×

The narrow fracture approximation by channeled flow. (English) Zbl 1273.76370

Summary: The singular problem of non-stationary Darcy flow in a region containing a narrow channel of width \(\mathcal O(\varepsilon)\) and high permeability \(O(\frac {1}{\varepsilon})\) is shown to be well approximated by a problem with flow concentrated on a weighted Sobolev space over a lower-dimensional interface. The convergence of the solution as \(\varepsilon \to 0\) is proved for both the stationary case and the corresponding initial-boundary-value problem. The structure of the limiting problems is dependent on the rate of taper of the channel at its extremities.

MSC:

76N99 Compressible fluids and gas dynamics
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, Robert A., Sobolev Spaces, Pure Appl. Math., vol. 65 (1975), Academic Press (a subsidiary of Harcourt Brace Jovanovich Publishers): Academic Press (a subsidiary of Harcourt Brace Jovanovich Publishers) New York, London · Zbl 0314.46030
[2] Cannon, John R.; Meyer, G. H., Diffusion in a fractured medium, SIAM J. Appl. Math., 20, 434-448 (1971) · Zbl 0266.35002
[3] Grisvard, P., Espaces interméiaires entre espaces de Sobolev avec poids, Ann. Sc. Norm. Super. Pisa (3), 17, 255-296 (1963) · Zbl 0117.08602
[4] Hung, Pham Huy; Sánchez-Palencia, Enrique, Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., 47, 284-309 (1974) · Zbl 0286.35007
[5] Kato, Tosio, Perturbation Theory for Linear Operators, Classics Math. (1995), Springer-Verlag: Springer-Verlag Berlin, reprint of the 1980 edition · Zbl 0836.47009
[6] Martin, Vincent; Jaffré, Jérôme; Roberts, Jean E., Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., 26, 5, 1667-1691 (2005), (electronic) · Zbl 1083.76058
[7] Meyer, Richard D., Some embedding theorems for generalized Sobolev spaces and applications to degenerate elliptic differential operators, J. Math. Mech., 16, 739-760 (1967) · Zbl 0149.09303
[8] Meyer, Richard D., Degenerate elliptic differential systems, J. Math. Anal. Appl., 29, 436-442 (1970) · Zbl 0169.44004
[9] Nield, Donald A.; Bejan, Adrian, Convection in Porous Media (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0924.76001
[10] Sánchez-Palencia, Enrique, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys., vol. 127 (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0432.70002
[11] Showalter, R. E., Degenerate evolution equations and applications, Indiana Univ. Math. J., 23, 655-677 (1973/74) · Zbl 0281.34061
[12] Showalter, R. E., Degenerate parabolic initial-boundary value problems, J. Differential Equations, 31, 3, 296-312 (1979) · Zbl 0416.35038
[13] Showalter, R. E., Hilbert Space Methods for Partial Differential Equations, Monographs and Studies in Mathematics, vol. 1 (1977), Pitman: Pitman London-San Farncisco, CA-Melbourne, Reprinted in Electronic Monographs in Differential Equations, Electronic Journal of Differential Equations, San Marcos, TX, 1994 (http://ejde.math.txstate.edu/) · Zbl 0364.35001
[14] Showalter, R. E., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr., vol. 49 (1997), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0870.35004
[15] Temam, Roger, Navier-Stokes Equations, Stud. Math. Appl., vol. 2 (1979), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0426.35003
[16] Weiler, Markus; McDonnell, J. J., Conceptualizing lateral preferential flow and flow networks and simulating the effects on gauged and ungauged hillslopes, Water Resour. Res., 43, W03403 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.