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Sensitivity and Resolution of Tomographic Pumping Tests in an Alluvial Aquifer

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## Sensitivity and Resolution of Tomographic Pumping Tests in an Alluvial Aquifer

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**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**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 Hydraulic Tomography Bohling**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 Field Site (GEMS) Bohling**Field Site Stratigraphy**From Butler, 2005, in Hydrogeophysics (Rubin and Hubbard, eds.), 23-58 Bohling**Tomographic Pumping Tests**Bohling**Transient Fit, Gems4S**Using K field for a = 0.025 with Ss = 9x10-5 m-1 Bohling**Drawdowns Relative to 3N**Bohling**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 Bohling**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 Bohling**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 Bohling**Sum Squared Sensitivity, All Tests**Sum of squared sensitivity of drawdown to K, Ss in each cell over all 23 tests, 6 obs points, 1-900 s Normalized sensitivities, so comparable Note difference in scales Bohling**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 = VpVp’ Bohling**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 Bohling**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 Bohling**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 Bohling**Singular Values of Jacobian, Steady-Shape**Jacobian for 280 K values Much clearer behavior than transient: Eigenvectors past first 115 represent unresolvable parameter combinations Bohling**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 Bohling**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) Bohling**Acknowledgment s**• 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 Bohling**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 Bohling