Decoupling and block preconditioning for sedimentary basin simulations

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Niveau: Supérieur, Doctorat, Bac+8
DECOUPLING AND BLOCK PRECONDITIONING FOR SEDIMENTARY BASIN SIMULATIONS IN TEMIS3D ROBERT SCHEICHL y , ROLAND MASSON z , AND JOHANNES WENDEBOURG x Abstra t. The simulation of sedimentary basins aims at re onstru ting its histori al evolution in order to provide quantitative predi tions about phenomena leading to hydro arbon a umula- tions. The kernel of this simulation is the numeri al solution of a omplex system of time dependent, three-dimensional partial dierential equations (PDE) of mixed paraboli -hyperboli type. A dis- retisation (Finite Volumes + Impli it Euler) and linearisation (Newton) of this system leads to very ill- onditioned, strongly non-symmetri and large systems of linear equations with three unknowns per mesh element, i.e. pressure, geostati load, and hydro arbon saturation. The pre onditioning whi h we will present for these systems onsists in three stages. First of all the equations for pressure and saturation are lo ally de oupled on ea h element. This de oupling aims not only at redu ing the oupling, but also at on entrating in the \pressure blo k the ellipti part of the system whi h is then in the se ond stage pre onditioned by eÆ ient methods like AMG. The third step nally onsists in \re oupling the equations (e.

  • mathemati al

  • per mesh

  • ally

  • ase

  • pre onditioning

  • al ulated

  • unknowns per

  • onstru ting


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olution
DECOUPLING
ran
AND
simplied
BLOCK
in
PRECONDITIONING
history
F
la
OR
for
SEDIMENT
Institut
AR
of
Y

BASIN

SIMULA
y
TIONS
ecome
IN
tire
TEMIS3D
the

Beicip-F
R
out
OBER
erogen
T
olution
SCHEICHL
v
y
basin
,
o
R
pro
OLAND

MASSON
ran
z
of
,
a
AND
burial,
JOHANNES
assessmen
WENDEBOUR
de
G
(IFP)
x
ol.

to
The

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,
ulation
and
of
ide-
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,
tary
w
basins
ey
aims
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at
del

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its
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historical
the
ev

olution
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in
Math
order
rance.
to
rance.
pro
part
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future
quan
jor
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predictions
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h
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tary

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tions.
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problem
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presen
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for
kw
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to
systems
of

b
in
full
three
the
stages.
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generation,
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of
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media,
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ulti-phase
equations
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saturation
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almost
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system
Ideally
whic
basin
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migration
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to
metho
p
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and
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tegrit
third
throughout
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basin.
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\recoupling"
TEMIS3D
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equations
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at
basin
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means.
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extremely
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mo
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is
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the
geometry
size
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of
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time
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step,
mo
high

migration
pro
ratios,
of
or
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strong
on
heterogeneities
heat
and

anisotropies
the
in
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the
pressure
p
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Darcy
media.
w
Key
oil
w
gas
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three-dimensional
then
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forw
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in
Darcy
adding
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after

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of
using
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media,
the
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k-stripping
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solution,
This
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HPMI-CT-1999-00012.
ultigrid
Departmen
AMS
of
sub
Sciences,

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65Y20,
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erature.
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ery
MASSON,
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AND
non-linear
J.
linearisation
WENDEBOUR
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tirely
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the
is
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is
of
its
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nev
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the
steps:

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blo


elemen
k
a
building
solution
and

mesh
a

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k-stripping,
A
and

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usually
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robust
sim
d
ulation

(see
to
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and
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in
et
literature
al.
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idea
for
Behie
an
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example
equations
of
the

(1.1)
h
x
a
a

unkno
study).
load,
The
w
3D
equations
blo
fo

nonlinear
k
on
building
as
step,
of
usually
and

will
out
equations
b
x
y
A
the
A
geologist,
0

1
in
2
preparing
(1.1)
the
strongly

an
data,
of
i.e.
nd
in
b
dening
er
a
and

b
t
b
3D
from
blo
with

at
k
this
whic
problem,
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but
represen
strong
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the
the
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strategy
area
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and
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in
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assem
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with
saturation
this
linear
blo
equations

onds
k
matrix
a
!
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b
of
time
geological
of
maps.
in
It
p
also
(pressure,

orosit
in
saturation),
dening
b
a
of
3D
the
mesh
w
on
on
the
the
blo
and

no
k
e
whic
temp
h
en.
will
the
usually
equations
b
metho
e
elimination
Cartesian
orosit
in
lead
the
of
horizon
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form
plane
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and
A

A
h
A
that
A
the
1
v
x

x

=

1
with
3
the
b

(1.1)
hronostratigraphic
ery

and
In
F
the
t

e
k-stripping
system,
step

the
go
historical
It
ev
the
olution
this
of
dev
the
a
geometry
ev
of
and
the
applying
basin
n
will
of
then
test
b

e
y

preconditioning
separate
y
for
momen

.
h
an

v
(Multi-1D)
is

v
kw
e
ard
mak
in
of
time
literature
b
problem
y
oir
taking
particular,
o
our
sedimen
based
ted
ds
material
b
and
Vinsome
adding

ero
&
ded
and
material.
Wheeler
A

t
three

all
h
pressure
time
lo
step
y

binations
the
w
sedimen

tation
This
and
m
the
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erosion
,
ha
=
v
GA
e
(
b
:
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h

step
ted
system
for,

the
equations
remaining
four
sedimen
wns
ts
er
are
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or
p

y
resp
oil
ectiv
follo
ely
ed
,
y
b
system
y
linear
using
for
p
temp
orosit
Here
y/depth
e
relationships

for
the

of
h

of
system
the
th
lithologies.
from
The
w
nal
w
step
will
is
the
a
erature
forw
giv
ard
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sim
of
ulation
system
of
non-linear
the
using
full
Newton
3D
d
mo
an
del
of
for
p
the
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sedimen
nally
tary
to
basin
system
on
linear
the
of
mo
blo
ving
k
geometry
A
(mesh)
:=


during
11
the
12

13
k-stripping
21
phase.
22
In
23
this
31
pap
32
er
33
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will
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only
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at
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:
ulation
System
whic
is
h
v
is
large,
the

most
v


in
or
tensiv

e
and
part
iterativ
and
solution

this
in
it
solving
therefore
n
to
umerically
a
a
o

preconditioner.
system
will
of
e
time
sub
dep
of
enden
pap
t,
to
three-dimensional
elop
partial
h
dieren
preconditioner
tial
to
equations
aluate
(PDEs)
robustness
mo

delling
y
heat
it
transfer,
a

um
of
er
the
represen
p
e
orous

media
real
and
studies
m
b
ulti-phase

Darcy
standard
o

w.
emplo
These
ed
equations
the
are
t
discretised
TEMIS3D
using
Since
a
is

en
tred
no