Jean-Laurent Mallet. Geomodeling
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548
CHAPTER
10.
DISCRETE SMOOTH PARTITION
From
the
same
set
of
data,
either
position
or
momentum
of
a
particle
can
be
measured
with
precision.
Within
the
framework
of
Membership Functions applied
to the
estimation
of
a
bounded
scalar
function
z(o)
presented above,
it can be
noted
that
there
lies
a
similar uncertainty principle:
• for
greater
precision
of
values
Zj(ct),
the
size
of the
intervals
{]zj,_i,
z
v
[
:
v =
l,n}
has to be
reduced
and,
as a
consequence,
the
associated
probabilities
{(p
v
(oL)
:
v =
l,n}
attached
to
these
classes
will
decrease;
• to
obtain
high
values
for the
probabilities
{(//'(a)
: v =
l,n}
attached
to the
intervals
{]zv-i,Zv[
'•
v
=
l,w},
it is
necessary
to
reduce
the
total
number
of
intervals,
thus
decreasing
the
precision
of the
values
Zj(a).
10.3
Structural
constraints
As
already mentioned,
in
addition
to the
mandatory intrinsic constraints
on
(f>,
it is
possible
to
take into account additional
DSI
constraints related
to
prior
information
on the
structure
of the
unknown partition
J-'.
In the
fol-
lowing
sections,
a
series
of
such possible constraints corresponding
to
types
of
information
encountered
in the
geosciences
are
presented:
10.3.1
Taking into account
a
priori
proportions
Notion
of
local
proportion
The
structural information under consideration
in
this section
is
related
to
local proportions
of
nodes
JP(F
v
\W(a}}
denned
as
follows
where
F
v
is a
part
of
the
unknown partition
T
to be
estimated:
In
practice,
as
suggested
in figure
(10.4),
W(a)
is a
"floating-window" centered
on
the
node
a and
JP(F
v
\W(a)}
describes
the
local percentage
of
facies
F
v
in
this
window.
14
Although
the set
F
v
is
unknown,
it may be
that
an
initial estimation
of
its
relative size
JP(F
V
W(a)}
in
W(a]
is
given.
For
example:
15
• If
(F
v
fl
W(a)
n L) is
large
enough,
then
a
decision
can be
made
to
estimate
JP(F
v
\W(a)}
as follows:
14
Note
that,
if
need
be,
W(a)
may be
independent
of a. In the
extreme,
W(a)
=
f2
may
even
be
chosen.
15
Remember
that
L is the set of
nodes
where
the
facies
is
known.
For
example,
L may
correspond
to
nodes located along well-paths.
10.3
Figure
10.4 Notion
of
local
proportion
JP(F
v
\W(o)~)
of
fades
F"
in the
neigh-
borhood
W(o)
of
a
node
a.
• If a
given
attribute
16
z(a]
depending linearly
on the
distribution
of
this
facies
F"
is
known
everywhere
in
Jl,
then
it is
likely
that
the
local average
z
w
(a)
of
z(a)
in
V^(o:)
will
be
approximately proportional
to
JP(F
v
\W(ot)}:
In
this case,
the
coefficient
k can be
estimated using equation (10.10)
at
locations
a £
L
corresponding
to
well
data.
• A
current practice
in the
modeling
of
reservoir heterogeneities consists
in
averaging
the
curvilinear projection
of all the
data
on the (u, v)
surface
or on
the w
axis
of a
curvilinear regular 3-grid:
—
As
suggested
in figure
(10.5),
the
curvilinear projection
of the
data
on
the
(u,
v)
surface
along
the w
axis enables
us to
construct
17
an
average
proportion
map
/™(w,
v)
such
that
In
this case,
W(a)
is
identical
to the
subset
W
U
v(oi)
of
nodes having
the
same
(ti,
v)
curvilinear coordinates
as a.
—
Similarly,
the
curvilinear projection
of the
data
on the w
axis along
the
(w,
v)
surface
enables
us to
construct
an
average proportion curve
fc(w)
such
that
In
this case,
W(a)
is
identical
to the
subset
W
w
(a)
of
nodes having
the
same
w
curvilinear coordinate
as a.
16
Such
as, for
example, seismic impedance.
17
For
this
purpose,
the
DSI
method
can be
used
for
interpolating
the
projected
data
in
the (u, v)
plane.
549
550
CHAPTER
10.
DISCRETE SMOOTH PARTITION
Figure
10.5
Notion
of
average
proportion
map
/™(w,
v)
and
average
proportion
curve
fc
(w)
in a
curvilinear
regular
3-grid.
Local
proportion
constraint
As
suggested
in figure
(10.4),
if
JP(F
v
\W(a})
is
known, then
it is
proposed
to
translate
this piece
of
information into
the
following
constraint
18
operating
on the
component
(p
v
of
<p:
This
constraint
is
equivalent
to the
DSI
constraint
c =
c(a,
z/,
W)
belonging
to the set
C~
as
denned
by
To
normalize this constraint (see equation
(4.4),
page 140),
it is
necessary
to
divide
both members
of the
above equation
by
^/|PF(o;)|:
18
As
usual,
the
number
of
elements contained
in the set
W(a)
is
noted
as
|W(o;)|.
10.3.
STRUCTURAL
CONSTRAINTS
The
coefficients
A^(f3]
and
b
c
associated with
the
canonic
form
of
this
DSI
constraint
are
then such
that
It is
easy
to
check
that
the
associated terms
{^(a)^T^(a\(p)}
occurring
in
the DSI
algorithm (see equation
(4.49)
page 163)
are
such
that
10.3.2 Taking into account associations with
a
subset
G
Let G be a
given
subset
of fi and let
JP(F
V
\G)
be the
relative proportion
of
nodes
of G
belonging
to the
.unknown
facies
F
v
:
It can be
observed
that
this relative proportion
TP(F
V
\G]
is no
more
than
a
conditional
probability
19
measuring
the
"association" between
the
given
subset
G and the
unknown subset
F
v
in the
following
sense:
• If
P(F
V
\G)
is
large, then,
on
average,
<f
v
(ex)
is
large when
a
scans
G;
• If
JP(F
V
\G)
is
small, then,
on
average,
(p"(a}
is
small when
a
scans
G.
In
other words,
if
JP(F
V
\G)
is
known,
this constitutes
a
piece
of
structural
information
that
it is
worthwhile taking into account when interpolating
the
Membership
Function
(p.
Although
the set
F
v
is
unknown,
it may be
that
an
initial estimation
of
F(F"\G)
is
given.
For
example,
20
if
(F
v
n G
n
L) is
large enough, then
a
decision
can be
made
to
estimate
IP(F
V
\G]
as
follows:
As
suggested
in
figure
(10.6),
if
JP(F
V
G) is
known, then
it is
proposed
to
translate
this
piece
of
information into
the
following
constraint operating
on
the
component
(p
v
of
(p:
From
a
mathematical point
of
view,
if we let
19
See
definition
on
page 446.
20
Remember
that
L is the set of
nodes
where
the
facies
is
known.
551
552
CHAPTER
10.
DISCRETE SMOOTH PARTITION
Figure
10.6
Notion
of
probability
of
association between
the
unknown fades
F
v
and a
given subset
G.
Figure
10.7
Nonlinear associations between
a
bounded unknown function
Z(oi)
to be
estimated
on
£1
and a
given function
X(o)
known everywhere
on
Q.
then
it can be
observed
that
the
"association" constraint described above
is
strictly identical
to the
"proportion"
DSI
constraint presented
in
section
(10.3.1).
However,
it
should
be
noted
that,
from
a
practical point
of
view,
these
two
versions
of the
same mathematical concept
are
used
to
model very
different
types
of
information.
10.3.
STRUCTURAL
CONSTRAINTS
Comment
To
reconsider
the
problem presented
in
section (10.2.3),
let us
assume
that
the
unknown bounded scalar
function
Z(a)
to
estimate
on
f2
is
linked,
in a
fuzzy
nonlinear way,
to a
bounded scalar function
X(a)
known everywhere
on
0.
In
this case,
as
suggested
in
figure
(10.7),
it is
wise
to
proceed
as
follows:
•
Choose
two
series
of
values
(XQ,
...,
x
m
}
and
{20,
• •
•,
z
n
}
to
build
a
2-dimensional
histogram
where
each
cell
{G
l
x
F"}
is
defined
by
• For
each
cell
{G
i
x
F"},
compute
the
number
N?
of
data points
{X(l),
Z(t)}
represented
as
black
dots
in figure
(10.7)
and
corresponding
to
nodes
i
G L of
$1
where
both
X(£)
and
Z(€)
are
known.
•
Estimate
the
probabilities
of
association
JP(F
l/
\G
l
)
as
follows:
• use
each
P(F"\G
1
}
to
build
an
association constraint
on the
Membership
Function
of the
partition
{F
1
,...,
F
v
}.
In
figure
(10.7)
the
fuzzy
link between
Z(a)
and
X(a)
is
represented
by a
curve
Z —
f(X),
but
this
is
only
one
very particular type
of
relationship. Note
that
the
constraint
presented above
can
account
for any
type
of
relationship
between
Z(a)
and
X(a),
even implicit equations with
the
following
general
form:
For
example,
if we
have
then
the
"cloud"
of
data
points
in the (x, z)
plane
of the
histogram
in
figure
(10.7)
is
approximately
a
circle
and
there
is no
function
Z
~
f(X)
linking
Z(a)
and
X(a}°,
however, contrary
to
traditional methods, even
in
such
an
extreme case
the
method presented above continues
to
work.
The
next section shows
how the
notion
of
association with
a
function
X(a)
known everywhere
can be
generalized
in
connection with
the
Moving-Centers
method presented
in
section (10.4.2)
and
(10.4.3):
10.3.3
Association with
a
function
X
Let
X(a]
be a
function
defined
and
known everywhere
on
il
and let us
assume
that
X(a)
takes
a
specific
range
of
values
on
each
of the n
unknown facies
{F^,...,
F
n
}.
It
makes sense
to
consider
X(a)
as a
"signature" characterizing
the
association
between
X and the
unknown facies
at
node
a. It is
proposed
to use the
following
method
for
taking this association into account:
553
554
CHAPTER
10.
DISCRETE SMOOTH PARTITION
•
define
Y(a)
as a
function
taking
integer
values
in the set
{1,2,...,
n} in
such
a way
that
• use the
method
based
on
Moving-Centers
presented
in
section
(10.4.2)
for es-
timating
the
conditional
Membership
Function
(f>(a\X)
as
denned
by
equation
(10.18);
•
define
the
probability
of
association
between
the
facies
(F
1
,...,
F"}
and the
function
X at
node
a as
being
equal
to
tp(a\X}.
In
this case
and to
take into account information held
by X on the
facies
{F",...,
F
n
},
it is
proposed
to
install,
on
each node
a 6 0, a
DSI
constraint
specifying
that
It can
easily
be
checked
that
each
of
these constraints
is a
fuzzy
Control-Point
DSI
constraint already introduced
in
section
(4.7).
10.3.4
Taking
into
account
anisotropies
Anisotropy
controlled
by a
structuring
window
Most
of the
time,
in
geoscience applications,
the
distribution
of the
facies
is
anisotropic
and
taking such
an
information into account
is of
paramount
importance.
A
very simple
and
efficient
way for
defining
the
anisotropy
of a
component
(p
v
of the
Membership Function consists
in
proceeding
as follows:
•
Choose
a
given
family
of
"structuring"
windows
{W^a)
: a
G
O}
where
each
window
W
v
(a)
consists
of a
subset
of
SI
containing
the
node
a and at
least
one
more
node
of
SI.
•
Specify
that,
for
each
node
a,
the
value
f
v
(oi}
is as
close
as
possible
from
the
average
value
of
</?"
in the
domain
W
l/
(a)
—
{a}:
In
practice, such
a
constraint
can
easily
be
transformed into
a
dynamic
fuzzy
DSI
Control-Node constraint
c =
c(a,
v,
W)(see
page 164)
specifying
that
^(ot]
should
be
such
that
In
this
constraint,
$
v
a
is
assumed
to be
dynamically
defined
as the
average
value
of
(f>
v
in the
domain
W
v
(ot]
—
{a}:
According
to
equations
(4.53)
the
associated terms
{^(a),T^.(a\(p)}
occurring
in
the DSI
algorithm (See equation
(4.49)
page 163)
are
such
that
10.3.
STRUCTURAL
CONSTRAINTS
555
Figure
10.8
Two
fades test example
for
anisotropy constraints:
the
probability
of
fades
F
1
is
coded
in
grey
and
part
(A)
shows
the
actual solution while part
(B)
shows
the
Control-Nodes. Parts
(D) and (C)
represent
the
result
of the
DSI
interpolation
of
(f>
1
with
and
without anisotropy constraints, respectively.
Comment
about
the
structuring
windows
It
should
be
noted
that
the
choice
of the
family
of
structuring
windows
{W^a)}
is
extremely
flexible:
• the
size
and the
shape
of
W
v
(ct)
may
vary
from
one
node
a to its
neighbors,
and
• a
different
family
of
structuring windows
can be
chosen
for
each component
of
the
Membership Function
(p.
In
practice,
the
shape,
the
size
and the
orientation
of the
structuring
windows
must
be
chosen
to
capture
the
style
of the
anisotropy.
For
example:
• if the
variogram
of a
variable depending
on the
facies
has
been estimated, then
the
range ellipsoid
of
this
variogram
can be
used
as a
structuring window,
and
•
if
the
facies
are
associated with channels whose direction
and
width
are
known,
then
the
shape
of
W(a)
can be
adapted locally
to
take into such
an
information
into
account.
Two
facies
test
example
Let
us
consider
the
two-dimensional
test
example
represented
in
figure
(10.8)-
A,
where
two
facies
F
1
and
F
2
correspond
to
three
horizontal
strips
covering
556
CHAPTER
10.
DISCRETE SMOOTH PARTITION
a
regular rectilinear 2-grid with
n
v
= 21
nodes
in the
vertical direction
and
n
u
= 21 in the
horizontal direction.
The
"observed"
data
consist
of a
series
of
Control-Nodes represented
in figure
(10.8)-B
specifying
that
•
F
1
is
given
and fixed on the
left-hand
side
of the
studied
domain,
and
• F
2
is
given
and fixed on the
right-hand
side
of the
studied
domain.
Interpolating
the
Membership Function
</?
with
DSI
without other constraints
than these Control-Nodes
and
intrinsic constraints produces
the
image repre-
sented
in figure
(10.8)-C,
which
is
quite
different
from
the
actual distribution
of
facies
represented
in figure
(10.8)-A.
Let
Aw and
Ai;
be the
absolute
values
of the
variations
of the
components
u and v of the
parameter
u of the
stratigraphic grid when moving
from
one
node
to its
closest neighbor
in the u or v
direction, respectively.
To
take
into account
the
anisotropy
of the
distribution
of the
facies,
consider
now the
family
of
structuring windows
defined
as
follows
in the 3D
parametric domain
of
the
stratigraphic grid:
As
can be
seen
in figure
(10.8)-C,
using anisotropy constraints based
on
this
family
of
structuring windows yields
a
solution closer
to the
actual solution
shown
in figure
(10.8)-A.
10.3.5
Taking
into
account
gradients
Notion
of a
gradient
of
facies
There
are
many cases where
the
construction
of the set of
facies
{F
l
...,
F"}
is
controlled
by one
main physical
parameter.
21
The
following
are
geological
examples:
• In a
sedimentary
basin,
the
type
of
rock
at a
given
location
a
depends
on
the
depth
of the sea
z(a)
at
this
location
at the
time
of
sedimentation;
for
example
when
one
moves
from
the
sea-shore
to the
off-shore,
the
facies
are
encountered
in the
following
order
controlled
by a
series
of
thresholds
of
depth
{^0,
Zl,
. . .,
Z
n
}-
conglomerate
—>
sandstone
—»
carbonate
—>
shales
• In a
metamorphic
region,
the
type
of
rock
at a
given
location
a
depends
mainly
on the
temperature
T(a)
at
this
location
at the
time
of
crystallization;
for
example,
when moving
from
low to
high
temperatures,
the
metamorphic
facies
are
encountered
in the
following
order,
which
is
controlled
by a
series
of
temperature
thresholds
{to,
ti,...,
t
n
}:
shist
—»•
gneiss
—>
migmatite
—>
granite
21
In
practice, this "controlling" parameter
can be a
synthetic
factor
merging
the
contri-
bution
of
several actual physical parameters.
10.3.
STRUCTURAL
CONSTRAINTS
More
generally,
if
there
is a
scalar
function
z(-)
defined
on
Q,
such
that
then,
one can
consider
that
this
function
z(-}
induces
an
order
on the set of
facies
composing
the
unknown partition
T
=
(F
1
,...,
F
n
}:
Most
of the
time,
the
values
of the
thresholds
{ZQ,
• •
••,
z
n
}
and the
variations
of
z(-)
on
O
are
unknown
and the
facies
of
T
cannot
be
built directly
from
equation (10.13). There
are
many situations, however, where
the
direction
g
of the
gradient
of
z(-)
can be
estimated over
a
region
A C
0:
The
next section shows
how
such
a
piece
of
information
can be
used
to
define
a
DSI
constraint
affecting
the
Membership Function
(p
associated with
the
unknown
partition
T
to be
estimated.
Facies-gradient
constraint
Let (a ®
l
s
)
be the
node
of
Q
reached when
one
moves
one
step
from
a on
the
graph
£(17,
TV)
in the
direction
g —
g(a):
Using
this notation,
it is
clear
that
the
unknown
function
z(-)
controlling
the
partition
T
is
such
that
In
other words,
if we are in
facies
number
v(a]
< n at
location
a €
£7,
then,
according
to
equations (10.13)
and
(10.14),
the
probability
of
being
in
facies
number
(v(a)
+ 1) at
location
(a ©
l
g
)
should increase:
It can be
concluded
that
this inequality induces
the
following
constraint
on
the
Membership Function
(p
associated with
the
partition
F
where
v(a)
is the
index
of the
most probable
facies
at
location
a
assumed
to
be
lower
than
n:
Such
a
constraint
can
easily
be
turned into
a
canonic inequality
DSI
constraint
c
=
c(a,
g)
belonging
to
C>
and
having
the
following
general
form
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