Development, Calibration and Validation of Physical Models1

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Development, Calibration and Validation of
Physical Models1

• Models tools to simulate behavior of physical
  systems
• Predict future evolution of system
• Interpretive tool to predict system dynamics
• Hint for data collection and experimental design
• Steps in modeling
• Goal definition
• Structure of model
• Formulating model equations
• Calibration
• Validation

1Giudici, M. Development, Calibration and
Validation of Physical Models. Ed. Clarke, K.,
B. Parks and M. Crane. Prentice Hall, 2002.
(basis for lecture, extensive quotations)
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Model development (real word to abstract models)

• Cross disciplinary effort
• Conceptual framework
• Processes
• Mathematical/statistical description
• An abstraction process
• Natural systems complex
• Impossible to fully represent complexity
• Modeling requires simplification and
  approximation
• Physicomathematical models permits the
  description of complex natural systems with
  powerful mathematical techniques
• Modeler links real world phenomena with physical
  and math abstractions

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Role of data in model development

• Provide connections between physiomathamatical
  model and real world
• Field measurements, information about the system
  and the physical process enter into all steps of
  modeling
• Inconsistencies that arise at any step of the
  modeling process require a revisit of the
  previous steps
• Types of data
• Quantitative (measurement of physical quantities)
• Hard directly measured (I.e. hydraulic
  conductivity)
• Soft - indirect relation (I.e. electrical
  resistively)
• Qualitative data
• Economic, social and financial data important but
  not linked to physical model
• Models not grounded in data often divorced from
  reality

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Modeling Goal

• Why is the model being developed?
• What phenomena is being described?
• Scientific/academic understand basic
  characteristics of phenomena
• Professional/ engineering give answers for to an
  application
• At what accuracy do we want to approximate the
  actual process?

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Model Structure

• Space and time domain of model
• Space and time scales of the model.
• Related to space and time scale of phenomena
  under investigation
• Consistency with field measurements
• Conceptual model
• Schema of essential features of physical system
• Hydro geological scheme for groundwater flow
  models
• Parameterization of heterogeneity and anisotropy
  of physical system
• Boundary conditions
• Interaction between model domain and rest of
  world
• Kind of sources ands sinks

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Physiocomathematical equations, Discrete
equations and computer code

• Modeler seeks equations that can represent the
  physical processes
• For example to estimate groundwater flow in a
  regional aquifer Darcys law1 combined with
  approximation of 2-D horizontal flow may be a
  reasonable approach
• Physiocomathematical equations are usually
  partial differential equations2 whose solutions
  are approximated by numerical techniques (e.g.,
  finite differences, finite elements, boundary
  elements).
• Element sizes function of level of data

1 Darcy's Law is a generalized relationship for
flow in porous media. It shows the volumetric
flow rate is a function of the flow area,
elevation, fluid pressure and a proportionality
constant. It may be stated in several different
forms depending on the flow conditions. Since
its discovery, it has been found valid for any
Newtonian fluid. Likewise, while it was
established under saturated flow conditions, it
may be adjusted to account for unsaturated and
multiphase flow.
2Partial derivative W 100 x2 y2 dw/dx -2x
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Example steady water flow in confined aquifer

• Model assumptions
• Aquifer is isotropic
• Flow is 2-D in horizontal direction and
  piezometric head is constant along the whole
  vertical dimension of the aquifer.
• Flow is stationary
• Darcys law is valid
• Differential equations are approximated using a
  finite differences discrete model. Mass
  conservation principles are applied to each cell.
• Sum of flow rates entering cell equal to the rate
  of water extraction from the cell
• Flow rates Sum of flow
  rates
• FNC FEC FSC FWC FC

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Example steady water flow in confined aquifer

• Flow rates Sum of flow
  rates
• FNC FEC FSC FWC FC
• FAB -Tab(hb-ha)
• Where FAB is the hydraulic gradient, (hb-ha) is
  the difference between piezometric heads at two
  adjacent nodes and Tab is internode
  transmissivity.
• By substitution
• TNC(hN-hC) TEC(hE-hC) TSC(hS-hC) TWC(hW-hC)
  FC
• This equation is the basis for the program
  MODFLOW one of the more popular programs used for
  ground water modeling.

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Computer Code

• Validation of code essential
• Compare with analytic solution to problem
• Tests finite method validity
• Computer code
• Compare with analytic solution to discrete
  problem
• Compare with other code
• Compare with field or lab measurements

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Calibration of the Model

• Determine numerical values of model parameters
• Obtained from direct field measurements.
• Direct least square techniques for determining
  best fit of parameters.
• Indirect Often an iterative process of model
  runs, comparison with field data, parameter
  adjustment and rerun until a minimum difference
  is found (e.g. maximum likelihood of minimizing
  differences).
• Calibrated parameters have uncertainty because of
  measurement error
• Sensitivity analysis
• How sensitive is the model to small changes in
  parameter settings
• Low small error in parameters small error in
  prediction
• High small error in parameters large error in
  prediction
• Blunder
• Effect of large changes on parameters
• Additional data in calibration
• Use different data corresponding to different
  conditions of physical systems to deal with
  instability and uncertainty of parameter values

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Model Validation

• Models are often too simplistic
• Model may be situation specific
• How does the model behave in different geographic
  regions
• Confidence in model requires careful validation
• What if model use is outside temporal calibration
  period?
• Unequivocal statement of model validation?
• Use data corresponding to a range of conditions
• Test on data not used in the calibration process
• Adjust model with real world feedback

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Evolving process of model development

• Model development improves the ability of model
  to mimic real world
• Complexity of process models and equations can
  increase predictability
• 20-80-20 rule
• Sometimes paradigm shift is need to improve model
  predictability

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Conclusion

• Development and application of model is complex
  work
• Each step is functionally dependent on the next
• Model is a tool to simulate behavior of real
  world at given space and time scales
• Success of the model is greatly effected by
  quality and quantity of data
• Model development an open ended process
• Modeling is a dynamic tool
• Beware Past is not always a good model for the
  future