university of oklahoma dynamic … of observation...university of oklahoma graduate college dynamic...

UNIVERSITY OF OKLAHOMA

GRADUATE COLLEGE

DYNAMIC OPTIMIZATION OF A WATER FLOOD RESERVOIR

A THESIS

SUBMITTED TO THE GRADUATE FACULTY

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE

By

JUDE NWAOZO Norman, Oklahoma

2006

ii

DYNAMIC OPTIMIZATION OF A WATER FLOOD RESERVOIR

A THESIS APPROVED FOR THE MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL

ENGINEERING

BY

__________________________ Dr. Dean Oliver

__________________________ Dr. Chandra Rai

__________________________ Dr. Dongxiao Zhang

iii

© Copyright by JUDE NWAOZO 2006 All Rights Reserved.

iv

Dedication

To my dad, mum and siblings

v

TABLE OF CONTENTS

DEDICATION..................................................................................................... IV

LIST OF FIGURES .........................................................................................VIII

LIST OF TABLES .............................................................................................XV

ABSTRACT...................................................................................................... XVI

1. INTRODUCTION..........................................................................................1

1.1 SMART WELLS ..........................................................................................................3

1.2 PRODUCTION OPTIMIZATION ....................................................................................5

2. LITERATURE REVIEW .............................................................................9

2.1 OIL AND GAS PRODUCTION OPTIMIZATION HISTORY ..............................................9

2.2 EOR PROCESS OPTIMIZATION ................................................................................16

2.3 OPTIMIZATION OF WELL PLACEMENT AND TYPE....................................................17

2.4 PRODUCTION OPTIMIZATION CONSIDERING UNCERTAINTY ...................................19

2.5 THIS OPTIMIZATION APPROACH .............................................................................20

3. METHODOLOGY ......................................................................................22

3.1 ENSEMBLE KALMAN FILTER (ENKF) .....................................................................22

3.2 OPTIMIZATION PROCEDURE....................................................................................23

4. RESERVOIR MODEL DESCRIPTION...................................................29

4.1 GRID........................................................................................................................30

vi

4.2 SCHEDULE ...............................................................................................................32

4.3 PVT PROPERTIES OF THE RESERVOIR FLUIDS ........................................................39

5. ECONOMICS ..............................................................................................42

5.1 NET PRESENT VALUE (NPV) ..................................................................................42

6. RESULTS AND ANALYSIS ......................................................................45

6.1 CASE 1.....................................................................................................................45

6.2 CASE 2.....................................................................................................................56

7. CONCLUSIONS ..........................................................................................71

NOMENCLATURE.............................................................................................73

REFERENCES.....................................................................................................74

A. APPENDIX – FORTRAN FLOW CHART ..............................................80

B. APPENDIX – RESULTS FROM CASE 3.................................................81

C. APPENDIX – DESCRIPTION OF FORTRAN CODE ...........................86

C.1 PERMEABILITY VALUES..........................................................................................86

C.2 FORWARD RUN.......................................................................................................87

C.3 REVENUE OPTIMIZATION .......................................................................................88

C.3.1 Optimum value of alpha .................................................................................90

C.4 FUNCTION OF ALPHA .............................................................................................91

C.5 PARAMETERS..........................................................................................................92

vii

D. APPENDIX – FORTRAN CODE...............................................................93

viii

LIST OF FIGURES

Figure 1-1: Top view of horizontal, 2-D reservoir model. The shaded zone

represents high permeability streak that is at right angles with the

injector and the producer.................................................................... 7

Figure 2-1: Shape of the oil-water front before breakthrough for the base

case (left) and for the optimized case (right). ................................... 14

Figure 2-2: Schematic of reservoir used by Lorentzen et al. ................... 15

Figure 2-3 (a) and (b): Development of optimized value.......................... 15

Figure 4-1: Top view of the reservoir model showing permeability field

distribution and well placements....................................................... 30

Figure 4-2: Histogram showing permeability field distribution.................. 31

Figure 4-3: Initial distribution of mean values of bottom hole pressure

profile for well P1.............................................................................. 33

Figure 4-4: Pressure profile generated using correlation range a = 5...... 35

Figure 4-5: Histogram for pressure realizations for well P1 ..................... 35


ix



Figure 4-9: Ten realizations of BHP profile for production well P1........... 38

Figure 4-10: Ten realizations of BHP profile for well P2. ......................... 38

Figure 4-11: Ten realizations of BHP profile of well P3. .......................... 39

Figure 4-12: Ten realizations of BHP profile of well P4. .......................... 39

Figure 4-13: Relative Permeability Curves .............................................. 40

Figure 6-1: Permeability field distribution for case 1. ............................... 45

Figure 6-2: Case 1 – Graph showing NPV for all realizations of pressure

profiles before and after optimization. .............................................. 46

Figure 6-3: Case 1 – Graph showing cumulative oil production for all

realizations of pressure profiles before and after optimization.......... 47

Figure 6-4: Case 1 – Graph showing 10 realizations of optimized pressure

profiles for well P1. ........................................................................... 47



x





Figure 6-8: Case 1 – Water saturation distribution a) before (top) and b)

after (bottom) optimization after 913 days. ....................................... 50

Figure 6-9 (a) and (b): Case 1 – Graphs showing water cuts from all wells

before (top) and after (bottom) optimization for realization 1 of BHP

profiles.............................................................................................. 51

Figure 6-10(a) and (b): Case 1 – Graphs showing water cuts from all wells


profiles.............................................................................................. 52


before (left) and after (right) optimization for realization 2 of BHP

profiles.............................................................................................. 53

Figure 6-12: Graph showing cumulative oil and water production before

and after optimization for realization 1 of BHP profile....................... 54

xi


and after optimization for realization 2 of BHP profile....................... 55

Figure 6-14: Graph showing oil and water production rates before and

after optimization for realization 1 of BHP profile.............................. 55


after optimization for realization 2 of BHP profile.............................. 56

Figure 6-16: Case 1 - Net Present Value vs. iterations............................ 57

Figure 6-17: Permeability field distribution for case 2. ............................. 58





Figure 6-20: Case 2 – Graph showing 10 realizations of optimized

pressure profiles for well P1. ............................................................ 60



xii






after (bottom) optimization after 913 days. ....................................... 62



profiles.............................................................................................. 64



profiles.............................................................................................. 65



profiles.............................................................................................. 66

Figure 6-28: Case 2 – Graph showing cumulative oil and water production

before and after optimization for realization 1 of BHP profile............ 67

xiii



Figure 6-30: Case 2 – Graph showing oil and water production rates




Figure 6-32: Case 2 – Net Present Value vs. iterations........................... 69

Figure B-1: Permeability distribution for case 3. ...................................... 81

Figure B-2: Case 3 – Graph showing NPV for all realizations of pressure


Figure B-3: Case 3 – Graph showing cumulative oil production for all


Figure B-4 (a), (b), (c), (d): Ten realizations of BHP for the optimized case

for all production wells ...................................................................... 82

Figure B-5: Case 3 – Graphs showing water cuts from all wells before

(left) and after (right) optimization for 3 realizations of BHP profiles. 83

xiv

Figure B-6 (a) and (b): Case 3 – Graphs showing cumulative oil and water

production before and after optimization for realizations 1 and 2. .... 84

Figure B-7 (a) and (b): Case 3 – Graphs showing oil and water production

rates before and after optimization for realizations 1 and 2 of BHP

profile................................................................................................ 84

Figure B-8: Case 3 – Water saturation distribution a) before (left) and b)

after (right) optimization after 913 days. ........................................... 85

Figure B-9: Case 3 – Net Present Value vs. iterations ............................ 85

Figure C-1: Graph of NPV as a function of alpha for 6 iterations............. 92

xv

LIST OF TABLES

Table 1: Summary of reservoir properties ............................................... 41

Table 2: Summary of results.................................................................... 70

xvi

ABSTRACT

It is of increasing necessity to produce oil and gas fields more efficiently

and economically because of the ever-increasing demand for petroleum

worldwide. Since most of the significant oil fields are mature fields and the

number of new discoveries per year is decreasing, the use of secondary

recovery processes is becoming more and more imperative. Waterflooding

is one of the most widely used secondary recovery means of production

after primary depletion energy has been exhausted. The use of smart

wells, which are typically wells that are equipped with downhole chokes

and other measuring instruments, is also gaining popularity in the industry

as a more efficient means of enhancing ultimate recovery. This research

presents a methodology of optimizing production or net present value from

a waterflood reservoir by controlling the bottom hole pressures of the

production wells with the use of smart well technology.

The optimization procedure involves maximizing the objective function

(e.g. cumulative oil produced or net present value) from a waterflood

reservoir by adjusting a set of controls (e.g. production wells’ bottom hole

pressure or flow rates). In this project, the pressure profile of the

production well that gives the maximum NPV is the solution to our

xvii

waterflood optimization problem. The production wells in this reservoir are

smart wells whose downhole chokes are automatically adjusted to meet

certain optimal requirements. The production well completions are also set

in such a way that they automatically shut off when certain economic limits

based on the watercut are reached.

Most efficient methods used in solving optimization problems require the

explicit knowledge of the underlying simulator equations for computation

of the gradient of the objective function with respect to the controls. As a

result of the large and complicated nature of reservoir models with large

number of unknowns and non-linear constraints, the software for gradient

calculations for practical optimization problems are very tedious and time

consuming to create. The approach presented in this study does not

require the solution of the adjoint equations. No knowledge of the

simulator equations is required and the simulator is run as a black box. In

this approach, a variant of the ensemble Kalman filter (EnKF) technique is

used in the optimization process. A relationship between the objective

function and the set of controls is obtained from the ensemble of

realizations of the state vector.

xviii

This research work also provides a validation of the optimization

methodology using various heterogeneous 2-D models with five-spot

pattern waterflood schemes. The forward run was carried out with the

Eclipse 100 (black oil) reservoir simulator. The results of the optimization

methodology presented in this study show an increase in net present

value of up to 9% and an increase in cumulative production of up to 12%

of the base case when the geology is known.

1

1. INTRODUCTION

In the past, a variety of secondary oil recovery methods have been

developed and applied to mature and depleted oil reservoirs. These

methods help to improve oil recovery compared to primary depletion1. The

oldest secondary recovery method is waterflooding since water is usually

readily available and inexpensive.

Fundamentally, waterflood involves pumping water through a well

(injector) into the reservoir. The water is forced through the pore spaces

and sweeps the oil towards the producing wells (producers). The

percentage of water in the produced fluids steadily increases until the cost

of removing and disposing of water exceeds the income from oil

production. After this point, it becomes uneconomical to continue the

operation and the waterflooding is stopped. Some wells remain

economical at a watercut of up to 99%2. On the average, about one-third

of the original oil in place (OOIP) is recovered, leaving two-thirds behind

after secondary recovery. Other secondary recovery methods include CO2

flooding and hydrocarbon gas injection, which requires a nearby source of

inexpensive gas in sufficient volume.

2

Waterflooding is most often used as a secondary recovery method of

increasing oil recovery in reservoirs where primary depletion energy has

been exhausted. It is responsible for the high production rates in the U.S.

and Canada3 where most of the fields are mature.

The number of new discoveries of significant oil fields per year is

decreasing worldwide and most of the existing major oilfields are already

at their mature stages. Consequently, it is becoming increasingly

necessary to produce these fields as efficiently as possible in order to

meet the global increase in demand for oil and gas4. For this reason,

waterflooding projects are very commonly found in most of these mature

fields. In many of these reservoirs however, water cuts from the

production wells are very high and sometimes uneconomical thereby

causing low ultimate recoveries5. This is because the injected water finds

its way through conductive fractures and high permeability zones within

the reservoir. Premature breakthrough mostly occurs in highly

heterogeneous reservoirs. As a result, many water injectors do not usually

achieve improved sweep efficiencies and a lot of the oil is by-passed.

Various methods of solving the problem of poor sweep efficiency have

3

been suggested. One method of mitigating this problem is by employing

smart production and injection wells6, 7, 8, 9.

1.1 Smart Wells

Smart well technology provides the opportunity to counteract the effects of

high permeability zones in a waterflood field by imposing a suitable

pressure or flow rate profile along the injection wells7. Smart wells are

development wells that contain permanent downhole measurement and

control equipments that enable significant improvement of oil production8

and increase the efficiency of injectors. Smart wells are equipped with a

battery of completion equipment designed to

a) Monitor well operating conditions downhole including flow rate,

pressure, temperature, phase composition, etc.

b) Control inflow and outflow rates of segregated segments of the

well.

c) Image the distribution of reservoir attributes away from the well.

These attributes may include resistivity, acoustic impedance, etc9.

A smart injector is an injector that has been divided into several intervals,

each of which can be independently controlled using inflow control valves

(ICVs) and open hole packers10. ICVs divide the injector into different

4

segments thereby making it possible to control water injection into

individual injection zones. With this type of controls in place, the injector

well can be used to selectively flood zones with more homogeneous

matrix or areas that result in less water production at the production wells.

This is achieved by opening and closing of the ICVs to flood desired

intervals. During injection, the water cuts from the production wells are

closely monitored to determine the optimal opening and closing of the

ICVs. When the water cuts from the production wells reach unacceptable

limits, the section of the horizontal injector contributing to high water cut at

the production well is isolated and shut-in using the ICVs. During this

period, the water injection continues via the matrix and other areas of

lower conductivity. This process is repeated over time until the sweep

efficiency of the injector is maximized and the water saturation is more

uniformly distributed across the injected zone8.

Smart well technology is presently undergoing the process of value

identification and quantification in the exploration and production industry9.

The advantages of this new technology to the industry include the

following:

5

1. Any sudden changes in the production or injection performance of

the well can be immediately observed and prompt response can be

carried out.

2. There is minimized down time and well interference leading to cost

savings.

3. More reserves can be drained per well due to improved well

management.

4. Improved well management also brings about increased ultimate

recovery.

1.2 Production Optimization

The best production schemes for oil and gas fields is being continually

sought after in order to maximize the production from these existing fields.

The objective of reservoir simulation is to determine the best production

design for a given field. This goal has been commonly achieved by trial

and error method. The reservoir engineer is left to decide what parameters

to change and how the changes are made to improve the results. This

imposes a high level of subjectivity to the optimization process. In the past

few decades, researches have been made to develop simulators that can

be used to determine the best production schemes. This can be

6

conceptually achieved by combining the existing reservoir simulators with

some numerical search algorithms.

The problem of production optimization requires the maximization or

minimization of some objective function g(x). In this optimization problem,

the objective function to be maximized is the net present value or

cumulative oil production. Here, x is a set of controls, which may include

bottom hole pressures, flow rates, choke size, etc and these controls may

be manipulated in order to achieve an optimum value at which the

objective function is maximized (or minimized). Optimization processes

result in the improvement of future performance of a reservoir and

therefore requires a simulation model of the real reservoir on which the

optimization is carried out. The simulation model is a dynamic model that

relates the objective function to the set of controls.

Consider a water injector and a producer in Figure 1-1. Let the objective

function g(x) be net present value and the total fluid production rates at

each of the production well completions be the set of controls, x. Changing

the production rates at each completion in turn changes the dynamic state

of the system (pressures and saturations). These changes subsequently

impact on the cumulative production and hence the objective function

7

(NPV). The controls, x are also subject to other constraints such as

surface production facilities, choke sizes, fracture limits, minimum

allowable bottom hole pressures, etc and these constraints determine

feasible values of the controls. These additional constraints pose major

problems and further complicate the solution of the optimization process.

Figure 1-1: Top view of horizontal, 2-D reservoir model. The shaded zone

represents high permeability streak that is at right angles with the injector

and the producer.

Two major categories of optimization algorithms exist in literature4:

gradient-based algorithms11,12,13 and stochastic algorithms. Gradient-

based algorithms require an efficient technique of calculating the gradient

8

of the objective function g(x) with respect to the controls x. The optimal

control theory is one of the most popular gradient-based algorithms.

The total number of controls to be adjusted is the product of the number of

controls to be updated in time (control steps) and the total number of wells

in the reservoir model. The number of controls could be very large even

for a simple reservoir model with a reasonable number of wells and control

steps, making the gradient estimation a very tedious process. Also,

another major drawback of the gradient-based method using adjoint

equations is that it requires explicit knowledge of the simulation model

equations used to describe the dynamic system.

On the other hand, the stochastic algorithms such as genetic

algorithms14,15 and simulated annealing16,17 require many forward model

evaluations but are capable of finding a global optimum with a sufficiently

large number of simulation runs. Unlike the gradient-based algorithms,

they do not require gradient estimations since the relationship between the

objective function and the controls can be obtained from several forward

models. However, the methods can be inefficient when the number of

variables is large.

9

2. LITERATURE REVIEW

The need for production optimization of reservoir fields has arisen as a

result of the global increase in demand for oil and gas. Several

applications of optimization algorithms have been developed and these

optimization techniques have proved to be beneficial in the various

problems of reservoir development, well testing and gas resource

distribution18. This chapter reviews the various optimization problems that

have been investigated by researchers and their methodologies to solving

the existing problems.

2.1 Oil and Gas Production Optimization History

Production optimization problems involving reservoir modeling with time

was first attempted by Lee and Aronofsky19. The purpose of their study

was to apply linear programming procedure to oil production scheduling

problems. The problem was to determine an oil production schedule from

5 different wells that will give the maximum profit over an eight-year

period. The constraints placed on the individual reservoir production rates

of the wells included well pressures and pipeline capacity. They solved

10

this problem using constant well interference coefficients as a substitute

for a real reservoir simulation model. Wattenbarger10, along with some

other researchers extended this study further with the use of real reservoir

simulation models for estimating the well interference coefficients.

Wattenbarger developed a method for maximizing withdrawals from a

natural gas storage reservoir.

Natural gas is commonly stored in underground reservoirs during the

summer months and then produced during the winter to meet seasonal

demands. This seasonal production can be maximized through optimal

scheduling of the individual wells. Wattenbarger10 proposed a method for

optimizing the withdrawal schedule problem using the linear programming

format. In his case, the withdrawal schedule was optimized in the sense

that no discretized withdrawal schedule can be specified for the finite-

difference model that will give greater total seasonal production while still

meeting the constraints placed on the problem. One of the constraints of

this problem requires that the wellbore pressure of each well not fall below

a minimum value. Also, the total reservoir withdrawal rate at any time is

limited to the demand rates.

11

All the work previously mentioned have been limited by the number of

phases, the phase behavior or by the geometry and size of the reservoir

model. An approach, which uses only the control variables explicitly for

numerical optimization has been developed. Asheim11,20 was involved in

the study of optimal control in water flood reservoirs using reservoir

simulation models. He developed a method for numerical optimization of

the net present value of a natural water drive and water drive by injection.

The method uses an areal two-phase reservoir simulator to calculate the

net present value (NPV) of a waterflooding scheme. In his study, the

variables subject to control were the well rates. The waterflooding scheme

that maximized the net present value was numerically obtained by

combining reservoir simulation with control theory practices of implicit

differentiation. He was able to achieve improved sweep efficiency and

delayed water breakthrough by dynamic control of the well flow rates. For

the reservoir models he considered, there was a net present value

improvement of up to 11%.

Brouwer and Jansen7 studied the optimization of water flooding with fully

penetrating, smart horizontal wells in 2-dimensional reservoirs with simple,

large-scale heterogeneities (Figure 1-1). They used optimal control theory

as an optimization algorithm for valve settings in smart wells. The

12

objective was to maximize the recovery or net present value of the

waterflooding process over a period of time.

In the study, they investigated the static optimization of waterflooding with

smart wells. Static implies that the injection and production rates in the

wells were kept constant during the displacement process, until water

breakthrough occurred. They observed significant improvements from

simple reservoir models. They however, observed that more

improvements could be achieved by dynamic optimization of the

production and injections. In a later study8, they addressed this same

problem using dynamic optimization in which case, the inflow control

valves in the wells were allowed to vary during the waterflooding process.

Waterflood was improved by changing the well profiles according to some

simple algorithm that move flow paths away from the high permeability

zones in order to delay water break-through. This was achieved by

calculating the productivity index (PI) for each segment. For each well, the

segments with the higher PI are shut-in and the rates are equally

distributed among the other segments that are open in order to maintain

the production rates. They repeated this process until the optimum flow

profile is obtained. This optimum flow profile was found to occur when the

13

ultimate oil recovery from a successive step is lower than that obtained

with the preceding flow profile.

Brouwer and Jansen7 investigated the optimization problem under two

different scenarios of well operating conditions – purely pressure-

constrained and purely rate-constrained operating environments. They

concluded that the benefit of smart wells under pressure-constrained

operating conditions was mainly the reduced amount of water production

rather than increased oil production. On the other hand, wells operating

under rate constraints gave an increased production and ultimate recovery

as well as reduced water production.

Their results show that water breakthrough is delayed from 253 days for

the base case to 658 days for the optimized case. Figure 2-1 shows

Brouwer and Jansen’s results for the oil and water saturation distribution

just before breakthrough for both the base case and the optimized case. It

can be observed that the sweep of the low permeability region is much

better for the optimized case, thereby improving the ultimate recovery.

14

Figure 2-1: Shape of the oil-water front before breakthrough for the base

case (left) and for the optimized case (right).

Lorentzen et al.21 also carried out a study on the dynamic optimization of

waterflooding using a different approach from those described above. He

carried out his optimization by controlling the chokes to maximize

cumulative oil production or net present value. Their new approach uses

the ensemble Kalman filter as an optimization routine. The ensemble

Kalman filter was originally used for estimation of state variables but has

been adapted to optimization in their work. In their optimization study, they

demonstrated the use of the ensemble Kalman filter as an optimization

routine on a simple 5-layer reservoir with different permeabilities. The

schematic of the reservoir used by Lorentzen et al. is shown in Figure 2-2.

The results from this approach are shown in Figure 2-3 a. and b.

15

Figure 2-2: Schematic of reservoir used by Lorentzen et al.

The above methodology provided by Lorentzen et al. avoids the use of the

optimal control theory since no adjoint equations were needed and the

model equations are treated as a “black box”.

Figure 2-3 (a) and (b): Development of optimized value

16

This methodology avoids one obvious disadvantage of the optimal control

approach when used as a solution to optimization problems – it entails the

construction and solution of an adjoint set of equations. These adjoint

equations require an explicit knowledge of the reservoir model equations

and also require extensive programming in order to implement them. This

has been shown by Sarma and Aziz4.

2.2 EOR Process Optimization

In 1984, Ramirez and Fathi12 applied the theory of optimal control to

determine the best possible injection policies for enhanced oil recovery

processes. Their study was motivated by the high operating costs

associated with EOR projects. The commercial application of new EOR

processes depends on whether economic projections indicate a decent

return on investment. The objective of their study was to develop an

optimization method to minimize injection costs while maximizing the

amount of oil recovered. The performance of their algorithm was

subsequently examined for surfactant injection as an EOR process in a

one-dimensional core flooding problem13. The control for the process was

the surfactant concentration of the injected fluid. They observed a

17

significant improvement in the ratio of the value of the oil recovered to the

cost of the surfactant injected from 1.5 to about 3.4. Optimal control was

also applied to steam flooding by Liu and Ramirez22 in 1993. They

developed an approach using optimal control theory to determine

operating strategies to maximize the economic attractiveness of steam

flooding process. Their objective was to maximize a performance index

which is defined as the difference between oil revenue and the cost of

injected steam. Their optimization methodology also obtained significant

improvement under optimal operation.

2.3 Optimization of Well placement and type

A great deal of research work has been carried out at Stanford University

to determine the optimum location, type and trajectory of wells to be drilled

in a field. The determination of a well location is a very complex problem

that depends on several variables which include reservoir and fluid

properties, well and surface equipment specifications, and economic

criteria. In 2002, a hybrid optimization technique based on genetic

algorithm (GA) was proposed by Baris et al.23 at Stanford University to

optimize placement of water-injection wells for an offshore field in the Gulf

of Mexico. The objective function used was NPV while the water injection

rates and well placements of up to four injectors were being optimized.

18

Their results showed an incremental NPV of $154 million with three

injectors after optimum placement has been achieved, compared to the no

injection case. Badru et al.24 also carried out a similar investigation using

the Hybrid Genetic Algorithm (HGA) to determine optimal well locations.

They used this technique to optimize both vertical and horizontal wells for

both gas injection and water injection projects using NPV as the objective

function. They compared the results obtained from the optimization of well

placements proposed by the HGA method with those selected by

engineering judgment. The optimized placement results obtained using

HGA showed a significant increase in cumulative production of about 70%

more than that proposed by engineering judgment. Burak et al.25 also at

Stanford University extended the research on well optimization process by

including well type and trajectory of nonconventional wells. This problem is

more complicated than other well optimization problems because of the

wide variety of possible well types that must be considered, which include

number of wells, location, and orientation of laterals. Their optimization

procedure entailed the use of GA in conjunction with other routines such

as artificial neural network. They observed a general increase in the

objective function relative to the reference case, up to 30% in some cases.

19

2.4 Production Optimization considering uncertainty

Naevdal, Brouwer and Jansen26 in 2005, developed a closed-loop control

approach where measurements from smart wells were used to

continuously update a waterflood reservoir model and an adjoint-based

optimal control strategy was computed based on the most recent update

of the reservoir model. The ensemble Kalman filter was used to obtain

frequent updates of the reservoir model. They demonstrated their

methodology on a simple reservoir model with one smart injector and

producer where the objective function was NPV and the total fluid

production rates were used as the controls. In a nut-shell, their

methodology is a combination of an optimal control for waterflood

optimization with automatic history matching of reservoir models using

ensemble Kalman filter to estimate the final permeability field. Naevdal et

al. observed that the results obtained using a closed-loop control starting

from an unknown permeability field, were almost as good as those

obtained assuming a priori knowledge of the permeability field.

Another closed-loop production optimization approach in a water flood

reservoir was presented by Sarma, Durlofsky and Aziz27 in 2005. In their

approach, a gradient-based optimization algorithm was used to determine

optimal control settings, while the parameter gradients are used for model

20

updating. The model-updating component of the closed loop is a problem

of inversion of production data (well pressures and flow rates) in order to

determine the reliable estimates of uncertain model parameters (porosity

and permeability). Their results showed substantial improvements in NPV

of up to 25% of the base case and very close to those obtained if the a

priori reservoir description was known.

Some other applications of optimization algorithms used in different

problems of reservoir development, oil production and well testing have

been surveyed by Virnovsky18. Asides added benefits in oil production

through the development of new waterflooding strategies, optimization

procedures have been successfully applied to gas distribution among a

group of gas lift wells. Mathematical programming algorithms in

conjunction with numerical simulation of the appropriate processes were

used to obtain the optimal solutions for each of the cases he presented.

2.5 This Optimization Approach

The new approach for production optimization of a waterflood reservoir

presented in this study is a variant of the ensemble Kalman filter

procedure. It does not require the explicit knowledge of the reservoir

21

simulation equations that are used to describe the dynamic state of the

reservoir system. Solutions to the adjoint equations are therefore not

required hence making software development less tedious.

As discussed earlier, Lorentzen et al.21 applied the ensemble Kalman filter

technique directly to his waterflood optimization problem as well. In his

application, he replaced the observed measurements by values

representing an upper limit for the possible NPV. The filter then returns

control settings that result in NPV as close to the predefined value. In this

new approach however, a predefined value of NPV is not required in the

optimization process. It simply optimizes NPV by maximizing an objective

function which includes the NPV and a penalty term that penalizes the

controls that are far away from the prior estimate. The gradient of the NPV

with respect to controls is obtained from the ensemble of control

realizations. A complete description of the methodology is presented in the

following chapter.

22

3. METHODOLOGY

The ensemble Kalman filter (EnKF) technique has been adapted to the

problem of NPV in this optimization study. In this study, an ensemble of 40

realizations of the controls (bottom hole pressure profiles) was generated

and continuously updated after each reservoir simulation run until nearly

optimum pressure profiles were obtained. The optimum pressure profile is

the profile at which the net present value is at its maximum. Since EnKF is

a Monte-Carlo approach, the final results will vary for each member of the

ensemble.

3.1 Ensemble Kalman Filter (EnKF)

The Kalman filter is typically used to estimate states in systems that

change with time28. The procedure consists of a forecast step and an

assimilation step in which variables that describe the state of the system

are corrected to honor the observations using a series of equations.

The ensemble Kalman filter (EnKF) is a modified form of the Kalman filter

that has been adapted to history matching in reservoir simulation26, 28. It is

a Monte-Carlo method in which an ensemble of initial reservoir state

23

vectors are generated by sampling from a probability density function and

kept up-to-date as data are assimilated sequentially. The reservoir state

vectors consist of all the reservoir variables that are uncertain and need to

be specified in order to run the reservoir simulator. The uncertainty of

reservoir state vectors is estimated from the ensemble28. The state vector

consists of two parts: model parameters (porosities, permeabilities,

saturations, and pressures) and the theoretical data (e.g. water-oil-ratios,

production rates, bottom hole pressures, etc.). If the reservoir state vector

is denoted by y, then the state vector for the reservoir model can be

written as

[ ]TTT dmy = 3-1

where m = model parameters

d = theoretical data

3.2 Optimization Procedure

The ensemble optimization process also consists of 2 steps – the forecast

(or forward) step and the update step. A numerical reservoir simulator is

used to perform the forecast step. The reservoir model is run for each

member of the ensemble of state vectors using Eclipse 100 for the forward

simulation. The reservoir state vector consists of all the control variables

24

that are uncertain and need to be optimized. In this project, the state

vector is made up of the bottom hole pressures for all the wells at every

time step, as well as the net present value obtained from running the

reservoir simulator with these controls. Since there are 4 wells in our

model and 20 time steps in total, the reservoir state vector for each

member of the ensemble is made up of (80 + 1) members. From the

forecast step, the net present value based on these controls is calculated

using the cumulative reservoir fluid production and the average estimated

oil price.

Let x be used to denote the number of controls on the wells for the time

periods. Then,

x = {x1, x2, x3… x80}

These controls could be choke settings, flow rates, bottom hole pressures,

etc. Also, let the net present value for the production period using controls

x be g(x). Assume also that we wish to penalize the control settings that

are far from our initial guess or that rapidly change with time.

The best control settings in this case will be the set x that maximizes the

following equation:

25

( ) ( )pX

T

p xxCxxxgxS −−−= −1

2)()(

α 3-2

where S(x) = objective function

g(x) = Net Present Value

α = weighting factor

x = new state vector

xp = prior state vector

Cx = covariance matrix of the control vector

A local quadratic approximation to S(x) at x = x’ is given as

xHxxxSxxF TT δδδγδ2

1)'()'( ++=+ 3-3

whereγ is equal to )(xS∇ , and the Hessian, H isTxS ))((∇∇ .

The value of xδ that maximizes the quadratic approximation to the objective

function is the extremum of this function and it occurs at 0=∇F or

0=+ xHδγ 3-4

The Newton equations for iteratively finding the extremum are

γδ −=xH 3-5

or l

l

l SxH −∇=+1δ 3-6

After computing1+lxδ from equation 3-6 above, the controls are updated

using the equation below.

26

11 ++ += lll xxx δ 3-7

The gradient of the objective function S(x) for the lth iteration is

( )p

l

X

lxxCxGS −−=∇ −1)( α 3-8

(assuming that xC is symmetric). In the above equation, )( lxG denotes the

matrix of the sensitivity coefficients or the derivatives of the objective

function with respect to the controls. The sensitivity coefficient is a

measure of how strongly the objective function, gi(x) is affected by a

change in the controls, x. The individual elements of the sensitivity matrix

are given by,

j

iji

x

gG

∂

∂=, 3-9

The approximate Hessian matrix is given by:

1)( −−∇≈ x

lCxGH α 3-10

Assuming that the second derivative of g(x) is negligible, equation 3-10

above becomes

1−−≈ xCH α 3-11

Therefore, substituting equation 3-8 and 3-11 in equation 3-6 gives

( )[ ]p

l

xl

l

X xxCGxC −−−=− −+− 111 αδα 3-12

After further manipulation, equation 3-12 becomes

27

( )p

l

lx

lxxGCx −−=+

αδ

11 3-13

Substituting the incremental controls 1+lxδ obtained in equation 3-13 into

equation 3-7;

p

l

lx

llxxGCxx +−+=+

α

11 3-14

Equation 3-14 reduces to

plx

lxGCx +=+

α

11 3-15

Equation 3-15 is used to calculate the updated state vector for the next

iteration step.

Let y be used to denote the state vector consisting of the controls (bottom

hole pressures) as well as the net present value obtained from using the

controls. The ensemble of state vectors can be written as

[ ]eNyyyyY ,...,,, 321= 3-16

Ne is used to denote the number of the ensemble members. The

covariance matrix for the state variables at any time can be estimated

from the ensemble using the standard statistical formula;

( )( )Te

Y YYYYN

C −−−

=1

1 3-17

28

whereY = Mean of state vector calculated across the ensemble.

SinceT

YlX MCGC = , then equation 3-15 becomes

p

T

Y

lxMCx +=+

α

11 3-18

The vector M is called the measurement operator and it relates the state

vector to the theoretical observation. Since the theoretical observation is

part of the state vector y, M is a simple matrix with 0 and 1 as its

components. The matrix M can be arranged as follows:

[ ]IM 0= 3-19

Where 0 is an Nd x (Ny – Nd) matrix with all 0s as entries and I is an Nd x

Nd identity matrix. Note that Nd is the number of measurements (Nd = 1 for

this project) and Ny is the number of variables in the state vector, y (Ny =

81 for this project).

29

4. RESERVOIR MODEL DESCRIPTION

As described earlier, the ultimate goal of a reservoir simulator is to

determine the optimum production scheme of an oil and gas field. This

can be achieved by combining a reservoir simulator with a numerical

optimization algorithm. The reservoir simulation phase of this study was

carried out with the use of Eclipse 100 – black oil option. The waterflood

optimization procedure developed in this research was tested on various

2-D Cartesian reservoir models consisting of 25 x 25 x 1 grid lattice. In this

study, a reservoir with no-flow boundaries on all sides was considered.

The phases present in the reservoir were oil and water. No free gas was

present. The model represents a 200-acre field (approximately 2950 ft x

2950 ft) with 1 vertical injector well (INJ) located at the center of the

reservoir (in grid block 13:13:1) and 4 vertical production wells (P1, P2,

P3, and P4) located at the corners of the field. Production well P1 is

located in block 1:1:1; well P2 is located in block 25:1:1; well P3 is located

in block 1:25:1; and well P4 is in block 25:25:1. Note that the well locations

are fixed and therefore not subject to optimization. The top view of one of

the reservoir models used in this study is shown in Figure 4-1. The wells

are drilled with a 40-acre spacing and are all brought to operation at the

30

same time. The depth of the top surface of the reservoir is 10,000 ft with a

net pay thickness of 50 ft.

Figure 4-1: Top view of the reservoir model showing permeability field

distribution and well placements.

4.1 Grid

The basic geometry of the simulation grid and various rock properties

(porosity, absolute permeability, etc) in each grid cell are specified in the

grid section. From these properties, the pore volumes of the grid blocks

and the inter-block transmissibilities are calculated by the simulator29. The

P1 P2

P3 P4

INJ

31

keywords used in this section usually depend on the geometry option

selected in the initialization section. In this case, we used the Cartesian,

block-centered geometry option. The porosity distribution in the reservoir

is assumed to be homogeneous with a porosity of 0.25 while the

permeability is heterogeneous with an average value of 60 md for the

base case. The permeability field was generated using the sequential

Gaussian simulation (SGS) algorithm in the Geostatical Software Library

(GSLIB). The orientation of the permeability correlation was set at an

angle of 45 degrees in GSLIB in order to achieve a diagonal permeability

trend.

50 100 150 200 250

Permeability HmdL0

20

40

60

80

100

120

rebmuN

fo

dirg

skcolb

Histogram showing permeability distribution in the reservoir

Figure 4-2: Histogram showing permeability field distribution

32

It should be noted here that the values obtained from the SGS simulation

are log permeabilities (lnk). They must first be converted to actual

permeability values by taking the exponential of the variable X before

applying them to the reservoir grid model. A histogram of the initial

permeability distribution is shown in Figure 4-2. The distribution appears to

be log normal.

The original fluids in place in the reservoir consist of water at a pore

volume saturation of 20% and undersaturated oil contained in 80% of the

pore volume. The residual oil and connate water saturation are 0.15 and

0.20, respectively. The initial reservoir pressure is 4500 psi.

4.2 Schedule

As said earlier, all the wells were drilled vertically and completed with 0.5

ft wellbore internal diameter to a depth of 10,050 ft and brought into

operation at the same time (1-Jan-1990). The wells were operated for a

10-year period with constant control settings for 6 months. In total, there

were 20 control settings for each well in the production period. The injector

well schedule had a rate controlled mode with an injection rate of 5000

stb/day. Bottom hole pressures in the production wells are the constraints

33

that were used for the optimization process. The minimum allowable

bottom hole pressure was set at 200 psi.

The following procedure was used to generate the 40 initial realizations of

pressure constraints for the production wells. These steps were used for

each well’s BHP profile.

Step 1: The mean value of each pressure profile was randomly selected

from a uniform distribution. This distribution characterizes a random

variable whose value is equally likely everywhere within the interval. The

upper limit of the uniform distribution was 3000 psi and the lower limit was

1000 psi. Figure 4-3 shows the distribution of the means of the pressure

profiles for well P1.

1500 2000 2500 3000

Mean BHP

0

2

4

6

8

10

rebmuN

fo

snoitazilaer

Histogram showing Mean BHPs for Well P1

Figure 4-3: Initial distribution of mean values of bottom hole pressure

profile for well P1.

34

Step 2: A gaussian covariance function30 with a practical range of about

2.5 years (5 time periods) was subsequently used to generate pressure

variations from the mean for the 20 time steps, which describe the bottom

hole pressure profile for each realization. The mean of this distribution is

the randomly selected value from step 1.

−−=

a

hhhC

ji

ji

)(3exp)( 2

, σ 4-1

where C(h i,j) = Covariance function

σ = Standard deviation

hi, hj = Random variables (Pressures)

a = Correlation range

Step 3: The generated pressure profiles were exported from mathematica

to a text file to be read into Eclipse schedule include file during the

simulation runs. Also the pressure profiles and histogram showing all the

realizations were plotted.

Step 4: Steps 1, 2 and 3 were repeated for all the 4 wells. A standard

deviation of 200 psi was used in the gaussian covariance model shown in

equation 4-1. Each of the 4 wells has a data file where pressure

realizations are stored. These files were also used to store updated

pressure profiles during each iteration process.

35

0 5 10 15 20

Timestep

1000

2000

3000

4000

mottoB

eloH

erusserP

HispL5 realizations of BHP profile for Well P1 using a = 5

Figure 4-4: Pressure profile generated using correlation range a = 5.

1000 1500 2000 2500 3000

Bottom Hole Pressure

0

10

20

30

40

50

60

rebmuN

fo

snoitazilaer

Histogram of BHP realizations for Well P1

Figure 4-5: Histogram for pressure realizations for well P1

Figure 4-5 – Figure 4-8 show histogram plots of initial pressure settings for

all 4 producing wells. Graphs of 10 realizations of the bottom hole

36

pressure profiles of production wells P1 – P4 used in the initial simulation

run is shown in Figure 4-9 - Figure 4-12.

1000 1500 2000 2500 3000


0

10

20

30

40

50

60

rebmuN

fo

snoitazilaer



1000 1500 2000 2500 3000


0

10

20

30

40

50

60

rebmuN

fo

snoitazilaer



37

1000 1500 2000 2500 3000 3500


0

10

20

30

40

50

60

rebmuN

fo

snoitazilaer



The economic limits of the production wells were set using the CECON

(Economic limits for production well connections) keyword from the list of

Eclipse keywords. If an individual connection (or group of connections)

violates one of the economic limits that have been set, it automatically

shuts off. In the case of this study, the maximum water cut of 0.93 is the

economic limit that has been set. Therefore, if any of the wells exceed a

water cut of 0.93, that well is automatically shut-off.

38

Bottom Hole pressure schedule for well P1 - Prior

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10

Figure 4-9: Ten realizations of BHP profile for production well P1.


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10

Figure 4-10: Ten realizations of BHP profile for well P2.

39


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10

Figure 4-11: Ten realizations of BHP profile of well P3.


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10

Figure 4-12: Ten realizations of BHP profile of well P4.

4.3 PVT Properties of the Reservoir Fluids

The reservoir fluids are oil and water. The oil contains a constant and

uniform concentration of 0.2 Mscf/stb of dissolved gas. The oil bubble

40

point pressure is assumed to be 400 psi. At a reference pressure of 4500

psi, the oil has a viscosity of 2.4 cp. The oil formation volume factor (Bo) is

0.972. At surface conditions, the oil is assumed to have a density of 56

lb/cuft while the density of water is assumed to be 62.4 lb/cuft. Water

compressibility is set at 3 x 10-6 psi-1, water formation volume factor (Bw) of

1.0034 rb/stb and viscosity of 0.96 cp at a reference pressure of 4500 psi.

The bulk compressibility of the rock was set at 4 x 10-6 psi-1.

The relative permeability curve used is shown in Figure 4-13 below. A

summary of the reservoir properties is shown in Table 1.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Water Saturation, Sw

Rela

tive P

erm

eabili

ty

Krw

Kro

Figure 4-13: Relative Permeability Curves

41

Table 1: Summary of reservoir properties

Number of grid blocks 25 x 25 x 1

Grid block size 118ft x 118ft x 50ft

Water injection rate 5000 STB/D

Reservoir thickness 50 ft

Porosity 25%

Actual reservoir area 200 acres

Initial Oil Saturation 0.8

Initial Water Saturation 0.2

Well Depth 10,000 ft

Initial reservoir pressure 4500 psi

Ave. Reservoir Temperature 284 F

Production period 10 years

Time step 6 months

42

5. ECONOMICS

The objective function used in this project is the net present value of the

waterflood operation for a given production period. The objective is to

maximize the net present value over the life of the reservoir and this is

achieved by adjusting a set of controls (bottom hole pressures or flow

rates). This chapter explains the concept of net present value and how it

can be determined.

5.1 Net Present Value (NPV)

Present value of money compares the value of a certain amount of money

today to the value of that same amount in the future and vice versa, taking

into consideration inflation and returns. Net present value (NPV) is the

difference between the present value of cash inflows and the present

value of cash outflows. Given an investment opportunity, NPV is used by

an organization to analyze the profitability of the project or investment and

to make decisions with regards to capital budgeting. It is sensitive to the

future cash inflows that an investment or project will yield.

NPV can be computed using the following formula31:

43

0

1 )1(C

r

CNPV

T

tt

t −+

=∑=

5-1

where t = Time step

Ct = Cash inflow after time t, $

r = Annual (or periodic) discount rate, fraction

T = Cumulative investment (or production) period

C0 = Initial investment

A conservative annual discount rate of 10% was used in this study in the

estimation of the present value of money and is based on the current rates

at which eligible institutions are charged to borrow short-term funds

directly from a Federal Reserve Bank (approximately 6.5%). Also, most oil

companies use this rate for evaluating the viability of proposed

investments.

Cash inflow is calculated from the oil and water production rates obtained

from each of the production wells or from the cumulative production from

the reservoir. The price of oil is pegged at $40 per barrel for the entire 10-

year production period while the cost of water disposal is $3 per barrel of

produced water. The total cash inflow for the entire production period is

given by,

44

( ) ( )watfwptbblfoptC $/$ ×−×= 5-2

where, C = Total cash inflow, $

$/bbl = Price of Oil per bbl, $

$wat = Cost of water disposal per bbl, $

fopt = Cumulative oil production, stb

fwpt = Cumulative water production, stb

The economic limit is determined by the time at which the cost of handling

the water exactly balances the income from selling the oil. The water cut

at which the economic limit is reached can be calculated thus;

%93340

40

$/$

/$≈

+=

+=

+=

watbbl

bbl

woprwwpr

wwprwct 5-3

where wct = Water cut, stb/stb

Therefore, each production well in the simulator has been set up in such a

way that the connection/perforation is automatically closed as soon as it

reaches an economic limit of 93% water cut.

45

6. RESULTS AND ANALYSIS

This chapter presents and analyzes the results obtained from this

research project. The codes developed were tested with three different

permeability fields which will be denoted as case 1 – case 3. Results from

case 1 and case 2 are presented in this chapter. Results from case 3 are

presented in Appendix B.

6.1 Case 1

The permeability field for case 1 is shown in Figure 6-1.

Figure 6-1: Permeability field distribution for case 1.

P1 P2

P3 P4

INJ

46

The increase in NPV and cumulative production after the optimization

process can be observed from Figure 6-2 and Figure 6-3, consecutively.

The percentage increase in NPV ranged between 2.4% and 8.7% while a

percentage increase of up to 9% was observed in the cumulative oil

production after optimization.

NPV for all realizations before and after optimization

128,000,000

130,000,000

132,000,000

134,000,000

136,000,000

138,000,000

140,000,000

142,000,000

144,000,000

146,000,000

148,000,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Realization Number

NP

V (

$)

Prior NPV Optimized NPV


profiles before and after optimization.

The optimized pressure profiles that give the highest net present value for

all four production wells are shown in Figure 6-4 – Figure 6-7 for ten

realizations. This can be compared to the initial BHP realizations shown in

section 4.2 of chapter 4 (see Figure 4-9 – Figure 4-12).

47

Cumulative Oil production for initial and optimized realizations

5,100,000

5,200,000

5,300,000

5,400,000

5,500,000

5,600,000

5,700,000

5,800,000

5,900,000

6,000,000

6,100,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Realization number

Cu

mu

lati

ve O

il P

rod

ucti

on

(S

TB

)

Initial cumulative production

Optimized cumulative production


realizations of pressure profiles before and after optimization.

Bottom Hole pressure schedule for well P1 after optimization

0

100

200

300

400

500

600

700

800

900

1000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10



48

Bottom Hole pressure schedule for well P2 - Optimized

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si





0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si




49


0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si




From the optimized pressure profiles shown in the figures above, NPV

optimization of case 1 requires that well P1 be produced at the minimum

bottom hole pressure for the duration of the production period. This is as a

result of the low permeability zone between the injector and the producer.

The profile of well P1 is however, contrary to the optimized pressure

profile of well P4 where pressures are continually increased to delay water

breakthrough. The effect of the optimized pressure profiles on the water

saturation distribution can be observed in Figure 6-9 below.

50


after (bottom) optimization after 913 days.

There is a considerable change in the distribution of water saturation

across the field at the time of the earliest water breakthrough. This change

can be observed in Figure 6-8. Figure 6-8a shows the water saturation

51

distribution before optimization while b) shows the distribution after

optimization. It can be observed that b) gives a more evenly distributed

water saturation across the field than a). This means that higher sweep

efficiency was attained after the optimization process.

Water cut from producing wells for schedule realization 1 before optimization

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb

Water cut from well P1




Water cut from producing wells for schedule realization 1 after optimization

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb





Figure 6-9 (a) and (b): Case 1 – Graphs showing water cuts from all wells


profiles

52

The graphs of water cuts from all four production wells using realization 1

of BHP profiles before and after optimization are shown in Figure 6-9. It

can be observed that the breakthrough times as well as the water cut

trends come closer to overlapping after the optimization process.


0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb







profiles

53

The above trend can also be observed in other realizations of BHP profile

(See Figure 6-10 and Figure 6-11).


0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb

Water cut from wellP1Water cut from wellP2Water cut from wellP3


before (left) and after (right) optimization for realization 2 of BHP profiles

54

Graphs of field cumulative production of oil and water with time before and

after optimization are shown in Figure 6-12 (for realization 1) and Figure

6-13 (for realization 2). The oil and water production rates for both

realizations are also shown in Figure 6-14 and Figure 6-15. It can be

observed that the optimization process sought to maximize the rates at the

early stages of production. Since the NPV is the objective function being

maximized, the early oil production contributes most to the NPV than the

later production.

Field cumulative production for schedule realization 1 before and after

optimization

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Cu

mu

lati

ve p

rod

ucti

on

, S

TB

Cum Oil prod - priorCum Oil Prod - OptimizedCum Water prod - priorCum Oil prod - Optimized


and after optimization for realization 1 of BHP profile.

55


optimization

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Cu

mu

lati

ve p

rod

ucti

on

, S

TB

Cum Oil prod - prior

Cum Oil Prod - Optimized

Cum Water prod - prior

Cum Water prod - Optimized


and after optimization for realization 2 of BHP profile.

Field production rates for schedule realization 1 before and after

optimization

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Pro

du

cti

on

rate

s, S

TB

/D

Oil prod rate - prior

Oil prod rate - Optimized

Water Prod rate - prior

Water prod rate - Optimized


after optimization for realization 1 of BHP profile.

56


optimization

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Pro

du

cti

on

rate

s, S

TB

/D






after optimization for realization 2 of BHP profile.

Finally, Figure 6-16 shows a plot of NPV against iterations. It is observed

that the optimized value of NPV continuously increases as a function of

iterations. The total increase in the mean NPV of the ensemble from the

initial to the optimized case is approximately 5.9%.

6.2 Case 2

The permeability field distribution for case 2 is shown in Figure 6-17. The

average permeability of the field is about 60 md. However, large patches

of very low permeability are observed between well P3 and the injector.

57

Mean NPV of ensemble as a function of iteration number

137,000,000

138,000,000

139,000,000

140,000,000

141,000,000

142,000,000

143,000,000

144,000,000

145,000,000

146,000,000

147,000,000

0 1 2 3 4 5 6 7 8 9 10

Iteration Number

NP

V (

$)

Prior Mean NPV for initial BHP realizations

Optimized Mean NPV

5.9 % increase

Figure 6-16: Case 1 - Net Present Value vs. iterations

After the optimization process, an increase in NPV and cumulative

production can be observed from Figure 6-18 and Figure 6-19,

consecutively. The percentage increase in NPV ranged between 3.7% and

7.6%. A percentage increase of up to 5% was observed in the cumulative

oil production after optimization.

58

Figure 6-17: Permeability field distribution for case 2.


130,000,000

132,000,000

134,000,000

136,000,000

138,000,000

140,000,000

142,000,000

144,000,000

146,000,000

148,000,000

150,000,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Realization Number

NP

V (

$)




P1 P2

P3 P4

INJ

59


5,300,000

5,400,000

5,500,000

5,600,000

5,700,000

5,800,000

5,900,000

6,000,000

6,100,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Realization number

Cu

mu

lati

ve O

il P

rod

ucti

on

(S

TB

)





The optimized pressure profiles for all four production wells are shown in

Figure 6-20 – Figure 6-23 for ten realizations. These optimized profiles

can be compared with the initial BHP realizations shown in section 4.2 of

chapter 4 (see Figure 4-9 – Figure 4-12).

From the optimized pressure profiles, NPV optimization of case 2 requires

that wells P2 and P3 be produced close to the minimum bottom hole

pressure for the duration of the production period. The profiles of well P1

and P4 have continually increasing pressure profiles with production time.

60


0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si



pressure profiles for well P1.


0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si




61


0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si





0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si




62

The effect of the optimized pressure profiles on the water saturation

distribution can be observed in Figure 6-24 below.


after (bottom) optimization after 913 days.

63

Figure 6-24a shows the water saturation distribution before optimization

while b) shows the distribution after optimization both for case 2. As

observed in case 1, Figure 6-24 b) gives a more evenly distributed water

saturation across the field than a). This means that higher sweep

efficiency was attained after the optimization process.

The graphs of water cuts from all four production wells using realization 1

of BHP profiles before and after optimization are shown in Figure 6-25. It

can be observed that the breakthrough times as well as the water cut

trends tend to overlap after the optimization process. This trend can also

be observed in other realizations of BHP profile (See Figure 6-26 and

Figure 6-27).

Graphs of field cumulative production of oil and water with time before and

after optimization are shown in Figure 6-28 (for realization 1) and Figure

6-29 (for realization 2). The oil and water production rates for both

realizations are also shown in Figure 6-30 and Figure 6-31. It can be

observed that the optimization process sought to maximize the rates at the

early stages of production. Since the NPV is the objective function being

maximized, the early oil production contributes most to the NPV than the

later production.

64


0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb







profiles

65


0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb







profiles

66


0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r cu

t, s

tb/s

tb







profiles

67


optimization

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Cu

mu

lati

ve p

rod

ucti

on

, S

TB



before and after optimization for realization 1 of BHP profile.


optimization

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Cu

mu

lati

ve p

rod

ucti

on

, S

TB







68


optimization

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Pro

du

cti

on

rate

s, S

TB

/D








optimization

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Pro

du

cti

on

rate

s, S

TB

/D







69

Figure 6-32 shows a plot of NPV against iterations. It is observed that the

optimized value of NPV continuously increases as a function of iterations.

The total increase in the mean NPV of the ensemble from the initial to the

optimized case is approximately 6.02%. The maximum number of

iterations used in the optimization process was 15. This limit was used to

minimize computational time.


138,000,000

139,000,000

140,000,000

141,000,000

142,000,000

143,000,000

144,000,000

145,000,000

146,000,000

147,000,000

148,000,000

0 2 4 6 8 10 12 14 16

Iteration Number

NP

V (

$)


Optimized Mean NPV

6.02 % increase

Figure 6-32: Case 2 – Net Present Value vs. iterations

A summary of the results obtained from this study is presented in table 2.

70

Table 2: Summary of results

REFERENCE OPTIMIZED

Cum. Oil

Production

NPV Cum. Oil

Production

Increase

in Cum.

Oil Prodn

NPV Increase

in NPV

Case

x106 STB ($Mil) x 106 STB ($Mil)

1 5.44 134.2 6.0 10.3% 146 8.8%

2 5.58 136.4 6.03 5.06% 147.9 8.4%

3 5.49 137.2 6.14 11.8% 147 7.1%

71

7. CONCLUSIONS

A new production optimization algorithm has been presented in this

project. The methodology borrows its concept from the ensemble Kalman

filter for continuous model update and has been successfully applied to

various heterogeneous waterflood reservoir models. The optimization

process showed remarkable improvement in net present value of up to 9%

from the initial base case as well as an improvement of cumulative

production of up to 8% from the base case. Also, the water saturation at

breakthrough was observed to be more uniformly distributed across the

reservoir after the optimization process as compared with the unoptimized

case.

The advantage of this methodology over the adjoint-based method is that

it does not require explicit knowledge of the simulator flow equations

thereby making it computationally less tedious. A commercial simulator

can easily be applied to this optimization technique without tampering with

its source code. Another advantage of this methodology over Lorentzen et

al’s is that it does not require a pre-selected NPV, which he used to

72

optimize controls. Rather, it optimizes the controls to the maximum

possible NPV by maximizing an objective function.

As a recommendation for future work, the optimization methodology

presented in this study can be used to optimize other objective functions

like cumulative oil production. Also, other controls including total fluid

production rates can also be used as constraints. The procedure may also

be applied to other waterflood patterns. This approach has been applied to

a simple 2-D heterogeneous reservoir with known geology. Further work

can also be carried out on reservoir geology while considering

uncertainties in the reservoir model parameters and also on large scale

field examples.

73

NOMENCLATURE

Ct = Cash inflow after time step, t

Cx = Covariance matrix of control vector

Cy = Covariance matrix of state vector

fopt = Cumulative oil production

fwpt = Cumulative water production

G = Sensitivity matrix

H = Hessian matrix

M = Measurement operator

Ne = Number of ensemble members

r = discount rate

S = Objective function

T = Cumulative production period

X = Control vector

$/bbl = Price of Oil per bbl

$wat = Cost of water disposal

α = weighting factor

γ = Gradient of objective function

74

REFERENCES

1. Sneider, R. M. and Sneider, J. S., ”New Oil in Old Places”,

prepared for presentation at the Pratt II Conference San Diego,

California January 12-15, 2000.

2. Lake, L. W., Schmidt, R. L., and Venuto, P. B., “A Niche for

Enhanced Oil Recovery in the 1990s”, Petroleum Engineer

International (January 1992): 55 – 61.

3. Craig, Forrest Jr.: “The Reservoir Engineering Aspects of

Waterflooding”, Society of Petroleum Engineers of AIME, 1971.

4. Sarma, P., Aziz, K., and Durlofsky, L. J., “Implementation of Adjoint

Solution for Optimal Control of Smart Wells”, paper SPE 92864

presented at the 2005 SPE Reservoir Simulation Symposium held

in Houston, Texas, 31 Jan – 2 Feb 2005.

5. Arenas, A. and Dolle, N., “Smart Waterflooding Tight Fractured

Reservoirs Using Inflow Control Valves”, paper SPE 84193

presented at the SPE Annual Technical Conference and Exhibition

held in Denver, Colorado, 5-8 October 2003.

6. Esmaiel, T. E. H., “Applications of Experimental Design in

Reservoir Management of Smart Wells”, paper SPE 94838

prepared for presentation at the SPE Latin American and

75

Caribbean Petroleum Engineering Conference held in Rio de

Janeiro, Brazil, 20 – 23 June 2005.

7. Brouwer, D. R., Jansen, J. D., Van Der Starre, S., van Kruijsdijk,

and Berentsen, C. W. J., “Recovery Increase through Waterflooding

with Smart Well Technology”, paper SPE 68979 presented at the

SPE European Formation Damage Conference held in the Hague,

The Netherlands, 21-22 May 2001.

8. Brouwer, D. R. and Jansen, J. D., “Dynamic Optimization of

Waterflooding with Smart Wells Using Optimal Control Theory”,

SPE Journal vol. 9, no. 4, Dec., pp. 391-402.

9. Glandt, C. A., “Reservoir Aspects of Smart Wells”, SPE Drilling &

Completion Journal, vol. 20, no. 4, December, pp. 281 – 288.

10. Wattenbarger, R. A., “Maximizing Seasonal Withdrawals from Gas

Storage Reservoirs”, Journal of Petroleum Technology, Aug 1970,

pp. 994-998.

11. Asheim, H., “Maximization of Water Sweep Efficiency by

Controlling Production and Injection Rates”, paper SPE 18365

prepared presented at the SPE European Petroleum Conference,

London, UK, October 16-19, 1988.

76

12. Ramirez, W. F., Fathi, Z., and Cagnol, J. L, “Optimal Injection

Policies for Enhanced Oil Recovery: Part 1 – Theory and

Computational Strategies”, SPE Journal, June 1984, 328-332.

13. Fathi, Z., and Ramirez, W. F., “Optimal Injection Policies for

Enhanced Oil Recovery: Part 2 – Surfactant Flooding”, SPE

Journal, June 1984, pp. 333-341.

14. Tavakkolian, M., Jalali, F., and Amadi, M. A., “Production

Optimization using Genetic Algorithm Approach”, paper SPE 88901

prepared for presentation at the 28th Annual SPE International

Technical Conference Exhibition in Abuja, Nigeria, August 2-4,

2004.

15. Harding, T. J., Radcliffe, N. J., and King, P. R., “Hydrocarbon

Production Scheduling with Genetic Algorithms” SPE Journal (June

1998), Vol. 3, no. 2, pp 99 – 107.

16. Zhou, C., Gao, C., Jin, Z., and Wu, X., “A Simulated Annealing

Approach to Constrained Nonlinear Optimization of Formation

Parameters in Quantitative Log Evaluation”, paper SPE 24723

prepared for presentation at the 67th Annual Technical Conference

and Exhibition in Washington DC, Oct 4-7, 1992.

17. Sen, M. K., Datta-Gupta, A., Stoffa, P. L., Lake, L. W., and Pope,

G. A., “Stochastic Reservoir Modeling Using Simulated Annealing

77

and Genetic Algorithms”, SPE Formation Evaluation Journal, March

1995, vol. 10, no. 1, pp. 49 – 56.

18. Virnovsky, G. A., “Optimization Techniques Application in Oil

Recovery Problems”, paper SPE 24281, prepared for presentation

at the SPE European Petroleum Computer Conference held in

Stavanger, Norway, 25-27 May 1992.

19. Lee, A. S. and Aronofsky, J. S., “A Linear Programming Model for

Scheduling Crude Oil Production”, Journal of Petroleum

Technology (July, 1958) vol. 10, No. 7, 51-54.

20. Asheim, H., “Optimal Control of Water Drive”, paper SPE 15978

provided to the Society of Petroleum Engineers for distribution and

publication in an SPE journal, July 21, 1986.

21. Lorentzen, R. J., Berg, M. A., Naevdal, G. and Vefring, E. H.,”A

New Approach for Dynamic Optimization of Water Flooding

Problems”, paper SPE 99690, prepared for presentation at the SPE

Intelligent Energy Conference and Exhibition held in Amsterdam,

The Netherlands, 11-13 April 2006.

22. Liu, W., Ramirez, W. F., and Qi, Y. F., “Optimal Control of

Steamflooding”, SPE Advanced Technology Series, July 1993, vol.

1, no. 2, pp. 73 – 82.

78

23. Baris, G., Horne, R. N., Rogers, L., and Rosenzweig, J. J.,

“Optimization of Well Placement in a Gulf of Mexico Waterflooding

Project”, SPE Reservoir Evaluation and Engineering, June 2002,

vol. 5, no. 3, pp. 229 – 236.

24. Badru, O., and Kabir, C. S., “Well Placement Optimization in Field

Development”, paper SPE 84191 prepared for presentation at the

SPE Annual Technical Conference and Exhibition held in Denver,

Colorado, 5 – 8 Oct, 2003.

25. Burak, Y., Durlofsky, L. J., and Aziz, K., “Optimization of

Nonconventional Well Type, Location and Trajectory”, SPE Journal

September 2003, vol. 8, no. 3, pp 200-210.

26. Naevdal, G., Brouwer, D. R., and Jansen, J. D., “Waterflooding

using Closed-loop Control”, Submitted to Computational

Geosciences, July 2005.

27. Sarma, P., Durlofsky, L. J., and Aziz, K., “Efficient Closed-loop

Production Optimization under Uncertainty”, paper SPE 94241

prepared for presentation at the SPE Europec/EAGE Annual

Conference held in Madrid, Spain, 13 – 16 June, 2005.

28. Gu, Y. and Oliver, D. S., “The Ensemble Kalman Filter for

Continuous Updating of Reservoir Simulation Models” Journal of

79

Energy Resources Technology, March 2006, Vol. 128, Issue 1, pp

79 – 87.

29. Schlumberger, “Eclipse Simulation Software Reference Manual”,

2004A.

30. Oliver, D. S., Petroleum Inverse Theory Class notes, MPGE

University of Oklahoma, fall semester 2005.

31. Luenberger, David G., “Investment Science”, Stanford University,

published in New York, 1998.

32. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W.

T., “Numerical Recipes – The Art of Scientific Computing” New

York, Cambridge University Press.

80

A. APPENDIX – FORTRAN Flow Chart

Start

Declare variables

Run Eclipse

Open Eclipse output files

Read out oil and water production for each

BHP realization

Compute NPV

Is new NPV greater than old

NPV?

Yes

No

Stop

Read in Permeability values to reservoir grid file

Read in pressures/state vector to schedule file

Compute Cy.HT

Optimize alpha

Compute new state vector

Call Forward Run Subroutine

Forward Run

Output data for results and plotting

parameters files

81

B. APPENDIX – Results from Case 3

Figure B-1: Permeability distribution for case 3.


132,000,000

134,000,000

136,000,000

138,000,000

140,000,000

142,000,000

144,000,000

146,000,000

148,000,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Realization Number

NP

V (

$)


Figure B-2: Case 3 – Graph showing NPV for all realizations of pressure


P1 P2

P3 P4

INJ

82


5,000,000

5,200,000

5,400,000

5,600,000

5,800,000

6,000,000

6,200,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Realization number

Cu

mu

lati

ve

Oil

Pro

du

cti

on

(S

TB

)



Figure B-3: Case 3 – Graph showing cumulative oil production for all



0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10


0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P,

psi

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10


0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10


0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500

Time, days

BH

P, p

si

Realization 1

Realization 2

Realization 3

Realization 4

Realization 5

Realization 6

Realization 7

Realization 8

Realization 9

Realization 10

Figure B-4 (a), (b), (c), (d): Ten realizations of BHP for the optimized case

for all production wells

83


0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r c

ut,

stb

/stb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wa

ter

cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wate

r c

ut,

stb

/stb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wa

ter

cu

t, s

tb/s

tb






0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wa

ter

cu

t, s

tb/s

tb





Water cut from producing wells for schedule realization

3 after optimization

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Wa

ter

cu

t, s

tb/s

tb





Figure B-5: Case 3 – Graphs showing water cuts from all wells before

(left) and after (right) optimization for 3 realizations of BHP profiles

84


optimization

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

10,000,000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Cu

mu

lati

ve p

rod

uc

tio

n, S

TB



optimization

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Cu

mu

lati

ve p

rod

ucti

on

, S

TB





Figure B-6 (a) and (b): Case 3 – Graphs showing cumulative oil and water

production before and after optimization for realizations 1 and 2.

Field production rates for schedule realization 1 before and after optimization

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Pro

du

cti

on

ra

tes

, S

TB

/D






optimization

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000 2500 3000 3500 4000

Time, days

Pro

du

cti

on

rate

s, S

TB

/D





Figure B-7 (a) and (b): Case 3 – Graphs showing oil and water production

rates before and after optimization for realizations 1 and 2 of BHP profile.

85

Figure B-8: Case 3 – Water saturation distribution a) before (left) and b)

after (right) optimization after 913 days.


139,000,000

140,000,000

141,000,000

142,000,000

143,000,000

144,000,000

145,000,000

146,000,000

147,000,000

0 2 4 6 8 10 12 14 16

Iteration Number

NP

V (

$)


Optimized Mean NPV

4.33 % increase

Figure B-9: Case 3 – Net Present Value vs. iterations

86

C. APPENDIX – DESCRIPTION OF FORTRAN CODE

The FORTRAN code developed in this study for the NPV optimization

process has been divided into 5 different subroutines. The various

sections include:

� Permeability values

� Forward Run

� Revenue Optimization

� Function of Alpha

� Parameters

The flow chart of the FORTRAN code is shown in appendix A. Each of the

sections of the code listed above is described in subsequent sections of

this chapter.

C.1 Permeability values

This section of the FORTRAN code is used to populate the grid blocks of

the reservoir model with the generated permeabilities. As described in

earlier chapters, the permeability values are initially generated using

GSLIB and exported to a data file from which this subroutine reads the

values into Eclipse grid data file. The Eclipse include file which contains

87

the grid properties was named RevOpt_gpro.INC (gpro is short for grid

properties). Figure 4-1 shows the top view of one of the permeability fields

used in this study.

C.2 Forward Run

This subroutine is used to run the reservoir simulator model for all 40

realizations of pressure profiles. It opens all the input and output files

(typically data files) as well as the schedule file. The input files are data

files containing the randomly generated bottom hole pressure realizations

(i.e. initial ensemble generated from mathematica) or the updated

pressures after each iteration process. The output files are the files

containing the following data: pressure, water cut, oil and water production

rates and cumulative production and net present values for the various

wells. These values are later used for plotting graphs.

The forward run routine updates the Eclipse schedule include file with the

pressures for the ensemble and runs the simulator model for each case.

The output data from each case is stored in the output files. This routine

also computes the net present value for each production well and the

cumulative net present values for the entire field. These values are

computed from the oil and water flow rates at the various time steps. The

88

rates and cumulative productions are read from Eclipse summary output

file with the RSM extension name.

C.3 Revenue Optimization

This is the main program in the code. In this section, the actual iteration

process is performed. If Y is used to represent the ensemble of the state

vector, then Y can be represented as follows:

=

)(

.

.

3

2

1

i

iNt

i

i

i

i

xg

x

x

x

x

Y C-1

where i = ith member of the ensemble

x = control variable, or BHP

)(xg = obective function, or NPV

Nt = Total number of controls (In this research, Nt = 80)

The mean of the ensemble of state vectorsY is calculated as follows:

89

=

∑

∑

∑

∑

∑

i

i

i

iNt

i

i

i

i

i

i

e

xg

x

x

x

x

NY

)(

.

.1

3

2

1

C-2

Ne represents the number of members in the ensemble of state vectors.

For the purpose of this study, Ne = 40.

Recall from chapter 3;

( )( )Te

Y YYYYN

C −−−

=1

1 C-3

SinceT

YlX MCGC = , then

( )( ) TT

e

T

YlX MYYYYN

MCGC −−−

==1

1 C-4

The product of the covariance matrix Cx and the sensitivity matrix lG can be

approximated using the above equation. The product ( lX GC ) gives a

column vector which is multiplied by a weighting factor (1/α) and then

added to the ensemble of the prior state vector used in the previous

iteration (see equation 3-15). The value of α is a variable that also has to

90

be optimized. The following section describes how this optimization

process is carried out.

C.3.1 Optimum value of alpha

A section of the revenue optimization subroutine also estimates an

optimum value of alpha to be used as a weighting factor. The alpha value

can be thought of as a way to control step length32.

In our problem, we wish to maximize the net present value from the

waterflood reservoir, while minimizing rapid changes in the controls. Alpha

is the independent variable that regularizes the control settings. The

solution to this problem is to find a value of alpha for which the net present

value is nearly maximized.

The procedure for finding the extremum of a NPV function is as follows:

1) Select a relatively large starting guess for alpha.

2) Solve equation 3-18 to obtain the updated estimate of optimal

control variables and then execute the simulator to compute the

NPV.

3) Reduce alpha and find the new NPV.

4) If the new NPV is greater than the old NPV, then repeat steps 2

and 3.

91

5) On the other hand, if the new NPV is less than the old NPV, then

the alpha with the highest NPV is the optimum alpha to be used for

the next iteration step.

An alpha value less than the optimum alpha obtained from the procedure

above is generally used for the next iteration step )1( +l . This is to reduce

the step size of the optimization process and thereby penalize the control

settings that change rapidly with time.

C.4 Function of Alpha

This subroutine is used to compute the net present value when given a

value of alpha. It first solves equation 5-18 to obtain the updated state

vector and exports the new state vectors to the pressure input files. Then

it calls the forward run subroutine from where NPV is obtained. A typical

graph of NPV as a function of alpha is shown in Figure C-1. The run time

for the optimization process can be reduced by truncating the optimization

process of alpha to fewer NPV values or by selecting a constant value of

alpha for each iteration step. This however, will result in reduce the

precision of the optimization process.

92

Figure C-1: Graph of NPV as a function of alpha for 6 iterations

C.5 Parameters

This section of the code is used to declare all the variables and

parameters that are used in the entire FORTRAN program. It also defines

the dimensions of the variables.

NPV as a function of alpha for all iterations

213,000,000

213,500,000

214,000,000

214,500,000

215,000,000

215,500,000

216,000,000

216,500,000

1.E+05 1.E+06 1.E+07

Alpha

NP

V (

$)

iteration 1

iteration 2

iteration 3

iteration 4

iteration 5

iteration 6


Optimized Mean NPV

93

D. APPENDIX – FORTRAN Code

94

1 Sneider, R. M. and Sneider, J. S., ”New Oil in Old Places”, prepared for

presentation at the Pratt II Conference San Diego, California January 12-

15, 2000.

2 Lake L W., Schmidt R. L., and Venuto P. B., “A Niche for Enhanced Oil

Recovery in the 1990s”, Petroleum Engineer International (January 1992):

55 – 61.

3 Craig, Forrest Jr.: “The Reservoir Engineering Aspects of Waterflooding”,

Society of Petroleum Engineers of AIME, 1971.

4 Sarma P., Aziz K., and Durlofsky L. J., “Implementation of Adjoint

Solution for Optimal Control of Smart Wells”, paper SPE 92864 presented

at the 2005 SPE Reservoir Simulation Symposium held in Houston,

Texas, 31 Jan – 2 Feb 2005.

5 Arenas A. and Dolle N., “Smart Waterflooding Tight Fractured Reservoirs

Using Inflow Control Valves”, paper SPE 84193 presented at the SPE

Annual Technical Conference and Exhibition held in Denver, Colorado, 5-

8 October 2003.

6 Esmaiel, T. E. H., “Applications of Experimental Design in Reservoir

Management of Smart Wells”, paper SPE 94838 prepared for presentation

95

at the SPE Latin American and Caribbean Petroleum Engineering

Conference held in Rio de Janeiro, Brazil, 20 – 23 June 2005.

7 Brouwer D. R., Jansen J. D., Van Der Starre S., van Kruijsdijk, and

Berentsen C. W. J., “Recovery Increase through Waterflooding with Smart

Well Technology”, paper SPE 68979 presented at the SPE European

Formation Damage Conference held in the Hague, The Netherlands, 21-

22 May 2001.

8 Brouwer D. R., Jansen J. D., “Dynamic Optimization of Waterflooding

With Smart Wells Using Optimal Control Theory”, paper SPE 78278

presented at the 2002 SPE European Petroleum Conference, Aberdeen,

U.K., 29-31 October.

9 Glandt C. A., “Reservoir Aspects of Smart Wells”, paper SPE 81107

presented at the SPE Latin American and Caribbean Petroleum

Engineering Conference held in Port-of-Spain, Trinidad, West Indies, 27-

30 April 2003.

10 Wattenbarger R. A., “Maximizing Seasonal Withdrawals from Gas

Storage Reservoirs”, paper SPE 2406 presented at SPE 44th Annual Fall

Meeting in Denver, Colorado, Sept. 28 – Oct 1, 1969.

11 Asheim, H., “Maximization of Water Sweep Efficiency by Controlling

Production and Injection Rates”, paper SPE 18365 prepared presented at

96

the SPE European Petroleum Conference, London, UK, October 16-19,

1988.

12 Ramirez, W. F., Fathi, Z., and Cagnol, J. L, “Optimal Injection Policies

for Enhanced Oil Recovery: Part 1 – Theory and Computational

Strategies”, SPE Journal, June 1984, pp. 328-332.

13 Fathi, Z., and Ramirez, W. F., “Optimal Injection Policies for Enhanced

Oil Recovery: Part 2 – Surfactant Flooding”, SPE Journal, June 1984, pp.

333-341.

14 Tavakkolian, M., Jalali, F., and Amadi, M. A., “Production Optimization

using Genetic Algorithm Approach”, paper SPE 88901 prepared for

presentation at the 28th Annual SPE International Technical Conference

Exhibition in Abuja, Nigeria, August 2-4, 2004.

15 Harding, T. J., Radcliffe, N. J., and King, P. R., “Hydrocarbon

Production Scheduling with Genetic Algorithms” Approved for publication

in SPE Journal (June 1998), Vol. 3, no. 2, pp 99 – 107.

16 Zhou, C., Gao, C., Jin, Z., and Wu, X., “A Simulated Annealing

Approach to Constrained Nonlinear Optimization of Formation Parameters

in Quantitative Log Evaluation”, paper SPE 24723 prepared for

presentation at the 67th Annual Technical Conference and Exhibition in

Washington DC, Oct 4-7, 1992.

97

17 Sen, M. K., Datta-Gupta, A., Stoffa, P. L., Lake, L. W., and Pope, G. A.,

“Stochastic Reservoir Modelling Using Simulated Annealing and Genetic

Algorithms”, Approved for publication in SPE Formation Evaluation

Journal, March 1995, vol. 10, no. 1, pp. 49 – 56.

18 Virnovsky, G. A., “Optimization Techniques Application in Oil Recovery

Problems”, paper SPE 24281, prepared for presentation at the SPE

European Petroleum Computer Conference held in Stavanger, Norway,

25-27 May 1992.

19 Lee, A. S. and Aronofsky, J. S., “A Linear Programming Model for

Scheduling Crude Oil Production”, Accepted for publication in Journal of

Petroleum Technology (July, 1958) vol. 10, No. 7, 51-54.

20 Asheim, H., “Optimal Control of Water Drive”, paper SPE 15978

provided to the Society of Petroleum Engineers for distribution and

publication in an SPE journal, July 21, 1986.

21 Lorentzen R. J., Berg M. A., Naevdal G. and Vefring E. H.,”A New

Approach for Dynamic Optimization of Water Flooding Problems”, paper

SPE 99690, prepared for presentation at the SPE Intelligent Energy

Conference and Exhibition held in Amsterdam, The Netherlands, 11-13

April 2006.

98

22 Liu, W., Ramirez, W. F., and Qi, Y. F., “Optimal Control of

Steamflooding”, SPE Advanced Technology Series, July 1993, vol. 1, no.

2, pp. 73 – 82.

23 Baris, G., Horne, R. N., Rogers, L., and Rosenzweig, J. J., “Optimization

of Well Placement in a Gulf of Mexico Waterflooding Project”, SPE

Reservoir Evaluation and Engineering, June 2002, vol. 5, no. 3, pp. 229 –

236.

24 Badru, O., and Kabir, C. S., “Well Placement Optimization in Field

Development”, paper SPE 84191 prepared for presentation at the SPE

Annual Technical Conference and Exhibition held in Denver, Colorado, 5 –

8 Oct, 2003.

25 Burak, Y., Durlofsky, L. J., and Aziz, K., “Optimization of

Nonconventional Well Type, Location and Trajectory”, SPE Journal

September 2003, vol. 8, no. 3, pp 200-210.

26 Naevdal G., Brouwer D. R., and Jansen J. D., “Waterflooding using

Closed-loop Control”, Submitted to Computational Geosciences, July

2005.

27 Sarma, P., Durlofsky, L. J., and Aziz, K., “Efficient Closed-loop

Production Optimization under Uncertainty”, paper SPE 94241 prepared

99

for presentation at the SPE Europec/EAGE Annual Conference held in

Madrid, Spain, 13 – 16 June, 2005.

28 Gu Yaqing and Oliver D. S., “The Ensemble Kalman Filter for

Continuous Updating of Reservoir Simulation Models” Journal of Energy

Resources Technology March 2006, .

29 Schlumberger, “Eclipse Simulation Software Reference Manual”,

2004A.

30 Oliver, D. S., Petroleum Inverse Theory Class notes, MPGE University

of Oklahoma, fall semester 2005.

31 Luenberger David G., “Investment Science”, Business, Economics and

Finance: New York, 1998.

32 Press W. H., Flannery B. P., Teukolsky S. A., and Vetterling W. T.,

“Numerical Recipes – The Art of Scientific Computing” New York,

Cambridge University Press.

university of oklahoma dynamic … of observation...university of oklahoma graduate college dynamic...

Documents