Advertisement

Hydrogeology Journal

, 16:1239 | Cite as

Efficient upscaling of hydraulic conductivity in heterogeneous alluvial aquifers

  • Jan H. FleckensteinEmail author
  • Graham E. Fogg
Paper

Abstract

An efficient method to upscale hydraulic conductivity (K) from detailed three-dimensional geostatistical models of hydrofacies heterogeneity to a coarser model grid is presented. Geologic heterogeneity of an alluvial fan system was characterized using transition-probability-based geostatistical simulations of hydrofacies distributions. For comparison of different hydrofacies architecture, two alternative models with different hydrofacies structures and geometries and a multi-Gaussian model, all with the same mean and variance in K, were created. Upscaling was performed on five realizations of each of the geostatistical models using the arithmetic and harmonic means of the K-values within vertical grid columns. The effects of upscaling on model domain equivalent K were investigated by means of steady-state flow simulations. A logarithmic increase in model domain equivalent K with increasing upscaling, was found for all fields. The shape of that upscaling function depended on the structure and geometry of the hydrofacies bodies. For different realizations of one geostatistical model, however, the upscaling function was the same. From the upscaling function a factor could be calculated to correct the upscaled K-fields for the local effects of upscaling.

Keywords

Upscaling Hydraulic properties Heterogeneity Geostatistics Numerical modeling 

Upscaling efficace de la conductivité hydraulique dans les aquifères alluviaux hétérogènes

Résumé

Une méthode efficace pour upscale la conductivité hydraulique (K) à partir de modèles géostatistiques 3D détaillés de l’hétérogénéité d’hydrofaciès vers un maillage de modèle plus grossier est présentée. L’hétérogénéité géologique d’un système de cône alluvial a été caractérisée en utilisant des simulations géostatistiques basées sur une probabilité d’évolution des distributions d’hydrofaciès. Pour comparer plusieurs compositions d’hydrofaciès deux modèles alternatifs avec des structures d’hydrofaciès et des géométries différentes et un modèle Gaussien multiple, tous avec la même moyenne et variance de K, ont été créés. L’upscaling a été réalisé sur cinq mises en œuvre de chacun des modèles géostatistiques en utilisant les moyennes arithmétiques et harmoniques des valeurs K au sein de colonnes verticales du maillage. Les effets de l’upscaling de l’équivalent K dans le domaine du modèle ont été étudiés au moyen de simulations en écoulement permanent. Un accroissement logarithmique de l’équivalent K dans le domaine du modèle avec un upscaling croissant, a été trouvé pour tous les domaines. La forme de cette fonction d’upscaling dépendait de la structure et de la géométrie des ensembles d’hydrofaciès. Pour différentes mises en œuvre d’un modèle géostatistique, toutefois, la fonction d’upscaling était la même. A partir de la fonction d’upscaling un facteur peut être calculé pour corriger les domaines de K upscaled des effets locaux de l’upscaling.

Sobre-escalado eficiente de la conductividad hidráulica en acuíferos aluviales heterogéneos

Resumen

Se presenta un método eficiente para el sobre-escalado de la conductividad hidráulica (K) a partir de modelos geoestadísticos tridimensionales de heterogeneidades de hidrofacies a modelos con grillas de mayor escala. La heterogeneidad geológica de un abanico aluvial se caracterizó usando probabilidad de transición basada en simulaciones de la distribución de las hidrofacies. Para la comparación de la arquitectura de las distintas hidrofacies, se crearon dos modelos alternativos con diferentes estructuras y geometrías de las hidrofacies y un modelo multi-gaussiano, con la misma media y varianza de K. El sobre-escalado se logró con cinco realizaciones de cada modelo geoestadístico usando las medias aritmética y armónica de los valores de K en cada columna vertical de la grilla. Los efectos del sobre-escalado se investigaron con simulaciones del flujo en estado estacionario. Se halló que un incremento en el sobre-escalado produce un incremento logarítmico en el dominio del modelo con K equivalente. La forma de la función de sobre-escalado depende de la estructura y geometría de los cuerpos de hidrofacies. Sin embargo, para diferentes realizaciones de un dado modelo geoestadístico, la función de sobre-escalado fue la misma. Esa función de sobre-escalado permite calcular un factor que corrige los campos de K sobre-escalados por efectos locales del sobre-escalado.

Notes

Acknowledgements

The authors would like to thank CALFED for their funding through the California Bay-Delta Authority's Ecosystem Restoration Program: Grants ERP No. 99-NO6 and ERP No. 01-NO1. The helpful comments and suggestions from the managing editor, three anonymous reviewers and Christen Knudby were also greatly appreciated.

References

  1. Ababou R, McLaughlin D, Gelhar LW, Tompson AFB (1989) Numerical-simulation of 3-dimensional saturated flow in randomly heterogeneous porous-media. Transp Porous Media 4(6):549–565CrossRefGoogle Scholar
  2. Carle SF (1999) TPROGS: transition probability geostatistical software, version 2.1, user manual. Hydrologic Sciences Graduate Group, University of California, Davis, USAGoogle Scholar
  3. Carle SF, Fogg GE (1996) Transition probability-based indicator geostatistics. Math Geol 28(4):453–476CrossRefGoogle Scholar
  4. Carle SF, Fogg GE (1997) Modeling spatial variability with one and multidimensional continuous-lag Markov chains. Math Geol 29(7):891–918CrossRefGoogle Scholar
  5. Carle SF, La Bolle EM, Weissmann GS Van Brocklin D, Fogg GE (1998) Conditional simulation of hydrofacies architecture: a transition probability/Markov approach. In: Fraser GS, Davis JM (eds) Hydrogeologic models of sedimentary aquifers. Concepts in Hydrogeology and Environmental Geology, SEPM Spec Publ 1:147–170Google Scholar
  6. Chen Y, Durlofsky LJ Gerritsen M, Wen XH (2003) A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv Water Resour 26:1041–1060CrossRefGoogle Scholar
  7. Cushman JH, Bennethum LS, Hu BX (2002) A primer on upscaling tools for porous media. Adv Water Resour 25:1043–1067CrossRefGoogle Scholar
  8. Dagan G (2001) Effective, equivalent and apparent properties of heterogeneous media. In: Aref H, Phillips JW (eds) Mechanics for a new millennium: Proceeding of the 20th International Congress on Theoretical and Applied Mechanics. Kluwer, Dordrecht, pp 473–485Google Scholar
  9. Dagan G, Neuman SP (1997) Subsurface flow and transport: a stochastic approach. University Press, CambridgeGoogle Scholar
  10. de Marsily G, Delay F, Teles V, Schafmeister MT (1998) Some current methods to represent the heterogeneity of natural media in hydrogeology. Hydrogeol J 6:115–130CrossRefGoogle Scholar
  11. de Marsily G, Delay F, Goncalves J, Renard P, Teles V, Violette S (2005) Dealing with spatial heterogeneity. Hydrogeol J 13(1):61–183CrossRefGoogle Scholar
  12. Desbarats AJ (1994) Spatial averaging of hydraulic conductivity under radial flow conditions. Math Geol 26:1–21CrossRefGoogle Scholar
  13. Deutsch CV, Journel AG (1998) GSLIB: geostatistical software library and user's guide. Oxford University Press, New YorkGoogle Scholar
  14. Eaton TT (2006) On the importance of geological heterogeneity for flow simulation. Sediment Geol 184(3–4):187–201CrossRefGoogle Scholar
  15. Fiori A, Jankovic I, Dagan G (2003) Flow and transport through two-dimensional isotropic media of binary conductivity distribution, part 1: numerical methodology and semi-analytical solutions. Stoch Environ Res Risk Assess 17:370–383CrossRefGoogle Scholar
  16. Fleckenstein JH (2004) Modeling river-aquifer interactions and geologic heterogeneity in an alluvial fan system, Cosumnes River, CA. PhD Thesis, Hydrologic Sciences Graduate Group, University of California, Davis, USAGoogle Scholar
  17. Fleckenstein JH, Niswonger RG, Fogg GE (2006) River-aquifer interactions, geologic heterogeneity, and low-flow management. Ground Water 44:837–852CrossRefGoogle Scholar
  18. Fogg GE (1986) Groundwater-flow and sand body interconnectedness in a thick, multiple-aquifer system. Water Resour Res 22(5):679–694CrossRefGoogle Scholar
  19. Fogg GE, Carle SF, Green CT (2000) Connected-network paradigm for the alluvial aquifer system. In: Zhang D, Winter CL (eds) Theory, modeling and field investigation in hydrogeology: a special volume in honor of Shlomo P. Neuman’s 60th birthday, Geological Society of America Spec Pap 348, pp 25–42Google Scholar
  20. Gomez-Hernandez JJ, Gorelick SM (1989) Effective groundwater model parameter values: influence of spatial variability of hydraulic conductivity, leakance and recharge. Water Resour Res 25(3):405–419CrossRefGoogle Scholar
  21. Harbaugh AW, Banta ER, Hill MC, McDonald MG (2000) MODFLOW-2000, The US Geological Survey Modular ground-water model: user guide to modularization concepts and the ground-water flow process, US Geol Surv Open-File Rep 00–92Google Scholar
  22. Harter T (2005) Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields. Phys Rev E Stat Nonlin Soft Matter Phys 72(2), 026120Google Scholar
  23. Indelman P (2003) Transient well-type flows in heterogenous formations. Water Resour Res 39(3), 1064.  10.1029/2002WR001407
  24. Jankovic I, Fiori A, Dagan G (2003) Effective conductivity of an isotropic heterogeneous medium of lognormal conductivity distribution. Multiscale Model Simul 1:40–56Google Scholar
  25. Journel AG (1996) Conditional simulation of geologically averaged block permeabilities. J Hydrol 183(1–2):23–35CrossRefGoogle Scholar
  26. Journel AG, Deutsch CV, Desbarats AJ (1986) Power averaging for block effective permeability. Paper presented at 56th California regional meeting of the Society of Petroleum Engineers, Oakland, CA, April 1986Google Scholar
  27. Knudby C (2004) On the concept of hydrogeologic connectivity. PhD Thesis, Technical University of Catalonia, SpainGoogle Scholar
  28. Knudby C, Carrera J (2005) On the relationship between indicators of geostatistical, flow and transport connectivity. Adv Water Resour 28(4):405–421CrossRefGoogle Scholar
  29. Knudby C, Carrera J, Bumgardner JD, Fogg GE (2006) Binary upscaling: the role of connectivity and a new formula. Adv Water Resour 29:590–604CrossRefGoogle Scholar
  30. Koltermann CE, Gorelick SM (1996) Heterogeneity in sedimentary deposits: a review of structure-imitating, process-imitating, and descriptive approaches. Water Resour Res 32(9):2617–2658CrossRefGoogle Scholar
  31. LaBolle EM, Fogg GE (2001) Role of molecular diffusion in contaminant migration and recovery in an alluvial aquifer system. Transp Porous Media 42(1–2):155–179CrossRefGoogle Scholar
  32. Lee SY (2004) Heterogeneity and transport: geostatistical modeling, non-Fickian transport, and efficiency of remediation methods. PhD Thesis, University of California, Davis, USA, 267 ppGoogle Scholar
  33. Lee SY, Carle SF, Fogg GE (2007) Geologic heterogeneity and a comparison of two geostatistical models: sequential Gaussian and transition probability-based geostatistical simulation. Adv Water Resour 30:1914–1932CrossRefGoogle Scholar
  34. McDonald MG, Harbaugh AW (1988) A modular three-dimensional finite-difference ground-water flow model. USGS Techniques of Water-Resources Investigations, Book 6, Chapter A1, US Geological Survey, Reston, VAGoogle Scholar
  35. Neuman SP, Di F (2003) Multifaceted nature of hydrogeologic scaling and its interpretation. Rev Geophys 41(3), 1014. DOI  10.1029/2003RG000130 CrossRefGoogle Scholar
  36. Noetinger B, Artus V, Zargar G (2005) The future of stochastic and upscaling methods in hydrogeology. Hydrogeol J 13(1):84–201CrossRefGoogle Scholar
  37. Paleologos EK, Neuman SP, Tartakovsky D (1996) Effective hydraulic conductivity of bounded, strongly heterogeneous porous media. Water Resour Res 32:1333–1341Google Scholar
  38. Pozdniakov S, Tsang CF (2004) A self-consistent approach for calculating the effective hydraulic conductivity of a binary, heterogeneous medium. Water Resour Res 40, W05105. DOI  10.1029/2003WR002617
  39. Renard P, de Marsily G (1997) Calculating equivalent permeability: a review. Adv Water Resour 20(5–6):253–278CrossRefGoogle Scholar
  40. Renard P, Le Loch G, Ledoux E, de Marsily G, Mackay R (2000) A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media. Water Resour Res 36(12):3567–3580CrossRefGoogle Scholar
  41. Ritzi RW, Jayne DF, Zahradnik AJ, Field AA, Fogg GE (1994) Geostatistical modeling of heterogeneity in glaciofluvial, buried-valley aquifers. Ground Water 32(4):666–674CrossRefGoogle Scholar
  42. Robin MJL, Gutjahr AL, Sudicky EA, Wilson JL (1993) Cross-correlated random field generation with the direct Fourier transform method. Water Resour Res 29(7):2385–2397CrossRefGoogle Scholar
  43. Rubin Y (1995) Flow and transport in bimodal heterogeneous formations. Water Resour Res 31:2461–2468CrossRefGoogle Scholar
  44. Sanchez-Vila X, Guadagnini A, Carrera J (2006) Representative hydraulic conductivities in saturated groundwater flow. Rev Geophys 44(3), RG3002. DOI  10.1029/2005RG000169
  45. Scheibe T, Yabusaki S (1998) Scaling of flow and transport behavior in heterogeneous groundwater systems. Adv Water Resour 22(3):223–238CrossRefGoogle Scholar
  46. Tartakovsky DM, Guadagnini A (2004) Effective properties of random composites. Siam J Sci Comput 26:625–635CrossRefGoogle Scholar
  47. Tartakovsky DM, Neuman SP (1998) Transient effective hydraulic conductivities under slowly and rapidly varying mean gradients in bounded three-dimensional random media. Water Resour Res 34:21–32Google Scholar
  48. Tompson AFB, Carle SF, Rosenberg ND, Maxwell RM (1999) Analysis of groundwater migration from artificial recharge in a large urban aquifer: a simulation perspective. Water Resour Res 35(10):2981–2998CrossRefGoogle Scholar
  49. Webb EK (1995) Simulation of braided channel topology and topography. Water Resour Res 31(10):2603–2611CrossRefGoogle Scholar
  50. Webb EK, Anderson MP (1996) Simulation of preferential flow in three-dimensional, heterogeneous conductivity fields with realistic internal architecture. Water Resour Res 32(3):533–545CrossRefGoogle Scholar
  51. Weissmann GS, Fogg GE (1999) Multi-scale alluvial fan heterogeneity modelled with transition probability geostatistics in a sequence stratigraphic framework. J Hydrol 226(1–2):48–65CrossRefGoogle Scholar
  52. Weissmann GS, Zhang Y, LaBolle EM, Fogg GE (2002) Dispersion of groundwater age in an alluvial aquifer system. Water Resour Res 38 (10), 1198. DOI  10.1029/2001WR000907
  53. Wen XH, Gomez Hernandez JJ (1996) Upscaling hydraulic conductivities in heterogeneous media: an overview. J Hydrol 183(1–2):9–32Google Scholar
  54. Wen XH, Durlofsky LJ, Edwards MG (2003) Use of border regions for improved permeability upscaling. Math Geol 35:521–547CrossRefGoogle Scholar
  55. Wen XH, Chen YG, Durlofsky LJ (2006) Efficient 3D implementation of local-global upscaling for reservoir simulation. SPE J 11:443–453Google Scholar
  56. Western AW, Blöschl G, Grayson RB (1998) How well do indicator variograms capture the spatial connectivity of soil moisture? Hydrol Process 12(12):1851–1868CrossRefGoogle Scholar
  57. Western AW, Blöschl G, Grayson RB (2001) Toward capturing hydrologically significant connectivity in spatial patterns. Water Resour Res 37(1):83–97CrossRefGoogle Scholar
  58. Yong Z (2004) Upscaling conductivity and porosity in three-dimensional heterogeneous porous media. Chin Sci Bull 49(22):2415–2423CrossRefGoogle Scholar
  59. Zappa G, Bersezio R, Felletti F, Giudici M (2006) Modeling heterogeneity of gravel-sand, braided stream, alluvial aquifers at the facies scale. J Hydrol 325(1–4):134–153CrossRefGoogle Scholar
  60. Zhang Y, Gable CW, Person M (2006) Equivalent hydraulic conductivity of an experimental stratigraphy: implications for basin-scale flow simulations. Water Resour Res 42(5), W05404. DOI  10.1029/2005WR004720
  61. Zhang Y, Person M, Gable CW (2007) Representative hydraulic conductivity of hydrogeologic units: insights from an experimental stratigraphy. J Hydrol 339:65–78CrossRefGoogle Scholar
  62. Zinn B, Harvey CF (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

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of HydrologyUniversity of BayreuthBayreuthGermany
  2. 2.Department of Land, Air and Water Resources and Department of GeologyUniversity of California, DavisDavisUSA

Personalised recommendations