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Mathematical Geosciences

, Volume 46, Issue 5, pp 625–645 | Cite as

Conditioning of Multiple-Point Statistics Facies Simulations to Tomographic Images

  • Tobias LochbühlerEmail author
  • Guillaume Pirot
  • Julien Straubhaar
  • Niklas Linde
Special Issue

Abstract

Geophysical tomography captures the spatial distribution of the underlying geophysical property at a relatively high resolution, but the tomographic images tend to be blurred representations of reality and generally fail to reproduce sharp interfaces. Such models may cause significant bias when taken as a basis for predictive flow and transport modeling and are unsuitable for uncertainty assessment. We present a methodology in which tomograms are used to condition multiple-point statistics (MPS) simulations. A large set of geologically reasonable facies realizations and their corresponding synthetically calculated cross-hole radar tomograms are used as a training image. The training image is scanned with a direct sampling algorithm for patterns in the conditioning tomogram, while accounting for the spatially varying resolution of the tomograms. In a post-processing step, only those conditional simulations that predicted the radar traveltimes within the expected data error levels are accepted. The methodology is demonstrated on a two-facies example featuring channels and an aquifer analog of alluvial sedimentary structures with five facies. For both cases, MPS simulations exhibit the sharp interfaces and the geological patterns found in the training image. Compared to unconditioned MPS simulations, the uncertainty in transport predictions is markedly decreased for simulations conditioned to tomograms. As an improvement to other approaches relying on classical smoothness-constrained geophysical tomography, the proposed method allows for: (1) reproduction of sharp interfaces, (2) incorporation of realistic geological constraints and (3) generation of multiple realizations that enables uncertainty assessment.

Keywords

Multiple-point statistics Multiple-point direct sampling Geophysical tomography Conditioning 

Notes

Acknowledgements

This research was funded by the Swiss National Science Foundation (SNF) and is a contribution to the ENSEMBLE project (grant no. CRSI22_132249). We would like to thank Philippe Renard for many helpful suggestions during the course of this study. We thank the three anonymous reviewers for their constructive comments that helped to improve the manuscript.

References

  1. Alharthi A, Lange J (1987) Soil water saturation: dielectric determination. Water Resour Res 23(4):591–595 CrossRefGoogle Scholar
  2. Alumbaugh D, Chang P, Paprocki L, Brainard J, Glass R, Rautman C (2002) Estimating moisture contents in the vadose zone using cross-borehole ground penetrating radar: a study of accuracy and repeatability. Water Resour Res 38(12):1309 Google Scholar
  3. Bayer P (2000) Aquifer-Analog-Studie in grobklastischen ‘braided river’ Ablagerungen: Sedimentäre/hydrogeologische Wandkartierung und Kalibrierung von Georadarmessungen—Diplomkartierung. Master’s thesis, University of Tübingen, Tübingen, Germany (in German) Google Scholar
  4. Bayer P, Huggenberger P, Renard P, Comunian A (2011) Three-dimensional high resolution fluvio-glacial aquifer analog: part 1: Field study. J Hydrol 405:1–9 CrossRefGoogle Scholar
  5. Caers J, Ma X (2002) Modeling conditional distributions of facies from seismic using neural nets. Math Geol 34(2):143–167 CrossRefGoogle Scholar
  6. Cardiff M, Kitanidis P (2009) Bayesian inversion for facies detection: an extensible level set framework. Water Resour Res 45(10):W10416 Google Scholar
  7. Coléou T, Poupon M, Azbel K (2003) Unsupervised seismic facies classification. Lead Edge 22(10):942–953 CrossRefGoogle Scholar
  8. Comunian A, Renard P, Straubhaar J, Bayer P (2011) Three-dimensional high resolution fluvio-glacial aquifer analog: part 2: Geostatistical modeling. J Hydrol 405:10–23 CrossRefGoogle Scholar
  9. Constable S, Parker R, Constable C (1987) Occam’s inversion: a practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics 52(3):289–300 CrossRefGoogle Scholar
  10. Cordua K, Hansen T, Mosegaard K (2012) Monte Carlo full-waveform inversion of crosshole GPR data using multiple-point geostatistical a priori information. Geophysics 77(2):H19–H31 CrossRefGoogle Scholar
  11. Dafflon B, Barrash W (2012) Three-dimensional stochastic estimation of porosity distribution: benefits of using ground-penetrating radar velocity tomograms in simulated-annealing-based or Bayesian sequential simulation approaches. Water Resour Res 48(5):W05553 Google Scholar
  12. Davis J, Annan A (1989) Ground-penetrating radar for high-resolution mapping of soil and rock stratigraphy. Geophys Prospect 37(5):531–551 CrossRefGoogle Scholar
  13. Day-Lewis F, Lane J Jr (2004) Assessing the resolution-dependent utility of tomograms for geostatistics. Geophys Res Lett 31(7):L07503 Google Scholar
  14. Day-Lewis F, Singha K, Binley A (2005) Applying petrophysical models to radar travel time and electrical resistivity tomograms: resolution-dependent limitations. J Geophys Res 110(B8):B08206 Google Scholar
  15. de Matos M, Osorio P, Johann P (2007) Unsupervised seismic facies analysis using wavelet transform and self-organizing maps. Geophysics 72(1):P9–P21 CrossRefGoogle Scholar
  16. Doser D, Crain K, Baker M, Kreinovich V, Gerstenberger M (1998) Estimating uncertainties for geophysical tomography. Reliab Comput 4(3):241–268 CrossRefGoogle Scholar
  17. Eppstein M, Dougherty D (1998) Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site. Water Resour Res 34(8):1889–1900 CrossRefGoogle Scholar
  18. Gómez-Hernández J, Wen X (1998) To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology. Adv Water Resour 21(1):47–61 CrossRefGoogle Scholar
  19. Heinz J, Kleineidam S, Teutsch G, Aigner T (2003) Heterogeneity patterns of quaternary glaciofluvial gravel bodies (SW-Germany): application to hydrogeology. Sediment Geol 158(1–2):1–23 CrossRefGoogle Scholar
  20. Hu L, Chugunova T (2008) Multiple-point geostatistics for modeling subsurface heterogeneity: a comprehensive review. Water Resour Res 44(11):W11413 Google Scholar
  21. Hubbard S, Chen J, Peterson J, Majer E, Williams K, Swift D, Mailloux B, Rubin Y (2001) Hydrogeological characterization of the South Oyster Bacterial Transport Site using geophysical data. Water Resour Res 37(10):2431–2456 CrossRefGoogle Scholar
  22. Hubbard S, Rubin Y (2000) Hydrogeological parameter estimation using geophysical data: a review of selected techniques. J Contam Hydrol 45(1–2):3–34 CrossRefGoogle Scholar
  23. Journel A, Posa D (1990) Characteristic behavior and order relations for indicator variograms. Math Geol 22(8):1011–1025 CrossRefGoogle Scholar
  24. Kitanidis P (1997) Introduction to geostatistics: applications in hydrogeology. Cambridge University Press, Cambridge CrossRefGoogle Scholar
  25. Klotzsche A, van der Kruk J, Meles G, Vereecken H (2012) Crosshole gpr full-waveform inversion of waveguides acting as preferential flow paths within aquifer systems. Geophysics 77(4):H57–H62 CrossRefGoogle Scholar
  26. Kostic B, Becht A, Aigner T (2005) 3-D sedimentary architecture of a quaternary gravel delta (SW-Germany): implications for hydrostratigraphy. Sediment Geol 181(3):147–171 CrossRefGoogle Scholar
  27. Krishnan S, Journel A (2003) Spatial connectivity: from variograms to multiple-point measures. Math Geol 35(8):915–925 CrossRefGoogle Scholar
  28. Künze R, Lunati I (2012) An adaptive multiscale method for density-driven instabilities. J Comput Phys 231:5557–5570 CrossRefGoogle Scholar
  29. Linde N, Binley A, Tryggvason A, Pedersen L, Revil A (2006) Improved hydrogeophysical characterization using joint inversion of cross-hole electrical resistance and ground-penetrating radar traveltime data. Water Resour Res 42(12):W04410 Google Scholar
  30. Mariethoz G, Renard P, Caers J (2010a) Bayesian inverse problem and optimization with iterative spatial resampling. Water Resour Res 46(11):W11530 Google Scholar
  31. Mariethoz G, Renard P, Straubhaar J (2010b) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46(11):W11536 Google Scholar
  32. Meerschman E, Pirot G, Mariethoz G, Straubhaar J, Meirvenne M, Renard P (2013) A practical guide to performing multiple-point statistical simulations with the direct sampling algorithm. Comput Geosci 52:307–324 CrossRefGoogle Scholar
  33. Meles G, Greenhalgh S, der Kruk J, Green A, Maurer H (2011) Taming the non-linearity problem in GPR full-waveform inversion for high contrast media. J Appl Geophys 73:174–186 CrossRefGoogle Scholar
  34. Menke W (1989) Geophysical data analysis: discrete inverse theory, vol 45. Academic Press, New York Google Scholar
  35. Moysey S, Caers J, Knight R, Allen-King R (2003) Stochastic estimation of facies using ground penetrating radar data. Stoch Environ Res Risk Assess 17(5):306–318 CrossRefGoogle Scholar
  36. Moysey S, Knight R, Jol H (2006) Texture-based classification of ground-penetrating radar images. Geophysics 71(6):K111–K118 CrossRefGoogle Scholar
  37. Moysey S, Singha K, Knight R (2005) A framework for inferring field-scale rock physics relationships through numerical simulation. Geophys Res Lett 32:L08304 Google Scholar
  38. Ory J, Pratt R (1995) Are our parameter estimators biased? The significance of finite-difference regularization operators. Inverse Probl 11:397–424 CrossRefGoogle Scholar
  39. Paige C, Saunders M (1982) LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans Math Softw 8(1):43–71 CrossRefGoogle Scholar
  40. Peterson J (2001) Pre-inversion corrections and analysis of radar tomographic data. J Environ Eng Geophys 6:1–18 CrossRefGoogle Scholar
  41. Podvin P, Lecomte I (1991) Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools. Geophys J Int 105(1):271–284 CrossRefGoogle Scholar
  42. Rankey E, Mitchell J (2003) That’s why it’s called interpretation: impact of horizon uncertainty on seismic attribute analysis. Lead Edge 22(9):820–828 CrossRefGoogle Scholar
  43. Rauber M, Stauffer F, Huggenberger P, Dracos T (1998) A numerical three-dimensional conditioned/unconditioned stochastic facies type model applied to a remediation well system. Water Resour Res 34(9):2225–2233 CrossRefGoogle Scholar
  44. Richardson J, Sangree J, Sneider R (1987) Applications of geophysics to geologic models and to reservoir description. J Pet Technol 39(7):753–755 CrossRefGoogle Scholar
  45. Roth K, Schulin R, Flühler H, Attinger W (1990) Calibration of time domain reflectometry for water content measurement using a composite dielectric approach. Water Resour Res 26(10):2267–2273 Google Scholar
  46. Sen P, Scala C, Cohen M (1981) A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics 46(5):781–795 CrossRefGoogle Scholar
  47. Singha K, Day-Lewis F, Moysey S (2007) Accounting for tomographic resolution in estimating hydrologic properties from geophysical data. In: Hyndman DW, Day-Lewis FD, Singha K (eds) Subsurface hydrology: data integration for properties and processes. Geophysical monograph series, vol 171. AGU, Washington, pp 227–242 Google Scholar
  48. Singha K, Moysey S (2006) Accounting for spatially variable resolution in electrical resistivity tomography through field-scale rock-physics relations. Geophysics 71(4):A25 CrossRefGoogle Scholar
  49. Straubhaar J, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43:305–328 CrossRefGoogle Scholar
  50. Straubhaar J, Walgenwitz A, Renard P (2013) Parallel multiple-point statistics algorithm based on list and tree structures. Math Geosci 45(2):1–17 CrossRefGoogle Scholar
  51. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21 CrossRefGoogle Scholar
  52. Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. Society for Industrial Mathematics CrossRefGoogle Scholar
  53. Tinga W, Voss W, Blossey D (1973) Generalized approach to multiphase dielectric mixture theory. J Appl Phys 44(9):3897–3902 CrossRefGoogle Scholar
  54. Tran T (1994) Improving variogram reproduction on dense simulation grids. Comput Geosci 20(7):1161–1168 CrossRefGoogle Scholar
  55. Vidale J (1988) Finite-difference calculation of travel times. Bull Seismol Soc Am 78(6):2062–2076 Google Scholar
  56. Western A, Blöschl G, Grayson R (2001) Toward capturing hydrologically significant connectivity in spatial patterns. Water Resour Res 37(1):83–97 CrossRefGoogle Scholar
  57. Williamson P (1991) A guide to the limits of resolution imposed by scattering in ray tomography. Geophysics 56(2):202–207 CrossRefGoogle Scholar
  58. Zhou H, Gómez-Hernández J, Li L (2012) A pattern-search-based inverse method. Water Resour Res 48(3):W03505 Google Scholar
  59. Zinn B, Harvey C (2003) When good statistical models of aquifer heterogeneity go bad: a comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields. Water Resour Res 39(3):1051 Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  • Tobias Lochbühler
    • 1
    Email author
  • Guillaume Pirot
    • 2
  • Julien Straubhaar
    • 2
  • Niklas Linde
    • 1
  1. 1.Applied and Environmental Geophysics Group, Faculty of Geosciences and EnvironmentUniversity of LausanneLausanneSwitzerland
  2. 2.Center for Hydrogeology and GeothermicsUniversity of NeuchâtelNeuchâtelSwitzerland

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