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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
July  2013, 18(5): 1253-1273. doi: 10.3934/dcdsb.2013.18.1253

## Analysis and numerical approximations of equations of nonlinear poroelasticity

 1 Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849, United States, United States

Received  August 2012 Revised  January 2013 Published  March 2013

The equations of quasi-static poroelasticity which model flow through elastic porous media are considered. It is assumed that the hydraulic conductivity depends nonlinearly on the displacement (the dilatation) of the medium. The existence of a weak solution is proved using the modified Rothe's method. Numerical approximations of solutions by the finite element method are considered. Error estimates are obtained and numerical experiments are conducted to illustrate the theoretical results, and the efficiency and accuracy of the numerical method.
Citation: Yanzhao Cao, Song Chen, A. J. Meir. Analysis and numerical approximations of equations of nonlinear poroelasticity. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1253-1273. doi: 10.3934/dcdsb.2013.18.1253
##### 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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