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Calculus Review Partial Derivatives 3 2.5 2 1.5 1 0.5 0 -0.5 S19 -1 S13 19 13 16 X 10 S7 7 4 1 -1.5 Y S1 • Functions of more than one variable • Example: h(x,y) = x4 + y3 + xy 2.5-3 2-2.5 1.5-2 1-1.5 0.5-1 0-0.5 -0.5-0 -1--0.5 -1.5--1 Partial Derivatives • Partial derivative of h with respect to x at a y location y0 • Notation dh/dx|y=y0 • Treat ys as constants • If these constants stand alone, they drop out of the result • If they are in multiplicative terms involving x, they are retained as constants Partial Derivatives 3 2.5 2 1.5 1 0.5 0 -0.5 S19 -1 S13 19 13 16 X 10 S7 7 4 1 -1.5 • Example: • h(x,y) = x4 + y3 + xy • dh/dx|y=y0 = 4x3 + y0 S1 Y 2.5-3 2-2.5 1.5-2 1-1.5 0.5-1 0-0.5 -0.5-0 -1--0.5 -1.5--1 Gradients • del C (or grad C) C C C i j x y • Diffusion (Fick’s 1st Law): J DC Numerical Derivatives • slope between points I P O L A Poisson Equation h q K x h h K x K x yb Rxy 0 x x x h x h R 2 x T 2 h x R x x x x T Analytical Solution to Poisson Equation h x R x T h R x T x •Incorporate flux BCs (including zero flux) here! • h/x|0 = 0; i.e., a no flow groundwater divide h R x c1 x T R h x c T 1 x R 2 h x c1 x c 2 2T Laplace Equation h 0 2 x 2 Poisson Equation h R 2 x T 2 Heat/Diffusion Equation Derivation z Jx|x y x + x x J x x Jx x x C yz xyz t C J D x Heat/Diffusion Equation Derivation C C D D x x x C yz xyz t x x C C D 2 t x 2 Fully explicit FD solution to Heat Equation t-t x +x x -x C/t|t-t/2 Estimate here C|x, t t x Fully explicit FD solution to Heat Equation C x ,t C x ,t t Dt 2 C x x ,t t 2C x ,t t C x x ,t t x • Need IC and BCs No diffusive flux BC • Fick’s law • If ∂C/∂x = 0, there is no flux • Finite difference expression for ∂C/∂x is C J D x C x x x / 2 C x x C x x C x x C x • Setting this to 0 is equivalent to • ‘Ghost’ points outside the domain at x + x • Then, if we make the concentration at the ghost points equal to the concentration inside the domain, there will be no flux • Often the boundary conditions are constant in time, but they need not be Closed Box • Finite system: C0 n h 2nl x h 2nl x C erf erf 2 n 4Dt 4Dt Superposition of original process and reflections C 0 x 0 500 t = 1000 450 400 C (mu lu -2) 350 300 t = 2000 250 200 t =11000 150 100 50 C x 0 -50 -30 -10 10 x (lu) 30 50 0 xmax Basic Fluid Dynamics Viscosity • • • • Resistance to flow; momentum diffusion Low viscosity: Air High viscosity: Honey Kinematic viscosity Reynolds Number • The Reynolds Number (Re) is a non-dimensional number that reflects the balance between viscous and inertial forces and hence relates to flow instability (i.e., the onset of turbulence) • Re = v L/n • L is a characteristic length in the system • Dominance of viscous force leads to laminar flow (low velocity, high viscosity, confined fluid) • Dominance of inertial force leads to turbulent flow (high velocity, low viscosity, unconfined fluid) Re << 1 (Stokes Flow) Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp. Separation Eddies and Cylinder Wakes S.Gokaltun Florida International University Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu) Eddies and Cylinder Wakes Re = 30 Re = 40 Re = 47 Re = 55 Re = 67 Re = 100 Re = 41 Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp. Poiseuille Flow Jean Léonard Marie Poiseuille; 1797 – 1869. From Sutera and Skalak, 1993. Annu. Rev. Fluid Mech. 25:1-19 Poiseuille Flow • In a slit or pipe, the velocities at the walls are 0 (no-slip boundaries) and the velocity reaches its maximum in the middle • The velocity profile in a slit is parabolic and given by: G 2 2 u( x ) (a x ) 2 • G can be gravitational pressure gradient (rg for example in a vertical slit) or (linear) pressure gradient (Pin – Pout)/L u(x) x=0 x=a Dispersion • Mixing induced by velocity variations • No velocity, no dispersion Geoffrey Ingram Taylor; 1886 - 1975. http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Taylor_Geoffrey.html Taylor Dispersion Taylor Dispersion C C 2C v D 2 t x x 1.2 1 C/C0 0.8 0.6 LBM Result Analytical Solution 0.4 0.2 0 0 50000 100000 150000 Time (time steps) 2 2 U W D Dm 210Dm 200000 250000 Taylor/Aris Dispersion Dispersion Coefficient (lu2 ts-1) 0.07 Analytical Solution: 2 2 D = U W /(210 Dm) + Dm 0.06 0.05 0.04 0.03 0.02 Distance of measured breakthrough curve from source: 50 lu 0.01 150 lu 300 lu 0.00 0 5 10 15 20 25 Width (lattice units) Stockman, H.W., A lattice-gas study of retardation and dispersion in fractures: assessment of errors from desorption kinetics and buoyancy, Wat. Resour. Res. 33, 1823 - 1831, 1997. 30 35 40 Diffusion in Poiseuille Flow Pore Volume Breakthrough Curves Influent Solution: Concentration C0 • ‘Piston’ Flow – no dispersion • Dispersed Flow • Retarded/ Dispersed Flow Effluent Solution: Concentration C Breakthrough Curve q = 1 m/y 100 Initial and Boundary Conditions: C (mg/l) 10 m 80 C(x,0) = 0 C(0,0<t<1) = 100 C(0,t>1) = 0 60 General Conditions: q = 1 m/year = 0.5 40 = 0.1 m 20 0 0 2 4 6 Time (years) 8 10 Continuous Source Pulse Source Peclet Number • Inside a pore, the dimensionless Peclet number (Pe ≡ vl/Dm, with l a characteristic length) indicates the relative importance of diffusion and convection; – large values of Pe indicate a convection dominated process – small values of Pe indicate the dominance of diffusion Dimensionless Diffusion-Dispersion Coefficient • The dimensionless diffusion-dispersion coefficient D* ≡ Dd/Dm reflects the relative importance of hydrodynamic dispersion and diffusion • For porous media with well-defined characteristic lengths (i.e., bead diameter in packed beds of uniformly sized glass beads), D* can be estimated from Pe based on empirical data Empirical relationship between dimensionless dispersion coefficient and Peclet number with data for uniformly sized particle beds. Adapted from Fried, JJ and Combarnous MA (1971) Dispersion in porous media. Adv. Hydroscience 7, 169-282. Classes of Behavior • Different classes of behavior proposed based on the observed relationship between Pe and D* – Class I: very slow flow, dominance of diffusion – Class II: transitional with approximately equal and additive hydrodynamic dispersion and diffusion – Class III: hydrodynamic dispersion dominates, but the role of diffusion is still non-negligible, – Class IV: diffusion negligible – Class V: velocity so high that the flow of many fluids is turbulent The process: • Measure grain size l • Look up Dm (10-5 cm2 s-1) – http://www.hbcpnetbase.com/ • Know mean pore water velocity from v = q/n • Compute Pe (= vl/Dm) • Take D* (=Dd/Dm) from graph • Compute Dd = D* Dm Ion Diffusion Coefficients in Water Organic Molecule Diffusion Coefficients in Water Large-scale Dispersion Neuman, 1995 Rule of Thumb: Dispersivity = 0.1 Scale Neuman, 1995 CDE z Jx|x C J x x J x xx yz t xyz * C J d D x y x + x x C C vC x D* vC x x D* x x x •Key difference from diffusion here! • Convective flux C yz xyz t x x C * C D x C v x x t 1st Order Spatial Derivative C x x x / 2 C x x C x x • Worked for estimating second order derivative (estimate ended up at x). • Need centered derivative approximation C xx C xx C x x 2x CDE Explicit Finite Difference C x ,t C x ,t t Dt vt 2 C x x ,t t 2C x ,t t C x x ,t t C x x ,t t C x x ,t t 2x x • Grid Pe = vL/D, where L is the grid spacing • Pe < 1, 4, 10 Isotherms • Linear: Cs = Kd Cw • Freundlich: Cs = Kf Cw1/n • Langmuir: Cs = Keq Cst Cw/(1 + Keq Cw) Koc Values • Kd = Koc foc Organic Carbon Partitioning Coefficients for Nonionizable Organic Compounds. Adapted from USEPA, Soil Screening Guidance: Technical Background Document. http://www.epa.gov/superfund/resources/soil/introtbd.htm Compound Acenaphthene mean Koc (L/kg) 5,028 Compound 1,4-Dichlorobenzene(p) mean Koc (L/kg) 687 Compound Methoxychlor mean Koc (L/kg) 80,000 Aldrin 48,686 1,1-Dichloroethane 54 Methyl bromide 9 Anthracene 24,362 1,2-Dichloroethane 44 Methyl chloride 6 1,1-Dichloroethylene 65 Methylene chloride trans-1,2-Dichloroethylene 38 Naphthalene 1,231 1,166,733 1,2-Dichloropropane 47 Nitrobenzene 141 76 1,3-Dichloropropene 27 Pentachlorobenzene 36,114 25,604 Pyrene 70,808 84 Styrene 912 Benz(a)anthracene Benzene Benzo(a)pyrene Bis(2-chloroethyl)ether Bis(2-ethylhexyl)phthalate Bromoform Butyl benzyl phthalate Carbon tetrachloride Chlordane Chlorobenzene Chloroform 459,882 66 114,337 126 14,055 158 51,798 Dieldrin Diethylphthalate 10 Di-n-butylphthalate 1,580 1,1,2,2-Tetrachloroethane Endosulfan 2,040 Tetrachloroethylene 272 Toluene 145 Endrin 11,422 260 Ethylbenzene 207 57 Fluoranthene 49,433 DDD 45,800 Fluorene DDE 86,405 DDT 792,158 Toxaphene 1,2,4-Trichlorobenzene 79 95,816 1,783 8,906 1,1,1-Trichloroethane 139 Heptachlor 10,070 1,1,2-Trichloroethane 77 Hexachlorobenzene 80,000 Trichloroethylene 97 Dibenz(a,h)anthracene 2,029,435 -HCH (-BHC) 1,835 o-Xylene 241 1,2-Dichlorobenzene(o) 390 b-HCH (b-BHC) 2,241 m-Xylene 204 g-HCH (Lindane) 1,477 p-Xylene 313 Retardation • Incorporate adsorbed solute mass r b Kd R 1 V R Vs Retardation C C C v D 2 R x x t 2 Kinetics • dC/dt = constant: zero order • dC/dt = -kC: first order C ( T ) C • Integrate: C (0) C T kt 0 C (T ) ln kT C (0) 1 ln 2 T 1/ 2 k Two-Site Conceptual Model Instantaneous Adsorption Sites Mobile Water Air Kinetic Adsorption Sites dC s 2 1 F KdC Cs 2 dt Two-site model • Selim et al., 1976; Cameron and Klute, 1977; and many more • Instantaneous equilibrium and kineticallylimited adsorption sites • Different constituents: • “Soil minerals, organic matter, Fe/Al oxides” • ‘F’ = Fraction of instantaneous sites • ‘’ = First-order rate constant Batch Sorption Kinetics 2600 Mean column = 0.06 hr-1 Concentration (dpm/ml) 2400 2200 First Order Model for All Data 2000 1800 1600 1400 First Order Model for t > 1 hour: = 0.11 hr-1 1200 1000 0 5 10 15 Time (hours) 20 25 Two-Region Conceptual Model Dynamic Soil Region Mobile Water Air Immobile Water Stagnant Soil Region dCim im 1 F rb Kd Cm Cim dt STANMOD • CXTFIT Toride et al.[1995] • For estimating solute transport parameters using a nonlinear least-squares parameter optimization method • Inverse problem by fitting a variety of analytical solutions of theoretical transport models, based upon the onedimensional convection-dispersion equation (CDE), to experimental results • Three different one-dimensional transport models are considered: – (i) the conventional equilibrium CDE; – (ii) the chemical and physical nonequilibrium CDEs; and – (iii) a stochastic stream tube model based upon the local-scale equilibrium or nonequilibrium CDE http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM STANMOD • CHAIN van Genuchten [1985] • For analyzing the convective-dispersive transport of solutes involved in sequential firstorder decay reactions. • Examples: – Migration of radionuclides in which the chain members form first-order decay reactions, and – Simultaneous movement of various interacting nitrogen or organic species http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM STANMOD • 3DADE Leij and Bradford [1994] • For evaluating analytical solutions for two- and three-dimensional equilibrium solute transport in the subsurface. • The analytical solutions assume steady unidirectional water flow in porous media having uniform flow and transport properties. • The transport equation contains terms accounting for – – – – solute movement by convection and dispersion, as well as for solute retardation, first-order decay, and zero-order production. • The 3DADE code can be used to solve the direct problem and the indirect (inverse) problem in which the program estimates selected transport parameters by fitting one of the analytical solutions to specified experimental data http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM STANMOD • N3DADE Leij and Toride [1997] • For evaluating analytical solutions of two- and three-dimensional nonequilibrium solute transport in porous media. • The analytical solutions pertain to multi-dimensional solute transport during steady unidirectional water flow in porous media in systems of semi-infinite length in the longitudinal direction, and of infinite length in the transverse direction. • Nonequilibrium solute transfer can occur between two domains in either the liquid phase (physical nonequilibrium) or the absorbed phase (chemical nonequilibrium). • The transport equation contains terms accounting – – – – solute movement by advection and dispersion, solute retardation, first-order decay zero-order production http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM 2- and 3-D Analytical Solutions to CDE Equation Solved: C C C C C v Dxx 2 Dyy 2 Dzz 2 R x x y z t 2 2 2 • Constant mean velocity in x direction only! •Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85. ‘Instantaneous’ Source • Solute mass only – M1, M2, M3 • Injection at origin of coordinate system (a point!) at t = 0 ‘Continuous’ Source • Solute mass flux – M1, M2, M3 = dM1,2,3/dt • Injection at origin of coordinate system (a point!) Instantaneous and Continuous Sources • 1-D 2 M1 x vt Ci exp 2 Dxxt 4 Dxxt xv exp x v erfc x vt M 1 exp 2D 2 D t xx 2 D xx xx Cc 2v xv x vt erfc exp 2 D t 2 D xx xx 2-D Instantaneous Source 2 2 M2 x vt y Ci exp 4 D yyt 4t Dxx Dyy 4 Dxxt 2-D Instantaneous Source Solution Dyy Dxx t = 25 t=1 Back dispersion Extreme concentration t = 51 3-D Instantaneous Source 2 2 2 M3 x vt y z Ci exp 3 3 4 Dyyt 4 Dzz t 8 t Dxx Dyy Dzz 4 Dxxt 3-D Instantaneous Source Solution Dzz Dyy Dxx t=1 t = 25 Back dispersion t = 51 Extreme concentration 3-D Continuous Source xv exp Rv erfc R vt M 3 exp 2D 2 D t xx 2 Dxx xx Cc 8R D yy Dzz R vt Rv exp erfc 2D 2 D t xx xx R Dxx 2 D xx x y z D yy Dzz 2 2 StAnMod (3DADE) • Same equation (mean x velocity only) • Better boundary and initial conditions • Leij, F.J., T.H. Skaggs, and M.Th. Van Genuchten, 1991. Analytical solutions for solute transport in three-dimensional semiinfinite porous media, Water Resources Research 20 (10) 2719-2733. Coordinate systems • x increasing downward z z r y y x x Boundary Conditions • Semi-infinite source z y -∞ -∞ x Boundary Conditions • Finite rectangular source z b -a a -b x y Boundary Conditions • Finite Circular Source z r=a y x Initial Conditions • Finite Cylindrical Source z y r=a x1 x2 x Initial Conditions • Finite Parallelepipedal Source z b y a x1 x2 x x2 x1 x r=a z y Comparing with Hunt • M3 = r2 (x1 – x2) Co (=1, small, high C) • Co = 1/[r2 (x1 – x2)] = 106 for r = x= 0.01 Wells? • Finite Parallelepipedal Source Pathlines Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation Primary Source: Ph.D. Dissertation David Benson University of Nevada Reno, 1998 Representative Elementary Volume (REV) From Jacob Bear Representative Elementary Volume (REV) • General notion for all continuum mechanical problems • Size cut-offs usually arbitrary for natural media (At what scale can we afford to treat medium as deterministically variable?) Conventional Derivatives r dx rx r 1 dx From Benson, 1998 Conventional Derivatives r dx rx r 1 dx From Benson, 1998 Fractional Derivatives The gamma function interpolates the factorial function. For integer n, gamma(n+1) = n! ( x) t e dt 0 x 1 t Fractional Derivatives (u 1) u q D x x (q u 1) q u From Benson, 1998 Standard Symmetric -Stable Probability Densities 0.35 0.30 f(x) 0.25 0.20 = 2 (Normal) = 1.5 0.15 0.10 = 1.8 0.05 0.00 -5 -4 -3 -2 -1 0 x 1 2 3 4 5 Standard Symmetric -Stable Probability Densities 1.0000 f(x) 0.1000 = 1.5 0.0100 = 1.8 0.0010 = 2 (Normal) 0.0001 -5 -4 -3 -2 -1 0 x 1 2 3 4 5 Standard Symmetric -Stable Probability Densities 1 0.1 f(x) 0.01 = 1.2 0.001 = 1.5 0.0001 0.00001 = 1.8 = 2 (Normal) 0.000001 0.0000001 1 10 x 100 Brownian Motion and Levy Flights Pr U u u D Pr U u 1, u 1 ln Pr U u D ln u ue ln PrU u D Monte-Carlo Simulation of Levy Flights Power Law Probability Distribution Uniform Probability Density 1 Pr(U>u) 0.8 x 0.6 0.4 0.2 0 Pr(x) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 D=1.7 D=1.2 0 5 10 u 15 FADE (Levy Flights) MATLAB Movie/ Turbulence Analogy 500 50 100 ‘flights’, 1000 time steps each Scaling and Tailing 1.0 C/C0 0.8 =0.12 Data FADE Fit ADE Fit 0.6 11 cm 17 cm 23 cm 0.4 0.2 0.0 0 20 40 60 80 100 120 140 Time (min) After Pachepsky Y, Benson DA, and Timlin D (2001) Transport of water and solutes in soils as in fractal porous media. In Phys ical and Chemical Processes of Water and Solute Transport/Retention in Soils. D. Sparks and M. Selim. Eds. Soil Sci. Soc. Am. Special Pub. 56, 51-77 with permission. Scaling and Tailing Depth (cm) Dispersion Coefficient 11 CDE 2 (cm /hr) 0.035 FADE 1.6 (cm /hr) 0.030 17 0.038 0.029 23 0.042 0.028 lbm Conclusions • Fractional calculus may be more appropriate for divergence theorem application in solute transport • Levy distributions generalize the normal distribution and may more accurately reflect solute transport processes • FADE appears to provide a superior fit to solute transport data and account for scale-dependence Continuous Time Random Walk Model Primary Sources: Berkowitz, B, G. Kosakowski, G. Margolin, and H. Scher, Application of continuuous time random walk theory to tracer test measurents in fractured and heterogeneous porous media, Ground Water 39, 593 - 604, 2001. Berkowitz, B. and H. Scher, On characterization of anomalous dispersion in porous and fractured media, Wat. Resour. Res. 31, 1461 - 1466, 1995. Mike Sukop/FIU Introduction • Continuous Time Random Walk (CTRW) models – Semiconductors [Scher and Lax, 1973] – Solute transport problems [Berkowitz and Scher, 1995] Introduction • Like FADE, CTRW solute particles move along various paths and encounter spatially varying velocities • The particle spatial transitions (direction and distance given by displacement vector s) in time t represented by a joint probability density function y(s,t) • Estimation of this function is central to application of the CTRW model Introduction • The functional form y(s,t) ~ t-1-b (b > 0) is of particular interest [Berkowitz et al, 2001] b characterizes the nature and magnitude of the dispersive processes Ranges of b b ≥ 2 is reported to be “…equivalent to the ADE…” – For b ≥ 2, the link between the dispersivity ( = D/v) in the ADE and CTRW dimensionless bb is bb = /L b between 1 and 2 reflects moderate nonFickian behavior • 0 < b < 1 indicates strong ‘anomalous’ behavior Fits 1.0 Data CTRW Fit ADE Fit C/C0 0.8 0.6 11 cm 17 cm 23 cm 0.4 0.2 0.0 0 20 40 60 80 100 Time (min) 120 140 Gas Phase Transport Principal Sources: VLEACH, A One-Dimensional Finite Difference Vadose Zone Leaching Model, Version 2.2 – 1997. United States Environmental Protection Agency, Office of Research and Development, National Risk Management Research Laboratory, Subsurface Protection and Remediation Division, Ada, Oklahoma. Šimůnek, J., M. Šejna, and M.T. van Genuchten. 1998. The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variablysaturated media. Version 2.0, IGWMC - TPS - 70, International Ground Water Modeling Center, Colorado School of Mines, Golden, Colorado, 202pp., 1998. Effective Diffusion • Tortuosity (T = Lpath/L) and percolation (2D) Macroscopic Gas Diffusion x D J C dC J D dx 2 D a 4/3 a D0 0.5 Maxwell (1873) Buckingham (1904) 0.4 Penman (1940) Marshall (1959) 0.3 D/Do Millington (1959) Wesseling (1962) 0.2 Currie (1965) WLR(Marshall): Moldrup et al (2000) 0.1 0 0 0.1 0.2 0.3 Volumetric Air Content 0.4 0.5 Total Mass • At Equilibrium: Henry’s Law Cg KH Cw • Dimensionless: • Common: atm m3 mol-1 Total Mass • At Equilibrium: MT ( z, t ) Cl ( )KH Cl rb Kd Cl VLEACH • Processes are conceptualized as occurring in a number of distinct, userdefined polygons that are vertically divided into a series of userdefined cells Voronoi Polygons/ Diagram • Voronoi_polygons – close('all') – clear('all') – axis equal – x = rand(1,100); y = rand(1,100); – voronoi(x,y) Chemical Parameters • Organic Carbon Partition Coefficient (Koc) = 100 ml/g • Henry’s Law Constant (KH) = 0.4 (Dimensionless) • Free Air Diffusion Coefficient (Dair) = 0.7 m2/day • Aqueous Solubility Limit (Csol) = 1100 mg/l Soil Parameters • • • • Bulk Density (rb) = 1.6 g/ml Porosity (f) = 0.4 Volumetric Water Content (q) = 0.3 Fraction Organic Carbon Content (foc) = 0.005 Environmental Parameters • Recharge Rate (q) = 1 ft/yr • Concentration of TCE in Recharge Water = 0 mg/l • Concentration of TCE in Atmospheric Air = 0 mg/l • Concentration of TCE at the Water Table = 0 mg/l Dispersion! • Dispersivity is implicit in the cell size (l) and equal to l/2 (Bear 1972) • Numerical dispersion but can be used appropriately Dispersion 100 VLEACH 0.1 m cells Initial and Boundary Conditions: VLEACH 1 m cells C(x,0) = 100 mg/l C(0,t) = 0 mg/l = 0.05 m = 0.5 m C (mg/l) 80 General Conditions: =5m 60 VLEACH 10 m cell q = 1 m/year = 0.5 CDE Flux-averaged concentrations (Dispersivity as shown) 40 VLEACH time step: 0.01 years 20 0 0 5 10 Time (years) 15 20 M.C. Sukop. 2001. Dispersion in VLEACH and similar models. Ground Water 39, No. 6, 953-954. Hydrus Hydrus • Solves – Richards’ Equation – Fickian solute transport – Sequential first order decay reactions Governing Equation w ,k 1, g ,k 1, and s,k 1 Provide linkage with preceding members of the chain Density-Dependent Flows Primary source: User’s Guide to SEAWAT: A Computer Program for Simulation of Three-Dimensional Variable-Density GroundWater Flow By Weixing Guo and Christian D. Langevin U.S. Geological Survey Techniques of Water-Resources Investigations 6-A7, Tallahassee, Florida2002 Sources of density variation • Solute concentration • Pressure • Temperature Freshwater Head • SEAWAT is based on the concept of equivalent freshwater head in a saline ground-water environment • Piezometer A contains freshwater • Piezometer B contains water identical to that present in the saline aquifer • The height of the water level in piezometer A is the freshwater head Converting between: Mass Balance • (with sink term) Density (and soon T!) Densities • • • • Freshwater: 1000 kg m-3 Seawater: 1025 kg m-3 Freshwater: 0 mg L-1 Seawater: 35,000 mg L-1 dr 1025 1000kg m 3 dC 35 kg m 3 0.714