Slide 1Slide 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 30Slide 31Slide 32Slide 33Slide 34Slide 35Water table contours from MODFLOW site model using flux boundary conditions extracted from analytic element (AE) modelParticle Tracking east of Trout LakeSlide 38Slide 39Slide 40Slide 41Grid design/boundary conditions and parameter selectionUSGS publication (on course website):Guidelines for Evaluating Ground-Water Flow ModelsScientific Investigations Report 2004-5038http://www.usgs.gov/NOTE:Same principles apply to the design of a finite element mesh.Finite Elements: basis functions, variational principle, Galerkin’s method, weighted residuals• Nodes plus elements; elements defined by nodes• Nodes located on flux boundaries• Flexibility in grid design: elements shaped to boundaries elements fitted to capture detail• Easier to accommodate anisotropy that occurs at an angle to the coordinate axis• Able to simulate point sources/sinks at nodes • Properties (K,S) assigned to elementsVariability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q)Considerations in selectingthe size of the nodal spacingin grid or mesh designKriging vs. zonationZonationZonationKrigingCurvature of the water tableVariability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q)Considerations in selectingthe size of the nodal spacingDesired detail around sources and sinks (e.g., rivers)Simulation ofa pumping wellCoarse GridFine GridShoreline featuresincluding streamsCurvature of the water tableVertical change in head (vertical grid resolution/layers)Variability of aquifer characteristics (K,T,S) (Kriging vs. zonation)Variability of hydraulic parameters (R, Q)Considerations in selectingthe size of the grid spacingDesired detail around sources and sinks (e.g., rivers)Hydrogeologic Cross SectionNeed anaverage Kx,Kzfor the cellK1K2K3Kx, KzField system hasisotropic layersModel cell is homogeneous andanisotropicSee eqns. 3.4a, 3.4b in A&W, p. 69, to computeequivalent Kx, Kz from K1, K2, K3.K2K1K2K14 mK1, K2Kx/Kz10, 1 3100, 1 251000, 1 25010,000, 1 2500anisotropy ratioSee Anderson et al. 2002, Ground Water 40(2)For discussion of high K nodes to simulate lakes2D model3D – 8 layer modelCapture zonesOrientation of the Grid• Co-linear with principal directions of KFinite difference grids are rectangular, which mayresult in inactive nodes outside the model boundaries.Finite element meshes can be fit to the boundaries.Boundary ConditionsBest to use physical boundaries when possible (e.g.,impermeable boundaries, lakes, rivers)Groundwater divides are hydraulic boundaries andcan shift position as conditions change in the field.If water table contours are used to set boundary conditionsin a transient model, in general it is better to specifyflux rather than head. > if transient effects (e.g., pumping) extend to the boundaries, a specified head acts as an infinite source of water while a specified flux limits the amount of water available. > You can switch from specified head to specified flux conditions as in Problem Set 6.Treating Distant Boundaries4 approachesGeneral Head Boundary ConditionTelescopic Mesh RefinementAnalytic Element Regional Screening ModelIrregular grid spacing out to distant boundarieshBLC = Conductance = K A/LK is the hydraulic conductivity of the aquifer between the model and the lake;A is the area of the boundary cell, perpendicular to flow.General Head Boundary (GHB)Q = C (hB - h)• Regular vs irregular grid spacingIrregular spacing may be used to obtain detailed head distributions in selected areas of the grid. Finite difference equations that use irregulargrid spacing have a higher associated error than FD equations that use regular grid spacing.Same is true for finite element meshes.Rule of thumb for expanding a finite difference grid:Maximum multiplication factor = 1.5e.g., 1 m, 1.5 m, 2.25 m, 3.375 m, etc.In a finite element mesh, the aspect ratio of elements ideally is close to one and definitely less than five.The aspect ratio is the ratio of maximum to minimumelement dimensions.Treating Distant BoundariesGeneral Head Boundary ConditionTelescopic Mesh RefinementAnalytic Element Regional Screening ModelIrregular grid spacing out to distant boundariesUsing a regional model to set boundary conditions for a site model• Telescopic Mesh Refinement (TMR) (USGS Open-File Report 99-238); a TMR option is available in GW Vistas.• Analytic Element Screening ModelUsing a regional model to set boundary conditions for a site model• Telescopic Mesh Refinement (TMR) (USGS Open-File Report 99-238); a TMR option is available in GW Vistas.• Analytic Element Screening Model• Analytical Solutions • Numerical Solutions • Hybrid (Analytic Element Method) (numerical superposition of analytic solutions) ReviewTypes of Models• Analytical Solutions Toth solution Theis equation etc… Continuous solution defined by h = f(x,y,z,t) ReviewTypes of Models• Numerical Solutions Discrete solution of head at selected nodal points. Involves numerical solution of a set of algebraic equations. ReviewTypes of ModelsFinite difference models (e.g., MODFLOW) Finite element models (e.g., MODFE: USGS TWRI Book 6 Ch. A3) See W&A, Ch. 6&7 for details of the FE method.Finite Elements: basis functions, variational principle, Galerkin’s method, weighted residuals• Nodes plus elements; elements defined by nodes• Nodes located on flux boundaries• Flexibility in grid design: elements shaped to boundaries elements fitted to capture detail• Easier to accommodate anisotropy that occurs at an angle to the coordinate axis• Able to simulate point sources/sinks at nodes • Properties (K,S) assigned to elementsInvolves superposition of analytic solutions. Heads are calculated in continuous space using a computer to do the mathematics involved in superposition. Hybrid Analytic Element Method (AEM)The AE Method was introduced by Otto Strack. A general purpose code, GFLOW, was developed byStrack’s student Henk Haitjema, who also wrote a textbook on the AE Method: Analytic Element Modeling of Groundwater Flow, Academic Press, 1995.Currently the method is limited to steady-state,two-dimensional,