# Mathematical and numerical analysis of a stratigraphic model

Véronique Gervais; Roland Masson

- Volume: 38, Issue: 4, page 585-611
- ISSN: 0764-583X

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topGervais, Véronique, and Masson, Roland. "Mathematical and numerical analysis of a stratigraphic model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.4 (2004): 585-611. <http://eudml.org/doc/245815>.

@article{Gervais2004,

abstract = {In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of $L$ lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main unknowns of the system are the sediment thickness $h$, the $L$ surface concentrations $c_i^s$ in lithology $i$ of the sediments at the top of the basin, and the $L$ concentrations $c_i$ in lithology $i$ of the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for $c_i^s$ with a linear advection equation for $c_i$ for which $c_i^s$ appears as an input boundary condition. For this coupled system, a weak formulation is introduced which is shown to have a unique solution. An implicit finite volume scheme is derived for which we show stability estimates and the convergence to the weak solution of the problem.},

author = {Gervais, Véronique, Masson, Roland},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {finite volume method; stratigraphic modelling; linear first order equations; convergence analysis; linear advection equation; unique weak solution; adjoint problem},

language = {eng},

number = {4},

pages = {585-611},

publisher = {EDP-Sciences},

title = {Mathematical and numerical analysis of a stratigraphic model},

url = {http://eudml.org/doc/245815},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Gervais, Véronique

AU - Masson, Roland

TI - Mathematical and numerical analysis of a stratigraphic model

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 4

SP - 585

EP - 611

AB - In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of $L$ lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main unknowns of the system are the sediment thickness $h$, the $L$ surface concentrations $c_i^s$ in lithology $i$ of the sediments at the top of the basin, and the $L$ concentrations $c_i$ in lithology $i$ of the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for $c_i^s$ with a linear advection equation for $c_i$ for which $c_i^s$ appears as an input boundary condition. For this coupled system, a weak formulation is introduced which is shown to have a unique solution. An implicit finite volume scheme is derived for which we show stability estimates and the convergence to the weak solution of the problem.

LA - eng

KW - finite volume method; stratigraphic modelling; linear first order equations; convergence analysis; linear advection equation; unique weak solution; adjoint problem

UR - http://eudml.org/doc/245815

ER -

## References

top- [1] R.S. Anderson and N.F. Humphrey, Interaction of Weathering and Transport Processes in the Evolution of Arid Landscapes, in Quantitative Dynamics Stratigraphy, T.A. Cross Ed., Prentice Hall (1989) 349–361.
- [2] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels ; théorèmes d’approximation ; application à l’équation de transport. Ann. Sci. École Norm. Sup. 3 (1971) 185–233. Zbl0202.36903
- [3] A. Blouza, H. Le Dret, An up-to-the boundary version of Friedrichs’ lemma and applications to the linear Koiter shell model. SIAM J. Math. Anal. 33 (2001) 877–895. Zbl1008.74057
- [4] R. Eymard, T. Gallouët, V. Gervais and R. Masson, Convergence of a numerical scheme for stratigraphic modeling. SIAM J. Numer. Anal. submitted. Zbl1096.35005MR2177876
- [5] R. Eymard, T. Gallouët, D. Granjeon, R. Masson and Q.H. Tran, Multi-lithology stratigraphic model under maximum erosion rate constraint. Int. J. Numer. Meth. Eng. 60 (2004) 527–548. Zbl1098.76618
- [6] P.B. Flemings and T.E. Jordan, A synthetic stratigraphic model of foreland basin development. J. Geophys. Res. 94 (1989) 3851–3866.
- [7] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996). Zbl0860.65075MR1410987
- [8] D. Granjeon, Modélisation Stratigraphique Déterministe: Conception et Applications d’un Modèle Diffusif 3D Multilithologique. Ph.D. Thesis, Géosciences Rennes, Rennes, France (1997).
- [9] D. Granjeon and P. Joseph, Concepts and applications of a 3D multiple lithology, diffusive model in stratigraphic modeling, in J.W. Harbaugh et al. Eds., Numerical Experiments in Stratigraphy, SEPM Sp. Publ. 62 (1999).
- [10] P.M. Kenyon and D.L. Turcotte, Morphology of a delta prograding by bulk sediment transport, Geological Society of America Bulletin 96 (1985) 1457–1465.
- [11] O. Ladyzenskaja, V. Solonnikov and N. Ural’ceva, Linear and quasilinear equations of parabolic type. Transl. Math. Monogr. 23 (1968). Zbl0174.15403
- [12] J.C. Rivenaes, Application of a dual lithology, depth-dependent diffusion equation in stratigraphic simulation. Basin Research 4 (1992) 133–146.
- [13] J.C. Rivenaes, Impact of sediment transport efficiency on large-scale sequence architecture: results from stratigraphic computer simulation. Basin Research 9 (1997) 91–105.
- [14] D.M. Tetzlaff and J.W. Harbaugh, Simulating Clastic Sedimentation. Van Norstrand Reinhold, New York (1989).
- [15] G.E. Tucker and R.L. Slingerland, Erosional dynamics, flexural isostasy, and long-lived escarpments: A numerical modeling study. J. Geophys. Res. 99 (1994) 229–243.

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