Thesis for the degree of doktor ingeniør Trondheim, June 2007 Norwegian University of Science and Technology Faculty of Information Technology, Mathematics and Electrical Engineering Department of Engineering Cybernetics Hans Petter Bieker Topics in Offshore Oil Production Optimization using Real-Time Data
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Thesis for the degree of doktor ingeniør
Trondheim, June 2007
Norwegian University ofScience and TechnologyFaculty of Information Technology, Mathematics and ElectricalEngineeringDepartment of Engineering Cybernetics
Hans Petter Bieker
Topics in Offshore Oil ProductionOptimization using Real-Time Data
i
Abstract
In all production systems, production optimization is important because
it can reduce the cost of operation and increase the production. This the-
sis is a contribution within the field of production optimization of off-
shore oil production systems using measured real-time data.
Four novel methods related to production optimization of such oil pro-
duction systems have been proposed. Using measured data, they are con-
tributing to maximize the total oil production rate or the expected total
oil production rate of the oil production system.
First, a method optimizing the total oil production rate from subsea wells
where a model of the pressure interconnection of a common flow line
must be included is proposed. The method uses a piecewise linear approx-
imation of the pressure drop in the flow lines and wells enabling global
optimization using a branch and bound mixed integer linear program-
ming solver.
Second, a method for optimizing the expected total oil production rate by
selecting wells for testing is proposed, using real-time data. The well test-
ing gives information on the gas oil ratios or the water cuts that is more
accurate allowing an improved prioritization of the wells compared to the
industry practice when a processing constraint is available. A method for
calculating stochastic distributions of the gas oil ratios or water cuts is
proposed.
Third, a method handling the uncertainties in the gas oil ratios or water
cuts explicitly for prioritizing the wells when a processing constraint is
available is proposed. The prioritization was found to depend on the
probability distribution of the gas oil ratios or water cuts, oil potential of
ii
each well, and processing capacity. The method is able to handle all these
uncertainties explicitly by using a user-provided probability distribution
for each of them.
Fourth, a method finding the optimal sequence to open the wells when a
limited flow change rate into the production separator and from each well
is required is proposed. The method may be used to find a ramp-up se-
quence after a shutdown. The excess treatment capacity is updated using
the measurements of the treatment utilization in each time step, allowing
the treatment capacity to be fully utilized.
iii
Acknowledgments
First, I would like to thank my supervisors Professor Dr Ing Tor Arne
Johansen and Dr Ing Olav Slupphaug. Tor Arne has been an invaluable
resource suggesting new ways of solving the challenges I studied. I appre-
ciate his constructive commenting of my manuscripts. Olav has been the
source of most of the industrial challenges studied in this work. He has
given me valuable and required background information on the operation
of oil production systems and challenges in production optimization. His
indefatigability commenting of my manuscripts has certainly improved
the quality of them. Without my supervisors, the thesis would not be
possible.
The Research Council of Norway, Norsk Hydro ASA, and ABB AS are
acknowledged for financing this work. In particular, I would like to thank
ABB AS for providing an inspiring working environment. It has been a
source of many of the challenges investigated in this thesis. Several of the
other professionals at ABB AS have been suggesting interesting chal-
lenges to study, and their help is also much appreciated.
Hans Petter Bieker
Oslo, June 2007
iv
v
Table of Contents
Abstract ........................................................................................................... i
Acknowledgments .......................................................................................... iii
Table of Contents ........................................................................................... v
thalpy, or production rates, etc. This will generally require on-line nonli-
near dynamic optimization. Current industry practice is, however, to do
this manually. In addition, binary decisions may be associated with
routing of different wells to different inlet separators to do load balancing
of these separators. These decisions are harder to make because they
make the optimization problem non-convex, and a global solver is re-
quired. Koninckx [42] gives an overview of many of these real-time pro-
duction optimization problems which have been developed for optimizing
operation of continuous processes.
38
Qin and Badgwell [43] give a review on the usage of Model Predictive
Control (MPC) for optimizing operation of continuous processes in gen-
eral. MPC is a closed loop control method where the decisions are found
by solving a dynamic constrained optimization problem on some finite
horizon into the future. The optimization problem includes a dynamic
model of the process, an objective function to be minimized, constraints
on states, and constraints on decision variable movements and values.
The objective function penalizes a predicted deviation from control objec-
tives. The decisions may be time variant within the optimization problem
(for instance one for each time step), and only the decisions of the first
time step are used. The next time step a new dynamic optimization prob-
lem is solved using new measurements, closing the loop. The control me-
thod requires that the state of the dynamic model be estimated. The con-
trol method is used in oil production optimization for problems including
load balancing.
2.3.3 Reservoir Planning
An important part of the reservoir planning is the injection strategy of
the reservoir. The production from an oil and gas production system is
largely driven by the pressure difference between the reservoir and the
surface. A typical strategy ensures that the pressure is maintained by in-
jecting roughly the same volume, under reservoir conditions, of water and
gas as the produced volumes of fluids. Some reservoirs are supported by
large aquifers, where as a result, the pressure in the reservoir is controlled
naturally.
The volume balance of the reservoir is however not the only important
property. Injecting close to the producer will typically increase the pres-
sure faster than injecting far from it. The permeability also makes a dif-
ference. This means that the pressure response between an injector and a
producer is not instant; it is a dynamic system. Because of this dynamics,
39
the water cut and the gas oil ratio from the wells will change slowly until
a water or gas breakthrough happens. An uneven phase interface is often
referred to as fingering, and it may result in a premature water break-
through if not accounted for properly in the reservoir model used for re-
servoir planning.
Sudaryanto et al. [44] studied strategies for water injection for a 2D re-
servoir with miscible fluids with the same mobility. The reservoir was a
heterogeneous porous media. The effect of gravity and dispersion were
neglected. The reservoir studied had one producer and multiple injectors.
They proposed to use optimal control theory to maximize the time of the
arrival of the water breakthrough constrained by a constant total injec-
tion rate. Multiple cases were studied where the geometry varied. In par-
ticular, the distance between the injectors and the producers varied. In
the cases studied, it was found optimal to inject all available water into
one injection well at the time, and that the optimal injector changed as
the waterfront moved in the reservoir. The typical optimal solution was
to start injecting the farthest off the producer and then switch to a new
injector at given times. The authors reported the bang-bang injection
strategy to delay the water breakthrough by the range from 13.8 % to
16 % for two particular geometries considered compared with constant
rate injection strategies.
Brouwer et al. [45] investigated the use of a simple heuristic algorithm for
delaying the water breakthrough by using smart injection and production
wells. Later, Brouwer [46] proposed to use optimal control theory on a
dynamic model to allocate rates of each water injector. A reservoir with
dimension 450×450×10 m was considered. Each block in the reservoir
model was 10×10×10 m. The reservoir had two horizontal wells with 45
segments. Each grid block penetrated by a well represented a segment.
The approach maximized the net present value with respect to volume
40
balance, rate, and pressure constraints. The result was twofold: Wells
operating on a bottom-hole pressure constraint benefited from reducing
water production, and rate-constrained wells gave accelerated production,
increased recovery, and reduced water production.
In Yeten et al. [47], the optimization of injection and production rates for
smart wells was studied. They used a gradient-based optimization algo-
rithm to find local optimal settings for the injection and production
chokes. The optimization algorithm was connected to a commercial dy-
namic reservoir simulator for the objective function evaluations. The ob-
jective function was the cumulative oil production within some specified
time interval, and the choke settings were kept constant within this time
interval. To forecast the development of the production system, the op-
timization was divided into periods. The choke settings for each period
were found by optimizing from the start of this period to the end of the
last period. The initial state of the reservoir for each period was the state
of the reservoir from the end of the previous period.
By the use of a history-matched streamline-based reservoir simulator,
Thiele and Batycky [48] proposed to compare the efficiencies of the injec-
tor-producer well pairs. A streamline-based reservoir simulator is able to
track the individual flows from injectors and producers. The injector-
producer well efficiency was defined as the increased oil production of a
well when injecting a particular amount of water. The efficiency of an
injector was defined as the sum of the injector-producer well efficiencies
of the injector for all producers. By iteratively increasing the water injec-
tion from low efficient injectors to high efficient injectors, the water in-
jection was optimized.
A reservoir model is typically subject to large parametric uncertainties in
reservoir geometry or permeability in the reservoir [49]. Therefore, Rag-
huraman et al. [49] proposed to calculate the value added by a smart
41
production well in a reservoir with uncertainty in the width of a high-
permeability zone in the reservoir and the inflow from an aquifer, see
Figure 2.6. Each of the uncertainties was divided into the possible out-
comes low, mean, and high. The reservoir consisted of three zones, and
the smart production well was able to control the flow from each zone
through chokes individually. The mean value of the net present value of
these nine combinations was maximized, finding an optimal setting for
each of the chokes. The economic gain of the smart production well was
found by comparing the net present values of the smart production well
and a conventionally completed well. The objective function also included
a risk aversion function enabling penalization of the variations in the
outcomes. Bailey et al. [50] extend this work to a full-field model of an
onshore production system with several uncertainties. The method found
an efficient frontier providing confidence bounds on the objective func-
tion.
Narayanan et al. [51] argue that the optimization of the exploration and
production of an oil and gas production system should be done in a
closed loop. The currently used method typically is stepwise where one
work group passes their results on to the next. Each work group tries to
optimize the net present value using detailed models. However, the mod-
els are not connected and the uncertainties across the models are not re-
flected. Thus, the uncertainties in the compound project are much larger
than the value calculated. By using Monte Carlo simulations, the uncer-
tainty in the optimized net present value is found. The work was later
extended by Cullick et al. [52] where a black box global solver was used
in a combination with the Monte Carlo simulations.
2.3.4 Model Updating
To reduce model complexity, models normally only consider a subsystem
of the production system, and only subsets of the input and output of the
42
system are considered. Their static and dynamic accuracy may also be
very different. Some assume a steady state and some can only accurately
predict changes. Because of this, a model may be good for one application
and may not be so good for other applications.
Models use parameters to describe specific processing facilities, the reser-
voirs, and wells. Most parameters may be set by design data. However,
because of wearing or just simplifications done in the model some para-
meters change with time. Therefore, it is important to update them to
make sure the model accurately describes the actual behavior of the
processing facilities, reservoirs, and wells.
2.3.4.1 Well
A well model provides the decision makers with predictions of the oil
production rates that can be used to decide from which wells to produce
and from which not to produce. For instance, if a production system is
constrained by its gas processing capacity, then it is crucial to know the
gas oil ratio. A similar relationship exists for water production. Some
wells may also be vulnerable to sand production. If so, it is necessary to
develop a relationship between some measured variables (for instance
pressures) and the sand production rate in such a way that the operator
can ensure that the constraint is not violated. Other parameters such as
the H2S concentration of the produced gas may be interesting to meet
product quality specification or for safety reasons.
Well tests may be undertaken to unveil the values mentioned above. De-
pending on how the production system is constrained and what type of
testing is done, a well test may or may not result in production losses
during testing. In some cases, the test separator will be used for produc-
tion when it is not used for testing. This means that some wells may
have to be choked back to let the wells producing to the test separator be
43
routed to one of the main separators. Even if the separation capacity is
not a limitation, testing may also result in losses because of transients;
operators cannot run the system on its limit during rerouting.
Well tests may be performed on a single rate or multiple rates. If a single
rate test is undertaken, only one choke setting or gas lift rate is used.
Multi-rate tests may be used to establish inflow performance relation-
ships or gas lift performance curves. Nevertheless, they are more expen-
sive because they take more time to run. For each change in the choke, it
is necessary to wait for the well or near well bore dynamics to settle.
When flow transmitters for multiphase flow are available, they may be
used to measure the gas oil ratio and the water cut, and they will reduce
the required frequency of well testing.
As an alternative to using flow transmitters for multiphase flow, the cur-
rent measurements from the production system can be used to estimate
the flow from the well using a model of the well and flow lines. Systems
such as Well Monitoring System3 and FlowManager4 estimate the pres-
sure and flow profiles of the well or pipe network by minimizing the devi-
ation between currently measured values and the pressure, temperature,
and flow profile in the simulator.
Few references that uses the theory of system identification [53] for up-
dating of well inflow models have been found. Such methods may conti-
nuously update the inflow models of the wells by using the available
measurements in the well.
3 ABB. 4 FMC.
44
2.3.4.2 Processing Facilities
During operation, different parts of the processing facilities may be worn
out or degraded. Thus, the capacity changes and the models should re-
flect this. The updating of available processing capacity is important to
ensure that the capacity is fully utilized.
Clay et al. [54] investigated the use of RTO systems on topside
processing facilities. Using a rigorous model of the natural gas liquid
(NGL) subsystem, it was optimized to give maximal NGL production or
“stabilizer bottoms”. Their calculations gave a 2 % potential production
increase using the optimization of this subsystem for the production sys-
tem considered. Furthermore, they suggested that booster compressors,
low temperature separators, stabilizers, MI compression, propane refrige-
ration, and crude blending could be applications for RTO.
2.3.4.3 Reservoir
In a work by Schulze-Riegert et al. [55] it was proposed to use a genetic
algorithm to do history matching of a reservoir. The solution found by
the genetic algorithm was used as an initial solution for a local optimiza-
tion algorithm to fine-tune the solution. The pre-solving using the genetic
algorithm may allow the local optimization algorithm to find a solution
closer to the global optimum. However, even if such a combination is
used, the method can at best guarantee that a local optimum solution is
found in finite time. In practice, the genetic algorithm would have to be
terminated after some predetermined time without a best bound on the
objective function.
The approach by Brouwer [46] described above was later refined by
Brouwer et al. [56] by including continuous state estimation of the reser-
voir. An ensemble Kalman filter was used to estimate the states. The fil-
45
ter utilized the production and injection rates as well as downhole pres-
sure gauges for each segment of the wells.
A data-driven reservoir management strategy was developed by Saputelli
et al. [57]. The strategy uses two levels. The upper level optimizes the net
present value and the lower level uses model predictive control to enforce
the results from the optimization layer. By using system identification
and state estimation, this becomes a “self learning reservoir management
system”. The concept was later elaborated [58] to a multi-level control
and optimization framework. The levels were separated by their domi-
nant time constants.
Kosmala et al. [59] investigated how the accuracy of a reservoir simula-
tion could be improved by including a production network simulator. The
two simulators were connected by a common bottom-hole pressure. Vari-
ous decisions were adjusted by an SQP algorithm to maximize the oil
production rate.
2.4 Challenges
The term “RTO” has recently found its way into the oil and gas indus-
try. However, Saputelli et al. [4] noticed that it is used more like a slogan
than a system that, in a mathematical sense, truly optimizes anything at
all. The technologies in the sections on “Production Planning” and “Re-
servoir Planning” offer optimization. Often the other references on model
updating or estimation somehow claim to optimize the production too.
This is hardly true, even though they support the optimization process.
To qualify to being an RTO system, the system must maximize or mi-
nimize some defined performance indicator. Furthermore, the method
should be systematic.
46
However, RTO is not just optimization. According to Figure 2.2, there
are four components in addition to the production system itself. If only
the model-based optimization process was included, the same result
would be produced repeatedly. The model-updating component ensures
that measurements from the production system are fed back to the mod-
el-based optimization component. Data validating and optimizer com-
mand conditioning components do pre-validation and post-validation of
data, ensuring reliability. An RTO system must consist of the model-
based optimization and the model updating as a minimum. Furthermore,
few, if any, results on systems with all four components of the RTO have
been published.
Usually, RTO uses a pure steady state model of the production system.
Thus, such RTO only makes sense if the near steady state periods are
long compared to the transient periods. An oil and gas production system
is never in a steady state because the drainage process changes the reser-
voir state, and there are always smaller or larger changes in the
processing capacity and wells available for production due to various rea-
sons. The former effect is accounted for in the reservoir planning prob-
lems, but all transient effects are more or less ignored in the other prob-
lems. Thus, the dynamics may be exploited in other planning problems in
order to increase performance [4]. Nevertheless, such time decomposition
of the optimization has been found to be useful in practice. This is prob-
ably because the time constants of the reservoir are very large compared
to the fast dynamics of the wells and processing facilities. In fact, a
framework for such decomposition was proposed by Saputelli et al. [57].
It was emphasized that:
To handle the complexity, multiple less complex RTO systems
should be used, each handling a sub-problem of the plant-wide
optimization problem.
47
Production data should be integrated for continuous learning of
key reservoir features.
The reservoir performance should be continuously optimized
without violating constraints.
Until now, few implementations of RTO exist on real offshore oil and gas
production systems. Fitting a steady state model to transient data can be
challenging and result in erroneous parameters. Some of the models in-
clude too many parameters to be fitted using only commonly available
measurements from the production system. Using simpler models will al-
low updating and optimizing more frequently because of less computa-
tional burden. Finding a model with the correct level of accuracy should
be addressed. Starting with a simple RTO system that solves small, but
significant, sub-problems robustly and later extending it to include new
features may be the way to go. Such systems tend to be more easily ac-
cepted by the management, engineers, and operators of the production
systems.
As the RTO typically uses a steady state model, it requires a stable pro-
duction system that is able to enforce decisions without violating con-
straints. For instance, gas-lifted wells are often over-injected to ensure
stability. By implementing stabilizing controllers, production can often be
increased without the help of RTO. By installing new or tuning existing
feedback control loops, the capacity of the production system can be in-
creased by enabling operation nearer alarm and shutdown levels without
increasing the risk of a shutdown. These improvements can be obtained
with or without an RTO system.
The RTO assumes that the production system is in a steady state, but
there will be transients caused by disturbances and changed recommend-
48
ed operation. None of the reviewed papers included an analysis of the
closed loop dynamics of closed loop RTO.
In production optimization, constraints are usually active at the optimal
operating conditions. This means that any change in these constraints
will affect the optimal operation. None of the reviewed methods consider
handling of the constraints of the production systems in a closed loop set-
ting. Thus, if the model-based recommended operation results in violated
constraints in the real production system caused by inaccuracies in the
production system model, ad hoc rules will be required to adjust the op-
eration. Such ad hoc rules will reduce the production and perhaps lead to
suboptimal production. If the recommended operation has some active
constraints, and these constraints do not become active in the real pro-
duction system operation caused by inaccuracies in the production sys-
tem model, then production can be increased by updating the constraints
in the model. An RTO scheme should have such a strategy. Other para-
meters should be updated as well. The handling of model uncertainty is a
key challenge for the success of RTO.
2.5 Conclusions
A vast number of optimization strategies for offshore oil and gas produc-
tion systems have been proposed in the literature. Most of the reviewed
strategies were designed for planning the operation of the production sys-
tem. Very few of the strategies were designed for on-line usage. Most
were designed to be used off-line to provide recommendations for the op-
eration.
RTO is not a replacement for the base control layer of a production sys-
tem, but utilizes the base control layer in its operation. RTO is a scheme
that uses a mathematical model of the production system to optimize the
production. The model is updated by using available measurements from
49
the production system. The scheme should update processing capacity
constraint parameters to avoid over-utilization and under-utilization due
to inaccuracies in the model of the production system model.
For reservoir planning, various strategies that use a dynamic model in
the optimization have been proposed. This is because of the dynamic na-
ture of the drainage process and the injection. For the more short-term
production planning, steady state models are dominating and few RTO
approaches have been proposed. The often varying and hard-to-measure
feed from the wells may explain why it is hard to reuse existing RTO ap-
proaches from the petrochemical industry. Thus focusing on identification
of well models and integration of steady state and dynamic models seems
to be central tasks in getting more RTO systems in operation in offshore
oil and gas production.
50
Storagecell
Reservoir Reservoir
Well stream
Seawater
Water injection
Gas export
Gas injection
Figure 2.1: An oil and gas production system includes components such
as reservoirs, production wells, injection wells, production manifolds,
flow lines, separators, heaters, coolers, compressors, scrubbers, and
pumps.
51
Production system
Data validation
Model updating
Optimizer command
conditioning
Model-based optimization
Figure 2.2: The components in a typical real-time optimization system
(RTO) are the production system, data validation, model updating,
model-based optimization, and optimizer command conditioning.
52
Production System
Data Acquisition
Data Storage
Reservoir Model Update
Well Model Update
Process Model Update
Reservoir Planning
Production Planning
OperatorControlStrategic Planning
Planning and Control
Data Aggrigation
Production System and Sensors
Figure 2.3: The decision loop in production optimization consists of the
production system, sensors, data aggregation, planning, and control.
53
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Decision variable
Obj
ectiv
e fu
nctio
n
Localmaxima
Globalmaximum
Figure 2.4: An objective function or optimization problem may have
multiple local maxima, and a global maximum is a local maximum with
no superior local maxima.
54
Gas lift rate
Oil
pro
ducti
on
ra
te
A good well with lift on
A well needs lift to flow
Figure 2.5: A gas lift performance curve relates the lift gas rate injected
into the well to the oil production rate from the well, and may typically
be used for finding the optimal gas lift rates for the wells [26].
55
Figure 2.6: The reservoir model has three isolated zones whose flow rates
can be independently controlled through valves [49].
56
57
3 Global Optimization of Multiphase Flow
Networks in Oil and Gas Production Sys-
tems
Based on
H.P. Bieker, O. Slupphaug, and T.A. Johansen,
submitted to
Computers & Chemical Engineering Journal,
presented at
2006 AIChE Annual Meeting
San Francisco, California, U.S.A., 12–17 November 2006
3.1 Introduction
In the daily operation of oil production systems many decisions have to
be taken that affects the volumes of oil produced. One of them is choos-
ing the settings to use for the chokes to maximize somehow the oil pro-
duction. Because of limited processing capacity, the optimal solution may
be to choke back some wells with high production of water or gas relative
to oil production.
To increase the production of oil, gas lift has been installed in many
wells. Gas lift reduces the pressure drop in the riser by reducing the av-
erage density of the fluid. The effect of gas lift reduces with the amounts
used because the gas also increases the friction. Furthermore, the gas has
to be processed by the production system’s compressors that are limited.
A challenging optimization problem then has to be solved in order to
maximize the production. The problem has been studied by many people,
58
including Fang and Lo [26]. In that paper, a scheme for solving the prob-
lem using gas lift performance curves was proposed using linear pro-
gramming. They pointed out that the method might not give a correct
solution if a well was not able to flow naturally, i.e. a well was not able
to produce with zero lift gas. They therefore proposed to use a mixed in-
teger solver in such cases. This was later studied by Wang [27] and oth-
ers. The problem was formulated as
o, ,max
i
i k i ki W k K
q
(3.1)
subject to:
gl gl,M, ,
i
i k i ki W k K
q q
(3.2)
, 1i
i kk K
i W
(3.3)
, 0 ,i k ii W k K (3.4)
,For each , at most two may be
positive, and they must be adjacent.
i ki (3.5)
The pair o gl, ,( , )i k i kq q is the oil and gas lift rate which makes up a point in
the gas lift performance curve, W is the set of wells, iK is the set of
points for well i , and gl,Mq is the maximal total gas lift rate. (3.4)-(3.5)
forms a special ordered set of type two [10], which is directly supported
by most modern mixed integer solvers [60]. It may also be formulated as
a pure mixed integer program [61, 62].
However, Wang [27] pointed out an important drawback with the me-
thods using gas lift performance curves. It assumed that the production
from each well was independent. In some sense, this is often true for
some offshore installations. Here, the blending point, called the produc-
tion manifold, is placed on the production platform itself. Due to the
short distance between the production manifold and the pressure con-
59
trolled production separator, the production manifold pressure is assumed
to be fixed.
However, the introduction of new subsea technologies has changed this
for the offshore oil production platforms. Wells far from the production
platforms are connected to a subsea template at the seabed, in which well
streams are blended. The blended well streams are sent through a flow
line to the production platform. Because of the long distance of this
shared flow line, the pressure drop may be large. Furthermore, the pres-
sure drop will typically be sensitive to the volumes flowing through the
flow line. Thus, adjusting the production from one well by changing the
lift gas rates or the production choke, will most certainly affect the other
wells.
The optimization of such a flow line network has therefore been studied
by several people. In [32] the optimal lift gas rates for one, two, and three
identical wells sharing a flow line was compared. In addition, larger field-
wide flow line networks were studied. By the use of Successive Quadratic
Programming (SQP), it was found that the optimal lift gas rates for each
well reduced as the number of wells increased. SQP was also used by
Wang [27] to solve a similar flow line network.
Instead of using SQP, Successive Linear Programming (SLP) was pro-
posed by [35]. The pressure drops in the flow lines were modeled using
standard nonlinear equations. In each iteration of the SLP algorithm, the
pressure drop is linearized in the flow lines were found. The inflow per-
formance of each well was modeled as a piecewise linear surface using li-
near inequalities, similar to [26]. According to the author, this reduced
the number of SLP iterations required.
All the above-proposed solutions use only local algorithms that at best
may guarantee that a local optimum is found. Because all the problems
60
formulated in general are non-convex, the algorithm may not find the
global optimum solution. Even worse, a feasible solution is not guaran-
teed to be found even if the problem is feasible for sure. Some parts of
the physics itself make the problem feature multiple local optima. For
fixed boundary pressures on a flow line, there may exist two different
flow rates satisfying the conditions; one low flow rate and one high flow
rate. If the wrong initial solution is used in the simulator, then the wrong
solution will be found. In an optimization, the problem will be the same,
but in a larger scale.
To be able to escape from only local optima, a genetic algorithm was
used by Stoisits et al. [39] to give a near global optimum solution in a
similar problem. Unfortunately, genetic algorithms still have some draw-
backs. They do include a guarantee for neither a local nor a global opti-
mum. Furthermore, the computational load is very high because little
structure of the problem is utilized.
The drawbacks for the above methods motivates for a new method for
solving such flow line networks that is able to find a proven global opti-
mum, do not require an initial solution to be provided, and has a reason-
able computational load in the optimization. In this work, such a method
will be presented.
3.2 Methodology
The work of Fang and Lo [26] allowed a global optimum to be found by
modifying the problem into a mixed integer problem. Unfortunately, it
can only be used for the simplest oil production systems due to the miss-
ing support for flow lines shared by multiple wells. In this work, this
model will be extended to include pressure drops in shared flow lines.
61
3.2.1 Well
The well model relates the oil rates, gas lift rates, and outlet pressure of
the well, i.e. the production manifold pressure, in some way. It is possible
to argue for different choices of independent and dependent variables, but
in this work the oil and gas lift rates were used as independent variables,
while the production manifold pressure was the dependent (calculated)
variable. This is because the flow in a pipe is calculated using integration
of the partial derivative of pressure. Thus, the pressure may be found
using a single integration, while a nonlinear equation set (including inte-
gration) would have to be solved to find the flow rates if the outlet pres-
sure was an independent variable. Using a mixed integer framework, the
outlet pressure equation
o lg( , )i i i ip p q q i W (3.6)
for well i will be modeled, where oiq is oil flow rate, lg
iq is the lift gas
rate, and ip is the outlet pressure of the well. Similar to the gas lift per-
formance curve, each of the independent variables will be defined into a
finite number of break points, for instance point in which a function
evaluation of O()ip will be performed. Call the break points o, oi kq and lg
lg
,i kq
for oil and lift gas rates, respectively. For the oil rate, the set oiK will
define the indices of the break points, while lgiK will have the same role
for lift gas (for each well i ). A function evaluation oo lg lgo lg,, , ,
: ( , )i ki k k i kp p q q
of the outlet pressure will be performed in each combination of those
points, thus
o lg o lg
o o lg lg, , , ,
i i
i i k k i k kk K k K
p p i W
(3.7)
The model should also include the oil and lift gas rates. To add them,
some auxiliary variables are defined
o o lg
lg lg
o o o, , ,
,i
ii k i k kk K
i W k K
(3.8)
62
lg o lg
o o
lg lg lg
, ,,
i
ii k i k kk K
i W k K
(3.9)
Using them, the oil and lift gas rates can be included
o o
o o
o o o o, , ,
i
i ii k i kk K
q q i W k K
(3.10)
lg lg
lg lg
lg lg lg lg
, ,,
i
i ii k i kk K
q q i W k K
(3.11)
The gas and water rates will also be handy, so they will also be defined
here
o o
o o
g g o o o, , ,
i
i i ii k i kk K
q r q i W k K
(3.12)
o o
o o
w w o o o, , ,
i
i i ii k i kk K
q r q i W k K
(3.13)
where gir is the gas oil ratio and w
ir is the water oil ratio (i.e. w : WC (1 WC )i i ir where WCi is the water cut of the well). Fur-
thermore, the convexity constraints are added similarly as in [26],
o lg
o o lg lg, ,
1i i
i k kk K k K
i W
(3.14)
lgo o lg lg
, ,0 , ,o i ii k ki W k K k K (3.15)
To ensure that neighbors are used in the interpolation, two more con-
strains have to be added
o
o,For each , at most two may be
positive, and they must be adjacent.
i ki (3.16)
lg
lg
,For each , at most two may be
positive, and they must be adjacent.i k
i (3.17)
The model of the well has now been completed.
3.2.2 Flow Line
The well model was an extension of previous work by Fang and Lo [26].
No similar model piecewise linear model of a flow line or pipe has been
found in the literature. The closest match was some work done by Litvak
63
and Darlow [63], who used a look up table of the pressure drop in a flow
line. They parameterized it in four independent variables: oil rate, gas
rate, water rate, and pressure. Thus, they assumed that the stream con-
sisted of only three linearly independent fluid compositions, and that the
temperature at the inlet was fixed. The same assumptions will be used in
this work. Tests done using a flow simulator for a real field showed little
change in temperature. However, it should be noted that the method it-
self does not restrict the inclusion of temperature or enthalpy. The as-
sumption is done to reduce the computational requirement. The tempera-
ture was included in a similar model [64] by the use of enthalpy.
As for the well, the outlet pressure will be described by piecewise linear
functions that is approximated. Thus,
o w I( , , , )gi i i i i ip p q q q p i F (3.18)
has to be modeled, where oiq is the gas rate, w
iq is the water rate, and Iip
is the inlet pressure of the flow line. Using the same notation as for wells,
the outlet pressure can be defined as
o w p o w p
o o g g w w p p, , , , , , , ,g g
i ii i
i i k k k k i k k k kk K k K k K k K
p p i F
(3.19)
Auxiliary variables are then defined
o o g w p
g g w w p p
o o o, , , , , ,
ii i
ii k i k k k kk K k K k K
i F k K
(3.20)
g o g w p
o o w w p p
g g g, , , , , ,
i i i
ii k i k k k kk K k K k K
i F k K
(3.21)
w o g w p
o o g g p p
w w w, , , , , ,
i i i
ii k i k k k kk K k K k K
i F k K
(3.22)
p o g w p
o o g g w w
p p p, , , , , ,
i ii
ii k i k k k kk K k K k K
i F k K
(3.23)
Using them, oil, gas, water, and inlet pressure can be included
o o
o o
o o o o o, , ,
i
i ii k i kk K
q q i F k K
(3.24)
64
g g
g g
g g g g g, , ,
i
i ii k i kk K
q q i F k K
(3.25)
w w
w w
w w w w w, , ,
i
i ii k i kk K
q q i F k K
(3.26)
p p
lg lg
I I p p p, , ,
i
i ii k i kk K
p p i F k K
(3.27)
Furthermore, the convexity constraints are added similarly to before,
o w p
o o g g w w p p, , , , 1g
i ii i
i k k k kk K k K k K k K
i F
(3.28)
o g w po o g g w w p p
, , , , 0 , , , ,i i i ii k k k k i F k K k K k K k K (3.29)
To ensure that neighbors are used in the interpolation, four more con-
strains have to be added
o
o,For each , at most two may be
positive, and they must be adjacent.
i ki (3.30)
g
g,For each , at most two may be
positive, and they must be adjacent.
i ki (3.31)
w
w,For each , at most two may be
positive, and they must be adjacent.
i ki (3.32)
p
p,For each , at most two may be
positive, and they must be adjacent.
i ki (3.33)
The model of the flow line has now been completed.
3.2.3 Choke
Wang [27] investigated how the pressure drop of the choke increased
when closing the choke for fixed flow rates. He utilized it to remove an
explicit model of the choke in the models used for optimizations. In his
work, it was attractive because some non-convex features of a typical
choke model.
For the piecewise linear model a model of the choke would introduce
more independent variables in the pressure drop equations (3.6) and
(3.18), thus requiring many new decision variables. Fortunately, because
65
of the property observed by Wang, this is not required. Instead, the mi-
nimal pressure drop of the choke will be included in the outlet pressure
ip of the wells and/or flow lines. This minimal pressure drop is found by
including the choke model in the calculation of the pressure drop in the
well and/or flow line with a choke opening set to it maximal opening,
typically position 1.0. Any reduction of the choke opening will give a
higher pressure drop, thus for any well or flow line i W F with a
choke
Oi .ip p (3.34)
And if a choke does not exist, then just use pressure equality
Oi .ip p (3.35)
It should be noted that the statement above is only true if the flow direc-
tion is given. If the flow changes direction, then the additional pressure
drop will have the opposite sign.
3.2.4 Outlet Boundary
A model of the outlet boundary of the system is included. This can be
the production manifold or the production separator. Nevertheless, it is
assumed that this outlet boundary i has a fixed inlet pressure Iip for all
i O where O is the set of outlet boundary nodes.
3.2.5 Connection
The flow lines have to be connected to other flow lines or wells at the
inlet. This is done by enforcing mass balance to be satisfied
o o ,i
i jj
q q i F B
(3.36)
g g ,i
i jj
q q i F B
(3.37)
w w .i
i jj
q q i F B
(3.38)
66
And the pressure equality at the node,
I O , ,i j ip p i F B j (3.39)
where i is the set of flow lines and wells connected to the inlet of flow
line or outlet boundary i .
3.2.6 Objective
The objective is to maximize the total oil production rate, which can be
formulated as
omax q .ii B (3.40)
This assumes that all production ends in an outlet boundary node i B .
3.2.7 Constraints
The stated optimization problem can easily be incorporated with con-
straints on flow rates and pressures. This is done by
o o ,i iq q i W F B (3.41)
g g ,i iq q i W F B (3.42)
w w ,i iq q i W F B (3.43)
w o l .i i iq q q i W F B (3.44)
And for pressure there is an upper bound
o o ,i ip p i W F B (3.45)
where oip denote the maximal outlet pressure and a lower bound
o o .i ip p i W F B (3.46)
3.3 Case Study
The proposed method was applied to data from a real oil field in the
North Sea. The oil field consists of a flow line (including a riser) configu-
67
ration with two subsea templates. Each subsea template has two wells.
The topology is shown in Figure 3.1.
Each independent variable in the piecewise linear functions was modeled
using 10 break points. This gave each flow line about 10,000 break
points. Each well was represented by 100 break points.
Various cases were studied where the constraints were varied. The com-
putational times were in a range from less than 1 second to a little below
100 seconds on a standard personal computer.
The optimization was conducted using the commercially available Xpress
MP 2004A optimization software, which is a solver for mixed integer op-
timization programming. The computer used was an Intel® Pentium®
M 1.7 GHz with 1 GB RAM.
The evaluation of the pressure drops in the flow lines and the wells were
done using the simulation capabilities of the commercial available virtual
flow metering software Well Monitoring Software [65].
A comparison was made on the total oil production rates for the same
choke openings using the simulation software and the approximated
model in the optimization model. It showed a difference in the total oil
production rate in the range from 1 % to 3 %. This can further be re-
duced by including more break points in the optimization problem.
3.4 Conclusions
In this work a method of calculating the optimal oil production rates for
an oil production system has been developed. The method uses a piece-
wise linear model to approximate the pressure drops in wells and flow
lines. By using this, it is possible to find the global optimal production
rates for each well in the oil field. Furthermore, the global optimum is
68
found, unlike other methods, without requiring the user to provide an
initial solution. In a combination, this makes the method robust for the
user. A further advantage of the method is the ability to terminate at
any time after a feasible solution is found and still provide a bound on
the global optimum.
However, the method does require the user or implementation to be able
to decide on ranges for some of the independent variables. Furthermore,
the distance between each point in the approximation must be carefully
chosen.
The inclusion of a pressure drop equation of the flow lines in the model
extends earlier work on piecewise linear gas lift performance curves, and
allows handling of cases where two or more wells shares a flow line.
The optimization itself was done within reasonable time (about ten
seconds). However, the generation of the lookup tables for the cases stu-
died nearly consumed a day of computations. Fortunately, generation of
new lookup tables is only required when changing geometry of the pipes,
reservoir pressure, or temperatures.
The proposed method satisfied the accuracy required for production by
being in the range from 1 % to 3 % of the rates predicted by the original
model. The accuracy can easily be further improved at the expense of the
computational load.
3.5 Further Work
The proposed method requires a large number of calculations in advance
to build pressure tables used in the optimization. Further work should
focus on how to reduce this number while maintaining the accuracy of
the model.
69
Currently, the method does not include any rules for defining the bounds
on the independent variables used, and the distance used when creating
the grid. Such a method should be developed.
The proposed method includes much structure. This structure can be uti-
lized to generate valid inequalities in order to provide tighter bounds for
the branch and cut or branch and bound solver.
3.6 Nomenclature
W Indexes of wells
F Indexes of flow lines
iK Indexes of the points in GLPC for well i oiK Indexes of the points in oil direction for well i lgiK Indexes of the points in lift gas direction for well i giK Indexes of the points in gas direction for well i wiK Indexes of the points in water direction for well i piK Indexes of the points in inlet pressure direction for well i
,i k Weight of point k in GLPC of well i
o,i k Weight of o,i kq of well i
lg,i k Weight of lg,i k
q of well i
oo,i k Weight of o
o,i kq of flow line i
gg,i k Weight of g
g,i kq of flow line i
ww,i k Weight of w
w,i kq of flow line i
pp,i k Weight of p
I,i kp of flow line i
o lg, ,i k k Weight of point o lg
o lg
,,,i ki k
q q of well i
o w p, , , ,gi k k k k Weight of point o g w po g w p
, , ,,, , ,i k i k i ki k
q q q q of flow line i
oiq Oil rate for well or flow line i o,i kq Oil rate for point k in GLPC of well i lgiq Gas lift rate for well or flow line i
70
wiq Water rate for well or flow line i
o lgg
, ,i k kq Gas rate for o lg
o lg
,,,i ki k
q q of well i
o lgw
, ,i k kq Water rate for o lg
o lg
,,,i ki k
q q of well i
giq Gas rate for well or flow line i lg,i kq Gas lift rate for point k in GLPC of well i lg,M,i kq Maximal available total gas lift rate
q Volumetric flow rate
ip Outlet pressure of well i , with open choke
()ip Evaluate outlet pressure of well or flow line i , with open choke Iip Inlet pressure of well i Oip Outlet pressure of well i
o lg, ,i k kp Outlet pressure at o lg
o lg
,,,i ki k
q q of well i , open choke
o w p, , , ,gi k k k kp Outlet pressure at o g w po g w p
, , ,,, , ,i k i k i ki k
q q q q of flow line i , open
choke
i Set of wells and/or flow lines at inlet of flow line
i Index of well or flow line
j Index of well or flow line
k Index of point in GLPC ok Index of point in oil direction lgk Index of point in gas lift direction gk Index of point in gas direction wk Index of point in water direction pk Index of point in inlet pressure direction
71
Flow line
Well 1 Well 2
Riser
Well 3 Well 4
Figure 3.1: The well topology of field studied.
72
73
4 Optimal Well-Testing Strategy for Pro-
duction Optimization: A Monte Carlo
Simulation Approach
Based on
H.P. Bieker, O. Slupphaug, and T.A. Johansen,
submitted to
SPE Journal,
presented at
2006 SPE Eastern Regional Meeting
Canton, Ohio, U.S.A., 11–13 October 2006
4.1 Introduction
In the daily operation of an oil production system, some means of optimi-
zation is used to increase the oil production rate. For example, Lo and
Holden [20] proposed a linear program to maximize the total oil produc-
tion rate for an oil production system constrained by maximal gas, water,
and liquid flow rates. It was assumed that the oil production rate of each
well was independent. Furthermore, it was assumed that the wells were
not coning gas or water. The proposed linear program is listed below us-
ing a modified notation.
The goal is to maximize the total oil production rate,
omax ii I
q (4.1)
where oiq denotes the oil production rate from well {1, , }i I n . The
oil production rate from each well is limited by the oil potential oiq in
each well
74
o o .i iq q i I (4.2)
Furthermore, the oil production rate from each well must be nonnegative,
o 0 .iq i I (4.3)
Due to limited capacity in the processing facilities, the total production
rate of gas, water, and liquid is limited by gq , wq , and lq , i.e.
g o g,i ii I
r q q
(4.4)
w o w,i ii I
r q q
(4.5)
w o l( 1) ,i ii I
r q q
(4.6)
where gir and w
ir are the gas and water oil ratios, respectively. The ra-
tios will later be referred to as the resource oil ratios.
Unfortunately, the coefficients in the linear program are time varying and
not known accurately. They are usually estimated by well tests. The tests
are not completely accurate, and more important, the uncertainty of the
estimate will increase with time. Wells are therefore retested from time to
time, to ensure that the estimate has the accuracy needed.
The testing is done by routing an individual well to a dedicated test se-
parator. The oil, water, and gas production rates at the outlet of the test
separator are then measured. Thus, important properties including the
gas oil ratio and the water oil ratio can be calculated. A well test may
take several hours, thus constraining the frequency at which the wells can
be tested. A policy is therefore required to decide the frequency to test
each individual well. One simple strategy would be to test all wells at the
same frequency. Others may want to test some well more than others do.
This may be due to higher uncertainty in some wells, or that some well
are more important than other wells (for instance a higher potential oil
production rate). Independently of the strategy used, the goal of the well
test is usually to give information that will enhance oil production.
75
Depending on how the oil production system is constrained, testing may
or may not result in production losses during well testing. For oil produc-
tion systems limited by production separator capacity, the test separator
is often used as a production separator when not testing a well. During a
well test, only the well being tested can be routed to the test separator.
This means that the wells previously routed to the test separator would
have to be rerouted to the production separator. However, the rerouting
may not be possible because of the limited production separator capacity.
Thus, some wells may have to be choked back or closed to complete the
well test. Even if the production separator capacity is not a limitation,
testing may also result in losses because of transients; operators cannot
run the oil production system on its limit during rerouting.
Cramer et al. [66] proposed a tool for optimization and automation of
well tests. A well test schedule is provided by the user, and the tool is
able to retrieve the required measurements. The tool is also able to de-
termine when to stop a well test by using on-line measurement data. The
method proposed in this work may be used as a part of well test automa-
tion tools such as the one proposed by Cramer et al.
While Lo and Holden [20] restricted their production optimization me-
thod to independent wells without gas lift or pumps, other works have
been conducted to solve various challenges in production optimization.
They include optimization of gas lift [26-30] or a network of wells [27, 37-
39]. This work will not focus on the optimization process itself. An intro-
duction to various production optimization techniques that can be used
on such oil production systems was given by Bieker et al. [67].
The existence of multiphase flow meters may reduce the need for testing
because they provide measurements of gas oil ratios and water oil ratios.
Unfortunately, the accuracy has not been shown to be good enough to
replace well testing. Soft sensors such as Well Monitoring System [65] or
76
FlowManager5 may also provide this information. However, most oil pro-
duction systems still relay on well tests to provide data.
In this work a method, for deciding which well test is expected to give
the highest oil production rate gain will be developed. The algorithm
used to predict the oil production rate is currently restricted to a single
processing facility constraint. Because of this, ir will denote the resource
oil ratio of the constraint considered, for instance gas, water, or liquid. q
will be the treatment capacity for this constraint.
4.2 Monte Carlo Simulation
Monte Carlo simulation is a powerful method for obtaining an approx-
imate distribution of any dependent value of stochastic variables. The
distributions of the stochastic variables are assumed to be known. Using
the distributions of the stochastic variables, a finite number of samples of
the stochastic variables are drawn. For each sample, the dependent value
is calculated using a function evaluation. Important properties such as
the standard deviation and average of the dependent variables estimated
using Monte Carlo simulation would converge to the correct value. The
data flow of the proposed method is shown in Figure 4.1.
The production optimization is normally based on an estimated resource
oil ratio ir . The estimate may be found using various techniques, but the
simplest is perhaps the last well test. As a result, this is also the most
commonly used technique.
The uncertainty in the resource oil ratio ir for well i can be described by
the distribution iD . Methods for estimating the distribution iD will be
discussed in a later section. Given these distributions, m samples
5 FMC.
77
1,1 ,1 1, ,, , , ,n m n mr r r r are drawn. ,i jr denotes the resource oil ratio for
well i and sample j . Furthermore, if a test of well k is conducted, the
estimate is updated. It is assumed that the well test is accurate, and
thus,
,
, | : ,else
i j
i j ki
r i kr
r
(4.7)
where , |i j kr is the estimate of the resource oil ratio of well i after well k
is tested using sample j .
Each sample j represents a vector of the resource oil ratios for the wells.
For each sample, the oil production rate will be calculated. Furthermore,
the oil production rate is calculated for each of the wells k I we con-
sider for testing. Because ,i jr is not known during operation of the oil
production system, the inaccurate estimate , |i j kr is used for the well pri-
oritization. This prioritization will be discussed in a later section. Let
o o, 1, , 1, | , | 1: ( , , , , , , , , , )k j j n j j k n j k nz f r r r r q q q (4.8)
be a prediction of the total oil production rate found in a calculation us-
ing sample j and testing well k . The expected total oil production rate
is
,1
1:
m
k k jj
z zm
(4.9)
given that well k is tested. The well to test can then be calculated by
* : arg max .k I kk z (4.10)
The described method draws a finite number of samples from a distribu-
tion, and evaluates each of the samples using a function ()f . This means
that n Monte Carlo simulations are conducted—one for each well test
candidate. Monte Carlo simulation is a simple but powerful method.
Most of the assumptions made here can easily be relaxed allowing other
78
strategies to be used. Usually there is some inaccuracy in the well test
performed, and this can easily be included by adjusting (4.7) to adding
noise for the case i k . Furthermore, the goal can easily be changed to
maximize the profit rate instead of the oil production rate by adjusting
the function ()f accordingly. Variance in total oil production rates for
the samples may be penalized by adjusting (4.9) accordingly.
4.3 Calculating Production
A calculation is required to predict the oil production rate of the oil pro-
duction system for a given set of physical parameters and the correspond-
ing estimates used for the operation of the oil production system. The
physical parameters may for instance be gas or water oil ratios of the
wells. One simple and commonly used method is the swing producer-
based method [67]. The method assumes that at most a single processing
constraint is active and that each well has a maximal oil production rate.
The goal is to maximize the oil production rate. The wells are operated
under the rule that the wells with the lowest (estimated) resource oil ra-
tios are opened at the expense of choking back wells with the highest re-
source oil ratio. In the end, there will be one well partly choked back.
The rest will be fully closed or opened. Algorithm 1 describes the calcula-
tion and can be used for predicting ,k jz . The algorithm is executed by
providing a well test candidate k , a set I of wells, samples , |i j kr of esti-
mated resource oil ratios, samples ,i jr of the resource oil ratios, a maxi-
mum total treatment capacity q of the resource, and a maximum oil
production rate oiq of each well. The output is the total oil production
rate ,k jz and the oil production rate oiq of each well.
Algorithm 4.1
v q
J I
79
while J
, |argmini J i j ki r
if ,o
i j ir q v o
,i i jq v r
else o oi iq q
end if o
,i j iv v r q
\{ }J J i
end while o
,k j ii Iz q
A key element here is the use of the sampled values ,i jr to ensure that
the physical constraints are not violated. The estimates are only used for
the well prioritization. For more complex oil production systems, Algo-
rithm 1 should be replaced with a more accurate prediction algorithm
that reflects the fashion the oil production system is operated.
4.4 Error Distribution of Oil Resource Ratio
The distribution of the resource oil ratio iD is generally not known.
Therefore, some method for estimating it will be required. In this work, it
will be assumed that the only available measurement is a set of historical
well tests. The historical well tests only include the resource oil ratio and
the time the well tests were conducted.
Assume that the variation in the resource oil ratio found in the historical
well tests indicate how much it changes from one test to another test.
One will expect that a new test will show little change if only small
changes have been observed in the past. Furthermore, old tests are ex-
pected to be less accurate than newer tests.
80
Assume that we can find the resource oil ratio between the first and the
last well test by using an interpolation algorithm such as a spline func-
tion [68] in Matlab. Then ( )ir will be completely defined for 0 1[ , ]i it t .
The time since the last well test will be denoted
1: ,i iT t t (4.11)
where t is the current time. The resource oil ratio : ( )i ir r t at time 1it t is the unknown value
1 1( ) ( , ),i i i i i ir r t e t T (4.12)
where 1( , )i i ie t T is the change in the resource oil ratio since previous well
test. The value of 1( , )i i ie t T is unknown. However, because historical
changes in the resource oil ratio are expected to indicate future changes, 1( , )i i ie t T will be looked at as a stochastic variable. A random sample of
the resource oil ratio will be found by
1, ,: ( ) ( , ),i j i i i i j ir r t e w T (4.13)
where ,i jw is a sample drawn from a uniform distribution on the interval 0 1[ , ]i i it t T . Thus, it is expected that the change in the resource oil ratio
from 1t to 1it t T is the same as from ,i jw to ,i j iw T for sample j ,
because , , ,( , ) ( ) ( )i i j i i i j i i i je w T r w T r w follows from (4.11)-(4.12).
,( )i i jr w and ,( )i i j ir w T are defined by said interpolation. The concept is
illustrated in Figure 4.2. If structural information about valid samples is
available, then (4.13) is modified to reflect this information. For gas oil
ratios, this means they are nonnegative,
1, ,: max{0, ( ) ( , )}.i j i i i i j ir r t e w T (4.14)
For liquid oil ratios, this means they are at least one,
1, ,: max{1, ( ) ( , )}.i j i i i i j ir r t e w T (4.15)
81
4.5 Case Study
The proposed method for choosing a well to test was simulated using his-
torical well test data from an offshore oil production system in the North
Sea. The simulation included 21 wells, and the total liquid production
rate was restricted to 10,000 Sm3/D. The length of the simulation was
180 days.
Each well was modeled in the simulation as a producer of oil and water.
The model consisted of a potential oil production rate and a liquid oil
ratio. The potential oil production rate for each well was defined by an
interpolation of the potential oil production rate found in the well test
data, making it a time variant function. To ensure a reasonable interpo-
lation, it was restricted to be non-negative. The liquid oil ratio was de-
fined as an interpolation of the liquid oil ratios found in the well test da-
ta, but restricted to be at least one.
The simulation started with the knowledge of all well tests done before
this day. The most recent well tests were used as estimates. For each
time a well test was found in the well test data, a well test was simu-
lated. The proposed method was used to select a candidate for testing,
and the liquid oil ratio from the interpolation was reported.
Figure 4.3 shows the total oil production rate as a function of time. The
cumulative oil production is shown in Figure 4.4. “No new tests” indicate
that estimated oil production rate will stay the same through the period.
No new information is utilized. “Field data” indicates that the well tests
are performed at the same time as found in the well test data. “Proposed
method” indicates that well tests are performed as proposed by the pro-
posed method. “Perfect information” indicates that all wells are tested
continuously. This gives an upper bound of the oil production rate. In
Figure 4.5, the estimates of the liquid oil ratio for one of the wells are
82
plotted as a function of time. “Proposed, samples” indicates by dots the
estimated distribution of the liquid oil ratio in the proposed method for
each time. The increased spread in the dots indicates that the quality of
the estimate decreases with time.
4.6 Conclusions
A method for finding the well to test in order to achieve the highest ex-
pected oil production rate has been developed. The method assumes that
the oil production rates from each well are independent and that the
treatment capacity constraint in the production can be described as a
single flow rate constraint in water, liquid, or gas. It is assumed that an
estimate of the resource oil ratio for each well is available. The well test
inaccuracy is neglected in this work. This assumption can however easily
be relaxed with the Monte Carlo framework used.
The method works by supplying the processing capacity of the oil pro-
duction system, the potential oil production rate of each well, and a list
of previous well test data for each well. The list should include the gas oil
ratios or the water oil ratios and the date of the test. For each well con-
sidered for testing, a Monte Carlo simulation is conducted using a proba-
bility distribution based on previous changes in the well tests. The Monte
Carlo simulations are only different in values of the estimates of the gas
oil ratio or water oil ratios. The well test giving the largest expected oil
production rate is recommended for implementation on the physical oil
production system.
A computational study using field data indicates a possible gain of using
this method. An increase of 7.5 % in the expected total oil production
rate from the oil production system has been shown over the currently
used method. The proposed method gave 96.3 % of the theoretical ex-
pected maximal oil production rate when perfect information is available
83
for the case studied. In the computational study, it assumed the pressure
interaction has been assumed neglectable, which makes the method most
useful for oil production systems having platform wells. The result also
requires that only one treatment constraint is active. Furthermore, the
method currently only uses well test data, and the well test data must
therefore reflect the future development of the gas oil ratio or water cut
in order to make the method work.
Because of the Monte Carlo simulation framework, the method is extens-
ible. The methods for calculating the oil production rate and estimating
the accuracy of the estimate may be replaced by other methods.
4.7 Further Work
The method should be extended to include available measurements that
can indicate changes in the gas oil ratio or water oil ratio. This includes
inaccurate samples of those variables and pressure and temperatures of
the well stream.
The method is currently restricted to single constraints. This should be
relaxed by developing a simulation method for multiple constraints. Fur-
thermore, support for more complicated oil production systems with
shares flow lines should be supported.
The parameters of the wells will also change after the well has been
tested. Currently, the optimization ignores this. The quality of the opti-
mization may be improved by changing the calculation from calculating
the rates at a single time to calculate the rate for a time interval of a few
days (using a stochastic development). Furthermore, the optimization
may be changed to decide a well test schedule instead of just proposing a
single well for testing. The decision-making can then be done in a reced-
ing horizon way.
84
4.8 Nomenclature
i Index of well
j Index of sample
k Well for testing *k Well recommended for testing
m Number of Monte Carlo simulations
n Number of wells
I Set of wells
J Working set of wells
ir Resource oil ratio for well i
( )ir Resource oil ratio for well i at time
ir Estimate of resource oil ratio for well i before test
, |i j kr Estimate of resource oil ratio for well i , well k tested and
sample j
,i jr Sample j of resource oil ratio for well i gir Gas oil ratio for well i wir Water oil ratio for well i oiq Oil production rate for well i oiq Maximal oil production rate for well i gq Maximum total gas production rate wq Maximum total water production rate lq Maximum total liquid production rate
v Unused capacity of resource
q Maximum total capacity of resource
( , )i ie t T Change in ( )ir when changing from t to it T
t Time when the well test is performed 0it Time of first well test of well i 1it Time of last well test of well i
iT Time since last well test on well i
Time
85
,i jw Time sample j for well i
iz Average oil production rate for all simulations when well i on
test
,i jz Total oil production rate for the simulation with sample j
when well i on test
86
1100 1150 1200 1250 1300 1350 1400 14502
4
6
8
10
12
14
16
Time [days]
Liq
uid
oil
ratio
[S
m3/S
m3]
1100 1150 1200 1250 1300 1350 1400 14501
1.5
2
2.5
Time [days]
Liq
uid
oil
ratio
[S
m3/S
m3]
0 2 4 6 8 10 12 14 16 180
2000
4000
6000
8000
10000
12000
14000
16000
18000
Liquid oil ratio ([Sm3/Sm
3]
Nu
mb
er
of
sam
ple
s
1.4 1.6 1.8 2 2.2 2.4 2.60
0.5
1
1.5
2
2.5
3
3.5
4x 10
4
3500 4000 4500 5000 5500 6000 6500 70000
1000
2000
3000
4000
5000
6000
7000
8000
9000
Oil rate [Sm3]
Nu
mb
er
of
sam
ple
s
3500 4000 4500 5000 5500 6000 6500 70000
2000
4000
6000
8000
10000
12000
Oil rate [Sm3]
Nu
mb
er
of
sam
ple
s
Maximum on average
Testing well 2
Testing well 1
Testing well 1
Generating samplesCalculating total oil
production using information from each test
Choosing the test giving highest total oil production
Obtaining well test data
Figure 4.1: The schematic data flow of the method is shown.
87
0 5 10 15 20 25 301.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Time [days]
Liq
uid
oil
ratio
[S
m3/S
m3]
Ti
ei(w
i,j,T
i)
wi,j
wi,j
+Ti
Figure 4.2: The estimate distribution is calculated by drawing time sam-
ples to obtain the change in the resource oil ratio.
Figure 4.5: The historical information is exploited to determine the accu-
racy of the current estimate.
91
5 Well Management under Uncertain Gas
or Water Oil Ratios
Based on
H.P. Bieker, O. Slupphaug, and T.A. Johansen,
submitted to
SPE Production & Operations Journal,
presented at
2007 SPE Digital Energy Conference and Exhibition
Houston, Texas, U.S.A., 11–12 April 2007
5.1 Introduction
In the daily operation of an oil production system, operators may use
various means of optimization to maximize the total oil production rate.
For example, Lo and Holden [20] proposed a linear program to maximize
the total oil production rate of an oil production system having a
processing facility constrained by maximum total gas, water, and liquid
production rates. The method assumed the oil potential for each well to
be independent of the oil potentials of the other wells (independent
wells). This is typically true if the wells do not share long flow lines. For
offshore oil production systems, so-called platform wells often satisfy this
assumption because the production manifold (a blending point) is placed
on the processing platform itself close to the wellheads, not on the
seabed. For such platform wells, the pressure drop in the shared piping
between the production manifold and the production separator may be
regarded as constant because of the short distance and typically large
diameters of the piping used. Furthermore, the method assumed that the
wells are not coning gas or water. Gas-lifted wells are generally not han-
92
dled by the method. The proposed linear program [20] is shown next us-
ing a modified notation. The goal is to maximize the total oil production
rate
omax ,ii I
q (5.1)
where oiq denotes the oil production rate from well i{1, , }i I n . The
oil production rate from each well is limited by the oil potential oiq in
each well,
o o .i iq q i I (5.2)
Furthermore, the oil production rate from each well must be nonnegative,
o 0 .iq i I (5.3)
Due to limited capacity in the processing facilities, the total production
of gas, water, and liquid are limited by gq , wq , and lq . Thus,
g o g,i ii I
r q q
(5.4)
w o w,i ii I
r q q
(5.5)
w o l( 1) ,i ii I
r q q
(5.6)
where gir is the gas oil ratio and w
ir is the water oil ratio. For simplicity, l w: 1i ir r will be used to denote the liquid oil ratio. The liquid is de-
fined as a combination of oil and water.
For supporting the operation of an oil production system, the linear pro-
gram stated above is rarely solved. The actual control handles are typi-
cally choke openings, not oil production rates assumed in the linear pro-
gram above, making the above method hard to implement. Furthermore,
the model does not handle uncertainties in the gas oil ratios gir , water oil
ratios wir , oil potentials o
iq , processing capacity gq of gas, processing ca-
pacity wq of water, or processing capacity lq of liquids. Instead, a simple
method able to handle a single, but uncertain, processing capacity con-
93
straint in the processing facilities in addition to choke openings as control
handles is typically used. The method works by creating a list of the
wells sorted in ascending order by the ratios gir , w
ir , or lir depending on
the type of capacity constraint. It follows from the definition of lir that
the sorted order of wir and l
ir will be the same. Starting with all wells
closed, the first well on the list is opened until the oil potential of the
well or the processing capacity is met. The method then continues to the
next well on the list. It can be guaranteed that the total oil production
will be maximal in the end if the prioritization list is sorted properly and
the uncertainty in the ratios is small enough to ensure that the priority
list will be the same for all combinations of the uncertainties. A more de-
tailed description of the algorithm can be found in Bieker et al. [69]. The
first well on the list is said to have the highest priority, and the list itself,
when sorted, is the prioritization list of the wells of the oil production
system. A well partly closed or partly opened is often called a swing pro-
ducer. The name is used because this well is used to control the produc-
tion so that the capacity of some processing resource is not over-utilized
or under-utilized.
Lo and Holden [20] restricted their production optimization method to
independent wells without gas lift or pumps, but others have conducted
work to solve challenges including the optimization of gas lift wells [26-
30] and networks of wells [27, 37-39, 70]. A survey on various production
optimization techniques that can be applied to such oil production sys-
tems was given by Bieker et al. [67].
The focus will be on a method for explicitly handling the uncertainties in
the optimization, and not on the solving of the associated optimization
problem.
Most optimization techniques use a mathematical model, and this model
has to be parameterized. For the model of Lo and Holden, this is the oil
94
potential of each well, processing capacities, gas oil ratios, and water oil
ratios. Unfortunately, these capacities and ratios are uncertain. In partic-
ular, the gas oil ratios and the water oil ratios of each well will generally
be changed as the reservoir is depleted and as water or gas injection
proceeds. Typically, they will also change with the oil production rates of
the wells because of the coning effect.
The gas oil ratio gir and the water oil ratio w
ir are typically found by well
tests. In some cases, the ratios are found by fluid sampling at the well-
heads, where the fluid sample is analyzed at the laboratory, or by using
multiphase flow meters. The validity of these well tests will decline with
time because of changed reservoir and well conditions, and eventually a
new well test or fluid sample will be required [69]. The decreased validity
of the well tests will be represented as increased uncertainty in the gas or
water oil ratios in this work.
In this paper, it is first shown by an example that if there is uncertainty
in these ratios, the order at which the wells should be prioritized might
be counterintuitive in that using the expected values of the ratios gives a
sub optimal solution. A generic method for choosing the order is then
presented, and a case study is conducted to identify the gains of the me-
thod. The method uses mixed integer linear optimization, and standard
commercially available solvers can solve the associated optimization
problem to a global optimum.
5.2 Uncertainty Matters
In this section, it will be shown that the uncertainties in the ratios mat-
ter. Consider a case with two wells with the same oil potential o o1 2q q of
100 Sm3/D. The liquid oil ratios for the wells have the possible combina-
tions shown in Table 5.1. Each combination is referred to as a sample. In
other works [49-52], the term scenario has been used. Well-1 has a liquid
95
oil ratio l1,1r of 2.0 or l
1,2r of 2.99. The probability of each outcome of the
liquid oil ratio is the same, and this gives the uncertainty in the optimi-
zation problem. Well-2 has a liquid oil ratio l l2,1 2,2r r of 2.5. Thus, the
expected value of the liquid oil ratio is the smallest for Well-1. Two cases
are considered, one case where the processing capacity of liquids is
200 Sm3/D and one case where it is 449 Sm3/D. If both wells produced
their oil potential, the total liquid production rate would be 450 Sm3/D
or 549 Sm3/D. Two cases are investigated where the processing capacity
is not large enough for both wells to produce at their oil potentials, and
at least one well must be choked back. Furthermore, because the liquid
oil ratio is not measured on-line, the well to choke back must be chosen
without using the actual liquid oil ratio. Some calculations will be per-
formed below to choose this well. The calculations will use the uncertain-
ty distributions of the liquid oil ratios.
5.2.1 Low Processing Capacity
The processing capacity of liquid is set to 200 Sm3/D. The processing ca-
pacity is enough to produce from only one well.
5.2.1.1 First Well-1, then Well-2
Well-1 is opened first, and the processing capacity is fully utilized when
the oil production rate for the well is given by l1,1r , the first sample,
lo 31,1 l
1,1
200100.00 Sm /D,
2.0
qq
r
and when using l1,2r , the second sample,
lo 31,2 l
1,2
20066.89 Sm /D.
2.99
qq
r
96
Because all the processing capacity is used, Well-2 will be closed, i.e. o o2,1 2,1q q is 0 Sm3/D. The expected value of the total oil production rate
for Well-1 and Well-2 is 83.44 Sm3/D.
5.2.1.2 First Well-2, then Well-1
Well-2 is opened first, and the processing capacity is fully utilized when
the oil production rate for the well is given by l1,1r , the first sample,
lo 32,1 l
2,1
20080.00 Sm /D,
2.5
qq
r
and when using l1,2r , the second sample,
lo 32,2 l
2,2
20080.00 Sm /D.
2.5
qq
r
Because all the processing capacity is used, Well-1 will be closed, i.e. o o1,1 1,2q q is 0 Sm3/D. The expected value of the total oil production rate
for Well-1 and Well-2 is 80.00 Sm3/D.
5.2.2 High Processing Capacity
The processing capacity of liquids is set to 449 Sm3/D. The processing
capacity is enough to produce fully from one and partly from a second
well.
5.2.2.1 First Well-1, then Well-2
Well-1 is opened first, and it will be producing at its oil potential o o o 31,1 1,2 1 100 Sm /Dq q q . If the liquid oil ratio of Well-1 is given by l
1,1r , the first sample, the oil production rate from Well-2 is
l o l1 1,1o 3
2,1 l2,1
449 100 2.099.60 Sm /D,
2.5
q q rq
r
and when using l1,2r , the second sample,
97
l o l1 1,2o 3
2,2 l2,2
449 100 2.9960.00 Sm /D.
2.5
q q rq
r
The expected value of the total oil production rate for Well-1 and Well-2
is 179.80 Sm3/D.
5.2.2.2 First Well-2, then Well-1
Well-2 is opened first, and it will be producing at its oil potential o o o 32,1 2,2 2 100 Sm /Dq q q . If the liquid oil ratio of Well-1 is given by l
1,1r , the first sample, the oil production rate from Well-1 is
l o l1 2,1o 3
1,1 l1,1
449 100 2.595.50 Sm /D,
2.0
q q rq
r
and when using l1,2r , the second sample,
l o l1 2,2o 3
1,2 l1,2
449 100 2.566.56 Sm /D.
2.99
q q rq
r
The expected value of the total oil production rate for Well-1 and Well-2
is 183.02 Sm3/D.
5.2.3 Comparison
Because the expected liquid oil ratio is less for Well-1 than Well-2, it is
intuitive to first open Well-1 and then open Well-2. For the low
processing capacity, this order gives the maximum expected total oil pro-
duction rate. However, for the high processing capacity, the expected to-
tal oil production rate is 179.80 Sm3/D for this order. By changing the
order to first open Well-2 and then open Well-1, the expected total oil
production rate increases to 183.02 Sm3/D, an increase of 3.23 Sm3/D or
1.80 %. Thus, the prioritization list should be Well-2 and then Well-1,
i.e. first open Well-2 and then Well-1 when having a high processing ca-
pacity. The expected total oil production rate for variable processing ca-
pacity for this particular example is shown in Figure 5.1. Similar changes
98
in the optimal prioritization list may easily be found by changing the oil
potential of a well.
In the next section, a generic method for choosing the prioritization order
under uncertainty in an optimal way will be developed.
5.3 Proposed Method
Assume that there is a finite number of possible combinations of values
of the liquid oil ratio of each well. Call each combination a sample. Each
sample will be denoted by an index k1, ,k K n . kn is a positive
integer denoting the number of samples. The method will find a prioriti-
zation list of the wells used to decide the order to choke back or open the
wells. The position in the list will be denoted by j1, ,j J n . The
length of the list is the same as the number of wells, thus j i:n n .
One of the decisions to be taken within the proposed method is the oil
production rate o,i i kq for each well i and for each sample k . ,i k is the
fraction of the oil potential of well i for sample k . This oil production
rate may be different for each sample because of the difference in the liq-
uid oil ratio. The objective of the optimization is to maximize the ex-
pected total oil production rate, or more specifically
k
o1,
, ,max .i i kno z
i I k K
q
(5.7)
The variables ,i jo are decision variables for the prioritization list, and the
variables ,j kz are used to define the prioritization strategy. The minimum
and maximum oil production rates for each well are limited by
,0 , ,i k i I k K (5.8)
, 1 , .i k i I k K (5.9)
The processing capacity of the liquid is limited to lq for each sample k
by
99
l o l, , ,i k i i k
i I
r q q k K
(5.10)
where l,i kr is sample k of the liquid oil ratio of well i . The constraints
are the same as those given in the work of Lo et al. [20] for each sample
and position in the prioritization list.
Each well is restricted to be placed only in one position in the prioritiza-
tion list. For this, a set of binary decision variables ,i jo will be used for
each well i and position j . Thus,
, 0,1 , ,i jo i I j J (5.11)
where , 1i jo means that well i is placed at position j in the prioritiza-
tion list. These variables make the optimization problem a mixed integer
linear program, and can be solved to a global optimum using a branch
and bound algorithm [14]. In order to ensure that each well is restricted
to exactly one position in the list,
, 1 .i jj J
o i I
(5.12)
A similar constraint is required to ensure that each position is restricted
to exactly one well. Thus,
, 1 .i ji I
o j J
(5.13)
The oil production rate is determined for each combination of a well and
a sample.
The prioritization list must be enforced. This means that for each sample,
the well at position j in the prioritization list must be fully opened if
any of the wells having a larger position is opened. This can be enforced
by using a binary variable
, 0,1 , .j kz j J k K (5.14)
100
A binary variable ,j kz can be defined to indicate if the production from a
well at position j of the prioritization list must be fully utilized. By the
use of an inequality
, , ,1 , ,j k i j i kz o i I j J k K , (5.15)
the production from well i is forced to be fully opened if at position j of
the prioritization list and , 1j kz . In fact, the inequality can be replaced
by a stronger inequality [14, 15]
, , ,1 , ,j k i j i kh Jh j
z o i I j J k K
. (5.16)
Furthermore, well i must be closed if it is not at position j , or less in
the prioritization list and , 0j kz . This means
, , ,1
, , .j
i k j k i hh
z o i I j J k K
(5.17)
The value of the processing capacity constraint lq and the oil potential oiq may be subject to uncertainty. This can be generalized by treating
them as sampled values ljq and o
,i jq . The added index j is used to indi-
cate that it is a sampled value.
The method presented above requires kn sample values of the liquid oil
ratio of each well to be available. Various methods can be used to select
them. For the example in the previous section, two samples were used.
Each sample represents a possible value in the distribution. However,
when there are uncertainties in more than one variable, the number of
samples grows rapidly. To reduce the computational load, the samples of
liquid oil ratios were chosen by randomly from their distributions using
Monte Carlo simulations. For such simulations, the number of samples kn is typically around 100 or 1000. The distributions of the liquid oil ra-
tios from which the samples were drawn were similar to Figure 5.2.
Moreover, the method is not restricted to oil production system with li-
101
mited liquid processing capacity. It can be used without modification on
systems limited by gas processing capacity, or some other processing ca-
pacity, by replacing all references to liquids by gas.
When there are only uncertainties in the processing capacities and the oil
potential, and not in the ratios, the simple method in the previous sec-
tion will give an optimal order for the proposed method.
It should be noted that the order from the proposed method might not be
unique even if the ratios are distinct and there are limited processing ca-
pacity, e.g. the first and the second well on the prioritization list may be
rearranged without affecting the maximum expected total oil production
rate when the third well on the prioritization list is not fully closed. The
wells that are fully closed may also be rearranged freely without affecting
the maximum expected total oil production rate.
5.4 Case Study
The proposed method was applied to field data from an oil production
system in the North Sea. The oil production system consists of 21 oil
production wells. The oil production was constrained by a liquid
processing capacity of 36,000 Sm3/D.
For each well, a probability distribution of the liquid oil ratio was calcu-
lated using the variations in the liquid oil ratios found in historical well
tests. The method used for calculating the distributions of the ratios can
be found in Bieker et al. [69]. The oil potential for each well was set to
the value from the last well test. k 100n samples from each distribution
were randomly drawn, populating the samples l,i jr with non-negative val-
ues. The oil potential of the wells varied from around 30 Sm3/D to just
over 2100 Sm3/D.
102
The XPress MP 2004A optimization software was used to solve the
mixed integer optimization problem, and it found a feasible integer solu-
tion giving 10,514.8 Sm3/D as the maximum expected total oil produc-
tion rate. The distribution of the total oil production rate is shown in
Figure 5.3. After 1482 seconds, the solver had processed 646 nodes (linear
sub problems) and had found 11 feasible integer solutions. An optimal
integer solution was still not proven after around 10,700 seconds or 1600
nodes. However, the solver gave an upper bound of the maximum ex-
pected total oil production rate of the samples as 10,565.3 Sm3/D. Thus,
at most 0.48 % is lost by terminating the solver at this time. The devel-
opment of the upper and lower bounds versus the number of nodes eva-
luated is shown in Figure 5.4. The computer used was an Intel® Pen-
tium® M 1.7 GHz with 1 GB RAM.
The 0.48 % loss is probably conservative. Seen from Figure 5.4, the upper
bound did not decrease during the solving. However, an alternative
branching strategy was tried that prioritized decreasing the upper bound.
The strategy was able to decrease the upper bound. Thus, only a small
further increase in the expected total oil production rate can be gained
by improving the solver software or the time the computer spent solving
the problem.
The method was compared to a standard way of doing optimization pri-
oritizing by the expected liquid oil ratio [67]. A list of the wells was made
where the wells were sorted by the expected value of the liquid oil ratios
(i.e. the average of the liquid oil ratios of the samples). For each sample,
a total oil production rate was calculated using this ordering and liquid
oil ratios from the samples. The expected total oil production rate was
then found to be 9,519 Sm3/D, giving the new method around a 10.1 %
higher expected total oil production rate on average than the rate found
when prioritizing the wells using the expected value of the liquid oil ratio
103
of each well. The increase found is limited to the case studied, but similar
result may be found in other case studies. It is assumed that the proba-
bility distributions are accurately known, which may not always be the
true. Furthermore, the result is an expected value and the actual increase
depends on the actual values of the water cuts.
5.5 Conclusions
A method for optimal well management explicitly handling uncertainty in
water or gas oil ratios has been developed. The method presented handles
a single, possibly uncertain, processing capacity constraint. Furthermore,
the method is restricted to wells having independent, possible uncertain,
oil potentials.
A case study based on field data showed an expected increase in the total
oil production rate of 10.1 % compared to a method where the expected
value of the probability distribution was used to prioritize the wells.
It has been shown by an example that the optimal prioritization list,
when having uncertain ratios, generally depends on the processing capaci-
ty of the production system. Furthermore, the optimal prioritization list
depends generally on the oil potential of the wells.
By using a commercially available branch and bound solver, the feasible
solution found in the example has been provn to be not more than
0.48 % less than the global optimum.
5.6 Further Work
This work should be extended to support multiple processing capacity
constraints. To be able to do this, the method of prioritization lists must
be generalized to multiple constraints.
104
A method for determining the distributions of the water oil ratios is re-
quired for this method, and additional research on improving the esti-
mates of the distributions may further improve the gains of the method.
This work used the method proposed by Bieker et al. [69].
The mathematical formulation of the optimization problem should be
elaborated in order to decrease the computational load of the algorithm.
Often, this means developing valid inequalities making the linear (con-
vex) relaxation of the optimization problem tighter to the integer pro-
gram itself, thus the required number of nodes to be evaluated is reduced.
Furthermore, a priori knowledge of the distributions of gas or water oil
ratios may be used to add rules forcing particular wells always to have
higher priority than other wells. Such rules may reduce in many cases the
number of required node evaluations in the branch and bound code, and
reduce the computational load.
Table 5.1: The liquid oil ratios for each of the samples are different for
Well-1. For Well-2, the values are the same.
Liquid oil ratio Sample 1 Sample 2
Well-1 2.00 2.99
Well-2 2.50 2.50
105
0 100 200 300 400 500 6000
20
40
60
80
100
120
140
160
180
200
Processing capacity of liquid, Sm3/D
Ex
pe
cte
d t
ota
l o
il p
rodu
ctio
n r
ate
, S
m3/D
Fist Well-1, then Well-2
First Well-2, then Well-1
Figure 5.1: The prioritization list giving the maximal expected total oil
production rate for the two-well example depends on the processing ca-
pacity.
106
1 2 3 4 5 6 7 8 90
2
4
6
8
10
12
Liquid oil ratio, Sm3/D
Fre
qu
en
cy in
dis
trib
utio
n
Figure 5.2: The liquid oil ratio of each well was represented by a multi-
tude of samples. The example shows the distribution of the water oil ra-
tio of a particular well from the case study.
107
0.9 0.95 1 1.05 1.1 1.15
x 104
0
2
4
6
8
10
12
14
Total oil production rate, Sm3/D
Fre
qu
en
cy in
re
su
lt
Figure 5.3: The total oil production rate for the optimal solution is a dis-
tribution.
108
0 200 400 600 800 1000 12001.044
1.046
1.048
1.05
1.052
1.054
1.056
1.058x 10
4
Node #
Av
era
ge
oil
rate
, S
m3/D
Lower bound
Upper bound
Figure 5.4: After 650 evaluated nodes, the gap between the upper and
lower bound of the objective function was only 0.48 %.
109
6 Optimal Start-up Scheduling of Produc-
tion Wells
Based on
H.P. Bieker, O. Slupphaug, and T.A. Johansen,
unpublished results
6.1 Introduction
At every instance of time, the operators of an oil production system are
challenged by decisions on maximizing the production of the oil produc-
tion system. The capacity of the processing equipment may change due
to reasons including maintenance, changed weather, and wear-out. Other
reasons to change the operation might be new data being available, or
changes in the reservoir or wells. A particular challenge arises during the
start-up of an oil production system.
When an oil production system is started after a shutdown, the operators
are usually supposed to open the wells so that they eventually will reach
optimal operational conditions, for instance the maximal total oil produc-
tion rate. Furthermore, this should be done as quickly as possible without
risking a new shutdown. For instance, the separators are usually con-
trolled by two level control loops and one pressure control loop. If the
feed to a separator is suddenly increased, then the gas pressure and the
interface levels in the separator will increase. The feedback controllers
will open the valves to counteract this. However, for large and sudden
changes, the valves might be opened too late, and the levels or pressure
will exceed the permitted values and an automatic shutdown will be trig-
110
gered. Thus, the rate at which the oil, gas, and water flows are changed
should be restricted.
Lo and Holden [20] proposed a linear program for finding the steady state
optimal operational conditions. They assumed that each well i I had
unconstrained rate streams oiq , w
iq , giq for oil, water, and gas, respec-
tively. I is the index set of wells. To find the optimal production, a li-
near program was formulated
omax ,i ii I
q (6.1)
subject to the constraints
o o,i ii I
q q
(6.2)
g g,i ii I
q q
(6.3)
w w,i ii I
q q
(6.4)
o w l( )i i ii I
q q q
(6.5)
where oq , wq , gq , lq was the field stream constrains. The decision va-
riables ,i i I represent the choke settings. 0 indicates a closed choke
and 1 indicates a fully opened choke, and thus the problem is further
constrained by
0 1, .i i I (6.6)
The problem with n nonnegative decision variables with upper bounds
and 4 general constraints can then easily be solved by the Simplex me-
thod [71] or similar methods. Later, Lo et al. [72] extended the model to
also support gas-lifted wells.
Both models of Lo et al. [20, 72] assumed that the production manifold
had a fixed pressure, and that the process capacities could be mapped to
flow rate constraints at this manifold. This assumption of independent
111
production rates from each well makes the methods unsuitable for oil
production systems where the manifold pressure changes with the pro-
duction of the wells. Numerous approaches have been proposed to handle
this nonlinearity [32-35, 37, 39, 40]. However, in this paper the assump-
tion of Lo et al. [20] will be used.
In the operation of most oil production systems, mathematical program-
ming is not used to choose the wells to open and the wells to close. In-
stead, priority lists having the water cut and gas oil ratio of each well are
used. When the gas processing capacity becomes limited, the non-closed
well having the highest gas oil ratio is choked back and the well not fully
opened having the lowest gas oil ratio is increased is opened. The liquid
processing capacity is handled a similar way using the water cut.
Example 1 Consider an oil production system with three wells A-7A, A-
23, and A-26 (Table 6.3). The gas and liquid production is limited to
400,000 Sm3/D and 2500 Sm3/D, respectively. The production is started
from zero production. Table 6.1 and Table 6.2 list the sequence at which
the chokes are opened. In Table 6.1, the chokes are opened starting with
the wells having a low gas oil ratio, and changing into priority on lower
water cut when it becomes an issue. In Table 6.2, the chokes are opened
starting with the wells having a low water cut. Neither strategy reaches the
maximal oil production of 468.5 Sm3/D where A-26 is closed. # denotes
the sequence number.
This example motivates for finding a strategy that ensuring the optimal
production is approached, which is the topic of the next section.
6.2 Short-Term Optimization
The optimization problem of Lo et al. [20] does not give any hints of how
to find a trajectory that eventually will lead to optimal operational con-
112
ditions. There will typically be some constraints on how fast each well
can be opened or closed, and how fast the total production can change to
reduce the risk of transients triggering a shutdown of the system. Because
of this restriction, it might be more interesting to find optimal operation-
al conditions at some specific time not too far in the future not violating
the restriction on how fast each well can be opened or closed.
The current time will be denoted 0t , and the goal will be to maximize the
oil production at the next time step 0 1t . This is what often is called
short-term optimization. The model of the previous section can then be
extended to fit this purpose by letting the decision variables i from the
previous section denote the choke setting of well i at time 0 1t .
High draw down in a well can damage the near well area. Therefore, the
rate at which flow can be changed will be restricted. By using i to spe-
cify the largest change in fractional opening of the choke that can be
done in each time step, this constraint can be modeled as
, 0, ,i i t i i I (6.7)
, 0, .i t i i i I (6.8)
The risk of triggering a shut down usually increases with the rate of
change in total production rates. Let , 0i t denote the choke setting for
well i at current time, then the rate of change in total production rates
can be restricted by
o o, 0
( ) ,i i t ii I
q
(6.9)
o o, 0
( ) ,i t i ii I
q
(6.10)
g g, 0
( ) ,i i t ii I
q
(6.11)
g g, 0
( ) ,i t i ii I
q
(6.12)
113
w w, 0
( ) ,i i t ii I
q
(6.13)
w w, 0
( ) ,i t i ii I
q
(6.14)
o w l, 0
( )( ) ,i i t i ii I
q q
(6.15)
o w l, 0
( )( ) .i t i i ii I
q q
(6.16)
The linear program (6.1)-(6.16) can then be solved and the calculated
changes of each well can be implemented to the wells. At time
0 0 1t t , the problem is solved again with updated information on
current values of , 0i t . New estimates of the process constraint parame-
ters oq , wq , wq , and lq should also be incorporated.
6.3 Full Horizon Optimization
In the previous section, only the next time step was considered during
optimization, giving the maximal oil production rate at this time step.
However, this does not ensure that the cumulative losses during the tran-
sient from zero production to full production are minimized. In this sec-
tion, the linear program will be revised to not only consider the time
0= 1t t , but any t on the horizon 0 0= { 1, , }T t t n where n is
some (finite) positive integer specifying the number of time steps in the
horizon.
The objective is to maximize the cumulative oil production, thus (6.1)
extends to
o,max i t i
t T i I
q (6.17)
where the subscript t is introduced to ,i t to denote the time step. Con-
tinuing the extension yields for (6.6)
,0 1, , .i t i I t T (6.18)
The processing capacities (6.2)-(6.5) must be fulfilled each point in time
114
o o, , ,i t i
i I
q q t T
(6.19)
g g, , ,i t i
i I
q q t T
(6.20)
w w, , ,i t i
i I
q q t T
(6.21)
o w l, ( ) , .i t i i
i I
q q q t T
(6.22)
Quality constraints on for instance H 2 S can be handled by
g c, ( ) , .i t i i
i I
c c q q t T
(6.23)
where cq is the quantity that can be removed by the process equipment,
ic is the H2S fraction of the gas from well i , and c is the permitted frac-
tion in the mixed gas.
The maximum permitted change per time step for each well in (6.7)-(6.8)
will similarly be extended to
, , 1 , , ,i t i t i i I t T (6.24)
, 1 , , , .i t i t i i I t T (6.25)
The maximum permitted change in total production rates in (6.9)–(6.16)
are extended to
o o, , 1( ) , ,i t i t i
i I
q t T
(6.26)
o o, 1 ,( ) , ,i t i t i
i I
q t T
(6.27)
g g, , 1( ) , ,i t i t i
i I
q t T
(6.28)
g g, 1 ,( ) , ,i t i t i
i I
q t T
(6.29)
w w, , 1( ) , ,i t i t i
i I
q t T
(6.30)
w w, 1 ,( ) , ,i t i t i
i I
q t T
(6.31)
o w l, , 1( )( ) , ,i t i t i i
i I
q q t T
(6.32)
o w l, 1 ,( )( ) , .i t i t i i
i I
q q t T
(6.33)
115
where o , g , w , and l specify the maximal change in the oil, gas,
water, and liquid for each time step, respectively.
Some of the previous inequalities depends on , 0i t , which is the fraction
of the well that is producing at initial time 0t . , 0i t should be treated as
a constant in the optimization problem.
Just like in the previous section, this optimization problem should be
solved repeatedly with updated parameters based on new measurements
at regular time intervals. After solving, the new calculated values , 0 1i t
should be used to open and close wells. The model proposed in this sec-
tion becomes the one in the previous section when = 1n .
By using such repeated optimization, this becomes a receding horizon
optimization approach similar to the Model Predictive Control [73]
(MPC). In MPC, the deviation from a reference on a horizon is mini-
mized by adjusting the input to the system. An estimate of the current
state of the system is used as an initial state. The calculated input from
the first time step is used. The process is then repeated by optimizing
again with an updated estimate of the current state. The states in the
proposed models are the decision variables ,i t for each time step t . The
system is a pure integrator of the inputs , , 1i t i t . The proposed ap-
proach is differentiated by the economical objective function used and the
way the constraints are updated.
If some measurement of the current excess capacity for the system is
known, then oq , gq , wq , and lq may be found by
o o o,, 0
= ,i t ii I
q q q
(6.34)
g g g,, 0
= ,i t ii I
q q q
(6.35)
116
w o w,, 0
= ,i t ii I
q q q
(6.36)
l l l,, 0
= ,i t ii I
q q q
(6.37)
where o,q , g,q , w,q , and l,q are the excess oil, gas, water, and oil ca-
pacities at time 0t , respectively. When updated information related to
the well capacities, oq , gq , and wq are available, this should be updated
as well.
6.4 Computational Results
In this section, three examples illustrate the benefit of the proposed linear
program of the previous section. The computations were performed using
field data from an oil production system in the North Sea. The details are
shown in Table 6.3. The processing capacity was 2,913,680 Sm3/D of gas
and 34,277 Sm3/D of liquid. The H2S scavenger was assumed to remove
60,887,970 PPM Sm3/D of H2S, and the gas had a quality constraint on
2.5 PPM of H2S. The length of the time steps was set to 60 seconds. Re-
sults from optimization with n equal to both 1 and 400 are presented.
For the case of = 1n , the model reduces to the one presented as short-
term optimization. With = 400n , the horizon is long enough for the
solver to reach steady state production for all the presented examples. In
this particular model, this means that no further increase of n will
change the trajectory. The model was simulated over 400 time steps, giv-
ing a potential steady state oil production potential of 2396.2 Sm3. Losses
will be defined as the difference from this number.
For all examples it was assumed that an equal fraction of the chokes
could be opened or closed every time step for all wells, giving
= , ,i j i j I . The permitted change of the choke position per time
step was set to = 0.01i , giving 6000 seconds in total opening time of a
well. In all examples, the production from A-43 was restricted to 25 % of
the listed production. The wells were set to an initial closed state. The
117
results from the examples are shown in Figure 6.1–Figure 6.4. The cumu-
lative oil production and the losses are listed in Table 6.4.
Example 2 Only the constraints specified above were used.
Example 3 The maximum change in total gas production rates for each
time step were set to 9000 Sm3/D.
Example 4 The maximum change in total oil, gas, water, and liquid pro-
duction rates for each time step were set respectively to 25 Sm3/D,
9000 Sm3/D, 80 Sm3/D, and 100 Sm3/D.
The difference between the short-term optimization, where = 1n , and
the long-term optimization, where = 400n , is largest for Example 4,
with a loss reduction of 14.1 Sm3 during start-up.
The linear programs were solved using the commercial linear program-
ming code XPress MP6. The number of rows, columns, non-zeros, dual
simplex iterations and times used is listed in Table 6.5. The low number
of dual simplex iterations for examples with = 1n might be due to pre-
solver included in XPress MP. Introducing three more constraints for
each time step in Example 4 increased the computational burden by
about 80 times.
6.5 Conclusions
A linear program for optimal scheduling of a start-up of an oil production
system has been proposed, maximizing the oil production on a horizon.
Each well is constrained by the permitted change rate of the choke for
6XPress MP 2003 from Dash Optimization, Inc.
118
opening or closing. Similarly, the total rates of changes of oil, gas, water,
and liquid production are also restricted. The proposed method has been
compared to a short-term optimization method, showing a 0–1 % increase
in cumulative production at the expense of lower production for a few
time steps. The increase should be considered small, and the contribution
of the method is the automation of the well management.
6.6 Further Work
A method should be developed to identify the various parameters used.
These parameters are the constraints on the change rates allowed for
each well and the processing equipment itself.
The constraints on the change rates should be extended to constraint the
changes in pressure drop in flow lines or pipes. Thus, the model may
have to be extended to include such components.
Table 6.1: The sequence used when having priority on wells with a low