1. Calculus Review

2. Partial Derivatives Functions of more than one variable Example: h(x,y) = x 4 + y 3 + xy

3. Partial Derivatives Partial derivative of h with respect to x at a y location y 0 Notation  h/  x| 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

4. Partial Derivatives Example: h(x,y) = x 4 + y 3 + xy  h/  x| y=y 0 = 4x 3 + y 0

5. Gradients del C (or grad C) Diffusion (Fick’s 1 st Law):

6. Numerical Derivatives slope between points

8. Poisson Equation

9. Analytical Solution to Poisson Equation Incorporate flux BCs (including zero flux) here!  h/  x| 0 = 0; i.e., a no flow groundwater divide

10. Laplace Equation

11. Poisson Equation

12. Heat/Diffusion Equation Derivation x +  x yy zz x Jx|xJx|x

13. Heat/Diffusion Equation Derivation

14. Fully explicit FD solution to Heat Equation C| x, t x x +  x  C/  t| t-  t/2 Estimate here t-  t t x -  x

15. Fully explicit FD solution to Heat Equation Need IC and BCs

16. No diffusive flux BC Fick’s law If ∂C/∂x = 0, there is no flux Finite difference expression for ∂C/∂x is 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

17. Closed Box Finite system: Standard ‘Bounce-back’ from solids boundary works for diffusion Superposition of original process and reflections

18. Basic Fluid Dynamics

19. Viscosity Resistance to flow; momentum diffusion Low viscosity: Air High viscosity: Honey Kinematic viscosity

20. 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/ 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)

21. Re << 1 (Stokes Flow) Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.

22. Separation

23. 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)

24. Eddies and Cylinder Wakes Re = 30 Re = 40 Re = 47 Re = 55 Re = 67 Re = 100 Re = 41 Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.

25. Poiseuille Flow Jean Léonard Marie Poiseuille; 1797 – 1869. From Sutera and Skalak, 1993. Annu. Rev. Fluid Mech. 25:1-19

26. 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: x = 0x = a u(x) G can be gravitational pressure gradient (  g for example in a vertical slit) or (linear) pressure gradient (P in – P out )/L

27. Dispersion Mixing induced by velocity variations No velocity, no dispersion

28. Taylor Dispersion Geoffrey Ingram Taylor; 1886 - 1975. http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Taylor_Geoffrey.html

29. Taylor Dispersion

30. Taylor/Aris Dispersion 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.

31. Diffusion in Poiseuille Flow

33. Pore Volume

34. Breakthrough Curves ‘Piston’ Flow – no dispersion‘Piston’ Flow – no dispersion Dispersed FlowDispersed Flow Retarded/ Dispersed FlowRetarded/ Dispersed Flow Influent Solution: Concentration C 0 Effluent Solution: Concentration C

35. Breakthrough Curve 10 m q = 1 m/y

36. Continuous Source

37. Pulse Source

38. Peclet Number Inside a pore, the dimensionless Peclet number (Pe ≡ vl/D m, 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

39. Dimensionless Diffusion-Dispersion Coefficient The dimensionless diffusion-dispersion coefficient D* ≡ D d /D m 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

40. 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.

41. 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

42. The process: Measure grain size l Look up D m (10 -5 cm 2 s -1 ) –http://www.hbcpnetbase.com/http://www.hbcpnetbase.com/ Know mean pore water velocity from v = q/n Compute Pe (= vl/D m ) Take D* (=D d /D m ) from graph Compute D d = D* D m

43. Ion Diffusion Coefficients in Water

44. Organic Molecule Diffusion Coefficients in Water

45. Large-scale Dispersion

46. Neuman, 1995

47. Rule of Thumb: Dispersivity = 0.1 Scale

48. CDE x +  x yy zz x Jx|xJx|x Key difference from diffusion here! Convective flux

49. 1 st Order Spatial Derivative Worked for estimating second order derivative (estimate ended up at x). Need centered derivative approximation

50. CDE Explicit Finite Difference Grid Pe = vL/D, where L is the grid spacing Pe < 1, 4, 10

51. Isotherms Linear: Cs = Kd Cw Freundlich: Cs = Kf Cw 1/n Langmuir: Cs = Keq Cst Cw/(1 + Keq Cw)

52. Koc Values Kd = Koc foc

53. 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 http://www.epa.gov/superfund/resources/soil/introtbd.htm Compoundmean Koc (L/kg)Compoundmean Koc (L/kg)Compoundmean Koc (L/kg) Acenaphthene5,0281,4-Dichlorobenzene(p)687Methoxychlor80,000 Aldrin48,6861,1-Dichloroethane54Methyl bromide9 Anthracene24,3621,2-Dichloroethane44Methyl chloride6 Benz(a)anthracene459,8821,1-Dichloroethylene65Methylene chloride10 Benzene66trans-1,2-Dichloroethylene38Naphthalene1,231 Benzo(a)pyrene1,166,7331,2-Dichloropropane47Nitrobenzene141 Bis(2-chloroethyl)ether761,3-Dichloropropene27Pentachlorobenzene36,114 Bis(2-ethylhexyl)phthalate114,337Dieldrin25,604Pyrene70,808 Bromoform126Diethylphthalate84Styrene912 Butyl benzyl phthalate14,055Di-n-butylphthalate1,5801,1,2,2-Tetrachloroethane79 Carbon tetrachloride158Endosulfan2,040Tetrachloroethylene272 Chlordane51,798Endrin11,422Toluene145 Chlorobenzene260Ethylbenzene207Toxaphene95,816 Chloroform57Fluoranthene49,4331,2,4-Trichlorobenzene1,783 DDD45,800Fluorene8,9061,1,1-Trichloroethane139 DDE86,405Heptachlor10,0701,1,2-Trichloroethane77 DDT792,158Hexachlorobenzene80,000Trichloroethylene97 Dibenz(a,h)anthracene2,029,435  -HCH (  -BHC) 1,835o-Xylene241 1,2-Dichlorobenzene(o)390  -HCH (  -BHC) 2,241m-Xylene204  -HCH (Lindane) 1,477p-Xylene313

54. Retardation Incorporate adsorbed solute mass

55. Retardation

56. Kinetics dC/dt = constant: zero order dC/dt = -kC: first order Integrate:

57. Two-Site Conceptual Model Instantaneous Adsorption Sites Mobile Water Air Kinetic Adsorption Sites 

58. Two-site model Selim et al., 1976; Cameron and Klute, 1977; and many more Instantaneous equilibrium and kinetically- limited adsorption sites Different constituents: “Soil minerals, organic matter, Fe/Al oxides” ‘F’ = Fraction of instantaneous sites ‘  ’ = First-order rate constant

59. Batch Sorption Kinetics First Order Model for All Data First Order Model for t > 1 hour:  = 0.11 hr -1 Mean column  = 0.06 hr -1

60. Two-Region Conceptual Model Dynamic Soil Region Mobile Water Air Immobile Water Stagnant Soil Region 

63. 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 one- dimensional 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

64. STANMOD CHAIN van Genuchten [1985] For analyzing the convective-dispersive transport of solutes involved in sequential first- order 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

65. 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

66. 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

67. 2- and 3-D Analytical Solutions to CDE

68. Equation Solved: Constant mean velocity in x direction only!

69. Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85.

70. ‘Instantaneous’ Source Solute mass only –M1, M2, M3 Injection at origin of coordinate system (a point!) at t = 0

71. ‘Continuous’ Source Solute mass flux –M 1, M 2, M 3 = dM 1,2,3 /dt Injection at origin of coordinate system (a point!)

72. Instantaneous and Continuous Sources 1-D

73. 2-D Instantaneous Source

74. 2-D Instantaneous Source Solution D yy D xx Back dispersion Extreme concentration t = 1 t = 25 t = 51

75. 3-D Instantaneous Source

76. 3-D Instantaneous Source Solution D zz D xx Back dispersion Extreme concentration t = 1 t = 25 t = 51 D yy

78. 3-D Continuous Source

79. 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 semi- infinite porous media, Water Resources Research 20 (10) 2719-2733.

80. Coordinate systems x increasing downward x z y x z y r

81. Boundary Conditions Semi-infinite source x z y -∞-∞ -∞-∞

82. Boundary Conditions Finite rectangular source x z y -b-b -a-a b a

83. Boundary Conditions Finite Circular Source x z y r = a

84. Initial Conditions Finite Cylindrical Source x z y r = a x1x1 x2x2

85. Initial Conditions Finite Parallelepipedal Source x z y x1x1 x2x2 b a

86. Comparing with Hunt M 3 =  r 2 (x 1 – x 2 ) C o (=1, small, high C) C o = 1/[  r 2 (x 1 – x 2 )] = 10 6  for r =  x= 0.01 x z y r = a x1x1 x2x2

87. Wells? Finite Parallelepipedal Source x z y x1x1 x2x2 b a

88. Pathlines

90. Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation Primary Source: Ph.D. Dissertation David Benson University of Nevada Reno, 1998

91. Representative Elementary Volume (REV) From Jacob Bear

92. 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?)

93. Conventional Derivatives From Benson, 1998

94. Conventional Derivatives From Benson, 1998

95. Fractional Derivatives The gamma function interpolates the factorial function. For integer n, gamma(n+1) = n!

96. Fractional Derivatives From Benson, 1998

97. Standard Symmetric  -Stable Probability Densities

100. Brownian Motion and Levy Flights

101. Monte-Carlo Simulation of Levy Flights

102. MATLAB Movie/ Turbulence Analogy FADE (Levy Flights) 100 ‘flights’, 1000 time steps each 50 500

103. Scaling and Tailing  =0.12 After Pachepsky Y, Benson DA, and Timlin D (2001) Transport of water and solutes in soils as in fractal porous media. In Physical 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.

104. Scaling and Tailing

105. lbm

106. 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

107. Continuous Time Random Walk Model Mike Sukop/FIU 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.

108. Introduction Continuous Time Random Walk (CTRW) models – Semiconductors [Scher and Lax, 1973] – Solute transport problems [Berkowitz and Scher, 1995]

109. 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  (s,t) Estimation of this function is central to application of the CTRW model

110. Introduction The functional form  (s,t) ~ t -1-  (  > 0) is of particular interest [Berkowitz et al, 2001]  characterizes the nature and magnitude of the dispersive processes

111. Ranges of   ≥ 2 is reported to be “…equivalent to the ADE…” –For  ≥ 2, the link between the dispersivity (  = D/v) in the ADE and CTRW dimensionless b  is b  =  /L  between 1 and 2 reflects moderate non- Fickian behavior 0 <  < 1 indicates strong ‘anomalous’ behavior

112. Fits

113. 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 variably- saturated media. Version 2.0, IGWMC - TPS - 70, International Ground Water Modeling Center, Colorado School of Mines, Golden, Colorado, 202pp., 1998.

114. Effective Diffusion Tortuosity (T = L path /L) and percolation (2D)

115. Macroscopic Gas Diffusion

117. Total Mass At Equilibrium:

118. Henry’s Law Dimensionless: Common: atm m 3 mol -1

120. Total Mass At Equilibrium:

121. VLEACH Processes are conceptualized as occurring in a number of distinct, user- defined polygons that are vertically divided into a series of user- defined cells

123. Voronoi Polygons/ Diagram Voronoi_polygons –close('all') –clear('all') –axis equal –x = rand(1,100); y = rand(1,100); –voronoi(x,y)

124. Chemical Parameters Organic Carbon Partition Coefficient (Koc) = 100 ml/g Henry’s Law Constant (K H ) = 0.4 (Dimensionless) Free Air Diffusion Coefficient (Dair) = 0.7 m 2 /day Aqueous Solubility Limit (Csol) = 1100 mg/l

125. 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

126. 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

127. Dispersion! Dispersivity is implicit in the cell size (  l) and equal to  l/2 (Bear 1972) Numerical dispersion but can be used appropriately

128. Dispersion M.C. Sukop. 2001. Dispersion in VLEACH and similar models. Ground Water 39, No. 6, 953-954.

129. Hydrus

130. Solves –Richards’ Equation –Fickian solute transport –Sequential first order decay reactions

132. Governing Equation Provide linkage with preceding members of the chain

134. Density-Dependent Flows Primary source: User’s Guide to SEAWAT: A Computer Program for Simulation of Three-Dimensional Variable-Density Ground- Water Flow By Weixing Guo and Christian D. Langevin U.S. Geological Survey Techniques of Water-Resources Investigations 6-A7, Tallahassee, Florida2002

135. Sources of density variation Solute concentration Pressure Temperature

136. 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

137. Converting between:

138. Mass Balance (with sink term)

139. Density (and soon T!)

140. Densities Freshwater: 1000 kg m -3 Seawater: 1025 kg m -3 Freshwater: 0 mg L -1 Seawater: 35,000 mg L -1