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
Modeling Approach
• Complex flux vector (qx, qy) instead of complex potential
vector (, )
• Solution using Cauchy’s Integral which solves both divergence
(·q=0) and curl (q=0) 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.
Cauchy Integral
Internal to domain:
Boundary:
Equivalent Vector Formation
Constant Boundary Conditions
Linear Interpolation
Ordinary Least Squares (OLS) Solution Strategy
Boundary Equation:
Over Determined ! !
(k > u)
Least Squares Solution - minimizes error on boundary:
Internal Points - once boundary fluxes are known:
Toth’s Model
Tóth’s Problem
Upper Boundary Condition:
Analytic Solution:
Stream function:
Flux vector:
Domenico and Palciauskas Solution
Upper Boundary Condition:
Analytic Solution:
Boundary Condition on Upper Surface
2
Water Surface Elevation
Tangential Flux
1.5
1
0.5
0
-0.5
-1
0
0.2
0.4
0.6
0.8
1
1.2
Distance
1.4
1.6
1.8
2
Flownet - Analytic
1
1
Velocity Contours - Analytic
41
1
1
Velocity Contours - Numeric
41
1
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
11
0
1
x
31
2
41
11
0
31
2
z
0.8
z
0.8
z
0.8
1
0
11
0
1
x
31
2
0
1
x
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)
Nawalany Solution
Upper Boundary Condition:
Analytic Solution:
Error Field
Conclusions
• Problems using only flux boundary conditions can be solved
directly using Cauchy’s 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 Tóth’s problem.
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.