Grid design/boundary conditions and parameter selection USGS publication (on course website): Guidelines for Evaluating Ground-Water Flow Models Scientific Investigations Report 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 elements

Variability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q) Considerations in selecting the size of the nodal spacing in grid or mesh design Kriging vs. zonation

Zonation

Kriging

Curvature of the water table Variability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q) Considerations in selecting the size of the nodal spacing Desired detail around sources and sinks (e.g., rivers)

Simulation of a pumping well Coarse Grid Fine Grid

Shoreline features including streams

Curvature of the water table Vertical 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 selecting the size of the grid spacing Desired detail around sources and sinks (e.g., rivers)

Hydrogeologic Cross Section Need an average Kx,Kz for the cell

K1K1 K2K2 K3K3 K x, K z Field system has isotropic layers Model cell is homogeneous and anisotropic See eqns. 3.4a, 3.4b in A&W, p. 69, to compute equivalent K x, K z from K 1, K 2, K 3.

K2K2 K1K1 K2K2 K1K1 4 m K 1, K 2 K x /K z 10, , , ,000, anisotropy ratio

See Anderson et al. 2002, Ground Water 40(2) For discussion of high K nodes to simulate lakes

2D model 3D – 8 layer model Capture zones

Orientation of the Grid Co-linear with principal directions of K Finite difference grids are rectangular, which may result in inactive nodes outside the model boundaries. Finite element meshes can be fit to the boundaries.

Boundary Conditions Best to use physical boundaries when possible (e.g., impermeable boundaries, lakes, rivers) Groundwater divides are hydraulic boundaries and can shift position as conditions change in the field. If water table contours are used to set boundary conditions in a transient model, in general it is better to specify flux 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 Boundaries 4 approaches General Head Boundary Condition Telescopic Mesh Refinement Analytic Element Regional Screening Model Irregular grid spacing out to distant boundaries

hBhB L C = Conductance = K A/L K 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 (h B - h)

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. Same is true for finite element meshes.

Rule of thumb for expanding a finite difference grid: Maximum multiplication factor = 1.5 e.g., 1 m, 1.5 m, 2.25 m, 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 minimum element dimensions.

Treating Distant Boundaries General Head Boundary Condition Telescopic Mesh Refinement Analytic Element Regional Screening Model Irregular 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 ); 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 ); 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) 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. Review Types of Models Finite 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 elements

Involves 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 by Strack’s student Henk Haitjema, who also wrote a textbook on the AE Method: Analytic Element Modeling of Groundwater Flow, Academic Press, Currently the method is limited to steady-state, two-dimensional, horizontal flow

How does superposition work? Example: The Theis solution may be added to an analytical 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. Apply principle of superposition by subtracting the drawdown calculated with the Theis solution from the head computed using an analytical solution for regional flow without pumping. (from Hornberger et al. 1998)

Using a regional model to set boundary conditions for a site model Telescopic Mesh Refinement (TMR) (USGS Open-File Report ); a TMR option is available in GW Vistas. Analytic Element Screening Model 

Trout Lake km N Example: An AEM screening model to set BCs for a site model of the Trout Lake Basin Outline of the site we want to model

Outline of the Trout Lake MODFLOW site model Analytical element model of the regional area surrounding the Trout Lake site Analytic elements outlined in blue & pink represent lakes and streams.

Flux boundary for the site model Results of the Analytic Element model using GFLOW

Water table contours from MODFLOW site model using flux boundary conditions extracted from analytic element (AE) model Trout Lake Flux boundaries

Particle Tracking east of Trout Lake Lake derived Simulated flow paths Allequash Lake Big Muskellunge Lake Terrestrial (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 might need 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.

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. [deLange (2006), Ground Water 44(1), p ]