Guide Objective Assisted Particle Swarm Optimization
and its Application to History Matching
Alan P. Reynolds
1
*,
Asaad
Abdollahzadeh
2
, David W. Corne
1
, Mike Christie
2
, Brian Davies
3
and Glyn Williams
3
1
School of Mathematical and Computer Sciences (MACS),
Heriot

Watt University, Edinburgh, Scotland
2
Institute of Petroleum Engineering (IPE),
Heriot

Watt University, Edinburgh, Scotland
3
BP
Optimization and separable problems
Fig
.
2
:
Standard
and
guided
PSO
updates,
minimizing
x
2
+
y
2
.
On
such
a
separable
problem,
the
best
values
for
x
(minimizing
x
2
)
and
y
(minimizing
y
2
)
provide
better
guidance
than
the
overall
best
solution
.
Motivation
References
1.
Kennedy,
J
.
,
Eberhart
,
R
.:
Particle
swarm
optimization
.
In
:
Proc
.
IEEE
Int
.
Conf
.
on
Neural
Networks
.
Vol
.
4
,
1942

1948
(
1995
)
2.
Kvasnicka
,
V
.
,
Pelikan
,
M
.
,
Pospichal
,
J
.:
Hill
climbing
with
learning
(an
abstraction
of
genetic
algorithm)
.
Neural
Network
World
6
(
5
),
773

796
(
1995
)
3.
Mohamed,
L
.
,
Christie,
M
.
,
Demyanov
,
V
.:
Reservoir
model
history
matching
with
particle
swarms
.
In
:
SPE
Oil
and
Gas
India
Conf
.
And
Exhibition
.
Mumbai,
India
(
2010
)
4.
Rosenbrock
,
H
.
H
.:
An
automatic
method
for
finding
the
greatest
or
least
value
of
a
function
.
The
Computer
Journal
3
(
3
),
175

184
(
1960
)
PSO and guide objectives
*A.Reynolds@hw.ac.uk
History
matching
is
the
improvement
of
parameterized
oil
reservoir
models
via
the
minimization
of
the
misfit
between
real
world
observations
and
those
obtained
through
simulation
.
We
wish
to
automate
the
history
matching
of
oil
reservoirs,
incorporating
reservoir
experts’
domain
knowledge
into
a
metaheuristic
.
However,
this
is
difficult
to
do
in
a
generally
applicable
way
.
We
note
that,
given
a
suitable
model
parameterization,
certain
model
parameters
will
affect
certain
misfit
components
to
a
greater
degree
than
others
.
This
suggests
that
the
problem
might
be
roughly
decomposed
,
with
subsets
of
misfit
components
being
used
to
create
a
guide
objective
for
subsets
of
the
model
parameters
.
We
show
how
PSO
can
be
adapted
to
use
both
the
guide
objectives
and
the
overall
objectives
in
a
single
optimization
run
.
Fig
.
1
:
The
PUNQ

S
3
case
study
If
we
have
a
separable
objective
function,
e
.
g
.
f(x,
y)
=
x
2
+
y
2
,
we
should
optimize
x
and
y
separately
.
Minimizing
f
directly
results
in
good
values
for
x
being
missed
when
coupled
with
poor
choices
for
y
and
vice
versa
.
Note
that
it
may
not
always
be
obvious
when
the
objective
can
be
separated,
e
.
g
.
f(x,
y)
=
x
4
+
2
x
2
y
2
+
y
4
.
Separate
optimization
is
also
appropriate
for
roughly
separable
problems,
e
.
g
.
minimizing
f(x,
y)
=
x
4
+
2
x
2
y
2
+
y
4
+
ε
x
3
y,
where
ε
is
small
.
A
near
optimal
solution
is
quickly
found
that
can
be
improved
further
by
optimizing
f
directly
if
desired
.
Here we refer to g(x) = x
2
and h(y) = y
2
as the
guide objectives
for x and y.
Basic PSO:
v
ij
←
wv
ij
+
α
r
1
(
p
ij
–
x
ij
) +
β
r
2
(
g
j
–
x
ij
) ,
v
ij
← min(
v
ij
,
V
max
, j
) ,
v
ij
← max(
v
ij
,

V
max
, j
) ,
x
ij
←
x
ij
+
v
ij
.
Velocity for particle
i
, component j.
Best position visited by
particle
i
(particle best)
Best position visited by
swarm (global best)
Particle position
v
ij
←
wv
ij
+
α
r
1
(
p
ij
(j)
–
x
ij
) +
β
r
2
(
g
j
(j)
–
x
ij
) .
Using guide objectives:
Particle best, according to guide
objective for decision variable j
Swarm best, according to
objective for decision variable j
Using guide objectives and the true objective:
v
ij
←
wv
ij
+
α
r
1
(
p
ij
–
x
ij
) +
β
r
2
(
g
j
–
x
ij
)
+
γ
r
1
(
p
ij
(j)
–
x
ij
) +
δ
r
2
(
g
j
(j)
–
x
ij
) .
Changing the values of
α
,
β
,
γ
and
δ
allows the influence of the guide objectives on
the search to be controlled.
This performs separate optimizations concurrently in a single run of PSO.
Test function
0
0.5
1
1.5
2
10
5
0
5
10
g(x)
x
Fig
.
3
:
The
highly
multimodal
function,
g(x),
of
Kvasnicka
et
al
.
Minimize:
20 variable
Rosenbrock
function
Roughly separable, with g(x
i
) acting as guide objective for x
i
.
3.55
3.60
3.65
3.70
3.75
0.2
0.5
0.8
1
1 → 0
Mean best
objective
Influence of guide objectives,
λ
Fig
.
4
:
Results
for
the
test
function,
with
95
%
confidence
intervals
.
Best
results
are
obtained
using
both
guide
objectives
and
the
true
objective
.
(Results
obtained
using
only
the
true
objective
are
of
considerably
poorer
quality
and
are
omitted
for
clarity
.
)
History matching: PUNQ

S3
Region
A
B
C
D
E
F
G
H
I
Guide wells
5
5, 12
5, 12
5, 12
4, 5, 12
1, 4, 15
1, 4, 11, 15
1, 11, 15
1, 11
Porosity
values
for
9
regions
in
5
layers
gives
45
decision
variables
.
The
guide
objective
for
a
variable
is
the
sum
of
misfits
for
a
subset
of
wells
associated
with
the
respective
region
.
Table
1
:
The
9
regions
in
the
PUNQ

S
3
reservoir
and
the
wells
most
likely
to
be
affected
by
changes
in
those
regions
.
1.5
1.7
1.9
2.1
2.3
2.5
2.7
0
0.2
0.5
0.8
1
1 → 0
Misfit (3000 evals.)
λ
2.5
3
3.5
4
4.5
5
0
0.2
0.5
0.8
1
1 → 0
Misfit (1000 evals.)
λ
Fig
.
5
:
Results
for
the
PUNQ

S
3
history
matching
problem
for
3000
and
1000
function
evaluations,
with
95
%
confidence
intervals
.
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