A structured approach to optimizing offshore oil and gas production with uncertain models

Abstract

(Freely downloadable from sites.google.com/site/steinarelgsaeter/)
Optimizing offshore production of oil and gas has received comparatively little attention despite the large scale of revenues involved. The complexity of multiphase flow means that any model for use in production optimization must be fitted to production data for accuracy, but the low information content of production data means that the uncertainty in the fitted parameters of any such model will be significant. Due to costs and risk the information content in production data cannot be increased through excitation unless the benefits are documented.

A structured approach is suggested which iteratively updates setpoints while documenting the benefits of each proposed setpoint change through excitation planning and result analysis. In simulations on an analog which mimics a real-world oil field and its typical low information content data the approach is able to realize a significant portion of the available profit potential while ensuring feasibility despite large initial model uncertainty.

Full text

A structured approach to optimizing oshore oil
and gas production with uncertain models
Steinar M. Elgsæter, Norwegian University of Science and Technology,
O.S Bragstads plass 2D, N-7491 Trondheim,Norway
Olav Slupphaug, ABB, Ole Deviks vei 10, N-0666 Oslo, Norway
Tor Arne Johansen, Norwegian University of Science and Technology,
O.S Bragstads plass 2D, N-7491 Trondheim, Norway
Computers and Chemical Engineering, Vol. 34, pp. 163-176, 2010
Abstract
Optimizing oshore production of oil and gas has received comparatively lit-
tle attention despite the large scale of revenues involved. The complexity of
multiphase ow means that any model for use in production optimization must
be tted to production data for accuracy, but the low information content of
production data means that the uncertainty in the tted parameters of any
such model will be signicant. Due to costs and risk the information content in
production data cannot be increased through excitation unless the benets are
documented.
A structured approach is suggested which iteratively updates setpoints while
documenting the benets of each proposed setpoint change through excitation
planning and result analysis. In simulations on an analog which mimics a real-
world oil eld and its typical low-information content data the approach is able
to realize a signicant portion of the available prot potential while ensuring
feasibility despite large initial model uncertainty.
Keywords: Uncertainty, Production optimization, Parameter estimation, Oil
and gas production, Excitation Planning, Result Analysis
1 Introduction
The potential for optimizing oshore oil and gas production may be signicant
as small increases in prots in relative terms may translate into large gains
due to scales of revenues involved (Elgsaeter, Slupphaug and Johansen, 2008
a
),
and as this topic has received little attention compared to optimization and
control of downstream processing facilities. Both modeling and measuring the
oshore production of oil and gas present challenges for optimization. Exploiting
the information in what measurements are available to t production models
is complicated by the low information content of production data (Elgsaeter,
Slupphaug and Johansen, 2007). Practitioners in the oil and gas industry are
risk averse as the scale of revenues mean cost and risks of implementing changes
1

in setpoints can be signicant. An approach to optimization which can quantify
these expected benets may have increased likelihood of industry acceptance.
This paper proposes a structured approach to optimizing production which takes
the nature of models, measurements and data into account, while the expected
monetary benets of each setpoint change are quantied.
Production
in the context of oshore oil and gas elds, can be considered
the total output of production wells, a mass ow with components including
hydrocarbons, in addition to water,
CO2
,
H2S
, sand and possibly other com-
ponents. Hydrocarbon production is for simplicity often lumped into oil and
gas. Production travels as multiphase ow from wells through ow lines to a
processing facility for separation. Water and gas injection is used for optimizing
hydrocarbon recovery of reservoirs. Gas lift can increase production to a certain
extent by increasing the pressure dierence between reservoir and well inlet.
Production is constrained by several factors, including: On the eld level,
the capacity of the facilities to separate components of production and the
capacity of facilities to compress gas. The production of groups of wells may
travel through shared ow lines or inlet separators which have a limited liquid
handling capacity. The production of individual wells may be constrained due to
slugging, other ow assurance issues or due to reservoir management constraints.
Multiphase ows are hard to measure and are usually not available for indi-
vidual ow lines in real-time, however measurements of total single-phase pro-
duced oil and gas rates are usually available, and estimates of total water rates
can often be found by adding dierent measured water rates after separation.
To determine the rates of oil, gas and water produced from individual wells,
the production of a single well is usually routed to a dedicated test separator at
intervals where the rate of each separated single-phase component is measured,
a
well test
. Well tests allow biases in models of individual wells to be updated,
and well tests which measure rates for dierent settings,
multi-rate
well tests,
also allow responses predicted by models to be validated.
In the production of oil and gas it is common to divide the task of op-
timization into subproblems on dierent time scales to limit complexity, and
to consider separately
reservoir management
, the optimization of reservoir in-
jection and drainage on the time scales of months and years, and
production
optimization
, the maximization of prot from the daily production of reservoir
uids (Saputelli, Mochizuki, Hutchins, Cramer, Anderson, Mueller, Escorcia,
Harms, Sisk, Pennebaker, Han, Brown, Kabir, Reese, Nuñez, Landgren, McKie
and Airlie, 2003). Reservoir management typically species constraints on pro-
duction optimization to link these problems.
Production optimization requires
production models
, equations which ex-
press the relationship between change in decision variables and resulting change
in production. Production models are usually nonlinear to describe signicant
nonlinear phenomena in production. Parameter estimation is adjustment of
tted parameters so that predictions of the production model match a set of re-
cent historical production data as closely as possible. Parameters of production
models should be tted to production data through parameter estimation to
compensate for un-modeled eects or disturbances and to set reasonable values
for physical parameters which cannot be measured directly or determined in
the laboratory. Erosion of production chokes is an example of a typical such
un-modeled disturbance. The term
information content
in this paper refers
the amount in variation in decision- and disturbance variables observed in pro-
2

duction data. Fitted parameters may be uncertain if tted to data with low
information content, as many parameter estimates may t data equally well for
a given model. If there is sucient information content in data a unique param-
eter estimate can be found robustly which matches data better than all other
estimates. A previous study concluded that uncertainty in tted parameters is
likely to result when production models are tted to production data describ-
ing normal operations, due to low information content (Elgsaeter et al., 2007).
Planned excitation
is variation in decision or disturbance variables introduced
deliberately for the purpose of exposing some aspect of production and reducing
parameter uncertainty. The concepts of information content and excitation used
in this paper are motivated by the eld of system identication (Ljung, 1999).
As planned excitation in the oshore production of oil and gas requires a tem-
porary reduction in production, planned excitation has a signicant associated
cost and completely eliminating parameter uncertainty may therefore not be
cost-eective.
As a consequence of the complexity of the process considered, of measure-
ment diculties, and of the low information content in data, production models
may be subject to signicant uncertainty that can be reduced against a cost or
not at all. As a result more than one setpoint can be the plausible optimum,
which raises several practical challenges. One challenge is that the result of a
setpoint change is uncertain, it may even be negative. Another challenge is that
ensuring that an implemented setpoint is
feasible
, i.e. that it obeys production
constraints, is non-trivial in the presence of uncertainty. A third challenge is
that when uncertainty can be reduced at a cost, some method of determining
how and when uncertainty reduction is cost-eective may be required.
The approach to optimization with uncertain models is motivated by cur-
rent practice in the optimization of oshore oil and gas production. In the
current practice, production optimization begins by implementing well tests, a
form of planned excitation. Which wells and when to test are chosen based
on the subjective insight of practitioners. Production models are updated by
tting against the most recent well test. An target setpoint is then calculated
by mathematical programming. If the target setpoint is expected to increase
prots signicantly, a setpoint change is implemented. The target setpoint is
not implemented instantaneously, instead production is moved toward the tar-
get setpoint by an operator in a series of smaller steps to limit transients and
to ensure feasibility. This type of gradual setpoint change to ensure feasibility
post-optimization will be referred to as an
operational strategy
in this paper.
Production optimization therefore involves excitation planning, model updat-
ing, risk analysis and operational strategy, and current industry practice does
not explicitly consider uncertainty in any of these steps. This is paradoxical
as the state of production data and complexity of multiphase ow dictate that
uncertainty will be signicant in each of these steps. The contribution of this
paper is to suggest how to uncertainty can be accounted for in a structured
manner in each of these steps.
3

1.1 Prior work
1.1.1 Modeling production and quantifying model uncertainty
Quantifying uncertainty in reservoir models has attracted much interest in re-
cent years. Authors have described tting multiple models to data (Griess,
Diab and Schulze-Riegert, 2006), describing measurement uncertainties (Little,
Fincham and Jutila, 2006) and sensitivity analysis using prior knowledge of pa-
rameter uncertainty (Costa, Schiozer and Poletto, 2006). Very few references
have been found which attempt to quantify uncertainty in models for produc-
tion optimization, which seems paradoxical as production models are tted to
much the same production data as reservoir models.
Monte-Carlo methods
are a class of computational algorithms, suitable for the
study of physical or mathematical systems with random or uncertain proper-
ties, which sample a probability distribution using a pseudo-random number
generator with uniform probability and observe the fraction of the numbers
obeying some property or properties (Metropolis and Ulam, 1949). A Monte-
Carlo method for planning single-rate well tests under uncertainty was explored
in (Bieker, Slupphaug and Johansen, 2006).
Bootstrapping
is a Monte-Carlo
type method designed to estimate the uncertainty in parameters tted to a set
of data through regression (Efron and Tibshirani, 1993).
Elgsaeter, Slupphaug and Johansen (2008
b
) considered a method for modeling
production for production optimization based on local valid linear and non-
linear stationary models, motivated by the concepts of system identication.
Bootstrapping methods were used to nd a large number of multiple parameter
estimates which describe the production data equally well. This set of multiple
parameter estimates expresses the consequences of low information content on
the tted parameters for the model structure chosen. The magnitude of esti-
mated parameter uncertainty of even simple models was found to be signicant.
Although this paper considered black-box type system identication models,
bootstrapping can be applied regardless of the choice of model, as bootstrap-
ping is a computational approach to quantifying uncertainty which makes no
assumptions about the chosen model structure. Elgsaeter et al. (2008
a
) esti-
mated the lost potential of production optimization caused by uncertainty in
tted parameters to be several percent of revenue for the case of a North Sea oil
and gas eld, based on scenario simulations. These earlier ndings motivate the
focus of this paper on handling quantied parameter uncertainty in production
optimization.
1.1.2 Optimization under uncertainty
Production optimization can be seen as a form of
real-time optimization system
,
a closed-loop controller which attempts to locate the optimal setpoint through a
series of steps, each consisting of a smaller setpoint change. Real-time optimiza-
tion systems, originally developed for chemical processes, consider an economic
objective function maximized subject to a rigorous steady-state nonlinear pro-
cess model and process constraints. Real-time optimization is a form of model-
based control, and obtaining the process model is considered the single most
dicult and time-consuming task in the application and maintenance of model-
based control and optimization (Andersen, Rasmussen and Jorgensen, 1991)
(Terwiesch, Agarwal and Rippin, 1994) (Zhu, 2006).
4

Some parameters in process models are typically tted to data in real-time
optimization. Ogunnaike (1995) has suggested that modeling for control should
be based on criteria related to the actual end use, and that tting equations
to data may be inadequate in the context of controller design. A constraint on
the maximum setpoint change is often applied in real-time optimization, and
smaller changes in setpoints implemented iteratively with re-identication and
re-optimization between each step, a
two-step approach
.
Uncertainty in real-time optimization falls into four main categories (Zhang,
Monder and Forbes, 2002):
market uncertainty
, the imprecise knowledge of
process economics,
process uncertainty
, the imprecise knowledge of operation
due to process disturbances or uncertain inputs,
measurement uncertainty
, the
imprecise knowledge of measured process variables due to sensor or transmission
errors, and
model uncertainty
, plant/model structural and parametric mismatch.
Explicit analysis of uncertainty has gained some attention in control design,
the methods of
robust control
address the problem of designing linear multivari-
able controllers that adhere to some robust stability and performance criterion
(Skogestad and Postlethwaite, 1996). Ensuring that a process is designed with
sucient exibility to operate under changing and uncertain uncertain parame-
ters represented by probability distributions has been suggested, usually under
the assumption of linear models and normal distributions, see for instance Pis-
tikopoulos and Mazzuchi (1990).
Approaches to handling uncertainty in optimization can be divided into
stochastic optimization, sensitivity analysis, back-o and robust optimization
methods.
Stochastic optimization
attempts to directly solve a problem given
uncertainty by formulating the stochastic optimization problem in determinis-
tic form using probabilities (Kall and Wallace, 1994). The
sensitivity analysis
approach consists of augmenting the objective function of the optimization prob-
lem with a penalty term intended to minimize parametric sensitivity (Becker,
Hall and Rustem, 1994). The
back-o
method adds a vector constraint on deci-
sion variables to ensure that operating points are feasible, and the back-o vector
is computed at intervals (Loeblein, Perkins, Srinivasan and Bonvin, 1999).
Ro-
bust optimization
optimizes the expected value for a chosen performance index
for a given level of risk, formulated in terms of worst-case, mean-variance and so
forth (Darlington, Pantelides, Rustem and Tanyi, 1999) (Mulvey and Vander-
bei, 1995). All these approaches introduce conservativeness to ensure feasibility
rather than exploiting measurements postoptimization.
Optimization methods which attempt exploit measurements fall into the
two main categories: modied two-step approaches and approaches which do
not update process models. Roberts and Williams (1981) suggests a modied
two-step approach which add a gradient modication term to the cost function
of the optimization problem to ensure that iterates converge to a point at which
the necessary conditions for optimality are satised. Methods which do not
require model updating can be classied into model-free and model-xed meth-
ods. Some model-free methods mimic various iterative numerical optimization
algorithms, such the Nelder-Mead simplex algorithm (Box and Draper, 1969).
Other model-free methods recast the nonlinear programming problem into a
problem of choosing decision variables whose optimal values are approximately
invariant to uncertainty.
Self-optimizing control
considers determining which
variables to keep at constant setpoints to keep the process acceptably close
to optimum (Skogestad, 2000).
Extremum-seeking control
are adaptive control
5

methods related to self-optimizing control which attempt to move process set-
points toward values which result in an extreme value (maximum or minimum)
of a measured output without a process model. Extremum-seeking control usu-
ally introduces ample excitation in setpoints to achieve this goal, for instance in
form of a low-frequency sinusoidal reference (Krstic, 2000).
NCO-tracking
is a
model-free method which uses o-line analysis of the process model to determine
functions for setpoint values at which the necessary conditions for optimum are
enforced, parameterized in terms of measured variables (François, Srinivasan
and Bonvin, 2005).
Fixed-model methods
utilize both the available measure-
ments and a process model for guiding the iterative scheme toward an optimal
operating point, but update constraint and cost function rather than model at
each iteration (Forbes and Marlin, 1994). Methods such as extremum-seeking or
those which mimic iterative numerical optimization algorithms introduce many
variations to setpoints. These methods may be unsuitable to the oshore pro-
duction of oil and gas where the costs and risk of each change in setpoint may
be signicant. So-called model-free methods still require models in the design
of controllers, and obtaining accurate models of the oshore production of oil
and gas oine may be challenging.
There is some precedence for analyzing the risk of implementing a setpoint
change. Miletic and Marlin (1998) have proposed
result analysis
using multi-
variable statistical hypothesis testing to determine whether the predicted in-
crease in prot from implementing changes in setpoints is statistically signicant
or a result of process noise.
A classic theoretic approach to uncertainty reduction is
dual control theory
,
which deals with the controller design for processes which are initially unknown
(Fel'dbaum, 1961
a
) (Fel'dbaum, 1961
b
). The theory is called dual as the ob-
jectives of such a controller are twofold, rstly to control the system as well as
possible based on current system knowledge, secondly to experiment with the
system so as to better learn how to control it in the future. The problem of
determining uncertainty reduction is also related to the eld of
reinforcement
learning
, an area of machine learning concerned with how an agent ought to take
actions in an environment so as to maximize some notion of long-term reward
(Sutton and Barto, 1998), focused on on-line implementation while making a
tradeo between
exploration
of uncharted territory and
exploitation
of current
knowledge. Reinforcement learning has seen some application in modeling of
batch and semi-batch chemical reactors (Martinez, 2000).
Optimal experiment
design
for control has focused on deriving an input signal that minimizes some
control-oriented measure of plant/model mismatch under a constraint on total
input power. Optimal experiment design is usually performed with the aim of
achieving control that is robust to disturbances (Gevers and Ljung, 1986).Yip
and Marlin (2003) suggested including excitation planning in real-time opti-
mization and to weight the cost of the excitation against benet, under the
assumption that the model parameters are initially known and that the occur-
rence of a disturbance which may necessitate excitation to re-t parameters can
be identied by measurements.
1.2 Problem formulation
The aim of this paper is to suggest and study a structured approach to optimiz-
ing the oshore production of oil and gas with uncertain production models. An
6

iterative two-step approach to optimization combined with post-optimization
feasibility assurance through an operational strategy is suggested. Bootstrap-
ping methods are used to quantify parameter uncertainty, which is exploited for
excitation planning and result analysis under uncertainty based on multivariate
Monte-Carlo-like methods.
The suggested approach is outlined in Section 2, a simulation case-study is
described in Section 3 before conclusions are drawn.
2 Production optimization under uncertainty
2.1 Casting uncertainty in production optimization in math-
ematical terms
This section suggest how uncertainty resulting from low information content
in production data can be cast in mathematical terms to allow a structured
treatment.Let
x
be a vector of internal variables and let
u
be a vector of decision
variables. This paper will consider a production optimization problem on the
form
[ˆu(θ) ˆx(θ)]= arg max
u,x M(x, u, d)
(1)
s.t 0 = f(x, u, d, θ)
(2)
0≤c(x, u, d),
(3)
where
θ
is a vector of parameters to be determined.
ˆu(θ)
and
ˆx(θ)
are the
optimal solution of (1)-(3) for a given
θ
.
d
is a vector of modeled and measured
disturbances independent of
u
.
M(x, u, d)
is a prot measure which is to be
maximized subject to a process model (2) and process constraints (3) for a
given parameter value
θ
. For oshore oil and gas production,
x
may be the
production rates of each uid from each well,
u
may be relative valve openings,
c
may describe constraints in total water and gas processing capacity and
M
may express total oil production.
In this paper we will consider response surface or performance curve
type production models (2), stationary, nonlinear, locally valid equations which
express production rates of each well in terms of gas-lift rates, production choke
openings, the most recent well test and tted parameters.
As production data are often local in nature, i.e. setpoints are only varied within
a narrow range of values, a model tted to data may only be
locally valid
, only
accurately able to describe production for a narrow range of setpoints. The
local nature of data and of any model tted to such data can be accounted for
by enforcing
max{umin, u0−Uprc ·U} ≤ u≤min{umax , u0+Uprc ·U}
(4)
when solving (1)(3) and iteratively re-updating the production model and re-
optimizing, an two-step approach.
U
denes the scale elements of
u
,
u0
is the
initial value of the decision variable,
Uprc <1
is a design parameter which limits
the magnitude of setpoint change at each step, and
umin, umax
are minimum
and maximum values of
u
, respectively.
This paper only considers
instantaneous
optimization, determining
(ˆu(θ),ˆx(θ))
7

at the current time.Let
y
be the vector of production measurements,
y
may
for instance be measured total rates of oil, gas and water.Let
ˆy(u, d, θ)
be an
estimate of process measurements based on the model (2). Let the
tuning data
set
be a set of historical process data spanning
N
time steps spanning the time
interval
t∈[1, N ]
ZN=[¯y(1)] ¯
d(1) ¯u(1) ¯y(2) ¯
d(2) ¯u(2) . . .
¯y(N)¯
d(N) ¯u(N)]
(5)
with residuals
ϵ(t, θ) = ¯y(t)−ˆy(u(t), d(t), θ)∀t∈ {1, . . . , N }.
(6)
Parameter estimation determines
ˆ
θ
by minimizing the sum of the squared resid-
uals for the tuning data set:
ˆ
θ= arg min
θ
N
∑
t=1
w(t)∥ϵ(t, θ)∥2
2+Vs(θ),
s.t.
cθ(θ)≥0
(7)
where
w(t)
is a user-specied weighting function,
Vs(θ)
is an optional soft-
constraint and
cθ(θ)
an optional constraint on
θ
. The process model (2) should
only be tted to historical process data that are
recent
, in the sense produc-
tion during the time interval spanned by the tuning set should be consistent
with current production and the production model. It may be impossible to
determine an accurate estimate
ˆ
θ
with (7) when the information content in
ZN
is low.
Extending the model structure to better describe more long-term eects
may enable the use of a longer tuning set, which may in turn reduce param-
eter uncertainty. Therefore parameter uncertainty is linked to other forms of
uncertainty such as structural uncertainty.
Let the
potential for production optimization
Po
be the increase in prot at-
tainable if the production was moved from the initial operating point
(x0, u0, d0)
to the globally optimal operating point
(x⋆(d0), u⋆(d0), d0)
:
Po
△
=M(x⋆(d0), u⋆(d0), d0)−M(x0, u0, d0)≥0.
(8)
When implementing
ˆu(θ)
Some of the potential
Po
may be remain unrealized
due to uncertainty, which motivates
potential of production optimization(
Po
)
△
=
realized potential (
Po,r
)
+
lost potential due to uncertainty (
Lu
)
.
(9)
By repeatedly solving (7) with bootstrapping methods, uncertainty in
ˆ
θ
re-
sulting from low information content in
ZN
can be quantied. In the inuence
of uncertainty in
ˆ
θ
on setpoints
ˆu(ˆ
θ)
and prots
M
can be assessed by Monte-
Carlo simulations of (1)(3). This idea was explored for estimation of
ˆ
Lu
in
Elgsaeter et al. (2008
a
).
8

2.2 Motivation for the chosen structured approach to op-
timization with uncertain models
The strategy considered in this paper, motivated by the above discussion, is
outlined in Figure 4. This paper considers the operational strategy explicitly as
a component of a structured approach to handling uncertainty, as such strate-
gies are expected to add robustness against some of the uncertainty considered.
Operational strategies can be considered a form of uncertainty handling through
feedback.
This paper considers planning of excitation in a single decision variable, which
will reduce only some of the present parameter uncertainty. The cost of such
planned excitation is associated with its resulting temporary reduction in pro-
duction and can be estimated in a fairly straightforward manner. The benet of
a planned excitation is much harder to quantify in advance, as it will depend on
the parameter value found, on the inuence of this updated parameter value on
the target setpoint found through production optimization, and on subsequent
implementation of the operational strategy while uncertainty in other parame-
ters still persist. This paper considers estimating the
benet of planned excita-
tion
, the marginal increase in prots that can be expected when re-optimizing
after the planned excitation, and this benet is estimated through stochastic
simulations. Not all reduction in parameter uncertainty will be protable, for
instance it should be intuitively clear that eliminating uncertainty against which
the operational strategy is robust has zero benet.
As the change in prots that result from implementing setpoint change toward
a target is uncertain and can even be negative, this paper considers basing the
choice of whether to implement setpoint change on the distribution of simulated
prot changes found through stochastic simulations of the operational strategy.
The choice of relating decisions to prots both in result analysis and excitation
planning is motivated by a desire to make the cost of uncertainty visible in
business terms as this is what ultimately drives industry decision making.
Stochastic simulations were chosen as the considered method for their concep-
tual simplicity, as they can be applied to a wide variety of operational strategies
without introducing simplifying assumptions, and as the frequency of decision
making in production optimization on the order of days or weeks allows compu-
tationally intensive methods such as these to remain practical. A formalization
of an operational strategy is discussed in detail in Section 2.3, result analysis in
Section 2.4, and excitation planning in Section 2.5.
2.3 Operational strategy
Example 1 illustrates how an operational strategy may operate.
Example 1
Consider a hypothetical oshore oil and gas eld, producing from
two wells with
u=[u1u2]T
, one decision variable aecting the production of
each well. Each well produces oil and water at rates
x=[q1
oq1
wq2
oq2
w]T
which depend nonlinearly on
u
, and the objective is to maximize total oil produc-
tion
qtot
o=q1
o+q2
o
while the total rate of produced water
qtot
w=q1
w+q2
w
must not
exceed a constraint. Suppose that a target setpoint has been determined by solv-
ing the production optimization problem. Suppose that at the current setpoint,
feasible and infeasible regions are as illustrated in Figure 1. The boundary be-
9

tween the feasible and infeasible regions is the set of setpoints at which all water
processing capacity is utilized, and parameter uncertainty may cause the posi-
tion of this boundary and the optimal setpoint to be uncertain. To ensure that
the implemented setpoint remains in the feasible region, an operational strategy
could alternately decrease
u1
and increase
u2
in smaller steps as illustrated in
Figure 1.
In this section the concepts illustrated in Example 1 are generalized and an op-
erational strategy is described formally for the purposes of analysis.In practice
a human operator would perhaps not be able to stringently reproduce the op-
erational strategy as stated in this paper, instead the algorithm could be used
more as a guideline.
The operational strategy attempts to implement changes in
u
toward a tar-
get
uT
by sequentially increasing and decreasing elements of
u
and monitoring
responses in prots and constraints.
uT
may be any desired value of
u
, for in-
stance that found by solving the production optimization problem numerically.
The operational strategy is intended as a postoptimization strategy for the
implementation of the new desired setpoint, and no production optimization or
parameter estimation is performed during the course of the operational strategy.
It is assumed that the sign of the response in prots and constraint utilization
to a change is setpoints is known. This will usually be the case in oshore oil
and gas production, where decision variables may for instance be choke open-
ings or gas-lift rates and the response to an increase in
u
is usually positive for
the range of setpoints considered. Let
U−
be the set of indices corresponding
the components of
uT−u0
which are negative, and let
U+
be the indices of
the components of corresponding the components of
uT−u0
which are positive.
uT
does not have to be feasible in practice, the operational strategy will ensure
feasibility of implemented setpoints.
The operational strategy should have a dened criterion for when to termi-
nate. If the termination criterion considers only the sign of prot change, the
operational strategy may not be able to proceed if the initial setpoint is close to
a local optimum. However, if the termination criterion considers the dierence
between monitored and predicted prot, the operational strategy can proceed
in the face of prot decrease, as long as this decrease is in reasonable agreement
with the model used in optimization. If prots decrease below the predicted val-
ues then the decision to terminate should weigh the magnitude of this decrease
against the predicted prot increase at the target. The rst scenario illustrated
in Figure 2 shows how a prot decrease
is
predicted by the model and it is
reasonable for the operational strategy to continue, while in the second scenario
the prot decrease
is not
predicted by the model and it is reasonable for the
operational strategy to terminate and return to the initial setpoint. These con-
siderations can be formulated in terms of estimated and measured prot change
and evaluated by comparing a cost function
JM
against a threshold value
JT
M
.
An operating strategy which is initiated at time
to
s
and terminated at time
to
e
, increases the realized potential by:
∆Po,r
△
=Po,r(to
s)−Po,r(to
e),
(10)
and in the section below it is suggested how a probability density function
f∆P r(∆ ˆ
Po,r)
can be found prior to implementing the operational strategy. The
10

operational strategy can continue altering setpoints beyond
uT
as long as each
step of the operational strategy causes increased prot. As the operational
strategy may overshoot the most protable setpoint, the nal step should be to
assess all steps and return to the most protable one. The suggested operational
strategy is outlined in Algorithm 1.
Algorithm 1 (An operational strategy)
Given a process that is initially at
a feasible setpoint
(u0, d0)
and given measured process constraints
¯c
, measured
prot
¯
M
and decision variables
¯u
, a target value
uT
and probability distributions
f∆P r(∆ ˆ
Po,r)
, let
0< S ≤1
denote step size.
1. Let the measured prot prior to implementing the operational strategy be
¯
M0
. Let
k
be the index of the current step, and set
k= 0
initially.
2. Determine sets
(U−, U +)
to correspond with
uT−u0
.
3. Repeat (3a),(3b),(3c)
(a) k = k+1
(b) decrease
ui, i ∈U−
by an amount proportional to
ui
T−ui
0, i ∈U−
,
so that
ui
k=ui
k−1−S·(ui
T−ui
0), i ∈U−
,
(c) increase
ui, i ∈U+
by an amount proportional to
ui
T−ui
0, i ∈U+
while observing
¯c
and
¯
M
, as long as all elements of
¯c
obey
¯c < 0
and
∂¯
M
∂u >0
, let the resulting measured prot be
¯
Mk
, and the resulting
setpoint
uk
,
while (
∥uk−u0∥>∥uT−u0∥
and
¯
Mk≥¯
Mk−1
) or (
∥uk−u0∥<∥uT−u0∥
and
JM(¯
Mk, fP(∆ ˆ
Po,r)) > J T
M
).
4. Implement the setpoint which resulted in the highest measured prot in
steps 1-3.
One of the operators' tasks is to ensure that the operational constraints are
obeyed, therefore it is reasonable to assume that
(u0, d0)
is feasible in Algo-
rithm 1. It is assumed that in practice cases of infeasible
(u0, d0)
are handled
operationally, i.e. an operator will apply process knowledge instead of mathe-
matical optimization to move the process to a feasible setpoint.
The magnitude of
S
is a user preference and preferably
S << 1
. The magni-
tude of
S
will also depend on how often production optimization is implemented.
Smaller
S
may result in an operational strategy which is more labor-intensive to
implement, but may in some cases cause the operational strategy to terminate
closer to optimum.
2.4 Result analysis
This section aims to investigate whether an estimate of the parameter uncer-
tainty can be exploited in result analysis. The approach chosen is simulating the
prot change
∆Po,r
that results from implementing a setpoint change suggested
by production optimization on the production model with dierent parameter
estimates drawn from the estimated parameter distribution
fθ(ˆ
θ)
in a Monte-
Carlo manner. The approach is outlined in Algorithm 2
11

Algorithm 2 (Estimating
f∆P r(∆ ˆ
Po,r)
)
Given an operational strategy, the
current operating point
(x0, u0, d0)
,
Uprc
and probability distributions
fθ(ˆ
θ)
and
fu(ˆu)
,
•
repeat
Nt
times

draw a sample
ˆ
θt
from
fθ(ˆ
θ)
,

obtain a point estimate of
∆Po,r
by simulating the operational strategy
running on a process described by the process model with negligible
structural model uncertainty and with parameters equal to
ˆ
θt
, while
enforcing the constraint (4).
•
The distribution of
Nt
point estimates
∆ˆ
Po,r
is an estimate of
f∆P r(∆ ˆ
Po,r)
.
As a copy of the production model cannot account for structural uncertainty
which will grow with the magnitude of
u−u0
, a constraint of the form (4) should
be enforced in simulations. The magnitude of estimates of
∆ˆ
Po,r
will depend on
the choice of
Uprc
, but a conservative choice of
Uprc
should yield conservative
estimates
ˆ
Po,r
on which result analysis could be based.
The aim of the result analysis is to determine whether to implement the
setpoint change suggested by production optimization. This decision could be
based on the evaluation of a cost function which depends on
f∆P r(∆ ˆ
Po,r)
:
Jr(f∆P r(∆ ˆ
Po,r)) ≥JT
r.
(11)
Jr(f∆P r(∆ ˆ
Po,r))
could for instance be the expected or worst-case value. The
threshold
JT
r
should ensure that setpoint changes are only performed when the
resulting increase in prots is expected to be signicant and to justify the risks
and operational costs which are associated with any setpoint change, such as
the risk of triggering an unplanned shutdown.
2.5 A cost/benet approach to excitation planning
The value of implementing a planned excitation prior to implementing produc-
tion optimization is that the excitation may reduce model uncertainty, allowing
production optimization and the operational strategy to nd a more protable
setpoint than otherwise possible. This paper investigates exploiting estimates
of parameter uncertainty
fθ(ˆ
θ)
found from recent historical production data to
form the basis of a structured approach to rank the benet of exciting dierent
decision variables. For simplicity this discussion is restricted to wells which are
decoupled
, i.e. where implementing change in the production of one well will not
inuence the production of other wells, which is often a reasonable assumption
for so-called platform wells as production from such wells travel through separate
owlines and join rst at production manifold. The discussion is restricted to
excitation of single decision variables, although simultaneously exciting several
variables is also conceivable.
The benet of excitation
BE
is proposed dened as the dierence between
the prot which would be realized if no excitation is performed,
∆Po,r
, and the
prot which would be realized after implementing an excitation,
∆PE
o,r
:
BE△
= ∆PE
o,r −∆Po,r.
(12)
12

The idea of the benet of excitation is illustrated in Example 2.
Example 2
Consider a hypothetical eld producing oil and water from two
decoupled wells. Well 1 produces
(q1
o, q1
w)
at rates which depend on
u1
, well
2 produces
(q2
o, q2
w)
at rates which depend on
u2
. The relationships between
(q1
o, q1
w)
and
u1
and between
(q2
o, q2
w)
and
u2
are uncertain, but a model which ex-
presses
ˆq1
o(u1, θ1),ˆq2
o(u2, θ2),ˆq1
w(u1, θ1),ˆq2
w(u2, θ2)
is available. Models for well
1
depend on parameters
θ1
and models for well
2
on
θ2
. Two parameter es-
timates
ˆ
θa=[ˆ
θ1
aˆ
θ2
a]
and
ˆ
θb=[ˆ
θ1
bˆ
θ2
b]
have been determined which result
in estimates of
qtot
o=q1
o+q2
o
and
qtot
w=q1
w+q2
o
which match measure-
ments equally well for a set of recent historical data. Optimization of the
objective
M= ˆq1
o(u1, θ1) + ˆq2
o(u2, θ2)
subject to the model and the constraint
ˆq1
w(u1, θ1) + ˆq2
w(u2, θ2)< qmax
w
is desired. If no excitation is implemented the
target
uT= ˆu(ˆ
θa)
is planned implemented using an operational strategy. If an
excitation could be performed which would reveal which of
ˆ
θa
and
ˆ
θb
better de-
scribe production, what would be the benet of that excitation?
If
ˆ
θa
describes production better
BE= 0
as
∆PE
o,r = ∆Po,r
. If
ˆ
θb
describes
production better
BE
will depend on how much prot increase the operational
strategy would be able to implement if given the target
uT= ˆu(ˆ
θb)
as opposed to
uT= ˆu(ˆ
θa)
. In the case that
ˆ
θb
describes production better while
uT= ˆu(ˆ
θa)
,
BE
could be found through simulations on the production models, provided that
structural model errors are small.
Example 2 illustrates that
BE
will depend on the interaction between uncer-
tainty in
ˆ
θ
, uncertainty in
ˆu(ˆ
θ)
and the ability of the operational strategy to
compensate for uncertainty in
ˆu(ˆ
θ)
. The benet of excitation will also depend
on what target the operational strategy will pursue if no excitation is performed.
In the oshore production of oil and gas, exciting all components of
u
is usu-
ally not feasible, so some method for ranking components of
u
to be excited is
desirable, as is illustrated in Example 3.
Example 3
Consider again the eld described in Example 2. Consider that
our aim is to choose whether to introduce excitation in
u1
, in
u2
or not at
all. Assume that an excitation of
u1
would allow us to distinguish between
ˆ
θ1
a
and
ˆ
θ1
b
, while
ˆ
θ2
would still be uncertain, and that an excitation of
u2
would
allow us to distinguish between
ˆ
θ2
a
and
ˆ
θ2
b
while
ˆ
θ1
would still be uncertain.
With the assumptions made it can be expected that an excitation of well 1 will
either result in implementing
uT= ˆu([ˆ
θ1
a,ˆ
θ2
a])
if production is described by
ˆ
θa
, or
implementing
uT= ˆu([ˆ
θ1
b,ˆ
θ2
a])
if production is described by
ˆ
θb
, as it was planned
to implement
uT= ˆu([ˆ
θ1
a,ˆ
θ2
a])
if no excitation was performed, and
ˆ
θ2
would still
be uncertain after excitation of
u1
. The outcome on the production models can
be simulated in both cases provided that structural model errors are small, which
would give two estimates of
ˆ
BE
. A structured approach to planning excitation
could then be to excite the well associated with the highest positive average
ˆ
BE
.
Examples 2 and 3 illustrate the principles of a Monte-Carlo simulation approach
to estimating
BE
for the simplest possible case of two wells, two parameter es-
timates and one constraint. In our case study the same principles are applied to
estimate
ˆ
BE
for a larger number of wells, constraints, and parameter estimates
ˆ
θ
.
13

Let
ˆ
θn
be the
nominal parameter estimate
, the parameter estimate found
by solving (7) directly. The estimation of
BE
will consider the case when the
operational strategy attempts to implement the target
uT= ˆu(ˆ
θn)
if no exci-
tation is performed. Note that although production of wells are decoupled, the
components of
ˆu(θ)
as calculated by (1)(3) are coupled, so updating models
describing one well may inuence suggested settings for other wells, and this
will need to be reected when estimating
ˆ
BE
. Simulations on the production
model assume that structural model uncertainty is negligible, and, as locally
valid models are considered, this assumption is reasonable as long as change
in decision variables is small, which motivates enforcing the constraint
(4)
in
simulations, and which should result in conservative estimates
ˆ
BE
.
The approach considered is outlined in Algorithm 3.
Algorithm 3 (Benet of excitation,
ui
)
Given that it is planned to imple-
ment
ˆu(ˆ
θn)
if no excitation is performed. Given
Nt
, the distribution
fθ(ˆ
θ)
, an
operational strategy, a process model which depends on parameters
θ
,
Uprc
, and
that the aim is to estimate the benet of exciting
ui
, a single decision variable
in the vector
u
.
•
repeat
Nt
times

draw a sample
ˆ
θt
from
fθ(ˆ
θ)
.

determine
∆ˆ
Po,r(ˆ
θt)
by simulating the operational strategy using
uT=
ˆu(ˆ
θn)
on the process model with parameters
ˆ
θt
while setpoint change
is limited by (4).

let
ˆ
θi
t
be parameters of
ˆ
θt
excited by
ui
. Let
θE(ˆ
θn,ˆ
θi
t)
be a vector of
parameters that is equal to
ˆ
θi
t
for parameters excited by
ui
and equal
to
ˆ
θn
for all other parameters.

determine
∆ˆ
PE,i
o,r (ˆ
θt)
by simulating implementing the operational strat-
egy using
uT= ˆu(θE(ˆ
θn,ˆ
θi
t))
on the process model with parameters
ˆ
θt
while setpoint change is limited by (4).

let
ˆ
BE,i (ˆ
θt) = ∆ ˆ
PE,i
o,r (ˆ
θt)−∆ˆ
Po,r(ˆ
θt)
.
•
the distribution of simulated values
ˆ
BE,i
is an estimate of the probability
density function for the benet of exciting
ui
,
fE
B(ˆ
BE,i )
A structured approach to excitation planning could be to excite those decision
variables
ui
which have highest
ˆ
BE,i
on average.
As components of
ˆu(θ)
as calculated by (1)(3) are coupled, excitation of
ui
can cause
ˆu(θE(ˆ
θn,ˆ
θi
t))
to dier from
ˆu(ˆ
θn)
in components other than
ui
.
If uncertainty for non-excited wells is signicant, implementing
ˆu(θE(ˆ
θn, θi
t))
may inadvertently cause lower prots than
ˆu(ˆ
θn)
, and in such cases estimates
ˆ
BE,i
may be negative. Negative
ˆ
BE,i
is an indication that the prot increase
resulting from excitation of
ui
is variant to uncertainty in models describing
other wells.
Except during startup it will in most cases be reasonable to assume that
production is initially utilizing at least one processing capacity fully, so that
any excitation will require production to back o from full capacity for a period
of time, incurring a cost. It is conceivable to simulate the cost of excitation
CE
14

in a Monte-Carlo fashion for dierent samples of
fθ(ˆ
θ)
. To limit the scope of this
paper, estimation of
CE
will not be considered. It is proposed that candidate
excitations can be ranked by the value of a function
JE(fE
B(ˆ
BE), f E
C(ˆ
CE)≥JE,T ,
(13)
where
JE,T
is a user-specied threshold and
JE(fE
B(ˆ
BE), f E
C(ˆ
CE)
is a user-
specied metric such as for instance the expected value of the dierence
ˆ
BE−
ˆ
CE
.
3 Case study
This section describes the application of the suggested approach to a case-study
modeled on the North Sea oil and gas eld and the set of real-world production
data considered previously in Elgsaeter et al. (2007), Elgsaeter et al. (2008
b
)
and Elgsaeter et al. (2008
a
).
3.1 Field description
The case study considered in this paper is motivated by a North Sea oil and
gas eld with 20 gas-lifted platform wells producing predominantly oil, gas and
water. The eld has a layout as depicted in Figure 3, with one production
separation train and one test separator. Measurements of the total rates of
produced oil, gas and water are available. The operator of the eld requested
that all data be kept anonymous, therefore all variables will be presented in
normalized form. The production data are characterized by little variation
in decision variables. The aim of production optimization on the eld is to
distribute available lift gas so as to maximize total oil production while keeping
total produced water and gas rates below capacity constraints.
3.2 Method
The cost and risks of implementing a trial of the suggested approaches on an
actual eld are signicant, and it may be dicult to compare strategies by
implementation on an actual eld as production will vary with time due to
disturbances. This motivates the choice of studying the suggested approach in
simulations.
Due to uncertainty there may be many plausible descriptions of production,
and an approach to optimization under uncertainty should ideally perform well
for all such plausible descriptions. This paper will consider simulating optimiza-
tion on a model that is plausible in the sense that it conforms with production
data and refer to this model as the
production analog
. When optimizing pro-
duction, the production analog is considered unknown, production optimization
can only infer knowledge of the production analog through measurements.
When modeling the response to change in gas-lift rate, a rst-order linear
kernel with a single tted parameter for each uid and phase with
dim(θ) = 63
(formulation A) will be compared with a second-order nonlinear kernel with two
tted parameters for each uid and phase and
dim(θ) = 123
(formulation B).
15

By comparing these formulations the signicance of model structure and param-
eterization on the suggested methodology can be assessed. Physical knowledge
is included through constraints and regularization on parameter estimation.
Both for formulations A and B parameters were re-tted and production
re-optimized according to the workow outlined in Figure 4 in four iterations.
Algorithms 2 and 3 were run for each iteration with
Nt= 200
, but no exci-
tations were implemented and setpoint change was implemented regardless of
risk-reward estimates. For comparison this process was repeated for two choices
of the design parameter
Uprc = 0.2
and
Uprc = 0.5
.
Further details on the simulation case study are given in Appendix A.
3.3 Results
A comparison of the predictions of the production analog against eld data
is shown in Figure 5. The production analog is shown along with models of
formulation A and B at iteration 1 in Figures 6 and 7. The distribution of
ˆu
found through Monte-Carlo simulation prior to iteration 1 are compared for
the two models in Figure 8. Figure 9 compares the realized prots
Po,r
of
formulations A and B at each iteration, for
Uprc = 0.2
and
Uprc = 0.5
, indicating
that between
30%
and
80%
of the prot potential was realized.
Figure 10 illustrates the changes in setpoint implemented by the operational
strategy for formulation B. Figures 11 and 12 compare the simulated benet of
excitation with estimates found with Algorithm 3 for iterations 1 and 4. Figure
13 compares
f∆P r(∆ ˆ
Po,r)
found with Algorithm 2 with realized prot for each
iteration of formulation A and B. Figure 14 compares target setpoints before
and after excitation for a particular strategy, well and iteration.
3.4 Discussion
The suggested optimization method can be classied as a two-step approach
and the operational strategy as postoptimization feasibility assurance. In the
simulation case considered, model uncertainty was signicant, as illustrated in
Figures 6 and 7, and as a result uncertainty in the setpoint
ˆu(θ)
suggested by
production optimization was highly uncertain, as illustrated in Figure 8. De-
spite this signicant uncertainty, the suggested approach was able to realize a
signicant prot increase with either model formulation and with either choice
of
Uprc
, as is illustrated in Figure 9.
(1)(3) is a nonlinear optimization problem and convergence of its solution to-
ward the global optimum cannot be guaranteed in general unless (1)(3) can
be shown to be a convex optimization problem and is solved with methods
of convex optimization (Boyd and Vandenberghe, 2004). Analytical methods
(Ljung, 1999) can only guarantee that the solution of (7) is the best description
of production if
N
approaches innity, if the reservoirs, wells and processing
facilities can be considered stationary, and if
ZN
is suciently informative to
distinguish between all solutions of (7) . Typically bootstrap estimates are not
exact but have an inherent error, while the bias in estimates is often small the
variance can often be quite large due to the nite amount of data and the nite
number of resamples (Efron and Tibshirani, 1993).
For the reasons listed above, no guarantees can be given that the implemented
setpoint will converge toward global optimum in practice when the suggested
16

approach is applied to processes such as these with low information content
data and limits on the planned excitation. We argue that lack of guaranteed
convergence is a result of the properties of the process considered and not of the
proposed solution, and processes with these properties are nevertheless inter-
esting for academic study. It is because of the lack of guaranteed convergence
that this paper has focused on identifying setpoint change which increase prots
with some measure of condence.
The methods suggested would remain sensible in the special case that all the
requirements stated above are satised and a unique parameter estimate
ˆ
θ
can
be found from (7) and (1)(3) is able to return the globally optimum setpoint.
In this case, bootstrapping (7) would return the same parameter estimate for
all resamples, all estimates of the benet of excitation found with Algorithm 3
would be zero, and and Algorithm 2 would predict a single prot change
∆ˆ
Po,r
with high condence.
The operational strategy suggested will always ensure that production is feasible
as long as production is initially at a feasible setpoint, constraints are measured,
transients are negligible and as long as the sign of change in constraint utiliza-
tion to change in decision variables are known. Constraints were enforced at all
times during the simulation case study, an example is shown in Figure 10. The
assumption of negligible transients may be relaxed by expanding the operational
strategy method, for instance by model-predictive control.
Excitation planning and result analysis suggested employ computational, Monte-
Carlo-like methods. These methods can be considered a form of non-parametric
multivariate analysis, as a sample of the entire parameter vector
ˆ
θ
is drawn at
each iteration, and co-variance between parameters is therefore accounted for in
the analysis. The fact that no assumptions about linearity of models or types
of probabilities are made are a strength of the methods, and this trait alone
separates these methods from the majority of rival approaches found in the lit-
erature.
The method for excitation planning suggested in this paper deals with identify-
ing the decision variable which when excited will increase the prot attained by
production optimization the most. This is done by analyzing the inuence of
the modeled uncertainty on prot function by stochastic simulations. The sug-
gested method dier from optimum experiment design in that the object is not
to achieve control that is robust to disturbance, but rather to increase prots by
as much as possible by targeting which decision variable to excite. In this case
the actual benet of excitation could be computed, as shown in Figures 11 and
12, and a correlation between large mean
ˆ
BE
and large actual
BE
is visible.
This nding of the simulation case study supports the assertion that testing
wells with large mean
ˆ
BE
is a sensible excitation planning approach, provided
estimated benets are signicant compared to estimated costs. Further work
could consider estimating the benet of simultaneously exciting a small num-
ber of decision variables simultaneously, as such excitation could be designed to
cause less temporary reduction in production and hence incur lower costs.
The suggested method for result analysis uses stochastic simulations to de-
termine a distribution for the change in prot that will result from changing
decision variables toward a target by means of the operational strategy. While
methods have been found in the literature which assess the signicance of esti-
mated prot change to noise, no reference was found to result analysis in the
presence of uncertainty in all tted parameters, either with or without post-
17

optimization feasibility assurance. Figure 13 illustrates that for Formulation B,
the potentially more accurate model with a higher number of parameters, esti-
mates of realized potential
ˆ
Po
are comparable to the actual realized potential
Po
and estimates decline with each iteration just as do actual
Po
. For the rst
iteration estimates
ˆ
Po
are lower than the actual
Po
due to the conservativeness
in the design of Algorithm 2, but the algorithm correctly identies a signicant
above-zero potential, which would have supported the decision to update set-
points toward the suggested target.
Much further work on design of alternative algorithms for quantifying uncer-
tainty, excitation planning, result analysis and operational strategies which t
the framework suggested here is possible. An element of this paper has not
considered in detail is how active decision variables can be chosen to manage
uncertainty, this is left for further work.
4 Conclusion
An approach for casting in mathematical terms the uncertainty in production
optimization arising from low information content in data has been suggested.
A structured approach to handling this uncertainty by combining an iterative
two-step approach to optimization and post-optimization feasibility assurance
was suggested and by uncertainty estimation, result analysis and excitation
planning based on multivariate Monte-Carlo-like methods.
In the simulation case study the method suggested realized between
30%
and
80%
of the available prot potential while feasibility was ensured at all times,
despite that models were tted to data with low information content similar to
that found on real-world oil elds.
Acknowledgments
The authors wish to acknowledge the Norwegian Research Council, StatoilHydro
and ABB for funding this research.
A Details of simulation case study
To ensure that the production analog is a plausible description of the eld
considered, parameters in (15) are estimated with bootstrapping methods. (15)
combined with one of the plausible parameters determined in this manner is
chosen as the the production analog for setpoint values similar to those observed
in the tuning set.
Let
qmax,i
gl
be the maximum values observed in the tuning set. To simulate
our lack of knowledge about process behavior for values of
u
outside those
observed in the tuning set, the kernel function
fgl
is replaced in (15) with
fi
u(qi
gl) = 1
2ci(qi
gl −qi,max
gl )2+bi(qi
gl −qi,max
gl ) + ai
(14)
when
qi
gl > qmax,i
gl
.
ai
and
bi
are chosen so that
qi(u)
is smooth and continuous
for
u
around
umax
, and the curvature coecient
ci
is chosen as a random nega-
tive value. Operating points
ql,i,a
of the production analog are chosen so that the
18

tuning set can be described by the production analog with bias terms. Figure 5
illustrates that the production analog is a plausible description of production,
as it matches production data well.
Production modeling of the eld considered based on the concepts of system
identication has been considered previously in (Elgsaeter et al., 2008
b
), and
similar methods are applied here. The oil, gas and water rates
qi
o, qi
g, qi
w
of each
well
i
are modeled as the product of two kernel functions, one describing the
eects of changes in production valve opening
zi
, and one describing the eects
of changes in gas lift rates
qi
gl
. Models are local around the most recent well
test.
The production model
ˆqi
p= max{0, ql,i
p·fi
z(zi, zl,i )·fi
gl(qi
gl, ql,i
gl )},
(15)
is considered, which is intended to be valid
locally
around the most recent well
test rates
ql,i
p∀i= 1, . . . , nw∀p∈ {o, g, w}
.
zi∈[0,1]
is the relative valve
opening of well
i
,
qi
gl
is the gas-lift rate of well
i
and
∆qi
gl
△
=qi
gl
ql,i
gl
−1
is the
normalized, relative gas lift rate for well
i
. Kernels are chosen as
fi
z(zi, zl,i ) = 1−(1 −zi)k
1−(1 −zl,i)k, k = 5
(16)
fi
gl(zi, zl,i ) = (1 + αi
p∆qi
gl +κi
p(∆qi
gl)2)
(17)
for all wells
i
.
The measurement vector
y(t) = [qtot
o(t)qtot
g(t)qtot
w(t)]T
is considered and an
attempt is made to nd
θ
so that estimates
ˆy(θ, t)△
= 1ˆx(θ, t) + ˆ
βy,
(18)
t measurements as close as possible for the tuning set, where
ˆ
βy
is the vector
of measurement biases due to calibration inaccuracies to be determined and
1
is a matrix of ones.
In addition an upper and lower constraint on
u
of the form (4) was enforced.
Some wells are in danger of slugging if gas-lift is decreased, and on these wells
a constraint which prohibits gas-lift from being decreased was implemented.
Based on the knowledge that the watercut is rate-independent, soft con-
straints which penalize deviation from
αo=αw
and
κo=κw
were added to
the objective function. A decline in
¯qtot
o
was visible in the tuning set, and it
is chosen to de-trend
¯qtot
o
and weigh older measurements less than newer ones
using the weighting-term
w(t)
in (7). In formulation B,
α
and
κ
for all wells and
for oil, gas and water, as well as
βy
are considered part of the vector of tted
parameters. Only the rst-order term in (17) is considered in formulation A.
19

The production optimization problem considered was of the form
ˆu(θ) = arg max
u∑
∀i∈Ia
ˆqi
o(u, d, θ) + bo(θ)
(19)
s.t.
(20)
ql,i
gl ≤qi
gl∀i∈ IMGS
(21)
max{umin, u0−Uprc ·U} ≤ u≤min{umax , u0+Uprc ·U}
(22)
∑
∀i∈Ia
ˆqi
g(u, d, θ)∑
∀iIB
qi
gl +bg(θ)≤qtot,c
g
(23)
∑
∀i∈Ia
ˆqi
w(u, d, θ) + bw(θ)≤qtot,c
w.
(24)
Ia
is the indices of all wells considered in production optimization. Biases
bo(θ), bg(θ), bw(θ)
were determined for each
θ
so that modeled and measured
production match at the time of optimization. For wells with indices
IMGS =
{1,4,10,13,14,16}
gas-lift could not be decreased without the risk of triggering
slugging.
ˆqi
o,ˆqi
g,ˆqi
w
are modeled rates.
The case study was implemented in
MATLAB
1
. Linear least-squares pa-
rameter estimation problems and linear real-time optimization problems were
solved using the
TOMLAB
2
solver
lssol
. Nonlinear production optimization
problems were solved using the sequential quadratic programming solvers based
on Schittkowski (1983).
The simulated benet of excitation of
ui
was found by replacing tted pa-
rameters excited by variation of
ui
with those that best describe the production
analog, and simulating production optimization and implementation with the
operational strategy.
Nomenclature
Subscript
o
,
g
and
w
indicate oil, gas and water, respectively. Numbered super-
scripts indicate well indices, the superscript
l
indicates a local operating point.
Bars (
¯.
) indicate measured variables, while hats (
ˆ.
) indicate estimated variables.
u
decision variables
x
internal variables
d
modeled and measured disturbances independent of
u
θ
parameters to be determined through parameter es-
timation
M
prot measure
c
production constraints
y
measured variables exploited in parameter estima-
tion
t
time
N
the number of time steps in the tuning data set
ZN
a tuning data set spanning
N
time steps
1
The Mathworks,Inc., version 7.0.4.365
2
TOMLAB Optimization Inc., version 5.5
20

ϵ
residuals in parameter estimation
Dy
normalization matrix for
y
w
weighting of residuals in parameter estimation
Vs
soft constraints on parameters
θ
cθ
hard constraints on parameters
θ
umin
the lower end of the range of possible values for
u
umax
the higher end of the range of possible values for
u
Uprc
a limit on the size of change in decision variables
U
scaling of u
x0, u0, d0x, u, d
at the time of optimization
fθ
probability function for parameters
θ
Po
the potential for production optimization
Po,r
realizable potential for production optimization
Lu
lost potential of production optimization
uT
a target value for an operational strategy
U−
the set of indices of
u
which are to be decreased as
an operational strategy approaches
uT
from
u0
U+
the set of indices of
u
which are to be increased as
an operational strategy approaches
uT
from
u0
JM
a cost function against which measured prot change
during implementation of an operational strategy is
to be compared to determine whether to terminate
the operational strategy before the target is reached
JT
M
a threshold value for
JM
to
s
the time at which the operational strategy is initiated
to
e
the time at which the operational strategy is termi-
nated
f∆P r
probability function for
∆Po,r
S
the user-specied size of setpoint change at each it-
eration of an operational strategy
fu
probability function for
u
Jr
a cost function for the expected prot attainable
from moving setpoints toward a new target
JT
r
a threshold value for
JM
BE
the benet of excitation
CE
the cost of excitation
θn
the nominal parameter estimate, in the context of
benet of excitation
JE
the cost-benet tradeo of an excitation
JE,T
a threshold value for
JE
qgl
gas-lift rate
z
relative production choke opening
Ia
the set of indices of all wells considered in optimiza-
tion
b
bias calculated over interval
[N−L, N ]
qtot,c
a capacity for total produced rates
IMGS
indices of wells with lower constraints on gas-lift rate
βy
bias estimated over interval
[1, N ]
α, κ
tted parameters in gas-lift models
21

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24

u2
u1
Infeasible
Feasible
ts
o
te
o
Figure 1: Example 1: Illustration of an operational strategy initiated at
to
s
and
terminating at
to
e
. Solid line illustrates the unknown border between feasible
and infeasible regions, the dotted lines illustrate the uncertainty of this border,
while the dashed line illustrates how an operational strategy might move set-
points in closed-loop. The circle illustrates the setpoint
u0
, and crosses illustrate
uncertain calculated optimal setpoints.
25

M(t)
t
t
M(t)
M0
M0
M +
0 o,r
ΔP
M +
0 o,r
ΔP
^
^
Scenario1
Scenario2
ta
ta
ts
ote
o
te
o
ts
o
increasingprocessmodeluncertainty
negativeprofitbias
Figure 2: The two plots illustrate how measured prot (solid lines) may
evolve compared to a calculated prediction interval (dotted lines) in two dif-
ferent cases.
M0
is the prot as the operational strategy is implemented, and
M0+ ∆ ˆ
Po,r
(dashed line) is the predicted increase in prot from implementing
the operational strategy.
26

Reservoir(s)
.... ....
Routing
....
Water
injection
....
Gas
injection
....
....
Separation
....
Separation
g
....
Gaslift
Export
Oil
Gas
Water
Platform
wells
Reinjection/Tosea
Liftgas/
gasinjection
Gaslift
Subsea
wells
Gaslift
Figure 3: A schematic model of oshore oil and gas production.
27

Perform excitation
planning
Perform production
optimization
Optionally: select
active decision
variables
Implement setpoint
change suggested
by production
optimization
according to
operational strategy
Is the cost/benefit
tradeoff of any
planned excitation
favorable?
Implement
planned
excitation
Yes
Update model:
Estimate parameters
and parameter
uncertainty
Is result analysis
favorable?
No
Yes
Wait until new data
becomes avialable
No
Perform result
analysis
Figure 4: Flowchart of the proposed structured approach to optimization of
oshore oil and gas production with uncertain models.
28

x13 x14 x15 x16 x17 x18
v07
v08
v09
v10
s09
x07 x08 x09 x10 x11 x12
v03
v04
v05
v06
s10
x01 x02 x03 x04 x05 x06
v01
v02
s11
s12
PSfrag replacements
qtot
o
qtot
g
qtot
w
t
[hours]
0
500
1000
1500
2000
2500
0
500
1000
1500
2000
2500
0
500
1000
1500
2000
2500
0.8
1
0.8
1
1.2
1.4
0.6
0.8
1
1.2
Figure 5: Case study: Production data (dotted), the production analog
(dashed), estimates with nominal nonlinear production model (solid).
29

152
153
154 155
156157158159160161162163164165166167168169170171172173
174
175
176 177
PSfrag replacements
qi
o
i= 1
qi
g
qi
w
i= 2
i= 3
i= 4
i= 5
i= 6
i= 7
i= 8
i= 9
i= 10
i= 11
i= 12
i= 13
i= 14
i= 15
i= 16
i= 17
i= 18
i= 19
qi
gl
i= 20
qi
gl
qi
gl
Figure 6: Case-study: Iteration
1
, formulation A. Each row shows modeled rates
of oil, gas and water for dierent parameter estimates found through bootstrap-
ping (dashed) compared with modeled rates for the nominal parameter estimate
(dotted) and the production analog (solid), global optimum (crosses) and cur-
rently implemented setpoint (circles).
30

152
153
154 155
156157158159160161162163164165166167168169170171172173
174
175
176 177
PSfrag replacements
qi
o
i= 1
qi
g
qi
w
i= 2
i= 3
i= 4
i= 5
i= 6
i= 7
i= 8
i= 9
i= 10
i= 11
i= 12
i= 13
i= 14
i= 15
i= 16
i= 17
i= 18
i= 19
qi
gl
i= 20
qi
gl
qi
gl
Figure 7: Case-study: Iteration
1
, formulation B. Each row shows modeled rates
of oil, gas and water for dierent parameter estimates found through bootstrap-
ping(dashed) compared with modeled rates for the nominal parameter estimate
(dotted) and the production analog (solid), global optimum (crosses) and cur-
rently implemented setpoint (circles).
31

139
140
141142143144145146147148149150151152153154155156157158
159
160
161
162
PSfrag replacements
i= 1
Formulation A
i= 2
i= 3
i= 4
i= 5
i= 6
i= 7
i= 8
i= 9
i= 10
i= 11
i= 12
i= 13
i= 14
i= 15
i= 16
i= 17
i= 18
i= 19
qi
gl
i= 20
Formulation B
qi
gl
Figure 8: Case-study: Distribution
fu(ˆu)
for formulations A and B shown for
each well after iteration
1
(bars). Constraints on
u
(dotted),
u0
(circle) and
uT
(stem), and optimal setpoint (crosses).
32

v01
v02
v03
v04
v05
v06
s03
s04
PSfrag replacements
∆Po,r
∆P∗
o,r ·100%
t
[iterations]
0
20
40
60
80
100
Figure 9: Case-study: The percentage of the initial potential
∆P⋆
o,r
that was
realized in simulations of formulation A (cirles) and B (squares) for
Uprc = 0.2
(line) and
Uprc = 0.5
(dashed).
33

v27
v28
v29
v30
s14
v23
v24
v25
v26
s15
v20
v21
v22
s16
x12 x13 x14 x15 x16
v12
v13
v14
v15
v16
v17
v18
v19
s17
s18
s22
s23
s25
s26
s27
s28
s29
s30
s31
s32
s33
s34
s35
s36
s37
s38
s39
s40
s41
s42
s43
s44
PSfrag replacements
qtot
o
qtot
g
qtot
w
∆qgl
ql
gl
t
[iterations]
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
2
3
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.985
0.99
0.995
0.98
0.985
0.99
0.995
1
1.01
1.02
1.03
Figure 10: Case study: An example of how the operational strategy imple-
ments setpoint changes while obeying process constraints for formulation B.
Top graphs show normalized prots (
qtot
o
), gas capacity utilization (
qtot
o
) and
water capacity utilization (
qtot
w
), lower graphs show normalized relative changes
in gas lift rates
qgl
for all wells.
34

139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
x04 x05 x06 x07
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
x01 x02 x03
201
202
PSfrag replacements
i= 1
Formulation A
0.69
i= 2
1.18
i= 3
0.10
i= 4
-0.01
i= 5
-0.16
i= 6
0.38
i= 7
0.97
i= 8
0.03
i= 9
1.17
i= 10
0.15
i= 11
0.28
i= 12
2.78
i= 13
0.03
i= 14
2.14
i= 15
0.69
i= 16
0.00
i= 17
0.24
i= 18
1.23
i= 19
-0.11
i= 20
BE(%)
0.86
Formulation B
0.29
1.16
0.49
-0.01
0.08
0.56
1.13
0.42
0.24
0.27
1.64
1.44
-0.10
0.97
-0.01
-0.00
-0.24
0.70
0.17
BE(%)
-0.36
-10
0
10
-10
0
10
20
Figure 11: Case-study: Distribution of
ˆ
BE
of exciting well
i
in isolation at
iteration
1
of the simulation, found with Algorithm 3 (bars),
BE
found by
simulating excitation on the production analog (stems). Numbers show average
ˆ
BE
as predicted by Algorithm 3. All values are expressed in percent of total
unrealized potential.
35

139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
x05 x06 x07
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
x01 x02 x03 x04
201
202
PSfrag replacements
i= 1
Formulation A
0.88
i= 2
-0.22
i= 3
0.25
i= 4
0.03
i= 5
-0.03
i= 6
0.85
i= 7
0.83
i= 8
0.62
i= 9
1.41
i= 10
0.28
i= 11
0.81
i= 12
5.16
i= 13
-0.12
i= 14
2.50
i= 15
0.59
i= 16
0.00
i= 17
0.02
i= 18
0.79
i= 19
-0.07
i= 20
BE(%)
1.19
Formulation B
1.80
6.40
0.38
0.18
-0.14
3.46
9.80
11.98
11.64
2.37
15.25
6.51
1.66
2.71
-0.01
0.12
0.82
18.32
2.88
BE(%)
1.35
-50
0
50
100
-20
0
20
Figure 12: Case-study: Distribution of
ˆ
BE
of exciting well
i
in isolation at
iteration
4
of the simulation, found with Algorithm 3 (bars),
BE
found by
simulating excitation on the production analog (stems). Numbers show average
ˆ
BE
as predicted by Algorithm 3. All values are expressed in percent of total
unrealized potential.
36

x19 x20
s27
s28
x16 x17 x18
s29
x14 x15
s30
x11 x12 x13
x09 x10
s31
x06 x07 x08
x04 x05
s32
s33 x01 x02 x03
s34
PSfrag replacements
it:1
Formulation A
Formulation B
it:2
it:3
it:4
∆Po,r(%)
∆Po,r(%)
0
1
2
0
2
0
1
2
0
2
0
1
2
0
2
0
1
2
0
2
Figure 13: Case-study: Estimated
f(∆Po,r)
for formulations A and B, one row
for each iteration. The increase in prots that was implemented in practice is
shown as stems.
37

s02s06s10s14s18s22s26s30s34s38s42s46s50s54s58s62s66s70s74
s77
s78
PSfrag replacements
i= 1
i= 2
i= 3
i= 4
i= 5
i= 6
i= 7
i= 8
i= 9
i= 10
i= 11
i= 12
i= 13
i= 14
i= 15
i= 16
i= 17
i= 18
i= 19
qi
gl
i= 20
Figure 14: Case-study: Formulation B, iteration 4, candidate excitation
u9
.
Upper and lower constraints on
u
(dashed), currently implemented setpoint
(circles), the target setpoint that would be implemented if no excitation is per-
formed of
u9
(stems with circles) and target setpoint after an excitation of
u9
is simulated (stems with squares).
38