Skip to main content
SpringerLink
Log in

Predictive Assessment of Groundwater Flow Uncertainty in Multiscale Porous Media by Using Truncated Power Variogram Model

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

Abstract

The spatial distribution of hydrogeological properties is essentially heterogeneous. Heterogeneity can be characterized quantitatively using geostatistics, which conventionally assumes that the stochastic process is stationary. However, growing evidence indicates that the spatial variability has the multiscale self-similarity characteristics and can be better characterized using non-stationary model but with statistically homogeneous increments. A general framework is developed in this work to conduct the uncertainty quantification analysis by using truncated power variogram model, which can explicitly account for measurement scale, observation scale, and window scale. The effect of the multiscale characteristics of the hydrogeological properties on the uncertainty and the consequential risk associated with the groundwater flow process is investigated. A synthetic two-dimensional saturated steady-state groundwater flow problem is used to evaluate the performance to predict the flow field distribution. For comparative purposes, the evaluation is based on both the truncated power and the traditional variogram models when the underlying porous medium is a random fractal field. The results show that the truncated power variogram model can perform the uncertainty quantification more accurately, and the adoption of traditional variogram model tends to result in a smoother estimation on the flow field and underestimate the uncertainty associated with the hydraulic head prediction. Upscaling is generally inevitable to avoid predictive uncertainty underestimation when the underlying random field exhibits multiscale characteristics.

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
Fig. 7

Similar content being viewed by others

References

  • Ahmadi, S.H., Sedghamiz, A.: Geostatistical analysis of spatial and temporal variations of groundwater level. Environ. Monit. Assess. 129(1–3), 277–294 (2007)

    Article  Google Scholar 

  • Armstrong, M., Galli, A.G., Loc’H, G.L., Geffroy, F., Eschard, R.: Plurigaussian Simulations in Geosciences. Springer, Berlin, Heidelberg (2011)

    Book  Google Scholar 

  • Borghi, A., Renard, P., Jenni, S.: A pseudo-genetic stochastic model to generate karstic networks. J. Hydrol. 414(2), 516–529 (2012)

    Article  Google Scholar 

  • Boufadel, M.C., Lu, S., Molz, F.J., Daniel, L.: Multifractal scaling of the intrinsic permeability. Water Resour. Res. 36(11), 3211–3222 (2000)

    Article  Google Scholar 

  • Carle, S.F., Fogg, G.E.: Transition probability-based indicator geostatistics. Math. Geol. 28(4), 453–476 (1996)

    Google Scholar 

  • Carlson, R.A., Osiensky, J.L.: Geostatistical Analysis and simulation of nonpoint source groundwater nitrate contamination: a case study. Environ. Geosci. 5(4), 177–186 (2010)

    Article  Google Scholar 

  • Chen, Y., Zhang, D.: Data assimilation for transient flow in geologic formations via ensemble Kalman filter. Adv. Water Resour. 29(8), 1107–1122 (2006)

    Article  Google Scholar 

  • Cohen, S., Istas, J.: Fractional Fields and Applications. Springer, Berlin, Heidelberg (2013)

    Book  Google Scholar 

  • Delhomme, J.P.: Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach. Water Resour. Res. 15(2), 269–280 (1979)

    Article  Google Scholar 

  • Desbarats, A.J., Bachu, S.: Geostatistical analysis of aquifer heterogeneity from the core scale to the basin scale: a case study. Water Resour. Res. 30(3), 673–684 (1994)

    Article  Google Scholar 

  • Deutsch, C.V., Journel, A.G.: GSLIB Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, Oxford (1998)

    Google Scholar 

  • Di Federico, V., Neuman, S.P.: Scaling of random fields by means of truncated power variograms and associated spectra. Water Resour. Res. 33(5), 1075–1085 (1997)

    Article  Google Scholar 

  • Di Federico, V., Neuman, S.P.: Transport in multiscale log conductivity fields with truncated power variograms. Water Resour. Res. 34(5), 963–973 (1998)

    Article  Google Scholar 

  • Dieker, T.: Simulation of fractional Brownian motion. M.Sc. thesis, University of Twente (2004)

  • Eggleston, J., Rojstaczer, S.: Inferring spatial correlation of hydraulic conductivity from sediment cores and outcrops. Geophys. Res. Lett. 25(13), 2321–2324 (1998)

    Article  Google Scholar 

  • Emery, X.: Simulation of geological domains using the plurigaussian model: new developments and computer programs. Comput. Geosci. 33(9), 1189–1201 (2007)

    Article  Google Scholar 

  • Eschard, R., Doligez, B., Beucher, H.: Using quantitative outcrop databases as a guide for geological reservoir modelling. In: Geostatistics Rio 2000. pp. 7–17. Springer, Dordrecht (2002)

  • Gabrovšek, F., Dreybrodt, W.: Karstification in unconfined limestone aquifers by mixing of phreatic water with surface water from a local input: a model. J. Hydrol. 386(1), 130–141 (2010)

    Article  Google Scholar 

  • Gaus, I., Kinniburgh, D.G., Talbot, J.C., Webster, R.: Geostatistical analysis of arsenic concentration in groundwater in Bangladesh using disjunctive kriging. Environ. Geol. 44(8), 939–948 (2003)

    Article  Google Scholar 

  • Gelhar, L.W., Welty, C., Rehfeldt, K.R.: A critical review of data on field-scale dispersion in aquifers. Water Resour. Res. 28(7), 1955–1974 (1992)

    Article  Google Scholar 

  • Gelhar, L.W.: Stochastic subsurface hydrology. Water Resour. Res. 22(9S), 135S–145S (1993)

    Article  Google Scholar 

  • Goncalvès, J., Violette, S., Guillocheau, F., Robin, C., Pagel, M., Bruel, D., Marsily, G.D., Ledoux, E.: Contribution of a three-dimensional regional scale basin model to the study of the past fluid flow evolution and the present hydrology of the Paris basin, France. Basin Res. 16(4), 569–586 (2004)

    Article  Google Scholar 

  • Guadagnini, A., Neuman, S.P.: Extended power-law scaling of self-affine signals exhibiting apparent multifractality. Geophys. Res. Lett. 38(13), 584–610 (2011)

    Article  Google Scholar 

  • Guadagnini, A., Neuman, S.P., Schaap, M.G., Riva, M.: Anisotropic statistical scaling of vadose zone hydraulic property estimates near Maricopa, Arizona. Water Resour. Res. 49(12), 8463–8479 (2013)

    Article  Google Scholar 

  • Guadagnini, A., Neuman, S.P., Schaap, M.G., Riva, M.: Anisotropic statistical scaling of soil and sediment texture in a stratified deep vadose zone near Maricopa, Arizona. Geoderma 214, 217–227 (2014)

    Article  Google Scholar 

  • Harbaugh, A.W., Banta, E.R., Hill, M.C., Mcdonald, M.G.: MODFLOW-2000, the U.S. geological survey modular ground-water flow model-User guide to modularization concepts and the ground-water flow process. U.S. Geol. Surv. Open File Rep., 00–92 (2000)

  • Heße, F., Prykhodko, V., Schlüter, S., Attinger, S.: Generating random fields with a truncated power-law variogram: a comparison of several numerical methods. Environ. Model. Softw. 55, 32–48 (2014)

    Article  Google Scholar 

  • Hoeksema, R.J., Kitanidis, P.K.: Prediction of transmissivities, heads, and seepage velocities using mathematical modeling and geostatistics. Adv. Water Resour. 12(2), 90–102 (1989)

    Article  Google Scholar 

  • Hosking, J.R.M.: Modeling persistence in hydrological time series using fractional differencing. Water Resour. Res. 20(12), 1898–1908 (1984)

    Article  Google Scholar 

  • Hu, L.Y., Chugunova, T.: Multiple-point geostatistics for modeling subsurface heterogeneity: a comprehensive review. Water Resour. Res. 44(11), 2276–2283 (2008)

    Google Scholar 

  • Hyun, Y., Neuman, S.P., Vesselinov, V.V., Illman, W.A., Tartakovsky, D.M., Federico, V.D.: Theoretical interpretation of a pronounced permeability scale effect in unsaturated fractured tuff. Water Resour. Res. 38(6), 281–288 (2002)

    Article  Google Scholar 

  • Jung, A., Aigner, T.: carbonate geobodies: hierarchical classification and database—a new workflow for 3D reservoir modelling. J. Petrol. Geol. 35(1), 49–65 (2011)

    Article  Google Scholar 

  • Kitanidis, P.K.: On the geostatistical approach to the inverse problem. Adv. Water Resour. 19(6), 333–342 (1996)

    Article  Google Scholar 

  • Linde, N., Renard, P., Mukerji, T., Caers, J.: Geological realism in hydrogeological and geophysical inverse modeling: a review. Adv. Water Resour. 86(3), 86–101 (2015)

    Article  Google Scholar 

  • Liu, H.H., Molz, F.J.: Discrimination of fractional Brownian movement and fractional Gaussian noise structures in permeability and related property distributions with range analyses. Water Resour. Res. 32(8), 2601–2605 (1996)

    Article  Google Scholar 

  • Liu, H.H., Molz, F.J.: Multifractal analyses of hydraulic conductivity distributions. Water Resour. Res. 33(11), 2483–2488 (1997)

    Article  Google Scholar 

  • Lopez, S., Cojan, I., Rivoirard, J., Galli, A.: Process-Based Stochastic Modelling: Meandering Channelized Reservoirs. Wiley, New York (2009)

    Google Scholar 

  • Mariethoz, G., Renard, P., Cornaton, F., Jaquet, O.: Truncated plurigaussian simulations to characterize aquifer heterogeneity. Ground Water 47(1), 13–24 (2009)

    Article  Google Scholar 

  • Matheron, G.: Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)

    Article  Google Scholar 

  • Meerschaert, M.M., Kozubowski, T.J., Molz, F.J., Lu, S.: Fractional Laplace model for hydraulic conductivity. Geophys. Res. Lett. 31(8), 1020–1029 (2004)

    Article  Google Scholar 

  • Michael, H.A., Li, H., Boucher, A., Sun, T., Caers, J., Gorelick, S.M.: Combining geologic-process models and geostatistics for conditional simulation of 3-D subsurface heterogeneity. Water Resour. Res. 46(5), 1532–1535 (2010)

    Article  Google Scholar 

  • Molz, F.J., Boman, G.K.: A fractal-based stochastic interpolation scheme in subsurface hydrology. Water Resour. Res. 29(11), 3769–3774 (1993)

    Article  Google Scholar 

  • Molz, F.J., Boman, G.K.: Further evidence of fractal structure in hydraulic conductivity distributions. Geophys. Res. Lett. 22(18), 2545–2548 (1995)

    Article  Google Scholar 

  • Moslehi, M., de Barros, F.P.J.: Uncertainty quantification of environmental performance metrics in heterogeneous aquifers with long-range correlations. J. Contam. Hydrol. 196, 21–29 (2017)

    Article  Google Scholar 

  • Neuman, S.P.: Relationship between juxtaposed, overlapping, and fractal representations of multimodal spatial variability. Water Resour. Res. 39(8), 1205–1213 (2003)

    Article  Google Scholar 

  • Neuman, S.P.: Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour. Res. 26(8), 1749–1758 (1990)

    Article  Google Scholar 

  • O’Malley, D., Cushman, J.H., O’Rear, P.: On generating conductivity fields with known fractal dimension and nonstationary increments. Water Resour. Res. 48, 1–6 (2012)

    Article  Google Scholar 

  • Painter, S.: Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formation. Water Resour. Res. 32(5), 1183–1195 (1996a)

    Article  Google Scholar 

  • Painter, S.: Stochastic interpolation of aquifer properties using fractional Lévy motion. Water Resour. Res. 32(5), 1323–1332 (1996b)

    Article  Google Scholar 

  • Painter, S.: Flexible scaling model for use in random field simulation of hydraulic conductivity. Water Resour. Res. 37(5), 1155–1164 (2001)

    Article  Google Scholar 

  • Prykhodko, V., Attinger, S.: Generating random fields with a truncated power-law variogram: a comparison of several numerical methods. Environ. Model Softw. 55(55), 32–48 (2014)

    Google Scholar 

  • Riva, M., Neuman, S.P., Guadagnini, A.: New scaling model for variables and increments with heavy-tailed distributions. Water Resour. Res. 51(6), 4623–4634 (2015)

    Article  Google Scholar 

  • Riva, M., Neuman, S.P., Guadagnini, A.: Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff. Stoch. Environ. Res. Risk Assess. 27(1), 195–207 (2013)

    Article  Google Scholar 

  • Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley, New York (2011)

    Book  Google Scholar 

  • Seifert, D., Jensen, J.L.: Using sequential indicator simulation as a tool in reservoir description: issues and uncertainties. Math. Geol. 31(5), 527–550 (1999)

    Article  Google Scholar 

  • Siena, M., Guadagnini, A., Riva, M., Neuman, S.P.: Extended power-law scaling of air permeabilities measured on a block of tuff. Hydrol. Earth Syst. Sci. Dis. 8(4), 29–42 (2011)

    Google Scholar 

  • Strebelle, S., Zhang, T.: Non-Stationary Multiple-Point Geostatistical Models. Springer, Dordrecht (2004)

    Google Scholar 

  • Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002)

    Article  Google Scholar 

  • Sun, N.Z., Yeh, W.W.G.: A stochastic inverse solution for transient groundwater flow: parameter identification and reliability analysis. Water Resour. Res. 28(12), 3269–3280 (2010)

    Article  Google Scholar 

  • Tennekoon, L., Boufadel, M.C., Lavallee, D., Weaver, J.: Multifractal anisotropic scaling of the hydraulic conductivity. Water Resour. Res. 39(7), 113–117 (2003)

    Article  Google Scholar 

  • Voss, R.F.: Characterization and measurement of random fractals. Phys. Scr. 13(T13), 27–35 (1986)

    Article  Google Scholar 

  • Wu, J., Boucher, A., Zhang, T.: A SGeMS code for pattern simulation of continuous and categorical variables: fILTERSIM. Comput. Geosci. 34(12), 1863–1876 (2008)

    Article  Google Scholar 

  • Yeh, T.C.J., Liu, S.: Hydraulic tomography: development of a new aquifer test method. Water Resour. Res. 36(36), 2095–2105 (2000)

    Article  Google Scholar 

  • Zimmerman, D.A., Marsily, G.D., Gotway, C.A., Marietta, M.G., Axness, C.L., Beauheim, R.L., Bras, R.L., Carrera, J., Dagan, G., Davies, P.B.: A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow. Water Resour. Res. 34(6), 1373–1413 (1998)

    Article  Google Scholar 

Download references

Acknowledgements

This work is funded by the Science Foundation of China University of Petroleum - Beijing (Grant No. 2462014YJRC038), the National Science and Technology Major Project (Grant No. 2016ZX05037003), National Natural Science Foundation of China (Grant No. 41602250) and China Geological Survey (Grant No. DD20160293).

Author information

Authors and Affiliations

  1. State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing, 102249, China

    Liang Xue

  2. Department of Oil-Gas Field Development, College of Petroleum Engineering, China University of Petroleum, Beijing, 102249, China

    Liang Xue & Diao Li

  3. Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, Nanjing, 210093, China

    Tongchao Nan & Jichun Wu

  4. State Key Laboratory of Pollution Control and Resources Reuse, Nanjing, 210093, China

    Tongchao Nan & Jichun Wu

Authors
  1. Liang Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tongchao Nan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jichun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tongchao Nan.

Ethics declarations

Conflict of interest

The authors declare no conflicts of interest or financial disclosures to report.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, L., Li, D., Nan, T. et al. Predictive Assessment of Groundwater Flow Uncertainty in Multiscale Porous Media by Using Truncated Power Variogram Model. Transp Porous Med 126, 97–114 (2019). https://doi.org/10.1007/s11242-018-1071-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-018-1071-9

Keywords

Search

Navigation