Analysis of tomographic pumping tests with regularized inversion
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Analysis of Tomographic Pumping Tests with Regularized Inversion. Geoffrey C. Bohling Kansas Geological Survey SIAM Geosciences Conference Santa Fe, NM, 22 March 2007. Simultaneous analysis of multiple tests (or stresses) with multiple observation points

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Analysis of Tomographic Pumping Tests with Regularized Inversion

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Analysis of tomographic pumping tests with regularized inversion

Analysis of Tomographic Pumping Tests with Regularized Inversion

Geoffrey C. Bohling

Kansas Geological Survey

SIAM Geosciences Conference

Santa Fe, NM, 22 March 2007


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

Hydraulic Tomography

Bohling


Forward and inverse modeling

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

Bohling


Regularizing the inverse problem

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

Bohling


Tikhonov regularization damped l s

Tikhonov Regularization (Damped L.S.)

Allow a large number of parameters (vector, m), but regularize by penalizing deviations from reference model, mref

Balancing residual norm (sum of squared residuals) against model norm (squared deviations from reference model)

Increasing regularization parameter, a, 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 a is an “L-curve” – used in selecting appropriate level of regularization

Bohling


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

Field Site (GEMS)

Bohling


Field site stratigraphy

Field Site Stratigraphy

From Butler, 2005, in Hydrogeophysics (Rubin and Hubbard, eds.), 23-58

Bohling


Tomographic pumping tests

Tomographic Pumping Tests

Bohling


Drawdowns from gems4s tests

Drawdowns from Gems4S Tests

From Bohling et al., 2007, Water Resources Research, in press

Bohling


Analysis approach

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 a

    • Steady-shape analysis

Bohling


Parallel synthetic experiments

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

Bohling


Synthetic aquifer 81 x 49 x 70

Synthetic Aquifer (81 x 49 x 70)

Bohling


Four grids

Four Grids

Bohling


L curves real and synthetic tests

L-curves, Real and Synthetic Tests

Bohling


Synthetic results

Synthetic Results

Bohling


Model norm relative to truth

Model Norm Relative to “Truth”

Bohling


Real results

Real Results

Bohling


Transient fit gems4s

Transient Fit, Gems4S

Using K field for a = 0.025 with Ss = 3x10-5 m-1

Bohling


Transient fit gems4n

Transient Fit, Gems4N

Using K field for a = 0.025 with Ss = 3x10-5 m-1

Bohling


Well locations

Well Locations

Bohling


Comparison to other estimates

Comparison to Other Estimates

Bohling


Conclusions

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

Bohling


Acknowledgment s

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

Bohling


Regularizing w r t stochastic priors

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

Bohling


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