Skip to main content
SpringerLink
Log in

Nonlinear Inversion of an Unconfined Aquifer: Simultaneous Estimation of Heterogeneous Hydraulic Conductivities, Recharge Rates, and Boundary Conditions

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

A new inverse method is developed to simultaneously estimate heterogeneous hydraulic conductivities, source/sink rates, and unknown boundary conditions for steady-state flow in an unconfined aquifer. Unlike objective function-based techniques, the new method does not optimize any data-model misfits. Instead, its formulation is developed by honoring physical flow principles as well as observation data at sampled locations. Under the Dupuit–Forchheimer assumption of negligible vertical flow, accuracy and stability of the new method are demonstrated using synthetic heterogeneous aquifer problems with increasingly complex flow: (1) aquifer domains without source/sink effects; (2) aquifer domains with a point sink (a pumping well operating under a constant discharge rate); (3) aquifer domains with constant or spatially variable recharge; (4) aquifer domains with constant or spatially variable recharge undergoing single-well pumping. For all problems, inversion yields stable solutions under increasing head measurement errors (up to \(\pm \)10 % of the total head variation in a problem), although accuracy of the estimated parameters degrades with the increasing errors. The inverse method is successfully tested on problems with high hydraulic conductivity contrasts—up to 10,000 times between the maximum and minimum values. In inverting several heterogeneous problems, if the aquifer is assumed homogeneous with a constant recharge rate, physically meaningful parameter estimates (i.e., equivalent conductivities and mean recharge rates) can be determined. Alternatively, if the inverse parameterization contains spurious parameters, inversion can identify such parameters, while the simultaneous estimation of non-spurious parameters is not affected. The method obviates the well-known issues associated with model “structure errors”, when inverse parameterization either simplifies or complexifies the true parameter field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Bear, J.: Dynamics of Fluids in Porous Media, vol. 764, 1st edn. Elsevier, New York (1972)

  • Bouwer, H., Rice, R.C.: A slug test for determining hydraulic conductivity of unconfined aquifers with completely or partially penetrating wells. Water Resourc. Res. 12(3), 423–428 (1976)

    Article  Google Scholar 

  • Cardiff, M.: W, B., Kitanidis, P., Malama, B., Revil, A., Straface, S., Rizzo, E.: A potential-based inversion of unconfined steady-state hydraulic tomography. Ground Water 47(2), 259–270 (2009)

    Article  Google Scholar 

  • Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: III. Application to synthetic and field data. Water Resourc. Res. 22(2), 228–242 (1986)

    Google Scholar 

  • Cooley, R.L., Christensen, S.: Bias and uncertainty in regression-calibrated models of groundwater flow in heterogeneous media. Adv. Water Resourc. 29(5), 639–656 (2005)

    Article  Google Scholar 

  • Cooper, H.H., Bredehoeft, J.D., Papadopulos, I.S.: Response of a finite-diameter well to an instantaneous charge of water. Water Resourc. Res. 3, 263–269 (1967)

    Article  Google Scholar 

  • Crisman, S.A., Molz, F.J., Dunn, D.L., Sappington, F.C.: Application procedures for the electromagnetic borehole flowmeter in shallow unconfined aquifers. Groundw. Monit. Remediat. 12(3), 96–100 (2007)

    Google Scholar 

  • Dagan, G.: A method determining the permeability and effective porosity of unconfined anisotropic aquifers. Water Resourc. Res. 3(4), 1059–1071 (1967)

    Article  Google Scholar 

  • Dagan, G.: A note on packer, slug, and recovery tests in unconfined aquifers. Water Resourc. Res. 14, 929–934 (1978)

    Article  Google Scholar 

  • Darnet, M., Marquis, G., Sailhac, P.: Estimating aquifer hydraulic properties from the inversion of surface streaming potential (SP) anomalies. Geophys. Res. Lett. 30(13), 1679 (2003). doi:10.1029/2003GL017631

    Article  Google Scholar 

  • Day-Lewis, F., Lane, J.W., Gorelick, S.M.: Combined interpretation of radar, hydraulic, and tracer data from a fractured-rock aquifer. Hydrogeol. J. 14(1–2), 1–14 (2006)

    Article  Google Scholar 

  • Dettinger, M.D.: Reconnaissance estimates of natural recharge to desert basins in Nevada, U.S.A., by using chloride-balance calculations. J. Hydrol. 106, 55–78 (1989)

    Article  Google Scholar 

  • Doherty, J., Welter, D.: A short exploration of structure noise. Water Resourc. Res. 46, W05525 (2010). doi:10.1029/2009WR008377

    Article  Google Scholar 

  • Fienen, M., Hunt, R., Krabbenhoft, D., Clemo, T.: Obtaining parsimonious hydraulic conductivity fields using head and transport observations: A Bayesian geostatistical parameter estimation approach. Water Resourc. Res. 45, W08405 (2009). doi:10.1029/2008WR007431

    Article  Google Scholar 

  • Gaganis, P., Smith, L.: Accounting for model error in risk assessments: Alternative to adopting a bias towards conservative risk estimates in decision models. Adv. Water Resourc. 31(8), 1074–1086 (2008)

    Article  Google Scholar 

  • Haitjema, H.M., Mitchell-Bruker, S.: Are water tables a subdued replica of the topography? Ground Water 43(6), 781–786 (2005)

    Google Scholar 

  • Hantush, M.S., Jacob, C.E.: Non-steady radial flow in an infinite leaky aquifer. Trans. Am. Geophys. Union 36(1), 95–100 (1955)

    Article  Google Scholar 

  • Harvey, C.F., Gorelick, S.M.: Mapping hydraulic conductivity: Sequential conditioning with measurements of solute arrival time, hydraulic head, and local conductivity. Water Resourc. Res. 31(7), 1615–1626 (1995)

    Article  Google Scholar 

  • Healy, R.W., Cook, P.: Using groundwater levels to estimate recharge. Hydrogeol. J. 10(10), 91–109 (2002)

    Article  Google Scholar 

  • Hill, M.C., Tiedeman, C.R.: Effective Groundwater Model Calibration: With Analysis of Data, Sensitivities, Predictions, and Uncertainty, vol. 480, 1st edn. Wiley-Interscience, Berlin (2007)

  • Irsa, J., Zhang, Y.: A new direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions. Water Resourc. Res. 48, W09526 (2012). doi:10.1029/2011WR011756

    Article  Google Scholar 

  • Jyrkama, M.I., Sykes, J.F., Norman, S.D.: Recharge estimation for transient ground water modeling. Ground Water 40(6), 638–648 (2002)

    Article  Google Scholar 

  • Keating, E.H., Doherty, J., Vrugt, J.A., Kang, Q.: Optimization and uncertainty assessment of strongly nonlinear groundwater models with high parameter dimensionality. Water Resourc. Res. 46, W10517 (2010). doi:10.1029/2009WR008584

    Article  Google Scholar 

  • Li, W., Englert, A., Cirpka, O.A., Vereecken, H.: Three dimensional geostatistical inversion of flowmeter and pumping test data. Ground Water 46(2), 193–201 (2008)

    Article  Google Scholar 

  • Lin, Y.F., Wang, J., Valocchi, A.: PRO-GRADE: GIS toolkits for ground water recharge and discharge estimation. Ground Water 47(1), 122–128 (2009)

    Article  Google Scholar 

  • Liu, G., Chen, Y., Zhang, D.: Investigation of flow and transport processes at the MADE site using ensemble Kalman filter. Adv. Water Resourc. 31, 975–986 (2008)

    Article  Google Scholar 

  • Liu, X., Kitanidis, P.: Large-scale inverse modeling with an application in hydraulic tomography. Water Resourc. Res. 47, W02501 (2011). doi:10.1029/2010WR009144

    Google Scholar 

  • Mao, D., Yeh, T.C.J., Wan, L., Wen, J.C., Lu, W., Lee, C.H., Hsu, K.C.: Joint interpretation of sequential pumping tests in unconfined aquifers. Water Resourc. Res. 49, 1782–1796 (2013)

    Article  Google Scholar 

  • Mao, D., Yeh, T.C.J., Wan, L., Hsu, K.C., Lee, C.H., Wen, J.C.: Necessary conditions for inverse modeling of flow through variably saturated porous media. Adv. Water Resourc. 52, 50–61 (2013)

    Article  Google Scholar 

  • McKenna, S., Poeter, E.: Field example of data fusion for site characterization. Water Resourc. Res. 31(12), 3229–3240 (1995)

    Article  Google Scholar 

  • Mishra, P.K., Neuman, S.P.: Improved forward and inverse analyses of saturated-unsaturated flow toward a well in a compressible unconfined aquifer. Water Resourc. Res. 46, W07508 (2010). doi:10.1029/2009WR008899

    Article  Google Scholar 

  • Mishra, P.K., Neuman, S.P.: Saturated-unsaturated flow to a well with storage in a compressible unconfined aquifer. Water Resourc. Res. 47(5), W05553 (2011). doi:10.1029/2010WR010177

    Article  Google Scholar 

  • Moench, A., Garabedian, S., LeBlanc, D.: Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, Cape Cod, Massachusetts. In: US Geological Survey Professional Paper, vol. 1629, pp. 1–51. US Geological Survey, Reston (2001)

  • Moench, A.F.: Importance of the vadose zone in analyses of unconfined aquifer tests. Ground Water 42(2), 223–233 (2004). doi:10.1111/j.1745

    Article  Google Scholar 

  • Neuman, S.P.: Theory of flow in unconfined aquifers considering delayed response of the water table. Water Resourc. Res. 8(4), 1031–1045 (1972)

    Article  Google Scholar 

  • Neuman, S.: Generalized scaling of permeabilities; validation and effect of support scale. Geophys. Res. Lett. 21(5), 349–352 (1994)

    Article  Google Scholar 

  • Neuman, S.P., Blattstein, A., Riva, M., Tartakovsky, D.M., Guadagnini, A., Ptak, T.: Type curve interpretation of late-time pumping test data in randomly heterogeneous aquifers. Water Resourc. Res. 43, W10421 (2007). doi:10.1029/2007WR005871

    Article  Google Scholar 

  • Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching, vol. 380, 1st edn. Cambridge Unviersity Press, Cambridge (2008)

  • Pan, L., Warrick, A.W., Wierenga, P.: Downward water flow through sloping layers in the vadose zone: time-dependence and effect of slope length. J. Hydrol. 199, 36–52 (1997)

    Article  Google Scholar 

  • Portniaguine, O., Solomon, D.: Parmater estimation using groundwater age and head data, Cap Cod. Massachusetts. Water Resourc. Res. 34(4), 637–645 (1998)

    Article  Google Scholar 

  • Reynolds, D.A., Marimuthu, S.: Deuterium composition and flow path analysis as additional calibration targets to calibrate groundwater flow simulation in a coastal wetlands system. Hydrogeol. J. 15, 515–535 (2007)

    Article  Google Scholar 

  • Russo, D., Zaidel, J., Lauter, A.: Numerical analysis of flow and transport in a combined heterogeneous vadose zone-groundwater system. Adv. Water Resourc. 24, 49–62 (2001)

    Article  Google Scholar 

  • Saiers, J.E., Genereux, D.P., Bolster, C.H.: Influence of calibration methodology on ground water flow predictions. Ground Water 42(1), 32–44 (2004)

    Article  Google Scholar 

  • Sakaki, T., Frippiat, C.C., Komatsu, M., Illangasekare, T.H.: On the value of lithofacies data for improving groundwater flow model accuracy in a three-dimensional laboratory-scale synthetic aquifer. Water Resourc. Res. 45, W11404 (2009). doi:10.1029/2008WR007229

    Google Scholar 

  • Sanchez-Vila, X., Carrera, J., Girardi, J.: Scale effects in transmissivity. J. Hydrol. 183, 1–22 (1996)

    Article  Google Scholar 

  • Scalon, B.R., Healy, R.W., Cook, P.G.: Choosing appropriate techniques for quantifying groundwater recharge. Hydrogeol. J. 10, 18–37 (2002)

    Article  Google Scholar 

  • Schulze-Makuch, D., Carlson, D., Cherkauer, D., Malik, P.: Scale dependency of hydraulic conductivity in heterogeneous media. Ground Water 37(6), 904–919 (1999)

    Article  Google Scholar 

  • Simmers, I.: Groundwater Recharge: An Overview Of Estimation “Problems” and Recent Developments, pp. 107–115. Geological Society of London, London (1998)

  • Tan, S., Shuy, E., Chua, L.: Regression method for estimating rainfall recharge at unconfined sandy aquifers with an equatorial climate. Hydrogeol. Process. 21, 3514–3526 (2007)

    Article  Google Scholar 

  • The Mathworks Inc.: Optimization Toolbox\(^{\rm TM}\) Users Guide. Mathworks, Natick (2012)

  • Tiedeman, C.R., Hill, M.C., D’Agnese, F.A., Faunt, C.C.: Methods for using groundwater model predictions to guide hydrogeologic data collection, with application to the Death Valley regional groundwater flow system. Water Resourc. Res. 39(1), 1010 (2003). doi:10.1029/2001WR001255

    Article  Google Scholar 

  • Wang, D., Zhang, Y., Irsa, J.: Proceeding of the 2013 AGU Hydrology Days (2013). http://hydrologydays.colostate.edu/Papers_13/Dongdong_paper.pdf. Accessed 16 August 2013

  • Zhang, Y., Gable, C.W., Person, M.: Equivalent hydraulic conductivity of an experimental stratigraphy–implications for basin-scale flow simulations. Water Resourc. Res. 42(7), W05404 (2006). doi:10.1029/2005WR004720

    Google Scholar 

  • Zlotnik, V.A., Zurbuchen, B.R.: Estimation of hydraulic conductivity from borehole flowmeter tests considering head losses. J. Hydrol. 281, 115–128 (2003)

    Article  Google Scholar 

Download references

Acknowledgments

This research is supported by the University of Wyoming Center for Fundamentals of Subsurface Flow (WYDEQ49811ZH). The author acknowledges helpful comments of two anonymous reviewers who helped to improve the content and organization of this paper.

Author information

Authors and Affiliations

  1. University of Wyoming, 1000 University Ave., Laramie, WY, USA

    Ye Zhang

Authors
  1. Ye Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ye Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Y. Nonlinear Inversion of an Unconfined Aquifer: Simultaneous Estimation of Heterogeneous Hydraulic Conductivities, Recharge Rates, and Boundary Conditions. Transp Porous Med 102, 275–299 (2014). https://doi.org/10.1007/s11242-014-0275-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-014-0275-x

Keywords

Search

Navigation