Multi-lithology stratigraphic model under maximum erosion

rate constraint

R. Eymard‡ T. Gallouet§ D. Granjeon†

R. Masson† Q.H. Tran†

March 23, 2011

Abstract

Non-linear single lithology or multi-lithology diffusion models have been widely usedby sedimentologists and geomorphologists in the field of stratigraphic basin simulationsto simulate the large scale depositional transport processes of sediments. Nevertheless, asnoticed by many authors, erosion and sedimentation processes are non symmetric. Soilmaterial must first be produced in situ by weathering processes prior to be transported bydiffusion. This is usually taken into account through a prescribed maximum erosion rateof the sediments, but no mathematical description of the coupling with the diffusion modelhas been proposed so far. In this paper, we introduce a new mathematical formulationfor the coupling of the weather limited erosion and the multi-lithology diffusion models,which appears as a non standard free boundary problem for a new variable acting as alimitor of the fluxes.

One of the main advantages of this formulation, compared to existing discrete cou-pling models, is to enable the definition of efficient discretization schemes. A finite volumescheme with implicit time integration is introduced which is proved to be unconditionallystable in the l∞ norm for the sediment thickness, the sediment concentrations in thelithologies, and the flux limitor variables. A Newton algorithm with an iterative com-putation of the saturated constraints is used to solve efficiently the non-linear systemresulting from the discretization. The efficiency of the model and the numerical schemeis illustrated on 2D and 3D basin simulation examples.

1 Introduction

In recent years, there has been a growing interest in the development of mathematical andnumerical models in stratigraphy and sedimentology in response to the need for quantitativemodeling in a traditionally qualitative science. In the field of sequence stratigraphy, forwardnumerical models have shown to be useful tools to study the effects of eustasy, tectonics, andsediment supply on facies distribution and stratal geometry of basins.

One of the most important process in basin evolution is the erosion and deposition mecha-nism of sediments. Authors usually distinguish between fluid flow and dynamic-slope modelsof the sedimentation erosion processes (see [R92], [R97]). The first ones use fluid flow equa-tions and empirical algorithms to simulate the transport of sediments in the hydrodynamicflow field (see e.g. [TH89]). They provide an accurate description of depositional processesfor small scales in time and space. At larger scale such as basin scales, fluid flow models are

†Institut Francais du Petrole, 1 et 4 av. de Bois Preau 92852 Rueil Malmaison Cedex‡Departement de Mathematiques, Universite de Marne la Vallee, 5 boulevard Descartes, Champs sur Marne,

F-77454, Marnes la vallee, Cedex 2.§LATP, Universite de Provence, 39 rue Frederic Joliot Curie, 13453 Marseille cedex 13.

1

computationally too expensive and dynamic-slope approaches are usually preferred.

These latter models act at a more macroscopic scale and use mass conservation equationsof sediments combined with diffusive transport laws that average over several processes (suchas river transport, creep, slumps and small slides). They provide a good description ofdepositional processes for time scales larger than, say, 105 yr and basin space scales (see[AH89], [R92], [TS94], [R97], [G97], [GJD98],[GJ99]).

In multi-lithology approaches, sediments are modeled as a mixture of several lithologiescharacterized by different grain size populations. The set of equations accounts for the massconservation of each lithology in the basin, knowing the surficial fluxes, a vertical compactionmodel (usually given by depth-porosity empirical laws for the lithologies), and an isostaticmodel for the lithosphere flexure (usually taken as a beam equation).

The main surficial transport process is a multi-lithology diffusion model introduced by[R92] for which fluxes are proportional to the slope of the topography as well as a lithologyfraction defined at the surface of basin (see also [G97], [GJ99], [QAD00]). Diffusion coeffi-cients are non-linear functions of the elevation (or equivalently the bathymetry) to model thetransition from non-marine to marine diffusion that can differ from up to one or two order ofmagnitude.

However, it is well known that sedimentation and erosion are non symmetric processes. Tobe transported by surficial processes, as described by [AH89], material must first be producedin situ by weathering processes depending on climate, elevation, compaction of sediments, ...This production is modeled in [AH89] by a weathering rate also called soil production ratedepending on the sediment depth with parameters function of climate. Then, on each cell ofthe discrete model, the sediment fluxes at the edges are constrained such that erosion cannotexceed the available soil thickness.

In [G97] or [GJ99], the model is slightly different in the sense that it directly prescribesa maximum erosion rate (defined as the partial time derivative of the sediment thickness)depending on climate, elevation, composition, and the age of the sediment.

As illustrated by [AH89], if the weathering rate is low, it can become the dominant pro-cess for erosion which is no longer governed by a diffusion transport while sedimentation isstill diffusion transport limited.

It appears that the coupling of the weather limited model and the diffusion transportmodel is an essential issue for modeling depositional processes. Nevertheless, this question isnot clearly addressed in [AH89] and is only referred to as “a diffusive transport with fluxeslimited by available sediment” without a detailed description of the coupling between bothprocesses. So far, we are not aware of any mathematical description of the coupling betweenweathering limited and diffusive transport models. The main objective of this paper is topropose such a mathematical model from which we are able to derive more efficient numericalalgorithms for stratigraphic simulations.

A model taking into account the dissymmetry between erosion and sedimentation is alsoproposed in [R97]. The author uses a correction of the diffusion coefficient by a porosity ratio(sediment porosity over the depositional porosity) equal to one for sedimentation and lowerthan one for erosion. This ratio acts as a flux limitor, but, being independent on the fluxes(or the slope), it will fail to correctly freeze the erosion at the maximum soil production rate.Similarly a pointwise dependency of the diffusion coefficient on the erosion rate will also failsince, as we shall see in the subsequent development, the flux limitor satisfying the erosion

2

rate constraint is actually a global function of the total flux.

Our approach starts from the numerical model developed at the discrete level in [G97]. Inthis model, the weather limited erosion is written as an inequality constraint on the partialtime derivative of the sediment thickness (erosion or sedimentation rate). Then, the mainidea is to threshold the edges output fluxes along the steepest descent paths in order to satisfythe maximal erosion rate constraint.

In this work, we propose an extension of this discrete model at the continuous level lead-ing to a mathematical formulation coupling the multi-lithology diffusion transport equationsto any weather limited model similar to the ones proposed by [AH89], [TS94] or [G97].

The basic idea of our mathematical formulation is first to impose the inequality constrainton the erosion rate and secondly to limit the fluxes by a factor λ constrained to be less thanone (i.e. the fluxes are lower or equal to the diffusion fluxes). Then, the coupling betweenboth models is obtained by imposing that the inequality constraints on the erosion rate and onthe flux limitor, satisfy complementarity conditions. In other words, whenever the maximalerosion rate constraint is not active, then fluxes are equal to diffusion fluxes (active constraintλ = 1), and conversely, whenever the flux limitor is strictly lower than one (inactive constraintλ < 1), then the maximal erosion rate constraint is active.

It will be shown at the discrete level that the flux limitor λ, satisfying the above comple-mentarity constraints, maximizes globally the fluxes such that the erosion rate cannot exceedthe soil production rate whereas the fluxes cannot exceed the diffusion fluxes.

This formulation will allow us to design an efficient numerical algorithm for the simulationof the model using a finite volume discretization in space, a fully implicit time integration,and a Newton algorithm for the non-linear system adapted to complementarity constraints.

The remaining of the paper outlines as follows. In section 2, we introduce the mathemat-ical model for the coupling of multi-lithology diffusive depositional processes with weatherlimited erosion models. In section 3, a finite volume discretization with implicit time inte-gration is derived which is shown to be unconditionally stable. In section 4, we detail theNewton algorithm for the treatment of complementarity constraints. Section 5 is devoted toa 2D and a 3D basin simulation examples.

2 Mathematical model

A basin model specifies the geometry defined by the basin horizontal extension (of dimen-sion two for 3D basin models and one for 2D basin models), the position of its base due tovertical tectonics displacements, and the sea level variations. It provides a description of thesediments (single lithology or multi-lithology models, porosity laws). Finally it specifies thesediment transport laws and their coupling, as well as the sediment fluxes at the boundary ofthe basin (boundary conditions). As they do not raise difficulties and for the sake of simplic-ity, we shall not consider in the sequel nor the tectonics displacements nor the compaction ofthe sediments.

The projection of the basin on a reference horizontal plane is considered as a fixed domainΩ ⊂ R

d of boundary ∂Ω, defining the horizontal extension of the basin, with d = 1 for twodimensional basin models and d = 2 for three dimensional models. We shall denote by z thevertical coordinate of a point in Ω × R with respect to the reference horizontal plane. For agiven time T > 0, let D denote the domain Ω×(0, T ), and let n be the normal outward to ∂Ω.

3

Let us denote by h(x, t) the vertical coordinate of the surface of the basin at time t ∈ (0, T )for all x ∈ Ω, which parameterizes the topography of the basin. The variable h also representsthe sediment thickness (positive or negative). Let Hm(t) denote the vertical coordinate ofthe sea level at time t and

b(x, t) = h(x, t) −Hm(t)

the bathymetry at the surface of the basin for (x, t) ∈ D.In order to clarify the presentation, the model will be described in three steps:

1 the diffusion model where sediments consist of a single lithology.

2 the weather limited diffusion model for which a maximum erosion rate constraint isadded to the previous model,

3 the full model for which sediments are in addition considered as a mixture of L litholo-gies.

2.1 Single lithology diffusion model

It is known from [AH89] that the evolution of the topography h in a sedimentary basin is welldescribed by non linear parabolic equation solutions. Let ψ(b) denote a strictly increasingfunction of the bathymeytry b, ϕ be the flux boundary condition at the boundary ∂Ω×(0, T ),and h0 the initial topography at time t = 0 defined on Ω. The flux f of sediments at thesurface of the basin (unit in m2/yr) is given by the diffusion law

f = −∇ψ. (1)

The topography h satisfies the following non-linear parabolic equation accounting for theconservation of the sediment thickness.

Given ϕ, h0, Hm, ψ, find h such that

∂th− div(∇ψ(b)) = 0 on D,−∇ψ(b) · n = ϕ on ∂Ω × (0, T ),

h|t=0 = h0 on Ω.

(2)

The non-linear dependence of ψ on the bathymetry b accounts for the fact that thediffusion coefficient k(b) = dψ

db(b) is usually of different orders of magnitude in marine (b < 0)

and continental environnements (b ≥ 0). Such a function is typically given by

ψ(b) =

km b if b < 0,kc b otherwise

where kc > 0 (resp. km > 0) is the continental (resp. marine) diffusion coefficient parameter.

An example of a 2D basin (d = 1) is sketched in figure 1 with Ω = (0, Lb). The figuredisplays at time t the main variables of the model i.e. the sediment thickness or topographyh, the sea level Hm, the bathymetry b = h −Hm, and the boundary flux ϕ (in this case anoutput flux ϕ(Lb, t) at x = Lb and an input flux ϕ(0, t) at x = 0).

4

x

z

bL0

ϕ b(x,t)

(t)mH

(L ,t)b

h(x,t)

(0,t)ϕ

Figure 1: Example of a 2D sedimentary basin (d = 1) for the single-lithology model

2.2 Weather limited model

Diffusion models assume that sedimentation and erosion are symmetric processes since thediffusion coefficient k(b) is independent on the sedimentation (∂th ≥ 0) or erosion (∂th < 0)rate ∂th. Actually, as stated in [AH89], sediments must first be produced in situ by weatheringprocesses prior to be transported by surficial erosion. This limitation is modeled in [G97]and [GJ99] through a maximum erosion rate −E ≤ 0 (unit in m/yr) such that

∂th ≥ −E.

For the sake of simplicity, we shall assume in the sequel that E is a non negative constantalthough in more realistic models E depends on other variables such as the bathymetry b.The model is termed weather limited when the constraint ∂th ≥ −E is active i.e. ∂th = −E.

The coupling of the weather limited and diffusion models is clearly an essential issue sinceboth diffusive sedimentation or erosion and weather limited erosion can clearly occur at thesame time in a basin.

Nevertheless, this question has not been addressed rigorously so far. Although coupledsingle lithology numerical simulations are performed in [AH89] and [TS94], the couplingmodel is never made explicit. The main explanation is that the coupling has only been de-veloped at the discrete level. This discrete coupling has been extended to multi-lithologymodels in [G97] and is detailed for a finite volume discretization in space and an explicit timeintegration scheme. The basic idea is to threshold the diffusion output fluxes at the edges ofthe finite volumes along the discrete steepest descent paths such that the limited fluxes aremaximum, and the solution satisfies the maximum erosion rate constraint on each cell.

The derivation of the continuous model mimics the discrete scheme defined in [G97]. Anew unknown λ is introduced, satisfying λ ≤ 1 and playing the role of a flux limitor leadingto the new definition of the flux

f = −λ ∇ψ(b)

and the new conservation equation of the sediment thickness h

∂th− div(

λ ∇ψ(b))

= 0 on D.

In order to close the model it is required that

5

(i) if the erosion rate constraint is inactive ∂th > −E then λ = 1 i.e. the flux f is equal tothe diffusion flux (1),

(ii) if λ < 1, then the constraint on the erosion rate is active i.e. ∂th = −E.

Such conditions are known as complementarity conditions for which we introduce the follow-ing notation: let U be any function from R

2 to R such that

U(A,B) = 0 iff

AB = 0,A ≥ 0,B ≥ 0,

for (A,B) ∈ R2. As illustrated in figure 2, the complementarity conditions U(A,B) = 0

equivalently mean that the point (A,B) is in the union of the two positive half axes.

00

A

B

Figure 2: Unilateral conditions U(A,B) = 0.

Then, the above closure conditions (i) and (ii) state that

U(

∂th+ E, 1 − λ)

= 0 on D. (3)

Let Σ− = (x, t) ∈ ∂Ω × (0, T ), ϕ(x, t) < 0 and Σ+ = (x, t) ∈ ∂Ω × (0, T ), ϕ(x, t) ≥ 0.On the input boundary Σ−, the boundary condition is unchanged. On the output boundaryΣ+ the boundary condition has to be adapted since the output flux ϕ may have to be limitedto satisfy the maximum erosion rate constraint. This is again achieved by imposing thefollowing complementarity conditions

U(

ϕ− f · n, ∂th+ E)

= 0 on Σ+, (4)

which lead to a switch from a flux boundary condition when ∂th + E > 0 to a Dirichletboundary condition on the time derivative of h when f · n < ϕ.

Below, we summarize the single-lithology model under maximum erosion rate constraint.

Given ϕ, h0, Hm, ψ, E, find h, λ such that

∂th− div(λ∇ψ(b)) = 0 on D,U(∂th+ E, 1 − λ) = 0 on D,

−λ∇ψ(b) · n = ϕ on Σ−,U(∂th+E,ϕ + λ∇ψ(b) · n) = 0 on Σ+,

h|t=0 = h0 on Ω.

(5)

6

2.3 Extension to the multi-lithology model

In such models, the sediments are described as a mixture of L lithologies characterized bytheir grain size population. Each lithology, i = 1, · · · , L, is considered as an uncompressiblematerial of constant grain density and null porosity (no compaction).

In addition to the evolution of the topography h, the model describes the composition ofthe sediments at each point of the basin domain (x, z), such that x ∈ Ω, z < h(x, t) for alltime t ∈ (0, T ).

The composition of the sediments inside the basin is modeled by their concentrationci(x, z, t) in the lithology i for all i = 1, · · · , L, defined on the domain

B = (x, z, t) such that (x, t) ∈ D, z < h(x, t).

and such that

ci ≥ 0 and

L∑

i=1

ci = 1 on B.

Since the compaction is not considered, there is no evolution in time of the concentrations ciinside the basin such that for all i = 1, · · · , L, the concentration ci satisfies the conservationequation

∂tci = 0 on B.

The initial composition of the basin is given by the concentrations c0i defined on the domainB0 = (x, z), such that x ∈ Ω, z < h0(x), such that c0i ≥ 0 for all i = 1, · · · , L, and∑L

i=1 c0i = 1.

The evolution of the concentration ci is governed for each i = 1, · · · , L by an inputboundary condition at the surface of the basin z = h(x, t) in the case of sedimentation∂th > 0. This input boundary condition is given by the new variable csi called surfaceconcentration defined on the domain D for each i = 1, · · · , L and such that

csi ≥ 0 and

L∑

i=1

csi = 1 on D.

All together this leads to the following first set of conservation equations for the multi-lithology model:

∂tci = 0 on B,ci|z=h = csi on D+,ci|t=0 = c0i on B0,

(6)

withD+ = (x, t) ∈ D such that ∂th(x, t) > 0.

In the case of sedimentation ∂th > 0, the surface concentrations csi , i = 1, · · · , L define thecomposition of the sediments deposited at the top of the basin and reduce to csi (x, t) =ci(x, h(x, t), t) for all i = 1, · · · , L and all (x, t) such that ∂th(x, t) > 0. However, in the ero-sive mode ∂th ≤ 0, the surface concentrations are no longer connected to the concentrationsci at the top of the basin. In that case, they represent the concentrations of the sedimentspassing through the surface as they appear in the following definition of the surficial fluxes.

In order to close the system, we need to extend the conservation equations (5) to themulti-lithology setting. This is achieved using the hillslope transport model introduced in

7

[R92]. Let ψi, i = 1, · · · , L be strictly increasing functions of the bathymetry b, then for eachi = 1, · · · , L,

fi = −csiλ∇ψi(b)

defines the surficial flux of the sediments in lithology i, proportional to the gradient of ψi andto the surface concentration csi , and limited by the flux limitor λ. We denote by ki(b) = dψi

db(b)

the diffusion coefficient of the lithology i. Then, the set of equations accounts, for eachlithology i = 1, · · · , L, for the conservation of the sediment thickness in lithology i definedfor all (x, t) ∈ D as

hi(x, t) =

∫ h(x,t)

0ci(x, z, t)dz,

with∑L

i=1 hi = h on D. It states that

∂thi + divfi = 0 on D for i = 1, · · · , L,∑L

i=1 csi = 1 on D,

(7)

together with the complementarity conditions (3). The initial condition is still defined byh|t=0 = h0. The boundary condition on the output boundary Σ+ is also given by (4) withthe total flux f =

∑Li=1 fi. The input boundary conditions are adapted by prescribing the

input fluxes fi · n = µiϕ on Σ− for all i = 1, · · · , L where µi, i = 1, · · · , L are given functionsdefined on Σ− such that µi ≥ 0 and

∑Li=1 µi = 1.

The 2D basin example is completed in figure 3 to include in addition to the previousvariables, the sediment concentrations ci, the surface sediment concentrations csi , and theboundary fluxes (in this case an output flux ϕ(Lb, t) at x = Lb and input fluxes µi(0, t)ϕ(0, t)for each lithology at x = 0).

Lb

z

x

µ

0

h(x,t)

b(x,t)

i (0,t)ϕ (0,t)

(t)

ϕ b(L ,t)

Hm

c (x,t)is

ic (x,z,t)

Figure 3: Example of a 2D sedimentary basin (d = 1) for the multi-lithology model

In summary, here is the multi-lithology model under maximum erosion rate constraint.

8

Givenϕ, h0, Hm, E, c

0i , µi, ψi for all i = 1, · · · , L

Findh, λ, ci, c

si for all i = 1, · · · , L

such that

Conservation equations on D∂thi − div(csiλ∇ψi(b)) = 0 on D,

∑Li=1 c

si = 1 on D,

U(∂th+ E, 1 − λ) = 0 on D,−csiλ∇ψi(b) · n = µiϕ on Σ−,

U(∂th+ E,ϕ+∑L

i=1 csiλ∇ψi(b) · n) = 0 on Σ+,

h|t=0 = h0 on Ω,Conservation equations on B

∂tci = 0 on B,ci|z=h = csi on D+,ci|t=0 = c0i on B0.

(8)

One of the main advantage of this formulation is to allow the definition of an efficientnumerical scheme using a fully implicit time integration and a Newton algorithm suited tocomplementarity constraints. This is the purpose of the next two sections.

3 Finite Volumes Discretization

3.1 The discrete scheme

The system (8) is discretized by a fully implicit time integration and a finite volume methodwith cell centered variables.

For the sake of simplicity, let us consider a rectangular domain Ω endowed with a Carte-sian mesh K (non structured meshes could be considered as well). Let |κ| denote the ddimensional volume of the cell κ ∈ K, and d(κ, κ′) be the distance between the centers of thecells κ and κ′ of K. Let Σκ denote the set of the edges of the cell κ, and |σ| be the d − 1dimensional measure (set to 1 for d = 1) of an edge σ ∈ Σκ.

The time discretization is denoted by tn, n ∈ N such that t0 = 0, and ∆tn+1 := tn+1−tn >0. In the following, the superscript n, n ∈ N, will be used to denote that the variables areconsidered at time tn.

Let us now introduce the primary variables of the discrete problem (see figures 4 and 5).For all κ ∈ K, and n ∈ N, we denote by hn+1

κ (resp. λn+1κ , and cs,n+1

i,κ for i = 1, · · · , L) the

approximation at time tn+1 and in the cell κ of the sediment thickness h (resp. of the fluxlimitor λ, and of the surface concentrations csi for i = 1, · · · , L). For all κ ∈ K, n ∈ N, andi = 1, · · · , L, cn+1

i,κ is a function defined on (−∞, hn+1κ ), which approximates at time tn+1 and

in the cell κ the concentration ci,κ.

9

κ

,iϕ

Σ

κ’’λn+1 ϕs,n+1

∋

∋

µκ’’

( )n+1

− κ

σ +n+1

Σ

σ diψ ( )

Cartesian mesh, Cell unknowns and Edge fluxes

κ

σn+1

− iκ ’, ’

n+1 ψ ( )bn+1

bc

κ

s,n+1,i

’

σ

n+1σ

κλ

s,n+1κi,κ

n+1κh n+1c |σ|,,

σ

κ

n+1σλ

’

’

Figure 4: Cartesian mesh K, cell unknowns (hn+1κ , λn+1

κ , cs,n+1i,κ ), and approximations of the

fluxes at the edge σ ∈ Σk ∩ Σk′ between two cells κ and κ′, at an input edge σ ∈ Σn+1− , and

at an output edge σ ∈ Σn+1+ ∩ Σk′′.

i,κ (z)nc

z zSedimentation Erosion

z

h

n+1

ci,κs,n+1

h

κ

κn

hκn+1

h

κ

i,κn+1

(z)c

n

ci,κn+1

(z)

Figure 5: Concentration cni,κ at time tn and update at time tn+1 of the concentration cn+1i,κ in

the sedimentation or erosion cases.

We suppose that, for all control volume κ ∈ K, the following initial values are defined:

1. h0κ is an approximation of h0 in the cell κ,

2. c0i,κ, for all species i, is a piecewise constant non negative approximation of c0i in the

cell κ defined on (−∞, h0κ) and such that

∑Li=1 c

0i,κ = 1.

The discretization of the conservation equations (7) is obtained by integration over thecell κ and between the times tn and tn+1

∫ tn+1

tn

∫

κ

∂thi dx dt +∑

σ∈Σκ

∫ tn+1

tn

∫

σ

−λ csi ∇ψi(b) · nκ dσ dt = 0,

where nκ is the normal to σ outward to κ, followed by approximations of the accumulationand flux terms detailed below.

As illustrated in figure 4, the flux

1

∆tn+1

∫ tn+1

tn

∫

σ

−λ csi ∇ψi(b) · nκ dσ dt (9)

10

at the edge σ ∈ Σκ ∩ Σκ′ between the cells κ and κ′ is approximated by

|σ|λn+1σ cs,n+1

i,σ

ψi(bn+1κ ) − ψi(b

n+1κ′ )

d(κ, κ′),

where bn+1κ = hn+1

κ −Hm(tn+1) is the bathymetry in the cell κ at time tn+1,

λn+1σ =

λn+1κ if bn+1

κ > bn+1κ′ ,

λn+1κ′ otherwise ,

and

cs,n+1i,σ =

cs,n+1i,κ if bn+1

κ > bn+1κ′ ,

cs,n+1i,κ′ otherwise .

This approximation is implicit in time and uses an upstream weighted evaluation of λ andcsi at the edge σ between the cells κ and κ′, and a two point approximation of the gradientof ψi in the normal direction. These choices are the key ingredients to obtain the stability ofthe numerical scheme discussed in the next section.

To define the approximation of the flux (9) for edges σ at the boundary ∂Ω, let us introducethe subset Σn+1

− (resp. Σn+1+ ) of

⋃

κ∈K Σκ such that Closurex, (x, tn+1) ∈ Σ− =⋃

σ∈Σn+1

−σ

(resp. Closurex, (x, tn+1) ∈ Σ+ =⋃

σ∈Σn+1

+

σ).

Then, for all σ ∈ Σn+1− ∩ Σκ, the flux (9) is given by the approximation

ϕn+1σ,i =

∫

σ

µi(x, tn+1)ϕ(x, tn+1) dσ

of the input boundary condition.

Let µs,n+1i,κ denote the fractional flow

cs,n+1

i,κ ki(bn+1κ )

PLj=1

cs,n+1

j,κkj(b

n+1κ )

. On the output boundary, for all

σ ∈ Σn+1+ ∩ Σκ, the flux (9) is approximated by

λn+1κ µs,n+1

i,κ ϕn+1σ with ϕn+1

σ =

∫

σ

ϕ(x, tn+1) dσ.

The approximation |κ|∆hn+1i,κ of the accumulation term

∫ tn+1

tn

∫

κ

∂thi(x, t) dx dt =

∫

κ

(

∫ h(x,tn+1)

0ci(x, z, t

n+1) dz −

∫ h(x,tn)

0ci(x, z, t

n) dz)

dx

is defined by

|κ|∆hn+1i,κ = |κ|

(

∫ hn+1κ

0cn+1i,κ (z) dz −

∫ hnκ

0cni,κ(z) dz

)

. (10)

The approximate concentration cn+1i,κ is the solution at time tn+1 of the conservation

equation

∂tci,κ(z, t) = 0, for all tn < t < tn+1, z < hnκ + (t− tn)hn+1κ −hn

κ

∆tn+1 ,

ci,κ(hκ(t), t)) = cs,n+1i,κ if hn+1

κ > hnκ,

ci,κ(z, tn) = cni,κ(z) for all z < hnκ.

(11)

The integration of this equation is straightforward considering separately the sedimentation(hn+1κ > hnκ) and erosion (hn+1

κ ≤ hnκ) cases. It leads to the update formulae (14) and (15)

11

illustrated in figure 5. We also deduce the new expression of the accumulation term ∆hn+1i,κ

given in (14) and (15).

The discretization of the complementarity constraints (3) is defined by

U(

hn+1κ −Hn+1

κ , 1 − λn+1κ

)

= 0,

where Hn+1κ is the solution hnκ −E∆tn+1 at time tn+1 to the erosion rate constraint equation

between tn+1 and tn: ∂tHκ(t) = −E, Hκ(tn) = hnκ.

Finally, the discrete problem summarizes as follows.

Given Hm, ψi(b), h0κ, and c0i,κ for all κ ∈ K, i = 1, · · · , L, ϕn+1

σ for all n ≥ 0, σ ∈ Σn+1+ , and

ϕn+1σ,i for all i = 1, · · · , L, n ≥ 0, σ ∈ Σn+1

− ,

Find hn+1κ , λn+1

κ , and cs,n+1i,κ , cn+1

i,κ for all κ ∈ K, n ≥ 0, i = 1, · · · , L such that for all k ∈ Kand i = 1, · · · , L:

Conservation of surface sediments:

∆hn+1i,κ

∆tn+1|κ| +

∑

σ∈Σκ∩Σκ′

λn+1σ cs,n+1

i,σ |σ|ψi(b

n+1κ ) − ψi(b

n+1κ′ )

d(κ, κ′)+

∑

σ∈Σκ∩Σn+1

−

ϕn+1σ,i + λn+1

κ µs,n+1i,κ

∑

σ∈Σκ∩Σn+1

+

ϕn+1σ = 0

(12)

L∑

i=1

cs,n+1i,κ = 1, (13)

Conservation of column sediments:

if hn+1κ ≥ hnκ (sedimentation)

∆hn+1i,κ = cs,n+1

i,κ (hn+1κ − hnκ)

cn+1i,κ (z) = cni,κ(z) for all z < hnκcn+1i,κ (z) = cs,n+1

i,κ for all z ∈ (hnκ, hn+1κ )

(14)

else (erosion)

∆hn+1i,κ =

∫ hn+1κ

hnκ

cni,κ(z)dz

cn+1i,κ (z) = cni,κ(z) for all z < hn+1

κ

(15)

Constraints:

U(

hn+1κ −Hn+1

κ , 1 − λn+1κ

)

= 0, (16)

Let us note that at each time step from tn to tn+1, the computation of the unknownshn+1κ , λn+1

κ , and cs,n+1i,κ decouples from the update of the column sediment concentrations

cn+1i,κ .

3.2 Stability properties of the finite volumes discretization

An essential topic is the study of the non-negativity of the limitors and of the concentrationssolutions to (12)-(16). It is possible to state such a result with a slight modification of thedefinition of the output boundary fluxes.

12

Proposition 3.1. Let us assume for all time tn+1, n ≥ 0, the new definition µs,n+1i,κ = cs,n+1

i,κ

for all κ ∈ K or alternatively ϕn+1σ = 0 for all σ ∈ Σn+1

+ .

Then, any solution of the scheme (12)-(16) satisfies λn+1κ ∈ [0, 1] and cs,n+1

i,κ ∈ [0, 1] forall i = 1, . . . , L, n ≥ 0, κ ∈ K, except for the degenerate points (κ, n).

The degenerate points (κ, n) are the points for which all the fluxes at the edges of thecontrol volume κ vanish, and hn+1

κ = hnκ. In that cases, the concentrations cs,n+1i,κ , i = 1, · · · , L

can be arbitrarily chosen such that their sum over the species is equal to 1, and the flux limitorλn+1κ is either arbitrary in the interval (−∞, 1] for E = 0 or equal to 1 for E > 0.

Proof: The proof is done by induction over n and over the control volumes κ ∈ K indecreasing topographical order.

For the highest topographical point(s) κ, the fluxes at the edges σ ∈ Σκ are either outputfluxes or input boundary fluxes ϕn+1

σ,i , σ ∈ Σκ ∩ Σn+1− , and therefore for all species i

∑

σ∈Σκ∩Σκ′ , bn+1κ <bn+1

κ′

λn+1σ cs,n+1

i,σ |σ|ψi(b

n+1κ ) − ψi(b

n+1κ′ )

d(κ, κ′)≤ 0. (17)

Let us consider a control volume κ ∈ K and assume that the proposition holds for all thelower cells at time tn+1 and all the previous times tl+1, 0 ≤ l < n. It results from thisinduction hypothesis and the upwinding of λ and csi that the inequality (17) also holds forall i = 1, · · · , L.

Hence, the proof will be obtained if the proposition is proved at time tn+1, assuming thatthe sediment concentrations cni,κ(z), κ ∈ K, z < hnκ are positive (which holds for n = 0) andthat the inequality (17) holds.

Let us proceed by assuming that hn+1κ − hnκ ≤ 0 (erosion). It results from the induction

hypothesis over n and over the control volumes that

λn+1κ cs,n+1

i,κ

(

∑

σ∈Σκ∩Σκ′ , bn+1κ ≥bn+1

κ′

|σ|ψi(b

n+1κ ) − ψi(b

n+1κ′ )

d(κ, κ′)+

∑

σ∈Σκ∩Σn+1

+

ϕn+1σ

)

≥ 0, (18)

for all species i. In equation (18), either the term inside the brackets is strictly positive for alli = 1, · · · , L or it vanishes for all i = 1, · · · , L. In the first case, it results from (13) that λn+1

κ

and cs,n+1i,κ , i = 1, · · · , L are non negative. In the second case, the point (κ, n) is a degenerate

point for which the concentrations cs,n+1i,κ , i = 1, · · · , L can be arbitrarily chosen such that

∑Li=1 c

s,n+1i,κ = 1. The flux limitor λn+1

κ can be also arbitrarily chosen in the interval (−∞, 1]if E = 0 but is equal to 1 from equations (16) if E > 0.

Let us now consider the sedimentation case for which hn+1κ − hnκ > 0. It results from (16)

that λn+1κ = 1. From equation (12) and the induction hypothesis we obtain that

cs,n+1i,κ

(hn+1κ − hnκ∆tn+1

|κ| +∑

σ∈Σκ∩Σκ′ , bn+1κ ≥bn+1

κ′

|σ|ψi(b

n+1κ ) − ψi(b

n+1κ′ )

d(κ, κ′)+

∑

σ∈Σκ∩Σn+1+

ϕn+1σ

)

≥ 0,

and hence cs,n+1i,κ ≥ 0, for all i = 1, · · · , L.

As a consequence of the non negativity of λn+1κ and of cs,n+1

i,κ ∈ [0, 1], n ≥ 0, κ ∈ K, itis possible to deduce that the sediment thickness satisfies a discrete maximum principle (see

13

[EGH00] for examples of such proofs).

The next proposition provides a characterization of the limitors as the coefficients maxi-mizing the edges total output fluxes (for given concentrations and sediment thicknesses) underthe constraints 1 − λn+1

κ ≥ 0 and hn+1κ ≥ Hn+1

κ . The proof is straightforward consideringthe cell conservation equations and complementarity constraints in decreasing topographicalorder.

Proposition 3.2. Let κl, l = 1, · · · ,#K be a decreasing topographical ordering of the cells(with #K denoting the cardinality of the set K). The limitors λn+1

κ , k ∈ K are given by

λn+1κl

= min(

1,ακl

βκl

)

for l = 1, · · · ,#K, with

ακl= E|κl| −

∑L

i=1

(

∑

σ∈Σκl∩Σκ′ , b

n+1κl

<bn+1

κ′

λn+1σ cs,n+1

i,σ |σ|ψi(b

n+1κl

) − ψi(bn+1κ′ )

d(κl, κ′)+

∑

σ∈Σκl∩Σn+1

−

ϕn+1σ,i

)

,

βκl=

∑L

i=1

∑

σ∈Σκl∩Σκ′ , b

n+1κl

≥bn+1

κ′

cs,n+1i,σ |σ|

ψi(bn+1κl

) − ψi(bn+1κ′ )

d(κl, κ′)+

∑

σ∈Σκl∩Σn+1

+

ϕn+1σ .

Note that for degenerate points (κ, n), λn+1κ is set to 1. This formula also characterizes the

unique solution of the following optimization problem: given the concentrations cs,n+1i,κ , i =

1, · · · , L and the sediment thicknesses hn+1κ , κ ∈ K, find λn+1 = (λn+1

κ )κ∈K maximum, in thesense of the vectorial ordering λ′ ≥ λ iff λ′κ ≥ λκ for all κ ∈ K, and under the inequalityconstraints

1 − λn+1κ ≥ 0, κ ∈ K,

∑L

i=1

(

∑

σ∈Σκ∩Σκ′

λn+1σ cs,n+1

i,σ |σ|ψi(b

n+1κ ) − ψi(b

n+1κ′ )

d(κ, κ′)+

∑

σ∈Σκ∩Σn+1−

ϕn+1σ,i

)

+

λn+1κ

∑

σ∈Σκ∩Σn+1

+

ϕn+1σ ≤ E|κ|, κ ∈ K.

Remark 3.1. From Proposition 3.2, the limitor λ exhibits a non local dependence on theedges total fluxes, which shows that a proper coupling of the weather limited and diffusionmodels could not be obtained with a limitor locally depending on the erosion rate.

4 Non-linear solver

The non-linear system (12)-(16) is solved using a Newton algorithm adapted to complemen-tarity constraints (see [EGH00]).

A binary phase index I = (Iκ)κ∈K ∈ 0, 1K is introduced where Iκ = 1 correspondsto the diffusion transport λn+1

κ = 1, hn+1κ ≥ Hn+1

κ , and Iκ = 0 corresponds to the weatherlimited transport hn+1

κ = Hn+1κ , λn+1

κ ≤ 1.For a fixed index phase I, we define the variable yn+1 such that for all k ∈ K

yn+1κ =

hn+1κ if Iκ = 1,λn+1κ if Iκ = 0.

Then, we denote by Rn+1 : RK × (RK)L → (RK)L × R

K, the residual of equations (12)-(13) as function of (yn+1, cn+1) for a fixed phase index I, where cn+1 stands for the vector

14

cs,n+1i,κ , κ ∈ K, i = 1, · · · , L.

The Newton phase index algorithm performs a sequence of Newton iterations applied tothe non-linear function Rn+1 followed by updates of the phase index in order to satisfy the

inequality constraints(

hn+1κ −Hn+1

κ

)

≥ 0 and(

1 − λn+1κ

)

≥ 0 for all κ ∈ K.

If the solution satisfies at convergence λn+1κ ≥ 0 and cs,n+1

i,κ ≥ 0, for all κ ∈ K, i = 1, · · · , L,as proved in proposition 3.1, this is no longer the case of the Newton iterates. Hence, theseconstraints are also imposed during the update step by projections.

Initialization:

In+1,0κ = 1, yn+1,0

κ = hnκ, κ ∈ K,cn+1,0 = cn.

Solve for q = 0, · · · ,until convergence

∂(yn+1,cn+1)Rn+1(yn+1,q, cn+1,q)(δy, δc)T = −Rn+1(yn+1,q, cn+1,q),

(yn+1,q+1, cn+1,q+1)T = (yn+1,q, cn+1,q)T + (δy, δc)T

if In+1,qκ = 1

if hn+1,q+1κ −Hn+1

κ < 0 then

In+1,q+1κ = 0,

hn+1,q+1κ = Hn+1

κ

else In+1,q+1κ = In+1,q

κ

if In+1,qκ = 0

if λn+1,q+1κ > 1 then In+1,q+1

κ = 1 and λn+1,q+1κ = 1,

else In+1,q+1κ = In+1,q

κ , λn+1,q+1κ := max(0, λn+1,q+1

κ ).

(

cs,n+1,q+1i,κ

)L

i=1:= Projd∈[0,1]L,

PLi=1 di=1

(

cs,n+1,q+1i,κ

)

i=1,··· ,L

End

The Jacobian ∂(yn+1,cn+1)Rn+1 is clearly singular in the two following cases corresponding

to non physical situations. Although these are excluded at convergence, they can occur duringthe Newton iterations.

The first case arises when, for a given cell κ, the erosion rate constraint is active In+1,qκ = 0

and all the fluxes at the edges σ ∈ Σκ are input fluxes. Then, it results from the upwindingof λ that the column ∂

yn+1κ

Rn+1 of the Jacobian vanishes.

Similarly, when for a given cell κ, hn+1,qκ − hnκ ≤ 0 (eroded cell) and all the fluxes at the

edges σ ∈ Σκ are input fluxes, then due to the upwinding of the surface sediment concentra-tions, the columns ∂

cs,n+1

i,k

Rn+1, i = 1, · · · , L, of the Jacobian contain only zero entries.

In such cases, we modify the Jacobian to avoid its singularity. Precisely, if∑L

i=1 ∂yn+1κ

Rn+1i,κ =

0, then we set ∂yn+1κ

Rn+11,κ = ǫ > 0. In practice, it is more efficient to combine this strategy

with a modification of the index phase and the Newton iterate during the update step whicheliminates partially these non physical degenerate cases.

Similarly for L > 1, if ∂cs,n+1

1,k

Rn+11,κ = 0, then we set ∂

cs,n+1

i,k

Rn+1i,κ = 1 for i = 1, · · · , L and

leave the concentrations cs,n+1i,k , i = 1, · · · , L unchanged at this Newton iteration.

4.1 Numerical examples

4.2 Delta Progradation

We consider the interval domain Ω = (0, Lb) with Lb = 200 km. The initial topography

is given by the function h0(x) = 25e−8 x

Lb + 10 m, and the eustatic sea level variations by

15

Hm(t) = 25 + 5 cos(6t) m.

The simulation is done with two lithologies and ψi(b) =∫ b

0 ki(u)du with the diffusioncoefficients ki(b) defined by

ki(b) =

ki,m if b ≤ −β,(

ki,c

ki,m

)b

2β(

ki,cki,m

)1

2

if − β < b < β,

ki,c if b ≥ β,

(19)

for i = 1, 2, with k1,c = 105 m2/yr, k1,m = 104 m2/yr, k2,c = 5.104 m2/yr, k2,m = 103 m2/yr,and β = 1 m.

The sediment supply is given by the input total flux ϕ = 2m2/yr and the fractional flowsµ1 = µ2 = 0.5 at x = 0, and the total output flux ϕ is set to zero at x = Lb. The basin initialcomposition is c01(x, z) = c02(x, z) = 0.5 on the domain z < h0(x), x ∈ Ω.

The stratigraphic layers of the basin at time T are defined as the isotime surfaces of thefunction

SL(x, t, T ) = mint≤t′≤T

h(x, t′), (x, t) ∈ D,

where the minimization over the interval (t, T ) accounts for the erosion of the layers betweenthe times t and T .

The mesh is uniform of step ∆x. The time stepping is adaptively computed as follows.We fix an objective maximal variation of the sediment thickness ∆hmax as well as a maximalvariation ∆Hm of Hm, a maximal time step ∆tmax, and an initial time step ∆t0. Then, thesuccessive time steps are computed by the following induction formula:

∆t0 = ∆t0,

∆tn+1 = min(

α∆tn,∆tmax,∆tm,∆hmax∆tn

(maxκ∈K |hnκ−h

n−1κ |)

)

(20)

with α > 1, and ∆tm such that |Hm(tn + ∆tm) −Hm(tn)| = ∆Hm.In case of non convergence of the Newton algorithm after a fixed maximum number of

iterations, the time step is restarted with a twice smaller value.

Figure 6 illustrates the influence of the maximum erosion rate parameter E on the strati-graphic layers and the sediment concentration c1 for x = 100 km. This effect is very noticeablefor decreasing sea level sequences. For E large, the erosion rate constraint is not active andthe continental topography is close to an horizontal straight line at the sea level (for largenon-marine diffusion coefficients ki,c). On the contrary, for E small, the non-marine ero-sion is constrained by the maximum erosion rate resulting in a much smoother continentaltopographical curve down to the sea level.

Note also, as illustrated on the bottom picture of figure 6, that the sediment concentrationsmay exhibit jumps resulting from erosion followed by sedimentation sequences.

16

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Z (

m)

X (100 km)

E = +Infinity m/Myr

SL(X,t,T): t = n*0.064, n=1,...,25 h0(X)

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Z (

m)

X (100 km)

E = 10 m/Myr

SL(X,t,T): t = n*0.064, n=1,...,25 h0(X)

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Z (

m)

X (100 km)

E = 5 m/Myr

SL(X,t,T): t = n*0.064, n=1,...,25 h0(X)

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Z (

m)

X (100 km)

E = 1 m/Myr

SL(X,t,T): t = n*0.064, n=1,...,25 h0(X)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

C1

Z (m)

SEDIMENT CONCENTRATION

E = 1 m/Myr E = 5 m/Myr

E = 10 m/Myr E = +Infinity m/Myr

Figure 6: Stratigraphic layers of the basin at time T = 1.6 Myr, and sediment concentrationsc1(x, z, T ), x = 100 km, for the Delta progradation test case with maximum erosion ratesE = 1, 5, 10, and +∞ m/Myr. The simulation is performed over a time span of 1.6 Myr,with a uniform time step ∆t = 0.0025 Myr, and the mesh size ∆x = 2 km.

The next figure 7 exhibits the plots of the sediment thickness h, the limitor λ, the totalflux f :=

∑Li=1 fi, the accumulation rate ∂th, and the surface sediment concentration cs1 at

times t = 0.5 Myr and t = 0.75 Myr. It illustrates the discontinuity of the limitor λ at thetransition from weather limited to diffusive transport processes, resulting in slope breaks ofthe sediment thickness.

The total flux f satisfies the constraint divf ≤ E. In one dimension d = 1, it results that

17

the total flux slope cannot exceed E which is clearly illustrated on the plots of figure 7.

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h (

m)

X (100 km)

SEDIMENT THICKNESS

t = 0.5 Myr t = 0.75 Myr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

LA

MB

DA

X (100 km)

LAMBDA

t = 0.5 Myr t = 0.75 Myr

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TO

TA

L F

LUX

(0.

1 m

2/yr

)

X (100 km)

TOTAL FLUX

t = 0.5 Myr t = 0.75 Myr

-20

0

20

40

60

80

100

120

140

160

180

200

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

dh/

dt (

m/M

yr)

X (100 km)

SEDIMENTATION RATE

t = 0.5 Myr t = 0.75 Myr

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

C1s

X (100 km)

SURFACE CONCENTRATION

t = 0.5 Myr t = 0.75 Myr

Figure 7: Sediment thickness h, limitor λ, total flux∑L

i=1 fi, sedimentation rate ∂th, andsurface sediment concentration cs1, at times t = 0.5 and t = 0.75 Myr, for the Delta progra-dation test case with maximum erosion rate E = 5 m/Myr, a uniform time step ∆t = 0.0025Myr, and the mesh size ∆x = 2 km.

Table 1 exhibits the performance of the Newton solver. The number of Newton iterationsappears sensitive to the mesh size. Various stabilizations of the algorithm for very largemeshes using Quasi Newton techniques or additional diffusion terms will be presented in aforthcoming paper.

18

∆x 2 km 1 km 0.5 km 0.25 km 0.125 km

Nnew / N∆t / Nfail 720/160/0 1003/160/0 1154/160/2 1434/160/3 2103/164/14

Table 1: Total number of Newton iterations/number of time steps/number of time stepfailures, for the Delta Progradation test case with maximum erosion rate E = 1 m/Myr, anddifferent mesh sizes ∆x. The initial and maximum time step is fixed to ∆t0 = ∆tmax = 0.01Myr, and ∆hmax = ∆Hm = +∞, α = +∞. The maximum number of Newton iterations is25, and the stopping criteria prescribes a relative residual lower than 10−5 in l2 norm.

4.3 Weather limited erosion of an island

We consider the square domain Ω = (0, Lb)× (0, Lb) with Lb = 20 km. The sea level is fixed

to Hm = 10 m, and the initial sediment thickness is set to h0(x1, x2) = 2 e2−(

x12Lb

−1)2−(x22Lb

−1)2.

The simulation is still done with two lithologies with diffusion coefficients given by (19) withk1,c = 103 m2/yr, k1,m = 2.102 m2/yr, k2,c = 5.102 m2/yr, k2,m = 50 m2/yr, and β = 0 m.

The sediment fluxes at the boundary ∂Ω are set to zero and the basin initial compositionis given by c01(x, z) = 1 − c02(x, z) = 0.7 on the domain z < h0(x), x ∈ Ω. The maximumerosion rate is fixed to E = 5 m/Myr

The mesh and the time stepping are uniform of sizes ∆x1 = ∆x2 = 0.4 km, and ∆t = 0.05Myr, and the simulation is performed over a time span of 1 Myr.

The sediment thickness solutions displayed Figure 8 illustrate the coupling between thedependence on the bathymetry of the diffusion coefficients ki(b) = dψi

db(b), i = 1, 2 and the

weather limited model. We can observe, in particular on the bottom picture, that the to-pography under the sea level results from a diffusive dominated transport with low marinediffusion coefficients, whereas the island is eroded under a weather limited mechanism.

19

x2 (10 km)x1 (10 km)

Initial basin at t=0 Myr

Sediment thickness h (m)

2

42

1.61.81.4

14

0.4

16

121086

01.41.210.80.6

0.20

1.6

1.210.80.6

1.8 20.20.4

x1 (10 km) x2 (10 km)

Sediment thickness h (m)

Basin at t=0.4 Myr

5

76

1.8

42

1.6

11

0.6

1312

1098

0.21.61.41.21

0 0.2 0.40.8

1.8

1.41.210.8

2 00.40.6

1

2

sea level

Sedi

men

t thi

ckne

ss h

(m)

t=0.2 Myr

1.8 21.61.4

x (10 km)

8

Cut of the basin at x =10 km

t=0 Myr

t=0.4 Myrt=0.6 Myr

t=1.0 Myrt=0.8 Myr

0

12

11

10

9

7

6

5

13

1.210.80.6

14

15

0.2 0.4

Figure 8: Initial sediment thickness at time t = 0, sediment thickness solution at timet = 0.4 Myr, vertical 2D cut at x2 = 10 km of the sediment thickness solutions at timest = 0, 0.2, 0.4, 0.6, 0.8 and 1 Myr.

5 Conclusion

Large scale mass conservation PDE models have become a powerful tool in the field of strati-graphic basin simulation to investigate the effects of eustasy, tectonics, and sediment supplyon facies distribution and stratal geometry of basins. In the petroleum industry these modelshave been successfully used to reconstruct the history of 3D basins. This is achieved by inver-sion of the parameters of the direct simulation, (input fluxes, sea level variations, tectonicsdisplacements, diffusion coefficients, and erosion rate), given sediment thickness seismic dataand bathymetry and sediment concentrations log data [GJD98], [GJ99].

We have introduced in this paper a new mathematical formulation for the coupling ofweather limited erosion and the multi-lithology diffusion models existing only at the dis-crete level so far. Both the multi-lithology and the coupled models appear as non standardformulations that remains to be analysed mathematically.

It has allowed the definition of a finite volume discretization scheme with implicit timeintegration which can be proved to be stable. This new scheme is computationally moreefficient than existing explicit schemes and enable the use of much larger time steps andmeshes. This computational efficiency and the better understanding of the coupled modelwill be helpful for a more efficient solution of the inverse problem.

A convergence analysis of the discrete scheme in a simplified case [EGGM03] as well as

20

the design of efficient non linear and linear solvers for large 3D basin models will be thesubject of forthcoming papers.

Acknowledgements

The authors are thankful to Professors R. Glowinski from the University of Houston, and G.Gagneux from l’Universit de Pau et des Pays de l’Adour for fruitful discussions during theelaboration of this work, and to Vronique Gervais from IFP for the numerical test in 3D.

References

[AH89] R. S. Anderson, N. F. Humphrey, “Interaction of Weathering and TransportProcesses in the Evolution of Arid Landscapes”, in Quantitative Dynamics Stratig-raphy, T.A. Cross, ed., Prentice Hall, pp. 349-361, 1989.

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