Characterization of the Amount of Swept and Unswept Fractions Between Two Wells in the Presence of Reservoir Anisotropy

Document Type : Research Paper


Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran


The reservoir heterogeneity controls interwell connectivity and affects reservoir dynamics. An approach is to use continuum percolation to study the flow behavior of low to intermediate net-to-gross reservoirs. In this study, reservoir models with a permeability contrast have been used, and the interwell connectivity between two wells and the remaining unswept oil has been determined. The percolation parameters, including the amount of recoverable oil connected between two wells and the amount of unswept oil (also referred to as dangling end fraction (that control fluid displacement (e.g. waterflooding) vary as a function of sand body size and reservoir size. These properties show a power-law function of net-to-gross (i.e. occupation fraction) with some exponents called critical exponents. There exist a few publications on the numerical values of these parameters. The main contribution of this study is to investigate the effects of reservoir anisotropy on the percolation parameters. To determine the swept (backbone) fraction connected between two wells, the flow-based criteria depending on the system size have been proposed. The results show that the critical exponents for the backbone and dangling ends are in the range of 0.3to 0.45 and -0.45 to -0.20.   It is notified that the limitation to perform simulations on infinite systems results in a range for these exponents, although there exist unique values for infinite systems. Moreover, a sensitivity analysis is implemented to find the correct flow-based criteria for the backbone. The results of this study extend the applicability of the percolation properties curves for anisotropic reservoirs. 


  1. King P R (1990) The connectivity and conductivity of overlapping sand bodies, in North Sea Oil and Gas Reservoirs—II, Springer, 353-362. ##
  2. Sadeghnejad S, Masihi M, King P R, Shojaei A, Pishvaei M (2010) Effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory. Physical Review E, 81, 6: 061119. ##
  3. Soltani A, Sadeghnejad S (2018) Scaling and critical behavior of lattice and continuum porous media with different connectivity configurations, Physica A: Statistical Mechanics and its Applications, 508: 376-389. ##
  4. King P R and Masihi M (2018) Percolation theory in reservoir engineering, 1st edition,: world scientific, 1-384. ##
  5. Chattopadhyay P B, Vedanti N (2016) Fractal characters of porous media and flow analysis, in Fractal Solutions for Understanding Complex Systems in Earth Sciences, Springer, 67-77. ##
  6. Hunt A G, Ghanbarian B (2016) Percolation theory for solute transport in porous media: Geochemistry, geomorphology, and carbon cycling, Water Resources Research, 52, 9: 7444-7459. ##
  7. Hunt A, Ewing R, Ghanbarian B (2014) Percolation theory for flow in porous media, Springer, 880: 3319037714. ##
  8. Sahini M, Sahimi M (1994) Applications of percolation theory, 1st edition, CRC Press, 1-276. ##
  9. Broadbent S R, Hammersley J M (1957) Percolation processes: I. Crystals and mazes, Cambridge University Press, 53, 3: 629-641. ##
  10. Stauffer D, Aharony A, Introduction to percolation theory (1992) Taylor and Francis, London, 2nd edittion, 1-192. ##
  11. Berkowitz B, Balberg I (1992) Percolation approach to the problem of hydraulic conductivity in porous media. Transport in Porous Media, 9, 3: 275-286. ##
  12. Berkowitz B (1995) Analysis of fracture network connectivity using percolation theory. Mathematical Geology, 27, 4: 467-483. ##
  13. Rintoul M D, Torquato S (1997) Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model. Journal of Physics A: Mathematical and General, 30, 16: 305-4470. ##
  14. Hoshen J, Berry M, Minser K (1997) Percolation and cluster structure parameters: The enhanced Hoshen-Kopelman algorithm. Physical Review E, 56, 2: 1455. ##
  15. Dokholyan N V, Lee Y, Buldyrev S V, Havlin S, King P R, Stanley H E (1998) Scaling of the distribution of shortest paths in percolation. Journal of Statistical Physics, 93, 3: 603-613. ##
  16. Berkowitz B, Balberg I (1993) Percolation theory and its application to groundwater hydrology, Water Resources Research, 29, 4: 775-794. ##
  17. Vogel E E, Lebrecht W, Valdés J F (2010) Bond percolation for homogeneous two-dimensional lattices. Physica A: Statistical Mechanics and its Applications, 389, 8: 1512-1520. ##
  18. Lee S B, Torquato S (1990) Monte Carlo study of correlated continuum percolation: Universality and percolation thresholds. Physical Review A, 41, 10: 5338. ##
  19. Lorenz C D, Ziff R M (2001) Precise determination of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algorithm, The Journal of Chemical Physics, 114, 8: 3659-3661. ##
  20. Masihi M, King P R, Nurafza P R (2005) Fast estimation of performance parameters in fractured reservoirs using percolation theory, Society of Petroleum Engineers, OnePetro. ##
  21. Masihi M, King P R (2007) A correlated fracture network: modeling and percolation properties, Water Resources Research, 43, 7: 0043-1397. ##
  22. Masihi M, King P R, Nurafza P R (2008) Connectivity prediction in fractured reservoirs with variable fracture size: analysis and validation, SPE Journal, 13, 1: 88-98. ##
  23. Sadeghnejad S, Masihi M, King P R, Shojaei A, Pishvaie M (2011) A reservoir conductivity evaluation using percolation theory, Petroleum Science and Technology, 29, 10: 1041-1053. ##
  24. Sadeghnejad S, Masihi M, Pishvaie M, King P R (2013) Rock type connectivity estimation using percolation theory, Mathematical Geosciences, 45, 3: 321-340. ##
  25. Tavagh-Mohammadi B, Masihi M, Ganjeh-Ghazvini M (2016) Point-to-point connectivity prediction in porous media using percolation theory. Physica A: Statistical Mechanics and its Applications, 460: 304-313. ##
  26. Sadeghnejad S, Masihi M (2016) Point to point continuum percolation in two dimensions. Journal of Statistical Mechanics: Theory and Experiment, 10: 103210. ##
  27. King P R, Buldyrev S V, Dokholyan N V, Havlin S, Lee Y, Paul, G., Stanley, H.E. Vandesteeg N (2001) Predicting oil recovery using percolation theory. Petroleum Geoscience, 7: 105-107. ##
  28. Nurafza P R, King P R, Masihi M (2006) Facies connectivity modelling: analysis and field study, Society of Petroleum Engineers, OnePetro. ##
  29. Wen H, King P R, Muggeridge A H, Vittoratos E S (2014) Using percolation theory to estimate recovery from poorly connected sands using pressure depletion, In ECMOR XIV-14th European Conference on the Mathematics of Oil Recovery, European Association of Geoscientists and Engineers, 1-13. ##
  30. Hoshen J, Berry M W, Minser K S (1997) Percolation and cluster structure parameters: The enhanced Hoshen-Kopelman algorithm. Physical Review E, 56, 2: 1455. ##
  31. Nurafza P R, King P R, Masihi M (2006) Connectivity modeling of heterogeneous systems: analysis and field study, Computational Methods in Water Resources, 19-22. ##