1. Analysis of Tomographic Pumping Tests with Regularized Inversion Geoffrey C. Bohling Kansas Geological Survey SIAM Geosciences Conference Santa Fe, NM, 22 March 2007

2. Santa Fe, 22 March 2007 Bohling 2 Hydraulic Tomography Simultaneous analysis of multiple tests (or stresses) with multiple observation points Information from multiple flowpaths helps reduce nonuniqueness But still the same inverse problem we have been dealing with for decades

3. Santa Fe, 22 March 2007 Bohling 3 Forward and Inverse Modeling Forward Problem: d = G(m)  Approximate: No model represents true mapping from parameter space (m) to data space (d)  Nonunique: BIG  m  small  d Inverse Problem: m = G -1 (d)  Effective: Estimated parameters are always “effective” (contingent on approximate model)  Unstable: small  d  BIG  m

4. Santa Fe, 22 March 2007 Bohling 4 Regularizing the Inverse Problem Groundwater flow models potentially have a very large number of parameters Uncontrolled inversion with many parameters can match almost anything, most likely with wildly varying parameter estimates Regularization restricts variation of parameters to try to keep them plausible Regularization by zonation is traditional approach in groundwater modeling

5. Santa Fe, 22 March 2007 Bohling 5 Tikhonov Regularization (Damped L.S.) Allow a large number of parameters (vector, m), but regularize by penalizing deviations from reference model, m ref Balancing residual norm (sum of squared residuals) against model norm (squared deviations from reference model) Increasing regularization parameter, , gives “smoother” solution Reduces instability of inversion and avoids overfitting data L is identity matrix for zeroth-order regularization, numerical Laplacian for second-order regularization Plot of model norm versus residual norm with varying  is an “L-curve” – used in selecting appropriate level of regularization

6. Santa Fe, 22 March 2007 Bohling 6 Field Site (GEMS) Highly permeable alluvial aquifer (K ~ 1.5x10 -3 m/s) Many experiments over past 19 years Induced gradient tracer test (GEMSTRAC1) in 1994 Hydraulic tomography experiments in 2002 Various direct push tests over past 7 years

7. Santa Fe, 22 March 2007 Bohling 7 Field Site Stratigraphy From Butler, 2005, in Hydrogeophysics (Rubin and Hubbard, eds.), 23-58

8. Santa Fe, 22 March 2007 Bohling 8 Tomographic Pumping Tests

9. Santa Fe, 22 March 2007 Bohling 9 Drawdowns from Gems4S Tests From Bohling et al., 2007, Water Resources Research, in press

10. Santa Fe, 22 March 2007 Bohling 10 Analysis Approach Forward simulation with 2D radial-vertical flow model in Matlab  Vertical “wedge” emanating from pumping well  Common 10 x 14 Cartesian grid of lnK values mapped into radial grid for each pumping well Inverse analysis with Matlab nonlinear least squares function, lsqnonlin  Fitting parameters are Cartesian grid lnK values  Regularization relative to uniform lnK (K = 1.5 x 10 -3 m/s) model for varying values of   Steady-shape analysis

11. Santa Fe, 22 March 2007 Bohling 11 Parallel Synthetic Experiments For guidance, tomographic pumping tests simulated in Modflow using synthetic aquifer Vertical lnK variogram for synthetic aquifer derived from GEMSTRAC1 lnK profile Vertical profile includes fining upward trend and periodic (cyclic) component Large horizontal range (61 m) yields “imperfectly layered” aquifer K values range from 4.9 x 10 -5 m/s (silty to clean sand) to 1.7 x 10 -2 m/s (clean sand to gravel) with a geometric mean of 1.2 x 10 -3 m/s

12. Santa Fe, 22 March 2007 Bohling 12 Synthetic Aquifer (81 x 49 x 70)

13. Santa Fe, 22 March 2007 Bohling 13 Four Grids

14. Santa Fe, 22 March 2007 Bohling 14 L-curves, Real and Synthetic Tests

15. Santa Fe, 22 March 2007 Bohling 15 Synthetic Results

16. Santa Fe, 22 March 2007 Bohling 16 Model Norm Relative to “Truth”

17. Santa Fe, 22 March 2007 Bohling 17 Real Results

18. Santa Fe, 22 March 2007 Bohling 18 Transient Fit, Gems4S Using K field for  = 0.025 with S s = 3x10 -5 m -1

19. Santa Fe, 22 March 2007 Bohling 19 Transient Fit, Gems4N Using K field for  = 0.025 with S s = 3x10 -5 m -1

20. Santa Fe, 22 March 2007 Bohling 20 Well Locations

21. Santa Fe, 22 March 2007 Bohling 21 Comparison to Other Estimates

22. Santa Fe, 22 March 2007 Bohling 22 Conclusions Synthetic results show that steady-shape radial analysis of tests captures salient features of K field, but also indicate “effective” nature of fits For real tests, pattern of estimated K probably reasonable, although range of estimated values may be too wide A lot of effort to characterize a 10 m x 10 m section of aquifer; perhaps not feasible for routine aquifer characterization studies Should be valuable for detailed characterization at research sites

23. Santa Fe, 22 March 2007 Bohling 23 Acknowledgments Field effort led by Jim Butler with support from John Healey, Greg Davis, and Sam Cain Support from NSF grant 9903103 and KGS Applied Geohydrology Summer Research Assistantship Program

24. Santa Fe, 22 March 2007 Bohling 24 Regularizing w.r.t. Stochastic Priors Second-order regularization – asking for smooth variations from prior model Fairly strong regularization here (α = 0.1) Best 5 fits of 50