Today’s Lecture: Grid design/boundary conditions and parameter selection. Thursday’s Lecture: Uncertainty analysis and Model Validation
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
Review Types of Models • Analytical Solutions • Numerical Solutions • Hybrid (Analytic Element Method) (numerical superposition of analytic solutions)
Review Types of Models • Analytical Solutions Toth solution Theis equation etc… Continuous solution defined by h = f(x, y, z, t)
Review Types of Models • Numerical Solutions Discrete solution of head at selected nodal points. Involves numerical solution of a set of algebraic equations. Finite difference models (e. g. , MODFLOW) Finite element models (e. g. , MODFE: USGS TWRI Book 6 Ch. A 3) 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 • Properties (K, S) assigned to elements • Nodes located on flux boundaries • Able to simulate point sources/sinks at nodes • 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
Hybrid Analytic Element Method (AEM) Involves superposition of analytic solutions. Heads are calculated in continuous space using a computer to do the mathematics involved in superposition. The AE Method was introduced by Otto Strack. A general purpose code, GFLOW, was developed by Strack’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, horizontal flow
How does superposition work? Example: Theis solution may be added to an analytic solution for regional flow without pumping to obtain heads under pumping conditions in a regional flow field. Theis solution assumes no regional flow. (from Hornberger et al. 1998)
Solution for regional flow. (from Hornberger et al. 1998) Apply principle of superposition by subtracting the drawdown calculated with the Theis solution from the head computed using an analytic solution for regional flow without pumping.
Example: An AEM screening model to set BCs for a site model of the Trout Lake Basin Trout Lake Outline of the site we want to model N 0 2 4 6 km
Analytical element model of the regional area surrounding the Trout Lake site Outline of the Trout Lake MODFLOW site model Analytic elements outlined in blue & pink represent lakes and streams.
Results of the Analytic Element model using GFLOW Flux boundary for the site model
Water table contours from MODFLOW site model using flux boundary conditions extracted from analytic element (AE) model Flux boundaries Trout Lake
Particle Tracking east of Trout Lake Allequash Lake Big Muskellunge Lake derived Terrestrial Simulated flow paths (Pint et. al, 2002)
Things to keep in mind when using TMR or an AEM screening models to set boundary conditions for site models • If you simulate a change in the site model that reflects changed conditions in the regional model, you should re-run the regional model and extract new boundary conditions for the site model.
Example: Simulating the effects of changes in recharge rate owing to changes in climate Flux boundary for the site model needs to be updated to reflect changed recharge rates.
Things to keep in mind when using TMR or an AEM screening models to set boundary conditions for site models • If transient effects simulated in the site model extend to the boundaries of the site model, you should re-run the regional model under those same transient effects and extract new boundary conditions for the site model for each time step. Example: Pumping in a site model such that drawdown extends to the boundary of the site model.
Treating Distant Boundaries Analytic Element Regional Screening Model Telescopic Mesh Refinement General Head Boundary Condition
TMR is increasingly being used to extract site models from regional scale MODFLOW models. For example: • Dane County Model • Model of Southeastern Wisconsin • RASA models Also there is an AEM model of The Netherlands that is used for regional management problems.
Considerations in selecting the size of the grid spacing Variability of aquifer characteristics (K, T, S) (Kriging vs. zonation) Variability of hydraulic parameters (R, Q) Curvature of the water table Vertical change in head Desired detail around sources and sinks (e. g. , rivers)
Grid Design and Boundary Conditions • Distant boundary conditions • Regular vs irregular grid spacing Irregular spacing may be used to obtain detailed head distributions in selected areas of the grid. Finite difference equations that use irregular grid spacing have a higher associated error than FD equations that use regular grid spacing.
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