1
P001 DEVELOPMENT AND APPLICATION OF A
MULTI-LITHOLOGY STRATIGRAPHIC MODEL
UNDER MAXIMUM EROSION RATE
*
V. Gervais , R. Masson, D. Granjeon
Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France
Abstract
We consider here a stratigraphic model coupling a multi-lithology
diffusive model with a constraint on the erosion rate. A finite volume
implicit in time discretization is derived from this model, leading to a
non-linear system, which is solved using a Newton algorithm adapted to
the constraint. However, this method can lack robustness, especially for
small mesh sizes. In this paper, we present a simulation of this model on
real input data and some techniques to improve the convergence of the
Newton method. Finally, we will describe a preconditioning strategy to
get a robust and efficient linear iterative solver in the Newton algorithm,
and so faster simulations.
I. Introduction
In the field of seismic- and sequence-stratigraphy, numerical models have been developed to
quantify the sedimentary basin infill. We consider here a stratigraphic model simulating largescale transport of sediments described as a mixture of non-miscible lithologies. Such models
have been successfully applied in the field of oil exploration ([5]). The surficial transport process
is a multi-lithology diffusion model for which fluxes are proportional to the slope of the
topography and to a lithology fraction at the surface of the basin, called surface concentration,
with non-linear depth-dependent diffusion coefficients. Then, the model accounts for the mass
conservation of each lithology in the basin, considering both the gravity-driven transport and the
limited availability in erodible sediments at the surface of the basin modeled by a constraint on
the erosion rate. The coupling between these two models is achieved by introducing a new
variable limiting the flux. Compared to existing couplings, only defined at a discrete level so far,
this new formulation enables to derive efficient discretization schemes.
An implicit in time finite volume scheme has been derived from this model (section III), for
which we have proved existence of stable discrete solutions. A Newton algorithm adapted to the
erosion rate constraint is used to solve the non-linear system resulting from this discretization.
The simulations provide stable and smooth solutions, what appears as a key asset to inverse the
model input parameters from seismic and well log data. However, the Newton method can lack
robustness, especially for small mesh sizes. Since we aim at simulating basins evolution on fine
grids, we have worked on an improvement of the convergence of this method. It is presented in
section IV. Finally, in order to get faster simulations, we have implemented preconditioned
linear iterative solvers in the Newton algorithm. The main ideas are first to decouple in the
preconditioner the sediment thickness variable from the other unknowns (the surface
concentrations), and secondly to use a topographical ordering of the concentration unknowns.
This will be detailed in section V.
II. Mathematical Model
A stratigraphic model describes the evolution through time of a sedimentary basin. More
especially, it takes into account the interaction between the three main processes involved in the
basin infill: the evolution of the accomodation space (subsidence, eustasy), the sediment supply
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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and the transport of the sediments in the basin. In this paper we consider the stratigraphic model
given in [4]. Let us denote by Ω ⊂ IRd (d=1 or 2) the horizontal extension of the basin (see figure
1). The sediments are modeled as a mixture of L non-miscible lithologies (such as sand or shale)
characterized by their grain size population and considered as incompressible materials of
constant grain density and null porosity. Then, the model describes the evolution of the sediment
thickness h, modeled as a function of x ∈ Ω and time t > 0 (see figure 1), and of the composition
of these sediments in each lithology. We assume that the following characteristics of the basin
are given:
•
•
•
the vertical coordinate H(x,t) of the base of the
basin (subsidence),
the sea level Hm(t),
and the sediment supplies at the boundary of the
basin.
In the following, let us denote by b(x,t) := Hm(t)[h(x,t)+H(x,t)] the bathymetry. We will first describe
the multi-lithology model and then the coupling of
this model with a constraint on the erosion rate.
Fig. 1 - 2D sedimentary basin
1. Multi-lithology model
Inside the basin, the sediments are described by the L concentrations c i ( x , ζ, t ) ≥ 0 in lithology i
on the domain B = {( x , ζ, t ) x ∈ Ω, t ∈ (0, T ), ζ ∈ (0, h ( x , t ))} (see figure 1) satisfying
∑i=1...L c i
= 1 . The sediments transported at the surface of the basin, i.e. deposited in case of
sedimentation or passing through the surface in case of erosion, are described by the L surface
concentrations c si ( x , t ) ≥ 0 defined on Ω × (0,T), and also satisfying ∑i =1...L c si = 1 . Since the
sediment transport only occurs at the surface of the basin, the composition inside the basin (i.e.
for ζ < h ( x, t ) ) remains constant: ∂ t c i = 0 on B. Then the evolution of this concentration is only
governed by the boundary condition at the top of the basin: c i
ζ =h
= c si in case of sedimentation
(∂th > 0). Thus we get
⎧
⎪∂ t c i = 0
⎪
s
⎨c i ζ =h = c i
⎪
0
⎪⎩c i t =0 = c i
on B = {( x , ζ, t ) x ∈ Ω, t ∈ (0, T), ζ ∈ (0, h ( x , t ))}
on D + = {( x , t ) ∈ Ω × (0, T) ∂ t h ( x , t ) > 0}
on {( x , ζ ) x ∈ Ω,ζ ∈ (0, h
(1)
}
t =0 ( x ))
where we have denoted by c i0 an initial condition prescribed to ci.
Then this system is coupled to the conservation equation of each lithology on the domain Ω ×
h ( x ,t )
(0,T), and more especially to the conservation of the fraction M i ( x , t ) = ∫ 0
c i ( x , ζ, t )dζ in
lithology i. The surficial transport process considered here is the multi-lithology diffusive model
introduced in [7] for which the flux ϕ i in lithology i is taken proportional to the slope and to the
surface concentration: ϕ i = − c si k i ∇(h + H). In this equation, we have denoted by ki the
diffusion coefficient of the lithology i, which reflects the capability of this lithology to be
3
transported by gravity. It is here modeled as a non-linear function of bathymetry:
⎪⎧k m if b ≥ 0
k i (b) = ⎨ ic
. Then, the conservation of the fraction Mi states that:
⎪⎩k i otherwise
⎧c i ζ =h ∂ t h + div [-csi k i (b) ∇(h + H)] = 0 on Ω × (0, T),
⎪
⎨
c si = 1
on Ω × (0, T).
⎪⎩i =∑
1...L
The equality c i
h: h
t =0 ( x )
ζ =h ∂ t h
(2)
= ∂ t M i derives formally from (1). An initial condition is also imposed to
= h 0 ( x ) on Ω. Finally, we prescribe Neumann boundary conditions: input fluxes
ϕ e,i ( x, t ) < 0 are imposed on Γe ( t ) for each lithology, and a global output flux ϕ s ( x , t ) is fixed
for ϕ = ∑i =1...L ϕ i on Γs ( t ) , with (Γe ( t ), Γs ( t )) a partition of the boundary of Ω at time t.
2. Coupling with the constraint on the erosion rate
However, this model does not take into account the dissymmetry between erosion and
sedimentation (see [1] for example). Indeed, transport can only apply to small grains, i.e. to the
sediments transported from upstream and to the sediments produced in situ by rock alteration
(weathering). This is modeled here by a constraint on the erosion rate (see [5]): ∂ t h ≥ − E , where
E ≥ 0 is the maximum erosion rate, function of bathymetry, climate… To couple this constraint
with the multi-lithology model, a variable λ ≤ 1 has been introduced to limit the global diffusive
flux ϕ, leading to the definition of the new fluxes fi = λϕi for all i=1…L. Then the closure of the
model is obtained by requiring that, if the constraint ∂ t h > − E is satisfied with the diffusive
model, λ is taken equal to one and the flux fi is equal to the diffusive flux ϕi ; otherwise, λ < 1
and the maximum erosion rate is reached: ∂ t h := − E. This can also be written is the following
way:
⎧(∂ t h + E)(1 − λ ) = 0,
⎨
⎩ ∂ t h + E ≥ 0, λ ≥ 0,
on Ω × (0, T).
(3)
Then, the complete model accounts for the conservation equations (1) and (2) with the new
fluxes fi instead of the diffusive ones, coupled to the constraint (3). Concerning the boundary
conditions, ϕ s ( x , t ) is now a maximum total output flux which may have to be limited to satisfy
the maximum erosion rate constraint (see [4] for details).
III. Fully implicit Finite Volume scheme
The system (1)-(3) is discretized by an Euler implicit method in time and a finite volume method
on Ω with cell centered variables. The scheme will not be detailed here but can be found in [4].
Let us denote by Κ an admissible finite volume mesh for Ω (see [4] for the definition of this
mesh). In the sequel, the superscript n (resp. the subscript κ) will be used to denote that the
discrete variable is considered at time tn (resp. on the cell κ ∈ Κ). Let us now highlight just two
of the main characteristics of the discretization:
•
First, the approximation of the fluxes at the edges of the control volumes uses an upstream
weighted evaluation of λ and c si : the value between two neighboring cells κ and κ' is taken
equal to the approximation of λ (or c si ) on κ if the sediments are transported from κ to κ'
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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(i.e. if the topography on κ is higher than on κ'), and to the approximation on κ' otherwise.
This choice, together with the implicit integration in time, leads to the stability of the scheme
(see [4] for the detailed proof).
•
Concerning the concentrations ci in the basin, their discretization is obtained by integrating
the equation ∂ t c i = 0 on each column of base κ between two discretization times t n and
t n +1 , with the approximation of the surface concentration c si,,κn +1 at time t n +1 as input
boundary condition at the surface in case of sedimentation. It amounts to add a cell,
corresponding to ζ ∈ (h nκ , h nκ+1 ) , at the top of the column in case of sedimentation, where the
concentration ci is taken equal to the surface concentration c si,,κn +1 , and to remove or truncate
cells in case of erosion down to ζ = h nκ+1 . Consequently, the concentrations ci can be
computed separately once the other variables have been determined.
Finally, we get at each time step a coupled non-linear system to solve with L+2 unknowns per
cell κ ∈ Κ, which are the discrete approximations of h, λ and c si .
IV. Non-linear Solver
The non-linear system resulting from the discretization is solved using a Newton algorithm
adapted to the constraint (see [4] for details). In this section, we will present some results
concerning the performance of this method and some new techniques to improve it.
1. Test case : Paris Basin (France)
The code implementing the implicit discretization scheme has been tested with real data. To
present our numerical results in the rest of this paper, we will use input data concerning the Paris
Basin (France) for eustasy, subsidence and sediment supply. The simulated basin is 500 km long
on 400 km large, filled with two lithologies (sand and shale), and the simulation is made over 40
My. The maximum erosion rate E is taken equal to 3 m/My, and the diffusion coefficients of the
c
m
c
lithologies
are:
k sand
= 10 km 2 /ky ,
k sand
= 1 km 2 /ky ,
k shale
= 200 km 2 /ky ,
and
m
k shale
= 4 km 2 /ky . On figure 2 are presented the bathymetry and the sand concentration in the
basin, with a time step ∆t equal to 0.5 My and a mesh size ∆x equal to 10 km.
2. Performance of the Newton algorithm
The Newton algorithm can have some difficulties to converge, especially for large time step and
small mesh sizes. These problems are due to the non-linearity induced by the diffusion
coefficients, to the constraint on the erosion rate, and to the particularity of the accumulation
term Mi in the sense that there is no time derivative of the surface concentrations in the model.
Thus we have worked on the stabilization of this algorithm.
3. Improvement of the robustness
3.1 Reduction of non-linearity
To improve the robustness of the Newton algorithm, we have tried to reduce the effect of the
non-linearity induced by the diffusion coefficients. First, a new function ψ i (b) has been
b
introduced for each lithology, defined by ψ i (b) = − ∫ 0 k i (u )du , in order to smooth the nonlinearities. The flux fi is then equal to − λc si ∇ψ i (b) and the diffusion coefficient is included in
5
Initial bathymetry
Final bathymetry
Final sand concentration
Fig. 2 - Simulation of the test case of section IV.1 - ∆t = 0.5 My, ∆x = 10 km
the gradient. This improves largely the convergence of the algorithm and the accuracy of the
solution.
Then, the diffusion coefficients have been smoothed by linearization in logarithmic scale on an
interval [−β, β], β > 0 (see figure 3). This improves the robustness of the algorithm, as can be
seen in table 1 for the test case of section IV.1, especially for small mesh sizes and high ratios
kc/km, but it can also lead to a smoothing of the surface of the basin near the shore for too large
values of β. An automatic calibration of this parameter has been carried out to obtain a
compromise between the decrease of the computation time and a limited change of the solution.
#Newton/∆t,
β=0
#Newton/∆t,
β=1m
10 km
6.7
5.3
5 km
9.3
6.6
2.5 km
30.9
15.7
Mesh size ∆x
Fig. 3 - Smoothing of the diffusion coefficients
Table 1 - Influence of the parameter β on the number of
iterations of the Newton algorithm per time step (stopping
criteria : relative residual < 10-5) - ∆t = 0.1 My
3.2 New discretization scheme
As mentioned previously, the constraint on the erosion rate makes the convergence of the
Newton algorithm more difficult. Suppose that the sediment thickness and the concentrations are
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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given. Summing equation (2) for i=1…L, we get ∂ t h = - div f with f = ∑i =1...L f i the global flux.
Then, using (3), the variable λ satisfies the problem:
⎧(div f − E )(1 − λ ) = 0,
⎨
⎩ div f − E ≤ 0, λ ≥ 0,
on Ω × (0, T ).
Applying this property at a discrete level, we can show, thanks to the upstream evaluation of λ in
the discrete fluxes, that an approximation of this variable can be computed explicitly by
induction on the cells sorted by decreasing topographical order (for a given approximation of the
sediment thickness and the concentrations, see [4] for details). We can thus define a new
discretization scheme, in which the sediment thickness and the surface concentrations are
implicit in time, while the variable λ is explicit in time. Some results are presented in table 2 for
the test case of section IV.1. They show an important improvement of the Newton convergence,
especially for large time steps. But this gain in time is balanced by an increasing difference of
the solution with the one given by the fully implicit scheme since the product (∂ t h ≥ −E )(1 − λ )
is not necessarily equal to zero with the new scheme.
∆t
#Newton/∆t
Fully implicit scheme
#Newton/∆t
λ explicit
0.5 My
17.3
4.3
0.1 My
6.6
3.5
0.02 My
4.0
2.9
Table 2 - Number of iterations of the Newton algorithm per time step for the two
discretization schemes (stopping criteria : relative residual < 10-5)-∆x = 5 km, β = 1 m
V. Linear iterative solvers with preconditioning
For fine grids, the number of unknowns of the non-linear problem becomes very large, and a
direct inversion of the linear system in the Newton algorithm is computationally too expensive.
For such large systems, linear iterative solvers appear as a useful tool. However in our problem
the jacobian is sparse, non-symmetric and ill-conditioned, and a good preconditioner must
therefore be defined to obtain an efficient and robust solution of the linear system. Our
preconditioning strategy consists in decoupling the variables in the preconditioner. This idea has
already been applied for example in [3] and [7] for the oil reservoirs, or in [8] for the simulation
of flows in porous media.
For the sake of simplicity, let us consider the case of two lithologies. Then at each Newton
iteration, we have to compute the vector X such that:
⎛H
J ⋅ X = ⎜⎜ 1
⎝H2
C1 ⎞ ⎛ X y ⎞ ⎛ Z y ⎞
⎟=⎜
⎟.
⎟⋅⎜
C 2 ⎟⎠ ⎜⎝ X c1 ⎟⎠ ⎜⎝ Z c1 ⎟⎠
In this equation, we have denoted by J the jacobian, in which the variables taken into account
are, on each cell κ, the concentration c1s,κ (using the equation ∑i =1...L c si = 1 , the variable c s2 can
easily be computed afterwards), and a variable y κ equal to h κ if the constraint on the erosion
rate is not reached ( λ κ is then equal to one), and to λ κ otherwise (in that case, h κ is given by
the equality ∂ t h = − E ). Then, in the jacobian, the first (resp. second) block row corresponds to
7
the conservation equation of the lithology 1 (resp. 2) on Ω × (0, T ) , and the first (resp. second)
block column to the variable y (resp. c1s ) on each cell κ. The idea of our preconditioning is to
decouple the sediment thickness and flux limitor variables from the surface concentrations in the
preconditioner of the jacobian. More precisely, we assume that the coupling between the
sediment thickness and the flux limitor on one side, and the surface concentrations on the other
⎛H
C1 ⎞
⎟ and
side is chiefly local. Thus, multiplying J (and Z) by a matrix G such that GJ = ⎜⎜ 1
⎟
H
C
2⎠
⎝ 2
the diagonal of C1 vanishes, we will consider that the block C1 can be neglected in the
preconditioner of GJ. Thus, the preconditioning strategy is the following: first, choose a matrix G
performing a local decoupling of the variables; and then, use a global preconditioner of the form
−1
~
⎛ H1 0 ⎞
~
~
−1
PBGS = ⎜⎜
~ ⎟⎟ for GJ, where we have denoted by H1 (resp. C 2 ) a preconditioner of the
⎝ H2 C2 ⎠
block H1 (resp. C 2 ). This preconditioner will be compared to a global preconditioner PG−1 of the
matrix GJ.
There only remains to define the matrix G. We have implemented three local decoupling, which
amount to (A) replacing the conservation equation of lithology 1 by the sum of the conservation
equations of the lithologies on each cell ; (B) replacing this equation by the sum of the
conservation equations weighted by the inverse of the diffusion coefficients; (C) using a Gauss
elimination to have the diagonal of C1 vanished. With these decouplings, the structure of the
block C 2 is the same as the one of C 2 : thanks to the upstream evaluation of the surface
concentrations at the edges of the control volumes in the discrete fluxes, these blocks are
triangular inferior if the cells are sorted by decreasing topographical order. Therefore, we use
this re-ordering and the Gauss-Seidel method to inverse exactly C 2 .
To test the robustness of the solvers, we have considered the number of iterations of the iterative
solver (BiCGStab here) necessary to decrease the initial residual by a factor of 10-6. The
implementation of the solver and the preconditioners was carried out with the PETSc library of
the Argonne National Laboratory, IL [2]. Some results for the test case of section IV.1 can be
found in tables 3 and 4. First, these results show a faster convergence with the preconditioner
−1
PBGS
than with a global preconditioner PG−1 (ILU(1) here). Then, the proposed local decouplings
appear equivalent. Concerning the preconditioning of the block H1 , we have tested several
techniques, but LU and ILU(p) show the best performances. It appears that LU is robust with the
time step and the mesh size, while ILU can lack robustness when the time step increases and the
mesh size decreases. For fine grids, it even lead to a big increase of the number of iterations.
Thus, depending on ∆t and ∆x, one of these two preconditioning methods for H1 seems to
perform the fastest simulations.
VI. Conclusion
The techniques presented in this paper enable to fasten the simulation of the model, even if some
−1
is
of them have to be used with care to keep an accurate solution. The block preconditioner PBGS
more robust and efficient than classical preconditioners such as ILU(p). It would be interesting in
the future to implement an Inexact Newton method with this block preconditioning technique.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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#BiCGStab it. / Newton it.
Time step ∆t
PG =ILU(1)
PBGS, LU
0.5 My
25.1
11.7
31.2
0.1 My
13.1
10.0
0.02 My
10.5
10.3
Time elapsed in solver / Newton it.
PBGS, ILU(1) PG =ILU(1)
PBGS, LU
PBGS,ILU(1)
0.43 s
0.27 s
0.34 s
16.3
0.30 s
0.25 s
0.18 s
12.6
0.22 s
0.25 s
0.15 s
Table 3 - Number of BiCGStab iterations and time elapsed in the iterative solver (Linux PC) per Newton iteration
with local decoupling (A), for various time steps, a global preconditioning PG of type ILU(1) and the
preconditioner PBGS with two preconditioning techniques for H1 ( LU and ILU(1)) - ∆x = 5 km, β = 1 m
#BiCGStab it. / Newton it.
Mesh size ∆x
PG =ILU(1)
PBGS, LU
10 km
12.4
13.0
18.1
5 km
13.1
10.0
2.5 km
48.5
18.1
Time elapsed in solver / Newton it.
PBGS, ILU(1) PG =ILU(1)
PBGS, LU
PBGS, ILU(1)
0.06 s
0.07 s
0.06 s
16.3
0.30 s
0.25 s
0.18 s
81.5
3.25 s
1.91 s
3.94 s
Table 4 - Number of BiCGStab iterations and time elapsed in the iterative solver (Linux PC) per Newton iteration
with local decoupling (A), for various mesh sizse, a global preconditioning PG of type ILU(1) and the
preconditioner PBGS with two preconditioning techniques for H1 (LU and ILU(1)) - ∆t = 0.1 My, β = 1 m
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