Determination of Groundwater Flow Velocities Using Complex Flux Boundary Conditions

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Determination of Groundwater Flow Velocities
Using Complex Flux Boundary Conditions

• Todd C. Rasmussen, Ph.D.
• Associate Professor of Hydrology
• Warnell School of Forest Resources
• University of Georgia, Athens GA 30602-2152
• www.hydrology.uga.edu
• Yu Guoqing
• Visiting Research Scientist
• Water Resources and Hydroelectric Power Institute
• Hohai University, Nanjing 210024 CHINA

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Modeling Approach

• Complex flux vector (qx, qy) instead of complex
  potential vector (?, ?)
• Solution using Cauchys Integral which solves
  both divergence (?q0) and curl (??q0) of flux
  vector
• Uses both normal and tangential components of
  boundary flux, but leads to extra equations.
• Overdetermined set of equations solved using a
  Complex Variable Boundary Equation Model (CVBEM)
  with Ordinary Least Squares (OLS)
• Two analytic solutions to the Tóth problem are
  compared with the CVBEM-OLS solution.

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Cauchy Integral
Internal to domain
Boundary
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Equivalent Vector Formation
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Constant Boundary Conditions
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Linear Interpolation
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Ordinary Least Squares (OLS) Solution Strategy
Boundary Equation
Over Determined ! ! (k gt u)
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Least Squares Solution - minimizes error on
boundary
Internal Points - once boundary fluxes are known
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Toths Model
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Tóths Problem
Upper Boundary Condition
Analytic Solution
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Stream function
Flux vector
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Domenico and Palciauskas Solution
Upper Boundary Condition
Analytic Solution
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Boundary Condition on Upper Surface
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Comments

• Nodal Equations
• 60 nodes total
• 120 total equations (2 equations per node)
• 64 known nodal values (overlap at corners)
• 56 unknown nodal vales
• CVBEM/OLS Solution
• Zero error if no boundary interpolation errors
• Fit is BLUE (Best Linear Unbiased Estimate)

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Nawalany Solution
Upper Boundary Condition
Analytic Solution
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Error Field
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Conclusions

• Problems using only flux boundary conditions can
  be solved directly using Cauchys Integral and
  the complex flux.
• Requires both the normal and tangential
  components of boundary fluxes.
• Complex solution solves both the divergence and
  curl equations
• An overdetermined set of equations results when
  both normal and tangential boundary conditions
  are specified at nodes.
• This overdetermined system of equations is
  readily solved using Ordinary Least Squares,
  which provides the best estimate of boundary
  conditions.
• The approach provides excellent predictions for
  two types of boundary conditions for Tóths
  problem.

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Unresolved Contour Lines

• Used brute force contouring method
• For complex potential, w h i s
• With one-to-one correspondence, we have
• Because w is known on the boundary
• z x i y can be found at any internal point
  for specified w values.