3. Scale needed for numerical accuracy

1
The Connection Between Conceptual Model Errors
and the Capabilities of Numerical Ground-Water
Flow ModelsMARY C. HILL1 and STEFFEN
MEHL1,2 1U.S. GEOLOGICAL SURVEY, BOULDER,
COLORADO, USA 2 DEPT. OF CIVIL, ENVIRONMENTAL
ARCHITECTURAL ENGINEERING, UNIVERSITY OF
COLORADO, BOULDER
A CHANGING WORLD New field data methods, like
tomography, reveal unprecedented subsurface
detail that can support smaller scale
conceptualizations. More powerful computers and
software make simulating these smaller scales
feasible. Are ground-water model CAPABILITIES
adequate to take advantage of these
opportunities? Here Do ERRORS and LIMITATIONS
in present model capabilities hold us back? Many
useful analyses have been done. Here we present
some we conducted using the popular and
relatively mature MODFLOW ground-water model.
Results are expected to apply to any discretized
modeling method unless noted. 2. Scale of
system features Hydraulic-conductivity field
Heterogeneity Example from Mehl and Hill
(2002a) Local features variation
changes shape of flow lines, but some bulk flow
and transport characteristics mimic those of a
simpler model. Persistent features can be very
important to transport. Need to consider data,
importance to predictions, and execution time to
determine what heterogeneity to explicitly
represent Model features can affect
conceptualization. MODFLOWs Hydrogeologic-Unit
Flow Package allows hydrogeologic units and
parameters to be defined separately from model
layers. Vertical grid refinement can be
implemented as warranted by observations and
predictions. Hydraulic-conductivity field
Anisotropy Many ground-water models are limited
in their ability to represent spatially variable
anisotropy. An analysis by Anderman and others
(in press) using a simple 2D problem developed
by John Wilson (NM Inst. of Tech.)
shows Implications up to 30
errors in local flows. With the new LVDA
Capability, horizontal and vertical anisotropy
now supported by MODFLOW, so this limitation
largely resolved for this model. Finite-element
models FEMWATER, SUTRA3D, FEHM, etc have even
greater flexibility. Important, for example, to
simulate a syncline.

• 3. Scale needed for numerical accuracy
• Even if the grid can closely reproduce features,
  accurate solution can require further refinement.
  Generally achievable by making the entire grid
  finer, but often this is not computationally
  feasible, or it may not be convenient if detail
  is needed in only a small part of a previously
  existing model application. Two basic approaches
  exist
• Gradual grid refinement
• Grid size refined gradually
• Advantage Numerically accurate.
• Disadvantage Inflexible.
• Local grid refinement
• Grid size refined abruptly.
• Advantage Flexible. Easy to refine
• part of an existing model.
• Disadvantage Numerical accuracy
• problematic for all model types, but
• new local grid refinement methods are
• improving the accuracy. The analysis
• below is for local grid refinement

• INTRODUCTION
• PROBLEM
• Conceptual Models
• Define characteristics and processes thought to
  be important to predictions of interest.
• Provide blueprints for model development
• Influenced by the capabilities, errors, and
  limitations of the mathematical model used?
• APPROACH
• CONSIDER ONLY SCALE ISSUES IN 3-D SYSTEMS
• Reasons
• 3-D important in all but very simple systems.
  Generally doable now or will be soon given modern
  computer capabilities. No need to compromise.
• SCALE issues that are spatial are important, as
  shown below. For numerical models, spatial scale
  and grid scale are interrelated.
• How do these issues fit into the larger picture?
• Red Issues considered here.
• Blue Math model abilities influence
  conceptual model
• Interpretation(s) of system features
• Scale of system features
• Commonly consider scales that can be simulated
• Consider scales important to purpose of model,
  but this can change as the model is used.
  Commonly subsequent analysis limited to features
  represented in the originally constructed model.

• Rivers and streams
• For stream-groundwater interactions, the
  sinuosity and width of a river
• relative to the grid size is key to what a model
  can represent.

Streamflow gains and losses simulated using the
405?405?9 grid. Streambed K Aquifer
Kv C(KvAriv)/M CStreambed conductance, Ariv
Area of the river in a cell, Mdepth to
block-centered finite-difference-cell node
N
Gradual grid refinement using triangular finite
elements
1222
(i)
(ii)
(iii)
Observation locations and head contours (s2Ln(K)
0)
Flow lines, vector field, and head contours
(s2Ln(K) 3.6)
1350
MODFLOW grid refined locally using shared nodes
of Mehl and Hill (2002a)
1260
0

• DISCUSSION How do the errors and discrepancies
  presented here relate to other types of error?
• Consider the method of evaluating model misfit of
  Hill (1998).
• Differences between observations and simulated
  values reflect
• many types of errors, including
• Errors in measurements used to obtain the
  observations
• Conceptual model errors (including
  parameterization errors)
• Numerical model errors
• Parameter value errors (often accommodate errors
  1 to 3)
• Etc.
• If observation weights are assigned based on an
  analysis of measurement errors (1) and the model
  fit is consistent with these errors, the standard
  error of regression equals 1 that is, s1.
• Commonly, sgt1, often 3 to 5, suggesting that the
  other types of error contribute about 2 to 4
  times more than measurement error to model
  misfit.
• So how big are measurement errors? Most of the
  errors cited in this poster relate to flows. For
  common flow measurements, errors are about 5-30.
  2 to 4 times more results in all other errors
  contributing 10- 120.

• CONCLUSION
• Existing capabilities are impressive, but
  enduring errors and limitations of mathematical
  ground-water models are significant and can
  affect conceptual models. The difficulties
• and limitations need to be understood,
  quantified, and, if possible, fixed.
• REFERENCES
• Anderman, E.R., K.L. Kipp, M.C. Hill, Johan
  Valstar, and R.M. Neupauer, in press,
  MODFLOW-2000, the U.S. Geological Survey modular
  ground-water model Documentation of the
  model-Layer Variable-Direction horizontal
  Anisotropy (LVDA) capability of the
  Hydrogeologic-Unit Flow (HUF) Package U.S.
  Geological Survey Open-File Report.
• Leake, S.A., and D.V. Claar, 1999, Procedures and
  computer programs for telescopic mesh refinement
  using MODFLOW, U.S. Geological Survey Open-File
  Report 99-238.
• Mehl, S.W. and M.C. Hill, 2002a, Development and
  evaluation of a local grid refinement method for
  block-centered finite-difference groundwater
  models using shared nodes Advances in Water
  Resources, v. 25, p. 497-511.
• Mehl, S.W. and M.C. Hill, 2002b, Evaluation of a
  local grid refinement method for steady-state
  block-centered finite-difference groundwater
  models eds. M. Hassanizadeh, proceedings of the
  Computer Methods in Water Resources Conference,
  June, 2002, Delft, the Netherlands.