Abstract
We consider the problem of modeling flow through naturally fractured porous media. In this type of media, various physical phenomena occur on disparate length scales, so it is difficult to properly average their effects. In particular, gravitational forces pose special problems. In this paper we develop a general understanding of how to incorporate gravitational forces into the dual-porosity concept. We accomplish this through the mathematical technique of formal two-scale homogenization. This technique enables us to average the single-porosity, Darcy equations that govern the flow on the finest (fracture thickness) scale. The resulting homogenized equations are of dual-porosity type. We consider three flow situations, the flow of a single component in a single phase, the flow of two fluid components in two completely immiscible phases, and the completely miscible flow of two components.
Similar content being viewed by others
References
Amaziane, B. and Bourgeat, A., Effective behavior of two-phase flow in heterogeneous reservoir, in Mary F. Wheeler (ed.),Numerical Simulation in Oil Recovery, The IMA Volumes in Mathematics and its Applications 11, Springer-Verlag, Berlin, New York, 1988, pp. 1–22.
Arbogast, T., Analysis of the simulation of single phase flow through a naturally fractured reservoir,SIAM J. Numer. Anal. 26 (1989), 12–29.
Arbogast, T., On the simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir,RAIRO Modél. Math. Anal. Numér. 23 (1989), 5–51.
Arbogast, T., Gravitational forces in dual-porosity models of single phase flow, inProceedings, Thirteenth IMACS World Congress on Computation and Applied Mathematics, Trinity College, Dublin, Ireland, July 22–26, 1991, pp 607–608.
Arbogast, T., Gravitational forces in dual-porosity systems II. Computational validation of the homogenized model,Transport in Porous Media 13 (1993), 205–220.
Arbogast, T., Douglas Jr., J., and Hornung, U., Derivation of the double porosity model of single phase flow via homogenization theory,SIAM J. Math. Anal. 21 (1990), 823–836.
Arbogast, T., Douglas Jr., J., and Hornung, U., Modeling of naturally fractured reservoirs by formal homogenization techniques, in R. Dautray, (ed.), Frontiers in Pure and Applied Mathematics, Elsevier, Amsterdam, 1991, pp. 1–19.
Barenblatt, G. I., Zheltov, Iu. P., and Kochina, I. N., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata],Prikl. Mat. Mekh. 24 (1960) 852–864;J. Appl. Math. Mech. 24 (1960) 1286–1303.
Bensoussan, A., Lions, J. L., and Papanicolaou, G.,Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978.
Bourgeat, A., Homogenization of two phase flow equations,Proc. Symp. Pure Math. 45 (1986), 157–163.
Douglas Jr., J. and Arbogast, T., Dual-porosity models for flow in naturally fractured reservoirs, in J. H. Cushman (ed.),Dynamics of Fluids in Hierarchical Porous Media, Academic Press, London, 1990, pp. 177–221.
Douglas Jr., J., Arbogast, T., and Paes-Leme, P. J., Two models for the waterflooding of naturally fractured reservoirs, Paper SPE 18425, inProceedings, Tenth SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, Dallas, Texas, 1989.
Douglas Jr., J., Hensley, J. L., and Arbogast, T., A dual-porosity model for waterflooding in naturally fractured reservoirs,Comp. Meth. in Appl. Mech. and Engng 87 (1991), 157–174.
Ene, H. I., Application of the homogenization method to transport in porous media, in J. H. Cushman (ed.),Dynamics of Fluids in Hierarchical Porous Media, Academic Press, London, 1990, pp. 223–241.
Hornung, U. and JÄger, W., A model for chemical reactions in porous media, inComplex Chemical Reaction Systems. Mathematical Modelling and Simulation, in J. Warnatz and W. JÄger (eds.), Springer series in Chemical Physics47, Springer-Verlag, Berlin, New York, 1987, pp. 318–334.
Hubbert, M. K.,The Theory of Ground-Water Motion and Related Papers, Hafner Publishing, New York, 1969.
Gilman, J. R. and Kazemi, H., Improvements in simulation of naturally fractured reservoirs,Soc. Petroleum Engr. J. 23 (Aug. 1983), 695–707.
Gilman, J. R. and Kazemi, H.,Improved calculations for viscous and gravity displacement in matrix blocks in dual-porosity simulators, Paper SPE 16010, in‘Proceedings, Ninth SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, Dallas, Texas, 1987, pp. 193–208.
Kazemi, H., Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution,Soc. Petroleum Engr. J. 9 (Dec. 1969), 451–462.
Pirson, S. J., Performance of fractured oil reservoirs,Bull. Amer. Assoc. Petroleum Geologists 37 (1953), 232–244.
Sanchez-Palencia, E.,Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, Berlin and New York, 1980.
de Swaan, A., Analytic solutions for determining naturally fractured reservoir properties by well testing,Soc. Petroleum Engr. J. 16 (June 1976), 117–122.
de Swaan, A., Theory of waterflooding in fractured reservoirs,Soc. Petroleum Engr. J. 18 (April 1978), 117–122.
Thomas, L. K., Dixon, T. N., and Pierson, R. G., Fractured reservoir simulation,Soc. Petroleum Engr. J. 23 (Feb. 1983), 42–54.
Warren, J. E. and Root, P. J., The behavior of naturally fractured reservoirs,Soc. Petroleum Engr. J. 3 (Sept. 1963), 245–255.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Science Foundation and by the State of Texas Governor's Energy Office.
Rights and permissions
About this article
Cite this article
Arbogast, T. Gravitational forces in dual-porosity systems: I. Model derivation by homogenization. Transp Porous Med 13, 179–203 (1993). https://doi.org/10.1007/BF00654409
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00654409