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

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


Forward and inverse modeling
Forward and Inverse Modeling multiple observation points

  • 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


Regularizing the inverse problem
Regularizing the Inverse Problem multiple observation points

  • 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


Tikhonov regularization damped l s
Tikhonov Regularization (Damped L.S.) multiple observation points

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


Field site gems

Highly permeable alluvial aquifer (K ~ 1.5x10 multiple observation points-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)


Field site stratigraphy
Field Site Stratigraphy multiple observation points

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


Tomographic pumping tests
Tomographic Pumping Tests multiple observation points


Drawdowns from gems4s tests
Drawdowns from Gems4S Tests multiple observation points

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


Analysis approach
Analysis Approach multiple observation points

  • 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


Parallel synthetic experiments
Parallel Synthetic Experiments multiple observation points

  • 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


Synthetic aquifer 81 x 49 x 70
Synthetic Aquifer (81 x 49 x 70) multiple observation points


Four grids
Four Grids multiple observation points


L curves real and synthetic tests
L-curves, Real and Synthetic Tests multiple observation points


Synthetic results
Synthetic Results multiple observation points


Model norm relative to truth
Model Norm Relative to “Truth” multiple observation points


Real results
Real Results multiple observation points


Transient fit gems4s
Transient Fit, Gems4S multiple observation points

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


Transient fit gems4n
Transient Fit, Gems4N multiple observation points

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


Well locations
Well Locations multiple observation points


Comparison to other estimates
Comparison to Other Estimates multiple observation points


Conclusions multiple observation points

  • 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


Acknowledgment s
Acknowledgment s multiple observation points

  • 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


Regularizing w r t stochastic priors
Regularizing w.r.t. Stochastic Priors multiple observation points

Second-order regularization – asking for smooth variations from prior model

Fairly strong regularization here (α = 0.1)

Best 5 fits of 50