PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Vector Form of Darcy’s LawSlide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Figure from Hornberger et al. (1998)Darcy’s data for two different sandsFigure from Hornberger et al. (1998)Range in hydraulic conductivity, K13 orders of magnitudeFigure from Hornberger et al. (1998)Figure from Hornberger et al. (1998)Generalization of Darcy’s column h/L = hydraulic gradientq = Q/AQ is proportionalto h/Lq is a vectorqxzqx 1qz 2zxzhKqyhKqxhKqzzyyxxq = Q/AIn general: Kz < Kx, KyzhKqyhKqxhKqzzyyxxq = - K grad hVector Form of Darcy’s Lawq = - K grad hq = specific discharge (L/T)K = hydraulic conductivity (L/T)grad h = hydraulic gradient (L/L)h = head (L)q = - K grad hK is a tensor with 9 components(three of which are Kx, Ky, Kz)q is a vector with 3 componentsh is a scalarScalar1 componentMagnitude Head, concentration, temperatureVector3 componentsMagnitude and directionSpecific discharge, (& velocity), mass flux, heat fluxTensor9 componentsMagnitude, direction and magnitude changing with directionHydraulic conductivity, Dispersion coefficient, thermal conductivityq = - K grad hDarcy’s lawgrad hq equipotential linegrad hqIsotropicKx = Ky = Kz = KAnisotropicKx, Ky, KzFigure from Hornberger et al. (1998)Linear flow paths assumed in Darcy’s lawTrue flow pathsAverage linear velocity v = Q/An= q/nn = effective porositySpecific dischargeq = Q/ARepresentative Elementary Volume(REV)REVEquivalent Porous Medium(epm)q = - K grad hLaw 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* Water balance equationInflow = OutflowRechargeDischargeSteady State Water Balance EquationTransient Water Balance EquationInflow = Outflow +/- Change in StorageOutflow - Inflow = Change in StorageFigures from Hornberger et al. (1998)Unconfined aquiferSpecific yield = SyConfined aquiferStorativity = SbhhStorage TermsS =  V / A  hS = Ss bSs = specific storage)( Wzqyqxqzyxx y z= change in storageOUT – IN = = - V/ tSs = V / (x y z h)V = Ss h (x y z)t tWREVS =  V / A  hSs = S/bhere b =  z)( WzqyqxqzyxOUT – IN = thSsWthSzhKzyhKyxhKxszyx)()()(zhKqyhKqxhKqzzyyxxLaw of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow --------------------------------------------------------------- div q = - Ss (h  t) +W (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = Ss (h  t) –WWthSzhKzyhKyxhKxszyx)()()(RthSyhhKyxhhKxyyx)()(2D unconfined:RthSyhTyxhTxyx)()(2D confined:(S = Ss b & T = K b)Figures from:Hornberger et al., 1998. Elements of Physical Hydrology,The Johns Hopkins Press, Baltimore, 302