Fluid flow in a layered medium
Authors:
G. W. Clark and R. E. Showalter
Journal:
Quart. Appl. Math. 52 (1994), 777-795
MSC:
Primary 76S05; Secondary 35Q35, 73B27
DOI:
https://doi.org/10.1090/qam/1306050
MathSciNet review:
MR1306050
Full-text PDF Free Access
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Abstract: A layered medium is modeled as a continuous distribution of relatively flat cells within a spatial region. Local flow within each cell as well as the exchange with the global flow over the region is modeled by a quasilinear parabolic system of partial differential equations, and the local geometry of the individual cells is included in the model. We introduce new terms to account for the secondary flux corresponding to either transverse flow across the cells or direct cell-to-cell diffusion driven by the global density gradient. The resulting initial-boundary-value problem is shown to be well-posed and to depend continuously on the parameter defining the type of interface condition on cell boundaries.
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975
T. Arbogast, Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM J. Numer. Anal. 26, 12–29 (1989)
T. Arbogast, The double porosity model for single phase flow in naturally fractured reservoirs, Inst. Math. Appl. 295 (1987)
T. Arbogast, J. Douglas, and U. Hornung, Modeling of naturally fractured petroleum reservoirs by formal homogenization techniques, in Frontiers in Pure and Applied Mathematics (R. Dautray, ed.), Elsevier, Amsterdam, 1991, pp. 1–19
G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech. 24, 1286–1303 (1960)
N. S. Boulton and T. D. Streltsova-Adams, Unsteady flow to a pumped well in an unconfined fissured aquifer, Advances in Hydroscience, 1978, pp. 357–423
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, 1955
U. Hornung and R. E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl. 147, 69–80 (1990)
E. Kasap and L. W. Lake, Calculating the effective permeability tensor of a gridblock, Society of Petroleum Engineers Formation Evaluation, June 1990, pp. 192–200
J. L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969
R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, Jour. Math. Anal. Appl. 66, 313–332 (1978)
J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29, 187–204 (1971)
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Company, New York, 1987
R. E. Showalter, Hilbert Space Method for Partial Differential Equations, Pitman, 1977
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (to appear)
R. E. Showalter and N. J. Walkington, Micro-structure models of diffusion in fissured media, Jour. Math. Anal. Appl. 155, 1–20 (1991)
R. E. Showalter and N. J. Walkington, Diffusion of fluid in a fissured medium with micro-structure, SIAM Jour. Math. Anal. 22, 1702–1722 (1991)
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975
T. Arbogast, Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM J. Numer. Anal. 26, 12–29 (1989)
T. Arbogast, The double porosity model for single phase flow in naturally fractured reservoirs, Inst. Math. Appl. 295 (1987)
T. Arbogast, J. Douglas, and U. Hornung, Modeling of naturally fractured petroleum reservoirs by formal homogenization techniques, in Frontiers in Pure and Applied Mathematics (R. Dautray, ed.), Elsevier, Amsterdam, 1991, pp. 1–19
G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech. 24, 1286–1303 (1960)
N. S. Boulton and T. D. Streltsova-Adams, Unsteady flow to a pumped well in an unconfined fissured aquifer, Advances in Hydroscience, 1978, pp. 357–423
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, 1955
U. Hornung and R. E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl. 147, 69–80 (1990)
E. Kasap and L. W. Lake, Calculating the effective permeability tensor of a gridblock, Society of Petroleum Engineers Formation Evaluation, June 1990, pp. 192–200
J. L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969
R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, Jour. Math. Anal. Appl. 66, 313–332 (1978)
J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29, 187–204 (1971)
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Company, New York, 1987
R. E. Showalter, Hilbert Space Method for Partial Differential Equations, Pitman, 1977
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (to appear)
R. E. Showalter and N. J. Walkington, Micro-structure models of diffusion in fissured media, Jour. Math. Anal. Appl. 155, 1–20 (1991)
R. E. Showalter and N. J. Walkington, Diffusion of fluid in a fissured medium with micro-structure, SIAM Jour. Math. Anal. 22, 1702–1722 (1991)
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Article copyright:
© Copyright 1994
American Mathematical Society
