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

1 Introduction ............................................................................................ 1

1.1 Offshore Oil Production System ....................................... 1

1.1.1 Reservoir ................................................................................. 1

1.1.2 Well......................................................................................... 2

1.1.3 Gathering Network ................................................................. 4

1.1.4 Processing Facilities ................................................................ 4

1.2 Motivation ........................................................................ 6

1.3 Summary and Contributions of Papers ............................ 10

1.3.1 Paper I: Real-Time Optimization of Oil and Gas

Production Systems: A Technology Survey .......................... 10

1.3.2 Paper II: Global Optimization of Multiphase Flow

Networks in Oil and Gas Production Systems ...................... 11

1.3.3 Paper III: Optimal Well-Testing Strategy for Production

Optimization: A Monte Carlo Simulation Approach ............ 12

1.3.4 Paper IV: Well Management under Uncertain Gas or

Water Oil Ratios ................................................................... 13

1.3.5 Paper V: Optimal Start-up Scheduling of Production

Wells ..................................................................................... 14

2 Real-Time Optimization of Oil and Gas Production Systems: A

Technology Survey ............................................................................... 19

2.1 Introduction ..................................................................... 19

2.2 Information Flow in Production Optimization ................. 22

2.2.1 Data Acquisition ................................................................... 22

2.2.2 Control .................................................................................. 23

2.2.3 Production Planning ............................................................. 23

vi

2.2.4 Operator ................................................................................ 24

2.2.5 Strategic Planning ................................................................. 24

2.2.6 Reservoir Planning ................................................................ 24

2.2.7 Well Model Updating ............................................................ 24

2.2.8 Processing Facility Model Updating ..................................... 25

2.2.9 Reservoir Model Updating .................................................... 25

2.3 Technology and Reference Cases ..................................... 25

2.3.1 Global Versus Local Optimization ........................................ 25

2.3.2 Production Planning ............................................................. 29

2.3.3 Reservoir Planning ................................................................ 38

2.3.4 Model Updating .................................................................... 41

2.4 Challenges ........................................................................ 45

2.5 Conclusions ...................................................................... 48

3 Global Optimization of Multiphase Flow Networks in Oil and Gas

Production Systems .............................................................................. 57

3.1 Introduction ..................................................................... 57

3.2 Methodology .................................................................... 60

3.2.1 Well....................................................................................... 61

3.2.2 Flow Line .............................................................................. 62

3.2.3 Choke .................................................................................... 64

3.2.4 Outlet Boundary ................................................................... 65

3.2.5 Connection ............................................................................ 65

3.2.6 Objective ............................................................................... 66

3.2.7 Constraints ............................................................................ 66

3.3 Case Study ....................................................................... 66

3.4 Conclusions ...................................................................... 67

3.5 Further Work ................................................................... 68

3.6 Nomenclature ................................................................... 69

4 Optimal Well-Testing Strategy for Production Optimization: A

Monte Carlo Simulation Approach ....................................................... 73

4.1 Introduction ..................................................................... 73

vii

4.2 Monte Carlo Simulation ................................................... 76

4.3 Calculating Production .................................................... 78

4.4 Error Distribution of Oil Resource Ratio ......................... 79

4.5 Case Study ....................................................................... 81

4.6 Conclusions ...................................................................... 82

4.7 Further Work ................................................................... 83

4.8 Nomenclature ................................................................... 84

5 Well Management under Uncertain Gas or Water Oil Ratios .............. 91

5.1 Introduction ..................................................................... 91

5.2 Uncertainty Matters......................................................... 94

5.2.1 Low Processing Capacity ...................................................... 95

5.2.2 High Processing Capacity ..................................................... 96

5.2.3 Comparison ........................................................................... 97

5.3 Proposed Method ............................................................. 98

5.4 Case Study ..................................................................... 101

5.5 Conclusions .................................................................... 103

5.6 Further Work ................................................................. 103

6 Optimal Start-up Scheduling of Production Wells ............................. 109

6.1 Introduction ................................................................... 109

6.2 Short-Term Optimization .............................................. 111

6.3 Full Horizon Optimization ............................................. 113

6.4 Computational Results ................................................... 116

6.5 Conclusions .................................................................... 117

6.6 Further Work ................................................................. 118

7 Conclusions ......................................................................................... 127

References ................................................................................................... 131

viii

1

1 Introduction

In this chapter, the work in this thesis is restricted, motivated, and the

contributions are placed in a wider perspective. The thesis is based on

five papers. One paper is accepted to a journal, four papers have been

presented on conferences and printed in the proceedings of the confe-

rences, and one paper is currently unpublished. A summary of each paper

is given. Furthermore, the major contributions of the individual papers

are outlined.

1.1 Offshore Oil Production System

In this section, a brief introduction to offshore oil production will be giv-

en. Most of the components and terminology used within the thesis will

be defined. Oil production is the extraction of oil and gas from the reser-

voir to the refinery [1]. Several disciplines are involved in the production

and planning.

1.1.1 Reservoir

A reservoir is a porous rock containing producible hydrocarbons such as

oil and gas. The reservoir will typically contain a mixture of hydrocarbon

components, water, and various contaminations.

The reservoir pressure is typically between the hydrostatic pressure (ap-

proximately 10,000 Pa/m) and the rock pressure (approximately

20,000 Pa/m) [1]. The reservoir pressure is reduced when the fluids are

extracted. The reservoir temperature increases with the depth of the re-

servoir, typically 0.03 K/m [1].

The hydrocarbon components and water will separate naturally in the

reservoir because of different fluid densities. At the top of the reservoir,

2

there will be a gas cap. The water will be on the bottom and the oil in

the middle. The ratio between the gas and liquid in the reservoir does

however depend on pressure, and much of the oil in the reservoir will be

gas at surface pressure conditions.

1.1.2 Well

The hydrocarbons in the reservoir are produced by a well into the reser-

voir. First, a wellbore is drilled into the reservoir by removing parts of

the rock along a path from the surface to the reservoir. The wellbore is

stabilized by a casing, which is a large-diameter pipe lowered and

mounted using cement into the wellbore. The hydrocarbons from the re-

servoirs do not flow in the casing, but in the tubing installed within the

casing. The space between the tubing and the casing is the annulus. A

packer isolates the annulus from the reservoir. The casing is perforated in

the reservoir allowing the fluids to be extracted. This part of the wellbore

is described as bottom-hole. The part of the subsurface wellbore is de-

scribed as downhole. Some wells may have downhole or bottom-hole pres-

sure or temperature measurement devices, which allow measuring the

temperatures or pressure at those locations. Some smart horizontal wells

even have downhole valves for controlling the inflow from multiple reser-

voir zones. The most important components of a well are shown in Figure

1.2.

The wellhead is the surface termination of the wellbore. It includes facili-

ties such as chokes for controlling the flow from the well. The choke is

similar to a valve. Typically, pressure and temperature measurement de-

vices are located both upstream and downstream the choke.

The extraction of the reservoir is driven by the pressure difference be-

tween the reservoir pressure and the pressure located upstream the

choke. As the reservoir is depleted, the production rates may decline be-

3

cause of a reduced reservoir pressure. Various artificial lift methods are

used to increase the oil production rates from the wells. These artificial

lift methods include pumps and gas lift.

Gas lift is an artificial lift method in which gas is injected into the tubing

to reduce the hydrostatic pressure drop of the well by decreasing the av-

erage fluid density. The reduced hydrostatic pressure drop decreases the

downhole pressure of the well allows the reservoir liquids to enter the

wellbore at a higher flow rate. The tubing-casing annulus is typically

used to transport the injection gas down to the lower part of the wellbore

at which there is a gas lift valve connecting the tubing-casing annulus

and the wellbore. There are two types of gas lift: intermittent and conti-

nuous gas lift. The continuous gas lift method injects gas at a continuous

basis. The intermittent gas lift method injects gas at a cyclical basis to

enable the buildup of liquids in the wellbore. The intermittent gas lift

method is used in relatively low productivity wells.

The extraction fluids from the reservoir will make the reservoir pressure

reduce, and the reduced reservoir pressure will reduce the production

rates from the wells. Gas or water injectors may be used to replace the

extracted fluid volumes by injecting gas or water at convenient locations

in the reservoir in order to support the pressure. In gas injection, sepa-

rated gas from the production wells or gas imported from other produc-

tion systems are injected into the reservoir. In fact, other gases such as

CO2 have also been tried. Water injection is popular in offshore oil pro-

duction because of good availability of seawater, which may be filtered

and treated inexpensively.

Coning is the change in oil-water or gas-oil interface profiles because of

drawdown pressures. The result of coning may be higher gas oil ratios or

water cuts because of perforations on the water or gas sides of the inter-

face levels.

4

1.1.3 Gathering Network

The production from the wells has to be gathered from the wells and

transported to the processing facilities of the production system. For this,

a set of flow lines is used. In offshore oil production systems the chokes

are typically located at surface. The riser, a special type of flow line, is a

part of the well. However, subsea wells have become more popular as the

technology has evolved. In subsea wells, the chokes are located in a sub-

sea facility and they share a flow line to the processing facilities. More

recent subsea facilities may even host some processing facilities separat-

ing gas and liquids to prevent slugging.

The production manifold is downstream to the chokes. The manifold is a

mixing point at which the well stream of each well is mixed. Typically, a

production system has one production manifold and one test manifold.

The production manifold mixes the well streams from the wells producing

to the production separator. The test manifold mixes the well streams

from the wells, typically one, producing to the test separator.

The instrumentation of subsea facilities may vary slightly—some may

include pressure and temperature measurement devices that communicate

with the rest of the production system. Some chokes of subsea facilities

are remotely controlled and some are not remotely controlled. The chokes

of subsea facilities that are not remotely controlled may require remotely

operated vehicles to adjust choke settings making changes very expensive.

1.1.4 Processing Facilities

The overall objective of the processing facilities of an offshore oil and gas

production system is to make the oil and gas from the reservoir trans-

portable. The oil is typically transported using tankers, which require

that the oil is stable at stock tank conditions. Furthermore, most of the

water is removed from the liquid to reduce the cost of transportation.

5

Gas is often exported using flow lines to remote gas terminals. The gas

must be dried to prevent liquid slugs to build up in the flow line, and

compressed to the required pressure.

In order to stabilize the oil, separators are used. The well streams enter

the separator, which is a horizontal tank, horizontally and hit a series of

perpendicular plates, causing the liquids to drop to the bottom and the

gas to rise to the top. Gravity separates the liquid of the well streams,

which is a mixture of oil and water, into oil and water layers at the bot-

tom of the separator. An abeam-vertical plate prevents water to enter the

part of the tank farther from the inlet, allowing the oil to be tapped here.

Water is tapped on the other side of the abeam-vertical plate. An outlet

is also located at the top of the tank for tapping of gas. A separator is

illustrated in Figure 1.1. The separators are typically serial-coupled to

improve the quality of separation, and a stage number distinguishes

them. A scrubber is a vertical separator designed to remove dirt, water,

foreign matter, or undesired liquids from a gas stream.

The oil-water and gas-oil interfaces have to be controlled to prevent oil

to enter the gas outlet, water to enter the oil outlet, or oil to enter the

water outlet. The interfaces are controlled using a control valve at each

outlet. An automatic feedback controller is typically used to maintain

each of the interfaces at their desired set points using a measurement of

the interface level. A similar controller is typically used to control the

separator pressure by adjusting the choke at the gas outlet and a separa-

tor pressure measurement.

A test separator, possibly equipped with special measurement devices, is

used to measure properties of the flow stream of a single well at the time.

The test separator enables the measurement of the gas oil ratio and the

water cut of each well using the flow rate measurements of the separator.

6

The separator may be effective, but the water from the separator will

still include oil after leaving the separator. A hydro cyclone can be used

for separating the remaining oil from the water. A hydro cyclone works

by hurling the oily water in the hydro cyclone with a large force (about

10,000 m/s2). Because of the different densities between oil and water,

the water will be hurled to the cyclone wall, while the oil will be in the

middle. The oil and water can then be tapped.

For each stage of separation, the pressure is dropped until the oil reaches

stock tank conditions. The gas, however, is exported or reinjected at a

higher pressure, and compression using gas compressors is required. Be-

cause compression increases the temperature of the gas, cooling is re-

quired.

Compression of the gas increases the gas temperature, demanding a coo-

ler of the gas downstream to the compressor. In offshore oil production

systems, seawater is used as a cooling medium for the heat exchangers.

Storage cells are used for storage of the produced oil until a tanker is

ready to pick it up.

1.2 Motivation

The world is experiencing an increased demand for petroleum in the be-

ginning of the 21st century, and many of the reservoirs of the existing oil

production systems are maturing reducing the oil production rates from

these systems. The increasing demand and reducing supply is materializ-

ing in raising oil prices are motivating development of new technologies

increasing the oil production. The new technologies are given many

names including the digital oil field, oil field of the future, and integrated

operation. Although the names and the content are different, the goal is

the same—to increase the oil production from the existing oil production

7

systems. The potential net present value of integrated operations on the

Norwegian Shelf is estimated to 250 billion NOK (approximately 40–45

billion USD) in a study for The Norwegian Oil Industry Association [2].

Most of the net present value is due to increased and accelerated produc-

tion owing to production optimization.

By changing the work flow of the decision-making in the operation to al-

low more interactions between disciplines, better decisions are supposed

to be taken. The silo thinking in operations is reduced by building colla-

boration rooms—rooms located onshore where professionals from multiple

disciplines are supposed to collaborate both within the room and with the

operators located offshore. The collaboration rooms are equipped with

large screens enabling teleconferences with the operators offshore. The

screens are also used for showing process measurements and calculations

based on these measurements targeting the professionals on the shared

objective.

Many of the companies operating the oil production systems are investing

in information systems making the process measurements available on-

shore to allow remote operations.

The decision-making in the collaboration rooms is related to the daily

operations of the oil production system. The goal is to maximize some

kind of performance measure, which typically is the total oil production

rate of the oil production system adjusted for the variable cost of opera-

tion. Many of the decisions are made using numerical simulations and

trends of process measurements, but mathematical optimization is rarely

used.

The demand for smarter operations makes mathematical programming

more of a topic. The increased availability of real-time process measure-

ments onshore is an enabler. The models used by the mathematical pro-

8

grams can be updated to fit the input-output behavior of the oil produc-

tion system. Many professionals study various applications of mathemati-

cal optimization in oil production optimization. This included the optimi-

zation of well placement, drilling operations, reservoir drainage, and daily

operations.

New gas and water injection strategies are currently being developed

where the goal is to maximize the recovery of the reservoir. By using

more process measurements including pressures, temperatures, and seis-

mic, the current state of the reservoir can be more accurately observed.

The more accurate information on the reservoir allows injecting gas and

water with reduced risk of a water breakthrough. Because of the delayed

water breakthrough, the processing equipments can produce more oil

with the same water treatment capacity. Smart wells, which are wells

equipped with downhole measurement devices and valves controlling the

flow from a multitude of reservoir zones, increase the degrees of freedom

available to enable more control of the extraction of oil and gas. Gas and

water injection strategies to the reservoir will however not be the focus of

the thesis.

Improved methods for operation of wells are also a topic currently devel-

oped. The methods include finding the optimal mixture of wells in order

to maximize the oil production rate without violating any constraints in

the processing equipments. Such constraints are typically related to ca-

pacity, quality or safety. The oil produced must not include more than a

specified amount of water in order to be accepted by the purchasers. The

water produced is often disposed or reinjected into the reservoir. If dis-

posed, local environmental regulations restrict the amount of oil and

chemicals that it may include. Reinjecting oily water may also be a prob-

lem because it may clog the injection well. Sand production may also be

an issue because of erosion in bends and chokes. Sand taking up space in

9

the separator, thus reducing the separator capacity, may also be an issue.

H2S may cause corrosion in the flow line, and the amount may be re-

stricted to prevent this. If exported or sent through flow lines, the gas

has requirements on the dryness to avoid liquid slugs. Furthermore, gas

typically has quality specification related to the gas composition, such as

the amount of H2S. Safety requirements may be related to design pres-

sures or temperatures of the processing equipments, or the piping or

valves connecting them. In order fully to utilize the limited capacity giv-

en by the processing equipments, the well mixture must be optimized to

consider the fluid composition from the wells. The focus of the thesis will

be to develop methods that can be used to find such optimal well mix-

tures using real-time data in day-to-day operation.

The reservoir is a dynamical system where oil is extracted from the re-

servoir through the well to the processing equipments. The extraction

affects the reservoir states by reducing the oil, water, and gas in the re-

servoir. Accordingly, the pressure is also reduced. The reduction of pres-

sure may be partly compensated by the injection of gas and water; how-

ever, the fluid compositions of the reservoir are changed. The focus of

this thesis will not consider effects on the reservoir, and the proposed so-

lutions will only try to maximize the current total oil production rates

rather than total recovery over the life cycle.

Oil production systems include many measurement devices, but the

numbers of states or values that are desired are even higher. In order to

find the optimal well mixture, accurate information about the fluid com-

position is required. The thesis will also focus on developing method for

optimally obtaining the desired information or measurements.

10

1.3 Summary and Contributions of Papers

In this section, a brief summary of each of the papers will be given. The

major contributions of each paper will be stated. The papers are con-

nected because they all propose production optimization schemes for

maximizing the total oil production rate. Two of the methods explicitly

handle the uncertainties by including it into the model.

1.3.1 Paper I: Real-Time Optimization of Oil and Gas

Production Systems: A Technology Survey

This paper is a non-critical survey of key literature in the field of real-

time optimization of offshore oil and gas production. The goal is to give

an overview of technologies that may be applied in a real-time production

optimization application. The concept of real-time production optimiza-

tion is also discussed. It is included as the first paper to function as an

introduction chapter in this thesis. The paper includes an information

flow description of the operation of an offshore oil and gas production

system. The elements in this description include data acquisition, data

storage, processing facility model updating, well model updating, reser-

voir model updating, production planning, reservoir planning, and stra-

tegic planning. Methods for well prioritization, gas lift optimization, gas

or water injection optimization, and model updating are reviewed in the

view of the information flow described. Challenges of real-time produc-

tion optimization are also discussed.

This paper contributes an overview and organization of existing technol-

ogies that may be used for real-time optimization applications in offshore

oil and gas production.

11

1.3.2 Paper II: Global Optimization of Multiphase Flow

Networks in Oil and Gas Production Systems

A mathematical program for finding the optimal oil production rates of

the wells in an oil production system is developed. Each well may be ma-

nipulated by injecting lift gas and adjusting a production choke. The oil

production from the wells may be restricted by multiple constraints in

the maximum oil flow rate, water flow rate, liquid flow rate, and gas flow

rate. The wells may also be restricted with a maximum total lift gas rate.

In oil production systems with subsea wells, flow lines are often shared

between two or more wells. The pressures in the production manifold in

such oil production systems are affected by the flow rates from the wells.

The commonly used models based on gas lift performance curves (GLPC)

no longer apply directly to these problems due to changing pressure con-

ditions in the production manifold. Because of this, a model of the flow

line is also required to get results that are more accurate. This work in-

corporates such a model. A piecewise linear approximation is proposed.

This makes it possible to find a proven global optimum, within the ap-

proximation, for the optimization problem. The problem is formulated as

a mixed integer linear program, and it is solved using a commercial

branch and cut solver.

This paper contributes a novel model of pressure drops in flow lines for

production optimization. A contribution of the paper is the use of piece-

wise linear models of the pressure drop in the common flow line. Further,

it is a contribution to solve this as a mixed integer program, which allows

for easy global optimization (of the approximate model).

The method is a refinement of the master thesis [3] of the author. The

method is improved by calculating the pressure drop from the source to

the sink instead of the opposite way. This eliminated the use of a numer-

12

ical solver to find the pressure drop, speeding up the calculation. Fur-

thermore, the method is modified to use special ordered sets of type two,

allowing faster convergence of the numerical solver.

A further contribution is the use of a branch and bound method for solv-

ing the production optimization problem of multiphase flow networks.

A case study is conducted using field data from a Norwegian offshore oil

production system comprising four subsea wells. The case study focuses

on the computational load of the proposed method. The method is able

to solve the optimization problem within ten seconds.

1.3.3 Paper III: Optimal Well-Testing Strategy for Pro-

duction Optimization: A Monte Carlo Simulation

Approach

Well testing may be performed to support many decisions including ones

related to production optimization of an oil production system. The in-

formation flow used for optimization of the system is described. In pro-

duction optimization, information such as the gas oil ratio and water cut

is used to decide, for example, on the wells to prioritize for choking back

or opening to avoid over-utilization or under-utilization of the production

capacity. Since the reservoir properties change with time, the uncertain-

ties of their estimated values increase with time, and eventually a new

well test will be required. The risk of prioritizing the wrong wells, giving

a lower total oil production rate than what is possible, increases as the

uncertainties in the estimates increase. A computer program is developed

to choose the well to test at a given time based on historical well test da-

ta. The program uses a Monte Carlo approach for identifying the well

test being more likely to lead to the highest increase in the total oil pro-

duction rate when the well test information is utilized to optimize the oil

13

production. The computer program is applied to field data quantifying

the benefits when applied to this specific field. The current implementa-

tion is limited to production systems where the pressure interaction

among wells may be neglected. Furthermore, the current implementation

assumes that a single treatment constraint is active.

This paper contributes a novel method for choosing wells for routing to

test separators. A contribution of the paper is the calculation of the ex-

pected total oil production rate using the measurements obtained in a

possible well test for choosing a well for testing. Furthermore, a contribu-

tion of the method is to use Monte Carlo simulations to calculate an ex-

pected total oil production rate using the possible outcomes of the mea-

surements in the well test. It is a further contribution to test the well

giving the maximal expected total oil production rate. The industry prac-

tice is to do well testing based on equal frequency for all wells, and to do

ad hoc testing when the measurements from a well look “suspicious”.

The paper contributes a method for finding a stochastic distribution of

the gas oil ratio or the water cut of a well using historical well test data.

1.3.4 Paper IV: Well Management under Uncertain Gas

or Water Oil Ratios

In the daily operation of an oil production system, it is often required to

choke back some of the oil production wells to ensure that the processing

capacity is not over-utilized. When the capacity of some processing re-

source is over-utilized, wells having large ratios between the consumption

of the resource and the oil production rate are choked back. When there

is free processing capacity, the chokes of the wells having small ratios are

opened. Often, the gas or water oil ratios (derived from the water cuts)

are used as such ratios. These ratios are uncertain. This paper proposes

to use information about the uncertainties of the gas or water oil ratios

14

to find the order of opening and closing the wells to maximize the ex-

pected total oil production rate from the wells. In a computational study

based on field data, the order was found to be different from the order

found using the expected value of the gas or water oil ratios.

This paper contributes a novel method for prioritizing wells producing to

a shared processing facility having a single processing constraint. A con-

tribution of the method is the use of stochastic distributions of the gas oil

ratios or water cuts to find the optimal wells to choke back or open max-

imizing the expected total oil production rate and not violating the

processing constraints. The industry practice is to regard the gas oil ra-

tios or water cuts as parameters without uncertainty, and to do prioriti-

zation using these presumably accurate values. The method further han-

dles uncertainties in the processing capacities and oil potentials explicitly.

It is a contribution that mixed integer linear programming is used for

finding such an order. Further, a contribution, in this context, is to use

values drawn from the stochastic distribution to approximate the sto-

chastic distributions themselves.

1.3.5 Paper V: Optimal Start-up Scheduling of Produc-

tion Wells

A linear program for finding the order to open wells after a shutdown is

proposed. The oil production over a horizon is maximized, thus minimiz-

ing the total losses during a start-up. The method is able to handle mul-

tiple constraints such as oil, gas, water, and liquid treatment capacities

as well as quality constraints on the gas. The method is shown to in-

crease cumulative production compared to a method using short-term

optimization only.

This paper contributes a novel method for handling the uncertainties in

the treatment capacities. A linear model of the oil production system is

15

optimized giving suggestions for changes to the chokes the model is up-

dated using measurements related to the excess treatment capacity. In a

closed loop, the operating point will therefore typically approach the

physical limitations of the system, and it will not just be on the con-

straint imposed by the uncertain model.

16

Water

Oil

Gas

LT

LTLC

PT

LC

PC

Well streams

Abeam-vertical plate

Series of perpendicular plates

Figure 1.1: A separator typically comprises two level control loops and a

pressure control loop.

17

Gas layer

Oil layer

Water layer

Perforation

Casing

Tubing

Downhole

Bottom-hole

Seabed

Packer

Annulus

Well head

Seawater

Rock

Figure 1.2: A well extracts fluids from a reservoir through tubing to the

surface.

18

19

2 Real-Time Optimization of Oil and Gas

Production Systems: A Technology Sur-

vey

Based on

H.P. Bieker, O. Slupphaug, and T.A. Johansen,

accepted for

SPE Production & Operations Journal,

presented at

2006 SPE Intelligent Energy Conference and Exhibition

Amsterdam, The Netherlands, 11–13 April 2006

2.1 Introduction

In the daily operation of an oil and gas production system, many deci-

sions (an element of a solution) have to be taken affecting the volumes

produced and the cost of production. These decisions are taken at differ-

ent levels in the organization, but eventually they will reach the produc-

tion system layout. Figure 2.1 gives an overview of a physical production

system. For such production systems, the decisions are typically related

to the choke or valve openings, compressor, and pump settings at every

instance of time.

An objective function is a single-valued and well-defined mathematical

function mapping the values of the decision variables into a performance

measure. Examples of such performances measures are the total oil pro-

duction rate, net present value (profit), or the recovery of the reservoir.

In the efforts towards better performance of the production system, a

question to be answered is which decisions are better in order to optimize

20

the objective function. In the process of making good decisions, informa-

tion about the production system is used. This information may be the

physical properties such as pipe diameters and lengths, or it may be mea-

surements from the production system.

The environment in which the production of oil and gas is obtained is

continuously changing. This will, therefore, affect the value of the per-

formance measure of the decisions being used. For example, if the cooling

capacity of the production system is an operational bottleneck at some

given time, this may no longer be the case if the seawater temperature

drops or another pump in the cooling system is started. Incidents in the

production system may also affect the value of the performance measure

of the decisions. A partial shutdown of the production system due to

maintenance may also affect system bottlenecks.

Real-Time Optimization (RTO) is a method for complete, or partial, au-

tomation of the process of making good or optimal decisions. The term

“optimal” is defined below. By continuously collecting and analyzing da-

ta from the production system, optimal decisions may be found. Either

these settings are then implemented directly in the production system or

they are presented to an operator or engineer for consideration. If the

settings are implemented directly, the RTO is said to be in a closed loop.

RTO defined by Saputelli et al. [4] reads: “a process of measure-calculate-

control cycles at a frequency, which maintains the system's optimal oper-

ating conditions within the time-constant constraints of the system”.

The main aim of RTO is to improve the utilization of the capacity of a

production system to obtain higher throughput or net present value. The

idea is to operate the production system, at every instant of time, as near

to the desired optimum as possible [5]. To achieve this, a model of the

production system is optimized to furnish an optimal solution. The model

is continuously being updated by measurements from the production sys-

21

tem to fit the actual input-output behavior of the processing facilities,

wells or network, and reservoir better.

A general RTO system used in, for example, downstream petrochemical

production systems consists of the following four components [6] shown in

Figure 2.2:

Data validation: The validated input and output data are vali-

dated using data reconciliation and signal processing techniques

(for instance using material and energy balances).

Model updating: The processing facility models, well models, flow

line network models, and reservoir models are updated to fit the

validated input and output data available the best.

Model-based optimization: An optimization problem based on the

updated models, objective function, and constraints is set up and

solved to obtain an optimal solution.

Optimizer command conditioning: A post-optimality analysis is

performed to check the validity of the computed solution before it

is implemented.

Although the definition of Saputelli et al. [4] was written with oil and gas

production systems in mind, it is general in the sense that it is not re-

strictive to some specific type of production system or method. The defi-

nition can be related to Figure 2.2.

Recently, SPE started a technical interest group that focuses on RTO for

oil and gas production systems. The driver behind this development is, as

in any industry, the demand for more profitable production systems. This

survey will help to organize previous work related to RTO. The focus will

22

be on offshore oil and gas production systems; however, relevant refer-

ences from other industries are also included.

A previous survey [7] was recently published. It focused on the organiza-

tional issues of using RTO. Because a survey on the organizational issues

is already given, this survey will focus on the existing software, tools, me-

thods, and approaches that can be applied for RTO. However, the survey

will not focus on the processing facilities. Furthermore, this is a non-

critical survey of key literature in the field.

This paper is organized as follows. A description of the information flow

associated with the optimization of offshore oil and gas production sys-

tems is given to relate the general RTO technology to this specific appli-

cation area. Technologies for optimization and model updating of such

production systems are reviewed, and reference cases will be presented.

Finally, key challenges are addressed and conclusions are stated.

2.2 Information Flow in Production Optimization

The operation of an oil and gas production system may be illustrated ac-

cording to Figure 2.3. The main components of the operation are de-

scribed below.

2.2.1 Data Acquisition

Modern production systems usually have good instrumentation. Level

(the height of oil-water or gas-oil interface in a separator), pressure, and

temperature transmitters are most common. In addition to required fiscal

meters, there are often also a few flow transmitters to measure flow rates

in gas, water, and oil pipes. Flow transmitters for multiphase flow may

also be available, but they are rare. Various off-line analyzers of parame-

ters including oil-in-water and other product qualities may also be avail-

able. The instrumentation varies considerably between different produc-

23

tion systems, typically with the age of the system and the country or re-

gion it is situated.

2.2.2 Control

A typical oil and gas production system has many (automatic) feedback

control loops to support an efficient production and meet the production

targets. A feedback control loop generates decisions, such as valve open-

ings, based on measurements from the production system. The simplest

form of such control is used to control levels and pressures in the separa-

tors. Centrifugal compressors are always protected by anti-surge control

loops. The control loops ensure that the compressors do not surge, and

prevent damage. Control is also used to balance the load among parallel-

coupled and serial-coupled processing units. A phenomenon that may be

observed in an oil and gas production system is severe slugging. The

pressure and flow rate in a well or flow line start oscillating, and the ef-

fective production capacity is reduced. This can sometimes be stabilized

by feedback control[8].

2.2.3 Production Planning

A typical oil and gas production system is operated by periodically gene-

rating a production and injection plan. This production and injection

plan lists the target production of oil, gas, and water for a specific period

for each individual well. Similarly, the injection of gas or water is stated

for the injection wells. The cycle time of the production and injection

plan depends on the policy of the field operator, but it will typically be

between a week and a month. The models and constraints of the

processing facilities and wells or networks are used together with con-

straints from the reservoir planning as inputs to the planning. Politics or

constraints from the strategic planning may also be enforced here.

24

2.2.4 Operator

Above all, the operators are responsible for ensuring safe operation. Fur-

thermore, they are responsible for implementing the recommendation

from the production and injection plan. When implementing the produc-

tion and injection plan, the operators have to meet the operational tar-

gets while obeying minimum and maximum limits on variables such as

pressures, temperatures, and rates.

2.2.5 Strategic Planning

The production and injection plan is somehow connected to the market

and the strategic considerations or policy of the company.

2.2.6 Reservoir Planning

The long-term reservoir drainage is planned here. This includes planning

of gas and water injection. The updated reservoir model is used for find-

ing proper draining strategies. Politics from the strategic planning may

also be enforced here.

2.2.7 Well Model Updating

To support making good decisions, models may be used to develop the

production plans. Typically, well tests are performed to determine the

gas oil ratio, water cut, and production rates of each individual well.

Well tests are performed by routing a well to a dedicated separator. This

separator will separate the three phases, and a flow transmitter is con-

nected to the outlet of each phase. The well model is then updated using

the measurements taken during the test. Fluid sampling may be used to

obtain the fluid composition including the water cut.

25

2.2.8 Processing Facility Model Updating

Typically, the processing facilities are modeled as constraints on oil, gas,

and water processing capacities. This means that the model is updated

whenever the capacity changes.

2.2.9 Reservoir Model Updating

To be able to conduct the reservoir planning, a reservoir simulator may

be used to evaluate different drainage strategies for the reservoir.

The initial state and parameters of the reservoir model must be updated

by measurements from the production system. The volumes produced,

volumes injected, and pressures are important measurements used in this

updating process. To ensure good accuracy of the model, parameters and

the initial state are fitted to longer series of historical production data.

The method is typically called history matching.

2.3 Technology and Reference Cases

Figure 2.3 shows an example of how decisions in production optimization

may be taken. Most or all the decisions are made with support by some

form of technology. This section will provide an outline of relevant tech-

nologies and reference cases from the industry that may be extended and

used as a part of an RTO system. More specifically, the technologies be-

longing inside the large rectangle of Figure 2.3 will be discussed here.

2.3.1 Global Versus Local Optimization

For all the planning activities, numerical optimization may be used to

find good or optimal feasible solutions (or decisions). This works by de-

fining an objective function to be minimized, or maximized, as a function

of decision variables. The feasible set of these decision variables is defined

by a set of equality and inequality constraints on the decision variables.

26

The objective function and the constraints define the optimization prob-

lem, or mathematical program.

A solver is used to find an optimal solution, and the solver should be

chosen using information about the problem structure. In particular, li-

near and convex quadratic programs [9] are often preferred because of

their convexity and the existence of mature algorithms for solving them.

In some cases, nonlinear constraint and objective functions may be re-

formulated to linear equivalents [10], and these mature algorithms may

be used.

A local optimal solution is defined as a feasible solution having a neigh-

borhood where no strictly superior feasible solution exists (in terms of the

objective function). A global optimal solution is defined as a feasible solu-

tion not having a strictly superior feasible solution (in terms of the objec-

tive function) in the feasible set, and hence a global optimal solution is

also a local optimal solution. The difference between a local and global

optimal solution are illustrated in Figure 2.4. Convex optimization prob-

lems are preferred because they guarantee that a local optimal solution is

also a global optimal solution (however a unique global optimal solution

is not necessarily guaranteed). Unfortunately, the term “optimal solu-

tion” is ambiguous, and it is used for both local and global optimal solu-

tions.

2.3.1.1 Local Solvers

Many local solvers use local information about the neighborhood of a cur-

rent solution to find a step that improves the objective function and

maintains the feasibility of the current solution. If a step is found, it is

used to update the current solution. If not found, the algorithm termi-

nates. Typically, there is a threshold on the improvement in the objective

function that should be satisfied for the current solution to be updated.

27

Examples of the local information used are the derivative of the objective

function and the constraints with respect to the decision variables eva-

luated at the current solution. Examples of solvers using derivative in-

formation are the Active Set Quadratic Programming, Successive Qua-

dratic Programming (SQP), and Successive Linear Programming (SLP)

methods [9]. The algorithms such as SQP and SLP may require an infi-

nite number of iterations to find a local optimum (i.e., they only con-

verge); however a termination criterion is used to terminate in finite

time.

2.3.1.2 Global Solvers

Global solvers are designed to handle multiple local optima [11, 12]. Ex-

amples of such solvers include the genetic algorithms and the branch and

bound method.

Genetic algorithms mimic the survival of the fittest [13]. A population of

solutions is maintained. The solutions are evaluated for fitness, meaning

for feasibility and the objective function value. Pairs of solutions are cho-

sen randomly from the population and recombined. The higher the fit-

ness, the higher the chance for reproduction. The recombination process

is done by combining random parts of each decision value. Mutations are

included in order to ensure a sufficient large variation in the population

for convergence to the optimum. The genetic algorithms do not use any

structural information on the optimization problem, and any black box

models may easily be applied. However, this is also the main drawback as

the method gives a bound of neither the global optimum nor the local

optimum on termination, and the computational load is usually large.

A general framework for global optimization is the branch and bound

method. The method terminates with an upper and lower bound of the

objective function. By iteratively dividing the optimization problem in

28

properly posed sub-problems, the upper and lower bounds converge. The

bounds are calculated using structural information on the optimization

problem, and the bounds become more accurate as the sub-problems are

divided.

The branch and bound framework has shown to be particularly useful for

solving (mixed) integer linear programs because an upper bound for max-

imization (or lower bound for minimization) may easily be calculated by

solving a linear program where the integer constraints are dropped. For

mixed integer linear programs with lower and upper bounds for the in-

teger variables, the number of sub-problems to be solved is finite and

bounded. Each sub-problem is a linear program that is solved in finite

and bounded time, and the complete mixed integer linear program is

solved in finite and bounded time; however, the bounded time grows gen-

erally exponentially with the size of the problem [14, 15].

The branch and bound framework may also be used for general nonlinear

programs; however, much work is typically required finding good and va-

lid bounding functions. Some solvers are able to find bounding functions

by analyzing the constraints and the objective function automatically.

However, this requires that the constraints and the objective function are

available in analytical, or symbolic, form to the solver. In practice, the

optimization problem often includes a black box model (for instance a

reservoir simulator), and such bounding functions may not be calculated

neither by hand nor automatically because the structure of the model is

unknown to both the user and the solver.

2.3.1.3 Hybrid Solvers

Global solvers such as genetic algorithms may terminate far from a local

optimum. By passing the solution as an initial value to a local solver, the

solution is improved to bring it close to a local optimum.

29

2.3.1.4 Proxy Models

Proxy models are simplified models that are used because they are faster

to evaluate [16] or have better numerical properties [17], and both are

properties that are very important in RTO systems. Artificial neural

networks have been successfully used as proxy models to reduce the com-

putational load in history matching of reservoirs [16, 18, 19]. The proxy

model is first typically fitted to a set of model evaluations, and then used

as the model in the optimization. The solution found by the solver using

the proxy model may be validated or used as an initial value for the orig-

inal model to improve the solution further.

The success of proxy models was illustrated by Cullick et al. [16] where

the number of reservoir model evaluations required was reduced by 25 %

using such a model in history matching. Each evaluation of the reservoir

model took six hours, and hundreds of evaluations were required for the

history matching.

2.3.2 Production Planning

The goal of this plan is typically to maximize the daily production rate,

for example of oil, and to inject gas and water according to some given

rules provided by the reservoir planning.

2.3.2.1 Well Prioritization

If the goal is to maximize the oil production rate, some method is re-

quired to find an optimal way to prioritize the wells to produce and the

rate to produce. It is often required to prioritize because the available

processing capacity is less than the combined flowing capacity of the

wells. The processing capacity constraints are related to satisfaction of

product quality specifications, safe operation, processing facility capaci-

ties, utilities capacities, etc.

30

When a processing capacity constraint is met, it is often related to the

processing of gas, water, or liquid. The operator will typically choke back

the well having the largest ratio of the consumption of the associated

processing capacity to the oil produced. Examples of such ratios are the

gas oil ratio, water-oil-ratio, and liquid-oil-ratio. By successively choking

back and opening the wells based on the ratios, the capacity is fully uti-

lized and the production system is assumed to give the maximum total

oil production rate. When the total oil production rate is maximal, one

well will be partly opened and the rest either fully closed or opened. The

method above has proven successful because it is unaffected by the un-

certainties in the flowing potentials, which are the maximal oil produc-

tion rates of wells, and processing capacities. The main drawbacks of the

method are its inability to handle multiple active processing constraints

and the assumption of the flowing potentials of the wells to be indepen-

dent.

The flowing capacities of a well may be regarded as independent when

changing the oil production rate from the well does not change the flow-

ing capacities from the other wells and the gas oil ratio and the water-oil-

ratio are invariant with respect to the oil production rate. An example of

such wells is platform wells with wellheads at the processing platform

and a short common large diameter flow line to the inlet separators.

Lo and Holden [20] used a linear program for finding which wells that

should be opened, partially opened, or closed. They assumed each well

could produce any oil production rate between zero and the flowing po-

tential, and that the water cut and gas oil ratio were the same for all

rates (i.e. not coning gas or water). The method is able to handle mul-

tiple constraints on oil, water, liquid, and gas production for groups, or

all, of the wells. However, uncertainties in the model are not handled.

31

A way of handling a gas compression-constrained production system un-

der gas coning conditions was proposed by Barnes et al. [21]. The method

is able to handle wells where the incremental gas oil ratio (IGOR) is mo-

notonically increasing with the oil production rate. A similar method was

proposed by Urbanczyk [22]. The idea is to increase the production from

the well with the lowest IGOR with unused capacity, and reduce the

production wells with the highest IGOR. At the optimum, all the wells

have the same IGOR or they are on a minimum or maximum oil produc-

tion rate constraint.

Naus et al. [23] investigated the use of a combination of a reservoir simu-

lator and real-time data could be used to maximize the daily production

of oil. The parameters of the reservoir simulator were continuously up-

dated to fit measurements from the production system as they became

available, and the reservoir simulator was used to find derivative infor-

mation. The cases consisted of a reservoir and a horizontal well with four

continuous inflow control valves to control the segments of the well. The

total water and gas processing capacities were constrained. An SLP algo-

rithm was used to solve the problem.

2.3.2.2 Gas Lift

Gas lift may be used to increase the productivity of wells having low gas

oil ratio. By injecting gas into the tubing, the density of the well bore

fluid is reduced and thus the pressure drop component resulting from

gravity is reduced. However, the gas lift also gives a larger pressure drop

component resulting from friction, giving some optimum lift gas rate for

the well. Because of friction, the optimum lift gas rate may be 0 Sm3/D.

Usually, the available lift gas is less than the sum of the individual opti-

mum lift gas rates. The gas lift optimization problem is to find the lift

gas rates for each well giving the maximum total oil production rate sub-

ject to a gas lift processing capacity constraint, and possibly other opera-

32

tional and processing constraints. A cost may also be associated with the

processing of gas, water, oil, and lift gas changing the problem into max-

imizing the profit.

Mayhill [24] generated a gas lift performance curve by plotting the oil

production rate or profit rate versus injected lift gas for each well. The

curve is duplicated in Figure 2.5. The performance curves were later used

in the equal slope method proposed by Kanu [25]. The equal slope me-

thod established a way of finding optimum lift gas rates. The name was

given because of the characteristics of the optimum solutions where the

effect of an infinitesimal increase of the lift gas rate would be the same

for all wells.

Fang and Lo [26] developed a method for finding optimum lift gas rates

using gas lift performance curves. Each curve was approximated by a fi-

nite number of break points, and the curve was assumed to be linear be-

tween any adjacent break points. The production of each well was formu-

lated as the convex combination of the break points, resulting in a linear

program. The method is able to handle oil, water, liquid, and gas produc-

tion constraints for groups of wells or all wells. The same is true for lift

gas. Wells with variable water cut or gas oil ratio are also handled. The

authors pointed out that the method has problems if some wells cannot

flow naturally. This can however be solved by using mixed integer pro-

gramming [27]. Figure 2.5 duplicates an illustration by Fang and Lo [26]

of gas lift performance curves for wells flowing naturally and wells requir-

ing lift to produce.

Buitrago et al. [28] proposed a multistart search algorithm to find the

optimal gas lift rates under a constrained total lift gas rate. The method

uses gas lift performance curves, and is able to handle wells that require

a finite non-zero lift gas rate to produce. A case study showed a reduc-

33

tion of 14 % of the total lift gas rate for the same total oil production

rate compared to the equal slope method.

Gómez [29] proposed to fit the points defining the gas lift performance

curve to a second order polynomial, and then solve the gas lift optimiza-

tion problem by quadratic programming. The method was later extended

by Alarcón et al. [30] to also include a logarithmic term for better fitting.

For naturally flowing wells, a global optimum can be proved to be found

for the resulting optimization problem because of the convexity of the

problem. A heuristic handling shut-ins of wells, which were not naturally

flowing, was proposed.

Vazquez et al. [31] stressed the fact that many of the proposed optimiza-

tion problems used for oil production optimization may have multiple

local optima, and that the solvers may easily be trapped in a local opti-

mum that is inferior to a global optimum. To elude this, they proposed

to describe the oil production rate of each well as a function of the gas

lift injection rate and the energy consumption. The total production was

described as the sum of individual production rates. By using a hybrid

solver consisting of a genetic algorithm and a Tabu search heuristic, near

global optimal values were found.

2.3.2.3 Network

Earlier in this section, it was assumed that the production of each well

was not dependent on the production from the other wells. What mat-

tered were only the choke position and the lift gas injection rate. This

may be true if the manifold pressure is practically constant and if the

reservoir state and parameters do not change. However, the introduction

of subsea templates in offshore production systems has changed this. A

few wells are connected to each template on the seabed, and a common

flow line connects the template to the platform or a different subsea tem-

34

plate. The manifold pressure at the template will depend on the flow

rates from each of the wells connected to it. Thus, if the production from

one well is changed, then the others are changed too because of the

changed pressure conditions. Increased flow from one well may actually

cause an increase or decrease in the production from the other wells de-

pending on the operating conditions. For instance, a high gas oil ratio

well may give a gas lift effect for the other wells in the riser. The oppo-

site effect may be observed if the production from a high water cut well

is increased.

Dutta-Roy and Kattapuram [32] compared the optimal lift gas rates for

one, two, and three identical wells sharing a common flow line. In addi-

tion, larger field-wide networks were studied. SQP was used to find the

optimal gas lift rates. It was noted that optimal lift gas rates for each

well reduced when the number of wells increased.

Wang [27] used SQP to optimize flow rates of gas-lifted wells in a gather-

ing network having a maximal total water production capacity con-

straint, and the results were compared to optimization using piecewise

linear gas lift performance curves ignoring the network. Both methods

was implemented on the same numerical simulator. The proposed method

included a mathematical model of the pressure drop along the gas lifted

wells and the gathering network, and was thus accounting for the pres-

sure interaction. The method using piecewise linear gas lift performance

curves comprised the steps of generating gas lift performance curves using

the numerical simulator for a given manifold pressure, solving a mixed

integer linear program to obtain optimal lift gas rates of the wells, and

implementing the lift gas rates of the wells to the optimal lift gas rates.

The proposed method and the piecewise linear method gave total oil pro-

duction rates of 1528 Sm3/D and 1413 Sm3/D, respectively, when imple-

mented in the numerical simulator. The 8 % difference indicated a need

35

for accounting for the pressure interaction. See also Wang et al. [33]. Lat-

er, Wang and Litvak [34] proposed to extend the SLP algorithm used to

solve a piecewise linear approximation in each iteration.

In a work by Handley-Schachler et al. [35] it was proposed to solve this

type of problem using a modified version of SLP. In SLP, a linearization

of the optimization problem is used in each iteration to calculate a new

step. The linearization is obtained using the derivative of the objective

function and the constraints evaluated at the current solution, and each

iteration is typically faster than for SQP because a Hessian is not re-

quired. The number of iterations may be higher because the linearization

does not give any hints of the optimal step length to use as opposed to

SQP. Handley-Schachler et al. modified the SLP code to include piece-

wise linear gas lift curves. The piecewise linear gas lift curves were formu-

lated using linear constraints and non-convex constraints enforcing inter-

polation between adjacent points of the curves only. According to the

authors, the modification resulted in faster convergence than standard

SLP. See also [36].

Kosmidis et al. [37, 38] studied the optimization of gas lift and the

routing of wells to manifolds and separators. A mixed integer nonlinear

program was proposed, and it was solved to a local optimum using a

modified version of SLP similar to Handley-Schachler et al. [35] except

for the routing. Each iteration in the modified version of SLP consists of

solving a mixed integer program with routing constraints and piecewise

linear gas lift performance curves. SLP was used to handle the nonlinear

equations describing the flow lines and associated network.

Stoisits et al. [39] proposed to use a genetic algorithm to find the optimal

gas lift rates and oil production rates for the wells. A large number of

simulations were fitted to an artificial neural network-based proxy model

36

to speed up function evaluations. The method was able to handle con-

straints in gas and water processing capacity.

Grothey and McKinnon [40] modeled the pressure drop in the gas gather-

ing and supply network of an oil and gas production system with gas-

lifted wells. The resulting optimization problem was non-convex. By us-

ing one integer variable for each well, the optimization problem was re-

formulated as a convex nonlinear optimization problem except for the

additional integer variables. The branch and bound framework was used

to solve the optimization problem using a continuous relaxation (for in-

stance relaxing the integer variables to continuous variables). To over-

come the high computational load experienced, the optimization problem

was decomposed (Benders’ decomposition [41]) into a master and a sub-

problem. The sub-problem calculated the maximum total oil production

from a subset of the wells for a given gas usage and supply. The master

problem optimally distributed the gas usage and supply between the sub-

sets of wells. It was not proven that the local optima for the sub-

problems also were global optima, and the resulting optimum solution

was only found to be a local optimum. However, an upper bound was

provided by a convex Lagrangian relaxation of the problem, and the me-

thod was able to give a bound on the global optimum.

Various software packages are commercially available. GAP1 allows the

user to find optimal solutions using SQP. Processing facilities may be in-

cluded in the model to provide better results. ReO2 may also be used for

finding optimal solutions. The software uses SLP for solving the optimi-

zation problem, and the method is described in [35, 36].

1 Petroleum Experts Ltd. 2 EPS Ltd.

37

2.3.2.4 Processing Facilities

The well prioritization and gas lift optimization problems reported in the

literature do not typically include detailed models of surface processing

facilities. In fact, it is often assumed that the processing facilities are able

to handle fixed oil, water, gas, liquid, and lift gas rates. This is of course

a simplified description; the capacity of each component cannot be inves-

tigated independently to what is being produced. For instance, the gas

compression capacity may be limited by the capacity of the cooling sys-

tem. If two well streams have different temperatures, this may make a

difference. This may justify an integrated optimization of the production

system including the processing facility system.

The processing facilities typically consist of both parallel- and serial-

coupled processing units. Regarding optimized operation, load balancing

becomes an issue in both cases. Load balancing is performed by changing

the operational conditions of the processing facilities by manipulating the

different pressure, temperature, flow, level, or speed controller set points.

Ideally, the load balancing should be carried out at a pace that compen-

sates for the disturbances to the processing facilities in the form of

changes in, say, the air temperature, seawater temperature, processing

facility efficiency, processing facility availability, fluid compositions, en-

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.

88

1460 1480 1500 1520 1540 1560 1580 1600 1620 1640 16604000

4500

5000

5500

6000

6500

7000

Time [days]

Oil

rate

[S

m3/D

]

No new tests

Field data

Proposed method

Perfect information

Figure 4.3: The simulated total oil production rates are shown for cases

where no new tests are performed, tests are performed as in field data,

tests are performed by the method, and tests are performed using per-

fect information.

89

1460 1480 1500 1520 1540 1560 1580 1600 1620 1640 16600

1

2

3

4

5

6

7

8

9

10x 10

5

Time [days]

Cu

mm

ula

tive o

il p

rod

uc

tion

[S

m3]

No new tests

Field data

Proposed method

Perfect information

Figure 4.4: The simulated cumulative total oil production are shown for

cases where no new tests are performed, tests are performed as from

field data, tests are performed as in field data, tests are performed by

the method, and tests are performed using perfect information.

90

1460 1480 1500 1520 1540 1560 1580 1600 1620 1640 16600

2

4

6

8

10

12

14

Time [days]

Liq

uid

oil

ratio

[S

m3/S

m3]

No new tests

Field data

Proposed method

Perfect information

Proposed, samples

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

gas oil ratio.

# A-7A, % A-23, % A-26, % Oil, Sm3/D Gas, Sm3/D Liquid, Sm3/D

1 0 0 0 0.0 0 0

2 0 76 0 341.1 36,835 2500

119

Table 6.2: The sequence used when having priority on wells with a low

water cut.

# A-7A, % A-23, % A-26, % Oil, Sm3/D Gas, Sm3/D Liquid, Sm3/D

1 0 0 0 0.0 0 0

2 100 0 0 150.0 390,000 100

3 100 0 5 160.0 400,000 157

4 98 20 5 467.5 400,000 2500

120

Table 6.3: Some properties of wells are listed.

Well Oil Rate, Sm3/D Gas oil ratio Water cut H2S, PPM

A-6A 400 98 0.77 200.0

A-7A 150 2600 0.30 2.0

A-8R2 450 108 0.82 10.0

A-10A 100 108 0.84 350.0

A-13 400 3400 0.75 1.0

A-14 200 100 0.50 41.0

A-16A 750 100 0.78 10.0

A-21A 450 153 0.77 8.0

A-23 450 108 0.88 2.2

A-26 200 1000 0.85 4.5

A-28 300 1000 0.90 70.0

A-29A 700 85 0.16 1.5

A-31 350 91 0.65 10.0

A-33 450 95 0.59 0.0

A-34A 350 109 0.90 450.0

A-36 200 500 0.90 40.0

A-37 250 85 0.85 20.0

A-39A 450 106 0.85 400.0

A-40 700 250 0.00 0.2

A-42 400 260 0.70 1.8

A-43 350 84 0.45 20.0

A-45 400 94 0.65 60.0

A-46T2 400 300 0.38 4.2

A-48B 550 1000 0.07 0.5

121

Table 6.4: Cumulative oil production and losses in the cases investigated.

Example n Cumulative oil production, Sm3 Lost oil production, Sm3

Example 2 1 2114.5 281.7

Example 2 400 2115.4 280.8

Example 3 1 1964.9 431.3

Example 3 400 1968.8 427.4

Example 4 1 1351.6 1044.6

Example 4 400 1365.7 1030.5

Table 6.5: Computational information for the first time step is listed.

Example n Rows Columns Non zeros Iterations Time, s

Example 2 1 76 53 219 0 0.0

Example 2 400 20,824 11,624 78,024 5,719 1.4

Example 3 1 77 53 291 1 0.0

Example 3 400 21,624 11,624 116,424 8,526 3.4

Example 4 1 83 53 575 3 0.0

Example 4 400 23,624 11,624 220,424 63,178 278.1

122

0 1 2 3 4 5 6 70

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time, h

Oil

Pro

du

ctio

n R

ate

, S

m3/D

Example 2 (n = 1)

Example 2 (n = 400)

Example 3 (n = 1)

Example 3 (n = 400)

Example 4 (n = 1)

Example 4 (n = 400)

Figure 6.1: The total oil production rate approaches the maximum total

oil production rate.

123

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3x 10

6

Time, h

Ga

s P

rod

uction

Ra

te,

Sm

3/D

Example 2 (n = 1)

Example 2 (n = 400)

Example 3 (n = 1)

Example 3 (n = 400)

Example 4 (n = 1)

Example 4 (n = 400)

Figure 6.2: The total gas production rate approaches the gas treatment

constraint.

124

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5x 10

4

Time, h

Liq

uid

Pro

ducti

on

Ra

te,

Sm

3/D

Example 2 (n = 1)

Example 2 (n = 400)

Example 3 (n = 1)

Example 3 (n = 400)

Example 4 (n = 1)

Example 4 (n = 400)

Figure 6.3: The total liquid production rate approaches the liquid treat-

ment constraint.

125

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

Time, h

H2S

Con

ce

ntr

atio

n,

PP

M

Example 2 (n = 1)

Example 2 (n = 400)

Example 3 (n = 1)

Example 3 (n = 400)

Example 4 (n = 1)

Example 4 (n = 400)

Figure 6.4: The H2S concentration in the total production is only a con-

straint during the start-up.

126

127

7 Conclusions

A method for optimizing the total oil production rate from an oil produc-

tion system with wells sharing a flow line has been developed. The me-

thod uses a piecewise linear model of the pressure drops in the flow lines

and the wells. The model structure enables the use of a mixed integer

linear program solver and the use of a commercial branch and bound

solver, and the optimization problem may be solved to a global optimum.

Furthermore, no initial solutions are required to be passed to the solver.

The method is able to solve the optimization problem within reasonable

time (about ten seconds). However, generating the lookup tables for the

cases studied consumed about one day. Fortunately, generation of new

curves is only required when changing the geometry of the pipes, reser-

voir pressure, or reservoir temperatures. The proposed method satisfies

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. The ad-

vantage of the method compared to other existing methods is the ability

to find a proven global optimum. The method may be terminated at any

time after a feasible solution is found and still provide a bound on the

global optimum.

A method for finding the well to test in order to achieve the highest ex-

pected total oil production rate has been developed. The method assumes

that the oil production rates from each well are independent and that the

processing 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 gas or water oil ratio for each well is available. The well

test inaccuracy at the time of testing is neglected in this work. This as-

sumption can however easily be relaxed with the Monte Carlo framework

used. The method works by supplying the processing capacity of the oil

128

production 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 or water oil ratios and the date of the test. For each well considered

for testing, a Monte Carlo simulation is conducted with a probability dis-

tribution based on previous changes in the well tests. The Monte Carlo

simulations are only different in values of the estimates of the gas oil ra-

tio or water oil ratios. The well test giving the largest expected oil pro-

duction rate is recommended for implementation on the physical oil pro-

duction system. A computational study using field data indicated an in-

crease of 7.5 % in the oil production rate from the oil production system

was estimated over a commonly used method. The proposed method gave

96.3 % of the theoretical maximal oil production rate when perfect infor-

mation is available for the case studied. The case study assumed that

previously mentioned assumption in this paragraph is satisfied.

A method for optimal well management under uncertain water or gas oil

ratios has been developed. The method was able to handle a single uncer-

tain processing capacity constraint. Furthermore, the method is restricted

to wells which flows can be independently controlled. A case study of a

model field showed an estimated increase in oil production rates of

10.1 % compared to a method where the expected value of the distribu-

tion was used. The optimization problem was solved using a commercial

branch and bound solver. The case study assumed that previously men-

tioned assumption in this paragraph is satisfied.

A linear program for optimal scheduling of a start-up of an oil production

system has been developed, maximizing the total oil production on a ho-

rizon. Each well is constrained by the rate the flow rate could be in-

creased and decreased. Similarly, the change in the total rates of oil, gas,

water, and liquid production is also restricted. The proposed method was

compared to a short-term optimization method, showing a 0–1 % increase

129

in cumulative production at the expense of a lower production rate for a

few time steps. The increase should be considered small, and the contri-

bution of the method is the automation of the well management.

All the presented methods are a part of a real-time production optimiza-

tion system by doing data validation, model updating, model-based op-

timization, or optimizer command conditioning. Each method optimizes a

particular subsystem of the oil production system. The methods use sim-

plified models that may easily be tuned to measurement data because of

the small number of model parameters used.

The case studies using field data show a particular large potential in-

crease in the expected total oil production rate by explicitly handling the

uncertainties in the model. By using probability distributions of each un-

certain parameter in the models, a probability distribution of the perfor-

mance measure, such as the total oil production rate, may be found. By

penalizing the variation of the performance measure, the risk in operation

may be reduced. Regardless of penalization, the risk in the operation is

revealed. Sampling of the probability distribution, inspired by Monte

Carlo simulations, has been found particularly useful for reducing the

computational load of the optimization.

130

131

References

1. Asheim, H., Petroleumsproduksjon og prosessering på plattformen.

1985, Otta, Norway: TANO. 449.

2. Potential value of Integrated Operations on the Norwegian Shelf.

2006, The Norwegian Oil Industry Association: Stavanger, Nor-

way. p. 31.

3. Bieker, H.P., Optimization of Oil Production Systems and Ma-

thematical Programming, Department of Engineering Cybernetics.

2004, Norwegian University of Science and Technology: Trond-

heim, Norway, Master Thesis. 108.

4. Saputelli, L.A., S. Mochizuki, L. Hutchins, R. Cramer, M.B. An-

derson, J.B. Mueller, et al. “Promoting Real-Time Optimization

of Hydrocarbon Producing Systems”. Offshore Europe 2003, paper

SPE 83978. 2003. Aberdeen, U.K.

5. Sequeira, S.E., M. Graells, and P. Luis, “Real-Time Evolution for

On-line Optimization of Continuous Processes”. Industrial and

Engineering Chemistry Research, 2002. 41(7): p. 1815–1825.

6. Xiong, Q. and A. Jutan, “Continuous Optimization Using a Dy-

namic Simplex Method”. Chemical Engineering Science, 2003.

58(16): p. 3817–2828.

7. Mochizuki, S., L.A. Saputelli, C.S. Kabir, R. Cramer, M.J. Loch-

mann, R.D. Reese, et al. “Real Time Optimization: Classification

and Assessment”. SPE Annual Technical Conference and Exhibi-

tion, paper SPE 90213. 2004. Houston, Texas, U.S.A.

8. Jansen, B., M. Dalsmo, L. Nøkleberg, K. Havre, V. Kristiansen,

and P. Lemetayer. “Automatic Control of Unstable Gas Lifted

Wells”. SPE Annual Technical Conference and Exhibition, paper

SPE 56832. 1999. Houston, Texas, U.S.A.

132

9. Nocedal, J. and S.J. Wright, Numerical Optimization. 1999, New

York, U.S.A.: Springer. 636.

10. Williams, H.P., Model Building in Mathematical Programming.

4th ed. 1999, Chichester, U.K.: Wiley. 354.

11. Horst, R. and H. Tuy, Global Optimization: Deterministic Ap-

proaches. 3rd ed. 1996, Berlin, Germany: Springer-Verlag.

12. Törn, A. and A. Žilinskas, Global optimization. 1989, Berlin,

Germany: Springer-Verlag. 255.

13. Radin, R.L., Optimization in Operations Research. 1998, Upper

Saddle River, New Jersey, U.S.A.: Prentice Hall. 919.

14. Wolsey, L.A., Integer Programming. 1998, New York, U.S.A.: Wi-

ley. 264.

15. Nemhauser, G.L. and L.A. Wolsey, Integer and Combinatorial

Optimization. 1988, New York, U.S.A.: Wiley. 763.

16. Cullick, A.S., D. Johnson, and G. Shi. “Improved and More-

Rapid History Matching With a Nonlinear Proxy and Global Op-

timization”. SPE Annual Technical Conference and Exhibition,

paper SPE 101933. 2006. San Antonio, Texas, U.S.A.

17. Zangl, G., T. Graf, and A. Al-Kinani. “Proxy Modeling in Pro-

duction Optimization”. SPE Europec/EAGE Annual Conference

and Exhibition. 2006. Vienna, Austria.

18. Lechner, J.P. and G. Zangl. “Treating Uncertainties in Reservoir-

Performance Prediction With Neural Networks”. SPE Euro-

pec/EAGE Annual Conference, paper SPE 94357. 2005. Madrid,

Spain.

19. Saputelli, L., H. Malki, J. Canelon, and M. Nikolaou. “A Critical

Overview of Artificial Neural Network Applications in the Con-

text of Continuous Oil Field Optimization”. SPE Annual Tech-

nical Conference and Exhibition, paper SPE 77703. 2002. San An-

tonio, Texas, U.S.A.

133

20. Lo, K.K. and C.W. Holden. “Use of Well Management Schemes

for Rate Forecasts”. SPE Western Regional Meeting, paper SPE

24071. 1992. Bakersfield, California, U.S.A.

21. Barnes, D.A., K. Humphrey, and L. Muellenberg. “A Production

Optimization System for Western Prudhoe Bay Field, Alaska”.

65th Annual Technical Conference and Exhibition of the Society

of Petroleum Engineers, paper SPE 20653. 1990. New Orleans,

Lousiana, U.S.A.

22. Urbanczyk, C.H. “Optimization of Well Rates Under Gas Coning

Conditions”. SPE Production Operations Symposium, paper SPE

21677. 1991. Oklahoma City, Oklahoma, U.S.A.

23. Naus, M.M.J.J., N. Dolle, and J.D. Jansen. “Optimization of

Commingled Production using Infinitely Variable Inflow Control

Valves”. SPE Annual Technical Conference and Exhibition, paper

SPE 90959. 2004. Houston, Texas, U.S.A.

24. Mayhill, T.D. “Simplified Method for Gas-Lift Well Problem

Identification and Diagnosis”. 49th Annual Fall Meeting of the

Society of Petroleum Engineers of AIME, paper SPE 5151. 1974.

Houston, Texas, U.S.A.

25. Kanu, E.P., J. Mach, and K.E. Brown, “Economic Approach to

Oil Production and Gas Allocation in Continuous Gas Lift”.

Journal of Petroleum Technology, 1981. 33(10): p. 1887-1892.

26. Fang, W.Y. and K.K. Lo. “A Generalized Well-Management

Scheme for Reservoir Simulation”. 13th SPE Symposium on Re-

servoir Simulation, paper SPE 29124. 1996. San Antonio, Texas,

U.S.A.

27. Wang, P., Development and Applications of Production Optimiza-

tion Techniques for Petroleum Fields, Department of Petroleum

Engineering. 2003, Stanford University: Stanford, California,

U.S.A., PhD Thesis. 181.

134

28. Buitrago, S., E. Rodríguez, and D. Espin. “Global Optimization

Techniques in Gas Allocation for Continuous Flow Gas Lift Sys-

tems”. Gas Technology Conference, paper SPE 35616. 1996. Cal-

gary, Alberta, Canada.

29. Gómez, V., Optimization of Continuous Flow Gas Lift Systems.

1974, University of Tulsa: Oklahoma, U.S.A., Master Thesis.

30. Alarcón, G.A., C.F. Torres, and L.E. Gómez, “Global Optimiza-

tion of Gas Allocation to a Group of Wells in Artificial Lift Using

Nonlinear Constrained Programming”. Journal of Energy Re-

sources Technology, 2002. 124(4): p. 262–268.

31. Vazquez, M., A. Suarez, H. Aponte, L. Ocanto, and F. Jos.

“Global Optimization of Oil Production Systems, A Unified Op-

erational View”. SPE Annual Technical Conference and Exhibi-

tion, paper SPE 71561. 2001. New Orleans, Lousiana, U.S.A.

32. Dutta-Roy, K. and J. Kattapuram. “A New Approach to Gas-Lift

Allocation Optimization”. SPE Western Regional Meeting, paper

SPE 38333. 1997. California, U.S.A.

33. Wang, P., M. Litvak, and K. Aziz. “Optimization of Production

Operations in Petroleum Fields”. SPE Annual Technical Confe-

rence and Exhibition, paper SPE 77658. 2002. San Antonio, Tex-

as, U.S.A.

34. Wang, P. and M. Litvak. “Gas Lift Optimization for Long-Term

Reservoir Simulations”. SPE Annual Technical Conference and

Exhibition, paper SPE 90506. 2004. Houston, Texas, U.S.A.

35. Handley-Schachler, S., C. McKie, and N. Quintero. “New mathe-

matical Techniques for the Optimization of Oil & Gas Production

Systems”. SPE European Petroleum Conference, paper SPE

65161. 2000. Paris, France.

36. McKie, C.J.N., E.A. Rojas, N.M. Quintero, J.R.C. Fonsesecan,

and N.J. Perozo. “Economic Benefits From Automated Optimiza-

135

tion of High Pressure Gas Usage in an Oil Production System”.

SPE Production and Operations Symposium, paper SPE 67187.

2001. Oklahoma City, Oklahoma, U.S.A.

37. Kosmidis, V.D., J.D. Perkins, and E.N. Pistikopoulos, “Optimiza-

tion of Well Oil Rate Allocations in Petroleum Fields”. Industrial

and Engineering Chemistry Research, 2004. 43(14): p. 3513–3527.

38. Kosmidis, V.D., J.D. Perkins, and E.N. Pistikopoulos, “A Mixed

Integer Optimization Formulation for the Well Scheduling Prob-

lem on Petroleum Fields”. Computers and Chemical Engineering,

2005. 29(7): p. 1523-1541.

39. Stoisits, R.F., K.D. Crawford, D.J. MacAllister, M.D. McCor-

mack, A.S. Lawal, and D.O. Ogbe. “Production Optimization at

the Kuparuk River Field Utilizing Neural Networks and Genetic

Algorithms”. SPE Mid-Continent Operations Symposium, paper

SPE 52177. 1999. Oklahoma City, Oklahoma, U.S.A.

40. Grothey, A. and K. McKinnon, Decomposing the Optimization of

a Gas Lifted Oil Well Network. 2000, University of Edinburgh,

Department of Mathematics and Statistics: Edinburgh, U.K.

Technical Report: MS-00-005.

41. Benders, J.F., “Partitioning Procedures for Solving Mixed-

Variable Programming Problem”. Numerische Mathematik, 1962.

4: p. 238-252.

42. Koninckx, J., On-line Optimization of Chemical Plants Using

steady State Models, Chemical & Nuclear Engineering Depart-

ment. 1988, University of Maryland, Master Thesis. 400.

43. Qin, S. and T. Badgwell, “A Survey of Industrial Model Predic-

tive Control Technology”. Control Engineering Practice, 2003.

11(7): p. 733-764.

44. Sudaryanto, B. and Y.C. Yortsos. “Optimization of Displace-

ments in Porous Media Using Rate Control”. SPE Annual Tech-

136

nical Conference and Exhibition, paper SPE 71509. 2001. New Or-

leans, Lousiana, U.S.A.

45. Brouwer, D.R., J.D. Jansen, S.v.d. Starre, C.P.J.W.v. Kruijsdijk,

and C.W.J. Berentsen. “Recovery Increase through Water Flood-

ing with Smart Well Technology”. SPE European Formation

Damage Conference, paper SPE 68979. 2001. The Hague, The

Netherlands.

46. Brouwer, D.R. “Dynamic Optimization of Water Flooding with

Smart Wells Using Optimal Control Theory”. SPE 13th European

Petroleum Conference, paper SPE 78278. 2002. Aberdeen, U.K.

47. Yeten, B., L.J. Durlofsky, and K. Aziz. “Optimization of Smart

Well Control”. SPE International Thermal Operations and Heavy

Oil Symposium and International Horizontal Well Technology

Conference, paper SPE 79031. 2002. Calgary, Alberta, Canada.

48. Thiele, M.R. and R.P. Batycky. “Water Injection Optimization

Using a Streamline-Based Workflow”. SPE Annual Technical

Conference and Exhibition, paper SPE 84080. 2003. Denver, Colo-

rado, U.S.A.

49. Raghuraman, B., B. Couët, and P. Savundararaj. “Value of

Technology and Information for Reservoir Risk Management”.

SPE Annual Technical Conference and Exhibition, paper SPE

77424. 2002. San Antonio, Texas, U.S.A.

50. Bailey, W.J., B. Couët, and D. Wilkinson, “Framework for Field

Optimization To Maximize Asset Value”. SPE Reservoir Evalua-

tion and Engineering, 2005. 8(1): p. 7-21.

51. Narayanan, K., A.S. Cullick, and M. Bennett. “Better Field De-

velopment Decisions from Multi-Scenario, Interdependent Reser-

voir, Well, and Facility Simulations”. SPE Reservoir Simulations

Symposium, paper SPE 79703. 2003. Houston, Texas, U.S.A.

137

52. Cullick, A.S., D. Heath, K. Narayanan, J. April, and J. Kelly.

“Optimizing Multiple-Field Scheduling and Production Strategy

with Reduced Risk”. SPE Annual Technical Conference and Ex-

hibition, paper SPE 84239. 2003. Denver, Colorado, U.S.A.

53. Söderstrom, T. and P. Stoica, System Identification. 1988, Upper

Saddle River, New Jersey, U.S.A.: Prentice-Hall.

54. Clay, R.M., R.F. Stoisits, M.D. Pritchett, R.C. Rodd, and J.R.

Cologgi. “An Approach to Real-Time Optimization of the Central

Gas Facility at the Prudhoe Bay Field”. SPE Annual Technical

Conference and Exhibition, paper SPE 49123. 1998. New Orleans,

Lousiana, U.S.A.

55. Schulze-Riegert, R.W., O. Haase, and A. Nekrassov. “Combined

Global and Local Optimization Techniques Applied to History

Matching”. SPE 13th European Petroleum Conference, paper

SPE 79668. 2003. Houston, Texas, U.S.A.

56. Brouwer, D.R., G. Nævdal, J.D. Jansen, E.H. Vefring, and

C.P.J.W. van Kruijsdijk. “Improved Reservoir Management

Trough Optimal Control and Continuous Model Updating”. SPE

Annual Technical Conference and Exhibition, paper SPE 90149.

2004. Houston, Texas, U.S.A.

57. Saputelli, L., M. Nikolaou, and M.J. Economides. “Self-Learning

Reservoir Management”. SPE Annual Technical Conference and

Exhibition, paper SPE 84064. 2003. Denver, Colorado, U.S.A.

58. Saputelli, L., M. Nikolaou, and M.J. Economides, “Real-Time Re-

servoir Management: A Multiscale Adaptive Optimization and

Control Approach”. Computational Geosciences, 2006. 10(1): p.

61-96.

59. Kosmala, A., S.I. Aanonsen, A. Gajraj, V. Briran, K. Brusdal, A.

Stokkenes, et al. “Coupling of a Surface Network With Reservoir

138

Simulation”. SPE Annual Technical Conference and Exhibition,

paper SPE 84220. 2003. Denver, Colorado, U.S.A.

60. Beale, E.M. and J.A. Tomlin. “Special Facilities in a General Ma-

thematical Programming System for Non-convex Problems Using

Ordered Sets of Variables”. Proc 5th IFORS Conference. 1970.

Tavistock, London, U.K.

61. Padberg, M.W., “Approximating Separable Nonlinear Functions

via Mixed Zero-One Programs”. Operations Research Letters,

2000. 27(1): p. 1–5.

62. Ogryczak, W., “A Note on Modeling Multiple Choice Require-

ments for Simple Mixed Integer Programming Solvers”. Comput-

ers and Operations Research, 1996. 23(2): p. 199–205.

63. Litvak, M.L. and B.L. Darlow. “Surface Network and Well Tu-

binghead Pressure Constraints in Compositional Simulation”. 13th

SPE Symposium on Reservoir Simulation, paper SPE 29125. 1995.

San Antonio, Texas, U.S.A.

64. EPS Training, EPS Ltd. Technical Report.

65. Melbø, H., S.A. Morud, B. Bringedal, R.v.d. Geest, and K. Ste-

nersen. “Software that Enables Flow Metering of Well Rates

With Long Tiebacks and With Limited or Inaccurate Instrumen-

tation”. Offshore Technology Conference, paper OTC 15363.

2003. Houston, Texas, U.S.A.

66. Cramer, R., C. Moncur, and L. Berendschot. “Well-Test Optimi-

zation and Automation”. SPE Intelligent Energy Conference and

Exhibition, paper SPE 99971. 2006. Amsterdam, The Netherlands.

67. Bieker, H.P., O. Slupphaug, and T.A. Johansen. “Real Time Pro-

duction Optimization of Offshore Oil and Gas Production Sys-

tems: A Technology Survey”. SPE Intelligent Energy Conference

and Exhibition, paper SPE 99446. 2006. Amsterdam, The Nether-

lands.

139

68. de Boor, C., A Practical Guide to Splines. 1978, New York,

U.S.A.: Springer-Verlag. 392

69. Bieker, H.P., O. Slupphaug, and T.A. Johansen. “Optimal Well-

Testing Strategy for Production Optimization: A Monte Carlo

Simulation Approach”. SPE Eastern Regional Meeting, paper

SPE 104535. 2006. Canton, Ohio, U.S.A.

70. Bieker, H.P., O. Slupphaug, and T.A. Johansen. “Global Optimi-

zation of Multiphase Flow Networks in Oil and Gas Production

Systems”. AIChE Annual Meeting. 2006. San Francisco, U.S.A.

71. Dantzig, G.B., Linear Programming and Extensions. 1963, Prin-

ceton, New Jersey, U.S.A.: Princeton University Press. 627.

72. Lo, K.K., G.P. Starley, and C.W. Holden. “Application of Linear

Programming to Reservoir Development Evaluations”. 68th An-

nual Technical Conference and Exhibition of the Society of Petro-

leum Engineers, paper SPE 26637. 1993. Houston, Texas, U.S.A.

73. Maciejowski, J.M., Predictive control: with constraints. 2002, Har-

low, U.K.: Prentice Hall. 331.