Sensitivity and Resolution of Tomographic Pumping Tests in an Alluvial Aquifer Geoffrey C. Bohling Kansas Geological Survey 2007 Joint Assembly Acapulco, Mexico, 23 May 2007

Acapulco, 23 May 2007 Bohling 2 Hydraulic Tomography Simultaneous analysis of multiple tests (or stresses) with multiple observation points Information from multiple flowpaths helps reduce nonuniqueness Presented regularized inversion before Now going back to look at sensitivity and resolution

Acapulco, 23 May 2007 Bohling 3 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 or 8 years

Acapulco, 23 May 2007 Bohling 4 Field Site Stratigraphy From Butler, 2005, in Hydrogeophysics (Rubin and Hubbard, eds.), 23-58

Acapulco, 23 May 2007 Bohling 5 Tomographic Pumping Tests

Acapulco, 23 May 2007 Bohling 6 K Field From Regularized Inversion

Acapulco, 23 May 2007 Bohling 7 Transient Fit, Gems4S Using K field for  = with S s = 9x10 -5 m -1

Acapulco, 23 May 2007 Bohling 8 Full Drawdown Record, Test 7, Gems4N

Acapulco, 23 May 2007 Bohling 9 Drawdowns Relative to 3N

Acapulco, 23 May 2007 Bohling 10 Sensitivity and Resolution Analysis Forward simulation with 2D radial-vertical flow model in Matlab (vertical wedge) Common 20 x 14 (1.5 m x 0.76 m) Cartesian grid of K, Ss values mapped into radial grid for each test (K=1x10 -3 m/s, Ss=1x10 -4 m -1 ) Finite-difference Jacobian matrix, J Model resolution matrix R = J † J, where J † is pseudo-inverse based on SVD Transient and steady-shape analyses

Acapulco, 23 May 2007 Bohling 11 Drawdown Sensitivity, Test 7, Gems4N K1, S1: r<10.3 m K2, S2: r>10.3 m S1 influences only early time data Later: Changes in drawdown controlled by K2, K1 and S2 together contribute constant term

Acapulco, 23 May 2007 Bohling 12 Drawdown Difference Sensitivity Looking at sensitivity of drawdown differences relative to 3N Almost entirely controlled by K1, that is, K within region of investigation (ROI) for these tests

Acapulco, 23 May 2007 Bohling 13 Sum Squared Sensitivity, All Tests Sum of squared sensitivity of drawdown to K, Ss in each cell over all 23 tests, 6 obs points, s Normalized sensitivities, so comparable Note difference in scales

Acapulco, 23 May 2007 Bohling 14 Singular Values of Jacobian, Transient 560 parameters: 280 K, 280 Ss Larger singular values associated with better resolved combinations of parameters (eigenvectors) Smaller singular values with more poorly resolved combinations R = J † J = V p V p ’

Acapulco, 23 May 2007 Bohling 15 Resolution, First 66 Eigenvectors R = 1 for perfectly resolved cells, 0 for unresolved Leading eigenvectors dominated by K in ROI Essentially no contribution of Ss to leading eigenvectors

Acapulco, 23 May 2007 Bohling 16 Resolution, First 145 Eigenvectors With 145 eigenvectors, resolve K in ROI quite well Some resolution of Ss in ROI (max R values of about 0.61) Properties outside ROI much harder to resolve

Acapulco, 23 May 2007 Bohling 17 K Sensitivity, Transient and Steady-Shape Root mean squared sensitivity to compensate for differing numbers of observations Similar patterns, but steady-shape focuses sensitivity on ROI

Acapulco, 23 May 2007 Bohling 18 Singular Values of Jacobian, Steady-Shape Jacobian for 280 K values Much clearer behavior than transient: Eigenvectors past first 115 represent unresolvable parameter combinations

Acapulco, 23 May 2007 Bohling 19 K Resolution, Transient and Steady-Shape Transient result using first 145 eigenvectors, as before Steady shape using first 115 eigenvectors So, steady shape resolution similar to “dominant” transient resolution

Acapulco, 23 May 2007 Bohling 20 Conclusions Transient analysis provides good resolution of K in ROI, some resolution of Ss in ROI Parameter variations outside ROI difficult to resolve Steady shape analysis focuses sensitivity on K in ROI and reduces or eliminates sensitivity to more poorly resolved parameters (K outside ROI, Ss anywhere)

Acapulco, 23 May 2007 Bohling 21 Acknowledgments Field effort led by Jim Butler with support from John Healey, Greg Davis, and Sam Cain Support from NSF grant and KGS Applied Geohydrology Summer Research Assistantship Program

Acapulco, 23 May 2007 Bohling 22 Regularizing w.r.t. Stochastic Priors Second-order regularization – asking for smooth variations from prior model Fairly strong regularization here (  = 0.1) Best 5 fits of 50