Determination of Groundwater Flow Velocities Using Complex Flux Boundary Conditions

1. 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

2. 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.

3. Cauchy Integral Internal to domain: Boundary:

4. Equivalent Vector Formation

5. Constant Boundary Conditions

6. Linear Interpolation

7. Ordinary Least Squares (OLS) Solution Strategy Boundary Equation: Over Determined ! ! (k > u)

8. Least Squares Solution - minimizes error on boundary: Internal Points - once boundary fluxes are known:

9. Toth’s Model

10. Tóth’s Problem Upper Boundary Condition: Analytic Solution:

11. Stream function: Flux vector:

12. Domenico and Palciauskas Solution Upper Boundary Condition: Analytic Solution:

13. Boundary Condition on Upper Surface

15. 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)

16. Nawalany Solution Upper Boundary Condition: Analytic Solution:

20. Error Field

21. 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.

22. 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.