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Grid design/boundary conditions and parameter selection

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Grid design/boundary conditions

and parameter selection

USGS publication (on course website):

Guidelines for Evaluating Ground-Water Flow Models

Scientific Investigations Report 2004-5038

http://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

- 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

Considerations in selecting

the size of the nodal spacing

in grid or mesh design

Variability of aquifer characteristics (K,T,S)

Variability of hydraulic parameters (R, Q)

Kriging vs. zonation

Zonation

Kriging

Considerations in selecting

the size of the nodal spacing

Variability of aquifer characteristics (K,T,S)

Variability of hydraulic parameters (R, Q)

Curvature of the water table

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

Shoreline features

including streams

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

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

Vertical change in head (vertical grid resolution/layers)

K1

K2

K3

Model cell is

homogeneous and

anisotropic

Field system has

isotropic layers

Kx, Kz

See eqns. 3.4a, 3.4b in A&W, p. 69, to compute

equivalent Kx, Kz from K1, K2, K3.

See Anderson et al. 2002, Ground Water 40(2)

For discussion of high K nodes to simulate lakes

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

Irregular grid spacing out to distant boundaries

Telescopic Mesh Refinement

Analytic Element Regional Screening Model

General Head Boundary (GHB)

L

Lake

Model

hB

Q = C (hB - h)

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.

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, 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 minimum

element dimensions.

Treating Distant Boundaries

General Head Boundary Condition

Irregular grid spacing out to distant boundaries

Telescopic Mesh Refinement

Analytic Element Regional 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

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

- AnalyticalSolutions
- Numerical Solutions
- Hybrid (Analytic Element Method)
- (numerical superposition of analytic solutions)

Review

Types of Models

- AnalyticalSolutions
- 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. 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

- 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

Theis solution assumes

no regional flow.

(from Hornberger et al. 1998)

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.

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 analytical solution for regional flow without pumping.

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 2 4 6 km

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

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.

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

Lakederived

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 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. 111-115]

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