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The use of connectivity clusters in stochastic modelling of highly heterogeneous porous media

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Abstract

Stochastic geostatistical techniques are essential tools for groundwater flow and transport modelling in highly heterogeneous media. Typically, these techniques require massive numbers of realizations to accurately simulate the high variability and account for the uncertainty. These massive numbers of realizations imposed several constraints on the stochastic techniques (e.g. increasing the computational effort, limiting the domain size, grid resolution, time step and convergence issues). Understanding the connectivity of the subsurface layers gives an opportunity to overcome these constraints. This research presents a sampling framework to reduce the number of the required Monte Carlo realizations utilizing the connectivity properties of the hydraulic conductivity distributions in a three-dimensional domain. Different geostatistical distributions were tested in this study including exponential distribution with the Turning Bands (TBM) algorithm and spherical distribution using Sequential Gaussian Simulation (SGSIM). It is found that the total connected fraction of the largest clusters and its tortuosity are highly correlated with the percentage of mass arrival and the first arrival quantiles at different control planes. Applying different sampling techniques together with several indicators suggested that a compact sample representing only 10% of the total number of realizations can be used to produce results that are close to the results of the full set of realizations. Also, the proposed sampling techniques specially utilizing the low conductivity clustering show very promising results in terms of matching the full range of realizations. Finally, the size of selected clusters relative to domain size significantly affects transport characteristics and the connectivity indicators.

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  1. https://www.iamg.org/documents/oldftp/VOL24/v24-1-7.zip

References

  • Adams EE, Gelhar LW (1992) Field study of dispersion in a heterogeneous aquifer, 2: spatial moment analysis. Water Resour Res 28(12):3293–3307

    Article  Google Scholar 

  • Awad MA, Hassan AE, Bekhit HM (2012) Propagating conceptual and parametric uncertainty from regional to local groundwater models. The IAHR Sixth International Groundwater Symposium. KISR, CRC Press

    Google Scholar 

  • Berkowitz B, Braester C (1991) Dispersion in sub-representative elementary volume fracture networks: percolation theory and random walk approaches. Water Resour Res 27(12):3159–3164

    Article  Google Scholar 

  • Berkowitz B, Balberg I (1992) Percolation approach to the problem of hydraulic conductivity in porous media. Transp Porous Media 9(3):275–286

    Article  Google Scholar 

  • Berkowitz B, Balberg I (1993) Percolation theory and its application to groundwater hydrology. Water Resour Res 29(4):775–794

    Article  Google Scholar 

  • Berkowitz B, Ewing RP (1998) Percolation theory and network modeling applications in soil physics. Surv Geophys 19:23–72

    Article  Google Scholar 

  • Bianchi M, Zheng C, Geoffrey RT, Gorelick SM (2008) Evaluation of Fickian and non-Fickian models for solute transport in porous media containing decimeter-scale preferential flow paths, in calibration and reliability in groundwater modelling: credibility of modelling proceedings of model CARE 2007 conference. IAHS Publ 320:9–14

    Google Scholar 

  • Bianchi M, Zheng C, Geoffrey RT, Gorelick SM (2011a) Investigation of small-scale preferential flow with a forced-gradient tracer test. Ground Water. https://doi.org/10.1111/j.1745-6584.2010.00746.x

  • Bianchi M, Zheng C, Wilson C, Geoffrey RT, Liu G, Gorelick SM (2011b) Spatial connectivity in a highly heterogeneous aquifer: from cores to preferential flow paths. Water Resour Res 47:W05524. https://doi.org/10.1029/2009WR008966

    Article  Google Scholar 

  • Boggs JM, Adams EE (1992) Field study of dispersion in a heterogeneous aquifer, 4, investigation of adsorption and sampling bias. Water Resour Res 28(12):3325–3336

    Article  Google Scholar 

  • Boggs JM, Beard LM, Long SE, McGee MP, MacIntyre WG, Antworth CP, Stauffer TB (1993) Database for the second macrodispersion experiment (MADE-2). Tech. Rep. TR-102072, Electric Power Res. Inst., Palo alto, California

  • Boggs JM, Young SC, Beard LM, Gelhar LW, Rehfeldt KR, Adams EE (1992) Field study of dispersion in a heterogeneous aquifer, 1, overview and site description. Water Resour Res 28(12):3281–3291

    Article  Google Scholar 

  • Broadbent SR, Hammersley JM (1957) Percolation processes, crystals and mazes. Proc Camb Philos Soc 53(629–641):1957

    Google Scholar 

  • Brandsaeter I, Wist HT, Naess A, Lia O, Arntzen OJ, Ringrose PS, Martinius AW, Lerdahl TR (2001) Ranking of stochastic realizations of complex tidal reservoirs using streamline criteria. Pet Geosci 7:S53–S63

    Article  Google Scholar 

  • Chin DA, Wang T (1992) An investigation of the validity of first-order stochastic dispersion theories in isotropic porous media. Water Resour Res 28(6):1531–1542

    Article  Google Scholar 

  • Dagan G (1986) Statistical theory of groundwater flow and transport: pore to laboratory, laboratory to formation, and formation to regional scale. Water Resour Res 22(9):120S–134S

    Article  Google Scholar 

  • Deutsch CV, Journel A (1992) GSLIB, Geostatistical Software Library. Oxford Univ. Press, New York

    Google Scholar 

  • Deutsch CV (1998) Fortran programs for calculating connectivity of three-dimensional numerical models and for ranking multiple realizations. Comput Geosci 24(1):69–76

    Article  Google Scholar 

  • Deutsch CV, Journel AG (1998) GSLIB Geostatistical Software Library and user’s guide, 2nd edn. Oxford Univ. Press, N Y, p 369

    Google Scholar 

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

    Article  Google Scholar 

  • Dogan M, Van Dam RL, Liu G, Meerschaert MM, Butler J, Bohling GC, Benson DA, Hyndman DW (2014) Predicting flow and transport in highly heterogeneous alluvial aquifers. GeophysRes Lett 41:7560–7565. https://doi.org/10.1002/2014GL061800

    Article  Google Scholar 

  • Freeze RA, Cherry JA (1979) Groundwater. Prentice-Hall, Englewood Cliffs, NJ, p 1979

    Google Scholar 

  • Flory PJ (1941) Molecular size distribution in three dimensional polymers. J Am Chem Soc 63:3083–3100

    Article  Google Scholar 

  • Gelhar LW (1986) Stochastic subsurface hydrology: from theory to applications. Water Resour Res 22(9):135S–145S

    Article  Google Scholar 

  • Gelhar LW (1993) Stochastic subsurface hydrology. Prentice-Hall, Old Tappan, N. J

  • Gorelick SM, Liu G, Zheng C (2005) Quantifying mass transfer in permeable media containing conductive dendritic networks. Geophys Res Lett 32:L18402. https://doi.org/10.1029/2005GL023512

    Article  Google Scholar 

  • Gwo JP, Toran LD, Morris M, Wilson G (1996) Subsurface stormflow modeling with sensitivity analysis using a latin-hypercube sampling technique. Ground Water. 34. https://doi.org/10.1111/j.1745-6584.1996.tb02075.x

  • Harbaugh AW, Banta ER, Hill MC, McDonald MG (2000) MODFLOW-2000, the U.S. Geological survey modular ground-water model—user guide to modularization concepts and the ground-water flow process. USGS Open-File Report 00-92. U.S. Geological Survey, Reston, p 121

  • Hassan AE, Cushman JH, Delleur JW (1998) A Monte Carlo assessment of Eulerian flow and transport perturbation models. Water Resour Res 34(5):1143–1163. https://doi.org/10.1029/98WR00011

    Article  Google Scholar 

  • Hird KB, Dubrule O (1998) Quantification of reservoir connectivity for reservoir description applications. SPE Reserv Eval Eng 1(1):12–17

    Article  Google Scholar 

  • Jussel P, Stauffer F, Dracos T (1994) Transport modeling in heterogeneous aquifers, 2, three-dimensional transport model and stochastic numerical tracer experiments. Water Resour Res 30(6):1819–1831

    Article  Google Scholar 

  • Kupfersberger H, Deutsch C (1999) Ranking stochastic realizations for improved aquifer response uncertainty assessment. J Hydrol 223:54–65

    Article  Google Scholar 

  • Koltermann C, Gorelick SM (1996) Heterogeneity in sedimentary deposits: a review of structure-imitating, process-imitating, and descriptive approaches. Water Resour Res 32(9):2617–2658

    Article  Google Scholar 

  • Knudby C, Carrera J (2005) On the relationship between indicators of geostatistical, flow and transport connectivity. Adv Water Resour 28(4):405–421. https://doi.org/10.1016/j.advwatres.2004.09.001

    Article  Google Scholar 

  • Lahkim M, Garcia L, Nuckols J (1999) Stochastic modeling of exposure and risk in contaminated heterogeneous aquifer, 2, application of Latin hypercube sampling. Env Eng Sc 16(5):329–343

    Article  Google Scholar 

  • Liu G, Zheng C, Gorelick SM (2004) Limits of applicability of the advection-dispersion model in aquifers containing connected high conductivity channels. Water Resour Res 40:W08308. https://doi.org/10.1029/2003WR002735

    Google Scholar 

  • Mantoglou A, Wilson JL (1982) The turning bands method for simulation of random fields using line generation by a spectral method. Water Resour Res 18:1379–1394. https://doi.org/10.1029/WR018i005p01379

    Article  Google Scholar 

  • Matheron G (1973) The intrinsic random functions and their application. Adv Appl Prob 5:439–468

    Article  Google Scholar 

  • McLennan J, and Deutch C (2005) Ranking geostatistical realizations by measures of connectivity. In: SPE/PS-CIM/CHOA 98168. PS2005–437; p 13

  • Mo H, Bai M, Lin D, Roegiers JC (1998) Study of flow and transport in fracture network using percolation theoy. Appl Math Model 22(1998):277–291

    Article  Google Scholar 

  • Neuman SP (1997) Stochastic approach to subsurface flow and transport: a view to the future. In: Dagan G, Neuman SP (eds) Subsurface flow and transport—a stochastic approach. International Hydrology Series, Unesco

    Google Scholar 

  • Pohll G, Hassan AE, Chapman J, Papelis C, Andricevic R (1999) Modeling groundwater flow and radioactive transport in a fractured aquifer. Groundwater 37(5):770–784

    Article  Google Scholar 

  • Pohlmann K, Hassan A, Chapman J (2000) Description of hydrogeologic heterogeneity and evaluation of radionuclide transport at an underground nuclear test. Contam Hydrol 44(3–4):353–386

    Article  Google Scholar 

  • Renard P, Allard D (2013) Connectivity metrics for subsurface flow and transport. Adv Water Resour 51:168–196

    Article  Google Scholar 

  • Scheidt C, Caers J (2009) Representing spatial uncertainty using distances and kernels. Math Geosci 41(4):397–419

    Article  Google Scholar 

  • Shirley C (2000) Evaluation of computationally efficient percolation cluster metrics as hydrologic predictors in geologic media exhibiting multiple scales of heterogeneity. In Computational Methods in Water Resources XIII, Bentley et al. (eds.), Balkema Pub., Rotterdam, 729–733

  • Stockmayer WH (1943) Theory of molecular sized distribution and gel formation in branched-chain polymers. J Chem Phys 11:45–55

    Article  Google Scholar 

  • Tyukhova AR, Kinzelbach W, Willmann M (2015). Delineation of connectivity structures in 2-D heterogeneous hydraulic conductivity fields. Water Resour Res

  • Yanuka A (1992) Percolation theory approach to transport phenomena in porous media. Transp Porous Media 7:265–282, 1992

    Article  Google Scholar 

  • Zheng C (1990) MT3D, a modular three-dimensional transport model for simulation of advection, dispersion, and chemical reactions of contaminants in groundwater systems. Report to the Kerr environmental research laboratory. US Environmental Protection Agency, Ada, OK

    Google Scholar 

  • Zheng D (2003) Stochastic methods for flow in porous media, coping with uncertainties. Academic Press, San Diego

    Google Scholar 

  • 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. https://doi.org/10.1029/2001WR001146

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Resources and Environment Department, Dar Group, Giza, Egypt

    Mohamed A. Awad

  2. Department of Irrigation and Hydraulics, Faculty of Engineering, Cairo University, Giza, Egypt

    Mohamed A. Awad, Ahmed E. Hassan & Hesham M. Bekhit

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  1. Mohamed A. Awad
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  2. Ahmed E. Hassan
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  3. Hesham M. Bekhit
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Correspondence to Mohamed A. Awad.

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Awad, M.A., Hassan, A.E. & Bekhit, H.M. The use of connectivity clusters in stochastic modelling of highly heterogeneous porous media. Arab J Geosci 11, 98 (2018). https://doi.org/10.1007/s12517-018-3432-7

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