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The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types
Analysis and numerical approximations of equations of nonlinear poroelasticity
1. | Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849, United States, United States |
References:
[1] |
Jean-Louis Auriault, Claude Boutin and Christian Geindreau, "Homogenization of Coupled Phenomena in Hetrogenous Media," ISTE Ltd., London, 2009. |
[2] |
H. Bryne and L. Preziosi, Modeling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366. |
[3] |
P. C. Carman, Permeability of saturated sands, soils and clays, Journal of Agricultural Science, 29 (1939), 263-273. |
[4] |
Robert Wayne Carroll and Ralph E. Showalter, "Singular and Degenerate Cauchy Problems," Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976. Mathematics in Science and Engineering, 127. |
[5] |
Olivier Coussy, "Poromechanics," John Wiley & Sons Ltd., Chichester, 2004. |
[6] |
Olivier Coussy, "Mechanics and Physics of Porous Solids," John Wiley & Sons Ltd., Chichester, 2010. |
[7] |
Alexandre Ern and Sébastien Meunier, A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems, M2AN Math. Model. Numer. Anal., 43 (2009), 353-375.
doi: 10.1051/m2an:2008048. |
[8] |
Lawrence C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010. |
[9] |
F. J. Gaspar, F. J. Lisbona and P. N. Vabishchevich, Finite difference schemes for poro-elastic problems, Comput. Methods Appl. Math., 2 (2002), 132-142. |
[10] |
F. J. Gaspar, F. J. Lisbona and P. N. Vabishchevich, A finite difference analysis of Biot's consolidation model, Appl. Numer. Math., 44 (2003), 487-506.
doi: 10.1016/S0168-9274(02)00190-3. |
[11] |
Vivette Girault and Pierre-Arnaud Raviart, "Finite Element Methods for Navier-Stokes Equations," 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and Algorithms.
doi: 10.1007/978-3-642-61623-5. |
[12] |
Kai Hiltunen, "Mathematical and Numerical Modelling of Consolidation Processes in Paper Machines," University of Jyväskylä Department of Mathematics, Jyväskylä, 1995. |
[13] |
J. Hudson, O. Stephansson, J. Andersson, C.-F. Tsang and L. Ling, Coupled t-h-m issues related to radioactive waste repository design and performance, International Journal of Rock Mechanics and Mining Sciences, 38 (2001), 143-161. |
[14] |
Jozef Kačur, "Method of Rothe in Evolution Equations," 80 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985. With German, French and Russian Summaries. |
[15] |
J.-M. Kim and R. Parizek, Numerical simulation of the noordbergum effect resulting from groundwater pumping in a layered aquifer system, Journal of Hydrology, 202 (1997), 231-243. |
[16] |
Kenneth L. Kuttler, Jr., Time-dependent implicit evolution equations, Nonlinear Anal., 10 (1986), 447-463.
doi: 10.1016/0362-546X(86)90050-7. |
[17] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, 1969. |
[18] |
N. Lubick, Modeling complex, multiphase porous media systems, SIAM News, 35 April (2002). |
[19] |
A. Naumovich and F. J. Gaspar, On a multigrid solver for the three-dimensional Biot poroe- lasticity system in multilayered domains, Comput. Vis. Sci., 11 (2008), 77-87.
doi: 10.1007/s00791-007-0059-8. |
[20] |
Phillip Joseph Phillips, "Finite Element Methods in Linear Poroelasticity: Theoretical and Computational Results," ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)-The University of Texas at Austin. |
[21] |
Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12 (2008), 417-435.
doi: 10.1007/s10596-008-9082-1. |
[22] |
Karel Rektorys, "The Method of Discretization in Time and Partial Differential Equations," 4 of Mathematics and Its Applications (East European Series), D. Reidel Publishing Co., Dordrecht, 1982. Translated from the Czech by the author. |
[23] |
Michael Renardy and Robert C. Rogers, "An Introduction to Partial Differential Equations," 13 of Texts in Applied Mathematics, Springer-Verlag, New York, second edition, 2004. |
[24] |
T. Roose, P. A. Netti, L. Munn, Y. Boucher and R. Jain, Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvascular Research, 66 (2003), 204-212. |
[25] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997. |
[26] |
R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251 (2000), 310-340.
doi: 10.1006/jmaa.2000.7048. |
[27] |
A. Smillie, I. Sobey and Z. Molnar, A hydro-elastic model of hydrocephalus, Technical report, Oxford University Computing Laboratory: Numerical Analysis Group, (2004). |
[28] |
Vidar Thomée, "Galerkin Finite Element Methods for Parabolic Problems," 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. |
[29] |
Herbert F. Wang, "Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrology," Princeton Series in Geophysics. Princeton University Press, Princeton, 2000. |
[30] |
A. Ženíšek, "Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations," Computational Mathematics and Applications. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1990. With a foreword by P.-A. Raviart. |
[31] |
Alexander Ženíšek, Finite element methods for coupled thermoelasticity and coupled consolidation of clay, RAIRO Anal. Numér., 18 (1984), 183-205. |
show all references
References:
[1] |
Jean-Louis Auriault, Claude Boutin and Christian Geindreau, "Homogenization of Coupled Phenomena in Hetrogenous Media," ISTE Ltd., London, 2009. |
[2] |
H. Bryne and L. Preziosi, Modeling solid tumour growth using the theory of mixtures, Mathematical Medicine and Biology, 20 (2003), 341-366. |
[3] |
P. C. Carman, Permeability of saturated sands, soils and clays, Journal of Agricultural Science, 29 (1939), 263-273. |
[4] |
Robert Wayne Carroll and Ralph E. Showalter, "Singular and Degenerate Cauchy Problems," Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976. Mathematics in Science and Engineering, 127. |
[5] |
Olivier Coussy, "Poromechanics," John Wiley & Sons Ltd., Chichester, 2004. |
[6] |
Olivier Coussy, "Mechanics and Physics of Porous Solids," John Wiley & Sons Ltd., Chichester, 2010. |
[7] |
Alexandre Ern and Sébastien Meunier, A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems, M2AN Math. Model. Numer. Anal., 43 (2009), 353-375.
doi: 10.1051/m2an:2008048. |
[8] |
Lawrence C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010. |
[9] |
F. J. Gaspar, F. J. Lisbona and P. N. Vabishchevich, Finite difference schemes for poro-elastic problems, Comput. Methods Appl. Math., 2 (2002), 132-142. |
[10] |
F. J. Gaspar, F. J. Lisbona and P. N. Vabishchevich, A finite difference analysis of Biot's consolidation model, Appl. Numer. Math., 44 (2003), 487-506.
doi: 10.1016/S0168-9274(02)00190-3. |
[11] |
Vivette Girault and Pierre-Arnaud Raviart, "Finite Element Methods for Navier-Stokes Equations," 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and Algorithms.
doi: 10.1007/978-3-642-61623-5. |
[12] |
Kai Hiltunen, "Mathematical and Numerical Modelling of Consolidation Processes in Paper Machines," University of Jyväskylä Department of Mathematics, Jyväskylä, 1995. |
[13] |
J. Hudson, O. Stephansson, J. Andersson, C.-F. Tsang and L. Ling, Coupled t-h-m issues related to radioactive waste repository design and performance, International Journal of Rock Mechanics and Mining Sciences, 38 (2001), 143-161. |
[14] |
Jozef Kačur, "Method of Rothe in Evolution Equations," 80 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985. With German, French and Russian Summaries. |
[15] |
J.-M. Kim and R. Parizek, Numerical simulation of the noordbergum effect resulting from groundwater pumping in a layered aquifer system, Journal of Hydrology, 202 (1997), 231-243. |
[16] |
Kenneth L. Kuttler, Jr., Time-dependent implicit evolution equations, Nonlinear Anal., 10 (1986), 447-463.
doi: 10.1016/0362-546X(86)90050-7. |
[17] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, 1969. |
[18] |
N. Lubick, Modeling complex, multiphase porous media systems, SIAM News, 35 April (2002). |
[19] |
A. Naumovich and F. J. Gaspar, On a multigrid solver for the three-dimensional Biot poroe- lasticity system in multilayered domains, Comput. Vis. Sci., 11 (2008), 77-87.
doi: 10.1007/s00791-007-0059-8. |
[20] |
Phillip Joseph Phillips, "Finite Element Methods in Linear Poroelasticity: Theoretical and Computational Results," ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)-The University of Texas at Austin. |
[21] |
Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12 (2008), 417-435.
doi: 10.1007/s10596-008-9082-1. |
[22] |
Karel Rektorys, "The Method of Discretization in Time and Partial Differential Equations," 4 of Mathematics and Its Applications (East European Series), D. Reidel Publishing Co., Dordrecht, 1982. Translated from the Czech by the author. |
[23] |
Michael Renardy and Robert C. Rogers, "An Introduction to Partial Differential Equations," 13 of Texts in Applied Mathematics, Springer-Verlag, New York, second edition, 2004. |
[24] |
T. Roose, P. A. Netti, L. Munn, Y. Boucher and R. Jain, Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvascular Research, 66 (2003), 204-212. |
[25] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997. |
[26] |
R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251 (2000), 310-340.
doi: 10.1006/jmaa.2000.7048. |
[27] |
A. Smillie, I. Sobey and Z. Molnar, A hydro-elastic model of hydrocephalus, Technical report, Oxford University Computing Laboratory: Numerical Analysis Group, (2004). |
[28] |
Vidar Thomée, "Galerkin Finite Element Methods for Parabolic Problems," 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. |
[29] |
Herbert F. Wang, "Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrology," Princeton Series in Geophysics. Princeton University Press, Princeton, 2000. |
[30] |
A. Ženíšek, "Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations," Computational Mathematics and Applications. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1990. With a foreword by P.-A. Raviart. |
[31] |
Alexander Ženíšek, Finite element methods for coupled thermoelasticity and coupled consolidation of clay, RAIRO Anal. Numér., 18 (1984), 183-205. |
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