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 & 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. doi: 10.1090/conm/446/08625. Google Scholar

[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. doi: 10.1142/S0219530506000759. Google Scholar

[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. doi: 10.1051/m2an:2006029. Google Scholar

[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. Google Scholar

[5]

S. N. Antontsev, G. Gagneux and G. Vallet, On some stratigraphic control problems,, Prikladnaya Mekhanika Tekhnicheskaja Fisika, 44 (2003), 85. doi: 10.1023/A:1026287705015. Google Scholar

[6]

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

[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. doi: 10.1137/0524082. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. doi: 10.1007/s00205-008-0164-y. Google Scholar

[15]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Travaux et Recherches Mathématiques, (1972). Google Scholar

[16]

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

[17]

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

[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. doi: 10.1002/nme.974. Google Scholar

[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. doi: 10.1137/S0036142903426208. Google Scholar

[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. doi: 10.1137/040605874. Google Scholar

[21]

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

[22]

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

[23]

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

[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. doi: 10.1051/m2an:2003032. Google Scholar

[25]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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. doi: 10.1016/j.nonrwa.2014.10.001. Google Scholar

[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. Google Scholar

[38]

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

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. doi: 10.1090/conm/446/08625. Google Scholar

[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. doi: 10.1142/S0219530506000759. Google Scholar

[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. doi: 10.1051/m2an:2006029. Google Scholar

[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. Google Scholar

[5]

S. N. Antontsev, G. Gagneux and G. Vallet, On some stratigraphic control problems,, Prikladnaya Mekhanika Tekhnicheskaja Fisika, 44 (2003), 85. doi: 10.1023/A:1026287705015. Google Scholar

[6]

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

[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. doi: 10.1137/0524082. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. doi: 10.1007/s00205-008-0164-y. Google Scholar

[15]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Travaux et Recherches Mathématiques, (1972). Google Scholar

[16]

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

[17]

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

[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. doi: 10.1002/nme.974. Google Scholar

[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. doi: 10.1137/S0036142903426208. Google Scholar

[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. doi: 10.1137/040605874. Google Scholar

[21]

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

[22]

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

[23]

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

[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. doi: 10.1051/m2an:2003032. Google Scholar

[25]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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. doi: 10.1016/j.nonrwa.2014.10.001. Google Scholar

[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. Google Scholar

[38]

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

[1]

Raul Borsche, Axel Klar, T. N. Ha Pham. Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 381-401. doi: 10.3934/nhm.2017017

[2]

Chengxiang Wang, Li Zeng, Yumeng Guo, Lingli Zhang. Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data. Inverse Problems & Imaging, 2017, 11 (6) : 917-948. doi: 10.3934/ipi.2017043

[3]

Larisa Beilina, Michel Cristofol, Kati Niinimäki. Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations. Inverse Problems & Imaging, 2015, 9 (1) : 1-25. doi: 10.3934/ipi.2015.9.1

[4]

Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024

[5]

Chenxi Guo, Guillaume Bal. Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields. Inverse Problems & Imaging, 2014, 8 (4) : 1033-1051. doi: 10.3934/ipi.2014.8.1033

[6]

Sarah Bailey Frick. Limited scope adic transformations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 269-285. doi: 10.3934/dcdss.2009.2.269

[7]

Mika Yoshida, Kinji Fuchikami, Tatsuya Uezu. Realization of immune response features by dynamical system models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 531-552. doi: 10.3934/mbe.2007.4.531

[8]

Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151

[9]

Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173

[10]

Jiaqing Yang, Bo Zhang, Ruming Zhang. Reconstruction of penetrable grating profiles. Inverse Problems & Imaging, 2013, 7 (4) : 1393-1407. doi: 10.3934/ipi.2013.7.1393

[11]

Jorge Tejero. Reconstruction of rough potentials in the plane. Inverse Problems & Imaging, 2019, 13 (4) : 863-878. doi: 10.3934/ipi.2019039

[12]

Aili Wang, Yanni Xiao, Huaiping Zhu. Dynamics of a Filippov epidemic model with limited hospital beds. Mathematical Biosciences & Engineering, 2018, 15 (3) : 739-764. doi: 10.3934/mbe.2018033

[13]

Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385

[14]

Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151

[15]

Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems & Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77

[16]

Horst Heck, Gunther Uhlmann, Jenn-Nan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems & Imaging, 2007, 1 (1) : 63-76. doi: 10.3934/ipi.2007.1.63

[17]

Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399

[18]

Pasquale Palumbo, Simona Panunzi, Andrea De Gaetano. Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 399-424. doi: 10.3934/dcdsb.2007.7.399

[19]

Yingli Pan, Ying Su, Junjie Wei. Bistable waves of a recursive system arising from seasonal age-structured population models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 511-528. doi: 10.3934/dcdsb.2018184

[20]

Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]