Skip to main content
SpringerLink
Log in

The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alabert, F. G. and Corre, B.: 1991, Heterogeneity in a coupled turbiditic reservoir: impact of field development, SPE, 22902. Proc. 67th Annual Tech. Conf. Exhibition, Dallas, Texas, October 1991.

  • Auriault J-L and Lewandowska J.:1997, Modelling of pollutant migration in porous media with interfacial transfer: local equilibrium/non-equilibrium, Mechanics of Cohesive Frictional Materials 2, 205-221.

    Google Scholar 

  • Barker, J. W. and Dupouy, P.: 1999, An analysis of dynamic pseudo relative permeability methods for oil-water flows, Petrol. Geosci. 5, 385-394.

    Google Scholar 

  • Barker, J. W. and Thibeau, S.: 1997, A critical review of the use of pseudorelative permeabilities for upscaling, SPE Reservoir Engng 12(5), 138-143.

    Google Scholar 

  • Blunt, M. J., Liu K. and Thiele M. R.: 1996, A generalized streamline method to predict reservoir flow, Petrol. Geosci. 2, 259-369.

    Google Scholar 

  • Chavent G.: 1976, A new formulation of diphasic incompressible flows in porous media, in: A. Dold and B. Eckman, (eds), Application of methods of functional analysis to problems in mechanics, Lecture Notes in Mathematics, 503 (Springer, Berlin, 1976), pp. 258-270.

    Google Scholar 

  • Chen, Z and Ewing, R. E.: 1997, Comparison of various formulations of three-phase flow in porous media, J. Computational Phys. 132, 362-373

    Google Scholar 

  • Christie, M. A., Mansfield, M., King, P. R., Barker, J.W. and Culverwell, I. D.: 1995, A renormalisation-based upscaling technique for WAG floods in heterogeneous reservoirs, SPE 29127, in: Proc. 13th Sym. Reservoir Simulation, San Antonio, Texas, 12-15 February 1995.

  • Coats K. H., Nielsen R. L., Terhune M. H., and Weber A. G.: 1967, Simulation of three-dimensional two-phase flow in oil and gas reservoirs, SPE J. 7(12), 377-388.

    Google Scholar 

  • Coats, K. H., Dempsey, J. R. and Henderson, J. H.: 1971, The use of vertical equilibrium in twodimensional simulation of three-dimensional reservoir performance, SPE J. 11(3), 63-71.

    Google Scholar 

  • Coll, C., Muggeridge, A. H. and Jing, X. D.: 1998, The impact of flow regime on the upscaling of heterogeneous media, Proc. 6th European Conf. Math. Oil Recovery, Peebles, Scotland, 8-11 September, 1998.

  • Coll, C., Muggeridge A. H. and Jing, X. D.: 2000, Regional Upscaling: A new method to upscale waterflooding in heterogeneous reservoirs for a range of capillary and gravity effects, SPE, 59337. Proc. 12th SPE/DOE IOR Symp. Tulsa, Oklahoma, 3-5 April, 2000

  • Craig F. F: 1957, A laboratory study of gravity segregation in frontal drives, Trans AIME, 210, 275-281.

    Google Scholar 

  • Darman, N. H., Sorbie, K. S. and Pickup, G. E., 1999: The development of pseudo functions for gravity-dominated immiscible gas displacements, SPE 51941. Proc. SPE Reservoir Simulation Symp. Houston, Texas, 14-17, February, 1999.

  • Dale, M., Ekrann, S., Myykkeltveit, J. and Virnovsky, G.: 1997, Effective relative permeabilities and capillary pressure for one-dimensional heterogeneous media, Transport in Porous Media 26, 229-260.

    Google Scholar 

  • Hewett T. A. and Behrens R. A.: 1993, Considerations affecting the scaling of displacements in heterogeneous permeability distributions, SPE Formation Evaluation 12, 258-266.

    Google Scholar 

  • Hewett, T. A. and Archer, R. A.: 1997, Scale-averaged effective flow properties for coarse-grid reservoir simulation, SPE 37988, in: Proc. 1997 SPE Reservoir Simulation Symp. Dallas, Texas, 8-11 June, 1997.

  • Huang, Y., Ringrose, P. S. and Sorbie, K. S.: 1995, Capillary trapping mechanisms in water-wet laminated rocks, SPE Reservoir Engng 10(11), 287-292.

    Google Scholar 

  • Jacks, H., Smith, O. J. E. and Mattax: C. C.: 1973, The modelling of a three-dimensional reservoir with a two-dimensional reservoir simulator, the use of dynamic pseudo functions, SPE J. 175-185.

  • Kumar, A. T. A. and Jerrauld, G. R.: 1996, Impacts of scale-up on fluid flow from plug to gridblock scale in reservoir rock, SPE 35452, Proc. Tenth SPE/DOE Symp. Improved Oil Recovery, Tulsa, Oklahoma, 21-24 April, 1996.

  • Kumar, A., Farmer, C. L., Jerrauld, G. R. and Li, D.: 1997, Efficient upscaling from cores to simulation models, SPE 38744, Proc. 1997 SPE Annual Technical Conference, San Antonio, Texas, 5-8 October, 1997.

  • Kocherber, S. and Collins, R. E.: 1990, Impact of reservoir heterogeneity on initial distributions of hydrocarbons, SPE 20547, in: Proc. 65th SPE Annual Technical Conference and Exhibition, New Orleans, USA, September, 1990.

  • Kortekaas, T. F. M.: 1985, Water/oil displacement characteristics in cross-bedded reservoir zones, SPE J. 25(6), 917-926.

    Google Scholar 

  • Kyte, J. R. and Berry, D. W.: 1975, New pseudo functions to control numerical dispersion, SPE J. 8, 269-275.

    Google Scholar 

  • Lake L. W.: 1989, Enhanced Oil Recovery, Prentice Hall, Englewoods Cliffs, N.J., U.S.A.

    Google Scholar 

  • Lemouzy, P., Kruel Romeu, L. and Morelon, I.: 1993, A new scaling-up method to compute relative permeability and capillary pressure for simulation of heterogeneous reservoirs, SPE 26660, in: Proc. 69th Annual Technical Conference Exhibition, October 3-6, 1993.

  • McDougall, S. and Sorbie, K. S.: 1995 The impact of wettability on waterflooding: Pore Scale Simulation, SPE Reservoir Engng 1(8), 208-213.

    Google Scholar 

  • Peaceman D. W.: 1965, Numerical solution of the nonlinear equations for two-phase flow through porous media, in: Proc. Symp. Methods of Solution of Non-Linear Partial Differential Equations, December, pp. 171-191.

  • Pickup G. E and Sorbie K. S.: 1996, The scaleup of two-phase flow in porous media using phase permeability tensors, SPE J. 12, 369-381.

    Google Scholar 

  • Pickup G. E. and Stephen K.D.: (1998) Steady state scaleup methods, in: Proc. 6th European Conf. Math. Oil Recovery, 8-11, September, 1998.

  • Rappoport L. A.: 1954, Scaling laws for use in design and operation of water-oil flow models, Tran. AIME 204, 143-150.

    Google Scholar 

  • Ringrose, P. S., Sorbie, K. S., Corbett, P. W. M. and Jensen, J. L.: 1993, Immiscible flow behaviour in laminated and cross-bedded sandstones, in: J. Petrol. Sci. Engng 9(2), 103-124.

    Google Scholar 

  • Saad, N., Cullick, A. S. and Honarpour, M. M.: 1995, Effective relative permeability in scale-up and simulation, SPE 29592, in: Proc. Joint Rocky Mountain Regional Meeting and Low Permeability Reservoirs Symp. Denver Colorado, 20-22 March, 1995.

  • Shook, M., Li, D. and Lake, L. W.: 1992, Scaling immiscible flow through permeable media by inspectional analysis, In-Situ 16, 311-349.

    Google Scholar 

  • Smith, E.: 1991, The influence of small-scale heterogeneity on average relative permeability, in: L. W. Lake, H. B. Carroll, Jr. and T. C. Wesson (eds) Reservoir Characterization II, Academic Press, San Diego, pp. 52-76.

    Google Scholar 

  • Sorbie K. S., Sheb, M., Hosseini, A. and Wat, R.M.S.: 1990, Scaled miscible floods in layered beadpacks investigating viscous crossflow, the effects of gravity and the dynamics of viscous slug breakdown, SPE 20520, in: Proc. 1990 SPE Annual Tech. Conf. Exhibition, New Orleans, LA.

  • Stone, H. L.: 1991, Rigorous black oil pseudo functions, SPE 21207, in: Proc. 1991 SPE Symp. Reservoir Simulation, Anaheim, California, 17-20 February, 1991.

  • Yortsos, Y. C. and Chang, J.: 1990, Capillary effects in steady state flow in heterogeneous cores, Transport in Porous Media 5, 399-420.

    Google Scholar 

  • Yortsos Y. C.: 1991, A theoretical analysis of vertical flow equilibrium, SPE 22612 in: Proc. 66th Annual Tech. Conf. Exhibition, Dallas, Oct. 6-9, pp. 185-197.

  • Zhou D., Fayers F. J. and Orr F. M. (1993) Scaling of multiphase flow in simple heterogeneous porous Media, SPE/DOE 27833, in: Proc. ASME Winter Meeting, New Orleans, November 28–December 3, 1993.

Download references

Author information

Authors and Affiliations

  1. Department of Petroleum Engineering, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, Scotland

    Karl D. Stephen, Gillian E. Pickup & Kenneth S. Sorbie

Authors
  1. Karl D. Stephen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gillian E. Pickup
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kenneth S. Sorbie
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stephen, K.D., Pickup, G.E. & Sorbie, K.S. The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow. Transport in Porous Media 45, 63–88 (2001). https://doi.org/10.1023/A:1011850618324

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011850618324

Search

Navigation