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December  2010, 5(4): 765-782. doi: 10.3934/nhm.2010.5.765

Groundwater flow in a fissurised porous stratum

Michiel Bertsch 1, and  Carlo Nitsch 2,

1. 

Istituto per le Applicazioni del Calcolo "M. Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma
2. 

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy

Received  February 2010 Published  November 2010

In [2] Barenblatt e.a. introduced a fluid model for groundwater flow in fissurised porous media. The system consists of two diffusion equations for the groundwater levels in, respectively, the porous bulk and the system of cracks. The equations are coupled by a fluid exchange term. Numerical evidence in [2, 8] suggests that the penetration depth of the fluid increases dramatically due to the presence of cracks and that the smallness of certain parameter values play a key role in this phenomenon. In the present paper we give precise estimates for the penetration depth in terms of the smallness of some of the parameters.
Keywords: Groundwater flow, system of partial differential equations, free boundary., fissurized porous medium.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35B30, 35Q35, 35R35, 76S0.
Citation: Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks & Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765
References:
[1]

G. I. Barenblatt, On some unsteady motions in a liquid or a gas in a porous medium, Prikladnaja Matematika i Mechanika, 16 (1952), 67-78.  Google Scholar

[2]

G. I. Barenblatt, E. A. Ingerman, H. Shvets and J. L. Vázquez, Very intense pulse in the gorundwater flow in fissurised-porous stratum, PNAS, 97 (2000), 1366-1369. doi: 10.1073/pnas.97.4.1366.  Google Scholar

[3]

M. Bertsch, R. Dal Passo and C. Nitsch, A system of degenerate parabolic nonlinear pde's: A new free boundary problem, Interfaces Free Bound, 7 (2005), 255-276.  Google Scholar

[4]

K. N. Chuen, C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. doi: 10.1512/iumj.1977.26.26029.  Google Scholar

[5]

R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomena for thin film equations, Ann. Scuola Norm. Sup. Pisa (4), 30 (2001), 437-463.  Google Scholar

[6]

R. Dal Passo, L. Giacomelli and G. Grün, "Waiting Time Phenomena for Degenerate Parabolic Equations - A Unifying Approach," in "Geometric Analysis and Nonlinear Partial Differential Equations" (S. Hildebrant and H. Karcher, eds.), Springer-Verlag, (2003), 637-648.  Google Scholar

[7]

R. Kersner, Nonlinear heat conduction with absorption: Space localization and extinction in finite time, SIAM J. Appl. Math., 43 (1983), 1274-1285. doi: 10.1137/0143085.  Google Scholar

[8]

Y. Shvets, "Problems of Flooding in Porous and Fissured Porous Rock," Ph.D. thesis, University of California, Berkeley, 2005, http://gradworks.umi.com/31/87/3187151.html. Google Scholar

[9]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  Google Scholar

[10]

G. Stampacchia, "Équationes Elliptiques Du Second Ordre à Coefficients Discontinus," Les presses de l'université de Montréal, 1966.  Google Scholar

[11]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

show all references

References:
[1]

G. I. Barenblatt, On some unsteady motions in a liquid or a gas in a porous medium, Prikladnaja Matematika i Mechanika, 16 (1952), 67-78.  Google Scholar

[2]

G. I. Barenblatt, E. A. Ingerman, H. Shvets and J. L. Vázquez, Very intense pulse in the gorundwater flow in fissurised-porous stratum, PNAS, 97 (2000), 1366-1369. doi: 10.1073/pnas.97.4.1366.  Google Scholar

[3]

M. Bertsch, R. Dal Passo and C. Nitsch, A system of degenerate parabolic nonlinear pde's: A new free boundary problem, Interfaces Free Bound, 7 (2005), 255-276.  Google Scholar

[4]

K. N. Chuen, C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. doi: 10.1512/iumj.1977.26.26029.  Google Scholar

[5]

R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomena for thin film equations, Ann. Scuola Norm. Sup. Pisa (4), 30 (2001), 437-463.  Google Scholar

[6]

R. Dal Passo, L. Giacomelli and G. Grün, "Waiting Time Phenomena for Degenerate Parabolic Equations - A Unifying Approach," in "Geometric Analysis and Nonlinear Partial Differential Equations" (S. Hildebrant and H. Karcher, eds.), Springer-Verlag, (2003), 637-648.  Google Scholar

[7]

R. Kersner, Nonlinear heat conduction with absorption: Space localization and extinction in finite time, SIAM J. Appl. Math., 43 (1983), 1274-1285. doi: 10.1137/0143085.  Google Scholar

[8]

Y. Shvets, "Problems of Flooding in Porous and Fissured Porous Rock," Ph.D. thesis, University of California, Berkeley, 2005, http://gradworks.umi.com/31/87/3187151.html. Google Scholar

[9]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  Google Scholar

[10]

G. Stampacchia, "Équationes Elliptiques Du Second Ordre à Coefficients Discontinus," Les presses de l'université de Montréal, 1966.  Google Scholar

[11]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

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