Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Giuseppe Maria Coclite [a] and Lorenzo di Ruvo [b]

The porous medium equation with capillary pressure effects

Pages: 173-190
Received: 31 March 2023
Accepted in revised form: 4 July 2023
Mathematics Subject Classification: 35G25, 35K55.
Keywords: Existence, uniqueness, stability, porous medium equation, Cauchy problem.
Authors address:
[a]: Politecnico di Bari, Dipartimento di Meccanica, Matematica e Management, Bari, Italy
[b]: Università di Bari, Dipartimento di Matematica, Bari, Italy

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). GMC has been partially supported by the Research Project of National Relevance ''Multiscale Innovative Materials and Structures'' granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017J4EAYB and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP - D94I18000260001).

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Abstract: We consider a third order equation, which includes pressure as a dissipative term, and describes the dynamics of two-phase flows in a porous media. It is a generalization of Benjamin-Bona-Mahony equation, which models long waves in a nonlinear dispersive system. We prove the well-posedness of the Cauchy problem, associated with this equation.

E. Abreu and J. Vieira, Computing numerical solutions of the pseudo-parabolic Buckley-Leverett equation with dynamic capillary pressure, Math. Comput. Simulation 137 (2017), 29-48. MR3624733
C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1989), no. 1, 1-49. MR1012198
J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions, Nonlinear Anal. 9 (1985), no. 8, 861-865. MR0799889
G. I. Barenblatt, M. Bertsch, R. Dal Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal. 24 (1993), no. 6, 1414-1439. MR1241152
G. I. Barenblatt, J. Garcia-Azorero, A. De Pablo and J. L. Vazquez, Mathematical model of the non-equilibrium water-oil displacement in porous strata, Appl. Anal. 65 (1997), no. 1-2, 19-45. MR1674579
G. I. Barenblatt, Iu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], J. Appl. Math. Mech. 24 (1960), 1286-1303. DOI
T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1972), no.1220, 47-78. MR0427868
G. M. Coclite and L. di Ruvo, On the convergence of the modified Rosenau and the modified Benjamin-Bona-Mahony equations, Comput. Math. Appl. 74 (2017), no. 5, 899-919. MR3689925
G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation, J. Math. Pures Appl. (9) 107 (2017), no. 3, 315-335. MR3609209
G. M. Coclite and L. di Ruvo, A note on convergence of the solutions of Benjamin-Bona-Mahony type equations, Nonlinear Anal. Real World Appl. 40 (2018), 64-81. MR3718975
G. M. Coclite and L. Di Ruvo, Existence results for the Kudryashov-Sinelshchikov-Olver equation, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 2, 425-450. MR4241285
G. M. Coclite and L. Di Ruvo, On the classical solutions for a Rosenau-Korteweg-deVries-Kawahara type equation, Asymptot. Anal. 129 (2022), no. 1, 51-73. MR4465912
C. Cuesta and J. Hulshof, A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves, Nonlinear Anal. 52 (2003), no. 4, 1199-1218. MR1941253
C. Cuesta, C. J. van Duijn and J. Hulshof, Infiltration in porous media with dynamic capillary pressure: travelling waves, European J. Appl. Math. 11 (2000), no. 4, 381-397. MR1790042
J. Garcia-Azorero and A. de Pablo, Finite propagation for a pseudoparabolic equation: two-phase non-equilibrium flows in porous media, Nonlinear Anal. 33 (1998), no. 6, 551-573. MR1635903
J. A. Goldstein and B. J. Wichnoski, On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear Anal. 4 (1980), no. 4, 665-675. MR0582535
B. Hayes and M. Shearer, Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 733-754. MR1718538
R. Helmig, A. Weiss and B. I. Wohlmuth, Dynamic capillary effects in heterogeneous porous media, Comput. Geosci. 11 (2007), no. 3, 261-274. MR2344202
J. Hulshof and J. R. King, Analysis of a Darcy flow model with a dynamic pressure saturation relation, SIAM J. Appl. Math. 59 (1999), no. 1, 318-346. MR1647829
D. Jacobs, B. McKinney and M. Shearer, Traveling wave solutions of the modified Korteweg-deVries-Burgers equation, J. Differential Equations 116 (1995), no. 2, 448-467. MR1318583
C.-Y. Kao, A. Kurganov, Z. Qu and Y. Wang, A fast explicit operator splitting method for modified Buckley-Leverett equations, J. Sci. Comput. 64 (2015), no. 3, 837-857. MR3377841
C. I. Kondo and C. M. Webler, The generalized BBM-Burger equations with non-linear dissipative term: existence and convergence results, Appl. Anal. 87 (2008), no. 9, 1085-1101. MR2463895
S. Manthey, S. M. Hassanizadeh, R. Helmig and R. Hilfer, Dimensional analysis of two-phase flow including a rate-dependent capillary pressure-saturation relationship, Advances in Water Resources 31 (2008), no. 9, 1137-1150. DOI
M. Meyvaci, Blow up of solutions of pseudoparabolic equations, J. Math. Anal. Appl. 352 (2009), no. 2, 629-633. MR2501907
M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982), no. 8, 959-1000. MR0668586
N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems, Nonlinear Anal. Real World Appl. 12 (2011), no. 5, 2625-2639. MR2813209
M. Shearer, K. R. Spayd and E. R. Swanson, Traveling waves for conservation laws with cubic nonlinearity and BBM type dispersion, J. Differential Equations 259 (2015), no. 7, 3216-3232. MR3360671
K. Spayd and M. Shearer, The Buckley-Leverett equation with dynamic capillary pressure, SIAM J. Appl. Math. 71 (2011), no. 4, 1088-1108. MR2823494
C. J. van Duijn, Y. Fan, L. A. Peletier and I. S. Pop, Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. Real World Appl. 14 (2013), no. 3, 1361-1383. MR3004506
C. J. van Duijn, L. A. Peletier and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal. 39 (2007), no. 2, 507-536. MR2338418
Y. Wang and C.-Y. Kao, Central schemes for the modified Buckley-Leverett equation, J. Comput. Sci. 4 (2013), no. 1-2, 12-23. DOI
P. A. Zegeling, An adaptive grid method for a non-equilibrium PDE model from porous media, J. Math. Study 48 (2015), no. 2, 187-198. MR3374383
H. Zhang and P. A. Zegeling, A numerical study of two-phase flow models with dynamic capillary pressure and hysteresis, Transp. Porous Media 116 (2017), no. 2, 825-846. MR3605205
H. Zhang and P. A. Zegeling, A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media, Commun. Comput. Phys. 22 (2017), no. 4, 935-964. MR3719254

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