Mathematical and numerical analysis of a stratigraphic model

Gervais, Véronique; Masson, Roland

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 4, pp. 585-611.

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.

[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. | Numdam | Zbl

[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. | Zbl

[4] R. Eymard, T. Gallouët, V. Gervais and R. Masson, Convergence of a numerical scheme for stratigraphic modeling. SIAM J. Numer. Anal. submitted. | MR | Zbl

[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. | Zbl

[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). | MR | Zbl

[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). | Zbl

[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.