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Profit and Risk Measures in Oil Production Optimization
Capolei, Andrea; Foss, Bjarne; Jørgensen, John Bagterp
Published in:
Proceedings of the 2nd IFAC Workshop on Automatic Control in Offshore Oil and Gas Production OOGP 2015
Link to article, DOI:
10.1016/j.ifacol.2015.08.034
Publication date:
2015
Document Version
Publisher's PDF, also known as Version of record
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Citation (APA):
Capolei, A., Foss, B., & Jørgensen, J. B. (2015). Profit and Risk Measures in Oil Production Optimization. In
Proceedings of the 2nd IFAC Workshop on Automatic Control in Offshore Oil and Gas Production OOGP 2015
(Vol. 48, pp. 214-220). (I F A C Workshop Series). DOI: 10.1016/j.ifacol.2015.08.034
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Proceedings of the 2nd IFAC Workshop on Automatic Control in
Proceedings
of the
2nd
IFAC Workshop on Automatic Control in
Offshore
Oil and
Gas
Production
Proceedings
of
2nd
IFAC
Proceedings
of the
the
2nd
IFAC Workshop
Workshop on
on Automatic
Automatic Control
Control in
in
Offshore
Oil
and
Gas
Production
May
27-29,
2015.
Florianópolis,
Brazil
Available
online
at www.sciencedirect.com
Offshore
Oil
and
Gas
Production
Offshore
Oil2015.
and Gas
Production
May
27-29,
Florianópolis,
Brazil
May
May 27-29,
27-29, 2015.
2015. Florianópolis,
Florianópolis, Brazil
Brazil
ScienceDirect
IFAC-PapersOnLine 48-6 (2015) 214–220
in
in
in
in
Profit
and
Risk
Measures
Profit
and
Risk
Measures
Profit
and
Risk
Measures
Profit
and RiskOptimization
Measures Oil
Production
Oil
Production
Optimization
Oil Production
Production Optimization
Optimization Oil
∗
Andrea Capolei ∗∗ Bjarne Foss ∗∗
∗∗ John Bagterp Jørgensen ∗
Andrea
Capolei
∗ Bjarne Foss ∗∗ John Bagterp Jørgensen ∗
∗ Bjarne Foss ∗∗ John Bagterp Jørgensen ∗
Andrea
Capolei
Andrea Capolei Bjarne Foss John Bagterp Jørgensen
∗
∗ Department of Applied Mathematics and Computer Science & Center
∗ Department of Applied Mathematics and Computer Science & Center
∗ for
Department
of
Mathematics
and
Science
&
Energy Resources
Engineering,
Technical
University
of Denmark,
Department
of Applied
Applied
Mathematics
and Computer
Computer
Science
& Center
Center
for
Energy
Resources
Engineering,
Technical
University
of
Denmark,
for
Energy
Resources
Engineering,
Technical
University
of
Denmark,
DK-2800
Kgs.
Lyngby,
Denmark.
(e-mail:
{acap,jbjo}@dtu.dk).
for
Energy
Resources
Engineering,
Technical
University
of
Denmark,
DK-2800
Kgs.
Lyngby,
Denmark.
(e-mail:
{acap,jbjo}@dtu.dk).
∗∗DK-2800 Kgs. Lyngby, Denmark. (e-mail: {acap,jbjo}@dtu.dk).
Department
of Lyngby,
Engineering
Cybernetics,
University of
∗∗DK-2800
Kgs.
Denmark.
(e-mail:Norwegian
{acap,jbjo}@dtu.dk).
∗∗ Department of Engineering Cybernetics, Norwegian University of
∗∗ Department
of
Engineering
Cybernetics,
Norwegian
University
of
Science
and Technology
(NTNU),
7491 Trondheim,
Norway.
(e-mail:
Department
of
Engineering
Cybernetics,
Norwegian
University
of
Science
and
Technology
(NTNU),
7491
Trondheim,
Norway.
(e-mail:
Science
and
Technology
(NTNU),
7491
Trondheim,
Norway.
(e-mail:
[email protected])
Science and Technology [email protected])
(NTNU), 7491 Trondheim, Norway. (e-mail:
[email protected])
[email protected])
Abstract: In oil production optimization, we usually aim to maximize a deterministic scalar
Abstract:
oil
optimization,
we
usually
aim
maximize
aa deterministic
scalar
Abstract: In
Inindex
oil production
production
optimization,
wethe
usually
aim to
to
maximize
deterministic
scalar
performance
such as the
profit overwe
expected
reservoir
lifespan.
However, scalar
when
Abstract:
In
oil
production
optimization,
usually
aim
to
maximize
a deterministic
performance
index
such
as
the
profit
over
the
expected
reservoir
lifespan.
However,
when
performance
index
such
as
the
profit
over
the
expected
reservoir
lifespan.
However,
uncertainty
inindex
the parameters
is profit
considered,
the profit
results
in a random
variable
thatwhen
can
performance
such
as
the
over
the
expected
reservoir
lifespan.
However,
when
uncertainty
in
the
parameters
is
considered,
the
profit
results
in
a
random
variable
that
can
uncertainty
in
the
parameters
is
considered,
the
profit
in
aaparameters.
random
variable
can
assume
a range
of values
depending
on the value
of theresults
uncertain
In thisthat
case,
a
uncertainty
in
the
parameters
is
considered,
the
profit
results
in
random
variable
that
can
assume
aa range
of
values
depending
on
the
value
of
the
uncertain
parameters.
In
this
case,
aa
assume
range
of
values
depending
on
the
value
of
the
uncertain
parameters.
In
this
case,
problem
reformulation
is
needed
to
properly
define
the
optimization
problem.
In
this
paper
assume
areformulation
range of values
depending
on the value
of the
the optimization
uncertain parameters.
InInthis
case,
a
problem
is
needed
to
properly
define
problem.
this
paper
problem
reformulation
is of
needed
to properly
properly
define
the
optimization
problem.
Inappropriate
this paper
paper
we
describe
the conceptis
risk and
we explore
howthe
to optimization
handle the risk
by usingIn
problem
reformulation
needed
to
define
problem.
this
we
describe
the
concept
of
risk
and
we
explore
how
to
handle
the
risk
by
using
appropriate
we
describe
the
concept
and
we
explore
to
the
using
risk
measures.
provideof
review
various
riskhow
measures
reporting
proby
and
consappropriate
for each of
we
describe
theWe
concept
ofaa risk
risk
andon
we
explore
how
to handle
handle
the risk
risk
by
using
appropriate
risk
measures.
We
provide
review
on
various
risk
measures
reporting
pro
and
cons
for
each of
risk
measures.
We
provide
a
review
on
various
risk
measures
reporting
pro
and
cons
for
of
them.
Finally,
among
the
presented
risk
measures,
we
identify
two
of
them
as
appropriate
risk
measures.
We
provide
a
review
on
various
risk
measures
reporting
pro
and
cons
for each
eachrisk
of
them.
Finally,
among
the
presented
risk
measures,
we
identify
two
of
them
as
appropriate
risk
them.
Finally,
among
the
presented
risk
measures,
we
identify
two
of
them
as
appropriate
risk
measures
when
minimizing
the
risk.
them. Finally,
among
the presented
risk measures, we identify two of them as appropriate risk
measures
when
minimizing
the
risk.
measures when
when minimizing
minimizing the
the risk.
risk.
measures
© 2015, IFAC (International
Federation
of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Keywords: Production Optimization, Oil production, Risk measures, Stochastic problems
Keywords:
Production
Optimization,
Oil production,
production, Risk measures,
measures, Stochastic problems
problems
Keywords:
Keywords: Production
Production Optimization,
Optimization, Oil
Oil production, Risk
Risk measures, Stochastic
Stochastic problems
1. INTRODUCTION
loss. When speaking of such a measure applied to the
1. INTRODUCTION
INTRODUCTION
loss.
When
speaking
of
such aa measure
applied
to
the
1.
loss.
When
speaking
of
applied
to
random
profit,
ψ, we have
in mind
that higher
outcomes
1. INTRODUCTION
loss.
When
speaking
of such
such
a measure
measure
applied
to the
the
random
profit,
ψ,
we
have
in
mind
that
higher
outcomes
random
ψ,
we
have
in
mind
that
higher
ψ areprofit,
welcome
while
lower
outcomes
are outcomes
disliked.
In oil production optimization, we are in general interested of
random
profit,
ψ,
we
have
in
mind
that
higher
outcomes
of
ψ
are
welcome
while
lower
outcomes
are
disliked.
In oil
oil production
production optimization,
optimization, we
we are
are in
in general
general interested
interested To
of ψ
ψreduce
are welcome
welcome
while
lower
outcomes
arelower
disliked.
the risk while
of losslower
then, outcomes
we seek to
the
In
in
anoptimization,
economic measure,
the profit
or the of
are
are
disliked.
In maximizing
oil production
we are like
in general
interested
reduce
the
risk
of
loss
then,
we
seek
to
lower
the
in
maximizing
an
economic
measure,
like
the
profit
or the
the To
reduce
the
risk
of
loss
then,
we
seek
to
lower
the
probability
of the
low
profits.
Certainly,
deviation
and
in
maximizing
an
economic
measure,
like
the
profit
or
net
present value
(NPV), over
the expected
reservoir
life To
To
reduce
the
risk
of
loss
then,
we
seek
to
lower
the
in
maximizing
an
economic
measure,
like
the
profit
or
the
probability
of
the
low
profits.
Certainly,
deviation
and
net
present
value
(NPV),
over
the
expected
reservoir
life
probability
of
the
low
profits.
Certainly,
deviation
and
risk
are
related
concepts
and
often
these
terms
are
used
net
present
value
(NPV),
over
the
expected
reservoir
life
time.
When value
uncertainty
isover
taken
into
account,
the profit
probability
of
the
low
profits.
Certainly,
deviation
and
net
present
(NPV),
the
expected
reservoir
life
are related
concepts and
often these
terms
are
used
time. When
When uncertainty
uncertainty is
is taken
taken into
into account,
account, the
the profit
profit risk
risk
and
these
terms
are
used
e.g. in finance,
we can
interpret
time.
is
not When
a singleuncertainty
quantity but
has a probability
distribution,
risk are
are related
related concepts
concepts
and often
often
these
terms the
areprofit
used
time.
is taken
into account,
the profit interchangeably,
interchangeably,
e.g.
in
finance,
we
can
interpret
the
profit
is
not
a
single
quantity
but
has
a
probability
distribution,
interchangeably,
e.g.
in
finance,
we
can
interpret
the
profit
volatility,
measured
by
the
standard
deviation,
as
risk.
is
not
a
single
quantity
but
has
a
probability
distribution,
i.e.
the
profit
is
described
by
a
random
variable
ψ.
An
interchangeably,
e.g.
in
finance,
we
can
interpret
the
profit
is not
a single
quantity
but has
a probability
distribution,
volatility,
measured
by
the
standard
deviation,
as
risk.
i.e.
the
profit
is
described
by
a
random
variable
ψ.
An
volatility,
measured
the
standard
deviation,
as
risk.
Following
this idea, by
Capolei
et al. (2015)
introduce
the
i.e.
the
profit
is
described
by
aa ψrandom
variable
ψ.
An
optimization
problem
involving
in terms
of a control
volatility,
measured
by
the
standard
deviation,
as
risk.
i.e.
the
profit
is
described
by
random
variable
ψ.
An
Following
this
idea,
Capolei
et
al.
(2015)
introduce
the
optimization
problem
involving
ψ
in
terms
of
a
control
Following
this
idea,
Capolei
et
al.
(2015)
introduce
the
mean-variance
criterion
for
production
optimization
and
optimization
problem
involving
ψ
in
terms
of
a
control
input u, must problem
express ψinvolving
as a scalarψ quantity.
this criterion
idea, Capolei
et al. (2015)
introduce and
the
optimization
in termsTraditionally,
of a control Following
mean-variance
for
production
optimization
input
u,
must
express
ψ
as
a
scalar
quantity.
Traditionally,
mean-variance
criterion
for
production
optimization
and
suggest
to
use
the
Sharpe
ratio
as
a
systematic
procedure
input
u,
must
express
ψ
as
a
scalar
quantity.
Traditionally,
this
single
quantity
is
the
expected
profit
(Van
Essen
mean-variance
criterion
for
production
optimization
and
input
u,
must
express
ψ
as
a
scalar
quantity.
Traditionally,
suggest
to
use
the
Sharpe
ratio
as
a
systematic
procedure
this single
single quantity
quantity is
is the expected
expected profit
profit (Van
(Van Essen
Essen suggest
to
the
as
procedure
to optimally
risk ratio
and return.
They interpret
the
this
et
2009;quantity
Capoleiisetthe
al.,expected
2013). By
using
onlyEssen
the suggest
to use
usetrade-off
the Sharpe
Sharpe
ratio
as aa systematic
systematic
procedure
thisal.,
single
profit
(Van
optimally
trade-off
risk
and
return.
They
interpret
the
et
al.,
2009; Capolei
Capolei et
etthe
al., 2013).
2013). By
By
using
only the
the to
to
optimally
trade-off
risk
and
return.
They
interpret
the
standard
deviation
as
a
measure
of
risk.
However,
et
al.,
2009;
al.,
using
only
expected
profit,
however,
we
are2013).
not able
tousing
include
others
to
optimally
trade-off
risk
and
return.
They
interpret
the
et
al.,
2009;
Capolei
et
al.,
By
only
the
standard
deviation
as
a
measure
of
risk.
However,
the
expected
profit,
however,
we
are
not
able
to
include
others
standard
deviation
as
a
measure
of
risk.
However,
the
mean-variance
approach
is
more
suited
to
reduce
the
profit
expected
profit,
however,
we
are
not
able
to
include
others
important
indicators,
that
shape
the
profit
distribution
ψ,
standard
deviation
as
a
measure
of
risk.
However,
the
expected
profit,
however,
we
are
not
able
to
include
others
mean-variance
approach
is
more
suited
to
reduce
the
profit
important
indicators,
that
shape
the
profit
distribution
ψ,
mean-variance
approach
is
more
suited
to
reduce
the
profit
uncertainty
than
to
reduce
the
risk
of
loss.
Fig.
1
illustrates
important
indicators,
that
shape
the
profit
distribution
ψ,
such
as
the
profit
deviation
and
the
risk
preference.
The
mean-variance
approach
is
more
suited
to
reduce
the
profit
important
indicators,
that
shape
the
profit
distribution
ψ,
uncertainty
than
to
reduce
the
risk
of
loss.
Fig.
1
illustrates
such as
as the
the profit
profit deviation
deviation and
and the
the risk
risk preference.
preference. The
The uncertainty
than
the
loss.
illustrates
two drawbacks
ofto
thereduce
mean-variance
used
such
role
a measure
deviationand
is tothe
quantify
the variability
than
to
reduce
the risk
risk of
offramework
loss. Fig.
Fig. 11when
illustrates
suchof
profitof
risk preference.
The uncertainty
two
drawbacks
of
the
mean-variance
framework
when
used
role
ofasaa the
measure
ofdeviation
deviation is
is to
to quantify
quantify
the variability
variability
two
drawbacks
of
the
mean-variance
framework
when
used
to
measure
risk
preferences.
First
of
all,
the
mean
variance
role
of
measure
of
deviation
the
of
a of
random
variable
ψ and the
uncertainty
in variability
ψ is often two
drawbacks
of
the
mean-variance
framework
when
used
role
a
measure
of
deviation
is
to
quantify
the
to
measure
risk
preferences.
First
of
all,
the
mean
variance
of
a
random
variable
ψ
and
the
uncertainty
in
ψ
is
often
to
measure
risk
preferences.
First
of
all,
the
mean
variance
approach
is
insensitive
to
the
profit
shape
distribution.
of
a
random
variable
ψ
and
the
uncertainty
in
ψ
is
often
measured
by
the
standard
deviation
of
ψ,
e.g.
in
classical
to
measure
risk
preferences.
First
of
all,
the
mean
variance
of
a
random
variable
ψ
and
the
uncertainty
in
ψ
is
often
is
insensitive
to
profit
distribution.
measured by
by the
the standard
standard deviation
deviation of
of ψ,
ψ, e.g.
e.g. in classical
classical approach
approach
is
insensitive
to the
thedifferent
profit shape
shape
distribution.
Fig.1a
is a is
sketch
representing
profit distributions
measured
portfolio
1959), the
insensitive
to
the
profit
shape
distribution.
measuredtheory
by the(Markowitz,
standard deviation
of standard
ψ, e.g. in
indeviation
classical approach
Fig.1a
is
aa sketch
representing
different
profit
distributions
portfolio
theory
(Markowitz,
1959),
the
standard
deviation
Fig.1a
is
sketch
representing
different
profit
distributions
having
the
same
values
for
mean
and
the
variance. In
portfolio
theory
(Markowitz,
1959),
the
standard
deviation
σ(ψ)
is
used
to
quantify
uncertainty
in
returns
of
financial
Fig.1a
is
a
sketch
representing
different
profit
distributions
portfolio
theory
(Markowitz,
1959), the
standardofdeviation
the
same
values
for
mean
and
the
variance.
In
σ(ψ)
is used
used
to quantify
quantify
uncertainty
in returns
returns
financial having
having
the
same
values
for
mean
and
the
variance.
In
mean-variance
framework
these distributions
yield the
σ(ψ)
is
uncertainty
in
portfolios.
Into
the
oil community,
Bailey
et of
al.financial
(2005); the
having
the
same
values
for
mean
and
the
variance.
In
σ(ψ)
is
used
to
quantify
uncertainty
in
returns
of
financial
the
mean-variance
framework
these
distributions
yield
the
portfolios.
In
the
oil
community,
Bailey
et
al.
(2005);
the
mean-variance
framework
these
distributions
yield
the
same
risk
preference.
In
Fig.1b
instead,
the
distributions
portfolios.
In
the
oil
community,
Bailey
et
al.
(2005);
Alhuthali
et
al.
(2010);
Yeten
et
al.
(2003)
propose
to
the
mean-variance
framework
these
distributions
yield
the
portfolios.
In
the
oil
community,
Bailey
et
al.
(2005);
same
risk
preference.
In
Fig.1b
instead,
the
distributions
Alhuthali et
et al.
al. (2010);
(2010); Yeten
Yeten et
et al.
al. (2003)
(2003) propose
propose to
to in
same
riskhave
preference.
Instandard
Fig.1b instead,
instead,
the distributions
distributions
blue
a lowerIn
deviation,
σ, than the
Alhuthali
reduce
the et
uncertainty
profit by
standard
risk
preference.
Fig.1b
the
Alhuthali
al. (2010);in
Yeten
et including
al. (2003)the
propose
to same
in
blue
have
a lower
standard
deviation,
σ, than
the
reduce
the
uncertainty
in
profit
by
including
the
standard
in
blue
have
lower
than
distribution
in aablack.
If standard
we use thedeviation,
standard σ,
deviation
as
reduce
the
in
profit
including
the
standard
deviation
inuncertainty
the cost function.
Inby
many
decision
problems
in
blue
have
lower
standard
deviation,
σ,
than the
the
reduce
the
uncertainty
in
profit
by
including
the
standard
distribution
in
black.
If
we
use
the
standard
deviation
as
deviation
in
the
cost
function.
In
many
decision
problems
in
black.
If
we
use
the
standard
as
adistribution
risk measure,
the blue
distributions
have deviation
a lower risk
deviation
in
the
cost
function.
In
many
problems
dealing
with
safety
and
reliability,
risk isdecision
often interpreted
distribution
in
black.
If
we
use
the
standard
deviation
as
deviation
in
the
cost
function.
In
many
decision
problems
aa risk
measure,
the
blue
distributions
have
a
lower
risk
dealing
with
safety
and
reliability,
risk
is
often
interpreted
risk
measure,
the
blue
distributions
have
a
lower
risk
than
the
black
profit
distribution,
no
matter
what
their
dealing
with
safety
and
reliability,
risk
is
often
interpreted
as
the
probability
of
a
dreadful
event
or
disaster
(Ditlevsen
a
risk
measure,
the
blue
distributions
have
a
lower
risk
dealing
with
safety
and
reliability,
risk
is
often
interpreted
than
the
black
profit
distribution,
no
matter
what
their
as the
the probability
probability of
of aa dreadful
dreadful event
event or
or disaster
disaster (Ditlevsen
(Ditlevsen expected
than the
the values
black profit
profit
distribution,the
no standard
matter what
what
their
are. Furthermore,
deviation
as
and
1996;
andorRoyset,
and than
black
distribution,
no
matter
their
as
theMadsen,
probability
of aRockafellar
dreadful event
disaster2010),
(Ditlevsen
values
are. Furthermore,
the
standard
deviation
and
Madsen,
1996;
Rockafellar
and
Royset,
2010),
and expected
expected
values
standard
deviation
as a measure
of are.
riskFurthermore,
is symmetric,the
which
means
that it
and
Madsen,
Rockafellar
and
Royset,
2010),
and
minimizing
the1996;
probability
of a highly
undesirable
event
expected
values
are.
Furthermore,
the
standard
deviation
and
Madsen,
1996;
Rockafellar
and
Royset,
2010),
and
aa measure
of
risk
is symmetric,
which
means
that it
minimizing the
the probability
probability of
of aa highly
highly undesirable
undesirable event
event as
as
of
risk
which
means
higher
and lower profits
minimizing
is
known as the
the probability
safety-first of
principle
(Roy,
1952). Inevent
this penalizes
as
a measure
measure
of profits
risk is
is symmetric,
symmetric,
which symmetrically.
means that
that it
it
minimizing
a highly
undesirable
penalizes
higher
profits
and
lower
profits
symmetrically.
is
known
as
the
safety-first
principle
(Roy,
1952).
In
this
penalizes
higher
profits
and
lower
profits
symmetrically.
This
last
shortcoming
have
been
recognised
by
Markowitz
is
known
as
the
safety-first
principle
(Roy,
1952).
In
this
paper
we
identify
the
risk
as
a
measure
of
the
risk
of
penalizes
higher
profits
and
lower
profits
symmetrically.
is
known
as
the
safety-first
principle
(Roy,
1952).
In
this
This
last
shortcoming
have
been
recognised
by
Markowitz
paper we
we identify
identify the
the risk
risk as aa measure
measure of the
the risk of
of (1959)
last
shortcoming
been
by
who
proposed have
to use
therecognised
semideviation
instead.
paper
This
last
shortcoming
have
been
recognised
by Markowitz
Markowitz
paper
identify
the risk as
as a measure of
of the risk
risk of This
This we
(1959)
who
proposed
to
use
the
semideviation
instead.
project is financially supported by the Danish
(1959)
who
proposed
to
use
the
semideviation
instead.
However,
even
by
using
the
semideviation,
we
still
do not
This research
(1959)
who
proposed
to
use
the
semideviation
instead.
research
project
is
financially
supported
by
the
Danish
This research
However,
even
by
using
the
semideviation,
we
still
do
not
Research
Council
for
Technology
and
Production
Sciences.
FTP
project
is
financially
supported
by
the
Danish
However,
even
by
using
the
semideviation,
we
still
do
not
This
research
project
is
financially
supported
by
the
Danish
have
common
properties
that
make
sense
both
for
a risk
Research Council for Technology and Production Sciences. FTP
However,
even properties
by using the
semideviation,
we still
do
not
have
common
that
make
sense
both
for
a
risk
Grant
no.
274-06-0284
and
the
Center
for
Integrated
Operations
in
Research
Council
for
Technology
and
Production
Sciences.
FTP
Research
for Technology
andfor
Production
FTP
have
common
properties
that
make
sense
both
for
a
risk
Grant
no. Council
274-06-0284
and the Center
IntegratedSciences.
Operations
in
have common properties that make sense both for a risk
the
Petroleum
Industryand
at NTNU.
Grant
no. 274-06-0284
the Center for Integrated Operations in
Grant
no. 274-06-0284
the Center for Integrated Operations in
the Petroleum
Industryand
at NTNU.
the
the Petroleum
Petroleum Industry
Industry at
at NTNU.
NTNU.
Copyright
© 2015,
2015 IFAC
214Hosting by Elsevier Ltd. All rights reserved.
2405-8963 ©
IFAC (International Federation of Automatic Control)
Copyright
© 2015 IFAC
214
Copyright
©
2015
IFAC
214
Peer
review
under
responsibility
of
International
Federation
of
Automatic
Copyright © 2015 IFAC
214Control.
10.1016/j.ifacol.2015.08.034
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215
measure like risk aversion and monotonicity, see Section
3.
2. PROBLEM FORMULATION
In this paper we follow an axiomatic approach to define
risk, i.e. we first define the principles that an appropriate
risk measure should have, then we select risk measures that
satisfies such principles. The risk axioms that we use are
the principles that define coherent averse risk measures as
introduced and defined in Artzner et al. (1999); Rockafellar
(2007); Krokhmal et al. (2011); Zabarankin and Uryasev
(2014). At our knowledge, this is the first time that such an
axiomatic approach is used in oil production optimization.
In oil production optimization, the profit can be visualized
as a function
ψ = ψ(u, θ)
(1)
of a decision vector u ∈ U ⊂ Rnu representing the control
vector, with U expressing linear decision constraints, and
a vector θ ⊂ Rm representing the values of a number
of parameters variables such as the permeability field,
porosity, economic parameters, etc. The function ψ usually
represents the NPV or some other performance index, and
its computation typically requires the use of a reservoir
simulator to solve the reservoir flow equations. If there
is no uncertainty in the parameters values θ, we can
maximize ψ by solving the following deterministic optimal
control problem (Brouwer and Jansen, 2004; Sarma et al.,
2005; Nævdal et al., 2006; Foss and Jensen, 2011; Capolei
et al., 2013)
max ψ(u, θ)
(2)
The paper is organized as follow. Section 2 formulates
the oil production optimization problem under uncertainty
as a risk minimization problem. Section 3 describes the
basic properties that we require from an appropriate risk
measure, while a number of risk measures are discussed in
Section 4. Conclusions are presented in Section 5.
u∈U
However, in oil problems there is a high uncertainty due
for example to the noisy and sparse nature of seismic data,
core samples, borehole logs, and future oil prices and plant
costs. Mathematically, we may represent model uncertainty by making the parameter vector θ a random variable
that has some probability distribution and that belongs to
some uncertainty space Θ. Consequently, the profit ψ is a
random variable. Due to the complexity of real oil reservoirs and the accompanying measurement problem, we
don’t know the probability distribution of θ, thus, we have
only incomplete informations about the uncertainty space
Θ. For these reasons, the traditional way of modeling the
uncertainty in oil production problems is to consider a finite set of possible scenarios for the parameters (Krokhmal
et al., 2011; Van Essen et al., 2009; Capolei et al., 2013,
2015). This means that we substitute Θ with the discretized space Θd := {θ1 , θ2 , . . . , θnd }. As a consequence,
a control input u, will correspond to a finite set of possible
profit outcomes ψ(u, θ1 ), . . . , ψ(u, θnd ), with probabilities
p1 , . .
. , pnd , respectively, where pi = P rob[θ = θi ] ∈ [0, 1]
nd
and i=1
pi = 1. Usually the possible realizations θi are
considered equiprobable, i.e. pi = 1/nd . It should be noted
that defining the uncertainty set Θd is a highly interdisciplinary exercise. Furthermore, uncertainty will be updated
as subsurface properties further reveal themselves from
measurement surveys and production data in addition to
new forecasts on oil price and costs.
(a)
∼ σ(ψ2 )
∼ σ(ψ1 )
When uncertainty is taken into account, the following
stochastic optimization problem can be written
(3)
max ψ(u, θ)
(b)
u∈U , θ∈Θd
Fig. 1. The mean-variance framework is indifferent to
shape distributions. Fig.1a is a sketch representing
different profit distributions having the same mean
and variance. In the mean-variance framework, these
distributions yield the same risk preference. In Fig.1b,
the distributions in blue have a lower standard deviation σ than the distribution in black. If we use
the standard deviation as a risk measure, this means
that the blue distributions have a lower risk than
the black profit distribution, no matter what their
expected values are.
215
However, this formulation is not well defined. The problem
is that the decision vector u must be chosen before the
outcome of the distribution of θ and consequently the value
of ψ, can be observed. To obtain a well defined problem,
we can substitute the random variable ψ by a functional
R : ψ → R that yields a scalar measure of ψ, and depends
both on u and θ. In this way we can reformulate problem
(3) as
(4)
min R(ψ(u, θ ∈ Θd ))
u∈U
Note that we have switched to a minimization problem
because we will interpret R as a risk measure to minimize.
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R is a surrogate for the distribution of ψ, thus, different
R expressions capture differents aspects of the profit
distribution. So far, different measures, R, have been
proposed in the oil community such as the expected profit,
i.e. R = −Eθ [ψ] (Van Essen et al., 2009) or the mean-
variance measure, i.e. R = − λEθ [ψ] − (1 − λ)σ 2 (ψ)
with λ ∈ [0, 1] (Capolei et al., 2015). However, for many
reasons, that we will discuss in Section 4, none of these
measure are satisfactory. Finally, we should stress that in
this paper we focus on single objective optimization only.
We do not consider important aspects that are connected
to multi-objective optimization, e.g. the trade-off between
long term vs short term profit (Van Essen et al., 2011).
However, our analysis on risk measures can be extended
to these cases. We can for example use the weighted sum
method (Liu and Reynolds, 2015) to trade-off a long term
profit measure R(ψ long ) versus a short term profit measure
R(ψ short ) by solving
min λ R(ψ short ) + (1 − λ) R(ψ long )
u∈U
(5)
for different λ ∈ [0, 1].
3. COHERENT AVERSE MEASURES OF RISK
Measures of risk have a crucial role in oil production optimization under uncertainty, especially in coping with the
losses in profit due to a too aggressive oil field development
plan. The role of a risk measure is to assign to the random
profit, ψ, a numerical value R(ψ) that can serve as a
surrogate for overall profit. The risk comparison of two
choices ψ � and ψ �� then reduces to comparing R(ψ � ) and
R(ψ �� ) and a decision maker, during the decision process,
should prefer solutions that minimize the risk or that
maintain the risk below a certain threshold. In this paper,
we consider as appropriate risk measures, the coherent and
averse measures of risk as defined by Artzner et al. (1999);
Rockafellar (2007); Krokhmal et al. (2011):
Definition 1. Coherent Averse measures of risk are functionals R : ψ → R satisfying
(A1) Risk aversion:
• R(c) = −c for constants c (constant equivalence)
• R(ψ) > −Eθ [ψ] for nonconstant ψ (averseness).
(A2) Positive homogeneity: R(λψ) = λR(ψ) when λ > 0.
(A3) Subadditivity: R(ψ � + ψ �� ) ≤ R(ψ � ) + R(ψ �� ) for all
ψ � and ψ �� .
(A4) Closure: ∀c ∈ R, the set {ψ|R(ψ) ≤ c} is closed.
(A5) Monotonicity: R(ψ � ) ≥ R(ψ �� ) when ψ � ≤ ψ �� .
These axioms require additional explanation. Axiom (A1)
formalizes the risk averse principle, see also Fig. 2. A riskaverse decision maker does not rely on the expected profit
exclusively and always prefers a deterministic payoff of
Eθ [ψ] over a non constant ψ. The risk of a deterministic
profit is given by its negative value, i.e. R(c) = −c. R(c) =
−c implies R(Eθ [ψ]) = −Eθ [ψ], so that the other condition
R(X) > −Eθ [ψ] can be restated as R(ψ) > R(Eθ [ψ])
for ψ �= c and constant c, which is the risk aversion
property in terms of R. The positive homogeneity axiom
(A2) ensures invariance under scaling, e.g. if the units of ψ
are converted from one currency to another, then the risk
is also simply scaled with the exchange rate. Finally, the
positive homogeneity enables the units of measurements of
R(ψ) to be the same as those of ψ. The subaddivity axiom
216
(A3) is a mathematical expression of the fundamental risk
management principle of risk reduction via diversification.
Also, it follows from the constant equivalence, i.e. R(c) =
−c, and axiom (A3) that
R(ψ + c) = R(ψ) − c
(6)
which is called translation invariance. This is explained by
its financial interpretation. If ψ is the payoff of a financial
position, then adding cash to this position reduces its risk
by the same amount; in particular one has
R(ψ + R(ψ)) = 0 = R(0)
(7)
i.e. adding a quantity of cash equal to the risk of ψ
to ψ reduces to the risk of getting a deterministic zero
payoff. The translation invariance principle, provides us
with a natural way of defining an acceptable risk (Artzner
et al., 1999; Rockafellar, 2007). We can consider a risk
acceptable if its value is lower than the risk of obtaining a
deterministic reference payoff cref , i.e.
(8)
R(ψ) ≤ R(cref ) = −cref
In oil problems, cref could come from the expected income
by investing in an alternative project. However, typically
we consider the risk as acceptable when its value is lower
than the risk of a deterministic zero profit, i.e. cref = 0
and (8) thus becomes
R(ψ) ≤ 0
(9)
The monotonicity axiom (A5) says that we consider ψ �
no less risky than ψ �� if every realization of ψ �� is no
smaller than every realization of ψ � , see Fig. 3. In the literature, risk measures that satisfy axioms (A1-A4) are called
averse measures of risk (Rockafellar, 2007; Krokhmal et al.,
2011). Risk measures that satisfy axioms (A2-A5) and
the constant equivalence property are called coherent risk
measures in the sense of Artzner (Artzner et al., 1999;
Krokhmal et al., 2011). A final note on risk measures principles is that the positive homogeneity and the subadditivity imply convexity of the risk measure R(.) (Rockafellar,
2007; Krokhmal et al., 2011). The convexity property is
important when we want to minimize the risk, R, because
it allows the optimizer to find solutions that are globally
optimal. Therefore, we would like to formulate convex
oil production optimization problems. In oil optimization
problems, however, ψ = J(u, θ) is computed by a function
J that is non convex with respect to the decision vector
u. As a consequence, R(J(u, θ)) is non convex and the
optimizer can only yield local minima solutions. Unfortunately, the non convexity originates from the physics of
the problem itself. However, future research in this field
may allow to efficiently approximate and reformulate the
problem by using convex surrogate models, e.g. piecewise
linear models where there, however, is a need to include
integer variables.
4. RISK MEASURES IN PRODUCTION
OPTIMIZATION
In this section, we review some traditional approaches of
measuring risk. For each approach we will discuss if and
how it adheres to the risk measure definition of Section 3.
4.1 Nominal profit
This approach is based on selecting one single realization,
θi , of the uncertainty space Θd , as outcome for the future
value of θ:
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217
R(c1 ) = −c1
R(c2 ) = −c2
R(c1 ) > R(c2 )
σ(c1 ) = σ(c2 ) = 0
(a) Constant equivalence property
15
10
10
5
5
ψ’
ψ’’
5
0
Eθ(ψ)
Eθ(Ψ)
ψ
10
0
0
−5
1
2
3 4 5 6 7
Realization #
−5
0
8
(b) Profit realizations
2
4
σ(ψ)
6
−5
−2
8
0
CVaR
2
(ψ)
4
0.4
(c) Standard deviation
(d) Coherent averse Risk measure
example
Fig. 2. Aversity axiom. The constant equivalence property in Fig. 2a states that for constant profit distributions, a risk
measure as defined in Section 3 yields a well defined risk preference. Instead, when we use the standard deviation,
σ, the risk preference is not well defined. Fig. 2b shows two profit distributions ψ and ψ . ψ yields negative profits
and also negative expected profit. ψ yields a negative profit only in the first realization and has positive expected
profit. Despite for the first realization results ψ < ψ , the probability of lower profits is much larger for the profit
distribution ψ than for ψ . However, if we use the standard deviation as a measure of risk, ψ would result with
lower risk, see Fig. 2c, because it has a lower standard deviation. Instead, a coherent averse risk measure would
allow to choice a risk preference for which ψ yields the lowest risk, see Fig. 2d.
10
15
8
5
6
1
2
3 4 5 6 7
Realization #
8
(a) Profit realizations
6
θ
4
0
−5
4
2
2
0
8
E (ψ)
ψ
Eθ(ψ)
10
10
ψ’
ψ’’
−4
−3
−2
−1
CVaR0.4(ψ)
0
(b) Coherent averse Risk measure
example
0
0
1
2
3
σ (ψ)
4
5
θ
(c) Standard deviation
Fig. 3. The monotonicity axiom states that if we have two profit distributions ψ and ψ , and for every realization
ψ ≤ ψ , see Fig.3a, then a risk measure should yield a higher risk for ψ than for ψ , see Fig.3b. For the standard
deviation, see Fig.3c, the monotonicity axiom does not hold.
R(ψ) = −ψ(u, θi ).
(10)
Despite the fact that this approach is coherent (not
averse), it is open to criticism since it relies on one parameter sample only. Thus, one may argue that this approach
is closer to guess work than analysis. With this approach,
a profit distribution is acceptable in the sense of (9) when
the profit for the chosen realization θi is greater than zero.
4.2 Certainty Equivalent profit
This approach is based on using the expected value of the
parameters, Eθ [θ], as outcome for the future value of θ:
217
R(ψ) = −ψ(u, Eθ [θ])
(11)
This approach was used in Capolei et al. (2013, 2015). The
idea is that the expected value of the parameters yields a
good representation of the model. However, there are some
problems with this approach. First of all, the expected
value of θ may not be a possible realization of Θ, i.e. it
could be that Eθ [θ] ∈
/ Θ. Secondly, as a risk measure it
is neither averse nor coherent. In fact, the aversity and
the monotonicity axioms are not valid because of the
nonlinearity and nonconvexity of ψ. With this approach,
a profit distribution is acceptable in the sense of (9) when
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the deterministic system with the expected value of θ for
the parameters, has a profit greater than zero.
4.3 Expected profit
R(ψ) = −Eθ [ψ]
(12)
This is a coherent measure of risk commonly used in oil
production optimization where it is called robust optimization (Van Essen et al., 2009; Capolei et al., 2013, 2015).
With this risk measure, a profit distribution is acceptable
in the sense of (9) when the expected profit is greater than
zero. Despite its widespread use, this risk measure has
some important drawbacks. These are its risk neutrality
and the fact that there is no control on the possible
outcome of very low profit realizations. The use of the
expected profit as risk measure in (4) is justified when the
following two assumptions hold (Krokhmal et al., 2011):
• Long run assumption: The control found as a solution
of the stochastic problem will be employed repeatedly
under identical or similar conditions.
• Low variability of the profit realizations. If the profit
is highly variable, then the possible outcome of very
low profit realizations could be disastrous.
However, in oil production optimization we do not apply
the control to many oil reservoirs that are in similar conditions and that share the same uncertainty distribution of
the parameters, i.e. we do not have repetition. So, the long
run assumption does not hold. Further, the variability in
the profit distribution ψ could be high due to high uncertainty in the parameters. For these reasons, the expected
value of the profit does not seems to be the right choice
for risk management in oil production optimization.
4.4 Worst-case scenario
R(ψ) = − inf {ψ}
θ
(13)
This is a coherent and averse measure of risk used
for example in Alhuthali et al. (2010). However, often
it provides a too conservative risk measure. In fact, its
major drawback is that it does not take into account the
probability distribution of ψ. With this risk measure, a
profit distribution is acceptable in the sense of (9) when
the lowest possible profit is greater than zero.
4.5 Standard deviation and variance
R(ψ) = −σθ (ψ), R(ψ) = −σθ2 (ψ)
(14)
By definition, σθ (ψ) measures the deviation of the profit
from the expected value. The risk preference with this risk
measure does not obey the constant equivalence principle.
In fact, taken two constant profit distributions c1 , c2 we
have σθ (c1 ) = σθ (c2 ) = 0, see Fig. 2a. Consequently, with
this risk measure it makes no sense to talk of acceptable
risk in the sense of (9). Further, the standard deviation
misses risk averseness and monotonicity (see the axioms
in Section 3). In general, this measure lacks aversity and
coherency, see Fig. 2 and Fig. 3. The standard deviation
was used as an uncertainty measure, for example in Yeten
et al. (2003); Bailey et al. (2005); Alhuthali et al. (2010);
Capolei et al. (2013) and interepreted as a risk measure
in Capolei et al. (2015). The variance, σ 2 , compared to σ,
misses also the positive homogeneity and subadditivity.
218
4.6 Safety margin
(15)
R(ψ) = − Eθ [ψ] − λ σθ (ψ) , λ > 0
This is an averse risk measure, but it is not coherent
because it does not satisfy the monotonicity axiom. With
this risk measure, a risk is acceptable, in the sense of (9),
when the expected value of ψ is λ units larger than the
standard deviation of ψ, i.e.
Eθ [ψ] ≥ λ σθ (ψ), λ > 0
(16)
The safety margin risk measure (15) was used by Yeten
et al. (2003); Bailey et al. (2005); Alhuthali et al. (2010).
However, they did not use (15) directly to measure the
risk. Rather, they used the λ parameter as a weight
to find solutions with different profit-uncertainty tradeoffs. By varying λ, they were able to compute different
(Eθ [ψ], σθ [ψ]) solution pairs.
4.7 Mean-Variance
R(ψ) = − λEθ [ψ] − (1 − λ) σθ2 (ψ) , λ ∈ [0, 1] (17)
For λ = 1 we find the expected profit measure (12) and for
λ = 0 we find the variance (14). For λ ∈ (0, 1), (17) satisfies
the risk aversion axiom, however, it does not satisfies the
positive homogeneity, the subadditivity, and the monocity
axioms. Then, for λ ∈ (0, 1), (17) is neither averse nor
coherent. Because (17) does not satisfies the translation
invariance principle (6), we cannot give a sense to the risk
acceptance constraint (9). The mean-variance risk measure
(17) was used by Capolei et al. (2015). However, they did
not use directly (17) to measure the risk. Rather, they
used the λ parameter as a weight to find Pareto solutions
with different profit-risk trade-offs. They used Eθ [ψ] as a
measure of profit and σθ2 (ψ) as a measure of risk.
4.8 Value at Risk (V aRα )
Value-at-Risk (V aR)) is one of the most widely used risk
measures in the area of financial risk management and is a
major competitor to the standard deviation measure (JP
Morgan, 1994; Jorion, 2006). Given a profit distribution
ψ, V aRα (ψ) is defined as the negative α-quantile
V aRα (ψ) = −qψ (α), α ∈ (0, 1)
(18)
where
qψ (α) = inf{z P rob[ψ ≤ z] > α}.
(19)
The quantile with α confidence level, denoted as qψ (α),
is the value for which the probability that the profit ψ
is lower than qψ (α) is no greater than α, see Fig. 4c.
The Value-at-Risk concept has its counterparts in the
form of probabilistic, or chance constraints, that were
introduced by Cooper and Symonds (1958), and since then
have been widely used in many disciplines as operations
research and stochastic programming, systems reliability
theory, reliability-based design and optimization, and others (Ditlevsen and Madsen, 1996; Rockafellar and Royset,
2010). With a chance constraint, we may declare that a
profit ψ should exceed a certain predefined level cref with
probability of at least 1 − α, with α ∈ (0, 1):
P rob[ψ ≥ cref ] ≥ 1 − α
(20)
whereas in the case of α = 0, constraint (20) reduces to the
worst case approach in Section 4.4. From a risk reduction
point of view, the probabilistic constraint (20) has a dual
aspect. One aspect is that for a fixed α, we would like to
find the highest value of cref such that (20) is satisfied.
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This ensures that with a probability greater than 1 − α,
the profit lower bound is the highest possible. On the
other hand for a fixed cref value, we would like to have
α as low as possible to increase the probability of having
profits larger than cref . The chance constraint (20) is also
called the failure probability constraint in reliability theory
(Ditlevsen and Madsen, 1996; Rockafellar and Royset,
2010). The probabilistic constraint (20) is equivalent to
P rob[ψ < cref ] ≤ α
(21)
and it can be expressed as a constraint on the Value-atRisk of ψ (Krokhmal et al., 2011; Zabarankin and Uryasev,
2014):
V aRα (ψ) ≤ −cref
(22)
One of the major deficiencies of V aRα is that it does not
take into account the tail of the profit distribution beyond
the α−quantile level. Even more importantly, V aRα does
not satisfies the subadditivity axiom (A3) (Artzner et al.,
1999). In addition, V aRα is discontinuous with respect
to the confidence level α, see Fig. 4c, meaning that small
changes in the values of α can lead to significant jumps
in the risk estimates provided by V aRα . Despite V aRα
does not satisfies the translation invariance principle (6),
because V aRα is not subadditive, we can still give a sense
to the risk acceptance constraint (9) by using the probabilistic interpretation (21). Then, a profit distribution is
acceptable in the sense of (21) when
V aRα (ψ) ≤ 0
(23)
The meaning of (23) is to limit with α the probability of
having negative profits (loss), i.e.
P rob[ψ < 0] ≤ α.
(24)
4.9 Conditional Value at Risk (CV aRα )
R(ψ) = CV aRα (ψ(u, θ))
(25)
As a measure of risk, V aRα (ψ) lacks continuity with
respect to α, provides no information of how significant
losses in the α-tail could be and it is not subadditive,
see axiom (A3) in Section 3. These V aR’s deficiencies are
resolved by the conditional value-at-risk (CV aR) (Rockafellar and Uryasev, 2002) defined as the average of V aR
on [0, α]:
1 α
CV aRα (ψ) =
V aRs (ψ)ds, α ∈ (0, 1)
(26)
α 0
Since CV aRα (ψ) dominates V aRα (ψ) i.e. CV aRα (ψ) ≥
V aRα (ψ) (Krokhmal et al., 2011; Zabarankin and Uryasev, 2014), we can approximate the chance constraint (20)
with
CV aRα (ψ) ≤ −cref .
(27)
CV aRα yields a convex upper bound approximation for
the failure probability. This was used in structural engineering problems by Rockafellar and Royset (2010). In the
oil community, CV aRα has been used by Valladao et al.
(2013) as a deviation measure. They used a λ parameter
as a weight to find Pareto solutions with different expected
profit - profit deviation trade-offs. They used Eθ [ψ] as
a measure of profit and D = Eθ [ψ] + CV aRα [ψ] as a
deviation measure. Note that this deviation measure D,
similarly to the standard deviation σ, does not satisfy the
risk aversion and the monotonicity axioms. In the limit
of α approaching zero, CV aRα reduces to the worst-case
measure (13). In fact, we have
lim CV aRα (ψ) = − inf {ψ}
(28)
+
α→0
θ
219
219
and in the limit of α approaching one we find the expectedprofit measure 4.3. In fact, we have
lim CV aRα (ψ) = −Eθ (ψ)
(29)
−
α→1
Finally, CV aRα for α ∈ (0, 1) satisfies all the axioms of a
coherent averse measure of risk, given in Definition 1. By
using CV aRα as a risk measure, a profit distribution is
acceptable in the sense of (9) when
CV aRα (ψ) ≤ 0
(30)
Considering that CV aRα is an approximation of the
chance constraint (21), the meaning of (30) is to limit with
α the probability of having negative profits (loss), i.e.
P rob[ψ < 0] ≤ α.
(31)
However, differently from V aRα , CV aRα considers information about the tail of the profit distribution beyond
the α−quantile level, and yields a smooth and convex
approximation of (20). In Fig. 4d we compute CV aRα for
a test profit distribution resulting from the scenario with
8 profit realizations of Fig. 4a.
5. CONCLUSIONS
When we look at the oil production optimization problem
as a risk minimization problem, we find that common risk
measures used in the oil community are non satisfactory.
Instead, we propose the conditional value at risk and the
worst-case scenario as appropriate risk measures for risk
minimization.
REFERENCES
Alhuthali, A.H., Datta-Gupta, A., Yuen, B., and
Fontanilla, J.P. (2010). Optimizing smart well controls
under geologic uncertainty. Journal of Petroleum Science and Engineering, 73, 107–121.
Artzner, P., Delbaen, F., Eber, J.M., and Heath, D. (1999).
Coherent measures of risk. Mathematical Finance, 9(3),
203–228.
Bailey, W.J., Couët, B., and Wilkinson, D. (2005). Framework for field optimization to maximize asset value. SPE
Reservoir Evaluation & Engineering, 8(1), 7–21.
Brouwer, D.R. and Jansen, J.D. (2004). Dynamic optimization of waterflooding with smart wells using optimal control theory. SPE Journal, 9(4), 391–402. SPE78278-PA.
Capolei, A., Suwartadi, E., Foss, B., and Jørgensen, J.B.
(2013). Waterflooding optimization in uncertain geological scenarios. Computational Geosciences, 17(6), 991–
1013.
Capolei, A., Suwartadi, E., Foss, B., and Jørgensen, J.B.
(2015). A mean-variance objective for robust production
optimization in uncertain geological scenarios. Journal
of Petroleum Science and Engineering, 125, 23–37.
Cooper, A. and Symonds, G. (1958). Cost horizons
and certainty equivalents: an approach to stochastic
programming of heating oil. Management Science, 4,
235–263.
Ditlevsen, O. and Madsen, H.O. (1996). Structural Reliability Methods. Wiley.
Foss, B. and Jensen, J.P. (2011). Performance analysis for
closed-loop reservoir management. SPE Journal, 16(1),
183–190. doi:10.2118/138891-PA. SPE-138891-PA.
IFAC Oilfield 2015
May 27-29, 2015. Florianópolis, Brazil
Andrea Capolei et al. / IFAC-PapersOnLine 48-6 (2015) 214–220
220
30
1
29
0.8
28
i
27
0.6
26
α
ψ (million USD)
Prob[ψ ≤ z]
Prob[ψ < z]
0.4
25
0.2
24
(−VaR ,α )
α
1
23
22
0
1
2
3
4
5
Realization #
6
7
24
25
26
Profit
27
28
29
30
31
27
26.5
28
26
−CVaRα(ψ)
27
α
1
23
(b) Probability functions and αquantile.
29
−VaR (ψ)
{ z | Prob[ψ ≤ z] > α }
22
8
(a) Profit for each realization of the ensemble.
26
25
25.5
25
24.5
24
23
0
1
24
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
23.5
0
1
(c) Value at Risk funtion
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
1
(d) Conditional Value at Risk
Fig. 4. Illustration of the concepts of quantile, value at risk and conditional value at risk by using an example of
8 realizations. Fig. 4a shows the profit for each realization in the ensemble. Fig. 4b illustrates the P rob[ψ ≤
z], P rob[ψ < z] functions. Fig. 4c depicts the discontinuous V aRα function while Fig. 4d shows the continuous
CV aRα function.
Jorion, P. (2006). Value at Risk: The New Benchmark for
Managing Financial Risk. McGraw-Hill, 3rd edition.
JP Morgan (1994). Riskmetrics. New York.
Krokhmal, P., Zabarankinb, M., and Uryasev, S. (2011).
Modeling and optimization of risk. Surveys in Operations Research and Management Science, 16, 49–66.
Liu, X. and Reynolds, A.C. (2015). Multiobjective optimization for maximizing expectation and minimizing
uncertainty or risk with application to optimal well
control. In SPE Reservoir Simulation Symposium, SPE173216-MS.
Markowitz, H.M. (1959). Portfolio selection: efficient
diversification of investments. John Wiley.
Nævdal, G., Brouwer, D.R., and Jansen, J.D. (2006).
Waterflooding using closed-loop control. Computational
Geosciences, 10, 37–60.
Rockafellar, R.T. and Royset, J.O. (2010). On buffered
failure probability in design and optimization of structures. Reliability Engineering and System Safety, 95,
499–510.
Rockafellar, R. (2007). Coherent Approaches to Risk in
Optimization Under Uncertainty, 38–61. Tutorials in
Operations Research INFORMS.
Rockafellar, R. and Uryasev, S. (2002). Conditional valueat-risk for general loss distributions. J. Bank. Financ.,
26(7).
220
Roy, A. (1952). Safety first and the holding of assets.
Econometrica, 20(3), 431–449.
Sarma, P., Aziz, K., and Durlofsky, L.J. (2005). Implementation of adjoint solution for optimal control of smart
wells. In SPE Reservoir Simulation Symposium, 31
January-2 Feburary 2005, The Woodlands, Texas, 67–
83.
Valladao, D.M., Torrado, R.R., Flach, B., and Embid, S.
(2013). On the stochastic response surface methodology
for the determination of the development plan of an
oil & gas field. In SPE Middle East Intelligent Energy
Conference and Exhibition, SPE-167446-MS.
Van Essen, G.M., Van den Hof, P.M.J., and Jansen,
J.D. (2011). Hierarchical long-term and short-term
production optimization. SPE Journal, 16(1), 191–199.
SPE-124332-PA.
Van Essen, G.M., Zandvliet, M.J., Van den Hof, P.M.J.,
Bosgra, O.H., and Jansen, J.D. (2009). Robust waterflooding optimazation of multiple geological scenarios.
SPE Journal, 14(1), 202–210. SPE-102913-PA.
Yeten, B., Durlofsky, L.J., and Aziz, K. (2003). Optimization of nonconventional well type, location, and
trajectory. SPE Journal, 8(3), 200–210.
Zabarankin, M. and Uryasev, S. (2014). Statistical Decision Problems. Springer.