grid design/boundary conditions and parameter selection usgs publication (on course website):...

Grid design/boundary conditions and parameter selection

USGS 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 elements

Variability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q)

Considerations in selectingthe size of the nodal spacingin grid or mesh design

Kriging vs. zonation

Zonation

Zonation

Kriging

Curvature of the water table

Variability of aquifer characteristics (K,T,S) Variability of hydraulic parameters (R, Q)

Considerations in selectingthe size of the nodal spacing

Desired detail around sources and sinks (e.g., rivers)

Simulation ofa pumping well

Coarse Grid

Fine Grid

Shoreline featuresincluding 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 selectingthe size of the grid spacing

Desired detail around sources and sinks (e.g., rivers)

Hydrogeologic Cross Section

Need anaverage Kx,Kzfor the cell

K1

K2

K3

Kx, Kz

Field system hasisotropic layers

Model cell is homogeneous andanisotropic

See eqns. 3.4a, 3.4b in A&W, p. 69, to computeequivalent Kx, Kz from K1, K2, K3.

K2

K1

K2

K1

4 m

K1, K2 Kx/Kz

10, 1 3

100, 1 25

1000, 1 250

10,000, 1 2500

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 mayresult 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 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 approaches

General Head Boundary Condition

Telescopic Mesh Refinement

Analytic Element Regional Screening Model

Irregular grid spacing out to distant boundaries

hB

L

C = 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 spacing

Irregular 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 = 1.5

e.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 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 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 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 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, horizontal flow

How does superposition work?

Example: The Theis solution may be added to an analyticalsolution for regional flow without pumping to obtain headsunder pumping conditions in a regional flow field.

Theis solution assumesno regional flow.

(from Hornberger et al. 1998)

Solution for regional flow.

Apply principle of superposition by subtracting the drawdowncalculated with the Theis solution from the head computedusing 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 99-238); a TMR option is available in GW Vistas.

• Analytic Element Screening Model

0 4 82 Kilometers

Trout Lake

0 2 4 6 km

N

Example: An AEM screening model to set BCs for a site model of the Trout Lake Basin

Outline of the sitewe want to model

Outline of the Trout LakeMODFLOW site model

Analytical element model

of the regional area surrounding

the Trout Lake site

Analytic elementsoutlined in blue& pink representlakes and streams.

Flux boundary

for the site model

Results of the Analytic Elementmodel using GFLOW

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

Trout Lake

Fluxboundaries

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 usingTMR or an AEM screening models toset boundary conditions for site models

• If you simulate a change in the site model that reflectschanged conditions in the regional model, you shouldre-run the regional model and extract new boundaryconditions for the site model.

Example: Simulating the effectsof changes in recharge rate owingto 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 usingTMR or an AEM screening models toset boundary conditions for site models

• If transient effects simulated in the site model extendto the boundaries of the site model, you should re-runthe regional model under those same transient effectsand extract new boundary conditions for the site model for each time step.

Example: Pumping in a site model such that drawdownextends to the boundary of the site model.

TMR is increasingly being used to extractsite models from regional scaleMODFLOW models.

For example:

• Dane County Model• Model of Southeastern Wisconsin• RASA models

Also there is an AEM model of The Netherlandsthat is used for regional management problems.[deLange (2006), Ground Water 44(1), p. 111-115]