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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:

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slide1
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.

slide2
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
slide3
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

slide5
Zonation

Kriging

slide6
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)

slide7
Simulation of

a pumping well

Coarse Grid

Fine Grid

slide8
Shoreline features

including streams

slide9
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)

slide11
Hydrogeologic Cross Section

Need an

average Kx,Kz

for the cell

slide12
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.

slide13
anisotropy ratio

K2

K1

4 m

K2

K1

slide14
See Anderson et al. 2002, Ground Water 40(2)

For discussion of high K nodes to simulate lakes

slide15
Capture zones

3D – 8 layer model

2D model

slide16
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.

slide17
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.

slide18
Treating Distant Boundaries

4 approaches

General Head Boundary Condition

Irregular grid spacing out to distant boundaries

Telescopic Mesh Refinement

Analytic Element Regional Screening Model

slide19
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.

slide20
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.

slide21
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.

slide22
Treating Distant Boundaries

General Head Boundary Condition

Irregular grid spacing out to distant boundaries

Telescopic Mesh Refinement

Analytic Element Regional Screening Model

slide23
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
slide24
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

slide25
Review

Types of Models

  • AnalyticalSolutions
  • Numerical Solutions
  • Hybrid (Analytic Element Method)
  • (numerical superposition of analytic solutions)
slide26
Review

Types of Models

  • AnalyticalSolutions
  • Toth solution
  • Theis equation
  • etc…
  • Continuous solution defined by h = f(x,y,z,t)
slide27
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.

slide28
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
slide29
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

slide30
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.

slide31
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.

slide32
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

slide33
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

slide34
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.

slide35
Flux boundary

for the site model

Results of the Analytic Element

model using GFLOW

slide36
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
Particle Tracking east of Trout Lake

Allequash Lake

Big Muskellunge Lake

Lakederived

Terrestrial

Simulated flow paths

(Pint et. al, 2002)

slide38
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.
slide39
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.

slide40
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.

slide41
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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