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Salinity and temperaturedependent groundwater flow in the Floridan aquifer system of South Florida
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by Joseph D. Hughes.
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2006.
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ABSTRACT: Densitydependent groundwater flow in the Floridan aquifer system (FAS) depends on chloride concentrations and fluid temperature. Previous studies addressing the role of chloride concentration and temperatures on groundwater flow in the FAS have relied on observation data or simplified twodimensional numerical models. A threedimensional hydrologic analysis of FAS in peninsular Florida was performed using a modified version of SUTRA (SUTRAMS) capable of simulating multispecies solute and heat transport. SUTRAMS was developed during this investigation and is capable of reproducing results for several problems with known solutions.The model was developed using available geometric and hydraulic parameter data and calibrated using hydraulic head, chloride concentrations, and temperatures representative of conditions prior to significant groundwater pumpage from the FAS. The calibrated model is capable of reproducing observed pressures and temperatures but in general ov ersimulates chloride concentrations. The inability of the model to simulate observed chloride concentrations suggests chloride concentrations in the FAS are not in equilibrium with current sea level. Previous hydrologic studies of the FAS have attributed anomalous chloride concentrations to incomplete flushing of relict seawater that entered the aquifer during previous sealevel highstands.Three hypothetical, sinusoidal sealevel changes occurring over 100,000years were used to evaluate how the aquifer responds to sealevel fluctuations. Model results indicate pressure equilibrates most rapidly and is followed by temperatures and then chloride concentrations. Confining unit thicknesses directly affect response times of pressure, temperature, and chloride concentrations in the FAS.Simulation of the system with ("geothermal case") and without ("isothermal case") the geothermal component reveals that the inflow of seawater from the Florida Straits would be similar without the heat f low but the distribution would differ significantly. The addition of heat flow also reduces the asymmetry of the circulation. Simulations evaluating aquifer responses to sealevel fluctuations and the thermal component indicate that the complicated threedimensional setting of the FAS is a key component of the groundwater flow system and steady state conditions may not exist for relatively thick coastal aquifers that have experienced multiple sealevel cycles.
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Dissertation (Ph.D.)University of South Florida, 2006.
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Text (Electronic dissertation) in PDF format.
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Adviser: H.L. Vacher, Ph.D.
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Densitydependent groundwater flow.
Floridan aquifer system.
Doublediffusive flow.
Hydrogeology.
numerical modeling.
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Dissertations, Academic
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x Geology
Doctoral.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.1625
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Salinityand TemperatureDependent Groundwater Flow in the Floridan Aquifer System of South Florida by Joseph D. Hughes A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Geology College of Arts and Sciences University of South Florida Major Professor: H.L. Vacher, Ph.D. Ward E. Sanford, Ph.D. Mark Stewart, Ph.D. Mark Rains, Ph.D. Eric Oches, Ph.D. Date of Approval: June 22, 2006 Keywords: densitydependent groundwater flow, Floridan Aquifer System, doublediffusive flow, hydrogeology, numerical modeling Copyright 2006, Joseph D. Hughes
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Acknowledgments I would like to thank my advisors H.L. Vacher and Ward Sanford for allowing me the opportunity to work on an interesting and challenging dissertation topic. I would also like to thank Paul Barlow, David Budd, Stephen Gingerich, John J. Hickey, Christian D. Langevin, Scott Pringle, Alden Provost, T.E. Reilly, Thomas M. Scott, Clifford I. Voss, and Richard Winston for their assistance, suggestions, comments, and reviews. Partial funding for this research was provided by th e U.S. Geological Survey Office of Ground Water.
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Note to Reader Several of the figures in Chapters 2, 3, 4, and 5 are in color and may be obtained as such from the University of South Florida. This dissertation is available in electronic format online via the University of South Florida library at http://www.lib.usf.edu.
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Table of Contents List of Tables viii List of Figures ix Abstract xvi 1 Introduction 1 1.1 Introduction 1 1.2 Numerical Approach 3 1.3 Dissertation organization 3 2 Development of the Numerical Code SUTRAMS 6 2.1 Abstract 6 2.2 Introduction 8 2.2.1 Purpose 9 2.2.2 SUTRA Fundamentals and Previous Applications 9 2.2.3 The Current Study 11 2.3 PhysicalMathematical Basis of SUTRAMS Modifications 12 2.3.1 General MassBalance Formulation 12 2.3.2 Fluid MassBalance Equation 13 2.3.3 Modified Form of the Fluid MassBalance Equation 14 2.3.4 Unified Energyand SoluteBalance Equation 16 2.3.5 Modified Form of the Unified Energyand SoluteBalance Equation 20 iv
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2.3.6 Modified Form of the EnergyBalance Equation used with GeometricMean Approximation for Bulk Thermal Conductivity 23 2.4 Numerical Methods 26 2.4.1 Numerical Approximation of SUTRAMS Fluid Mass Balance 26 2.4.2 Numerical Approximation of SUTRAMS Unified Energyand SoluteBalance Equation 29 2.4.3 Temporal Evaluation of Adsorbate Mass Balance 33 2.4.4 Solution Sequencing 36 2.5 Additional SUTRAMS Options 39 2.5.1 Simple TimeVarying Boundary Conditions 40 2.5.2 Specified User Output Times 41 2.5.3 Simple Automatic TimeStepping Algorithm 42 2.5.4 Specified Observation Locations 44 2.5.5 Specification of Hydraulic Parameters Using Zones 45 2.6 SUTRAMS Simulation Examples 48 2.6.1 Densitydependent flow, heat transport, and solute transport, Solution for multicomponent fluid flow in a saline aquifer system 48 2.6.2 Solution for doublediffusive finger convection induced by different fluid dispersivities and viscosities 61 2.6.3 Densitydependent flow with transport of a nonreactive tracer and zeroorder production and transport of a solute to simulate groundwater age 71 3 Numerical simulation of doublediffusive finger convection 80 3.1 Abstract 80 3.2 Introduction 80 v
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3.3 Governing equations 83 3.4 Numerical Approximation 85 3.5 Experimental doublediffusive HeleShaw study of Pringle et al. (2002) 86 3.6 Numerical modeling 90 3.6.1 Spatial discretization and model parameters 90 3.6.2 Boundary and initial conditions 93 3.7 Results and Discussion 96 3.7.1 Simulated numerical results 96 3.7.2 Influence of discretization on numerical results 102 3.7.3 Comparison of the diffusion rates of dye, sodium chloride, and sucrose 106 3.8 Concluding Remarks 108 4 Temporal response of temperature and salinity to sealevel changes, Floridan aquifer system, USA. 114 4.1 Abstract 114 4.2 Introduction 114 4.3 General Description of the Study Area 118 4.4 Geologic and hydrogeologic setting of the Florida platform 119 4.5 Regional Flow System 123 4.6 Simulation of the Groundwater Flow System 126 4.6.1 Model Construction 126 4.6.2 Model Calibration 131 4.7 Response to Sealevel Changes 142 4.8 Summary and Conclusions 152 vi
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5 Threedimensional flow in the Florida Platform: Theoretical Analysis of Kohout Convection at its Type Locality 154 5.1 Abstract 154 5.2 Introduction 155 5.3 Numerical Model 157 5.4 The Pattern of Geothermal Circulation 159 5.5 The Pattern of Isothermal Circulation 160 5.6 Quantifying the effect of heat flow on the saltwater circulation 162 5.7 Implications of the effect of heat flow on dolomitization time 163 5.8 Conclusion 164 6 Summary 166 References 169 Appendix A: Notation 180 Appendix B: SUTRAMS Input Data 191 Appendix C: SUTRAMS HeleShaw Numerical Methods Description 275 About the Author End Page vii
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List of Tables Table 1. Discretization requirements for several aquifer aspect ratios (). 60 Table 2 HeleShaw cell experiment parameters 87 Table 3 Fluid properties and Rayleigh numbers 88 Table 4 Numerical Parameters used 94 Table 5 Model Parameters 127 Table 6 Calibration statistics for the hydrostratigraphic units of the FAS 140 viii
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List of Figures Figure 1 Location of the Floridan aquifer system and the Florida Platform. 2 Figure 2 Comparison of bulk thermal conductivity as a function of porosity, fluid saturation, and solid matrix thermal conductivity using a volumetric average and geometricmean approximations. 24 Figure 3 Simple representation of the differences in memory requirements for hydraulic parameters that are discretized by elements for SUTRA and SUTRAMS using zones. 46 Figure 4 Finite element mesh (=1) and pressure (gray), concentration (blue), and temperature (red) boundary conditions for Henry and Hilleke (1972) solution. 51 Figure 5 Match of percentseawater contours and the SUTRAMS flow field for N=3, NC=10, and =1 Henry and Hilleke numerical solution (0.5percent seawater concentration only) (solid red line), HST3D code solution (dashed black line), and SUTRAMS solution (colored). 55 Figure 6 Match of percentseawater contours and the SUTRAMS flow field for N=3, NC=10, and =0.10 Henry and Hilleke numerical solution (0.5percent seawater concentration only) (solid red line), HST3D code solution (dashed black line), and SUTRAMS solution (colored). 56 Figure 7 Match of percent seawater contours, SUTRAMS flow field, and match of isotherms for N=3, NC=10, NT=1, and =1 Henry and Hilleke numerical solution (0.5 percent seawater concentration and isotherm only) (solid red line), HST3D code solution (dashed black line), and SUTRA solution (colored) with DM=8.333106 and DT=8.333105. 58 ix
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Figure 8 Match of percentseawater contours, SUTRAMS flow field, and match of isotherms for N=3, NC=10, NT=1, and =0.10 Henry and Hilleke numerical solution (0.5percent seawater concentration and isotherm only) (solid red line), HST3D code solution (dashed black line), and SUTRA solution (colored) with DM=8.333108 and DT=8.333107. 59 Figure 9 Boundary and initial concentration conditions and finiteelement mesh (every 16th element) used to simulate the Pringle and others (2002) HeleShaw experiment. 63 Figure 10 Absolute viscosity and fluid density relationships with NaCl and sucrose concentrations used in all SUTRAMS simulations. Data are from Weast (1986). 66 Figure 11 Observed results from Pringle and others (2002) at (A) t* = 4.06105, (B) t* = 1.29104, (C) t* = 3.96104, (D) t* = 3.35104, (E) t* = 4.35104, (F) t* = 5.36104, (G) t* = 6.03104, and (H) t* = 7.37104, (I) t* = 8.04104, (J) t* = 1.04103, (K) t* = 1.78103, and (L) t* = 3.19103. 67 Figure 12 Simulated results SUTRAMS results at (A) t* = 4.06105, (B) t* = 1.29104, (C) t* = 3.96104, (D) t* = 3.35104, (E) t* = 4.35104, (F) t* = 5.36104, (G) t* = 6.03104, and (H) t* = 7.37104, (I) t* = 8.04104, (J) t* = 1.04103, (K) t* = 1.78103, and (L) t* = 3.19103. 68 Figure 13 Normalized vertical length, h*=h/H, and mass transfer across the center line, M*=M/Mo, as a function of time showing regions of steady growth for the original HeleShaw experiment (open gray circles) and the SUTRAMS simulation (solid black circles and solid black line). 70 Figure 14 SUTRA finiteelement mesh and boundary conditions. 73 x
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Figure 15 Simulated percentseawater contours from the 2D SUTRAMS simulation after (A) 13.89 and (B) 27.78 days and the (C) 3D SUTRAMS simulation after 27.78 days. 76 Figure 16 Simulated nonreactive tracerconcentration contours from (A) 2D SUTRAMS, (B) 3D SUTRAMS in XZ Plane at y=40 m, and (C) 3D SUTRAMS in XY Plane at z=50 m. 78 Figure 17 Simulated groundwater age, in days, from SUTRAMS. 79 Figure 18 Location of the HeleShaw experiment (closed circle) within Rayleigh parameter space with an R value of 1.22 (Modified from Pringle et al., 2002). 89 Figure 19 Observed results from Pringle et al. (2002) at (A) t* = 4.03105, (B) t* = 1.31104, (C) t* = 2.21104, (D) t* = 3.22104, (E) t* = 4.23104, (F) t* = 5.24104, (G) t* = 6.04104, and (H) t* = 7.25104, (I) t* = 7.85104, (J) t* = 1.03103, (K) t* = 1.77103, and (L) t* = 3.17103. 92 Figure 20 Absolute viscosity and fluid density relationships with sodium chloride and sucrose concentration used in all simulations. 95 Figure 21 Simulated numerical results for the dye at (A) t* = 4.03105, (B) t* = 1.31104, (C) t* = 2.21104, (D) t* = 3.22104, (E) t* = 4.23104, (F) t* = 5.24104, (G) t* = 6.04104, and (H) t* = 7.25104, (I) t* = 7.85104, (J) t* = 1.03103, (K) t* = 1.77103, and (L) t* = 3.17103. 98 Figure 22 Normalized vertical length, h*=h/H, mass transfer across the center line, M*=M/Mo, and normalized horizontal length scale at the center line, *=/H, as a function of time showing regions of steady growth for the original HeleShaw experiment (closed gray circles) and the numerical simulation. 100 xi
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Figure 23 Normalized horizontal length scale, *=/H, at 0.75H and 0.25H as a function of time showing apparent asymmetry in finger evolution above and below the center line for simulated and observed dye concentrations. 101 Figure 24 Normalized vertical length, h*=h/H, mass transfer across the center line, M*=M/Mo, and normalized horizontal length scale at the center line, *=/H, as a function of time, showing regions of steady growth for the 41,984, 167,936, and 671,744element simulations. 105 Figure 25 Conceptual (A) sucrose concentrations, (B) sodium chloride concentrations, (C) dye concentrations, and (D) fluid density relative to the initial average fluid density showing the effect of different diffusivities on component concentrations and fluid density. 110 Figure 26 Simulated numerical results for sodium chloride at (A) t* = 4.03105, (B) t* = 1.31104, (C) t* = 2.21104, (D) t* = 3.22104, (E) t* = 4.23104, (F) t* = 5.24104, (G) t* = 6.04104, and (H) t* = 7.25104, (I) t* = 7.85104, (J) t* = 1.03103, (K) t* = 1.77103, and (L) t* = 3.17103. 111 Figure 27 Simulated sucrose concentrations at (A) t* = 5.24104, (B) t* = 7.85104, and (C) t* = 3.17103. 112 Figure 28 Normalized vertical length, h*=h/H, mass transfer across the center line, M*=M/Mo, and normalized horizontal length scale at the center line, *=/H, as a function of time showing regions of steady growth for the experiment, simulated sodium chloride concentrations, and simulated dye concentrations. 113 Figure 29 Extent of the Floridan aquifer system, study area, and hydrostratigraphic data sources used in the study. 116 xii
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Figure 30 Altitude of (a) the 50percent seawater concentration in the Floridan aquifer system (adapted from Sprinkle, 1989) and (b) the top of the Floridan aquifer system relative to sea level. 117 Figure 31 Bathymetry and topography in the study area. 119 Figure 32 Hydrostratigraphy of the Floridan Aquifer in peninsular Florida, and hydraulic parameters in the calibrated model (modified from Johnston and Bush, 1988). 122 Figure 33 Thickness of the Floridan aquifer system and extent of the Boulder Zone in peninsular Florida. 123 Figure 34 Predevelopment UFA potentiometric surface (m) and location of springs in the study area. 124 Figure 35 Model mesh used in the study area. 128 Figure 36 Intrinsic permeability of the a) ICU/IAS, b) UFA, c) MCU, and d) LFA/Boulder Zone used in the model. 132 Figure 37 Net recharge (in/yr) applied to the surficial aquifer in the steadystate model. 133 Figure 38 Simulated fluxes (in/yr) between the Miocene IAS/ICU and the UFA in the emergent part of the platform. 134 Figure 39 Crosssections locations where simulated results for the calibrated model are evaluated. 135 Figure 40 Simulated percent seawater chloride concentrations (A) and temperatures (B) along crosssection line A to A shown on Figure 39. 136 Figure 41 Simulated percent seawater chloride concentrations (A) and temperatures (B) along crosssection line B to B shown on Figure 39. 137 Figure 42 Simulated percent seawater chloride concentrations (A) and temperatures (B) along crosssection line C to C shown on Figure 39. 138 xiii
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Figure 43 Simulated and observed hydraulic heads, percent seawater, and temperatures for the calibrated model. 141 Figure 44 Simulated sealevel changes applied to the calibrated model to evaluate the response of the FAS to a variety of sealevel fluctuations. 143 Figure 45 Simulated extent of flooding between select ranges of the amplitude of sealevel change (+70 m) and the location of selected observation wells evaluated. 145 Figure 46 Simulated response of the FAS to sealevel changes at the selected observation well at location 1 (west coast). 147 Figure 47 Simulated response of the FAS to sealevel changes at the selected observation well at location 2 (center of the emergent platform). 148 Figure 48 Simulated response of the FAS to sealevel changes at the selected observation well at location 3 (east coast). 149 Figure 49 Simulated response of the FAS to sealevel changes at the selected observation well at location 4 (south Florida). 150 Figure 50 Simulated percent seawater chloride concentrations at the top of the UFA at a) 0, b) 50,000, c) 70,000, and d) 100,000 years in response to the simulated 70 m sealevel event with a 100,000 year duration. 151 Figure 51 Conceptual crosssection of cyclic flow of saltwater induced by geothermal heating (Kohout Convection) in Florida (adapted from Kohout et al., 1977). 155 Figure 52 a) Study area, predevelopment FAS potentiometric surface (CI = 3 m), generalized predevelopment FAS flow directions, and location of conceptual crosssection of Kohout et al. (1977). 157 Figure 53 Simulated results for crosssection AA. 161 xiv
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Figure 54 Simulated distribution of oceanaquifer exchanges in the Atlantic Ocean and the Gulf of Mexico for a) the geothermal case and b) the isothermal case. 162 Figure 55 Minimization of bandwidth by careful numbering of nodes (Fig. 7.1 Voss and Provost, 2002). 227 Figure 56 Allocation of sources and boundary fluxes in equalsized elements (Fig. B.1 Voss and Provost, 2002). 242 xv
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Salinityand TemperatureDependent Groundwater Flow in the Floridan Aquifer System of South Florida Joseph D. Hughes ABSTRACT Densitydependent groundwater flow in the Floridan aquifer system (FAS) depends on chloride concentrations and fluid temperature. Previous studies addressing the role of chloride concentration and temperatures on groundwater flow in the FAS have relied on observation data or simplified twodimensional numerical models. A threedimensional hydrologic analysis of FAS in peninsular Florida was performed using a modified version of SUTRA (SUTRAMS) capable of simulating multispecies solute and heat transport. SUTRAMS was developed during this investigation and is capable of reproducing results for several problems with known solutions. The model was developed using available geometric and hydraulic parameter data and calibrated using hydraulic head, chloride concentrations, and temperatures representative of conditions prior to significant groundwater pumpage from the FAS. The calibrated model is capable of reproducing observed pressures and temperatures but in general oversimulates chloride concentrations. The inability of the model to simulate observed chloride concentrations suggests chloride concentrations in the FAS are not in equilibrium with current sea level. Previous hydrologic studies of the FAS have attributed anomalous chloride concentrations to incomplete flushing of relict seawater that entered the aquifer during previous sealevel highstands. Three hypothetical, sinusoidal sealevel changes occurring over 100,000years were used to evaluate how the aquifer responds to sealevel fluctuations. Model results indicate pressure equilibrates most rapidly and is followed by temperatures and then xvi
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chloride concentrations. Confining unit thicknesses directly affect response times of pressure, temperature, and chloride concentrations in the FAS. Simulation of the system with (geothermal case) and without (isothermal case) the geothermal component reveals that the inflow of seawater from the Florida Straits would be similar without the heat flow but the distribution would differ significantly. The addition of heat flow also reduces the asymmetry of the circulation. Simulations evaluating aquifer responses to sealevel fluctuations and the thermal component indicate that the complicated threedimensional setting of the FAS is a key component of the groundwater flow system and steady state conditions may not exist for relatively thick coastal aquifers that have experienced multiple sealevel cycles. xvii
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Chapter 1 1 Introduction 1.1 Introduction The Floridan aquifer system (FAS) is one of the major sources of groundwater in the southeastern United States and underlies all of Florida, southern Georgia, and parts of Alabama and South Carolina (Figure 1). The FAS of peninsular Florida is made up of several Tertiary carbonate formations that are hydraulically connected to form a regional hydrologic unit. Although the FAS is composed of hydraulically connected carbonate units there are significant horizontal and vertical differences in the hydraulic properties, water chemistry, temperatures, and flow characteristics. In general, the FAS is thickest in central and south Florida where the Miocene Hawthorn Formation is present and acts as a semiconfining to confining unit. The Miocene Hawthorn Formation thickens from 0 m in westcentral Florida to more than 200 m in south Florida. Increased chloride concentrations are typically also found where the Hawthorn Formation is thick and close to the Gulf of Mexico or Atlantic Ocean. Apparent 14C ages, and 234U/238U alphaactivities suggest that some portions of the lower portions of the FAS in south Florida have a close connection to the Atlantic Ocean. These areas also correspond to locations where inverted temperature profiles have been observed (Kohout, 1965; Meyer, 1989a). Kohout (1965) hypothesized that thermally induced convective circulation was occurring in lower portions of the FAS in south Florida and inflow of cold seawater to the LFA from the Straits of Florida was responsible for the observed temperature anomalies. Numerical investigations by Kohout et al. (1977), using a simplified model of the FAS in south Florida, supported Kohouts hypothesis of thermally forced circulation (Kohout Convection). Since the work of Kohout et al. (1977), Kohout Convection has been used to explain massive dolomitization of carbonate platforms (e.g., Simms, 1984). 1
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Figure 1 Location of the Floridan aquifer system and the Florida Platform. Several investigators (e.g., Meyer, 1989a, Reese and Memberg, 2000) have also noted that chloride concentrations in the FAS of south Florida appear to be in disequilibrium with current sea level as a result of incomplete flushing of seawater that invaded during Pleistocene sealevel highstands. In an analysis of the effects of rising sea levels during the Holocene Transgression (last 18,000 years) in south Florida, Meyer (1989a) concluded that sealevel rise would likely be accompanied by rising water tables in emergent parts of Florida even if rainfall remained about the same, because sea level represents a boundary condition for the freshwater flow system. Meyer (1989a) also estimated that that seawater moved inland about 1 mile for each 1ft rise in sea level and is probably still moving slowly inland. Previous numerical studies of the FAS have generally ignored the effect of chloride concentrations on the groundwater flow system (e.g., Sepulveda, 2002) or considered it in a simplified twodimensional sense (e.g., Kohout et al., 1977) or 2
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analytically (e.g., Meyer, 1989a). Furthermore, most studies have not quantitatively dealt with heat flow or the effect of sealevel changes on the FAS. In this dissertation, threedimensional, densitydependent groundwater flow in the Floridan aquifer system (FAS) is evaluated using numerical methods. A threedimensional model is developed because of the complex geometry of the FAS, the aquifer heterogeneity, and the boundary conditions. The developed threedimensional model is used to evaluate the effect of sealevel changes and heat flux on the groundwater flow system. 1.2 Numerical Approach A numerical code (SUTRAMS) is developed to (1) represent the threedimensional nature of the FAS in an efficient and accurate manner, (2) explicitly simulate the effect of temperature and/or salinity on groundwater flow and transport, and 3) simulate the effect of timevarying sea levels. Simulated results from SUTRAMS are compared to three benchmark problems with known solutions. The benchmark problems included the salinityand temperaturedependent problem of Henry and Hilleke (1972), the experimental HeleShaw results of Pringle et al. 2002, and the classic Henry (Henry, 1964) problem in twoand threedimensions. A steadystate, threedimensional SUTRAMS model of the FAS was developed from published data, calibrated using observed pressure (head), chloride concentration, and temperature data, and used to evaluate the effects of sealevel changes and heat flow on the FAS. 1.3 Dissertation organization This dissertation investigates the relative importance of heat and salinity on densitydependent groundwater flow in the FAS of south Florida through application of numerical methods. It also addresses the response of the FAS to sealevel changes using numerical methods. Chapter 2 and 3 are peerreviewed papers: 3
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Hughes J.D., and W.E. Sanford, 2004, SUTRAMS: A version of SUTRA modified to simulate heat and multiplespecies solute transport, U.S. Geological Survey OpenFile Report 20041207, U.S. Geological Survey, Reston, Virginia, 156 p. Hughes, J.D., Sanford, W.E., and Vacher, H.L., 2005, Numerical simulation of doublediffusive finger convection: Water Resources Research, v. 41, no. 1, W01019 10.1029/2003WR002777. Chapters 2 and 3 are almost identical to the publications. Figure, table, and equation numbers have been changed to be consistent with the remainder of the dissertation. To maintain consistency, abstracts for chapters 4 and 5 are also included. The components of the work are divided into the following chapters. Chapter 2: Development of the Numerical Code SUTRAMS The governing flow equations and numerical approximation of the governing equations used by SUTRAMS are presented. Results for three benchmark problems are presented and indicate that the numerical code is accurately simulating groundwater flow and transport in variabledensity, nonisothermal environments. Chapter 3: Numerical simulation of doublediffusive finger convection Results for the HeleShaw benchmark problem of Pringle et al. (2002) are presented in Chapter 2. Concentration data was collected in the HeleShaw experiment using optical techniques and a dye to map NaCl concentrations. Numerical results indicate differences in the fluid properties of the dye and NaCl are significant, the dye does map the NaCl as closely indicated in Princle et al. (2002), and vertical mass transfer of the dye is greater than NaCl. Chapter 4: Temporal response of temperature and salinity to sealevel changes, Floridan aquifer system, USA. A steadystate, threedimensional model of the FAS is developed using existing data and calibrated using hydraulic head, chloride concentrations, and temperatures. Model results indicate observed chloride concentrations in the FAS are not in equilibrium 4
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with current sea level. Evaluation of the effect of sealevel changes conditions in the FAS indicates that pressures and temperatures generally equilibrate quickly but chloride concentrations can take a significant amount of time to equilibrate. Results also show that increases in chloride concentrations propagate from current recharge and discharge areas as a result of reduction of topographic head gradients as the Florida platform is flooded. Chapter 5: Threedimensional flow in the Florida Platform: Theoretical Analysis of Kohout Convection at its Type Locality The effect of heat flux on the FAS is evaluated using the threedimensional model of the FAS developed in Chapter 4. Simulated results show that the flow system is significantly different with and without consideration of heat transport. Simulated results indicate there are differences in calculated times for complete dolomitization of the FAS. 5
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Chapter 2 2 Development of the Numerical Code SUTRAMS 2.1 Abstract A modified version of SUTRA is introduced that is capable of simulating variabledensity flow and transport of heat and multiple dissolved species through variably to fully saturated porous media. The original version of SUTRA is capable or simulating variabledensity flow and transport of either heat or one dissolved species through variably to fully saturated porous media. This modified version was developed because of the importance of temperature and solute concentrations in many variabledensity flow environments and the desire to implicitly simulate the transport of multiple dissolved species that may or may not affect fluid density. Users familiar with SUTRA should have little difficulty applying this version of SUTRA to multispecies transport problems. All modifications to SUTRA are general, the number of dissolved species that can be simulated is unrestricted by the program, any of the simulated species can affect fluid density and or viscosity, and simulation of heat transport is unrestricted by the number of simulated dissolved species. The model assumes density and viscosity are linear functions of solute concentration. For simulation of energy transport, the model assumes density is a linear function of temperature but the relation between temperature and viscosity is nonlinear. A limitation of the current temperatureviscosity relationship is that it can be scaled only with userspecified parameters unless the code is modified and recompiled. However, alternative temperatureviscosity relationships can be easily incorporated in the source code. In addition to modifications that allow for multispecies transport, SUTRA has been modified to allow use of a spatially distributed solid matrix thermal conductivity and the option to use a geometricmean approximation for bulk thermal conductivity that accounts for partial saturation. The geometricmean approximation for bulk thermal conductivity was added because of supporting empirical evidence (Sass and others, 1971). 6
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In addition to modifications to the numerical algorithms in SUTRA, a number of optional functions have been added to minimize solution nonconvergence, minimize user coding to simulate timevarying boundary conditions, reduce output file size, allow specification of hydraulic parameters using zones, and specify observation locations with spatial coordinates. A simplified automatic timestep algorithm has been added that reduces the timestep length, if the number of iterations exceeds a userspecified value, and can rerun time steps if the specified maximum number of iterations is exceeded. A simple algorithm has been added that allows any boundary condition to be time varying without the need for userprogrammed functions, with the limitation that boundary conditions between the times when conditions are changed (at the beginning of stress periods) are not interpolated. Time steps are reduced to the minimum time step when new boundarycondition values are read. A simple technique has been implemented that permits the user to specify the exact time to write data to the nodal and elemental output files. Functionality has been added that allows hydraulic parameters to be specified using zones in order to reduce the memory requirements for large twoand threedimensional simulations and to better facilitate the use of modelindependent parameterestimation codes (e.g., UCODE, Poeter and Hill, 1998; PEST, Doherty, 1994). A simple routine that allows observation locations to be specified using coordinates rather than node numbers also has been included. The modified version of SUTRA has retained all the functionality of the original SUTRA and can solve the flow and transport equations in either two or three dimensions. This version of SUTRA is backward compatible with standard twoand threedimensional SUTRA single species input data sets. Multispecies data sets are different from standard data sets only when additional data are required for each simulated species. Three examples are presented that demonstrate the ability of the modified version of SUTRA to simulate multispecies transport. Two of these examples demonstrate the ability to simulate fluid density affected by more than one species, and one example shows several additional applications of the modified version of SUTRA. Where possible, the modified code is compared to observed data, the standard version of SUTRA (Voss and Provost, 2002), or other codes capable of simulating variabledensity 7
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flow and heat and solute transport. Comparisons indicate the modified version of SUTRA is comparable to HST3D (Kipp, 1997) for a coupled heat and solute variabledensityflow problem, and is capable of simulating a coupled variabledensity multispecies HeleShaw experiment. 2.2 Introduction Groundwater in coastal environments, areas of high evaporation rates, and urban and industrial areas may have dissolved constituents that can affect fluid density. For example, in coastal aquifers, chloride concentrations can vary from 0 to 18,500 parts per thousands and have fluid densities ranging from 1,000 to 1,025 kg/m3 (kilograms per cubic meter) respectively. The range in fluid density of chloride concentrations from freshwater to seawater represents only a 2.5percent increase in fluid density, but this small difference has been shown to have significant effects on groundwaterflow rates and patterns. Typically, multiple dissolved constituents are present in variabledensity flow environments, but density variations usually can be related to a single constituent (i.e., chloride concentrations, total dissolved solids.) because of the positive correlation of each constituent with all the other constituents. Fluid temperature variations also affect fluid density. A 10C increase in fluid temperature of freshwater results in a reduction in fluid density of approximately 4 kg/m3 (~0.5percent reduction). The small change in fluid density over the range of fluid temperatures found in many shallow coastal systems allows temperature effects on fluid density to be ignored (Konikow and Reilly, 1999). In some natural systems, however, temperature variations have a significant effect on the flow system (e.g., hydrothermal systems, thick and/or deep aquifers). Similarly, fluid density may be a function of more than one dissolved species in some natural systems (Pringle and others, 2002). Furthermore, lateral and vertical aquifer heterogeneities in many natural systems cause the variabledensityflow system to be three dimensional in nature. A variety of numerical approaches have been used to simulate variabledensityflow problems (for a summary of available methods, see Sorak and Pinder, 1999). The methods that have been used include finite differences (e.g., INTERA, 1979), finite 8
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elements (e.g., Voss, 1984; Huyakorn and others, 1987; and Diersch, 1988), finite volumes (e.g., Kipp, 1987; Oldenburg and Pruess, 2000), analytical elements (Strack, 1995), and hybrid EulerianLagrangian finite differences/finite elements (e.g., Sanford and Konikow, 1985; Yeh, 1987 and 1990). These approaches have included twodimensional and threedimensional implementations of the available methods. This report describes a modified version of SUTRA capable of simulating variabledensity flow that is dependent on multiple dissolved constituents and temperature. SUTRA was selected for modification because of its wide use and modular nature that easily accommodates additional functionality. 2.2.1 Purpose The principal objectives of this chapter are to (1) introduce a general methodology that incorporates multiple dissolved species into the set of partial differential equations that describe variabledensity flow and solute transport, (2) present a numerical model, SUTRAMS (S aturatedU nsaturated Tra nsport of M ultiple S pecies), that utilizes this methodology; (3) benchmark the computer code against existing numerical codes, previous numerical solutions, and experimental data; and (4) use SUTRAMS results to demonstrate some of the processes that are important in the simultaneous transport of multiple species. 2.2.2 SUTRA Fundamentals and Previous Applications SUTRA uses a hybridized finite element and implicit finitedifference technique to solve the fluid massbalance equation and unified energyand solutebalance equation for variabledensity, singlephase, saturatedunsaturated flow and singlespecies transport. Solving the equations in the time domain is accomplished by the implicit finitedifference method for nonflux terms (e.g., time derivatives and sources) in the region surrounding each specified node. All flux terms and associated parameters are discretized on an element basis and are solved by a modified Galerkin method. SUTRA uses bilinear quadrilateral elements in two dimensions and trilinear hexahedrons in three dimensions. 9
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SUTRA uses a modified form of the standard Galerkin finiteelement method to allow for calculation of a consistent vertical velocity for each element. This is a significant but important modification for variabledensityflow simulations. In the standard Galerkin finiteelement method, spurious vertical velocities are generated everywhere there is a vertical concentration gradient within the finiteelement mesh, even if a hydrostatic pressure distribution is used (Voss, 1984; Voss and Souza, 1987). These spurious vertical velocities make it impossible to simulate a narrow transition zone between fluids with significant density differences (e.g., freshwater and seawater) with the standard finiteelement method, regardless of the specified dispersivity for the system. The modification used in SUTRA provides for a vertical velocity calculation within each finite element based on consistent spatial variability of the pressure gradient, p, and the buoyancy term, g, in the variabledensity form of Darcy's equation. SUTRA is a very general code that was developed using a modular coding convention (Voss, 1999). The modular nature of SUTRA has allowed it to be modified for specific projects and applications. Some of the modifications have included addition of equilibrium chemical reactions (Lewis and others, 1986), addition of kinetic reactions (e.g., Smith and others, 1997; Sahoo and others, 1998), and calculation of path lines (Cordes and Kinzelbach, 1992). SUTRA was initially released as a twodimensional code that could solve constantdensity areal problems or constantto variabledensity crosssectional problems. A 3D version of SUTRA was developed and tested in 1982. After running several test problems with the 3D version of SUTRA, the threedimensional simulation capabilities were removed because of the limited computer resources that most users had access to in the early 1980s. Threedimensional problems that were sufficiently discretized required access to supercomputer facilities and had limited use to a small number of users (Voss, 1999). The decision to release a 2D version of SUTRA was appropriate and has allowed SUTRA to be widely used and applied to a variety of problems. The 2D version of SUTRA has been applied to a number of problems including: (1) unsaturated flow studies; (2) contaminant/tracer studies; (3) energytransport studies; 10
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(4) inverse modeling and optimization studies; (5) solutedependent variabledensity flow; and (6) temperaturedependent variabledensity flow. A summary of published SUTRA applications is given in Voss (1999). Recent improvements in computer processors and associated hardware have allowed an updated and improved 3D version of SUTRA to be released (Voss and Provost, 2002). Advances in techniques to solve numerical problems have allowed incorporation of robust, iterative solvers in the 3D version of SUTRA and made developing and solving 3D variabledensityflow problems practical. The 3D version of SUTRA retains all the original functionality of the 2D version and has been applied successfully to contaminanttransport problems and variabledensityflow problems. 2.2.3 The Current Study This chapter outlines the development of and some applications for a modified version of SUTRA: SUTRAMS. The principal modification in SUTRAMS is an extension of existing numerical methods to solve for the transport of multiple species and allow for dependence of density and viscosity on any of the simulated species. All the features of the original version of SUTRA have been retained in SUTRAMS, including variably saturated to fully saturated flow, advection, and production and decay of simulated species. Other modifications include: capability to simulate a spatially varying solid matrix thermal conductivity, capability to use either a volumetric average or geometricmean approximation for bulk thermal conductivity, ability to simulate simple timevarying boundary conditions that do not need to be user programmed, capability to output nodaland elementsimulation data at userspecified times, ability to use a simplified automatic timestepping algorithm to adjust time steps based on solution convergence, including the capability to rerun a time step if userspecified parameters are not achieved, 11
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ability to enter observation locations using spatial coordinates instead of node numbers, capability to specify hydraulic parameters using zones in order to reduce memory requirements for large twoand threedimensional problems and facilitate the use of generic parameterestimation codes, and use of Fortran 95 intrinsic functions or programming standards where beneficial to program execution times, maintainability, or sourcecode clarity. The current modifications do not place an arbitrary limit on the number of species that can be simulated. The number of species that can be simulated is a function of the problem size and the amount of available random access memory (RAM). Furthermore, the modifications are such that more complicated relationships between solute concentration and system state (e.g., density, etc.) can be easily incorporated. 2.3 PhysicalMathematical Basis of SUTRAMS Modifications This section summarizes the governing equations used in SUTRA to simulate fluid flow and solute transport in constantdensity and variabledensity environments. The modifications to the original equations are summarized here to provide background for the numerical implementation discussed in Section 2.4. To allow for simulation of multispecies transport with possible fluiddensity and viscosity dependence on solute concentrations and temperature, all equations that include fluid density, fluid viscosity, velocitydependent dispersion, molecular dispersion, thermal conductivity, solute adsorption, solute production and decay, solute or temperature boundary conditions, solute concentrations, or temperature require modification. Notations used in the equations are explained after their first use and also are summarized in appendix 1. 2.3.1 General MassBalance Formulation SUTRAMS is a modified version of SUTRA capable of simulating unsaturated to saturated, variabledensity fluid flow dependent on the transport of heat and multiple 12
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dissolved species. Because the modifications required to allow SUTRA to simulate transport of more than one species, including temperature, may affect both the flow and transport equations solved by SUTRA, a brief overview of the general massbalance equations used in SUTRA is given below. A more complete discussion of SUTRA fundamentals is given in Voss and Provost (2002). 2.3.2 Fluid MassBalance Equation The fluid massbalance equation, which is usually referred to as the groundwaterflow equation, is p r w w opwQ p k t U U p S t p p S SS g k )() ( Eq. 1 where Sw is the water saturation [], is the porosity [], p is the fluid pressure [M/LT2], t is time [T], U is either temperature [ C] or solute mass fraction [Msolute/Mfluid], k is the permeability tensor [L2], kr is the relative permeability for unsaturated flow [], is the fluid viscosity [M/LT], is the fluid density [M/L3], g is the gravitational vector [L/T2], Qp is a fluid mass source [M/L3T], and Sop is the specific pressure storativity [M/LT2]1. Specific pressure storativity, Sop, is defined as 1opS Eq. 2 13
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where is the compressibility of the porous matrix [M/LT2]1, and is the compressibility of the fluid [M/LT2]1. 2.3.3 Modified Form of the Fluid MassBalance Equation In SUTRAMS, the generalized fluid massbalance equation (Eq. 1) has been modified to allow for more than one species to affect fluid density. The modified fluid massbalance equation is p r k NS k k w w opwQ p k t T T p t C C p S t p p S SS g k 1) ( Eq. 3 where NS is the number of dissolved species simulated, = 1 if heat transport is being simulated and = 0 if otherwise, Ck is the solute concentration of species k [Msolute/Mfluid], and T is the temperature of the solution [ C]. The modified form of the fluid density, equation used in SUTRAMS is o NS k kok k oTT T CC C 1 Eq. 4 where o is the fluid density at the base concentration of all simulated species at the base solute concentrations and temperature [Msolute/L3 solute], Cko is the base solute mass fraction of species k [Msolute/Mfluid], and To is the base temperature [ C]. 14
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Equation 4 assumes / Ck and / T are constant over the range of simulated mass fractions, and temperatures and density relationships are linear and additive. This is an adhoc relationship but it is consistent with relationships used by others for similar applications ( e.g., Kipp, 1987). This adhoc relationship should be evaluated prior to use and modified if it is determined to be inappropriate for a particular application. The modified form of the fluid viscosity, used in SUTRAMS is NS k kok k oCC C T1 Eq. 5 where a userdefined base fluid viscosity for isothermal flow at the base mass fraction for each species [M/LT], or 239.4 x 10 7 1024837 T 13315 for temperaturedependent flow [M/LT]. If temperaturedependent flow is simulated, the units of o(T) can be converted to those desired using a scaling factor in the program input data. Equation 5 assumes / Ck is constant over the range of simulated mass fractions, the specified nonlinear viscositytemperature relationship is appropriate for the range of simulated temperatures, and viscosity massfraction relationships are linear and are not affected by species or temperature interaction. This is an adhoc relationship based on the concentrationdensity relationship described above. This adhoc relationship should be evaluated prior to use and modified if it is determined to be inappropriate for a particular application. 15
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2.3.4 Unified Energyand SoluteBalance Equation SUTRA uses a unified energyand solutebalance equation to solve both energy and solute transport. A unified energysolute balance is possible because of the similarity of the mechanisms affecting the fluxes of energy and solute mass in solution when energy and solute transport equations are formulated in terms of energy per unit volume and total species mass per unit volume, respectively. In fact, of the transport processes represented in SUTRA, only nonlinear sorption processes and firstorder production of solute and adsorbate mass have no analogy in energy transport. A massconservative form of the unified balance equation is solved by SUTRA and is derived from a general form of the equation in Voss and Provost (2002). The derivation of the massconservative form of the unified balance equation involves removing terms accounted for in the fluid massbalance equation (fluid saturation and pressurechange contribution to energy and solute balances) and terms related to change in the solid matrix density with time. The reader is referred to Voss and Provost (2002) for a more complete description of the derivation of the massconservative form of the unified balance equation. The massconservative form of the unified balance equation is Swcw1scs Ut SwcwvUcwSwwID1sIUQpcwU*USw1wU1s1sUsSwow1sos Eq. 6 where cw is the fluid specific heat [E/MC] for heat transport or one (1) for solute transport, s is the density of the solid matrix [M/L3], cs is solid matrix specific heat [E/MC] for heat transport or the sorption coefficient [] for solute transport, v is the fluid velocity [L/T], 16
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w is the fluid thermal diffusivity [L2/T] for heat transport or molecular diffusivity (Dm) for solute transport [L2/T], I is the identity tensor, D is the dispersion tensor [L2/T] for heat or solute transport, s is the solid matrix thermal diffusivity [L2/T] for heat transport or zero (0) for solute transport, U* is the temperature of the source fluid [C] for heat transport or the solute concentration of the source fluid [M solute/M fluid] for solute transport, w1 is zero (0) for heat transport or the firstorder massproduction rate of solute [T1] for solute transport, s1 is zero (0) for heat transport or the firstorder massproduction rate of adsorbate [T1] for solute transport, Us is the specific energy content of the solid matrix [E/M] for heat transport (but does not contribute to the unified balance equation because is zero (0)) or the concentration of the adsorbate on the solid matrix (Cs) for solute transport [M adsorbate/M solid matrix], s1 w0 is the zeroorder energyproduction rate in the fluid [E/MT] for heat transport or the zeroorder massproduction rate of solute [(M solute/M fluid)T1] for solute transport, and s0 is the zeroorder energyproduction rate in the solid matrix [E/MT] for heat transport or the zeroorder massproduction rate of adsorbate [(M adsorbate/M solid matrix and adsorbate)T1] for solute transport. Fluid thermal diffusivity, w, in Equation 6 is defined as wwwc Eq. 7 17
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where w is the fluid thermal conductivity [E/TLC]. Solid matrix thermal diffusivity, s, in Equation 6 is defined as wssc Eq. 8 where s is the solid thermal conductivity [E/TLC]. Three equilibrium sorption models, which specify cs in Equation. 6, are possible in SUTRA. These three sorption models assume fluid density is constant. The three sorption models are Linear equilibrium sorption model UUos 1 Eq. 9 tUtUos 1 Eq. 10 osc 11 Eq. 11 where 1 is the linear distribution coefficient [L3 fluid/M solid and adsorbate], and 1 is the first general sorption coefficient. 18
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Freundlich equilibrium sorption model 211UUos Eq. 12 tUUcdtUooss22121 Eq. 13 22121211 Ucos Eq. 14 where 1 is a Freundlich distribution coefficient [L3 fluid/M solid and adsorbate], and 2 is the Freundlich coefficient []. Langmuir equilibrium sorption model UUUoos 211 Eq. 15 tUUtUoos 2211 Eq. 16 22111Ucoos Eq. 17 where 1 is a Langmuir distribution coefficient [L3 fluid/M solid and adsorbate], and 2 is the Langmuir coefficient []. 19
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2.3.5 Modified Form of the Unified Energyand SoluteBalance Equation In SUTRAMS, the massconservative form of the unified balance equation (Eq. 6) has been modified to allow for simultaneous, sequential simulation of more than one species. The three forms of the sorption equations (Eqs. 9, 10, 11, 12, 13, 14, 15, 16, 17) also have been modified to allow simulation of multiple species, each of which can have different adsorption isotherms. The modified massconservative form of the unified balance equation is ksoskwowskksskkwwkkwkpkskkwkwwkkwkwkskswkwSUUSUUcQUScUcStUccS111111*IDIv Eq. 18 where cwk is the fluid specific heat [E/MC] for heat transport if species k is ENERGY or one (1) for solute transport of species k, csk is the solid matrix specific heat [E/MC] for heat transport if species k is ENERGY or the sorption coefficient [] for solute transport of species k Uk is either fluid temperature [C] if species k is ENERGY or concentration [M solute/M fluid] for solute transport of species k, Dk is the thermaldispersion tensor [L2/T] if species k is ENERGY or the solute transport dispersion tensor or [L2/T] of species k, sk is the solid matrix thermal diffusivity [L2/T] if species k is ENERGY or zero (0) for solute transport of species k, wk is the fluid thermal diffusivity [L2/T] if species k is ENERGY or molecular diffusivity, DM, [L2/T] of species k, 20
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*kU is the temperature of the source fluid [C] if species k is ENERGY or the solute concentration of the source fluid [M solute/M fluid] for species k, kw1 is zero (0) if species k is ENERGY or the firstorder massproduction rate of species k solute [T1], ks1 is zero (0) if species k is ENERGY or the firstorder massproduction rate of species k adsorbate [T1], Usk is the specific energy content of the solid matrix [E/M] if species k is ENERGY (does not contribute to the unified balance equation because ns1 is zero (0)) or the concentration of species k adsorbate on the solid matrix [M solute/M solid matrix], kw0 is the zeroorder energyproduction rate in the fluid [E/MT] if species k is ENERGY or zeroorder massproduction rate of species k solute [(M solute/M fluid)T1], and ks0 is the zeroorder energyproduction rate in the solid matrix [E/MT] if species k is ENERGY or zeroorder massproduction rate of species k adsorbate [(M adsorbate/M solid matrix and adsorbate)T1]. The modified forms of the three equilibrium models, which specify csk in Equation 18, are Linear equilibrium sorption model okkskc 11 Eq. 19 where 1k is the linear distribution coefficient [L3 fluid/M solid and adsorbate] of species k, and k1 is the first general sorption coefficient of species k. 21
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Freundlich equilibrium sorption model kkkkokkkskUc22121211 Eq. 20 where 1k is a Freundlich distribution coefficient [L3 fluid/M solid and adsorbate] for species k, and 2k is the Freundlich coefficient [] for species k. Langmuir equilibrium sorption model 22111kokokkskUc Eq. 21 where 1k is a Langmuir distribution coefficient [L3 fluid/M solid and adsorbate] for species k, and 2k is the Langmuir coefficient [] for species k. Equations 19 through 21 also assume fluid density, o, is constant and aqueous reactions among the simulated species are limited and can be ignored. The appropriateness of this assumption should be evaluated before using SUTRAMS for a particular application. Although sorption of multiple species has been implemented in a simple fashion, SUTRAMS is modular and a more sophisticated method of sorption could be implemented in the current program if required. More sophisticated sorption methods that account for aqueous interactions among dissolved species that also adsorb 22
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to the solid matrix have been implemented in a constantdensity version of SUTRA, SATRACHEM (Lewis and others, 1986), and could be used as a template for modifying SUTRAMS. 2.3.6 Modified Form of the EnergyBalance Equation used with GeometricMean Approximation for Bulk Thermal Conductivity The thermal conductivity of upper crustal materials generally varies by less than a factor of 5 whereas permeability can vary over 16 orders of magnitude (Ingebritsen and Sanford, 1998). SUTRA uses a volumetric average approximation for bulk thermal conductivity (Eq. 22). swwS 1 Eq. 22 where is the bulk thermal conductivity. Empirical evidence shows that bulk thermal conductivity is well modeled using the geometric mean of the rock conductivity and fluid conductivity (Sass and others, 1971). The geometricmean approximation of bulk thermal conductivity modified for variably saturated media is )1()1(sSwwSwA Eq. 23 where A is the thermal conductivity of air. The thermal conductivity of air is a function of temperature but varies by less than 15 percent (0.025 to 0.027 W/mC) over typical appropriate temperature ranges for 23
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SUTRAMS simulations (between 5 and 50 C). Because the thermal conductivity of air is approximately 1 to 2 orders of magnitude less that the thermal conductivity of water and geologic materials, respectively, a constant value of 0.026 W/m C is used in bulk thermalconductivity calculations when the geometricmean approximation is used. 00.20.40.60.81 Porosity 0.01 0.1 1 10Bulk The r mal Conductivity (W/moC) 00.20.40.60.81 Porosity 0.01 0.1 1 10Bulk Thermal Conductivity (W/moC) B = SWW+(1)SB = (A (1SW)W SW)S (1) s=5 00.20.40.60.81 Porosity 0.01 0.1 1 10Bulk The r mal Conductivity (W/moC) 00.20.40.60.81 Porosity 0.01 0.1 1 10Bulk Thermal Conductivity (W/moC) 00.20.40.60.81 Po r osit y 0.01 0.1 1 10Bulk The r mal Conductivity (W/moC) 00.20.40.60.81 Po r osit y 0.01 0.1 1 10Bulk Thermal Conductivity (W/moC) s=4 s=3 s=5 s=4 s=3Fluid Saturaion Sw1.0 0.8 0.6 0.4 0.2 0.0 Figure 2 Comparison of bulk thermal conductivity as a function of porosity, fluid saturation, and solid matrix thermal conductivity using a volumetric average and geometricmean approximations. 24
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Use of a geometricmean approximation for bulk thermal conductivity typically results in a reduction of approximately 15 percent in bulk thermal conductivity from volumetric average bulk thermal conductivities at typical porosities and solid matrix thermal conductivities. The differences in bulk thermal conductivities calculated using the volumetric average and geometricmean approximations generally increase with increasing porosity and/or decreasing fluid saturation (Figure 2). When the geometric mean thermalconductivity approximation is used, bulk thermal diffusivity in Equation. 6 is defined as wswSwwSAbgc11 Eq. 24 where bg is the bulk thermal diffusivity [L2/T]. Equation 2.18 is modified for energy transport in the following manner when using a geometricmean approximation for bulk thermal conductivity ksoskwowskksskkwwkkwkpkkwbgkwkkwkwkskswkwSUUSUUcQUScUcStUccS 11111*DIv Eq. 25 where bgk is the bulk thermal diffusivity [L2/T] if species k is ENERGY or molecular diffusivity, DM, [L2/T] of species k. 25
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2.4 Numerical Methods As a result of the modified fluid massbalance equation (Eq. 3) and the modified unified energyand solutebalance equation (Eq. 18), the discretized, weightedresidual relations approximating these equations have been modified in SUTRAMS. Complete development of the discretized, weighted residual approximations of Equations 1 and 6 are given in Voss and Provost (2002). Explanation of the notation used in the equations developed in Section 2.4 is given after its first use in an equation and also is summarized in Appendix A. 2.4.1 Numerical Approximation of SUTRAMS Fluid Mass Balance The modified unified energyand solutebalance equation (Eq. 3) is approximated numerically through nodewise, elementwise, and cellwise discretization. The weighted residual relation that allow for solution of pressures at nodes at the end of the present time step is NN1,i 1111*11111111NSkniknikninnininiBCpininjNNjijpinijnijnidtdUCFptAFDFpQpBFtAF Eq. 26 where iiwopwniVpSSSAF1 Eq. 27 iikwnikVUSCF1 Eq. 28 26
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dxdydz k BFi xyz j r L n ij 1 k Eq. 29 dxdydz k DFi xyz r L n i *1 g k Eq. 30 ij is the Kronecker delta: ji jiij if1 if 01ntis the length of the current time step, 1n ipis the pressure in cell i at the end of the current time step, n ipis the pressure in cell i at the end of the previous time step, 1n iQis the total mass source [M/T] to cell i for the current time step, pi is the pressurebased conductance [LT] for the specified pressure source in cell i (zero (0) for all nodes that are not specified pressure nodes), 1 n BCipis the specified pressure in cell i,(zero (0) for all nodes that are not specified pressure nodes), n ikdt dU is the rate of change in solute concentration of species k in cell i during the previous time step n n ik n ikt UU1, Vi is the volume of cell i, j is the symmetric bilinear (2D) or trilinear (3D) basis function in global coordinates for node j i is the asymmetric weighting function in global coordinates for node i, Lk is the permeability tensor [L2] that is discretized elementwise, and 27
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*gis an elementwise discretization of (g) that is consistent with the discretization of p. For twodimensional simulations, equations 29 and 30 are written as dxdyB k BFi xy j r L ij k Eq. 31 dxdyB k DFi xy r L i g k Eq. 32 where the thickness of the mesh, B(x,y) is evaluated at each Gauss point according to the following nodewise discretization NN i iiyxByxB1),( Eq. 33 The superscript involving (n) or ( n+1) indicates the value is evaluated at the end of the previous time step and the end of the current time step, respectively. The only integrals requiring Gaussian integration are BFij and DFij. These integrals are evaluated in an elementbyelement manner in the SUTRAMS subroutine ELEMN2 (for 2D) or ELEMEN3 (for 3D). The other terms, except for those involving pi, are evaluated cellwise (one for each node) by the SUTRAMS subroutine NODAL. The specified pressure terms are evaluated by subroutine BC. Equation 26 is assembled for each node in the model mesh in a given time step and is solved using either Gaussian elimination or an iterative solver. For the iterative solvers, an approximate solution is achieved when the residual error (summation of Eq. 26 for each node) is less than a user28
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specified convergence criteria. Details of the flow and transport solution scheme used in SUTRAMS is given in Section 2.4.4. 2.4.2 Numerical Approximation of SUTRAMS Unified Energyand SoluteBalance Equation The modified unified energyand solutebalance equation (Eq. 18) is approximated numerically through nodewise, elementwise, and cellwise discretization. The weighted residual relations that allow for solution of concentration or temperature at nodes at the end of the present time step are NS1,k NN1,i 1 1 1 1 1 1 1 *11 1 1 1 1 1 1 *1 1 1 n ik n n ik n iks n ik n IN n UBCU n BC n BC n ik n iwk n ik NN j ijwk n BC n i n iks n ik U n ijk n ijk n ij n ikU t AT TRG ET U UQUQc U cQQTLGGT BT DT t ATik ik ik ik i i ik Eq. 34 where i i sks wkw n ikVc cSAT 11 Eq. 35 dxdydz cS DTi xyz j wkw n ijk *1 v Eq. 36 dxdydz Sc BTi xyz j L sk k wkw wk n ijk 11 I DI Eq. 37 29
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iiwkwnikVSGT11 Eq. 38 iiLksksniksVsTLG111 Eq. 39 iiRksksniksVsTRG111 Eq. 40 dScinLskwkwwknikINnIDI111 Eq. 41 iiskswkwnikVSET0011 Eq. 42 1nikU is the concentration or temperature of species k in cell i at the end of the current time step, nikU is the concentration or temperature of species k in cell i at the end of the previous time step, niBCQ is the total solute [M/MT] or energy source [E/T] of species k to cell i due to a specified pressure for the previous time step, Uik is the concentration [LT] or temperaturebased conductance [LT] for the specified concentration or temperature of species k in cell i (zero (0) for all nodes that are not specified pressure nodes) for the current time step, sLk is the sorption isotherm contribution to the lefthand side of equation 34 for species k (discussed further in section 2.4.3), sRk is the sorption isotherm contribution to the righthand side of equation 34 for species k (discussed further in section 2.4.3), 1nikBCU is the specified concentration or temperature of species k in cell i for inflow due to a specified pressure (zero (0) for all nodes that are not specified pressure nodes) for the current time step, n is the unit outward normal vector, 30
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j is the symmetric bilinear (2D) or trilinear (3D) basis function in global coordinates for node j, is the external boundary area of the simulated region [L2], L s is the solid matrix thermal diffusivity [L2/T] that is discretized elementwise, and is an elementwise discretization of porosity. For twodimensional simulations, equations 36 and 37 reduce to dxdyB cS DTi xy j wkw n ijk *1 v Eq. 43 dxdyB Sc BTi xy j L sk k wkw wk n ijk 11 I DI Eq. 44 where the thickness of the mesh, B(x,y), is evaluated at each Gauss point according to equation 33. For threedimensional simulations, when the geometricmean approximation for thermal conductivity is used, equation 37 for energy transport is modified to dxdydz S c BTi xyz j kw L bgk wk n ijk 1 D I Eq. 45 31
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where Lbgk is the bulk thermal diffusivity [L2/T], which includes the constants A and w, and is approximated using a geometric mean and is discretized elementwise if species k is ENERGY or molecular diffusivity (DM) of species k for solute transport, which is a constant value for all elements. For twodimensional simulations, the equivalent modification to equation 44 for simulation of heat transport using a geometricmean approximation for thermal conductivity is dxdyBScBTixyjkwLbgkwknijk 1DI Eq. 46 Equations 37 and 44 are unmodified for solute transport. Equation 34 is assembled for each node in the model mesh for a given species and time step. The only integrals requiring Gaussian integration are BTij and DTij. These integrals are evaluated in an elementbyelement manner in the SUTRAMS subroutine ELEMN2 (for 2D) or ELEMEN3 (for 3D). The other terms, except for those involving Uik, are evaluated cellwise (one for each node) by the SUTRAMS subroutine NODAL. The specified concentration terms are evaluated by subroutine BC. The matrix assembled for a given species is solved using either Gaussian elimination or an iterative solver. For the iterative solvers, an approximate solution is achieved when the residual error (summation of eq. 34 for each node) is less than a specified convergence criterion for a given species. After an approximate solution is achieved for a given species, equation 34 is assembled for the next species. This process is continued until an approximate solution is achieved for each species simulated. Details of the flow and transport solution scheme used in SUTRAMS are given in Section 2.4.4. 32
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2.4.3 Temporal Evaluation of Adsorbate Mass Balance The terms in the modified unified energyand solutebalance equation (eq. 34) that stem from the adsorbate mass balance () require particular temporal evaluation because some are nonlinear. For solute transport, the coefficient, in becomes 111 and ,nikniknikETGTAT iskc 1nikAT ink11 The relation that defines ink11 is given by either equation 19, 20, or 21, depending on the sorption isotherm simulated. The variable, insU1 is expressed in terms of the concentration of adsorbate inskC1 in a form given by RkLkskssCsCU Eq. 47 where Csk is the concentration of the adsorbate for species k, and all other terms are as previously defined. The parameters in equation 47, sLk and sRk, are defined in this section based on the simulated sorption isotherm (eq.19, 20, or 21). For linear sorption, all terms and coefficient related to adsorbate mass are linear and are evaluated at the new time level and strictly solved for at this level: inkokinskinskCCU1111 Eq. 48 okinkinskC1111 Eq. 49 okLks 1 Eq. 50 0 Rks Eq. 51 33
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For Freundlich sorption, the adsorbate concentration is split into a product of two parts for temporal evaluation. The first part is treated as a firstorder term, such as linear sorption, evaluated at the new time level, and solved for on each iteration or time step. The last part is evaluated as a known quantity, either based on a projected value of the initial solute concentration at the end of the time step on the first iteration, or based on the most recent concentration, Cik, on any subsequent iteration. Freundlich adsorption is determined using: 122121111nikkiprojkkokinskinskCCCU Eq. 52 okinkinskC1111 Eq. 53 221211kiprojkkokLkCs Eq. 54 0 Rks Eq. 55 where the coefficient, is evaluated using the projected or most recent value of Cki, depending on the iteration. 11nk For Langmuir sorption, the form used preserves the dependence on a linear relationship to Cki. The linear relationship is appropriate only at low solute concentrations. At high concentrations, the adsorbate concentration approaches (1k2k). Therefore, one temporal approximation for low concentrations and one temporal approximation for high concentrations are used. When 2oCki << 1, the following approximation for low values of C, referred to as is used 0skC 34
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projkokprojkoknkokskCCCC2211011 Eq. 56 When 2oCki >> 1, the following approximation for high values of C, referred to as is used skC projkoknkokkskCCC2121111 Eq. 57 Thus inskC1 can be defined as skkskkinskinskCWCWCU0011 Eq. 58 where the weights Wo and W are projkokprojkokkCCW221 Eq. 59 kkWW 10 Eq. 60 By substituting equations. 56, 57, 59, and 60 into equation 58, the following temporal evaluation of inskC1 is obtained after algebraic manipulation 22212211111projkiokprojkiokprojkiokprojkioknkiokinskCCCCCC Eq. 61 The coefficient, is defined as 11nk 35
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2211111projkiokokinkinskCC Eq. 62 2211projkiokokLkCs Eq. 63 222211projkiokprojkiokkRkCCs Eq. 64 The first term in equation 61 is solved every iteration, and the second term is treated as a known value. In equations 61, 62, 63, and 64, is based on a projection for the first time step and is the most recent value of Cki on subsequent iterations for the time step. projkiC 2.4.4 Solution Sequencing On any given time step, the matrix equations are created and solved in the following order: (1) set up the matrix equation for the fluid mass balance, (2) set up the transportbalance matrix equation for the first species, (3) solve for pressure, (4) solve for concentration or temperature of the first species, and (5) set up the transportbalance matrix equation for the second species and solve for each of the remaining species (k=2,NS+). The balances for fluid mass transport for the first species are set up at the same time to limit elementwise calculations to a single pass. Elementwise calculations for each of the remaining species are done individually, after a solution is achieved for the preceding species. Fluid flow and transport of all the species (NS+) are not solved in a single pass in order to keep storage requirements reasonable for multispecies simulations. Functionality exists in SUTRAMS to set up and solve either the solute mass balance or fluid mass balance only every few time steps in a cyclic manner based on parameters NPCYC and NUCYC. This functionality is derived from SUTRA, and the 36
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values NPCYC and NUCYC represent the solution cycle in time steps. Currently, a unique NUCYC value cannot be specified for each species but could be easily implemented. Examples include setting up and solving for both flow and transport for three species each time step (NPCYC = NUCYC = 1): Time step 1 2 3 4 5 6 7 Assemble equations for { p U1 p U1 p U1 p U1 p U1 P U1 p U1 Solve for { p U1 p U1 p U1 p U1 p U1 p U1 p U1 Assemble equation for U2 U2 U2 U2 U2 U2 U2 Solve for U2 U2 U2 U2 U2 U2 U2 Assemble equation for U3 U3 U3 U3 U3 U3 U3 Solve for U3 U3 U3 U3 U3 U3 U3 or solving for flow every three time steps and transport for three species each time step (NPCYC = 3 and NUCYC = 1): Time step 1 2 3 4 5 6 7 8 9 10 11 12 13 Assemble equations for { p U1 U1 P U1 U1 U1 p U1 U1 U1 p U1 U1 U1 p U1 U1 Solve for { p U1 U1 P U1 U1 U1 p U1 U1 U1 p U1 U1 U1 p U1 U1 Assemble equation for U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 Solve for U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 Assemble equation for U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 Solve for U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 U3 37
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However, either flow or transport must be solved on each time step and requires setting either NPCYC or NUCYC to one (1). For a simulation with steady flow and transient transport of three species, the sequencing is: Time step 0 1 2 3 4 5 Assemble equations for { p U1 U1 U1 U1 U1 U1 Solve for { p U1 U1 U1 U1 U1 U1 Assemble equation for U2 U2 U2 U2 U2 U2 Solve for U2 U2 U2 U2 U2 U2 Assemble equation for U3 U3 U3 U3 U3 U3 Solve for U3 U3 U3 U3 U3 U3 For a simulation with steady flow and steady transport of three species, the sequencing is: Time step 0 1 Assemble equations for { p U1 Solve for { p U1 Assemble equation for U2 Solve for U2 Assemble equation for U3 Solve for U3 38
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The only exception to the cycling is that for nonsteady cases, both unknowns are solved on the first time step, as shown in the case for NPCYC = 3 and NUCYC = 1, above, and on the last time step, regardless of the values of NPCYC and NUCYC. It is computationally advantageous to avoid unnecessarily reconstructing the transport equation and, when the direct solver is used, to avoid the transport matrix decomposition steps by allowing for back substitution only. This is begun on the second time step by solving for transport only after the time step on which both fluid mass balance and transport are solved. To do this, the matrix coefficients (including the time step) must remain constant. Thus, nonlinear variables and fluid velocity are held constant with values used on the first time step for transport after the step for flow and transport. An example is when NPCYC = 6 and NUCYC = 1 for a simulation with two species: Time step 1 2 3 4 5 6 7 8 9 10 11 12 Assemble equations for { p U1 U1 U1 U1 U1 p U1 U1 U1 U1 U1 U1 p U1 Solve for { p U1 U1 U1 U1 U1 p U1 U1 U1 U1 U1 U1 p U1 Assemble equation for U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 Solve for U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2 U2  constant values   constant values  backsubstitute  backsubstitute  Note that flow and transport solutions must be set to occur on time steps when relevant boundary conditions, such as sources or sinks, are set to change in value. 2.5 Additional SUTRAMS Options This section outlines the additional options available in SUTRAMS that have not been previously discussed. A general discussion of each option is given and how it is 39
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implemented in SUTRAMS. Details on input data required to execute these options are given in Appendix B. 2.5.1 Simple TimeVarying Boundary Conditions Simple functionality has been included in SUTRAMS that allows timevarying boundary conditions to be used without the requirement that they be user programmed in the subroutine BCTIME. It is assumed that boundary conditions are constant between userspecified times (timevarying boundary conditions are step functions). The simple functionality comprises a single subroutine (RDTBCS) that reads the time when boundary conditions are modified and the updated boundarycondition data. An internal logical parameter controls whether the subroutine should process time data or boundarycondition data. When reading boundarycondition data, the user has complete flexibility in determining which boundary conditions are modified. There is no requirement that the type or number of boundary conditions modified be the same for every boundarycondition modification time period (stress period). For example, the following could be implemented with the simple timevaryingboundary condition option: Boundary Condition PBC1 PBC2 PBC3 PBC4 U(1)BC1 U(1)BC2 U(1)BC3 U(2)BC1 QfBC1 QU(2)BC1 Stress Period 2 Stress Period 3 Stress Period 4 Stress Period 5 Stress Period 6 Where PBCi is specified pressure boundary condition i with specified temperatures and/or solute concentrations, U(n)BCi is specified concentration i for species n, QfBCi is specified fluid flux i, and QU(n)BCi is specified solute or heat flux i for species n. The bullets 40
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indicate that the specified boundary condition is modified at the beginning of the specified time step. Boundary conditions that are not modified retain values from the previous time step or the initial values, depending on the current stress period. Unlike previous versions of SUTRA, a negative node number should not be specified for transient boundary conditions that use the simple timevaryingboundary condition routine. The simple timevaryingboundary condition routine can be used in conjunction with the standard userprogrammable routine BCTIME. Time data are read as simulation time, and boundary conditions for the next stress period are read once the specified simulation time is reached. The timestep length is automatically adjusted, if necessary, to ensure that the beginning of each stress period coincides with the beginning of a time step. In order to accurately simulate earlytime transient effects when boundary conditions are modified, the timestep interval is reduced to the minimum timestep length (initial time step length). In subsequent time steps, the time step is allowed to increase according to the userspecified timestep multiplier and number of time steps between increases in time steps. Error checking has been included in the subroutine RDTBCS to ensure that there are no formatting errors in the transient boundarycondition data set. 2.5.2 Specified User Output Times Simple functionality has been included in SUTRAMS that allows simulation output to be saved at userspecified times. This functionality has been added because the standard version of SUTRA prints output at a userspecified timestep interval. This sometimes makes it difficult to get output at a specific time, especially if a timestep multiplier is used, and can result in large output files with unwanted or unnecessary data. A single subroutine processes output times (RDPRNT), and output of simulation results at specified times is done within the SUTRA subroutine. Whether output is printed for a particular item (i.e., velocity in the xdirection) depends on whether this item has 41
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been included in the output control for the listing file, the nodal data file, or the elemental data file, and whether this file has been included in SUTRA.FIL. Standard SUTRA output printing at fixed time steps is disabled when the userspecified output time option is implemented. Time data are read as simulation time, and output is written at the end of the time step when the specified output time is reached. The timestep length is automatically adjusted, if necessary, to ensure that time steps coincide with specified output times. 2.5.3 Simple Automatic TimeStepping Algorithm A simple method to reduce the timestep length automatically, as needed, has been included in SUTRAMS. A single subroutine has been added to process input data for the option (RDATS), and all logic to control the option is contained in the SUTRA subroutine. This option has been added to minimize termination of SUTRAMS simulations due to nonconvergence. This was found to be beneficial during development of SUTRAMS because of the long run times experienced with several of the sample problems developed during code testing and the need for a reduced timestep length for select periods in the simulations. The algorithm monitors the maximum number of iterations required for convergence of the pressure and transport solutions. If the number of iterations exceeds a userspecified criteria, the time step is reduced using the following equation tNEW = tn+1 / DTMULT2 Eq. 65 where tNEW is the revised timestep length [sec], t(N+1) is the original timestep length for the current time step [sec], and DTMULT is the userspecified timestep multiplier [ ]. 42
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The square of DTMULT is used to calculate tNEW in order to reduce the timestep length to the value that was used prior to the last update (two previous timestep cycle changes (ITCYC)). This increases the likelihood that the reduced timestep length is sufficient to overcome convergence problems resulting from inappropriate timestep cycle change parameters and/or maximum timestep lengths. A userspecified minimum timestep length is used as the lower limit on the timestep length. If tNEW is less than the specified minimum timestep length, then it is set to the minimum timestep length. No operations are done on the timestep length if it is already equal to the minimum value. The form of the equation used to reduce the simulation timestep length (eq. 65) was chosen for its simplicity and its general appropriateness for most applications. The implementation of the simple automatic timestepping algorithm is general enough that another method for reducing the time step (e.g., based on Courant criteria, etc.) could easily be implemented. Two options are available that control how the simulation proceeds after the userspecified maximum iteration criterion is exceeded. The first and simplest option is to reduce the timestep length and proceed to the next time step. The second option is more involved and allows the timestep length to be reduced and the solution to be rerun using the new timestep length and results from the previous time step. This option requires resetting the current solution results (n+1) to the solution results from the previous time step (n) and results from the end of the previous time step (n) to results from the time step prior to the previous time step (n1). This requires additional storage to save the pressure and transport solutions from the previous time step (n), the pressure and transport solutions from the time step prior to the previous time step (n1), the density, adsorbate mass for all species, and saturation from the previous time step (n). To prevent excessive recalculation of the current time step when the second option is selected the timestep length is reduced and the solution is rerun until a userspecified number of reruns are completed or the userspecified maximum number of iterations criterion is satisfied. Increases in timestep length are handled by standard SUTRA timestepadjustment algorithms. The simple automatic timestepping algorithm in combination 43
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with standard SUTRA timestep algorithms affords the user great flexibility in tailoring the solution scheme to specific problems. However, because the simple automatic timestepping algorithm requires the maximum allowable iteration criterion be set by the user, this parameter should be adjusted to provide the optimal value. A maximum iteration criterion that is excessively small will result in excessive timesteplength reductions and may unnecessarily increase run times. Conversely, a maximum iteration criterion that is excessively large will reduce the effectiveness of the algorithm and may allow nonconvergent solutions to occur. For optimal performance, DTMULT and the number of time steps in a timestep cycle change, ITCYC, used by the standard SUTRA timestep algorithms, should be adjusted in combination with tuning of simple automatic timestepping parameters for best results. It should be noted that this algorithm does not ensure that convergence will be achieved for all model setups. Chronic nonconvergence may indicate problems with the data set, discretization that does not meet stability requirements (Peclet criteria), timestep lengths that exceed stability criteria (Courant criteria), or incorrect boundary conditions. 2.5.4 Specified Observation Locations Simple functionality has been included in SUTRAMS that allows specification of observation locations using actual coordinates. This functionality has been added because the standard version of SUTRA requires observation locations be specified using node locations. This additional functionality allows definition of observations to be mesh independent. Specified observation location input data are processed by a single subroutine (Allo_Rd_SobData), which also allocates the required storage, and a single subroutine to calculate the closest node to the specified location (CalcObsNode). The correspondence of a specified observation location to a specific node is printed in the standard output file (.lst), and information also is given if the specified observation location is outside the mesh. If a specified observation location is outside the mesh, the closest node is used; a warning message is written to the output file, but the simulation proceeds. 44
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2.5.5 Specification of Hydraulic Parameters Using Zones Functionality has been included in SUTRAMS that allows hydraulic parameters to be specified using zones. This additional functionality can significantly reduce memory requirements through use of data structures dimensioned based on the number of hydraulic zones rather than vectors of hydraulicparameter data dimensioned based on the number of nodes and elements. The ability to specify hydraulic parameters using zones reduces the effort required to set up and use SUTRAMS with parameterestimation codes (i.e., UCODE or PEST). In order to reduce the memory requirements for problems using zones, the original data structure for hydraulic parameters was modified from vectors dimensioned by nodes or elements to a number of data structures dimensioned by zones. A simple representation of the differences between the modified data structure and the original SUTRA structure is shown in Figure 3. A separate data structure has been established for nodewise and elementwise discretized hydraulic data. Currently, the nodewise data structure includes porosity, matrix compressibility, and matrix density. The elementwise data structure includes permeability, permeability angle, dispersivity, and matrix thermal conductivity. Except in the case where the number of node and element zones is equal to the number of nodes and elements, memory requirements are less for problems that specify hydraulic parameters using zones. When using zones, all hydraulicparameter data are specified in the .zon file, and the spatial distribution of nodal and elemental zones are specified using the existing NREG and LREG vectors that were used previously to specify the distribution of unsaturated zone properties. Addition of matrix compressibility and matrix density to nodewise parameters that can be varied on a zone basis allows more flexibility with storage and matrix density properties than was possible with SUTRA. This means that transient responses due to storage differences and spatial differences in adsorption resulting from differences in aquifer materials can be accommodated directly without additional modifications to the source code. 45
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Elements26Zones2ElementPMAXPMINANGLEXALMAXALMINATMAXATMINSIGMAS*1A1A2A3A4A5A6A7NA2B1B2B3B4B5B6B7NA...........................26Z1Z2Z3Z4Z5Z6Z7NABytes2082082082082082082081Total Bytes1457Zone12PMAXA1B1PMINA2B2ANGLEXA3B3ALMAXA4B4ALMINA5B5ATMAXA6B6ATMINA7B7SIGMASA8B8Bytes6464Total Bytes128* SIGMAS is constant for all nodes in SUTRANA Not Applicable Figure 3 Simple representation of the differences in memory requirements for hydraulic parameters that are discretized by elements for SUTRA and SUTRAMS using zones. The data structure used to specify hydraulic parameter zones has been made general so that it can be easily extended to add additional parameters. For example, it could easily be extended to allow specification of unsaturated zone properties and would eliminate the need to use the user programmed UNSAT subroutine for unsaturated flow problems. All input processing is handled by a single subroutine (RdZoneData), which also includes some functionality to write zone information to the standard output file (.lst). The modifications resulting from implementation of the zone module are widespread 46
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throughout the code. Every location where an element of a hydraulicparameter vector was called for has been replaced with the equivalent zone datastructure element. An example of the modification is: PERMXX (L) ElemData(ElemMap(L))%permxx where, PERMXX (L) is the permeability of element L in the XX direction, ElemData() is the hydraulic permeability data structure for elementwise discretized hydraulicparameter data, ElemMap(L) is the zone number for element L (LREG(L) in SUTRA), and ElemData(ElemMap(L))%permxx is the permeability of zone ElemMap(L) in the XX direction. Error checking has been included in the subroutine RdZoneData to ensure that there are no formatting errors in the node and element zone data sets. In addition, a subroutine (MkZoneSOP) to calculate the specific storage for each zone replaces the standard calculation in subroutine INDAT1. 47
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2.6 SUTRAMS Simulation Examples This section outlines three example problems that demonstrate some of the specific capabilities of SUTRAMS. The examples show results that are compared with numerical solutions from SUTRA or other numerical codes. In several cases, the examples demonstrate some interesting applications possible with SUTRAMS. Additional examples of the capabilities of SUTRA are contained in Voss and Provost (2002). 2.6.1 Densitydependent flow, heat transport, and solute transport, Solution for multicomponent fluid flow in a saline aquifer system 2.6.1.1 Physical Setup This example considers seawater intrusion into both an isothermal and a nonisothermal confined aquifer studied in cross section under steadystate conditions (Henry and Hilleke, 1972). Freshwater recharge flows from an inland boundary over more saline water derived from a seaward boundary and discharges in the upper portions of a vertical sea boundary. The problem is nonlinear and is solved by gradually approaching steady state with a series of time steps. Initially the aquifer has hydrostaticpressure conditions with either isothermal and 0.0 percent seawater concentrations or linearly varying percent seawater concentrations and temperatures. At time zero, heat begins to be transported inward from the top, bottom, and left boundaries, and seawater begins to intrude the freshwater system by moving in laterally under the freshwater from the right (seaward) boundary. Seawater intrusion in the aquifer is primarily the result of the greater density of the seawater. Lateral temperature variations at the top and bottom of the aquifer and vertical temperature variations at the freshwater boundary increase vertical freshwater fluid movement and intrusion of seawater at the base of the aquifer when compared to a similar isothermal case. 48
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Dimensions of the problem were selected to allow comparison with the steadystate dimensionless solution of Henry and Hilleke (1972). Two different aspect ratios are evaluated to assess the impact of the aquiferaspect ratio (ratio of aquifer length to height {}) on simulation results. Two different total simulation times (3.3 and 33.5 hours) were used for the two aspect ratios evaluated; both simulation times are sufficient for the simulations to reach steadystate conditions. 2.6.1.2 Simulation Setup Two different meshes were used in order to evaluate the effect of aquiferaspect ratios on the simulation results. The first mesh consists of 40 by 40 elements, each of size 0.025 m (meters) by 0.025 m, and was used for =1. The second mesh consists of 50 by 501 elements, each of size 0.02 m by 0.02 m, and was used for =0.10. An example of the mesh geometry for =1 is shown on Figure 4. The thickness was 1.0 m in the ydimension for both meshes. A constant timestep length of 60 and 30 seconds was used for =1 and =0.10, respectively. A total of 200 and 4,022 time steps were taken for the =1 and 0.10 simulations. The number of time steps was increased for the =0.10 simulations because the problem is an order of magnitude larger in the xdimension than the =1 problems. A reduced timestep length also was used for the =0.10 simulations to reduce numerical oscillations. Pressure, concentration, and temperature were solved on each time step (NPCYC=NUCYC=1). 49
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2.6.1.3 Parameters k = 1.02041109 [m2] based on K = 0.01 [m / s] b = 1.0 [m] = 0.35 [] g = 9.8 [m / s2] CS = 0.0357 [kgdissolved solids / kgseawater] CO = 0.000 [kgdissolved solids / kgseawater] S = 1024.99 [kg / m3] O = 1000. [kg / m3] SOLID = 2600. [kg / m3] QIN = 8.333102 [kg /s] distributed over 41 nodes or 8.333103 [kg /s] distributed over 51 nodes C = 700. [kgseawater2 / (kgdissolved solids m3) ] = 1103 [ kg / (m s)] T = 0.375 [kgseawater / (C m3) ] L = T = 0.0 [m] DSOLUTE = 2.381105 [m2 / s] or = 2.381107 [m2 / s] DTEMPERATURE = 2.381104 [m2 / s] or = 2.381106 [m2 / s] W = 995.7342 [J / (m C s)] or = 9.957 [J / (m C s)] SOLID = 0.0 [J / (m C s)] cW = 4182. [J / (kg C)] cSOLID = 0.0 [J / (kg C)] TMIN = 5 C TMAX = 50 C 50
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Figure 4 Finite element mesh (=1) and pressure (gray), concentration (blue), and temperature (red) boundary conditions for Henry and Hilleke (1972) solution. 2.6.1.4 Boundary Conditions Noflow conditions were maintained across the top and bottom boundaries. A freshwater source, with a concentration of 0.000 kgdissolved solids/kgfreshwater for all cases and a temperature that varies linearly from 50C at y=0 m to 38.75C at y=1 m for the nonisothermal case, was implemented by using fluid source nodes at the left vertical boundary. The right vertical boundary was held at hydrostatic pressure, assuming a constant density equal to that of seawater, through use of specified pressure nodes. Any water entering through these nodes has the concentration, CS, of seawater for all cases and a temperature, TMIN, of 5C for the nonisothermal case. Temperatures were not assigned to the specified flux nodes on the left boundary or the specified pressure nodes on the right boundary in the isothermal case. 51
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In the original Henry and Hilleke (1972) simulation, fluid concentrations were fixed on the left and right boundaries and temperatures were fixed at all the external boundaries. In order to replicate the original results, specified concentration nodes were assigned to the right and left boundaries for the isothermal and nonisothermal cases. In the nonisothermal case, specified temperature nodes were assigned to the right, left, top, and bottom boundaries. Specified temperature nodes were not assigned in the isothermal case. Use of specified concentration and temperature nodes at the boundaries also allow the SUTRAMS simulations to be compared directly with results using HST3D (Kipp, 1987). Concentrations of 0.0357 and 0.0000 kgdissolved solids/kgseawater were assigned to the specified concentration nodes along the right and left boundaries, respectively. In the nonisothermal case, a constant temperature of 5C was assigned to the specified temperature nodes along the right boundary, and temperatures ranging from 50 to 38.5C were assigned to specified temperature nodes along the left boundary. Additionally, in the nonisothermal case, a specified temperature varying from 50 to 5C and 38.5 to 5C in the direction of increasing x was assigned to specified temperature nodes on the bottom and top boundaries, respectively.. The assigned pressure and concentration boundary conditions are shown graphically on Figure 4A. Assigned temperature conditions for the nonisothermal case are shown graphically on figure 5.1.1B. 2.6.1.5 Initial Conditions Hydrostatic pressures based on an initial solute concentration of 0.000 kgdissolved solids /kgseawater and temperature of 5C were used as initial conditions for =1 simulations. Hydrostatic pressures based on initial solute concentrations and temperatures linearly varying from 0.000 to 0.0357 kgdissolved solids/kgseawater and 44.375 to 5.000C from x = 0.0 to 10.0 m were used for =0.10 simulations. Linear variations in initial solute concentrations and temperatures were used for =0.10 simulations in an effort to provide initial conditions that were closer to steadystate conditions and to minimize numerical instabilities. 52
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2.6.1.6 Results Henry and Hillekes (1972) solution assumes that total solute dispersion can be accounted for through use of a large solute moleculardiffusion coefficient rather than a combination of velocitydependent dispersivity and molecular diffusion. Similarly, thermal dispersion is simulated using a large fluid thermal conductivity and insulated solid matrix. Use of an insulated solid matrix is an appropriate simplification because use of a large fluid thermal conductivity is equivalent to the net effect of a smaller, more reasonable fluid thermal conductivity and a physically realistic solid matrix thermal conductivity. The total solute and thermal dispersion coefficient of Henry and Hilleke (1972), DM and DT, are equivalent to the product of the porosity and solute and thermal molecular diffusivity, respectively (Eqs. 66 and 67). DM = DSOLUTE Eq. 66 DT = DTEMP = WWc Eq. 67 Henry and Hillekes results are given for two of their nondimensional simulations: N = 3 and NC = 10 for an isothermal case and N = 3, NC = 10, and NT = 1 for a nonisothermal case. The dimensionless parameters N, NC, and NT are defined in equations 68, 69, and 70, respectively. N = TQKbR' Eq. 68 NC = MTDQ Eq. 69 53
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NT = TTDQ Eq. 70 where K is the hydraulic conductivity [L/T], b is the aquifer thickness [L], R is the fractional difference in freshwater and seawater density = 0.025, = Lb = 1 or 0.1, and Eq. 71 QT is the total freshwater inflow per unit width [L3/LT]. In order to match the nondimensional Henry and Hilleke parameters listed above, values of DSOLUTE = 2.381105 m2 / s, DM = 8.333106 m2 / s, DTEMP = 2.381104 m2 / s, and DT = 8.333105 m2 / s are required for =1. For =0.10, DSOLUTE = 2.381107 m2 / s, DM = 8.333108 m2 / s, DTEMP = 2.381106 m2 / s, and DT = 8.333107 m2 / s are required. Results for the isothermal problem and the =1 mesh using SUTRAMS and HST3D are compared with the Henry and Hilleke solution for the 0.5percent seawater contour (Figure 5). The simulated results from SUTRAMS and HST3D compare favorably at the base of the model but differ slightly at the top of the model for the 0.5, 0.2, and 0.05percent seawater contours. Neither SUTRAMS nor HST3D compare favorably to the original Henry and Hilleke (1972) solution using the =1 mesh. 54
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Figure 5 Match of percentseawater contours and the SUTRAMS flow field for N=3, NC=10, and =1 Henry and Hilleke numerical solution (0.5percent seawater concentration only) (solid red line), HST3D code solution (dashed black line), and SUTRAMS solution (colored). 55
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Results for the isothermal problem and the =0.10 mesh using SUTRAMS and HST3D are shown in Figure 6 and are compared with the Henry and Hilleke solution for the 0.5percent seawater contour. The simulated results from SUTRAMS and HST3D are in very close agreement with each other except at the base of the model where the 0.95and 0.8percent seawater contours differ slightly. SUTRAMS and HST3D solutions are closer to the original Henry and Hilleke (1972) solution using the =0.10 mesh than they are with the =1 mesh. Figure 6 Match of percentseawater contours and the SUTRAMS flow field for N=3, NC=10, and =0.10 Henry and Hilleke numerical solution (0.5percent seawater concentration only) (solid red line), HST3D code solution (dashed black line), and SUTRAMS solution (colored). Results for the nonisothermal problem using SUTRAMS, HST3D, and the =1 mesh are shown in Figure 7 and are compared with the Henry and Hilleke solution for the 0.5 percent seawater contour. The simulated results from SUTRAMS and HST3D 56
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compare favorably at the 0.95, 0.8, 0.5, and 0.2percent seawater contour except at the top of the model where the 0.2and 0.05percent seawater contours differ slightly. SUTRAMS and HST3D isotherms compare favorably at all temperatures. Neither SUTRAMS nor HST3D solute concentrations compare favorably to the original Henry and Hilleke (1972) solution with the =1 mesh. Simulated temperatures for SUTRAMS and HST3D compare favorable to the original Henry and Hilleke (1972) solution at the top and bottom of the model but not in the center of the model. Results for the nonisothermal problem using SUTRAMS, HST3D, and the =0.10 mesh are shown in Figure 8 and are compared with the Henry and Hilleke solution for the 0.5percent seawater contour. The simulated results from SUTRAMS and HST3D compare favorably at the top of the model but are slightly off for the 0.95, 0.8, and 0.5percent seawater contours at the base of the model. SUTRAMS and HST3D isotherms compare favorably at all temperatures. SUTRAMS and HST3D percentseawater concentrations do not compare favorably to the original Henry and Hilleke (1972) solution with the =0.10 mesh but are closer than simulations that used the =1 mesh. Simulated temperatures for SUTRAMS and HST3D compare favorable to the original Henry and Hilleke (1972) solution. 57
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Figure 7 Match of percent seawater contours, SUTRAMS flow field, and match of isotherms for N=3, NC=10, NT=1, and =1 Henry and Hilleke numerical solution (0.5 percent seawater concentration and isotherm only) (solid red line), HST3D code solution (dashed black line), and SUTRA solution (colored) with DM=8.333106 and DT=8.333105. 58
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Figure 8 Match of percentseawater contours, SUTRAMS flow field, and match of isotherms for N=3, NC=10, NT=1, and =0.10 Henry and Hilleke numerical solution (0.5percent seawater concentration and isotherm only) (solid red line), HST3D code solution (dashed black line), and SUTRA solution (colored) with DM=8.333108 and DT=8.333107. The original Henry and Hilleke (1972) solutions were based on a finitedifference model that used a simplified form of the variabledensity flow and transport equation. The simplified form approximates the full equation when <<1. The finitedifference 59
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model was developed based on an aquifer length of 268 km (kilometers) and an aquifer depth of 0.762 km (<<1). The simulations presented for the =1 and =0.10 meshes indicate that a closer correspondence to the original Henry and Hilleke (1972) solution could be obtained with meshes having aspect ratios () less than 0.10. Aspect ratios less than 0.10 were not simulated because of the spatial discretization required to satisfy numerical stability criteria (Table 1). Table 1. Discretization requirements for several aquifer aspect ratios (). See Appendix A for symbol definitions. Mesh Geometry x (m) y (m) width (m) K (m/s) () o (kg/m3) Q (m3/ms) 1 1 1 1 0.01 0.35 1000 1 8.333E05 2 10 1 1 0.01 0.35 1000 0.1 8.333E06 3 100 1 1 0.01 0.35 1000 0.01 8.333E07 4 1000 1 1 0.01 0.35 1000 0.001 8.333E08 5 10000 1 1 0.01 0.35 1000 0.0001 8.333E09 Mesh Geometry QIN (kg/s) DS (m2/s) DT (m2/s) w vx (m/s) x from Pex=2 and DT (m) x from Pex=2 and DS (m) nxMAX 1 8.333E02 2.381E05 2.381E04 9.957E+02 2.381E04 2.00 0.2 5 2 8.333E03 2.381E07 2.381E06 9.957E+00 2.381E05 0.20 0.02 500 3 8.333E04 2.381E09 2.381E08 9.957E02 2.381E06 0.02 0.002 50000 4 8.333E05 2.381E11 2.381E10 9.957E04 2.381E07 0.002 0.0002 5000000 5 8.333E06 2.381E13 2.381E12 9.957E06 2.381E08 0.0002 0.00002 500000000 Henry and Hilleke (1972) indicated that their solution had erratic values in the upper righthand corner of the model, and convergence was dependent on using initial conditions that were reasonably close to the final solution. Henry and Hilleke (1972) suggested the erratic values in the upper righthand corner were a result of coarse discretization at the outflow face, and they reset calculated concentrations and temperatures to zero or one if numerical results exceeded the minimum or maximum 60
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values of zero and one, respectively. These same numerical issues are present in the SUTRAMS and HST3D results for the =0.10 simulations but were not resolved, because instabilities were restricted to the outflow face, did not propagate a significant distance into the problem domain, and would require finer discretization at the outflow face that would increase numerical overhead and simulation times substantially. 2.6.2 Solution for doublediffusive finger convection induced by different fluid dispersivities and viscosities 2.6.2.1 Physical Setup The intent of this example is to verify the accuracy of the SUTRAMS in representing bulk fluid flow initiated by differences in the dispersivity and viscosity of two solute species (NaCl and sucrose). Unlike the other examples presented in this report, this problem is driven exclusively by internal changes in fluid density and viscosity and not by external boundary conditions. In order to compare simulated and experimental results, a conservative dye solution also is simulated that approximately maps the evolving NaCl field. The mapping of the NaCl field is approximate because the dispersivity of the conservative dye is less than the dispersivity of NaCl by a factor of about 2.5. Initially, the two solutes were unmixed and vertically stratified in a stable density configuration. To initiate flow, the interface between the unmixed fluids was randomly perturbed. A conservative dye tracer, initially coincident with the NaCl solution, also was randomly perturbed at the interface. The initial perturbations of the species that affect fluid density form the nucleus for the developing fingers. The dimensions of the problem were selected to be identical to those of the original HeleShaw experiment by Pringle and others (2002) in order to allow for a direct comparison of simulated and observed results. The total simulation time was 16 hours, which is identical to the duration of the original experiment. 61
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2.6.2.2 Simulation Setup A nonintrusive lighttransmission technique and a chargecoupleddevice (CCD) camera were used to measure the light intensity field transmitted through the HeleShaw cell. The evolution of the flow field was monitored with a dye tracer (Warner Jenkins FD&C Blue #1) added to the NaCl solution. The HeleShaw cell was 0.2541 m wide and 0.1625 m high and the progression of the experiment was recorded using images captured with the CCD camera and converted to concentrations using calibration curves obtained from a series of dye concentrations in a base salt solution. Captured images had a resolution of 1650 x 1055 pixels with a pixel size of 1.54104 m. The simulation mesh consisted of 1,024 by 656 square elements that are 2.48104 m by 2.48104 m. The pixel resolution in the experimental setup of Pringle and others (2002) was a factor of 1.7 times smaller than the cell size used in the numerical simulation. The selected numerical discretization was a compromise between computer run times and desired numerical accuracy. A simplified version of the simulation mesh is shown in Figure 9. A variable time step, ranging from 0.1 to 10 seconds, and a timestep multiplier of 1.05 were used. Small time steps were required to achieve convergence during early stages of the simulation when mass transfer rates and vertical velocities were higher. Pressure, NaCl concentration, sucrose concentration, and dye concentration were solved on each time step (NPCYC=NUCYC(k)=1), and an iterative solution was used to resolve nonlinearities during each time step (ITRMAX=10). 62
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Figure 9 Boundary and initial concentration conditions and finiteelement mesh (every 16th element) used to simulate the Pringle and others (2002) HeleShaw experiment. 2.6.2.3 Parameters k = 2.61109 [m2] b = 1.0 [m] = 1.0 [] g = 4.14 [m / s2] based on 9.8[m / s2] sin (25) CO(NaCl) = 0.000 [kgNaCl / kgfluid] CO(Sucrose) = 0.000 [kgSucrose / kgfluid] CMAX(NaCl) = 0.03463 [kgNaCl / kgfluid] CMAX(Sucrose) = 0.05234 [kgSucrose / kgfluid] CO(Dye) = 0.000 [kgDye / kgfluid] CMAX(Dye) = 0.00025 [kgDye / kgfluid] O = 998. [kg / m3] O = 1103 [kg / (m s)] NaClC = 689. [kgfluid2 / (kgNaCl m3)] SucroseC = 371. [kgfluid2 / (kgSucrose m3)] NaClC = 1.59103 [kgfluid kg / (kgNaCl m s)] SucroseC = 2.75103 [kgfluid kg / (kgSucrose m s)] DyeC = 0.00 [kgfluid2 / (kgDye m3)] DyeC = 0.00 [kgfluid kg / (kgDye m s)] DNaCl = 1.477109 [m2 / s] DSucrose = 4.8781010 [m2 / s] DDye = 5.6701010 [m2 / s] L = T = 0.0 [m] 63
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2.6.2.4 Boundary Conditions Noflow conditions occured across the top, bottom, left, and right boundaries except at the upper right and left nodes. (blue circles on Figure 9). The top right and top left boundaries were specified to have a 0.0 [Pa] pressure, 0.00000 kgNaCl / kgfluid, 0.05234 kgsucrose / kgfluid, and 0.00000 kgDye / kgfluid] in order to prevent calculation of negative pressures. No external sources of fluid were used. 2.6.2.5 Initial Conditions Hydrostatic pressures based on the initial concentrations of NaCl and sucrose shown on Figure 9 were used as initial conditions. The conservative dye tracer was restricted initially to the lower half of the HeleShaw cell, coincident with the initial distribution of NaCl. In order to simulate the smallscale perturbations that existed in the original experiment, the concentrations of NaCl, sucrose, and the conservative dye tracer were randomly perturbed at the interface. The interface was randomly perturbed using the following procedure. Random numbers, ranging from 0 to 1, were generated for concentrations of NaCl, sucrose, and the conservative dye tracer for all nodes located at the interface between the NaCl and sucrose solutions. If the random number generated for a specific node and solute species was less than 0.5, the initial concentration of the solute species was reduced. Conversely, if the random number generated for a specific node and solute species was greater than or equal to 0.5, the concentration of the solute species was not perturbed. A maximum perturbation of 1 percent of the maximum solute concentrations was used for all the SUTRAMS simulations. Hydrostatic pressures were reset based on the perturbed concentrations developed at the interface. 2.6.2.6 Results Concentrationdependent density and viscosity relationships for NaCl and sucrose were derived from Weast (1986). As shown in Figure 10, NaCl and sucrose show 64
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approximately linear concentrationdensity and concentrationviscosity relationships over the range of concentrations used in the HeleShaw experiment. Fluid density and viscosity are assumed to be independent of the dye concentration. Furthermore, it is assumed that interaction of NaCl, sucrose, and the dye tracer was negligible, and the net effect of the three individual solute species is equal to the sum of the effect of the three components separately. The observed results from the HeleShaw experiment (Pringle and others, 2002) are shown in Figure 11. Observed and simulated time is presented as dimensionless and calculated as t* = t DNaCl / H2 Eq. 72 where t is the elapsed simulation time [sec], DNaCl is the molecular diffusion of the NaCl solution, and H is the height of the HeleShaw cell. t* is based on the dimensionless governing equations of Nield and Bejan (1998). Fluid concentrations are presented as a ratio of the concentration at time t*, C, and the maximum concentration, CO, varying from 0.0 to 1.0, for both observed and simulated results. 65
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00.010.020.03CONCENTRATION OF SOLUTE R EL A TIVE TO W A TE R ,k g / k g 0.9911.011.021.03103 DENSITY, kg/m3 11.011.021.031.041.051.06103 ABSOLUTE VISCOSITY, kg m1s1 00.010.020.03CONCENTRATION OF SOLUTERELATIVE TO WATER, kg/kg SODIUM CHLORIDE AT 20OCSODIUM CHLORIDE AT 20OCR2=0.996385R2=0.999939 00.010.020.030.040.05CONCENTRATION OF SOLUTE R EL A TIVETO W A TE R kg / kg 0.99511.0051.011.0151.02103 DENSITY, kg/m3 11.051.11.15103 ABSOLUTE VISCOSITY, kg m1s1 00.010.020.030.040.05CONCENTRATION OF SOLUTERELATIVE TO WATER, kg/kg SUCROSE AT 20OCSUCROSE AT 20OCR2=0.999902R2=0.999591 Figure 10 Absolute viscosity and fluid density relationships with NaCl and sucrose concentrations used in all SUTRAMS simulations. Data are from Weast (1986). 66
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Figure 11 Observed results from Pringle and others (2002) at (A) t* = 4.06105, (B) t* = 1.29104, (C) t* = 3.96104, (D) t* = 3.35104, (E) t* = 4.35104, (F) t* = 5.36104, (G) t* = 6.03104, and (H) t* = 7.37104, (I) t* = 8.04104, (J) t* = 1.04103, (K) t* = 1.78103, and (L) t* = 3.19103. Color sequence blackbluegreenyelloworangered depicts normalized NaCl concentration from 0 to 1. 67
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Figure 12 Simulated results SUTRAMS results at (A) t* = 4.06105, (B) t* = 1.29104, (C) t* = 3.96104, (D) t* = 3.35104, (E) t* = 4.35104, (F) t* = 5.36104, (G) t* = 6.03104, and (H) t* = 7.37104, (I) t* = 8.04104, (J) t* = 1.04103, (K) t* = 1.78103, and (L) t* = 3.19103. Color sequence blackbluegreenyelloworangered depicts normalized dye concentrations from 0 to 1. 68
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Simulated results compare well until t*=3.19103 although the rate of finger development appears to be slightly slower than that observed (Figure 12). Two quantitative measures were used to compare simulated SUTRAMS concentration fields to observed concentrations. The first quantitative measure is the normalized vertical length scale that is calculated as h* = (h/H) Eq. 73 where h is the vertical distance between the horizontally averaged 0.05 and 0.95 normalized dye concentration (C/CO), and H is the height of the HeleShaw cell. The normalized length scale, h*, is a measure of the evolving finger structure. h* becomes undefined when fingers reach the top and bottom boundaries and concentration profiles invert. The second quantitative measure is the normalized mass transfer of dye upward across the centerline of the cell in time that is calculated as M* = (M/MO) Eq. 74 where M is the total mass of dye above the centerline at time t*, and MO is the mass of dye below the centerline at t*=0. 69
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Quantitative comparisons of the evolution of simulated and observed dye concentrations are shown in Figure 13. 1x1061x1051x1041x1031x102t* 1x1021x1011x100h* Simulated Observed 1x1061x1051x1041x1031x102t* 1x1021x1011x100M* Figure 13 Normalized vertical length, h*=h/H, and mass transfer across the center line, M*=M/Mo, as a function of time showing regions of steady growth for the original HeleShaw experiment (open gray circles) and the SUTRAMS simulation (solid black circles and solid black line). Figure 13 quantitatively demonstrates that SUTRAMS is capable of duplicating the general behavior observed in the HeleShaw experiment. Some key aspects that are captured in the SUTRAMS simulation are the linear increases in h* at early and mature stages and the timing of the initiation of the rundown stage observed in the M* data. A noticeable difference is that the simulated normalized mass transfer, M*, is less than that observed in the HeleShaw experiment. 70
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2.6.3 Densitydependent flow with transport of a nonreactive tracer and zeroorder production and transport of a solute to simulate groundwater age 2.6.3.1 Physical Setup This problem involves seawater intrusion into a confined aquifer studied in cross section under steadystate conditions (Henry, 1964) with the release of a nonreactive tracer after steady flow conditions are established. Groundwater age is simulated using a third species that is generated internally using a zeroorder production rate (Goode, 1996). Freshwater recharge from inland sources flows over seawater and discharges at a vertical sea boundary. This problem demonstrates the utility of SUTRAMS to simulate multiple species, some of which have no effect on fluid density. With SUTRA, simulation of more than one transport species (solute or heat) could only be approximated only for variabledensityflow problems, even under steady flow conditions. Approximation of conservative tracer transport in a variabledensity flow field required simulation of steady variabledensityflow conditions and use of variabledensityflow results as the initial conditions for a separate, subsequent SUTRA simulation. SUTRA and SUTRAMS use the velocity field from the previous time step or the velocity field based on the initial conditions for the first time step to simulate solute transport for the current time step. Because the velocity field used for solute transport is lagged one time step, a simulation that uses the final velocity field from a variabledensity problem and a single large time step can be used to simulate conservative transport of a nondensitydependent species. Application of this technique is limited: only a single time step can be simulated because tracer concentrations would alter the variabledensity velocity field on the second time step. Results from the crosssection SUTRAMS simulation (twodimensional simulation) also are compared to a threedimensional version of the problem where the model domain has been extended a factor of 20 in the ydimension. The comparison of twoand threedimensional SUTRAMS results is meant to show differences in non71
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reactive tracer concentrations that occur when dispersion in the ydimension is permitted (threedimensional simulation). The intrusion problem is nonlinear and is solved by gradually approaching steady state with a series of time steps. Initially, there is no saltwater in the aquifer. At time zero, saltwater begins to intrude the freshwater system by flowing under the freshwater from the sea boundary. The intrusion of saltwater is caused by differences in saltwater and freshwater density and hydrostatic seawater boundary conditions specified at the sea boundary. Dimensions of the twodimensional problem are based on the SUTRA setup described in Voss (1984), but length scales have been increased by a factor of 100. A stabilization simulation time of 13.89 days was selected because this is sufficient time for the variabledensity flow field and chloride concentrations to reach steady state at the scale simulated. A nonreactive solute is released at 13.89 days from three constant concentration nodes. A total tracertransportsimulation time of 13.89 days is sufficient time for the nonreactive solute to move more than half the total length of the problem in the xdirection. 2.6.3.2 Simulation Setup The mesh for the twodimensional problem consists of 100 by 50 elements, each of size 2.0 m by 2.0 m (Figure 14). The mesh for the threedimensional problem consists of 100 by 50 by 20 elements, each of size 2.0 m by 2.0 m by 2.0 m. The threedimensional problem uses the same discretization in the xz plane as shown in Figure 14 and has a total length of 40m in the ydirection. A constant timestep length of 100 minutes was used for all SUTRAMS simulations, and 400 time steps were taken. Fluid flow and solute transport were solved during each time step in the twoand threedimensional SUTRAMS simulations (NUCYC=NPCYC=1). 72
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Figure 14 SUTRA finiteelement mesh and boundary conditions. 2.6.3.3 Parameters k = 1.02041109 [m2] based on K = 0.01 [m / s] B = 2.0 [m] = 0.35 [] g = 9.8 [m / s2] CS = 0.0357 [kgdissolved solids / kgfluid] CO = 0.000 [kgdissolved solids / kgfluid] S = 1024.99 [kg / m3] O = 1000. [kg / m3] SOLID = 2600. [kg / m3] QIN = 6.6 [kg / s m] SOLUTEC = 700. [kgfluid2 / (kgNaCl m3)] = 1103 [kg / (m s)] LSOLUTE = 0.0 [m] TSOLUTE = 0.0 [m] TracerC = 0.00 [kgfluid2 / (kgTracer m3)] CTracer = 100.0 [kgTracer / kgseawater] LTracer = 0.5 [m] TTracer = 0.5 [m] GWAgeC = 0.00 [kgfluid2 / (kgGWAge m3)] CGWAge = 0.0 [kgGWAge / kgseawater] LGWAge = 0.000 [m] TGWAge = 0.0000 [m] DSOLUTE = 18.8571104 [m2 / s] DTracer = 0.0 [m2 / s] DGWAge = 18.8571104 [m2 / s] PRODF0GWAge = 1.667102 [1 / s] 2.6.3.4 Boundary Conditions Noflow conditions occurred across the top and bottom boundaries. A source of freshwater was implemented by using source nodes that injected freshwater at rate QIN with concentration CO at the left vertical boundary. A total freshwater source of 13.2 kg/s 73
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(kilograms per second), distributed over 51 nodes, was applied at the left vertical boundary of the twodimensional problem. A total freshwater source of 264 kg/s, distributed over 1,071 nodes, was applied at the left vertical face of the threedimensional problem. The right vertical boundary was held at hydrostatic pressure, assuming a constant density equal to seawater, through use of specified pressure nodes. Any water entering through these nodes has the concentration, CS, of seawater. Five specified concentration nodes at x = 0.0 m and z = 47.0 to 52.0 m were used to simulate the release of the nonreactive tracer in the twodimensional SUTRAMS problem (red arrows on Figure 14). In the threedimensional problem, five specified concentration nodes at x = 0.0 m, y = 40.0 m, and z = 47.0 to 52.0 m were used to simulate the release of the nonreactive tracer. In both problems, specified concentration nodes had a concentration of 0.0 kgdissolved solids/kgfluid from 0 to 13.89 days and 100.0 kgdissolved solids/kgfluid from 13.89 to 27.78 days. A high tracer concentration was simulated after 13.89 days so that the total tracer mass was sufficient to develop a plume that extended a significant distant from the inflow boundary in both problems. No boundary conditions were specified for species simulating groundwater age using zeroorder production, except that the age of the incoming fluid was specified to be zero. 2.6.3.5 Initial Conditions Hydrostatic pressures based on an initially freshwater aquifer with a constant concentration of CO were used for all simulations. Initial concentrations for the nonreactive tracer and the species simulating groundwater age were set to zero in all simulations. 2.6.3.6 Results In Henrys solution, dispersion is represented by a constant, large diffusion coefficient, rather than by a velocitydependent dispersivity. Two different values of diffusivity have been used to test simulators against Henrys solution (Voss, 1984). The 74
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total dispersion coefficient of Henry (1964), D, is equivalent to the product of porosity and molecular diffusivity in SUTRA (see Eq. 66). Henrys results are given for his nondimensional parameters: =2.0, b=0.1, a0.264 (Henry, 1964, p. C80, fig. 34). In order to match Henrys parameters using simulation parameters as listed above, values of D = 6.6104 m2/s and DSOLUTE = 18.8571104 m2/s are required. Because this example is not a comparison of results with simulators other than SUTRAMS, D and DSOLUTE values evaluated are limited to the values listed above. Figure 15 shows that percentseawater contours are essentially identical at 13.89 and 27.78 days in the twodimensional SUTRAMS simulation, and the variabledensity flow field has essentially achieved steadystate conditions by 13.89 days. Because the variabledensity flow field has achieved steady conditions by 13.89 days, use of the simulated pressure field is appropriate for simulation of conservative tracer transport in a steady variabledensity flow field. Percentseawater contours at 27.78 days for the threedimensional SUTRAMS simulation also are also shown in Figure 15 and are comparable to contours in the twodimensional SUTRAMS simulation. 75
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Figure 15 Simulated percentseawater contours from the 2D SUTRAMS simulation after (A) 13.89 and (B) 27.78 days and the (C) 3D SUTRAMS simulation after 27.78 days. 76
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A velocitydependent dispersivity was used for the nonreactive tracer. Molecular dispersion was assumed to be zero (0) for the nonreactive tracer. A longitudinal and transverse dispersivity of 0.5 and 0.5 m, respectively, were used for the nonreactive tracer. Simulated SUTRAMS nonreactive tracerconcentration contours for the twoand threedimensional simulations are shown in Figure 16. Nonreactive tracerconcentration contours for the threedimensional SUTRAMS simulation are significantly more dispersed in the ydimension and slightly less dispersed in the zdimension than in the twodimensional SUTRAMS simulation. As a result, the nonreactive tracer does not extend as far into the problem domain in the threedimensional SUTRAMS problem. Figure 17 shows the simulated groundwater age after 27.78 days for the twodimensional SUTRAMS simulation. This is done to further illustrate the utility of SUTRAMS. Groundwater age was simulated with a third species (that does not affect fluid density) having a zeroorder production rate of 1.667102 sec1 (1 min1). 77
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Figure 16 Simulated nonreactive tracerconcentration contours from (A) 2D SUTRAMS, (B) 3D SUTRAMS in XZ Plane at y=40 m, and (C) 3D SUTRAMS in XY Plane at z=50 m. 78
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Figure 17 Simulated groundwater age, in days, from SUTRAMS. 79
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Chapter 3 3 Numerical simulation of doublediffusive finger convection 3.1 Abstract A hybrid finiteelement, integrated finitedifference numerical model is developed for the simulation of doublediffusive and multicomponent flow in two and three dimensions. The model is based on a multidimensional, densitydependent, saturatedunsaturated transport model (SUTRA), which uses one governing equation for fluid flow and another for solute transport. The solutetransport equation is applied sequentially to each simulated species. Density coupling of the flow and solutetransport equations is accounted for and handled using a sequential implicit Picard iterative scheme. Highresolution data from a doublediffusive HeleShaw experiment, initially in a densitystable configuration, is used to verify the numerical model. The temporal and spatial evolution of simulated doublediffusive convection is in good agreement with experimental results. Numerical results are very sensitive to discretization and correspond closest to experimental results when element sizes adequately define the spatial resolution of observed fingering. Numerical results also indicate that differences in the molecular diffusivity of sodium chloride and the dye used to visualize experimental sodium chloride concentrations are significant and cause inaccurate mapping of sodium chloride concentrations by the dye, especially at late times. As a result of reduced diffusion, simulated dye fingers are better defined than simulated sodium chloride fingers and exhibit more vertical mass transfer. 3.2 Introduction A wide range of convective structures resulting from local hydrodynamic instabilities can develop in fluids containing multiple species that affect fluid density and viscosity. Convection can occur in systems that are initially in densitystable configurations and contain two or more components with different diffusivities that make opposing contributions to vertical density gradients. Diffusivity differences can result in 80
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buoyant instabilities capable of initiating convection. The nature of the convective flow is strongly dependent on the initial distribution and concentration of each component. In cases when the lower diffusivity component is on top, a parcel of fluid perturbed downward across the interface takes on solute mass from the surrounding fluid faster that it diffuses solute mass so the parcel continues to fall. This mode of convection is termed doublediffusive or multicomponent finger convection and is characterized by convection driven by long, narrow columns (fingers) of rising and falling fluid. Alternatively, when the higher diffusivity component is on top, a parcel of fluid perturbed downward across the interface diffuses mass outward to the surrounding fluid more rapidly than it gains mass from the lower diffusivity solute. The parcel of fluid then becomes less dense than the surrounding fluid, moves upward, and overshoots its original position before repeating the motion. This mode of convection is termed oscillatory doublediffusive convection and can lead to wellmixed convecting layers separated by sharp contrasts in fluid density. Convection features are much different from those that develop in stable advective, dispersive, and/or diffusive processes, and can significantly increase mass transfer rates. Doublediffusive or multicomponent finger convection is an important mixing process in the open ocean and can be an important process in contaminant transport in porous media (Green, 1984; Imhoff and Green 1988; Cooper et al. 1997). Laboratory methods used to quantify convective systems have ranged from intrusive methods that sample portions of the developing convective system (e.g.,Turner, 1967; Imhoff and Green, 1988) to nonintrusive optical methods (e.g., Lambert and Demenkow, 1971; Kazmierczak and Poulikakos, 1989). These experimental methods have increased understanding of doublediffusive and multicomponent processes but have not been used directly in numerical simulations because of the limited amount of data collected in typical HeleShaw experiments. Recent application of nonintrusive light transmission techniques to doublediffusive HeleShaw experiments has allowed collection of pointwise concentration measurements over entire flow fields at relatively high spatial (~ 0.015 cm) and temporal (< 1 sec) resolution (e.g., Cooper et al., 1997, 2001; Pringle et al., 2002). These experiments have investigated doublediffusive flow field evolution for systems with 81
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different initial departures from equilibrium and have increased understanding of doublediffusive processes by allowing fullfield determination of fluid mass transfer rates and the temporal evolution of fluid mass transfer. Additionally, these highresolution datasets are ideal for comparison with numerical models capable of simulating doublediffusive flow processes. In this study, experimental HeleShaw results collected by Pringle et al. (2002) using the nonintrusive light transmission techniques are used to verify a variabledensity and variableviscosity numerical code (based on SUTRA (Voss and Provost, 2002)) capable of simulating multicomponent variabledensity fluid flow and transport (Hughes and Sanford, 2004). The dataset of Pringle et al. (2002) is wellsuited for code verification of doublediffusive and multicomponent flow and transport numerical models because, unlike most previous experimental HeleShaw datasets, it is of sufficient spatial and temporal resolution to allow accurate comparisons of simulated and observed convective fingering. In some cases, when an appropriate observed dataset was unavailable, numerical codes have been tested against other numerical codes. Consistent model results were often considered sufficient to demonstrate that a numerical code accurately represents the physics of a given problem. Verification of the numerical code presented in this paper would have been difficult without the dataset of Pringle et al. (2002) because no generic numerical codes capable of simulating doublediffusive and multicomponent flow and transport in porous media are widely available (Sorak and Pinder, 1999). HeleShaw experiments are appropriate for verification of variabledensity numerical codes because they evolve in response to internal conditions and are not forced by external boundary conditions. Dependence of many standard variabledensity test cases on external boundary conditions often reduces the ability to isolate numerical inadequacies (Voss and Souza, 1987). In addition, the spatial and temporal resolution of the data of Pringle et al. (2002) are sufficient to improve our understanding of simulating nonlinear fingering processes. The fullfield images from the experimental dataset allow qualitative comparison of the evolving flow field and quantitative comparison of mass transfer rates. The ability to 82
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compare mass transfer rates allows the level of discretization required to accurately simulate nonlinear fingering and the mass transfer rates of each component to be evaluated. 3.3 Governing equations The mass balance of fluid per unit aquifer volume, assuming fully saturated flow and negligible contribution of solute dispersion to the mass average flux of the fluid, is given by Bear (1979): p k NS k kQ p t C Ct p g k 11 Eq. 75 where (x,y,z,t) is the fluid density [M/L3]; (x,y,z) is the effective porosity [dimensionless]; is the compressibility of the porous matrix [M/LT2]1; is the compressibility of the fluid [M/LT2]1; p (x,y,z,t) is the fluid pressure [M/LT2]; NS is the number of dissolved species simulated; Ck(x,y,z,t) is the solute concentration of species k [Msolute/Mfluid]; k(x,y,z) is the intrinsic permeability tensor [L2]; (x,y,z,t) is the fluid viscosity [M/LT]; g is the gravitational acceleration vector [L/T2]; and Qp(x,y,z,t) is a fluid mass source [M/L3T]. Fluid density is approximated as a linear function of solute concentration, expressed as 83
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NSkkokkoCCC1 Eq. 76 where o is the fluid density at the base concentration of all simulated species; Cko is the base solute mass fraction of species k; and Ck is a constant coefficient of density variability for each species. Fluid viscosity is approximated as a linear function of solute concentration: NSkkokkoCCC1 Eq. 77 where o is the fluid viscosity at the base concentration of all simulated species; Cko is the base solute mass fraction of species k; and Ck is a constant coefficient of viscosity variability for each species. Darcys law gives the average fluid velocity at a point as gkvp Eq. 78 The solute mass balance per unit volume of a variable density fluid containing more than one dissolved species that may affect density is given by Bear (1979): kkpkmkCCQCDCtC*DIv Eq. 79 where v(x,y,z,t) is the fluid velocity [L/T]; Dmk is the molecular diffusivity of species k, [L2/T]; I is the identity tensor; 84
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Dk(x,y,z,t) is the mechanical dispersion tensor of species k, [L2/T]; and Ck*(x,y,z,t) is the solute concentration of the source fluid of species k, [Msolute/Mfluid]. A generalized version of mechanical dispersion in isotropic homogeneous porous media is used to account for dispersion in an anisotropic porous medium. A detailed description of the generalized mechanical dispersion model used is given in Voss and Provost (2002). 3.4 Numerical Approximation The basic framework of the code to approximate Equations (Eq. 75) and (Eq. 79) is based on a version of SUTRA capable of simulating twoand threedimensional problems (Hughes and Sanford, 2004). Equations (Eq. 75) and (Eq. 79) are approximated using a weighted numericalresidual method that combines Galerkin finiteelement and integrated finitedifference techniques. Time derivative and source terms in equations (Eq. 75) and (Eq. 79) are discretized cellwise and all other terms are discretized elementwise. The Galerkin finiteelement method allows geometric flexibility in mesh design and gives robust directionand anisotropyindependent representation of fluid and solute fluxes. The integrated finitedifference representation for the spatial integration of all nonflux terms in the governing equations provides an economical alternative to the Galerkin method while giving accuracy sufficient for any mildly nonlinear simulation problem. The hybrid weightedresidual and integrated finitedifference method applied to the fluid mass balance (Eq. 75) and solute mass balance for each species (Eq. 79) is discussed in Appendix C. 85
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3.5 Experimental doublediffusive HeleShaw study of Pringle et al. (2002) Pringle et al. (2002) used a HeleShaw cell to explore the temporal and spatial distribution of doublediffusive finger convection of two fluids initially in a densitystable configuration with a mean interface thickness of ~ 1103 m. The evolution of the resulting convection system was monitored using a quantitative light transmission technique. This quantitative light transmission technique allowed acquisition of fullfield concentration data at a spatial and temporal resolution of 1.54104 m and 1 second, respectively. Full details of the experimental procedure are given in Pringle et al. (2002). The relevant features of the experiment are briefly discussed here and form the basis of the present numerical study. The mean aperture width () of the HeleShaw cell was 1.77104 m. Other relevant physical properties of the HeleShaw cell are given in Table 2. The equivalent intrinsic permeability (k) is 2.61109 m2 based on the relationship k = 2/12 (Bear 1988). The HeleShaw cell was filled with a sucrose solution (S) over a sodium chloride solution (T). The fluid properties of the sucrose and sodium chloride solutions are given in Table 3. The dimensionless buoyancy ratio, R = TT/SS, is near neutral gravitational stability at 1.22, where T is the maximum initial sodium chloride concentration, S is the maximum initial sucrose concentration, T = (1/o)( / T) and S = (1/o)( / S) are the concentration expansion coefficients (Nield & Bejan 1998) of sodium chloride and sucrose, respectively. The dimensionless solutal Rayleigh numbers, with permeability included in the scaling of viscous influences, are defined as RT = TTgHk / DT and RS = SSgHk / DS, where g is the gravitational acceleration in the plane of the cell, H is the cell height, k is the intrinsic permeability, DT is the molecular diffusivity of sodium chloride, DS is the molecular diffusivity of sucrose, and is the mean kinematic viscosity of the fluid. The HeleShaw cell was inclined at an angle of 25 relative to horizontal and gravitational acceleration in the plane of the cell is 4.14 m/s2 based on gsin. The location of the HeleShaw experiment in Rayleigh parameter space is shown in Figure 18. 86
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To visualize sodium chloride concentrations and quantify convective motion, a dye tracer with a concentration of 0.00025 kg/kg was mixed with the sodium chloride solution. The dye had a negligible effect on fluid density. Sodium chloride concentrations are not mapped perfectly by the dye because the diffusivity of sodium chloride is approximately 2.5 times greater than the diffusivity of the dye (Table 3). Because the motion is convective through most of the experiment, Pringle et al. (2002) suggested the diffusivity differences had little impact on the mapping of sodium chloride concentrations over the length of time of the experiment. Table 2 HeleShaw cell experiment parameters Length (L) 0.2541 m Height (H) 0.1625 m Cell Angle Relative to Horizontal 25 Light Acquisition Data Points 1650 pixels x 1055 pixels Pixel Size 1.54104 m Aperture width () 1.77104 m Intrinsic permeability (k) 2.61109 m2 To minimize initial perturbations, the inclined HeleShaw cell was initially saturated with water, then the sucrose and sodium chloridedye solutions were flushed through the cell at the upper and lower corners of one side, respectively, and flushed out of the center at the opposite side. After ~100 HeleShaw cell volumes of each fluid were flushed, the inflow and outflow valves were closed, and the instability was allowed to evolve naturally from the rest state. Pringle et al. (2002) presented images of the flushing process and concentrations prior to initiating of the experiment that indicated the upper and lower portions of the HeleShaw cell contained sucrose and sodium chloride in an initially stratified configuration, respectively, and the interface was free of discernable 87
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perturbations. The initial solution interface had a mean thickness of ~0.001 m and was determined by single pixelwide vertical transects across the cell. Minimization of artificial perturbations is important because these perturbations can influence fluid motion and make interpretation of experimental data difficult. Table 3 Fluid properties and Rayleigh numbers T A 0.03463 (kgsolute/kgfluid) S B 0.05234 (kgsolute/kgfluid) Dye 0.00025 (kgsolute/kgfluid) T 0.6892 (kg/kg) S 0.3719 (kg/kg) DT 1.477109 m2s1 DS 4.8781010 m2s1 DdyeC 5.6701010 m2s1 T 1.033106 m2s1 S 1.125106 m2s1 RT 26,460 RS 21,579 A T and subscript T used to denote sodium chloride B S and subscript S used to denote sucrose C From Detwiler et al. (2000) 88
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3000020000100000RSODIUM CHLORIDE 0100002000030000RSUCROSE Line of NeutralGravitational StabilityStability Boundary[least constrained; Nield (1968)]STABLE DOUBLEDIFFUSIVEFINGER CONVECTIONBuoyancy Ratio = 1.22 Pringle et al. (2002) UNSTABLEGravitationalInstability Figure 18 Location of the HeleShaw experiment (closed circle) within Rayleigh parameter space with an R value of 1.22 (Modified from Pringle et al., 2002). Light images were transformed by Pringle et al. (2002) into normalized concentration fields (C/Co) using calibration curves developed using solutions of known dye and sodium chloride concentrations. Normalized concentrations (C/Co) are referred to as concentration in the remainder of the paper. A total of 300 images of the evolving concentration field were collected over the duration of the 16hour HeleShaw experiment. Images were collected at 20second intervals during early times and at 10minute intervals at late times. A sequence of concentration fields from the experiment is shown in Figure 19 and was chosen by Pringle et al. (2002) because the images captured upward masstransfer of sodium chloride and evolution of the convection system during the early, mature, and rundown stages of the experiment. Time is presented as dimensionless (t* = tDT / H2) based on the dimensionless governing equations in Nield and Bejan (1998). 89
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Starting from the rest state, distinct fingers develop quickly and grow rapidly in unison. As convection proceeds, smallscale fingers continuously develop from the region of the initial solution interface. The complexity of the fingers increases after t* of 2.21104 (Figure 19C) and includes coalescing of fingers and generation of new finger pairs at the tips of some upward and downward growing fingers. At t* of 4.23104 (Figure 19E) the fastestgrowing fingers reach the top and bottom boundaries of the cell and begin to spread laterally forming more and less dense plumes of fluid at the bottom and top of the cell, respectively. The plumes migrate to the center of the upper and lower boundary in Figure 19FH and new fingers continue to form at the initial solution interface located at the centerline in regions with nearinitial fluid concentrations. At late times, the finger structure exhibits a branching pattern with greater lateral travel than observed at early times. Diffusion begins to dominate the flow field at late times which slowly begins to decrease spatial variations in fluid concentrations and results in a well mixed concentration field at t* > 1.77103. 3.6 Numerical modeling 3.6.1 Spatial discretization and model parameters The HeleShaw cell was discretized using 671,744 elements. A uniform mesh, with square elements 2.48104 m along each side, was used throughout the model domain. A uniform mesh was used because convection did not appear to occur in preferential locations during the original HeleShaw experiment. Transport of sodium chloride, sucrose, and the dye was simulated along with fluid flow. The relationship of fluid density and viscosity to sodium chloride and sucrose concentrations was developed from data in Weast (1986). Fluid density and viscosity are approximately linearly related to sodium chloride and sucrose concentrations over the range of concentrations used in the HeleShaw experiments (Table 3) as shown in Figure 20. The coefficient of determination (R2) for linear regression of all sodium chloride and sucrose concentrationdensity and concentrationviscosity data was greater than 0.99 90
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(Figure 20). The effects of sodium chloride and sucrose concentrations on fluid density and viscosity were assumed to be independent of each other and additive (Eqs. 76 and 77). Dye concentrations were assumed to have no effect on fluid density or viscosity. Except for dispersion in the flow direction resulting from nonuniform velocity profiles across the width of the HeleShaw cell (Taylor dispersion), which is not represented in the numerical model and is typically less than molecular diffusion (Detwiler et al., 2000), mechanical dispersion does not occur in a HeleShaw cell. As a result, longitudinal and transverse dispersivities were set to zero and molecular diffusion was the only dispersive component simulated. Additional flow and solute transport parameters are summarized in Table 3 and Table 4. 91
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Figure 19 Observed results from Pringle et al. (2002) at (A) t* = 4.03105, (B) t* = 1.31104, (C) t* = 2.21104, (D) t* = 3.22104, (E) t* = 4.23104, (F) t* = 5.24104, (G) t* = 6.04104, and (H) t* = 7.25104, (I) t* = 7.85104, (J) t* = 1.03103, (K) t* = 1.77103, and (L) t* = 3.17103. Color sequence blackbluegreenyelloworangered depicts normalized dye concentrations from 0 to 1. 92
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3.6.2 Boundary and initial conditions Noflux conditions were assumed at all external boundary faces except the upper left and right corners of the HeleShaw cell. Specified pressure boundary conditions with a pressure of 0.00 Pa and associated sodium chloride and dye concentrations of 0.0 kg/kg and sucrose concentrations of 0.05234 kg/kg were assigned at the upper left and right corners of the model to minimize numerical problems that can occur when boundary conditions are not specified. Specifiedpressure boundary conditions are treated as pressuredependent boundary conditions (Cauchy boundary condition) and solute mass is added to the model only if an inward gradient from the specified pressure boundary condition to the upper left or right nodes is calculated. No fluid inflow occurred during the simulation and only a small amount of fluid left the model through the specified pressure nodes (< 2%). Although Pringle et al. (2002) indicated that significant effort was expended to minimize initial perturbations and the thickness of the initial solute interface was small (~0.001 m), some perturbations existed at the start of the experiment and were seeds for initial finger development. Model error (roundoff error) can create perturbations that vary spatially and temporally in an uncontrollable fashion (Simmons et al., 1999). Small changes in dispersion parameters and spatial and temporal discretization can cause different perturbations to occur. To control initial seeds for finger development, random noise with a mean of zero and maximum amplitude of 0.5% of maximum initial concentrations was applied to the sodium chloride and sucrose concentrations at the initial solution interface. The sensitivity of model results to the magnitude and amplitude of perturbations was not evaluated in this study. 93
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Table 4 Numerical Parameters used x = y 2.48104 m Number of elements in x direction 1024 Number of elements in z direction 656 Gravitational acceleration (g sin25) 4.14 m/s2 Porosity () 1.0 Cell Thickness (b) 1.0 m Base fluid density (o) 998.0 kg/m3 Base fluid viscosity (o) 1.00103 kg / m s Coefficient of fluid density change to sodium chloride concentration ( / CNaCl) 689. kgfluid2 / kgNaCl m3 Coefficient of fluid density change to sucrose concentration ( / CSucrose) 371. kgfluid2 / kgSucrose m3 Coefficient of fluid density change to dye concentration ( / CDye) 0.00 kgfluid2 / kgDye m3 Coefficient of fluid viscosity change to sodium chloride concentration ( / CNaCl) 1.59103 kgfluid kg / kgNaCl m s Coefficient of fluid viscosity change to sucrose concentration ( / CSucrose) 2.75103 kgfluid kg / kgNaCl m s Coefficient of fluid viscosity change to dye concentration ( / CDye) 0.00 kgfluid kg / kgNaCl m s Longitudinal dispersivity (L) 0.00 m Transverse dispersivity (L) 0.00 m 94
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00.010.020.03CONCENTRATION OF SOLUTERELATIVE TO WATER, kg/kg 0.9911.011.021.03103 DENSITY, kg/m3 11.011.021.031.041.051.06103 ABSOLUTE VISCOSITY, kg m1s1 00.010.020.03CONCENTRATION OF SOLUTERELATIVE TO WATER, kg/kg SODIUM CHLORIDE AT 20OCSODIUM CHLORIDE AT 20OCR2=0.996R2=0.999 00.010.020.030.040.05CONCENTRATION OF SOLUTERELATIVE TO WATER, kg/kg 0.99511.0051.011.0151.02103 DENSITY, kg/m3 11.051.11.15103 ABSOLUTE VISCOSITY, kg m1s1 00.010.020.030.040.05CONCENTRATION OF SOLUTERELATIVE TO WATER, kg/kg SUCROSE AT 20OCSUCROSE AT 20OCR2=0.999R2=0.999 Figure 20 Absolute viscosity and fluid density relationships with sodium chloride and sucrose concentration used in all simulations. Data are from Weast (1986). 95
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To develop initial perturbations, two random numbers between 0 and 1 were generated for each node located at the initial solution interface (z = 0.08125 m). If the first random number (RN1) was less than 0.5, then the sodium chloride concentration for the node was reduced by 0.01RN1CNaCl(max), where CNaCl(max) is the maximum initial sodium chloride concentration (Table 3). If the second random number (RN2) was less than 0.5, then the sucrose concentration for the node was reduced by 0.01RN2CSucrose(max), where CSucrose(max) is the maximum initial sucrose concentration (Table 3). If either RN1 or RN2 was greater than or equal to 0.5, sodium chloride and/or sucrose concentrations for the node remained unchanged and equal to maximum values defined in Table 3. Initial dye concentrations at the interface were not perturbed. A consistent initial pressure field was developed using the perturbed concentration fields and the HeleShaw cell experiment parameters and fluid parameters summarized in Table 2 through Table 4. 3.7 Results and Discussion 3.7.1 Simulated numerical results Simulated numerical results at the same dimensionless times of Pringle et al. (2002) are shown in Figure 21. The concentrations shown in Figure 21 are simulated dye concentrations. Qualitatively, the numerical results are very similar to the experimental results until t*=1.03103, although experimental vertical finger evolution appears to be slightly ahead of simulated fingers. Pringle et al. (2002) presented three measures that were used to quantitatively assess finger movement, vertical mass flux, and average horizontal finger width. The third measure can be applied to simulated results and used to determine the correspondence of finger development and evolution in the numerical model and the HeleShaw experiment. For the quantitative measures used to assess finger movement and vertical mass flux, dye concentration fields were horizontally averaged. Concentrations along a horizontal traverse of the HeleShaw cell were used to calculate average horizontal finger widths. 96
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The first quantitative measure is the normalized vertical length scale, h*=h/H, which is a measure of evolving finger structure where h is the vertical distance in the plane of the HeleShaw cell between the horizontally averaged 0.05 and 0.95 dye concentration at time t* and H is the height in the plane of the HeleShaw cell. Both h and h* become undefined as fingers invert at their ends after reaching the top and bottom boundaries. The second quantitative measure is the normalized mass transfer of dye, M*=M/Mo, upward across the centerline of the cell where M is the dye mass above the centerline of the cell at time t* and Mo is the total dye mass in the cell. The third quantitative measure is the normalized horizontal length scale, *=/H, along a horizontal traverse of the HeleShaw cell where is the average horizontal length scale and H is the height of the HeleShaw cell. The average horizontal length scale, was determined by dividing the cell length by the number of upward and downward concentration pairs along a horizontal traverse of the cell. This measure can be thought of as twice the normalized average finger width or a normalized wavelength and will have different values at different vertical traverses. Differences in simulated and observed experimental values are presented as percent errors: obsobssimCCCError 100% Eq. 80 where Csim is the simulated concentration and Cobs is the observed experimental concentration. Positive percent errors indicate the model is oversimulating concentrations and negative percent errors indicate the model is undersimulating concentrations. Simulated data is available for all t* values shown on Figure 21 and are used to calculated simulated h*, M*, and values. 97
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Figure 21 Simulated numerical results for the dye at (A) t* = 4.03105, (B) t* = 1.31104, (C) t* = 2.21104, (D) t* = 3.22104, (E) t* = 4.23104, (F) t* = 5.24104, (G) t* = 6.04104, and (H) t* = 7.25104, (I) t* = 7.85104, (J) t* = 1.03103, (K) t* = 1.77103, and (L) t* = 3.17103. Color sequence blackbluegreenyelloworangered depicts normalized dye concentration from 0 to 1. 98
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Comparison of simulated and observed values of h*, M*, and are shown in Figure 22. Observed values from all experimental images analyzed by Pringle et al. (2002) are presented. Simulated results were only saved at t* values indicated in Figure 19 to minimize data storage requirements. In general, numerical results compare well to observed values of h* at all compared times. h* percent errors range from +6% at early times to 9% at late times and had an average value of 4% for all times compared. Numerical results compare reasonably well to observed values of M* until t*=1103. After t*=1103, simulated mass transfer is less than observed mass transfer. M* percent errors range from +53% at early times to 17% at late times and had an average value of 8% for all times compared. Large percent errors at early times are an artifact of small M* values and represent small absolute differences in mass transfer (e.g., 0.011 observed and 0.017 simulated). Reduced simulated mass transfer is also evident in Figure 21L where fingers are more defined than in observed results (Figure 19L). More intense convection at late times would move additional mass vertically and would lead to better mixing of the sucrose and sodium chloride and reduction/elimination of the finger structure observed at late times in the simulated results. Simulated values compare well to observed values in the finger generation zone at the centerline of the cell during the mature stage of finger development. Only two simulated and observed values are coincident but percent errors range from 1%. Temporally, the mature stage simulated exhibits a power law relationship with an exponent of 0.52 for 4.03105 t* 1.03103. Pringle et al. (2002) did not present values for t* > 2.0104 because of the increasingly diffuse nature of the concentration swings in the finger generation zone resulted in inaccurate measurements. Simulated values for t* > 2.0104 are presented because they appear consistent with mature stage data for t* values less than 2.0104. The simulated power law exponent compares favorably well with the observed mature stage exponent of 0.57 and the experimental work of Cooper et al. (2001) which yielded an exponent of ~0.5. 99
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1x1061x1051x101x1031x102t* 1x1021x1011x100h* 1x1061x1051x101x1031x102t* 1x1031x1021x101* 1x1061x1051x101x1031x102t* 1x1021x1011x100M* Simulated Observed Figure 22 Normalized vertical length, h*=h/H, mass transfer across the center line, M*=M/Mo, and normalized horizontal length scale at the center line, *=/H, as a function of time showing regions of steady growth for the original HeleShaw experiment (closed gray circles) and the numerical simulation. Observed data included for all 300 images taken by Pringle et al. (2002) of the HeleShaw cell experiment. 100
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1x103t* 1x101* 1x103t* 1x101* Y = 0.04063 mabove center lineY = 0.04063 mbelow center line Simulated Observed Figure 23 Normalized horizontal length scale, *=/H, at 0.75H and 0.25H as a function of time showing apparent asymmetry in finger evolution above and below the center line for simulated and observed dye concentrations. Visually, finger dimensions (vertical and horizontal lengths) below the centerline appear to be larger than finger dimensions above the centerline when observed and simulated dye concentrations on Figure 19 and Figure 21 are compared, suggesting finger evolution is faster below the finger generation zone at the centerline. Differences in finger evolution above and below the centerline can be explained by the differences in the molecular diffusivity of sodium chloride and sucrose. Because the molecular diffusivity of sucrose is a factor of three less than sodium chloride, density instabilities below the centerline will gain sodium chloride mass at a much greater rate than sucrose mass is lost and density instabilities above the centerline will lose sodium chloride mass faster than sucrose mass is gained. To quantitatively evaluate vertical finger evolution away from the finger generation zone, simulated and observed values at 0.75H (y = 0.1219 m) and 0.25H (y = 0.0406 m) have been calculated and are shown in Figure 23. values are undefined for t* values less than 2.21104 because the fingers have not penetrated to 0.25H from the centerline. Because of the generally diffuse nature of the fingers during the mature 101
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and rundown stages, average values and average percent errors in simulated and observed values at 0.75H and 0.25H are compared. Furthermore, values have not been calculated for t* values greater than 7.85104 when there is significant lateral movement of solute at the top and bottom of the HeleShaw cell. Observed average values at 0.75H and 0.25H are 5.52102 and 6.16102 at t* values greater than 3.2104 and supports the qualitative assessment that finger evolution is greater below the centerline. Simulated average values at 0.75H and 0.25H are 5.05102 and 5.42102 at t* values greater than 3.2104 and indicate simulated finger evolution is faster below the centerline. Average percent errors in values at 0.75H and 0.25H at t* values greater than 3.2104 are % and %, respectively. It is suspected that reduced simulated finger evolution at 0.75H and 0.25H is a result of decreased vertical mass transfer rates in the numerical model. A possible explanation for decreased mass transfer rates is discussed below in the following section. 3.7.2 Influence of discretization on numerical results Mass transfer resulting from doublediffusive flow is a function of density differences that are affected by finger dimensions. Because most mass transfer in a doublediffusive flow system is a result of convection, small finger dimensions may increase vertical mass transfer by allowing larger lateral and vertical density gradients to be maintained and permit more fingers to develop in a given lateral distance. Simulation of solute transport can often be difficult when truncation errors and/or grid size exceed the length scale of the concentration field being simulated. Truncation errors are commonly referred to as numerical dispersion and cause artificial spreading of concentration gradients. Numerical dispersion is a function of fluid velocity, element size, and the numerical technique used to solve the solute transport equation (Lantz, 1971). When a centered in time (CIT) and center in space (CIS) numerical technique is used, no numerical dispersion is introduced, but dispersion, discretization, and timestep lengths must be sufficient to satisfy Peclet and Courant number criteria and can be numerically expensive. When less expensive CITbackward in space (BIS), backward in 102
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time (BIT)CIS, or BITBIS numerical techniques are used numerical dispersion is vxx/2, vx2t/2, and vxx/2 + vx2t/2, respectively (Lantz, 1971; Kipp, 1987). An implicit assumption in these equations is that a single element is smaller than the dimensions of any solute features of interest (i.e., solute fronts are larger than a single element). If solute fronts are less than a single element, minimum concentration front widths are x/2 regardless of the fluid velocity and the numerical technique used. This is particularly true for finiteelement codes that use linearbasis functions to develop the matrix equations used to solve the flow and transport equations. A BITCIS numerical technique has been used in these numerical simulations and solute fronts (fingers) can be sharp relative to the element sizes used. As a result, the minimum finger width that can be resolved in these simulations is x/2. The finest spatial discretization used in the numerical simulations was a factor of 1.6 larger than the pixel size of the HeleShaw experiment and may not be sufficient to represent density gradients in the real system and resulting mass transfer. The effect of spatial discretization on simulated results was quantified using coarser discretizations of 41,984 and 167,936 elements (a factor of 16 and 4 fewer elements, respectively). The sides of the elements of the 41,984 and 167,936 element problems were 4.96104 m and 9.92104 m, respectively. Comparison of h*, M*, and values for the 41,984, 167,936, and 671,744element simulations are shown on in Figure 24. Comparison of h*, M* and values for the three levels of discretization shows that normalized length scales, mass transfer, and normalized horizontal length scales are a function of element size. The percent error of h* values for the 41,984element simulation was 8% at early times to 41% at late times and had an average of 22%. The percent error of M* values for the 41,984element simulation ranged from +105% at early times to 28% at late times and had an average value of 21%. The percent error of values for the 41,984element simulation ranged from +146% to +23%. The percent error of h* values for the 167,936element simulation was +48% at early times to % at late times and had an average of 1%. The percent error of M* values for the 167,936element simulation ranged from +87% at early times to 24% at 103
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late times and had an average of 16%. The percent error of values for the 167,936element simulation ranged from +44% to +0%. Percent errors were calculated in all comparisons using Equation 80 and simulated and observed experimental values. Large percent differences at early times are a result of small values of h* and M* and represent relatively small differences in the normalized length scale and normalized mass transfer. These results suggest that coarser discretization and subsequently coarser fingers move more mass vertically at early stages of finger development, but after t*2104 (mature finger stage) fingers are less defined both horizontally and vertically as a result of numerical dispersion and are unable to induce as much vertical mass transfer (as simulated in the 671,744element simulation and in the observed in the HeleShaw experiment). Enhanced mass transfer in the early stages of finger development is likely a result of initial perturbations that are effectively more concentrated at a coarser discretization, resulting in larger areas of density contrast at the solution interface. These results also suggest that a mesh with more than 671,744 elements would likely result in simulated normalized length scales and mass transfer closer to values calculated for the HeleShaw experiment. During the course of this study it was determined that it was not feasible to reduce the grid size by a factor of two (2) because of the resulting excessive run times and RAM requirements of ~28 days and ~3.8 GB for a 2,686,976 element mesh, respectively, for such a simulation. 104
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1x1061x1051x1041x1031x102t* 1x1021x1011x100h* 1x1061x1051x1041x1031x102t* 1x1031x1021x101* 1x1061x1051x1041x1031x102t* 1x1021x1011x100M* 671,744 elements 167,936 elements 41,984 elements Figure 24 Normalized vertical length, h*=h/H, mass transfer across the center line, M*=M/Mo, and normalized horizontal length scale at the center line, *=/H, as a function of time, showing regions of steady growth for the 41,984, 167,936, and 671,744element simulations. 105
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3.7.3 Comparison of the diffusion rates of dye, sodium chloride, and sucrose The molecular diffusivity of sodium chloride is a factor of 2.5 larger than the molecular diffusivity of the dye used to visualize sodium chloride concentrations. Similarly, the molecular diffusivity of sodium chloride is a factor of 3.0 larger than the molecular diffusivity of sucrose. The effects of the different molecular diffusivities on normalized sodium chloride, sucrose, and the dye concentrations and the resulting density profile relative to average initial column densities in a onedimensional column are shown graphically on Figure 25. The curves shown on Figure 25 at time to and ti represent the initial vertical profiles and vertical profiles after some period of time using a onedimensional form of the diffusion equation that assumes diffusion is Fickian, does not account for the effect of vertical fluiddensity differences on fluid flow, uses fluid parameters identical to the original experiment (Table 3), and spatial dimensions similar to the HeleShaw experiment. As shown in Figure 25, sodium chloride diffuses upward faster than sucrose diffuses downward from the initially stable density configuration. This difference in diffusion rates causes the development of instabilities in fluid density (Figure 25D) that will initiate convection when the destabilizing buoyancy forces exceed the stabilizing effects of diffusion. Convection will continue until diffusion of sodium chloride (the destabilizing component) eliminates density instabilities/gradients and/or fingers reach the top and bottom boundaries of the HeleShaw cell. Mass transfer typically decreases once fingers reach the top and bottom boundaries of the HeleShaw cell unless the horizontal length scale is relatively large and subsequent horizontal masstransfer rates are high. Vertical fluid flow caused by density instabilities will transport fluid with and without dye upward and downward, respectively, but differences in the molecular diffusivities of sodium chloride and dye will cause imperfect mapping of chloride concentrations by the dye. The discrepancy between sodium chloride and dye concentrations will increase as the simulation progresses as differences in mass diffusion increase and may significantly affect calculated quantitative measures of finger development and vertical mass transfer. Furthermore, dye concentrations should map the 106
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downward movement of sucrose better than the upward movement of sodium chloride because of the similarity of the molecular diffusivities of sucrose and dye (Figure 25A and Figure 25C). Unlike the HeleShaw experiment where sodium chloride and sucrose concentrations are not available independent of observed dye concentrations, the numerical model allows qualitative and quantitative comparisons of sodium chloride, sucrose, and dye concentrations. Figure 26 shows simulated sodium chloride concentrations at the same t* values evaluated in Figure 21. Figure 27 shows normalized sodium chloride, sucrose, and dye concentrations at t* = 5.24104, t* = 7.85104, and t* = 3.17103. Fluid density at t* = 5.24104, t* = 7.85104, and t* = 3.17103 are also shown in Figure 27. In general, sodium chloride concentrations are horizontally more dispersed (wider) than simulated dye concentrations (Figure 21) and visually it appears that vertical mass transfer of sodium chloride is less than that of the dye. Sucrose concentrations appear to map the inverse of dye concentrations reasonably well and exhibit similar finger lengths and widths. Development of a stable density configuration as the system transitions from a mature stage when convection is active to a rundown stage when diffusion dominates is shown in Figure 27J through Figure 27L. Comparison of the vertical length scales (h*), vertical mass transfer (M*), and normalized horizontal length scales (*) of dye and sodium chloride are shown in Figure 28. Vertical length scales (h*) vertical mass transfer (M*), and normalized horizontal length scales (*) have not been calculated for sucrose because these measures were not calculated by Pringle et al. (2002) because of the inability to measure sucrose concentrations. Percent errors for h* values for sodium chloride ranged from 19% at early times to 46% at late times and averaged 36% of experimental dye values. Percent error for M* values for sodium chloride were +25% at early times to 41% at late times and averaged 35% of experimental dye values. The percent error of values for the 167,936 element simulation ranged from +14% to +7% when compared to experimental dye values. Calculated h* and M* values for sodium chloride were less than calculated dye values at 16 out of 17 values of t* compared and had a difference in average h* and 107
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M* values of % and %, respectively. In all comparisons, percent errors were calculated using Equation 80 and simulated and observed experimental values. Increased discrepancies between sodium chloride and dye concentrations over the length of the experiment were noted by Pringle et al. (2002) and were attributed to the dominance of diffusion in the rundown stages of the experiment (t* 9.00104) and interaction of sodium chloride, sucrose, and dye. The maximum error caused by using the dye to map sodium chloride was estimated by Pringle et al. (2002) for the end of the experiment to be 5%, which is significantly less than that indicated using the numerical model. 3.8 Concluding Remarks The numerical model presented is capable of simulating high resolution HeleShaw doublediffusive finger convection. The numerical model does a reasonable job of simulating the temporal and spatial evolution of convection except at late times (t* 3.19103). Average percent errors in normalized vertical length scales (h*) and normalized mass transfer (M*) are less than 10% over the length of the simulation. At late times, simulated fingers remained more defined than were observed during the experiment and simulated mass transfer was a maximum of 8% less than observed. Numerical results are very sensitive to discretization and are closest to experimental results using when elements that are 2.48104 m along each side. Mass transfer was enhanced at early times and reduced at late times using meshes coarser than 2.48104 m. Enhanced mass transfer at early times was a result of effectively larger initial instabilities (fewer nodes over the same length). Reduced mass transfer at late times was a result of increased lateral numerical dispersion. The finest discretization used was a factor of 1.6 larger than the resolution of experimental results and discretization sensitivity results suggest that numerical results would likely improve if cell sizes were significantly smaller than the average finger width. Finer discretization was not attempted in this study because of excessive RAM requirements (>3 GB) and run times (>>7 days) required to satisfy numerical stability requirements. Parallelization of 108
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the model code may be necessary to make simulations with finer meshes practical at this time. Differences in the molecular diffusivity of sodium chloride and the dye used to visualize sodium chloride concentrations result in an average difference of 27% in calculated dye and sodium chloride vertical mass transfer over the entire simulation length. The effect of differences in fluid properties could not be addressed in the experimental results because sodium chloride concentrations in the convection system could not be determined independent of dye concentrations. Comparison of the numerical model to experimental HeleShaw data is a robust test of the models ability to simulate multicomponent variable density (doublediffusive) flow because the convective system is complex and driven exclusively by differences in fluid properties. These simulated results suggest it would also be possible in the future to apply the numerical model to additional HeleShaw doublediffusive convection datasets (e.g., Cooper et al., 2001) with different buoyancy ratios (R). 109
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00.20.40.60.81CS/maxCS 00.20.40.60.81CT/maxCT 42024Relative (kg/m3) 0.0030.0020.00100.0010.0020.003Position Relative to Initial Interface (m) 00.20.40.60.81CDye/maxCDye ABC D toti Figure 25 Conceptual (A) sucrose concentrations, (B) sodium chloride concentrations, (C) dye concentrations, and (D) fluid density relative to the initial average fluid density showing the effect of different diffusivities on component concentrations and fluid density. Concentrations calculated at time ti using a onedimensional form of the diffusion equation with diffusivities and spatial dimensions consistent with the HeleShaw experimental setup. Note that the vertical density profile is unstable at time ti. 110
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Figure 26 Simulated numerical results for sodium chloride at (A) t* = 4.03105, (B) t* = 1.31104, (C) t* = 2.21104, (D) t* = 3.22104, (E) t* = 4.23104, (F) t* = 5.24104, (G) t* = 6.04104, and (H) t* = 7.25104, (I) t* = 7.85104, (J) t* = 1.03103, (K) t* = 1.77103, and (L) t* = 3.17103. Color sequence blackbluegreenyelloworangered depicts normalized sodium chloride concentration from 0 to 1. 111
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Figure 27 Simulated sucrose concentrations at (A) t* = 5.24104, (B) t* = 7.85104, and (C) t* = 3.17103. Simulated sodium chloride concentrations at (D) t* = 5.24104, (E) t* = 7.85104, and (F) t* = 3.17103. Simulated dye concentrations at (G) t* = 5.24104, (H) t* = 7.85104, and (I) t* = 3.17103. Simulated fluid density at (J) t* = 5.24104, (K) t* = 7.85104, and (L) t* = 3.17103. Color sequence blackbluegreenyelloworangered depicts normalized concentrations from 0 to 1 (AI) and fluid density (JL). 112
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1x1061x1051x1041x1031x102t* 1x1021x1011x100h* 1x1061x1051x1041x1031x102t* 1x1031x1021x101* 1x1061x1051x1041x1031x102t* 1x1021x1011x100M* DyeObserved NaCl Figure 28 Normalized vertical length, h*=h/H, mass transfer across the center line, M*=M/Mo, and normalized horizontal length scale at the center line, *=/H, as a function of time showing regions of steady growth for the experiment, simulated sodium chloride concentrations, and simulated dye concentrations. 113
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Chapter 4 4 Temporal response of temperature and salinity to sealevel changes, Floridan aquifer system, USA. 4.1 Abstract Threedimensional densitydependent flow and transport modeling of the Floridan aquifer system shows that current chloride concentrations are not in equilibrium with current sealevel and, second, that the geometric configuration of the aquifer has a significant effect on system responses. In detail, the modeling shows that pressure equilibrates most rapidly and is followed by temperatures and then chloride concentrations. The model was constructed using a modified version of SUTRA capable of simulating multispecies heat and solute transport and calibrated using hydraulic heads, chloride concentrations, and temperatures. Three hypothetical, sinusoidal sealevel changes occurring over 100,000years were used to evaluate how the aquifer responds to sealevel changes. Pressure responses lag behind sealevel changes only where the thickness of the Miocene Hawthorn confining unit exceeds 100 m. Temperatures equilibrate quickly except where the Hawthorn confining unit is more than 200 m thick and the duration of the sea level event is long. Response times for chloride concentrations are shortest near the coastline and where the aquifer is unconfined; chloride concentrations do not change significantly over the 100,000year simulation period where the Hawthorn confining unit is thick. 4.2 Introduction The Floridan aquifer system (FAS) is one of the major sources of groundwater in the southeastern United States, and underlies all of Florida, southern Georgia, and parts of Alabama and South Carolina (Figure 29). The FAS of peninsular Florida is made up of several Tertiary carbonate formations that are hydraulically connected to form a regional hydrogeologic unit. Although the FAS is hydraulically connected throughout, 114
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there are significant variations in its hydraulic properties, water chemistry and flow characteristics. Comparison of the 50% seawater isochlor (Sprinkle 1989) and the top of the FAS (Miller 1986) shows that the relationship between the two is not simple (Figure 30). The FAS in peninsular Florida can contain groundwater with seawater or nearseawater compositions because of the close proximity to the Gulf of Mexico or Atlantic Ocean. The FAS can also contain seawater or nearseawater concentrations where the degree of confinement is high, recharge is relatively low, and the top of the FAS is deep. Apparent 14C ages and 234U/238U alphaactivities of groundwater suggest that some portions of the deep, saline FAS have a close connection to the Atlantic. Exchange between the FAS and the Atlantic Ocean is thought to occur via solution features or outcrops of the FAS in the Straits of Florida (Meyer 1989a). These areas also correspond to locations where inverted temperature profiles are present (Kohout, 1965; Meyer, 1989a). Outcrops of Florida Platform carbonates are also present in the Gulf of Mexico (Paull et al., 1991). Apparent 14C ages and noble gas analyses in groundwater in south Florida suggest that the connection of the upper FAS to the Atlantic Ocean is much more restricted than the lower FAS (Lukasiewicz, SFWMD, personal communication). Relatively high chloride concentrations and/or decreasing chloride concentrations with depth occur in the upper FAS in south Florida on the east and west coast suggesting that flushing of seawater from Pleistocene sealevel highstands has not been complete (Reese, 1994; Reese, 2000; Reese and Memberg, 2000). Because current chloride concentrations in portions of the FAS may reflect past sealevel conditions, it is important to quantify the extent to which the aquifer has a memory. To date there are only a few quantitative studies of the effects of sealevel changes on groundwater flow and chloride concentrations. Examples include the North Atlantic Coastal Plain, USA (Meisler et al. 1984); Galveston Bay, Texas, USA (Leatherman 1984); the PotomacRaritanMagothy aquifer system, New Jersey, USA (Lennon et al. 1986); the FAS of South Florida, USA (Meyer 1989a); and the Netherlands (Oude Essink 1999). Many of these studies did not evaluate the effects of 115
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sealevel changes on groundwater chloride concentrations using densitydependent groundwaterflow and transport models. Figure 29 Extent of the Floridan aquifer system, study area, and hydrostratigraphic data sources used in the study. 116
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Figure 30 Altitude of (a) the 50percent seawater concentration in the Floridan aquifer system (adapted from Sprinkle, 1989) and (b) the top of the Floridan aquifer system relative to sea level. 117
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The purpose of this study was to explore the effect of sealevel change on the pressures, chloride concentrations, and temperatures in the FAS. A densitydependent groundwater flow and transport model of the Florida Platform is developed and calibrated to conditions prior to significant groundwater pumpage. Simulated results from the calibrated model are used as initial conditions for simulations that utilized timevarying boundary conditions to represent a hypothetical rising and falling sea level. The simulated sealevel rise is not meant to represent any specific past sealevel event, but its magnitude is consistent with rises to levels that are marked by the preQuaternary terraces of Florida (Healy, 1975). 4.3 General Description of the Study Area Peninsular Florida has a humid, subtropical climate and is characterized by warm, normally wet summers and mild, dry winters. Rainfall ranges from 1,270 mm to 1,520 mm per year, with more than half falling between June and September as a result of frequent highintensity thundershowers resulting from convective storms. Annual evapotranspiration ranges from 570 to 1,080 mm for natural vegetation and up to 1,450 mm for open water with no emergent vegetation (Bidlake et al. 1996; Sumner 1996; Lee and Swancar 1997; and German 2000). Topographic elevations in peninsular Florida range from sea level along the coast to 70 m on the Lakeland Ridge in central Florida. The slope break in the Gulf of Mexico is approximately 200 km offshore from the west coast; as a result, water depths are relatively shallow in the Gulf of Mexico (Figure 31). Along the Atlantic coast, the shelf break ranges from just offshore near Miami, Florida, to approximately 100 km offshore near Jacksonville, Florida (Figure 31). Seaward of the shelf break, water depths can exceed 3,000 m in the Gulf of Mexico and the 800 m in the Straits of Florida. Soils in peninsular Florida are generally sandy and highly permeable. The combination of high soil permeability and low topographic relief results in low runoff rates and high groundwater recharge rates (Miller 1997). As a result, the FAS is one of the most prolific aquifer systems in the United States (Johnston and Bush 1988). Recharge directly to the upper carbonate units of the FAS occurs over approximately 118
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55% of the state and varies from less than 1 mm per year to more than 250 mm per year (Berndt et al. 1998). Figure 31 Bathymetry and topography in the study area. 4.4 Geologic and hydrogeologic setting of the Florida platform The Coastal Plain physiographic province of the southeastern United States is underlain by a thick sequence of unconsolidated to semiconsolidated sedimentary rocks that range in age from Jurassic to Holocene. These rocks generally thicken seaward from 119
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where they crop out against older metamorphic and igneous rocks of the Piedmont and Appalachian provinces (Figure 29). Coastal Plain rocks generally dip gently toward the Atlantic Ocean or the Gulf of Mexico, except where they are tilted or faulted on a local to subregional scale. The thickness of the coastal plain sediments and rocks is greatest in south Florida and can exceed 7,600 m (Johnston and Bush 1988). Coastal Plain sediments were laid down on an eroded surface developed on igneous intrusive rocks, lowgrade metamorphic rocks, metamorphosed Paleozoic sedimentary rocks, and grabenfill sedimentary deposits of Triassic to Early Jurassic Age (Barnett 1975; Neathery and Thomas 1975; Chowns and Williams 1983). Coastal Plain sediments and rocks in peninsular Florida can be separated into two general facies: (1) predominantly clastic rocks, containing minor amounts of limestone, and (2) a thick continuous sequence of shallowwater platform carbonate rocks. Tertiary carbonate rocks form most of the FAS and crop out or are located near the surface (subcrop) in portions of westcentral peninsular Florida (Figure 30). Although the defined FAS ends at the present day coastline, timeequivalent carbonate units extend some distance into the Atlantic and a substantial distance into the Gulf of Mexico. The FAS has been subjected to physical erosion, chemical alteration, and permeability enhancement during sealevel lowstands and deposition of additional sediments during sealevel highstands. Prior to the Miocene, carbonate deposition was the dominant sedimentary process during sealevel highstands. Filling of a channel in northern Florida (Suwannee Straits) during the Miocene allowed deposition of intermixed sand, clays, and carbonate sediments to occur throughout most of peninsular Florida and effectively ended largescale carbonate production in the peninsula except in southernmost Florida. The intermixed sand, clay, and carbonate sediments of the resultant siliciclastic part of the Hawthorn Group (Miocene) thicken to more than 150 meters in south Florida and form the upper confining unit for the FAS in most of peninsular Florida except in westcentral Florida where postMiocene erosion has exposed the FAS (Scott 1988). PostMiocene erosion of the Miocene Hawthorn Group siliciclastics has exposed the FAS in some areas of westcentral Florida. 120
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The FAS is a sequence of hydraulically connected carbonate rocks ranging from Paleocene to early Miocene in age (Figure 32). The FAS thickens to the south to more than 900 m with a maximum measured thickness of about 1,050 m in southwest Florida (Figure 33). The Hawthorn Group is a confining unit for the FAS throughout most of peninsular Florida. The base of the FAS is defined as the top of the bedded anhydrite unit in the Paleocene Cedar Keys Formation. The FAS is generally subdivided into an upper Floridan aquifer (UFA) and a lower Floridan aquifer (LFA) where middleEocene carbonates are significantly less permeable than upper and lower Eocene carbonates. The UFA and LFA are defined on the basis of permeability, and their boundaries locally do not coincide with those of either timestratigraphic or rockstratigraphic units. Where present, the lesspermeable middle Eocene age carbonates are termed the middle confining to semiconfining unit (MCU) (Miller 1986). The MCU ranges from dense gypsiferous limestone in the central portion of peninsular Florida to soft chalky limestone in the Florida Keys (Johnston and Bush 1988). The UFA is a very permeable unit that is composed of the Oligocene Suwannee Limestone, the upper Eocene Ocala Limestone, and the upper part of the middle Eocene Avon Park Formation. The UFA is used extensively as a source of potable water, and its properties are relatively well known. The LFA consists largely of middle Eocene to Upper Paleocene carbonate beds (Johnston and Bush 1988). The LFA has not been extensively utilized as a source of potable water except in northern parts of the peninsula where it is relatively shallow and the MCU is thin to absent (Johnston and Bush 1988; Miller 1986). The heavily fractured dolomitic zones of the LFA in south Florida has received considerable attention because of its use as an injection zone for municipal waste and oilfield brines and its unique thermal and geochemical properties (Kohout 1965; Meyer 1989a; Meyer 1989b). This heavily fractured dolomitic zone is typically referred to as the Boulder Zone (BZ) and its extent, based on Puri and Winston (1974), is shown on Figure 33. 121
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Figure 32 Hydrostratigraphy of the Floridan Aquifer in peninsular Florida, and hydraulic parameters in the calibrated model (modified from Johnston and Bush, 1988). 122
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Figure 33 Thickness of the Floridan aquifer system and extent of the Boulder Zone in peninsular Florida. 4.5 Regional Flow System In peninsular Florida, the general flow direction in the UFA is toward the Gulf of Mexico and Atlantic Ocean from central inland areas (Figure 34). The predevelopment potentiometric surface indicates that recharge occurs in inland areas in the north to northcentral portions of the peninsula and discharge occurs in coastal areas (Johnston et al. 1980). The degree of confinement of the UFA is the major feature that influences the 123
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distribution of recharge from the overlying surficial aquifer and discharge. Most of the recharge and discharge occurs in areas where the UFA is unconfined or semiconfined (Figure 34). Figure 34 Predevelopment UFA potentiometric surface (m) and location of springs in the study area. Relatively high temperature springs are shown as red circles and are located in southwest Florida. Other springs are shown as black circles and are located in discharge areas. Estimated Floridan aquifer system recharge rates (in/yr) of Aucott (1988) are also shown. 124
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Discharge from springs accounts for a high percentage of the total discharge from the FAS (Scott et al. 2002). Nearly all of the springs occur in unconfined and semiconfined parts of the aquifer system in Florida (Figure 34). In the 1980s, the combined average discharge from about 300 known UFA springs ranged from 355 to 370 m3/s. Spring discharge has been estimated to be 88 percent of all groundwater discharge (540 m3/s of 610 m3/s total FAS discharge) from the FAS prior to extensive groundwater withdrawals (Johnston and Bush 1988). Kohout (1965) hypothesized a convective circulation with an inflow of cold seawater to the LFA from the Straits of Florida. Numerical investigations by Kohout et al. (1977) using a simplified model of the FAS in south Florida supported Kohouts hypothesis of thermally forced circulation (Kohout Convection). Relatively low temperatures and inverted temperature profiles in the LFA on the east coast of Florida from Stuart to Miami are further evidence for a connection between the FAS and the Atlantic Ocean in the Straits of Florida. The lowest observed temperature (10.3 C) was measured in a deep disposal well (G2334) at Fort Lauderdale (Meyer, 1989a). Temperatures generally increase from the Straits of Florida inland toward the center of the Florida Plateau, which is actually located along the west coast of the peninsula. 14C and 234U/238U alphaactivity ratios in the LFA are consistent with the temperature data and suggest seawater enters the FAS in the Straits of Florida (Meyer, 1989a). Plummer (1977) estimated apparent 14C age of water in the UFA as 36,000 years in southwest Florida. More recent work in the UFA and LFA in south Florida suggests that the apparent 14C age is about 35,000 years in the UFA and about 15,000 years in the LFA (Lukasiewicz, SFWMD, personal communication). The smaller apparent 14C age in the LFA is consistent with Kohouts hypothesis that seawater from the Atlantic enters the LFA, flows inland and upward to the UFA, and eventually discharges to the ocean from the UFA. Surface discharge of relatively hightemperature saline groundwater at several locations along the Gulf coast in south Florida has long been considered evidence of thermal upwelling from deeper portions of the Floridan aquifer (e.g., Kohout et al.1977; Clausen et al. 1979; Fanning et al. 1981). Relatively hightemperature discharge locations (Figure 34) include Warm Mineral Springs (32 C), Little Salt Springs (24.4 125
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C), HumbleLowndesTreadwell No. 1 (36 C), and the Mud Hole Submarine Springs Complex (~40 C). Recent geochemical data from Warm Mineral and Little Salt Springs suggests the sources of water for these springs are the Ocala LimestoneAvon Park Formation and Suwannee Limestone stratigraphic units of the UFA, respectively (Hutchinson 1992; Sacks and Tihansky 1996). 4.6 Simulation of the Groundwater Flow System 4.6.1 Model Construction We use the USGS program SUTRAMS (Hughes and Sanford 2004) to simulate coupled heat transport, salt transport, and groundwater flow in the FAS of peninsular Florida. SUTRAMS uses a weighted numericalresidual method that combines Galerkin finiteelement and integrated finitedifference techniques to simulate densitydependent groundwater flow and transport. The Galerkin finiteelement method allows geometric flexibility in mesh design and gives robust directionand anisotropyindependent representation of fluid and solute fluxes. The integrated finitedifference representation for the spatial integration of all nonflux terms in the governing equations provides an economical alternative to the Galerkin method and is capable of giving sufficient accuracy for mildly nonlinear problems. The equations used in SUTRAMS have been described in detail elsewhere (Voss and Provost, 2002; Hughes and Sanford, 2004; Hughes et al., 2005). Parameters simulated in the model include fluid pressure, chloride, and temperature. Fluid and spatially constant matrix parameters are summarized in Table 5. Chloride concentration results are presented as percent seawater chloride concentrations, based on an assumed seawater chloride concentration of 19,400 mg/L, but are referred to as chloride concentrations. Each aquifer and confining unit shown in Figure 32 is vertically discretized into five element layers. We discretized the study area shown in Figure 29 using a total of 10,556 nodes horizontally and 36 nodes vertically, using the horizontal mesh shown in Figure 35. Thus the total volume is represented using 380,016 nodes and 362,250 126
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elements. In the emergent part of the Florida platform the top nodes correspond to the average elevation of the water table based on watertable depth data from Sepulveda (2002). Table 5 Model Parameters C0 = Cfw = 0.000 [kgdissolved solids / kgseawater] Csw = 0.0357 [kgdissolved solids / kgseawater] T0 = 20 C TLand = 23.5 C Tsw = ze00150085.25 fw = 1,000 [kg / m3] at 20 C sw = 1,024.99 [kg / m3] at 20 C SOLID = 2600. [kg / m3] C = 700. [kgseawater 2 / (kgdissolved solids m3) ] T = 0.375 [kgseawater / ( C m3) ] = 15133 37248 71010394.2Tx [kg / (m s)] L = 1,250 or 500 [m] T = 125 or 50 [m] DM = 1 109 [m2 / s] W = 0.6 [J / (m C s)] cW = 4,182. [J / (kg C)] cS = 840 [J / (kg C)] = 1.3 106 [kg / m s2]1 = 4.4 1010 [kg / m s2]1 g = 9.8 [m / s2] 127
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We used data generated as part of the U.S. Geological Survey Regional AquiferSystem Analysis (RASA) program in Florida (Miller, 1988) to develop the framework of the UFA, MCU, and LFA on the emergent part of the Florida Platform. Seismic data from Mullins et al. (1988) and Jee (1993) was used to develop the hydrogeologic framework in the Gulf of Mexico for offshore carbonate units that are time equivalent to carbonate units composing the ICU, FAS, and Paleocene basal confining units. Mullins et al. (1988) and Jee (1993) correlated seismic reflectors to timeequivalent units using a combination of piston core data and offshore lithologic, biostratigraphic, and geophysical data collected by the petroleum industry. Data sources used to define the hydrogeologic framework in the study are shown graphically on Figure 29. Figure 35 Model mesh used in the study area. 128
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We used data from Sepulveda (2002) to develop the initial distribution of hydraulic parameters. The data in Sepulveda (2002) was supplemented with data from Tibbals (1990) for northeast Florida; Johnston and Bush (1988), Merritt (1997), Reese (2000), Reese and Memberg (2000), and USEPA (2003) for South Florida; Hutchinson (1992) for Sarasota and Charlotte Counties; Ryder (1985) and Budd and Vacher (2004) for westcentral Florida; and Spechler and Halford (2001) for central Florida. The final distribution of horizontal intrinsic permeability values used in the model is shown in Figure 36. Final hydraulic parameters ranges are also summarized in Figure 32. A constant horizontal intrinsic permeability of 412 m2 is defined for surficial deposits on the emergent part of the platform. Spatially varying horizontal intrinsic permeabilities are assigned to the ICU/IAS and UFA based on data from Merritt (1997) and Sepulveda (2002). Constant horizontal intrinsic permeabilities are assigned to the MCU, LFA, and BZ and were based on data from Merritt (1997) and Sepulveda (2002). A constant intrinsic permeability of 51017 m2 is assigned to the Cedar Keys Formation and is representative of lowpermeability dolomite and massive anhydrite beds. The high horizontal, and corresponding vertical, intrinsic permeability for the ICU/IAS shown in the Straits of Florida in Figure 36a is meant to represent fracture porosity and karst features thought to exist offshore and allow direct exchange of seawater between the FAS and the Straits of Florida. Matrix thermal conductivities (s) are calculated from endmember values of 1.3 J/mCs for gypsum, 2.3 J/mCs for marine sediments/sands, 3.3 J/mCs for limestone, 4.0 J/mCs for dolomite, and 5.0 J/mCs for anhydrite (Birch and Clark, 1940; Herrin and Clark, 1956; Clauser and Huenges, 1995). The marine sediments/sands s end member has been applied to surficial deposits and the ICU. The IAS is assigned a composite s value based on an assumed mixture of 33% limestone and 66% marine sediments/sands. The limestone s end member is applied to the UFA. The MCU is assigned a composite s value based on an assumed mixture of 3.5% gypsum and 96.5% limestone in areas where gypsum is present (Figure 36c) and the limestone s value elsewhere. The LFA and BZ are assigned a composite s value based on an assumed 129
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mixture of 50% limestone and 50% dolomite. The Cedar Keys Formation is assigned a composite s value based on a mixture of 50% anhydrite and 50% dolomite. A geometric mean of the rock conductivity and fluid conductivity is used to define the bulk thermal conductivity of the modeled hydrogeologic units (Sass et al. 1971), Eq. 81 )1(sw where w is the thermal conductivity of water, (x,y,z) is the porosity, and s is the thermal conductivity of the solid matrix. An effective porosity of 21% and 35% based on effective porosity values used in Vacher et al. (2006) is specified for the FAS and the Boulder Zone of the FAS, respectively. Horizontal and vertical dispersivities are specified as 500 and 50 m in areas with a 1km node spacing and 1,250 and 125 m in areas with a 5km node spacing, respectively. Specified dispersivities satisfy mesh Peclet number numerical stability criteria, but are also consistent with the scaledependent nature of dispersivity (Gelhar, 1986). Testing the effect of smaller values of dispersivity was not computationally feasible. Side and bottom boundaries are specified to be noflow, and a heat flux of 60 mW/m2, based on the data of Griffin et al. (1977) and Smith and Lord (1997), is assigned to the lower boundary of the model (within the massive anhydrite beds of the Cedar Keys Formation). Oceanboundary temperatures are specified using a temperature vs. depth relationship developed from Atlantic Ocean and Gulf of Mexico observation data (see Table 5). The salinity at the ocean boundary in the Atlantic Ocean and the Gulf of Mexico is set at 36 o/oo (unmodified seawater). Fluid pressures at the ocean boundaries are calculated based on the density of the overlying column of seawater. For the emergent part of the platform, net groundwater recharge is applied to the upper surface nodes to represent groundwater recharge in recharge areas and groundwater discharge in 130
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discharge areas. Initial estimates of net groundwater recharge were derived from data of Aucott (1988) and augmented with modelderived values of Langevin (2001). Specified salinity concentrations of 0 o/oo (freshwater) and temperatures representative of the annual average air temperature for the study area (23.5C) are applied to all surface nodes in the emergent part of the platform. 4.6.2 Model Calibration We calibrated a steadystate model using 315 observation points that provided 81 head measurements, 244 chloride concentration observations, and 128 temperature observations. Observations representative of conditions prior to significant groundwater pumpage were selected for model calibration and in most cases correspond to the earliest available measurements. The locations of observations in the ICU/IAS, UFA, MCU, and LFA are shown in Figure 36. Overall, the model reproduces the gross temperature and salinity patterns in the Florida Platform, although in detail there are poor fits to a number of local observations. 4.6.2.1 Simulated Results The initial estimate of net groundwater recharge in the emergent part of the Florida Platform was adjusted during model calibration to match estimated water table elevations (the top of the model) and the location of aquifer discharge locations (e.g. major rivers). The final net recharge applied in the steadystate model is shown in Figure 37 and corresponds to the general pattern of recharge and the discharge in south Florida and along the major rivers (Hillsborough River, Kissimmee River, Peace River, St. Johns River, and the Withlacoochee River) shown in Figure 34. Simulated fluxes between the ICU/IAS and the UFA shown in Figure 38 can be directly compared to Figure 34. Simulated UFA recharge and discharge rates are consistent with the data of Aucott (1988). The simulated UFA potentiometric surface (Figure 38) shows that the calibrated model captures key features and elevations of the predevelopment potentiometric surface (Johnston et al. 1980). 131
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Figure 36 Intrinsic permeability of the a) ICU/IAS, b) UFA, c) MCU, and d) LFA/Boulder Zone used in the model. The location of calibration wells used in each hydrostratigraphic unit is also shown. The location of the unconfined portion of the FAS, the location of the IAS, the location of the Avon Park Formation with Gypsum, and the extent of the Bolder Zone are also shown. 132
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Figure 37 Net recharge (in/yr) applied to the surficial aquifer in the steadystate model. Major rivers are also shown to highlight discharge areas. 133
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Figure 38 Simulated fluxes (in/yr) between the Miocene IAS/ICU and the UFA in the emergent part of the platform. The simulated potentiometric surface for the UFA is also shown. Crosssection A to A shown in Figure 39 is identical to the crosssection evaluated by Kohout et al. (1977) and simulated chloride concentrations and temperatures along crosssection A to A are shown in Figure 40. The location of the 50%seawater chloride concentration is located in the IAS to UFA on the west side of the crosssection and in the MCU on the eastern side of the crosssection. The location of the 50%134
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seawater chloride concentration is consistent with Reese (2000) and Reese and Memberg (2002). Simulated temperatures show that temperatures are highest on the west side of the crosssection and decrease to temperatures close to temperatures in the Straits of Florida (5C) on the east side of the crosssection. Simulated temperatures also show an inverted temperature gradient from the MCU to the BZ on the east side of the crosssection. Simulated horizontal temperature gradients are consistent with the observations of Kohout et al. (1977). Figure 39 Crosssections locations where simulated results for the calibrated model are evaluated. 135
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Figure 40 Simulated percent seawater chloride concentrations (A) and temperatures (B) along crosssection line A to A shown on Figure 39. Crosssection B to B (Figure 39) corresponds to the axis of the UFA predevelopment potentiometric surface. The calibrated model shows that freshwater extends to the Cedar Keys Formation in recharge areas located at the north end of the study area where the MCU is thin (Figure 41). The FAS contains a higher proportion of seawaterlike water as the ICU/IAS thickens to the south and the distance from direct recharge areas increases. The elevation of the 50%seawater concentration is consistent with Sprinkle (1989). The results also show that the highest temperatures along crosssection B to B are found in the center of the crosssection. This location corresponds to an area of known elevated temperatures (Kohout et al. 1977) and is approximately 100 km east of the center of the Florida Platform that corresponds to the area with the highest observed temperatures. The lowest simulated temperatures are found adjacent to the Florida Straits. 136
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Figure 41 Simulated percent seawater chloride concentrations (A) and temperatures (B) along crosssection line B to B shown on Figure 39. Crosssection C to C shown in Figure 39 corresponds to the axis of the Sarasota Arch. Simulated chloride concentrations show that the 50%seawater contour is within the Cedar Keys Formation in the center of the crosssection and along the axis of the UFA predevelopment potentiometric high (Figure 42). The elevation of the 50%seawater chloride concentration is close to or at the top of the model on the west side of 137
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the crosssection where the crosssection is parallel to the Tampa Bay coastline. The 50%seawater chloride concentration elevation is close to the surface on the east side of the crosssection near Cape Canaveral. Simulated temperatures are high on the both sides of the crosssections. The high temperatures on the west side of the crosssection correspond to the temperature high simulated in the vicinity of Ft. Meyers. The high temperature on the east side of the crosssection is a result of the geometry of the FAS in the northeastern area of the study area and the distance from a source of lowtemperature water (freshwater recharge and cold Atlantic Ocean waters). Figure 42 Simulated percent seawater chloride concentrations (A) and temperatures (B) along crosssection line C to C shown on Figure 39. 4.6.2.2 Analysis of Fit The steadystate model was manually calibrated from initial estimates of aquifer parameters because forward simulation times were too long (10 to 20 hours) to make significant use of parameterestimation techniques. The computer code UCODE (Poeter and Hill 1998) was used early in the calibration process to identify parameters and model areas with significant sensitivity. Subsequent calibration efforts were focused on parameters and areas determined to be sensitive to adjustment. 138
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Simulated pressures were converted to groundwater heads to evaluate model calibration using zphfTg Eq. 82 where hT(x,y,z,t) is the total hydraulic head, p(x,y,z,t) is the fluid pressure, f (x,y,z,t) is the fluid density, g is gravitational acceleration, and z(x,y) is the elevation of the node. Simulated temperatures and percent seawater values were compared directly to observation data. The root mean square error (RMSE) for each hydrostratigraphic unit and the entire FAS is shown in Table 6. The range of heads, chloride concentrations, and temperatures for each hydrostatic unit and the entire FAS are also shown in Table 6. The root mean square error was calculated using nobssimRMSEniii12 Eq. 83 where simi is the simulated value at location i, obsi is the observed value at location i, and n is the number of observations. The observed hydraulic heads plotted against simulated heads show a generally reasonable fit (Figure 43a). The poorest fit of hydraulic heads occurs in the MCU and the 139
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LFA. The poor fit in the MCU may be a result of simplification of this hydrostratigraphic unit. Recent evaluation of geophysical data by Reese and Richardson (2004) indicates that the hydrostratigraphy of the MCU in south Florida is more complicated than presented by Miller (1986). Table 6 Calibration statistics for the hydrostratigraphic units of the FAS Number of Observations Range of Values RMSE Unit Total Head (m) Percent Seawater Temp. (C) Total Head (m) Percent Seawater Temp. (C) Total Head (m) Percent Seawater Temp. (C) ICU 8 12 9 6.32 10% 5.40 4.06 28% 2.64 IAS 9 45 14 15.31 23% 6.90 6.42 22% 8.63 UFA 57 90 80 20.89 97% 6.70 7.25 32% 3.39 MCU 6 31 8 20.13 98% 13.20 3.59 32% 3.54 LFA 1 55 7 13.11 92% 5.80 49% 5.46 BZ 0 11 10 93% 20.40 41% 7.23 A ll Units 81 244 128 24.91 98% 32.4 6.68 35% 4.73 The observed chloride concentrations plotted against simulated chloride concentrations (Figure 43b) show a larger amount of scatter than the hydraulic heads. The poorest fit occurs in the LFA and the BZ; in general, simulated chloride concentrations tend to be slightly larger than observed. The LFA and the BZ in South Florida generally contain saline water but 18 of 55 and 2 of 11 chloride observations evaluated, respectively, have values that are less than 25% of seawater chloride concentrations; relatively low chloride concentrations in the LFA and BZ are found in areas where anomalous chloride concentrations have been noted (e.g., Reese and Memberg 2000; Reese 2004). Furthermore, the poorer fit to the percentseawater data was expected because: (1) heads represent a smoothly varying potential field that can be easily fit to the solution of the flow equation, but solution of the transport equation for chloride concentrations is a function of the first derivatives of the groundwaterpotential field (i.e., velocity) and are more difficult to fit (Sanford et al. 2004); (2) transport 140
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phenomena are dependent on localscale features (Gelhar, 1986) that are typically poorly constrained and difficult to represent in regionalscale models; and (3) chloride concentrations are highly dependent on the specified initial conditions that are a function of past conditions which are poorly constrained. All of these factors can contribute to the total simulated error and result in a poorer fit in comparison to the fit of head data. The lack of a good fit led us to investigate the role of transport lagtime and sealevel changes. Figure 43 Simulated and observed hydraulic heads, percent seawater, and temperatures for the calibrated model. 141
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The observed temperatures plotted against simulated temperatures (Figure 43c) show a reasonable fit (R = 0.68). The poorest fit was observed in the IAS and the BZ. The relatively high calculated RMSE of the IAS indicates model performance is relatively poor in this hydrogeologic unit and may be a result of the assumed percentage of limestone or siliciclastic sediments in the IAS and the assigned endmember thermal conductivities. Although RMSE for the BZ indicates simulated temperatures are relatively poor in the hydrogelogic unit, the model is generally within the range of observed temperatures (10.3 to 36.1C). Errors in simulated temperatures may be a result of using a constant specified heat flux value or constant thermal properties for each hydrostratigraphic unit except the MCU. Spatiallyvarying calculated heat flux values and thermal conductivities have been reported by Griffin et al. (1977) and Smith and Lord (1997). 4.7 Response to Sealevel Changes We used the calibrated model in three simulations to investigate the effect of sealevel changes of varying duration on simulated pressures, chloride concentrations, and temperatures at several locations in the study area. Each simulation used a sinusoidal sealevel change starting from the presentday sea level [at sin(/2)] and rising to a maximum elevation of +70 m [at sin(/2)]. The three differed in the timing of the maximum: 5,000 years, 15,000 years, and 50,000 years. In each case, sealevel was returned to its presentday position along a continuation of the sine function [to sin(3/2)]. Then sealevel was held constant at the presentday level for the remainder of the 100,000year simulation period. The simulated 70m sealevel rise is not meant to represent any specific past sealevel condition but rather the general magnitude of largescale glacioeustatic fluctuations. The magnitude and duration of the three hypothetical sealevel curves applied to the calibrated model is shown in Figure 44. Falling sealevel conditions are not evaluated because they would have required simulation of unsaturated 142
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conditions and increased discretization of the upper units of the model; both of the these requirements made simulation of falling sealevel conditions computationally infeasible. Figure 44 Simulated sealevel changes applied to the calibrated model to evaluate the response of the FAS to a variety of sealevel fluctuations. The simulation used transient specified seawater boundaries to represent the sealevel oscillation. Specified pressure boundary conditions were applied to all nodes with elevations less than the simulated sealevel elevation at a specific time. Specified pressures were calculated assuming that conditions are hydrostatic; the overlying water is seawater; and the depthtemperature relationship is as indicated on Table 5. Specified boundary conditions for temperature, based on the depthtemperature relationship as indicated on Table 5, and salinity (unmodified seawater) were also specified to all nodes with elevations less than the simulated sealevel elevation at a specific time. Net recharge values shown in Figure 37 were applied to areas with surface elevations exceeding the simulated sealevel elevation. Final results from the calibration model were used as 143
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initial conditions for the model used to evaluate the effect of sealevel changes on pressures, chloride concentrations, and temperatures in the FAS. To facilitate evaluation of the response of the FAS to the hypothetical sealevel fluctuations we present results for four locations (Figure 45). They are: (1) a location on the west coast, (2) the center of the state, (3) the east coast, and (4) south Florida. The topography has also been classified into four categories in Figure 45 to show areas that are flooded at particular positions in the simulated sealevel rise. The positions are at the quarters of the halfcycle rise from the starting minimum to the 70m maximum. We will refer to these positions as the 0%, 25%, 50% and 75% positions of sea level. Location 1 (west coast) is located in the unconfined portion of the UFA and is flooded at the 0% sealevel position (Figure 46). The response of the simulated pressures coincides directly with the timing of sealevel changes. Transient chloride concentrations achieve maximum values at approximately the same time as maximum sea levels during the 10,000and 30,000year oscillations. Maximum chloride concentrations do not occur until approximately 70,000 years for the 100,000year oscillation. The simulated change in percent seawater values is approximately 3% for the 10,000and 30,000year oscillations but almost 10% for the 100,000year oscillation. Temperatures decrease slightly when sea level is at +70 m. Temperatures increase slightly at the beginning and end of the sealevel oscillation because of the depthdependent temperatures applied to model areas below sea level and transport of heat in the system. The magnitude of temperature change is similar for all three sealevel cycles. Pressures and temperatures return to initial values by the end of the simulation but percent seawater values remain elevated. Location 2 (central Florida) is in the confined portion of the UFA at the southern end of the recharge area and is not flooded until just before the 75% sealevel position (Figure 47). Response of the simulated pressures is slightly delayed for all three oscillations. Chloride concentrations do not change significantly during any of the oscillations, but simulated temperatures do increase. The magnitude and time to maximum temperatures increase with increasing duration of the oscillation. Pressures and temperatures return to initial values by the end of the simulation. 144
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Figure 45 Simulated extent of flooding between select ranges of the amplitude of sealevel change (+70 m) and the location of selected observation wells evaluated. Location 3 (east coast) is in the confined portion of the UFA on the east coast and is flooded before the 25% sealevel position (Figure 48). The response of the simulated pressures coincides directly with the sealevel changes. Chloride concentrations do not change significantly during the 10,000and 30,000year oscillations but increase slightly (~1%) during the rise in the100,000year oscillation and remain elevated at the end of the 100,000year simulation. Simulated temperatures increase in response to sealevel 145
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changes. The magnitude of the temperature rise increases with duration of the oscillation. The maximum temperature changes occur after the sea level reaches +70 m in the 10,000and 100,000year oscillations. The maximum temperatures occur before sea level reaches +70 m in the 30,000year oscillation. Pressures and temperatures return to initial values by the end of the simulation. The large peak observed at approximately 80,000years during the 100,000year oscillation at Location 3 is the result of development of relatively high temperatures offshore from Daytona Beach (see Figure 45 for the location) in the Atlantic Ocean. The high temperatures offshore of Location 3 develop as a result of the relatively low intrinsic permeability of aquifer units offshore that reduce aquiferocean exchange and heat transport. Location 4 (south Florida) is in the confined portion of the UFA where the ICU is approximately 200 m thick and is flooded before the 25% sealevel position (Figure 49). Maximum observed increases in pressure occur after sea level reaches +70 m, and the time to maximum pressure increase with increasing duration of the simulated oscillation. Chloride concentrations do not change significantly for any of the hypothetical sealevel curves, but simulated temperatures increase in response to the sealevel changes. The magnitude of temperature changes is similar for all three oscillations, and the time to maximum temperatures increase with increasing period of the oscillations. Simulated pressures return to initial values by the end of the simulation but temperatures remain elevated. Temperatures are approximately at maximum values for the 100,000year oscillation at the end of the simulation. Chloride concentrations at the top of the UFA at 0, 50,000, 70,000, and 100,000years for the 100,000year oscillation are shown in Figure 50. At 50,000years the largest changes in chloride concentrations occur at the northern end of the study area where the ICU is thin or missing. In South Florida, chloride concentrations are only slightly higher after 50,000years and reflect the loss of lateral freshwater recharge. Transient chloride concentration changes proceed in the simulation in a manner that is comparable to the current pattern of recharge, discharge, and groundwater ages in the Floridan aquifer. Since transport phenomena control the rate at which chloride concentrations propagate through the system, and chloride would be transported in the 146
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same manner as freshwater in the presentday aquifer, the minimum time for complete salinization of Floridan aquifer is constrained by aquifer residence times (~35,000 years), assuming mixing of salt water with freshwater currently in the aquifer does not occur during flooding events. Mixing of saltwater and freshwater would increase the time for complete salinization of the FAS. Figure 46 Simulated response of the FAS to sealevel changes at the selected observation well at location 1 (west coast). This observation well is located in an area that is flooded quickly and where the FAS is unconfined. 147
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Figure 47 Simulated response of the FAS to sealevel changes at the selected observation well at location 2 (center of the emergent platform). This observation well is located in an area where flooding occurs relatively late and the FAS is confined. Simulated percent seawater values do not change significantly at this location. 148
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Figure 48 Simulated response of the FAS to sealevel changes at the selected observation well at location 3 (east coast). This observation well is located in an area that is flooded quickly and where the FAS is confined. Simulated percent seawater values change at this location but the simulated response is less than observed in the unconfined portion of the FAS on the west coast (see Figure 45). 149
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Figure 49 Simulated response of the FAS to sealevel changes at the selected observation well at location 4 (south Florida). This observation well is located in an area that is flooded quickly and the FAS is confined by several hundred meters of the Miocene Hawthorn Formation. Simulated percent seawater values do not change significantly at this location. 150
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Figure 50 Simulated percent seawater chloride concentrations at the top of the UFA at a) 0, b) 50,000, c) 70,000, and d) 100,000 years in response to the simulated 70 m sealevel event with a 100,000 year duration. 151
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4.8 Summary and Conclusions A steady model of the peninsular portion of the Floridan platform was developed using existing data from the United States Geological Survey RASA evaluation of the FAS and data developed in subsequent investigations. The model was calibrated with 315 observation wells that included a combination of head, chloride concentrations, and temperature data for the hydrostratigraphic units of the FAS. The model is capable of simulating hydraulic heads; simulated temperatures represent the range of observed temperatures and are consistent with the observations of Kohout et al. (1977). The calibrated model tends to overestimate chloride concentrations but this result not surprising because of the regional nature of the model and its inability to account for localscale features that would affect solute transport. The model is able to produce vertical fluxes between the ICU/IAS and the UFA that are similar to those in Aucott (1988) as well as a UFA potentiomentric surface that compares favorably to the predevelopment potentiometric surface prepared by the USGS (Johnston et al. 1980). Because the calibrated model is capable of simulating reasonable heads, chloride concentrations, and temperatures in the FAS, we used it to evaluate how the FAS would respond to hypothetical sealevel rises. The calibrated model was subjected to three sealevel curves with a +70m maximum rise and oscillation periods of 10,000, 30,000, and 100,000years. Results from these analyses show that the FAS responds quickly in areas that are unconfined and located close to the presentday coast and more slowly in areas that are well confined and farther from the presentday coast. In cases where the thickness of the ICU/IAS was less than 100 m, pressures responded quickly to sealevel changes. Pressures returned to initial values by the end of the 100,000year simulation for all cases. Chloride concentrations changed only at areas that were located close to the present coastline and retained elevated values at the end of the simulation. The slow response of chloride concentrations is a result of the length of time it takes for dissolved solutes to be transported throughout the FAS (> 35,000years) and mixing of saltwater with freshwater currently in the aquifer. The magnitude of temperature changes and the time to maximum temperature changes increased with increasing cycle duration but 152
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returned to initial values by the end of the 100,000year simulation except in areas with ICU/IAS thicknesses that exceeded 200m. These simulations of sealevel changes show that pressure, chloride concentrations, and temperatures respond at different rates. The results indicate that pressure changes are rapid, temperature changes are intermediate, and salinity changes are very slow. Temperature changes are complicated by the effect that increased pressures and groundwater recharge have on heat transport. Not only was the response of chloride concentrations the slowest but, in several cases, flooding on the 100,000year cycle was insufficient to change chloride concentrations in areas where the FAS is well confined. These simulations also show that the threedimensional configuration of the aquifer plays a key role in controlling the rate at which information propagates through the system. In the FAS, response times for chloride concentrations are long. Long responses times are common for many geologic processes (e.g., heat flow measurements affected by atmospheric temperatures) and can make it difficult to generate estimates of current conditions for areas without significant observation data without an understanding of past conditions. Thus, the term steadystate may not be appropriate for complicated densitydependent groundwater flow and transport environments that have been exposed to timevarying boundary conditions such as highfrequency Pleistocene sealevel fluctuations. 153
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Chapter 5 5 Threedimensional flow in the Florida Platform: Theoretical Analysis of Kohout Convection at its Type Locality 5.1 Abstract Kohout convection is the name given to the circulation of saline groundwater deep within carbonate platforms as first proposed for south Florida by F.A. Kohout in the 1960s. It is now seen as a Mg2+ pump for dolomitization by seawater. As proposed by Kohout, cold seawater is drawn into the Florida platform from the deep Florida Straits as part of a geothermally driven circulation in which the seawater then rises in the interior of the platform to mix and exit with the seawarddischarging meteoric water of the Floridan Aquifer System. Simulation of the asymmetrically emergent Florida Platform with the new, threedimensional, finiteelement, groundwater flow and transport model SUTRAMS that couples salinitydependent and temperaturedependent density variations allows analysis of how much of the cyclic flow is due to geothermal heating (free convection) and how much is due to mixing with meteoric water discharging to the shoreline (forced convection). Simulation of the system with (geothermal case) and without (isothermal case) an assumption of geothermal heating reveals that the inflow of seawater from the Florida Straits would be similar without the heat flow but the distribution would differ significantly. The addition of heat flow reduces the asymmetry of the circulation: it decreases seawater inflows on the Atlantic side by 8% and halves seawater inflows on the Gulf side. The study illustrates the complex interplay of freshwater/saltwater mixing, geothermal heat flow, and projected dolomitization in complicated threedimensional settings with asymmetric boundary conditions and realistic vertical and horizontal variations in hydraulic properties. 154
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5.2 Introduction Some 40 years ago, Francis A. Kohout proposed that the deep, saline groundwater beneath Florida is not static. Rather (K ohout, 1965, 1967), geothermal heat flow causes a regional circulation in which cold seawater enters the lower Florida Aquifer System (FAS) from the Straits of Florida, then rises because of thermally induced buoyancy and mixes with seawardflowing meteoric groundwater from the upper FAS (Figure 51). The evidence was temperature anomalies (inverse geothermal gradients) in oil exploration wells. The location was the southern tip of the Florida peninsula (Figure 52a), in the vicinity of Miami and the Everglades. 20Co 15Co 10Co 9Co8Co7Co 6Co6C 15 2535 65 95TEMPERATURECo25Co>24Co>24Co>8 >8Co7Co 8Co GEOTHERMALHEATFLOW TEMPERATUREPROFILE APPROXIMATEBELOW1,200m CEDARKEYSANHYDRITEPRINCIPALARTESIAN ZONE BOULDERZONE FLORIDAN AQUIFERINTERMEDIATE CONFINING UNIT GULFOFMEXICOSINKHOLE REGIONPOTENTIOMETRICSURFACEBISCAYNEBAY MIAMIMIAMIBEACHSTRAITSOFFLORIDABIMINI40MILE BEND FELDA SEAWATER FRESHWATER 225 150 75 0 75 APPROXIMATEDISTANCEINKILOMETERS 0 500 1000 1500 2000DEPTHNMETERS,BELOWM.S.L.A A Figure 51 Conceptual crosssection of cyclic flow of saltwater induced by geothermal heating (Kohout Convection) in Florida (adapted from Kohout et al., 1977). Soon after Kohout proposed and promoted the concept of regional, cyclic flow of saltwater within the Florida Platform (e.g., Kohout et al., 1977), Simms (1984) labeled the phenomenon Kohout convection and proffered it as a Mg2+ pump to explain largescale dolomitization. With or without the name, Kohout convection has been invoked as a conceptual model for dolomitization in numerous modern carbonate platforms 155
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including the Bahamas (Whitaker at al., 1994; Caspard et al., 2004), Enewetak Atoll (Saller, 1984), Mururoa (Buigues, 1997), French Polynesia (Rougerie et al., 1997), and other atolls (Leclerc et al., 1999). Numerical simulation of twodimensional flow in generalized carbonate platforms has suggested that geothermal heat flow can indeed cause convective circulation at depths as great as several kilometers (e.g., Kohout et al. 1977, Sanford et al., 1998; Wilson et al., 2001). As Kohout well knew, however, geothermal heating is not a necessary condition for cyclic flow of saltwater at a coastline. Cooper (1959) had proposed that mixing in the brackish freshwatersaltwater transition zone induces a cyclic flow of saltwater beneath the freshwater discharge, a phenomenon that Bear (1960) attributed to conservation of salt. Coopers work was coupled to Kohouts own study of the Biscayne Aquifer of Miami (Kohout, 1960, 1964). The question, then, is: How much of Kohouts cyclic flow of saltwater deep within the Florida Platform is due to geothermal heating (free convection) and how much is due to the simple mixing caused by discharge of meteoric water to the shoreline (forced convection)? In other words, how much difference does the geothermal heat flow make to the circulation of seawater? Flow systems of carbonate platforms can be expected to be more complicated than shown in generalized crosssections such as Figure 51 and modeled in published twodimensional simulations (e.g., Kohout et al., 1977; Sanford et al., 1998; Wilson et al., 2001). In particular, the threedimensional nature of the Florida problem is obvious because the classic Kohout crosssection is located across the southern tip of the peninsula where flow paths diverge radially away a potentiometric high in the center of the emergent platform (Figure 52a). The new, finiteelement, variabledensity groundwater flow and transport model SUTRAMS (Hughes and Sanford, 2004) is now capable of coupling salinitydependent and temperaturedependent density variations in complicated, threedimensional coastal and platform settings such as the asymmetrically emergent Florida Platform (Figure 52). In this paper, we use SUTRAMS to simulate and compare the flow in the south Florida system with (geothermal) and without (isothermal) heat flow across the Cedar Keys Anhydrite hotplate (in the language of 156
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Kohout et al., 1977). The difference isolates the effect of geothermal heat flow on Kohout convection at its type locality. Figure 52 a) Study area, predevelopment FAS potentiometric surface (CI = 3 m), generalized predevelopment FAS flow directions, and location of conceptual crosssection of Kohout et al. (1977). b) Generalized hydrostratigraphy in south Florida at the conceptual crosssection of Kohout et al. (1977); vertical exaggeration is 178. 5.3 Numerical Model We model the central and southern portion of the entire Florida Platform (Figure 52a) in order to reduce uncertainties resulting from specifying arbitrary boundaries at the shoreline. The model includes the surficial aquifer system (unconsolidated sedimentary watertable aquifer) to allow for realistic simulation of groundwater recharge, and the upper portion of the Cedar Keys Formation to represent the geothermal flux at the base of 157
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the FAS. The top of the model is the estimated watertable elevation in the emergent part of the platform and the sea floor in the Atlantic Ocean and the Gulf of Mexico. The model is composed of threedimensional hexahedral elements with eight nodes, a horizontal node spacing ranging from 1 to 5 km, and an eightunit hydrostratigraphic model. Each hydrostratigraphic unit consists of five elements vertically with thicknesses ranging from 5 to 160 m. The total model domain includes 380,016 nodes and 362,250 elements. The threedimensional hydrostratigraphic model for the mesh of finite elements was derived from datasets of Miller (1986), Mullins et al. (1988), and Jee (1993). Initial aquifer parameters were developed from datasets of Meyer (1989), Hutchinson (1992), Sepulveda (2002), USEPA (2003), and Budd and Vacher (2004) and were calibrated using 315 observation points that provided 85 head measurements, 251 salinity concentration observations, and 133 temperature observations. Calibrated FAS horizontal permeabilities (kh) and kh / kv ratios ranged from 3.614 to 2.39 m2 and 1 to 1,000, respectively. The highest kh values were assigned to the Boulder Zone (2.39 m2) and unconfined regions in westcentral Florida (3.610 m2). The lowest kh values were assigned to the Middle Confining Unit (3.614 m2) and isolated portions of the Upper Floridan aquifer in central Florida (3.612 m2). Relatively low kh values were assigned to the Upper Floridan aquifer in south Florida (1.111 m2). An effective porosity of 21% and 35% was specified for the FAS and the Boulder Zone of the FAS, respectively. The effective porosity was based on an assumption that effective porosity is 70% of total porosity and a total porosity of 35% for the FAS and 50% for the Boulder Zone (Vacher et al. 2006). Horizontal and vertical dispersivities that satisfy mesh Peclet number numerical stability criteria were specified as 500 and 50 m in areas with a 1km node spacing to 1,250 and 125 m in areas with a 5km node spacing, respectively, and are consistent with the scaledependent dispersivity values of Gelhar (1986). Simulated time step lengths were constrained to restrict transport of simulated fronts to a small fraction of an element per time step (Voss and Provost, 2002). Side and bottom boundaries were specified to be noflow, and a heat flux of 60 mW/m2, based on the data of Griffin et al. (1977) and Smith and Lord (1997), was assigned to the basal 158
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boundary of the Cedar Keys Formation. Oceanboundary temperatures were specified using a temperature vs. depth relationship developed from data in the U.S. Navy Generalized Digital Environmental Model (Teague et al., 1990) for the Atlantic Ocean and Gulf of Mexico observation data {minimum of (25e0.0015 z, 5C)}. The salinity at the ocean boundary in the Atlantic Ocean and the Gulf of Mexico was set at 36 o/oo (unmodified seawater). Fluid pressures at the ocean boundaries were assumed to be hydrostatic. For the emergent part of the platform, net groundwater recharge was applied to the upper surface nodes to represent groundwater recharge in recharge areas and groundwater discharge in discharge areas. Initial estimates of net groundwater recharge were derived from data of Aucott (1988) and augmented with modelderived data of Langevin (2001). Specified salinity concentrations of 0 o/oo (freshwater) and temperatures representative of the average annual air temperature in the study area (23.5C Florida Department of Natural Resources, 1974) were applied to all surface nodes in the emergent part of the platform. 5.4 The Pattern of Geothermal Circulation We ran the simulation of coupled temperatureand salinitydependent groundwater flow and transport in transient mode for 100,000 years starting from initial conditions based on presentday observation data. Results are shown in Figure 53, at the line of crosssection used by Kohout et al. (1977). As shown in the crosssection, the transition zone occurs within the Middle Confining Unit (MCU) and Lower Floridan Aquifer (LFA) on the east side of the crosssection and within the Upper Floridan Aquifer (UFA) on the western side of the crosssection; temperatures vary from low to high in a easttowest direction, consistent with Kohouts observations; horizontal flow dominates in the highly permeable, socalled Boulder Zone (BZ); and vertical flow dominates above the BZ except in the UFA and Intermediate Aquifer System (IAS) (Figure 53). Flow in the FAS is highest where the IAS is present in the western part of the crosssection and generally low where the Intermediate Confining Unit (ICU) is thick. Vertical flow and recharge to the BZ are relatively large beneath the Straits of Florida. In general, the pattern of oceanaquifer exchange (Figure 54a) is complicated in the Gulf of 159
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Mexico and the northern part of the Atlantic Ocean and a general inflow area exists in the Straits of Florida along the lower east coast area. 5.5 The Pattern of Isothermal Circulation To eliminate temperature effects, we removed the dependence of fluid properties on temperature by using a viscosity corresponding to 20C (0.001 kg/m s), a temperaturedensity and a temperatureviscosity coefficient of 0 (kg) / (m3C) and 0 (kg) / (m secC), respectively, and we assumed isothermal, hydrostatic seawater boundary conditions with a pressuretemperature coefficient of 0 (kg) / (m sec2 C). Otherwise we used the same initial and boundary conditions for the isothermal case as for the geothermal circulation. The simulation time was the same, 100,000 years. It produced the spatial distribution of oceanaquifer exchanges shown in Figure 54b. Comparison of the geothermal and isothermal cases in Figure 54 shows that there are significant differences in the distribution of oceanaquifer exchange of the two circulations. Some noticeable differences are: the area of coastal outflow is greater in the geothermal case; localized areas of inflow and outflow develop in the geothermal case; inflow dominates along the lower east coast in the geothermal case; and the isothermal case is dominated by inflow except in the Atlantic Ocean below a depth of 100 m. In the geothermal case, effects related to the ocean temperature vs. depth relationship and geothermal heat flux cause localized areas of inflow and outflow to develop as a result of sloping density surfaces and convection (Wilson, 2005). Localized inflow and outflow areas offshore (Figure 54) call attention to the threedimensional nature of convection induced by the sloping density surfaces. A significant quantity of inflow occurs along the lower east coast as a result of the increased connection with the FAS in this area and higher relative pressures in the deep portions of the Florida Straits. 160
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Figure 53 Simulated results for crosssection AA. a) simulated chloride concentrations (in percent seawater) for the geothermal case, b) simulated temperatures for the geothermal case, c) simulated velocity magnitude for the geothermal case, and d) simulated areas of vertical flow and general flow directions for the geothermal case. For the location of the crosssection see Figure 52a. Vertical exaggeration is 178. 161
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Figure 54 Simulated distribution of oceanaquifer exchanges in the Atlantic Ocean and the Gulf of Mexico for a) the geothermal case and b) the isothermal case. Positive values represent seawater inflow and negative values represent aquiferseawater exchanges. The localized inflow and outflow areas in the geothermal case is a result of dipping density surfaces that cause convection and develop in response to seafloor temperature differences and geothermal heat flux. 5.6 Quantifying the effect of heat flow on the saltwater circulation It is difficult to discern the effect of heat flow on the saltwater circulation by comparing the oceanaquifer exchange maps of Figure 54. To further evaluate Kohouts hypothesis, we examine total simulated oceanaquifer exchange rates for the Atlantic Ocean and Gulf of Mexico. There are striking differences. Taking the geothermal case as the baseline, seawater inflow exceeds aquifer discharge for both the Atlantic Ocean (87 vs. 32 Mg/sec [megagrams of fluid per second]) and Gulf of Mexico (189 vs. 92 Mg/sec). The net magnitudes of the ocean162
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aquifer exchanges (inflows outflows) are less than a factor of two different on the two sides (55 Mg/sec for the Atlantic and 97 Mg/sec for the Gulf). Percent differences (% Difference), calculated as 100 (Isothermal Rate Geothermal Rate) / Geothermal Rate, describe the relative difference between the two cases in oceanaquifer exchanges. Removal of the heat flow effectively increases the seawater inflow from the Atlantic Ocean by about a tenth (8% difference to 94 Mg/sec) and doubles the seawater inflow from the Gulf of Mexico (113% difference to 402 Mg/sec). Meanwhile removal of the heat flow increases the aquifer outflow from the Atlantic by about a sixth (16% difference to 38 Mg/sec) and decreases the aquifer outflow from the Gulf of Mexico by about twothirds (85% difference to 14 Mg/sec) and reflects a change in onshore and offshore discharge areas. As a result, the asymmetry of flow in the isothermal case is greater than in the geothermal case. Net oceanaquifer exchange in the Atlantic Ocean is approximately equivalent in the geothermal case (55 Mg/sec) and the isothermal case (56 Mg/sec), but net oceanaquifer exchange in the Gulf of Mexico is a factor of 4 less in the geothermal case (97 Mg/sec) than in the isothermal case (388 Mg/sec). The asymmetry in the oceanaquifer exchange rates in the geothermal case reflects the disparate seawater boundary pressures and temperatures resulting from differences in sea floor bathymetry bounding the platform. Furthermore, the addition of heat decreases seawater inflows because it contributes to fluid buoyancy and counteracts the effect of chloride concentrations on fluid density. Increased pressures and lower temperatures in the deep Straits of Florida and the proximity to the BZ allow for increased seawater inflows in the lower east coast area, as pointed out by Kohout (1967). 5.7 Implications of the effect of heat flow on dolomitization time The disparate oceanaquifer exchange rates imply an asymmetry in dolomitization times for the FAS. Following Caspard et al. (2004), we can calculate the time for complete dolomitization by assuming that (1) dolomitization occurs as a replacement process, and (2) dolomitization is limited by Mg2+ inflow rates that are a function of seawater inflow rates and seawater Mg2+ molarity (0.045 moles/liter). For the geometry 163
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of our case, we further assume that (3) the axis of the predevelopment upper FAS potentiometric surface (Figure 52a) represents the flow divide for the entire FAS, and (4) inflowing seawater is distributed equally throughout the oceanaquifer boundaries. With these assumptions, a weighted total porosity of 30%, a molar volume of 6.434 5 m3/mole, and total volumes of 4.5 13 and 1.6 14 m3 for the Atlantic Ocean and Gulf of Mexico portions of the platform, we find that the slight decrease in seawater inflow rate due to the geothermal heat flow would increase the time for complete dolomitization of the Atlantic Ocean portion of the FAS by 4 m.y (from 41 m.y. to 45 m.y.). Furthermore, the decrease in seawater inflow rate due to the geothermal heat flow would increase the time for complete dolomitization of the Gulf of Mexico portion of the FAS by 37 m.y. (from 33 m.y. to 70 m.y.). We note that the massively bedded dolomite boulder zone as defined by Miller (1986) is qualitatively more extensive on the Atlantic Ocean side of the platform than on the Gulf of Mexico side. Because seawater inflows are not equally distributed throughout the oceanaquifer boundaries (Figure 54a) the calculated dolomitization times probably represent minimum times for complete dolomitization of the FAS. 5.8 Conclusion Although Kohout convection is routinely considered to be synonymous with geothermally driven convection in carbonate platforms, a geothermal heat flux is not a necessary condition for the cyclic flow of seawater in an environment like south Florida, where meteoric water flushes through the emergent portion of the platform. In the case of the type locality of Kohout convection, our simulations show that the inflow of seawater from the Atlantic would increase by a tenth if there were no heat flow, and, even more striking, inflow of seawater from the Gulf of Mexico would double without the heat flow. Furthermore, our simulations show that the distribution of inflow and outflow are significantly different without the heat flow and simulations with heat flow are more consistent with the observations. The study illustrates the complex interplay of freshwater/saltwater mixing and geothermal heat flow in complicated threedimensional settings with asymmetric boundary conditions and realistic vertical and horizontal 164
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variations in hydraulic properties. The complexities do not lend themselves to easy generalizations from simplified twodimensional models of geologically complicated platforms. In other words, geometry matters. 165
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Chapter 6 6 Summary This dissertation investigates the threedimensional nature of flow in the FAS of peninsular Florida. Evaluation of the FAS required development of a model that was capable of representing the complex threedimensional nature of the geometry and aquifer parameters, the effect of chloride concentrations and/or temperatures on fluid density, and could be used to represent variable sealevel conditions. The ability to represent these features and understand how they affect groundwater flow in the FAS is important because it can provide a guide for future studies of the FAS and other variabledensity coastal aquifers. In particular, current and projected population demands in Florida are causing regulators to consider utilization of saline portions of the FAS. Therefore, understanding of the important components that need to be represented in a numerical evaluation of groundwater resources is crucial for interested parties to have confidence in predictions of the impact of proposed activities on the FAS. Chapter 1 provides a brief introduction to the FAS and discusses the problems to be addressed in the dissertation. Chapter 2 details the development of the numerical code (SUTRAMS) used to represent the FAS in later chapters. The governing equations and numerical representation of the governing equations are detailed, and the code is benchmarked against three problems with known solutions. Benchmark tests indicate SUTRAMS is capable of simulating groundwater flow in variabledensity environments where chloride concentrations and/or temperature contribute to fluid density. Chapter 3 further develops the HeleShaw benchmark problem presented in Chapter 2 to illustrate the advantages of applying a numerical code to this problem. A dye, with fluid properties very similar to the fluid it was mapping, was used to map NaCl and fluid components to evaluate evolution of the flow system. Numerical results indicate that the mapping of the dye and NaCl is not exact and differences in fluid properties lead to significant discrepancies in calculated vertical flux rates. The 166
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simulated evolution of the dye compares very favorably with the observed experimental results that are based on dye concentrations. Development of a threedimensional SUTRAMS model of the FAS is described in Chapter 4. The calibrated steadystate model is capable of simulating hydraulic heads and temperatures but not chloride concentrations. Because of the poor fit to chloride concentrations, the model is used it to evaluate how the FAS responds to hypothetical sealevel rises. Simulations of sealevel changes show that pressure, chloride concentrations, and temperatures respond at different rates. In general, pressure changes are rapid, temperature changes are intermediate, and salinity changes are very slow. Temperature changes are complicated by the effect that flooding of the Florida platform has on heat transport in the FAS. Not only is the response of chloride concentrations the slowest but, in several cases, flooding on the 100,000year cycle is insufficient to change chloride concentrations in areas where the FAS is well confined. These simulations also show that the threedimensional configuration of the aquifer plays a key role in controlling the rate at which information propagates through the system. As a result, the term steadystate may not be appropriate for complicated densitydependent groundwater flow and transport environments that have been exposed to highfrequency timevarying boundary conditions such as Pleistocene sealevel fluctuations. The calibrated model developed in Chapter 4 is used to evaluate the role of heat transport in the presentday FAS in Chapter 5. Although Kohout convection is routinely considered to be synonymous with geothermally driven convection in carbonate platforms, simulated results show that a geothermal heat flux is not a necessary condition for the cyclic flow of seawater in an environment like south Florida, where meteoric water flushes through the emergent portion of the platform. In the case of the type locality of Kohout convection, simulations show that the inflow of seawater from the Atlantic would increase by a tenth if there were no heat flow and inflow of seawater from the Gulf of Mexico would double without the heat flow. Furthermore, simulations show that the distribution of inflow and outflow are significantly different without the heat flow and simulations with heat flow are more consistent with the observations. The study 167
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illustrates the complex interplay of freshwater/saltwater mixing and geothermal heat flow in complicated threedimensional settings with asymmetric boundary conditions and realistic vertical and horizontal variations in hydraulic properties. The complexities do not lend themselves to easy generalizations from simplified twodimensional models of geologically complicated platforms. Aquifer geometry has a significant effect on the flow system and is consistent with the conclusions of Chapter 4. 168
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Mullins, H.T., Gardulski, A.F., Hine, A.C., Melilli, A.J., Wise, S.W. Jr., and Applegate, J., 1988, Threedimensional sedimentary framework of the carbonate ramp slope of central west Florida: A sequential seismic stratigraphic perspective: Geological Society of America Bulletin, v. 100(4), p. 514533. Neathery, T.L., and Thomas, W.A., 1975, PreMesozoic basement of rocks of the Alabama Coastal Plain, Gulf Coast Association of Geological Societies Transactions v. 25, p. 8699. Nield, D.A., 1968, Onset of thermohaline convection in a porous medium: Water Resources Research, v. 4, p. 553560. Nield, D.A., and Bejan, A., 1998, Convection in porous media: SpringerVerlag, 496 p.. Oldenburg, C.M., and Pruess, K., 2000, Simulation of propagating fronts in geothermal reservoirs with the implicit Leonard total variation diminishing scheme: Geothermics, v. 29, p. 125. Oude Essink, G.H.P, 1999, Impact of sea level rise in the Netherlands, in, Bear, J., Cheng, A.H.D., Sorek, S., Ouazar, D., and Herrera, I., eds., Seawater Intrusion in Coastal Aquifers Concepts, Methods and Practices, Kluwer Academic Publishers, Netherlands, p. 507530. Paull, C.K., Twichell, D.C., Spiess, F.N., and Curray, J.R., 1991, Morphological development of the Florida Escarpment: observations on the generation of time transgressive unconformities in carbonate terrains, Marine Geology, v. 101, p.181201. Phillips, O.M., 1991, Flow and Reactions in Permeable Rocks, Cambridge Univ. Press, New York, 285 p. Plummer, L.N., 1977, Defining reactions and mass transfer in part of the Floridan aquifer. Water Resources Research, v. 13, p. 801812. Poeter, E.P., and Hill, M.C., 1998, Documentation of UCODE, a computer code for universal inverse modeling: U. S. Geological Survey WaterResources Investigations Report 984080, 116 p. Pringle, S.E., Glass, R.J., and Cooper, C.A., 2002, Doublediffusive finger convection in a HeleShaw cellAn experiment exploring the evolution of concentration fields, length scales and mass transfer: Transport in Porous Media, v. 47, no. 2, p. 195214. Puri, H.S., and Winston, G.O., 1974, Geologic framework of the high transmissivity zones in south Florida, State of Florida Department of Natural Resources, Special Publication No. 20. 175
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Appendix A: Notation 180
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Appendix A: (Continued) 181 This appendix contains a summary of the notation used in the equations in Chapters 2 and 3. The notation summary includes a reference to the first occurence of the variable in an equation and the dimensions of the variable, where appropriate. A summary of generic units and special mathematical notation used is also given. Generic Units [] dimensionless [1] unity implies dimensionless [0] zero implies dimensionless [E] energy units or [ML2/T2] [L] length units [] fluid volume 3fL [] solid grain volume 3GL [M] fluid mass units [MG] solid grain mass units (matrix and adsorbate) [MS] solute mass units [T] time units Units [C] degrees Celsius Special Notation d t d t or time derivative of v = ivx + jvy + kvz vector v with components in the i, j, and k directions zyxkji gradient of scalar zzvyyvxxvv divergence of vector v NNNNi,...,4,3,2,1,1 index i takes on all integer values between one and NN discrete change in value of
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Appendix A: (Continued) 182 (e.g., = 1 2) o initial value or zeroth value of BC value of at a boundary condition node i or j value of at node or cell i or j IN value of in inflow L value of in element L vx value of vector v in x direction vy value of vector v in ydirection vz value of vector v in zdirection L value of in element L n value of at time step n n+1 value of at time step n+1 (36) [] elementwise discretization of porosity kL (29) [L2] permeability tensor that is discretized elementwise Ls (41) [L2/T] elementwise discretization of solid matrix thermal diffusivity for heat transport [0] not used for for solute transport of species k Lbgk (45) [L2/T] elementwise discretization of bulk thermal diffusivity [L2/T] that is approximated using a geometric mean and includes the constants A and w [L2/T] constant molecular diffusivity (DM) of species k for solute transport g* (30) [M/L2T2] elementwise discretization of (g) that is consistent with the discretization of p v* (36) [M/T] elementwise discretization of v that is consistent with the discretization of p NNiNNi1321... summation
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Appendix A: (Continued) 183 Roman Lowercase b (68) [L] aquifer thickness cs (6) [E/MC] solid matrix specific heat for heat transport [] sorption coefficient for solute transport csk (18) [E/MC] solid matrix specific heat for heat transport [] sorption coefficient for species k for solute transport cw (6) [E/MC] fluid specific heat for heat transport [1] for solute transport cwk (18) [E/MC] fluid specific heat for heat transport [1] for solute transport of species k k (1) [L2] permeability tensor kr (1) [] relative permeability for unsaturated flow g (1) [L/T2] gravitational vector n (41) [] unit outward normal vector p (1) [M/LT2] fluid pressure 1nip (26) [M/LT2] fluid pressure at node i at the end of the current time step 1niBCp (26) [M/LT2] specified fluid pressure at node i at the end of the current time step (zero for all nodes that are not specified pressure nodes) sLk (39) [1] is the sorption isotherm contribution to the lefthand side of unified solute mass balance equation sRk (40) [Msolute/ Mfluid] is the sorption isotherm contribution to the righthand side of unified solute mass balance equation t (1) [T] time v (6) [L/T] fluid velocity
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Appendix A: (Continued) 184 Roman Uppercase 1niAF (26) matrix coefficient of pressure time derivative 1nikAT (34) matrix coefficient of U time derivative for species k B(x,y) (31) [L] aquifer thickness 1nijBF (26) matrix coefficient in pressure equation 1nijkBT (34) matrix coefficient in U equation for species k Ck (3) [Msolute/ Mfluid] fluid solute concentration of species k Cko (4) [Msolute/ Mfluid] base solute concentration of species k Csk (47) [Madsorbate/ Msolid] is the concentration of adsorbate for species k 1nskC (48) [Madsorbate/ Msolid] is the concentration of adsorbate for species k at the end of the current time step inskC1 (61) [Madsorbate/ Msolid] is the concentration of adsorbate for species k at node i at the end of the current time step PROJskC (56) [Madsorbate/ Msolid] is the concentration of adsorbate for species k based on a projection for the first time step or at the end of the last time step iPROJskC (61) [Madsorbate/ Msolid] is the concentration of adsorbate for species k at node i based on a projection for the first time step or at the end of the last time step 0skC (56) [Madsorbate/ Msolid] is the approximation used for concentration of adsorbate for species k at low fluid concentrations of species k skC (57) [Madsorbate/ Msolid] is the approximation used for concentration of adsorbate for species k at high fluid concentrations of species k 1nijCF (26) matrix coefficient of U time derivative in pressure equation
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Appendix A: (Continued) 185 Csk (45) [Msolute/ Msolid matrix] concentration of the adsorbate for species k D (6) [L2/T] solute dispersion tensor [L2/T] thermal dispersion tensor Dk (18) [L2/T] solute dispersion tensor for species k [L2/T] thermal dispersion tensor for species k DM (66) [L2/T] total solute dispersion which is the product of the molecular dispersion of NaCl and porosity DNaCl (72) [L2/T] molecular dispersion of NaCl DSOLUTE (66) [L2/T] molecular diffusivity of NaCl DT (67) [L2/T] total thermal dispersion which is the product of the thermal diffusivity and porosity DTEMP (67) [L2/T] thermal diffusivity DTMULT (65) [] time step multiplier 1nijDF (26) element of vector on right side of pressure equation 1nijkDT (34) element of vector on right side of U equation for species k 1nikET (34) element of vector on right side of U equation for species k 1niksTLG (34) element of vector on left side of U equation for species k 1niksTRG (34) element of vector on right side of U equation for species k 1nikGT (34) element of vector on left side of U equation for species k I (6) [1] identity tensor K (68) [L/T] hydraulic conductivity NC (69) [] nondimensional parameter of Henry and Hilleke (1972) for relationship of advective flux to dispersion of salt NT (70) [] nondimensional parameter of Henry and Hilleke (1972) for relationship of advective flux to dispersion of heat
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Appendix A: (Continued) 186 N (68) [] nondimensional parameter of Henry and Hilleke (1972) for relationship of porous media tranmissibility to advective flux NE number of elements in mesh NN (26) number of nodes in mesh NS (3) [] number of simulated of solute species niBCQ (34) [E/T] total energy source of species k to cell I for previous time step [Msolute/MfluidT] total concentration of species k to cell I for previous time step Qi (26) [M/T] total mass source to cell i 1niQ (26) [M/T] total mass source to cell i for the current time step Qp (1) [M/L3T] fluid mass source (including pure water mass plus solute mass dissolved in source water) QT (68) [L3/LT] total freshwater inflow per unit width R (68) [] fractional difference between the density of seawater and freshwater (0.025) Sop (1) [M/LT2]1 specific pressure storativity Sw (1) [] fractional water saturation To (4) [C] base fluid temperature Tc (5) [] U (1) [C] fluid temperature for heat transport or [Msolute/ Mfluid] fluid solute concentration Uk (18) [C] fluid temperature of species k for heat transport [Msolute/ Mfluid] fluid solute concentration of species k for solute transport nikU (34) [C] temperature of cell i at the end of the previous time step [Msolute/ Mfluid] concentration of species k in cell i at the end of the previous time step 1nikU (34) [C] temperature of cell i at the end of the current time step [Msolute/ Mfluid] concentration of species k in cell i at the end of the current time step
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Appendix A: (Continued) 187 1nikUBCU (34) [C] specified temperature of cell i at the end of the current time step [Msolute/ Mfluid] specified concentration of species k in cell i at the end of the current time step Uik (26) [C] fluid temperature of species k at node i for heat transport [Msolute/ Mfluid] fluid solute concentration of species k at node i for solute transport Us (6) [E/M] specific energy content of the solid matrix for heat transport [Msolute/Msolid matrix] concentration of the adsorbate on the solid matrix for solute transport Usk (18) [E/M] specific energy content of the solid matrix [E/M] for heat transport [Msolute/Msolid matrix] concentration of the adsorbate on the solid matrix for species k for solute transport U* (6) [C] temperature of the source fluid for heat transport [Msolute/ Mfluid] solute concentration of the source fluid for solute transport *kU (18) [C] temperature of the source fluid for heat transport [Msolute/ Mfluid] solute concentration of the source fluid for species k for solute transport Vi (27) [L3] cell volume at node I kW0 (59) [] low solid concentration weight used for Langmuir isotherm for species k kW (60) [] high solid concentration weight used for Langmuir isotherm for species k Greek Lowercase (2) [M/LT2] compressibility of the porous matrix (2) [M/LT2]1 compressibility of the fluid w1 (6) [0] not used for heat transport [T1] firstorder mass production rate of solute for solute transport
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Appendix A: (Continued) 188 kw1 (18) [0] not used for heat transport [T1] firstorder fluid mass production rate of species k for solute transport s1 (6) [0] not used for heat transport [T1] firstorder mass production rate of adsorbate for solute transport, ks1 (18) [0] not used for heat transport [T1] firstorder mass production rate of species k adsorbate for solute transport w0 (6) [E/MT] zeroorder energy production rate in the fluid for heat transport [(Msolute/Mfluid)T1] zeroorder mass production rate of solute for solute transport kw0 (18) [E/MT] zeroorder energy production rate in the fluid for heat transport [(Msolute/Mfluid)T1] zeroorder fluid mass production rate of species k for solute transport s0 (6) [E/MT] zeroorder energy production rate in the solid matrix for heat transport [(Madsorbate/Msolid)T1] zeroorder mass production rate of solute for solute transport ks0 (18) [E/MT] zeroorder energy production rate in the solid matrix for heat transport [(Madsorbate/Msolid)T1] zeroorder mass production rate of solute for species k for solute transport ij (26) [0 or 1] Kronecker delta (1) [] porosity 1 (11) [M/MG] first general sorption coefficient 1k(19) [M/MG]first general sorption coefficient of species k (22) [E/TLC] bulk thermal conductivity A (23) [E/TLC] thermal conductivity of air s (8) [E/TLC] solid thermal conductivity w (7) [E/TLC] fluid thermal conductivity (1) [M/LT] fluid viscosity (5) [M/LT] fluid viscosity at the base solute concentration for each species or a temperatureviscosity relationship used for nonisothermal conditions
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Appendix A: (Continued) 189 (71) [] aquifer aspect ratio ratio of aquifer thickness to aquifer length pi (26) [LT] pressurebased conductance for the specified pressure source in cell i Uik (34) [LT] concentrationor temperaturebased conductance for the specified concentration or temperature of species k source in cell i (1) [M/L3] fluid density s (6) [M/L3] density of the solid matrix o (4) [M/L3] is the fluid density at the base mass fraction and temperature, s (6) [L2/T] solid matrix thermal diffusivity for heat transport [0] not used for for solute transport sk (18) [L2/T] solid matrix thermal diffusivity for heat transport if species k is ENERGY [0] not used for for solute transport of species k w (6) [L2/T] fluid thermal diffusivity for heat transport [L2/T] molecular diffusivity (DM) for solute transport wk (18) [L2/T] fluid thermal diffusivity for heat transport if species k is ENERGY [L2/T] molecular diffusivity (DM) of species k for solute transport bg (24) [L2/T] geometricmean approximation of bulk thermal diffusivity for heat transport bgk (25) [L2/T] geometricmean approximation of bulk thermal diffusivity for heat transport if species k is ENERGY [L2/T] molecular diffusivity (DM) of species k for solute transport j (29) symmetric bilinear (2D) or trilinear (3D) basis function in global coordinates for node j jk (36) symmetric bilinear (2D) or trilinear (3D) basis function in global coordinates for species k at node j
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Appendix A: (Continued) 190 ik (37) symmetric bilinear (2D) or trilinear (3D) basis function in global coordinates for species k at node i (3) [0 or 1] value equals 1 if heat transport is being simulated or 0 if heat transport is not being simulated 1 (9) [/ MG] linear distribution coefficient 3fL 1 (12) [/ MG] Freundlich distribution coefficient 3fL 2 (13) [] Freundlich coefficient 1 (15) [/ MG] Langmuir distribution coefficient 3fL 2 (15) [] Langmuir coefficient 1k (19) [/ MG] linear distribution coefficient for species k 3fL 1k (20) [/ MG] Freundlich distribution coefficient for species k 3fL 2k (20) [] Freundlich coefficient for species k 1k (21) [/ MG] Langmuir distribution coefficient for species k 3fL 2k (21) [] Langmuir coefficient for species k i (29) asymmetric weighting function in global coordinates for node i Greek Uppercase (41) external boundary of simulated region tNEW (65) [T] modified revised time step length for simple automatic time stepping algorithm tn+1 (26) [T] time step length for the current time step 1nikIN (34) [E/T] total energy source to boundary of cell i due to diffusion or dispersion for current time step [Msolute/MfluidT] total concentration source of species k to cell i due to diffusion or dispersion for current time step
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Appendix B: SUTRAMS Input Data 191
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Appendix B: (Continued) This appendix contains the data input format for all required SUTRAMS files. It also includes the input data format of all options available in SUTRAMS. SUTRA Input Data List List of Input Data for the File Assignment Input File, SUTRA.FIL Model Series: SUTRAMS Model Version: 2D3DMS.1 The file SUTRA.FIL contains file assignments (one line for each assignment) in the following format: Variable Type Description FTYPE Character File type. Valid values are as follows: INP = .inp input file ICS = .ics input file LST = .lst output file RST = .rst output file NOD = .nod output file ELE = .ele output file OBS = .obs output file TBC = .tbc input file OTM = .otm input file ATS = .ats input file SOB = .sob input file ZON = .zon input file IUNIT Integer FORTRAN unit number to be assigned to the file. FNAME Character Full name of the file. 192
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Appendix B: (Continued) Note: Assignments for the .nod, .ele, .obs, .tbc, .otm, .ats, .sob, and .zon files are optional. If any of these assignments are omitted, the corresponding output files will not be created by SUTRA or the optional SUTRAMS functionality will not be used. Assignments for the remaining input and output files are required. Assignments may be listed in any order. Example: INP 50 project.inp ICS 55 project.ics LST 60 project.lst RST 66 project.rst NOD 70 project.nod ELE 80 project.ele OBS 90 project.obs TBC 91 project.tbc OTM 92 project.otm ATS 93 project.ats SOB 94 project.sob ZON 95 project.zon 193
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Appendix B: (Continued) General Format of the .inp, .ics, .tbc, .otm, .ats, .sob, and .zon Input Files SUTRA reads the .inp, .ics, .tbc, .otm, .ats, .sob, and .zon input files in a listdirected fashion (except for DATASET 1 of the .inp file): a. Input data appearing on the same line should be spaceor tabseparated. b. Any data that are not optional must be given values in the input file (blanks are not sufficient). c. Enclose input variables of character type in single quotation marks (unless specified otherwise) to provide maximum compatibility across computing platforms. d. Comment lines may be placed within the .inp, .ics, .tbc, .otm, .ats, .sob, and .zon files, subject to the following restrictions: Comment lines must have a pound sign, #, in the first column. Comments lines can be placed before or after any data set. Comments lines may not be placed within a data set (such as in the middle of a list of specified pressures). For this purpose, data sets with letter designations (such as 2A and 2B) count as distinct data sets comment lines may be placed between them. Comments lines may not be placed within any of the restart information that follows DATASET 3 when a .rst (restart) file is used as a .ics (initial conditions) file. Comments (or any text) can be appended to the end of any line of input data, provided all the required parameters have first been entered on that line. Be sure to leave at least one space or tab between the last required parameter and the beginning of the comment. All input lines unique to SUTRAMS are listed in italic. 194
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Appendix B: (Continued) List of Input Data for the .inp (Main Input) File Model Series: SUTRAMS Model Version: 2D3DMS.1 DATASET 1: Output Heading (two lines) Variable Type Description TITLE1 Character First line of heading for the input data set. TITLE2 Character Second line of heading for the input data set. The first 80 characters of each line are printed as a heading on SUTRA output. In this data set, the character inputs need not be enclosed in quotation marks. 195
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Appendix B: (Continued) DATASET 2A: Simulation Type (one line) Variable Type Description SIMULA Character Two to three words. The first word must be SUTRA. The second word indicates the type of transport, and must be either ENERGY, SOLUTE, ENERGY SOLUTE, or MULTI SOLUTE. Any additional words are ignored by SUTRA. Examples: For energy transport simulation, one may write the following: SUTRA ENERGY TRANSPORT For solute transport simulation, one may write the following: SUTRA SOLUTE TRANSPORT For multispecies energy and solute transport, one may write the following: SUTRA ENERGY SOLUTE TRANSPORT or SUTRA SOLUTE ENERGY TRANSPORT For multispecies solute transport, one may write the following: SUTRA SOLUTE MULTI TRANSPORT or SUTRA MULTI SOLUTE TRANSPORT In these examples, the word TRANSPORT is ignored by SUTRAMS but is included to make the input more readable. 196
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Appendix B: (Continued) DATASET 2B: Mesh Structure (four lines) Variable Type Description Line 1: MSHSTR Character Two words. The first word indicates the dimensionality of the mesh, and must be either D or D. The second word indicates the regularity of the mesh, and must be either REGULAR, BLOCKWISE, or IRREGULAR. Any additional words are ignored by SUTRA. (D IRREGULAR meshes are not currently supported.) NN1 Integer For a REGULAR mesh, the number of nodes in the first numbering direction. May be omitted if the mesh is IRREGULAR. NN2 Integer For a REGULAR mesh, the number of nodes in the second numbering direction. May be omitted if the mesh is IRREGULAR. NN3 Integer For a REGULAR, 3D mesh, the number of nodes in the third numbering direction. May be omitted if the mesh is IRREGULAR and/or 2D. Omit lines 2 4 if mesh is NOT BLOCKWISE. Line 2: NBLK1 Integer Number of blocks in the first numbering direction. LDIV1 Integer A list of the number of elements into which to divide each of the NBLK1 blocks along the first numbering direction. 197
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Appendix B: (Continued) Line 3: NBLK2 Integer Number of blocks in the second numbering direction. LDIV2 Integer A list of the number of elements into which to divide each of the NBLK2 blocks along the second numbering direction. Line 4: NBLK3 Integer Number of blocks in the third numbering direction. LDIV3 Integer A list of the number of elements into which to divide each of the NBLK3 blocks along the third numbering direction. Note: A BLOCKWISE mesh is a special type of REGULAR mesh that is created by the preprocessor SUTRAPREP. The additional data associated with a BLOCKWISE mesh are not currently used by SUTRA and are included only for postprocessing purposes. Examples: For a 3D, regular (logicallyrectangular), 10x20x30node mesh, one may write the following: D REGULAR MESH 10 20 30 For a 2D, irregular mesh, one may write the following: D IRREGULAR MESH In these examples, the word MESH is ignored by SUTRAMS but is included to make the input more readable. 198
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Appendix B: (Continued) DATASET 3A: Simulation Control Numbers (one line) Variable Type Description NN Integer Exact number of nodes in finiteelement mesh. NE Integer Exact number of elements in finiteelement mesh. NPBC Integer Exact number of nodes at which pressure is a specified constant value or function of time. MNUBC Integer Maximum number of nodes at which temperature or concentration is a specified constant value or function of time for each simulated species. NSOP Integer Exact number of nodes at which a fluid source/sink is a specified constant value or function of time. MNSOU Integer Maximum number of nodes at which an energy or solute mass source/sink is a specified constant value or function of time for each simulated species. NOBS Integer Exact number of nodes at which observations will be made. Set to zero for no observations or if using the optional specified observation locations (.sob) data set. NSPE Integer Number of species simulated (includes energy and all solute species). O M I T for singlespecies transport simulations. 199
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Appendix B: (Continued) DATASET 3B: Species Control Numbers (one line for each of species specified in DATASET 3A, plus one line) O M I T when only one transport species is being simulated (NSPE = 1). Variable Type Description Lines 1 to NSPE: NSN Integer Species Number. SPNAME Character Tencharacter descriptive name for species. The energy transport species must be named ENERGY, if energy transport is being simulated. NUBC Integer Exact number of nodes at which temperature or concentration is a specified constant value or function of time for simulated species NSN. Must be less than or equal to MNUBC specified in DATASET 3A. NSOU Integer Exact number of nodes at which an energy or solute mass source/sink is a specified constant value or function of time for simulated species NSN. Must be less than or equal to MNSOU specified in DATASET 3A. Last line: Integer Placed immediately following all DATASET 3B lines. Line must begin with the integer 0. Examples: For a data set to simulate two transport species (e.g., Energy and Brine), one may write the following: ## ## DATASET 3B: Species Names and number of boundary conditions for each species ## [NSN] [SPNAME] [NUBC] [NSOU] 1 'ENERGY' 63 0 2 'BRINE' 0 10 0 200
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Appendix B: (Continued) DATASET 4: Simulation Mode Options (one line) Variable Type Description CUNSAT Character One word. Set to UNSATURATED for simulation of unsaturated/saturated groundwater flow. Set to SATURATED for simulation of only saturated flow. When UNSATURATED flow is allowed, the unsaturated flow functions must be programmed by the user in subroutine UNSAT. CSSFLO Character One word. Set to TRANSIENT for simulation of transient groundwater flow. Set to STEADY for simulation of steadystate flow. If fluid density is to change with time, then TRANSIENT flow must be selected. CSSTRA Character One word. Set to TRANSIENT for simulation of transient solute or energy transport. Set to STEADY for simulation of steadystate transport. Note that steadystate transport requires a steadystate flow field. So, if CSSTRA = STEADY, then also set CSSFLO = STEADY. CREAD Character One word. Set to COLD to read initial condition data (.ics file) for a cold start (the very first time step of a simulation or series of restarted simulations). Set to WARM to read initial condition data (.ics file) for a warm restart of a simulation using data that were previously stored by SUTRA in a .rst file. ISTORE Integer To store results for each ISTORE time step in the .rst file for later use as initial conditions on a 201
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Appendix B: (Continued) restart, set to +1 or greater value. To cancel storage, set to 0. This option is recommended as a backup for storage of results of intermediate time steps during long simulations. Should the execution halt unexpectedly, it may be restarted with initial conditions consisting of results of the last successfully completed time step stored in the .rst file. When ISTORE > +1, results are always stored in the .rst file after the last time step of a simulation regardless of whether the step is an even multiple of ISTORE. Any extra words included in the character variables in this data set are ignored by SUTRA. Example: To simulate saturated, steadystate groundwater flow with transient solute or energy transport from a cold start, storing intermediate results every 10 time steps, one may write the following: 'SATURATED FLOW' 'STEADY FLOW' 'TRANSIENT TRANSPORT' 'COLD START' 10 In this example, the words FLOW, TRANSPORT, and START are ignored by SUTRA but are included to make the input more readable. 202
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Appendix B: (Continued) DATASET 5: Numerical Control Parameters (one line) Variable Type Description UP Real Fractional upstream weight for stabilization of numerical oscillations due to highly advective transport or unsaturated flow. UP may be given any value from 0.0 to +1.0. UP=0.0 implies no upstream weighting (Galerkin method). UP=0.5 implies 50% upstream weighting, UP=1.0 implies full (100%) upstream weighting. Recommended value is zero. WARNING: Upstream weighting increases the local effective longitudinal dispersivity of the simulation by approximately UP*(L)/2 where L is the local distance between element sides along the direction of flow. Note that the amount of this increase varies from place to place depending on flow direction and element size. Thus a nonzero value for UP actually changes the value of longitudinal dispersivity used by the simulation, and also broadens otherwise sharp concentration, temperature or saturation fronts. GNUP Real Pressure boundary condition, conductance. A high value causes SUTRA simulated and specified pressure values at specified pressure nodes to be equal in all significant figures. A low value causes simulated pressure to deviate significantly from specified values. The ideal value of GNUP causes simulated and specified pressures to match in the largest six or seven significant figures only, and deviate in the rest. Trialanderror is required to determine an ideal GNUP value for a given simulation by comparing values specified with those calculated at the appropriate nodes for different values of GNUP. An initial guess of 0.01 is suggested. 203
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Appendix B: (Continued) GNUU Real Concentration/temperature boundary condition factor. A high value causes SUTRA simulated values and specified values at specified concentration/temperature nodes to be equal in all significant figures. A low value causes simulated values to deviate significantly from specified values. The ideal value of GNUU causes simulated and specified concentrations or temperatures to match in the largest six or seven significant figures only, and deviate in the rest. Trialanderror is required to determine an ideal GNUU value for a given simulation by comparing specified values with those calculated at the appropriate nodes for different values of GNUU. An initial guess of 0.01 is suggested. For multispecies transport simulations, enter a GNUU value for each species. GNUU values must be entered in order for species 1 to NSPE, as specified in DATASET 3B. Examples: For a data set to simulate two transport species (e.g., Energy and Brine), one may write the following: ## ## DATASET 5: Numerical Control Parameters ## [UP] [GNUP] [GNUU1] [GNUU2] 0.0 0.01 0.01 0.01 or for a data set to simulate a single transport species, one may write the following: ## ## DATASET 5: Numerical Control Parameters ## [UP] [GNUP] [GNUU] 0.0 0.01 0.01 204
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Appendix B: (Continued) DATASET 6: Temporal Control and Solution Cycling Data (one line) Variable Type Description ITMAX Integer Maximum allowed number of time steps in simulation. DELT Real Duration of initial time step. [s] TMAX Real Maximum allowed simulation time. [s] SUTRA time units are always in seconds. Other time measures are related as follows: [min] = 60. [s] [h] = 60. [min] [d] = 24. [h] [week] = 7. [d] [mo] = 30.4375 [d] [yr] = 365.250 [d] ITCYC Integer Number of time steps in timestep change cycle. A new timestep size is begun at time steps numbered: 1+ n*(ITCYC). DTMULT Real Multiplier for timestep change cycle. New time step size is: (DELT)*(DTMULT). DTMAX Real Maximum allowed size of time step when using timestep multiplier. Timestep size is not allowed to increase above this value. NPCYC Integer Number of time steps in pressure solution cycle. Pressure is solved on time steps numbered: n*(NPCYC), as well as on the initial time step. NUCYC Integer Number of time steps in temperature/concentration solution cycle. Transport equation is solved on time steps numbered: n*(NUCYC) as well as on the initial time step. Either NPCYC or NUCYC must be set to 1. 205
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Appendix B: (Continued) DATASET 7A: Iteration Controls for Resolving Nonlinearities (one line) Variable Type Description ITRMAX Integer Maximum number of iterations allowed per time step to resolve nonlinearities. Set to a value of +1 for noniterative solution. Noniterative solution may be used for saturated aquifers when density variability of the fluid is small, or for unsaturated aquifers when time steps are chosen to be small. RPMAX Real Absolute iteration convergence criterion for pressure solution. Pressure solution has converged when largest pressure change from the previous iterations solution of any node in mesh is less then RPMAX. May be omitted for noniterative solution. RUMAX Real Absolute iteration convergence criterion for transport solution. Transport solution has converged when largest concentration or temperature change from the previous iterations solution of any node in mesh is less than RUMAX. May be omitted for noniterative solution. For multispecies transport simulations, enter a RUMAX value for each species. RUMAX values must be entered in order for species 1 to NSPE, as specified in DATASET 3B. Examples: For a data set to simulate two transport species (e.g., Energy and Brine), one may write the following: ## ## DATASET 7A: Iteration and Matrix Solver Controls ## [ITRMAX] [RPMAX] [RUMAX1] [RUMAX2] 100 0.1 0.1 0.1 or for a data set to simulate a single transport species, one may write the following: 206
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Appendix B: (Continued) ## ## DATASET 7A: Iteration and Matrix Solver Controls ## [ITRMAX] [RPMAX] [RUMAX] 100 0.1 0.1 207
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Appendix B: (Continued) DATASET 7B: Matrix Solver Controls for Pressure Solution (one line) Variable Type Description CSOLVP Character Name of solver to be used to obtain the pressure solution. Valid values are as follows: 'DIRECT' = Banded Gaussian elimination (direct) 'CG' = ICPreconditioned Conjugate Gradient (CG) 'GMRES' = ILUPreconditioned Generalized Minimum Residual (GMRES) 'ORTHOMIN' = ILUPreconditioned ORTHOMIN If the DIRECT solver is used, it must be used for both the pressure and the transport solution ; if either CSOLVP or CSOLVU (DATASET 7C) is set to DIRECT, then the other must also be set to DIRECT. If an ITERATIVE solver is used, the recommended value is CG if no upstream weighting is used (UP=0.). If upstream weighting is used (UP>0.), use GMRES or ORTHOMIN. ITRMXP Integer Maximum number of solver iterations during pressure solution. May be omitted if the DIRECT solver is used. ITOLP Integer Type of convergence criterion to be applied to solver iterations during pressure solution. Recommended value is 0. (For other options, see SLAP solver documentation.) May be omitted if the DIRECT solver is used. TOLP Real Convergence tolerance for solver iterations during pressure solution. May be omitted if the DIRECT solver is used. 208
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Appendix B: (Continued) NSAVEP Integer For GMRES and ORTHOMIN solvers, the number of direction vectors to save and orthogonalize against during pressure solution. Recommended value is 10. (For other options, see SLAP solver documentation.) For CG solver, set to any integer value. May be omitted if the DIRECT solver is used. 209
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Appendix B: (Continued) DATASET 7C: Matrix Solver Controls for Transport Solution (one line or NSPE lines plus one for multispecies transport simulations) Variable Type Description Lines 1 to NSPE: NSN Integer Species Number. O M I T if a single species transport simulation. CSOLVU Character Name of solver to be used to obtain the transport solution for species NSN. Valid values are as follows: 'DIRECT' = Banded Gaussian elimination (direct) 'CG' = ICPreconditioned Conjugate Gradient (CG) 'GMRES' = ILUPreconditioned Generalized Minimum Residual (GMRES) 'ORTHOMIN' = ILUPreconditioned ORTHOMIN If the DIRECT solver is used, it must be used for both the pressure and the transport solution ; if either CSOLVU or CSOLVP (DATASET 7B) is set to DIRECT, then the other must also be set to DIRECT. If an ITERATIVE solver is used, the recommended values are GMRES and ORTHOMIN. ITRMXU Integer Maximum number of solver iterations during transport solution for species NSN. May be omitted if the DIRECT solver is used. ITOLU Integer Type of convergence criterion to be applied to solver iterations during transport solution for species NSN. Recommended value is 0. (For other options, see SLAP solver documentation.) May be omitted if the DIRECT solver is used. TOLU Real Convergence tolerance for solver iterations during transport solution for species NSN. May be omitted if the DIRECT solver is used. 210
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Appendix B: (Continued) NSAVEU Integer For GMRES and ORTHOMIN solvers, the number of direction vectors to save and orthogonalize against during transport solution for species NSN. Recommended value is 10. (For other options, see SLAP solver documentation.) For CG solver, set to any integer value. May be omitted if the DIRECT solver is used. STARTU Real Starting time of initiation of transport of species NSN. O M I T if a single species transport simulation and O P T I O N A L if simulating multiple species. [s] Last line: Integer Placed immediately following all DATASET 7B lines. Line must begin with the integer 0. O M I T if a single species transport simulation. Examples: For a data set to simulate two transport species (e.g., Energy and Brine), one may write the following: ## DATASET 7C: Matrix Solver Controls for Transport Solution ## [NSN] [CSOLVU] [ITRMXU] [ITOLU] [TOLU] [NSAVEU] [STARTU] 1 'ORTHOMIN' 100 0 0.1 10 0.0 2 'ORTHOMIN' 100 0 0.1 10 1000.0 0 For a data set to simulate a single transport species, one may write the following: ## DATASET 7C: Matrix Solver Controls for Transport Solution ## [CSOLVU] [ITRMXU] [ITOLU] [TOLU] [NSAVEU] 'ORTHOMIN' 100 0 0.1 10 211
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Appendix B: (Continued) DATASET 8A: Output Controls and Options for .lst (Main Output) File (one line) Variable Type Description NPRINT Integer Printed output is produced on time steps numbered: n (NPRINT), as well as on first and last time step. CNODAL Character A value of N cancels printout of node coordinates, nodewise element thicknesses, and nodewise porosities. Set to Y for full printout. CELMNT Character A value of N cancels printout of elementwise permeabilities and elementwise dispersivities. Set to Y for full printout. CINCID Character A value of N cancels printout of node incidences and pinch node incidences in elements. Set to Y for full printout. CVEL Character Set to a value of Y to calculate and print fluid velocities at element centroids each time printed output is produced. Note that for nonsteady state flow, velocities are based on results and pressures of the previous time step or iteration and not on the newest values. Set to N to cancel option. CBUDG Character Set to a value of Y to calculate and print a fluid mass budget and energy or solute mass budget each time printed output is produced. A value of N cancels the option. 212
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Appendix B: (Continued) DATASET 8B: Output Controls and Options for .nod File (Nodewise Results Listed in Columns) (one line) Variable Type Description NCOLPR Integer Printed output of nodewise solution data are produced on timestep multiples of NCOLPR, as well as on first and last time step. K5COL Character List of names of variables to be printed in columns in the .nod file. The ordering of columns corresponds to the ordering of variable names in the list. Valid names are as follows: N = node number (if used it must appear first in list) X = Xcoordinate of node Y = Ycoordinate of node Z = Zcoordinate of node (3D only ) P = pressure (or head) U = concentration or temperature (all species) S = saturation  = end of list (any names following  are ignored) The symbol  (a single dash) must be used at the end of the list. Example: To output the 3D node coordinates, pressure, and solute concentration in columns in the .nod file every 5 time steps, write the following: 5 X Y Z P U  213
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Appendix B: (Continued) DATASET 8C: Output Controls and Options for .ele File (Velocities at Element Centroids Listed in Columns) (one line) Variable Type Description LCOLPR Integer Printed output of elementwise solution data are produced on timestep multiples of LCOLPR, as well as on first and last time step. K6COL Character List of names of variables to be printed in columns in the .ele file. The ordering of columns corresponds to the ordering of variable names in the list. Valid names are as follows: E = element number (if used it must appear first in list) X = Xcoordinate of element centroid Y = Ycoordinate of element centroid Z = Zcoordinate of element centroid (3D only ) VX = Xcomponent of fluid velocity VY = Ycomponent of fluid velocity VZ = Zcomponent of fluid velocity  = end of list (any names following  are ignored) The symbol  (a single dash) must be used at the end of the list. Example: To output the 3D element centroid coordinates and velocity components in columns in the .ele file every 10 time steps, write the following: 10 X Y Z VX VY VZ  214
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Appendix B: (Continued) DATASET 8D: Output Controls and Options for .obs File (Observation Node Results Listed in Columns list directed input with no comments between entries) O M I T when there are no observation nodes or if using the optional specified observation locations file (.sob). Variable Type Description NOBCYC Integer Printed output of observation node data are produced on timestep multiples of NOBCYC, as well as on first and last time step. INOB Integer List of node numbers of observation nodes. Enter a value of zero as an extra observation node number following the last real observation node to indicate the end of the list. Example: To output pressures/heads, concentrations/temperatures, and saturations at nodes 1, 22, 333, and 4444 in columns in the .obs file every 100 time steps, write the following: 100 1 22 333 4444 0 215
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Appendix B: (Continued) DATASET 9A: Base Fluid Properties (one line) Variable Type Description COMPFL Real Fluid compressibility, p. M/(Ls2)1. Note, specific pressure storativity is: Sop = (1) + CW Real Fluid specific heat, cw. [E/(MC)] (May be left blank for solute transport simulation.) BSIGMAW Real Base fluid diffusivity, w. For energy transport represents fluid thermal conductivity, w. [E/(LCs)]. For solute transport represents molecular diffusivity of solute in pure fluid, Dm [L2/s]. May be decreased from pure value to account for tortuosity of fluid paths. RHOW Real Density of fluid at base concentration or temperature. [M/L3]. BURHOW Real Base value of solute concentration (as mass fraction) or temperature of fluid at which base fluid density, RHOW is specified. [Ms/M] or [C]. BDRWDU Real Base fluid coefficient of density change with concentration (fraction) or temperature: = RHOW + DRWDU (UURHOW). [M2/(L3Ms)] or [M/(L3C)] BVISC Real For solute transport: base fluid viscosity, [M/Ls]. For energy transport and multispecies energy and solute transport, this value is a scale factor. It multiplies the viscosity that is calculated internally in units of [kg/ms]. BVISC may be used for energy transport to convert units of [kg/ms] to desired units of viscosity. Examples: See the examples given below for DATASET 9B to see how DATASET 9A and 9B are related for multispecies transport simulations. 216
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Appendix B: (Continued) DATASET 9B: Fluid Properties (Up to NSPE lines plus one line). DATASET 9B data only needs to be entered for those species for which fluid properties are different from the base values specified in DATASET 9A. O M I T when only one transport species is being simulated (NSPE = 1). Variable Type Description Maximum Lines NSPE: NSN Integer Species Number. SIGMAW Real Fluid diffusivity, w,, for species NSN. For energy transport represents fluid thermal conductivity, w. [E/(LCT)]. For solute transport represents molecular diffusivity of solute in pure fluid, Dm [L2/T]. May be decreased from pure value to account for tortuosity of fluid paths. URHOW Real Value of solute concentration (as mass fraction) or temperature of fluid of species NSN at which base fluid density, RHOW is specified. [Msolute/Mfluid] or [C]. DRWDU Real Fluid coefficient of density change with concentration (fraction) or temperature for species NSN: = RHOW + DRWDUn (UnURHOWn) (See Eq. 2.4). [Mfluid2/(L3Msolute)] or [Mfluid/(L3C)] DVSDU Real Fluid coefficient of viscosity change with concentration (fraction) or temperature for species NSN: = BVISC + DVSDUn (UnURHOWn). For energy transport species DVSDU can be any real value because it is internally replaced with BVISC, BVISC is subsequently set to 0, and the nonlinear viscosity temperature relationship is used (see eq. 2.5). [Mfluid2/(LTMsolute)] or [Mfluid/(LTC)] Last line: Integer Placed immediately following all DATASET 9B lines. Line must begin with the integer 0. 217
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Appendix B: (Continued) Examples: For a data set to simulate four transport species of which only two have fluid properties that are different from the base values in DATASET 9A (e.g., Energy and Brine), one may write the following: ## ## DATASET 9A: Base Fluid Properties ## [COMPFL] [CW] [SIGMAW] [RHOW0] [URHOW0] [DRWDU] [BISC0] 0.00D0 4182.D0 1.000D9 1000.0 0.00 0.00 1.0 ## ## DATASET 9B: Multiple Species Fluid Properties ## [NSN] [SIGMAW] [URHOW0] [DRWDU] [DVSDU] 2 6.6D6 0.00 7.00D2 0.0 1 0.6 25.5 0.15D0 0.0 0 218
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Appendix B: (Continued) DATASET 10: Solid Matrix Properties (one line) Variable Type Description COMPMA Real Solid matrix compressibility, =(1)1 / p. [M/(Ls2)]1 CS Real Solid grain specific heat, cs. [E/(MC)] (May set to any real value for solute transport simulation.) SIGMAS Real Solid grain diffusivity, s. For energy transport represents thermal conductivity of a solid grain. [E/(LCs)] (May be set to any real value for solute transport simulation.) Set SIGMAS to a negative value if spatially varying thermal conductivities are specified in DATASET 15D and/or to use geometric mean bulk thermal conductivities in a simulation.  SIGMAS  is used as the base thermal conductivity for all elements not modified in DATASET 15D. If the hydraulic parameters are defined using the optional zone definition file spatially varying thermal conductivities can be specified without specifying a negative SIGMAS value and inclusion of DATASET 15D in the SUTRAMS *.inp file. However, a negative SIGMAS value must be specified to use geometric mean thermal conductivities even if the optional zone definition file is used. In this case, a zero value can be entered on a line immediately after the THERMEQ value for DATASET 15D. RHOS Real Density of a solid grain, s. [M/L3] 219
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Appendix B: (Continued) DATASET 11: Adsorption Parameters (up to NSPE plus one line). Adsorption parameters must be defined for single species transport simulations even if ADSMOD is NONE but do not have to be defined for multispecies transport simulations if ADSMOD is NONE for all species. All adsorption parameters are initialized to zero prior to reading DATASET 11. DATASET 11 must be terminated with a zero NSN value for multispecies transport simulations. Variable Type Description Up to a Maximum of NSPE Lines: NSN Integer Species Number. O M I T for single species transport simulations (NSPE=1). ADSMOD Character For no sorption or for energy transport simulation, set to NONE and leave rest of line blank. For linear sorption model, set to LINEAR. For Freundlich sorption model, set to FREUNDLICH. For Langmuir sorption model, set to LANGMUIR. CHI1 Real Value of linear, Freundlich, or Langmuir distribution coefficient for species NSN, depending on sorption model chosen in ADSMOD, 1. /MG. 3fL CHI2 Real Value of Freundlich or Langmuir coefficient for species NSN, depending on sorption model chosen in ADSMOD. Set to any real value for linear sorption. 2. 1 for Freundlich. /Ms for Langmuir. 3fL 220
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Appendix B: (Continued) Last line: Integer Placed immediately following all DATASET 11 lines. Line must begin with the integer 0. O M I T if a single species transport simulation. Examples: For a data set to simulate two transport species of which only one is not equal to NONE, one may write the following: ## ## DATASET 11: Adsorption Parameters ## [NSN] [ADSMOD] [CHI1] [CHI2] 1 NONE 0.0 0.0 2 'LINEAR' 1.0D1 0.0 0 or ## ## DATASET 11: Adsorption Parameters ## [NSN] [ADSMOD] [CHI1] [CHI2] 2 'LINEAR' 1.0D1 0.0D0 0 or for a data set to simulate a single transport species, one may write the following: ## ## DATASET 11: Adsorption Parameters ## [ADSMOD] [CHI1] [CHI2] NONE 0.0 0.0 221
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Appendix B: (Continued) DATASET 12: Production of Energy or Solute Mass (up to NSPE plus one lines). Production rates must be defined for single species transport simulations but do not have to be defined for all species in multispecies transport simulations. All production rates are initialized to zero prior to reading DATASET 12. DATASET 12 must be terminated with a zero NSN value for multispecies transport simulations. Variable Type Description Up to a Maximum of NSPE Lines: NSN Integer Species Number. O M I T for single species transport simulations (NSPE=1). PRODF Real Zeroorder rate of production for species NSN in the fluid [(E/M)/s] for energy production, [(Ms/M)/s] for solute mass production. wo PRODS Real Zeroorder rate of production for species NSN in the immobile phase, [(E/MG)/s] for energy production, [(Ms/MG)/s] for adsorbate mass production. so PRODF1 Real Firstorder rate of solute mass production for species NSN in the fluid, [s1]. Set to any real value for energy transport. wi PRODS1 Real Firstorder rate of adsorbate mass production for species NSN in the immobile phase, [s1]. Set to any real value for energy transport. si Last line: Integer Placed immediately following all DATASET 12 lines. Line must begin with the integer 0. O M I T if a single species transport simulation. 222
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Appendix B: (Continued) Examples: For a data set to simulate two transport species, one may write the following: ## ## DATASET 12: Production of Energy or Solute Mass ## [NSN] [PRODF0] [PRODS0] [PRODF1] [PRODS1] 1 0.0D0 0.0D0 0.0D0 0.0D0 2 3.16881D08 0.0D0 0.0D0 0.0D0 0 or ## ## DATASET 12: Production of Energy or Solute Mass ## [NSN] [PRODF0] [PRODS0] [PRODF1] [PRODS1] 2 3.16881D08 0.0D0 0.0D0 0.0D0 0 or for a data set to simulate a single transport species, one may write the following: ## ## DATASET 12: Production of Energy or Solute Mass ## [PRODF0] [PRODS0] [PRODF1] [PRODS1] 3.16881D08 0.0D0 0.0D0 0.0D0 223
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Appendix B: (Continued) DATASET 13: Orientation of Coordinates to Gravity (one line) Variable Type Description GRAVX Real Component of gravity vector in +xdirection. [L/s2] GRAVX = g ( ELEVATION/x), where g is the total acceleration due to gravity in [L/s2]. GRAVY Real Component of gravity vector in +ydirection. [L/s2] GRAVY= g ( ELEVATION/y), where g is the total acceleration due to gravity in [L/s2]. GRAVZ Real Component of gravity vector in +zdirection. [L/s2] GRAVY= g ( ELEVATION/y), where g is the total acceleration due to gravity in [L/s2]. Set to any real value for 2D problems. 224
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Appendix B: (Continued) DATASET 14A: Scale Factor for Nodewise Data (one line) Variable Type Description Character Line must begin with the word NODE. SCALX Real The scaled xcoordinates of nodes in DATASET 14B are multiplied by SCALX in SUTRA. May be used to change from map to field scales, or from English to SI units. A value of 1.0 gives no scaling. SCALY Real The scaled ycoordinates of nodes in DATASET 14B are multiplied by SCALY in SUTRA. May be used to change from map to field scales, or from English to SI units. A value of 1.0 gives no scaling. SCALZ Real For 3D problems, the scaled zcoordinates of nodes in DATASET 14B are multiplied by SCALZ in SUTRA. May be used to change from map to field scales, or from English to SI units. A value of 1.0 gives no scaling. For 2D problems, the scaled element (mesh) thicknesses at nodes in DATASET 14B are multiplied by SCALZ in SUTRA. May be used to easily change entire mesh thickness or to convert English to SI units. A value of 1.0 gives no scaling. PORFAC Real The scaled nodewise porosities of DATASET 14B are multiplied by PORFAC in SUTRA. May be used to easily assign a constant porosity value to all nodes by setting PORFAC=porosity, and all POR(II)=1.0 in DATASET 14B. 225
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Appendix B: (Continued) DATASET 14B: Nodewise Data (one line for each of NN nodes) Variable Type Description II Integer Number of node to which data on this line refers, i. NREG(II) Integer Unsaturated flow property region number to which node II belongs. Set to any integer value when flow simulation is saturated only and hydraulic parameters are not defined by zones. The node region number is usually the same as the region number of the elements it appears in. When the node is to be at the boundary of two regions, it may be assigned to either region. If nodewise data are defined using zones then NREG defines the spatial distribution of the zones specified in the SUTRAMS zone file. X(II) Real Scaled xcoordinate of node II, xi. [L] Y(II) Real Scaled ycoordinate of node II, yi. [L] Z(II) Real For 3D problems, scaled zcoordinate of node II, zi. [L] For 2D problems, scaled thickness of mesh at node II. [L] In order to simulate radial crosssections, set THICK(II) = (2)(radiusi), where radiusi is the radial distance from the vertical center axis to node i. POR(II) Real Scaled porosity value at node II, i. [1] Note: The order in which the nodes are numbered affects the bandwidth of the global banded matrix which in turn affects computational and storage efficiency when the DIRECT solver is used. In this case, the user should take care to number the nodes to minimize the problem bandwidth. SUTRA sets the problem bandwidth equal to one plus twice the maximum difference in node numbers in the element containing the largest node number difference in the mesh. See Figure 55 for an example. When 226
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Appendix B: (Continued) an iterative solver is used, it is still advantageous to minimize the problem bandwidth, although not as critical as in the case of the DIRECT solver. Figure 55 Minimization of bandwidth by careful numbering of nodes (Fig. 7.1, Voss and Provost, 2002). In this 2D example, the same mesh has been numbered two different ways. In the first numbering scheme, the largest difference between node numbers in a single element is 15, giving a bandwidth of 1+2(15)=31. In the second numbering scheme, the largest difference between node numbers in a single element is 5, giving a bandwidth of 1+2(5)=11. The same principle applies to 3D meshes: the bandwidth equals one plus the maximum difference between node numbers in the element that contains the largest node number difference in the mesh. 227
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Appendix B: (Continued) DATASET 15A: Scale Factors for Elementwise Data (one line) Variable Type Description Character Line must begin with the word ELEMENT. PMAXFA Real The scaled maximum permeability values, PMAX, in DATASET 15B are multiplied by PMAXFA in SUTRA. May be used to convert units or to aid in assignment of maximum permeability values in elements. PMIDFA Real The scaled middle permeability values, PMID, in DATASET 15B are multiplied by PMIDFA in SUTRA. May be used to convert units or to aid in assignment of maximum permeability values in elements. Omit for 2D problems. PMINFA Real The scaled minimum permeability values, PMIN, in DATASET 15B are multiplied by PMINFA in SUTRA. May be used to convert units or to aid assignment of minimum permeability values in elements. ANG1FA Real The scaled angles ANGLE1 in DATASET 15B are multiplied by ANG1FA in SUTRA. For 2D problems, may be used to easily assign a uniform direction of anisotropy by setting ANG1FA=angle, and all ANGLE1(L)=1.0 in DATASET 15B. ANG2FA Real The scaled angles ANGLE2 in DATASET 15B are multiplied by ANG2FA in SUTRA. Omit for 2D problems. ANG3FA Real The scaled angles ANGLE3 in DATASET 15B are multiplied by ANG3FA in SUTRA. Omit for 2D problems. ALMAXF Real The scaled longitudinal dispersivities in the maximum permeability direction, ALMAX, in DATASET 15B are multiplied by ALMAXF in SUTRA. May be used to convert units or to aid in assignment of dispersivities. 228
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Appendix B: (Continued) ALMIDF Real The scaled longitudinal dispersivities in the middle permeability direction, ALMAX, in DATASET 15B are multiplied by ALMAXF in SUTRA. May be used to convert units or to aid in assignment of dispersivities. Omit for 2D problems. ALMINF Real The scaled longitudinal dispersivities in the minimum permeability direction, ALMIN, in DATASET 15B are multiplied by ALMINF in SUTRA. May be used to convert units or to aid in assignment of dispersivities. ATMXF Real The scaled first transverse dispersivities in the maximum permeability direction, AT1MAX, in DATASET 15B are multiplied by AT1MXF in SUTRA. May be used to convert units or to aid in assignment of dispersivity. ATMDF Real The scaled first transverse dispersivities in the middle permeability direction, AT1MID, in DATASET 15B are multiplied by AT1MDF in SUTRA. May be used to convert units or to aid in assignment of dispersivity. Omit for 2D problems. ATMNF Real The scaled first transverse dispersivities in the minimum permeability direction, AT1MIN, in DATASET 15B are multiplied by AT1MNF in SUTRA. May be used to convert units or to aid in assignment of dispersivity. 229
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Appendix B: (Continued) DATASET 15B: Elementwise Data (one line for each of NE elements) Variable Type Description L Integer Number of element to which data on this line refers. LREG(L) Integer Unsaturated flow property region number to which this element belongs. Set to any integer value when flow simulation is saturated only and hydraulic parameters are not defined by zones. If elementwise data are defined using zones then LREG defines the spatial distribution of the zones specified in the SUTRAMS zone file. PMAX(L) Real Scaled maximum permeability value of element L, kmax(L). [L2] PMID(L) Real Scaled middle permeability value of element L, kmid(L). [L2] Isotropic permeability requires: PMID(L)=PMAX(L). Omit for 2D problems. PMIN(L) Real Scaled minimum permeability value of element L, kmin(L). [L2] Isotropic permeability requires: PMIN(L)=PMAX(L). ANGLE1(L) Real Scaled angle within the x,yplane, measured counterclockwise from +xdirection to maximum permeability direction in element L, L. [] In 3D this is the yaw of the principal permeability axes relative to the x, y, and z coordinate axes; arbitrary if, after application of scale factors, PMID(L)=PMAX(L), ALMID(L)=ALMAX(L), AT1MID(L)=AT1MAX(L), and AT2MID(L)=AT2MAX(L). In 2D arbitrary if, after application of scale factors, PMIN(L)=PMAX(L) and ALMIN(L)=ALMAX(L), and AT1MIN(L)=AT1MAX(L). 230
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Appendix B: (Continued) ANGLE2(L) Real Scaled angle measured (after ANGLE1 has been applied) upward from x,yplane to maximum permeability direction in element L, L. [] In 3D, this is the pitch of the principal permeability axes relative to the x, y, and z coordinate axes; arbitrary if, after application of scale factors, PMIN(L)=PMAX(L), ALMIN(L)=ALMAX(L), AT1MIN(L)=AT1MAX(L), and AT2MIN(L)=AT2MAX(L). Omit for 2D problems. ANGLE3(L) Real Scaled angle about the axis of maximum permeability, measured (after ANGLE1 and ANGLE2 have been applied) looking down the positive half of the axis toward the origin, and counterclockwise from the horizontal to the middle permeability direction in element L, L. [] In 3D, this is the roll of the principal permeability axes relative to the x, y, and z coordinate axes; arbitrary if, after application of scale factors. PMIN(L)=PMID(L), ALMIN(L)=ALMID(L), AT1MIN(L)=AT1MID(L), and AT2MIN(L)=AT2MID(L). Omit for 2D problems. If the permeability ellipsoid is initially oriented with the maximum, middle, and minimum permeability axes aligned in the x, y, and zdirections, respectively, then the "yaw, pitch, and roll" convention is equivalent to rotating the ellipsoid by ANGLE3 about the xaxis, by ANGLE2 about the yaxis, and by ANGLE1 about the zaxis, in that order. ALMAX(L) Real Scaled longitudinal flow dispersivity value of element L for flow in the direction of maximum permeability PMAX(L), Lmax(L). [L] ALMID(L) Real Scaled longitudinal flow dispersivity value of element L for flow in the direction of middle permeability PMID(L), Lmid(L). [L] 231
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Appendix B: (Continued) Omit for 2D problems. ALMIN(L) Real Scaled longitudinal dispersivity value of element L for flow in the direction of minimum permeability PMIN(L), Lmin(L). [L] ATMAX(L) Real Scaled first transverse dispersivity value of element L for flow in the direction of maximum permeability PMAX(L), T1max(L). [L] ATMID(L) Real Scaled first transverse dispersivity value of element L for flow in the direction of middle permeability PMID(L), T1mid(L). [L] Omit for 2D problems. ATMIN(L) Real Scaled first transverse dispersivity value of element L for flow in the direction of minimum permeability PMIN(L), T1min(L). [L] Notes: Note that the convention for determining the 2D transverse dispersivity, T, differs from the one used in versions of SUTRA prior to version 2D3D.1 (Voss, 1984). See section 2.5 of Voss and Provost (2002). The following notes are provided to assist the user in understanding the meaning of the ellipsoids and angles used in 3D SUTRA simulations. The dispersivities ALMAX(L), ALMID(L), and ALMIN(L) represent the radii of the longitudinal dispersivity ellipsoid in the maximum, middle, and minimum permeability directions, respectively, for element L. The effective longitudinal dispersivity is the radius of this ellipsoid measured in the direction of flow. Thus, ALMAX(L), ALMID(L), and ALMIN(L) are the effective longitudinal dispersivities for flow in the maximum, middle, and minimum permeability directions, respectively. Note that MAX, MID, and MIN do not refer to the size of the dispersivities, but rather to direction. 232
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Appendix B: (Continued) The dispersivities ATMAX(L), ATMID(L), and ATMIN(L) represent the radii of the transverse dispersivity ellipsoid in the maximum, middle, and minimum permeability directions, respectively, for element L. The two effective transverse dispersivities are the principal radii of the ellipse formed by the intersection of this ellipsoid with the plane that passes through the center of the ellipsoid and is perpendicular to the flow direction. Thus, each of the values ATMAX(L), ATMID(L), and ATMIN(L) is an effective transverse dispersivity for flow within a plane containing two of the three principal permeability directions: For all flow directions within the (MAX,MID)plane, ATMIN(L) is the effective dispersivity that controls transverse dispersion in the MIN direction. For all flow directions within the (MAX,MIN)plane, ATMID(L) is the effective dispersivity that controls transverse dispersion in the MID direction. For all flow directions within the (MID,MIN)plane, ATMAX(L) is the effective dispersivity that controls transverse dispersion in the MAX direction. It follows that when the flow direction coincides with one of the principal permeability directions, the effective transverse dispersivities are those corresponding to the remaining two principal permeability directions: For flow in the MAX permeability direction, the effective transverse dispersivities are ATMID(L) and ATMIN(L). For flow in the MID permeability direction, the effective transverse dispersivities are ATMAX(L) and ATMIN(L). For flow in the MIN permeability direction, the effective transverse dispersivities are ATMAX(L) and ATMID(L). The angles ANGLE1(L), ANGLE2(L), and ANGLE3(L) may be thought of, in aeronautical terms, as the yaw, pitch, and roll of the permeability ellipsoid for element L with respect to the x, y, and zcoordinate axes. That is, if the maximum, middle, and minimum permeability axes of this ellipsoid are initially aligned with the x, 233
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Appendix B: (Continued) y, and zaxes, the ellipsoid is oriented by rotating it by ANGLE1(L) within the x,yplane, by ANGLE2(L) upward out of the x,yplane, and by ANGLE3(L) about the maximum permeability axis, in that order. Note that this is equivalent to rotating the ellipsoid by ANGLE3(L) about the xaxis, by ANGLE2(L) about the yaxis, and by ANGLE1(L) about the zaxis, in that order. In 3D simulations, ANGLE3(L) is arbitrary if the permeability and dispersion tensors are isotropic within the (MID,MIN)plane, that is, if, after the application of scale factors, PMIN(L)=PMID(L), ALMIN(L)=ALMID(L), and ATMIN(L)=ATMID(L). All three angles, ANGLE1(L), ANGLE2(L), and ANGLE3(L), are arbitrary if the permeability and dispersion tensors are completely isotropic, that is, if, after the application of scale factors, PMIN(L)=PMID(L)=PMAX(L), ALMIN(L)=ALMID(L)=ALMAX(L), and ATMIN(L)=ATMID(L)=ATMAX(L). In 2D simulations, ANGLE1(L) is arbitrary if the permeability and dispersion tensors are isotropic, that is, if, after application of scale factors, PMIN(L)=PMAX(L), ALMIN(L)=ALMAX(L), and ATMIN(L)=ATMAX(L). 234
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Appendix B: (Continued) DATASET 15C: Dispersivity Multipliers (up to NSPE plus one line). Multipliers are applied to element dispersivity values specified in DATASET 15B for each species in a multispecies transport simulation. Only species that differ from the values in DATASET 15B need to be specified. O M I T when only one transport species is being simulated (NSPE = 1). Variable Type Description Up to a Maximum of NSPE Lines: NSN Integer Species Number. ATSPMULT Real Dispersivity multiplier for species NSN. LMAX(NSN) = ALMAX ATSPMULT(NSN) LMID(NSN) = ALMID ATSPMULT(NSN) LMIN(NSN) = ALMIN ATSPMULT(NSN) TMAX(NSN) = ATMAX ATSPMULT(NSN) TMID(NSN) = ATMID ATSPMULT(NSN) TMIN(NSN) = ATMIN ATSPMULT(NSN) Last line: Integer Placed immediately following all DATASET 15C lines. Line must begin with the integer 0. Examples: For a data set to simulate two transport species, one may write the following: ## ## DATASET 15C: DISPERSIVITY MULTIPLIERS ## [NSN] [ATSPMULT] 1 0.1 2 1.0 0 235
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Appendix B: (Continued) or ## ## DATASET 15C: DISPERSIVITY MULTIPLIERS ## [NSN] [ATSPMULT] 1 0.1 0 236
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Appendix B: (Continued) DATASET 15D: Solid grain diffusivities (thermal conductivity) (up to NE plus one line). Multipliers applied to element dispersivity values specified in DATASET 15B for each species in a multispecies transport simulation. Only elements that differ from the  SIGMAS  value in DATASET 10 need to be specified. O M I T when energy transport is not being simulated and/or the SIGMAS value specified in DATASET 10 is greater than or equal to zero. Variable Type Description THERMEQ Character Bulk thermal conductivity equation to use for energy transport or multispecies transport and energy transport simulations (eq. 2.16 or 2.17). O P T I O N A L parameter. If specified must be AVERAGE or GEOMETRIC. If THERMEQ is not specified then the volumetric mean thermal conductivity equation is used (eq. 2.16). Up to a Maximum of NE Lines: L Integer Element Number LAMBDAS Real Solid grain diffusivity, s. For energy transport represents thermal conductivity of a solid grain. [E/(LCs)] Last line: Integer Placed immediately following all DATASET 15D lines. Line must begin with the integer 0. 237
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Appendix B: (Continued) Examples: For a data set to simulate heat transport using a geometric mean bulk thermal conductivity, one may write the following: ## ## DATASET 15D: SOLID GRAIN DIFFUSIVITIES ## THERMEQ GEOMETRIC ## [L] [LAMBDAS] 1 4.77 2 4.77 . 10 5.22 11 5.22 0 238
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Appendix B: (Continued) DATASET 16: NO LONGER USED 239
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Appendix B: (Continued) DATASET 17: Data for Fluid Source and Sinks (one line for each of NSOP fluid source nodes as specified in DATASET 3, plus one line) O M I T when there are no fluid source nodes Variable Type Description IQCP Integer Number of node to which source/sink data on this line refers. Specifying the node number with a negative sign indicates to SUTRA that the source flow rate or concentration or temperature of the source fluid vary in a specified manner with time. Information regarding a timedependent source node must be programmed by the user in Subroutine BCTIME. QINC Real Fluid source (or sink) which is a specified constant value at node IQCP, QIN. M/s A positive value is a source of fluid to the aquifer. May be omitted if this value is specified as timedependent in Subroutine BCTIME (IQCP<0). Sources are allocated by cell as shown in Figure 56 for equalsized elements. For unequalsized elements, sources are allocated in proportion to the cell length, area, or volume over which the source fluid enters the system. UINC Real Temperature or solute concentration (mass fraction) of fluid entering the aquifer which is a specified constant value for a fluid source at node IQCP, UIN. C or Ms/M Enter a UINC value for each species for a multispecies transport simulation. UINC values must be entered in order from 1 to NSPE as specified in DATASET 3B. May be omitted if this value is specified as timedependent in Subroutine BCTIME (IQCP<0) or if QINC0. 240
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Appendix B: (Continued) Last line: Integer Placed immediately following all NSOP fluid source node lines. Line must begin with the integer 0. Example: For a data set to simulate two transport species, one may write the following: ## ## DATASET 17: Data for Fluid Source and Sinks ## [IQCP] [QINC] [UINC1] [UINC2] 1 8.25D+02 3.875E+01 0.00D+00 22 1.50D+03 3.875E+01 0.00D+00 . 673 8.25D+02 5.000E+01 0.000D+00 0 or for a data set to simulate a single transport species, one may write the following: ## ## DATASET 17: Data for Fluid Source and Sinks ## [IQCP] [QINC] [UINC1] 1 8.25D+02 3.875E+01 22 1.50D+03 3.875E+01 . 673 8.25D+02 5.000E+01 0 241
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Appendix B: (Continued) Figure 56 Allocation of sources and boundary fluxes in equalsized elements (Fig. B.1 Voss and Provost, 2002). The top four panels pertain to 2D areal and 3D meshes. The bottom four panels pertain to 2D crosssectional meshes. 242
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Appendix B: (Continued) DATASET 18: Data for Energy or Solute Mass Sources and Sinks. For multispecies simulations, source/sink data for each species can be specified in any order as long as the total number of nodes for each species is equal to the NSOU values specified for each species in DATASET 3B. (one line for each NSOU energy or solute source nodes as specified in DATASET 3B, plus one line) O M I T when there are no energy or solute source nodes Variable Type Description Lines 1 to NSOU): NSN Integer Species Number. O M I T for single species transport simulations. IQCU Integer Number of node to which source/sink data on this line refers. Specifying the node number with a negative sign indicates to SUTRA that the source rate varies in a specified manner with time. All information regarding a timedependent source node must be programmed by the user in Subroutine BCTIME. Sources are allocated by cell as shown in figure 6.3.2 for equalsized elements. For unequalsized elements, sources are allocated in proportion to the cell length, area, or volume over which the source energy or solute mass enters the system. QUINC Real Source (or sink) which is a specified constant value at node IQCU, IN. E/s for energy transport, Ms/s for solute transport. A positive value is a source to the aquifer. May be omitted if IQCU is negative, and this value is specified as timedependent in Subroutine BCTIME. 243
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Appendix B: (Continued) Last line: Integer Placed immediately following all NSOU energy or solute mass source node lines. Line must begin with the integer 0. Examples: For a data set to simulate two transport species, one may write the following: ## ## DATASET 18: Data for Energy or Solute Mass Sources and Sinks ## [SP] [IQCU] [QUINC] 2 650 1.0D01 1 671 2.0D01 0 or for a data set to simulate one transport species, one may write the following: ## ## DATASET 18: Data for Energy or Solute Mass Sources and Sinks ## [IQCU] [QUINC] 650 1.0D01 671 1.0D01 0 244
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Appendix B: (Continued) DATASET 19: Data for Specified Pressure Nodes (one line for each of NPBC specified pressure nodes as indicated in DATASET 3B, plus one line) O M I T when there are no specified pressure nodes Variable Type Description Lines 1 to NPBC: IPBC Integer Number of node to which specified pressure data on this line refers. Specifying the node number with a negative sign indicates to SUTRA that the specified pressure value or inflow concentration or temperature at this node vary in a specified manner with time. Information regarding a timedependent specified pressure node must be programmed by the user in Subroutine BCTIME. PBC Real Pressure value which is a specified constant at node IPBC. M/(Ls2) May be omitted if this value is specified as timedependent in Subroutine BCTIME. UBC Real Temperature or solute concentration of any external fluid which enters the aquifer at node IPBC. UBC is a specified constant value. [C] or Ms/M Enter a UBC value for each species for a multispecies transport simulation. UBC values must be entered in order from 1 to NSPE as specified in DATASET 3B. May be omitted if this value is specified as timedependent in Subroutine BCTIME. Last line: Integer Placed immediately following all NPBC specified pressure lines. Line must begin with the integer 0. 245
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Appendix B: (Continued) Example: For a data set to simulate two transport species, one may write the following: ## ## DATASET 19: Data for Specified Pressure Nodes ## [IPBC] [PBC] [UBC1] [UBC2] 1 0.00D+00 5.00E+01 3.57D02 22 0.00D+00 5.00E+01 3.57D02 . 673 1.00D+06 5.00E+01 3.57D02 0 or for a data set to simulate a single transport species, one may write the following: ## ## DATASET 19: Data for Specified Pressure Nodes ## [IPBC] [PBC] [UBC] 1 0.00D+00 5.00E+01 22 0.00D+00 5.00E+01 . 673 1.00D+06 5.00E+01 0 246
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Appendix B: (Continued) DATASET 20: Data for Specified Concentration or Temperature Nodes. For multispecies simulations, source/sink data for each species can be specified in any order as long as the total number of nodes for each species is equal to the NUBC values specified for each species in DATASET 3B. (one line for each NUBC specified concentration or temperature nodes indicated in DATASET 3B, plus one line) O M I T when there are no specified concentration or temperature nodes Variable Type Description Lines 1 to NUBC): NSN Integer Species Number. O M I T for single species transport simulations. IUBC Integer Number of node to which specified concentration or temperature data on this line refers. Specifying the node number with a negative sign indicates to SUTRA that the specified value at this node varies in a specified manner with time. This timedependent concentration or temperature must be programmed by the user in Subroutine BCTIME. UBC Real Temperature or solute concentration value which is a specified constant at node IUBC. C or [Ms/M] May be omitted if IUBC is negative and this value is specified as timedependent in Subroutine BCTIME. Last line: Integer Placed immediately following all NUBC specified temperature or concentration lines. Line must begin with the integer 0. 247
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Appendix B: (Continued) Examples: For a data set to simulate two transport species, one may write the following: ## ## DATASET 20: Data for Specified Concentration or Temperature Nodes ## [NSN] [IUBC] [UBC] 1 631 5.000E+01 2 631 0.357E02 . 1 692 7.250E+00 2 692 0.000E+00 0 or for a data set to simulate one transport species, one may write the following: ## ## DATASET 20: Data for Specified Concentration or Temperature Nodes ## [IUBC] [UBC] 631 5.000E+01 . 692 7.250E+00 0 248
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Appendix B: (Continued) DATASET 21: NO LONGER USED 249
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Appendix B: (Continued) DATASET 22: Element Incidence Data (one line, plus one line for each of NE elements) Variable Type Description Line 1: Character Line must begin with the word INCIDENCE. Lines 2 to NE+1: LL Integer Number of element to which data on this line refers. IIN Integer Node incidence list; list of corner node numbers in element LL, beginning at any node. For 2D problems, the four nodes are listed in an order counterclockwise about the element. For 3D problems, the eight nodes are listed as follows: Approach the element from any of its six sides. On the face farthest from you (the back face), list the four nodes in an order counterclockwise about the face. Then, on the face closest to you (the front face), again list the four nodes counterclockwise, starting with the node directly in front of the node that was listed first. (This convention assumes a righthanded coordinate system.) ________________________________________________________________________ End of Input Data List for .inp File ________________________________________________________________________ 250
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Appendix B: (Continued) List of Input Data for .ics (Initial Conditions) File Model Series: SUTRAMS Model Version: 2D3DMS.1 Data in the .ics input file need be created by the user only for cold starts of SUTRA simulation (i.e., for the very first time step of a given simulation or series of restarted simulations). SUTRA will optionally store final results of a simulation in a .rst file, which is directly useable as a .ics file for later restarts. The restart options are controlled by CREAD and ISTORE in DATASET 4 of the .inp file. 251
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Appendix B: (Continued) DATASET 1: Simulation Starting Time (one line) Variable Type Description TSTART Real Elapsed time at which the initial conditions for simulation specified in the .ics file are given. [s] This sets the simulation clock starting time. Usually set to a value of zero for a cold start. 252
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Appendix B: (Continued) DATASET 2: Initial Pressure Values at Nodes (two lines; second line can be broken up over multiple lines) Variable Type Description Line 1: CPUNI Character One word. Set to UNIFORM to specify a uniform pressure for all nodes. Set to NONUNIFORM to specify a separate pressure for each node. Line 2: PVEC Real For UNIFORM pressure specification, a single value of initial (starting) pressure to be applied at all NN nodes at time TSTART. [M/(Ls2)] For NONUNIFORM pressure specification, a list of values of initial (starting) pressures at time TSTART, one value for each of NN nodes, in exact order of node numbers. [M/(Ls2)] If the STEADY (steadystate) flow option in DATASET 4 of the .inp file has been chosen, PVEC serves as an initial guess for the pressure solution when an ITERATIVE solver is used, and is ignored when the DIRECT solver is used. Initial hydrostatic or natural pressures in a crosssection may be obtained by running a single steadyflow time step with the store option. Then the natural pressures are calculated and stored in the .rst file, and may be copied to the cold start .ics file without change in format, to be used as initial conditions for a transient run. 253
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Appendix B: (Continued) DATASET 3: Initial Temperature or Concentration Values at Nodes. For multispecies simulations, DATASET 3 must be repeated NSPE times. (two lines; second line can be broken up over multiple lines) Variable Type Description Line 1: CUUNI Character One word. Set to UNIFORM to specify a uniform temperature to solute concentration for all nodes. Set to NONUNIFORM to specify a separate value for each node. CTSPE Character One word. Set to SPECIES NSN to specify the temperature or solute concentration on Line 2 is for species NSN, where NSN varies from 1 to NSPE. O M I T if a single species transport simulation data set (NSPE=1). Line 2: UVEC Real For UNIFORM temperature or solute concentration specification, a single initial (starting) value to be applied at all NN nodes at time TSTART. [C] or [Ms/M] For NONUNIFORM temperature or solute concentration specification, a list of initial (starting) values at time TSTART, one value for each of NN nodes, in exact order of node numbers. [C] or [Ms/M] 254
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Appendix B: (Continued) Examples: For a data set to simulate two transport species, one may write the following: ## ## DATASET 3: Initial Temp or Conc at Nodes [SPECIES 1] 'UNIFORM' 'SPECIES 1' 3.875D+01 ## ## DATASET 3: Initial Temp or Conc at Nodes [SPECIES 2] 'UNIFORM' 'SPECIES 2' 0.0357D+00 or for a data set to simulate one transport species one may write the following: ## ## DATASET 3: Initial Temp or Conc at Nodes 'UNIFORM' 3.875D+01 ________________________________________________________________________ End of Input Data List for .ics File ________________________________________________________________________ 255
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Appendix B: (Continued) List of Input Data for .tbc (Simple Transient Boundary Conditions) File Model Series: SUTRAMS Model Version: 2D3DMS.1 Data in the .tbc input file need be created by the user only if the simple transient boundary conditions option is required and TBC is specified in the File Assignment Input File (SUTRA.FIL). This package is compatible with userprogrammed boundary conditions in BCTIME. If a simple boundary condition is specified for a node with a userprogrammed boundary condition (specified with a negative node number in the main SUTRAMS input file) or does not correspond to a node specified in DATASETS 17, 18, 19, or 20 contained in the main SUTRAMS data set (.inp), the item is ignored. Repeat Data Set TBC1, TBC 2, TBC 3A, TBC 3B, TBC 3C, and TBC 3D as many times as required. DATASET TBC1: Starting Time of Next Simple Transient Boundary Conditions (one line) Variable Type Description DNTIME Real Starting time of New Transient Boundary Conditions. Starting time is the elapsed time from the start of the simulation. [s] 256
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Appendix B: (Continued) DATASET TBC2: Transient Boundary Condition Control Parameters (one line) Variable Type Description NTPBC Integer Number of transient specified pressure nodes. NTUBC Integer Number of transient specified concentration or temperature nodes. NTSOP Integer Number of transient specified fluid source or sink nodes. NTSOU Integer Number of transient specified fluid solute or heat source or sink nodes. Note: An error condition will occur if the NTPBC, NTUBC, NTSOP, or NTSOU exceed NPBC, MNUBC, NSOP, or MNSOU specified in DATASET 3A, respectively. 257
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Appendix B: (Continued) DATASET TBC3A: Data for Transient Specified Pressure Nodes (one line for each of NTPBC specified pressure nodes as indicated in DATASET 2, plus one line) O M I T when there are no transient specified pressure nodes Variable Type Description Lines 1 to NTPBC: ITPBC Integer Number of node to which specified pressure data on this line refers. TPBC Real Pressure value which is a specified constant at node ITPBC. M/(Ls2) TUBC Real Temperature or solute concentration of any external fluid which enters the aquifer at node ITPBC. TUBC is a specified constant value. [C] or Ms/M Enter a TUBC value for each species for a multispecies transport simulation. TUBC values must be entered in order from 1 to NSPE as specified in DATASET 3B. Last line: Integer Placed immediately following all NTPBC specified pressure lines. Line must begin with the integer 0. 258
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Appendix B: (Continued) DATASET TBC3B: Data for Transient Specified Concentration or Temperature Nodes. For multispecies simulations, source/sink data for each species can be specified in any order as long as the total number of nodes for each species is equal to the NTUBC values specified for each species in DATASET 2. (one line for each of NTUBC specified concentration or temperature nodes indicated in DATASET 2, plus one line) O M I T when there are no transient specified concentration or temperature nodes Variable Type Description Lines 1 to NTUBC): NSN Integer Species Number. O M I T for single species transport simulations. ITUBC Integer Number of node to which specified concentration or temperature data on this line refers. TUBC Real Temperature or solute concentration value which is a specified constant at node ITUBC. C or [Ms/M] Last line: Integer Placed immediately following all NTUBC specified temperature or concentration lines. Line must begin with the integer 0. 259
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Appendix B: (Continued) DATASET TBC3C: Data for Transient Fluid Source and Sinks (one line for each of NTSOP fluid source nodes as specified in DATASET 2, plus one line) O M I T when there are no transient fluid source nodes Variable Type Description ITQCP Integer Number of node to which source/sink data on this line refers. TQINC Real Fluid source (or sink) which is a specified constant value at node ITQCP, QIN. M/s Sources are allocated by cell as shown in figure 6.3.2 for equalsized elements. For unequalsized elements, sources are allocated in proportion to the cell length, area, or volume over which the source fluid enters the system. TUINC Real Temperature or solute concentration (mass fraction) of fluid entering the aquifer which is a specified constant value for a fluid source at node ITQCP, UIN. C or Ms/M Enter a UINC value for each species for a multispecies transport simulation. UINC values must be entered in order from 1 to NSPE as specified in DATASET 3B. Last line: Integer Placed immediately following all NTSOP fluid source node lines. Line must begin with the integer 0. 260
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Appendix B: (Continued) DATASET TBC3D: Data for Transient Energy or Solute Mass Sources and Sinks. For multispecies simulations, source/sink data for each species can be specified in any order as long as the total number of nodes for each species is equal to the NTSOU values specified for each species in DATASET 2. (one line for each NTSOU energy or solute source nodes as specified in DATASET 2, plus one line) O M I T when there are no transient energy or solute source nodes Variable Type Description Lines 1 to NTSOU): NSN Integer Species Number. O M I T for single species transport simulations. ITQCU Integer Number of node to which source/sink data on this line refers. Sources are allocated by cell as shown in figure 6.3.2 for equalsized elements. For unequalsized elements, sources are allocated in proportion to the cell length, area, or volume over which the source energy or solute mass enters the system. TQUINC Real Source (or sink) which is a specified constant value at node ITQCU, IN. E/s for energy transport, Ms/s for solute transport. A positive value is a source to the aquifer. May be omitted if ITQCU is negative, and this value is specified as timedependent in Subroutine BCTIME. Last line: Integer Placed immediately following all NTSOU energy or solute mass source node lines. Line must begin with the integer 0. 261
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Appendix B: (Continued) DATASET TBC4: Transient Boundary Condition Termination Control Parameter (one line) Variable Type Description CTERM Character One word. Must be set to END to terminate reading transient boundary condition data. Examples: For a data set to simulate transient boundary conditions that change three times (including initial plus two during the simulation) for a simulation with two transport species, one may write the following: ## ## DATASET TBC1 ## DNTIME 1 18000.0 ## ## DATASET TBC2 ## NTPBC,NTUBC,NTSOP,NTSOU 0 1 0 0 ## ## DATASET TBC3A ## SPECIFIED PRESSURES ## ***NONE*** ## ## DATASET TBC3B ## SPECIFIED CONCENTRATION ##[NSN] [ITUBC] [UTBC] 2 6 10.0 0 ## ## DATASET TBC3C ## SPECIFIED FLUID FLUXES ## ***NONE*** ## ## DATASET TBC3D ## SPECIFIED SOLUTE OR TEMPERATURE FLUXES ## ***NONE*** ## ## DATASET TBC1 ## DNTIME 2 19000.0 ## DATASET TBC2 ## NTPBC,NTUBC,NTSOP,NTSOU 262
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Appendix B: (Continued) 0 2 0 0 ## ## DATASET TBC3A ## SPECIFIED PRESSURES ## ***NONE*** ## ## DATASET TBC3B ## SPECIFIED CONCENTRATION ##[NSN] [ITUBC] [UTBC] 2 6 9.0 1 100 1.0 0 ## ## DATASET TBC3C ## SPECIFIED FLUID FLUXES ## ***NONE*** ## ## DATASET TBC3D ## SPECIFIED SOLUTE OR TEMPERATURE FLUXES ## ***NONE*** ## ## DATASET TBC4 ## DATA SET TERMINATION 'END' ## ________________________________________________________________________ End of Input Data List for .tbc File ________________________________________________________________________ 263
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Appendix B: (Continued) List of Input Data for .otm (Specified User Output Times) File Model Series: SUTRAMS Model Version: 2D3DMS.1 Data in the .otm input file need be created by the user only if the specified user output times option is required and OTM is specified in the File Assignment Input File (SUTRA.FIL). If this optional package is used then output control options (NPRINT, CNADAL, and CELEMNT) specified in DATASET 8A in the main SUTRAMS input file (.inp) are not used for output control for the .lst, .nod, and .ele output files. All other output control items in the main SUTRAMS input file (.inp) are still used. Repeat DATASET OTM1 as many times as required. DATASET OTM1: Print Time for output to the .lst, .nod, and .ele output files (one line) Variable Type Description DPTIME Real Starting time of New Transient Boundary Conditions. Starting time is the elapsed time from the start of the simulation. [s] 264
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Appendix B: (Continued) DATASET OTM2: Specified User Output Times Termination Control Parameter (one line) Variable Type Description CTERM Character One word. Must be set to END to terminate reading transient boundary condition data. Examples: For a data set to output simulation data to the .lst, .nod, and .ele at three specified times, one may write the following: ## ## Dataset OTM1DPTIME 1 150.0 ## ## Dataset OTM1DPTIME 2 18000.0 ## ## Dataset OTM1DPTIME 3 24000.0 ## ## DATASET OTM2 ## DATA SET TERMINATION 'END' ## ________________________________________________________________________ End of Input Data List for .otm File ________________________________________________________________________ 265
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Appendix B: (Continued) List of Input Data for .ats (Simple Automatic TimeStepping Algorithm) File Model Series: SUTRAMS Model Version: 2D3DMS.1 Data in the .ats input file need be created by the user only if the specified user output times option is required and ATS is specified in the File Assignment Input File (SUTRA.FIL). If this optional package is used then the simulation timestep is reduced if userspecified iteration criteria are not satisfied. If the userspecified iteration criteria are not satisfied, the user has the option to rerun the current solution with a reduced timestep length. If the user elects to rerun the current solution with a reduced time step, the user specifies the maximum number of times the current solution is to be rerun if the userspecified iteration criteria are not met before continuing on to the next time step. All other timestep control items specified in DATASET 6 of the main SUTRAMS input file (.inp) are still used to increase the time step. DATASET ATS1: Control Parameters for the ATS option (one line) Variable Type Description imaxtarget Integer Maximum iteration allowed for the current time step before implementing a timestep length reduction. dtmin Real Minimum simulation time step [s]. LContinue Logical Logical parameter used to control rerun of current time step. If true, the current time step is not rerun if imaxtarget is exceeded. 266
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Appendix B: (Continued) If false, the current time step is rerun if imaxtarget is exceeded for either the flow or any transport solution. DATASET ATS2: Control parameters for rerunning time steps (one line) O M I T when LContinue is true Variable Type Description NconvMax Integer Maximum number of times to rerun the current time step. Examples: For a data set to reduce the time step if a userspecified iteration criteria is exceeded and rerun the current timestep, one may write the following: ## ## INPUT DATA FOR AUTOMATIC TIME STEP (ATS) ALGORITHM ## UNIT K12 ## ## Dataset ATS1 ## [imaxtarget] [dtmin] [LContinue] 40 0.1 F ## Dataset ATS2 ## [NConvMax] 10 ## ## END OF ATS DATA ________________________________________________________________________ End of Input Data List for .ats File ________________________________________________________________________ 267
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Appendix B: (Continued) List of Input Data for .sob (Specified Observation Locations) File Model Series: SUTRAMS Model Version: 2D3DMS.1 Data in the .sob input file need be created by the user only if the specified observation locations are specified by spatial coordinates rather than node numbers and SOB is specified in the File Assignment Input File (SUTRA.FIL). If this optional package is used then the specified coordinates (x, y, and zlocation) are specified for each observation location instead of node number (DATASET 8D). DATASET SOB1: Control Parameters for the SOB input file (one line) Variable Type Description NOBS Integer Exact number of nodes at which observations will be made. NOBCYC Integer Printed output of observation node data are produced on timestep multiples of NOBCYC, as well as on first and last time step. 268
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Appendix B: (Continued) DATASET SOB2: Coordinates of observation locations (one line for each NOBS specified concentration or temperature nodes indicated in DATASET SOB2) Variable Type Description X(N) Real Unscaled xcoordinate of observation location N, xn. [L] Y(II) Real Unscaled ycoordinate of observation location N, yn. [L] Z(II) Real For 3D problems, unscaled zcoordinate of observation location N, zn. [L] O M I T for 2D problems. Examples: For a data set to observe simulated results at five (5) specified locations in a 2D problem one may write the following: ## ## Specified Observation Locations (.SOB) file ## Dataset SOB1 ## NOBS NOBCYC 5 1 ## ## Dataset SOB2 ## x y 0.00 0.00 0.50 0.75 0.75 0.50 1.00 0.25 2.00 1.00 ## ## END OF SOB DATA ## ________________________________________________________________________ End of Input Data List for .sob File ________________________________________________________________________ 269
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Appendix B: (Continued) List of Input Data for .zon (Node and Element zone parameters) File Model Series: SUTRAMS Model Version: 2D3DMS.1 Data in the .zon input file need be created by the user only if the user wants to specify nodal and elemental hydraulic parameters by zones and ZON is specified in the File Assignment Input File (SUTRA.FIL). If this optional package is used then the nodal and hydraulic parameters usually specified in DATASETS 14B and 15B in the main SUTRA input file (.inp) are specified in the .zon file. Zones are defined using the NREG and LREG variables for nodes and elements in DATASETS 14B and 15B, respectively. DATASET ZON1: Control Parameters for the ZON option (one line) Variable Type Description NodeZones Integer Number of zones used for hydraulic parameters with nodewise discretization (i.e., porosity, matrix compressibility, and solid matrix density. ElemZones Integer Number of zones used for hydraulic parameters with elementwise discretization (i.e., permeability, permeability angles, dispersivity, and solid matrix thermal conductivity. 270
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Appendix B: (Continued) DATASET ZON2: Hydraulic parameter for the exact number of nodewise zones specified in DATASET ZON1 (NodeZones+1 line) Variable Type Description NodeZones Lines: POR(II) Real Scaled porosity value at node II, i. [1] COMPMA Real Solid matrix compressibility, =(1)1 / p. [M/(Ls2)]1 RHOS Real Density of a solid grain, s. [M/L3] Last line: Integer Placed immediately following all DATASET Z2 lines. Line must begin with the integer 0. 271
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Appendix B: (Continued) DATASET ZON3: Hydraulic parameter for the exact number of elementwise zones specified in DATASET ZON1 (ElemZones+1 line) Variable Type Description ElemZones Lines: PMAX(L) Real Scaled maximum permeability value of element L, kmax(L). [L2] PMID(L) Real Scaled middle permeability value of element L, kmid(L). [L2] Isotropic permeability requires: PMID(L)=PMAX(L). Omit for 2D problems. PMIN(L) Real Scaled minimum permeability value of element L, kmin(L). [L2] Isotropic permeability requires: PMIN(L)=PMAX(L). ANGLE1(L) Real Scaled angle within the x,yplane, measured counterclockwise from +xdirection to maximum permeability direction in element L, L. [] ANGLE2(L) Real Scaled angle measured (after ANGLE1 has been applied) upward from x,yplane to maximum permeability direction in element L, L. [] Omit for 2D problems. ANGLE3(L) Real Scaled angle about the axis of maximum permeability, measured (after ANGLE1 and ANGLE2 have been applied) looking down the positive half of the axis toward the origin, and counterclockwise from the horizontal to the middle permeability direction in element L, L. [] Omit for 2D problems. ALMAX(L) Real Scaled longitudinal flow dispersivity value of element L for flow in the direction of maximum permeability PMAX(L), Lmax(L). [L] 272
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Appendix B: (Continued) ALMID(L) Real Scaled longitudinal flow dispersivity value of element L for flow in the direction of middle permeability PMID(L), Lmid(L). [L] Omit for 2D problems. ALMIN(L) Real Scaled longitudinal dispersivity value of element L for flow in the direction of minimum permeability PMIN(L), Lmin(L). [L] ATMAX(L) Real Scaled first transverse dispersivity value of element L for flow in the direction of maximum permeability PMAX(L), T1max(L). [L] ATMID(L) Real Scaled first transverse dispersivity value of element L for flow in the direction of middle permeability PMID(L), T1mid(L). [L] Omit for 2D problems. ATMIN(L) Real Scaled first transverse dispersivity value of element L for flow in the direction of minimum permeability PMIN(L), T1min(L). [L] LAMBDAS Real Solid grain diffusivity, s. For energy transport represents thermal conductivity of a solid grain. [E/(LCs)] Last line: Integer Placed immediately following all DATASET Z3 lines. Line must begin with the integer 0. Notes: See input instructions for DATASET 15B for a complete description of the hydraulic parameters for DATASET Z3. 273
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Appendix B: (Continued) Examples: For a 2D data set with one (1) nodewise and two (2) hydraulic parameter zones, one may write the following: ## ## Zone file ## Dataset ZON1 ## NodeZones, ElemZones 1 2 ## ## Node data ## Dataset ZON2 ## ## zone, por, compma, rhos 1 0.20 0.0 1. 0 ## ## Element data ## Dataset ZON3 ## ## zone, pmax, pmin,anglex, almax, almin, atmax, atmin, lambdas 1 2.e4 2.e4 0.0 500.0 500.0 100.0 100.0 0.0 2 2.e10 2.e10 0.0 500.0 500.0 100.0 100.0 0.0 0 ________________________________________________________________________ End of Input Data List for .zon File ________________________________________________________________________ 274
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Appendix C: SUTRAMS HeleShaw Numerical Methods Description 275
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Appendix C: (Continued) 276 This appendix contains a modified description of the numerical methods developed in Chapter 2 and referenced in Chapter 3. The modified numerical methods are appropriate for simulation of double diffusive flow under fully saturated conditions. The hybrid weighted residual and integrated finitedifference method applied to the fluid mass balance (1) results in NN relations: NN1,i 1111*11111111NSkniknininnininBCpininjNNjijpinijnijnidtdUCFptAFDFpQpBFtAFi Eq. C1 where ij is the Kronecker delta, NN is the number of nodes in the finiteelement mesh, pi is the pressurebased conductance for the specified pressure source in cell i, terms with the superscript n are at the end of the last time step, terms with the superscript n+1 are at the end of the current time step, and , and are matrices given by the following: 1niAF 1niBF 1niCF 1niDF iiniVAF11 Eq. C1a ikniVUCF1 Eq. C1b
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Appendix C: (Continued) 277 dV BFV j L n ij k1 Eq. C1c dV DFV L n i *1g k Eq. C1d where Vi is the volume of cell i, j is the symmetric basis function in global coordinates for node j, Lk is the intrinsic permeability tensor, and *g is a discretization of (g) that is consistent with the discretization of p. Consistent discretization of (g) relative to the discretization of p is required to eliminate numerical dispersion of sharp concentration fronts in variable density fluids (Voss and Souza, 1987). The hybrid weighted residual and integrated finitedifference method applied to the fluid mass balance (1) results in NN relations for each simulated species: NS1,k NN1,i 1 1 1 1 1 *11 1 1 1 1 *1 1 1 n ik n n ik n ik IN n ik UBC ik U n ik BC n i BC n ik n iwk n jk NN j ijw n i BC n i ik U n ijk n ijk n ij n ikU t AT U UQUQc UcQQ BT DT t AT Eq. C2 where ij is the Kronecker delta; NN is the number of nodes in the finiteelement mesh; NS is the number of simulate species; Uik is the concentrationbased conductance for the specified solute source in cell i; is the concentration of species k in cell i; is the ikUn i BCQ
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Appendix C: (Continued) 278 mass of species k to cell i provided by a specified fluid source, for the previous time step; is the specified concentration of species k in cell i; terms with the superscript n are at the end of the last time step; and terms with the superscript n+1 are at the end of the current time step. , and are matrices given by 1nikBCU 1niAT 1niBT 1niDT 1nINik iiwknikVcAT1 Eq. C2a dVcDTVjwknijk**1v Eq. C2b dVcBTiVjkwkwknijkDI1 Eq. C2c dUccinkwkwwnINiknDI11 Eq. C2d where Vi is the volume of cell i, j is the symmetric basis function in global coordinates for node j, is a elementwise discretization of porosity, *v is a discretization of velocity that is consistent with the discretization of p, n is the unit outward normal vector, and is the external boundary area of the simulated region. Time derivatives are discretized using a backwards finitedifference approximation having the general form
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Appendix C: (Continued) 279 nnniniittdtd 11 Eq. C3 where i can be the pressure or concentration of a simulated species at node i. Integration of terms in the fluid and solute massbalance equations are calculated on an element by element basis after transforming the quadrilateral element to a local coordinate system in which each element is a square in two dimensions or a cube in three dimensions. The numerical Gauss integration uses 2d Gauss points in the local coordinate system where d is the spatial dimension of the problem (2D or 3D). Integration is only performed on the terms in the fluid mass balance and solute massbalance equations that are discretized in an elementwise fashion. The matrix systems for fluid flow (A1) and multispecies transport (A2) are linked by the fluid density (2), fluid viscosity (3), and fluid velocity (4). The densityand viscositycoupling makes doublediffusive problems nonlinear. The degree of nonlinearity is dependent on buoyancy effects. To efficiently solve large two and threedimensional problems, robust iterative solvers, which use efficient matrix storage methods, have been included in the model for solution of the flow and transport equations. Iterative methods that have been included include the generalized minimum residual (GMRES) and ORTHOMIN method with incomplete LU preconditioning.
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Appendix C: (Continued) 280 A sequential solution procedure for fluid flow and transport and an implicit Picard iterative scheme is used. Time extrapolation formulas based on initial values or values from the previous time step are used to extrapolate nodal values of pressure and concentration at the end of the current time step. To eliminate the need to specify a time step that is less than or equal to the minimum value required for convergence of all time steps in a numerical simulation, an automatic time step algorithm is used to reduce timestep lengths when the prescribed number of iterations is exceeded.
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About the Author Joseph Hughes received a Bachelor of Science Degree in Geology from The University of South Florida in 1992 and a Master of Science Degree in Geology from The University of Michigan in 1994. He worked as a consulting hydrogeologist until until he entered the Ph.D. program at the University of South Florida in 1998. While in the Ph.D. program at the University of South Florida, Mr. Hughes continued to be employed as a consulting hydrogeologist conducting numerical groundwater and surface water studies for a variety of state and federal agencies. Mr. Hughes currently works as a numerical modeler for the Danish Hydraulic Institute in Tampa, Florida and is a Professional Geologist in the State of Florida.
