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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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