PowerPoint PresentationSubsurface HydrologySlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44tcRxcvxcD22Subsurface HydrologyUnsaturated Zone HydrologyGroundwater Hydrology (Hydrogeology)Water TableR = P - ET - ROPEETETRORGroundwaterwasteProcesses we need to model• Groundwater flowcalculate both heads and flows (q)• Solute transport – requires information on flow (velocities) calculate concentrations v = q /  = K I /Darcy’s lawTypes of Models• Physical (e.g., sand tank)• Analog (electric analog, Hele-Shaw)• MathematicalTypes of Solutions of Mathematical Models• Analytical Solutions: h= f(x,y,z,t) (example: Theis eqn.)• Numerical SolutionsFinite difference methodsFinite element methods• Analytic Element Methods (AEM)Finite difference modelsmay be solved using:•a computer programs (e.g., a FORTRAN program)•a spreadsheet (e.g., EXCEL)Components of a Mathematical Model• Governing Equation • Boundary Conditions• Initial conditions (for transient problems)The governing equation for solute transport problems is the advection-dispersion equation. In full solute transport problems, we have twomathematical models: one for flow and one for transport.Flow Code: MODFLOW  USGS code finite difference code to solve the groundwater flow equation•MODFLOW 88•MODFLOW 96•MODFLOW 2000Transport Code: MT3DMS Univ. of Alabama finite difference code to solve the advection-dispersion eqn. •Links to MODFLOWThe pre- and post-processorGroundwater Vistaslinks and runs MODFLOW and MT3DMS.Introduction to solute transport modelingandReview of the governing equation for groundwater flowConceptual ModelA descriptive representationof a groundwater system that incorporates an interpretation of the geological, hydrological, and geochemical conditions, including information about the boundaries of the problem domain.Toth ProblemImpermeable RockGroundwater divideGroundwater divideHomogeneous, isotropic aquifer2D, steady stateHead specified along the water tableHomogeneous, isotropic aquiferToth Problem with contaminant sourceImpermeable RockGroundwater divideGroundwater divide2D, steady stateContaminant sourceProcesses to model1. Groundwater flow2. Transport(a) Particle tracking: requires velocities and a particle tracking code. calculate path lines(b) Full solute transport: requires velocites and a solute transport model. calculate concentrationsTopo-DriveFinite element model of a version of the Toth Problem for regional flow in cross section. Includes a groundwater flow model with particle tracking.Toth Problem with contaminant sourceImpermeable RockGroundwater divideGroundwater divideadvection-dispersion eqn2D, steady stateContaminant sourceProcesses we need to model• Groundwater flowcalculate both heads and flows (q)• Solute transport – requires information on flow (velocities) calculate concentrations v = q/n = K I / nRequires a flow model and a solute transport model.Groundwater flow is described by Darcy’s law.This type of flow is known as advection.True flow pathsLinear flow paths assumed in Darcy’s lawFigures from Hornberger et al. (1998)The deviation of flow paths fromthe linear Darcy paths is knownas dispersion.In addition to advection, we need to consider two other processes in transport problems.• Dispersion• Chemical reactionsAdvection-dispersion equation with chemical reaction terms.DispersionAdvectionChemicalReactionsSource/sink termChange in concentrationwith time is porosity;D is dispersion coefficient;v is velocity.Allows for multiplechemical species*)()()( WzhKzyhKyxhKxthSzyxsadvection-dispersion equationgroundwater flow equation*)()()( WzhKzyhKyxhKxthSzyxsadvection-dispersion equationgroundwater flow equationthSxhT221D, transient flow; homogeneous, isotropic, confined aquifer; no sink/source termtcRxcvxcD22Flow Equation:Transport Equation:Uniform 1D flow; longitudinal dispersion;No sink/source term; retardationthSxhT221D, transient flow; homogeneous, isotropic, confined aquifer; no sink/source termtcRxcvxcD22Flow Equation:Transport Equation:Uniform 1D flow; longitudinal dispersion;No sink/source term; retardationAssumption of theEquivalent Porous Medium(epm)Representative Elementary VolumeREVDual Porosity MediumFigure from Freeze & Cherry (1979)Review of the derivation of thegoverning equation forgroundwater flow*)()()( WthSzhKzyhKyxhKxszyxGeneral governing equationfor groundwater flowSpecific StorageSs = V / (x y z h)Kx, Ky, Kz are componentsof the hydraulic conductivitytensor.Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow --------------------------------------------------------------- div q = - Ss (h  t) +R* (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = Ss (h  t) –R*Figure taken from Hornberger et al. (1998)Darcy column h/L = grad hq = Q/AQ is proportionalto grad hq = - K grad hK is a tensor with 9 componentsK = Kxx Kxy KxzKyx Kyy KyzKzx Kzy KzzPrincipal components of Kq = - K grad hDarcy’s lawgrad hq equipotential linegrad hqIsotropicKx = Ky = Kz = KAnisotropicKx, Ky, KzzhKyhKxhKqzhKyhKxhKqzhKyhKxhKqzzzyzxzyzyyyxyxzxyxxxq = - K grad hxzx’z’global localKxx Kxy KxzKyx Kyy KyzKzx Kzy KzzK’x 0 00 K’y 00 0 K’z[K] = [R]-1 [K’] [R]zhKyhKxhKqzhKyhKxhKqzhKyhKxhKqzzzyzxzyzyyyxyxzxyxxxzhKqyhKqxhKqzzyyxxLaw of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow --------------------------------------------------------------- div q = - Ss (h t) +R* (Law of Mass Balance) q = - K grad h (Darcy’s Law)