On the simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 23 (1989) no. 1, p. 5-51
@article{M2AN_1989__23_1_5_0,
author = {Arbogast, Todd},
title = {On the simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {Dunod},
volume = {23},
number = {1},
year = {1989},
pages = {5-51},
zbl = {0668.76131},
mrnumber = {1015918},
language = {en},
url = {http://www.numdam.org/item/M2AN_1989__23_1_5_0}
}

Arbogast, Todd. On the simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 23 (1989) no. 1, pp. 5-51. http://www.numdam.org/item/M2AN_1989__23_1_5_0/

[1] T. Arbogast, Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM J. Numer. Anal. 26 (1989) (to appear). | MR 977946 | Zbl 0668.76130

[2] T. Arbogast, The double porosity model for single phase flow in naturally fractured reservoirs, in Numerical Simulation in Oil Recovery, M. F. Wheeler, ed., The IMA Volumes in Mathematics and its Applications 11, Springer-Verlag, Berlin and New York, 1988, pp. 23-45. | MR 922955 | Zbl 0701.76094

[3] T. Arbogast, Simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir, Ph. D. Thesis, The University of Chicago, Chicago, Illinois, 1987.

[4] T. Arbogast, J. Jr. Douglas and J. E. Santos, Two-phase immiscible flow in naturally fractured reservoirs, in Numerical Simulation in Oil Recovery, M. F. Wheeler, ed., the IMA Volumes in Mathematics and its Applications 11, Springer-Verlag, Berlin and New York, 1988, pp. 47-66. | MR 922956 | Zbl 0699.76103

[5] G. I. Barenblatt, Iu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], Prikl. Mat. Mekh., 24 (1960), pp. 852-864 ; J. Appl. Math. Mech., 24 (1960), pp.1286-1303. | Zbl 0104.21702

[6] F. Brezzi, J. Jr. Douglas, R. Durán, and M. Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), pp. 237-250. | MR 890035 | Zbl 0631.65107

[7] F. Brezzi, J. Jr. Douglas, M. Fortin, and L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables, R.A.I.R.O. Modél. Math. Anal. Numér., 21 (1987), pp. 581-604. | Numdam | MR 921828 | Zbl 0689.65065

[8] J. Jr. Douglas, Numerical methods for the flow of miscible fluids in porous media, in Numerical Methods in Coupled Systems, R. W. Lewis, P. Bettes, and E. Hinton, eds., John Wiley and Sons Ltd, London, 1984, pp. 405-439. | Zbl 0585.76138

[9] J. Jr. Douglas, R. E. Ewing, and M. F. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O. Anal. Numér., 17 (1983), pp. 17-33. | Numdam | MR 695450 | Zbl 0516.76094

[10] J. Jr. Douglas, R. E. Ewing, and M. F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, R.A.I.R.O. Anal. Numér., 17 (1983), pp. 249-265. | Numdam | MR 702137 | Zbl 0526.76094

[11] J. Jr. Douglas, P. J. Paes Leme, T. Arbogast, and T. Schmitt, Simulation of flow in naturally fractured reservoirs, Paper SPE 16019, in Proceedings, Ninth SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, Dallas, Texas, 1987, pp. 271-279.

[12] J. Jr. Douglas, and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), pp. 39-52. | MR 771029 | Zbl 0624.65109

[13] J. Jr. Douglas, and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), pp. 871-885. | MR 672564 | Zbl 0492.65051

[14] J. Jr. Douglas, M. F. Wheeler, B. L. Darlow, and R. P. Kendall, Self-adaptive finite element simulation of miscible displacement in porous media, Comp. Meth. Appl. Mech. Eng., 47 (1984), pp. 131-159. | Zbl 0545.76128

[15] J. Jr. Douglas, and Y. Yuan, Numerical simulation of immiscible flow in porous media based on combining the method of characteristics with mixed finite element procedures, in Numerical Simulation in Oil Recovery, M. F. Wheeler, ed., The IMA Volumes in Mathematics and its Applications 11, Springer-Verlag, Berlin and New York, 1988, pp. 119-131. | MR 922962 | Zbl 0699.76105

[16] R. Durán, On the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method, SIAM J. Numer. Anal., 25 (1988), pp. 989-1001. | MR 960861 | Zbl 0661.76096

[17] R. E. Ewing, T. F. Russell, and M. F. Wheeler, Simulation of miscible displacement using mixed methods and a modified method of characteristics, Paper SPE 12241, in Proceedings, Seventh SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, Dallas, Texas, 1983, pp. 71-81.

[18] R. E. Ewing, T. F. Russell, and M. F. Wheeler, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comp. Meth. Appl. Mech. Eng., 47 (1984), pp. 73-92. | MR 777394 | Zbl 0545.76131

[19] R. E. Ewing and M. F. Wheeler, Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal., 17 (1980), pp. 351-365. | MR 581482 | Zbl 0458.76092

[20] R. E. Ewing and M. F. Wheeler, Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case, in Mathematical Methods in Energy Research, K. I. Gross, ed., Society for Industrial and Applied Mathematics, Philadelphia, 1984, pp. 40-58. | MR 790511 | Zbl 0551.76079

[21] H. Kazemi, Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution, Soc. Pet. Eng. J. (1969), pp. 451-462.

[22] J. C. Nedelec, Mixed finite elements in R3, Numer. Math., 35 (1980), pp. 315-341. | MR 592160 | Zbl 0419.65069

[23] D. W. Peaceman, Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement, Soc. Pet. Eng. J. (1966), pp. 213-216.

[24] P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, Springer-Verlag, Berlin and New York, 1977, pp. 292-315. | MR 483555 | Zbl 0362.65089

[25] T. F. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal., 22 (1985), pp. 970-1013. | MR 799124 | Zbl 0594.76087

[26] P. H. Sammon, Numerical approximations for a miscible displacement process in porous media, SIAM J. Numer. Anal., 23 (1986), pp. 508-542. | MR 842642 | Zbl 0608.76084

[27] F. Sonier, P. Souillard, and F. T. Blaskovich, Numerical simulation of naturally fractured reservoirs, Paper SPE 15627, in Proceedings, 61st Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Society of Petroleum Engineers, Dallas, Texas, 1986.

[28] A. De Swaan O., Analytic solutions for determining naturally fractured reservoirs properties by well testing, Soc. Pet. Eng. J. (1976), pp. 117-122.

[29] J. E. Warren and P. J. Root, The behavior of naturally fractured reservoirs, Soc. Pet. Eng. J. (1963), pp. 245-255.

[30] M. F. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723-759. | MR 351124 | Zbl 0232.35060