Mathematical Geosciences

, 40:753 | Cite as

Identifying the Representative Elementary Volume for Permeability in Heterolithic Deposits Using Numerical Rock Models

  • Kjetil NordahlEmail author
  • Philip S. Ringrose


Using a range of realistic 3D numerical lithofacies (dm-scale) models of ripple laminated sandstone intercalated with mudstone we evaluate how single-phase permeability varies as a function of sample support. The models represent a range of mudstone content which is typical for tidal deposits. Furthermore, the spatial distribution of flow barriers (i.e. mudstone) is not random, but governed by sedimentological rules giving a variable anisotropy ratio as a function of mudstone content. Both vertical and horizontal permeability are found to vary at small sample volumes, but these fluctuations reduce as the sample volume increases. The vertical permeability increases while the horizontal permeability is nearly constant as a function of sample support for small mudstone contents. For higher mudstone content, the horizontal permeability decreases while the vertical permeability is nearly constant as a function of sample support. We propose a criterion, based on a normalised standard deviation, to determine the Representative Elementary Volume (REV). The size of the REV is dependent on both the property measured (vertical and horizontal permeability) and the correlation lengths of the lithological elements (i.e. lithofacies). Based on this we identify three flow upscaling regimes that each require a different method for upscaling: (1) layered systems where the arithmetic and harmonic averages are appropriate, (2) systems close to the percolation threshold where a percolation model should be used, and (3) discontinuous systems where an effective medium method provides the best estimate of permeability. The work gives, by using numerical experiments on a range of heterogeneous systems, a new insight in determination of the REV for permeability at the lithofacies scale and its relation to sedimentological parameters.


Permeability Representative elementary volume Heterolithic sandstone 


  1. Anguy YM, Ehrlich R, Prince CM, Riggert VL, Bernard D (1994) The sample support problem for permeability assessment in sandstone reservoir. In: Yarus JM, Chambers RL (eds) Stochastic modeling in geostatistics: principles, methods, and case studies. AAPG computer applications in geology, vol 3, pp 37–54 Google Scholar
  2. Baas JH (1994) A flume study on the development and equilibrium morphology of current ripples in very fine sand. Sedimentology 41:185–209 CrossRefGoogle Scholar
  3. Barton M, Van der Vlugt F, Slik P (2003) Modeling uncertainty in discount factors for turbidite channel reservoirs (abs.). Presented at the AAPG annual meeting 2003, Salt Lake City, UT, USA, 11–14 May 2003 Google Scholar
  4. Baveye P, Sposito G (1984) The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers. Water Resour Res 20:521–530 CrossRefGoogle Scholar
  5. Bear J (1972) Dynamics of fluids in porous media. American Elsevier, New York Google Scholar
  6. Bear J (1988) Dynamics of fluids in porous media. Dover, New York Google Scholar
  7. Begg SH, King PR (1985) Modelling the effects of shales on reservoir performance: calculation of effective vertical permeability. Society of petroleum engineers paper 13529, presented at SPE symposium on reservoir simulation, Dallas, 10–13 February 1985 Google Scholar
  8. Begg SH, Carter RR, Dranfield P (1989) Assigning effective values to simulator grid-lock parameters heterogeneous reservoirs. In: SPE reservoir engineering, pp 455–463; Society of petroleum engineers paper 16754. Google Scholar
  9. Bernabé Y, Mok U, Evans B, Herrmann FJ (2004) Permeability and storativity of binary mixtures of high- and low-permeability materials. J Geophys Res 109:B12207 CrossRefGoogle Scholar
  10. Brandsæter I, Wist HT, Næss A, Lia O, Arntzen OJ, Ringrose PS, Martinius AW, Lerdal TR (2001) Ranking of stochastic realisations of complex tidal reservoirs using streamline simulation criteria. Spec Publ Pet Geosci 7:S53–S63 Google Scholar
  11. Corbett PWM, Jensen JL (1992) Estimating the mean permeability: how many measurements do we need? First Break 10(3):89–94 Google Scholar
  12. Corbett PMW, Ringrose PS, Jensen JL, Sorbie KS (1992) Laminated clastic reservoirs; the interplay of capillary pressure and sedimentary architecture. Society of petroleum engineers paper 24699, presented at the 67th annual technical SPE conference, Washington DC, 4–7 October 1992 Google Scholar
  13. Corbett PWM, Jensen JL, Sorbie KS (1998) A review of up-scaling and cross-scaling issues in core and log data interpretation and prediction. In: Harvey PK, Lovell MA (eds) Core-log integration. Geological society special publication, vol 136, London, pp 9–16 Google Scholar
  14. Corbett P, Anggraeni S, Bowen D (1999) The use of the probe permeameter in carbonates—addressing the problems of permeability support and stationarity. Log Analyst 40(5):316–326 Google Scholar
  15. Dagan G (1979) Models of groundwater flow in statistically homogeneous porous formations. Water Resour Res 15(1):47–63 CrossRefGoogle Scholar
  16. Desbarats A (1987) Support effects and the spatial averaging of transport properties. Math Geol 21(3):383–389 CrossRefGoogle Scholar
  17. Deutch C (1989) Calculating effective absolute permeability in sandstone/shale sequences. SPE Form Eval 4(3):343–348 Google Scholar
  18. Durlofsky LJ (1991) Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour Res 27(5):699–708 CrossRefGoogle Scholar
  19. Elfenbein C, Husby Ø, Ringrose PS (2005) Geologically-based estimation of kv/kh ratios: an example from the garn formation, Tyrihans Field, Mid-Norway. In: Dore AG, Vining B (eds) Petroleum geology: North-West Europe and global perspectives. Proceedings of the 6th petroleum geology conference. The Geological Society, London Google Scholar
  20. Enderlin MB, Hansen DKT, Hoyt BR (1991) Rock volumes: considerations for relating well log and core data. In: Lake LW, Carroll HB, Wesson TC (eds) Reservoir characterization II. Academic Press, San Diego, pp 277–288 Google Scholar
  21. Gomez-Hernandez JJ, Journel AG (1990) Stochastic characterization of grid-block permeabilities: from point values to block tensors. In: Guérillot D, Guillon O (eds) 2nd European conference on the mathematics of oil recovery. Institute François du Pétrol, Paris Google Scholar
  22. Haldorsen HH (1986) Simulator parameter assignment and the problem of scale in reservoir engineering. In: Lake LW, Caroll HB (eds) Reservoir characterization. Academic Press, Orlando, pp 293–340 Google Scholar
  23. Haldorsen HH, Chang DM (1986) Notes on stochastic shales; from outcrop to simulation mode. In: Lake LW, Caroll HB (eds) Reservoir characterization. Academic Press, Orlando, pp 445–485 Google Scholar
  24. Hartkamp-Bakker CA, Donselaar ME (1993) Permeability patterns in point bar deposits: Tertiary Loranca Basin, central Spain. In: Flint S, Bryant ID (eds) The geological modeling of hydrocarbon reservoirs and outcrop analogues. Special publication No. 15 of the international association of sedimentologist, pp 157–168 Google Scholar
  25. Hassanizadeh M, Gray WG (1983) General conservation equations for multi-phase systems, 1: averaging procedure. In: Pinder GF (ed) Flow through porous media, recent developments: a computational mechanics publications, pp 1–16 Google Scholar
  26. Henriette A, Jacquin CG, Adler PM (1989) The effective permeability of heterogeneous porous media. Phys Chem Hydrodyn 11(1):63–80 Google Scholar
  27. Hurst A (1993) Sedimentary flow units in hydrocarbon reservoirs: some shortcomings and a case for high-resolution permeability data. In: Flint S, Bryant ID (eds) The geological modeling of hydrocarbon reservoirs and outcrop analogues. Special publication No. 15 of the international association of sedimentologists, pp 191–204 Google Scholar
  28. Jackson MD, Muggeridge AH, Yoshida S, Johnson HD (2003) Upscaling permeability measurements within complex heterolithic tidal sandstones. Math Geol 35(5):446–454 CrossRefGoogle Scholar
  29. Jackson MD, Yoshida S, Muggeridge AH, Johnson HD (2005) Three-dimensional reservoir characterization and flow simulation of heterolithic tidal sandstones. Am Assoc Pet Geol Bull 89(4):507–528 Google Scholar
  30. Jensen JL, Lake LW, Corbett PWM, Goggin DJ (2000) Statistics for petroleum engineers and geoscientists, 2nd edn. Elsevier, Amsterdam Google Scholar
  31. Kløv T, Øren P-E, Stensen JÅ, Lerdahl TR, Berge LI, Bakke S, Boassen T, Virnovsky G (2003) Pore-to-field scale modeling of WAG. Society of petroleum engineers paper 84549, presented at the SPE annual technical conference and exhibition, Denver, CO, USA, 5–8 October 2003 Google Scholar
  32. Koltermann CE, Gorelick SM (1996) Heterogeneity in sedimentary deposits: a review of structure imitating, process-imitating and descriptive approaches. Water Resour Res 32(9):2617–2658 CrossRefGoogle Scholar
  33. Li D, Lake LW (1995) Scaling fluid flow through heterogeneous permeable media. SPE Adv Technol Ser 3(1):188–197 Google Scholar
  34. McKinley JM, Lloyd CD, Ruffel AH (2004) Use of variography in permeability characterization of visually homogeneous sandstone reservoir with examples from outcrop studies. Math Geol 36(7):761–779 CrossRefGoogle Scholar
  35. Nio A-D, Yang C-S (1991) Diagnostic attributes of clastic tidal deposits: a review. In: Smith DG, Reinson GE, Zaitlin BA, Rahmani RA (eds) Clastic tidal sedimentology. Canadian society of petroleum geologist, memoir 16, pp 3–28 Google Scholar
  36. Nordahl K (2004) A petrophysical evaluation of tidal heterolithic deposits: application of a near wellbore model for reconciliation of scale dependent well data. PhD thesis, Norwegian University of Science and Technology, Trondheim, Norway Google Scholar
  37. Nordahl K, Ringrose PS, Wen R (2005) Petrophysical characterization of a heterolithic tidal reservoir interval using a process-based modeling tool. Pet Geosci 11:17–28 Google Scholar
  38. Norris RJ, Lewis JJM (1991) The geological modeling of effective permeability in complex heterolithic faces. Society of petroleum engineers paper 22692, presented at the 66th annual technical conference and exhibition, Dallas, TX, USA, 6–9 October 1991 Google Scholar
  39. Noetinger B (1994) The effective permeability of a heterogeneous porous medium. Transp Porous Media 15:99–127 CrossRefGoogle Scholar
  40. Noetinger B, Jacquin C (1991) Experimental tests of a simple permeability composition formula. Society of petroleum engineers paper 22841 Google Scholar
  41. Olea RA (1991) Geostatistical glossary and multilingual dictionary. International association for mathematical geology studies in mathematical geology, vol 3. Oxford University Press, London Google Scholar
  42. Oost AP, Baas JH (1994) The development of small scale bedforms in tidal environments: an empirical model for unsteady flow and its applications. Sedimentology 41:883–903 CrossRefGoogle Scholar
  43. Pickup GE, Ringrose PS, Jensen JL, Sorbie KS (1994) Permeability tensors for sedimentary structures. Math Geol 26(2):227–250 CrossRefGoogle Scholar
  44. Pickup GE, Ringrose PS, Corbett PWM, Jensen JL, Sorbie KS (1995) Geology, geometry and effective flow. Pet Geosci 1:37–42 Google Scholar
  45. Pickup GE, Ringrose PS, Sharif A (2000) Steady-state upscaling: from lamina-scale to full-field model. SPE J 5(2):208–217 Google Scholar
  46. Rapaport L (1955) Scaling laws for use in design and operation of water-oil flow models. Trans AIME 204:143–150 Google Scholar
  47. Reineck H-E, Singh IB (1980) Depositional sedimentary environments. With reference to terrigenous clastics, 2nd edn. Springer, Berlin Google Scholar
  48. Reineck HE, Wunderlich (1968) Classification and origin of flaser and lenticular bedding. Sedimentology 11:99–104 CrossRefGoogle Scholar
  49. Renard Ph, de Marsily G (1997) Calculating equivalent permeability: a review. Adv Water Resour 20(5–6):253–278 CrossRefGoogle Scholar
  50. Ringrose PS, Pickup GE, Jensen JL, Forrester M (1999) The Ardross reservoir gridblock analogue: sedimentology, statistical representative and flow upsclaing. In: Schatzinger R, Jordan J (eds) Reservoir characterization—recent advances. Am assoc petrol geol memoir, vol 71, pp 265–276 Google Scholar
  51. Ringrose PS, Skjetne E, Elfenbein C (2003) Permeability estimation functions based on forward modeling of sedimentary heterogeneity. Society of petroleum engineers paper 84275, presented at the SPE annual technical conference and exhibition, Denver, CO, USA, 5–8 October 2003 Google Scholar
  52. Ringrose P, Nordahl K, Wen R (2005) Vertical permeability estimation in heterolithic tidal deltaic sandstones. Pet Geosci 11:29–36 CrossRefGoogle Scholar
  53. Rubin DM (1987) Cross-bedding, bedforms, and paleocurrents. Concepts in sedimentology and paleontology, vol 1. Society of Economic Paleontologists and Mineralogists, p 187 Google Scholar
  54. Scaglioni P, Ruvo L, Cozzi M (2006) Implicit net-to-gross in the petrophysical characterization of thin-layered reservoirs. Pet Geosci 12:325–333 Google Scholar
  55. Sahimi M (1995) Flow and transport in porous media and fractured rock. From classical methods to modern approaches. VCH, Weinheim Google Scholar
  56. Stauffer D, Aharony A (1992) Introduction to percolation theory, 2nd edn. Taylor & Francis, London Google Scholar
  57. Theting TG, Rustad AB, Lerdahl TR, Stensen JÅ, Boassen T, Øren P-E, Bakke S, Ringrose P (2005) Pore-to-field multi-phase upscaling for a depressurization process. Presented at the 13th European symposium on improved oil recovery, Budapest, Hungary, 25–27 April 2005 Google Scholar
  58. Tidwell VC, Wilson JL (1999) Permeability upscaling measured on a block of Berea Sandstone: results and interpretation. Math Geol 31(7):749–769 CrossRefGoogle Scholar
  59. Tidwell VC, Wilson JL (2000) Heterogeneity, permeability patterns, and permeability upscaling: physical characterization of a block of Massillon Sandstone exhibiting nested scales of heterogeneity. SPE Res Eval Eng 3(4):283–291 Google Scholar
  60. Weber KJ (1982) Influence of common sedimentary structures on fluid flow in reservoir models. J Pet Technol 44:665–672 Google Scholar
  61. Wen R, Martinius AW, Næss A, Ringrose P (1998) Three-dimensional simulation of small-scale heterogeneity in tidal deposits—a process-based stochastic simulation method. In: Buccianti A, Nardi G, Potenza R (eds) Proceedings of the 4th annual conference of the international association of mathematical geology (IAMG), Ischia, pp 129–134 Google Scholar
  62. White CD, Horne RN (1987) Computing absolute transmissibility in the presence of fine-scale heterogeneity. Society of petroleum engineers paper 16011, presented at the 9th SPE symposium of reservoir simulation, San Antonio, TX, 1–4 February 1987 Google Scholar
  63. Worthington PF (1994) Effective integration of core and log data. Marine Pet Geol 11(4):457–466 CrossRefGoogle Scholar
  64. Worthington PF (2003) The effect of scale on the petrophysical estimation of intergranular permeability. Presented at the 44th SPWLA annual logging symposium, Paper A Google Scholar
  65. Zhang P, Langeland H, Ringrose P, Nordahl K, Elfenbein C, Næss A (2004) Permeability rescaling and near-wellbore modeling of heterogeneities in Lower Jurassic tidal-influenced Tilje Formation, Heidrun Field. Presented at the 45th annual logging symposium, Noordwijk, The Netherlands, 6–9 June 2004 Google Scholar

Copyright information

© International Association for Mathematical Geology 2008

Authors and Affiliations

  1. 1.StatoilHydro Research CentreTrondheimNorway

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