April  2016, 9(2): 457-474. doi: 10.3934/dcdss.2016007

A geological delayed response model for stratigraphic reconstructions

1. 

LaSIE, UMR-CNRS 7356, Université de La Rochelle, Avenue Michel Crépeau, 17000 La Rochelle, France, France

Received  March 2015 Revised  October 2015 Published  March 2016

We are interested by a nonlinear single lithology diffusion model adapted from ideas originally developed by the Institut Français du Pétrole (IFP). The geological stratigraphic modeling has to describe transports of sediments, erosion and sedimentation processes by taking into account a limited weathering condition; the method by which the history of a sedimentary basin is revealed relies on knowledge of both initial and final data and can be generalized to multiple lithology. For this purpose, we introduce a relaxation time related to a delayed response for establishing equilibrium states; this approach introduces regularizing effects according to the ideas of G.I. Barenblatt - S. Sobolev and J.-L. Lions - O. A. Oleinik. New well-posedness results are presented.
Citation: Gérard Gagneux, Olivier Millet. A geological delayed response model for stratigraphic reconstructions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 457-474. doi: 10.3934/dcdss.2016007
References:
[1]

L. Ambrosio, C. De Lellis and J. Maly, On the chain rule for the divergence of BV-like vector fields: Applications, partial results, open problems, Contemporary Mathematics, 446 (2007), 31-67. doi: 10.1090/conm/446/08625.

[2]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, A non-standard free boundary problem arising from stratigraphy, Analysis and Applications, 4 (2006), 209-236. doi: 10.1142/S0219530506000759.

[3]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, New unilateral problems in stratigraphy, ESAIM: Mathematical Modelling and Numerical Analysis, 40 (2006), 765-784. doi: 10.1051/m2an:2006029.

[4]

S. N. Antontsev, G. Gagneux, A. Mokrani and G. Vallet, Stratigraphic modelling by the way of a pseudoparabolic problem with constraint, Advances in Mathematical Sciences and Applications, 19 (2009), 195-209.

[5]

S. N. Antontsev, G. Gagneux and G. Vallet, On some stratigraphic control problems, Prikladnaya Mekhanika Tekhnicheskaja Fisika, 44 (2003), 85-94 (in Russian). English version: Journal of Applied Mechanics and Technical Physics, 44 (2003), 821-828. doi: 10.1023/A:1026287705015.

[6]

V. R. Baker and D. F. Ritter, Competence of rivers to transport coarse bedload material, Geological Society of America Bulletin, 86 (1975), 975-978.

[7]

G. I. Barenblatt, M. Bertsch, R. D. 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 Journal on Mathematical Analysis, 24 (1993), 1414-1439. doi: 10.1137/0524082.

[8]

J. Blum, G. Dobranszky, R. Eymard and R. Masson, Identification of a stratigraphic model with seismic constraints, Inverse problems, 22 (2006), 1207-1225. doi: 10.1088/0266-5611/22/4/006.

[9]

A. Cimetière, F. Delvare and F. Pons, Une méthode inverse d'ordre un pour les problèmes de complétion de données, Comptes rendus mécanique, 333 (2005), 123-126.

[10]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, An inversion method for harmonic functions reconstruction, International journal of thermal sciences, 41 (2002), 509-516.

[11]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570. doi: 10.1088/0266-5611/17/3/313.

[12]

S. Clain, Elliptic operators of divergence type with Hölder coefficients in fractional Sobolev spaces, Rend. Mat. Appl, 17 (1997), 207-236.

[13]

I. Csato, D. Granjeon, O. Catuneanu and G. R. Baum, A three-dimensional stratigraphic model for the Messinian crisis in the Pannonian Basin, eastern Hungary, Basin Research, 25 (2013), 121-148.

[14]

J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology, Archive for Rational Mechanics and Analysis, 194 (2009), 75-103. doi: 10.1007/s00205-008-0164-y.

[15]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, Dunod, Paris, 1972.

[16]

L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 14 (2004), 1599-1620. doi: 10.1142/S0218202504003763.

[17]

L. C. Evans, A survey of entropy methods for partial differential equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 409-438. doi: 10.1090/S0273-0979-04-01032-8.

[18]

R. Eymard, T. Gallouët, D. Granjeon, R. Masson and Q. H. Tran, Multi-lithology stratigraphic model under maximum erosion rate constraint, International Journal for Numerical Methods in Engineering, 60 (2004), 527-548. doi: 10.1002/nme.974.

[19]

R. Eymard, T. Gallouët, V. Gervais and R. Masson, Convergence of a numerical scheme for stratigraphic modeling, SIAM Journal on Numerical Analysis, 43 (2005), 474-501. doi: 10.1137/S0036142903426208.

[20]

R. Eymard and T. Gallouët, Analytical and numerical study of a model of erosion and sedimentation, SIAM Journal on Numerical Analysis, 43 (2006), 2344-2370. doi: 10.1137/040605874.

[21]

R. Eymard and T. Gallouët, A partial differential inequality in geological models, Chinese Annals of Mathematics, Series B, 28 (2007), 709-736. doi: 10.1007/s11401-006-0215-3.

[22]

G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modèles non Linéaires de L'ingénierie Pétrolière (Vol. 22), Springer, 1995.

[23]

G. Gagneux and G. Vallet, Sur des problèmes d'asservissements stratigraphiques. A tribute to J.-L. Lions, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 715-739. doi: 10.1051/cocv:2002055.

[24]

G. Gagneux, R. Masson, A. Plouvier-Debaigt, G. Vallet and S. Wolf, Vertical compaction in a faulted sedimentary basin, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 373-388. doi: 10.1051/m2an:2003032.

[25]

V. Gervais and R. Masson, Mathematical and numerical analysis of a stratigraphic model, ESAIM: Mathematical Modelling and Numerical Analysis, 38 (2004), 585-611. doi: 10.1051/m2an:2004035.

[26]

D. Granjeon, Q. Huy Tran, R. Masson and R. Glowinski, Modèle Stratigraphique Multilithologique Sous Contrainte de Taux D'érosion Maximum, Institut Français du Pétrole. Internal report, 2000.

[27]

Z. Gvirtzman, I. Csato and D. Granjeon, Constraining sediment transport to deep marine basins through submarine channels: The Levant margin in the Late Cenozoic, Marine Geology, 347 (2014), 12-26.

[28]

N. Hawie, R. Deschamps, F. H. Nader, C. Gorini, C. Müller, D. Desmares ... and F. Baudin, Sedimentological and stratigraphic evolution of northern Lebanon since the Late Cretaceous: implications for the Levant margin and basin, Arabian Journal of Geosciences, (2013), 1-27.

[29]

E. Leroux, M. Rabineau, D. Aslanian, D. Granjeon, L. Droz and C. Gorini, Stratigraphic simulations of the shelf of the Gulf of Lions: testing subsidence rates and sea-level curves during the Pliocene and Quaternary, Terra Nova, 2014.

[30]

J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles (Vol. 1), Paris: Dunod, 1968.

[31]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (Vol. 31), Paris: Dunod, 1969.

[32]

Y. Mualem and G. Dagan, A dependent domain model of capillary hysteresis, Water Resources Research, 11 (1975), 452-460.

[33]

J. Nečas and S. Mas-Gallic, Écoulements de Fluide: Compacitè par Entropie (Vol. 10), Masson, 1989.

[34]

A. Poulovassilis and E. C. Childs, The hysteresis of pore water: the non-independence of domains, Soil Science, 112 (1971), 301-312.

[35]

J. C. Rivenaes, Application of a dual-lithology, depth-dependent diffusion equation in stratigraphic simulation, Basin Research, 4 (1992), 133-146.

[36]

S. Shmarev and G. Vallet, Local in time solvability of a nonstandard free boundary problem in stratigraphy: A Lagrangian approach, Nonlinear Analysis: Real World Applications, 22 (2015), 404-422. doi: 10.1016/j.nonrwa.2014.10.001.

[37]

R. Tolosana-Delgado and H. von Eynatten, Grain-size control on petrographic composition of sediments: Compositional regression and rounded zeros, Mathematical Geosciences, 41 (2009), 869-886.

[38]

G. Vallet, Sur une loi de conservation issue de la géologie, Comptes Rendus Mathématiques, 337 (2003), 559-564. doi: 10.1016/j.crma.2003.08.012.

show all references

References:
[1]

L. Ambrosio, C. De Lellis and J. Maly, On the chain rule for the divergence of BV-like vector fields: Applications, partial results, open problems, Contemporary Mathematics, 446 (2007), 31-67. doi: 10.1090/conm/446/08625.

[2]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, A non-standard free boundary problem arising from stratigraphy, Analysis and Applications, 4 (2006), 209-236. doi: 10.1142/S0219530506000759.

[3]

S. N. Antontsev, G. Gagneux, R. Luce and G. Vallet, New unilateral problems in stratigraphy, ESAIM: Mathematical Modelling and Numerical Analysis, 40 (2006), 765-784. doi: 10.1051/m2an:2006029.

[4]

S. N. Antontsev, G. Gagneux, A. Mokrani and G. Vallet, Stratigraphic modelling by the way of a pseudoparabolic problem with constraint, Advances in Mathematical Sciences and Applications, 19 (2009), 195-209.

[5]

S. N. Antontsev, G. Gagneux and G. Vallet, On some stratigraphic control problems, Prikladnaya Mekhanika Tekhnicheskaja Fisika, 44 (2003), 85-94 (in Russian). English version: Journal of Applied Mechanics and Technical Physics, 44 (2003), 821-828. doi: 10.1023/A:1026287705015.

[6]

V. R. Baker and D. F. Ritter, Competence of rivers to transport coarse bedload material, Geological Society of America Bulletin, 86 (1975), 975-978.

[7]

G. I. Barenblatt, M. Bertsch, R. D. 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 Journal on Mathematical Analysis, 24 (1993), 1414-1439. doi: 10.1137/0524082.

[8]

J. Blum, G. Dobranszky, R. Eymard and R. Masson, Identification of a stratigraphic model with seismic constraints, Inverse problems, 22 (2006), 1207-1225. doi: 10.1088/0266-5611/22/4/006.

[9]

A. Cimetière, F. Delvare and F. Pons, Une méthode inverse d'ordre un pour les problèmes de complétion de données, Comptes rendus mécanique, 333 (2005), 123-126.

[10]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, An inversion method for harmonic functions reconstruction, International journal of thermal sciences, 41 (2002), 509-516.

[11]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570. doi: 10.1088/0266-5611/17/3/313.

[12]

S. Clain, Elliptic operators of divergence type with Hölder coefficients in fractional Sobolev spaces, Rend. Mat. Appl, 17 (1997), 207-236.

[13]

I. Csato, D. Granjeon, O. Catuneanu and G. R. Baum, A three-dimensional stratigraphic model for the Messinian crisis in the Pannonian Basin, eastern Hungary, Basin Research, 25 (2013), 121-148.

[14]

J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology, Archive for Rational Mechanics and Analysis, 194 (2009), 75-103. doi: 10.1007/s00205-008-0164-y.

[15]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, Dunod, Paris, 1972.

[16]

L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 14 (2004), 1599-1620. doi: 10.1142/S0218202504003763.

[17]

L. C. Evans, A survey of entropy methods for partial differential equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 409-438. doi: 10.1090/S0273-0979-04-01032-8.

[18]

R. Eymard, T. Gallouët, D. Granjeon, R. Masson and Q. H. Tran, Multi-lithology stratigraphic model under maximum erosion rate constraint, International Journal for Numerical Methods in Engineering, 60 (2004), 527-548. doi: 10.1002/nme.974.

[19]

R. Eymard, T. Gallouët, V. Gervais and R. Masson, Convergence of a numerical scheme for stratigraphic modeling, SIAM Journal on Numerical Analysis, 43 (2005), 474-501. doi: 10.1137/S0036142903426208.

[20]

R. Eymard and T. Gallouët, Analytical and numerical study of a model of erosion and sedimentation, SIAM Journal on Numerical Analysis, 43 (2006), 2344-2370. doi: 10.1137/040605874.

[21]

R. Eymard and T. Gallouët, A partial differential inequality in geological models, Chinese Annals of Mathematics, Series B, 28 (2007), 709-736. doi: 10.1007/s11401-006-0215-3.

[22]

G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modèles non Linéaires de L'ingénierie Pétrolière (Vol. 22), Springer, 1995.

[23]

G. Gagneux and G. Vallet, Sur des problèmes d'asservissements stratigraphiques. A tribute to J.-L. Lions, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 715-739. doi: 10.1051/cocv:2002055.

[24]

G. Gagneux, R. Masson, A. Plouvier-Debaigt, G. Vallet and S. Wolf, Vertical compaction in a faulted sedimentary basin, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 373-388. doi: 10.1051/m2an:2003032.

[25]

V. Gervais and R. Masson, Mathematical and numerical analysis of a stratigraphic model, ESAIM: Mathematical Modelling and Numerical Analysis, 38 (2004), 585-611. doi: 10.1051/m2an:2004035.

[26]

D. Granjeon, Q. Huy Tran, R. Masson and R. Glowinski, Modèle Stratigraphique Multilithologique Sous Contrainte de Taux D'érosion Maximum, Institut Français du Pétrole. Internal report, 2000.

[27]

Z. Gvirtzman, I. Csato and D. Granjeon, Constraining sediment transport to deep marine basins through submarine channels: The Levant margin in the Late Cenozoic, Marine Geology, 347 (2014), 12-26.

[28]

N. Hawie, R. Deschamps, F. H. Nader, C. Gorini, C. Müller, D. Desmares ... and F. Baudin, Sedimentological and stratigraphic evolution of northern Lebanon since the Late Cretaceous: implications for the Levant margin and basin, Arabian Journal of Geosciences, (2013), 1-27.

[29]

E. Leroux, M. Rabineau, D. Aslanian, D. Granjeon, L. Droz and C. Gorini, Stratigraphic simulations of the shelf of the Gulf of Lions: testing subsidence rates and sea-level curves during the Pliocene and Quaternary, Terra Nova, 2014.

[30]

J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles (Vol. 1), Paris: Dunod, 1968.

[31]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (Vol. 31), Paris: Dunod, 1969.

[32]

Y. Mualem and G. Dagan, A dependent domain model of capillary hysteresis, Water Resources Research, 11 (1975), 452-460.

[33]

J. Nečas and S. Mas-Gallic, Écoulements de Fluide: Compacitè par Entropie (Vol. 10), Masson, 1989.

[34]

A. Poulovassilis and E. C. Childs, The hysteresis of pore water: the non-independence of domains, Soil Science, 112 (1971), 301-312.

[35]

J. C. Rivenaes, Application of a dual-lithology, depth-dependent diffusion equation in stratigraphic simulation, Basin Research, 4 (1992), 133-146.

[36]

S. Shmarev and G. Vallet, Local in time solvability of a nonstandard free boundary problem in stratigraphy: A Lagrangian approach, Nonlinear Analysis: Real World Applications, 22 (2015), 404-422. doi: 10.1016/j.nonrwa.2014.10.001.

[37]

R. Tolosana-Delgado and H. von Eynatten, Grain-size control on petrographic composition of sediments: Compositional regression and rounded zeros, Mathematical Geosciences, 41 (2009), 869-886.

[38]

G. Vallet, Sur une loi de conservation issue de la géologie, Comptes Rendus Mathématiques, 337 (2003), 559-564. doi: 10.1016/j.crma.2003.08.012.

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