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Sensitivity and Resolution of Tomographic Pumping Tests in an Alluvial Aquifer Geoffrey C. Bohling...
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Sensitivity and Resolution of Tomographic Pumping Tests in an Alluvial Aquifer
Geoffrey C. BohlingKansas Geological Survey2007 Joint AssemblyAcapulco, Mexico, 23 May 2007
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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
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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
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Field Site Stratigraphy
From Butler, 2005, in Hydrogeophysics (Rubin and Hubbard, eds.), 23-58
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Tomographic Pumping Tests
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K Field From Regularized Inversion
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Transient Fit, Gems4SUsing K field for = 0.025 with Ss = 9x10-5 m-1
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Full Drawdown Record, Test 7, Gems4N
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Drawdowns Relative to 3N
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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
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Drawdown Sensitivity, Test 7, Gems4NK1, 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
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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
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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
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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’
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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
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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
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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
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Singular Values of Jacobian, Steady-Shape
Jacobian for 280 K values
Much clearer behavior than transient: Eigenvectors past first 115 represent unresolvable parameter combinations
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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
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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)
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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
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Regularizing w.r.t. Stochastic PriorsSecond-order regularization – asking for smooth variations from prior model
Fairly strong regularization here ( = 0.1)
Best 5 fits of 50