Grid%20design/boundary%20conditions

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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.
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Finite Elements basis functions, variational
principle, Galerkins 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

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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
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Zonation
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Zonation
Kriging
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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)
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Simulation of a pumping well
Coarse Grid
Fine Grid
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Shoreline features including streams
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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)
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Hydrogeologic Cross Section
Need an average Kx,Kz for the cell
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Model cell is homogeneous and anisotropic
Field system has isotropic layers
Kx, Kz
See eqns. 3.4a, 3.4b in AW, p. 69, to
compute equivalent Kx, Kz from K1, K2, K3.
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anisotropy ratio
K1, K2 Kx/Kz
10, 1 3
100, 1 25
1000, 1 250
10,000, 1 2500
K2
K1
4 m
K2
K1
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Capture zones
3D 8 layer model
2D model
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• 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.
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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.
gt 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. gt You can switch from
specified head to specified flux
conditions as in Problem Set 6.
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Treating Distant Boundaries 4 approaches
General Head Boundary Condition
Irregular grid spacing out to distant boundaries
Telescopic Mesh Refinement
Analytic Element Regional Screening Model
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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.
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• 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.
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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.
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Treating Distant Boundaries
General Head Boundary Condition
?
Irregular grid spacing out to distant boundaries
?
Telescopic Mesh Refinement
Analytic Element Regional Screening Model
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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

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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

?
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Review Types of Models

• Analytical Solutions
• Numerical Solutions
• Hybrid (Analytic Element Method)
• (numerical superposition of analytic
  solutions)

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Review Types of Models

• Analytical Solutions
• Toth solution
• Theis equation
• etc
• Continuous solution defined by h f(x,y,z,t)
  

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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 WA, Ch. 67
for details of the FE
method.
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Finite Elements basis functions, variational
principle, Galerkins 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

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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 Stracks 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
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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.
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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.
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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

?
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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
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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.
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Results of the Analytic Element model using GFLOW
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Water table contours from MODFLOW site model
using flux boundary conditions extracted from
analytic element (AE) model
Flux boundaries
Trout Lake
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Particle Tracking east of Trout Lake
Allequash Lake
Big Muskellunge Lake
Lake derived
Terrestrial
Simulated flow paths
(Pint et. al, 2002)
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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.

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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.
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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.
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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