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Risk-cost-benefit framework for the design of dewatering systems in open pit mines Sperling, Tony 1990

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R I S K - C O S T - B E N E F I T F R A M E W O R K F O R T H E D E S I G N O F D E W A T E R I N G S Y S T E M S I N O P E N P I T M I N E S By Tony Sperling BA.Sc, The University of British Columbia, 1983 M A . S c , The University of British Columbia, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY T H E FACULTY OF G R A D U A T E STUDIES Department of Geological Sciences We accept this thesis as conforming to the required standard T H E UNIVERSITY OF feRTTTSH COLUMBIA April 1991 ® Tony Sperling, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of &fDlOG>tlM SOC^Cfe The University of British Columbia Vancouver, Canada Date APrtJl yD}  DE-6 (2/88) ABSTRACT Control of groundwater plays an important part in operations at many open pit mines. Selection of an efficient and cost effective dewatering program that will improve slope stability of the pit walls is frequently complicated by the complex and somewhat uncertain hydrogeologic environment found at most mine sites. This dissertation describes a risk-cost-benefit (RCB) framework that can be used to identify the most effective dewatering strategy under such conditions, because the stochastic framework explicitly accounts for uncertainty in hydrogeologic and shear strength parameters in the groundwater flow, slope stability and economic analyses. In the framework, the monetary worth of each design alternative is measured in terms of an economic objective function. This function is defined in terms of a discounted stream of benefits, costs and risks over the operational life of the mine. Benefits consist of revenue generated from the sale of mineral concentrate. Costs include normal operating and dewatering expenses. Monetary risks are defined as the economic consequences associated with slope failure of the pit wall, multiplied by the probability of such a failure occurring. Selection of the best design strategy from a specified set of alternatives is achieved by determining the economic objective function for each design and then selecting the alternative that yields the highest value of the objective function. Estimation of the probability of slope failure requires an accurate assessment of the level of uncertainty associated with each input parameter, a forecast of how dewatering efforts are expected to affect pore pressures in the pit wall in light of the uncertain hydrogeologic environment, and an evaluation of the effect that the pore pressure reductions will have on improving stability of the pit wall. Prediction of the pore pressure response to dewatering efforts is achieved with SG-FLOW, a steady state, saturated-unsaturated finite element model of groundwater flow. Slope stability is evaluated with SG-SLOPE, a two dimensional, limit equilibrium stability model based on the versatile Sarma method of stability analysis. To account for input parameter uncertainty, both the groundwater flow stability models are invoked in a conditional Monte-Carlo simulation that is based on a geostatistical description of the level of uncertainty inherent in the available hydrogeological and geotechnical data. Besides documenting the methodology implemented in the framework to conduct the geostatistical groundwater flow and economic analyses of the objective function, this dissertation also presents a sensitivity analysis and a case history study that demonstrate the application of the RCB framework to design problems typically encountered in operating mines. The sensitivity study explores how each set of input parameters, including hydrologic data, shear strength parameters, slope angles of the pit wall and dewatering system specifications impact on the profitability of the mining operation. The study utilized a base case scenario that is based on overburden conditions at Highland Valley Copper; therefore, the conclusions cannot be applied blindly at other sites. However, the framework can be used to formulate site specific conclusions for other large base-metal open pit mines. After the objective function was calculated for the base case, the aforementioned input parameters were systematically perturbed in turn to study how each parameter impacts on profitability of the mine. The sensitivity study showed that in the particular case analyzed changes in the slope angle and dewatering efforts can improve profitability by many millions of dollars. In particular, steep slope angles can be utilized in the early stages of mine development while the pit walls are relatively low, and then flattened as the pit wall height increases and the monetary consequences of slope failure become more pronounced. Furthermore, the sensitivity results indicated that pit dewatering is likely to be effective over a range of hydraulic conductivities from lxlO"8 m/s to lxlO'5 m/s and that accurate estimation of the mean hydraulic conductivity is much more important than estimating other statistics that describe the hydraulic conductivity field, including the variance and the range of correlation. Results of the sensitivity study clearly demonstrate that the RCB framework can be used effectively to identify the most effective dewatering strategy given a limited amount of geologic and hydrologic information. Also, it is shown that the framework can be used to identify the most important input parameters for each specific dewatering problem and to establish the approximate monetary worth of data collection. ii The case history study documents how the RCB framework was applied at Highland Valley Copper (HVC). Groundwater control is recognized as an important component of mining operations at this mine site; dewatering measures utilized on the property involve both high capacity dewatering wells and horizontal drains. The benefits of pit dewatering include improved slope stability, drier operating conditions in the pit, and a convenient production water supply. These benefits do not come cheaply, H V C is expecting to spend in excess of six million dollars on groundwater control in the next ten years. Before investing such large sums in groundwater control, mine management should be confident that the capital investment is justified, i.e. that the resulting economic benefits will significantly exceed the costs of the dewatering effort. Using historical data provided by H V C , the case history study documented in this dissertation shows how the RCB framework is used to identify the most profitable combination of slope geometry and groundwater control in design sector R3 of HVC's Valley Pit. By considering three possible slope angle and groundwater control options it is shown that by continuing to implement an aggressive dewatering program, H V C can expect to reduce operating costs by as much as nine million dollars in this design sector. Graduate Supervisor: R. Allan Freeze Department of Geological Sciences iii TABLE OF CONTENTS Page ABSTRACT ii T A B L E OF CONTENTS iv LIST OF FIGURES viii LIST OF TABLES xii A C K N O W L E D G E M E N T xiii CHAPTER 1 INTRODUCTION 1 1.1 PROBLEM DESCRIPTION 1 1.2 RESEARCH OBJECTIVES 2 1.3 SCOPE 2 1.4 THESIS OVERVIEW 3 CHAPTER 2 GROUNDWATER C O N T R O L IN O P E N PIT MINES 6 2.1 INTRODUCTION 6 2.2 CANADIAN MINING INDUSTRY 6 2.3 OPEN PIT MINING OPERATIONS 7 2.3.1 PRE-PRODUCTION PLANNING 7 23.2 PRODUCTION C Y C L E 8 2.3.3 DECOMMISSIONING AND RECLAMATION 8 2.4 SLOPE STABILITY CONSIDERATIONS 8 2.5 GROUNDWATER IMPACTS ON MINE PRODUCTION 10 2.6 DEWATERING ALTERNATIVES 11 2.7 SUMMARY 14 CHAPTER 3 RISK-COST-BENEFIT ANALYSIS 15 3.1 INTRODUCTION 15 3.2 DECISION ANALYSIS 15 3.3 FORMULATION OF OBJECTIVE FUNCTION 17 3.3.1 SYSTEM BENEFITS 17 3.3.2 SYSTEM COSTS 18 3.3.3 SYSTEM RISKS 19 3.4 COMPONENTS OF THE RISK COST BENEFIT FRAMEWORK 20 J.J SOFTWARE MODULES 22 3.6 SUMMARY 25 iv TABLE OF CONTENTS T A B L E O F C O N T E N T S Page C H A P T E R 4 G E O S T A T I S T I C S 26 4.1 OVERVIEW 26 4.2 BASIC STATISTICS 29 4.2.1 GEOSTATISTICAL TERMINOLOGY DEFINED 30 4.2.2 RELATIONSHIP BETWEEN y(h) & C(h): 34 4.3 VARIOGRAM MODEL 35 43.1 EXPERIMENTAL SEMI-VARIOGRAM 35 4.3.2 VARIOGRAM FUNCTIONS 38 4.4 ESTIMATION 41 4.4.1 PHILOSOPHY OF KRIGING 41 4.4.2 KRIGING IS B L U E 43 4.4.3 MECHANICS OF KRIGING 43 4.4.4 PROPERTIES OF KRIGING 44 4.4.5 ESTIMATION ERROR 47 4.4.6 VERIFICATION 50 4.4.7 CASE HISTORY & FITTING SEMI-VARIOGRAM 52 4.5 SIMULATION 58 4.5.1 HISTORICAL OVERVIEW 59 4.5.2 UNCONDITIONAL SIMULATION, CONDITIONAL SIMULATION A N D INVERSE 61 4.5.3 REVIEW OF SIMULATION METHODS 63 4.5.5 L U MATRIX DECOMPOSITION M E T H O D 67 4.5.5 VERIFICATION OF SIMULATION RESULTS 70 4.6 GEOSTATISTICS AND DESIGN 76 4.7 SUMMARY 79 C H A P T E R 5 G R O U N D W A T E R H Y D R O L O G Y 81 5.1 OVERVIEW 81 5.2 BOUNDARY VALUE PROBLEM 82 5.2.1 FLOW EQUATION AND BOUNDARY CONDITIONS 83 5.2.2 T H E FINITE ELEMENT M E T H O D 84 5.2.3 REPRESENTATION OF WELLS A N D HORIZONTAL DRAINS 84 5.3 FLOW IN THE UNSATURATED ZONE 86 5.3.1 FREE SURFACE APPROACH 86 5.3.2 SATURATED/UNSATURATED APPROACH 87 5.4 IMPORTANCE OF GEOLOGY AND Ksxr 90 5.5 STOCHASTIC GROUNDWATER FLOW MODELLING • 92 5.6 SUMMARY 92 v TABLE OF CONTENTS TABLE OF CONTENTS Page CHAPTER 6 SLOPE STABILITY 94 6.1 OVERVIEW 94 6.2 LIMIT EQUILIBRIUM ANALYSIS 95 6.3 METHODS OF LIMIT EQUILIBRIUM ANALYSIS 98 63.1 FELLENIUS M E T H O D 100 63.2 BISHOP'S RIGOROUS M E T H O D 100 633 BISHOP'S SIMPLIFIED M E T H O D 101 63.4 JANBU'S GENERALIZED PROCEDURE 102 6.33 M E T H O D OF MORGENSTERN AND PRICE 102 63.6 SPENCER'S M E T H O D 104 63.7 SARMA'S M E T H O D 105 6.3.8 SELECTION OF M E T H O D FOR INCORPORATION IN FRAMEWORK 106 6.4 MECHANICS OF SARMA'S METHOD 107 6.4.1 GOVERNING EQUATIONS 107 6.4.2 NUMERICAL SOLUTION PROCEDURE 109 6.4.3 ITERATION PROCEDURE TO C A L C U L A T E F 110 6.5 PROBABILITY OF FAILURE 124 6.6 INCORPORATING PORE PRESSURES IN ANALYSIS 126 6.7 SUMMARY 127 CHAPTER 7 SENSITIVITY STUDY 128 7.1 OVERVIEW 128 7.2 BASE CASE PARAMETERS 131 7.3 BASE CASE ANALYSIS 140 7.4 IMPACT OF GEOTECHNICAL DESIGN DECISIONS 147 7.5 SENSITIVITY TO HYDROLOGIC PARAMETERS 153 7.5.1 SENSITIVITY T O M E A N HYDRAULIC CONDUCTIVITY 154 7.5.2 SENSITIVITY TO STANDARD DEVIATION OF HYDRAULIC CONDUCTIVITY 157 7.5.3 SENSITIVITY TO R A N G E OF CORRELATION 160 7.6 SENSITIVITY TO SHEAR STRENGTH PARAMETERS 162 7.6.1 SENSITIVITY TO M E A N FRICTION A N G L E 162 7.6.2 SENSITIVITY T O STANDARD DEVIATION IN FRICTION A N G L E 165 7.6.3 SENSITIVITY TO M E A N COHESION 166 7.6.4 SENSITIVITY TO STANDARD DEVIATION IN COHESION 168 7.6.5 EVALUATION OF SENSITIVITY RESULTS 169 7.7 SENSITIVITY TO DEWATERING DESIGN 173 7.7.1 IMPACT OF DEWATERING ON ULTIMATE PIT STABILITY 173 7.7.2 STEEPENING SLOPE A N G L E IN EARLY PITS 175 7.8 SENSITIVITY TO FIELD DATA 177 7.8.1 EFFECT OF D A T A ON CONFIDENCE IN T H E STABILITY ASSESSMENT 177 7.8.2 MONETARY WORTH OF A SITE INVESTIGATION P R O G R A M 182 7.8.3 USING D A T A TO IMPROVE DEWATERING DESIGN 184 7.9 SUMMARY 188 vi TABLE OF CONTENTS T A B L E O F C O N T E N T S Page C H A P T E R 8 C A S E HISTORY 192 8.1 OVERVIEW 192 8.2 SITE ORIENTATION 194 8.3 GEOLOGIC INTERPRETATION 197 8.3.1 B E D R O C K G E O L O G Y 197 8.3.2 O V E R B U R D E N G E O L O G Y 199 8.4 HYDROGEOLOGY 202 8.4.1 S U R F A C E W A T E R H Y D R O L O G Y 202 8.4.2 G R O U N D W A T E R H Y D R O L O G Y 204 8.4.3 EST IMATES O F K F R O M P U M P TESTS 204 8.4.4 EST IMATES O F K F R O M L O N G T E R M W E L L RESPONSE 206 8.4.5 EST IMATES O F K F R O M L A B O R A T O R Y T R I A X I A L TESTS 207 8.4.6 EST IMATES O F K BASED O N G R A I N S IZE A N A L Y S E S 207 8.4.7 S E M I - V A R I O G R A M IN V E R T I C A L D IRECT ION 208 8.4.8 S E M I - V A R I O G R A M IN H O R I Z O N T A L P L A N E 209 8.4.9 H Y D R A U L I C C O N D U C T I V I T Y M O D E L O N SECT ION R-3 212 8.4.10 S E L E C T I O N O F B O U N D A R Y CONDITIONS W ITH S IMPLE A Q U I F E R M O D E L 214 8.5 CURRENT PIT DESIGN AND DEWATERING STRATEGY 216 8.5.1 S L O P E DES IGN 216 8.5.2 G R O U N D W A T E R C O N T R O L OBJECTIVES A N D C U R R E N T D E W A T E R I N G P L A N 217 8.6 STOCHASTIC MODEL OF GROUNDWATER FLOW 218 8.6.1 DESCR IPT ION O F B O U N D A R Y V A L U E P R O B L E M 218 8.6.2 G E N E R A T I N G REAL IZAT IONS O F T H E H Y D R A U L I C C O N D U C T I V I T Y F I E LD 219 8.6.3 M O D E L L I N G O F 1987 C O N D r n O N S - A VER I F ICAT ION 221 8.6.4 M O D E L L I N G O F T H E L O N G T E R M P O R E PRESSURE R E S P O N S E 221 8.7 SLOPE STABILITY EVALUATION 224 8.7.1 S H E A R S T R E N G T H D A T A 224 8.7.2 DETERMIN IST IC A N A L Y S E S / CR IT ICAL F A I L U R E M O D E S 227 8.7.3 STOCHAST IC A N A L Y S E S / PROBAB I L ITY O F F A I L U R E 229 8.8 RISK-COST-BENEFIT ANALYSIS OF EXISTING MINE PLAN 231 8.9 IMPROVEMENTS TO SLOPE DESIGN AND DEWATERING STRATEGY 233 8.9.1 T O E D R A I N A G E O N L Y 233 8.9.2 F L A T T E N I N G SLOPE O N L Y 234 8.9.3 F L A T T E N I N G T O E A N D PROVISIONS FOR T O E D R A I N A G E 235 8.9.4 E V A L U A T I O N O F DES IGN OPTIONS 235 8.10 SUMMARY 236 C H A P T E R 9 S U M M A R Y A N D C O N C L U S I O N S 240 9.1 REVIEW AND SUMMARY OF DISSERTATION 240 9.2 SUMMARY OF PRINCIPAL ASSUMPTIONS 252 LIST O F S E L E C T E D R E F E R E N C E S 254 APPt/VDiC£S 2bO v i i T A B L E OF CONTENTS LIST O F F I G U R E S Page Figure 2.1 Mine Planning and the Daily Production Cycle 7 Figure 2.2 Relationship Between Slope Angle and Stripping Costs 9 Figure 3.1 Decision Tree for Example Design Problem 16 Figure 3.2 Flow Chart Illustrating Key Steps in Risk-Cost-Benefit Framework 21 Figure 3.3 Flow Chart Illustrating Software Modules 23 Figure 4.1 Hydraulic Conductivity Profile in Borehole 27 Figure 4.2 Organization Chart - Geostatistics 27 Figure 4.3 Log Hydraulic Conductivity Contours 30 Figure 4.4 Hydraulic Conductivity Field Discretized into Cells 30 Figure 4.5 Normal PDF - Mean & Variance Defined 31 Figure 4.6 Characteristics of Covariance & Semi-Variogram 32 Figure 4.7 Semi-Variogram & Covariance Function with Nugget Effect 34 Figure 4.8 Experimental Semi-Variogram for Demonstration Profile 36 Figure 4.9 Semi-Variogram Behaviour at Origin & Large Distances 37 Figure 4.10 Standard Variogram Models 40 Figure 4.11 Location of Estimation Volume and Supporting Measurement Points 41 Figure 4.12 Location of Hydraulic Conductivity Measurement Points - Packer Tests & Test Pits 45 Figure 4.13 Kriged Estimate of Hydraulic Conductivity Field 45 Figure 4.14 Smoothing Effect of Kriging 46 Figure 4.15 Calculating Average Covariance C J X m ,Xm] 47 Figure 4.16 Increased Uncertainty due to Duplication of Information 48 Figure 4.17 Contour Plan of Variance of Estimation Errors for Point Estimates 49 Figure 4.18 Contour Plan of Percent Variance Reduction for 10x10 m Block Estimates 49 Figure 4.19 Histogram of Normalized Estimation Errors at Measurement Points 51 Figure 4.20 Histogram of Normalized Estimation Errors at all Grid Points 51 Figure 4.21 Location of Transmissivity Measurements - Avra Valley 52 Figure 4.22 Experimental Semi-Variogram Structure and Six Possible Models 53 Figure 4.23 Results of Verification Analysis 55 Figure 4.24 Standard Deviations of Estimation Errors for 1x1 Mile Blocks - Avra Valley 57 Figure 4.25 Contours of Kriged Transmissivity Estimates - Avra Valley 58 Figure 4.26 Surface Representations of Hydraulic Conductivity Field 62 Figure 4.27 Differences between Standard and Lu Conditioning Techniques 69 Figure 4.28 Results of Simulation Verification - Example Problem 72 Figure 4.29 Sensitivity of Simulation Statistics to Range/Length Domain Ratio 74 Figure 4.30 Sensitivity of Simulation Statistics to Number of Cells in Domain 75 Figure 5.1 3D Schematic of Typical Dewatering Design 82 Figure 5.2 2D Representation of Boundary Value Problem 83 Figure 5.3 Illustration of Finite Element Mesh 84 Figure 5.4 Plan View Showing Well Locations and Piezometric Surface 85 Figure 5.5 Piezometric Profiles on Section A-A' . . 85 Figure 5.6 Cause of Reduced Hydraulic Conductivity in Unsaturated Zone 87 viii TABLE OF CONTENTS LIST O F F I G U R E S Page Figure 5.7 Characteristic Curves Defining Unsaturated Behaviour (after Freeze & Cherry, 1978) . . . 88 Figure 5.8 Boundary Value Problem Used to Test Sensitivity to Characteristic Curve 89 Figure 5.9 Influence of Changing K^, on Position of Water Table 90 Figure 5.10 Influence of Geologic Structure on Pore Pressure Distribution 91 Figure 6.1 Modes of Pit Wall Failure 95 Figure 6.2 Failure Block Divided Into Slices 97 Figure 63 Body Forces Acting on Slice 97 Figure 6.4 Force Distribution - Fellenius Method 100 Figure 6.5 Force Distribution - Bishop's Rigorous Method 100 Figure 6.6 Force Distribution - Bishop's Simplified Method 101 Figure 6.7 Force Distribution - Janbu's Generalized Method 102 Figure 6.8 Force Distribution - Morgenstern -Price Method 103 Figure 6.9 Function f(x) Relating Interslice Shear and Normal Forces 103 Figure 6.10 Force Distribution - Spencer's Method 104 Figure 6.11 Relationship Between Factor of Safety and Resultant Angle 8 104 Figure 6.12 Force Distribution - Sarma's Method 105 Figure 6.13 Unknowns Associated with Slice i 107 Figure 6.14 Known Forces and Points of Application - Slice i 107 Figure 6.15 Functional Form Relating K and F 110 Figure 6.16 Geometry of Spoil Pile Failure I l l Figure 6.17 Magnitude of Reaction Forces with Changing FOS 112 Figure 6.18 Force Vector Diagram For Slice 1 112 Figure 6.19 Graph of Reaction Force Summation 113 Figure 6.20 F vs. K,. Scatter Plot 114 Figure 6.21 Error in Estimate of F 114 Figure 6.22 Force Vectors, Dutchman's Ridge with Strong Crest . . . 116 Figure 6.23 Force Vectors, Toe Region of Dutchman's Ridge 116 Figure 6.24 Force Vectors, Dutchman's Ridge with Reduced Pore Pressures at Crest 117 Figure 6.25 Force Vectors, Embankment with Strong Toe Region 118 Figure 6.26 Force Vectors, Circular Failure in Embankment above Retaining Wall 118 Figure 6.27 Negative Stresses Caused by Wrong Slice Orientation 119 Figure 6.28 Normal Stress vs. Angle of Slice Side 120 Figure 6.29 Factor of Safety vs. Angle of Slice Side 120 Figure 6.30 Force Vectors for Irregular Failure Surface 121 Figure 6.31 Coal Mine Geometry with Narrow Slice at Toe 123 Figure 6.32 Coal Mine Geometry that Satisfies Moment Acceptability 123 Figure 6.33 Probability Distributions Showing Relationship Between POF and F 125 Figure 6.34 Determining Water Forces Acting on Slice 126 Figure 7.1 Organization Chart for Chapter 7 129 Figure 7.2 Schematic of Ore Deposit 131 Figure 7.3 Geologic Section Showing Ore Grade and Ovb. Bedrock Contact 132 Figure 7.4 Section Showing Digability Characteristics of Subsurface 132 ix TABLE OF CONTENTS LIST O F F I G U R E S Page Figure 15 Section Showing Pit Limits at 5 Year Intervals 133 Figure 7.6 Plan View of Well Locations for Base Case Pit Design 137 Figure 7.7 Section Showing Location of Wells Relative to Pit Limits 137 Figure 7.8 Section Showing Approximate Location Critical Failure Surfaces 139 Figure 7.9 Realization of Hydraulic Conductivity Field (log10 scale) 140 Figure 7.10 Boundary Value Problem - Ultimate Pit Configuration 141 Figure 7.11 Computed Hydraulic Head Distribution 141 Figure 7.12 Contour Plot of Standard Deviation in Pressure Head 142 Figure 7.13 Histogram of K,. Coefficients for Ultimate Pit Failure 143 Figure 7.14 Probability of Failure of Pit Wall - 5 year stages 143 Figure 7.15 Net Incomes per Push-Back, Conditional and Expected 145 Figure 7.16 Projected Net Income - Importance of Probability of Failure 147 Figure 7.17 Probabilities of Failure for each Slope Design 148 Figure 7.18 Expected Net Income per Push-Back, Four Different Scenarios 149 Figure 7.19 Expected Cumulative Net Income for Four Design Scenarios 150 Figure 7.20 Expected Cumulative Net Income as a Function of Slope Angle 151 Figure 7.21 Range of Values of Hydraulic Conductivity for Different Geologic Environments (after Freeze & Cherry, 1979) 154 Figure 7.22 Avg. Position of Water Table - Varying Hydraulic Conductivity 155 Figure 7.23 FOS vs. Mean Hydraulic Conductivity 155 Figure 7.24 Sensitivity of POF and Monetary Risk to Changes in Mean Hydraulic Conductivity 156 Figure 7.25 FOS Distributions for Several Levels of Standard Deviation in K 158 Figure 7.26 Sensitivity of P.O.F. and Monetary Risk to Changes in Standard Deviation of Hydraulic Conductivity 158 Figure 7.27 Contours of Standard Deviation in Pressure Head for o=0.1 159 Figure 7.28 Contours of Standard Deviation in Pressure Head for a = 1.41 159 Figure 7.29 FOS Distribution for Several Values of Range 161 Figure 7.30 Sensitivity of POF and Monetary Risk to Changes in Range of Correlation of Hydraulic Conductivity Field 162 Figure 7.31 Sensitivity of Mean FOS to Changes in Friction Angle 164 Figure 7.32 Sensitivity of P.O.F. and Risk to Changes in Mean Friction Angle 164 Figure 7.33 FOS Distribution for Several Levels of Uncertainty in Friction Angle 165 Figure 7.34 Sensitivity of POF "and Risk to Changes in Standard Deviation of <|> 166 Figure 7.35 Sensitivity of Mean FOS to Changes in Cohesion 167 Figure 7.36 Sensitivity of POF and Risk to Changes in Mean Cohesion 167 Figure 7.37 FOS Distribution for Several Levels of Uncertainty in Cohesion 168 Figure 7.38 Sensitivity of POF and Risk to Changes in Standard Deviation of Cohesion 168 Figure 7.39 Response of Standard Deviation in FOS to Changes in Uncertainty of Input Parameters 170 Figure 7.40 Response of FOS Distribution to Combined Increases in Uncertainty of K, <J>, and c . . . 171 Figure 7.41 Sensitivity of POF and Monetary Risk to Combined Changes in Standard Deviation of K, <{> and c 171 Figure 7.42 Average Water Table Positions for Test Dewatering Design Programs 174 Figure 7.43 Expected Operating Costs as a Function of Dewatering Budget 174 x TABLE OF CONTENTS LIST O F F I G U R E S Page Figure 7.44 Expected Net Income per Push-Back, Steep Design & Two Other Scenarios 176 Figure 7.45 Expected Cumulative Net Income for Steep Design and Two Other Scenarios 176 Figure 7.46 Location of Hydraulic Conductivity Measurements & Observed Values 178 Figure 7.47 Variance in Estimating K for Three Levels of Measurements 179 Figure 7.48 Contours of Standard Deviation in Estimated Pressure Head for Three Levels of Measurement 180 Figure 7.49 FOS Distributions for Three Levels of Measurement Networks 181 Figure 7.50 Concept of Regret 182 Figure 7.51 Monetary Worth of Three Site Investigation Strategies 183 Figure 7.52 Estimate of Hydraulic Conductivity Field Based on Low Density Testing Program 186 Figure 7.53 Hydraulic Head Distributions for Dewatering Designs A to D 187 Figure 8.1 Map of British Columbia Showing Location of Highland Valley Copper 194 Figure 8.2 Topographic Map of Highland Valley and Surroundings 195 Figure 8.3 Geologic Cross Section of Valley Pit, Looking Northwest 196 Figure 8.4 Plan Showing Outline of 1988 Pit, Existing Wells, and Design Sector R3 196 Figure 8.5 Geologic Section Showing Location of High Grade Ore Zones 198 Figure 8.6 Section Showing Digability and Hardness Characteristics on Section R3 198 Figure 8.7 Contour Plan Showing Elevation of Overburden/Bedrock Contact 199 Figure 8.8 Overburden Geology Interpretation - Design Sector R3 200 Figure 8.9 Average Monthly Precipitation - Ashcroft Weather Station 203 Figure 8.10 Piezometric Contours in Main Aquifer (1988) 205 Figure 8.11 Log(K) vs. Geology Coefficient C g 209 Figure 8.12 Semi-Variograms for Piezometers DP15-DP20 210 Figure 8.13 Semi-Variograms for Boreholes Without Proper Geologic Logging 210 Figure 8.14 Hydraulic Conductivity Measurements and Kriged Map of K Field 211 Figure 8.15 Semi-Variogram in Horizontal Direction 212 Figure 8.16 Stochastic Model of Hydraulic Conductivity Field 213 Figure 8.17 Comparison of Predicted and Observed Piezometric Levels - 1987 215 Figure 8.18 Development of Pit in 5 Year Increments - Section R3 216 Figure 8.19 Boundary Value Problem For Section R-3 219 Figure 8.20 Typical Realization of Hydraulic Conductivity 220 Figure 8.21 Contours of Variance of Estimation Errors 220 Figure 8.22 Contour Plan of Mean Hydraulic Head on Section R-3 for 1987 222 Figure 8.23 Contour Plan of the Standard Deviation in Predicted Pressure Head - 1987 222 Figure 8.24 Contour Plan of Mean Hydraulic Head on Section R-3 for Ultimate Pit 223 Figure 8.25 Mohr-Coulomb Strength Envelopes for Lower Sequence 226 Figure 8.26 Identification of Critical Failure Surface 228 Figure 8.27 Histograms of FOS Distributions for 1987, 92, 97 and Ultimate Push-backs 230 Figure 8.28 Cumulative Net Incomes by Push-Back, Conditional and Expected 232 Figure 8.29 Contour Plan of Mean Hydraulic Head on Section R-3 with Horizontal Toe Drains . . . . 234 TABLE OF CONTENTS LIST O F T A B L E S Page Table 2.1 Summary Statistics for Groundwater Control Programs that Utilize Dewatering Wells. . . . 12 Table 2.2 Approximate Costs of Dewatering with Most Common Methods 14 Table 4.1 Kriging Weights for Sampling Array 42 Table 4.2 Results of Verification Calculations at Measurement Points 51 Table 43 Coefficients for Semi-Variogram Models 54 Table 4.4 Results of Verification Analysis 56 Table 4.5 Performance Parameters for Simulation Methods 67 Table 4.6 Desired Statistics For Simulation 70 Table 6.1 Unknown Forces and Available Equations 98 Table 6.2 Methods of Slices and Associated Assumptions 99 Table 6.3 System of Linear Equations for Sarma Method 109 Table 7.1 Input Parameter List for RCB Analysis 134 Table 7.2 Base Case Dewatering Costs for Sensitivity Study 138 Table 7.3 List of Unit Costs for Well Diameter Independent Items 138 Table 7.4 List of Unit Costs for Well Diameter Dependent Items 139 Table 7.5 Projected Tonnages for Each Push-Back, With & Without Failure 144 Table 7.6 Projected Conditional Costs and Revenue, with & without Failure 144 Table 7.7 Monetary Values of Expected Net Income per Push-Back 149 Table 7.8 Monetary Values of Cumulative Expected Net Income - Four Designs 150 Table 7.9 Typical Properties for Soil and Rock (Modified after Table 1, Hoek & Bray, 1981) 163 Table 7.10 Standard Deviations and Confidence Intervals in Input Parameters 169 Table 7.11 Design Parameters and Costs for Several Dewatering Programs 173 Table 7.12 Slope Angles, Dewatering Costs and Net Incomes for Design Alternatives Analyzed . . . . 175 Table 7.13 Reductions in Uncertainty Due to Measurements 178 Table 7.14 Cost of Data Collection and Posterior Worth of Data 183 Table 7.15 Costs and Factors of Safety for Dewatering Designs A to D 186 Table 8.1 Overburden Stratigraphy and Material Composition 201 Table 8.2 Run-Off Coefficients (After Ontario Dept. of Highways Report 1979-08-10 203 Table 8.3 Hydraulic Conductivity Estimates of Aquifer Horizons 206 Table 8.4 Hydraulic Conductivity Estimates of Lower Silt and Clay Sequence based on Rate of Consolidation 207 Table 8.5 Hydraulic Conductivity Estimates Based on Sieve Analyses 208 Table 8.6 Summary of Hydraulic Conductivity Information for Each Geologic Horizon 212 Table 8.7 Results of Triaxial Shear Tests by Golder Associates 225 Table 8.8 Shear Strength Parameters Adopted in This Study 227 Table 8.9 Summary of Slope Stability Analysis Results 229 Table 8.10 Projected Tonnages to be Mined During Each Push-Back 231 Table 8.11 Projected Conditional Costs and Revenue per Push-Back 232 Table 8.12 Projected Expected Costs, Revenue and Net Income For Current Design 232 Table 8.13 Summary of Results for R-C-B Analysis of Design Options 235 xii TABLE OF CONTENTS A C K N O W L E D G E M E N T S Foremost, I would like to acknowledge the guidance and support provided by Allan Freeze in his role as thesis supervisor. His enthusiasm for widespread use of risk based engineering design in hydrogeology provided inspiration throughout all phases of this research effort. Also, I would like to extend my appreciation to Chuck Brawner, Bill Caselton, Drummond Cavers and Les Smith for serving on the advisory committee. In particular, I wish to acknowledge the encouragement and advice in the art of geotechnique provided by Drummond Cavers, who took valuable time out from his consulting practice to review, discuss and constructively criticize at each opportunity. Also, Chuck Brawner deserves a special thank you for encouraging me to apply the research in practice, and especially for opening the doors to Highland Valley Copper's direct involvement in the case history study. The most rewarding aspect of my research has been the application of the framework to the design of dewatering faculties at Highland Valley Copper. To this end, I would like to thank the management at Highland Valley Copper, and especially W.K. Munro and F. Amon of the Mine Engineering Department, for providing the opportunity to become directly involved in all aspects of their dewatering operations. Financial support provided by the National Research Council of Canada Scholarships and The University Graduate Fellowships during the course of my studies has been very appreciated. Many thanks are extended to the Department of Hydrogeology and Water Resources at the University of Arizona, and in particular Schlomo Neuman and Tom Maddock, for their hospitalities during the author's six month visit. Among the graduate students of UBC's Groundwater Group, I wish to acknowledge Bruce James, Lorrie Cahn, and Craig Forster for their interest, insight and warm generosity. Special gratitude is due to Dave Osmond and John Gartner of Gartner Lee Limited for providing financial support, freedom and encouragement to complete writing of the dissertation during the past year. In closing, I would like to thank my parents, Aljeh Chambers and numerous friends for their continued interest in my progress and their encouragement to see the job completed. It is a joy to be able to say that finally, the thesis is done. xiii CHAPTER 1 INTRODUCTION 1.1 PROBLEM DESCRIPTION 1.2 RESEARCH OBJECTIVES. 1.3 SCOPE 1.4 THESIS OVERVIEW 3 2 2 1 This thesis introduces a risk-cost-benefit framework for the design of groundwater control systems at open pit mines and demonstrates how the framework has been used successfully to identify the optimum slope configuration and dewatering design at Highland Valley Copper, one of the largest copper mines in the world. Open pit mining is a very costly endeavour. Excavation and haulage costs alone typically range from 4 to 40 million dollars per year, depending on the size of the mining operation. Steepening of the ultimate pit wall will reduce mining costs provided that stability of the pit wall can be maintained. The cost reductions are realized because less waste material has to be moved to expose and extract the ore body. A steep pit wall is also desirable during the first few pit expansions because it will lead to increased cash flow during the critical early years of mine production. For these reasons mine operators continually strive to design the pit walls as steeply as possible while maintaining stability. However, if a pit wall is over-steepened and one or more large failures develop, then the mine operator will experience severe economic consequences, especially if the failure impedes or curtails normal production. Stability of the pit wall is controlled by geologic conditions, groundwater conditions, blasting, and slope design. Stability can often be improved by a properly engineered groundwater control program, especially if high groundwater pore pressures are anticipated. In many instances, reduction of pore pressures will improve stability sufficiently to permit steepening of the pit walls by several degrees without increasing the risk of slope failure. Unfortunately, groundwater control is also an expensive undertaking. Prior to committing to a dewatering program, mine management must ascertain whether the benefits of improved stability and a steeper slope design will be sufficient to justify the extra costs of groundwater control. Even when it appears obvious that some form of groundwater control program will be beneficial, identifying the optimum dewatering strategy is a formidable task, especially when detailed knowledge of actual subsurface conditions is not available. A satisfactory solution to the problem of selecting the most appropriate slope angle and optimum dewatering strategy under conditions of geologic and hydrologic uncertainty will benefit the mining community because a wrong design decision can result in operating cost increases of tens of millions of dollars over the life of a large open pit mine. Providing a satisfactory solution requires an integrated, multi-disciplined analysis. The analysis must involve: • An assessment of subsurface geologic conditions, the objective being to accurately estimate both geotechnical and hydrogeologic parameters in the pit wall. The effect of uncertainty in parameter estimates must be recognized and considered in the assessment. • An analysis of groundwater flow, the objectives being to predict pore pressures on the failure surface and to determine the likely impact of the dewatering program being considered. • An evaluation of slope stability. The stability evaluation must utilize the pore pressure estimates obtained above, as well as account for uncertainty in strength parameters. 1.1 PROBLEM DESCRIPTION Chapter 1 1 INTRODUCTION • An economic evaluation of the slope design and dewatering strategy. The evaluation must consider revenues, production costs, costs of groundwater control and costs of data collection, as well as the monetary consequences of a pit wall failure, and the likelihood of such a failure developing. 1.2 RESEARCH OBJECTIVES The primary objective of this research effort is to develop a comprehensive design tool, based on the risk-cost-benefit framework, that can address the tasks listed above in an integrated, technically sound, and economically defensible manner. A number of secondary goals were also identified at the start of the research effort. These objectives are listed below in order of importance: • To develop a practical framework that could be applied widely in industry to any number of site specific design problems. • To incorporate the latest engineering approaches to the analysis of each of the four design problems encountered in this framework: geostatistical interpretation, groundwater modelling, slope stability analysis, and decision analysis. • To demonstrate that the framework can be applied successfully at an operating mine, and used to make actual dewatering design decisions that will increase profitability of the mining operation. • To develop the framework as a flexible, user-friendly, graphically intensive software package that can be executed on low cost personal computers, yet will have sufficient power to tackle realistic design problems. 1.3 SCOPE The research objectives were achieved in sequential fashion through literature review, algorithm development, computer programming, sensitivity analysis, geotechnical and hydrogeologic site investigation, and engineering analysis and design. The detailed literature review, conducted in 1985-86, focused on identifying methods of estimating subsurface conditions from limited field data, of modelling groundwater flow, of evaluating slope stability, and of assessing the monetary value of various design alternatives. The goal of the literature review was to identify techniques for each of these tasks that could be successfully integrated into a unified framework. Upon completion, it was concluded that the risk-cost-benefit framework could be successfully developed if the following analytical methods were adopted: 1) geostatistical methods for the task of parameter estimation, 2) a two dimensional saturated/unsaturated finite element technique for simulation of groundwater flow, 3) the Sarma two dimensional limit equilibrium approach for slope stability, and 4) an expected value decision analysis technique for the economic impact assessment. Development of the four principal computer modules (1. geostatistics, 2. groundwater, 3. slope stability, and 4. economic analysis) was completed in 1986-1988. All of the computer modules were developed by the author. The modules were based on solution strategies documented in the literature, and coupled with advanced, graphic intensive data entry and output display modules, also developed by the author. When developing the software modules, the intent was to prepare a user friendly software package, one that would be capable of analyzing complex design problems quickly and efficiently. Fortran-77, and Quick-Basic were selected for the programming effort so that the resulting software would be portable, and fully compatible with most IBM compatible personal computers. Chapter 1 2 INTRODUCTION Upon completion of the software development, the risk-cost-benefit framework was utilized to conduct a sensitivity study that evaluated the importance of groundwater control in open pit mines. As well, the sensitivity study examined the effect of changes to the slope angle and the dewatering budget, and the potential impact of the number and accuracy of hydraulic conductivity and shear strength measurements on mine economics. A major part of this research effort involved the application of the risk-cost-benefit framework to evaluate the groundwater control system at Highland Valley Copper. The design effort is documented as a case history. It involved two distinct components. First, during the spring and summer months of 1987 and 1988 the author was on site at Highland Valley Copper, where he worked closely with mine engineering personnel and drilling contractors to establish the large data base required for the analysis. The data acquisition program entailed detailed geotechnical logging, piezometer and well monitoring, pump testing, and shear strength testing. All new data, as well as previously collected information, was entered into a computer data base and analyzed to establish the best possible values for input parameters required for the risk-cost-benefit analysis. Second, the new framework was used to conduct a detailed assessment of the current overburden dewatering strategy at Highland Valley Copper. Several alternative dewatering strategies were also evaluated with the risk-cost-benefit framework in an attempt to identify the most cost effective dewatering design for Highland Valley Copper. 1.4 THESIS OVERVIEW This thesis is comprised of nine chapters and five appendices. This chapter and Chapter 2 provide an introduction to the thesis research and document why the research is beneficial to Canada's mining industry. Chapters 3 through 6 focus on the four technical components of the framework, including the decision framework in Chapter 3, the geostatistics methodology in Chapter 4, the groundwater flow model in Chapter 5, and the slope stability analysis in Chapter 6. Chapters 7 & 8 describe how the new framework has been used to improve pit dewatering strategies in general, and at Highland Valley Copper in particular. Chapter 9 serves as a summary of the key findings presented in this thesis, placing emphasis on research contributions to the mining, hydrogeological, and geotechnical communities. Appendices A through C provide a more detailed discussion of the theory underlying the analytical techniques utilized in this thesis, Appendix D presents the results of the numerous sensitivity studies, and Appendix E provides background information about the Highland Valley case history. The paragraphs that follow provide a brief overview of the material discussed in each chapter. Chapter 2 describes the numerous ways groundwater can impact on the open pit mining operation. The chapter begins with a brief description of modern open pit mining methods and a discussion of how the mining operation can be affected by pit wall instability. The most common methods of improving slope stability are then reviewed, especially the various techniques of groundwater control. The chapter concludes with a brief cost comparison that illustrates that groundwater control can be a very cost effective method of improving slope stability in many circumstances. Chapter 3 presents the risk-cost-benefit framework. First, the chapter introduces decision analysis and describes how this tool can be used to evaluate the expected worth of a number of different dewatering and slope design strategies. The discussion then focuses on the objective function, explaining why expected net income provides a suitable measure of the monetary value of each design strategy. Each of the four terms that comprise the objective function are then examined in detail; the terms include benefits, production costs, dewatering costs, and risks. The assessment of benefits and costs is relatively straightforward; however, the calculation of monetary risk is much more complex because it is dependent on the probability of failure. Chapter 1 3 INTRODUCTION Probability of failure is a measure of slope stability. It is a function of the slope design, of the effectiveness of dewatering strategy, and of the hydrologic and the shear strength conditions in the pit wall. Both shear strength and hydrologic conditions are always uncertain to some degree. The remainder of Chapter 3 describes the new methodology that has been developed as part of this thesis to evaluate the probability of slope failure. The methodology is different from traditional design approaches in that it explicitly accounts for uncertainty in both hydrogeologic and shear strength parameters. The probability of failure assessment involves a geostatistical analysis of subsurface conditions, followed by Monte-Carlo analyses of groundwater pore pressures and slope stability to obtain the probability of failure, and finally, an economic evaluation to determine the objective function. Chapter 3 shows how these components come together to form the risk-cost-benefit framework; Chapters 4 through 6 provide more detailed descriptions of the various techniques that are necessary in order to successfully apply the framework to solve actual field problems. Chapter 4 introduces the geostatistical tools that are utilized in this framework to generate the large number of realizations of the hydraulic conductivity field required for Monte Carlo simulation. The chapter begins with a review of the basic geostatistical concepts and terminology. A number of semi-variogram models are then presented, as are several efficient techniques for fitting the semi-variogram model to the experimental semi-variogram. The discussion then focuses on Kriging, a parameter estimation method that results in the Best Linear Unbiased Estimate (BLUE). Kriging is used in this framework to make an estimate of hydraulic conductivities everywhere in the flow domain based on a limited number of measurements, and to determine the estimation error associated with each hydraulic conductivity prediction. Finally, Chapter 4 examines a number of simulation techniques that can be used to generate the large number of hydraulic conductivity realizations required for the Monte-Carlo analysis of groundwater flow. Chapter 5 describes the computer modelling approach that is used to predict groundwater pore pressures in the pit wall. The discussion begins with a description of the unsaturated groundwater flow system that exists in open pit walls, and how the system can be described in terms of a boundary value problem that can be solved by numerical modelling. The finite element method is then introduced; the presentation is very brief and focuses primarily on the iterative solution strategy that is used to account for flow through the unsaturated zone and to predict the location of the water table. The theme then switches to more pragmatic issues: the important effect of stratigraphy and how it can affect the pore pressure field, the role of boundary conditions, and the importance of hydraulic conductivity measurements. Probability of failure is the most important and complex factor in the assessment of monetary risk due to pit wall failure. Chapter 6 describes how limit equilibrium stability analysis and Monte-Carlo simulation techniques are used to estimate this essential parameter. Beginning with a review of available limit equilibrium methods, the discussion quickly focuses on Sanaa's method of slices, the technique utilized in this framework. The mechanics of Sarma's method are then explored, with special attention directed to the factor of safety convergence problem experienced by previous users. The cause of this iteration problem is identified and a new iteration strategy is introduced that overcomes the convergence problem in most circumstances. The effect of groundwater on slope stability is then explored, the objective being to illustrate why dewatering can lead to dramatic improvements in stability of the pit wall. Finally, the discussion turns to probabilistic methods of slope stability analysis that, unlike the popular factor-of-safety approach, explicitly consider the effect of uncertainty and variability of geologic and hydrologic parameters in the stability assessment of pit walls. Because the probability of failure can also be incorporated directly into the economic evaluation model, it is the preferred stability criterion in this thesis. Chapter 1 4 INTRODUCTION Chapter 7 presents the results of a detailed sensitivity study that utilized the new framework to explore how each of the many input parameters and decision variables (e.g. hydrologic data, shear strength parameters, pit angles, dewatering design, etc.) impact on the overall economics of the mining operation. Key issues that are investigated include: • How to assess the economic impact of changes to the pit-wall angle, in particular, how the optimum slope angle can be identified. • How to predict the influence of hydrogeologic conditions in the subsurface on the effectiveness of a dewatering program, in particular, the effect of the mean hydraulic conductivity, the variance in hydraulic conductivity, and the correlation range. • How to determine the value of shear strength conditions, especially whether shear strength measurements are more or less important than hydraulic conductivity data. • How to identify the optimum level of expenditure for the groundwater control effort. • How to identify the monetary worth of hydraulic conductivity measurements, especially when to stop additional site investigation efforts. Chapter 8 documents how the new framework was used to evaluate the current overburden dewatering strategy at Highland Valley Copper and to identify dewatering design modifications that could increase profitability of the mining operation. The case history includes a description of the input parameters that are required to conduct the analysis, including geologic conditions, hydrologic conditions, the current pit design, and the current dewatering strategy. Emphasis is placed on describing how the various types of geologic and hydrologic information were integrated into a unified geostatistical description of the hydraulic conductivity field. After evaluating the current slope design and dewatering strategy, the framework is used to evaluate several alternative dewatering designs in an attempt to identify a more efficient groundwater control plan. However, the case history presented in Chapter 8 is not a dewatering design study. More field work (especially direct shear testing), transient flow modelling and stability analyses that consider the full spectrum of possible failure modes will be required before results of this RCB framework can be used with confidence to justify modifications to the current slope design and dewatering plan. Chapter 9 provides a global summary of the thesis. The contents and key conclusions of each chapter are briefly reviewed. Also, the principal assumptions that have been adopted in the framework are summarized. Chapter 1 5 INTRODUCTION CHAPTER 2 GROUNDWATER CONTROL IN OPEN PIT MINES 2.1 INTRODUCTION 6 2.2 CANADIAN MINING INDUSTRY 6 2.3 OPEN PIT MINING OPERATIONS 7 2.4 SLOPE STABILITY CONSIDERATIONS 8 2.5 GROUNDWATER IMPACTS ON MINE PRODUCTION 10 26 DEWATERING ALTERNATIVES 11 27 SUMMARY 14 11 INTRODUCTION This chapter serves as an introduction to the subject of groundwater control in open pit mines. Because optimizing the design of modern open pit mines has become a complex undertaking that requires a coordinated effort amongst specialists trained in geology, geostatistics, geotechnics, mine operations, mineral processing, economics and marketing, the discussion is not limited to the narrow subject of groundwater control from a hydrogeologic perspective, rather, it examines the impacts of groundwater control on all aspects of mine operations. The chapter begins with a brief introduction to open pit mining. Subjects that are discussed include typical pit sizes, production rates, mine design concepts and components of the mining sequence. The issue of slope stability is examined next, with emphasis on the motivation for steep slopes, the consequences of failure, and the accepted deterministic approach to slope design. The subject of groundwater is then introduced, first identifying the potential impacts of groundwater on mine operations and then presenting a review of the various control methods. The chapter concludes with a few words on the economics and suitability of various dewatering options. 22 CANADIAN MINING INDUSTRY Over the past 30 years, Canada has consistently remained in an enviable position as one of the world's largest and lowest cost mineral producers. Mineral production in Canada has been increasing steadily, today the Canadian mining industry generates over $15 billion in revenue per year from mineral exports that include gold, coal, copper, zinc, iron ore, and a host of other mineral commodities. In 1987, Canada ranked first as a world producer of uranium and zinc, second for potash, nickeL sulphur, asbestos and gypsum, third for titanium, cadmium, aluminum, platinum and gold, and fourth for copper, molybdenum, lead and cobalt. A large percentage of these mineral commodities are produced at open pit mines. In the late 1980's demand for many metal commodities returned to pre-recession levels, but prices remained depressed as a result of adequate supplies and strong international competition. The Mining Association of Canada predicts that the mining industry can adjust to the new, more competitive international market, but it will require careful management, cost control, and continued productivity improvements. The association believes that in the long term, increased productivity and cost control can only be achieved by technological innovation, effective marketing, and the discovery of new, high quality mineral deposits. Based on results of risk-cost-benefit studies presented later in the sensitivity and case history chapters of this thesis, it appears that a high technology approach to dewatering design may provide the industry with one such alternative for reducing operating costs and gaining a competitive advantage in international markets. Chapter 2 6 OPEN PIT MINES Figure 2.1 Mine Planning and the Daily Production Cycle 2.3 OPEN PIT MINING OPERATIONS Open pit mining methods are used to extract ore from economic deposits situated close to surface and for large volume, low grade ore bodies, e.g. the large copper porphyry deposits found in British Columbia. The size of pits varies from small glory holes several hundred metres in diameter and less than 100 m in depth to pits over 2 km in diameter and 1 km in depth. The mill size is usually selected so that the mine will operate over a period 10 to 20 years. Production rates of 10,000 to 100,000 tons per day are typically required to provide adequate feed to the mill. Mining operations can be broken down into three broad categories: 1) pre-production planning and infrastructure development, 2) production cycle and 3) decommissioning and reclamation. The following paragraphs provide a brief introduction to the activities conducted during each phase of operations. The normal sequence of these activities are portrayed in Figure 2.1. 23.1 Pre-production Planning Pre-production planning begins shortly after exploration activities identify a mineral deposit of sufficient size and grade to be potentially economic. The first stage of this planning process normally involves drilling on tight spacings to prove ore reserves, detailed geologic interpretation of the drill results, and geostatistical modelling to predict grades. If the results continue to be encouraging then the planning process advances to include concentrate recovery studies, geotechnical investigations, preliminary pit design, equipment selection, environmental impact evaluations and economic studies. If these feasibility studies continue to show that the mine will be physically and economically viable then the planning process advances once more to the development stage. Markets are established, financing is arranged and construction begins on mill facilities, maintenance shops, tailing disposal facilities, and transportation corridors. Chapter 2 7 OPEN PIT MINES 232 Production Cycle Once the mine and mill are brought on stream, a routine daily production cycle is established. The cycle begins with short term planning to identify which benches will be worked in order to provide a balanced flow of ore and waste from the pit and to produce a mill feed with relatively constant grade and hardness properties. Blast patterns are laid out, drilled, loaded and detonated. Drill cuttings are sampled and analyzed for mineral content to differentiate between zones that should be mined as ore and those to be shipped as waste. The blasted rock is excavated with large electric shovels and loaded into haulage trucks for transport out of the pit. Ore is sent to the primary crusher where it is reduced to a coarse gravel size for feed into the mill. Waste is hauled to adjacent waste dumps. In the mill, the ore is processed through several grinding and floatation circuits to separate the economic mineralization from the tailings. The concentrate is then dried and shipped to smelters while the tailings are pumped to the tailings pond for disposal. 233 Decommissioning and Reclamation Once all economic mineralization is removed from the pit, the mine is closed. In the past, the decommissioning process involved nothing more than salvaging any usable equipment and closing all doors. Abandoned mine sites full of interesting relics are frequently encountered in the back country. Recent amendments to the Mines Act require operators to thoroughly reclaim disturbed ground. All facilities must be removed, waste dumps and tailings facilities must be contoured and reseeded, and in the case of acid mine drainage (AMD) generating mines, adequate abatement measures must be incorporated in the reclamation plan to ensure that the environment will not be affected by A M D in perpetuity. 2.4 SLOPE STABILITY CONSIDERATIONS There is a strong economic incentive for designing pit walls as steeply as possible, especially during the final push-back. By adopting a steep slope angle, the mine will be able to leave more waste undisturbed while fully exploiting the ore body. On account of the reduced stripping requirements, the mine will benefit from reduced operating costs, less waste rock to dispose of on dumps, and more rapid exploitation of the ore body. Figure 2.2 provides a conceptual illustration of the relationship between slope angle, increased stripping and increased production costs. The left half of the figure shows a cross-section through a cylindrical ore body and several possible slope angles. The right half of the figure shows total waste tonnage and stripping cost as function of the slope angle. By steepening the pit from 30° to 35° for example, the mine would stand to reduce total stripping costs from $2.23 billion to $1.75 billion, based on a typical mining cost of $1.25 per tonne. Steepening of the pit wall will usually result in decreased slope stability, and in most cases, an increase in the probability of failure. If a pit wall is over-steepened and one or more large failures develop, then the mine will experience severe economic consequences, especially if the failure impedes or curtails normal production. The consequences of slope failure may include: • Clean up costs of removing failed material from pit. Costs may be somewhat higher than excavating material in situ if the rubble breaks into large blocks that require additional blasting. • Lost production, especially if the slope failure affects a primary haul road or conveyor line, or if the failure buries an active mining area (e.g. bottom of pit) • Lost ore, if the failure block contains ore that cannot be separated from waste within the failure rubble or the costs of clean-up are sufficiently high to render further mining uneconomic. • Reduced cash-flow, especially when a primary haul road is affected. • Damaged equipment and services, e.g. shovels, water pipe lines, conveyor systems. • Physical risk to workers and portable equipment in pit. Chapter 2 8 OPEN PIT MINES Figure 2.2 Relationship Between Slope Angle and Stripping Costs Only costs of clean-up, lost production and lost ore are considered in this risk-cost-benefit formulation because these items generally represent the most costly consequences of slope failure. It is also possible to assign a fair monetary value to these consequences directly, based on ore grade and slope geometry data. Cash-flow and equipment damage considerations have not been incorporated in the formulation as yet, because these consequences are believed to be economically less significant than consequences 1 to 3 and because estimation of these parameters would require a detailed economic study that is beyond the scope of this thesis. The potential hazard to mine workers as a result of slope failure is also not addressed explicitly in this study because it is assumed that for the large volume slides considered, modern slope monitoring methods will provide sufficient advance warning of impending slope failure to ensure that all men and equipment are removed from the pit well in advance of any life threatening slope movement. In the process of selecting the most suitable slope geometry, mine engineering staff usually attempt to identify the steepest slope angle that will have an acceptable factor of safety against failure. The traditional deterministic approach to this problem has been to design large pit slopes to a factor of safety of 1.1 to 1.5. The higher factor of safety is adopted when there is uncertainty regarding shear strength parameters or when it is especially important to maintain stability in a particular design sector (e.g. haul road or in-pit crusher located below wall). In the risk-based design approach proposed in this thesis economic considerations dictate the optimum probability of failure. The objective is to balance the benefits of a steeper slope against the monetary risks of failure. During the limited number of sensitivity and case history analyses conducted as part of this thesis research it was found that the optimum slope design was always associated with a low probability of failure, typically in the range of 0 to 5%. In the same studies, it was also observed that the mean factor of safety for the optimum slope design usually exceeded a factor of safety of 1.1, the minimum deterministic stability criterion. Additional research and more case history studies of the risk-cost-benefit approach to slope design are required before suitable design guidelines can be drawn-up in terms of probability of failure and monetary risk. In the interim, it is recommended that the risk-cost-benefit framework proposed in this thesis be applied in parallel with the traditional factor of safety approach, and that all designs exceed a minimum factor of safety criterion of 1.1. Chapter 2 9 OPEN PIT MINES 25 GROUNDWATER IMPACTS ON MINE PRODUCTION Groundwater can impact on open pit mine operations in many ways. The most common impacts include: Each of these impacts are discussed briefly in the following paragraphs. Slope Stability: The effect of groundwater pore pressures on slope stability is explained by the principle of effective stress. Given that the total stress remains constant, any increase in pore pressure will result in an equal decrease in effective stress and a loss in the frictional component of shear strength. The underlying theory is reviewed more completely in Section 6.6 of this thesis, additional references are also cited in that section. Recently, many open pit mines have introduced groundwater control programs to improve slope stability, and especially, to stabilize problem areas that have experienced some deformation. Slope Erosion: Groundwater seepage can lead to gradual berm deterioration in areas where the pit wall is excavated in overburden or highly weathered bedrock. The deterioration can be caused by the following mechanisms. Rapid excavation or freezing at the seepage face can induce pore pressures that exceed the confining stress and result in berm scale spalling failures. Accumulation of water within the loose sluff found at the toe of overburden berms can lead to liquefaction failures. Continued seepage can wash-out fine grained soils from the berm face, resulting in progressive over-steepening and eventual collapse. Trafficability: In overburden and highly weathered bedrock, uncontrolled seepage can turn operating surfaces into a quagmire that bogs down equipment and results in an increased frequency of mechanical breakdowns, especially the front drives on shovel undercarriages. In problem areas, mine operators place a 1 to 2 m thick layer of waste rock to establish a bearing pad capable of sustaining vehicle traffic. This double handling of waste rock can increase mining costs by 10 to 20%. Blasting: On account of it's low cost, ANFO (Ammonium Nitrate Fuel Oil) is the explosive of choice in open pit mining applications. However, ANFO is unstable in water, so more expensive emulsion based slurry explosives must be utilized in wet blast holes. Because of the difference in the price of the two explosives ($24.90/m loaded with ANFO vs. $59.30/m loaded with slurry), blasting costs typically increase by several million dollars per year when wet conditions requiring slurry explosives are consistently encountered. Production Water: Immense quantities of production water are required by modern high tonnage mills. For example, the mill complex at Highland Valley Copper requires 291 million litres of water per day. Although much or the production water is continuously recycled from the tailings pond, 10 to 20% of the daily demand must be replenished on a continuous basis. Although many mines in British Columbia can draw on local surface water supplies to provide all of their water requirements at nominal cost, other mines must pump water from far away sources at substantial cost. In the later situation, groundwater generated by the dewatering program provides a cost effective substitute to pumping from a remote source. Environmental Impacts: Without proper regard for environmental issues, operating mines can have a detrimental impact on surface and groundwater hydrology, affecting both water quality and quantity. Due to growing public awareness in environmental issues, legislation is being introduced to ensure that all future mining operations will be conducted in a responsible manner. As a result of these developments, the additional costs and risks of environmental protection will have to be incorporated in future risk-cost-benefit evaluations of the groundwater control plans. Slope Stability Slope Erosion Trafficability Blasting Production Water Environmental Impacts Chapter 2 10 OPEN PIT MINES The most severe environmental problem that affects a number of producing and abandoned mines in British Columbia is Acid Mine Drainage (AMD). The problem develops when gangue sulphide minerals within the waste rock piles and tailings ponds oxidize to generate strongly acidic groundwater. Toxic metals, including copper, lead and zinc, are easily mobilized under these conditions. Environmental damage occurs when the contaminant loadings enter the biosphere, most commonly as groundwater discharge to a streams or lakes, where the toxins can gradually poison aquatic organisms for several kilometres downstream. Cyanide, a toxic chemical used in the recovery of gold, can lead to similar environmental impacts. Open pit mines can also have negative impact on surface and groundwater supplies. The most common of these impacts, especially in the coal fields of eastern United States and Europe, is aquifer depressurization. In this situation, seepage into the pit, or pumping from high capacity dewatering wells will induce a large drop in the piezometric surface that may leave surrounding domestic water supply wells dry (Bair, 1981). Surface settlement may also be experienced as a result of the depressurization. Utilization of a large portion of surface and groundwater resources within a watershed can also impact downstream users. 26 DEWATERING ALTERNATIVES A number of groundwater control methods have evolved in the mining industry to improve stability and operating conditions in the pit. This section reviews the most widely used drainage methods, including horizontal drains, wells, and drainage galleries. The review of each option includes a discussion on the advantages and disadvantages of the method, on recent developments in the technology, and a rough estimate of installation and operating costs for each system. Horizontal Drains: This dewatering approach provides the most effective and economical technique for dewatering both bedrock and low permeability overburden slopes. The dewatering program usually involves one or more rows of horizontal drains along berms in potentially unstable areas. If movement is not occurring, individual drains are normally drilled a short distance beyond the anticipated slip surface to facilitate depressurization on both sides of the discontinuity. If deformation has occurred then it may be possible to reduce damage to the drains by completing them before they penetrate the slip surface, thus avoiding potential drilling difficulties and subsequent shearing of drains. The drains are usually inclined slightly upward (2 to 5°) to facilitate flow once water enters the drain. Horizontal spacings of 3 to 10 m are utilized, a fan like drilling pattern is often adopted to reduce the number of drill moves. Installation costs of $50 to $150 per m are typical, depending on ground conditions and site location. Water collection systems can be incorporated in the design, especially for drains situated in the upper 2/3 of the slope. This precaution prevents groundwater discharges from seeping back into the pit wall and increasing water pressures down-slope. Temporarily placing the horizontal drains under vacuum to increase gradients and flow rates (i.e. turning drains into "horizontal wells") is a new' development that has resulted in significantly improved drain performance at a number of mine sites (Brawner, 1987). The technique is useful for increasing the rate of dewatering on active slides during the critical first few weeks of stabilization efforts. Due to accessibility and maintenance difficulties, the vacuum technique is not practical for long term groundwater control. The main advantages of horizontal drains are: • Relatively low cost. • No maintenance (except collection systems) • Depressurization occurs along the failure surface, where it can provide the greatest benefit. Chapter 2 11 OPEN PIT MINES The disadvantages are: • Water collection systems are susceptible to damage from ravelling rock. Continued access to the drains for repair purposes may pose a serious hazard. • Drains may be sheared off as result of continued deformation. • Presence of drains and collection systems in the pit may impede normal operations (e.g. scaling). Steel pipe installed during earlier pit expansions will result in production problems during subsequent push-backs. Horizontal drains have been utilized widely in the mining industry, successful programs have been completed at Highland Valley Copper, Afton Mines and Gibraltar Mines in British Columbia, Syncrude in Alberta and Canadian Johns-Mannville in Quebec (Brawner, 1987). At present, most horizontal drains are installed as remedial stabilization measures after slope deformation is recognized. However, the findings of the risk-cost-benefit studies presented later in this thesis indicate that preventative groundwater control is also cost effective in many circumstances. Dewatering Wells: Historically, dewatering wells have been the most widely applied method of groundwater control at open pit mines. Typical applications of dewatering wells include slope stabilization at Highland Valley Copper (Sperling, 1988), control of pit flooding and freezing problems at Pine Point Mines (Calver, 1969) and prevention of pit floor heave at Syncrude and Suncor (Fair, 1987, PurcelL 1987). Conventional water well rigs are used to install most dewatering wells. Based on a review of the literature, there appears to be no preferred drilling technology, dewatering projects have been successfully completed with air rotary, mud rotary and cable tool equipment. Wells in competent bedrock usually remain uncased, except when poor ground is encountered. Although drilling in overburden usually progresses faster than drilling in bedrock, well casing and screens are generally required to prevent caving of the hole. As a result of the extra development time and hardware required, the costs of developing wells in overburden are approximately the same as in bedrock. Drilling and development costs of $1000 to $2000 per m are typical for large diameter dewatering wells, while annual operating, maintenance and monitoring costs range from $10,000 to $20,000 per well. Table 2.1 provides a summary of dewatering statistics for three Canadian open pit mines that utilize wells for groundwater control. Table ZI Summary Statistics for Groundwater Control Programs that Utilize Dewatering Wells. Mine Site Year No. Wells Avg. Depth (m) Dev. Cost ($ million) Cost per m Pine Point 1968 21 50 2.1 $2,000 Gibraltar 1976 13 83 2.4 $2,224 Highland Valley 1988 21 100 2.6 $1,238 All costs converted to 1988 dollars, assuming an annual inflation rate of 5%. Because in-pit wells interfere with normal operations, mine management prefers to situate dewatering wells beyond the ultimate pit perimeter. However, in large open pits where the horizontal distance between toe and crest frequently exceeds 500 m, perimeter wells will have little impact on water pressures in the lower reaches of a deep seated failure, and it is in this area that depressurization provides the greatest benefits. From a hydrologic perspective, wells located in the pit or horizontal drains will provide a more functional dewatering alternative. A co-operative design approach that involves both the mine planning team and the hydrogeologist is encouraged to ensure that the recommended dewatering design meet all geotechnical objectives while resulting in minimum impact on mine operations. Chapter 2 12 OPEN PIT MINES The main advantages of dewatering wells are: • Wells can be situated outside pit limits (if pit is relatively small, i.e. less than 100 m deep). • Depressurization can be initiated in advance of mining activity. • Depressurization proceeds quickly. • In arid locations the dewatering yields can be used to augment other more expensive production water supplies. The disadvantages are: • Development and operating costs are high. • Wells interfere with normal operations if situated within the pit. • Daily monitoring and periodic adjustment of wells is frequently required to maintain all wells at peak efficiency. • Dewatering wells are not effective in geologic horizons that are either highly pervious or relatively impermeable. Drainage Adits: This groundwater control method requires the development of one or more tunnels just beneath the slip surface of a potential failure. The smallest possible adit profile is usually selected to keep costs to a minimum. A 2x3 m opening is considered a minimum size for ease of access and machine assisted excavation. Development costs are strongly dependent on geologic and groundwater conditions, costs of $2000 to $3600 per m are considered typical (CO. Brawner and D. Moore, personal communication). Because development costs of drainage adits are high when compared on a per m basis, drainage adits are utilized much less frequently than horizontal drains or wells. Applications that justify the expense usually involve very large slope failures in excess of 10 million m3. A drainage adit solution can also be very cost effective if old underground working exist below the potential failure surface, and the workings can be re-opened at nominal cost, and in third world countries where costs of labour are low while drilling costs are high. Advantages: • A very large effective diameter compared to wells and horizontal drains. • Radial drain holes can be drilled from inside adit to intercept a much larger number of water bearing discontinuities. • More effective than horizontal drains or wells because the full length of the adit is usually situated just below the failure surface, where it contributes most to slope stability. Disadvantages: • Practical only in fair to good bedrock conditions. Support problems preclude the use of this technology in overburden or badly broken ground. • Location of the slip surface should be well defined so that the drainage adit can be positioned for maximum effect. • Development time will be much longer than installation of horizontal drains. Cost Comparison: The total cost of a groundwater control program will depend on the drainage method selected, size of the area to be dewatered, ground conditions, complexity of the hydraulic conductivity distribution, the time span over which dewatering will be required, and long term stability of the pit wall. For comparative purposes, Table 2.2 presents the range of costs that one might experience when dewatering a single 500 m wide design sector with each method. Horizontal drains generally provide the lowest cost solution, followed by dewatering wells, and ultimately drainage adits. Chapter 2 13 OPEN PIT MINES Table 2.2 Approximate Costs of Dewatering with Most Common Methods Dewatering Technology Cost per m No. Required Length (m) Total Cost ($ million) Horizontal Drains $60 - $100 100-200 50-150 0.3-3.0 Vacuum Option 0.05-0.2 Wells $1,000-$2,000 5-15 50-150 0.25-4.5 Drainage Adit $2,000-$3,600 1-2 500-1500 1.0-5.4 Vacuum Option 0.05-0.1 27 SUMMARY This chapter has provided a brief overview of open pit mining operations and how those operations can be affected by groundwater seepage into the pit. Potential impacts that were addressed include slope stability, berm erosion, trafficability, blasting, production water and environmental concerns. The discussion then turned to a review of modern dewatering methods, including an evaluation of the advantages, liabilities and typical installation costs associated with each. It was concluded that horizontal drains frequently provide the most cost effective dewatering solution; however, the costs associated with all three drainage methods are high, $0.5 to $2.5 million is usually spent by mines to depressurize a single design sector. When addressing the issue of whether or not to invest in some form of groundwater control program, mine management and supporting engineering staff are faced with a technically complex and risky decision. The risk-cost-benefit approach, introduced in the following chapter, provides a rational framework for analyzing this complex problem that has been shown to involve many related and inter-disciplinary issues. Chapter 2 14 OPEN PIT MINES CHAPTER 3 RISK-COST-BENEFIT ANALYSIS 3.1 3.2 3.3 3.4 3.5 3.6 INTRODUCTION DECISION ANALYSIS FORMULATION OF OBJECTIVE FUNCTION COMPONENTS OF THE RISK COST BENEFIT FRAMEWORK SOFTWARE MODULES SUMMARY 15 15 17 20 22 25 3.1 INTRODUCTION This chapter provides an overview of the complete risk-cost-benefit framework. The objective is to show how decision analysis is used to identify the most cost effective design strategy from a limited set of proposed slope design and dewatering options. The various components of the framework, that include geostatistics, analysis of groundwater flow, evaluation of slope stability and economic assessment, are also described briefly to show how each component fits into the global scheme of the framework. Having laid the foundation for the framework in this chapter, Chapters 4 through 6 will then examine the technical aspects of the principal framework components. Decision analysis is a tool used in many engineering disciplines to make design and operating decisions in complex situations where the best design or operating strategy is not intuitively obvious (Massmann and Freeze, 1987, Baecher, 1980, Whitman, 1984). The analysis involves the definition of an economic objective function, which is usually taken as the summation over an engineering time horizon of the net present value of all system risks, costs and benefits. The objective function is used as a measure of the worth of various design options. The optimal design strategy is identified as the strategy that maximizes the value of the objective function. The decision analysis approach adopted in this framework is based on earlier research at The University of British Columbia by Massmann and Freeze. To avoid duplication of material in an area where this dissertation makes no research contribution, only the basic elements of decision analysis logic will be presented in this section. The interested reader is referred to Chapter 2 of Massmann's (1987) dissertation for a more detailed treatment of the subject matter. A decision problem occurs whenever there is a choice between two or more alternative courses of action that result in different consequences.. Consider a typical open pit design problem involving three possible design alternatives: 1) leave slope design unchanged, 2) flatten slope and 3) introduce a dewatering system. Complete characterization of this decision problem usually involves four components: • Decision Variables define the list of possible design alternatives, e.g. flatten slope or leave slope unchanged, and introduce dewatering program or do not dewater. • Consequences describe the final outcome of the decision, e.g. slope fails or slope remains stable. • State Variables describe parameters that will impact on the consequences, but are beyond the control of the decision maker. Site conditions are typically considered state variables, e.g. hydraulic conductivity distribution, shear strength parameters, etc. 3.2 DECISION ANALYSIS Chapter 3 15 RISK-COST-BENEFIT ANALYSIS Figure 3.1 Decision Tree for Example Design Problem I , , . , DESIGN OPERATION Conditional Net—Income ($ million) Cost of Dewatering: $5 million Cost of Extra Stripping Due to Flattening: $25 million • Constraints may be imposed on the decision problem to limit the number of viable design variables (e.g. overall slope must be shallower than 45° to prevent ravelling), or to guarantee that certain unacceptable consequences do not materialize under any circumstances (e.g. probability of failure must be less than 20%). For the decision problem considered in this thesis, the relationship between decision variables, consequences, state variables and constraints, as well as the selection process used to identify the optimum design alternative is best illustrated with a simple decision tree, as shown in Figure 3.1. The decision problem typically involves a limited number of design alternatives, each alternative being represented by a single branch emanating from the square decision node or trunk, found at the extreme left of the decision tree. Each design branch ends at chance node, where nature would determine the actual site conditions, and ultimately, whether the particular design will result in stability or failure. Since site conditions are usually uncertain to some degree, design performance cannot be predicted exactly a priori. However, it is possible, given statistical information about each state variable, to predict the probability that a particular design will result in stability or failure. In the illustrative example considered in Figure 3.1, the 30% probability of failure for the original design is reduced to 5% by dewatering, and to 2% by slope flattening. Clearly, slope flattening is the best design in terms of stability. However, an economic analysis is required to identify which of the three design options will maximize profit for the mine operator. Formulation of the economic model will be the theme of the following section. Chapter 3 16 RISK-COST-BENEFIT ANALYSIS 3.3 FORMULATION OF OBJECTIVE FUNCTION Before a comparison study can be conducted to identify the optimum design alternative, it is necessary to express the consequences associated with each design in terms that can be used to compare the relative value of each option in an unbiased and consistent manner. In the field of system optimization, this frame of reference is known as the objective function. In an open pit mine, the primary objective of virtually all activities is to maximize production and achieve the greatest possible profit, subject of course to constraints regarding worker safety and damage to the environment. Therefore, it is natural to express the objective function in terms of profit. In the most general sense, mine profitability is influenced by three categories of parameter uncertainty. The first category considers ore grades and recoveries, as these can differ significantly from exploration projections. The second category consists of uncertain economic factors. Included in this group are world metal prices, interest rates, tax laws, changes in marketing contracts, and fluctuations in salaries and commodities such as explosives, fuel, electric power and tires. The third category of parameter uncertainty is comprised of geotechnical factors that will impact on slope stability. These factors include shear strength parameters and hydrologic conditions. Because the objective of this dissertation is to investigate the importance of groundwater control on mine profitability, only geotechnical factors will be treated in a stochastic manner. Ore grades and economic factors will be treated as deterministic parameters; with each deterministic parameter assigned a single "most representative" value. Because all economic factors are held constant, it is more convenient to express the objective function in terms of expected net income, rather than profit. In this context, expected net income is defined as the difference between revenue generated by the sale of concentrate, less all operating costs required to produce the concentrate, less monetary risks associated with slope failure. Expected Net Income = S Benefits - S Costs - S Risks Equation 3.1 Unlike profit, net income does not include adjustments for capital financing, taxes, etc.; therefore, it provides a simpler, crisper definition for the objective function of this study. The following sub-sections identify the many economic components considered within the benefit, cost and risk categories. 33.1 SYSTEM BENEFITS The stream of benefits realized from a particular design alternative is the revenue received from the sale of mineral concentrate mined over the design period. Revenue is a function of: amount of ore mined, ore grade, recovery realized by the mill, and price received for the concentrate. The amount of ore produced from the open pit will depend on the size and shape of the ore deposit, the maximum depth of the pit, and the pit wall angle. Because ore grades, recoveries and metal prices are factors over which the geotechnical engineer has no control, they are treated as fixed deterministic coefficients in the calculations. In this framework, all economic calculations are performed automatically with computer program SG-RCB. The following algorithm is utilized in the computer program to estimate revenue: • A single vertical section is selected to represent the design sector. It is assumed that the entire section has a constant width in the third dimension and that all properties remain constant in that dimension. The section is then divided into a finite number of equal size blocks, typically 10x10 m to 25x25m in size, depending on the size of the pit. • A user friendly data interface in SG-RCB is used to associate each block with an ore grade, the push-back during which it will be mined, and digability characteristics. Chapter 3 17 Risk-Cost-Benefit Analysis • Using the above information, the computer calculates the total tonnage in each block, whether the block constitutes ore, how much economic mineralization is present within the block, how much mineralization can be recovered, and what monetary value should be assigned to the block. The total revenue per push-back is obtained by summing the block values for all blocks that will be mined during the particular pit expansion. 332 SYSTEM COSTS A large number of costs are incurred by the mine operator during production. It is useful to divide these costs into two broad categories: operating costs and dewatering costs. Operating costs are defined as all costs of mining and processing the ore. The costs include: • engineering design • blasting • excavation • transport to crusher and waste dumps • milling of ore • environmentally sound disposal of waste rock and tailings • numerous ancillary support tasks The task specific operating costs are not tracked individually in this framework. Instead, operating costs are obtained based on total mining and milling costs per tonne because these statistics are more readily available than the detailed unit costs. Total mining costs include all costs associated with the extraction of ore and waste from the pit to the primary crusher. Total milling costs cover all expenses incurred to produce concentrate once the ore passes through the primary crusher. Recall that operating costs are very dependent on the pit wall angle; due to the large increase in the amount of waste rock that must be blasted, excavated, transported out of the pit and disposed of as the pit wall angle is flattened. Dewatering costs are defined as all costs associated with pit wall depressurization. The costs include: • site investigation • engineering design • purchase of required equipment • installation and development of horizontal drains or wells • pumping costs • monitoring and maintenance costs • costs of water treatment Dewatering costs will reflect design decisions made regarding the amount of site investigation performed to define hydrologic parameters, the type of dewatering system selected, the ground conditions, the spacing and location of the drains and/or wells, and the amount of water pumped. Chapter 3 18 Risk-Cost-Benefit Analysis 333 SYSTEM RISKS In the formulation of the objective function, monetary risk is defined as the expected cost of slope failure. In other words, risk is the product of the conditional cost that must be borne by the mine operator in the event of a slope failure, multiplied by the probability of the slope failure occurring. Recall from Section 2.4 that the consequences of slope failure may include: • Clean up costs of removing failed material from pit. • Lost production due to a blocked haul road or buried ore. • Lost ore within the failure block that can no longer be separated from waste. • Reduced cash-flow. • Damaged equipment and services. • Physical hazard to workers and mobile equipment in pit. Estimates of the monetary consequences to be attributed to each of these factors will vary from design sector to design sector. Input from the mine planning department will usually be required to identify all potential impacts of slope failure in the design sector and realistically forecast the costs associated with those impacts. Having described how benefits, operating costs and costs of failure are estimated in this framework, only probability of failure remains to be discussed before the definition of the objective function is complete. As estimation of the probability of failure is the key step that lies at core of this risk-cost-benefit framework, this issue is addressed separately in the following section. Before moving on to the discussion of probability of failure, it is important to note that the expected value formulation does not predict the true net income that will be realized by the mine operator; it only provides a weighted average estimate of net income; the average being calculated over the two possible outcomes (i.e. stable wall or failure). The net income that will actually be achieved by the mine operator is conditional, it will depend on whether or not a slope failure is experienced. If no failure is experienced, net income will likely be significantly higher than in the case a failure does develop. While conducting the decision analysis, it is worthwhile to consider the conditional net income projections in the analysis, as these projections indicate how the mine will prosper if the pit wall remains stable, or alternately, if failure does develop. If the costs of failure are sufficiently high to create economic hardship for the company (e.g. premature mine closure), then mine management may perceive that the true consequences of failure are much higher than indicated by the narrow scope of this analysis. Although not considered in this dissertation, utility theory (cf. Lindley, 1971; Fischoff, et al, 1981; Crouch and Wilson, 1982; Freeze et al, 1990) could be introduced to make corrections for this risk-averse behaviour. It should also be noted that in many situations where decision analysis has been applied in the past, the formulation of an objective function was not straightforward. Complications frequently encountered include multiple objectives, an adversarial decision involving several parties (Massmann and Freeze, 1987), and risks that are not easily quantified in terms of monetary value, e.g. a threat to human life or welfare (Baecher, 1980). Although some external issues can also affect operating decisions at open pit mines (e.g. regulatory agencies, unions, Workmen's Compensation Board, etc.), the design problem considered here is almost ideal in that the formulation of the objective function is straightforward; the mine operator is the primary party to make and to be affected by the consequences of decisions, and most decisions are made to achieve the primary objective, to maximize profitability at the mine site. Chapter 3 19 RISK-COST-BENEFIT ANALYSIS 3.4 COMPONENTS OF THE RISK COST BENEFIT FRAMEWORK In this framework, probability of failure is calculated using a Monte-Carlo simulation approach. This section presents a global overview of the methodology, subsequent chapters will address the technical details of the individual framework components that include geostatistics, groundwater flow and slope stability. Stability of a pit wall is influenced by slope configuration, shear strength properties of individual geologic horizons through which the failure surface will pass, and pore pressure conditions along the failure surface1. In order to properly evaluate how each of these factors will impact on the probability of failure, each step of the stability analysis must be orchestrated so that all necessary input will always be available at the completion of the preceding step. Figure 3.2, a flow chart of the risk-cost-benefit framework, illustrates the sequence of analytical steps adopted in this framework. Data Collection and Interpretation: As shown in Figure 3.2, the analysis begins with collection and interpretation of field data required to construct the groundwater flow, slope stability and economic models. Because the list of necessary parameters is extensive, individual items will not be addressed here. Section 7.2 of the sensitivity study and Appendix E provide two practical examples of the types and quantities of information that are required to conduct a risk-cost-benefit analysis of dewatering options. . i Geostatistical Analysis: Formulation of a reasonable stochastic description of the hydraulic conductivity field is the most demanding aspect of the data collection and interpretation effort. In order to obtain a sufficiently large data base that will yield meaningful statistics, it is usually necessary to collect hydraulic conductivity information from all available sources, e.g. pump tests, slug tests, grain-size analyses, consolidation tests, geologic logs, etc. Once all local measurements are evaluated, a geostatistical analysis is conducted to estimate the mean, variance and semi-variogram function of the hydraulic conductivity field. Whenever some form of preferentially oriented fabric is detected during subsurface investigations (e.g. bedding, joint sets, foliation, etc.), it is also necessary to identify the degree of anisotropy that is introduced within the auto-correlation structure. During the case history study at Highland Valley Copper for example, it was shown that the hydraulic conductivity field was correlated over much larger distances parallel to bedding than across bedding. Simulation of Hydraulic Conductivity Field: Having quantified all necessary input parameters, the Monte-Carlo simulation begins. The first step in the simulation involves generating a large number of realizations of the hydraulic conductivity field. The objectives are: 1) to generate each realization so that it duplicates all important features of the geologic environment being modelled, 2) to effectively reproduce the specified geostatistical structure in each realization, and 3) to reproduce the appropriate level of prediction uncertainty at each estimation point over the ensemble of realizations. A number of simulation techniques have been pioneered in recent years to achieve this objective. The available options, and their distmgnishing attributes, are explored in Chapter 4. Prediction of Pore Pressures: A two dimensional, saturated/unsaturated, finite element model of groundwater flow is used to predict the pore pressure distribution in the pit wall for each realization of the hydraulic conductivity field. Given the complete hydraulic conductivity distribution, boundary conditions and pumping strategy, the model uses an iterative procedure based on the free surface approach pioneered by Neuman (1973) to solve for the hydraulic head and pore pressure at each finite element node in the flow domain. The theory, unique features, and capabilities of the stochastic groundwater flow model are presented in Chapter 5. Dynamic effects due to blasting or earthquake loading can also be important, but are not be considered in this dissertation. Chapter 3 20 RISK-COST-BENEFIT ANALYSIS Figure JL2 Flow Chart Illustrating Key Steps in Risk-Cost-Benefit Framework. Data Col lect ion, Analysis and Ident i f icat ion of Model Pa ramete rs Hyd rologic E conomic Shea r S t rength Geosta t i s t i ca l Analysis of Hydrau l i c Conduct i v i t y F ie ld Monte Car lo S imu la t i on to Obta in POF •< Calcu la te P robab i l i t y of Fai lure as: N u m b e r of Real izat ions Result ing in Fa i lure Tota l N u m b e r of Real izat ions E conomic Analysis of Objective Func t i on Se lec t ion of Best Slope Design and Dewater ing Opt ions Generate Rea l iza t ion of W Hydrau l i c Conduc t i v i t y F ie ld ; Calculate Pore P ressu res in Flovr Doma in Us ing F in i te E l emen t Model Yes Generate Shea r S t r e n g t h Pa r ame te r s For C u r r e n t Rea l iza t ion Determine S tab i l i t y of Slope for C u r r e n t Shea r S t r e n g t h a n d Pore P ressure Rea l iza t ion Evaluate Whether Rea l iza t ion Resulted in Stable Slope (Kc>0) o r Fa i lure (Kc<0) Note: Maximum refers to the maximum number of realizations to be performed during the Monte Carlo Simulation. Simulation of Shear Strength Parameters: Having defined the pore pressure field, the next step in the analysis involves the generation of shear strength parameters for the current realization. Using mean and standard deviation statistics for cohesion and friction angle that must be specified for each geologic horizon, a Gaussian random number generator is used to select a unique cohesion and friction angle for each geologic horizon that will apply for the current realization. Analysis of Slope Stability: A two dimensional method slices based on Sarma's (1979) analysis technique is used to determine whether the pit wall design will remain stable or result in failure under the current realization of shear strength and pore pressure. The Sarma method simultaneously solves the equations of horizontal and vertical force equilibrium, moment equilibrium and a failure criterion for each slice to obtain Ka the critical horizontal acceleration factor that indicates whether or not the resulting slope will be stable. If Kc is positive, the slope will remain stable; if Kc is negative, the realization will result in failure. Chapter 6 provides a more complete discussion of the theory underlying Sarma's method and documentation that outlines how pore pressures predicted by the finite element model are incorporated in the stability analysis. Estimation of Probability of Failure: Having completed the stability evaluation of each realization and determined whether the particular realization will result in stability or failure, the probability of failure is calculated as the ratio of the number of realizations resulting in failure, divided by the total number of realizations in the ensemble. Economic Analysis: In this step of the analysis the benefit, cost and risk terms are assessed for the design under consideration, and substituted into Equation 3.1 to solve for the expected net income. Recall from Section 3.3 that in this framework, expected net income serves as the objective function that is used to measure the worth of each design option. Chapter 3 21 Risk-Cost-Benefit Analysis Figure 3-1, the decision tree illustration presented earlier in this chapter, provides a concise example of the calculations involved. First, one design alternative is selected from the available options. Conditional net incomes are then calculated for both stable and failure scenarios. As demonstrated in the figure, the monetary worth of each possible combination of decision variable, state variable and consequence can be determined by simply following the branches of the decision tree to the desired terminus and summing up all benefits, costs and risks along the route. For example, assuming the slope remains stable, the conditional net income for the dewatering design is $115 million. The expected net income is then obtained by multiplying each of the two conditional net incomes by the corresponding probabilities of failure, and summing the two results (e.g. expected net income for the dewatering only option is $1123 million). Selection of Best Design: In order to identify the best slope configuration and dewatering alternative, the entire calculation process outlined above is repeated for each design under consideration. Once a value of the objective function is computed for each design, selection of the best design strategy simply involves choosing the design that results in the highest value of expected net income. Given the specified level of uncertainty in field parameters, this design alternative is expected to generate the greatest profit from mining activity in the design sector, on average. In the example presented in Figure 3-1, the dewatering only alternative is expected to result in the highest net income of the three options considered ($112.5 million), given the likelihood of failure associated with each design. 3.5 SOFTWARE MODULES Personal computer software is utilized extensively in each step of the risk-cost-benefit framework, from field data interpretation to economic evaluation. A major part of the research effort summarized in this dissertation involved the development of custom software modules to carry out the four principal tasks that comprise the risk cost benefit framework, i.e. SG-Stat for geostatistical analysis, SG-Flow to model groundwater flow, SG-Slope to analyze slope stability and SG-RCB to conduct the economic analysis. Five other software modules were later incorporated in the framework to facilitate various aspects of data management and interpretation, especially during the case history study of the dewatering design at Highland Valley Copper. This section provides a very brief overview of each module, and an indication where the program fits in the global scheme of the risk-cost-benefit framework. Figure 3.3 identifies the various data analysis and modelling functions that comprise the risk-cost-benefit framework. The software modules that are used to conduct each aspect of the analysis are also shown (highlighted in double boxes). SG-CoreLog was developed to efficiently process large amounts of geologic data normally generated during exploration drilling programs. Once information is entered in the data base, SG-CoreLog can also be used to generate graphic strip logs and geologic cross-sections. The program was used extensively by the author during the case history study of Highland Valley Copper to interpret the complex overburden stratigraphy that exists at the site. SG-Pump has been developed as a practical working tool to analyze drawdown data from a large number of pump tests conducted at Highland Valley Copper. Based on the Jacob-Cooper semi-log method of analysis, the program has been enhanced to also permit analysis of stepped drawdown tests, i.e. where the pumping rate is periodically increased as the test progresses. An aquifer response module has also been incorporated in the program. This module, based on the Theis (1935) solution, is used to predict the drawdown response for any user specified pumping rate once transmissivity and storativity parameters are known. Chapter 3 22 Risk-Cost-Benefit Analysis Figure 3.3 Flow Chart Illustrating Software Modules. Slope Design Geology Data Pump Test Data Well Monitoring Piezometer Monitoring • • * |SG-Pump SG- Well SG-Piezo • Shear Strength Data Dewatering Design | Geologic Interpretation | | Hydrologic Interpretation SG-Stat Geostatistical Description of Hydraulic Conductivity Field i Conditional Realizations of Hydraulic Conductivity Field >! SG-Flow Realizations of Pore Pressure Distribution in Pit Wall •> SG-Slope * Economic Data SG-BudgetH Cost of Dewatering Program Dewatering Budget Probability of Failure > SG-RCB Evaluation of Objective Function Selection of Best Design Alternative SG-WeUser\ds as a data base and graphic display module to store and analyze large amounts of well monitoring data that are generated as part of daily or weekly monitoring programs. The module is capable of storing an unlimited number of monitoring records, accessing any stored information in seconds, and displaying the data in a number of easy to interpret graphic formats (e.g. time graphs, bar graphs, X Y graphs). The module was developed to organize and interpret the thousands of well monitoring records that have been collected at Highland Valley Copper since 1984. SG-Piezo serves essentially the same function as SG-Well, except that it is used to store and interpret water level observations from piezometer monitoring networks. SG-Budget was developed to provide a fast and accurate method of forecasting the costs of future dewatering programs at Highland Valley Copper. Given an outline of the dewatering plan, including the completion depth of each well, well diameter, expected flow rate, etc.) the program automatically calculates the development and operating costs for each installation. A year by year summary of costs and flow rates is also provided for planning purposes. Chapter 3 23 Risk-Cost-Benefit Analysis SGStat is a modular program used to conduct all aspects of the geostatistical analysis of hydraulic conductivity data. Capabilities of the program include: • Structural analysis to identify the mean, variance and semi-variogram statistics of raw field data. • Kriging estimation to obtain the best linear unbiased estimate of hydraulic conductivity at each prediction node and the associated estimation error based on available field data. • Stochastic simulation to generate hydraulic conductivity realizations that possess the desired auto-correlation structure as well as reflecting an appropriate amount of variability at each prediction point over the ensemble of realizations (Le. simulated values located close to measurement points should show little variability over the ensemble of realizations while simulated values far from supporting data should possess somewhat more). • Ability to simulate complex geologic deposits that may possess several distinct geologic horizons (e.g. layered aquifer/aquitard sequence or distinct fault zone) and sedimentary stratification that may result in an anisotropic auto-correlation structure. Chapter 4 provides a complete description of SG-Stat, the underlying theoretical concepts, and numerous examples of the software capabilities. SG-Flow is the two dimensional saturated/unsaturated finite element model of groundwater flow that has been developed to predict pore pressures in the pit wall. Given a description of the hydraulic conductivity field, the pit wall configuration, and the location of all pumping wells or horizontal drains, the program predicts the steady state pore pressure distribution that will be established. Interfaces have been developed to SGStat and SG-Slope so that hydraulic conductivity realizations generated by SG-Stat are automatically imported into the program and pore pressure realizations are automatically accessed from SG-Slope. SG-Slope is a two dimensional slope stability program based on Sarma's (1979) method of slices. The program is capable of conducting both the stochastic method of analysis to obtain a probability of failure, as well as the more conventional deterministic analysis to obtain factors of safety. The effects of groundwater on stability can be evaluated in one of three ways: 1) drained analysis, 2) Dupuit assumptions and 3) pore pressures from finite element model. Chapter 6 provides a more detailed description of the underlying theory and unique capabilities provided by this program. SG-RCB is the final program in the framework. It is used to conduct the economic risk-cost-benefit analysis once all of the necessary input data has been compiled (i.e. probability of failure, costs of dewatering program, etc.). Given the ore grade distribution, the pit development plan, the critical failure geometry for each pit expansion, the associated probability of failure, the dewatering plan and a gamut of economic parameters, SG-RCB evaluates the revenue, the operating costs and monetary risks that will be realized during each push-back. Expected net income, is then reported for each push-back and for the entire mining cycle. Besides using the computed objective function to identify the best pit design and dewatering strategy, statistics calculated by SG-RCB can also be used to quantify the economic consequences of slope failure during each pit expansion so that mine management can accurately judge the importance of stable pit walls in the design sector. A user friendly software interface and easy to interpret colour graphics have been incorporated in all nine software modules. Time saving features of the user interface include on screen menus, on-line help and graphic data entry. Graphic data entry involves painting input data (e.g. hydraulic conductivity field, boundary conditions, slice geometry) directly on the screen rather than keying the information into a data file. The guiding motivation behind the extra programming effort was to disseminate the contributions made in this research not only through this dissertation and subsequent technical papers, but through practical software that future researchers and Chapter 3 24 Risk-Cost-Benefit Analysis practitioners will be able to apply to similar design problems. To this end, several of the software modules are already being used in other graduate research at The University of British Columbia, as practical analytical tools in a number of consulting firms and at Highland Valley Copper, and as teaching tools in the core curriculum of the geological engineering program at U.B.C. All of the software modules that constitute the RCB framework, including SGStat, SG-Flow, SG-Slope and SG-Risk have been circulated in the public domain. Diskettes containing the software, including both source code and executable code have been provided to the UBC Geology Library. However, the software user must accept all responsibility for the accuracy of the modules for his particular design problem. Also, technical support for the software is not provided. 3.6 SUMMARY This chapter has provided an overview of the complete risk-cost-benefit framework and a summary of the decision making processes that are used to identify the most cost effective slope design and dewatering strategy from a limited number of design options. The discussion began with an introduction to the decision analysis approach adopted in the framework. A decision tree was used to illustrate the basic decision making logic. Expected net income was then identified as the preferred objective function, one that can be used to compare the relative economic value of each design option in an unbiased and consistent manner. It was shown that net income can be calculated as the sum of all system benefits, less operating costs, less monetary risks of failure. The various economic components that must be considered in the calculation of the benefits, costs and risks were then briefly listed. Of the three terms, monetary risk was shown to be the most difficult to evaluate because it requires estimates of both economic consequences of failure and probability of failure. The risk-cost-benefit framework was then broken down into nine basic steps, including: • Data collection and interpretation • Geostatistical Analysis • Simulation of Hydraulic Conductivity Field • Prediction of Pore Pressures • Simulation of Shear Strength Parameters • Analysis of Slope Stability • Estimation of Probability of Failure • Economic Analysis • Selection of Best Design The tasks that are conducted during each of the above analysis steps were reviewed briefly to provide the reader with an understanding of the global framework before addressing the various task specific issues, including geostatistics and simulation, groundwater flow modelling and slope stability in following chapters. Finally, the four computer software modules that are needed to carry out this risk-cost-benefit analysis (SG-Stat, SG-Flow, SG-Slope and SG-RCB), as well as a number of auxiliary data interpretation programs, were introduced in the last section of this chapter. Chapter 3 25 RISK-COST-BENEFIT ANALYSIS CHAPTER 4 GEOSTATISTICS 4.1 OVERVIEW 4.2 BASIC STATISTICS 4.3 VARIOGRAM MODEL 4.4 ESTIMATION 4.5 SIMULATION 4.6 GEOSTATISTICS AND DESIGN 4.7 SUMMARY 26 29 35 41 58 76 79 4.1 OVERVIEW Subsurface conditions are variable in space, and frequently highly uncertain. When designing a pit wall the primary objective of the geotechnical team performing the site investigation is to characterize subsurface conditions in a satisfactory manner that will result in a reliable and economic design. An equally important objective is to quantify the uncertainty associated with each parameter estimate, and to predict how the uncertainty will impact on the economics of the pit wall design. Index properties commonly recorded during geotechnical investigations may include grain size, fracture spacing, RQD, and hydraulic conductivity measurements. Observations of these properties on surface exposures, drill cores and cutting samples suggest that physical properties of the soil or rock mass are spatially variable because measurements tend to fluctuate as observations are taken along a traverse or bore hole (see Figure 4.1). In most cases the physical properties are also spatially autocorrelated; measurements taken close to each other are more strongly correlated with one another than measurements taken far apart. Geostatistics provides a powerful set of tools for estimating hydraulic conductivity at selected points in an autocorrelated flow domain based on a limited set of field data. The data may include pump tests, slug tests, grain size analyses and other soft data such as geophysical results. The geostatistical tools can also be used to estimate the degree of uncertainty associated with each hydraulic conductivity prediction. This chapter provides a summary of existing geostatistical tools that have been adopted in this Risk-Cost-Benefit framework to generate realizations of the hydraulic conductivity field that are required for Monte Carlo Simulation. Figure 4.2 is an organization chart that outlines the topics that will be addressed on the following pages and how they are related. For the most part, the technical material presented in this chapter is drawn from Journel and Huijbregts (1978), De Marsily (1984) and Clifton and Neuman (1982). Section 4.2 introduces essential statistical concepts that provide the basic building blocks for geostatistics. These include: mean, variance, covariance and semi-variogram. Unique terminology has evolved in the domain of mining geostatistics to describe the semi-variogram. The meaning of several important terms including: nugget effect, sill, and range is also given. Section 4.3 explains how the experimental semi-variogram is constructed from field data, and how it is used to describe the natural variability of the hydraulic conductivity field. Emphasis is placed on exploring how the shape of the semi-variogram reflects the correlation structure of the hydraulic conductivity field. A small set of simple mathematical functions have been developed to model the experimental semi-variogram. Section 4.3 presents the most common of these functions and suggests simple rules of thumb for quickly selecting model coefficients that will result in a good fit to the experimental semi-variogram. Chapter 4 26 GEOSTATISTICS Figure 4.1 Hydraulic Conductivity Profile in Borehole -4.0 -5.0 — o -6.0 — -7.0 r i i -j 1 — 20 30 40 50 DEPTH FROM COLLAR (metres) 60 ~I 70 Figure 4L2 Organization Chart - Geostatistics DATA COMPILATION ESTIMATION K r i f i i n s Geologic Interp. Inverse Distance Others SIMULATION I Analytical Spectral Methods Monte—Carlo ' M l l i u l . l t l t i l l First & Second Moment Analysis  Nearest Neighbour Model (HT) Turning Bands I.U M a i i ix DocuinpoMlion Chapter 4 27 GEOSTATISTICS Section 4.4 introduces Kriging, a popular method of parameter estimation. Kriging is an averaging method that results in the Best Linear Unbiased Estimate. The meaning and significance of BLUE is reviewed briefly as are the fundamental assumptions on which Kriging is based. Each Kriging estimate is associated with an estimation error. Section 4.4 also examines the physical meaning of each term in the Kriging Error equation and shows how estimation uncertainty is affected by quantity, location, size, relative orientation and measurement error of available data. Kriging variance distributions are prepared using a number of different sampling strategies to illustrate the most significant relationships between geology, sampling geometry, and uncertainty. Section 4.5 introduces simulation, a tool that is used to generate a large number of hydraulic conductivity realizations from available statistics. The realizations serve as input for the Monte Carlo analysis of groundwater flow, slope stability, and ultimately, probability of pit wall failure. Conditional and unconditional simulation methods are defined and practical guidelines are given as to when each simulation method should be used. A number of simulation techniques have evolved for generating realizations. Section 4.5 provides a brief review of the available techniques. An expanded discussion of the theory underlying the Fast Fourier Transform (FFT) and LU Matrix Decomposition simulation methods that are utilized in this framework is presented in Appendices B and C. The examples in Section 4.5 demonstrate that the FFT simulation method has many advantages over the other simulation approaches commonly used by groundwater researchers today. Therefore, application of the FFT method to the generation of auto-correlated hydraulic conductivity fields is one of the key research contributions of this chapter. A few remarks about verification and practical examples of grid size and flow domain selection in relation to the semi-variogram model complete the discussion in this section. The quantity and quality of information increases dramatically as a project matures from conceptualization, through site investigation, to design and construction. With the aid of an illustrative example, Section 4.6 shows how the geostatistical model presented in this chapter can be applied at each stage of the design process to evaluate all available information and to assist in making logical design decisions as new information is revealed. Because the worth of additional data, and in particular, a suitable stopping rule that will indicate when enough field data has been collected to complete the pit wall design safely, are both intimately linked to the economic component of the Risk-Cost-Benefit analysis, a discussion of data worth is not presented until the sensitivity discussion in Chapter 7. At that point all of the necessary tools will be in place to evaluate and compare the various sampling strategies with the Risk-Cost-Benefit framework. Chapter 4 28 GEOSTATISTICS 4.2 BASIC STATISTICS Before introducing the techniques of kriging and simulation, and showing how these methods can be applied to dewatering system design, it is necessary to review a few of the underlying statistical foundations. In this section the terms mean, variance, standard deviation, covariance, variogram, semi-variogram, stationarity, intrinsic hypothesis, sill, range, and nugget effect will be presented. To maintain a coherent focus, the presentation will be organized as a statistical analysis of a geologic profile. Figure 4.3 is a contour map of the log hydraulic conductivity field that will be referenced in this section. Geologic materials are naturally variable because they are deposited in dynamic environments that undergo change throughout the period of deposition. Once deposited, the materials are further subjected to complex stress histories that are controlled by irregular tectonic and climatic events. It is for these reasons that geologic materials exhibit variability. The irregular hydraulic conductivity fields such as the field portrayed in Figure 4.3 are reflections of variability in the physical composition and structure of the deposit. Furthermore, the actual value of hydraulic conductivity at any given point cannot be predicted with certainty unless a measurement is taken nearby or the hydraulic conductivity field is sufficiently homogeneous to allow extrapolation of distant data. Today, deterministic models of groundwater flow such as USGS-MODFLOW and PC-SEEP are used widely to predict pore pressures in complex engineered structures such as open pit walls and earth dams (Sumner 1987, Krahn, 1987). The deterministic models require that a single value of hydraulic conductivity be specified for each cell in the flow domain, even if that value is not known with confidence. Because it is assumed that all input data is known exactly, output from deterministic simulations does not yield any information about the possible error of model predictions. Over the past fifteen years researchers, and more recently practitioners, have adopted statistical methods for characterizing variability of the hydraulic conductivity field and incorporating the parameter uncertainty into stochastic models of groundwater flow (Freeze, 1975, Clifton & Neuman 1982, Delhomme, 1983). In stochastic groundwater literature hydraulic conductivity, Y(xJ, is frequently considered as a spatially autocorrelated, log-normally distributed random variable, instead of working with Y(xJ directly, it becomes more convenient to define ZfxJ as log10fY(xJJ since the new random variable will be normally distributed. At each point in the flow domain the random variable Z(xJ is not known for certain; but the range of likely values can be described by a probability density function. The geostatistical techniques described in this chapter require that the probability density function be a Gaussian Normal distribution specified by mean m and standard deviation a. The hydraulic conductivity field is autocorrelated because Z(xJ changes gradually from point to point, unless a discontinuity such as a geologic contact or major fault is present. In this framework, Monte Carlo simulation provides the vital link that incorporates parameter uncertainty into the groundwater flow model. Monte Carlo simulation generates a large number of realizations of the desired output variable, the hydraulic head field. The uncertainty in hydraulic head prediction at each point in the flow domain, calculated by statistical analysis over all realizations, gives the analyst an indication of how input parameter uncertainty ultimately influences reliability of the groundwater flow model predictions. Such uncertainty predictions cannot be achieved with deterministic analysis. The biggest advantage of the stochastic approach is realized when probabilistic output from the groundwater flow analysis is utilized as input into an economic decision making component of the risk-cost-benefit framework to compare merits of different dewatering strategies. The first step in this demonstration is to discretize the hydraulic conductivity field into an array of gridded point values. This step, illustrated in Figure 4.4, is analogous to conducting a permeability testing program on a regular grid. The next step is geostatistical characterization of the observed correlation structure. Chapter 4 29 GEOSTATISTICS Figure 4.3 Log Hydraulic Conductivity Contours a 40 a o S 20 120 140 160 180 Horizontal Distance (m) 200 Figure 4.4 Hydraulic Conductivity Field Discretized into Cells a o E 50 40 30 20 10 0 - 4 • - 5 - 6 - 8 - 7 - 6 - 4 - 4 - 3 - 3 - 2 - 2 - 2 - 3 - 4 - 5 - 5 - 6 - 5 - 6 - 5 - 3 - 4 - 6 - 8 - 7 - 6 - 6 - 4 - 3 - 3 - 2 -1 -1 - 2 - 3 - 3 - 4 - 4 - 3 - 4 - 5 "3 - 3 - 4 - 7 - 6 - 5 - 5 - 5 - 3 - 2 -1 -1 - 2 - 2 - 3 - 4 - 5 - 3 - 2 - 2 - 4 -1 - 2 - 4 - 7 - 6 - 5 - 4 - 4 - 3 - 2 -1 - 2 - 3 - 4 - 4 - 6 - 6 - 4 - 3 - 4 - 5 -1 - 2 - 4 - 5 - 5 - 5 - 4 - 3 - 4 - 3 - 2 - 4 - 5 - 5 - 6 - 7 - 5 - 5 - 4 - 4 -6 20 40 60 80 100 120 140 160 Horizontal Distance (m) 180 200 42.1 GEOSTATISTICAL TERMINOLOGY DEFINED Ensemble: An ensemble is a set of realizations of the random field (e.g. hydraulic conductivity). Because each realization in the ensemble is generated from one set of statistics, on average all realizations will have the correlation structure, but actual parameter values at a fixed point in space will vary from realization to realization in accordance with the level of estimation uncertainty associated with each estimation point. Mean: This parameter is defined as the expectation of the random variable, Z(xJ: M = E[Z(xJ] Equation 4.1 where E[] denotes the expected value operator. For the problem at hand, log hydraulic conductivity, Z(xJ, is not known everywhere in the flow domain. Therefore, the expectation E[Z(xJ] cannot be determined and the mean must be estimated from available hydraulic conductivity measurements, z(xj: Chapter 4 30 GEOSTATISTICS Figure 4.5 Normal PDF - Mean & Variance Defined Relatively Homogeneous Environment m=-7.5 0.50 - i 0.40 o § 0.30 fl fc, £ 0.20 -xt a •g H PU 0.00 CT=0.5 Heterogeneoi Environment d) -10 -8 - 6 - 4 - 2 Log H y d r a u l i c C o n d u c t i v i t y m = S zfo) i - l Equation 4.2 The mean of the discretized geologic profile in Figure 4.4, calculated with Equation 4.2, is -3.98. Variance: Variability of the log hydraulic conductivity field about its mean value is described by its variance, a2. Variance is defined as the expected value of the squared difference between a local value of the random variable Z(x) and the mean value M, when averaged over the entire domain: a2 = E[{Z(xJ-M}-{Z(x)-M}] Equation 4.3 The variance operator, given by the right hand side of Equation 4.3, is frequently abbreviated as VAR[Z(xJJ. For a set of / discrete data points such as the set portrayed in Figure 4.4, the variance can be calculated by. i " a2 = 2 {z(x) - m}2 Equation 4.4 i - l I Standard deviation: Defined as the square root of variance, this parameter is useful for relating variability to the shape of the PDF. If data is normally distributed, 69% of samples will fall within ±1 standard deviation of the mean, 95% within 2a, and 99.7% within 3a (see Figure 4.5). Sedimentary deposits that exhibit a high degree of heterogeneity such as glaciofluvial sediments will have a relatively large a and a broad PDF, while homogeneous strata (e.g. evaporite deposits) will have a smaller a and a tighter PDF. Statistics compiled from the literature by Freeze (1975) suggest that the standard deviation of log hydraulic conductivity ranges from 0.2 for very homogeneous strata to 2.0 for highly heterogeneous strata. The standard deviation of the discretized geologic profile in Figure 4.4 is 1.66. Chapter 4 31 GEOSTATISTICS Figure 4.6 Characteristics of Covariance & Semi-Variogram xi —^* u v o a m •c > o o 1.20 c(0) A 0.80 0.40 -0.00 Covariance Model 10 20 30 Lag h 50 ^ 1.20 6 u o u > I •r-1 8 <u co 0.80 -0.40 -0.00 Semi-Variogram Model co 10 20 30 Lag h 40 50 Covariance: This parameter is a measure of spatial correlation over a separation vector1 h. Symbolized by C[h], the covariance is defined as: C[h] = Ef{Z(xJ - m}-{Z(Xi + h)-m}] Equation 4.5 When two points xt are strongly correlated, log hydraulic conductivity Z(Xj) will be very similar to Z(Xj). Therefore, the product {ZfxJ - m}-{Z(x^) -m} will always be positive and C[h], the expectation of the product will be significantly greater than 0. On the other hand, for longer lags over which points are no longer correlated, some product terms {Z(xJ - m}-{Z(xi + h) - m} will be negative, others will be positive, and the net result will be a covariance value very close to 0. When h-0 Equation 4.5 simplifies to the expression for variance, hence variance is often designated C[0J. The left side of Figure 4.6 presents a graph of a typical covariance function. The maximum value, C[0], occurs at the origin, the function then decays to 0 at some distance over which hydraulic conductivity ceases to be correlated. Variogram: Represented by 2y[h], the variogram is defined as the variance of first order increments2: 2y[h] = VAR[Z(x+h) - Z(x)J Equation 4.6 The variogram can be interpreted as the estimation variance of predicting Z(x) by Z(x+h). Semi-Variogram: Defined as one half of the variogram, the semi-variogram function y(h) can also be used to characterize the observed correlation structure. Indeed, it will be shown below that the semi-variogram and covariance are so closely related to be equally applicable when dealing with stochastic fields that do not exhibit spatial drift. As shown on the right side of Figure 4.6 the semi-variogram function begins at the origin and then increases to a maximum value of C[0] at the maximum distance over which the data is correlated. 1 Separation vector h is frequently referred to as lag in the stochastic literature. 2 First order increment is the difference [Z(x+h)-Z(x)J. Chapter 4 32 GEOSTATISTICS Before the relationship between the semi-variogram and covariance can be explored in detail it is necessary to review three commonly applied assumptions about the degree of stationarity exhibited by the log hydraulic conductivity field. The discussion is very brief, additional details are provided in Appendix A.l. Slationarity: This condition requires that all statistical properties, including mean, variance, covariance and higher order moments be constant and independent of location x. 2nd Order Stationarity: Less restrictive than strict stationarity, this condition requires only that the first two moments of the random variable be stationary. The mean and variance of the log hydraulic conductivity field must be constant over the flow domain and the covariance must be dependent only on the length and orientation of the separation vector h, not on the location x. Intrinsic Hypothesis: Less stringent than the hypothesis of 2nd Order Stationarity, the intrinsic hypothesis requires only that the variance of first order increments be finite and that the increments themselves be second order stationary, la Appendix A. 1 it is shown that the intrinsic hypothesis is useful for analyzing problems where variability in the log hydraulic conductivity field continues to increase with increasing size of the sampling domain and a finite variance C[0] cannot be defined over the size of the flow domain being analyzed. If geostatistics is to provide meaningful simulation results at least one of the above hypotheses must be applicable to the random variable Z(x). In practice, the hypothesis of 2nd order stationarity can be applied to most field problems. The intrinsic hypothesis is required only for situations where the dimension of the flow domain is shorter than the distance over which the random variable is correlated or a hidden trend is present that cannot be filtered out prior to the geostatistical analysis. Silt For log hydraulic conductivity fields that are 2nd order stationary the experimental semi-variogram will increase to a sill value equal to the variance C[0] and then remain at that plateau for all increasing lags (see Figure 4.6). When the sill is not observed, C[0] cannot be defined, and the log hydraulic conductivity field will not be 2nd order stationary at the scale of the flow domain. In that case, only methods based on the intrinsic hypothesis can be utilized to analyze the data. Range: This parameter is defined as the lag at which the sill is first encountered and the covariance function vanishes. As shown in Figure 4.6, the range is the lag distance at which the hydraulic conductivity field ceases to be correlated. Nugget Effect: Recognized as a discontinuity in the semi-variogram or covariance function at the origin, the nugget effect can be caused by variability in the log hydraulic conductivity field at a scale smaller than the sampling interval or by random measurement errors. The term is originally derived from analysis of gold bearing deposits where samples with very high gold grades occur directly adjacent to samples with background gold values. The experimental semi-variogram for such deposits, illustrated in Figure 4.7, appears discontinuous near the origin, jumping from 0 at the origin to a finite nugget variance at the first non-zero experimental lag point (equal to sample interval). Chapter 4 33 GEOSTATISTICS Figure 4.7 Semi-Variogram & Covariance Function with Nugget Effect Lag h 422 RELATIONSHIP BETWEEN y(h) & C(h): If the log hydraulic conductivity field is second order stationary then the variance C[0] exists and a relationship between the semi-variogram and covariance can be derived. Starting with a definition of the variogram given by Equation 4.6, writing the variance operator explicitly, and recognizing that the expectation of the first increment is 0 since the field is second order stationary yields: 2y(h) = E[{Z(x+h) - Z(x)}-fZ(x+h) - Z(x)}] Equation 4.7 Expanding the product, taking the expectation of each term, recognizing that E[Z(x)2J is equal to E[Z(x+h)2] since the hydraulic conductivity field is second order stationary gives: y(h) = E[{Z(x)}2] - EfZ(x)-Z(x+h)J Equation 4.8 Introducing m2 and -m2 on the right side of the equation yields the desired expression. y(h) = fE[{Z(x)}2] + m2} - {EfZ(x)-Z(x+h)J + m2} Equation 4.9 Substituting the expressions for C[0] and C[h] given by Equations 4.3 and 4.4 results in the relationship between the semi-variogram and the covariance function: y[h] = C[0] - C[h] Equation 4.10 Equation 4.10 states that the semi-variogram is a mirror image of the covariance function, shifted upward by the constant value C[0J. The concept is illustrated in Figure 4.7. When C[0J exists, it is a trivial matter to switch from one parameter to the other. Chapter 4 34 GEOSTATISTICS 4.3 VARIOGRAM MODEL Analysis of the correlation structure begins with construction of an experimental semi-variogram or covariance function from field data. A mathematical function that duplicates the essential features of the experimental semi-variogram is then selected to represent the observed correlation structure. A number of parametric estimation techniques including maximum likelihood and restricted maximum likelihood can be used to automate the parameter selection process (Kitanidis, 1983, Kitanidis and Lane, 1985, Wagner and Gorelick, 1989, Jury and Russo, 1989). However, experiments conducted by this author (see Section 4.4.7) have demonstrated that simulations of the types of correlated hydraulic conductivity fields being considered here are not sensitive to the exact shape of the semi-variogram function as long as the function is reasonable. Since, in this case, the additional accuracy of parametric estimation methods is not required, a simpler manual model matching approach to semi-variogram fitting is adopted. This section presents the algorithm utilized in SG-STAT3 to calculate the experimental semi-variogram. Several paragraphs are devoted to explaining irregular behaviour of the experimental semi-variogram that is frequently observed at lags approaching the maximum dimension of the flow domain. 43.1 EXPERIMENTAL SEMI-VARIOGRAM The experimental semi-variogram is constructed by defining 10 to 20 contiguous lag intervals Hk at which the semi-variogram function will be evaluated. Each lag interval is selected so that it spans approximately 1/10 to 1/20 of the longest dimension in the flow domain. If data is collected on a regular grid pattern, a lag interval equal to the distance between adjacent samples is selected. Since the demonstration data in Figure 4.4 is discretized on a 10x10 m grid, semi-variogram intervals are selected as follows: Hy 5~15, H2 15->25, H2 25-35 ... H21 205->215. For each interval if„ the average of all point semi-variograms Ypotoij = {Z(xJ - Z(xp)}>{Z(xd - Z(xJ} Equation 4.11 whose lag falls within the interval Hk is taken to represent the value of the experimental semi-variogram for lag Hb applied at the midpoint of the interval. Figure 4.8 is a graph of the experimental semi-variogram calculated for the demonstration data in Figure 4.4. The sill value, selected to pass through the middle of the sinuous semi-variogram trace observed at longer lags, is approximately equal to the sample variance, 2.78. The range is picked at 50 m, the lag at which the experimental semi-variogram is first equal to the sill. The correlation distance indicated by the range is consistent with the distance over which hydraulic conductivity appears to be correlated in Figure 4.3. The shape of the experimental semi-variogram reveals a wealth of information about the correlation structure of the corresponding hydraulic conductivity field. Figure 4.9 illustrates some frequently observed semi-variogram characteristics. The physical significance of each pattern is discussed below:4 Parabolic Behaviour at Origin: Characteristic of very regular spatial variability, parabolic behaviour is observed when adjacent samples are strongly correlated. Linear Behaviour at Origin: Frequently observed, this behaviour occurs when the correlation structure is somewhat irregular or when the sampling interval is too coarse to capture details of the correlation structure. 3 SG-STAT is a geostatistical software package developed by the author as part of this thesis research to analyze hydrogeologic data. 4 Journel (1978) provides a more complete treatise on this subject. Chapter 4 35 GEOSTATISTICS Figure 4.8 Experimental Semi-Variogram for Demonstration Profile 5.0 - i Lag h ( m e t r e s ) Discontinuity at Origin: Indicates presence of nugget effect; even samples taken very close together are not perfectly correlated. Nugget effect can be caused by measurement errors or small scale variability in the log hydraulic conductivity field. Pure Nugget Effect: Indicates total lack of correlation between adjacent log hydraulic conductivity measurements. Equivalent to white noise phenomenon in physics. Encountered in hydrogeology when permeability tests are performed using a sampling interval that is longer than the range of the geologic deposit. SiO: Confirms that hydraulic conductivity field is 2nd order stationary. No trend is present. Semi-Variogram Increasing at Large Lags: Suggests that a trend exists in the hydraulic conductivity field or the range is significantly larger than the maximum dimension of the domain sampled. Sinuosity: Also observed in Figure 4.8, sinuosity reflects the waviness or periodicity of the hydraulic conductivity field. For example, the semi-variogram peak at 70 m in Figure 4.8 occurs because many of the point semi-variograms Y point ij for this lag are taken between the extreme low conductivity zone in the upper left corner of Figure 4.3 and the very high conductivity zone in the central portion of the geologic profile. Sinuosity of the semi-variogram becomes less pronounced when samples are collected over a larger area since local periodicity effects cancel and any two maxima/minima cannot dominate. Domain Size Effects: Can lead to significant distortion of the experimental semi-variogram at large lags approximately equal to the maximum distance of the flow domain. For example, hydraulic conductivities in both the extreme left and extreme right of Figure 4.3 are close to the mean value of -4. Since there appears to be strong correlation between these points the experimental semi-variogram is very low. On the other hand, if a maximum is present at one end of the domain, and a minimum at the other, the experimental semi-variogram will shoot up well above the sill. As a practical rule of thumb, the semi-variogram model should be fitted only to experimental semi-variogram points that are based on at least 30 supporting pairs of point-semi variograms. As an additional safeguard, the semi-variogram model should not be considered reliable beyond V6 D ^ , the maximum dimension of the flow domain (JourneL 1978). Chapter 4 36 GEOSTATISTICS [Figure 4.9 Semi-Variogram Behaviour at Origin & Large Distances P a r a b o l i c B e h a v i o u r a t O r i g i n S i l l L i n e a r B e h a v i o u r a t O r i g i n D i s c o n t i n u i t y a t O r i g i n ( N u g g e t E f f e c t ) P u r e N u g g e t E f f e c t S e m i - V a r i o g r a m I n c r e a s i n g a t L a r g e L a g s S i n u o s i t y B o u n d a r y E f f e c t s I Chapter 4 37 GEOSTATISTICS 432 VARIOGRAM FUNCTIONS To apply information about the nature of the correlation structure contained in the experimental semi-variogram to problems of estimation and simulation, it is necessary to describe the experimental semi-variogram in terms of a continuous mathematical function that can be evaluated at any lag A. Geostatistical practitioners have found that in most cases a reasonable fit to the experimental semi-variogram can be obtained by selecting the corresponding model from a limited set mathematical functions that includes: linear, spherical, exponential, Gaussian, power, and nugget models. The standard semi-variogram models can be combined to fit complex experimental variograms for which a satisfactory match cannot be obtained with a single semi-variogram model. For example, the nugget model could be combined with a spherical model to reproduce the semi-variogram structure illustrated in Figure 4.7. Not all functions f(h) provide suitable semi-variogram or covariance models. To be acceptable, -y(h) must be conditionally positive definite and if C[h] exists, it must be positive definite. Consider making an estimate of the true log hydraulic conductivity, Z(x), by any linear combination of / measured data, z(xj: i Z'(x) = S Xi'zfxJ Equation 4.12 To be physically meaningful, the variance of the estimate must be greater than or equal to zero. i i VAR[Z"(x)] = E. S.A-i Xi • (-yfcj) z 0 Equation 4.13 Appendix A.2 provides the derivation of positive definite conditions and additional insight. Equations for the six standard semi-variogram models are presented below; suggestions for initial parameter settings are also given. Needless to say, each of the standard models satisfy the conditionally positive definite requirement. Linear Model- y(h) = A-h A £ 0 The linear model fits well to experimental semi-variograms that are intrinsic only, without a well defined sill. The model can also be used in combination with other semi-variogram models to simulate an increasing experimental semi-variogram at large lags. An increase in coefficient A steepens the semi-variogram profile. A suitable starting point is a 2 / / )^ . Spherical Model y(h) = a2 • (1.5-h/a - 0.5 • c3/^3) 0 <.h <. a y(h) = a 2 a > h The spherical model provides a suitable match for experimental semi-variograms with a well developed sill CfOJ. It is the only standard model that does not continue to increase at large lags by definition. When fitting this model, a 2 should be set to the observed sill and a to the observed range. Exponential Model: y(h) = a2 • (1 - exp(-h/a)) a > h The exponential model can be used to simulate most experimental semi-variograms that increase rapidly at the origin and then gradually flatten out. Although the exponential model attains a 2 only at infinity; for all practical purposes, a sill occurs at h~3-a, where the variogram is already equal to 95% of the sill value. If the experimental semi-variogram exhibits a sill a suitable starting point can be achieved by setting a 2 to the observed sill value and a to 1/3 of the range. If a sill is not present, set a to 1/3 and a2 to the maximum value of the experimental semi-variogram. Gaussian Model: y(h) = a2-(l - exp(-hl/a-)) a> h Chapter 4 38 GEOSTATISTICS The Gaussian model increases very gradually at the origin and then rises quickly to a sill at h "2a. The model is suited to simulating experimental semi-variograms that exhibit parabolic behaviour at the origin, suggesting that a very strong and regular correlation structure is present. Setting a 2 equal to the experimental sill and a equal to V2 the range generally results in a good starting point for the fitting procedure. PowerModet y(h) = a-hb a > h The power model is generally reserved for experimental semi-variograms that do not attain a sill over the lag distances sampled. The power coefficient, b, controls how quickly the semi-variogram model levels off to a sill. Values of b less than 0.6 are generally required if the experimental variogram levels off at large lags. Also, b must always be less than 2 since values larger than 2 result in a variogram model that is not positive definite. Reasonable starting values for this semi-variogram are: b=0.5, a=a2/(Dma)v'. Nugget Effect Modet y(h) = C n u g g t t The nugget model is used almost exclusively in conjunction with other semi-variogram models to simulate a discontinuity at the origin of the experimental semi-variogram. When a discontinuity is present, the nugget variance C n u g g e t is set equal to the y-intercept of the experimental semi-variogram. One of the other standard models is then added to the nugget semi-variogram to simulate the experimental behaviour at non-zero lags. Figure 4.10 illustrates the characteristic shapes of the standard semi-variogram models. The standard model that best matches the shape of an experimental semi-variogram is selected to represent the correlation structure of the sampled log hydraulic conductivity field. Once a suitable model is selected, the model parameters are adjusted until the model results in a good fit to the experimental semi-variogram points. In some instances, semi-variogram models have to be combined in a composite model in order to achieve a good fit. In Section 4.6 it will be shown that estimation and simulation of hydraulic conductivity fields is not sensitive to the exact shape of the semi-variogram, as long as it captures the dominant features of the experimental data. Chapter 4 39 GEOSTATISTICS [Figure 4.10 Standard Variogram Models Is a tS u ao o <a > 1 a L I N E A R MODEL 3 1 , 2 0 Is a an o "C d > I 1 0 0 0.00 0.80 0.40 H n 50 Lag h S P H E R I C A L MODEL 1.20 40 50 E X P O N E N T I A L MODEL 1.20 —i 0.80 0.40 0.00 40 ~1 50 Is G A U S S I A N M O D E L .a Is a ID t. 00 o • c > I 1 S) CO 1.20 0.80 0.40 -i 0.00 i^s a ao o > i a CO 1.20 - i 0.80 H 0.40 H 0.00 u ao o "C a) > I a a co 1.20 -i 0.80 0.40 H 0.00 ~T~ 40 POWER M O D E L 50 20 30 10 40 Lag h N U G G E T M O D E L 50 10 I 20 30 Lag h 40 50 Chapter 4 40 GEOSTATISTICS 4.4 ESTIMATION Numerical models of groundwater flow, such as the finite element model used in this framework to predict pore pressures in the pit wall, require that the hydraulic conductivity field be specified at every point in the flow domain. Pump tests, slug tests, and virtually all other subsurface measurements provide only local parameter values that are valid at discrete points or over small volumes relative to the size of the flow domain. The primary objective of such measurements however, is not to ascertain the exact value of hydraulic conductivity at a point; rather, it is to provide hard data that can be used to estimate hydraulic conductivity values everywhere in the flow domain. Today, the hydrologist can select from a large number of averaging methods that have been designed to estimate a continuous function from a set of such discrete measurements. The methods include hand contouring, method of minimum curvature, method of polygons, inverse distance weighting and kriging. Of these, kriging is the estimation method of choice. Unlike other estimation strategies that assign weights based exclusively on the relative geometry between sample and prediction point, kriging weights are influenced by both sample geometry and the correlation structure of the parameter being predicted. This section begins with an introduction to the philosophy of kriging, followed by a brief discussion of why it is the Best Linear Unbiased Estimator. The mechanics of kriging are then introduced, together with a summary of some of the strengths and weaknesses of the method, and suggestions as to when the method should and should not be used. One of the biggest advantages of kriging is that the variance of estimation errors, calculated as part of the kriging process, indicates the degree of uncertainty associated with each kriging estimate. This section develops an expression for the variance of estimation errors and examines how the equation is affected by changes in sampling strategy. A verification of the Kriging program developed as part of this thesis research and several practical examples based on the familiar demonstration problem first introduced in Section 4.2 and on the Avra Valley Aquifer in southern Arizona are presented in the final subsection of this section. 4.4.1 PHILOSOPHY OF KRIGING Given a set of log hydraulic conductivity measurements z(xj, the objective of estimation is to use the data to estimate as accurately as possible actual log hydraulic conductivity values, Z(X) at locations where no measurements exist given available supporting data. The more sophisticated estimation methods define the estimator, Z'(X), as a weighted combination of available data: Z"(X) = A.J -zfrj + X2 •z(x^) +X3-z(X3) + X^(xO Equation 4.14 Figure 4.11 Location of Estimation Volume and Supporting Measurement Points 30-, S 20 -1 j w 10-5 • 3 1 • Xm 2 4 A 0 0 10 20 30 40 50 60 70 Horizontal Distance Chapter 4 41 GEOSTATISTICS The difference between the various estimation methods lies in how the weights, Xt are determined. Presented below is a list of desirable attributes that should be possessed by the estimation method if it is to yield the best possible estimates. Figure 4.11 portrays an array of sample locations xt surrounding a central prediction volume centred on xm. This figure will be referenced on several occasions to illuminate the most important concepts. • The method should be unbiased. On average, the estimation error, {Z'(x) - Z(x)}, should be equal to zero. • The method should honour data. Measured values z(xj should be reproduced by the estimate Z*(xJ. • The weighting scheme should be influenced by the distance between measurements and the prediction volume. Measurements closest to the prediction volume should be assigned larger weights (e.g. X1 > X2 on Figure 4.11). • The weighting scheme should be influenced by the correlation structure of the log hydraulic conductivity field, as indicated by the shape of the experimental semi-variogram. More weight should be assigned to a measurement in a strongly correlated medium than to an equi-distant measurement in a medium exhibiting weak correlation. • The weighting scheme should preserve symmetry of sample locations, XA = Xs. • The weighting scheme should be influenced by relative geometry of the measurement points relative to each other. If two measurements are equi-distant from the prediction volume and one sample is isolated while the other is in proximity to other data points, the isolated sample should be weighted more heavily since each sample in the cluster carries much the same information, e.g. X2 > X3. • Samples separated from the prediction volume by a distance greater than the range are not correlated with therefore, they should not influence the prediction, except when contributing to the global mean. Since x6 is well beyond the range (25 m), X6 should be approximately equal to 0. • The method should be able to quantify the uncertainty associated with each estimate. Estimation weights provided by Kriging satisfy each of these requirements. As a preliminary example, Table 4.1 lists the kriging weights for each support point in Figure 4.11. The negative value of X3 appears erroneous at first glance; however, it is correct. Section 4.4.4 explains why kriging assigns negative weights for certain sample configurations. Table 4.1 Kriging Weights for Sampling Array SAMPLE LOCATION: xl x2 x3 x< x5 x6 DISTANCE TO Xm: 5.00 20.00 20.00 20.61 20.61 35.00 KRIGING WEIGHT: 0.8003 0.1436 -0.0486 0.0041 0.0041 0.0963 Chapter 4 42 GEOSTATISTICS 4.42 KRIGING IS BLUE The acronym BLUE stands for Best Linear Unbiased Estimator. This section explores why kriging is BLUE and why kriging generally results in estimates that are more accurate than those of other estimation methods. Kriging is a linear estimator because Z-(X) is formed from a linear combination of supporting data5: i Z'(X) = S X{-z(x) Equation 4.15 i - l The resulting estimates are unbiased because one of the conditions imposed by kriging when calculating the optimal set of estimation weights is that on average, the prediction error should be equal to zero: E[ZT(X) - Z(X)J = 0 Equation 4.16 Kriging is the best linear estimator because it yields estimates that have the smallest possible variance of estimation error, VAR[Z'(X) - Z(X)J. In the following section it will be shown that the unknown kriging weights are determined by solving a system of simultaneous linear equations known as the kriging system. A brief explanation of why the kriging system guarantees the minimum variance of estimation errors will also be presented. 4.43 MECHANICS OF KRIGING This section presents a brief overview of the key steps that lead to the development of the kriging system. For completeness, the full derivation is presented ia Appendix A. As was pointed out in the previous section, the objective of kriging is to find a set of weights X{ that will result in an unbiased estimate Z'(XJ) with the minimum variance of estimation errors. In Appendix A3 it is shown that the unbiased condition leads to the constraint: i S\ X, = 1 Equation 4.17 By invoking the assumption that the log hydraulic conductivity field is 2— order stationary and the definition of covariance given by Equation 4.5, Appendix A.4 derives an expression for the estimation variance at prediction point Xm: i i J o m 2 = CJX* XJ-2{ S A, • CJX^ xj} + S S X{ Xj • {CJXmi xmJ + • a, • Oj/ Equation 4.18 i=l i=l j=l From Equation 4.18 it is seen that am 2 is a function of: • Average covariance, Cm[Xm fX^, over prediction volume Vm. • Average covariance, Cw[Xm ^x^J, between Vm and each supporting data point x^. • Average covariance between the individual data points, C^Jx^ jcmJ • Variance of measurement error at each data point, a*. ' The unknown kriging weights Nonlinear estimators that include higher order terms such as z(x)2 in the expression for Z'(X) are available, but are not used extensively in practice because they are complicated and require more information about the correlation structure of the parameter being estimated. Chapter 4 43 GEOSTATISTICS Each of the average covariance terms above can be evaluated provided the semi-variogram model and the location, size and shape of the prediction and support volumes are known. The desired kriging weights are calculated by solving for the global minimum of o m 2 with respect to each X„ subject to the non-bias constraint given by Equation 4.17. Using the method of Lagrange Multipliers, the non bias constraint can be introduced into Equation 4.18. A set of equations that can be solved for Xt and u is obtained by taking the 1+1 partial derivatives d/dXJa^) and d/d\i(aj), and setting each derivative equal to 0. This procedure results in 1+1 simultaneous linear equations known as the kriging system: i = constant j5i X> ' C"fx<* »*««/ + V + H = CJXm Equation 4.19 imax S X; = 1 Equation 4.20 The kriging system can be expressed in matrix form. For example, for J=3: [cUi.X!) + <7,2 C<x„x 2) C(x l fx 3) 1 l I fc(Xm.x1) ] jccxj.x,) C(x2,x2) + <x22 C(x2 fx3) 1 j • |12 | = |C(Xm,x2) j Equation 4.21 |C(x3,x1) CtXj .Xj ) C(x3,X3> + a 3 2 1 | j l j | |C(Xm,x3) j I 1 1 1 0 J M I 1 J Equation 4.21 is then solved for the unknowns, Xt and u with any linear equation solver subroutine. When the calculated kriging weights are re-introduced into Equation 4.15, the resulting expression for Z'(XtJ becomes the best linear unbiased estimate of the true log hydraulic conductivity over the prediction volume based on available support. To demonstrate how kriging is applied to analysis of real data, imagine that a permeability measurement program is conducted on the geologic profde presented in Figure 4.4. The program consists of packer tests at 15 & 35 m elevations in six vertical bore holes and eleven penneability estimates based on grain size analysis of test pit samples obtained just below the ground surface. Since the open pit mine will be located on the left side of the profde, hole spacings are tighter in that area to provide better geologic control. Figure 4.12 portrays the resulting data array. The shape of the experimental semi-variogram constructed only from sample data resembles the semi-variogram in Figure 4.6 that was constructed from the entire data array. A spherical model with a sill at 0.278 and a range of 50 m provides a close fit to the observed experimental structure. A contour map of the resulting kriged estimate of the log hydraulic conductivity field is illustrated in Figure 4.13. The contour pattern of the estimated field reproduces the actual profile provided in Figure 4.3 reasonably well, especially on the left half of the profile where measurements are closely spaced together, at distances approximately equal to Vi of the range. 4.4.4 PROPERTIES OF KRIGING Smoothing Behaviour: Kriging is a smoothing process. To minimize estimation variance, the kriged surface passes through the central or expected value at each prediction point, never through values that deviate significantly from the mean. Therefore, a kriged log hydraulic conductivity field will always show much less variability than the actual field from which the samples were collected. Figure 4.14 illustrates the smoothing behaviour in one dimension. The solid line represents the true log hydraulic conductivity profile in a bore hole, the dots indicate locations where point measurements were conducted, and the dashed line indicates the resulting kriged profile. Chapter 4 44 GEOSTATISTICS Figure 4.12 Location of Hydraulic Conductivity Measurement Points - Packer Tests & Test Pits o -4 -6 -7 -4 -3 -2 -2 -4 -5 -5 -5 -4 -8 -6 -4 -3 i l l 1 | | H I i l l -2 i l l -4 i l