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

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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.

Keywords

Multiscale Random fractals Observation scale Truncated power variogram Geostatistics 

Notes

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).

Compliance with Ethical Standards

Conflict of interest

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

References

  1. Ahmadi, S.H., Sedghamiz, A.: Geostatistical analysis of spatial and temporal variations of groundwater level. Environ. Monit. Assess. 129(1–3), 277–294 (2007)CrossRefGoogle Scholar
  2. Armstrong, M., Galli, A.G., Loc’H, G.L., Geffroy, F., Eschard, R.: Plurigaussian Simulations in Geosciences. Springer, Berlin, Heidelberg (2011)CrossRefGoogle Scholar
  3. Borghi, A., Renard, P., Jenni, S.: A pseudo-genetic stochastic model to generate karstic networks. J. Hydrol. 414(2), 516–529 (2012)CrossRefGoogle Scholar
  4. Boufadel, M.C., Lu, S., Molz, F.J., Daniel, L.: Multifractal scaling of the intrinsic permeability. Water Resour. Res. 36(11), 3211–3222 (2000)CrossRefGoogle Scholar
  5. Carle, S.F., Fogg, G.E.: Transition probability-based indicator geostatistics. Math. Geol. 28(4), 453–476 (1996)CrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. Chen, Y., Zhang, D.: Data assimilation for transient flow in geologic formations via ensemble Kalman filter. Adv. Water Resour. 29(8), 1107–1122 (2006)CrossRefGoogle Scholar
  8. Cohen, S., Istas, J.: Fractional Fields and Applications. Springer, Berlin, Heidelberg (2013)CrossRefGoogle Scholar
  9. Delhomme, J.P.: Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach. Water Resour. Res. 15(2), 269–280 (1979)CrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. Deutsch, C.V., Journel, A.G.: GSLIB Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, Oxford (1998)Google Scholar
  12. 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)CrossRefGoogle Scholar
  13. Di Federico, V., Neuman, S.P.: Transport in multiscale log conductivity fields with truncated power variograms. Water Resour. Res. 34(5), 963–973 (1998)CrossRefGoogle Scholar
  14. Dieker, T.: Simulation of fractional Brownian motion. M.Sc. thesis, University of Twente (2004)Google Scholar
  15. Eggleston, J., Rojstaczer, S.: Inferring spatial correlation of hydraulic conductivity from sediment cores and outcrops. Geophys. Res. Lett. 25(13), 2321–2324 (1998)CrossRefGoogle Scholar
  16. Emery, X.: Simulation of geological domains using the plurigaussian model: new developments and computer programs. Comput. Geosci. 33(9), 1189–1201 (2007)CrossRefGoogle Scholar
  17. 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)Google Scholar
  18. 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)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. Gelhar, L.W.: Stochastic subsurface hydrology. Water Resour. Res. 22(9S), 135S–145S (1993)CrossRefGoogle Scholar
  22. 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)CrossRefGoogle Scholar
  23. Guadagnini, A., Neuman, S.P.: Extended power-law scaling of self-affine signals exhibiting apparent multifractality. Geophys. Res. Lett. 38(13), 584–610 (2011)CrossRefGoogle Scholar
  24. 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)CrossRefGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 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)Google Scholar
  27. 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)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. Hosking, J.R.M.: Modeling persistence in hydrological time series using fractional differencing. Water Resour. Res. 20(12), 1898–1908 (1984)CrossRefGoogle Scholar
  30. 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
  31. 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)CrossRefGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. Kitanidis, P.K.: On the geostatistical approach to the inverse problem. Adv. Water Resour. 19(6), 333–342 (1996)CrossRefGoogle Scholar
  34. 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)CrossRefGoogle Scholar
  35. 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)CrossRefGoogle Scholar
  36. Liu, H.H., Molz, F.J.: Multifractal analyses of hydraulic conductivity distributions. Water Resour. Res. 33(11), 2483–2488 (1997)CrossRefGoogle Scholar
  37. Lopez, S., Cojan, I., Rivoirard, J., Galli, A.: Process-Based Stochastic Modelling: Meandering Channelized Reservoirs. Wiley, New York (2009)Google Scholar
  38. Mariethoz, G., Renard, P., Cornaton, F., Jaquet, O.: Truncated plurigaussian simulations to characterize aquifer heterogeneity. Ground Water 47(1), 13–24 (2009)CrossRefGoogle Scholar
  39. Matheron, G.: Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)CrossRefGoogle Scholar
  40. 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)CrossRefGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. Molz, F.J., Boman, G.K.: A fractal-based stochastic interpolation scheme in subsurface hydrology. Water Resour. Res. 29(11), 3769–3774 (1993)CrossRefGoogle Scholar
  43. Molz, F.J., Boman, G.K.: Further evidence of fractal structure in hydraulic conductivity distributions. Geophys. Res. Lett. 22(18), 2545–2548 (1995)CrossRefGoogle Scholar
  44. 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)CrossRefGoogle Scholar
  45. Neuman, S.P.: Relationship between juxtaposed, overlapping, and fractal representations of multimodal spatial variability. Water Resour. Res. 39(8), 1205–1213 (2003)CrossRefGoogle Scholar
  46. Neuman, S.P.: Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour. Res. 26(8), 1749–1758 (1990)CrossRefGoogle Scholar
  47. 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)CrossRefGoogle Scholar
  48. Painter, S.: Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formation. Water Resour. Res. 32(5), 1183–1195 (1996a)CrossRefGoogle Scholar
  49. Painter, S.: Stochastic interpolation of aquifer properties using fractional Lévy motion. Water Resour. Res. 32(5), 1323–1332 (1996b)CrossRefGoogle Scholar
  50. Painter, S.: Flexible scaling model for use in random field simulation of hydraulic conductivity. Water Resour. Res. 37(5), 1155–1164 (2001)CrossRefGoogle Scholar
  51. 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
  52. 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)CrossRefGoogle Scholar
  53. 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)CrossRefGoogle Scholar
  54. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley, New York (2011)CrossRefGoogle Scholar
  55. 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)CrossRefGoogle Scholar
  56. 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
  57. Strebelle, S., Zhang, T.: Non-Stationary Multiple-Point Geostatistical Models. Springer, Dordrecht (2004)Google Scholar
  58. Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002)CrossRefGoogle Scholar
  59. 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)CrossRefGoogle Scholar
  60. Tennekoon, L., Boufadel, M.C., Lavallee, D., Weaver, J.: Multifractal anisotropic scaling of the hydraulic conductivity. Water Resour. Res. 39(7), 113–117 (2003)CrossRefGoogle Scholar
  61. Voss, R.F.: Characterization and measurement of random fractals. Phys. Scr. 13(T13), 27–35 (1986)CrossRefGoogle Scholar
  62. 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)CrossRefGoogle Scholar
  63. Yeh, T.C.J., Liu, S.: Hydraulic tomography: development of a new aquifer test method. Water Resour. Res. 36(36), 2095–2105 (2000)CrossRefGoogle Scholar
  64. 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)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.State Key Laboratory of Petroleum Resources and ProspectingChina University of PetroleumBeijingChina
  2. 2.Department of Oil-Gas Field Development, College of Petroleum EngineeringChina University of PetroleumBeijingChina
  3. 3.Department of Hydrosciences, School of Earth Sciences and EngineeringNanjing UniversityNanjingChina
  4. 4.State Key Laboratory of Pollution Control and Resources ReuseNanjingChina

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