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The role of groundwater flow in the genesis of stratabound ore deposits : a quantitative analysis Garven, G. 1982

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THE ROLE OF GROUNDWATER FLOW IN THE GENESIS OF STRATABOUND ORE DEPOSITS : A QUANTITATIVE ANALYSIS by Grant Garven B.Sc. The University of Regina, 1976 M.S. The University of Arizona, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENTJOF:' THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in ' THE FACULTY OF GRADUATE STUDIES (Department of Geological Sciences) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © Grant Garven, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ^la.ojlo^oco-^. ^cie^n CJL^ The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Oof-ob^ U , tf??. nrc-6 n/sn TO AUDREY i i ABSTRACT Many conceptual models have been proposed to explain the flu i d - f l o w mechanism responsible for the o r i g i n of carbonate-hosted lead-zinc deposits such as those in the Mi s s i s s i p p i Valley and at Pine Point. This study i s devoted to the quantitative investigation of one ore-genesis mechanism : gravity-driven groundwater-flow systems. Numerical modeling techniques are used to develop a self-contained computer code for two-dimensional simulation of regional transport processes along cross sections through sedimentary basins. The finite-element method i s applied to solve the steady-state, f l u i d - f l o w and heat-transport equations, and a moving-particle random-walk model i s developed to predict the dispersion and advection of aqueous components. The program EQ3/EQ6 i s used to compute possible reaction-path scenarios at the ore-forming s i t e . F u l l integration of geochemical calculations into the transport model i s currently impractical because of computer-time l i m i t a t i o n s . Results of a s e n s i t i v i t y analysis indicate that gravity-driven ground-water-flow systems are capable of sustaining favorable f l u i d - f l o w rates, temperatures, and metal concentrations, for ore formation near the thin", edge of a basin. Dispersive processes render long-distance transport of metal and sulfi d e i n the same f l u i d an unlikely process i n the genesis of large ore deposits, unless metal and sulf i d e are being added to the f l u i d along the flow path. The transport of metal i n sulfate-type brines i s a more defensible model, i n which case the presence of reducing agents control the location of ore deposition. Hydrodynamic conditions that could result.--in ore formation through mixing of two f l u i d s are rare. i i i i v The theoretical approach i s a powerful t o o l for gaining insight into the r o l e of f l u i d flow i n ore genesis and i n the study of s p e c i f i c ore d i s t r i c t s . A preliminary model of the Pine.Point deposit suggests paleoflow rates on the order of 1.0 to 5.0 m3/m2 y r , paleoconcentrations of zinc on the order of 1.0 to 5.0 mg/kg • H^ O, and paleotemperatures i n the range 60°C to 100°C. Under these conditions, the time required for the formation of Pine Point would be on the order of 0.5 to 5.0 m i l l i o n years. TABLE OF CONTENTS Page LIST OF TABLES , . v i i LIST OF ILLUSTRATIONS i x ACKNOWLEDGEMENTS . xv 1. INTRODUCTION 1 2. CONCEPTUAL MODELS FOR THE ORIGIN OF CARBONATE-HOSTED LEAD-ZINC DEPOSITS OF THE MISSISSIPPI VALLEY TYPE 7 Geologic Setting 9 Southeast Missouri D i s t r i c t 9 Pine Point D i s t r i c t 13 Conceptual Models of Ore Genesis 18 Fl u i d Flow Mechanism 18 Fl u i d Composition and Temperature 28 Source, Transport and Pre c i p i t a t i o n of Metals „ 29 Source of Sulfur 33 Timing and Paths of F l u i d Migration 35 Summary •. 36 3. FUNDAMENTALS OF TRANSPORT PROCESSES IN ORE GENESIS 38 Governing Equations 39 Flu i d Flow 40 Heat Transport 45 Mass Transport 48 Equations of State 52 Geochemical Equilibrium and Reaction Paths 56 Coupling of Transport Phenomena 64 Summary of Equations and Assumptions 65 4. DEVELOPMENT OF THE NUMERICAL MODELS 69 Numerical Formulation of the Governing Equations 71 Fl u i d Flow 72 Heat Transport 77 Mass Transport 78 Geochemical Equilibrium and Reaction Paths 85 Solution Procedure 89 The Complete Transport Model. 89 A Simplified Model 92 Model V e r i f i c a t i o n . . . . . . . . . 98 Simulation Example . . . . . . . . . . 103 v v i TABLE OF CONTENTS — Continued Page 5. EVALUATION OF TRANSPORT PROCESSES IN ORE GENESIS : QUANTITATIVE RESULTS . . . . . . . . . . . . . . . . . . . 118 Factors Controlling Fluid-Flow Patterns and Velocity . . . . . 120 Hydraulic Conductivity 121 Temperature and S a l i n i t y Gradients 135 Basin Geometry 152 Factors Controlling Subsurface Temperatures 171 Hydraulic Conductivity 175 Thermal Conductivity . 179 S a l i n i t y 183 Thermal Dispersivity . . . . . . . . . . . . 184 Geothermal Heat Flow 186 Climate 192 Basin Geometry 192 Factors Controlling Mass Transport 198 Hydraulic Conductivity 201 Dispersivity 211 Geology 217 Geochemical Models of Transport and Pre c i p i t a t i o n 223 Metal-Sulfate Brine Models . . 22~5 Metal-Sulfide Brine Models . . . 242 A Preliminary Application to the Pine Point Deposits _ 251 6. SUMMARY AND CONCLUSIONS 278 REFERENCES ' 291 LIST OF TABLES Table Page 1. Characteristics of carbonate-hosted lead-zinc deposits (after Snyder, 1968, and Macqueen, 1976). . 8 2. Equations governing chemical equilibrium i n a system containing t elements, s species, and X minerals (after Wolery, 1979a) 60 . 3. Governing equations 66 4-. Modeling decisions 70 5. Model parameter data for simulation example I l l 6. Outline of s e n s i t i v i t y analysis 119 7. Hydraulic conductivity and i n t r i n s i c permeability range of geologic materials 123 8. Porosity values of geologic materials 124 9. Summary of the velocity components at the reference s i t e for various temperature and salinity-dependent flow models 14-9 10. Thermal conductivity range of geologic materials 180 11. Longitudinal d i s p e r s i v i t y range of geologic materials . . . 213 12. Model parameter data for simulation problem of a reef • structure at depth ::222 13. Geochemical models for stratabound ore genesis. '. (after Anderson, 1978 and Sverjensky, 1981)-. . . . . . . 224 14-. I n i t i a l conditions for the metal-sulfate brine model. . . . 227 15. I n i t i a l conditions for the metal-sulfide brine model. . . . 244 16. Mass of sphalerite precipitated i n reaction with dolomite as a function of s a l t concentration 247 17. Mass of sphalerite precipitated i n reaction with dolomite as a function of temperature . . . . . . . . . 247 v i i v i i i LIST OF TABLES — Continued Table Page 18. Relative masses of minerals p r e c i p i t a t e d i n cooling from 100°C to 60°C (.per kg«H 20) 249 19. Pine Point simulation : input parameter data. . . 263 LIST OF ILLUSTRATIONS Figure Page 1. Geologic structure of mid-continent region and location of cross sections (.after Oetking, Feray and Renfro, 1966) 10 2. Regional cross sections through mid-continent area (after Oetking, Feray and Renfro, 1960; and Bennison, 1978) 12 3. Geologic structure of the Western Canada sedimentary basin (after Bassett and Stout, 1967) 14 4. Regional cross sections through the Western Canada sedimentary basin (after Gussow, 1962; and Douglas, et a l . , 1973) 16 5. Schematic representation of the youthful and mature stages i n the evolution of a sedimentary basin (after Levorsen, 1967) 19 6. Conceptual model of gravity-driven f l u i d flow i n sedimentary basins 23 7. Effect of topography on regional groundwater flow (after Freeze and Witherspoon, 1967 ) 24 8. Effect of geology on regional groundwater flow (after Freeze and Witherspoon, 1967 ) 25 9. Variation i n water density with temperature and NaCl concentration at a depth of 1 km 54 10. Variation i n water v i s c o s i t y with temperature and NaCl concentration at a depth of 1 km 55 11. Vector diagram showing particle-transport components during a single time step At - 82 12. Region of flow and finite-element mesh for two-dimensional analysis of transport processes in sedimentary basins 90 13. Flow chart for s i m p l i f i e d transport code. (.Dashed l i n e s indicate route i s optional.) . . . . . . . . . 94 i x X LIST OF ILLUSTRATIONS — Continued Figure Page 14. Mesh used i n testing moving-particle code against a n a l y t i c a l solution at A-A' 102 15. Comparison of moving-particle random-walk solutions of three-different treatments of p a r t i c l e s crossing no-flow boundaries with a n a l y t i c a l solution 104 16. Simulation example 10 5 17. Finite-element mesh and basin dimensions for s e n s i t i v i t y analysis 122 18. Homogeneous-isotropic basin showing relationship of f l u i d v e l o c i t y and hydraulic conductivity 125 19. Effect of anisotropy on f l u i d flow K x x / K z z = 1 0 0 • • • • 128 20. Effect of layering on f l u i d flow 130 21. Effect of aquifer lens i n upstream end of basin . . . . . 133 22. Effect of aquifer lens i n downstream end of basin . . . . 134 23. Fluid-flow pattern where density i s a function of temperature alone 136 24. Fluid-flow.pattern where v i s c o s i t y i s a function of temperature alone . . . . . 138 25. Effect of temperature-dependent f l u i d properties on regional f l u i d flow 14-0 26. Velocity p r o f i l e i n discharge end of basin showing the effect of temperature-dependent f l u i d flow . . . 14-1 27. Fluid-flow pattern where density i s a function of s a l i n i t y alone 14.3 28. Fluid-flow pattern where v i s c o s i t y i s a function of s a l i n i t y alone. . . . . . . 144 29. Effect of salinity-dependent f l u i d properties on regional f l u i d flow 14-5 30. Velocity p r o f i l e i n discharge end of basin showing the effect of salinity-dependent f l u i d flow. . . . . 14-6 x i LIST OF ILLUSTRATIONS — Continued Figure Page 31. Effect of combined temperature and salinity-dependent f l u i d properties on regional f l u i d flow. . . . . . . 148 32. Variation i n hydraulic-head patterns and f l u i d v e l o c i t y as a function of s a l i n i t y gradient 150 33. Dimensionless cross section showing the relationship between regional f l u i d flow and basin size 153 34. Hydraulic-head patterns as a function of the length-to-depth r a t i o of a 3 km-thick basin 155 35. Hydraulic-head patterns and f l u i d v e l o c i t y as a function of the length-to-depth r a t i o of a 300 km-long basin 157 36. Regional f l u i d flow i n a flat-bottom basin 160 37. Regional flow patterns i n a basin containing a basement arch with a r e l i e f of 500 m 161 38. Regional flow patterns i n a basin containing a basement arch with a r e l i e f of 1000 m .^ 163 39. Effect of basement structure on the f l u i d velocity i n a basal aquifer 164 40. Regional f l u i d flow with a break i n the water-table slope near the basin margin 166 41. Regional f l u i d flow with a water-table ridge at the basin margin 167 42. Regional f l u i d flow with a water-table ridge i n the basin i n t e r i o r 169 43. Regional f l u i d flow with a water-table depression i n the basin i n t e r i o r 170 44. Regional f l u i d flow with an irregular water-table configuration 172 45. Steady-state temperature patterns showing the effect of convection for various hydraulic conductivities . 176 46. Effect of hydraulic conductivity on the temperature along a basal aquifer. . . . 178 x i i LIST OF ILLUSTRATIONS — Continued Figure Page 47. Temperature patterns i n a thermally homogeneous basin for various values of thermal conductivities . . . . . . . . . . . . . . 181 4-8. Temperature patterns i n a thermally nonhomogeneous basin with two layers 182 49. Temperature patterns i n a two-layer basin for longitudinal thermal d i s p e r s i v i t y values of 1 m and 1000 m . . . , 185 50. Effect of longitudinal d i s p e r s i v i t y on the numerical s t a b i l i t y of high-convection regimes 187 51. Effect of transverse d i s p e r s i v i t y on subsurface temperatures 188 52. Effect of the geothermal heat f l u x on subsurface temperature;. 190 53. Temperature p r o f i l e i n the shallow end of the basin as a function of the geothermal heat f l u x 191 54. Effect of the prescribed water-table temperature on regional temperatures 193 55. Control of basin size and thickness on subsurface temperature.-1 194 56. Control of basement structure on subsurface temperature 196 57. Temperature along the basal aquifer as a function of basement structure 19 7 58. Control of water-table configuration on subsurface temperature 199 59. Mass transport pattern i n a two-layer basin . . . . . . . 203 60. Relationship between the average basin-residence time for a metal-bearing pulse and the hydraulic conductivity of the basal aquifer, i n a two-layer basin 204 61. Maximum solute concentration i n the basal aquifer as a function of transport distance . . . . . . . . . . 207 x i i i LIST OF ILLUSTRATIONS — Continued Figure Page 62. Effect of hydraulic conductivity contrast between aquifer and aquitard on the mass-discharge pattern at the water table . . . . . . . . 209 63. Copper content of the Nonesuch Shale and conceptual model of f l u i d flow i n the Lake Superior basin at the time of formation of the White Pine copper deposit (modified after White, 1971). . . . . 210 64. Effect of isotro p i c and homogeneous d i s p e r s i v i t y on the mass-transport patterns i n a two-layer basin . 215 65. Effect of longitudinal d i s p e r s i v i t y on mass transport patterns . . . . 216 66. Maximum concentration i n the basal aquifer as a function of longitudinal d i s p e r s i v i t y 218 67. Idealized cross sections of several, geologic. configurations of possible importance to stratabound ore formation 220 68. Transport simulation showing the influence of a reef structure at depth 221 69. S t a b i l i t y f i e l d s of sulfur species as a function of pH and oxidation state f (after Anderson, 1978) 229 70. Sulfate-sulfide concentration relationship as a function of oxygen fugacity and temperature (after Helgeson, 1969) . . 231 71 Lead and zinc concentrations i n a 3.0 m NaCl brine as a function of sulfide concentration and pH (after Anderson, 1973) . . . . . 232 72. Effect of s a l i n i t y on zinc concentration (.after Anderson, 1973 and Barnes, 1979) 233 73. Effect of temperature on sphalerite s o l u b i l i t y (modified from Anderson, 1973) . . . . 235 74. EQ3/EQ6 simulation of the reaction of a metal-sulfate brine with H„S . . . . . . . . . . 237 xiv LIST OF ILLUSTRATIONS — Continued Figure Page 75. EQ3/EQ6 simulation of the reaction of a metal-sulfa'te brine with CH^ . . . . . . . . . . . . . . . 239 76. EQ3/EQ6 simulation of the reaction of a metal-sulfate brine with, p y r i t e . . 241 77. EQ3/EQ6 simulation of the reaction of a metal-sul f i d e brine with dolomite 245 78. EQ3/EQ6 simulation showing the e f f e c t Q o f cooling the metal-sulfide brine from 100 C to 60 C . . . . . 79. Map of the Great Slave Lake region showing the location of Pine Point, the Keg River b a r r i e r , and the section l i n e s A-A" and B-B". Geology i s after Law (.1971) . . . . . . . . . 253 80. Cross section A-A"*, transverse to the barrier complex (after Macqueen et a l . , 1975). . . . . . . . 254 81. Cross section B-B", longitudinal to the barri e r complex from Fort Nelson, B.C. to Pine Point, N.W.T 255 82. Pine Point model : (a) Hydrostratigraphy,. (b) Finite-element mesh, (c)-(e) Fluid-flow r e s u l t s , (f)-(h) Heat-transport r e s u l t s , (.i )-(.&) Mass-transport results . . . . . . . . . . . 257 83. Hydraulic-head pattern i n north-central Alberta , ( (.after Hitchon 1969a, 1974) 266 84. F l u i d v e l o c i t y and temperature at the Pine Point s i t e as a function of the horizontal hydraulic conductivity of the barrier complex 269 85. Geothermal gradient i n central Alberta showing the effects of regional f l u i d flow (after Deroo et a l . , 1977) 271 86. Relationship of s p e c i f i c discharge, zinc precipitated, and duration of mineralization for the Pine Point deposit. ..." 276 ACKNOWLEDGEMENTS It i s a great pleasure to express my appreciation to Allan Freeze for his invaluable advice, continuous encouragement, and generous support. I am p a r t i c u l a r l y indebted to him for suggesting the research t o p i c , and for the day-to-day interactions that we have shared. Many hours of discussion were shared with Gordon J.amieson, Keith Loague, Tom Nicols, and Jennifer Rulon, which are also sincerely appreciated. It i s also a pleasure to acknowledge my discussion with Craig Bethke, Jon C o l l i n s , Colin Godwin, Roger Macqueen, and Denis Norton. Brian Hitchon of the Alberta Research Council provided s a l i n i t y cross sections of Alberta and Keith Williams of the Geological Survey of Canada supplied useful geologic information on the D i s t r i c t of Mackenzie. I especially wish to acknowledge the help of Thomas Wolery of the Lawrence Livermore Laboratory for providing access to his geochemical models EQ3/EQ6. The manuscript benefited from the valuable suggestions of Tom Brown, John Sharp J r . , and Le s l i e Smith. This research was supported by a grant to Allan Freeze from the Natural Science's and Engineering Research Council of Canada. The University of B r i t i s h Columbia Computing Centre contributed substantial computer time that helped complete the project. I am also grateful for the scholarship support provided over the period 1979-1982 by the Natural Science and Engineering Research Council of Canada. xv CHAPTER 1 INTRODUCTION Numerical modeling i s used i n t h i s thesis to gain insight into the role of groundwater flow i n the genesis of ore deposits i n sedimentary basins. Although t h i s approach to the problem i s new, the idea that warm, meteoric-derived f l u i d s are responsible for the o r i g i n of many types of ore deposits has been around for over a century. Daubree (1887) concluded that hot water was the most important agent i n the formation of ore deposits. He believed these waters were not always of magmatic o r i g i n , but could be meteoric waters that become heated at depth. His theories presuppose the existence of large-scale groundwater-flow systems. Since the early theories regarding ore genesis, the orig i n s of hydrothermal f l u i d s have been long debated. Three possible sources exist for the warm, saline brines that are known to have formed ore . deposits: magmatic, connate, and meteoric. Connate waters refer to those f l u i d s that are trapped i n sediments at the time of sedimentation. Compaction of sedimentary column may eventually cause the expulsion of these f l u i d s i n subsiding, sedimentary basins. Meteoric waters usually refer to those f l u i d s that are an active part of the earth's hydrologic cycle, and have originated at the land surface. Both connate and meteoric waters can take on a wide variety of chemical compositions, which depend on the o r i g i n a l f l u i d composition, rate of flow, and mineralogy of the porous media, among other factors. The term groundwater i s used here to describe any subsurface water, without reference to o r i g i n or s a l i n i t y . 1 2 White (1968) concluded that, whatever the f l u i d o r i g i n , an ore-forming, hydrothermal system requires four major characteristics: (1) A dispersed source of ore constituents, whether i t be a sedimentary rock sequence, volcanic flows, or igneous intrusives. (2) A f l u i d capable of acquiring ore-forming elements from source beds. (3) Transport of the ore-forming f l u i d may require large distances. (4-) A suitable environment of deposition to s e l e c t i v e l y precipitate ore. Much research has been devoted to the geology and geochemistry implications of these four c h a r a c t e r i s t i c s , and a large suite of case studies and data have been accumulated. Groundwater i s the fundamental f l u i d genetically r e l a t i n g a l l stages of ore deposition and yet modern theories of hydro-geology have not seen widespread application i n the f i e l d of economic geology. Few studies consider the hydrogeologic influence on the transport of ore-forming f l u i d s , and fewer yet assess the flow rates needed to produce an ore deposit. This study relates to a s p e c i f i c type of ore occurrence known as an epigenetic, stratabound deposit. The o r i g i n of such deposits i s widely known to involve the flow of warm, saline f l u i d s through porous media. The term epigenetic means that the f l u i d s deposited the ore minerals after the rocks were consolidated, and the term stratabound i s used because the deposits are generally confined to s p e c i f i c beds i n a stratigraphic succession. The ores do not occur as concentrated layers, but are found as pore-space f i l l i n g s , cave and solution-cavity l i n i n g s , and i n f i l l e d fracture or breccia zones (Stanton, 1972). Small-scale textures and structures genetically distinguish these ores from other types of mineral deposits that occur i n sedimentary beds. The stratabound ore deposits of epigenetic o r i g i n can be grouped into f i v e major associations: 3 (1) Carbonate-hosted lead-zinc (2) Sandstone-hosted uranium-vanadium-copper (3) Shale-and sandstone-hosted copper (4) Sandstone-hosted lead-zinc (5) Conglomerate-hosted gold-uranium In many cases, the ore-bearing strata indicate clear signs that mineraliza-tion i s largely controlled by permeability and porosity. Mineralogy i s usually simple and nondiagnostic. Sphalerite and galena are the dominant sulfides i n many ore deposits, but chalcopyrite, p y r i t e , marcasite and chalcocite are also important. In other deposits, oxides of uranium, vanadium, and copper dominate; and yet i n others native copper, gold, and s i l v e r form major mining d i s t r i c t s . Gangue minerals associated with the ore vary from one group to another. The absence of extensive host-rock a l t e r a t i o n , simple structure and mineralogy, and pore-space f i l l i n g textures provide ample evidence that deposition occurred from aqueous solutions i n an environment of r e l a t i v e l y low temperature and pressure. The association of deposits with major paleoaquifers also suggests that large volumes of ore-forming f l u i d s played an important role i n the formation of the deposits. The carbonate-hosted lead-zinc association and sandstone-uranium association are by far the most abundant stratabound ore deposits. Important North American lead-zinc ores occur i n the upper and middle areas of the Mi s s i s s i p p i Valley, the t r i - s t a t e area of Missouri-Kansas-Oklahoma, the southern Appalachians, the Pine Point area of the Great Slave Lake region, and the Cornwallis Island; region of the Canadian A r c t i c . Sandstone-uranium-type d i s t r i c t s are best represented by the deposits of the Colorado Plateau, the Powder River basin i n Wyoming, and the Grants d i s t r i c t of New Mexico. Both of these ore associations have 4 r e c e i v e d l a r g e amounts of g e o l o g i c a l and geochemical r e s e a r c h , which i n t u r n makes them i d e a l s u bjects f o r h y d r o l o g i c s t u d i e s . For the purposes of t h i s study, however, only the carbonate-hosted l e a d - z i n c deposits w i l l be discussed i n f u r t h e r d e t a i l . The i n t e r e s t e d reader i s r e f e r r e d to Stanton (1972) and Wolf (1976) f o r general references on stratabound d e p o s i t s , and the papers by F i s c h e r (1968, 1970) and Adler (1976) f o r f u r t h e r d e t a i l s on the sandstone-uranium a s s o c i a t i o n s . In s p i t e of the l a r g e amount of f i e l d data on carbonate-hosted l e a d -z i n c d e p o s i t s , the r o l e of f l u i d flow i n ore genesis has not been c l e a r l y e s t a b l i s h e d . I f p a l e o a q u i f e r systems acted as conduits f o r long-distance t r a n s p o r t of ore-forming c o n s t i t u e n t s from wide areas, many questions a r i s e as t o the mechanism of b r i n e formation and movement, l o c a t i o n and area of source beds, p o r o s i t y and p e r m e a b i l i t y e f f e c t s on concentration p a t t e r n s , t i m i n g of m i n e r a l i z a t i o n , and the flow r a t e s r e q u i r e d f o r the d e p o s i t i o n of major ore bodies. There are a l s o f l u i d - r e l a t e d questions w i t h respect t o geochemical supply and geothermal gradients i n sedimentary b a s i n s . T h e o r e t i c a l foundations and numerical modeling techniques are w e l l e s t a b l i s h e d i n both groundwater hydrology and low-temperature, aqueous geochemistry. I t should be p o s s i b l e , t h e r e f o r e , t o perform q u a n t i t a t i v e s t u d i e s of various ore-genesis t h e o r i e s , perhaps even simulate or p r e d i c t ore occurrence. To the w r i t e r ' s knowledge, no one has yet f u l l y q u a n t i f i e d the r o l e of groundwater flow i n stratabound ore formation. Norton and Cathles (1979) have s t u d i e d the c o o l i n g h i s t o r i e s o f i n t r u d e d plutons using f i n i t e - d i f f e r e n c e modeling of f l u i d flow and heat t r a n s p o r t . Their work i s d i r e c t e d t o high-temperature environments accompanying porphyry copper d e p o s i t s . The f i r s t concerted e f f o r t t o apply groundwater-type c a l c u l a t i o n s i n stratabound ore genesis modeling was by 5 White (1971). His analysis consisted of making one-dimensional estimates of the f l u i d flow needed to form the White Pine copper deposit, Michigan. White found that a gravity-based flow system provided more than enough flow to deposit the ore body. Sharp (1978) introduced the power of numerical techniques by solving the coupled equations f o r one-dimensional momentum and energy transport i n compacting basins. He showed that compaction of the Ouachita basin, with subsequent f a u l t i n g , could have generated s u f f i c i e n t flow rates to account for lead-zinc mineralization i n southeastern Missouri. The r e l a t i o n between groundwater flow and uranium mineralization i n the Colorado Plateau has just begun to be assessed through numerical modeling, as demonstrated by Ortiz et a l . (1980) and Sanford (1982). The purpose of t h i s thesis i s to quantitatively evaluate the role of groundwater flow i n the genesis of stratabound ore deposits. Special attention i s paid to lead-zinc deposits i n carbonate rocks for the purpose of i l l u s t r a t i o n , but i t i s f e l t that the results of the study have wide application to other types of stratabound ores. The main goal i s to establish fundamentals on the physics of ore genesis and the possible constraints imposed on the system by hydrodynamics. Numerical modeling i s used to solve the equations governing the flow of variable-density groundwater, heat transport, mass transport of metal or sulfur,:, and multi-component aqueous e q u i l i b r i a . Because of the l i m i t a t i o n of one-dimensional analyses, two-dimensional modeling i s u t i l i z e d . Simulations are f i r s t performed i n simple, hypothetical cross sections so as to id e n t i f y the effects of various model parameters. The complexity of the basin i s then gradually increased, u n t i l i t eventually begins to mirror r e a l sedimentary basins. The role of groundwater flow i s c l a r i f i e d by documenting how variations in certain parameters constrain the flow rates, subsurface 6 temperatures, and concentrations of metal in brines moving through a representative basin. An a d d i t i o n a l goal i s to estimate the necessary and s u f f i c i e n t conditions f o r the formation of a large ore deposit. It i s hoped that t h i s approach w i l l also provide quantitative insight into the v a l i d i t y of a model of gravity-driven groundwater flow f o r stratabound ore deposits. F i n a l l y , the t h e s i s presents a preliminary hydrogeologic analysis of the Pine Point ore deposit, Northwest T e r r i t o r i e s . CHAPTER 2 CONCEPTUAL MODELS FOR THE ORIGIN OF CARBONATE-HOSTED LEAD-ZINC DEPOSITS OF THE MISSISSIPPI VALLEY TYPE Lead-zinc deposits i n carbonate rocks constitute some of the oldest known ore occurrences i n North America and have received a tremendous amount of geologic research. Exploration remains very active, especially i n south-eastern Missouri, Pine Point, N.W.T., and i n the Canadian A r c t i c . According to Stanton (1972), the carbonate-hosted ore deposits are the p r i n c i p a l source of lead and zinc i n the United States and Europe, and contribute a s i g n i f i c a n t supply i n Canada and northern A f r i c a . The number of papers devoted to the so-called M i s s i s s i p p i Valley-type deposits are numerous. Recent reviews on the geology and o r i g i n can be found i n Ohle (1980), and Anderson and Macqueen ( i n press). Individual deposits d i f f e r i n many d e t a i l s , but several features are common to a l l . Table 1 summarizes the important characteristics of carbonate-hosted lead-zinc deposits. The majority of the ore bodies are found i n limestone and dolomite rocks, or nearby sandstone and shale beds. The carbonate host usually occurs as part of a reef structure or bar r i e r complex where dolomitization has played a major role i n c o n t r o l l i n g ore deposition by enhancing the permeability and porosity of the strata. The ore bodies commonly occur i n undeformed s t r a t a , although f o l d structures and breccia zones can act to l o c a l i z e mineralization. Igneous int r u s i v e s , which could be potential sources of ore solutions, are absent. Most deposits are found at shallow depths (less than 1000 m) and near the margins of large, sedimentary basins. 7 8 Table 1. Characteristics of carbonate-hosted lead-zinc deposits (after Snyder, 1968 and Macqueen, 1976) (1) Ore occurrence i n host rock. In limestone and dolomite. Near margins of sedimentary basins. Over basement highs. (2) Structural patterns of d i s t r i c t s . Ore i s not related to major f a u l t s . Minor fractures, dissolution and subsidence control mineralization patterns. (3) Types of ore-bearing structures. Stratigraphic traps. Reefs, bioherms, and associated structures. Breccias, often associated with solution collapse. Fractures, solution channels, and caves. Size of ore body related to size of entraping structures i n carbonate s t r a t a . (4) Mineralogical features. Open-space f i l l i n g s , replacements less important or absent. Galena, sphalerite, b a r i t e , c a l c i t e , dolomite, and quartz are most abundant. F l u o r i t e , c e l e s t i t e , p y r i t e , chalcopyrite, native s u l f u r , and bitumen occur in minor quantities. Mineral zonation i s d i s t i n c t i v e i n some deposits. Sulfur-isotope studies indicate shallow crustal sources, probably biogenic o r i g i n . Paragenetic studies d i f f i c u l t to interpret because of changing f l u i d s with time. F l u i d inclusions are mainly Na-Ca-Cl brines. Temperatures of pr e c i p i t a t i o n range from 50° to 150°C. Solution of limestone prior and during ore pr e c i p i t a t i o n i s common and probably a necessary feature for deposition. 9 In some regions, the location of ore bodies are c l e a r l y related to basement structures such as Precambrian arches. Mineralogy i s simple, with galena, sphalerite, b a r i t e , c a l c i t e , dolomite, and quartz dominating. Secondary i n abundance are f l u o r i t e , c e l e s t i t e , p y r i t e , marcasite, chalcopyrite, native s u l f u r , and bitumen. Fluid-inclusion studies of the s u l f i d e minerals indicate that the ore-forming f l u i d s were basinal brines with s a l i n i t i e s of 10% to 30% equivalent weight as NaCl and at temperatures of 50° to 150°C. The geochemical character of the f l u i d s i s , i n many respects, very s i m i l a r to present-day o i l - f i e l d brines (Hitchon, 1977). Geologic Setting To demonstrate the detailed nature of a carbonate-hosted lead-zinc deposit requires an examination of individual ore d i s t r i c t s . Perhaps more important from the point of view of understanding the role of basinal-brine flow i s the regional setting of the ore deposit with respect to the sedimentary basin. For these purposes, we w i l l now take a b r i e f look at two of the larger deposits i n North America: Southeast Missouri and Pine Point, Northwest T e r r i t o r i e s . Southeast Missouri D i s t r i c t The Southeast Missouri lead deposits occur near the sides of a Precambrian high that crops out as the St. Francois Mountains, about 100 km south of St. Louis. Figure 1 shows the regional geology and location of the mining d i s t r i c t , as well as the associated tectonic features of the mid-continent region of the United States. The deposits are found i n Cambrian and Ordovician strata that overlie a domal structure known as the Ozark u p l i f t . North of the d i s t r i c t , these Paleozoic strata are covered by younger beds that become part of the I l l i n o i s basin i n the northeast, 10 Figure 1. Geologic s t r u c t u r e of mid-continent region and l o c a t i o n of cross s e c t i o n s ( a f t e r Oetking, Feray, and Renfro, 1966). and part of the Forest City basin i n the northwest. In the south, the strata extend into the str u c t u r a l complex sections of the Ouachita Mountains, which were u p l i f t e d i n l a t e Pennslyvanian-Permian (Sharp, 1978). The general stratigraphy of the mid-continent region i s depicted on the cross sections given i n Figure 2. Notice should be made of the shape and thickness of the sedimentary cover. Good descriptions of the geology and o r i g i n of the deposits i n the Southeast Missouri d i s t r i c t can be found i n Snyder and Gerdemann (1968), Gerdemann and Myers (1972), and Davis (1977). The ore-host strata consist of mainly the Bonneterre Formation and underlying Lamotte Formation. The Lamotte Fm. i s a quartz-rich sandstone unit that unconformably rests on the Precambrian surface. I t i s fine-grained,, arkosic, permeable, and attains a thickness of about 100 m. The Bonneterre overlies the Lamotte sandstone beds and i t consists predominantly of dolomite. Reef facies and associated structures'are the main hosts for the lead deposits. Ore bodies range i n siz e up to tens of mi l l i o n s of tons, which are distributed over two major s u b d i s t r i c t s , the Viburnum Trend and Old Lead Belt. Both subdistricts are s p a t i a l l y related to pinchouts, buried ridges, bar reefs, and other features associated with the barrier reef, which formed an island complex i n Late Cambrian time. The Bonneterre Formation varies i n thickness from 100 m to 400 m i n southeast Missouri. About 250 m of dolomite and shale presently overlie the Bonneterre. Galena i s the dominant sulfide mineral i n the ore d i s t r i c t , but sphalerite and chalcopyrite can become l o c a l l y abundant. Galena c h i e f l y occurs as sheets along bedding planes, pore-space f i l l i n g s , and as fracture f i l l i n g s . Davis (1977) estimates that over 35 m i l l i o n tons of lead was deposited i n southeast Missouri. Grades of 10-15% metal are common and Cretaceous Precambrian Nemaha Ridge OZARK UPLIFT ST FRANCOIS MTS. A' Cretaceous , Pennsylvanian B \ Arkoma Basin Devonian-Mississippian B' Q Carboniferous Ordovician -Oevonian Carboniferous C THE OZARKS - A -1 —1— t —' -. -— / Cambrian Ozark Uplift / H Ordovician - Devonian o -i m t- -1000 5 Figure 2. Regional cross sections through mid-continent area (after Oetking, Feray, and Renfro 1960; and Bennison, 1978). ro brecciated zones may exceed 30%. Dolomite i s the most abundant gangue mineral while c a l c i t e and quartz are present i n smaller amounts. Textural data suggest that the ore minerals were precipitated and dissolved r e p e t i t i v e l y during ore formation (Sverjensky, 1981). According to Snyder and Gerdemann (1968), metal-bearing brines were probably driven out of the adjacent sedimentary basins sometime after compaction of the sediments. Flow was through the very permeable Lamotte Sandstone towards the pinchout edge, where flow was eventually forced up into permeable sections of the Bonneterre Formation. Metals precipitated as sulfides on encountering hydrogen s u l f i d e gas i n the porous, carbonate-reef complex. Evidence from f l u i d inclusion studies suggest temperatures of formation of 80° to 120°C (Roedder, 1967; Sverjensky, 1981), and brine s a l i n i t i e s equivalent to 20-30% NaCl. Recently, Sverjensky (1981) has shown that various l i n e s of geologic evidence suggest that base metals were transported together with reduced sulfur i n the same solution to the sit e s of deposition. P r e c i p i t a t i o n could have been caused by an increase i n pH on reaction of brines with the carbonates or possibly because of a temperature drop. Pine Point D i s t r i c t Figure 3 shows the geologic setting of the Pine Point ore deposits i n r e l a t i o n to the Western Canada sedimentary basin. Pine Point i s located on the south shore of Great Slave Lake, about 800 km north of Edmonton, Alberta. As i n the case of the Missouri ores, numerous a r t i c l e s have been written on the geology and o r i g i n of the Pine Point deposits. Some of the c l a s s i c references on t h i s deposit include Beales and Jackson (1966), Campbell (1967), Jackson and Beales (1967), and S k a l l (.1975). Recent work by Kyle (1977, 1980, 1981) are the most current contributions 14-A laska KILOMETERS Figure 3. Geologic structure of the Western Canada sedimentary-basin ( a f t e r Bassett and Stout, 1967). i n the l i t e r a t u r e . The Pine Point d i s t r i c t occurs near the present-day, eastern edge of the Western Canada sedimentary basin where over 500 m of Paleozoic strata rest on c r y s t a l l i n e rocks of the Precambrian basement. Norris (1965) reported that up to 100 m of lower Paleozoic (Ordovician?) red beds and evaporitic dolomite overliethe g r a n i t i c basement i n the Great Slave Lake region. Above the red beds i s a complex succession of Middle Devonian evaporites and carbonates, which include the ore-bearing dolomite fa.cies of a ba r r i e r complex (Figure 3). C o l l e c t i v e l y , these beds can reach a thicknes i n excess of 400 m. A l l of the Paleozoic formations dip gently to the west into much thicker sections of the basin, as shown by the two regional sections i n Figure 4. The l i n e s of the cross section are given i n Figure 3. To the west, the Middle Devonian strata become overlain by thick shale beds of Upper Devonian age, which i n turn are buried by Late Paleozoic and Cretaceous st r a t a . Descriptions regarding the geology of western Canada can be found i n McCrossan and Glaister (1964). The Keg River Barrier separated a deep-water environment of the Mackenzie basin from the back-reef evaporite-forming conditions of the Elk Point basin during the Devonian period. S k a l l (1975) provides a detailed interpretation of the b a r r i e r at Pine Point, and Williams (1981) has prepared several geologic maps of the complex, facies patterns associated over the entire length of the b a r r i e r . At Pine Point, the barrier-complex strata overlie about 100 m of anhydrite and gypsum of the Chinchaga Formation The platform of the barrier i s b u i l t on a 50 m t h i c k , uniform dolomite known as the Keg River Formation. The carbonate-barrier complex i s represented by several d i s t i n c t facies that are c o l l e c t i v e l y known as the Pine Point Group ( S k a l l , 1975). This i s the main ore-bearing zone and i t reaches a thickness of 150 m. The upper portion of the Pine Point Group contains a - 3 0 0 0 J Figure 4-. Regional cross sections through the Western Canada sedimentary basin (after Gussow, 1962; and Douglas, et a l . , 1973). CJ> highly cavernous f a c i e s , better known as the Presqu'ile Facies. Over 50 massive and tabular ore bodies have been found i n the bar r i e r complex, over an area of 1000 km2 (Kyle, 19.80). Ore-control i s largely influenced by paleo-solution features i n the k a r s t i c facies. Above the Pine Point Group are about 75 m of shale and dolomite of the Watt Mountain and Slave Point Formations. Sphalerite i s the predominant sul f i d e mineral at Pine Point, followed by galena, pyrite and marcasite. Dolomite i s ubiquitous and other gangue minerals include c a l c i t e , c e l e s t i t e , anhydrite, and gypsum. Native sulfur i s present, and pockets of heavy o i l and bitumen are commonly encountered. Sphalerite occurs as encrustations i n c a v i t i e s , as colloform features, and disseminations (Jackson and Beales, 1967). Galena generally tends to form coarser, cavity f i l l i n g s than sphalerite. Kyle (.1977) believes the sphalerite and galena precipitated together, but with the sphalerite probably s t a r t i n g s l i g h t l y e a r l i e r . Approximately 100 m i l l i o n tons of lead-zinc ore i s present at Pine Point, and metal grades range around an average of 9% (Kyle, 1980). Jackson and Beales (1967) proposed that the barrier complex acted as the main conduit for transporting metal-rich brines from shales compacting in the Mackenzie basin, and sulfur from the evaporites of the Elk Point basin. Reduction of sulfur to H^S^as) caused p r e c i p i t a t i o n of the ore i n the reef complex near Pine Point, because of the abundant supply of organic matter i n the area. This scenario of ore genesis f i t s quite well with the observed f i e l d data of mineralization at Pine Point. The work recently presented by Kyle (1980, 1981) also supports the basinal brine hypothesis advanced by Jackson and Beales. Kyle suggests that ore occurrence i s strongly controlled on a l o c a l scale by the location of breccia zones that allowed mixing between paleokarst aquifers. Roedder (1968) has estimated, from f l u i d - i n c l u s i o n studies, that the ore-forming brines were at temperatures of 51° to 91°C and had s a l i n i t i e s 10%-23% NaCl. Conceptual Models of Ore Genesis The topic of ore genesis i n sedimentary basins has been the subject of much controversy ever since studies of stratabound deposits began. Many conceptual models have been proposed to explain the o r i g i n of the ores, especially the carbonate-hosted lead-zinc deposits. Ridge C1976) and Davis (1977) outline the chronological development of the various theories. The general model of warm, metal-rich brines flowing through paleoaquifers and depositing ore at si t e s where reduced sulfur was available i s widely accepted. F l u i d inclusion and stable-isotope data, along with many geologic observations, support t h i s basinal-brine model. In spite of the general concensus of t h i s model, Ohle C1980) points out that disagreement s t i l l exists as to: the or i g i n of the brine, the driving mechanism for flow, the role of hydro-dynamics, the means by which metals and sulfur are transported i n solution, the source of reduced s u l f u r , the nature of the source beds of metal, the causes for p r e c i p i t a t i o n , the timing of mineralization, and the reasons why deposits occur where they are now. Ohle's comments emphasize that there i s plenty of room for further research along several avenues. However, there has been progress i n placing constraints on certain aspects of the basinal brine model, and t h i s progress i s now summarized. Fl u i d Flow Mechanism Consider the schematic cross section of a sedimentary basin shown i n Figure 5. It represents two stages i n the evolution of a l a t e r a l l y composite basin. The upper section A-B depicts the structure of the basin DEVELOPMENT OF A LATERALLY COMPOSITE BASIN i - 100 km Figure 5. Schematic representation of the youthful and mature stages i n the evolution of a sedimentary basin ( a f t e r Levorsen, 1967). i—1 to at a youthful stage when sediment loading causes general subsidence. Shallow, ep e i r i c seas encompass the entire region i n the youthful stage. At some la t e r time, gradual and widespread u p l i f t causes the basin to emerge as seas regress from the region. Orogenic movements eventually u p l i f t the "western" end of the basin, which results i n deformation of the sedimentary p i l e and creation of a mountain belt through overthrusting. As orogenesis continues, substantial downwarping may occur, along with deposition of thick continental-type sediments. Section A'-B' i n Figure 5 depicts a period i n geologic time when tectonic a c t i v i t y i n the area i s quiescent and the basin has evolved to a mature stage i n i t s history. A topographic gradient i s established across the basin and surface-drainage systems are well developed. This model of basin evolution applies to a large number of sedimentary basins that Levorsen (.1967) c l a s s i f i e s as l a t e r a l l y composite i n o r i g i n . He discusses several f i e l d examples from a l l over the world, and t h e i r importance as petroleum producers. Many of the sedimentary basins associated with stratabound ore formation are of t h i s type. The sections presented i n Figures 2 and 4 serve as good examples for demonstrating the general stratigraphic character and wedge-shaped nature of these basins. The flow of f l u i d s through sedimentary basins can be caused by the gradients i n a variety of energy potentials (van Everdingen, 1968). The potential gradients most important to f l u i d flow, however, are those components of elevation and pressure that contribute to the f l u i d p o t e n t i a l . Hubbert (194-0) showed that the f l u i d potential represents the mechanical energy per unit mass of f l u i d , and that flow i s always from regions of high energy towards regions i n which i t i s lower. Hubbert also showed that the f l u i d potential at any point i n a basin i s simply the hydraulic head multi-p l i e d by the acceleration due to gravity. Hydraulic head consists of two 21 components: the elevation of the point of i n t e r e s t , and the pressure head at that point. Measurement of the f i e l d d i s t r i b u t i o n of hydraulic head w i l l determine, therefore, the f l u i d - f l o w direction i n porous media. Hitchon (1976) has reviewed the history of the development of f l u i d -flow models of sedimentary basins, and t h e i r implications to stratabound ore genesis. In the youthful-basin stage (Figure 5), compaction of sediments i s the primary mechanism for f l u i d movement. Most of the f l u i d - p o t e n t i a l losses by compaction are incurred soon after deposition, with f l u i d flow directed upwards to the sea bottom (or downwards with reference to the depositional surface). Continued sedimentation, basin subsidence, and compaction results in further f l u i d expulsion with b u r i a l . Excess pore-fluid pressures may develop i n permeable sandstone or carbonate strata i f f l u i d flow becomes r e s t r i c t e d by the rapid deposition of low-permeability clays, or other causes. The p o s s i b i l i t y then arises that f a u l t i n g could bleed-off t h i s excess pressure by allowing f l u i d flow up into shallower stratigraphic l e v e l s . The development of excess f l u i d pressures i s a temporary phenomena that w i l l eventually dissipate with time, provided f a u l t i n g does not interrupt the compaction process. A compaction-driven flow model for driving deeply-buried . metal-rich brines into shallow aquifers and out to the basin margin i s commonly cal l e d upon to explain the o r i g i n of carbonate-hosted lead-zinc deposits. Noble (1963) introduced the concept to o r i g i n a l l y explain a wide variety of stratabound ores, and Dozy (1970) l a t e r expanded on the hypothesis for the carbonate-hosted lead-zinc deposits. Jackson and Beales (1967) popularized i t s possible role i n the formation of the Pine Point deposit. Quantitative modeling of compaction i n the Ouachita basin by Sharp (1978) has shown that compaction-driven flow would have been s u f f i c i e n t to account for the ore deposition i n southeast Missouri, provided several pulses of compaction-expelled brines reached the edge of the basin. In the mature stage of a sedimentary basin's evolution, the effects of compaction are l i k e l y to be negligible as the basin i s now u p l i f t e d and a l l of the strata are well indurated (Figure 5). Any transient processes i n i t i a t e d through b u r i a l are also l i k e l y to have long dissipated. F l u i d flow i s now primarily driven through the potential gradient created by the water-table configuration, which i s commonly a subdued r e p l i c a of the basin topography. Groundwater flow i s from areas of higher elevation to regions of lower elevation. The exact character of the flow paths depends upon the size of the basin and i t s shape, the r e l i e f on the water table, and the nature of permeability changes in the subsurface. Figure 6 demonstrates the conceptual picture of the gravity-driven f l u i d - f l o w system i n a large basin. F l u i d flow i s directed across the thick shale sequence and focused into an extensive basal carbonate aquifer, which discharges at the basin margin. Unlike the compaction-driven model, regional gravity-driven flow i s most appropriately treated as a steady-state system. So long as the topography of the basin i s not changing appreciably, compared to the t o t a l thickness of the basin, then the case of dynamic equilibrium (steady state) i s a v a l i d approximation to the f l u i d flow regime (Freeze and Cherry, 1979, p. 194). This condition i s probably s a t i s f i e d i n mature basins, at least for time intervals shorter than a few m i l l i o n years. Quantitative studies of steady-state groundwater flow systems was begun by Toth (1962, 1963), and l a t e r expanded upon by Freeze and Witherspoon (1966, 1967, 1968) who used numerical models to investigate a wide variety of hypothetical cross sections and f i e l d problems. Both studies sought to examine the effects of topography and geology on groundwater flow. Figures 7 KILOMETERS m - f c O J ro — o — ro oj 0.2S-1 (c) DIMENSIONLESS DISTANCE Figure 7. Effect of topography on regional groundwater flow (after Freeze and Witherspoon, 1967). ,^  Figure 8. E f f e c t of geology on regional groundwater flow ( a f t e r Freeze and Witherspoon, 1967), and 8 show some of the modeling results given by Freeze and Witherspoon (1967). The flow nets shown here are of a true-to-scale basin section, which i s of dimensionless length S and thickness of 0.1S. The dimensionless form of the models allow the results to be applied to any size of sedimentary basin with s i m i l a r length-to-depth r a t i o . The dashed li n e s (Figures 7, 8) represent hydraulic-head contours, and the direction of f l u i d flow i s indicated by flowlines. The upper surface of the region of flow represents: the water table, while the bottom represents an impermeable basement. The two v e r t i c a l boundaries are also impermeable and coincide with regional groundwater divides. These occur because of the symmetry imposed on flow systems by valleys and highlands. Fl u i d flow i s from areas of high f l u i d potential (hydraulic head) to regions of lower f l u i d p o t e n t i a l . In Figure 7, the effect of topography i s demonstrated with three models, one with a constant slope on the water table, one with a break i n slope near the right side of the basin, and another with an irregular watery table configuration. A l l of the sections assume a homogeneous and isotropic porous medium. The uniform slope of Figure 7a creates a single flow system across the entire basin. The change i n slope i n Figure 7b produces two flow systems; a regional system and a l o c a l flow system, both of which discharge into the valley depression. Numerous recharge and discharge subsystems are superimposed on the regional system when an irregular topography exists in a basin (Figure 7c). Depending on i t s o r i g i n a l location i n the basin, a water p a r t i c l e containing metal may be discharged after a short distance of transport or carried into the large flow system. Figure 8 shows the effect of geology on regional flow patterns. The r e l a t i v e hydraulic conductivity (permeability) i s indicated by the dimensionless number i n each layer. The main effect of a buried aquifer i s to focus flow down across lower permeability strata i n recharge areas, and discharge flow up across the low permeability beds i n discharge areas. Increasing the permeability contrast between layers increases the hydraulic-head gradients i n the overlying aquitard. Notice i n Figure 8b that the presence of a pinchout causes a discharge of flow near the center of the basin, which would not have been interpreted from considering topography alone (Figure 7b). The combined effect of irregular topography and geology i s shown i n Figures 8c and 8d. Topographic effects w i l l not always dominate flow control i n a basin, especially i f a highly permeable aquifer i s present (Figure 8d). More complicated flow patterns can also arise i n the case of dipping beds (Figure 8e), and i f the rocks are anisotropic with respect to hydraulic conductivity (Figure 8f). The concept of gravity-driven f l u i d - f l o w regimes i n stratabound ore genesis has not received the same attention as that given to the compaction-driven flow mechanism over recent years, at least i n North America. This model proposes that small amounts of ore-forming constituents are leached from the sedimentary section and transported by gravity-driven flow systems to s i t e s of deposition, perhaps at the discharge end of the basin. I t i s clear from Figure 6, that the large volumes of low-permeability shales would constitute potential source beds and at the same time serve to focus f l u i d flow down into carbonate or sandstone aquifers. The action of water-rock reactions, membrane f i l t r a t i o n , and presence of evaporites would easily transform the shallow meteoric waters into brines as they flow through deeper parts of the basin. The main requirement i s that some form of paleotopo-graphic gradient exist i n order to drive the gravity-based flow systems. Cox (.1911) and Siebenthal (.1915) advocated t h i s mechanism for the or i g i n of lead-zinc ores of the Mi s s i s s i p p i Valley d i s t r i c t , and i t i s commonly cit e d to explain the genesis of sandstone-hosted uranium deposits (Butler, 1969; Adler, 1976). Regional hydrodynamic studies by Hitchon (1969a, 1969b) and Toth (1978) of Alberta confirm the importance of gravity-driven cross-formational flow i n sedimentary basins. Toth (1980) has also demonstrated that gravity-induced flow i s the p r i n c i p a l agent i n the transport and accumulation of hydrocarbons i n geologically mature basins. We can conclude, therefore, that both compaction-and gravity-driven flow systems are important mechanisms of flow. The par t i c u l a r model that dominates depends on the stage of basin evolution. Hitchon (1976, 1977) i s the one researcher who has emphasized the importance of both mechanisms i n the o r i g i n of o i l f i e l d s and ore deposits. He believes that compaction may be a more e f f i c i e n t generator of hydrocarbons, and that gravity flow may be more e f f i c i e n t for metals. Models of the type given i n Figures 7 and 8 should help to quantify the importance of groundwater flow systems i n ore genesis. This subject i s treated in greater d e t a i l i n the chapters to follow. F l u i d Composition and Temperature A large amount of f l u i d - i n c l u s i o n data has proven beyond l i t t l e doubt that the carbonate-hosted lead-zinc deposits were formed by metal-bearing brines of sodium-calcium-chloride composition with s a l i n i t i e s of 10-30% NaCl (Roedder, 1976). Some inclusions contain high concentrations of metal and methane, and droplets of o i l are not uncommon. Concentration of reduced sulfur i s generally low. The saline character of f l u i d inclusions has led a number of authors to make comparisons of present-day o i l - f i e l d brines to ancient ore-forming solutions. The work by B i l l i n g s et a l . (1969), Carpenter et a l . (1974), and Hitchon (1977) support the hypothesis that basinal brines are i n fact the ore-forming f l u i d s . The o r i g i n of present-day brines i n sedimentary basins can be attributed to a number of subsurface processes. Suffice to say that the presence of evaporites, water-rock interactions, and membrane f i l t r a t i o n are mainly responsible for the occurrence of brines. Bredehoeft et a l . (1963) discuss a mathematical model for membrane f i l t r a t i o n , which i s capable of producing brine s a l i n i t i e s similar to those observed i n the deep subsurface. Other types of processes may be equally important. The temperature of mineralization can also be estimated from inclusion data. The majority of carbonate-hosted lead-zinc deposits f a l l i n the range of 50°C to 150°C, with temperatures around. 100°C accepted as the average or representative value of t h i s ore-forming environment (Roedder, 1967). At least a few kilometers of sediments are needed to a t t a i n these temperatures under the average geothermal gradients existing today. The dilemma facing us here i s that most of the ores occur at shallow depth, and based on geologic evidence were probably formed at depths less than 2 km. Locally high geothermal gradients at the depositional s i t e could be caused by a heat source, but heat r e d i s t r i b u t i o n due to f l u i d flow from deep i n the basin i s l i k e l y a better explanation (White, 1974; Hanor, 1979). Source, Transport, and Pr e c i p i t a t i o n of Metals Fundamental to the basinal-brine model i s the concept of an o r i g i n a l source bed from which metal species can be expelled into aqueous solutions, either by compaction flow or gravity-based flow. Eventually the metals are transported through aquifers and precipitated at the depositional s i t e . Jackson and Beales (1967) argue that shales make good source rocks because they usually contain more Zn and Pb than other l i t h o l o g i e s . They favored a mechanism where the metal ions would be released from the clays upon compaction of the basin. Macqueen (1976) and Kyle (.1980) make the point that early compaction of sediments result i n large volumes of low s a l i n i t y waters, which are poorer carriers of metals than highly saline brines. Granite, carbonates, evaporites, feldspathic sandstone, and red-bed cements have also been considered as suitable source beds by various authors. Each lit h o l o g y has i t s pros and cons, depending on the geology of the basin i n question. For carbonate-hosted lead-zinc deposits, shales and organic-r i c h carbonates remain the most a t t r a c t i v e source beds for metals. Carpenter et a l . (1974-) concluded that cross-formational flow of potassium-r i c h brines across marine shales i s an important mechanism for the o r i g i n of metal-rich brines. The exact mechanism by which metals are released from source rocks i s unclear. According to Anderson and Macqueen ( i n press), the release of metals may take place through desorption phenomena, r e c r y s t a l l i z a t i o n processes, and thermal alte r a t i o n of metal-organic complexes. Whatever the mechanism, f i e l d observations and experimental studies confirm that chloride-n r i c h brines can leach trace amounts of metals from various rock types , ( E l l i s , 1968; Bischoff et a l . , 1981; and others). The geochemistry of metal transport and p r e c i p i t a t i o n has received a large amount of theoretical and experimental research. Because of t h i s , the geochemistry of hydrothermal solutions has become the most quantitative branch of ore-genesis studies. Excellent review papers on the geochemistry of the carbonate-hosted lead-zinc deposits are provided by Anderson (1978), Barnes (1979), Skinner (1979), and Anderson and Macqueen ( i n press). Skinner (1979) summarizes the recent findings on the general chemistry of hydrothermal f l u i d s . The formation of complex ions i n solution i s now known to be the primary mechanism for transporting metal species i n ore-depositing.flow, systems. Complexing substantially increases the s o l u b i l i t i e s of metals, while also i n h i b i t i n g s u l f i d e p r e c i p i t a t i o n . This provides a means.for the long-distance transport of metals i n f l u i d flow systems. Complexes also show greater increases i n metal s o l u b i l i t y with increasing temperatures. Both chloride complexes and sulfide complexes have been called upon to explain the transport of metals i n ore genesis. Chloride complexes form i n the presence of high concentrations of chloride ion and can carry large amounts of lead and zinc i n solution, so long as s u l f i d e (H^SCaq), HS ) concentrations are low. The brine must be acidic i n character (pH<4) i f appreciable amounts of metal and sulfide are transported i n the same solution, otherwise immediate pr e c i p i t a t i o n r e s u l t s . The important role of chloride complexing i n ore genesis was pioneered by Helgeson (1964). Barnes and Czamanske (1967) proposed that complexes of reduced sulfur provide an alternative model to metal-chloride brines, which would overcome the problem of transporting acidic-type solutions through carbonate aquifers. Sulfide complexes require large amounts of reduced sulfur i n solution to be stable with r e l a t i v e l y low, metal concentrations. Alkaline pH conditions, however, can exist i f metals are transported as b i s u l f i d e complexes. The r e l a t i v e importance of these two types of complexes i s s t i l l debated by geochemists. In both cases, one of the ore-forming components must be at a low concentration in order to transport large quantities of the other component over large distances. Fluid-inclusion data and modern brines almost always indicate low su l f i d e content, but high chloride compositions. These observations, along with e q u i l i b r i a calculations have led a number of researches, including Anderson (19 73), Giordano and Barnes (1979), and Sverjensky (1981), to conclude that metal-bisulfide complexing i s not a geologically s i g n i f i c a n t transport mechanism for the genesis of carbonate-hosted lead-zinc deposits. Anderson (1973, 1975) believes that metal transport i s attributed to chloride complexing i n sulfide-free brines. Sverjensky (1981) argues, however, that metals and sulfide can be transported together i n a solution at equilibrium with carbonates, although metal concentrations w i l l be low. Recent data presented by Giordano and Barnes (1981) indicates that organic-based complexes may also play an important role i n metal transport. The pre c i p i t a t i o n of su l f i d e minerals can be attributed to several mechanisms, which Skinner (.1979) groups into four main . •'. categories: (1) Temperature changes. (2) Pressure changes. (3) Water-rock interaction. (4) Mixing of different solutions. The cooling of an ore-forming brine causes pre c i p i t a t i o n by reducing s o l u b i l i t i e s of metal sulfides and lowering the s t a b i l i t y of the complex ions transporting the metals. A temperature drop of 20°C or more i s needed to remove a l o t of mass out of solution (Skinner, 1979). Temperature drops would have to occur over very short distances i n order to concentrate an ore body. Pressure changes also cause mineral s o l u b i l i t i e s to vary, but pressure effects are probably not important i n the formation of stratabound ore deposits at shallow depths. Chemical reactions between the brine and porous media constitute an important means of pr e c i p i t a t i n g ore deposits. For example, the dissolution of carbonates and hydrolysis of feldspars w i l l cause the pH i n aci d i c brines to r i s e , which reduces the s t a b i l i t y of complexes and results i n p r e c i p i t a t i o n of sul f i d e minerals. The presence of r^S o r other forms of reduced sulfur w i l l also react with metal-rich brines to deposit galena 33 and sphalerite. The effects of carbonaceous compounds are equally important in p r e c i p i t a t i n g metals, mainly through a change i n oxidation state of the f l u i d . They may also take part i n the reduction of sulfate by anaerobic bacteria to produce H S. Other forms of reducing agents such, as methane or pre-existing pyrite can cause ore deposition. The type of water-rock mechanism cal l e d upon to form an ore deposit w i l l depend on the type of geochemical model that i s favored. I f metal and reduced sulfur are trans-ported together, then pH r i s e or temperature drop are l i k e l y candidates. I f metals are transported alone or with sulfate i n a brine, then addition of reduced sulfur or sulfate reduction i s required at the ore-forming s i t e (Anderson, 1978). Mixing of two separate f l u i d s i s a mechanism that avoids the equilibrium constraints imposed on a single-solution model where metal and su l f i d e t r a v e l together. This mechanism could produce large quantities of ore i n a short period of time. Pre c i p i t a t i o n could be caused by an increase i n s u l f i d e , temperature drop, s a l i n i t y changes, or pH r i s e when the f l u i d s mix. I t w i l l be shown l a t e r that special geologic situations are needed to develop hydrodynamic patterns conducive to mixing of - one f l u i d carrying metal and another carrying s u l f i d e . Source of Sulfur It i s generally agreed upon that hydrogen su l f i d e gas i s the l i k e l y source of su l f i d e for many cases of ore p r e c i p i t a t i o n . What remains i n controversy i s how the H^ S arrived on the scene and where i t came from. A number of researchers have drawn attention to the close genetic r e l a t i o n -ships between o i l f i e l d s and ore deposits (e.g. Hitchon, 1977). The evolution of hydrocarbons and t h e i r migration may be important to the 34 deposition of ore at s i t e s where metal-rich brines mixed with H^ S derived from petroleum reservoirs. Anderson (1975) showed that a continuous supply of H S i s required, because r e a l i s t i c i n s i t u volumes could not account for the deposition of a major ore body. Brine-derived sulfate could be reduced to H^ S at s i t e s containing anaerobic bacteria and organic matter, which would overcome the supply problem of petroleum reservoirs. One s t r i k e against t h i s model i n some ore environments i s that sulfate-reducing bacteria become inactive above:temperatures of 80°C (Skinner, 1979). Sulfate can be reduced inorganically i n the presence of methane, as suggested by Barton (1967), but Anderson (1975) points out that t h i s would be gradual and i t would not explain the dominant association of lead-zinc ores to carbonate rocks. Another interesting process for reducing sulfate at temperatures above 75°C i s presented by Orr (1974) to explain the generation of large volumes of H S i n deep petroleum reservoirs. It involves the reaction of sulfate with small amounts of E^S to produce elemental s u l f u r , which i n turn reacts with organic matter to produce more hydrogen s u l f i d e . Macqueen and Powell ( i n press) found that t h i s method of sulfate reduction helps explain some of the features observed i n the Pine Point ore deposits. They believe the reduced sulfur was produced i n s i t u by small amounts of pre-existing H S i n the reef complex. The sulfate-reduction process continued as warm (75°-100°C), metal-bearing brines passed through the organic-rich reef environment. We seem to be given three possible sources for reduced sulfur: (1) Migration of petroleum-derived H^ S over long distances and through aquifers isolated from metal-brine trans-port paths. 35 (2) Transport of small amounts of dissolved H S and metal i n the same brine. (3) Sulfate reduction of metal-sulfate brines at sites containing abundant organic matter and anaerobic bacteria. The f i r s t model requires two f l u i d s converging at a common discharge area. It has the advantage that high metal or sulf i d e concentrations could exist i n each aquifer, thereby allowing large amounts of ore to deposit i n a short period of time. As mentioned e a r l i e r , a two-fluid mixing model demands ;a special hydrogeologic condition. The second model requires large volumes of s l i g h t l y a c i d i c brine, and probably long periods of f l u i d flow. It could be appropriate for the Southeast Missouri d i s t r i c t where the extensive Lamotte sandstone aquifer acted as a major conduit for delivering brines to overlying carbonate reefs or other permeable zones i n the Bonneterre Formation. The t h i r d model can tolerate a wider range of geochemical conditions for the metal-bearing brine (Anderson, 1973), but ore occurrence i s controlled by the location of suitable supply of organic material and bacteria. At environments such as Pine Point, sulfate reduction provides a very plausible mechanism. The occurrence of uniform mineralogical features (Ohle, 1980), and general absence of extensive well-rock a l t e r a t i o n suggests that i n s i t u sulfate reduction may be the major cause of ore deposition i n many basins. Timing and Paths of Flui d Migration The geology of carbonate-hosted lead-zinc deposits give few clues when i t comes to interpreting c r i t i c a l genetic relationships such as the age of the deposit, duration of mineralization, or paths of f l u i d movement. Fi e l d estimates of l o c a l f l u i d flow directions have been made through mineralogical studies, but they seldom give insight on the regional hydro-dynamic system. In d i s t r i c t s such as Pine Point and Southeast Missouri, the 36 i d e n t i f i c a t i o n of l i k e l y paleoaquifers i s made easy by the presence of extensive porous dolomite and sandstone formations. A two-fluid mixing model demands that the brines be hydraulically i s o l a t e d , u n t i l the point of mixing near the basin margin. Long distance transport of brines i n a sedimentary basin i s cer t a i n l y possible, but to avoid mixing through cross-format ional flow or dispersive processes may require r e s t r i c t i v e hydraulic constraints. The age and duration of mineralization of lead-zinc ore deposits i s currently unknown except that they are p o s t - l i t h i f i c a t i o n of the host rock. Estimates of the time needed to form an ore deposit can be made by assuming representative values of metal concentrations and flow rates. Roedder (1976) believes the possible geologic time span i s wide, ranging from 1000 years to 10 m i l l i o n years. Based on a hydrologic model, White (1971) estimated that a period of 1 m i l l i o n years to 100 m i l l i o n years i s geologically reasonable for the White Pine copper deposit i n northern Michigan. Sverjensky (1981) concluded that the entire Southeast Missouri lead d i s t r i c t could have formed i n a period of 67,000 to 1 m i l l i o n years, assuming lead 3 2 concentrations of 1 to 18 ppm and flow rates of about 30 m /m /yr across an area of 100 m by 10 km. These estimates indicate that the duration of ore deposition may be r e l a t i v e l y short i n comparison to other geologic processes. Summary It i s clear that many problems regarding the o r i g i n of carbonate-hosted lead-zinc deposits remain to be completely solved. Only some of these problems were i d e n t i f i e d here. Ohle (1980), and Anderson and Macqueen ( i n press) present much more thorough reviews of the problems at hand and suggestions for future research. The primary concern of t h i s study i s to develop a quantitative understanding of the role of the gravity-driven groundwater flow model i n ore formation. The flow nets of Figures 7 and 8 show that a wide variety of subsurface flow patterns can e x i s t , and suggest that further consideration of the factors c o n t r o l l i n g f l u i d flow w i l l be useful i n constraining conceptual models of ore genesis. The types of questions that t h i s analysis w i l l t r y to answer include the following: (1) What i s the exact nature of gravity-driven f l u i d - f l o w systems in sedimentary basins? ( 2 ) What factors influence flow patterns in the subsurface, and the rates of f l u i d movement through large basins? (3) How i s the subsurface temperature pattern of a basin affected by f l u i d flow? (4) Are ordinary geothermal conditions adequate to sustain temperatures of ore formation near the edge of a basin? ( 5 ) Can flu i d - f l o w systems effect the concentrations of metal-bearing brines as they flow through a basin? (6) Can fl u i d - f l o w models and reaction-path models constrain the geochemical models of transport and deposition? (7) Can gravity-driven flow systems produce major ore deposits i n reasonable periods of geologic time? Numerical models are developed i n the next two chapters for the purpose of seeking quantitative answers to these questions. CHAPTER 3 FUNDAMENTALS OF TRANSPORT PROCESSES IN ORE GENESIS In order to evaluate the conceptual models of ore genesis i n a quantitative way, four main types of information are needed: (1) Flow patterns and f l u i d v e l o c i t i e s . (2) Temperatures. (3) Concentration patterns of metal or s u l f i d e . (4) Masses of minerals dissolved and precipitated. Fluid-flow patterns and v e l o c i t i e s can be determined by analyzing the forces causing flow i n porous media. For the purpose of t h i s study, the porous medium i s assumed to be r i g i d , and the variation i n a gravity-based f l u i d potential i s the only mechanism causing f l u i d transport. Subsurface temperatures depend on thermal conduction and the convection effects of f l u i d flow. The concentrations of transported chemical species w i l l be controlled by f l u i d v e l o c i t i e s , flow patterns, mechanical dispersion, and molecular d i f f u s i o n . I f the rocks are chemically reactive, mass transfer between the aqueous phase and mineral phases w i l l occur v i a water-rock reactions. The mass of minerals produced or destroyed w i l l depend on e q u i l i b r i a constraints. This study w i l l treat the theory of f l u i d flow, heat transport, and mass transport i n a r e l a t i v e l y complete way; the theory of mass-transfer processes and chemical equilibrium i s reviewed i n a shorter fashion. The principles of transport processes are based on the laws of f l u i d dynamics, thermodynamics, and heat and mass transport. Common to a l l types of processes i s the transport of some quantity or property, a driving 38 39 force, and a general move toward equilibrium. The property being transported may be mass, momentum, thermal energy, e l e c t r i c a l charge, or any other physical quantity. A l l of these transport processes can be described by mathematical expressions, many of which are developed i n t h i s chapter. The theory given here i s not new; many references on the physics of transport theory are available, including the well-known texts of Bi r d , Stewart, and Lightfoot (1960), Bear (1972), and Slattery (1972). An excellent review of the l i t e r a t u r e concerning transport theory and geologic processes i s given by Domenico (1977). Governing Equations The treatment of transport phenomena i n porous media requires a continuum approach where average values of media parameters are introduced, i n order to pass from the microscopic, pore-size scale to a macroscopic, continuum l e v e l . This transformation i s straightforward i n unconsolidated sediments or sedimentary rocks, but there are inherent d i f f i c u l t i e s i n formations with abundant fractures or cavernous openings. In the continuum approach, the scale of the transport problem i s taken at a l e v e l such that average media properties are representative. This avoids the more compli-cated analysis of flow through ind i v i d u a l fractures. For the regional study of f l u i d flow i n sedimentary basins, a continuum porous-media model i s certainly j u s t i f i e d . The equations governing f l u i d flow, heat transport, mass transport, and geochemical equilibrium are now presented and discussed i n d i v i d u a l l y . Simplifications are made to each transport equation, so as to s a t i s f y the needs of a preliminary assessment of the role of f l u i d flow i n ore genesis. Many of the equations, except those governing geochemical equilibrium, are written i n v e c t o r i a l notation. For the purpose of t h i s study, only two 4-0 s p a t i a l dimensions (x, z) are used i n modeling the transport processes. I t i s a r e l a t i v e l y easy task to extend the equations to t h e i r three-dimensional form. F l u i d Flow The equation of conservation of mass for a single-phase f l u i d i s written by Bear C1972) as 3(<|>p) + V'(pq) = 0 (3-1) 3t where cj) = porosity (dimensionless fraction) 3 p = f l u i d density [M/L ] . 3 2 q = (q: ;, q_z), s p e c i f i c discharge vector or Darcy ve l o c i t y [L /L *t] t = time [t] V = (_3 l + 3 k) 3x 3z The f i r s t term represents the net rate of mass accumulation within a unit control volume, while the second term accounts for the net mass flu x out of the control volume. This continuity equation assumes that the porous media i s r i g i d , and that the mass f l u x carried by the average f l u i d velocity i s much greater than any flow due to vel o c i t y fluctuations along the flow path. The media i s also assumed to be f u l l y saturated with a l i q u i d phase only and no f l u i d sources or sinks are present. The s p e c i f i c discharge vector q i s given by the following general form of Darcy's law q = k(Vp + pgVZ) (3-2) y 4-1 where k = i n t r i n s i c permeability (.tensor) [L 2] u = dynamic v i s c o s i t y [M/Lt] p = f l u i d pressure [M/Lt 2] g = gravitational acceleration constant [L/t 2] Z = elevation of reference point above datum [L] Equation (3-2) applies to homogeneous or inhomogeneous f l u i d s (variable density )> and i t assumes that f l u i d flow i s laminar i n nature. I t i s possible to derive t h i s form of Darcy's law from a formal analysis of conservation of f l u i d momentum, as expressed by the Navier-Stokes equations, (e.g. Bear, 1972; Dagan, 1972; Sorey, 1978), although the s i m p l i f i e d form of t h i s relationship was o r i g i n a l l y presented empirically by Darcy on the basis of laboratory experiments. For homogeneous f l u i d s , the concept of hydraulic head can be i n t r o -duced, which represents the energy per unit weight of f l u i d . h = _p_ + Z (3-3) -p. tig where h = hydraulic head [L] pg = reference density (constant) [M/L3] Equation (.3-2) then reduces to the f a m i l i a r form of Darcy's law with q = KVh (.3-4) and K = k p 0g (3-5) where K = hydraulic conductivity (tensor) [L/t] y0 = reference dynamic v i s c o s i t y [M/Lt] Hubbert (.1940) shows that the hydraulic head i s s t r i c t l y v a l i d only under isothermal and adiabatic conditions where the f l u i d i s of homogeneous composition. For these reasons, pressure-based equations rather than head-based equations are t r a d i t i o n a l l y used i n petroleum and geothermal reservoir analysis. The formation of stratabound ore deposits i s dependent on the flow of variable-density brines i n nonisothermal aquifers. It i s necessary, therefore, to employ Equation (3-2) rather than (3-4) i n our f l u i d - f l o w calculations. One problem with Equation (3-2), however, i s that s t a t i c pressure i s included with the component responsible for f l u i d movement, the dynamic pressure increment. Bear (1972) and Frind (.1980) recommend replacing the pressure variable with an equivalent fresh-water hydraulic head, as defined by Equation (.3-3). Frind (1980) suggests t h i s formulation reduces inaccuracy i n the numerical solution of the f l u i d - f l o w equation. Upon substituting the hydraulic-head r e l a t i o n of (3-3) into (3-2), they arrive at the following expression (Bear, 1972, p. 654) q = 4c p 0g Vh - R (P-Po)g VZ (3-6) y y The s t a t i c pressure component i s eliminated and the Darcy equation i s now reduced to the two driving forces: the hydraulic gradient and a buoyancy term. I t must be recognized that the hydraulic head i n (3-6) i s no longer a potential function as i t i s in the case of homogeneous f l u i d flow. In f a c t , mathematically there doesn't exist a unique hydraulic potential i n inhomogeneous f l u i d s because of rotations caused by density gradients. The subject of inhomogeneous f l u i d flow i s treated i n greater d e t a i l by de Josselin de Jong (1960, 1969), Yih (1961), Lusczynski (1961), Bear (1972), and Dagan (1972). 43 Further transformations can be made on Equation (3-6 ) i n order to eliminate i n t r i n s i c permeability i n favor of hydraulic conductivity. After multiplying both sides of (3-6_)_ by Po/prj and PQ/PO, and c o l l e c t i n g terms q = -k.pQg y 0 V h -k" P . O S C P - P Q ) P O V Z (3-7) or P O vi P O P O P q = - K y 0 [Vh + C P - P Q ) v Z ] (3-8) P P O This equation can be si m p l i f i e d further by introducing r e l a t i v e v i s c o s i t y and density terms, such that q = K y (Vh + p VZ) (3-9) r r where y^ = no, r e l a t i v e v i s c o s i t y (dimensionless) P p^ = (p-Po)s r e l a t i v e density (dimensionless) P O " The f l u i d continuity equation [Equation (.3-1)] now becomes 3(<j>p) + V- [-K py (Vh + p VZ)] = 0 (3-10) The transient term on the f a r l e f t can be expanded using the chain rule of d i f f e r e n t i a t i o n i n the usual manner (Freeze and Cherry, 1979) to obtain V« [R py (Vh + p VZ)] = pS 3h (3-11) r r S 3t where S g = s p e c i f i c s t o r a t i v i t y [1/L] Two types of fl u i d - f l o w problems can be solved with Equation (3-11): steady-state flow and transient (nonsteady) flow. Steady-state conditions 44 occur when at any given point i n the flow f i e l d the magnitude and direction of s p e c i f i c discharge (Darcy velocity) are constant with time. In transient flow, the magnitude and direction of the Darcy velocity vary with time. Problems concerning regional groundwater flow are commonly treated i n steady-state format (Freeze and Cherry, 1979). This implies that the hydraulic-head d i s t r i b u t i o n does not change s i g n i f i c a n t l y with time, and that the s p a t i a l d i s t r i b u t i o n of f l u i d density i s r e l a t i v e l y constant. For steady-state flow, the transient side of Equation (3-11) i s set to zero and the continuity expression now becomes (Bear, 1972; Dagan, 1972) V« [Kpy (Vh + p VZ)] = 0 (3-12) r r The use of steady-state theory i n the modeling of regional f l u i d flow i n large sedimentary basins i s approximately s a t i s f i e d by the time span involved i n ore formation. The basic assumption being made i s that the configuration of the water table (or topography) remains nearly constant over time, or that fluctuations i n the water-table configuration are small r e l a t i v e to the thickness of the sedimentary basin (Freeze and Witherspoon, 1966). Solving Equation (3-12) for the hydraulic head requires that suitable boundary conditions are provided, along with material and f l u i d properties of the flow system. The porous medium can be heterogeneous and anisotropic with respect to hydraulic conductivity, and porosity can vary i n space. The f l u i d can be inhomogeneous and nonisothermal, such that density and vi s c o s i t y are functions of pressure, temperature and solute concentration. The terms i n the hydraulic conductivity tensor are computed for a x-z coordinate system as K = K 1cos 26 + K„sin26 (3-13a) xx 1 . 2 K = K nsin 20 + K cos 26 (3-13b) zz 1 2 K = K = (K -K )sin6cos0 (3-13c) xz zx 1 2 where K2 - p r i n c i p a l hydraulic c o n d u c t i v i t i e s [L/t] 0 = angle between and x-coordinate (radians) In most of the modeling r e s u l t s presented here, the p r i n c i p a l d i r e c t i o n s of hydraulic conductivity w i l l be oriented p a r a l l e l to the x-z axi s , i n which case 6=0 and K = K l 9 K = K ? , K = K =0. XX 1 zz ^ xz zx With the boundary-value problem f u l l y s p e c i f i e d , the hydraulic-head sol u t i o n can then be used to cal c u l a t e the s p e c i f i c discharge (Darcy v e l o c i t y ) from Equation (3-9). The average l i n e a r v e l o c i t y (Freeze and Cherry, 1979) i s given by v = q (3-14) •Q>. where <j> i s the porosity. It represents the volumetric f l u i d f l u x across the actua l unit c r o s s - s e c t i o n a l area of porous medium. Dividing by the porosity corrects f o r the fact that flow i s i n r e a l i t y through the pore spaces only, and not the f u l l c r o s s - s e c t i o n a l area of continuum. The specific-discharge vector q i s used to compute the e f f e c t s of convection i n heat transport, while the average l i n e a r - v e l o c i t y vector v i s used to predict the movement of solutes i n mass transport. Heat Transport An expression f o r the conservation of thermal energy i n a saturated porous medium can be written as [cf>pc c + (l-<j>)p.c ] 9T + pc j- q«VT = V* [icVT] +V« [tfrlLvT] (3-15) Vf S VS -ir— Vf ^  H ot where c = s p e c i f i c heat of f l u i d [L 2/t 2T] c = s p e c i f i c heat of solids [L 2/t 2T] vs p g = density of solids [M/L3] T = temperature [T] k' = effective thermal conductivity (tensor) [ML/t3T] = coefficient of thermal dispersion (tensor) [ML/t3T] It i s possible to write the heat equation i n various other forms. References in the groundwater l i t e r a t u r e include Stallman (1963), Bear (1972), Dagan (1972), Domenico and Palciauskas (1973), Witherspoon et a l . (1975), Sorey (.1978), and Faust and Mercer (1979). The mathematical and physical content of these various forms i s nearly i d e n t i c a l . The f i r s t term on the l e f t side of Equation (3-15) represents the net rate of thermal energy gain per unit volume of porous medium. It assumes thermal equilibrium between the pore f l u i d and rock, or i n other words the temperature of the f l u i d i s the same as the s o l i d framework. The second term on the left-hand side describes the amount of heat f l u x due to convection. On the right side of (3-15), the f i r s t term i s the net rate of heat loss per unit volume by conduction, and the l a s t term gives the net heat transfer by the spreading effects of mechanical dispersion. Terms accounting for viscous dissipation of heat, energy increases by compression, and heat sinks or sources are neglected. The production and loss of heat caused by organic or inorganic reactions are also neglected. The bulk thermal conductivity of the s o l i d - l i q u i d matrix i s much easier to measure than the ind i v i d u a l thermal conductivities of the f l u i d .K.£ and solids < (Dagan, 1972). Therefore an effective thermal conductivity '•'k i s commonly u t i l i z e d , which can be defined as a geometric mean 47 J.< ( l - 6 ) (3-16) f s Other forms of t h i s expression can be implemented (see Woodside and Messmer, 1961 or Bear, 1972), but Equation (.3-16) i s s u f f i c i e n t for t h i s study. Notice that the thermal conductivity tensor of Equation (3-15) i s now reduced to a scalar by assuming the media i s isotr o p i c with respect to thermal conductivity. Bear (1972) states that the coeffi c i e n t of thermal dispersion i s sim i l a r to the coeffi c i e n t of mechanical dispersion used i n mass-transport theory. The functional form of the thermal dispersion tensor i n the x-z plane of a thermally isotr o p i c media becomes (Rubin, 1974; Tyvand, 19 77; Sorey, 1978; Pickens and Grisak, 1979) * ( V x x = P°vf ^ x 2 + £ T Q ( 3 ~ 1 7 a ) q • C V z z = P C v f S^x2 + £lO ( 3" 1 7 B ) 4>(D„) = 4>(D ) = PC , Ce—e ) -q .: q (3-17c) H xz H zx vf L T x z where <{> = porosity e = longitudinal thermal d i s p e r s i v i t y [L] = transverse thermal d i s p e r s i v i t y [L] q = ( q x 2 + q z 2 ) ^ ' vector resultant of the s p e c i f i c discharge [L/t] For small-scale heat transport problems, thermal dispersion i s small compared to conduction. On the regional scale, however, thermal dispersion can become si g n i f i c a n t (Mercer and Faust, 1981). Mathematical analysis have shown that thermal dispersion increases the s t a b i l i t y of a f l u i d - f l o w f i e l d and i n h i b i t s the start of convection currents i n free-convection heat-flow 48 problems (Weber, 1975; Tyvand, 1981). With these d e f i n i t i o n s , the heat-transport equation can now be rewritten as V'(EVT) - Pc _ q VT = [<j>pc . + (1 - tj>)p c ] 3T (3-18) vf vf s vs -r— 9t where the combined thermal conduction-dispersion tensor E i s given by E = k + CJJIL (3-19) rl Given appropriate i n i t i a l and boundary conditions, the solution of the heat equation produces the s p a t i a l temperature d i s t r i b u t i o n as a function of time. For the case of regional heat flow, a steady-state solution i s usually adequate, provided the f l u i d - f l o w pattern i s steady and boundary conditions are not changing appreciably with time (Domenico and Palciauskas, 1973). Under steady-state conditions, Equation (3-18) reduces to V'(fVT) - p c q V T = 0 (3-20) vf As i n the case of regional groundwater flow, the use of (3-20) i n ore-genesis modeling i s also partly j u s t i f i e d because of the time scale involved. Any thermal perturbations caused by sudden u p l i f t or climatic changes would probably dissipate after several thousand years. A r e l a t i v e l y stable f l u i d -flow and heat-flow regime are envisaged i n the mature stage of basis evolution. Mass Transport The equation for conservation of mass of the s-th constituent (component) i n a multicomponent system i s written as (Bear, 1972; Schwartz and Domenico, 1973; Carnahan, 1975) 9(.<|>pc ) + :E v R = V«(<j>pDVc ) - v-Upvc ) (3-21) s sr r s s §t— r = 1 49 i where s = subscript index, s = 1, 2,.... § § = t o t a l number of components i n s o l u t i o n c s= mass concentration of component s per unit mass of f l u i d (dimensionless) v = stoichiometric c o e f f i c i e n t of s i n reaction r (dimensionless) sr R::' = rate of reaction r [M/L 3t] r r = reaction index, r = 1, 2, .... f" £ = t o t a l number of reactions D = dispersion c o e f f i c i e n t (tensor) [ L 2 / t ] cf> = porosity ( f r a c t i o n ) p = f l u i d density [M/L3] v = average l i n e a r v e l o c i t y [L/t] This expression i s commonly known as the advection-dispersion equation i n groundwater theory. In petroleum engineering, the process in v o l v i n g dispersion and advective transport i s c a l l e d miscible displacement (Garder, 1964). Derivations of Equation (3-21) are found i n Reddell and Sunada (1970), Rumer (1972), Bear (1972), Konikow and Grove (1977), and others. The transient ( f i r s t ) term i n (3-21) describes the net rate of mass accumulation within a s p e c i f i e d control volume. Geochemical reactions are represented under the summation symbol, the terms of which are negative f o r the addition of a chemical component and p o s i t i v e f o r removal. Any type of r e a c t i o n can be involved, including p r e c i p i t a t i o n , d i s s o l u t i o n , complexing, radioactive decay, and ion exchange. Discussion of equilibrium geochemistry i s postponed u n t i l a l a t e r section. The f i r s t term on the right-hand side of Equation (3-21) represents the net mass f l u x from an elemental volume by the processes of mechanical dispersion and chemical d i f f u s i o n . The l a s t term i n the equation r e f l e c t s 50 the mass transport due to advectipn. In a multicomponent aqueous system, there i s one mass balance equation for each transported component. The dispersion tensor D has been studied i n great d e t a i l by various investigators, both from a theoretical point of view and with an experimental approach. The coefficient of dispersion depends on the flow v e l o c i t y , on the molecular d i f f u s i v i t y of the aqueous species, and the medium char a c t e r i s t i c s . The functional form i s usually written as (Bear, 1972) D = a v v + TTK (3-22) x z d v where a = d i s p e r s i v i t y coefficient (fourth-rank tensor) [L] f = tortuosity c o e f f i c i e n t (sec.ond-rank tensor) (dimensionless) = apparent di f f u s i o n c o e f f i c i e n t i n porous media [L 2/t] The d i s p e r s i v i t y tensor i s a characteristic of the porous medium alone, which contains 81 components i n three-dimensional space (Bear, 1972). I f the porous medium i s iso t r o p i c with respect to dispersion, then i t can be shown that D has i t s p r i n c i p a l components along the direction of flow and transverse to i t . In the isotrop i c case, the d i s p e r s i v i t y tensor reduces to two constants: the longitudinal d i s p e r s i v i t y ct and the transverse Li d i s p e r s i v i t y a . For p r a c t i c a l purposes, the tortuosity tensor f i s usually assumed to be a constant and set equal to unity. Under these conditions, the hydrodynamic dispersion tensor i s written as D = a v 2 + a v 2 + D (3-23a) xx L x T z d v v D = a v 2 + a v 2 + D, (3-23b) zz T x L z d -v v D = D = (a - a_) v v (3-23c) xz zx L T x z This s i m p l i f i e d formulation of the dispersion tensor appears to be widely accepted (Bear, 1972; Bredehoeft and Pinder, 1972; Pickens and Lennox, 1976; and others). For media that are anisotropic with respect to mass-transport parameters, researchers have been unable to derive any p r a c t i c a l form of the dispersion tensor. Poreh (.1965) derived an equation for axisymmetric media, which i s li m i t e d to s p e c i f i c f l u i d - f l o w conditions. De Josselin de Jong (19 69) concluded that the mass-transport equation developed for the i s o -tropic case can th e o r e t i c a l l y only be extended to the anisotropic case i f the dispersion coeff i c i e n t i s expressed by an i n f i n i t e series. Some studies have made use of Equation (.3-23) along with an anisotropic f l u i d - f l o w model in order to match f i e l d observations (Segal and Pinder, 1976; Schwartz and Crowe, 1980). An alternative approach i s to use a stochastic analysis as done by Smith and Schwartz (1980). To date, however, i t i s f a i r to say that the problem of dealing with anisotropy i n mass transport parameters has yet to be resolved. Bear (.1972) and Ahlstrom et a l . (1977) show that the mass-transport equation can be si m p l i f i e d i n the case of incompressible f l u i d flow where the t o t a l mass density of the f l u i d remains nearly constant and areal change i n porosity i s also small. In t h i s case, the advection-dispersion equation becomes r 8C + £ v R = V J(D VC ) - vVC (3-24) s n rs r s s r=l 3t where s = aqueous components, s = 1, 2, .... s C g = mass concentration per unit volume [M/L3] This form of the equation i s the most common style used i n hydrology and so i t w i l l be adopted here. Konikow and Grove (1977) warn that i t i s only approximate because the presence of sinks or sources and density variations introduce terms into the advection part of the expression, which are not accounted for i n (.3-24-). The solution of Equation (3-24) i n theory w i l l provide the s p a t i a l and time-varying concentrations of each aqueous component i n the f l u i d - f l o w system. It i s shown l a t e r that i n practice the solution of mass-transport equations i n reactive aqueous systems i s currently impractical. S i m p l i f i c a -t i o n s , such as uncoupling geochemical reactions and l i m i t i n g the number of modeled components to less than a few, help a l l e v i a t e the problem. Equations of State F l u i d density and v i s c o s i t y can vary s i g n i f i c a n t l y with temperature, solute concentration, and to a lesser degree with pressure, over the range of conditions encountered i n sedimentary basins. In order to model the flow of basinal brines, we need functional relationships of the form p = p(p, T, C) (3-25a) and u = y(p, T, C) (3-25b) Equations of t h i s format can be obtained from engineering handbooks (e.g. Perry and Chilton, 1973), and from geothermal studies (e.g. Mercer and Pinder, 1974; Huyakorn and Pinder, 1977; Sorey, 1978). Unfortunately, none of these references present equations of state that are complete enough to model the f u l l range of l i k e l y temperatures and s a l i n i t i e s encountered i n the formation of stratabound ore deposits. Fortunately, experimental data on the density and v i s c o s i t y of aqueous NaCl solutions has been recently collected and correlated by Kestin et a l . (1978), and Kestin, Khalifa and Correia (1981).. E a r l i e r compilations on the density of sodium-chloride solutions can be found i n Rowe and Chou (.1970), and Haas (.1976). The Kestin equations w i l l be used here because of th e i r completeness. These equations cover a temperature range from 20° to 150°C, a NaCl concentration range from 0.0 to 6.0 molal (.about 25% weight equivalent), and a pressure range from 0.1 to 35.0 MPa. The experimental results have an estimated uncertainty of t 0.5% (.Kestin, et a l . , 1981). The f l u i d density equation i s of the following form 1 = A - Bp - Cp 2 + wD -f w2E - wFp - w2Gp - Hp2w (.3-26) where p =. p (.p, T, Wj^p-,),. f l u i d density (kg/m3) "NaCl p = pressure T = temperature (°C) w = mass fra c t i o n of sodium-chloride solution (.dimensionless) The functions A, B, C, D, E, F, G, and H are written as f i t t e d polynomial expressions which are temperature dependent. The correlation equation for dynamic v i s c o s i t y i s given by (Kestin, et a l . , 1981) y = y° (1 + Bp) (.3-27) where y = y(.p, T, C ) dynamic vi s c o s i t y (Pa«s) 'NaCl y° = hypothetical zero-pressure v i s c o s i t y (Pa«s) (3 = pressure coeffi c i e n t of vis c o s i t y ClO^Pa) p = f l u i d pressure (.109Pa) The polynomial expressions used i n evaluating y° and 3 are l i s t e d i n Kestin et a l . (1981). Figures 9 and 10 show plots of the variation i n density and vi s c o s i t y with temperature, and for selected values of NaCl molality (-raNaC1)» a s 54 TEMPERATURE (°C) Figure 9. V a r i a t i o n i n water density with temperature and NaCl concentration at a depth of 1 km. 55 Figure 10. Va r i a t i o n i n water v i s c o s i t y with temperature and NaCl concentration at a depth of 1 km. calculated from Kestin's equations of.state. Both.density and v i s c o s i t y exhibit a strong dependency on temperature and s a l i n i t y . The range of conditions depicted on these graphs are similar to the subsurface environ-ments expected i n the genesis of stratabound ore deposits.-At a depth of 1 km ( i . e . hydrostatic pressure of 10 MPa - 100 bars), f l u i d density changes from 1003 kg/m^ at 20 C to 979 kg/irr at 75 C, assuming r r i ^ ^ = 0.0. The effect of higher s a l i n i t y i s to increase density. For example, a 3.0 m NaCl brine has a density of 1083 kg/nr at 75 C (Figure 9). Viscosity shows more dramatic variations than density, especially with temperature (Figure 10). Dynamic v i s c o s i t y drops from 998 micro Pa*s at 20°C to 381 micro Pa«s at 75°C, for fresh water at 10 MPa pressure (1 centipdi-s = 1000 micro Pa*s at a temperature of 75°C and 1 km depth ). The effect of pressure on density and v i s c o s i t y i s small, over the temperatures and depths suggested for stratabound ore formation. In the pressure range of 0.1 to 35.0 MPa (about 4 km depth), fresh-water density Q O changes from 998 to 1003 kg/m , at a temperature of 50 C. Viscosity increases s l i g h t l y from 547 to 553 micro Pa*s under the same pressure range and temperature (Kestin, et a l . , 1981). Both the f l u i d density and vi s c o s i t y equations of state are needed to solve the coupled equations of f l u i d flow and heat transport. I f the modeling also c a l l s upon for the transport of Na and C l , then the advection-dispersion equations for these components would be coupled to the fl u i d - f l o w and heat-transport equations through the f l u i d density and v i s c o s i t y effects. Geochemical Equilibrium and Reaction Paths The calculation of chemical equilibrium between aqueous solutions and minerals i s essential i n quantitatively evaluating ore formation i n a porous medium. In theory, i t also provides the reaction terms i n the mass-transport 57 equation. The reaction of an aqueous phase with a mineral assemblage causes mass to be transferred among the phases, perhaps creating new minerals, dissolving existing minerals, or changing the f l u i d composition as the chemical system i s driven to o v e r a l l equilibrium. In porous media, the extent to which a reaction takes place w i l l depend on the composition of the f l u i d , rock mineralogy, and flow rate (.Helgeson, 1979). Thermodynamic constraints such as pressure, temperature, and fixed chemical potentials of the system components provide the most useful c r i t e r i a i n modeling geochemical reactions. Some reactions w i l l also depend on k i n e t i c constraints, but unfortunately k i n e t i c data are unknown or unobtainable for many reactions of interest i n ore genesis. The geochemical modeling approach taken i n t h i s study i s based purely on equilibrium thermodynamics, and the techniques pioneered by Helgeson (.1968) and Helgeson et a l . (.1970). It u t i l i z e s the thermodynamic constraints that are required to achieve equilibrium i n order to compute mass transfer and d i s t r i -bution of chemical components during water-rock reactions. Helgeson (.1979) b r i e f l y reviews some of the other theoretical approaches that have been applied to modeling geochemical mass transfer. An expression r e l a t i n g the change i n the energy of a closed thermo-dynamic system, as a result i n a change i n i t s chemical composition, i s given by Gibb's equat ion (Helgeson, 1968; Van Zeggeren and Storey, 1970) dG = E y*dn. l where dG = Gibb's free energy increment y# = chemical potential of the i - t h component dn^ = i n f i n i t e s i m a l change i n the number of moles of the i - t h component. This equation assumes that pressure and temperature are held constant. The condition of chemical equilibrium occurs at the minimum free-energy l e v e l of the system, where dG=0. The problem of computing the d i s t r i b u t i o n of compo-58 nents or species i n the aqueous phase at equilibrium reduces therefore to solving (3-29) for the set of n^ values that minimize G, subject to the additional constraint of mass conservation. Several approaches can be used to solve t h i s minimization problem, but two main types have prevailed i n computing geochemical equilibrium. One technique involves the direct application of optimization theory, while the second method formulates the problem as a set of nonlinear equations based on equilibrium constants (Van Zeggeren and Storey, 1970). The l a t t e r approach involves the substitution of expressions for the chemical potentials i n terms of a c t i v i t i e s . The result i s a set of mass-action and mass-balance equations, which constrain the equilibrium d i s t r i b u t i o n of components among a l l phases (Helgeson, 1968). As an example, consider the problem of calculating an equilibrium model for a single f l u i d phase i n which there are J components and I associ-ated e q u i l i b r i a constraints i n reactions. Wolery and Walters (1975) studied t h i s problem from a numerical point of view. Their system can be described by J mass-balance equations and I mass-action equations, including the condition of electro-neutrality (charge balance) 9 where r mT,1 = m + E v m j = l, 2 , .... J (3-29) i = l J J m = ' i q n m V ] 1 i = 1, 2 , . . . . I ( 3 - 3 0 ) j=l : J I £ z. m. + E z. m. = 0 ( j=l 1 1 i = l 1 1 ( 3 m . = t o t a l concentration of component j (moles/kg HO) m_. = concentration of j - t h free ion (moles/kg H^ O) nu - concentration of associated complex (moles/kg H^ O) 59 v_.^  = number of moles of j - t h component per mole of i - t h complex (..dimensionless) K? = apparent thermodynamic association constant for i - t h complex (.dimensionless) z.. ,z^ = net e l e c t r i c charge on species Thermodynamic data are tabulated i n the l i t e r a t u r e and values of K" are known for many reactions (Robie and Waldbaum, 1968; Helgeson, 1969; Helgeson and Kirkham, 1974a). A c t i v i t i e s may be used instead of molal concentrations, i n which case a = ym (.3-32) but the a c t i v i t y c o e f f i c i e n t y demands an estimate of the ionic strength I of the solution, which i s calculated from (Wolery and Walters, 1975) J I I = h [ I m. z. 2 + £ m. z. 2] (.3-33_) j=l : 3 i = l 1 1 ' The a c t i v i t y c o e f f i c i e n t i s then computed from the Debye-Huckel equation or other means (Garrels and Christ, 1965; Helgeson and Kirkham, 1974b). Simultaneous solution of Equations (3-29) through (3-33) provides a complete description of the d i s t r i b u t i o n of aqueous species i n equilibrium. The set of governing equations for a system containing an aqueous phase and minerals i s more involved. Table 2 shows the f u l l set of governing equations, as written by Wolery (1978, 1979a). It assumes a closed system E Cthe equilibrium subsystem) where the t o t a l elemental compositions n^ are specified at a fixed temperature and pressure. Only pure minerals are treated in Table 2. Solid-solution reactions for minerals of variable composition are deleted from Wolery*s o r i g i n a l set of equations, so as to simplify the presentation given here. The equations for equilibrium are expressed 60 Table.2. Equations governing chemical e q u i l i b r i u m i n a system containg t elements, s species and A* minerals Carter Wolery, 1979a) Mass balance f o r each element C.e = 1, S) § X n = E T n + E T ^ n. e es s , N SA A s=l X=l Charge balance i n the aqueous phase Cs = 2, §) § 0 = E Z.;n s=2 S S Mass a c t i o n f o r intra-aqueous r e a c t i o n s ( r = 1, V , V % S V r s " r s r s K = a f n TT.. a s - -0 2 s' = I s where s" > s r e f e r s t o species destroyed m the r - t h r e a c t i o n Mass a c t i o n f o r minerals of f i x e d composition CA = 1, X) vi.& s V K: = f n X s a r s X °2 s=l S A c t i v i t y - c o n c e n t r a t i o n r e l a t i o n f o r aqueous species (s = 2,cis") a = Y m s s s D e f i n i t i o n of equivalent s t o i c h i o m e t r i c i o n i c strength ^ v u + = Na or C l 1 = E T + m where e s=2 £ " S  D e f i n i t i o n of t r u e i o n i c s t r e n g t h s s s I = h E m_ z 2 s = 2 A c t i v i t y of l i q u i d water l o g a i r = -2 I $^ where = osmotic c o e f f i c i e n t (.Helgeson et a l . , 1970) w e .2.303 e 61 Table 2. (Continued) A c t i v i t y c o e f f i c i e n t s of aqueous species log Y = f (I) w h e r e  & s s f (I) = function depending on i o n i c strength and type of species (Wolery, 1979a) s Molal concentration of species rn = w n s s n w Glossary of symbols E = chemical element subscript e = t o t a l number of elements s = aqueous species subscript w = water subscript (s = 1) s aqueous species i n basis set § = t o t a l number of species s number of basis species = t = s - 1 S" = subscript of species being destroyed i n r-th reaction r = reaction subscript f number of independent reactions = S - % = pure mineral subscript X = number of minerals *e osmotic c o e f f i c i e n t n = mass of species, i n moles t n e t o t a l mass of e - th element z s = e l e c t r i c a l charge on species s m s molal concentration (moles/kg H 20) A s = a c t i v i t y c o e f f i c i e n t of species s a = thermodynamic a c t i v i t y I = true i o n i c strength I = equivalent i o n i c strength \ fugacity of oxygen i n aqueous phase OJ = water constant = 55.51 T = composition c o e f f i c i e n t (moles'element per formula we V r s = reaction or stoichiometric c o e f f i c i e n t (moles of s in K' = equilibrium constant for reaction 62 i n a format that assumes a s p e c i f i c ordering. Wolery has chosen to write the f i r s t s species (s = 1 denotes H^ 'O) as corresponding to a one-to-one basis with the elements and associated mass-balance equations. Oxygen fugacity i s used as the redox parameter i n writing the equations, such as (Wolery, 1978) OX- F e S + - h H20 + F e 2 + + l i + + k 0 2(g, aq) (3-34) The species O^tg, aq) marks the redox parameter and i s denoted as the s-th species (s = s = 1). I t can also be assigned with the charge-balance equation (Wolery, 1978). The remaining aqueous species, from s = s + 1 to s, are represented by the r aqueous reactions i n which they are involved. Wolery c a l l s the f i r s t s species the basis species and shows that a l l reactions can be written i n terms of the basis species and t h e i r associated species for the reaction. Further detailed explanation of Table 2 i s given i n Wolery (1978, 1979a) and need not be repeated here. Suffice to say that the unknown masses of species are ea s i l y found by simultaneous solution of these equations. A discussion of factors such as the effect of pressure and temperature on equilibrium constants., the calculation of the a c t i v i t y of water, the calculation of a c t i v i t y c o e f f i c i e n t s , and treatment of solid-solution phenomena are also omitted„here. The evaluation of these factors i s presented elsewhere (e.g. Helgeson, 1968; Helgeson et a l . , 1970). With the d i s t r i b u t i o n of aqueous species i n hand, we can now proceed to consider the prediction of mass transfer as a function of reaction progress. Helgeson (1979) presents an excellent review of reaction-path theory, and summarizes the application of the theory i n studying ore-forming hydro-thermal systems through the numerical approach f i r s t introduced by Helgeson (1968) and Helgeson et a l . (1970). Predicting the exact path of a reaction and the amounts of mass transfer i n a system requires introducing a reaction-progress variable , such that d£* = - dn = R (3-35) dt dt where moles of substance reacted n mass of substance present t time R rate expression for the reaction Rather than define an ind i v i d u a l progress variable for each reaction, Helgeson (1968) assigns an o v e r a l l reaction-progress variable for the geo-chemical process being modeled. I t generally refers to the i r r e v e r s i b l e reaction of the aqueous phase with one of the minerals i n the reacting porous media. The rates at which other minerals react with the f l u i d can be expressed i n r e l a t i o n to the o v e r a l l progress variable by specifying r e l a t i v e rates of reactions (Helgeson et a l . , 1970; Brown, 1977). The numerical solution of Equation (3-35) can be accomplished i n more than one way. Wolery's (.1978, 1979a) approach i s based on the techniques developed by Helgeson (1968) and Helgeson et a l . (1970), but his governing equations are formulated d i f f e r e n t l y . A comparison of the methods i s given by Wolery (1979a). In Wolery's models, the reaction path i s traced t 'Xi by changing the mass-balance constraints (n^) on the closed system E and by specifying a set of reactant-tracking coe f f i c i e n t s £, which correspond to a set of reactants f\. The tracking coefficients are interpreted as r e l a t i v e -rate constants, and as such Equation (3-35) i s recast for the n—th reactant i n a power series (Wolery, 1979a) The change i n the bulk-elemental composition of the system E as a function of (3-36) 64 i s given by (Wolery, 1979a) t « dn = - £ x (dn ) (3-37) e e r i n d F n _ 1 Upon i n t e g r a t i o n , t t ^ \ - n J , n + \ T e n [hr>W + + Vu*)3] (3-38) • 5"=o . n=i 2 — g — The r e s u l t of solving (3-38) i s a continuous de s c r i p t i o n of the changes i n f l u i d composition and amounts of minerals produced or destroyed during the reaction process, without reference to space or time. Temperature and pressure changes can also be accounted for i n these c a l c u l a t i o n s , as can the e f f e c t s of a flow-through system (Helgeson et a l . , 1970; Wolery, 1979a). Coupling of Transport Phenomena In t h i s chapter, a l l of the transport equations are written i n terms of i n d i v i d u a l fluxes caused by independent d r i v i n g forces that are represented by gradients i n pressure, density, temperature, solute concentration, or chemical p o t e n t i a l . Although the equations may be p h y s i c a l l y "coupled" through interdependency on f l u i d properties and v e l o c i t y , the p o s s i b i l i t y of i n t e r a c t i o n between the d r i v i n g forces has yet to be addressed. Freeze and Cherry (1979) show that these d r i v i n g forces may i n t e r a c t with each other to produce cross-transport phenomena (coupled processes). For example, i n addition to hydraulic-head gradients, the flow of f l u i d s through a porous medium may be caused by temperature gradients, e l e c t r i c a l gradients, or s a l i n i t y gradients. Similar r e l a t i o n s h i p s may also e x i s t f o r the flow of heat and mass transport i n porous media. Bear (1972, p. 85) and Domenico (1977) discuss the theory behind cross-transport phenomena, which i s based on a thermodynamic analysis r e f e r r e d to as Onsager's theory. 65 In practice, however, there are few data on the suite of phenomeno-l o g i c a l c o e f f i c i e n t s needed to quantitatively evaluate.the role of cross effects i n groundwater flow. The data which are available suggests that cross effects are small. Dagan (.1972) estimated that the cross effects caused by the s a l i n i t y gradient on heat flow (Dufour e f f e c t ) , and the effects of a temperature gradient on mass transport (Soret effect) are about three-orders of magnitude smaller than the direct driving forces. Carnahan (.1975) presents a detailed derivation of coupled phenomena i n porous media through a thermodynamic analysis. He calculated that the effect of a temperature gradient to cause f l u i d flow i s small under normal conditions, the r a t i o of the thermal effect to Darcy flow being about 3 x 10~ 3. Carnahan (1975) also shows that geochemical reaction rates are not thermodynamically coupled to other transport processes, although there i s coupling between the reactions themselves. Based on the available data, i t i s safe to conclude that coupled phenomena are r e l a t i v e l y unimportant when compared to the primary transport processes involved i n stratabound ore genesis. The effects of cross terms, therefore, are neglected i n t h i s study. Summary of Equations and Assumptions The mathematical treatment of the physics of transport processes i s now complete. Equations have been presented that describe inhomogeneous f l u i d flow, heat transport, and mass transport i n reactive porous media. The purpose of t h i s discussion i s to provide a theoretical framework for modeling subsurface f l u i d v e l o c i t i e s , temperatures, transport patterns, and geochemical reactions i n the formation of carbonate-hosted lead-zinc deposits of the Mis s i s s i p p i Valley type. Table 3 gives the complete set of si m p l i f i e d equations. Before continuing on to the next chapter, which deals Table.3. Governing Equations Darcy 1s law q = -Ry. (Vh + p VZ) = vcj> u r r F l u i d Flow V* (pq) = 0 Heat Transport V* (EVT) - p c v f • qVT = 0 Mass Transport 9G> + £ v R• = V'CDVG ) - w e s , rs r s s r=l 3t Equations of state P = P ( P ' T ' y = y (p, T, C N a C 1 ) Geochemical equilibrium and reaction mass tr a n s f e r t 8- * mass balance: n = E x n + E T . n , e •. es s , £ A A s=l X=l charge balance: 0 = E z n s=2 3 S mass action f o r aqueous species: V V <\, S V K- = (a .,.) r S " ( f . , ) r S ir a r s °r s=l S mass action f o r minerals: v , „ s v, X ° 2 s=l S reaction t r a c k i n g : t t A n = n I + E x • [t (5") + 5 ! eg*=o n=i e n l n with the subject of solution procedure, we should r e f l e c t on the main assumptions and l i m i t a t i o n s behind the governing equations. They can be summarized i n point form as follows. ( 1 ) Transport processes are assumed to occur i n a f u l l y saturated porous medium, and under conditions of laminar f l u i d flow. The equation models fractured" media as a continuum. ( 2 ) The porous medium can be heterogeneous and anisotropic with respect to hydraulic conductivity. (3) The porous medium i s r i g i d and well indurated. Stress-strain phenomena such as compaction are not considered. (4) Changes i n porosity due to geochemical reactions are assumed to not effect regional f l u i d - f l o w patterns or v e l o c i t i e s . (5) Gravity-driven hydraulic-head gradients are the only driving forces causing f l u i d flow. The f l u i d - f l o w equation, however, does take into account the effects of variable temperature and s a l i n i t y . (6) F l u i d density and v i s c o s i t y are nonlinear functions of pressure, temperature, and s a l i n i t y . Local density gradients are assumed to be small. (7) The f l u i d - f l o w regime i s at steady state. (8) Temperature of the pore f l u i d and host rock are equal, indicating thermal equilibrium. (9) The porous medium can be heterogeneous and anisotropic with respect to thermal properties. Thermal conductivity, however, i s more easily handled as an isotro p i c parameter. (10) Both heat conduction and convection are accounted for i n the heat-transport equation. The influence of thermal dispersion i s also included. ( 1 1 ) The heat-flow regime i s at steady state. ( 1 2 ) The mechanical dispersion and dif f u s i o n of a solute are modeled as a Fickian-type process. In the absence of dispersion, mass i s advected through, porous media at the average l i n e a r v e l o c i t y of the f l u i d . (13) It i s assumed that the porous medium i s isotropic with respect to dispersion processes. Under t h i s condition, the dispersion tensor can be reduced to two terms: a longitudinal and transverse dispersion coefficient (14) Mass transport processes are treated as transient problems. (15) Geochemical mass transfer through reactions appears as a source-sink term i n the advection-dispersion equation. These terms are evaluated through the thermodynamic constraints imposed by equilibrium, and reaction path modeling. (16) Reaction-path theory i s assumed to provide the best approach to modeling reaction progress and multicomponent mass transfer. I f more than one reactant are present, r e l a t i v e reaction rates must be specified. (17) The effect of thermodynamic coupling between the transport processes are considered to be small when compared to the main driving forces. The problem ahead of us now i s to solve the system of equations (Table 3) for the dependent variables of f l u i d flow, heat transport, and mass transport i n a chosen two-dimensional domain, given a set of i n i t i a l and boundary conditions. Although cross-transport phenomena have been eliminated from the analysis, the equations are s t i l l p hysically coupled because of the interdependency between pressure, temperature, and concentration i n the equations, and because of the influence of these f i e l d s on the f l u i d properties. The solution of t h i s complex, boundary-value problem i s treated i n the following chapter. CHAPTER 4 DEVELOPMENT OF THE NUMERICAL MODELS The procedure of mathematical modeling evolves along an organized route of analysis. Freeze (1978) l i s t s a four-step process: (1) I d e n t i f i c a t i o n of the physical problem and formulation of a conceptual model. (2) Determination of the governing equations and formulation of a mathematical model. (3) Solution of the mathematical model, given i n i t i a l and boundary conditions. (4) Interpretation of the model results i n l i g h t of geologic data and model c a l i b r a t i o n . Table 4 summarizes many of the modeling decisions that must be made in order to construct a mathematical model of transport processes i n str a t a -bound ore formation. We have already described the problem of stratabound ore genesis, and presented a set of equations that govern transport processes i n two-dimensional representations of sedimentary basins along cross sections. Simplifications have been made to some of these equations in order to reduce the complexity of the analysis. The conceptual model presented e a r l i e r (Figure 6) also gives the general size and shape of the cross sections i n which we want to simulate areal variations i n f l u i d v e l o c i t y , temperature, concentration of species, and reaction progress. The next step i n the analysis i s to develop a method for solving the mathematical model. The purpose of t h i s chapter i s to describe the 69 70 Table 4. Modeling decisions F l u i d Flow Heat Transport Mass Transport Geochemistry Media One, two, or three dimensions Single phase, multiphase Saturated, unsaturated Homogeneous, inhomogeneous f l u i d Gravity drive, compaction drive Steady, transient One, two, or three dimensions Convection, conduction, dispersion Thermal equilibrium, nonequilibrium Steady, transient One, two, or three dimensions Single species, multicomponent Advection, mechanical dispersion, d i f f u s i o n Steady, transient Lumped, distributed (time, space) Reactions, retardation Relative reaction rates, k i n e t i c s Closed, open, flow-through Porous media, fractured rock (discrete, continuum) Rigid, unconsolidated Reactive, nonreactive Homogeneous, heterogeneous Isotropic, anisotropic Region of Flow and Scale of Study Regional, l o c a l Sedimentary basin, aquifer Idealization of Geology Stratigraphy (general, detailed) Parameter values I n i t i a l and Boundary Conditions Prescribed value Constant, time-varying Numerical Solution F i n i t e difference F i n i t e element Integrated f i n i t e difference Random walk (Monte Carlo) Method of characteristics Optimization methods Newton-Raphson i t e r a t i o n , direct inversion Taylor-series expansions 71 s o l u t i o n procedure and to construct a p r a c t i c a l , quantitative t o o l f o r studying ore genesis i n sedimentary basins. Numerical Formulation of the Governing Equations Mathematical models (problems) can be solved through a n a l y t i c a l techniques or by numerical methods. The a n a l y t i c a l techniques provide a closed-form s o l u t i o n to the problem, which i n theory can be exact, so long as the region of flow i s very simple and the equations can be l i n e a r i z e d . The main advantage of numerical methods i s t h e i r a b i l i t y to solve mathematical problems i n which a n a l y t i c a l solutions are too d i f f i c u l t to obtain. In modeling multidimensional f l u i d flow and other transport processes i n basins, t h i s i s almost always the case. The coupled and nonlinear nature of our governing transport equations (Table 3) demand a numerical method of s o l u t i o n . The so l u t i o n technique must allow the media properties, the general geology, and the si z e of the study region to be e a s i l y varied. A n a l y t i c a l techniques could be applied, but t h i s approach would require major s i m p l i f i c a t i o n s to the problem at hand. It w i l l be shown l a t e r , however, that the numerical approach also has l i m i t a t i o n s i n solving t h i s complex set of equations. The numerical method used i n solving p a r t i a l d i f f e r e n t i a l equations involves d i v i d i n g the flow domain into a g r i d or mesh cons i s t i n g of many small blocks or c e l l s c a l l e d elements. In t h i s d i s c r e t i z a t i o n of the flow domain, each element i s defined by corner points c a l l e d nodes. Several techniques are then a v a i l a b l e to replace the p a r t i a l d i f f e r e n t i a l equation by a system of algebraic equations. Properties of the porous medium, such as permeability, porosity, thermal conductivity, d i s p e r s i v i t y and others, are s p e c i f i e d f o r each element i n the mesh. Suitable modifications of the algebraic equations are also made to s a t i s f y i n i t i a l and boundary conditions. These systems of equations can.be solved by matrix-inversion .methods for the value of the dependent variable at each nodal point i n the mesh, thereby producing a s p a t i a l solution over the flow domain. In theory, the numerical solution w i l l approach the exact solution as the number of elements increases. A different method of numerical analysis i s needed to solve the highly non-li n e a r system of equations representing chemical equilibrium and reaction progress. Instead of d i s c r e t i z i n g s p a t i a l coordinates, i t i s necessary to d i s c r e t i z e the reaction-progress variable. In t h i s study, the finite-element method i s used to solve the equations governing f l u i d flow and heat transport. The problem of multi-component mass transport i s solved through the development of a code that uses moving-particle random-walk theory. Geochemical reactions are simulated through the EQ3/EQ6 code, which was written by Wolery (1979a). It applies Newton-Raphson i t e r a t i o n along with f i n i t e - d i f f e r e n c e theory to predict reaction paths. General discussions on each of the numerical methods used in t h i s study are given below. Further reference on mathematical modeling in hydrogeology can be found i n Freeze and Cherry (1979), Mercer and Faust (1981), and Wang and Anderson (1982). A review of numerical modeling i n aqueous geochemistry i s given i n Nordstrom, et a l . (1979). F l u i d Flow The Galerkin-based finite-element method has been applied to numerous problems of flow and transport i n porous media. Finite-element theory has received much attention i n hydrogeology, and therefore only the highlights of i t s formulation are presented here. The interested reader i s referred to the texts of Bathe and Wilson 5(1976)Pinder and Gray (1977), and Zienkiewicz (1977) for more d e t a i l on the method. The region of study i s f i r s t subdivided into a set of f i n i t e elements, with the corners of the elements represented by nodes. The dependent variable of interest (hydraulic head) i s approximated over the elements, usually i n a l i n e a r fashion so that the numerical value of the dependent variable i s easily interpolated at any given point. The size and shape of the elements are a r b i t r a r y , but generally the mesh i s made fine enough to model complicated boundaries and lithology changes. Triangular-shaped elements are commonly used because of t h e i r s i m p l i c i t y and ease of application i n two-dimensional problems. The next step i s to rewrite the equation of f l u i d flow (Table 3) into an operator form as L(h) = V- [K pu r(vh + p^VZ)] = 0 (4-1) The problem i s to f i n d an approximate solution to t h i s equation. According N . to the Galerkin method, the approximate solution h i s given by the following interpolation formula (Neuman, 1973) N ' h = h = E h C (4-2) m m m=l where h = hydraulic head (exact solution) N = t o t a l number of nodes i n the mesh h = hydraulic-head coefficient.at node m m e E = l o c a l coordinate function or Chapeau function m associated with the m-th node and e-th element The coefficients h^ are computed to s a t i s f y the boundary conditions and the necessary requirement of orthogonality, namely that the r e l a t i o n 74 N p p £ J7 L(.h ) • 5 dR = 0 ; n = 1, 2, N (4-3) e e n K i s v a l i d throughout the entire two-dimensional flow domain (R ) of the e elements. The l o c a l coordinate functions E, are chosen to have a value of n e unity at node n and are zero at other nodes:'.in..the .mesh. I f 5 i s assumed to vary l i n e a r l y across an element, then the Chapeau function can be written as E,B = (a + b x + c z) (4-4) n n io n 2 A 6 e where A i s the area of the element. The constants a , b , c , and the area n n n' e A are e a s i l y computed from the nodal coordinates. For a t r i a n g l e with nodes at each vertex, these constants are given by b l = Z l - Z3 C l = X 3 " X2 b2 = Z 3 " Z l C2 = X l - X 3 ( 4 ' 5 ) b 3 = Z l " Z2 C3 = X2 " X l and 2A = b ± • c 2 - c ± ' b 2 where (x^, x 2, x^) and(z , z 2 , z g ) are the l o c a l coordinates f o r the nodes n = 1, 2, 3 i n the t r i a n g l e . The de r i v a t i v e s of the Chapeau function are e a s i l y computed from Equation (4-4). After s u b s t i t u t i n g Equations (4-1) and (4-2) into (4-3), and using Green's f i r s t i d e n t i t y (Neuman, 1973), the r e s u l t i s a system of equations 0 A • h + B = Q ; m, n = 1, 2, .... N (4-6) nm m n n ' ' ' ' where A i s an N x N matrix, and B and Q are vectors of siz e N. The nm n n conductance ( s t i f f n e s s ) matrix A i s given by 75 A = E A® = E ff R PP V£ 8 • V § 6 dR 8 nm e nm e r n m K = E pp [K b b + K ( c b + b c ) + K c c ] (4-7) e r xx n m xz " n m n m zz n m 4A~ It should be noted that the conductance matrix i s symmetric, sparse, and banded. I t i s also diagonally dominant and p o s i t i v e d e f i n i t e . The buoyancy vector i s written as B = E B 8 = E ff Kpp p VZ • V^ 6 dR 6 n e n e e .. j r r n R = E pp p ( K b + K c ) (4-8) e r r xz n zz n L a t e r a l f l u i d f l u x i s lumped into the vector Q , which i s represented by Q = E Q S = - E / q 5 S dr S n e n e ^e u n = - E q»L e (4-9) 6 2 e e where T s i g n i f i e s element boundaries and L i s the length of the element side experiencing a f l u x q. The vector vanishes i f no boundaries i n the finite-element mesh are of the prescribed-flux type, or i f the prescribed f l u x i s zero, as i s the case on an impermeable boundary. The s p e c i f i c discharge (Darcy v e l o c i t y ) i s calculated f o r each element using the formulation given by Pickens and Grisak (1979), 3 q S = - E (K u 3§ S • h + K p d g e • h + K p p ) TI xx r m m xz r m m xz r r m=l -r 3x, dz 3 3 3 = - ;E (K p b • h + K p C vh + K p p ) (4-10) xx r _m m xz r _m- m xz r r m _ 1 2A 2A 3 and 76 3 q 8 = - E (K p .H6 h + K y 95 S h + K y. p ) n z ., zx r m m zz r m m zz r r m=l - T T — — — 7 , — 9x 9z 3 3 = - E ( . K y b h + K y c h + K y p ) (4-11) zx r _m m zz r _m m zz r r 2A 2A 3 These two vectors are needed to solve the equations governing heat and mass transport. The system of equations represented by Equation (4-6) can be solved using a direct method such as Gaussian elimination. Computational speed i s increased by taking advantage of the symmetric and banded nature of the conductance matrix. Only half of the matrix band needs to be stored i n the computer because of symmetry. I f the hydraulic head i s known ( i . e . prescribed) for the p-th node i n the g r i d , then the p-th equation i n (4-6) i s replaced by a dummy expression A • h = known head value (4-12) PP P The coeffi c i e n t A =1.0 and the constant head value i s substituted into PP other equations i n the system where h^ i s present. The known terms are then transferred to the right-hand side of Equation (4-6) i n order to preserve symmetry. The treatment of prescribed-flux nodes involves modifying the vector, as discussed e a r l i e r . Upon solving (4-6) for the hydraulic heads,, the Darcy v e o l c i t i e s are computed using Equations (4-10) and (4-11). Flow rates w i l l be constant over ind i v i d u a l elements, but discontinuous across element boundaries. A continuous velocity f i e l d can be obtained numerically by solving Equations (4-3), (4-10), and (4-11) simultaneously (Pinder and Gray, 1977), but the amount of computation increases dramatically. The finite-element method i s generally preferred over other numerical techniques, such as the f i n i t e - d i f f e r e n c e method, for several 77 reasons. F i r s t , the finite-element technique i s capable of accurately modeling complex boundary shapes. Second, anisotropic media properties are easily treated, while other methods have d i f f i c u l t i e s . Third, boundary conditions of prescribed f l u x are better accounted for i n the finite-element method. F i n a l l y , the numerical accuracy i s generally greater than the f i n i t e - d i f f e r e n c e method. Other types of interpolation schemes, beside the l i n e a r model, can also be implemented to achieve greater accuracy with fewer elements. The improved accuracy of the finite-element method becomes especially important when i t comes to solving the heat-transport equation, because of the added d i f f i c u l t y associated with the convection term. Heat Transport The finite-element method i s also used i n t h i s study to solve the steady-state heat equation. Using the same mesh, the operator i n t h i s case i s written as L(T) = V- [EVT] - p c v f q VT = 0 (4-13) Once again the approximate solution i s written i n terms of an interpolation formula - N T = T = E T £ S (4-14) m m m=l e • where £ are the same Chapeau functions defined for f l u i d flow, and T are m ^ ' m the temperature coefficients to be determined. As before, we set E ff L(T N) • £ S dR8 = 0 ; n = 1, 2, N (4-15) e Re Substituting Equations (4-13) and (4-14) into (-15), and integrating gives S T = F ; m, n = 1, 2, N (4-16) nm m n where S i s an N x N matrix and F i s a vector of size N. The thermal nm n 78 conductance matrix S i s expressed as nm S = E S 8 = E // (1 V£ S • VE 6 + pc q £ S • V£e) dR6 nm e nm e e n m vf u n m K = E [ 1 ( E b b + E Cb c + c b ) + E c c ) e -q^ " xx n m •:• . xz n m n m zz n m + pc _ (q :•. b + q c )] (4-17) vf TC v m z m 6 ' .. This' matrix i s also banded, diagonally dominant, and positive d e f i n i t e , but asymmetric. The asymmetry i s caused by convection terms i n (4-17). The l a t e r a l f l u x vector i s represented by F = E F S = - E / J • £ 6 dr 6 n e n e p e n = - E J L e (4-18). e . e where T s i g n i f i e s element boundaries and L i s the length of the element side experiencing a heat f l u x J. The term F^ vanishes i f the prescribed f l u x i s zero, such as on insulated (no heat flow) boundaries. The systems of equations represented by (4-16) i s easily solved using Gaussian elimination. In t h i s case, however, the f u l l band of the matrix must be stored because of i t s asymmetry. Mass Transport In groundwater hydrology, the equation of mass transport i s usually solved by the method of characteristics and finite-element techniques (e.g. Reddell and Sunada, 1970; Pinder and Cooper, 1970: Pinder, 1973). F i n i t e -difference models have also been applied, but they commonly encounter numerical dispersion problems, which are caused by truncation errors (Anderson, 1979). An alternative numerical technique i s the moving-particle random-walk method. Instead of approximating numerically the p a r t i a l d i f f e r e n t i a l equations, the random-walk simulates the p h y s i c a l behavior of the mass-transport system. This i s accomplished by tracking a large number of p a r t i c l e s as they are advected and dispersed through porous media. The p a r t i c l e s are disc r e t e quantities of mass that represent the chemical components dissolved i n the f l u i d . The advection of mass i s simulated by p h y s i c a l l y moving each p a r t i c l e a s p e c i f i e d distance i n space. This distance i s calculated from the l o c a l , average l i n e a r v e l o c i t y of the f l u i d and s p e c i f i e d time-step s i z e . Processes such as mechanical dispersion and d i f f u s i o n are simulated by allowing the p a r t i c l e s to take a random step i n space. The magnitude of the random step i s a function of the time-step s i z e , the dispersive properties of the media, and a random v a r i a b l e . The random-walk simulation, l i k e the other numerical methods, i s also performed on a d i s c r e t i z e d mesh. P r o b a b i l i t y theory i s usred to transform the p a r t i c l e d i s t r i b u t i o n over the mesh into concentrations of the transported aqueous species. Random-walk theory i s presented elsewhere i n the l i t e r a t u r e ( F e l l e r , 1961; Haji-Sheikh and Sparrow, 1966; Pipes and H a r v i l l , 1970), and need not be given here. In f a c t , i t i s n ' t necessary to understand the d e t a i l e d mathematics behind t h i s form of Monte Carlo simulation i n order to be convinced of i t s a b i l i t y to solve the mass-transport equation. Bear (1969) evaluates a simple, one-dimensional random-walk problem to prove t h i s point. Extensions can also be made up to three-dimensional space, but through a more complicated analysis (see Bear, 1972). The a p p l i c a t i o n of random-walk modeling i n hydrogeology i s not new, although the use of i t i s r e l a t i v e l y recent when compared to the other numerical methods. It was introduced for solving groundwater-pollution problems by Ahlstrom (1975) and Schwartz (19 75). The report by Ahlstrom 80 et a l . (.1977) was the f i r s t detailed explanation of the numerical approach. Since then, other particle-type codes have been published (e.g. Schwartz and Crowe, 1980; P r i c k e t t , Naymik, and Lonnquist, 1981). Random-walk modeling has been applied i n a variety of studies that range from f i e l d investigations (e.g. Naymik and Barcelona, 1981) to th e o r e t i c a l analyses of dispersion phenomena (e.g. Schwartz, 1977; Evenson and Dettinger, 1979; Smith and Schwartz, 1980). Ahlstrom et a l . (1977) provide an excellent discussion of the advantages and disadvantages of p a r t i c l e codes over other numerical methods. They conclude that the main advantage i s i t s a b i l i t y to minimize numerical dispersion i n transport models. P a r t i c l e codes are also easy to program, regardless of the number of chemical components i n the transport system. Their main disadvantage i s the large amount of computer time and storage required, especially for highly accurate solutions. Ahlstrom et a l . (1977) show that four times as many pa r t i c l e s must be used to double the accuracy of a random-walk simulation. In spite of t h i s disadvantage, the moving-particle random-walk method i s probably one of the better approaches for modeling multicomponent, mass transport i n sedimentary basins. The numerical implementation of the random-walk method i s simple. The aqueous components are represented by a f i n i t e number of p a r t i c l e s that are assumed to move independently from each other, and are of zero si z e . Each p a r t i c l e can be tagged with more than one aqueous component, i f we assume p a r t i c l e movement i s not affected by the type of species transported. Associated with each p a r t i c l e i s a set of s p a t i a l coordinates (x , z, ) for k k the k-th p a r t i c l e , and a set of discrete quantities of mass n for the s-th K component i n the k-th p a r t i c l e . During a single time step At, each p a r t i c l e i s separately advected 81 and dispersed, as shown i n Figure 11. The new lo c a t i o n of a p a r t i c l e a f t e r advection i s simply calculated from x * = x* + v « A t (4-19, K K X The l o c a l v e l o c i t y vector (v , v ) i s supplied from the finite-element s o l u t i o n to the f l u i d - f l o w equation [see Equations (4-10) and (4-11)]. step i s r e l a t e d to the root-mean-square distance (2D • A t ) 2 , which i s derived from the Gaussian d i s t r i b u t i o n . The random walk i s divided into a component co-incident-'.; with the flow vector v, and a component traverse to the flow path (Figure 11). The step length i s generated i n a random fashion by sampling a p r o b a b i l i t y d i s t r i b u t i o n having the proper root-mean-square value and zero arithmetic mean. Ahlstrom et a l . , u t i l i z e a uniform d i s t r i b u t i o n and derive the following expressions f o r the random-walk components z. * - z, + v • At k k z (4-19b) Ahlstrom et a l . (1977) show that the magnitude of the random-walk (4-20a) Az" = (24 D, i v i : . T At) 2 • [0.5 - (cp) ] (4-20b) 0 where D = l o n g i t u d i n a l dispersion c o e f f i c i e n t [ L / t ] Li D„ = transverse dispersion c o e f f i c i e n t [ L 2 / t ] > 1 ('$) = random number between 0 and 1 0 The dispersion c o e f f i c i e n t s are computed from L • v + D d (4-21a) D T = a T • v + D d (4-21b) where Figure 11.. Vector diagram showing p a r t i c l e - t r a n s p o r t component during a si n g l e time step At. • a = l o n g i t u d i n a l d i s p e r s i v i t y [L] a - transverse d i s p e r s i v i t y [L] D^ = e f f e c t i v e d i f f u s i v i t y [ L 2 / t ] v = average l i n e a r v e l o c i t y [L/t] The new p o s i t i o n of a p a r t i c l e at the end of a time step i s now known. It can be translated from the Cx"*, z') coordinate system (Figure 11) to the base coordinates (x, z) by using the formulas x n t +- = x* + (Ax' • cos $ - Az'-sin 3) (4-22a) z * + 1 = z* + (Ax' • s i n 3 + Az'-cos g) (4-22b) X K where 3 i s the angle between the flow vector and the x a x i s , and (x*, z*) k k are the coordinates of the p a r t i c l e a f t e r the advective step (Figure 11). The same c a l c u l a t i o n s are performed f o r each p a r t i c l e and repeated u n t i l the end of the simulation. S p a t i a l concentration patterns are obtained by superimposing a d i s c r e t i z e d mesh over the p a r t i c l e d i s t r i b u t i o n , and then tabulating the number of p a r t i c l e s i n each c e l l of the mesh. The t o t a l mass of each chemical component i n a c e l l i s found by summing the masses over a l l p a r t i c l e s present. Knowing the volume and porosity of the c e l l allows the mass to be expressed as a concentration value. These c a l c u l a t i o n s are only needed at the end of those time steps f o r which a s p a t i a l concentration pattern i s desired. A moving-particle simulation can be run i n two d i f f e r e n t modes. In one type of simulation, a single set of p a r t i c l e s i s released at a point source, and then i t s movement i s tracked through the mesh. This i s analogous to watching an instantaneous slug of dye that has been injec t e d into a flow f i e l d , and i s eventually dispersed as i t passes through the system. The second type of simulation involves a continuous release of new p a r t i c l e s at the source location f o r a l l time. This type of simulation could represent a source bed from which mass i s being continuously released as a heated brine passes through. After a s u f f i c i e n t l y - l o n g simulation of continuous release, the concentration pattern w i l l s t a b i l i z e to a steady form. Both types of simulation w i l l show an equal concentration change as mass i s transported through the groundwater-flow system. The presence of sources or sinks are accounted for by s e l e c t i v e l y removing or adding p a r t i c l e s , or by changing the component masses associated with existing p a r t i c l e s i n the mesh (Alhstrom et a l . , 1977).. Chemical reactions w i l l require that mass quantities be redistributed within each c e l l i n which reactions occur. To accomplish t h i s mass transfer, the mass of a given component for a given p a r t i c l e i s adjusted so that equilibrium and reaction path constraints are s a t i s f i e d at the end of a time step. Ahlstrom et a l . , (1977) suggest modifying the mass of each p a r t i c l e i n the reaction c e l l L by the r a t i o of the new concentration to the old concentration s,t+l r. _s,t+l •, s,t ., n R ' = [ C L' ] • n k ' (4-23) s t+1 where n^' represents the s-th component mass i n p a r t i c l e k at the new time l e v e l t t 1. The numerical procedure for calculating the new equilibrium concentrations i s presented i n the next section. Boundary conditions impose certain r e s t r i c t i o n s on the transport of-mass i n a fl u i d - f l o w system. Impermeable boundaries may demand that mass i s not allowed to cross them. Therefore, a p a r t i c l e model must check at each time step that mass i s not transported past no-flow boundaries during the advection or dispersion steps. A similar.accounting procedure must be made 85 along free-flow boundaries. P a r t i c l e s can be created at inflow boundaries (e.g. recharge areas), and allowed to disappear from the basin at natural discharge boundaries. Other types of phenomena may occur at boundaries that may require p a r t i c l e paths to be truncated, l e f t undispersed, or adsorbed (Evenson and Dettinger, 1979). In t h i s study, p a r t i c l e s are either reflected on impermeable boundaries, or allowed to enter or exit f r e e l y on free-flow boundaries such as the water table. The treatment of impermeable boundaries w i l l be discussed i n greater d e t a i l i n a l a t e r section. Geochemical Equilibrium and Reaction Paths Geochemical modeling of water-rock interaction has been an active area of research ever since Helgeson (1968) and Helgeson et a l . (1970) i n t r o -duced the theory of reaction-path modeling and demonstrated t h e i r computer program PATH. Many other .numerical models have been developed, over the past decade, to predict speciation and reaction progress i n aqueous systems. Nordstrom et a l . , (1979) and Wolery (1979a) present detailed discussions on the o r i g i n of the various computer programs used i n geochemistry, and review the special advantages and disadvantages of each code. Wolery's (19 78, 1979a) geochemical program EQ3/EQ6 i s currently the most v e r s a t i l e code that has been documented. I t offers a number of simula-ti o n options, some of which include the calculation of reaction processes under a temperature gradient and i n a flow-through environment. EQ3/EQ6 i s used i n t h i s study, and the discussion, that follows i s based e n t i r e l y on Wolery's approach to solving the equations governing chemical equilibrium (Table 2). The solution of t h i s set of equations i s achieved i n a series of calculations involving the following steps (Wolery, 1979a): (.1) The bulk composition of the chemical system i s specified by the number of moles of each element (n^, e = 1, £). (2) The governing equations are reformulated so as to reduce the number of operating variables to a set of unknowns, which are expressed i n a log-.' arithmic format. (3) An i n i t i a l calculation of the d i s t r i b u t i o n of species i s made using the method of monotone sequences (Wolery and Walters, 1975). (4) A very small amount of reactant i s introduced into the system. A new di s t r i b u t i o n of species i s then determined by solving the system of equations by Newton-Raphson i t e r a t i o n . (5) The value of each operating variable at a new point of reaction progress (£*) i s estimated from a Taylor's series expansion. The derivatives of t h i s expansion are determined from f i n i t e - d i f f e r e n c e calculations. (6) The aqueous system i s r e - e q u i l i b r i a t e d using Newton-Raphson i t e r a t i o n on the new set of mass constraints, thereby correcting the predicted estimates made i n step (5). (7) Mineral phases that become saturated with respect to the f l u i d phase are precipitated using a variety of c r i t e r i a l i s t e d by Wolery (1979a). Steps (5) to (7) are repeated u n t i l a l l the reactant i s destroyed or the system becomes saturated with respect to a reactant. The formulation of the governing equations i n terms of common logarithms i s given i n Wolery (1978, 1979a). The set of operating variables includes the masses of an independent basis set of aqueous species, the oxygen fugacity, the ionic strength parameters, and masses of minerals present i n the system. These operating variables are written as a vector z z = U w , £ s = 2 * s =', log f Q . , log I 1, log I, I i x = J..)(4-24) 2 where the symbols are the same as those defined i n Table 2, except & denotes a logarithmic mass (log ). I n i t i a l conditions are determined for z by means of the montone-sequence algorithm described by Walters and Wolery (1975). This i t e r a t i v e technique i s less sensitive to i n i t i a l conditions than other methods and handles a wide variety of input data. Newton-Raphson i t e r a t i o n i s used to solve the equilibrium equations during reaction-path calculations because i t i s usually fast to converge. Once an estimate has been made for z, the error can be determined from.the residual vector a, which contains the equilibrium equations. The Newton-Raphson method computes a correction vector 6 from the equation [J] • 6 = -a (4-25) where [J] i s the Jacobian matrix (9cu/9z_.), and i , j range over the elements i n z (Carnahan, Luther, and Wilkes, 1969, p. 319). The correction vector 6 i s solved d i r e c t l y from Equation (4-25) by using Gaussian elimination. A new estimate of the basis variables i s then made for the next i t e r a t i o n (m + 1) from z = z +6 (4-26) m+1 m m The i t e r a t i v e process converges as a approaches a value of zero. Wolery (1979a) discusses the several methods he uses to aid convergence i n EQ3/EQ6. Starting estimates of the operating variables at new points of reaction progress can now be determined from a Taylor's series expansion for the k-th component, which i s expressed i n terms of previously computed points as + ... (4-27) Z ] <| = z k | + (A£") • dz v, + (AE*)2 • d 2; '1 d£* 3 '1 d£*2 where the subscript 1 denotes the most recent point of reaction progress. Wolery (1979a) shows that the derivatives can be written i n f i n i t e - d i f f e r e n c e form. The accuracy of the reaction-path model w i l l depend on the step size Ag*, although truncation errors may accumulate even where terms as high as the sixth-order are used i n the Taylor's series (Wolery, 1979a). In EQ3/EQ6, the step s i z e i s repeatedly cut u n t i l convergence i s obtained i n Newton-Raphson i t e r a t i o n and the error i s reduced to a desired tolerance. At t h i s stage, the geochemical model consists of a closed system i n which the r e l a t i v e rates of d i s s o l u t i o n of a set of reactants are assumed to be the only r a t e - l i m i t i n g functions. Other types of reaction-path models are also considered by Wolery (1979a, 1980). The one of most p r a c t i c a l i n t e r e s t to ore genesis i s the flow-through model. It allows product minerals to be p e r i o d i c a l l y removed from the reacting system so as to simulate compositional changes i n the f l u i d as i t flows through the rea c t i n g media. These t r a n s f e r s do not, however, take into account f l u i d - f l o w rates nor are they associated with s p e c i f i c s p a t i a l coordinates. EQ3/EQ6 can also be used to simulate the e f f e c t of a temperature gradient on the composition of a f l u i d packet. (Wolery (1979a, 1979b) tr e a t s t h i s problem by modeling the temperature as a function of reaction . progress T = T 0 + Q± + © 2 ( £ * ) 2 + 0 3 ( S * ) 3 (4-28) where T0'= temperature (°C) at = 0 0 , 0 , 0 • - temperature-tracking c o e f f i c i e n t s X Z. o As i n the case of r e a c t i o n tracking, the v a r i a t i o n i n temperature i s an a r b i t r a r y " function and f u n c t i o n a l expressions other than a power serie s could be applied. In p r a c t i c a l a p p l i c a t i o n s , the reaction-progress v a r i a b l e i s usually set equal to the temperature ( i . e . 6 = 1 and 0 = 0 = 0). Wolery's codes contain a thermodynamic base that allows c a l c u l a t i o n s at any temperature value between 0°C and 35'0°C. The f i n a l result of a geochemical simulation i s a continuous description of the concentration of aqueous species and masses of minerals precipitated or dissolved during the reaction process. Solution Procedure The next step i s to design a solution method for combining the algorithms of the numerical techniques. The nature of t h i s procedure can be simple or complex, depending on the degree of coupling between the physical processes. The Complete Transport Model Consider a flow region as shown i n Figure 12. Suppose t h i s cross section represents a sedimentary basin for which the areal variation i n f l u i d v e l o c i t y , temperature, concentration and reaction-path geochemistry are to be modeled. Superimposed on the section i s a finite-element mesh. The lower boundary (A-B) i s the base of the sedimentary rock sequence,..which uncon-formably rests on a c r y s t a l l i n e basement. This surface i s impermeable to f l u i d flow and mass transport. A constant geothermal f l u x occurs along the basal boundary. The two l a t e r a l boundaries (A-D and B-C) are also imperm-eable to f l u i d flow and mass transport. However, they are imaginary flow boundaries that occur at groundwater divides, as discussed for Figures 7 and 8. The l a t e r a l boundaries are also chosen to be insulative to heat flow (see Domenico and Palciauskas, 1973), and coincide with the groundwater divides. A water-table surface (C-D) forms the upper boundary to the region of flow. It i s assumed to remain i n a constant position over geologic periods of at least a few m i l l i o n years. The temperature along the water table i s the mean annual a i r temperature and i t i s assumed to remain nearly constant. Chemical species are free to enter or leave the flow domain only along the FINITE ELEMENT MESH (Vertical exaggeration = 20 :1) Kilometers Figure 1 2 . Region of flow and finite-element mesh f o r two-dimensional analysis of transport processes i n sedimentary basins. 91 water table, but concentrations can be modified through reactions along the flow paths. The components of f l u i d flow, heat transport, and mass transport normal to the p r o f i l e are assumed to be n e g l i g i b l e , thereby j u s t i f y i n g the two-dimensional representation of the basin. In other words, the t h i r d dimension normal to the cross section i s of i n f i n i t e extent. Our goal i s to predict multicomponent mass transport i n the geologic section. S a l i n i t y and temperature gradients exist i n the basin, and they must be determined i n the numerical solution. The fl u i d - f l o w and heat-transport problems are coupled through t h e i r common dependence on density and v i s c o s i t y . F l u i d properties w i l l also depend on the concentration of dissolved s a l t s , especially NaCl i n brine-flow systems. The f l u i d - f l o w equation and heat equation are therefore coupled to the mass-transport equations. The number of chemical components and the possible reactions w i l l depend on the character of the pore f l u i d and types of mineral assemblages encountered as water flows through the basin. Variation i n temperature and s a l i n i t y w i l l also require that these effects be incorporated i n the geo-chemical calculations. In summary, the coupling between a l l of the transport equations demands that they be solved simultaneously, or at least successively i n some type of i t e r a t i v e technique. The equations of f l u i d flow and heat transport ought to be solved together at every time step of the mass-transport simulation. S i m i l a r l y , geochemical mass transfer ought to be computed for each c e l l i n the mesh at each time step, so as to maintain l o c a l equilibrium. The simulation would be finished after the desired number of time steps have been completed, or perhaps when a steady state has been achieved. This f u l l y integrated transport has not yet been solved. Several authors have developed various sets of governing equations, but no one has designed a p r a c t i c a l technique for simulating simultaneous inhomogeneous f l u i d flow, heat transport, and multicomponent mass transport with dispersion i n reactive porous media. The. studies by.Schwartz and Domenico (.1973), Fletcher and Vidale (1975), and Norton and Taylor (1979) represent the state-of-the-art i n the geological sciences for modeling mass transport i n reactive porous media. A l l three use the reaction-path approach i n computing chemical mass transfer along one-dimensional flow paths. None of these f i n i t e -difference codes take into account effects of dispersion, although the work by Fletcher and Vidale (1975) does include molecular d i f f u s i o n . The mass-transport code of Ahlstrom et a l . , (.1977) also contains an equilibrium sub-model, however, i t does not predict p r e c i p i t a t i o n or dissolution of mineral masses. Chapman (.1982) has succeeded i n coupling a geochemical model for simulation of mass transport i n r i v e r s . He uses an a n a l y t i c a l solution to represent one-dimensional mass transport of a nonreactive pulse of mass, and then modifies the pulse concentration for p r e c i p i t a t i o n , adsorption, and sedimentation processes as i t travels down stream. Other types of models have been developed to simulate mass transport i n reactive media, although they are commonly lim i t e d to a few reactions for which k i n e t i c data are available ( H i l l , 1978; L i , 1980). Mass-balance models of the type introduced by Plummer and Back (1980) may prove to be the most p r a c t i c a l approach i n modeling reactions i n regional flow systems. A Simplified Model The moving-particle random-walk model can be used to simulate mass transport through reactive media. To calculate geochemical mass transfer, flow rate, and temperature for each c e l l i n a mesh, and at every time step, i s a formidable and currently impractical task. I t i s clear that an alternative numerical procedure must be employed that w i l l simplify the problem to a manageable form. This i s especially desirable for the purpose of running the large number of simulations required by a s e n s i t i v i t y analysis. Figure 13 outlines the numerical procedure chosen i n t h i s study. One major s i m p l i f i c a t i o n i s to l i m i t the number of transported components to a single solute, such as metal or s u l f i d e . Furthermore, the mass-transport equation i s uncoupled from the computation of geochemical equilibrium. Geo-chemical reactions are assumed, therefore,to not affect the concentration or the path of the solute reaching the ore-forming s i t e . The amounts of minerals and speciation during ore p r e c i p i t a t i o n can be computed for any c e l l i n the mesh, but t h i s is. ^ accomplished'': out side of the- transport code by running the EQ3/EQ6 program separately. For the purpose of t h i s study, only the temperature and nonreacted concentration of a proposed depositional c e l l are used as input to EQ3/EQ6. The geochemical model i s only used to provide possible scenarios of ore p r e c i p i t a t i o n at the depositional s i t e , and not reactions along the flow path. A second major s i m p l i f i c a t i o n i s to assume, that s a l i n i t y gradients are constant i n space and time. This allows the mass-transport equation for a s a l t component to be eliminated from the problem, which uncouples the transient dependence of f l u i d properties on s a l t concentrations. A s a l i n i t y p r o f i l e can s t i l l be inserted to make the f l u i d - f l o w model more r e a l i s t i c of basinal brines, but i t s effects are only f e l t i n the f l u i d - f l o w and heat-flow solutions. Solute concentrations along the flow path w i l l be influenced i n d i r e c t l y by s a l i n i t y because of the change s a l i n i t y w i l l cause on f l u i d v e l o c i t y . The variation i n s a l i n i t y i n the basin can be specified by the program user as either a l i n e a r gradient with depth or as an arbitrary d i s t r i b u t i o n based on f i e l d data. Referring once again to Figure 13, control parameters are read i n 94 P R I N T DATA E C H O P R I N T N O D A L C O O R D I N A T E S 7-7 -^ S T A R T ^ R E A D IN D A T A G E N E R A T E M E S H E S ZP R I N T H E A D S 7 _ J FOR N O D E S / * " | / P R I N T E L E M E N T ' D A R C Y V E L O C I T Y Z P R I N T 7 T E M P E R A T U R E S / • FOR N O D E S / P L O T M E S H E S J D E T E R M I N E F L U I D P R O P E R T I E S C O M P U T E H Y D R A U L I C H E A D S P L O T C O N T O U R E D H E A D S C A L C U L A T E D A R C Y V E L O C I T I E S P L O T V E L O C I T Y V E C T O R S , C O M P U T E T E M P E R A T U R E S P L O T C O N T O U R E D T E M P E R A T U R E S L N 0 C H E C K FOR X ( C O N V E R G E N C E ) \ ^ IF N E E D E D J Y E S T R A N S P O R T P A R T I C L E S : A D V E C T , R A N D O M WALK P L O T P A R T I C L E D I S T R I B U T I O N (' A L L T I M E " \ ^_ S T E P S D O N E ? / " * P L O T C O N T O U R E D C O N C E N T R A T I O N S Y E S C O M P U T E INITIAL S P E C 1 A T ION : E 0 3 C O M P U T E R E A C T I O N P A T H : E O 6 ^ S T O P -Figure 13. Flow chart f o r s i m p l i f i e d transport code. (Dashed l i n e s indicate route i s optional.) f i r s t which set up the simulation options, the size and shape of the mesh, the geologic configuration of the s t r a t i g r a p h i c u n i t s , the material properties, and reference f l u i d properties. The computer program then generates two meshes. One of the grids i s composed of t r i a n g u l a r elements (Figure 12), and i s used i n solving the coupled f l u i d - f l o w and heat-transport equations. The other mesh i s constructed by combining t r i a n g l e s to form quadrilateral-shaped c e l l s , which are used i n c a l c u l a t i n g concentrations and tracking p a r t i c l e s i n the random-walk sol u t i o n to the mass transport equation. Both grids have the same nodal coordinates. Boundary conditions of prescribed head and temperature, as well as prescribed f l u x , are accounted f o r i n the code. For the transport model, the number of input p a r t i c l e s and t h e i r i n i t i a l locations are also included as input data. At the s t a r t of a simulation, the steady-state hydraulic-head d i s t r i b u t i o n i s computed f i r s t by assuming there are no s a l i n i t y or tempera-ture gradients present. Using the nodal values of the computed hydraulic head, average l i n e a r v e l o c i t i e s are then computed f o r each t r i a n g u l a r element i n the mesh. The steady-state heat equation i s solved next to f i n d the temperature pattern. With the new values of pressure and temperature, and inputted s a l i n i t y p r o f i l e , f l u i d d e n s i t i e s and v i s c o s i t i e s are ca l c u l a t e d from the equations of state. These four steps are repeated u n t i l the i t e r a t i o n s converge to a stable temperature s o l u t i o n . Convergence i s achieved when the maximum temperature change between i t e r a t i o n s i s les s than a s p e c i f i e d tolerance f o r a l l nodes i n the mesh. Most problems solved i n t h i s study required l e s s than 5 i t e r a t i o n s , f o r a tolerance l e v e l of 1°C. Afte r obtaining the steady-state v e l o c i t y d i s t r i b u t i o n , the next step i s to solve the mass-transport problem f o r a nonreactive solute. The source bed of the in j e c t e d mass i s s p e c i f i e d by the user. P a r t i c l e movement 96 i s composed of an advected and random-walk component. At the end of each displacement, a check i s made to ensure that p a r t i c l e s are properly conserved i f they encounter any boundaries. The coordinates of each p a r t i c l e , and the tags that i d e n t i f y the c e l l l o c ation, are continuously updated through the simulation. The size of the time step i s constant, and i t must be chosen caref u l l y so as to ensure p a r t i c l e s reside i n any given c e l l for at least a few time steps. I f the p a r t i c l e motion per time step i s too large, the s t a t i s t i c a l accuracy of the solution w i l l weaken and the p a r t i c l e s may become very d i f f i c u l t to track without performing time-consuming grid searches. Two basic types of information are provided by the mass-transport model: the path of transport, and the concentration decrease caused by the d i l u t i o n effect of dispersion. The easiest way to obtain these results i s to release a certain mass of p a r t i c l e s at a chosen source lo c a t i o n , and then follow the pulse as i t moves through the basin. Given enough time steps, a l l of the p a r t i c l e s w i l l eventually leave the mesh along discharge zones at the water table. P a r t i c l e s crossing no-flow barriers are reflected back into the mesh i n order to meet t h i s boundary condition. The number of p a r t i c l e s leaving free-flow boundaries ( i . e . water table) are tabulated, along with t h e i r exit p o s i t i o n , for l a t e r reference. P a r t i c l e motion from one hydro-stratigraphic unit to another requires special handling because of the problems involved when f l u i d v e l o c i t i e s change suddenly between c e l l s . The technique used i n t h i s study i s to modify the p a r t i c l e step size by the r a t i o of the v e l o c i t i e s between c e l l s . I f a steady-state p a r t i c l e d i s t r i b u t i o n i s sought, then mass must be continuously released from the source bed u n t i l a steady pattern emerges. For the scale of the problem modeled here, the cost of a steady-state . solution i s prohibitive because of the large number of p a r t i c l e s (tens of 97 thousands) that must be tracked to obtain an accurate solution. In t h i s study, only the single-pulse method i s u t i l i z e d i n pre-di c t i n g the transport path. The pulse of p a r t i c l e s i s released from a single c e l l l o c a t i o n , i n any part of the basin. An i n i t i a l concentration of Co i s assigned to the source c e l l . As the mass i s transported along the flow path, dispersive processes reduce the concentration of the pulse, which i s simulated by the spreading of the p a r t i c l e cluster. Relative concentrations C/Co can be computed at any specified time step i n the simulation period by simply tabulating the mass of p a r t i c l e s i n each c e l l , as discussed e a r l i e r . The concentration value of the c e l l i s then assigned to a f i c t i t i o u s point i n the center of the c e l l for contouring purposes. Experience with the random-walk model provides the only c r i t e r i a for selecting the number of p a r t i c l e s needed and the best size of time step. Reasonably accurate solutions are obtained with 1000 to 2000 p a r t i c l e s , although as few as 500 p a r t i c l e s can be used i n some situations. The p a r t i c l e model gives the areal and transient variation i n the concentration of one of the ore-forming components, such as zinc, lead, or reduced su l f u r . This concentration pattern i s based on the assumption that advection and dispersion are the only processes effecting the concentration of the component between source bed and depositional s i t e . Reaction scenarios at the proposed ore-forming s i t e can be made by making further assumptions regarding the general composition of the brine carrying the component to the s i t e . Usually t h i s i s not d i f f i c u l t because the general compositions of ore-forming brines are well known from f l u i d - i n c l u s i o n data and present-day basinal brines. With the ore-forming component concentration i n hand, and reasonable assumptions made for brine composition, EQ3/EQ6 i s then used to assess a 98 variety of possible p r e c i p i t a t i o n mechanisms for the depositional s i t e . This program package consists of two separate codes, EQ3 and EQ6, and supporting thermodynamic data f i l e s . About 180. aqueous species and 175 minerals exist i n the data f i l e s , which are supported i n equilibrium calculations up to 350°C. Accurate modeling of saline aqueous solutions i s li m i t e d to ionic strengths of less than about one molal i n EQ3/EQ6 (Wolery, 1979a). As shown i n Figure 13 and discussed e a r l i e r , EQ3 calculates an i n i t i a l d i s t r i b u t i o n of species, which i s then used as an input model for EQ6. The simulation then proceeds by dissolving the specified set of reactants, while tracking the composition of the f l u i d and predicting the amounts of secondary minerals precipitated. EQ6 finishes a simulation when the t o t a l mass of the reactants are consumed, the specified temperature change i s achieved, or when the system becomes saturated ( e q u i l i b r i a t e s ) with respect to the reactants. The description of the simulation procedure i s now completed. Execution times for the transport model and EQ3/EQ6 vary with the size of the finite-element mesh, stratigraphic complexity, number of p a r t i c l e s transported, and geochemical nature of the reaction-path simulation. An average transport simulation commonly takes about one minute of Central Processing Unit (CPU) time to process on the Amdahl 4-70 V8 computer at the University of B r i t i s h Columbia. A single reaction-path simulation also takes about one minute of CPU time, for the types of simple simulations to be presented l a t e r . Longer execution times may be required i n complicated problems. Model V e r i f i c a t i o n The transport code developed for t h i s study has received considerable t e s t i n g , as i s outlined i n the following pages. The f l u i d - f l o w part of the 99 code was v e r i f i e d by running several test-data decks i n another f i n i t e -element routine, which was written by Pickens and Lennox (1976). Comparisons were only made f o r the case of an isothermal, fresh-water basin. S i m i l a r l y f o r the heat-transport subroutine, r e s u l t s of several simulations were compared to model r e s u l t s presented elsewhere, i n t h i s case by Betcher (1977). Only the mass-transport subroutine was tested i n further d e t a i l , mainly because the use of moving-particle random-walk codes i s r e l a t i v e l y new i n hydrogeologic modeling. Solute-transport models are commonly v e r i f i e d by comparing the numerical answer to a n a l y t i c a l solutions of the advection-dispersion equation. Bear (1972, p. 627) presents several a n a l y t i c a l expressions that can be used fo r t e s t i n g purposes. One of the simplest problems i s to solve f or the spread of a concentration front i n a semi-infinite, column. The f l u i d v e l o c i t y i s assumed to be uniform and the solute can e i t h e r be inje c t e d as a sin g l e slug or continuously released at one end of the column. Schwartz (1975, 1978) and Ahlstrom et a l . (1977) compare t h e i r particle-code solutions to the a n a l y t i c a l s o l u t i o n of t h i s one-dimensional problem. Both studies show that the accuracy of a p a r t i c l e model depends on the number of p a r t i c l e s used, the time step, and the c e l l s izes i n the mesh. They found that a few thousand p a r t i c l e s are needed to produce highly accurate solutions i n many transport problems. The same type of one-dimensional t e s t was applied to the p a r t i c l e code developed i n t h i s study. Several simulations were conducted to e s t a b l i s h the working a b i l i t y of the algorithm. The r e s u l t s and conclusion duplicate those of Schwartz (1975) and Ahlstrom et a l . , (1977). The model was also compared against a two-dimensional simulation given by Schwartz (1978). The problem with one-dimensional t e s t s i s that they do not v e r i f y the numerical treatment of the two-dimensional boundary conditions, which are 100 encountered i n modeling sedimentary basins. P a r t i c l e models commonly treat impermeable boundaries as being r e f l e c t i v e , but the theory isn't clear on why p a r t i c l e s have to be reflected. Evenson and Dettinger (1977) suggest that p a r t i c l e s could a l t e r n a t i v e l y be adsorbed on boundaries, or t h e i r paths truncated. A new boundary-value problem was therefore designed to help investigate t h i s question, and to test the accuracy of the code in two dimensions. The remainder of t h i s section discusses these te s t s . Consider the transport of a nonreactive solute i n a uniform vel o c i t y f i e l d v, which i s directed along the +' axis i n the x - z plane. The flow region i s bounded i n the z - direction by p a r a l l e l , impermeable boundaries at z = L and z = - L, and i s of i n f i n i t e extent i n the x - dire c t i o n . At time t = 0, a slug of solute with mass M i s instantaneously injected into the flow f i e l d at the coordinate o r i g i n (0, 0). The porous medium i s i s o -t r o p i c and homogeneous, with porosity <j>. The problem at hand i s to solve for the transient concentration pattern as the solute i s advected and dispersed through the flow region. Mathematically, the concentration d i s t r i b u t i o n i s found by solving the following p a r t i a l d i f f e r e n t i a l equation v 3C dx (4-29a) subject to the conditions that C (x, z, o) = M 6(x) • 6(z) for t - o (4-29b) 3C - o at z - +L, - L 3z for t > o (U-29c) C (x, z, t ) i s bounded for . x (4-29d) where 101 C = concentration of solute [M/L3] D = longitudinal dispersion coeff i c i e n t [L 2/t] Li D = transverse dispersion coeff i c i e n t [L 2/t] .v - f l u i d v e l o c i t y [L/t] <j> = porosity 6 = dirac-delta function The a n a l y t i c a l solution of t h i s problem i s too lengthy to present here, but the derivation can be obtained from the author upon writing. In short, i t involves the use of Green's functions and the method of images, both of which are described i n advanced mathematics texts Ce.g. Carslaw and Jaeger, 1959). The f i n a l result i s C(x, z, t ) = (M/(j)) • exp[ -(.x-vt) 2 ] • { exp[ - z 2 ] + exp [ - U - 2 L ) 2 ] 47rt (D BT)h 4 D_"t 4D Tt 4D Tt + exp[ -(z-2L) 2 ]•} (4-30) M> t The l a s t two exponential terms i n t h i s equation account for the boundary effects i n the transport problem. U n t i l the solute mass spreads out enough to be affected by the boundaries, the influence of these terms i s small. The numerical solution of Equation (4-29) i s completed with the moving-particle model and the mesh shown i n Figure 14. Notice that the v e r t i c a l scale i s expanded to twice the horizontal scale so as to i l l u s t r a t e d e t a i l s of the g r i d . The top and bottom boundaries are represented as no-flow b a r r i e r s , while the two l a t e r a l borders allow p a r t i c l e s to enter or leave fr e e l y . The flow region, i s 460 m long by 75 m wide, and the porous medium has the following parameter values: <j> = 0.10, v = 1.0 m/yr, and DL = DT = 3 m2/'yr. The o r i g i n a l amount of mass released at point Q (Figure 14) i s M = 1.0 x 10 4 mg. At t = 0, 5000 p a r t i c l e s are instantaneously no-flow boundary LINE OF PLOTTED SOLUTION V=l m/yr AZ=5m{ r N \ \ \ \ 1 1 1 0 1 \ 1 \ 1 \ \ 1 \ 1 1 1 \ \ \ \ \ V > \ \ V \ \ V V \ \ \ ^ \ \\ 1 V v \ \ \ \ ^ \ I . ... J \ \ \ I V V 1 A X = 20 m (VERTICAL EXAGGERATION - 2) A" ORIGIN OF SOLUTE MASS free-flow boundary Figure 14. Mesh used i n t e s t i n g moving-particle code against a n a l y t i c a l s o l u t i o n at.'.AVA'. o 103 released with an :assigned mass of 2 mg each. A time-step of 0.5 yr i s .used i n running the simulation over a time i n t e r v a l of 100 years. As the cloud of par t i c l e s move across the f i e l d , p a r t i c l e s are dispersed both along the flow path and transverse to i t . The p a r t i c l e cloud eventually encounters the two bar r i e r boundaries at z = +L, -L. Three separate simulations are made to test three different ways of treating p a r t i c l e s that cross no-flow boundaries. The f i r s t method r e f l e c t s p a r t i c l e s back into the mesh, the second method truncates t h e i r path at the b a r r i e r , and the t h i r d method a r b i t r a r i l y takes the ejected p a r t i c l e and relocates i t at the center of the c e l l from which i t l e f t . Figure 15 i l l u s t r a t e s the results of the three simulations, for the traverse l i n e A-A"* marked i n Figure 14-. This position of the solute pulse corresponds to t = 100 years. The concentrations i n each c e l l along A-A' are plotted on the ordinate, and distance from the center l i n e on the abscissa The graphs also contain the computed a n a l y t i c a l solution from Equation (4-30), with x = 100 m and t = 100 yr. These results v e r i f y that a r e f l e c t i v e type boundary condition i s the proper technique to use i n simulating no-flow b a r r i e r s . Truncating the path of the p a r t i c l e or relocating the p a r t i c l e are un j u s t i f i e d from a mathematical point of view. Both the truncation and relocation techniques over estimate the amount of mass near the center of the distribution,.and under estimate the concentration near the r e a l barriers (Figure 15). Simulation Example In order to demonstrate the f u l l use of the s i m p l i f i e d numerical procedure, a sample simulation i s present i n Figure 16. I t i s meant to show the modeling approach only and should not be taken as a f i n a l r e s u l t . The simulation does, however, give a t y p i c a l selection of the types of computer C mg/l (a ) REFLECTED igure 15. Comparison of moving-particle random-walk solutions of t h r e e - d i f f e r e n t treatments of p a r t i c l e s crossing no-flow boundaries with a n a l y t i c a l s o l u t i o n . Figure 16. Simulation example. H Y D R A U L I C H E A D S Kilometers (f) Figure 16. (Continued) 4n I — Release Origin 4-i 2-{ <x> a> E o i n I l I—r OJ E * i o o MASS CONCENTRATIONS Time = 40,000 years MASS CONCENTRATIONS Time = 60,000 years 1 1 " r l 1 1 1 1 1 J | 1 1 I 1 1 1 1 1 ( h ) MASS CONCENTRATIONS Time = 70,000 years i i i i i i—i 1 — i — i — i — i — i — 50 100 150 Kilometers —i 1—i 1 1 1—i 1 — i — i 1 200 250 300 (i) Figure 16. (Continued) 4 n 5 3 I 2 4 MOVING PARTICLE MESH 50 100 1 I 1 1 1 150 Kilometers — | 1 — i — i — i — | — i 1 — i — r 200 I 250 300 i ( j ) / PARTICLE DISTRIBUTION TIME =70000 years . 1 4 7 3 7 1 2 5 3 4 6 7 3 5 1 4 2 4 4 2 5 2 6 2 3 3 4 1 5 4 4 I I 4 ~ ~ I 1 I I 1 6 3 4 5 1 6 1 5 2 8 5 5 7 5 5 4 8 5 0 3 9 1 6 1 6 I I 8 2 3 1 1 2 4 4 2 5 5 9 1 2 1 3 1 4 2 9 2 3 2 6 3 5 2 4 4 5 4 4 6 8 5 1 7 6 6 7 8 1 7 9 7 9 6 9 9 5 4 2 5 4 1 7 1 5 r 2 km H I km 250 km 1 300 km Figure 16. (Continued) ' H O CO 109 Figure 16. (Continued) plots generated by the code. Table 5 l i s t s values of the parameters used i n t h i s simulation. The numbers chosen for these parameters are j u s t i f i e d i n the next chapter. For now i t i s s u f f i c i e n t to say that they are somewhat ar b i t r a r y , but the numbers are representative of media properties and brine compositions that occur i n sedimentary basins. Approximately 50 seconds of CPU time was needed to run t h i s transport problem (Figures 16a to 16h), and 65 seconds was used i n predicting the reaction-path model (Figure 1 6 i ) . Figure 16a i s a schematic cross section of a sedimentary basin. The wedge-shape p r o f i l e i s designed to r e f l e c t the major features considered i n the conceptual model of Figure 6. The length of the basin i s 300 km and the stratigraphic sequence varies i n thickness from 3000 m at x = 0 km, down to 900 m at x = 300 km. In t h i s section, the v e r t i c a l scale i s expanded to 10 times that of the horizontal scale for i l l u s t r a t i v e purposes. The geology of. the basin i s very simple. I t consists of three main stratigraphic u n i t s , which have different hydraulic properties (Table 5). The lower boundary forms the contact of the permeable, sedimentary sequence with an impermeable, c r y s t a l l i n e basement. The basement surface dips gently to the l e f t side of the diagram at a rate of about 3.3 m/km. The beds marked and i n Figure 16a produce an extensive basal aquifer, representative of a karstic-type carbonate unit which passes down dip into a less permeable limestone-dolomite facies. Overlying the basal aquifer i s a thick sequence of low-permeability mudstones (K^). The K • unit i s the most permeable unit i n the section (Table 5). It has a hydraulic conductivity that i s 100 times greater than the K beds and 5 times greater than the hydraulic conductivity of the K-, beds. An anisotropy r a t i o (K /K ) of 100:1 i s assumed for a l l 2 J xx zz strata i n the basin. This value i s chosen to represent the inherent aniso-tropy within i n d i v i d u a l sedimentary beds, and the layered heterogeneity that Table 5. Model parameter data for simulation example Parameter Symbol Hydrostratigraphic Unit 2 3 F i e l d Units F l u i d Flow Porosity Horizontal hydraulic conductivity V e r t i c a l hydraulic conductivity Reference f l u i d density Reference f l u i d v i s c o s i t y S a l i n i t y gradient Heat Transport Thermal conductivity of f l u i d Thermal conductivity of so l i d s S p e c i f i c heat capacity of f l u i d S p e c i f i c heat capacity of s o l i d s Longitudinal thermal d i s p e r s i v i t y Transverse thermal d i s p e r s i v i t y Geothermal heat flux at base Temperature at water table xx K zz PO vf J To 0.20 0.15 500.0 100.0 5.0 1.0 998.2 ( a l l units) 1 . 0 x l 0 ~ 3 ( a l l units) 0.005 ( a l l units) 0.63 ( a l l units) 3.0 3.0 4187.0 ( a l l units) 1005.0 ( a l l units) 100.0 100.0 1.0 1.0 70.0 (constant) 20.0 (constant) 0.10 5.0 0.05 2.0 1.0 1.0 f r a c t i o n m/yr m/yr kg/m3 kg/m-s or Pa's % NaCl/m W/m°C W/m°C J/kg°C J/kg°C mW/m^  (Continued on next page) Mass Transport Longitudinal d i s p e r s i v i t y Transverse d i s p e r s i v i t y Apparent d i f f u s i o n c o e f f i c i e n t 100.0 1.0 3.0x10 " 3 100.0 1.0 m 1.0 1.0 m 3.0xl0" 3 3.0xl0" 3 m 2/yr Geochemical Reactions/Mass Transfer I n i t i a l equilibrium constraints: Species Na C l Zn Pb C0„ Concentration (per kg H 20) 1.0 moles lo0 moles 0.50 mg 0.01 mg 0.01 moles pH Eh T 6.8 +0.58 vo l t s C a l c i t e , Dolomite and Quartz are i n i t i a l l y i n equilibrium with the f l u i d SO,. 100 Reactant: 1.0 x 10~ 3 moles of dissolved H 2S(g) 35 mg/kg H 20 reactant-tracking c o e f f i c i e n t s 5 = 0 ? = 1 C, 113 exis t s at a f i n e r scale than the stratigraphy depicted i n a regional representation. The boundary conditions are the same as those described for Figure 12. The bottom i s impermeable to f l u i d flow and mass f l u x , and a constant geothermal f l u x J i s prescribed along i t s length. The two l a t e r a l boundaries are barriers to f l u i d flow, heat flow, and mass transport. The upper surface i s the water table, which i s composed of two linear-slope segments. I t i s meant to r e f l e c t a basin with a gentle topographic gradient. Mass i s allowed to leave the basin where groundwater discharges across the water table. The temperature along the upper boundary i s fix e d at the mean annual a i r temperature. Figure 16b shows the finite-element mesh used i n solving the f l u i d -flow and heat-transport equations. I t consists of 252 nodes and 410 triangular elements. The same nodes are u t i l i z e d i n constructing quadrilateral-shaped c e l l s that are used for tracking p a r t i c l e movement and computing concentrations. Figures 16c, 16d, 16e, and 16f display the numerical solution to the coupled problem of inhomogeneous f l u i d flow and heat transport. In Figures 16c and 16d, the hydraulic-head solutions are plotted with a 50 m contour-i n t e r v a l . By d e f i n i t i o n , the hydraulic head i s equal to the elevation at the water table. Therefore, the contour values take on the elevation of the point at which i t intersects the water-table surface. The hydraulic-head pattern in Figure 16c i s derived from the f i r s t i t e r a t i o n of calculations, i n which there are no s a l i n i t y or temperature gradients present. Upon completing s i x iterations of solving the f l u i d - f l o w and heat equations, the coupled solutions converge to the patterns shown i n Figures 16d and 16f. A l i n e a r s a l i n i t y gradient of 0.005% NaCl/m i s assumed to exist with depth, but only below the 2 km elevation l e v e l . Above t h i s datum the water i s assumed to be 114 fresh. This s a l i n i t y gradient produces a maximum s a l i n i t y of 10% NaCl at the thick end of the wedge-shape basin. The v e l o c i t y f i e l d of the basin i s e a s i l y calculated upon knowing the hydraulic-head solution, s a l i n i t y p r o f i l e , and temperature d i s t r i b u t i o n . Figure 16e .shows the average-linear v e l o c i t y vectors for the simulation example. The orientation of the arrows indicate the direction of flow and the length of the arrows indicate the magnitude of the vel o c i t y value. A velo c i t y value i s calculated for each triangular element i n the mesh (Figure 16b), but only h a l f of the vectors are plotted for c l a r i t y . Notice that the presence of the basal carbonate aquifers causes f l u i d flow to be directed down across the shale beds i n the elevated end of the basin ( i . e . x = 0 to x = 150 km), and discharged upwards at the basin margin. The influence of permeability changes i s best i l l u s t r a t e d by the sharp bends i n the hydraulic-head contours, but these changes are also marked by the velocity-vector pattern. V e l o c i t i e s vary i n the basin from an average of 0.2 m/yr i n the K unit to over 10.0 m/yr i n the K beds. Heat transport i s affected by both thermal convection and conduction, as shown by the temperature pattern i n Figure 16f. Much of the bending and spacing of the isotherms i s due to the thermal conductivities of the s t r a t i -graphic units and the geothermal f l u x imposed along the base of the section. However, s i g n i f i c a n t convection effects are also included i n the numerical solution. This i s especially evident i n the recharge end of the basin where downward f l u i d flow depresses the temperature, and i n the discharge end of the basin where upward f l u i d flow elevates the temperature (Figure 16f). In t h i s simulation, temperatures easily exceed 100°C at the very end of the section, primarily because of the influence of f l u i d flow. Conduction alone would not produce these warm temperatures at such a shallow 115 depth, under normal heat flow conditions. Figure 16g shows the results of the random-walk solution to the advection-dispersion equation. The purpose of t h i s simulation i s to predict the movement and dispersion of a metal-bearing f l u i d packet, which i s released from an arbitrary source location i n the shale beds, at the recharge end of the basin (see Figure 16g). Only the concentration patterns at four-different times are plotted. The contour values represent 3%, 6%, 9%, and 12% of the o r i g i n a l concentration Co released at the source. Notice that as the pulse travels through the basin, dispersion causes the plume to spread out with time as concentrations weaken. The maximum concentration of solute reaching the edge of the basin i s approximately 15% of the source Co. Figure 16'.g shows the detailed p a r t i c l e d i s t r i b u t i o n at t = 70000 yr. These numbers represent the actual number of p a r t i c l e s counted in each c e l l at that s p e c i f i c time step. Also plotted i s the cumulative p a r t i c l e discharge at the water table. I f a simulation i s run long enough, a l l of the p a r t i c l e s w i l l discharge to the surface. The particle-discharge d i s t r i b u t i o n along the water table provides qua l i t a t i v e information as to the nature of mass transport i n the basin. The use of t h i s information i s shown l a t e r . So far we have not specified what type of metal species was trans-ported or. the o r i g i n a l source concentration. The main assumption i s that the solute i s nonreactive along the flow path. Because the concentrations are expressed as a dimensionless fraction (percent) of the source value, the simulation can be used for any concentration l e v e l of interest. I f the source for example, i s sustaining a zinc concentration of 3 mg/kg H^ O, then about 0.5 mg/kg H^ O w i l l reach the aquifer zone near the edge of the basin (x - 250 km). We can then use t h i s value as input to geochemical reaction-116 path models. Suppose the ore-forming s i t e i s located at a depth of about 1 km i n the unit and near x = 250 km (Figure 16a). The temperature at t h i s depth i s approximately 75°C and the s a l i n i t y i s 7% NaCl (about 1.0 molal). Let's assume that approximately 35 mg/kg H20 (10 3 m) of H^ S (gas) i s being continuously made available through sulfate reduction i n the organic-rich carbonate aquifer. Figure 16k shows the r e s u l t i n g reaction path and f l u i d speciation as the metal-bearing brine encounters H^ S at the proposed ore-forming s i t e . The i n i t i a l composition assumed for the brine i s l i s t e d i n Table 5. Only a selected group of the aqueous species present are plotted i n Figure 16k. As H^ S begins to dissolve, the brine f i r s t precipitates dolomite, which l a t e r dissolves, but addition of further H2S causes sphalerite and galena to precipitate. The oxidation state becomes highly reduced and the pH becomes more acidic as the reaction progresses. Approximately 0.72 mg of sphalerite and 0.01 mg of galena are produced per kg of water. At the same time 0.97 mg of dolomite and 35.0 mg of H S are dissolved. The time required to deposit an average size ore body can now be made by using the flow rates computed e a r l i e r . The average l i n e a r v e l o c i t y near t h i s s i t e i s about 9.0 m/yr (Figure 16e), therefore, the s p e c i f i c d i s -charge (flow rate per unit area) i s about 1.8 m3/m2*yr. The volume flow rate can be calculated by assuming an area through which the flow i s taking place. A short calculation shows that for an e f f e c t i v e discharge area 200 m thick by 10 km wide, about 2600 kg/yr of sphalerite would be precipitated at the proposed ore s i t e . To deposit 100 m i l l i o n tons of 5% Zn w i l l require therefore, about 2.5 m i l l i o n years under t h i s flow rate and brine-reaction scenario. 117 I t i s clear that the transport models and reaction-path simulations can provide a s i g n i f i c a n t amount of information about the role of f l u i d flow i n ore genesis, and i t s effects on subsurface temperatures, mass-transport patterns, and deposition rates i n the ore-forming process. I t should be possible to learn much more about the factors affecting ore genesis i n sedimentary basins by continuing t h i s analysis i n greater d e t a i l , and by comparing simulated models to actual f i e l d situations. CHAPTER 5 EVALUATION OF TRANSPORT PROCESSES IN ORE GENESIS: QUANTITATIVE RESULTS The purpose of t h i s chapter i s to carry out a detailed s e n s i t i v i t y analysis with the numerical models. The goal of these experiments i s to determine the influence of various factors on the transport processes involved i n the formation of carbonate-hosted lead-zinc deposits. Through the simu-l a t i o n s , the model of a gravity-driven flow system w i l l be f u l l y tested, and a comprehensive set of conditions and parameter constraints w i l l emerge to help establish the role of regional groundwater flow in ore genesis. A s e n s i t i v i t y analysis can take one of two different routes, depending on the purpose of the study. One approach i s ca l l e d s i t e - s p e c i f i c because the modeling i s designed for a p a r t i c u l a r geologic basin or s p e c i f i c ore deposit. It can y i e l d detailed information, provided an adequate amount of data can be collected to calibrate the model. The second approach i s known as generic because the simulations are carried out on hypothetical' basins... Such models are designed to r e f l e c t many of the geologic features observed i n r e a l sedimentary basins, but are purposely kept simple i n order to be of general use i n understanding the phenomena being studied. For the generic approach, a range of r e a l i s t i c input data i s chosen to assess general mechanisms and to assess the s e n s i t i v i t y of the model results to a change i n the numerical value of any p a r t i c u l a r parameter. Generic modeling forms the main basis of analysis i n this istudy. The simulation results given below are divided into several sections, as outlined i n Table 6. F i r s t , the factors affecting f l u i d flow, heat 118 Table 6. Outline of s e n s i t i v i t y analysis Simulation Parameter Figure F l u i d Flow Isotropic hydraulic conductivity 18 Anisotropic hydraulic conductivity 19 Layering ' 20 Discontinuous layers (pinchouts) 21,22 Temperature-dependent flow 23,24,25,26 Salinity-dependent flow 27,28,29,30 Combined temperature and s a l i n i t y e f f e c t s 31 S a l i n i t y gradient 32 Basin size 33,34,35 'Basement structure 36,37,38,39 Water-table configuration (topography) 40,41,42,43,44 Heat Transport Hydraulic conductivity 45,46 Thermal conductivity ( i s o t r o p i c ) 47 Layering 48 Thermal dispersion 49,50,51 Geothermal f l u x 52,53 Surface temperature (climate) 54 Basin s i z e 55 Basement structure 56,57 Water-table configuration (topography) 58 Mass Transport Hydraulic conductivity Maximum pulse concentration Layering Isotropic d i s p e r s i v i t y Longitudinal d i s p e r s i v i t y Geologic configuration Geochemical Equilibrium and Reaction Paths Aqueous s u l f u r species s t a b i l i t y f i e l d s 69,70 Galena and sphalerite s o l u b i l i t y as a function of: a) pH and reduced s u l f u r content 71 b) s a l i n i t y 72 c) temperature 73 Metal-sulfate brine model: a) reaction with hydrogen s u l f i d e 74 b) reaction with methane 75 c) reaction with p y r i t e 76 Metal-sulfide brine model: a) reaction with dolomite 77 b) e f f e c t of cooling 78 59,60 61 62 64 65,66 67,68 120 transport, and mass transport are independently and systematically examined. A l l of these simulations are performed i n a simple, two-layer basin model. A variety of more complex geologic configurations are also introduced, although the discussion i s kept b r i e f . Following the transport modeling, a review i s made of the factors influencing metal concentrations i n brines. Using some of the conceptual models proposed for ore p r e c i p i t a t i o n , several reaction-path simulations are then conducted and evaluated. To close the chapter, a preliminary simulation i s f i n a l l y presented of the Pine Point lead-zinc deposits, Northwest T e r r i t o r i e s . Factors Controlling Fluid-Flow Patterns and Vel o c i t i e s There has been much work i n hydrogeology on the effect of variations in controlling parameters on regional groundwater flow. The work by Toth (1962, 1963) and' Freeze and Witherspoon (1966, 1967, 1968) were already discussed e a r l i e r (see Figures 7 and 8). They found that the most important factors affecting flow patterns are: (1) depth to length r a t i o of the basin. (2) topography or water-table configuration (3) stratigraphy and the res u l t i n g variations i n the hydraulic conductivity From the point of view of defining the r o l e of hydrodynamics i n ore genesis, some of the most interesting results of these e a r l i e r studies are those that demonstrate how basin-wide aquifers serve to direct flow across less permeable beds and focus flow toward major discharge areas. In t h i s section, some of these con t r o l l i n g factors are examined i n greater d e t a i l . Although part of the analysis i s similar to that given by Freeze and Witherspoon (1967), i t i s included here for the purpose of complete-ness and review. Differences also exist because of the scale of the basin 121 being modeled, and the fact that s a l i n i t y and temperature effects are accounted for i n t h i s study. As a sta r t i n g point, consider a sedimentary basin of the shape and size depicted by the finite-element mesh shown i n Figure 17. The region of flow i s based on the same c r i t e r i a as those specified for the simulation example (Figure 16). This mesh i s used throughout the s e n s i t i v i t y analysis, unless otherwise specified. I t consists of 410 triangular elements and 252 nodes. Topographic r e l i e f (water-table configuration) varies l i n e a r l y with a t o t a l drop of 1000 m over the 300 km length of the basin. The v e r t i c a l scale of the plot (Figure 17) i s exaggerated by a factor of 10:1 i n order to show the simulation r e s u l t s . The true-to-scale diagram of the basin i s plotted d i r e c t l y above the mesh to i l l u s t r a t e the nature of the v e r t i c a l exaggeration. Hydraulic Conductivity The quantity of f l u i d that i s able to flow through a basin to form an ore deposit depends on many factors. The basic relationship governing the flow i s Darcy's law (Table 3). For fixed f l u i d properties and a constant hydraulic-head gradient, the magnitude of flow i s solely determined by the hydraulic conductivity of the porous medium. The range of values of hydraulic conductivity of geologic materials i s very wide. Table 7 shows the range established for a variety of l i t h o l o g i e s , as l i s t e d by Freeze and Cherry (1979). Porosity i s also important i n that i t i s used to calculate the average l i n e a r v e l o c i t y of the f l u i d . Table 8 l i s t s the porosity range of values for the common rock types, which i s based on data given by Davis (1969) and Freeze and Cherry (1979). The simplest hydrogeologic model of a sedimentary basin i s the case of a homogeneous and isotropic section. Figure 18 shows the results of a FINITE ELEMENT MESH Kilometers (Vertical Exaggeration = 10:l) Figure 17. Finite-element mesh and basin dimensions f o r s e n s i t i v i t y a n a l y s i s . Table 7. Hydraulic c o n d u c t i v i t y and i n t r i n s i c p e r m e a b i l i t y range of geologic m a t e r i a l s . SAND AND GRAVEL [ KARST LIMESTONE AND DOLOMITE I FRACTURED BASALT | I ' FRACTURED CRYSTALLINE ROCKS I 1 LIMESTONE AND DOLOMITE I I SANDSTONE I I SHALE I I CRYSTALLINE ROCKS I I EVAPORITES i 1 1 r 1 1 1 1 1 1 1 1 1 1 IO"7 IO"6 IO"5 10"* 10"' IO"2 10"' I 10 IOJ 10' 10' 10' 10* HYDRAULIC CONDUCTIVITY (m/yr) I 1 1 1 1 1 1 1 1 1 1 1 1 1 IO"6 10"' IO"4 I0"J 10"' 10"' I 10 10* IO5 10* 10s IO6 I0T INTRINSIC PERMEABILITY ( millidarcys) i I 1 1 1 1 1 1 1 1 1 1 1 1 1 10"* 10"" 10"'° 10"' IO"8 IO"7 IO' 6 10"' 10"* IO"5 I0"J 10"' I 10 HYRAULIC CONDUCTIVITY (cm/s) K3 OJ Table 8. Porosity values of geologic materials. I 1 CLAY | ~| SAND AND GRAVEL | FRACTURED BASALT | KARST DOLOMITE AND LIMESTONE | SANDSTONE ] DOLOMITE AND LIMESTONE SHALE | | FRACTURED CRYSTALLINE ROCK | | CRYSTALLINE ROCK 0 EVAPORITES O.IO 0.20 030 0.40 0.50 0.60 0.70 0.80 POROSITY (FRACTION) 4 - | V 3-0) 6 2 i o 0-HYDRAULIC HEADS I I i I i i 1 1 1 1 1 1 r ~] i i i 1—I 1 1 1 1 1— i 1 — i — r (a) a> 1 2H o 0-VELOCITY VECTORS 1 l i i I—I—i 1 1 — | 1 — r 50 100 " | — i 1— i 1 — | r 150 200 Kilometers ( b ) n 1—i 1 — r — r _ « — i 250 300 Figure 18. Homogeneous-isotropic basin showing r e l a t i o n s h i p of f l u i d v e l o c i t y and hydraulic conductivity. 126 Figure 18. (Continued) 127 simulation of t h i s case. The e f f e c t s of temperature and s a l i n i t y are excluded from the f l u i d model f o r the time being. Flow i s uniform and h o r i z o n t a l , except at the extreme ends of the basin.' The t h i n nature of the section ( i n true to s c a l e ) , l i n e a r slope i n the water t a b l e , and homogeneity of the basin f i l l produces v e r t i c a l hydraulic-head contours. Under these conditions, the f l u i d v e l o c i t y w i l l be a l i n e a r function of hydraulic conductivity. Figure 18c shows the r e s u l t s of changing the permeability of the basin medium, f o r the reference point marked A on the cross section (Figure 18a). Porosity i s set equal to a constant 0.15 f o r each model. Sedimentary rocks are r a r e l y i s o t r o p i c with respect to hydraulic conductivity. Anisotropy a r i s e s from l i t h o l o g i c a l features within beds, and from the layered heterogeneity that e x i s t s on a regional scale. The r a t i o of ho r i z o n t a l to v e r t i c a l hydraulic conductivity, K /K , commonly reaches Z Z values of 100.0 f o r regional s i t u a t i o n s (Freeze and Cherry, 1979). Even higher r a t i o s may occur f o r the scale of the basin being modeled here. Figure 19 i l l u s t r a t e s the e f f e c t of a representative anisotropy r a t i o of 100:1 on the f l u i d - f l o w regime. The main e f f e c t i s a more pronounced delinea t i o n of recharge and discharge areas, as shown by the f l e x i n g of the hydraulic-head contours at the ends of the section. Average l i n e a r v e l o c i t i e s vary along the base of the section, from about 1.0 m/yr near x = 50 km to 2.2.m/yr near x = 250 km. The r a t i o i n the h o r i z o n t a l and v e r t i c a l components of the v e l o c i t y vector w i l l depend on the anisotropy r a t i o , among other f a c t o r s . In Figure 19, the ho r i z o n t a l hydraulic conducti-v i t y i s K = 100 m/yr and the v e r t i c a l hydraulic conductivity i s K = 1 m/y: ZZ These p a r t i c u l a r values give a h o r i z o n t a l v e l o c i t y v =2.3 m/yr and an upward v e r t i c a l v e l o c i t y v^ = 5.<+ x 10 3 m/yr at reference mark A, using a porosity value of 0.15. HYDRAULIC HEADS Figure 19. E f f e c t of anisotropy on f l u i d flow. '00 129 Several numerical experiments were run to test the s e n s i t i v i t y of the modeling results to dipping strata. The effect of i n c l i n i n g the p r i n c i p a l directions of hydraulic conductivity appears to be negligible for bedding dips of 0.3° (.3 m/km). Flow patterns are effected, however, i f the beds dip at gradients exceeding 5°. It would be advisable to account for t h i s factor i n any studies that consider the formation of ore bodies i n s t r u c t u r a l l y complex stratigraphic sequences. For the purpose of t h i s study, only horizontal or gently-dipping strata are considered further. In a steady-state flow system, the porosity i s required only i n computing the average l i n e a r velocity terms from the values of s p e c i f i c d i s -charge (Table 3). Porosity does not affect the flow pattern or the volume flow rate. For example, the sp e c i f i c discharge at the reference point A i n Figure 19 i s q = 0.33 m 3^i 2 ,yr. The average line a r velocity w i l l be 33.0 m/yr, 3.3 m/yr, and 1.65 m/yr for assumed porosities of 0.01, 0.10, and 0.20 respectively. Homogeneous sections are not very r e a l i s t i c representations of the geologic character of most sedimentary basins. Although some foreland basins might approach the homogeneous case, i t i s safe to conclude that heterogeneous systems are more common (see Figures 2 and 4). Hydraulically, a layered sedimentary basin i s very conducive to the focusing of flow as we have already seen i n the e a r l i e r flow models (Figures 7, 8, and 16). Focusing i s not present i n the homogeneous basin model of Figure 18. Consider the simple, two-layer model shown i n Figure 20a. The bottom layer i s a carbonate 'aquifer' with a hydraulic conductivity K = 100 m/yr XX and a porosity <j> = 0.15. The upper layer represents a less permeable shale u n i t , i n which K = 10 m/yr and <j) = 0.10. Anisotropy i n both layers i s assigned a representative of 100:1. F l u i d flow i s now directed through the Kilometers (d) Figure 20. E f f e c t of layering on f l u i d flow. H CO o 131 Figure 20. (Continued) 132 shale beds down into the aquifer i n the recharge uplands, flows l a t e r a l l y across the basin, and i s eventually forced up along the discharge end of the basin. The r a t i o of hydraulic-conductivity values of the two layers controls the degree of v e r t i c a l f l u i d f l u x recharging the basal aquifer. A greater permeability contrast between aquitard and aquifer produces a sharper bending of the hydraulic-head contours, as shown by the contrast r a t i o K = 100 i n Figure 20'c. Groundwater flow i n the shale formation (.K ) i s mostly downward i n the l e f t - h a l f of the basin and upwards i n the r i g h t - h a l f of the section. Notice that the areas of highest flow are located i n the basal aquifer and near the basin margin. Figure 20e shows a graph of the horizontal to v e r t i c a l v e l o c i t y r a t i o as a function of the hydraulic contrast of the two-layer model. I t summarizes the results at the reference s i t e A (Figure 20a) for several simulations. The hydraulic conductivity of the shale aquitard i s kept at a constant 10 m/yr. The v e r t i c a l f l u i d f l u x i s nearly equivalent to the horizontal f l u x for r a t i o s of K^/K^ exceeding 100. We w i l l see i n l a t e r simulations the importance of the permeability contrast i n influencing mass transport patterns. The effects of pinchouts on flow systems are shown i n Figures 21 and 22. Both models assume horizontal hydraulic conductivities of 1000 m/yr and 10 m/yr for the aquifer and aquitard respectively. The horizontal permeability i s taken to be 100 times the v e r t i c a l permeability. In Figure 21, the average veloc i t y i n the aquifer i s 4 m/yr, while i n Figure 22 the vel o c i t y i s about 7 m/yr. An expanded picture of the shallow-aquifer lens (Figure 22 ) :. i l l u s t r a t e s a s i t u a t i o n where shallow-derived H^S-bearing ;waters would continuously recharge the p a r t i a l basal aquifer from above while metal-bearing brines are also recharging the aquifer at the upstream end of the lens. A si m i l a r type of flow regime also exists i n Figure 16 (Chapter 4). HYDRAULIC HEADS (a) VELOCITY VECTORS —•4.4m/yr Kilometers (C) Figure 21. E f f e c t of aquifer lens i n upstream end of basin. i—• CO CO HYDRAULIC HEADS 4n 3-VELOCITY VECTORS —»• 7.3 m/yr e o 5 I 2H • 5 a 1 wi •9 * » a I - * . » 1 I i i | 1—i 1 1—| 1 1 1 1 1 1 1 r 50 100 150 I ~I r 200 ~i 1—i 1—r 250 Kilometers I J 300 7.3 m/yr —• t a * t r 2 I- • I 200 250 L 0 300 Kilometers Figure 22. E f f e c t of aquifer lens i n downstream end of basin. CO -F 135 Other types of layered stratigraphy can be studied through s i m i l a r modeling. A number of e a r l i e r studies, such as Freeze and Witherspoon (1967) and Freeze (.1969), give many examples of the e f f e c t s of l a y e r i n g on f l u i d -flow patterns. Although many of these are of i n t e r e s t , space does not permit an examination of every case. The basin-wide aquifer model treated above i s simple and we l l - s u i t e d f o r a preliminary analysis of ore genesis. We w i l l only proceed, therefore, to examine the two-layer model i n greater d e t a i l Temperature and. S a l i n i t y Gradients The flow models presented so f a r have assumed that no temperature or s a l i n i t y gradients exist i n the basin. This assumption i s commonly made i n the modeling of shallow groundwater systems, but i t i s not r e a l i s t i c i n the analysis of flow i n large sedimentary basins. The purpose of t h i s section i s to look at the e f f e c t of v a r i a b l e density and v i s c o s i t y i n the development of r e g i o n a l flow systems i n sedimentary basins. Figures 9 and 10 (Chapter 3) show the r e l a t i o n s h i p s of water density and v i s c o s i t y as a function of temperature and s a l i n i t y . As noted e a r l i e r , these equations of state are numerically incorporated i n t o Equation (3-9), which i s the more complete form of Darcy's law. Because no previous studies have assessed the r o l e of temperature and s a l i n i t y gradients i n regional flow, a somewhat de t a i l e d evaluation w i l l be made. The f i r s t set of simulations to be presented assume that there i s only a temperature gradient across the two-layer basin. Figure 23 shows the temperature f i e l d and r e s u l t i n g flow diagrams f o r the case i n which f l u i d density i s allowed to vary with temperature alone. The temperature pattern i s computed from the numerical so l u t i o n of the heat-transport equation. The discussion of f a c t o r s c o n t r o l l i n g heat flow i s reserved f o r l a t e r , and f o r now we w i l l j ust present the so l u t i o n without explanation. The hydraulic TEMPERATURES g 3+ o VELOCITY VECTORS —^2.2 m/yr -» >=S * S •== y • • •» • . » » * 0 0 50 | r 100 "1 r 150 | i i i 1 1 — i 1 — i — i 1 200 250 300 Kilometers ( 0 Figure 23. Fluid-flow pattern where density i s a function of temperature alone. 137 conductivity [see Equation (.3-5)] of the basal carbonate aquifer i s K = 100 m/yr, while that of the shale layer i s 10 m/yr. P o r o s i t i e s are 0.15 and 0.10 r e s p e c t i v e l y . Both hydrostratigraphic units have an anisotropy r a t i o of 100:1. It should be pointed out that one can no longer use the r u l e whereby flow vectors are drawn perpendicular to hydraulic-head contours (e.g. Freeze and Cherry, 1979, Chapter 5), when modeling inhomo-geneous f l u i d flow. For the case of inhomogeneous f l u i d s , the hydraulic-head pl o t only represents a fresh-water equivalent pressure head, not a true hydraulic p o t e n t i a l . However, i t w i l l s t i l l be a u s e f u l v a r i a b l e i n the following diagrams f o r comparison purposes. The main feature introduced by allowing density to vary with temperature i s an upstream bending of the hydraulic-head contours i n the aquifer (compare Figures 20a and 23b). Q u a l i t a t i v e l y , the increase in temperature with depth lowers the f l u i d density, and thereby increases the upward component of flow. For the marked reference point A i n Figure 23b, the downward v e l o c i t y decreased from 1.0 x 10 3 m/yr i n the uncoupled problem (Figure 20a) to 3.2 x 10 4 m/yr f o r the coupled problem (Figure 23b and 23c). At reference point B i n Figure 23b, the average l i n e a r v e l o c i t y changes only s l i g h t l y from the uncoupled to coupled problem. Figure 24- gives the r e s u l t s of a model that allows the v i s c o s i t y of water to vary with temperature alone, while keeping density constant. The temperature e f f e c t on v i s c o s i t y i s s i g n i f i c a n t , as could be expected from Figure 10. Warmer temperatures decrease f l u i d v i s c o s i t y which i n turn creates l a r g e r flow r a t e s , r e l a t i v e to the isothermal case at 20°C. Because v i s c o s i t y e f f e c t s are introduced into the flow equations as a l i n e a r m u l t i p l i c a t i o n f a c t o r [see Equation (3-8)], i t i s the magnitude of flow that i s a f f e c t e d more so than the flow pattern. This feature i s well i l l u s t r a t e d by comparing TEMPERATURES VELOCITY VECTORS — 5 . 2 m/yr 300 Kilometers (c) Figure 24. Fluid-flow pattern where v i s c o s i t y i s a function of temperature a l Lone. 139 Figures 20 and 24-. In the coupled-flow problem, the warm temperatures ' i n the basal aquifer cause the system to behave as i f the aquifer i s more permeable than the uncoupled model (Figure 20). The e f f e c t of a lower v i s c o s i t y i s to permit flow v e l o c i t i e s of 4.7 m/yr at point A (Figure 24), which i s more than double the v e l o c i t y at the same point i n the uncoupled case (Figure 20a). Flow rates double, i n f a c t , along the ent i r e length of the basal aquifer. Although i t i s i n t e r e s t i n g to look at the i n d i v i d u a l e f f e c t s of temperature on density and v i s c o s i t y , i n r e a l i t y the presence of a geothermal gradient a f f e c t s both f l u i d properties simultaneously. A simulation showing t h i s phenomena i s given i n Figure 25. None of the media parameters have been changed from the two previous models. Flow i s now influenced by both buoyancy e f f e c t s and higher v e l o c i t i e s i n the warmer parts of the basin. At the reference s i t e A (Figure 25b), the h o r i z o n t a l component of v e l o c i t y i s 4.7 m/yr, and the v e r t i c a l component i s 1.9 x 10 2 m/yr, which i s greater than the uncoupled v e r t i c a l v e l o c i t y of 7.8 x 10 3 m/yr. Figure 26 summarizes the quantitative r e s u l t s found i n modeling the e f f e c t of temperature on f l u i d flow i n a large sedimentary basin. The p o s i t i o n of the v e l o c i t y p r o f i l e i s marked by the points Z-Z' i n Figure 25a. The increase i n the v e r t i c a l - v e l o c i t y component i s more subdued i n the basal aquifer because of the large h o r i z o n t a l component of flow. With respect to the e f f e c t of temperature on v i s c o s i t y , as i l l u s t r a t e d by the h o r i z o n t a l component of flow, the influence i s greater i n the aquifer than i n the shale because of the higher temperatures. Phenomena of these types are obviously important when considering the modeling of basinal-brine flow i n ore genesis. In t h i s p a r t i c u l a r example, the volume of flow i n the basal aquifer would be out by a f a c t o r of at l e a s t 2.0 i f the e f f e c t of temperature i s not included TEMPERATURES z' 300 Kilometers ( 0 Figure 25. E f f e c t of temperature-dependent f l u i d properties on regional f l u i d flow. Figure 26. Velocity p r o f i l e i n discharge end of basin showing the e f f e c t of temperature-dependent f l u i d flow. 142 i n the c a l c u l a t i o n . The next question to ask would be: what e f f e c t does s a l i n i t y have on the f l u i d - f l o w system? Figures 27, 28, 29, and 30 show the r e s u l t s of some numerical experiments that were conducted to help answer t h i s question. A l l of these models assume there i s no temperature gradient present. The two-layer stratigraphy and material properties are the same as i n the previous case. The models presented here assume that water above the 2000 m elevation i s f resh and therefore i s assigned a zero s a l i n i t y . Below the datum, s a l i n i t y increases l i n e a r l y with depth. S a l i n i t y i s expressed as an equivalent weight percent of sodium ch l o r i d e (% NaCl). At the deepest point i n the model basin (2 km below datum, Figure 27a), a gradient of 0.01 % NaCl/m w i l l produce a s a l i n i t y of 20% NaCl (about 250,000 mg/liter or 4.3 molal). This s a l i n i t y gradient i s a r b i t a r y , but r e a l data could e a s i l y be inserted i n actual s i t e -s p e c i f i c , f i e l d simulations. Temperature i s assumed t o be constant at 20°C throughout'.the basin. Figure 27 i s a simulation of the case where density alone i s a function of s a l i n i t y . The s a l i n i t y d i s t r i b u t i o n i s also p l o t t e d f o r reference. By comparing Figures 20a and 27, i t can be seen that the most obvious e f f e c t of s a l i n i t y i s the change i n the hydraulic-head pattern. High s a l i n i t i e s i n the deeper parts of the basin cause an increase i n f l u i d density which leads to greater buoyancy forces that act downward (Equation 3-8). In order to drive the f l u i d through the basin, higher gradients in hydraulic head are required than those of the fresh-water-basin model (Figure 20a). At s i t e A i n Figure 27b, the v e r t i c a l v e l o c i t y increase to 1.5 x 10 3 m/yr r e l a t i v e to the 5.5 x 10 4 m/yr of the fresh-water model. This e f f e c t i s more subdued i n shallower l e v e l s of the basin. The e f f e c t of s a l i n i t y on v i s c o s i t y alone i s shown i n Figure 28. As i n the case of temperature, v i s c o s i t y influences flow rates i n a l i n e a r SALINITY PROFILE 1 1 1 ' r | i 1 1 1 1 1 1 1 1 j 1 — i 1 1 j — r »0 100 150 200 250 Kilometers (c) Figure 27. Fluid-flow pattern where density i s a function of s a l i n i t y alone. 4n 3 2 SALINITY PROFILE Gradient =0.01 % NaCI/m 15% •I0%-5 % 0% i i 3 T — i — i 1 — j — i r (a) n r i 1 r n 1 r T r HYDRAULIC HEADS (COUPLED) \K2 = I0 \ \ \ \ \ \ V V 1 1 I \ \ \ \ \ \ \ — 1 IK, =ioo 1 1 l l *—1 1 — 1 1 L_! . A ' I I I | i i i i j i i i i | i i i i j — i — i — i — i — | — i — i — i — i — | (b) 4n 3 0 VELOCITY VECTORS —^1.8 m/yr * • •m J» 0 I ~ 1 1 1 1 r 0 50 |— 100 - i 1 . r 150 Kilometers T — i — | — i — r 200 T 1 r 250 300 (c) Figure 28. Fluid-flow pattern where v i s c o s i t y i s a function of s a l i n i t y alone. SALINITY PROFILE Gradient =0.01% NaCI/m Kilometers (c) cn Figure 29. E f f e c t of salinity-dependent f l u i d properties on re g i o n a l f l u i d - f l o w . VERTICAL COMPONENT HORIZONTAL COMPONENT %NaCI 0 5 10 15 VELOCITY (m/yr) Figure 30. V e l o c i t y p r o f i l e i n discharge end of basin showing the e f f e c t of salinity-dependent •p 147 fashion. At s i t e A in Figure 28b, velo c i t y decreases to.1.8 m/yr, r e l a t i v e to the 2.2 m/yr i n the fresh-water case of Figure 20b. The hydraulic-head pattern i s not s i g n i f i c a n t l y altered by s a l i n i t y effects on v i s c o s i t y . The f i n a l set of diagrams i n t h i s group are shown i n Figures 29 and 30. This simulation includes the effects of both density and v i s c o s i t y . For reference point A i n Figure 29b, the ov e r a l l response i s an increase i n the v e r t i c a l v e l o c i t y to 1.1 x 10 3 m/yr as compared to the 5.0 x 10 4 m/yr i n the -fresh^water problem. The effect at the discharge end of the basin i s plotted i n Figure 30 for the p r o f i l e l i n e Z-Z"* i n Figure 29a. S a l i n i t y lowers the volume of flow that passes through a basal aquifer at the basin margin, when compared to' a fresh-water (uncoupled) system. I t i s an important factor, therefore, i n the analysis of hydrodynamics i n ore formation. With the indi v i d u a l assessments of temperature and s a l i n i t y effects on flow completed, we can advance to a more r e a l i s t i c basin model. Figure 31 shows the results of such a model where both temperature and s a l i n i t y affect the f l u i d properties. A s a l i n i t y gradient of 0.01 % NaCl/m i s assumed, and the media properties are the same as i n the preceding cases. The average l i n e a r v e l o c i t y at the reference s i t e A i s 3.8 m/yr. Table 9 compares t h i s v e l o c i t y value with the previous simulations. . I t i s clear that temperature gradients affect the flow system i n a basin to a greater degree than do s a l i n i t y gradients. However, the s a l i n i t y effect i s s i g n i f i c a n t and i t w i l l be included i n subsequent models. The importance of s a l i n i t y i s demonstrated by the effect of different s a l i n i t y gradients i n Figure 32. Gradients of 0.005, 0.010, 0.015, and 0.02 % NaCl/m are modeled. The largest gradient produces a brine with 40% NaCl at the deepest point i n the cross section. At the reference point A (Figure 32), the s a l i n i t y i s 6%, 12%, 18%, and 24% for the respective TEMPERATURES T 1 i i | i i i r 300 Figure 31. E f f e c t of combined temperature and salinity-dependent f l u i d properties on regional f l u i d flow. 149 Table;9. Summary of the v e l o c i t y components at the reference s i t e f o r various temperature-and salinity-dependent flow models Simulation V e l o c i t y Components at S i t e 'A* Horizontal (m/yr) V e r t i c a l (m/yr) 1. No temperature or s a l i n i t y gradients 2.2 7.8 x 10~ 3 2. Temperature gradient only 4.7 1.9 x 10~ 2 3^ S a l i n i t y gradient only 6.3 x 10 3 4. Temperature and s a l i n i t y 3.8 1.5 x 10 " - See Figure 31 f o r l o c a t i o n HYDRAULIC HEADS Kilometers (d) Figure 32. Variation i n hydraulic-head patterns and f l u i d v e l o c i t y as a function of s a l i n i t y gradient. 0 1 o 151 5 0 0.0 0.005 0010 0015 0.020 SALINITY GRADIENT (%NaCI/m) • (e) Figure 32. (Continued) 152 s a l i n i t y gradients. Figure 32e shows how the v e l o c i t y components decrease as higher s a l i n i t i e s are assumed. In the simulations, to follow, we w i l l employ the 0.01 % NaCl/m gradient as being a representative value for the basin. Basin Geometry Fluid-flow simulations can also be conducted to review the influence of basin s i z e , structure, and topography on the groundwater regime. Toth (1963) and Freeze and Witherspoon (1967) provide a more thorough analysis of the r o l e of these factors than w i l l be presented i n t h i s evaluation, but as before, the attention here w i l l be directed to the problem of flow i n large sedimentary basins. A l l of the simulations w i l l model a basin c o n s i s t i n g of two l a y e r s . In the preceding simulations the s i z e of the basin was kept a constant 300 km i n length and 3 km thick at the thickest point of the wedge. The choice of these dimensions i s based on r e a l sedimentary basins, such as those of Figures 2 and 4. It i s obvious that basin s i z e may vary: some flow systems may operate over distances of a few kilometers, while others may involve distances of several hundred kilometers.. Toth (1963) and Freeze and Witherspoon (1967) demonstrated with t h e i r models that the t o t a l continuous length of a r e g i o n a l flow system depends on topographic r e l i e f , basin thickness, and subsurface hydraulic c o n d u c t i v i t i e s . I f the basin i s reasonably t h i c k , r e l i e f i s gentle, and aquifers exist at depth, then continuous flow systems can develop on a basin-wide scale. Numerous f i e l d studies have documented the existence of very long, gravity-type flow systems i n sedimentary basins (e.g. van Everdingen, 1968; Hitchon, 1969a, 1969b; Bond, 1972; Toth, 1978). Figure 33 i l l u s t r a t e s the e f f e c t of basin s i z e on subsurface flow, over a possible range of dimensions. The length of the basin i s TL' kilometers, and the v e r t i c a l scale i s exaggerated by a f a c t o r of 10 i n order to i l l u s t r a t e HYDRAULIC HEADS (UNCOUPLED) • K x / K z = IOO K,= IO K,--(IOO 1 XLXLTLXCiss i 1 1 ~1 1 1 VELOCITY VECTORS — - V = 2 .2m/yr 0 OIL 0.2L 0.3L 04L 0.5L 0.6L 0.7L 0.8 L 0.9L IOL Dimensionless length (Vertical Exaggeration -10;l) Case I • L = 300 km, D = 3 km Case 2 : L = 150 km, D = 1.5 km Case 3 : L =30 km, D =0.3 km / Figure 33. Dimensionless cross section showing the r e l a t i o n s h i p between r e g i o n a l f l u i d flow and basin s i z e . , 0 1 CO 154 the length-to-depth r a t i o of 100:1. The slope of the water table i s assumed to follow the surface topography with a l i n e a r gradient of .0.003. The effect of temperature and s a l i n i t y are excluded from Figure 33, because i t i s obvious that a basin 300 m thick w i l l not have the same range of temperature as a basin 3 km thick. The hydraulic-head contours are dimensionless because the length-to-depth r a t i o , water-table slope, and hydraulic conductivities are the same for each basin si z e . The contour i n t e r v a l represents 5% of the t o t a l potential drop across the basin, which i s equivalent to 5 m, 25m and 50 m for basin lengths of L = 30 km, 150 km, and 300 km respectively. Whatever the basin size i n the uncoupled case, the flow rates remain the same through-out the basin. These results r e f l e c t the mathematical concept of dimensional si m i l i t u d e . This concept recognizes that for si m i l a r r a t i o s of co n t r o l l i n g parameters, such as topography, geology, and length-to-depth r a t i o , the quantitative results are s i m i l a r . Basin similitude allows one to extrapolate modeling results to different system scales under these c r i t e r i a . Other examples of dimensionless flow nets are shown i n Figures 7 and; 8. Although i t i s mathematically very useful to report modeling results i n terms of dimensionless parameters, the data interpretation can become confusing for anyone not experienced i n similitude analysis. The subject also becomes very complicated when considering the coupling of heat and mass transport. Dimensionless analysis i s not used i n t h i s study for these reasons. Further discussions of the general application of similitude theory can be found i n Kline (1965), and i n Bear (1972) for reference to porous-media problems. The length-to-depth r a t i o of a basin has s i g n i f i c a n t influence on f l u i d - f l o w patterns. Figure 34 shows the results of three simulations where the r a t i o L/D i s varied as: (a) 10:1, (b) 50:1, and (c) 100:1. Depth, i s f i x e d 155 Figure 34. Hydraulic-head patterns as a function of the length-to-depth r a t i o of a 3 km - thick basin. 156 at a constant 3 km. The configuration of the water table i s l i n e a r , with gradients of .0.030, 0.007, and 0.003.for the three cases. Temperature and s a l i n i t y are excluded from the models. The media properties are as before with <K = 0.15, <j>n = 0.10, K = 100 m/yr, K n = 10 m/yr, and K /K 100. 1 ' 2 ' 1 2 J xx zz Halving.the basin length from 300 km to 150 km (Figure 34b) and doubling the topographic slope increases the hydraulic gradients i n the low-permeability unit. F l u i d . v e l o c i t i e s increase throughout the basin, especially in the v e r t i c a l d i r e c t i o n . I f the basin length i s shortened by a factor of ten (Figure 34a), a considerable amount of energy i s required to force f l u i d flow to the same depth, but over a shorter length. As a result the hydraulic-head equipotentials exhibit large gradients i n the shale unit and very low gradients throughout the basal aquifer. Basins with small length-to-depth r a t i o s will-be e f f i c i e n t at producing large v e r t i c a l fluxes over broad areas, but unless moderately high topographic r e l i e f exists across the basin, flow rates w i l l be s i g n i f i c a n t l y smaller than those basins with larger length-to-depth r a t i o s . Changing the thickness of a basin of constant length produces similar r e s u l t s , as shown by the hydraulic-head, plots i n Figure 35. The thicker the basin, the steeper the hydraulic-head gradient becomes i n the shale aquitard due to the smaller length-to-depth r a t i o . Notice that as the maximum thick-ness (at x = 0 km) i s changed from 1.5. km i n Figure 35b to 6 km i n Figure 35d, the model assumes that the water-table slope changes from 0.0017 to 0.0070. This i s purposely done so as to avoid constructing new finite-element meshes for each new model i n Figure 35. Cases (a) and (e) are less common examples, but are included for comparison. Flow becomes horizontal i n basins of very large length-to-depth r a t i o s (Figure 35a). I t i s unlikely that such a flow system as depicted here would exist i n mature basins because the influence of 157 Figure 35. Hydraulic-head,patterns and f l u i d v e l o c i t y as a function of the length-to-depth r a t i o of a 300 km long basin. 158 DEPTH (KILOMETERS) I i 1 1 O 0.0033 0.0066 0.010 TOPOGRAPHIC GRADIENT (f)-Figure 35. (Continued) 0 159 topography.would ea s i l y p a r t i t i o n the continuous s t r i p into numerous, l o c a l flow systems. This type of basin would not be conducive to long-distance transport of metal-bearing brines i n ore formation. Figure 32f summarizes the relationship between velo c i t y at reference s i t e A (Figure 35a), basin thickness, and topographic gradient. Host of the veloc i t y change i s caused by the change i n the water-table slope. Q u a l i t a t i v e l y , i t may prove useful i n future studies to consider a suite of steady-state models, such as Figure 35, i n assessing the change i n flow patterns as a basin evolves to the mature stage. Few studies have addressed the problem of how basement structure might affect flow patterns and v e l o c i t i e s i n a basin. Figures 36 and 37 show the effects of a flat-bottom basement and basement arch on the hydrodynamic picture. Comparison of these re s u l t s can be made with the sloping-bottom basement of Figures 20 and 31. The length-to-depth r a t i o , water-table slope, and.material properties are the same for each case. Figure 36a i l l u s t r a t e s the finite-element mesh used to compute the hydraulic-head patterns i n Figures 36b and 36c. The f l a t basement produces a more symmetric flow pattern than the sloping-basement case. Groundwater recharge occupies the l e f t h a l f of the basin and discharge dominates the right h a l f of the basin. The area of highest flow s h i f t s from the shallow shelf of the basin i n the sloping-basement basin (Figure 31) to the central part of the basin i n the flat-bottom case.(Figure 36d). The effect.of a sloping basement i s to cause the higher flow rates along the thinning shelf, than would be encountered at the same position i n a flat-bottom basin. For the reference s i t e A (Figures 31 and 36), the veloc i t y changes from 3.2 m/yr i n the f l a t -bottom basin to 4.2 m/yr i n the sloping-basement model. An accompanying f i v e -f o l d increase i s f e l t i n the v e r t i c a l v e l o c i t y component. Figure 36. Regional f l u i d flow i n a flat-bottom basin, CTI O l e i Figure 37. Regional flow patterns i n a basin containing a basement arch with a r e l i e f of 500 m. 162 The flow system becomes s l i g h t l y more complicated when the impermeable bottom surface i s perturbed by a large basement high. Figure 37 considers the two-layer case where a basement arch i s present i n the r i g h t - c e n t r a l part of the basin. It i s over 50 km i n length, and.total r e l i e f i s 500 m. Figure 37 shows how the basement arch forces f l u i d flow over the high, thereby creating a zone of large flow rate immediately above, the structure. Velocity i s greatest on the upstream side of the ridge where v =3.3 m/yr and v = X z 4.8 x 10 2 m/yr. The presence of the ridge has created an appreciable v e r t i c a l flow of f l u i d i n t h i s region. Increasing the r e l i e f of the basement arch has an even greater effect on flow as shown i n Figure 38. In t h i s model, the t o t a l r e l i e f i s about 1000 m. F l u i d v e l o c i t y over the ridge reaches a maximum of 3.2 m/yr, which i s the highest i n the basin. Temperature and s a l i n i t y effects are accounted for i n Figures 38c: and 38d. A rather steep s a l i n i t y gradient of 0.015 % NaCl/m i s assumed i n t h i s model. Vel o c i t i e s for the coupled simulation reach values of v = 5.8 m/yr and v = 0.1 m/yr. The accompanying temperature patterns JC z for both Figures 37 and 38 are presented l a t e r . I t i s apparent from these simple models that the presence of a base-ment arch impedes the flow of brines moving through basal aquifers to the edge of a basin. On the other hand, phenomena of t h i s type could be important in explaining the genetic association of stratabound ore deposits and base-ment highs. The conceptual effect of basement structures has been recognized by geologists for a long time, but the simulations given here quantitatively confirm the effect. Figure 39 summarizes the influence of basement structure on f l u i d flow, for the four models given here. The vel o c i t y d i s t r i b u t i o n i n the basal aquifer of each case i s plotted as a function of distance along the basin. FINITE ELEMENT MESH Figure 38. Regional flow patterns i n a basin containing a basement arch with a r e l i e f of 1000 m. H cn CO 3.5 3.0 H 2.5H J 2.0H o o Ul I i.oH 0.5 H 0.0-Sloping Bottom -i 1 r ~| 1 1 1 1 1 1 1 1 r 50 100 150 KILOMETERS i — 1 — ' — ' — 1 — r 200 250 300 Figure 39. Ef f e c t of basement structure on the f l u i d v e o l c i t y i n a basal aquifer. CD -P 165 For comparison purposes, a l l four of the simulations assume there are no temperature or s a l i n i t y gradients i n the basin. Most of the features exhibited i n Figure 39 were discussed above and are self-explanatory. The mechanism responsible for driving the gravity-based flow system i s the water-table gradient, which i n t h i s study i s assumed to be a subdued r e p l i c a of the topography. As we have seen, the slope of the water-table plays a major role i n determining the flow rates i n a basin. In t h i s f i n a l section on f l u i d - f l o w analysis, we w i l l also see that the configuration of the water-table exerts a very strong control on the geometry of flow systems. The influence of t h i s factor has been documented for fresh-water, isothermal basins by Toth (1963) and Freeze and Witherspoon (1967), as explained i n Chapter 2 (see Figures 7 and 8). The simulations that follow review the effect of topography, with respect to flow i n large sedimentary basins with temperature and s a l i n i t y effects included. Once again a simple, two-layer section i s studied, and the material properties are kept the same as before. Figure 4-0 shows the effect of a change i n water-table slope near the basin margin (x - 280 km). The water table now consists of two lin e a r segments, the long part with a regional slope of 3 m/km, and the short part with a slope of 8 m/km. What happens i n smaller basins i s that a zone of recharge i s created d i r e c t l y below the point of slope change (see Figure 7b, Chapter 2). Due to the large length-to-depth r a t i o of the basin i n Figure 4-0, t h i s re-charge feature i s not observed. The only influence of the break i n slope i s to s l i g h t l y reduce the upward component of flow i n the discharge end of the basin. The r e s u l t s i s that discharge i s better focused near the basin margin, instead of being spread out over a wide area. In Figure 40c, a s a l i n i t y gradient of .0.015 % NaCl/m i s assumed to be present. The second topographic model, Figure 41, shows the effect of a r i s e FINITE ELEMENT MESH 4-, 3 2 K2= 10 7 T i 1—i 1 1—i 1 — i — | — i 1—r M HYDRAULIC HEADS I) UNCOUPLED z K, = 100 ' 1 1 I 1 1 -| 1 1 1 1 1 1 1 1 r 1 1 1 1 1 1—i 1 1 2" I 0 4 3 2-—-- , y / / I 1—— 2) COUPLED 1—1 1 i i r— 1 1 1 1 1 1 r (c) VELOCITY VECTORS -—+• 3.7 m/yr -| 1 r 50 100 150 Kilometers (d) — i > i i — i — | — i — i — i — i — | 200 250 300 Figure 40. Regional f l u i d flow with a break i n the water-table slope near the basin margin. 167 Figure 41. Regional f l u i d flow with a water-table ridge at the basin margin. 168 i n the water-table elevation at the end of a basin. This ridge has a t o t a l r e l i e f of about 260 m, as compared to.the 800 m r e l i e f of the regional slope. Its influence i s strong enough to create a l o c a l flow system that pushes against the long, regional system, and forms a groundwater divide at the base of the ridge. This type of flow system would be capable of bringingttogether, for example, metal-bearing.brines from the regional flow system and H^S-bearing f l u i d s from the l o c a l system. Mixing would occur over a r e l a t i v e l y l o c a l area (Figure 41e), provided hydrodynamic dispersion occurs. I t w i l l be shown l a t e r that the upward discharge, i n the.area.also causes the temperature to be elevated over the normal geothermal gradient. I f the water-table ridge i s shifted toward the basin i n t e r i o r , a different type of flow system develops, as shown by Figure 42. This ridge causes the development of a l o c a l flow system i n the shale beds, which serves to recharge the basal aquifer and creates two small discharge zones (Figure 42e), but the direction of flow i n the basal aquifer i s unchanged. Notice that maximum flow v e l o c i t i e s have changed i n the basal aquifer from about 3.5 m/yr i n the smooth water-table configuration (Figure 40) to about 5.0 m/yr i n Figure 42. The presence of a major valley (water-table depression) i n the basin i s much more effective at interrupting the flow of brines i n a basal aquifer.. Figure 43 depicts a depression of the water-table with an approxi-mate depth of 300 m. F l u i d flow i n the basal aquifer i s l a t e r a l across most of the basin, u n t i l the influence of the valley i s reached, and then flow i s forced upward to discharge into the depression. Local groundwater flow near the edges of the depression also discharges into the v a l l e y , which may d i l u t e the brine flow from greater depths. F l u i d v e l o c i t i e s i n the basal aquifer ::• reach a maximum of .6.7 m/yr below the water-table depression. On the down-169 4 - | e 3 QJ E o 5 I FINITE ELEMENT MESH g g g g 2 2 2 2 2 2 Z Z > ^ g g E g g g g g l z g E g K p a a i i i i i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — | — i — i — i — i — | , <<») HYDRAULIC HEADS 1) UNCOUPLED ;--7>VKi7 I / K, =100  1 i 1—r—r—-i—i—^—i—i—i— i — i — i — i — > ^ 1 i i i i — i — i — i — i — i — i — i i — i — i — i — i — i — i — | (b). 2) COUPLED E o i2 4 n 2 3 a» o E * o s 1 (c) VELOCITY VECTORS —»-5.0 m/yr 0 1 — r — r 5.0 m/yr 200 km . (e) 250 km Figure 42. Regional f l u i d flow with a water-table ridge i n the basin i n t e r i o r . 1 7 0 FINITE ELEMENT MESH 300 6.7 m/yr -i r 200 km (e) 250 km Figure 43. Regional f l u i d flow with a water-table depression i n the basin i n t e r i o r . 171 stream end of the v a l l e y , v e l o c i t y i n the aquifer becomes quite small. The cross-marked velocity vector i n Figure 43e (at x =2-2Bkm), denotes an area of flow where the v e l o c i t y i s less than one-hundreth of the maximum recorded velo c i t y (6.7 m/yr). The f i n a l topographic model i s presented i n Figure 44. I t i s a model of a basin with an irregular water-table configuration: a gentle, r o l l i n g p l a i n i n the uplands area and a rugged, h i l l y set of perturbations i n the discharge end of the basin. For the most part, f l u i d flow i s r e l a t i v e -l y undisturbed i n the basal aquifer and flow rates are roughly the same as the model with a l i n e a r water-table configuration. Superimposed on the regional system i s a set of complex, l o c a l flow systems, but t h e i r depth of influence i s l i m i t e d to the low-permeability shale unit. The influence of the water-table i r r e g u l a r i t i e s would be greater for features of greater r e l i e f , or i f the hydraulic conductivity contrast between the basal carbon-ate unit and shale unit was less than the 10:1 r a t i o assumed here. Factor Controlling Subsurface Temperatures Most'quantitative modeling of.heat transport and f l u i d flow has been carried out f o r the purpose of assessing geothermal reservoirs, heat storage i n aquifers, radioactive-waste disposal, or the intrusion of plutons. The effect of regional groundwater flow on steady-state temperature patterns has also received study from a number of authors. Stallman (1963) and Bredehoeft and Papadopoulos (1965) show how groundwater v e l o c i t i e s can be estimated from observed temperature data. The influence of groundwater flow systems on heat flow i n an idealized cross section was investigated mathematically by Domenico and Palciauskas (1973). Parsons (1970) looked at a similar problem using f i n i t e - d i f f e r e n c e models. Both of these studies quantitatively demonstrate that convective heat transport causes geothermal gradients to 172 FINITE ELEMENT MESH i — i — | 1—i 1—i—|—i 1 r HYDRAULIC HEADS 1) UNCOUPLED - — K 2 = I O / \ / v r ^ r ^ 7 " < \ ^ — i 1 K / ' I O O —I r — i — i — i — i — i — i — i — i — i — i — i i i 1 i • ' i 2) C O U P L E D 300 I 200 km -r— 1 1 r -i 1 r (e) 250 km 300 km Figure 4-4-. Regional fluid..flow with an i r r e g u l a r water-table s configuration. 173 decrease i n groundwater areas and increase i n discharge areas, r e l a t i v e to the case of pure conduction. Betcher (.1977). performs a s e n s i t i v i t y analysis of many of the parameters affecting heat flow i n a hypothetical 10 km-long groundwater basin using finite-element models. He presents some of the general relationships between hydraulic and thermal parameters of t h i s basin. The mathematical analysis of free convection i n porous media has also received considerable study. Some of the better-known works are those of Nield (1968), Rubin (1973, 1981), Weber (.1975), Tyvand (1977, 1981) and Straus and Schubert (1977). Most of t h i s research i s theoretical i n nature, and concerns s t a b i l i t y c r i t e r i a for heat and mass transport with f l u i d flow i n single aquifers. The main purpose of t h i s new assessment i s to review some of the controlling factors of heat transport, but on the scale of a large sedimentary basin. Simulations are presented to i l l u s t r a t e the effect.of the fl u i d - f l o w regime, the boundary conditions, the thermal properties of the porous media, and the basin geometry on the'temperature d i s t r i b u t i o n i n large basins. Some of the models repeat the general concepts analyzed i n previous studies, but t h e i r presentation here i s j u s t i f i e d on the basis of t h e i r worth to the sp e c i f i c problem of ore formation and for reasons of completeness. The results are based on the steady-state solution of the coupled f l u i d and heat-flow equations. To begin the analysis, l e t us consider the hydraulic effects on the temperature pattern i n the two-layer basins that we have been studying thus fa r . For example, Figure 31a depicts the t y p i c a l temperature pattern for the case of the li n e a r water-table slope. I t assumes that a constant geothermal heat f l u x of 60 mW/m2 occurs along the entire length of the basement boundary. 174 This heat flow value i s representative of continental platform areas. The two l a t e r a l boundaries are assumed to be insulated barriers across which there i s no heat flow. The water-table surface i s an isothermal boundary, where temperature i s assigned a value of 20°C to represent the mean a i r temperature. The lower layer is. a dolomite aquifer with a horizontal hydraulic conductivity of 100 m/yr, a porosity of 0.15, an anisotropy r a t i o of 100 m/yr, and a thermal conductivity of 3.0 W/m°C. A value of 0.63 W/m°C i s assumed for the thermal conductivity of the f l u i d throughout the basin. The upper layer i s a shale aquitard with a horizontal hydraulic conductivity of 10 m/yr, a porosity of 0.10, an anisotropy r a t i o . o f 100:1, and a thermal conductivity of 2.0 W/m C. Specific heat capacities of 1005 J/kg C for the rocks and 4187 J/kg°C for the f l u i d are assumed to remain constant. Notice i n Figure 31a that the isotherms are not perfectly horizontal nor evenly spaced. The depression of the contours i n the recharge end of the basin i s caused by f l u i d flow convecting heat down into the basal aquifer. Upward flow i n the downstream end of the basin causes, a warping of the isotherms near x = 200 km. Thermal conduction exerts major influence on the temperature f i e l d , i n spite of the groundwater flow system. This i s evident i n the near-horizontal character of the isotherms and the spacing control on the isotherms caused by variations i n thermal conductivity. The effect of variable density and v i s c o s i t y are accounted for i n Figure 31a. Some of the e a r l i e r simulations given here also considered heat flow i n t h e i r f l u i d - f l o w solutions. In Figure 23a, the vi s c o s i t y i s held constant, but density i s allowed to vary with temperature. The temperature o f i e l d attains a maximum of 83 C near the 100 km-distance marker. I f the density i s fixed at a constant value, while the vi s c o s i t y varies with temperature, the change i n pattern i s pronounced, as shown by Figure 24a. 175 The effect of v i s c o s i t y alone i s to cause higher rates of groundwater flow i n the basal aquifer, and thereby s h i f t i n g the area of highest temperature up dip i n the basin by about 50 km. Temperature m t h i s model a:s: below 75 C. The point to be emphasized here i s that heat-flow patterns i n sedimentary basins are the r e s u l t of two important transport processes: conduction and convection. I t i s also important that temperature and s a l i n i t y effects on water density and v i s c o s i t y be included i n the numerical modeling of heat transport, so as to properly account for t h e i r influence on convection. With these introductory observations made,,we can now turn to an examination of the possible ranges of model parameters, and t h e i r control on temperatures i n ore-forming environments near basin margins. Hydraulic Conductivity I n t u i t i v e l y i t makes sense that the process of heat convection w i l l become stronger i n a basin as fluid-flow.rates increase. Conversely, the effect of thermal conduction dominates the thermal regime i f flow rates are small. As we have seen, higher flow rates can be obtained either by increasing the hydraulic conductivity or by increasing the topographic gradient. Coupling f l u i d flow with a temperature-dependent v i s c o s i t y (e.g. Figure 24) also pro-vides a means for increasing f l u i d flow i n a basin. Figure .45 shows the quantitative results of some numerical experiments that were run to demonstrate the effect of hydraulic conductivity on heat flow. To i s o l a t e the e f f e c t s , the thermal conductivities of the two hydrostratigraphic units are set equal to 3.0 W/m°C, and the thermal d i s p e r s i v i t i e s are set equal J to zero. The hydraulic conductivity i s increased by the same, factor i n each layer so as to maintain the permeability contrast of 10:1 i n each simulation. An anisotropy r a t i o of 100:1 for hydraulic conductivity i s also held constant. In Figure 45a, the hydraulic conductivity ( K ) i s set very low at TEMPERATURES K, =IO~"m/yr 300 Kilometers (d) Figure 4 - 5 . Steady-state temperature patterns showing the e f f e c t of convection f o r various hydraulic c o n d u c t i v i t i e s . 177 10 1 1 m/yr, which ensures that heat transport w i l l be dominated by conduction. Increasing the hydraulic conductivity of the basal unit (K^) to 10 m/yr and that of the upper unit (K,_,) to 1 m/yr (..Figure 45b) causes some heat con-vection, but the temperature pattern change i s negligible from case (a). However, when = 100 m/yr (Figure 45c) the change i n the temperature pattern i s s i g n i f i c a n t . The temperature contours are now depressed i n the recharge end of the basin, but shifted upward i n the discharge end of the basin. Notice that the area of highest temperature i s no longer i n the deepest part of the section as the conduction models, but now i s shifted updip i n the basin. The l a s t diagram,. Figure 45d, shows what happens when the hydraulic conductivity of the basal aquifer i s assigned a reasonably high value of 500 m/yr. Convection causes a major depression of subsurface temperature i n the recharge end of the basin, while elevating temperatures along the shallow platform. Phenomena of t h i s type may explain the association of high temperatures at shallow depths with the occurrence of carbonate-hosted lead-zinc deposits near a basin margin. Figure 46 i s a plot of average temperature i n the basal aquifer as a function of distance along the basin and hydraulic conductivity. These temperature p r o f i l e s are from the models shown i n Figure 45. Isothermal conditions rar e l y occur i n regional aquifers, although t h i s situation may be approached when flow rates are high enough. Higher values of flow reduce the temperature i n recharge areas and raise the temperature i n discharge areas, r e l a t i v e to the case of conduction alone. Increasing the hydraulic conduct-i v i t y or flow rates has the effect of s h i f t i n g the 'hinge point' downstream from point A to point B i n Figure 46. I f flow rates w e r e u n r e a l i s t i c a l l y high, heat transport would be completely dominated by convection, and basin temperatures may become, nearly isothermal. Figure 46. E f f e c t of hydraulic conductivity on the temperature along a basal . aquifer. 179 Thermal Conductivity In the absence of convection, temperature gradients are controlled by the boundary conditions on the thermal regime, and by the thermal conduct-i v i t y of the porous media. Table 10 gives the range of thermal conductivities for several types of geologic materials. These have been compiled from the l i s t i n g s of Clark C1966), Kappelmeyer and Haenel (.1974), and Sass et a l . (1981). Recall from Equation (3-16) that the effective thermal conductivity of the porous medium i s a function of the f l u i d thermal conductivity, the rock thermal conductivity, and the porosity. Temperature also affects the thermal conductivity of rocks and water, but the influence i s small for temperatures below 150°C. Porosity i s set equal to a constant i n the simula-tions given below, and only the thermal conductivity of the rock i s varied. Figure 47 shows the results of s i x simulations i n which the thermal conductivity of the stratigraphic section i s varied between 1.0 and 6.0 W/m°C. The basin i s homogeneous with respect to the thermal properties, and a constant geothermal heat fl u x of 60 mW/m2 i s prescribed on the basement boundary. The hydraulic properties are the same as those of Figure 45c, namely that K = 100 m/yr, K = 10 m/yr, K /K = 100, <}> = 0.15, and <j) = 0.10. _L A XX Z Z _L 2. Temperatures i n the downstream end of the basin appear to meet ore-genesis requirements of T > 50°C for basin thermal conductivities less than 3.0 W/m°C. It i s apparent from these simulations and Table 10 that the presence of a thick sequence of shale of limestone i s conducive to ore formation at r e l a t i v e -l y shallow depths. The effect of a heterogeneous thermal conductivity i s displayed i n Figure 48. In part (a), the two-layer basin has a top unit with a thermal conductivity K =.2.0 W/m°C. Temperatures i n the basal aquifer reach a maximum of 75°C, which i s a 10°C r i s e over the homogeneous model of Figure 47(c). Table 10. Thermal conductivity range of geologic materials. I Clay Coal ]] Crystalline rock " "1 Quartzite Basalt H Dolomite Limestone Sandstone Shale 3 Evaporites Jo 7o To *0 M 6.'0 7.0 8X> 9.0 (W/mK) —, x—. r — I 1 1 1 1 ' ' I 5.0 10-0 (I0"3cal/cm s°C) — — i 1 1 1 1 1 — — 15.0 2Q0 , 1 - " 1 r -To 2.0 3.0 4.0 5.0 (Btu./ft hr °F) THERMAL CONDUCTIVITY 1 8 1 Figure 4 7 . Temperature patterns i n a thermally homogeneous basin f o r various values of thermal co n d u c t i v i t i e s , H CO 183 One would expect even higher temperatures for a basin with a thinner aquifer and thicker shale aquitard. Reversing the thermal conductivity of the layers, but not the hydraulic properties, produees.. the temperature pattern i n Figure 48b. Now the isotherms are more closely spaced i n the basal layer than i n the overlying layer. This sit u a t i o n would probably be less common i n nature because aquifer l i t h o l o g i e s are commonly sandstone or dolomite, both of which are more conductive than low-permeability shale or limestone (Table 10). S a l i n i t y The effect of a s a l i n i t y p r o f i l e on f l u i d - f l o w v e l o c i t i e s was observed e a r l i e r to be r e l a t i v e l y small (Figure 30), unless very high s a l i n i t y gradients were present (Figure 32e). The change i n temperature patterns can be expected to be s i m i l a r l y small. To v e r i f y t h i s expectation, three simula-tions were run with s a l i n i t y gradients of 0.005, 0.010, and 0.015 % NaCl/m i n the model basin of Figure 45d. Temperatures were nearly i d e n t i c a l i n a l l three simulations, the largest difference being about 1°C i n the deepest part of the basin. One aspect about s a l i n i t y that may be more important i s i t s effect on the thermal properties of the f l u i d . Bear.(1972, p. 649) reports a thermal conductivity for a 25% NaCl solution that i s ten times less than that of fresh water. The presence of very saline brines, therefore, may cause a 20% drop i n the effective thermal conductivity of the porous media, r e l a t i v e to the case of less saline water. This type of coupling between the heat equation and s a l i n i t y p r o f i l e i s not included i n the present analysis, but the effect can be estimated q u a l i t a t i v e l y by comparing the change i n temperatures between Figure 47e and 47d. 184 Thermal D i s p e r s i v i t y The c o e f f i c i e n t s of thermal d i s p e r s i v i t y , e and e , are the l e a s t -Li 1 known parameters i n the heat-transport equation. Thermal d i s p e r s i o n r e f e r s t o the a d d i t i o n a l heat t r a n s p o r t i n a porous medium that i s caused by v a r i a t i o n s i n f l u i d v e l o c i t y along the f l o w path. I t can be viewed upon as a process of mixing t h a t . d i s p e r s e s heat as the f l u i d packet flows along i t s tortuous path. The concept i s , i n f a c t , borrowed from mass t r a n s p o r t theory. D i s p e r s i v i t y data i s very scarce, but l o n g i t u d i n a l d i s p e r s i v i t y e probably ranges between 1 and 500 m and t r a n s v e r s e d i s p e r s i v i t y between 0.1 and 50 m, f o r the r e g i o n a l s c a l e of a sedimentary b a s i n . These numbers are estimated from the compilation given by Anderson (.1979). Figure 4-9 shows how. the extreme magnitudes of the l o n g i t u d i n a l d i s p e r s i v i t y parameter a f f e c t the temperature p a t t e r n i n a two-layer b a s i n , where •= 100 m/yr and K = 10 m/yr. Transverse d i s p e r s i v i t y i s assumed a constant value of 1 m i n both l a y e r s . The l o n g i t u d i n a l d i s p e r s i v i t y i s f i x e d at 1 m i n the shale l a y e r , but i s changed from 1 m t o 1000 m i n the b a s a l a q u i f e r . A heat f l u x of 60 mW/m2 i s assumed f o r the bottom boundary, and thermal c o n d u c t i v i t i e s of 3.0 W/m°C and 2.0 W/m°C are assigned f o r the lower and upper l a y e r s . S a l i n i t y increases with depth at a gradient of 0.015 I NaCl/m. I t i s c l e a r from t h i s s i m u l a t i o n t h a t subsurface temperatures are not a f f e c t e d by l o n g i t u d i n a l thermal d i s p e r s i o n . This r e s u l t i s not s u r p r i s i n g i n t h a t the l a r g e s t component of heat t r a n s p o r t i s i n the v e r t i c a l plane, which i s orthogonal t o the main f l u i d - f l o w d i r e c t i o n i n the b a s i n . S e v e r a l s i m u l a t i o n s , however, show that l o n g i t u d i n a l d i s p e r s i v i t y does help s t a b i l i z e the coupled f l u i d flow-heat t r a n s p o r t s o l u t i o n . Under c o n d i t i o n s of l a r g e flow r a t e s , problems a r i s e i n the numerical s o l u t i o n of the steady-state system and i n s t a b i l i t i e s ( i n the numerical s o l u t i o n ) cause TEMPERATURES Figure 49. Temperature patterns i n a two-layer basin f o r l o n g i t u d i n a l thermal d i s p e r s i v i t y values of 1 m and 1000 m. CO 186 convergence problems with the simple i t e r a t i v e technique used i n t h i s study. Numerical experiments show that the range of s t a b i l i t y can be extended when longitudinal thermal d i s p e r s i v i t y i s set greater than 10 m. Figure 50 gives the results of one experiment where the longitudinal d i s p e r s i v i t y of the basal aquifer i s set equal to 500 m. The numerical solution converged to a numeri-c a l l y stable temperature f i e l d for hydraulic conductivities as high as 1000 m/yr. Previous simulations, with longitudinal d i s p e r s i v i t y less than 10 m could not achieve a stable solution for hydraulic conductivities above 500 m/yr. The influence of t h i s parameter may be s i g n i f i c a n t where numerical models are designed spe c i f i c a l l y , to simulate heat transport i n highly-fractured and karst aquifers. Although longitudinal d i s p e r s i v i t y has a weak influence on temperatures where v e r t i c a l heat flow, i s dominant, transverse d i s p e r s i v i t y can be a sig n i f i c a n t parameter. With reference to Figure 51, the transverse d i s p e r s i v i t y i n both layers i s varied over the range of 1 - 50 m. In these pa r t i c u l a r simulations, = 200 m/yr, = 10 m/yr, and the basal heat f l u x J = 70 mW/m2. The longitudinal d i s p e r s i v i t y i s set equal to the transverse d i s p e r s i v i t y i n a l l three cases. Notice that as d i s p e r s i v i t y i s changed to higher values, the amount of thermal dispersion r i s e s , and temperatures are s i g n i f i c a n t l y effected. The most pronounced effect occurs i n the basal aquifer because f l u i d - f l o w v e l o c i t i e s are much greater than i n the shale. Geothermal Heat Flow The simulations presented so far have considered a prescribed basal heat f l u x of 60 - 70 mW/m2 as being representative of continental p l a t -forms. F i e l d measurements of regional heat flow may vary from about 4-0 mW/m2 i n stable, g r a n i t i c shields to over 120 mW/m2 i n tee t o n i c a l l y active regions (Sass et a l . , 1981). Heat flow can exceed 200 mW/m2 i n some geothermal TEMPERATURES K, = 500 m/yr 70 °C 70°C ~i i i 1 1 1 1 1 1 r 150 200 Kilometers 60 °C 300 ' ( c ) Figure 50. E f f e c t of l o n g i t u d i n a l d i s p e r s i v i t y on the numerical s t a b i l i t y of high-convect regimes. i o n H CO TEMPERATURES T = 20°C C L = lm € T=lm 189 environments. Figure 52 shows the effect of s i x different values of heat fl u x on basin temperature patterns. The hydraulic and thermal properties of the two-layer basin are as follows: K = 100 m/yr, K = 10 m/yr, K /K = 100, <j>_ = 0.15, i>n = 0.10, K =.3.0 W/m°C, K = 2.0 W/m°C, and e = e = 0.0. A 1 2 1 . 2 Li 1 s a l i n i t y gradient of 0.015 % NaCl/m i s also imposed across the section. In Figure 52a, the basal heat f l u x i s 50 mW/m2 and the maximum O 9 temperature observed i s 70 C. Doubling the heat flow to 100 mW/mz at the base causes temperatures to r i s e up to 100°C i n the central part of the basin (Figure 52f). The graph of Figure 53 summarizes these r e s u l t s , with the temperature p r o f i l e along section Z-Z' (Figure 52b) plotted as a function of depth. Also l i s t e d on the diagram are the f l u i d v e l o c i t i e s i n the basal aquifer from each of the heat-flux simulations. Intermediate heat fluxes between 50-100 mW/m2 are probably the most r e a l i s t i c i n stratabound ore-forming systems. The simulations i n Figure 52 show that rather high geothermal fluxes would be needed i n t h i s basin model to approach the upper temperature range of 150°C, which i s observed for some Mis s i s s i p p i Valley-type ore deposits. For lower temperature lead-zinc ores, heat flows of 60 - 80 mW/m2 appear to be adequate for the size and thickness of. the basin i n Figure 52. Jessop and Lewis (1978) predicted that the maximum heat flow that a stable, continental region could sustain i s about 80 mW/m2, based on r a d i o a c t i v i t y considerations alone. This places an upper constraint, therefore, on geologically reasonable values of geothermal f l u x i n our basin model. Unless heat convection i s greater than i n Figure 52, temperatures w i l l not exceed 80°C near the basin margin (Figure 53). 190 Figure 52. E f f e c t of the geothermal heat f l u x on subsurface temperature. 1 9 1 Figure 53. Temperature p r o f i l e i n the shallow end of the basin as a function of the geothermal heat f l u x . 192 Climate Climatic conditions are included i n the heat-flow model through the s p e c i f i c a t i o n of the temperature. T Q on the water-table boundary. For si m p l i c i t y , the same mean annual surface temperature i s prescribed over the ent i r e length of the basin. Figure 54 demonstrates the control of the water-t a b l e temperature on the simulated temperature pattern. The geothermal f l u x i s 60 mW/m2, and other model parameters are the same as the preceding model. Changing the surface temperature by 10°C causes a general numerical change i n each isotherm by 10°C, although the e f f e c t i s not p e r f e c t l y l i n e a r . I f we are w i l l i n g to accept a heat flow of 80 mW/m2 as an upper l i m i t , then the maximum temperature i n the basin model ought to range between 85°C and 105°C f o r the surface-temperature range of 10°C to 30°C. Climatic conditions were apparently favorable f o r t h i s surface temperature range during Devonian to Tertiary'time i n the North American continent. Basin Geometry In t h i s section, a review of the e f f e c t s of basin s i z e , structure, and water-table configuration are presented. Some of the temperature p l o t s are obtained from the same simulations that were presented i n the preceding section on f l u i d - f l o w . Reference to Figures 33 to 44 can be made f o r the accompanying f l u i d - f l o w patterns. A basal geothermal f l u x of 60 mW/m2 and a constant surface, temperature of 20°C are assumed i n most of the models given below. The hydraulic and thermal properties of the two-layer basin model have not been changed. The e f f e c t of basin si z e on subsurface temperature i s shown i n Figure 55. The length-to-depth r a t i o of each basin i s 100:1. Unlike the f l u i d - f l o w problem of Figure 33, the v e l o c i t y f i e l d i s no longer the same i n each basin because of the coupling between temperature, s a l i n i t y , and the f l u i d TEMPERATURES 7 = i o ° c Surface Temperature, T 0=IO°C i i • i | i i i i | i i i — i — | — i — i — i i — j 1—i 1 1 — | — i 1 — i — i 1 50 100 150 200 250 300 Kilometers ( c ) Figure 5 4 . E f f e c t of the prescribed water-table temperature on re g i o n a l temperat TEMPERATURES 55. Control of basin size and thickness on subsurface temperat 195 properties. V e l o c i t i e s i n the basal aquifer of Figures 55a, 55b, and 55c are a l l around 2.5 m/yr, but v e l o c i t i e s reach values as high as 7.2 m/yr i n the 600 km-long basin (Figure 55d). Temperatures easily exceed 1Q0PC at the downstream end of t h i s large basin. Perhaps i t i s t h i s scale of flow system, with f l u i d c i r c u l a t i o n to depths of 6 km, that i s responsible for the formation of lead-zinc ores at the high end of t h e i r temperature range (100° - 150°C), as in the.Mississippi Valley d i s t r i c t . A higher geothermal gradient of 80 mW/m2 and surface temperature of 30°C could push temperatures above 150°C i n the discharge end of the basin. The influence of basement structure on the thermal, regime i s i l l u s t r a t e d i n Figure 56. Once again the two-layer basin i s used for the demonstration, and the material properties are the same as before. Three cases are modeled; a f l a t bottom, sloping bottom, and basement arch on a sloping bottom. Temperatures i n the flat-bottom model are much higher at.the discharge edge of the basin than the sloping, wedge-shape basin because the stratigraphic section i s nearly twice as thick. The presence of. a basement arch forces heat to be convected up over the structure, and thereby produces a steep, geothermal gradient on the upstream side of the ridge and a lower gradient on the down-stream side. Higher r e l i e f on the structure accentuates the thermal anomaly. A l l of these features are more c l e a r l y i l l u s t r a t e d by the graph of Figure 57. It i s a plot of temperature i n the basal aquifer as a function of distance along the basin. Notice how effective the basement arch i s at reducing the convective transport of heat from the thick part of the basin to the t h i n basin margin. This feature bears out the importance of long, continuous flow systems i n the development of adequate temperatures at shallow depths and near the margin of a basin. On the other hand, basement arches i Figure 56. Control of basement structure on subsurface temperature. to CD 0 50 100 150 200 250 300 KILOMETERS Figure 57. Temperature along the basal aquifer as a function of basement structures. 198 between major sedimentary basins may themselves serve to form suitable conditions for ore genesis, (.e.g. Figure 56d). Remember that the absolute temperatures i n Figure 57 could be 10°C to 30°C greater, under warmer climatic conditions or higher geothermal heat flow. Figure 58 displays the effect of topography on temperature patterns in the sloping-bottom basin. The sl i g h t break i n topographic slope i n Figure 58a appears to have l i t t l e influence on the thermal regime. Intro-ducing a water-table ridge at the end of the basin has a greater effect on bending the isotherms near the topographic depression. Upward discharge of groundwater i s well i l l u s t r a t e d by the thermal perturbations shown i n Figures 58c and 58d. Major topographic lows serve to focus groundwater..discharge, which can create anomalously high temperatures at shallow depths. A compli-cated water-table configuration, i s reflected by irregular perturbations on the isotherms, as shown i n Figure 58e. It i s apparent from these simulations that major topographic features within a basin serve to form zones of downward heat convection i n recharge areas and upward heat convection i n discharge areas. As i n the case of f l u i d flow, the heat-transport system can be weakly or strongly modified, depending on topographic r e l i e f , basin size and structure, and material properties of the hydrostratigraphic units. The genetic association between anomalously high temperatures and lead-zinc ores i s c l e a r l y linked to groundwater-discharge features. Factors Controlling Mass Transport As a metal-bearing brine flows through a basin the concentration w i l l vary along the flow path. This variation can be caused by a number of geo-chemical and hydraulic processes. In the absence of mixing or geochemical reactions, the solute w i l l t r a v e l at a rate equal to the average l i n e a r 199 TEMPERATURES T = 20°C 50 100 150 200 250 300 Kilometers (e) Figure 58. C o n t r o l of water-table c o n f i g u r a t i o n on subsurface temperature. 200 v e l o c i t y of the groundwater and the concentration will.remain unchanged. Solute transport i n porous media i s not, however, t h i s simple. Even i n a nonreactive medium, the processes of mechanical mixing and molecular di f f u s i o n casue the solute to spread away from the flow path and decrease the o r i g i n a l concentration of the aqueous component. This spreading phenomena i s better known as hydrodynamic dispersion, which was mathematically described i n previous chapters. This study i s not the f i r s t to consider the general factors affecting mass transport i n groundwater-flow systems. Pickens and Lennox (.1976), Schwartz (.1977), and others have established the importance of dispersion through s e n s i t i v i t y analyses of modeling parameters. The simulation results given here provide a review of the factors affecting mass transport, but on the scale of a large sedimentary basin and i n r e l a t i o n to ore genesis rather than groundwater contamination. Many of the concepts, which are d i r e c t l y dependent on the advection patterns of mass, are already self-evident from the f l u i d - f l o w patterns presented e a r l i e r . For t h i s analysis, attention i s directed.to the s p e c i f i c problem of understanding how the concentration of an ore-forming component reaching a depositional s i t e depends on the o r i g i n a l concentration i n the source-bed area, the length of the flow path, the rate of flow, and the degree of d i l u t i o n caused by dispersion along the flow path. . This type of information i s needed when making reasonable estimates of the metal concentrations i n a brine passing through an ore-forming s i t e . The transport models given here do not simulate the geochemical process of releasing metals from source bed to the f l u i d s . Instead, they simply assume that a certain metal concentration exists at a specified location and i n i t i a l time. The goal i s to follow the path of a metal-bearing 201 f l u i d packet as i t t r a v e l s through the.basin to a discharge area, and show how the concentration is. affected by the f l u i d - f l o w system. Although, when reference i s made to the solute i t i s usually meant to describe a metal species, the modeling r e s u l t s are equally v a l i d f o r any aqueous species, including s u l f i d e . A l l of the simulations previously shown are, i n one way or another, useful to the understanding of mass transport i n sedimentary basins. The flow p a t t e r n s . t e l l us how metal species would be advected through the basin, and therefore place constraints on where an ore deposit could form or' from what source beds an aquifer could 'drain' an adequate supply of metals. The possible combination of f l u i d - f l o w f a c t ors a f f e c t i n g the advection of a solute are numerous. Figures 7 and 8, and 16 through 44 give a wide s e l e c t i o n of the types of flow patterns that can exist i n sedimentary basins, and how they can regulate the possible locations of ore deposition. The flow v e l o c i t i e s from these simulations add quantitative c r i t e r i a to the assessment by determining t r a v e l times, mass-flux r a t e s , and the time required to deposit a large ore body f o r a given concentration value. In t h i s s e c t i o n , the emphasis i s placed upon the quantitative aspects of mass transport. Three c o n t r o l l i n g factors are examined i n d e t a i l : hydraulic conductivity, permeability contrasts i n layered basins, and dispersion. Hydraulic Conductivity Consider the simple, two-layer basin with a sloping basement and l i n e a r water-table slope. The f l u i d - f l o w and heat-flow solutions were given i n Figure 31. Groundwater flow i s directed downward across the shale beds i n the elevated end of the basin; i t flows l a t e r a l l y through the basal aquifer and shale aquitard i n the mid-section of the basin; and i t i s forced 202 upward across the shale unit near the basin margin. The flow pattern remains nearly the same for any fixed contrast i n the hydraulic conductivities between the layers, but the f l u i d v e l o c i t y depends on the actualvvalue of•:.:"the'..\hy.dr.aulic conductivities and the size of the basin, as shown e a r l i e r . The effect of hydraulic conductivity on transport times and concentra-t i o n patterns i s easily determined.. Figures.59 and 60 summarize the results of some simulations designed for t h i s purpose. For the model shown i n Figure 59 the parameter values used are as follows: the carbonate aquifer - = 80 m/yr, <j), = 0.15, K /K 100, K = 3.0 W/m°C, £T = 10 m, e = 1 m, a = 10 m, 1 xx zz 1 ' L T L a = 1 m, and the shale aquitard - K = 4- m/yr, $ = 0.10, K /K = 100, 1 2. 2 ZZ K = 2„0 W/m°C, e - 10 m, e = 1 m, a = 1 m, a_ = 1 m. A s a l i n i t y gradient of 0.10 % NaCl/m i s also present. A surface temperature of 10°C i s placed on the water-table and a geothermal f l u x .of 70 mW/m2 i s prescribed on the bottom boundary. The d i s p e r s i v i t y coefficients a and a also affect mass transport, Li 1 but the effects of these parameters are reserved for l a t e r consideration. Only the r e l a t i v e solute concentrations at four-different time steps and the steady-state velocity-vector f i e l d are shown i n Figure 59. The r e l a t i v e concentration values of 5%, 10%, and 15% of the source are represented by the contours i n Figure 59b. That part of. the plume with r e l a t i v e concentra-tions greater than 15% of the source value i s shaded. The units of concentra-t i o n are ar b i t r a r y . At the source location, a pulse of metal species i s introduced so as to produce an i n i t i a l concentraion of i n the brine at time t = 0. The position and concentration pattern of the pulse i s shown at times of 50,000,775 ,000, and;'lOOgOOO„yr. . By-t = 200,000 y r , most of the o r i g i n a l mass released has been transported through the basin, and i t has been discharged to the land surface. It should be emphasized that Figure 59 represents a single numerical 4 - 1 VELOCIT.Y VECTORS 3 — * 3.4 m/yr 2H ° 1 ' ' 1 I ' 1 1 1 I 1 1 ' ' | • • • i | ; , r— r (a) 4T 3-0-t - 50000 yr • \ SOURCE r = 0 t = 75000 yr CONCENTRATIONS C/Co t= 100000 yr i I i" ~1 l 1 1 1—| " — i — i — i — | — i r 50 100 150 I Kilometers ' T — r — r — r 200 I 1 1 — ' 1 2 5 0 3 0 0 2 , 2 n 9 g CUMULATIVE PARTICLE FLUX t = 200,000 yr. 72 67118 200 Km 250 Km ( c ) 300 Km Figure 59. Mass tr a n s p o r t p a t t e r n i n a two-layer b a s i n . 204 1 0 0 0 HYDRAULIC CONDUCTIVITY OF BASAL AQUIFER, K x x(m/yr) Figure 60. Relationship between the average basin-residence time f o r a metal-bearing pulse and the hydraulic conductivity of the basal aquifer, i n a two-layer basin. 205 event of metals entering the flow system and being transported through the basin. This i s not to imply that ore formation i s the results of a single pulse of metal-bearing brine, or even a set of discrete pulses. The representation i n Figure 59 i s simply a mathematical device for estimating the concentration levels i n a f l u i d as i t continuously flows through the basin. As f l u i d passes through the shale bed and into the basal aquifer, i n the recharge end of the basin, a substantial amount of mixing occurs. After 50,000 y r , the pulse has t r a v e l l e d nearly one-third of the basin length, mostly through the aquifer. The maximum concentration i n the pulse C x i s now only about 40% of the source concentration C The pulse i s at a mid-position i n the basin at 75,000 y r , and C /C. has dropped to about 35%. As ' max 0 c the brine moves updip toward the shallow end.of the basin, upward f l u i d d i s -charge begins to influence concentrations. This i s marked by the t r a i l i n g ' t a i l ' of the pulse at t = 100,000 yr.. Leakage of mass up into the low-permeability shale i s responsible for t h i s feature. Mass i n the shale unit moves much slower than i n the aquifer. (Figure 59a) and as a result the concentration plume takes on a stretched-out appearance. However, a large portion of the mass continues i t s route through the permeable aquifer and eventually i s forced to surface at the end of the basin. The leakage of mass into the.overlying shale produces a dispersed zone of mass discharge between x = 200 km and x = 300 km. The nature of the discharge d i s t r i b u t i o n i s best documented by Figure 59c, which expands the view of the discharge end of the basin at t = 200,000 yr. As explained i n the l a s t chapter, the numbers plotted on the top of the water table represent the cumulative p a r t i c l e flow leaving the mesh through the upper row of c e l l s . This cumulative p a r t i c l e d i s t r i b u t i o n i s obtained by t a l l y i n g the location of the p a r t i c l e as i t i s naturally 206 ejected from the mesh i n the moving-particle random-walk numerical procedure. Much of the p a r t i c l e f l u x from the c e l l s between x = 250. km and x = 300 km are transported.mainly through the aquifer and are discharged r e l a t i v e l y e a r l y , as compared to the r e s t of the p a r t i c l e d i s t r i b u t i o n , which a r r i v e d much l a t e r because of i t s longer path through the shale aquitard. Although the time d i s t r i b u t i o n of p a r t i c l e f l u x i s not shown, the cumulative d i s t r i b u t i o n does produce a type of s p a t i a l histogram that shows how mass i s discharged at the basin margin. Keeping other parameters constant while varying hydraulic conductivity causes transport times to increase or decrease according to the d i r e c t i o n of change i n the hydraulic conductivity. Doubling the permeability means that the t r a n s i t time of a pulse w i l l be halved. Figure 60 shows the nearly l i n e a r r e l a t i o n s h i p obtained f o r the two-layer basin model. Hydraulic conductivity i s p l o t t e d against the observed time needed f o r the center of mass of the pulse to break through to the water-table surface at the downstream end of the basin. Concentrations i n the metal-bearing pulse decrease along the flow path (Figure 59b). Higher flow rates should lead to greater dispersion i n a flow system, but l i t t l e change i s observed i n the mass-transport, pattern when hydraulic conductivity i s varied over the range shown i n Figure 60. Figure 61 displays the r e s u l t s of several simulations from which, the maximum pulse concentration i n the basal aquifer C. /C„ i s p l o t t e d as a function of u max 0 r distance. The s o l i d curve i s drawn to indicate the general trend,of the solute concentration i n a schematic form. It i s obvious that dispersion can s i g n i f i c a n t l y reduce the amount of metal reaching an ore-forming s i t e near the edge of a basin. In the two-layer basin, about 20% of the source-bed concentration i s present i n the f l u i d as i t passes through the aquifer at Figure 61. Maximum solute concentration in the basal aquifer as a function of transport distance. 208 the basin margin. The degree of hydraulic-conductivity contrast between the layers determines the v e r t i c a l flow rates and the area over which, cross-formation flow occurs, as shown e a r l i e r i n Figure 20. One would expect i n an aquifer-aquitard system, with a high r a t i o of hydraulic conductivity contrast K^/K^, that the discharge of mass would be spread over a narrow area at the downstream end of a basin because of the enhanced focusing of flow into the aquifer. Figure 62 shows some simulation results that demonstrate the s e n s i t i v i t y of the mass-discharge pattern to the hydraulic-conductivity r a t i o K / l ^ . These histograms are constructed from the type of p a r t i c l e - f l u x d i s t r i b u t i o n given in Figure 59c. The p a r t i c l e mass counted at the water table i s simply expressed as a percentage of the t o t a l mass of the original.pulse. High K^/K^ r a t i o s appear to l i m i t the area of mass discharge, r e l a t i v e to low r a t i o s , but the influence i s not strong. In Figure 62c, the mass i s more evenly distributed i n the discharge-half of the basin than i n the cases of lower ratios (Figure 62a and 62b). Leakage of mass into the overlying shale i s less i n the low-ratio cases because the component of upward flow i s smaller. It i s d i f f i c u l t to assess the importance of the mass-discharge d i s t r i -bution i n cont r o l l i n g ore-precipitation patterns i n carbonate-hosted lead-zinc deposits, mainly because ore occurrence i s commonly confined to k a r s t i c aquifers. However, modeling of t h i s type may have application i n other types of stratabound ore deposits. As an example, Figure 63 i s a schematic cross section through the Lake Superior basin at the time of formation of the White Pine copper deposit. The geology and copper content of the Nonesuch Shale are obtained from the study by White (1971). The conceptualized f l u i d - f l o w pattern i s based on an interpretation of the gravity-driven groundwater model proposed by White (1971) to explain copper mineralization i n the Nonesuch (a) K , / K 2 = 10 1 — — ^ (c) K, / K 2 -100 150 ~i r" 200 250 DISTANCE (Km) 300 Figure 62. E f f e c t of hydraulic conductivity contrast between aquifer Kj_ and aquitard K2 on the mass-discharge pattern at the water t a b l e . NORTH SOUTH meter 1 0 0 DISTANCE (Km) Figure 63. Copper content of the Nonesuch Shale and conceptual model of f l u i d flow i n the Lake Superior basin at the time of formation of the White Pine copper deposit (modified after White, 1971). 211 Shale at White Pine, Michigan. Copper-bearing f l u i d s flowed through the basal conglomerate aquifer toward the edge of the Lake Superior basin, and were eventually forced upward and through the shale beds. Copper p r e c i p i t a -t i o n was caused by replacement of pre-existing iron-sulfide minerals i n the shale (Brown, 1971). Notice that the copper-content p r o f i l e (Figure 63) i s comparable to the mass-discharge patterns of Figure 62. The presence of a widespread reactant ( i . e . pyrite) allows the development of a well-defined mineralization pattern, with maximum.deposition where upward flow rates were probably the greatest. The l o c a l i z a t i o n at White Pine may be due to the presence of the basement arch, which served to focus upward discharge, as i n Figures 37 and 38. I t could also be argued that the wide spread of the copper mineralization may indicate that the hydraulic conductivity contrast between the two main hydrostratigraphic units i n the basin, the Copper Harbor Conglomerate' and the Freda Sandstone, was not great. Dispersivity Dispersion i s a mixing process that results i n the spreading of a solute i n both a longitudinal and l a t e r a l direction with respect to a flow path. I t i s caused by the combined action of molecular d i f f u s i o n , microscopic variations i n f l u i d v e l o c i t y , and changes i n vel o c i t y from bed to bed. In geologic systems such as large sedimentary basins, i t i s the large-scale dispersive processes that are most s i g n i f i c a n t . The mathematical expression for dispersion lumps a l l of these processes together as two c o e f f i c i e n t s : the longitudinal dispersion coeff i c i e n t D and the transverse dispersion c o e f f i c i e n t L D . Equation (4-21) shows how each c o e f f i c i e n t consists of two components, a term representing the medium d i s p e r s i v i t y times the v e l o c i t y , and a term representing the effective molecular d i f f u s i v i t y . The d i s p e r s i v i t y i s a 212 function of the medium and the aqueous species. Diffusion coefficients for aqueous species commonly range from 10 9 to 10 1 2 m2/s i n porous media. (Freeze and Cherry, 1979). Compared to mechanical dispersion, diffusion i s several orders of magnitude smaller. Its effect on mass transport i s r e l a t i v e l y small, unless hydraulic conductivity i s very low. A constant value of d i f f u s i v i t y . o f 10 1 0 m2/s i s used i n t h i s study, but i n a l l cases i t s effect i s n e g l i g i b l e . In preceding simulations, a longitudinal d i s p e r s i v i t y a of 10 m and Li a transverse d i s p e r s i v i t y a of 1 m are assumed for the aquifer layer. Dis-p e r s i v i t y i s set at a constant 1 m for both components i n the overlying shale. These numbers are assumed to be representative of t h i s basin. Unfortunately, there i s l i t t l e data available.on d i s p e r s i v i t y coefficients i n natural environments. Observed and model-calibrated data collected to date indicate a d i s p e r s i v i t y range of 0.01 to 100 m (Anderson, 1979). The actual value of d i s p e r s i v i t y chosen usually r e f l e c t s the nature and scale of heterogeneities in the flow system. Representative values for the basin being modeled i n t h i s study are thought to range between 10 m and 500 m for the longitudinal co-e f f i c i e n t . The transverse d i s p e r s i v i t y i s expected to be lower i n value by a factor of 1 to 100. Further d e t a i l s on dispersion i n porous media can be found i n Anderson (1979).- Table 11 summarizes her findings for the range of d i s p e r s i v i t y of common geologic materials. To test the s e n s i t i v i t y of the two-layer basin to the d i s p e r s i v i t y parameters, a series of simulations have been run where these coefficients are varied. A l l other parameters are kept, constant. The basal aquifer i s assumed to have a horizontal hydraulic conductivity of 200 m/yr and a porosity of 0.15. The overlying aquitard has a horizontal hydraulic conductiv-i t y of 10 m/yr and a porosity of 0.10. An anisotropy r a t i o of horizontal to Table 11. Longitudinal d i s p e r s i v i t y range of geologic materials. J I I GLACIAL DEPOSITS 1 | ALLUVIAL SEDIMENTS FRACTURED BASALT | | | DOLOMITE | ] LIMESTONE | 1 SANDSTONE I I SHALE r ~ I i i 1 r 1 1 1 1 1 0 10 20 30 40 50 60 70 80 90 100 LONGITUDINAL DISPERSIVITY (m) 214 v e r t i c a l hydraulic conductivity of 100:1 i s once again assumed for both hydro-stratigraphic units. The f i r s t set of simulations consider the homogeneous and i s o t r o p i c d i s p e r s i v i t y case where the d i s p e r s i v i t y i s constant throughout the basin. Mass i s instantaneously released from source beds at a point deep i n the recharge end of the basin. Figure 64 shows the position and shape of the concentration plumes at t = 20,000 yr and t = 40,000 yr. Four separate simulations are given i n Figure 64, with d i s p e r s i v i t y values of 1 m, 5 m, 10 m, and 50 m respectively. Concentration contours of 5%, 10%, and 15% C/C^ have been plotted. I f there i s l i t t l e dispersion i n a flow system, a dissolved metal species w i l l move along a flow path as plug-type flow with l i t t l e change i n concentration between the points of entrance and e x i t . In Figure 64b, the maximum concentration of the pulse averages nearly 35% C /C at t = 40.000 yr. J max 0 Increasing the d i s p e r s i v i t y causes further d i l u t i o n of the solute. The C /C„ max 0 values decrease to about 15%, 10%, and 5% for a = 5 m, 10 m, and 50 m. Clearly the chance of having a concentrated, metal-bearing brine reach an ore-forming s i t e near the end of a basin i s unlikely i f transport involves long distances i n a highly dispersive flow system. A more r e a l i s t i c f i e l d situation occurs when the transverse disper-s i v i t y i s much less than the longitudinal d i s p e r s i v i t y . Figure 65 shows some of the modeling res u l t s from a series of simulations designed to examine the effect of the a /a r a t i o . In the shale unit a = a = 1 m, and i n the L 1 Li I aquifer a - 1 m, but a i s varied. The hydraulic properties and v e l o c i t y -1 L vector diagram are the same as Figure 64a. The results show that increasing the longitudinal d i s p e r s i v i t y i n the aquifer causes the concentration contours to stretch out along the flow 215 4-» • * 4 — — — — VELOCITY VECTORS i —-8.5 m/yr Kilome -0 4-e 3-at I I i i | i i i i | • t« 20000yr-i r i | i i (a) — i — i — | — i — i — i — i — | — i — i — i — i — | CONCENTRATIONS C/Co • * L **r* 1 m t« 40000 m/yr Kilometi — ro SOURCE j 0 4-£ 3-i i i i 1 • i i i i 1 i • i > , i i (b) I 1 | 1 T I 1 j 1 I I I | * V « T « 5 m | 2-1 - i 1 1 1 1 r- - i 1 — i r -(c) " £ L * * T * 1 0 m • I I i i I r i 1 1 1 1 1 — i 1 1 1 1 1 — i — r S 3-(d) "S.'V 50m • 50 100 150 200 2 50 300 Kilometers (e) Figure 64-. Effect, of i s o t r o p i c and homogeneous d i s p e r s i v i t y on the mass-transport patterns i n a two-layer basin. t = 20000 yr CONCENTRATIONS C/Co r = 40000yr ° < : L = IOm <*T = lm Figure 65. E f f e c t of l o n g i t u d i n a l disp T — i — i — i — 1 ~ 150 200 Kilometers ( c ) e r s i v i t y on mass transport patterns. H CO 217 path. Passing from a = 10 m to.a = 1000 m produces a nearly three-fold expansion of the plume at t = 4-0,000 yr. Comparison of Figures 64d and 65a i l l u s t r a t e s the effect of transverse d i s p e r s i v i t y on the 'height' of the mass-transport plume. The increased mixing by systems with larger a coefficients i s shown in Figure 66. For the same time t •=. 40,000 y r , an a = 10 m reduces the Li pulse concentration to about 30% of C , while an a - 1000 m causes the con-centration to drop to about 5% C/C . Existing f i e l d data support longitudinal d i s p e r s i v i t y coefficients of 10 m to 100 m for regional-scale problems. The amount of hydrodynamic d i l u t i o n that can occur between source beds and possible depositonal sit e s can be estimated from Figure 66. Based on the modeling r e s u l t s , a maximum of about 20-30% of the source concentration l e v e l would be present i n a metal-bearing brine that has t r a v e l l e d over 250 km to an ore-forming s i t e . I f a geochemical model requires a minimum concentration of 3 mg/kg'H^O Zn to precipitate enough' sphalerite at the ore s i t e , then the o r i g i n a l concentration of zinc i n the source-bed area would have to be at least 10 to 15 mg/kg'H^O. Although t h i s transport model i s greatly s i m p l i f i e d because it.does not account for the gain or loss of zinc through reactions along the flow path, i t does provide a better model of the r o l e of f l u i d flow i n ore formation than a simpler plug-flow calculation. Geology A variety of geologic configurations and structures i n a basin may control the formation and position of stratabound ore deposits. So far we have seen the effects of some.simple basin structures and.water-table config-urations on f l u i d - f l o w patterns. The two-layer basin has f i t the needs of t h i s preliminary analysis of an ore-forming system by s p l i t t i n g the geology into i t s simplest components of aquifer and aquitard l i t h o l o g i e s . Numerous MAXIMUM PULSE CONCENTRATION IN AQUIFER Cmax/Co (%) TI P' era c CD <7> H O 3 OP P-r t c a. P-3 O O 3 H O CD CL 3 r t 4 CD rr 4 CO P-< P* r t P-O 3 P-3 r t 3" CD tr fu w pj SU c P-l-h CD fl) W Hi 3 O r t P-O 3 O Hi 219 modifications of t h i s simple configuration are possible. Whatever the nature of the stratigraphy, however, aquifers serve to focus flow such that f l u i d s w i l l follow the route of least energy loss. From t h i s point of view, the two-layer basin analysis i s j u s t i f i e d i n order to understand the role of the cont r o l l i n g factors at t h e i r simplest l e v e l . I t would be impractical to cover a l l the possible geologic configura-tions of interest i n ore formation, even i n an idealized format. However, over twenty transport simulations have been conducted on specialized geologic configurations•with a variety of stratigraphic and structural features. Each of these models displays a unique set of f l u i d - f l o w patterns, subsurface temperature d i s t r i b u t i o n , and concentration patterns that arise from the sp e c i f i c geologic configuration. Figure 67 considers some of the hypothetical configurations that are modeled. The existence of discontinuous aquifers generally causes an upward movement of flow near the margin of the aquifer, whether or not the pinchout i s due to a facies change, f a u l t i n g , basement structures, or unconformities (Figures 67a, 67b, 67c, and 67d). Upward flow also brings with i t higher temperatures at these pinchout locations. The effect of a broadly-warped basin (Figure 67d) i s not much different from an undeformed basin of the same size. Figure 67f shows the configuration of a high-permeability carbonate-reef structure resting near the top of a basal sandstone aquifer. The upper unit surrounding the reef consists of shale and limestone. In t h i s model, f l u i d flow i s mostly through the basal aquifer, although the presence of the high-permeability reef also focuses downward flow from the shale beds into the reef, much as i n the manner shown i n Figure 22. Space does not permit the presentation of a l l these modeling r e s u l t s , but one example w i l l be shown. Figure 68 i s an idealized section of a basin with a basal carbonate aquifer (.Unit 1) overlain by a thick sequence of 220. Figure 67. Idealized cross sections of several geologic configurations of possible importance to stratabound ore formation. HYDROSTRATIGRAPHY E 2H o 0 4-1 3 1,71. I LIMESTOME \ 2 \ SHALE | 3 | DOLOMITE I i (a) E o H T 1 > i 1 i 1 1 r—i 1 1 VELOCITY VECTORS 0 r * r Kilometers (d) Figure 6 8 . Transport simulation showing the influence of a r e e f structure at depth. ro ro Table 12. Model parameter data for simulation problem of a reef structure at depth Hydrostratigraphic Unit Parameter Symbol 1 2 3 Units F l u i d Flow Porosity Horizontal hydraulic conductivity V e r t i c a l hydraulic conductivity S a l i n i t y gradient. 0.20 0.10 200.0 10.0 2.0 0.1 0.010 ( a l l units) 0.30 500.0 5.0 (fr a c t i o n ) m/yr m/yr %NaCl/m Heat Transport Thermal conductivity of rock Longitudinal thermal d i s p e r s i v i t y Transverse thermal d i s p e r s i v i t y Geothermal heat flux at base Temperature at water table Mass Transport Longitudinal d i s p e r s i v i t y Transverse d i s p e r s i v i t y Apparent d i f f u s i o n c o e f f i c i e n t 70.00 T Q = 20.0 3.0 100.0 1.0 2.0 1.0 1.0 100. 1.0 1.0 1.0 3.0x10"3 ( a l l units) 3.0 100.0 1.0 100.0 1.0 W/m°C m mW/m2 o„ m 2/yr 223 mudstones (Unit 2). Near the basin margin, a karstic-reef feature (Unit 3) i s present, which i s f i f t y - t i m e s more permeable than the shale beds and twenty-times more permeable than the carbonate aquifer. Table 12 l i s t s some of the set of parameter values assumed i n the simulation. The karst-reef structure does not create major alterations i n the regional flow pattern of the two-layer basin model. It has the main effect of causing l a t e r a l flow over a larger area of the basin, and l i m i t s the zone of upward discharge to that part of the basin updip from the structure. Flu i d v e l o c i t i e s are the highest i n the basal part of the reef at 9.0 m/yr. These high flow rates are also..responsible for maintaining a nearly constant temperature along the basal aquifer unit. The strong influence of convection on the thermal regime i s marked by the parallelism of the isotherms with the basement slope. Subsurface temperatures do not exceed 85°C in the reef structure. Figure 68d shows the mass-transport solutions at t = 20,000 yr and t = 40s000 yr for a pulse of mass released just above the carbonate aquifer i n the recharge end of the basin. Only the 5%, 10%, and 15% C/C^ concentration levels are contoured.' Approximately 20% of the source-bed concentration l e v e l reaches the karst-reef structure at the other end of the basin. Geochemical Models of Transport and Pr e c i p i t a t i o n Of a l l the processes effecting ore genesis i n sedimentary basins, the geochemical factors that control metal s o l u b i l i t i e s and ore p r e c i p i t a t i o n have received the most quantitative research. With reference to the carbonate-hosted lead-zinc deposits, recent studies that review the general theories include Anderson (.1978), Anderson and Macqueen (1982), Giordano and Barnes (.1981), and Sverjensky (1981). Some of these concepts have been introduced e a r l i e r i n Chapter 2. Table 13 contains a summary of the three basic geo-chemical models proposed i n the l i t e r a t u r e . Table 13. Geochemical models f o r stratabound ore genesis (a f t e r Anderson, 1978 and Sverjensky, 1981) Transport Model Reasons f o r Deposition Metals transported i n low-sulfur f l u i d a) Mixing with H 2S-bearing solutions b) Replacement of iron s u l f i d e s c) Thermal degradation of organics I I . Metals transported i n sulfate-bearing f l u i d a) Sulfate reduction by b a c t e r i a l destruction of organics to produce H 2S b) Sulfate reduction by non-bacterial destruction of organics to produce elemental su l f u r c) Sulfate reduction i n t e r n a l l y due to methane d) Sulfate reduction externally on encountering petroleum e) Sulfate reduction due to reaction with iron-bearing minerals I I I . Metals and s u l f i d e transported together i n the same f l u i d a) Reaction with host rock and r i s e i n pH b) Temperature drop c) Decreased s t a b i l i t y of chloride complexes due to mixing 225 The purpose of t h i s section i s to provide a quantitative analysis of two of the most l i k e l y geochemical models, through the use of reaction-path modeling with EQ3/EQ6. F i r s t to be considered are basinal-brine models that support metal transport i n a sulfate-type f l u i d ( I I , Table 13). Secondly, the geochemical conditions necessary for simultaneous transport of metal and sulfide i n the same f l u i d ( I I I , Table 13) are examined and possible p r e c i p i t a -t i o n mechanisms tested. Mixing-type models that assume transport of metal i n one f l u i d and sulfi d e i n another are not e x p l i c i t l y assessed i n t h i s evaluation. We have already seen from the fl u i d - f l o w and mass-transport modeling that the transport of two immiscible f l u i d s , without mixing through dispersion, i s highly unlikely i n long-distance brine transport. On the other hand, some hydrogeologic conditions are capable of supplying to mixing areas one or the other ore-form-ing components:":from different parts of the basin. This i s accomplished through the development of l o c a l flow systems, which are superimposed on regional systems (e.g. Figures 22 and 41). The p o s s i b i l i t y of a mixing-type model i s s t i l l open, provided f i e l d evidence can be found to c l e a r l y support i t over the simpler hydrodynamic situation of a single I ore-forming f l u i d . Metal-Sulfate Brine Models Consider the model of a metal-bearing brine containing sulfate S0^-.„ I f i t reaches an area where organic material i s present, then the p o s s i b i l i t y exists for the reduction of sulfate to s u l f i d e , and pr e c i p i t a t i o n of the metals from solution. Several mechanisms are available i n sedimentary basins for the reduction of sulfate to reduced sulfide (H^S or HS ) i n solution, as l i s t e d i n Table 13. Organic material i s abundant i n many carbonate-hosted lead-zinc deposit environments,.and i t s destruction through bacterial-reduction processes i s a primary source of H' S at temperatures less than 80°C. Table 14 outlines the composition of a hypothetical;2.0 m (molal) o NaCl brine at a temperature of 75 C. The concentration values are chosen to be representative of a deep, basinal brine i n equilibrium with carbonate aquifers. Comparison can be made with present-day o i l - f i e l d brines and f l u i d - i n c l u s i o n analyses for data of t h i s type. Numerous compilations have been published that l i s t brine compositions, including White (1968), Hitchon, B i l l i n g s and Klovan (1971), Carpenter, Trout, and Pickett (1974), and Land and Prezbindowski (1981). The actual range of compositions that could be modeled i s wide, but Table 14 i s suitable for t h i s study. Perhaps the most sensitive parameter i n Table 14, and the poorest known, i s the pH. The value chosen i s c r i t i c a l to mineral and aqueous species s t a b i l i t i e s , yet the natural range of pH i n brines can be wide. Anderson (1973, 1975) demonstrated that the minimum pH of an ore-forming f l u i d i s about 5.7 at 100°C, assuming a p a r t i a l pressure of CC^ of 1 bar and e q u i l i -brium with c a l c i t e and dolomite.. Sverjensky (.1981) has estimated, however, that the pH of basinal brines can be as low as 4.3, and s t i l l be within two log units of n e u t r a l i t y . At temperatures of 50°C and 100°C, n e u t r a l i t y occurs at pH = 6.63 and 6.12, respectively. He assumed a maximum concentration of dissolved CC^ '[as H2CC>3 and C0 2(aq)] of 0.05 molal, and reported that McLimans (1977) measured 0.02 to 0.14 m CO^ i n f l u i d inclusions from ores i n the Upper Mis s i s s i p p i Valley d i s t r i c t (see Figure 1 for location). Hitchon (198.1) has studied modern o i l - f i e l d brines i n Alberta, and has found p a r t i a l pressures of CO^ as high as 6.5 bars. In t h i s study, a maximum dissolved CO^ (as H^CO^) of 0.04 m i s assumed, and pH i s allowed to range between 5.0 and 6.8. There appears to be l i t t l e doubt from experimental data and f i e l d Table 14. I n i t i a l conditions f o r the metal-sulfate brine model T = 75°C pH =5.0 f02 = 1 0 " 5 0 fC02 = 5 b a r S (Eh = + 0.014 v o l t s ) Component Concentration Totals (moles/kg) (mg/kg) Na 1.46 30,030 K 5.0 x 10" 2 1,746 XCa 2.4 x 1 0 _ 1 8,710 2Mg 9.2 x 10~ 3 200 SiO (aq) 2.0 x 10-i* 5 CI 2.0 ' 63,330 SO^ 5.0 x 10" 3 145 3H 2C0 3(app) 4.2 x I O - 2 2,160 Fe 5.0 x 10" 9 2.5 x 10-1* 4Zn Pb 1.0 x I O - 4 1.0 x 10" 6 6.0-( 0.2 C a l c i t e saturation 2 Dolomite saturation 3 . H 2C0 3(app) = C0 2(aq) + H^COg(aq) 4 . Total zinc content = Zn + Z n C l + + ZnCl 2 +< ZnCl" + ZnCl 2' 228 observations that metal transport i n brines i s accomplished mainly through chloride complexing (Helgeson, 1964-). Fluid-inclusion studies by Roedder (1967) have also shown that the bulk of ore-forming brines, i n carbonate-hosted lead-zinc ore environments, were sodium-calcium-chloride f l u i d s , the reduced-sulfur content was low, and the s a l i n i t y ranged between 1.0 m and 5.0 NaCl (10% to 30% NaCl by weight), but averaged around.2.0 m NaCl. Other types of complexing, such as b i s u l f i d e and organic, may have been important i n certain systems (Giordano and Barnes, 1981). For the purpose of t h i s study, only chloride complexing i s considered. The s o l u b i l i t y of galena or sphalerite i n brines i s a function of the reduced sulfur content, s a l i n i t y , and temperature. I f reduced sulfur (H^S or HS ) i s low, metal transport i s easily accomplished. In fresh water, s o l u b i l i t i e s are less than 1 ppm (parts per m i l l i o n ) , but chloride complexing can allow concentrations up to thousands of parts per m i l l i o n (Anderson, 1978). Factors c o n t r o l l i n g the s o l u b i l i t i e s of sphalerite and galena have been studied with both experimental and the o r e t i c a l approaches (see Barnes, 1979). We w i l l b r i e f l y review some of these factors below. Figure 69 i l l u s t r a t e s the s t a b i l i t y f i e l d s of the sulfur species as a function of the solution pH and the oxidation state, which i s represented by the oxygen gas fugacity. The temperature i s 100°C, and the brine has a s a l i n i t y of 3.0 m NaCl. Total sulfur content of sulfate and su l f i d e i s set at 10 2 m, which i s thought to be representative of p r e c i p i t a t i o n conditions i n the M i s s i s s i p p i Valley-type deposits (Anderson, 1973). Also plotted on the graph are the 10 5 m concentration curves of t o t a l zinc and lead i n equilibrium with sphalerite and galena. Reference to these curves are made la t e r i n the chapter. At a pH of 5.0 to 6.0, the dominant sulfur species i s either SO^2 Figure 69. S t a b i l i t y f i e l d s of sulfur species as a function of pH and oxidation state f (after Anderson, 1978) A 'to 230 or H^SCaq),.depending on the reducing conditions (Figure 69) and temperature Helgeson (1969) calculated the.concentration r a t i o of sulfate to su l f i d e as a function of oxygen fugacity, temperature, and pH. Figure 70 charts t h i s relationship for a fixed pH = 5.0. Sulfate-bearing brines tend to occur O -cc at temperatures less than 100 C, at least for f ^ > 10 3 . Higher temperatures or more reducing conditions w i l l generally favor sulfide-bearing brines. For example, approximately ten times more su l f i d e exists at 150°C than at 100°C. The temperature regime of a sedimentary basin, therefore, w i l l place strong constraints on the type of sulfate species involved i n metal transport. The effects of reduced sulfur concentration ori the concentrations of lead and zinc are shown i n Figure 71. The temperature i s 100°C and the brine i s a 3.0 m NaCl solution. Three sets of curves are plotted for each metal at pH =.4.0, 5.0, and 6.0. Notice that metal concentrations can easily exceed 10 5m(2 ppm Pb, 1 ppm Zn) i n brines where reduced sulfur i s less than 10 5 m (0.3 ppm H^S). Decreasing the pH from 5.0 to 4.0 has a strong influence on metal concentrations, causing an increase by a factor of ten i n stoichiometric solutions. S a l i n i t y also exerts a s i g n i f i c a n t control on the s o l u b i l i t y of metal su l f i d e s . Figure 72 demonstrates the change i n zinc concentration due to a r i s e i n s a l i n i t y from 1.0 m to 3.0 m NaCl (5 - 15% wt. equiv. NaCl). Curves for pH = 4.0 and 6.0 are plotted, and. temperature i s 100°C. The 3.0 m NaCl curves are from Anderson (1973, Figure 2), and the 1.0 m NaCl curves are from Barnes (1979, Figure 8.10). At a pH 3 4.0, the change i n s a l i n i t y from 1.0 m to. 3.0 m NaCl causes the concentration of zinc i n equilibrium with sphalerite to r i s e from about 2 ppm to 15 ppm (Figure 72). The metal content of brines i n t h i s range of s a l i n i t y i s almost e n t i r e l y due to chloride complexes. The o — o — O O species PbCl^ and.ZnCl u dominate m the temperature range of 50 C to 150 C Figure 70. Sulfate-sulfide concentration r e l a t i o n ship as a function of oxygen fugacity and temperature (after Helgeson, 1 9 6 9 ) 232 CONCENTRATION H 2 S + H S _ (LOG MOLALITY) (a) -10 - 8 - 6 -4 - 2 0 CONCENTRATION H 2 S + H S " (LOG MOLALITY) (b) Figure 71. Lead and zinc concentrations i n a 3.0 m NaCl brine as a function of s u l f i d e concentration and pH • ( a f t e r Anderson, 1973). 233 1 Figure 72. E f f e c t of s a l i n i t y on zinc concentration ( a f t e r Anderson, 1973 and Barnes, 1979). 234 (Helgeson, 1969, Figure 15). Higher temperatures enhance the effect of the chloride complexes on metal s o l u b i l i t y , as shown i n Figure 73, for a, 3.0 m NaCl solution at pH = 5.0. Total zinc concentration i n equilibrium with sphalerite i s expressed i n ppm (mg/kg • H20) and plotted as a function of temperature (°C). At 50°C the sphalerite s o l u b i l i t y i s about 0.2 ppm, but t h i s increases to 5.5 ppm at 150°C. This data i s roughly estimated from the o r i g i n a l diagram given by Anderson (1973) by dividing the numbers at pH = 4.0 by a factor of ten (Anderson, 1973). The minimum concentration of Pb or Zn needed to form a major ore deposit i s uncertain, but a range, of 10 5 m to 10 m i s commonly stated i n the l i t e r a t u r e (Anderson, 1973; Barnes, 1979). Natural concentrations of base metals i n brines vary from less than one to several-hundred mg/kg • H^ O (Anderson and Macqueen, 1982). Chemical analyses of f l u i d inclusions indicate a concentration range of 10 5 m to 10 2 m f o r ores of the M i s s i s s i p p i Valley region (Sverjensky, 1981). Although metal concentrations do not commonly exceed a few mg/kg • H^ O i n present-day basinal brines, Anderson (1978) believes these brines are well q u a l i f i e d as examples of the ore-forming f l u i d s of the past. The sulfate content of subsurface brines i s variable, but concentra-tions of at least a few hundred ppm are common (e.g. Hitchon, 1977). Total dissolved sulfur contents of f l u i d inclusions range i n t h e i r maximum values from 1000 ppm (Roedder, 1976) to 4000 ppm (McLimans, 1977). Most brines appear to contain more than enough sulfur as sulfate (10 2 m) to precipitate s u l f i d e minerals, provided some form of mechanism i s available to reduce the sulfate at the ore-forming s i t e . Hydrogen sul f i d e gas i s a common constituent of many o i l - f i e l d brines, but unfortunately the concentration range of H ^ S C a q ) i s not well documented. Reasonable concentration values probably range between 10 4 and 10 2 m, 235 40 60 80 100 120 140 160 TEMPERATURE (°C) Figure 73. E f f e c t of temperature^on sphalerite s o l u b i l i t y (modified a f t e r Anderson, 1973). 236 according to Anderson (.1977).,. F i e l d data reported by Giordano and Barnes (1981) places H^S concentrations i n the range from 10. 5 m to 10 2 m f o r o i l -f i e l d brines i n M i s s i s s i p p i . Hitchon (.19 81) reports p a r t i a l pressures of H^S as high as 15 bars or about 0.6 m H^S i n natural gas re s e r v o i r s i n Alberta. The problem with known sulfide-type brines, with respect, to ore genesis, i s that those containing even a small amount of H^S are usually very low i n metal content (Anderson and Macqueen, 1982). It i s evident from the above discussion that a wide v a r i e t y of geo-chemical conditions w i l l allow a metal-sulfate brine to carry s u f f i c i e n t lead or zinc to form a major ore deposit, provided some sort of sulfate-reduction process occurs. Three reaction-path simulations are now presented to demon-st r a t e some of the p o s s i b i l i t i e s . The f i r s t experiment to.be.considered involves the addition of H^S to the model brine of Table 14. It i s assumed that s u l f a t e i n the f l u i d has been reduced at the ore s i t e to H^S. The actual process of s u l f a t e reduction i s not simulated, only that a source of H^S i s suddenly a v a i l a b l e . Sulfate reduction can be accomplished through the anaerobic-bacterial decay of organic matter, or the i n t e r n a l reduction due to the presence of methane (Table 13). The amounts of lead and zinc chosen here are 10 6 m Pb and 10 - t + m Zn. These numbers are not meant to simulate a metal r a t i o from any p a r t i c u l a r known ore environment, but are simply assumed as reasonable concentrations, based on the s o l u b i l i t y data given above. Figure 74 shows the r e s u l t s of a reaction-path simulation with. EQ3/EQ6 where 10~ 3 moles/kg • (34 mg/kg) of H 2S(g) are dissolved into the metal-sulfate brine. Only a selected group of aqueous species are plo t t e d i n the speciation diagram. The brine i s assumed to be i n i t i a l l y i n equilibrium with dolomite and limestoneat a temperature of 75°C. As H 0S di s s o l v e s , the 237 Figure 74. EQ3/EQ6 simulation of the reaction of a metal-s u l f a t e brine with H S. 238 aqueous sul f i d e species gradually become more concentrated. Dolomite i s the f i r s t mineral phase to p r e c i p i t a t e , but l a t e r begins to dissolve with further addition of H^ S and p r e c i p i t a t i o n of sphalerite. The pH remains constant throughout the reaction process, while the oxygen fugacity drops from 10 5 0 to 10 5 7 . P r e c i p i t a t i o n of the sulfide minerals i s also reflected i n the change i n concentrations of the free-metal species and metal-chloride. Most of the metal content of the f l u i d i s exhausted by the point at which 10 moles of H- S have been dissolved. Galena i s the next mineral to precipitate after sphalerite, and pyrite i s deposited l a s t . Approximately 10 ^ moles (10 mg) of sphalerite, 10 6 moles (0.3 mg) of galena, and 10 8 moles (0.001 mg) of pyrite are precipitated per.kg of H^ O. These figures can be compared to the simulation example given i n the previous chapter (Figure 16), i n which the brine i s at a pH = 6.8 and the zinc concentration.is an.order of magnitude lower. Reacting a basic type of sulfate brine with H^ S produced about twenty times more dolomite dissolution, than the example of Figure 74. The addition of other strong reducing agents, such as methane or hydrogen gas, to a metal-bearing brine can also cause sul f i d e mineral p r e c i t a -t i o n . Methane i s a common component of basinal brines, and f l u i d inclusion data indicate that substantial amounts of methane occurred i n ore-forming f l u i d s , even as high as 800 mg/kg • H^ O (Roedder, 1976). Figure 75 shows the effects when the brine of Table 14 encounters methane along i t s flow path. In t h i s example, 10 3 moles (16 mg). per kg • H^ O i s permitted to react with the metal-bearing f l u i d . Addition of 10 7 moles of methane causes an abrupt drop i n the oxidation state of the f l u i d , as marked by the pe curve (Figure 75). The s h i f t to a more reducing environment changes the sulfate to s u l f i d e r a t i o (Figure 70), as noticed i n the sudden increase i n H S(aq) and HS concentrations. Toward the end of sphalerite p r e c i p i t a t i o n , dolomite 239 - 1 0 - 9 - 8 - 7 - 6 - 5 -4 - 3 - 2 r l 0 CH 4(g) DISSOLVED (LOG MOLES)' Figure 75. EQ3/EQ6 simulation of the reaction of a metal-s u l f a t e brine with CH,-.. 4 240 that was precipitated e a r l i e r begins to dissolve. As the zinc concentrations drop o f f , the addition of more methane causes galena to p r e c i p i t a t e , and then p y r i t e . Sulfide p r e c i p i t a t i o n ceases after 10 ^ moles of CH^ has dissolved, but further addition of methane causes the oxidation state to drop further and large amounts of c a l c i t e and dolomite to precipitate. Late-stage carbon-ate mineralization i s a common feature of many lead-zinc ore deposits. In t h i s p a r t i c u l a r model, the same amounts of sulfide minerals precipitated as i n the H S model, but about 75 mg of c a l c i t e and 10 mg of dolomite are deposited per kg • H^ O. Dissolved methane may also reduce sulfate i n brines i n t e r n a l l y , but the rate of reaction i s thought to be too slow for ore-genesis (Barton, 1967). On the other hand, t h i s feature could allow CH^ to be carried metastably i n solution f o r long distances, perhaps to an ore-forming s i t e where bacteria could enhance the reaction rate (Skinner, 1979). Another p o s s i b i l i t y of reducing a sulfate-bearing brine i s through replacement-type reactions with pre-existing sulfide minerals. Figure 76 considers the case where the model brine of Table 13, containing 10 ^ m Zn and 10 5 m Pb, passes through a pyrite-bearing carbonate bed. The amount of pyrite (FeS2) available for reaction i s 10 3 moles (120 mg). Compared to the preceding models, the reaction with pyrite appears to be a less e f f i c i e n t mechanism for ore p r e c i p i t a t i o n at these metal concentrations. Only 10 6 moles (0.1 mg) of sphalerite i s precipitated i n t h i s model before the pyrite comes to equilibrium with the brine. Subsequent simulations showed that greater amounts of ore w i l l precipitate i f the i n i t i a l zinc and lead concen-trations are higher than those of Table 14. For example, using a zinc concentration of 10 3 moles/kg • H^ O resulted i n the p r e c i p i t a t i o n of 38 mg of sphalerite per kg of water. Approximately 2 x 10 4 moles (27 mg) of 24-1 co LU —I O -A-\ CD Q CO CO < < or LU LU > - 6 H - 7 • - 8 H -9-T = 75°C 2.0 m NaCl DOLOMITE / Pyrife f ^ ^ - ^ Saturates SPHALERITE -10 - 8 - 7 - 6 - 5 - 4 - 3 - 2 PYRITE, F e S 2 DISSOLVED (LOG MOLES) , -i Figure 76. EQ3/EQ6 simulation of the reaction of a metal-s u l f a t e brine with p y r i t e . 242 p y r i t e i s destroyed before pyrite saturates with respect to the brine. A l l three of the reducing models given above have shown the v i a b i l i t y of a metal-sulfate brine to produce ore mineralization when H S i s added to the f l u i d or the oxidation state i s modified. Geologic evidence as to the r e l a t i v e importance of the various sulfate-reduction mechanisms has not been cl e a r l y established. I t i s known that bacteria-assisted reduction i s unlikely i n ore environments where temperatures are above 80°C; however, the inorganic mechanism suggested by Orr (1974) i s apparently adequate for systems of 80°C to 120°C in organic-rich strata. Reactions with methane or pyrite are equally reasonable from geologic evidence. Mineral.:textures commonly show evidence to support the geochemical model of H2S addition to brines. At Pine Point, for example, sulfide minerals demonstrate c l e a r l y that r e l a t i v e l y rapid p r e c i p i t a t i o n took place, which supports H^ S addition (Beales, 1975). Anderson and Macqueen (1982) point out, however, that slow addition of H^ S to a brine may as easily produce mineral textures that usually are thought to be characteristic of deposition from a metal-bearing sulfide-type f l u i d . Metal-Sulfide Brine Models We have seen that sulfate-reduction mechanisms allow a wide variety of geochemical conditions for the brine carrying the metal to a depositional s i t e . Lead and zinc concentrations can easily exceed 10 4 moles/kg • H^ O, even at pH values greater than 5.0, so long as reduced sulfur a c t i v i t i e s are low (Figure 71). In a brine transporting both metal and aqueous s u l f i d e , the geochemical conditions are much more r e s t r i c t i v e , especially i n regard to the amount of metal that can be carried i n solution with an equal amount of s u l f i d e . Figures 71 and 72 show that about 10 5 m i s the maximum concentration of metal sulfide that can be carried i n the same solution, at a pH of 5.0 and a 243 o temperature of 100 .C. S l i g h t l y higher concentrations can be achieved i n more aci d i c brines, depending on the s a l i n i t y (.Figure 72). Two simulation r e s u l t s are now presented to demonstrate the r e l a t i v e importance of two-different pre c i p i t a t i o n mechanisms. In the f i r s t simulation, a metal-sulfide brine i s saturated with respect to sphalerite and galena. Pre-c i p i t a t i o n of sulfide minerals occurs when the brine encounters a dolomite bed. In the second simulation, a drop i n temperature causes sulfide p r e c i p i t a t i o n from a brine o r i g i n a l l y i n equilibrium with sphalerite, galena, and quartz. Consider the model-brine composition of Table 15. The solution i s i n equilibrium with quartz, sphalerite, and galena at a temperature of 100°C. Besides the higher temperature and more reducing conditions, the f l u i d i s otherwise of sim i l a r composition to Table 14. In t h i s p a r t i c u l a r model, the sphalerite s o l u b i l i t y gives a zinc concentration of 6.0 x 10 6 moles/kg • H^ O in a 2.0 m NaCl solution. An equal amount of reduced sulfur i s also present. The f l u i d i s undersaturated with respect to c a l c i t e and dolomite. Figure 77 shows the results of an. EQ3/EQ6 simulation i n which the metal-sulfide brine encounters a dolomite bed. About 1.0 mole (185 g) of dolomite i s allowed to react with the fl u i d . . An increase i n pH from 5.0 to 5.3 causes.the brine, to become saturated with respect to sphalerite f i r s t , and then l a t e r with galena. This i s accompanied by an increase i n concentra-t i o n of the carbonate-type aqueous species, and a decrease i n metal and su l f i d e concentrations. The solution eventually becomes saturated with respect to dolomite, after about 6 x 10 4 moles (115 mg) have dissolved. The t o t a l quantity of sphalerite precipitated i s 3.1 x 10 5 moles (0.3 mg), and the quantity of galena precipitated is.4.6 x 10 8 moles (.0.01 mg), a l l per kg • H20. Varying the s a l i n i t y or the temperature of the brine w i l l a l t e r ^ thef Table 15. I n i t i a l conditions f o r the metal-sulfide brine model 100°C pH = 5.0 \ - 10"55 fco2 = 5 b a r s (Eh = - 0.168 vo l t s ) Component Concentration Totals (moles/kg) (mg/kg) Na 1.85 38,000 K 5.0 x 10." 2 1,745 Ca 5.0 x 10~ 2 1,790 Mg 1.0 x 10~3 20 Si0 2(aq) 8.0 x 10~K 20 CI 2.0 63,330 1 SO^ 1.7 x 1 0 " 1 0 1.5 x 10 H 2C0 3(app) 2.8 x 10- 2 1,585 Fe 5.0 x 10~9 2.5 x 1 0 _ 8 2 Zn 6.0 x 10~ 6 0.36 3 Pb 1.0 x 10"7 0.02 t S ( r ) 6.0 x 10"6 0.20 Quartz saturation 2 Sphalerite saturation 3 Galena saturation 4. Total reduced s u l f u r S ( r ) = H 2s(aq) + HS~ 245 Figure 77. EQ3/EQ6 simulation of the reaction of a metal-s u l f ide brine with dolomite. 246 the s o l u b i l i t y of sphalerite and galena, and therefore the amount of ore pre c i p i t a t i o n due to the pH s h i f t . Table 16 l i s t s the results of s i x EQ3/EQ6 simulations i n which the NaCl concentration was varied. As expected, maximum deposition occurs from the most saline case. The effect of temperature on the amount of sphalerite formed i s shown i n Table 17. Increasing the temperature from 100°C to 150°C creates a tenfold increase i n the amount of metal depositied, for a pH s h i f t from.-5.0 to 5.3. I t i s clear that high temperatures and s a l i n i t i e s enable the metal-sulfide brine model to produce greater amounts of ore for a given pH change. The second reaction mechanism to be tested i s the effect of cooling. Figure 78 charts.the mass-transfer results in. which the model brine of Table 15 i s cooled from 100°C to 60°C. The solution.is i n i t i a l l y saturated with respect to.quartz, sphalerite, and galena. Over the 40°C temperature drop, about 5.0 x 10~ 6 moles (0.46 mg) of sphalerite, 9.0 x 10~ 8 moles (0.021 mg) of galena, and 5.0 x 10 4 moles (29.4 mg) of quartz are precipitated per kg • H20. The temperature drop also caused the oxygen fugacity to vary from 10 5 5 to 10 6 5 bars, while pH i s s l i g h t l y modified from 5.0 to.4.9. Pre c i p i t a t i o n of the secondary minerals forces an accompanying decrease i n the concentration of most of the metal-chloride complexes, sulfide species, and aqueous s i l i c a . The change i n temperature and.oxidation state are very effective at s h i f t i n g the sulfate to sulfide r a t i o of the solution, as witnessed by the SO^2 - decrease (Figure 78). Within the f i r s t 10°C of cooling, about 35% of the f i n a l mineral masses have precipitated. It takes roughly 30°C drop i n temperature to achieve a recovery of at least 85% of the sulfide minerals. Further observations of t h i s type are l i s t e d i n Table 18. Whether or not a cooling of 40°C i s geologically reasonable i s debatable. To l o c a l i z e enough ore Table 16. Mass of sphalerite p r e c i p i t a t e d i n reaction with dolomite as a function of s a l t concentration Case * NaCl Concentration Mass Prec i p i t a t e d (molality) 1.0 3 x I O - 7 2.0 3 x I O - 6 C 3.0 D 4.0 (moles/kg) (mg/kg) 0.03 0.30 4 x I O - 6 0.35 5 x 10" 6 0.50 8 x 10" 6 0.75 E 5.0 F 6.0 1 x 10" 5 1.10 (- - T = 100°C, pH = 5.0 Table 17. Mass of sphalerite p r e c i p i t a t e d in reaction with dolomite as a function of temperature Case " Temperature Mass Pre c i p i t a t e d (°C) (moles/kg) (mg/kg) A 25 3 x 10" 7 0.03 B 50 6 x I O - 7 0.06 C 75 8 x 10" 7 0.07 D 100 3 x 10~ 6 0.30 E 125 1 x 10" 5 1.00 F 150 3 x I O - 5 3.00 (* - 2.0 tn NaCl, pH = 5.0) 248 100 95 90 85 80 75 70 65 60 TEMPERATURE ( ° C ) Figure 78. EQ3/EQ6 simulation showing the e f f e c t of cooling the metal-sulfide brine from 100°C to 60°C. Table 18. Relative masses of minerals precipitated i n cooling from 100°C to 60°C (.per kg HO) Temperature Drop Product Minerals * (from 100°C) Galena Sphalerite Quartz io uc 43% 37% 32% 20°C •71% 65% 59% 30°C 89% 85% 81% 40°C 100% 100% 100% U-''(O.V021t;mg) (0.45; rr.g) (29.4 mg) (* - Mineral masses at T = 60 C are defined as 100% reference points.) 250 through t h i s s i z e of temperature change would require extremely high geothermal gradients. Based on the heat-transport r e s u l t s , modest changes i n temperature of 10°C are more l i k e l y to occur. The r e s u l t s of Figure 78 indicate that even i f large changes i n temperature do occur, large flow rates and long periods of time are needed to form a major ore deposit by the cooling mechanism. Diff e r e n t geochemical conditions than assumed here w i l l change the effectiveness of pH s h i f t and cooling i n ore p r e c i p i t a t i o n . Anderson (1977, 1975) also evaluated the e f f e c t of pH s h i f t and cooling on the p r e c i p i t a t i o n of galena and sphalerite.- Figure 69 summarizes some of his r e s u l t s i n which the s o l u b i l i t y contours of 10 5 m of Pb, Zn, and reduced s u l f u r S, , are (r) superimposed over the s u l f u r speciation f i e l d . Anderson's papers demonstrated that both processes can deposit appreciable quantities of metal i f the f l u i d i s a c i d i c . As one example, a pH change from 4.0 to 4-. 3 can produce about 7 mg/kg F^O galena, while a temperature change from 100°C to 90°C can r e s u l t i n about 4 mg/kg H^ O galena. The range of conditions over which these mechanisms are e f f e c t i v e , however, are l i m i t e d i n the case of lead to brine with pH l e s s than 4.0 and f Q of 10~ 5° to 10~ 5 1 bars (Figure 69). Anderson suggests t h i s a c i d i t y l e v e l i s not common i n b a s i n a l brines, and therefore concluded that transport of both metal and s u l f i d e i n the same f l u i d i s not ge o l o g i c a l l y important i n the formation of M i s s i s s i p p i Valley-type deposits. Sverjensky (1981) has recently found a possible exception to Anderson's conclusion. Sulfur-and lead-isotope data from the Viburnum Trend of southeast Missouri support a transport model where s u l f u r i s c a r r i e d with metal to the s i t e of deposition (Sverjensky, Rye, and Doe, 1979). Various l i n e s of geologic and geochemical evidence l e d Sverjensky (1981) to conclude that s u l f a t e reduction.could not have been an important mechanism i n t h i s region. He re c a l c u l a t e d the s o l u b i l i t i e s of galena and sph a l e r i t e using new 251 thermodynamic data, and found that concentrations could be greater than those of Figure 71. As one example, Sverjensky computes s o l u b i l i t i e s of 6 x 10 6 m (.1.2 ppm) f o r lead and 4 x 10 5 m (2.6 ppm) f o r z i n c , at a pH = 5.0 and a temperature of 100°C.' This data suggests that the sulfide-type brine model may be an important factor i n stratabound. ore genesis. Large flow rates would s t i l l be required, however, i f changes i n pH and temperature are responsible f o r ore p r e c i p i t a t i o n . A Preliminary Application to the Pine Point Deposits Up to now we have seen the numerical r e s u l t s for a wide v a r i e t y of i d e a l i z e d geologic configurations-. These models can provide much insi g h t i n t o the f a c t o r s c o n t r o l l i n g f l u i d v e l o c i t i e s , subsurface temperatures, metal concentrations i n brines, and possible scenarios of s u l f i d e p r e c i p i t a t i o n at the ore-forming s i t e . The ultimate p r a c t i c a l use of the models, however, i s to assess ore-genesis theories of gravity-driven flow i n s p e c i f i c basins and ore d i s t r i c t s . The object of t h i s f i n a l section i s to demonstrate the s i g n i f i c a n c e of the modeling r e s u l t s with a quantitative analysis of a f i e l d example. For t h i s purpose, the Pine Point lead-zinc ore d i s t r i c t i s examined i n relation, to r e g i o n a l paleoflow systems i n the Western Canada sedimentary basin of Alberta and the Northwest T e r r i t o r i e s . The treatment i s r e l a t i v e l y b r i e f at t h i s stage. Emphasis i s f i r s t placed on understanding the possible r o l e of ancient transport processes on a regional scale, not the actual p r e d i c t i o n of l o c a l ore d i s t r i b u t i o n at the depositional s i t e . More s t r a t i g r a p h i c and hydrologic d e t a i l i s needed before a reasonable thorough assessment can emerge. The r e s u l t s given below are preliminary i n nature and should be regarded as such. The f i r s t step i n constructing a quantitative model of f l u i d flow i s 252 to i d e n t i f y the geometry and geologic configuration of the basin. The general geology of Pine Point and i t s s t r u c t u r a l s e t t i n g i n the Western Canada sedimentary basin was introduced i n Chapter 2 (see Figures 3 and 4). Figure 79 shows further d e t a i l s of the regional s e t t i n g of Pine Point, the Keg River b a r r i e r (Middle Devonian), and the accompanying s t r u c t u r a l features i n the Great Slave Lake region. Two cross-section l i n e s are marked on the map: section A-A' crosses the basin i n a northwest-southeast d i r e c t i o n that i s transverse to the b a r r i e r complex, and section B-B' crosses the basin i n a southwest-northeast direction, that i s p a r a l l e l to the b a r r i e r complex. The terms b a r r i e r complex or Keg River b a r r i e r are synonymous f o r the carbonate s t r a t a of Middle Devonian age that extend from the Keg River Formation to the Slave Point Formation i n c l u s i v e , as defined by Williams (1981). The stratigraphy along section A-A" i s shown i n Figure 80, and the stratigraphy along section B-B" is.shown i n Figure 81. L i t h o l o g i c a l f a c i e s are represented i n a s i m p l i f i e d form i n both sections, p a r t i c u l a r l y f o r the long p r o f i l e B-B". Detailed descriptions of these s t r a t i g r a p h i c units appear i n several p u b l i c a t i o n s , the most recent of which are Williams (1978, 1981). In general, the subsurface succession includes: a basal unit of Middle Devonian carbonates and evaporites, a t h i c k sequence of Upper Devonian shales, more shales and carbonates.of Mississippian age, and shales and sandstones of Cretaceous age which o v e r l i e the older s t r a t a unconformably. A gravity-driven f l u i d - f l o w model requires a topographic gradient to develop b a s i n a l flow systems. Geologic evidence suggests that the western margin of the Western Canada basin has been topographically higher than the eastern platform of the basin since the early Mesozoic. This was due to a gradual orogenic u p l i f t i n the C o r d i l l e r a , which culminated with the formation of the Rocky Mountains i n Eocene time (McCce.ossan and G l a i s t e r , 1964-). Toth 253 Figure 79. Map of the Great Slave Lake region showing the l o c a t i o n of Pine Point, the Keg River b a r r i e r , and the section l i n e s A-A- and B-B'. Geology i s a f t e r Law (1971). 20 —t— 32 40 Miles - t 64 Kilometers \+ + + + T + + + + + , + " + - » + + + + + + < V + + + + * - + + + V + + 4 + + H-yW Precambrian L E G E N D 1 Hay River Formation, shale 2 Horn River Formation, limestone & shale 3 Slave Point, Sulphur Point Formations, Limestone 4 Middle Devonian carbonate barrier complex 5 Pine Point Formation, limestone 6 Bituminous Member 7 Muskeg Formation, evaporites 6 Lonely Bay Formation, dolomite (Keg River Fm.) 9 Chinchaga Formation and older Devonian evaporites h S e a Level (datum) ^ 1 0 0 0 Figure 80. Cross section A-A", transverse to the b a r r i e r complex (a f t e r Macqueen et a l . , 1975). l\3 -P B (SW) + 1000 -SEA . L E V E L -1000—H - 2 0 0 0 -- BRITISH COLUMBIA • - ALBERTA- • NORTHWEST TERRITORIES -Pleurocen* Cometon (till) H , l l s B' (NE) Pine Point \— +1000 , SEA ' L E V E I --I000 -2000 Figure 81. Cross section B-B**, longi t u d i n a l to the b a r r i e r comples from Fort Nelson, B.C. to Pine Point, N.W.T. Cn 256 C1978) has studied paleoflcw systems i n a large area of northern Alberta, and he found that the e x i s t i n g flow patterns of the b a s i n a l brines owe t h e i r o r i g to Pliocene topography. Furthermore, T6th contends that much of northern Alberta became g e o l o g i c a l l y and topographically mature i n the period between Late Miss i s s i p p i a n and Early Cretaceous, a' period of about 175 m i l l i o n years. A well-developed, topography-controlled g r a v i t y flow system operated during t h i s time i n t e r v a l , with major f l u i d flow from southwest to northeast across the emerging basin (see Toth, 1978). The present-day existence of large flow systems from the Rocky Mountains to the Canadian Shield i s also documented i n other studies, including those of van Everdingen (1968) and Hitchon (1969a, 1969b). Based upon the discussion above, we can reasonably assume that the major d i r e c t i o n of f l u i d flow through the Keg River b a r r i e r was l i k e l y to the northeast ( p a r a l l e l to section B-B") f o r much of the Mesozoic and Cenozoic eras. According to Kyle (1980), there i s no geologic evidence to constrain the date of lead-zinc mineralization at Pine Point, except that i t i s post Devonian. The cross section i n Figure 81 can be used to form a hydrogeologic configuration f o r a two-dimensional representation of the basin. This w i l l provide a good approximation of the flow f i e l d because of the very long dimension of the Western Canada basin i n the northwest-southeast d i r e c t i o n ( Figure 3). Hitchon (1971) has mapped the present-day hydraulic-head d i s t r i b u t i o n i n the b a r r i e r complex, and he found that the flow pattern i s mostly l o n g i t u d i n a l to the b a r r i e r , with f l u i d flow updip toward the north-east and Pine Point. Figure 82 shows an example of one of the hydrogeologic simulations that have been performed along section•B-B'". In Figure 82a, the geologic configuration of Figure 81 i s s i m p l i f i e d to four major hydrostratigraphic HYDROSTRATIGRAPHY (EARLY CRETACEOUS) I 100 ~> i i " i — i — | — i — i — i — i — | -200 3Q0 400 Kilometers (a) L E G E N D 4 I Shale and Sandstone (Lower Cretaceous) 3 | Shale and Limestone (Mississippian) 2 | Shale (Upper Devonian) I Dolomite (Middle Devonian) 500 FINITE ELEMENT MESH ~i 1 1 r 200 300 Kilometers (b) 400 500 Figure 82. Pine Point model. ro cn < 1 Kilometers O — tn * _l I 1 1 Kilometers TJ H« Oq C 4 fD oo ro T) P-3 . CD T) O H* 3 H-3 O CD O O 3 rt H-3 CD < m r -O o 1 H • -< ro o < 3 m o «< — I O — ZD c n Kilometers 8S2 TEMPERATURES K, = 1000 m/^r K X / K H = 100 Kilometers ( h ) Figure 82. Pine Point model. (Continued) ro cn ID Cumulative Percentage 092 261 units. The basin i s 5Q0 km i n length, and varies in thickness along the wedge from 3000 m at x = 0 km to 10Q0 m at the groundwater divide at x = 500 km. A low topographic r e l i e f i s reflected i n a gentle water-table gradient of .. 2 m/km. The lower boundary represents the base of the deepest permeable formation, which i n t h i s case i s the Keg River barrier.complex (Figure 81). The hydraulic conductivity and porosity of evaporites are very low (Tables 7 and 8), therefore, the top of the Chinchaga Formation makes a convenient place to s i t e the impermeable base of the model. This impermeable boundary has a gentle slope, which r i s e s 2 m/km toward the northeast. In the east, a ground-water-flow divide i s a r b i t r a r i l y selected at a distance of 50 km east of the existing location of Pine Point (Figure 82a). The formations probably thinned to a feather edge somewhere beyond t h i s region. The western flow divide i s located i n an area to the east of Fort Nelson (Figure 79), near the edge of the present-day F o o t h i l l s of the Rocky Mountains. The stratigraphic section thickens considerably (^  6 km maximum) toward.the thrusted section of the basin, west of the modeled section. Figure 82a i s meant to depict the geologic setting of the basin i n the Early Cretaceous, about 125 m i l l i o n years ago. At t h i s period i n time the basin was geologically mature and gravity-flow systems were probably well developed, due to u p l i f t i n the Rocky Mountain region. The exact thickness of the rocks overlying the Pine Point s i t e at t h i s time are unknown, mainly because the l a t e r Paleozoic and Mesozoic strata are now absent i n the region. Williams (1979) estimates that a maximum of 750 m of Upper Devonian shale, 300 m of Carboniferous s t r a t a , and 0-1500 m of Jurassic-Cretaceous e l a s t i c s may have overlain the Pine Point area, but are now missing because of erosion. These observations are consistent with the b u r i a l history made by Deroo et a l . (.1977) for Devonian rocks i n Alberta. They, concluded that the 262 maximum depth of b u r i a l was probaly less than 1500 m on the eastern shelf of the Western Canada sedimentary basin. Based on coal-rank studies, they also estimated that approximately 1800 m of sediments were eroded from the basin, after the u p l i f t of the Rocky Mountains i n the Eocene. It i s probable, therefore, that the Pine Point area was covered by at least 1000 m of strata over the period from the Permian to the Tertiary. For the purpose of t h i s model (Figure 82a),'500.m of Upper Devonian shale (Unit 2) are assumed to be present at Pine Point, which are unconform-ably overlain by about 400 m of Jurassic-Cretaceous e l a s t i c s (Unit 4). Over 300 m of Mississippian strata are truncated further west i n the basin. Approximately 350 m of Middle Devonian carbonates (Keg River Fm. to Slave Point Fm. inclusive) are represented at the Pine Point s i t e . A l l four of the hydrostratigraphic units thicken downdip i n the basin. Table 19 l i s t s the set of material properties and model parameters used, i n the Pine Point simulation.. Porosity data are estimated from Table 8 and the values compiled by Hitchon (1968). Representative values of hydraulic conductivity are chosen from Table 7 and the re s u l t s from the preceding s e n s i t i v i t y simulations. Toth (1978) has analyzed the present-day permeability data for a large area of study about 300 km south of Pine Point. His data, which was compiled from d r i l l s t e m tests and core analyses, can also be used as a guide for selecting the possible range of hydraulic conductivities i n t h i s model. In terms of ore formation, the most c r i t i c a l permeability para-meter i s the value of hydraulic conductivity assumed for the barrier complex. A value of 1000 m/yr (about 3 darcy). i s chosen i n Table 19 for the p r i n c i p a l hydraulic conductivity i n the x-direction. I t i s a conservative number for a karst-type aquifer, but probably representative of an aquifer i n the deep subsurface. Vogwill (1976) has performed extensive aquifer tests on the Table 19. Parameter Pine Point Symbol Simulation : input parameters data Hydrostratigraphic Unit Units F l u i d Flow 1 2 3 4 Porosity 0.25 0.10 0.10 0.15 f r a c t i o n '•Horizontal hydraulic conductivity K XX 1000.00 20.00 50.00 25.00 • m/yr '•Vertical hydraulic conductivity K zz 10.00 0.20 0.50 0.25 m/yr Reference f l u i d density Po 998.2 ( a l l u nits) kg/m3 Reference f l u i d v i s c o s i t y PO 1.0 x 10" 3 ( a l l units) Pa • S '''Salinity gradient - 0.01 ( a l l units) %NaCl/m Heat Transport Thermal conductivity of f l u i d K f 0.63 ( a l l units) W/m • K Thermal conductivity of rock s o l i d s K s . 3.0 1.9 2.5 2.0 W/m • K Sp e c i f i c heat capacity of f l u i d C v f 4187.0 ( a l l units) J/kg • K Sp e c i f i c heat capacity of rock s o l i d s 4 C vs 1005.0 ( a l l units) J/kg • K Longitudinal thermal d i s p e r s i v i t y £ L 50.0 10.0 10.0 ' 10.0 m Transverse thermal d i s p e r s i v i t y £T 1.0 1.0 1.0 1.0 m ''Geothermal heat f l u x at base J 70.0 (constant) mWm2 •'Temperature at water table To 20.0 (constant) °C Mass Transport ''Longitudinal d i s p e r s i v i t y U L 50.0 10.0 10.0 ' 10.0 m Transverse d i s p e r s i v i t y °r • 50.0 1.0 1.0 1.0 m Apparent d i f f u s i o n c o e f f i c i e n t D d 3 xlO" 3 ( a l l units) m 2/yr (" - Parameters varied for d i f f e r e n t simulations«) K 3 CD CO 264 barrier-complex rocks at Pine Point, and based on his data, the present-day hydraulic conductivity reaches a maximum of about 5000 m/yr. A s a l i n i t y gradient of 0.010 % NaCl/m i s assumed.to have existed at t h i s stage i n geologic time. It produces a s a l i n i t y of about 10% NaCl (.2.0 molal) at the Pine Point s i t e and a maximum s a l i n i t y of 20% NaCl (.4.3 molal) at the western end of the basin. Increasing the proposed s a l i n i t y p r o f i l e w i l l decrease f l u i d v e l o c i t y (Figure .32. e), but for an increase to 15% NaCl (at the ore-forming s i t e ) the change w i l l be small. Several f i e l d studies support t h i s assumed s a l i n i t y range. For example, f l u i d - i n c l u s i o n studies by Roedder.(l968) indicate that brine s a l i n i t y at Pine Point was between 10% and .25% NaCl. Present-day s a l i n i t y variations In the Devonian strata of western Canada range from 100,000 to 200,000 mg/liter as total-dissolved solids (Hitchon and Horn, i n press). Formational f l u i d s i n contact with the Chinchaga evaporites commonly exceed 200,000 mg/liter t o t a l s o l i d s . Brine samples from the Keg River barrier have been analyzed by B i l l i n g s , Kesler, and Jackson (1969), at the point marked 'S' i n Figure 79. They found that t o t a l dissolved solids average around 140,000 mg/liter, which also supports the modeled s a l i n i t y gradient. A l l of the heat and mass-trasnport coef f i c i e n t s are chosen as representative values, based on the simulations presented e a r l i e r and the type o f . l i t h o l o g i e s involved. Thermal conductivities are obtained from Table 10, and the d i s p e r s i v i t y c o e f f i c i e n t s are based on the range given i n Table 11. The geothermal heat fl u x of 70 mW/m2 and assumed surface tempera-ture of 20°C are also defensible as geologically reasonable. The thermal conductivities and heat flow are comparable to present-day data reported by Majorowicz and Jessop (1981) for western Canada. The finite-element mesh for the Pine Point model i s shown i n 265 Figure 82b. I t i s of similar shape to the mesh used i n the s e n s i t i v i t y analysis (Figure 17), but now i t contains twice as many elements. This provides a greater degree of accuracy, which i s needed when progressing from simple basins to more r e a l i s t i c and complicated geologic configurations. Because so many elements are present, only the quadrilateral c e l l s , and not the triangular elements, are drawn. Also notice that the v e r t i c a l exaggeration of the sections i n Figure 82 are now 20:1, which i s double the exaggeration used i n the sensi-t i v i t y analysis. The hydraulic-head d i s t r i b u t i o n and resu l t i n g flow pattern are shown in Figures 82c, 82d, and 82e. Maximum fl u i d - f l o w v e l o c i t i e s of 20 m/yr occur i n the barrier-complex u n i t , near the downstream end of the basin. This k a r s t i c aquifer i s fi f t y - t i m e s more permeable than the overlying shales of Unit 2 (Fort Simpson Fm.), and because of the high contrast i t has the greatest influence on the basin flow systems.. F l u i d flow i s directed down across the less-permeable beds i n the recharge end of the basin and upward i n the d i s -charge end of the basin. Cross-formational flow could easily provide a mechanism for the transport, of metals into the Keg River barrier i n the re-charge part of the flow system. F l u i d flow i n the midsections of the basin i s e s s e n t i a l l y l a t e r a l and updip. Comparison.can be made with some of the simulations given i n the s e n s i t i v i t y analysis to conceptualize how the flow pattern i n Figure 82d would be different for other configurations of topo-graphy, basement structure, or hydraulic-conductivity structure. Let us digress from Figure 82 for a moment and examine Figure 83. The purpose of showing t h i s diagram i s to demonstrate the correlation between the flow systems generated by the numerical model and actual flow systems observed i n r e a l sedimentary basins. The upper diagram of Figure 83 shows the present day hydraulic-head d i s t r i b u t i o n along a west-east cross section 266 H Y D R A U L I C H E A D S WEST EAST I 1 z 0 o p - I LU -2H Lu - 3 - 1 O H e z o _i LU SEA LEVEL DATUM SEA LEVEL DATUM - 2 - » 50 km J 500— Hydraulic Head (meters) — _ D i r e c t i o n of Flow Vertical Exaggeration = 40:1 Figure 83. Hydraulic-head pattern i n north-central Alberta ( a f t e r Hitchon 1969a, 1974). 267 through north-central Alberta, as mapped by Hitchon (1969a). The lower diagram i s a more detailed and contour expansion of the flow regime, which i s modified from Hitchon (1974). The location and geology of t h i s cross section can be found i n Figures 3 and 4, where i t i s marked as section B-B"*. The zones of low hydraulic head coincide with major topographic lows and i n the highly permeable reef carbonates i n the Upper Devonian strata. The presence of t h i s high-permeability layer causes the focusing of f l u i d flow that i s shown i n Figure 83. Besides the influence of an irregular topography, many s i m i l a r i t i e s exist between the present-day hydraulic heads and the flow patterns from the th e o r e t i c a l model of Figure 82d. One point of special interest i s the increase i n hydraulic head with depth i n the recharge end of the basin (Figure 82c). This feature was shown e a r l i e r to be caused by the s a l i n i t y effect on f l u i d density (Figure 27). Future studies of hydrodynamics of r e a l sedimentary basins w i l l have to consider t h i s phenomena i n order not to misinterpret the true f l u i d - f l o w directions.; The simulated temperature pattern for the Pine Point basin model i s given i n Figure 82f. It shows that temperatures of at least 80°C are possible, at the depth of the assumed Pine Point deposit (see Figure 82a). Remember from the s e n s i t i v i t y models that t h i s maximum temperature value i s highly dependent on the prescribed boundary conditions, among other factors. A higher geothermal f l u x or higher surface temperature w i l l y i e l d even greater subsurface temperatures than the 80°C - 100°C range shown here. Fl u i d flow causes the geothermal gradient to be depressed where the Keg River barrier i s recharged, and the gradient i s elevated i n the discharge end of the flow system. Figures 82g and 82h i l l u s t r a t e the temperature changes that are possible i f the hydraulic conductivity of the formations are modified. In Figure 82g, the anisotropy r a t i o of horizontal to v e r t i c a l hydraulic conducti-268 v i t y i s increased to 4-00 : .1. This allows less v e r t i c a l flow of water, which i n turn allows a smaller, amount of v e r t i c a l thermal convection, r e l a t i v e to the 100 : 1 r a t i o of Figure 82f. The r e s u l t s i s a l e s s disturbed temperature f i e l d i n the recharge and discharge ends of the basin. A second example of the e f f e c t of convection i s shown i n Figure 82h. In t h i s simulation,, the hydraulic conductivity of the b a r r i e r complex i s increased to 5000 m/yr, which causes a f i v e - f o l d increase i n the v e l o c i t i e s of Figure 82d. Convective heat t r a n s f e r i s now quite large, the r e s u l t being that temperatures are s u b s t a n t i a l l y lowered at the proposed depositional s i t e to about 60°C. On the other hand, i f flow rates are too low, thermal con-vection w i l l not be s u f f i c i e n t to carry enough heat from the deeper parts of the basin up into the thinner s h e l f . Nor would the flow volumes be adequate to form a major ore deposit, over a reasonable period of time. Figure 84 presents the r e l a t i o n s h i p between the h o r i z o n t a l hydraulic conductivity of the barrier-complex aquifer and the r e s u l t i n g average l i n e a r v e l o c i t y and temperature. The curves are p l o t t e d f o r the reference s i t e of Pine Point (Figure 82a), which i s at a depth of about 1200 m below the water-t a b l e . The heat flow on the basal boundary i s held at a constant 70 mW/m2, and the temperature at the water.table i s 20°C. It i s c l e a r that a s a c r i f i c e i n f l u i d v e l o c i t y i s required to maintain a warm temperature, or vic e versa. V e l o c i t i e s up to 25 m/yr are possible i n t h i s model, with accompanying temperatures above 70°C. F l u i d - i n c l u s i o n work by Roedder (1968) and Kyle. (.1977) indicates that the ore-forming, f l u i d s at Pine Point ranged i n temperature from 51°C to 99°C. The thermal, models given above are therefore consistent with the f i e l d data. Geothermal. gradients i n these simulations (.Figure 82f, g, and h) are as low as 20°C/km i n the recharge end of the basin, and generally increase to about HYDRAULIC CONDUCTIVITY OF BARRIER-COMPLEX (m/yr) Figure 84. F l u i d v e l o c i t y and temperature at the Pine Point s i t e as a function of the hor i z o n t a l hydraulic conductivity of the b a r r i e r complex. ^ 270 60°C/km i n the downstream section of the basin. Very high gradients of 80°C/km or more are possible near f l u i d - f l o w d i v i d e s . Present geothermal gradients i n the Western Canada, sedimentary basin range from les s than 25°C/km to over 50°C/km, and heat flow varies between 50 mW/m2 and 100 mW/m2 (Majorowicz and Jessop, 1981). Figure 85 i s a map of c e n t r a l Alberta from Deroo et a l . (1977) that shows a trend of increasing geothermal gradient across the basin, from le s s than 30°C/km i n the F o o t h i l l s region of the Rocky Mountains to over 50°C/km near the mid-section of the basin. This pattern i s probably the r e s u l t of downward f l u i d flow into the basal carbonate s t r a t a , and northeasterly movement of f l u i d s updip, as i n Figure 83. Several closed-contour highs and lows are also present on the geo-thermal gradient map, and they appear to be r e l a t e d to topography-induced convection between major r i v e r v a l l e y s , much as i n the manner depicted i n Figure 58d. Present-day temperature data i n the barrier-complex i t s e l f are scarce. Majorowicz and Jessop (1981) report very high values of heat flow (100 mW/m2) in the northeast corner of B r i t i s h Columbia (see Figure 79), and they also report temperatures i n excess of 90°C f o r basement rocks below the b a r r i e r complex. It i s safe to conclude from the discussion above that e x i s t i n g geothermal data in western Canada support the t h e o r e t i c a l heat-flow i n t e r p r e t a t i o n s . Figures 82i - 82£ show the r e s u l t s of a mass-transport simulation f o r the Pine Point model. The purpose of t h i s simulation i s to document the influence of dispersion on d i l u t i n g the metal concentration of a brine flowing through source beds i n the western side of the basin to the eastern-platform edge. The f l u i d - f l o w pattern i s that of Figure 82d. Contours representing 5%, 10%, and 15% of the i n i t i a l concentration Co have been p l o t t e d . Also p l o t t e d above the plume i n each time diagram i s the approximate maximum •'.o;.;.e.;..'...t 271 Figure 85. Geothermal gradient i n c e n t r a l Alberta showing the e f f e c t s of regional f l u i d flow ( a f t e r Deroo et a l . , 1977). 272 concentration predicted i n the Keg River, b a r r i e r . Areas with.relative concentrations greater than 15% C/Co are shaded. A l a t e r a l d i s p e r s i v i t y of 50 m i s assumed for the k a r s t i c aquifer (.Unit 1), and a value of 10 m i s assigned to the overlying strata. Transverse d i s p e r s i v i t y i s a constant 1 m in a l l four layers (Table 19). By time t = 30,000 y r , the mass has t r a v e l l e d about 300 km and only 15% of the source-bed concentration l e v e l i s present i n the basal aquifer. Mass i s also s t a r t i n g to be advected upward into the Devonian shales at t h i s stage. At t = 40,000 y r , the center of mass of the metal-bearing pulse has just passed the proposed Pine Point location. Based on t h i s simulation r e s u l t , i t appears that 5 - 10% of the metal concentration leaving the source beds deep i n the basin would reach the Pine Point s i t e . Nearly a l l of the mass i s discharged upward i n the basin beyond x = 400 km, as shown by the p a r t i c l e - d i s t r i b u t i o n histogram of Figure 82£. The l a s t bar i n t h i s histogram represents the f r a c t i o n of mass that passed through the entire length of the aquifer, while the l a t e r a r r i v i n g , bell-shaped section of the histogram represents the mass that i s discharged up into the shale beds r e l a t i v e l y early. We have seen that metal concentrations are dependent on the amount of dissolved s u l f i d e , s a l i n i t y , and temperature.. I f aqueous su l f i d e i s low, then substantial metal concentrations can e x i s t . To deposit ore, however, requires that either sulfide i s provided externally at the s i t e , or that accompanying sulfate i s reduced to sulfide at the s i t e . Small amounts of metal and sulfide can be carried together in equilibrium, but the pH must be s l i g h t l y acidic and the temperature should be greater than about 100°C. Unless both metal and sulfide are continually added to the f l u i d along the flow path, long-distance transport can be e f f e c t i v e l y ruled out for t h i s transport model because of the 273 d i l u t i n g effects caused by dispersion. I t was also shown that even i f d i s -persion i s small, the p r e c i p i t a t i o n mechanisms of pH s h i f t and cooling require that large flow rates occur over a long period of time to form a major ore deposit. The metal concentrations of the brines that passed through Pine Point and the p r e c i p i t a t i o n mechanism causing ore deposition are not known. Geologic evidence appears to indicate that the addition of H^ S to a metal-bearing brine at the s i t e caused ore deposition. Jackson and Beales (1967) proposed that destruction of abundant organic matter in the barrier-complex rocks by sulfate-reducing bacteria was a probable source of H^S. Bacterial degradation of organic matter can also generate large volumes of methane through anaerobic fermentation, which could have acted as an important reducing agent. I t i s also noted, however, that hydrogen sulfide i s more abundant when the organic matter i s r i c h i n s u l f u r , as i s often the case.with carbonate-evaporite sequences (T.issot and Welte, 1978). Abundant organic matter, heavy o i l , and bitumen occur within the bar r i e r complex at Pine Point (Macqueen and Powell, i n press). Hitchon (1977) and Macqueen et a l . (1975) have also shown that many of the shales, carbonates, and evaporites i n the Keg River ba r r i e r region are r e l a t i v e l y r i c h i n metals, p a r t i c u l a r l y zinc. Present-day brines i n the ba r r i e r complex are noted to be metal-rich. B i l l i n g s , Kesler, and Jackson (1969) report an average zinc con-centration of 19 mg/liter for brine samples taken downdip i n the b a r r i e r , about 350 km southwest of Pine Point (at point 'S' i n Figure 79). Hitchon (1981) reports an average zinc concentration of 1.4 mg/liter i n Alberta formation waters from Devonian s t r a t a , but also reports that up to 91 mg/liter have been found i n some locations. The l i k e l y range of metal concentrations i n ancient ore-forming brines i s debated. Ranges of 10 - 1000 mg/kg • Ho0 are 274 commonly defended (.Sverjensky, 1981), but maximum concentrations were probably l e s s than a few hundred mg/kg • H^ O (Anderson, 1978). Based on the present-day brine data, a maximum l e v e l of 50 - 100 mg/kg • H^ O seems reason-able f o r the Western Canada sedimentary basin. Macqueen and Powell ( i n press) conclude that the exact genetic r e l a t i o n -ship between lead-zinc mineralization and indigenous organic matter remains un-c l e a r at Pine Point. They also conclude, however, that i n - s i t u generation of H S by s u l f a t e reduction was probably the primary mechanism f o r ore p r e c i p i t a -t i o n . They suggest that sulfate-type brines could have reacted with small amounts of p r e - e x i s t i n g H^S i n the b a r r i e r to produce elemental s u l f u r , which i n turn reacted with organic matter to produce more H^S. This abiogenic reduction mechanism has an advantage over b a c t e r i a l reduction through i t s a b i l i t y to operate at temperatures above 80°C. Whatever type of s u l f a t e reduction took place, the continued supply of H S was apparently s u f f i c i e n t to deposit about 100 m i l l i o n tons of 9% lead-zinc ore (Kyle, 1980). Although the Pine Point model of Figure 82 i s preliminary i n nature, i t can be used to estimate the possible duration of mineralization required to account for the known ore reserves. We w i l l assume that the ore-forming f l u i d leached s u f f i c i e n t amounts of metal from the shale bed area deep i n the basin (Figure 821). As a f i r s t estimate, a zinc concentration of about 20 mg/kg '•• H^ O i s assumed to e x i s t i n the source beds. Figure 82k shows that only about 5 - 10% of t h i s i n i t i a l concentration or 1 - 2 mg/kg • H^ O would be present near the end of the flow system, because of dispersion along the flow path. This ore-forming concentration i s i n the range of minimum values suggested by Anderson (1975) and Barnes (.1979). The amount of metal p r e c i p i t a t e d per kg of brine w i l l of course depend on the l o c a l geochemical environment and how conditions v a r i e d through time. R e l a t i v e l y r a p i d deposition i s indicated at Pine Point (Anderson, 275 1978; and others), and therefore i t i s l i k e l y that almost a l l of the metal i n solution was deposited on encountering H^S, as i n Figure -74. The general paragenetic sequence of sphalerite followed by. galena i s observed, and both carbonate pr e c i p i t a t i o n and dissolution are common i n the mineralization process (.Kyle, 1980). Figure 84- can be used to estimate the possible flow rates involved i n ore deposition. To convert these average l i n e a r v e l o c i t i e s to volume-flow rates ( i . e . s p e c i f i c discharge), the vel o c i t y i s multiplied by the porosity value (0.25) of the barrier aquifer. Knowing the t o t a l cross-sectional area of flow, one can then compute the t o t a l volume of brine flowing through the bar r i e r per unit.time. The barrier complex i s at least 50 km wide (Figure 79) and approximately 300 m thick (Figures 80, 81). According to Kyle (1980, 1981), the most permeable sections at Pine Point occurred i h the dolomite ;.facies of the Pine Point Group, which i s over 10 km wide. The forty-some ore bodies are scattered over a stratigraphic i n t e r v a l of 200 m. Based' on these dimensions, a conservative estimate of the effective cross-sectional flow area would be 10 km wide by 100 m t h i c k , or 1.0 x 10 6 m2. For an average velo c i t y of 20 m/yr (Figure 84-), the temperature i s about 75°C and the s p e c i f i c discharge rate i s 5.0 m3/m2 • yr. Total discharge p a r a l l e l to the b a r r i e r , therefore i s about 5.0 x 10 6 m3/yr. I f only 1 mg Zn i s transported per kg of brine (density = 1050 kg/m3), then the zinc mass-flux rate i s about 5250 kg/yr (5.8 tons/yr). To deposit the t o t a l zinc content of 5 million.tons at Pine Point (Kyle, 1981) would require approximately 865,000 years. Figure 86 summarizes the relationship between s p e c i f i c discharge, zinc precipitated, and time for deposition of :5:..million tons i n the Pine Point model. The style of the graph i s based on a design f i r s t presented by Roedder (.I960). The boxed-in region represents what are believed to be the most representative 2 7 6 Figure 86. Relationship of s p e c i f i c discharge, zinc p r e c i p i t a t e d , and duration of mineralization f o r the Pine Point deposit. 277 range of variables that could have.led to the formation of Pine Point. With s p e c i f i c discharges of 0.5-7.0 m3/m2 • yr and zinc p r e c i p i t a t i o n of 1 - 100...mg/ kg • H^ O, the duration of mineralization would range from 10 5 to 10 7 yr. In the case of long-distance transport, zinc concentrations would not l i k e l y have exceeded 5 mg/kg • H^O. For t h i s range of zinc precipitated, the ore deposits at Pine Point would have formed in.0.5 to 5.0 m i l l i o n years, based on flow rates of 1.0 to 5.0 m3/m2 • yr. Comparable time spans have been estimated for other stratabound ore deposits (e.g. White, 1971; Sverjensky, 1981). The main conclusion to be drawn from t h i s preliminary analysis of Pine Point, however, i s that gravity-driven groundwater flow systems are quantitatively capable of forming large ore deposits i n sedimentary basins. 278 CHAPTER 6 SUMMARY AND CONCLUSIONS kn.a£y&<u> coYi^AJuni) what i>implz common, i>o,Ki>2, tmaheA o 5 , namzZy, tko, coHA&ctnteA o£ judgment* AJ> <U much. moh,z ph.oba.bto, o i the. judgej, axo, moh.o, numeAoui) and.mono. ZYiLLghtmid. Laplace, 1820. , The o r i g i n of.stratabound ore formation i s known to be dependent on the flow of f l u i d s through porous media. Carbonate-hosted lead-zinc deposits have received the largest amount of geologic research of any of the st r a t a -bound ores, and yet the fundamental role of flui d - f l o w systems has been poorly understood. On the other side, groundwater hydrologists have only recently become aware of the potential applications of modern fl u i d - f l o w theory and modeling techniques i n addressing geologic processes of t h i s type, A quantitative model that simulates the role of groundwater flow i n . the formation of stratabound ore deposits has been presented. The numerical modeling has been used to assess the role of gravity-driven flow systems. The various factors effecting f l u i d - f l o w patterns, subsurface temperatures, mass-transport, patterns , and ore pr e c i p i t a t i o n have been studied, and generic modeling of hypothetical sedimentary basins has been used to establish the r e l a t i v e importance of various model parameters and geologic environments. The findings of t h i s quantitative study are summarized below. Geologic Considerations 1. Lead-zinc deposits of the Mis s i s s i p p i Valley type belong to a group of low-temperature ores that are c l a s s i f i e d as stratabound and epigenetic i n 2 7 9 o r i g i n . Although they were deposited sometime after the sediments were l i t h i f i e d , stratabound ores are now considered to be genetically related to the normal evolution of a sedimentary basin. Only the carbonate-hosted lead-zinc ores are discussed i n t h i s study. 2. These deposits generally occur i n f l a t - l y i n g dolomite s t r a t a , which are commonly associated with porous, barrier-reef complexes and adjacent facies. On a regional scale, carbonate-hosted lead-zinc deposits are found near the margins of large sedimentary basins, and are presently at shallow depths. Because many of the deposits are hosted by Paleozoic s t r a t a , the configuration of the Precambrian basement can bear some influence on ore location. More important, however, i s the role of large, extensive aquifers. In southeast Missouri, an extensive basal sandstone probably served as a paleo-aquifer, while at Pine Point, N.W.T., the presence of a k a r s t i c b a r r i e r played a major role i n ore formation. Dolomitization and k a r s t i f i c a t i o n were also important processes i n the development of permeable conduits i n many ore d i s t r i c t s . 3. Mineralogy i s dominated by various combinations of galena, sphalerite, p y r i t e , b a r i t e , dolomite, c a l c i t e , and quartz. Secondary i n abundance are f l u o r i t e , c e l e s t i t e , chalcopyrite, native s u l f u r , and bitumen. Ore mineraliza-t i o n i s p r i n c i p a l l y due to pre c i p i t a t i o n of sulfides i n small, pore-space openings, but solution features t y p i c a l l y l o c a l i z e the ore into high-grade zones.. Fluid-inclusion studies provide s i g n i f i c a n t data concerning the nature of the mineralizing fl u i d s , , which were mainly Na-Ca-Cl brines. Temperatures of 50° - 150°C and s a l i n i t i e s of 10% - 30% NaCl are indicated by the f l u i d -inclusion studies. Furthermore, the geochemical character of these ore-forming f l u i d s i s comparable to present-day o i l - f i e l d brines. 4. Disagreement exists as to the o r i g i n of the driving mechanism for f l u i d flow. Although the concept of l a t e r a l secretion i s generally agreed upon, the 280 r e l a t i v e importance of basin compaction as compared to gravity-driven flow systems i s uncertain. It i s l i k e l y that compaction and gravity-based flow are both important as end members i n the hydrodynamic evolution of a basin. The computer models designed i n t h i s study are s p e c i f i c a l l y b u i l t to t e s t the gravity-driven flow mechanism i n a mature sedimentary basin. 5. The geochemical.environments of metal transport and ore p r e c i p i t a t i o n are well known, but widely debated. Geologic evidence i n some ore d i s t r i c t s support a geochemical model where both metal and s u l f i d e species are trans-ported i n the same so l u t i o n , and ore p r e c i p i t a t i o n i s caused by cooling or pH change. Other ore d i s t r i c t s indicate that only metal i s transported i n the brine, and accompanying s u l f a t e i s reduced at the ore s i t e to deposit the s u l f i d e minerals. It has also been suggested that the ore deposits could represent s i t e s at which two-different f l u i d s mixed. The implications that can be drawn from t h i s study with respect to these various theories of ore genesis are summarized l a t e r . The Numerical' Models 1. The transport of f l u i d , heat, and chemical components through porous media can be described mathematically, according to ph y s i c a l laws of f l u i d flow, heat and mass transport, and thermodynamics. The r e s u l t i n g set of governing equations can take on various l e v e l s of complexity depending on the nature of the problem and the degree of coupling between the processes being studied. 2. For the purposes of modeling f l u i d - f l o w , temperature, and metal trans-port and p r e c i p i t a t i o n , a number of assumptions and s i m p l i f i c a t i o n s must be made. Many of these s i m p l i f i c a t i o n s are necessary i n order to f i n d a p r a c t i c a l method f o r the numerical s o l u t i o n of the equations. This i s e s p e c i a l l y true when i t comes to modeling f l u i d flow i n chemically r e a c t i v e porous media. 2 8 1 3. The model developed here i s for a two-dimensional v e r t i c a l cross-section through a basin, i n which the configuration of the topography and geology are assumed to remain r e l a t i v e l y constant over a period of at least a few m i l l i o n years. The basin i s assumed to be f u l l y saturated and r i g i d , and f l u i d s a l i n i t y i s a prescribed function of depth. 4. The equations of f l u i d flow and heat transport are coupled through the f l u i d density and v i s c o s i t y , and are solved i t e r a t i v e l y . Both equations ' are represented i n t h e i r steady-estate forms,and are solved with the f i n i t e -element technique. The movement and concentration of a metal component i n the brine i s modeled with the moving-particle random-walk solution to the transient mass-transport equation. Multicomponent modeling with t h i s technique i s t h e o r e t i c a l l y possible, but the computer time needed for such simulations i s currently impractical. For these reasons, only a single component i s tracked through the basin i n any given simulation. A wide variety of basin shapes, topographic expressions, and geologic configurations can be modeled. Output i s i n the form of the contoured steady-state hydraulic-head f i e l d , the temperature f i e l d , and the velocity-vector f i e l d . Concentra-t i o n patterns are also contoured on the p r o f i l e s , and plots can be generated for any specified time. 5. Geochemical speciation and reaction-path computations are made extern-a l l y to the transport models with the EQ3/EQ6 code written by Wolery (.1978, 1979a). Given the bulk composition of a f l u i d , i t calculates the complete d i s t r i b u t i o n of species and then predicts the consequences of allowing react-ants to be dissolved into the modeled f l u i d . Newton-Raphson i t e r a t i o n In conjunction with f i n i t e - d i f f e r e n c e methods are used to solve the highly non-linea r equations governing chemical equilibrium. Output i s i n the form of species concentrations and masses of minerals produced or destroyed per kg of 282 H^O. The code can model through-flow systems, closed systems, and allows temperature to vary as a function of reaction progress. 6. Unfortunately, the computer time required to perform geochemical c a l c u l a t i o n s i n one c e l l of a finite-element mesh i s approximately equal to the t o t a l computational time required f o r the f l u i d - f l o w , the heat transport, and the mass-transport simulations combined. The cost of running a f u l l y -t r ansient transport model with geochemical reactions is. therefore impractical with current computer c a p a c i t i e s . Nevertheless, the reaction-path technique.-, i s very useful i n pr e d i c t i n g metal concentrations and scenarios of s u l f i d e p r e c i p i t a t i o n at possible ore-forming s i t e s . Modeling Results F l u i d Flow 1. The factors effecting steady-state f l u i d - f l o w patterns i n sedimentary basins are well known i n the f i e l d of groundwater hydrology. Variations i n subsurface p e r m e a b i l i t i e s , basin geometry, and water-table configuration are the main c o n t r o l l i n g f a c t o r s . F l u i d v e l o c i t i e s are determined by the hydraulic gradient imposed on the basin by the water-table slope. 2. The presence of a basal aquifer i n a basin causes groundwater flow to be focused down into the aquifer i n the higher-elevation recharge areas, and focused upward out of the aquifer, i n the lower-elevation discharge areas. The hydraulic-conductivity contrast between the aquifer and overlying s t r a t a controls the degree of focusing into the aquifer. Contrast r a t i o s as small as 10:1 are s u f f i c i e n t to allow s i g n i f i c a n t cross-formational flow from shale beds into major aqu i f e r s . L a t e r a l v a r i a t i o n s i n hydraulic conductivity can cause profound disruptions i n a r e g i o n a l flow system, the e f f e c t s of which i n some cases can create l o c a l flow systems. These l o c a l flow paths conceivable could provide a mechanism f o r the gradual supply of a component such as aqueous H 0S 283 to a.metal-bearing brine flowing through a regional aquifer. 3. Geothermal gradients are coupled to fl u i d - f l o w patterns and flow v e l o c i t i e s through t h e i r influence on f l u i d density and v i s c o s i t y . Higher temperatures cause greater flow rates by lowering the v i s c o s i t y of a basinal f l u i d . Near the l a t e r a l edge of a basin, t h i s effect could cause f l u i d v e l o c i t i e s calculated with an uncoupled analysis to be too small by a factor of two. Higher temperatures also cause the density to decrease with depth, thereby leading to upward buoyancy forces. This can increase v e r t i c a l f l u i d v e l o c i t i e s by a factor of about three. 4. The pronounced effect of s a l i n i t y on f l u i d flow can be observed i n field-measured hydraulic-head patterns. With an increase i n s a l i n i t y with depth, the hydraulic head may actually increase at deeper levels i n the re-charge end of a basin, rather than decrease as i t does i n fresh-water flow systems. Higher s a l i n i t y increases the density and v i s c o s i t y of a basinal f l u i d , which generally reduces both horizontal and v e r t i c a l components of the v e l o c i t y vector. Near the less saline edge of a basin, the s a l i n i t y effect on f l u i d v e l o c i t y i s r e l a t i v e l y small, causing a decrease of about 20% over a computed fresh-water model. 5. The effects of temperature and s a l i n i t y do not cancel each other out. Although thermal effects are greater, both factors should be taken into account when modeling f l u i d flow i n sedimentary basins. 6. The length of a flow system i n a basin depend on topographic r e l i e f , basin thickness, and permeability variations. I f the basin i s a few kilometers t h i c k , r e l i e f i s gentle, and aquifer exist at depth, then continuous flow systems can develop over distances of several-hundred kilometers. Major v a l l e y s , however, can interrupt the flow paths by focusing f l u i d flow into the valley bottom. Paleotopography, therefore, can have an important influence on the 284 location of ore formation. 7. In t h i s study, most of the s e n s i t i v i t y simulations are carried out i n a wedge-shape basin that i s 300 km i n length and varies i n thickness from 3000 m in.the higher-elevation recharge end to 1000 m i n the lower-elevation d i s -charge end. The basement surface i s assumed to slope at a rate of 3 m/km updip to the basin margin. In a wedge-shape basin, f l u i d v e l o c i t i e s i n a basal aquifer are at a maximum near the shallow, updip end of the basin. Basement structure also effects flow patterns and v e l o c i t i e s i n other types of basins. For basins of r e l a t i v e l y constant thickness ( i . e . f l a t bottom), the flow pattern i s more symmetric and velocity i s highest near the mid-section of the basin. Major basement arches can r e s t r i c t f l u i d flow to a th i n stratigraphic i n t e r v a l , which w i l l concentrate upward discharge and high flow rates above the arch. This feature may explain the association of some ore deposits with base-ment highs. Heat Transport '1. A l l of the factors that effect f l u i d , flow also effect subsurface temper-ature because flow rates determine the amount of heat convection i n a porous medium. Both conduction and convection are equally important processes i n the thermal regime. Under conditions of r e l a t i v e l y slow f l u i d - f l o w rates, thermal conduction i s the dominant mechanism of heat transport. In a two-layer basin with a hydraulic conductivity r a t i o between layers of 10:1 and an anisotropy r a t i o of 100:1 i n both layers, the effects of convection begin to appear i n the temperature pattern when the hydraulic conductivity of the basal layer approaches a value of lOm/yr. 2. Flu i d flow i n a sedimentary basin causes temperature to be lowered i n recharge areas and raised i n discharge areas. The d i f f e r i n g gradients are causes by the convection of heat away from the deeper parts of the basin and 285 updip to the discharge area of the regional f l u i d - f l o w system near the basin margin. Depending on the hydraulic conductivity and i t s anisotropy, tempera-ture can be raised as much as 30°C i n discharge areas, over the.temperature;: expected under conduction alone. 3. On the basis of the effects of t h e i r thermal conductivities on heat and f l u i d - f l o w patterns, i t can be shown that the presence of overlying sequences of shale and limestone i s conducive to the development of tempera-tures i n a basal aquifer that iss characteristic of ore-forming environments. 4. Thermal dispersion has a weak influence on the temperature pattern i n a basin. Transverse d i s p e r s i v i t i e s greater than about 50 m are needed before temperatures are affected by more than 10°C i n large, f l u i d - f l o w systems. However, the co e f f i c i e n t of transverse d i s p e r s i v i t y i s normally less than 10 m. 5. Based on the representative range of continental heat flows of 50 - 80 mW/m2 and surface temperatures of 10° - 30°C, maximum temperatures near the shallow edge of a 300 km long, wedge-shape basin can range from about 50°C to 95°C at a depth of less than 1000 m. Temperatures of 100°C to 150°C are possible i n larger basins with dimensions of several kilometers maximum thick-ness by several-hundred kilometers long. There i s no need, therefore, to invoke deep-seated heat sources or sudden pulses of hot brine from depth to explain the occurrence of warm temperatures i n stratabound ore formation. 6. Groundwater discharge zones, whether they are caused by stratigraphic pinchouts, topographic expressions, or basement arches, can s i g n i f i c a n t l y perturb the subsurface temperature pattern. In some cases, the high geothermal gradients l o c a l l y created by these features may play a role i n l o c a l i z i n g ore pre c i p i t a t i o n through cooling. 286 Mass Transport 1. In the absence of geochemical reactions and hydrodynamic dispersion between source beds and depositional s i t e s , the transport of an aqueous metal i s solely governed by the f l u i d - f l o w regime. Mass that i s released from a source bed. w i l l t r a v e l as an undisturbed plug along the flowline and with the average line a r v e l o c i t y of the groundwater. This type of mass transport, however, rarely occurs i n nature.. 2. Hydrodynamic dispersion includes the processes of mechanical dispersion and molecular d i f f u s i o n . Mechanical dispersion i s a mixing process that results i n the spreading of a solute i n both a longitudinal and l a t e r a l direction with respect to the flow path. Molecular d i f f u s i o n i n a porous medium also results i n the reduction of concentration gradients, but i t s effects are usually several orders of magnitude smaller than mechanical dispersion. 3. The influence of dispersion on long-distance transport of metals i s severe. For transport distances of 300 km, dispersion can d i l u t e the metal concentration of a f l u i d to a value that i s 10 - 20% of the o r i g i n a l concentra-tion of the f l u i d i n the source-bed area. I f 1 mg/kg • H^ O i s taken as the minimum metal..concentration, needed to form ore at the depositional s i t e , then approximately 5 - 1 0 mg/kg • U^O must be present, .in :the source-bed area, again for transport distances of 300 km. This.assumes there are no other sources or sinks for the metal along the flow path. 4. On a regional scale, the gradual increase i n upward flow beyond the mid-point of the basin causes mass to spread out of a basal aquifer. The area over which t h i s occurs depends on the hydraulic-conductivity r a t i o between the aquifer and surrounding strata and the configuration of the water table. 5. A variety of geologic configurations, such as pinchouts and the l i k e , can create l o c a l flow systems that lead to the convergence of flow paths from 287 d i f f e r e n t s t r a t i g r a p h i c l e v e l s . I t i s u n l i k e l y , i n the presence of dispersion,, that mixing of two-different f l u i d s would occur over a small-enough area to l o c a l i z e an ore deposit, unless flow i s r e s t r i c t e d to di s c r e t e fracture zones. Geochemical Transport and Ore P r e c i p i t a t i o n 1. The main factors e f f e c t i n g the s o l u b i l i t i e s of galena and sph a l e r i t e i n b a s i n a l brines include the temperature, the s a l i n i t y , and the pH and oxida-t i o n state. Concentrations of several hundred mg/kg • H^ O are pos s i b l e , pro-vided that the s u l f i d e content of the brine i s very low. Sul f i d e can be supplied at the ore-forming s i t e , e i t h e r through reduction of s u l f a t e i n the metal-bearing brine, or by addition of H^S through reactions with the surround-ing rocks. Other reducing agents such as methane or pr e - e x i s t i n g i r o n - s u l f i d e minerals can also cause ore deposition. When s u l f i d e i s supplied at the s i t e of deposition, the composition of the metal-bearing brine can be of many p o s s i b i l i t i e s . 2. Metal and aqueous s u l f i d e can be c a r r i e d i n the same so l u t i o n at equilibrium, but a combination of high s a l i n i t y , high temperature, and a c i d i c conditions (pH = 3-5) i s required to achieve concentrations of even a few mg/kg '-HO. P r e c i p i t a t i o n of ore can be accomplished through a pH s h i f t or cooling. The pH change i s usually assumed to come from the d i s s o l u t i o n of carbonates. However, neither of these mechanisms are very e f f e c t i v e at depositing s u l f i d e s when the pH i s greater than 5.0, the temperature i s l e s s than 80°C, and the s a l i n i t y i s below 3.0 m NaCl. Furthermore, a rather narrow range of pH and oxygen fugacity conditions r e s t r i c t the possible composition range of a so l u t i o n carrying metal and s u l f i d e through carbonate rocks. 3. The r e l a t i v e importance of these two geochemical models may vary from one basin environment to another. Geologic evidence from some ore d i s t r i c t s i n d i c a t e a sulfate-reducing mechanism f o r ore p r e c i p i t a t i o n , while other 288 d i s t r i c t s support a model with transport of metal and s u l f i d e together. 4. Long-distance transport of metal and s u l f i d e i n solution i s ruled out because the d i l u t i o n e f f e c t of dispersion would reduce these small concentra-tions even lower. Continuous addition of both metal and s u l f i d e along the flow path, or much higher i n i t i a l concentrations due to organic complexing may overcome the d i l u t i o n problem. Large flow rates and long periods of time would s t i l l be required i f changes i n pH and temperature are responsible f o r ore p r e c i p i t a t i o n . Pine Point Model 1. The a b i l i t y of a gravity-driven f l u i d - f l o w system to produce a major ore deposit can be q u a n t i t a t i v e l y v e r i f i e d by studying known ore d i s t r i c t s . For t h i s purpose, a preliminary cross-section model along the Keg River b a r r i e r complex provides i n s i g h t into the formation of lead-zinc mineralization at Pine Point. 2. The age of min e r a l i z a t i o n at Pine Point i s unknown, except that i t occurred i n post-Devonian time. Southwest to northeast gravity-driven flow systems could have operated i n the Western Canada sedimentary basin over much of the 175 million-year period between the Late Mississippian and the T e r t i a r y . About 1000 m of sedimentary cover would have ov e r l a i n the Pint Point area i n the E a r l y Cretaceous. 3. Reasonable estimates of the model parameters needed f o r studying Pine Point can be made through geologic considerations, the r e s u l t s of the s e n s i t i -v i t y a n a l y s i s , and the present-day data on hydraulic conductivity, heat flow, and brine geochemistry i n the basin. 4. The modeling r e s u l t s show that the barrier-complex aquifer was "the major f a c t o r i n focusing f l u i d flow across the Upper Devonian shales i n the t h i c k , western part of the section, and i n discharging metal-bearing brines 289 updip to the basin margin i n the east. Average l i n e a r v e l o c i t i e s probably-reached the range of .4.0 - 20.0 m/yr i n the basin margin area, which leads to specific-discharge rates of about ; 1.0 - 5.0 m3/m2 • yr through the barrier rocks at Pine Point. 5. Temperatures probably averaged around 80°C at the assumed 1000 m depth of the Pine Point s i t e . Depending on flow rates and geothermal parameters, temperatures could have ranged from 60°C to 100°C. These predictions are con-sistent with temperature estimates made from f l u i d - i n c l u s i o n data. 6. Geologic and hydrodynamic c r i t e r i a support a sulfate-reduction model for the deposition of ore at Pine Point. Metal was brought to the ore-forming s i t e i n a sulfate-type brine, and the action of sulfate-reducing bacteria, aided by abiogenic reaction with H S, caused ore pr e c i p i t a t i o n in the more permeable, k a r s t i c sections of the ba r r i e r . I t i s also possible that l a t e r a l -permeability changes downdip or topographic effects near the end of the basin caused l o c a l flow systems to develop, which i n turn assisted i n the supply of H S through reduction of sulfate at shallow depth with flow down into the karst aquifer. 7. I f transport of metals was from shales deep i n the basin, then minimum metal concentrations of about 20 mg/kg • H^ O near the source, were needed to sustain a concentration of at least 1 mg/kg • H^ O near the basin margin. Closer source beds or addition of metal along the flow path would not require the same i n i t i a l concentration, because the d i l u t i o n caused by dispersion would be l e s s . 8. An estimated time period of 0.5 to 5.0 m i l l i o n years i s necessary to account for the metal tonnage at Pine Point, based on a probable pr e c i p i t a t i o n range of 1 - 5 mg/kg • HO and flow rates of 1.0 - 5.0 m3/m2 • yr. 290 This study has shown that gravity-driven groundwater flow systems are capable of forming large ore deposits over r e l a t i v e l y short i n t e r v a l s of geologic time. Numerical modeling provides a powerful t o o l f o r t e s t i n g ore-genesis hypotheses and f o r developing an understanding of the conditions necessary f o r ore deposition i n sedimentary basins. As i t stands ,,.hydrogeo-l o g i c modeling can be used to r e f i n e exploration models of sedimentary basins or s p e c i f i c ore d i s t r i c t s , but on a r e g i o n a l s c a l e . 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