PowerPoint PresentationSlide 2Slide 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 30A descriptive representation of a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget.Conceptual Modela set of equations that describesthe physical and/or chemical processesoccurring in a system.Mathematical Model• Governing Equation • Boundary Conditions• Specified head (1st type or Neumann) constant head• Specified flux (2nd type or Dirichlet) no fluxComponents of a Mathematical Model• Initial Conditions (for transient conditions)Mathematical Model of the Toth ProblemLaplace Equation2D, steady state02222zhxh0xh0xh0zhh = c x + zoTypes of Solutions of Mathematical Models• Analytical Solutions: h= f(x,y,z,t) (example: Theis eqn., Toth 1962)• Numerical SolutionsFinite difference methodsFinite element methods• Analytic Element Methods (AEM)Toth ProblemzxAnalytical SolutionNumerical Solution02222zhxhcontinuous solution discrete solution0xh0xh0zhh = c x + zoMathematicalmodelToth ProblemzxAnalytical SolutionNumerical Solution02222zhxhh(x,z) = zo + cs/2 – 4cs/2 …zxcontinuous solution discrete solution(eqn. 2.1 in W&A)0xh0xh0zhh = c x + zoMathematicalmodelToth ProblemzxAnalytical SolutionNumerical Solution02222zhxhh(x,z) = zo + cs/2 – 4cs/2 …hi,j = (hi+1,j + hi-1,j + hi,j+1 + hi,j-1)/4zxcontinuous solution discrete solution(eqn. 2.1 in W&A)0xh0xh0zhh = c x + zoMathematicalmodel433243ECDDDExample of spreadsheet formulaHinge lineAdd a water balance& compute waterbalance errorINOUTOUT – IN = 0Hinge lineQ= KIAHinge lineAdd a water balance& compute waterbalance error1 mxQ = KIA=K(h/z)(x)(1)Ax=z Q = K hzzx(x/2) x x(x/2)Mesh centered grid: area needed in water balanceNo FlowBoundarywatertablenodesx=z Q = K hxxBlock centered grid: area needed in water balanceNo flow boundarywatertablenodesK as a Tensordiv q = 0q = - K grad hSteady state mass balance eqn.Darcy’s lawgrad hq equipotential linegrad hqIsotropic AnisotropicKx = KzKx  Kzzxdiv q = 0q = - K grad hsteady state mass balance eqn.Darcy’s lawScalar1 componentMagnitude Head (h)Vector3 componentsMagnitude and directionq & grad Tensor9 componentsMagnitude, direction and magnitude changing with directionHydraulic conductivity (K)0222222zhyhxhdiv q = 0q = - K grad hsteady state mass balance eqn.Darcy’s lawAssume K = a constant(homogeneous and isotropic conditions)Laplace Equation0)()()( zhKzyhKyxhKxzyxGoverning Eqn. for TopoDrive2D, steady-state, heterogeneous, anisotropicxzx’z’global localKxx Kxy KxzKyx Kyy KyzKzx Kzy KzzK’x 0 00 K’y 00 0 K’zbedding planesKxx 0 00 Kyy 00 0 Kzzqxqyqz= - zhyhxhq = - K grad hxhKqxxxyhKqyyyzhKqzzzK = Kxx Kxy KxzKyx Kyy KyzKzx Kzy KzzKxx ,Kyy, Kzz are the principal components of KK is a tensor with 9 componentsq = - K grad hKxx Kxy KxzKyx Kyy KyzKzx Kzy Kzzqxqyqz= - zhyhxhq = - K grad hzhKyhKxhKqxzxyxxxzhKyhKxhKqzhKyhKxhKqzhKyhKxhKqzzzyzxzyzyyyxyxzxyxxxWthSzhKyhKxhKzzhKyhKxhKyzhKyhKxhKxszzzyzxyzyyyxxzxyxx)()()(WthSzhKzyhKyxhKxszyx)()()(storagein change)( WzqyqxqzyxthSsThis is the form of the governing equation used in MODFLOW.xzx’z’global localKxx Kxy KxzKyx Kyy KyzKzx Kzy KzzK’x 0 00 K’y 00 0 K’zbedding planesxzx’z’globallocal0xhzhKqxzxgrad h q’zhKqzzz)''('xhKqxx0)'(''zhKqzzKz’=0Assume that there is no flow across impermeable bedding planesqzhKyhKxhKqzhKyhKxhKqzhKyhKxhKqzzzyzxzyzyyyxyxzxyxxxxzx’z’global localKxx Kxy KxzKyx Kyy KyzKzx Kzy KzzK’x 0 00 K’y 00 0 K’z[K] = [R]-1 [K’] [R]bedding