1,1.0,6.11,10, 10,1,,6, 11,1.1,0,7,12.11. 12,1,1.0,8.12,7. 13,1,1.0,9,14,13, 14,1,1,0,10,14,9, 15,1,1,0,10,15,14, 16, 1 , 1 ,0, 1 1 , 15, 10, 17,1,1.0.11,16,15. 18,1,1.0,12,16.11. ROCK, 9.8.0.,0.. 5.0.5, 1 , 1,0..0.0001,1., 1,4., 2,4. , 3,4. , 4,4. , 5,3. , 6,3. . 7,3. , 8,3. ,  100  L i s t i n g of WAND, a t 12:02:29 on JAN 30, 1985 f o r CC1d=BRAW Page 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  1  ^ E x e c u t i o n begins 6 MAXIT= 5 ********** Total Total Node 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  NODAL INFORMATION  **********  number of nodes= 16 number of unknowns 3  Coordinates X Y 0.0 4.000 0.0 2.000 0.0 1.000 0.0 0.0 2.000 3.700 2.000 2.000 2.000 1.000 2.000 0.0 4.000 3.360 4.000 2.000 4.000 1.000 4.000 0.0 6.000 3.000 6.000 2.OOO 6.000 1.000 6.000 0.0 **********  8  Known c o n d i t i o n 4 4 4 4 5 0 0 0 5 0 0 0 4 4 4 4  1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1  ELEMENT INFORMATION  **********  T o t a l number of e l e m e n t 18 Maximum bandw1dth= 6 Number of q u a d r l l a t e r a l s = 0 Number of t n a n g l e s = 18 Element types: 1-Triangle 2-QuadMlateral 3  Element 1 2 3 4 5 6 .7 8 9 10 1 1 12 13 14 15 16  Material Type 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  Element Type 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  Movable c o n d i t i o n  Nodes A s s o i a t e d with Element Code • 1 6 5 2 1 6 2 7 6 3 7 2 3 7 8 4 3 8 5 10 9 6 10 5 6 1 1 10 7 11 6 7 12 1 1 8 12 7 9 14 13 9 10 14 10 15 14 1 1 15 10  101  it Ing of WANO. a t 12:02:29 on JAN 30, 1985 f o r CC id=BRAW Page 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 11 1 112 113 1 14 1 15 1 16 117 118  1 1  17 18  1 1  **«*«*****Mater1al  1 1 16 12 16  15 11  propert1es**********  U n i t weight of water************ 9.80 M a t e r i a l type Conductivity Conductivity Number in X d i r e c t i o n in Y direction 1 0.00010 0.00010 IT= 1 ************** + * * * * * p RATIO OF Ky/Kx IS 1.00000 0 u t  DRAIN LENGTH  3  U t  ^nformation *********  0.0 (M)  NODE NUMBER HYDRAULIC HEAD(m) WATER PRESSUI 1 4 .000 0 .0 4 .000 19 .600 2 3 4 .000 29 .400 4 4 .000 39 .200 5 3 .720 0 . 193 6 3 .695 16 .613 7 3 .688 26 .343 8 36 .121 3 .686 9 3 .392 0 .312 10 3 .366 13 .383 11 3 .357 23 .095 12 3 . 354 32 .868 13 3 .000 0 .0 14 3 .000 9 .800 15 3 .000 19 .600 16 29 .400 3 .000 Vx THIT ELEMENT Vy 1 0.000013801424 -0.000001439324 -5. 95373 2 0.000015240745 -0.0 0. 0 0.OOOO15240745 -O.OOO00O711495 -2. 67283 3 4 0.OOOO15596495 -0.0 0. 0 5 0.OOOO15596495 -0.000000222816 -0. 81848 6 0. 0 O.000015707890 -O.O 7 0.000016062913 -0.000001928770 -6. 84705 -4 .99172 8 0.000016478938 -0.000001439324 9 0.000016478923 -0.000000901329 -3 .13071 0.000016573846 •-0.000000711495 -2. 45812 10 11 0.000016573846 -0.000000273438 -0. 94519 12 0.000016599151 -0.000000222816 -0. 76905 0.OOOO19591855 -O.O O.O 13 14 0.000018280305 -0.000001928770 -6. 02302 15 0.000018280291 -0.0 0. 0 16 0.000017829632 -0.000000901329 -2. 89396 17 0.000017829632 -0.0 0. 0 18 0.000017692902 -0.000000273438 -0. 88541 *** UBC P l o t  Subroutines - End of P l o t t i n g * * *  Storage 0.0  102  Listing 119 120 121 122 123 124 125 126  of  WAND, a t  12:02:29 o n  JAN  30,  1985  f o r CC1d=BRAW  Page  3  Number o f p l o t f r a m e s g e n e r a t e d = 1 I f t h i s p l o t 1s q u e u e d f o r p l o t t i n g 1t w i l l t a k e a p p r o x i m a t e l y 2 m i n u t e s t o p l o t a t an a p p r o x i m a t e c o s t of 0.11 d o l l a r s ( U n i v e r s i t y r a t e s ) and use 26 i n c h e s o f p a p e r . A p p r o x i m a t e l y 3 6 % o f t h e t i m e w i l l b e s p e n t p l o t t i n g w i t h the pen raised. * * * * * * * * * * C A L C U L A T I O N END ********** /^Execution t e r m i n a t e d  


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