Advertisement

Transport in Porous Media

, Volume 92, Issue 1, pp 187–212 | Cite as

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

  • F. Hussain
  • Y. CinarEmail author
  • P. Bedrikovetsky
Article

Abstract

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.

Keywords

Two phase flow Porous medium Semi-analytical model Coreflood Relative permeability 

List of Symbols

D

Velocity of the shock front

f

Fractional flow of water phase

\({{f}_{\rm s}^{{\prime}}}\)

Derivative of fractional flow w.r.t. saturation

Fw_wet

Fraction of the water-wet rock surface

J

Leverett function

k

Permeability

kro

Oil relative permeability

kro_max

Maximum oil relative permeability

krw

Water relative permeability

krw_max

Maximum water relative permeability

L

Core length

n

Exponent for Corey-type power law

p

Pressure

Pc

Capillary pressure

s

Water phase saturation

si

Initial water saturation

sir

Irreducible water saturation

s0

Maximum water saturation, equals to unity minus residual/critical oil saturation

t

Time

T

Dimensionless time, p.v. injection

Tp

Pore volumes produced

Top

Pore volumes of oil produced

Twp

Pore volumes of water produced

U

Velocity

x

Linear coordinate, from injectors towards producers

X

Dimensionless linear coordinate

λ

Total mobility of the oil-water fluid

Ψ

Potential for capillary forces

\({\varepsilon}\)

Capillary-viscous ratios

\({\xi}\)

Self-sharpening large scale (slow) coordinate

\({\omega}\)

Travelling wave fast coordinate near to the displacement front

\({\zeta}\)

Fast coordinate near to the core outlet

\({\phi}\)

Porosity

μ

Fluid viscosity

σ

Interfacial tension

θ

Contact angle

Subscripts and Superscripts

W, O

Water, oil

i

Initial (of water saturation)

0

Boundary value on the injector (saturations, flux)

BL

Buckley–Leverett

BT

Breakthrough

SZ

Stabilized zone

ee

End effect

min

Minimum

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alizadeh, A.H., Keshavarz, A., Haghighi, M.: Flow rate effect on two-phase relative permeability in Iranian carbonate rocks. In: SPE middle east oil and gas show and conference, Bahrain, 11–14 Mar 2007Google Scholar
  2. Amyx J., Bass D., Whiting R.: Petroleum reservoir engineering: physical properties. McGraw-Hill College, New York (1960)Google Scholar
  3. Anderson W.: Wettability literature survey part 5: the effects of wettability on relative permeability. J. Petroleum Technol. 39, 1453–1468 (1987)Google Scholar
  4. Bacri J.C., Leygnac C., Salin D.: Evidence of capillary hyperdiffusion in two-phase fluid flows. J. Phys. Lett. 46, 467–473 (1985)CrossRefGoogle Scholar
  5. Barenblatt G., Entov V., Ryzhik V.: Theory of fluid flows through natural rocks. Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  6. Basbug, B., Karpyn, Z.: Determination of relative permeability and capillary pressure curves using an automated history-matching approach. In: SPE eastern regional/AAPG eastern section joint meeting, Pittsburgh, 11–15 Oct 2008Google Scholar
  7. Batycky J., Mccaffery F., Hodgins P., Fisher D.: Interpreting relative permeability and wettability from unsteady-state displacement measurements. Old SPE J. 21, 296–308 (1981)Google Scholar
  8. Bedrikovetsky P.: Mathematical theory of oil and gas recovery: with applications to ex-USSR oil and gas fields. Kluwer Academic Publishers, Boston (1993)Google Scholar
  9. Bedrikovetsky, P., Rodrigues, J., Britto, P.: Analytical model for the waterflood honouring capillary pressure (with applications to laboratory studies). In: SPE Latin America/Caribbean petroleum engineering conference, Port-of-Spain, 23–26 Apr 1996Google Scholar
  10. Buckley S., Leverett M.: Mechanism of fluid displacement in sands. Trans. AIME. 146, 107–116 (1942)Google Scholar
  11. Burdine N., Gournay L., Reichertz P.: Pore size distribution of petroleum reservoir rocks. Trans. AIME. 189, 195–204 (1950)Google Scholar
  12. Chardaire-Riviere C., Chavent G., Jaffre J., Liu J., Bourbiaux B.: Simultaneous estimation of relative permeabilities and capillary pressure. SPE Form. Evaluation 7, 283–289 (1992)Google Scholar
  13. Christiansen, R.: Two-phase flow through porous media. Colorado School of Mines, Golden (2001)Google Scholar
  14. Civan F., Donaldson E.: Relative permeability from unsteady-state displacements with capillary pressure included. SPE Form. Evaluation 4, 189–193 (1989)Google Scholar
  15. Donnez P.: Essentials of reservoir engineering. Technip Editions, Paris (2007)Google Scholar
  16. Erdélyi A.: Asymptotic expansions. Dover Pubns, New York (1956)Google Scholar
  17. Gel’fand I.: Some problems in the theory of quasi-linear equations. Uspekhi Matematicheskikh Nauk 14, 87–158 (1959)Google Scholar
  18. Hamming R.: Numerical methods for scientists and engineers. Dover Publications, Mineola (1986)Google Scholar
  19. Honarpour M., Koederitz L., Harvey A.: Relative permeability of petroleum reservoirs. CRC Press, Boca Raton (1986)Google Scholar
  20. Huang D., Honarpour M.: Capillary end effects in coreflood calculations. J. Petroleum Sci. Eng. 19, 103–118 (1998)CrossRefGoogle Scholar
  21. Hussain, F., Cinar, Y., Bedrikovetsky, P.: Comparison of methods for drainage relative permeability estimation from displacement tests. In: SPE IOR Symposium, Tulsa (2010)Google Scholar
  22. Islam, M., Bentsen, R.: A dynamic method for measuring relative permeability. JCPT, Jan–Feb, pp. 39–50 (1986)Google Scholar
  23. Johnson E., Bossler D., Naumann V.: Calculation of relative permeability from displacement experiments. Trans. AIME. 216, 370–372 (1959)Google Scholar
  24. Jones S., Roszelle W.: Graphical techniques for determining relative permeability from displacement experiments. J. Petroleum Technol. 30, 807–817 (1978)Google Scholar
  25. Kalbus, J., Christiansen, R.: New data reduction developments for relative permeability determination. In: ATC & Exhibition, Dallas, 22–25 Oct 1995Google Scholar
  26. Kerig P., Watson A.: Relative-permeability estimation from displacement experiments: an error analysis. SPE Reserv. Eng. 1, 175–182 (1986)Google Scholar
  27. Kevorkian J., Cole J., John F.: Perturbation methods in applied mathematics. Springer-Verlag, New York (1981)Google Scholar
  28. Krause M., Perrin J.C., Benson S.: Recent progress in predicting permeability distributions for history matching core flooding experiments. Energy Procedia 4, 4354–4361 (2011)CrossRefGoogle Scholar
  29. Nayfeh A.: Perturbation methods. Wiley, New York (1973)Google Scholar
  30. Nordtvedt, J., Urkedal, H., Ebeltoft, E., Kolltveit, K., Petersen, E., Sylte, A., Valestrand, R.: The significance of violated assumptions on core analysis results. In: International symposium of the society of core analyts, Golden (1999)Google Scholar
  31. Odeh A., Dotson B.: A method for reducing the rate effect on oil and water relative permeabilities calculated from dynamic displacement data. J. Petroleum Technol. 37, 2051–2058 (1985)Google Scholar
  32. Perrin J.C., Benson S.: An experimental study on the influence of sub-core scale heterogeneities on CO2 distribution in reservoir rocks. Transp. Porous Med. 82, 93–109 (2010)CrossRefGoogle Scholar
  33. Poulsen, S., Skauge, T., Dyrhol, S., Stenby, E.H., Skauge, A.: Including capillary pressure in simulations of steady state relative permeability experiments. In: International symposium of the society of core analyts, Abu Dhabi (2000)Google Scholar
  34. Purcell W.: Capillary pressures–their measurement using mercury and the calculation of permeability therefrom. Trans. Am. Inst. Min. Metall. Pet. Eng. 186, 39–48 (1949)Google Scholar
  35. Qadeer S.: Techniques to handle limitations in dynamic relative permeability measurements. Stanford University, California (2001)Google Scholar
  36. Qadeer, S., Brigham, W., Castanier, L.: Techniques to handle limitations in dynamic relative permeability measurements. National petroleum technology office, Tulsa, Prepared for US Department of Energy Assistant Secretary for Fossil Energy, p. 142 (2002)Google Scholar
  37. Qadeer, S., Dehghani, K., Ogbe, D., Ostermann, R.: Correcting oil/water relative permeability data for capillary end effect in displacement experiments. In: SPE California regional meeting, Long Beach, 23–25 Mar (1988)Google Scholar
  38. Ramakrishnan T., Cappiello A.: A new technique to measure static and dynamic properties of a partially saturated porous medium. Chem. Eng. Sci. 46, 1157–1163 (1991)CrossRefGoogle Scholar
  39. Rapoport L., Leas W.: Properties of linear waterfloods. Trans. AIME. 198, 139–148 (1953)Google Scholar
  40. Richmond P., Watsons A., Texas A.: Estimation of multiphase flow functions from displacement experiments. SPE Reserv. Eng. 5, 121–127 (1990)Google Scholar
  41. Sigmund P., Mccaffery F.: An improved unsteady-state procedure for determining the relative-permeability characteristics of heterogeneous porous media. Soc. Petroleum Eng. J. 19, 15–28 (1979)Google Scholar
  42. Subbey S., Monfared H., Christie M., Sambridge M.: Quantifying uncertainty in flow functions derived from SCAL data. Trans. Porous Med. 65, 265–286 (2006)CrossRefGoogle Scholar
  43. Tikhonov A., Arsenin V., John F.: Solutions of ill-posed problems. Vh Winston, Washington (1977)Google Scholar
  44. Toth J., Bodi T., Szucs P., Civan F.: Convenient formulae for determination of relative permeability from unsteady-state fluid displacements in core plugs. J. Petroleum Sci. Eng. 36, 33–44 (2002)CrossRefGoogle Scholar
  45. Tsakiroglou, C., Avraam, D., Payatakes, A.: Simulation of the immiscible displacement in porous media using capillary pressure and relative permeability curves from transient and steady-state experiments. In: International symposium of the society of core analysts, Abu Dhabi, UAE (2004)Google Scholar
  46. Udegbunam E.: A FORTRAN program for interpretation of relative permeability from unsteady-state displacements with capillary pressure included. Comput. Geosci. 17, 1351–1357 (1991)CrossRefGoogle Scholar
  47. Van Dyke M.: Perturbation methods in fluid mechanics/Annotated edition. NASA STI/Recon Tech. Report A. 75, 46926 (1975)Google Scholar
  48. Virnovsky, G., Skjaeveland, S., Surdal, J., Ingsoy, P.: Steady-state relative permeability measurements corrected for capillary effects. In: SPE annual technical conference and exhibition, Dallas, 22–25 Oct 1995Google Scholar
  49. Welge H.: A simplified method for computing oil recovery by gas or water drive. Trans. Am. Inst. Min. Metall. Petrol. Eng. 195, 91–98 (1952)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of New South WalesSydneyAustralia
  2. 2.University of AdelaideAdelaideAustralia

Personalised recommendations