Aliyev2011

USE OF HYBRID APPROACHES AND METAOPTIMIZATION

FOR WELL PLACEMENT PROBLEMS

A THESIS SUBMITTED TO THE DEPARTMENT OF

ENERGY RESOURCES ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE

Elnur Aliyev

May 2011

c© Copyright by Elnur Aliyev 2011

All Rights Reserved

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I certify that I have read this thesis and that in my opinion it is fully

adequate, in scope and in quality, as partial fulfillment of the degree of

Master of Science in Energy Resources Engineering.

Louis J. Durlofsky (Principal Adviser)

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Abstract

In the context of oil field development, determining the well locations that maximize

cumulative oil production or net present value is an important problem. A variety of

optimization methods can be used to find the optimum well locations in the reservoir.

In this study, gradient-free methods are considered. Both global (stochastic) and

local (deterministic) methods are applied. A hybrid procedure that combines these

two types of algorithms is developed. In addition, a metaoptimization technique is

applied to determine the optimum way to combine different algorithms.

For the global optimization algorithm, different families of particle swarm op-

timization (PSO) are investigated. Explorative PSO families, such as centered-

progressive (CP-PSO) and progressive-progressive (PP-PSO), in addition to the stan-

dard PSO algorithm, are considered. The local optimization algorithm used is Hooke-

Jeeves direct search (HJDS). The hybrid algorithm entails some number of function

evaluations (reservoir simulations) using a PSO method. The best solution found is

then used as the initial guess for HJDS. The overall algorithm takes advantage of the

broad search provided by PSO and the fast convergence to a local optimum provided

by HJDS. The hybrid algorithm is run for different PSO families and the results are

compared to those using standalone PSO, and in some cases to standalone HJDS.

Three cases, involving optimizing the locations of vertical wells in two-dimensional

heterogeneous reservoir models, are considered. In general, the hybrid algorithms

outperform the standalone methods, sometimes by a substantial margin.

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Metaoptimization is applied to determine the best PSO-HJDS hybrid algorithm.

The parameters determined by metaoptimization are the number of PSO function

evaluations and the PSO family type. The metaoptimization runs are very expensive,

but they provide the best results for the three cases considered. The results achieved

by the best PSO-HJDS hybrid are, however, very close to those from metaoptimiza-

tion.

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Acknowledgments

I would like to express my sincere gratitude to my adviser Prof. Lou Durlofsky for

his time, confidence, support and guidance. None of this would have been possible

without him. His comments and suggestions were extremely valuable to my research.

I would also like to thank Prof. Juan L.F. Martinez and Prof. Tapan Mukerji for

their useful input regarding PSO families. Dr. David Echeverria and Dr. Honggang

Wang also provided helpful comments regarding the use of optimization algorithms.

Special thanks also go to Dr. Jerome Onwunalu for providing the initial PSO code

and for useful discussions.

I thank Obi Isebor, a current Ph.D. student in the department, for his significant

help on this work.

I am grateful to the Smart Fields industrial affilliates program for financial support

for this research.

I am grateful to my family and friends who supported me during my study.

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Contents

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Stochastic Optimization Methods . . . . . . . . . . . . . . . . 3

1.1.2 Deterministic Methods . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Hybrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.4 Metaoptimization Approaches . . . . . . . . . . . . . . . . . . 5

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Optimization Methods for Well Placement 8

2.1 Particle Swarm Optimization (PSO) . . . . . . . . . . . . . . . . . . 8

2.1.1 PSO Families . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Hooke-Jeeves Direct Search . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Hybrid Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Hybrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Application of Metaoptimization to Hybrid Method . . . . . . 15

3 Results for Standalone and Hybrid Algorithms 17

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3.1 Case 1: Five Wells in a 20 × 20 Model . . . . . . . . . . . . . . . . . 18

3.1.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.2 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 20




3.3 Case 3: Ten Wells in a 100 × 100 Model . . . . . . . . . . . . . . . . 33



3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Metaoptimization Results 41



4.1.2 Comparison of Metaoptimization to Standalone and Hybrid Re-

sults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


4.3 Case 3: Ten Wells in a 100 × 100 Model . . . . . . . . . . . . . . . . 46

4.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Summary and Future Work 49

5.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Bibliography 53

x

List of Tables

3.1 Simulation parameters for Case 1 . . . . . . . . . . . . . . . . . . . . 20

3.2 Economic parameters for Case 1 . . . . . . . . . . . . . . . . . . . . . 20

3.3 Optimization results for Case 1∗ . . . . . . . . . . . . . . . . . . . . 25



3.6 Optimization results for Case 2∗ . . . . . . . . . . . . . . . . . . . . . 33



3.9 Optimization results for Case 3∗ . . . . . . . . . . . . . . . . . . . . . 40

3.10 Best algorithms for each case (based on best average NPV and best

individual run from the three runs) . . . . . . . . . . . . . . . . . . . 40

4.1 Standalone and hybrid optimization results for Case 1∗ . . . . . . . . 43

4.2 Comparison of all optimization methods for Case 1∗ . . . . . . . . . . 43





4.7 Optimum hybrid parameters found by metaoptimization for all three

cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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List of Figures

2.1 PSO velocity calculation for particle xi in a two-dimensional search

space (from [11]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 PSO neighborhod topologies (from [11]) . . . . . . . . . . . . . . . . 10

2.3 Clouds defining different PSO families (data from [19, 20]) . . . . . . 11

2.4 Illustration of exploratory and pattern moves in HJDS (from [1]) . . . 12

2.5 Schematic of the hybrid algorithm . . . . . . . . . . . . . . . . . . . . 14

2.6 Schematic of metaoptimization procedure . . . . . . . . . . . . . . . . 16

3.1 Permeability field for Case 1 . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Relative permeability curves for Case 1 . . . . . . . . . . . . . . . . . 19

3.3 Results for standalone standard PSO, standalone HJDS and standard

PSO-HJDS hybrid (Case 1) . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Best well locations from standard PSO and hybrid algorithms (Case 1) 22

3.5 Areal sweep at the end of the simulation for standard PSO and hybrid

algorithms (Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 Results for standalone CP-PSO, standalone HJDS and CP-PSO-HJDS

hybrid (Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Results for standalone PP-PSO, standalone HJDS and PP-PSO-HJDS

hybrid (Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


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3.9 Relative permeability curves for Case 2 . . . . . . . . . . . . . . . . . 27

3.10 Results for standalone standard PSO and standard PSO-HJDS hybrid

(Case 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.11 Best well locations from standard PSO and hybrid algorithms (Case 2) 29

3.12 Areal sweep at the end of the simulation for standard PSO and hybrid

algorithms (Case 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.13 Results for standalone CP-PSO and CP-PSO-HJDS hybrid (Case 2) . 30

3.14 Results for standalone PP-PSO and PP-PSO-HJDS hybrid (Case 2) . 31

3.15 Best well locations from PP-PSO and hybrid algorithms (Case 2) . . 32

3.16 Areal sweep at the end of the simulation for PP-PSO and hybrid algo-

rithms (Case 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


3.18 Results for standalone standard PSO and standard PSO-HJDS hybrid

(Case 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.19 Best well locations from PP-PSO and hybrid algorithms (Case 3) . . 37

3.20 Areal sweep at the end of the simulation for PP-PSO and hybrid algo-

rithms (Case 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.21 Results for standalone CP-PSO and CP-PSO-HJDS hybrid (Case 3) . 38

3.22 Results for standalone PP-PSO and PP-PSO-HJDS hybrid (Case 3) . 39

4.1 Well locations and the areal sweep at the end of the simulation for the

metaoptimization solution (Case 1) . . . . . . . . . . . . . . . . . . . 44





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Chapter 1

Introduction

As demand for oil increases and as production operations become more challenging

and expensive, the efficient development of oil reservoirs becomes essential. A key

decision engineers must make is where to drill wells in the reservoir to maximize net

present value (NPV), hydrocarbon recovery, or some other objective. In practice,

simulation models are run to quantify the performance of various field development

plans.

As the number of wells to be drilled increases, the number of possible solutions

becomes very large, and the use of a computational optimization procedure is required.

Different optimization methods can be used to determine the optimum well locations

in a reservoir. The optimization problem is nonlinear and generally contains multiple

local minima. Gradient-free optimization algorithms are commonly used for well

placement problems. The different optimization algorithms that are applied for this

problem have different search strategies. Some of the algorithms search globally,

though they can be expensive because they require a large number of simulation runs.

Some of the algorithms search locally and thus require fewer function evaluations,

though they converge to a local optimum. The aim of this study is to evaluate

different optimization algorithms for well placement problems, and to determine the

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2 CHAPTER 1. INTRODUCTION

combination of algorithms that provides the best performance.

1.1 Literature Review

Production optimization is often divided into well placement and well control opti-

mization problems. In well placement problems, the locations of wells are optimized,

while in well control problems, well settings such as flowrates or bottom hole pres-

sures (BHPs) are optimized. Many optimization methods have been presented for

well control problems. Gradient-based methods are commonly used for these prob-

lems, though gradient-free approaches have also been applied [1, 2]. Some of these

techniques may also be suitable for well placement problems.

In gradient-based optimization algorithms, the derivative of the objective function

with respect to optimization variables is required. Gradient-based methods for the

well placement problem are described in [3, 4]. Sequential linear interpolation (SLI) [5]

has also been applied for well placement. SLI uses approximate gradient information

determined by performing function evaluations over a simplex search space. Since well

placement problems typically display many local minima, gradient-based methods

cannot be expected to provide the global optimum. In practice, the algorithm can be

run several times to improve the search.

Gradient-free methods suitable for use in well placement optimization include sim-

ulated annealing [6], genetic algorithms [7, 8, 9], particle swarm optimization [10, 11],

generalized pattern search [1, 2], Hooke-Jeeves direct search [12], mesh adaptive direct

search [1, 2] and polytope [7]. Gradient-free techniques can be divided into global and

local search methods. Simulated annealing, genetic algorithms and particle swarm op-

timization methods are global search methods. These methods require more function

evaluations compared to local search methods. On the other hand, these methods can

often be easily parallelized, which greatly reduces elapsed time. Generalized pattern

1.1. LITERATURE REVIEW 3

search, Hooke-Jeeves direct search and mesh adaptive direct search are deterministic

local optimizers and are dependent on the initial guess.

1.1.1 Stochastic Optimization Methods

The most common stochastic method applied to date for well placement problems

appears to be the genetic algorithm (GA). For example, Guyaguler [7] and Yeten

et al. [8, 9] used GA to optimize well placement. Morales et al. [13] applied GA

to optimize horizontal well locations in gas condensate reservoirs. Bangerth et al.

[14] compared GA performance with other optimization methods for well placement

problems. They found that the relative performance of the various methods is prob-

lem specific. Emerick et al. [15] applied GA for well placement optimization with

nonlinear constraints. Recently, Onwunalu and Durlofsky [16, 11] and Onwunalu [10]

applied standard particle swarm optimization (PSO) as an alternative to GA for well

placement problems. They showed that, on average, PSO provided better results

than GA for the problems considered [11].

The performance of PSO is dependent on the parameters used in the algorithm.

There are three key parameters to be selected in PSO, which impact the so-called

inertia, cognitive and social components of particle velocity. By varying these param-

eters the PSO algorithm can be made more explorative or more exploitative (meaning

less exploration but faster convergence). Martinez and Gonzalo [17, 18, 19] identi-

fied several PSO variants. They suggested selecting these parameters from prescribed

“clouds” and changing parameters at every PSO iteration rather than taking constant

parameter values for the entire optimization. They defined different PSO families,

such as centered-progressive (CP-PSO), progressive-progressive (PP-PSO), centered-

centered (CC-PSO), regressive-regressive (RR-PSO), and generalized (G-PSO), based

on the parameter spaces used. CP-PSO and PP-PSO are considered to be explorative


PSO families. The associated parameter spaces were calculated numerically for math-

ematical functions such as the Rosenbrock and Griewank functions [17, 18, 19]. Since

the well placement problem is multimodal, its behavior may resemble that of the

Griewank function. As noted above, Onwunalu and Durlofsky [11, 16] applied the

standard PSO approach for well placement optimization. Onwunalu [10] also used

metaoptimization to determine PSO parameters. This approach is described below

in Section 1.1.4.

1.1.2 Deterministic Methods

As indicated above, deterministic local-search methods can also be applied to well

placement problems. GPS, HJDS and MADS are stencil based optimization methods

that share some similarities with one another. GPS and MADS have the advantage

that they can be parallelized easily. Refer to [20] for more details on the implementa-

tion of GPS and MADS. In terms of number of function evaluations, HJDS was found

to perform better than GPS and MADS for the well control problems considered in

[1, 2], though it is not clear if this finding holds for well placement problems. HJDS

includes an opportunistic move that makes it difficult to parallelize. SLI methods

evaluate the objective function at the vertices of simplexes in search space, which

can be done in parallel. The gradient can then be approximated and used in the

search (note that SLI is essentially a gradient based method). SLI was applied for

well placement optimization in [5].

1.1.3 Hybrid Methods

Hybrid approaches entail the combined use of two or more optimization algorithms.

They can be useful as they can exploit the relative advantages of different methods.

Bittencourt and Horne [21] implemented a hybrid algorithm by combining a polytope

1.1. LITERATURE REVIEW 5

method with GA. Guyaguler [7] integrated a proxy model into the hybrid algorithm

developed by [21]. Yeten et al. [8, 9] implemented a hybrid procedure by combining

GA with a hill climber algorithm. They used near-well upscaling and an artificial

neural network (ANN) to reduce the number of actual simulation runs in the hybrid

method. Wetter and Wright [22] hybridized HJDS with PSO for use in optimizing

supply air temperature in a building. In their approach, PSO is run for some number

of iterations, and then the best PSO particle is selected and used as the initial guess

in HJDS. According to Wetter and Wright [22], the best results were obtained by the

hybrid algorithm. In this study we will develop a hybridization of PSO and HJDS

for well placement problems. Consistent with [22], we will see that this algorithm

generally outperforms either standalone algorithm.

1.1.4 Metaoptimization Approaches

Metaoptimization procedures are used to optimize the parameters that appear in

optimization algorithms. Metaoptimization approaches consist of two optimization

stages. Onwunalu [10] developed a metaoptimization procedure that uses PSO for

both stages. The first stage is called the superswarm. At this stage PSO is run

to optimize the PSO parameters. In the second stage, which involves the so-called

subswarm, the PSO parameters obtained from the superswarm are used to solve the

specific optimization problem. Onwunalu [10] applied metaoptimization to optimize

the PSO parameters associated with the inertia, social, and cognitive components

of particle velocity. According to [10], use of metaoptimization improved PSO per-

formance. However, metaoptimization is more costly than standard procedures. We

note that different optimization algorithms can be used to optimize PSO parameters,

and the use of another algorithm might accelerate the computations.

Metaoptimization can be applied in two general ways. Under the first approach, we

optimize a simple well placement problem involving one or two wells and determine


optimal parameters. Then, we use these parameters in more realistic (multiwell)

optimization problems. In the second approach we apply metaoptimization directly

to the problem under study [23]. Onwunalu [10] obtained better results using the

second approach, though the optimizations were very expensive.

1.2 Thesis Outline

This thesis aims to assess the performance of different gradient-free optimization

algorithms for well placement problems. Both standalone and hybrid procedures will

be considered. Metaoptimization will also be applied to determine the optimum way

of hybridizing PSO and HJDS.

In Chapter 2, different gradient-free stochastic and deterministic optimization al-

gorithms are described. Standard PSO and different PSO families are discussed and

the main differences between the various PSO families are presented. Then, a deter-

ministic method, Hooke-Jeeves direct search (HJDS), is described. The advantages

and disadvantages of each method are discussed, and approaches for hybridizing these

methods are presented. Finally, the use of metaoptimization for determining an op-

timal hybrid procedure will be described.

In Chapter 3, optimization results using standalone PSO and HJDS approaches

as well as hybrid methods are presented. In the hybrid method, PSO is run for a

specified number of iterations and the best solution is used as an initial guess in

HJDS, as in [22]. The hybrid method is run using the two most explorative PSO

families, CP-PSO and PP-PSO, as well as with standard PSO. Performance of the

various algorithms will be seen to be problem specific, though the hybrid method

generally outperforms the standalone methods.

In Chapter 4, metaoptimization is applied to define parameters in the hybrid

algorithm. The parameters optimized are the number of PSO iterations and the PSO

1.2. THESIS OUTLINE 7

family to be used. The metaoptimization approach requires only one run but this run

entails a large number of function evaluations. It will be shown to provide the best

solutions. Very similar results are achieved, however, by performing multiple runs for

a sequence of hybrid algorithms.

A summary of our findings and suggestions for future work are presented in Chap-

ter 5.

Chapter 2

Optimization Methods for Well

Placement

As discussed in Chapter 1, there are a number of different gradient-free optimization

techniques that can be used for well placement problems. In this work we consider

particle swarm optimization (PSO), a stochastic global search technique, and Hooke-

Jeeves direct search, a local deterministic procedure.

2.1 Particle Swarm Optimization (PSO)

PSO is a stochastic global optimization method developed by Kennedy and Eberhardt

[24]. It is based on the social behaviors observed in swarms of animals. PSO uses

a set of candidate solutions at every iteration. Each of these candidate solutions is

called a particle and the collection of particles is called the swarm. Like GA, PSO is a

population based algorithm. The candidate solutions are updated at every iteration

by adding a velocity term to the current position (in search space) of each particle.

The position at iteration k + 1 for particle i, designated xi(k + 1) is calculated as:

8

2.1. PARTICLE SWARM OPTIMIZATION (PSO) 9

xi(k + 1) = xi(k) + vi(k + 1)∆t, (2.1)

where vi is the velocity of particle i and ∆t is taken to be equal to 1.

Figure 2.1: PSO velocity calculation for particle xi in a two-dimensional search space(from [11])

Figure 2.1 shows the PSO velocity calculation and particle position update for

particle xi in a two-dimensional search space. The velocity term consists of three

components, which are called the inertia, cognitive and social components. Velocity

10 CHAPTER 2. OPTIMIZATION METHODS FOR WELL PLACEMENT

at iteration k + 1 is given by:

vi(k + 1) = ωvi(k) + c1D1(k) · vci (k) + c2D2(k) · vs

i (k), (2.2)

where D1(k) and D2(k) are diagonal matrices whose elements are uniformly dis-

tributed random variables between (0, 1), and ω, c1 and c2 are parameters, discussed

in more detail in the next section. The quantities vci and vs

i are the cognitive and

social components of velocity, defined as vci = xpbest

i (k) - xi(k) and vsi = xnbest

i (k)

- xi(k) respectively. Here xpbesti (k) is the best position found by particle i up to

iteration k and xnbesti (k) is the position of the best particle in the ‘neighborhood’

of particle i. There are different neighborhood topologies such as star, cluster, ring

and random, and these determine which particles a given particle can interact with.

Several neighborhood topologies are shown in Figure 2.2. In the star topology all

particles interact. In the cluster and ring topologies, interactions are more limited.

In the random topology, particles in the neighborhood are selected randomly. See

[10] for more details. Throughout this work we consistently use the star topology.

Figure 2.2: PSO neighborhod topologies (from [11])

2.1. PARTICLE SWARM OPTIMIZATION (PSO) 11

Figure 2.3: Clouds defining different PSO families (data from [19, 20])

2.1.1 PSO Families

Different PSO families are defined by the values used for the three PSO parameters

ω, c1 and c2. These parameters determine whether the behavior of the search is more

explorative or more exploitative. Several authors have suggested different values for

ω, c1 and c2. Standard PSO uses ω = 0.721 and c1 = c2 = 1.193. These values are

based on the numerical experiments performed by [25]. Martinez and Gonzalo [17, 18]

suggested selecting the PSO parameters from certain ranges of values (clouds) and

changing parameters at every PSO iteration. They defined different PSO families

such as centered-progressive (CP-PSO), progressive-progressive (PP-PSO), centered-

centered (CC-PSO), regressive-regressive (RR-PSO), and generalized (G-PSO) based

on the parameter clouds. Martinez and Gonzalo [17, 18] demonstrated that CP-PSO

and PP-PSO are more explorative than other families.


Four PSO methods are displayed on Figure 2.3. The clouds were determined by

Martinez and Gonzalo [17, 18] by studying the convergence of different PSO families

for the Griewank function. Here φ is defined as φ = (c1 + c2)/2. For CP-PSO, φ

varies between 0 and 4, and it is generally greater than φ = 1.193, which is the value

used in standard PSO. Larger φ values can cause PSO particles to jump past xnbesti (k)

and xpbesti (k), which explains why CP-PSO is more explorative. Standard PSO, by

contrast, has more of a tendency to pull particles closer to xnbesti (k) and xpbest

i (k). For

PP-PSO, we cannot analyze particle movement in the same way because the position

update is done differently. Specifically, velocity is updated at every second iteration

in PP-PSO rather than at every iteration as in all other PSO families.

Different stopping criteria can be used in the PSO algorithm. One approach is

to stop the algorithm when all particles collapse to one solution. In this case the

velocity of each particle becomes close to zero. Another approach is to terminate the

optimization when a specified maximum number of function evaluations is reached.

This is the stopping criterion used in this study.

Figure 2.4: Illustration of exploratory and pattern moves in HJDS (from [1])

2.2. HOOKE-JEEVES DIRECT SEARCH 13

2.2 Hooke-Jeeves Direct Search

Hooke-Jeeves direct search (HJDS) method [12] is a gradient-free, deterministic local

optimization technique. HJDS is a stencil based optimization method. There are two

moves performed by HJDS: exploratory and pattern moves. Figure 2.4 shows how the

search proceeds in a two-dimensional search space. The exploratory move resembles

a gradient estimation: the objective function is evaluated in coordinate directions as

shown in Figure 2.4. If the algorithm finds a better point, then the exploratory move

is performed again. If two exploratory moves are successful, an aggressive pattern

move is performed as shown in Figure 2.4 (red arrow). If the aggressive pattern move

fails, then the exploratory move is performed. When no improvement is achieved, the

stencil size is reduced. For more details about HJDS refer to [1, 2, 26, 12].

There are two stopping criteria used in the HJDS algorithm. The first is a maxi-

mum number of function evaluations, while the second is that a minimum stencil size

is reached. Since we apply HJDS to well placement problems in this study, which is a

discrete optimization problem, the minimum stencil size used is 1. When the stencil

size becomes less than 1, HJDS is stopped and convergence is achieved.

One of the main disadvantages of HJDS is that it is a sequential optimization

technique and cannot be parallelized. Another drawback of HJDS is that it is a local

optimization technique and the result is thus very dependent on the initial point.

2.3 Hybrid Methodology

Two hybridization techniques are considered in this study. In the first approach, PSO

and HJDS are combined in specified ways. In the second approach, metaoptimization

is applied to find the best way of combining PSO and HJDS.


2.3.1 Hybrid Method

One of the main advantages of PSO is that it has broad (global) explorative charac-

teristics. PSO performance should not be as highly dependent on the initial solution

as local optimization techniques. Another advantage of PSO is that it is naturally

parallelizable and allows us to run using a computer cluster. As discussed earlier,

there are different PSO families, some of which are explorative and some exploitative.

In our hybridization, we use the two most explorative PSO families, CP-PSO (cen-

tered progressive) and PP-PSO (progressive progressive), along with standard PSO.

Figure 2.5: Schematic of the hybrid algorithm

In hybridization, the idea is to exploit the advantages of both global and local

search algorithms. If we analyze PSO results we see that after some number of

function evaluations there is little improvement in the objective function. Also, since

PSO is a global search method, it may converge slowly to nearby solutions. When

2.3. HYBRID METHODOLOGY 15

this occurs it may make sense to switch to a local optimizer (HJDS), which is what

we do in the hybrid scheme. Our specific approach is to run PSO for some number

of iterations and then use the best particle in the swarm as an initial guess in HJDS.

Figure 2.5 shows a schematic of the hybrid method. This approach is similar to that

used previously by Wetter and Wright [22] for optimizing supply air temperature in

a building. We note that other local optimization algorithms, such as GPS, MADS

or SLI could be used in place of HJDS. These algorithms might be better choices if a

large number of processors is available since they parallelize naturally.

2.3.2 Application of Metaoptimization to Hybrid Method

In the hybrid technique described above, there are several parameters that must be

specified. One of these parameters is when to switch from PSO to HJDS. This decision

is problem dependent. In some problems PSO should be run many iterations before

switching to HJDS, while in other cases relatively few PSO iterations will suffice.

Another (categorical) parameter that must be specified is the PSO family to use.

In this work, we apply metaoptimization to determine when to switch from PSO to

HJDS and which PSO family to use. As noted previously, metaoptimization consists

of two stages. In the first stage, the hybrid parameters are optimized, and PSO is

used for this purpose. In the second stage, the parameters determined in the first

stage are used to optimize the well placement problem of interest. The first stage

involves the superswarm and the second stage involves the subswarm. Figure 2.6

shows a sketch of the metaoptimization method.

There are two general ways to use metaoptimization methods. We can apply

metaoptimization to small, benchmark optimization problems and determine hybrid

optimization parameters. These parameters can then be used in more realistic op-

timization problems, with the hope that they are optimal or at least reasonable.

The second approach is to apply metaoptimization directly to each new optimization


Figure 2.6: Schematic of metaoptimization procedure

problem. This is expensive but should provide better results because the hybrid opti-

mization parameters are problem specific. According to Onwunalu [10] and Meissner

et al. [23], the use of the second approach was found to provide the best results.

Metaoptimization is very costly and requires many function evaluations. Compu-

tational expense can be decreased by using a smaller number of particles and fewer

iterations in the superswarm. This is reasonable in our problem because the number

of optimization parameters in the superswarm problem is very small.

Chapter 3

Results for Standalone and Hybrid

Algorithms

The algorithms described in the previous chapter are applied to three different two-

dimensional heterogeneous reservoir models. Results for standalone and hybrid algo-

rithms are presented. Several different PSO families are considered. All optimization

runs include minimum distance constraints between wells. A maximum water cut con-

straint of 0.9 is specified for production wells, so they can be considered to operate

in a ‘reactive control’ mode.

The objective function is undiscounted net present value (NPV) in all cases. The

objective function is maximized by determining the locations (x and y coordinates)

of the production and injection wells. The objective function J(x) is given by

J(x) = pprodo Qprod

o (x)− pprodw Qprod

w (x)− pinjw Q

injw (x)− Cdrill, (3.1)

where pprodo indicates the price of oil ($/STB), pprod

w and pinjw are the costs of produced

and injected water ($/STB) respectively, Qprodo and Qprod

w are the cumulative oil and

water produced, and Qinjw is cumulative water injected (STB). These quantities are

17

18 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS

obtained from the reservoir simulation output. Stanford’s General Purpose Research

Simulator (GPRS) [27] is used in all simulation runs. The drilling and completion

cost, Cdrill, is the same for every solution in a given problem since it depends only on

the number of wells drilled.

3.1 Case 1: Five Wells in a 20 × 20 Model

3.1.1 Problem Specification

The reservoir model used for Case 1 is shown in Figure 3.1. The locations of three

production and two injection wells are to be optimized in this problem. Porosity varies

from cell to cell; the average porosity is 0.14. Permeability varies from 1 md to 2000

md. The system contains oil and water. The relative permeability curves are shown

Figure 3.1: Permeability field for Case 1

in Figure 3.2. Capillary pressure is neglected in all simulation runs. Other simulation

parameters are given in Table 3.1. The model contains 20 × 20 grid blocks, with

each block of dimensions 200 × 200 × 50 ft3. Initial water saturation is 0.2. Injector

and producer bottom hole pressures (BHPs) are 6000 psia and 1000 psia respectively,

3.1. CASE 1: FIVE WELLS IN A 20 × 20 MODEL 19

Figure 3.2: Relative permeability curves for Case 1

and they are kept constant throughout the simulation. The total simulation run time

is 1000 days. The economic parameters used in the NPV computation are given in

Table 3.2.

For standalone runs, the PSO swarm size is 20 and the number of iterations is

also 20, so the total number of function evaluations is 400. For HJDS, the maximum

number of function evaluations is also set to 400. For hybrid algorithms, PSO is run

for 300 function evaluations (swarm size of 20 and 15 maximum iterations). The best

particle in the swarm is then used as an initial guess in Hooke-Jeeves direct search.

HJDS is run for 100 function evaluations. The PSO run is also continued for five more

PSO iterations, so both approaches use a total of 400 function evaluations. Both stan-

dalone and hybrid runs are performed with different PSO families, namely standard

PSO, centered-progressive PSO and progressive-progressive PSO. Each optimization

procedure is run three times using different (random) initial guesses.


Table 3.1: Simulation parameters for Case 1

Grid cell dimensions 200 × 200 × 50 ft3

Initial pressure, Pi 4800 psiAverage porosity 0.14cr at atm pressure 3 × 10-6 psi-1

µo at atm pressure 1.2 cpµw at atm pressure 0.31 cpρo 49.1 lbm/ft3

ρw 64.79 lbm/ft3

Bo at atm pressure 1.03 RB/STBBw at atm pressure 1.04 RB/STBInjector BHP 6000 psiProducer BHP 1000 psiSimulation duration 1000 days

Table 3.2: Economic parameters for Case 1

Drilling cost, Cdrill 100 × 106 ($)Oil price, pprod

o 80 ($/STB)Water production cost, pprod

w 5 ($/STB)Water injection cost, pinj

w 5 ($/STB)Discount rate, r 0.00

3.1.2 Optimization Results

Figure 3.3 presents results for standalone standard PSO, standalone HJDS, and stan-

dard PSO-HJDS hybrid algorithms. The thin lines correspond to different runs and

the bold lines are the arithmetic averages of the three runs. On average, HJDS finds

a local minimum and stops after 310 function evaluations. The hybrid method gives

better results on average than either standalone method for Case 1. In addition, the

hybrid algorithm improves upon the standard PSO solution in two out of three runs.


The best well locations found by standard PSO after 300 function evaluations are

plotted in Figure 3.4(a). This solution is used as the initial guess for HJDS, and the

resulting well locations are plotted in Figure 3.4(b). It is evident that the wells shift

only slightly, though this leads to an increase in NPV of $12 ×106. Figure 3.5 shows

areal sweep at 1000 days for the two cases. The hybrid solution shows somewhat

better sweep. For this case it is possible that a more cost effective solution could be

achieved with a single injection well, though the number of injectors and producers

was not treated as an optimization variable. This is a topic for future work.

Figure 3.3: Results for standalone standard PSO, standalone HJDS and standardPSO-HJDS hybrid (Case 1)

We now consider the use of different PSO families, CP-PSO and PP-PSO, both

standalone and hybridized. Results are shown in Figures 3.6 and 3.7, respectively, for

CP-PSO and PP-PSO. The standalone HJDS results (black curves) in these figures

are the same as those shown in Figure 3.3. From Figure 3.6 we see that running HJDS


(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations

Figure 3.4: Best well locations from standard PSO and hybrid algorithms (Case 1)


Figure 3.5: Areal sweep at the end of the simulation for standard PSO and hybridalgorithms (Case 1)


after CP-PSO does not improve NPV significantly. For this example, on average CP-

PSO finds the best solution. From Figure 3.7, where PP-PSO is used, we see that

running HJDS after PP-PSO improves the results significantly. From the results

above we can conclude that the variuos hybrid algorithms perform differently. For

standard PSO and PP-PSO, running HJDS after PSO improves the results, especially

for PP-PSO. However, for CP-PSO, hybridization does not lead to improvement on

average.

Figure 3.6: Results for standalone CP-PSO, standalone HJDS and CP-PSO-HJDShybrid (Case 1)


Figure 3.7: Results for standalone PP-PSO, standalone HJDS and PP-PSO-HJDShybrid (Case 1)

Table 3.3 shows the results for each standalone PSO and hybrid algorithm. We see

that the highest average NPV is obtained by running CP-PSO standalone. However,

the highest NPV for a single run ($3.09 ×108) is obtained with the CP-PSO-HJDS

hybrid. Another observation is that it is important to consider different PSO families

because, at least for this example, the best solutions are not obtained by standard

PSO.


Table 3.3: Optimization results for Case 1∗

run 1 run 2 run 3 averageStandard PSO 2.86 2.67 2.91 2.81CP-PSO 2.89 3.03 3.02 2.98PP-PSO 2.75 2.80 2.36 2.64Standard PSO-HJDS 2.98 2.63 2.95 2.85CP-PSO-HJDS 2.92 3.09 2.82 2.94PP-PSO-HJDS 2.76 2.89 2.79 2.81

∗all entries should be multiplied by $108




The reservoir model used for Case 2 is shown in Figure 3.8. This problem involves

optimizing the locations of three production and two injection wells. Porosity varies






ρw 62.4 lbm/ft3

Bo at atm pressure 1.00 RB/STBBw at atm pressure 1.00 RB/STBInjector BHP 6000 psiProducer BHP 1000 psiSimulation duration 10 years







Figure 3.9: Relative permeability curves for Case 2

from cell to cell and the average porosity is 0.22. Permeability varies from 1 md to

1000 md. Oil and water are present in the model. The relative permeability curves

are shown in Figure 3.9. Simulation and economic parameters are given in Tables 3.4

and 3.5.

For this case we run only the standalone PSO algorithms and the PSO-HJDS

hybrids (standalone HJDS is not run). The settings for the optimization algorithms

are the same as in Case 1.


The standalone standard PSO results are compared to standard PSO-HJDS hybrid

results in Figure 3.10. We see that running HJDS after standard PSO for this case

does not improve NPV values significantly.


Figure 3.10: Results for standalone standard PSO and standard PSO-HJDS hybrid(Case 2)

Figure 3.11 shows the best solution found by the standalone PSO and hybrid algo-

rithms. We see that the well locations obtained by PSO after 300 function evaluations

change very little after applying HJDS. The areal sweep for the two solutions (Figure

3.12) is also very similar.



Figure 3.11: Best well locations from standard PSO and hybrid algorithms (Case 2)


Figure 3.12: Areal sweep at the end of the simulation for standard PSO and hybridalgorithms (Case 2)


Figure 3.13: Results for standalone CP-PSO and CP-PSO-HJDS hybrid (Case 2)

Figure 3.13 compares hybrid CP-PSO-HJDS to CP-PSO. We see that the use of

HJDS does not improve CP-PSO performance significantly. However, in Figure 3.14

we see that there is a significant difference between standalone PP-PSO and hybrid

PP-PSO-HJDS results. This result is similar to that obtained in Case 1. This increase

in NPV occurs because PP-PSO is very explorative, so applying HJDS after PP-PSO

has a large impact because it directs the search toward the local optimum.

Figure 3.15(a) shows the best well locations obtained by PP-PSO after 300 func-

tion evaluations and Figure 3.15(b) shows the well locations found by the subsequent

application of HJDS. It is evident that the wells have shifted by a considerable amount.

The improvement from the hybrid optimization is apparent in the areal sweep for the

two cases, shown in Figure 3.16. This example demonstrates the importance of run-

ning HJDS after PSO when using the PP-PSO family.


Figure 3.14: Results for standalone PP-PSO and PP-PSO-HJDS hybrid (Case 2)

Table 3.6 shows the results obtained by the standalone PSO families and hybrid

procedures for Case 2. We see that the best average NPV is obtained by the hybrid

CP-PSO-HJDS algorithm. As in Case 1, in Case 2 the best individual solution is

obtained by CP-PSO-HJDS.



Figure 3.15: Best well locations from PP-PSO and hybrid algorithms (Case 2)


Figure 3.16: Areal sweep at the end of the simulation for PP-PSO and hybrid algo-rithms (Case 2)

3.3. CASE 3: TEN WELLS IN A 100 × 100 MODEL 33




3.3 Case 3: Ten Wells in a 100 × 100 Model


The reservoir model used for Case 3 is shown in Figure 3.17. The locations of seven

production and three injection wells are optimized. Average porosity is 0.15 and

permeability varies from 1 md to 2000 md. The relative permeability curves are the

same as in Case 1. Simulation and economic parameters are given in Tables 3.7 and

3.8. The algorithmic settings are the same as in previous cases (400 total function

evaluations for both standalone PSO and hybrid algorithms).






ρw 64.79 lbm/ft3

Bo at atm pressure 1.03 RB/STBBw at atm pressure 1.04 RB/STBInjector BHP 6000 psiProducer BHP 1000 psiSimulation duration 1000 days









Results for the standalone standard PSO and the hybrid algorithm are shown in

Figure 3.18. It is evident that running HJDS after standard PSO increases NPV

significantly in all three runs.

The optimum well locations are shown in Figure 3.19. The areal sweep at the

end of the simulation is presented in Figure 3.20. We see that the wells are shifted

significantly by HJDS, and this clearly impacts sweep effiency. Sweep for both cases

would be increased if the simulations were run for longer times (those simulations are

run for only 1000 days).


Figure 3.18: Results for standalone standard PSO and standard PSO-HJDS hybrid(Case 3)

Standalone PSO and hybrid algorithms are also run for the CP-PSO and PP-PSO

families. Results are shown in Figures 3.21 and 3.22. From the figures we see that

running HJDS after both CP-PSO and PP-PSO improves results significantly. For

this example, we can conclude that the hybrid algorithms give consistently better

results than the standalone PSO procedures.



Figure 3.19: Best well locations from PP-PSO and hybrid algorithms (Case 3)


Figure 3.20: Areal sweep at the end of the simulation for PP-PSO and hybrid algo-rithms (Case 3)


Figure 3.21: Results for standalone CP-PSO and CP-PSO-HJDS hybrid (Case 3)

Table 3.9 shows the results for the standalone PSO and hybrid algorithms. We

see that the highest average NPV is achieved by hybrid CP-PSO-HJDS. However, the

highest individual NPV value is obtained with the standard PSO-HJDS hybrid. Note

that, although the NPV values obtained by standalone CP-PSO display the lowest

average, the hybrid CP-PSO-HJDS has the highest NPV average. This demonstrates

the potential impact of hybridization.

3.4. DISCUSSION 39

Figure 3.22: Results for standalone PP-PSO and PP-PSO-HJDS hybrid (Case 3)

3.4 Discussion

Considering our findings for Cases 1-3, it is evident that the relative performance of

the various algorithms is problem specific. The best algorithms for each case (based

on best average NPV and best run) are presented in Table 3.10. We see that CP-PSO

is often the PSO method of choice. This shows the importance of considering PSO

families other than standard PSO. In addition, running HJDS after PSO generally

improves the solution (for the same number of total function evaluations), though the

best approach (on average) for Case 1 was standalone CP-PSO.

There are two key questions that remain to be addressed in the hybrid method.

First, how many PSO iterations should be run before switching to HJDS. And second,

which PSO family should we use for the hybrid algorithm. To address these questions,

metaoptimization will be applied. This is the subject of Chapter 4.





Table 3.10: Best algorithms for each case (based on best average NPV and bestindividual run from the three runs)

Best average NPV Best runCase 1 CP-PSO CP-PSO-HJDSCase 2 CP-PSO-HJDS CP-PSO-HJDSCase 3 CP-PSO-HJDS Standard PSO-HJDS

Chapter 4

Metaoptimization Results

In this chapter, we apply metaoptimization to enhance the PSO-HJDS hybrid. This

approach represents an alternative to running each PSO family standalone and in a

PSO-HJDS hybrid. As we described previously, metaoptimization entails a two-stage

algorithm. In the first stage (superswarm), PSO is used to optimize the two hybrid

parameters, PSO family type, which is a categorical variable, and the number of PSO

function evaluations. These parameters are then used in the hybrid algorithm to

optimize well locations.

Because the optimum number of PSO iterations and the optimal PSO family

are problem specific, metaoptimization is applied to the problem at hand. This

type of metaoptimization is, however, very expensive since it requires many function

evaluations. Computational expense can be decreased by using small swarm sizes and

fewer iterations in the superswarm PSO. This is reasonable because there are only

two optimization variables in the superswarm problem.

In this chapter, metaoptimization is applied to the three reservoir models consid-

ered in Chapter 3. In all three cases, metaoptimization will be seen to provide the

best results compared to running all of the standalone and hybridized PSO families,

for the same number of total function evaluations. However, the improvement offered

41

42 CHAPTER 4. METAOPTIMIZATION RESULTS

by metaoptimization over the best performing hybrid procedure will be seen to be

small.



Case 1 is the same as in Chapter 3. Reservoir and economic parameters are given in

Tables 3.1 and 3.2 respectively.

The optimization parameters for standalone PSO are the same as in Chapter 3

(total number of function evaluations is 400, swarm size and number of iterations are

20). Hybrid optimization parameters are also the same as in Chapter 3 (number of

function evaluations for PSO and HJDS are 300 and 100 respectively). The only dif-

ference in these runs from those in Chapter 3 is that we now perform five optimization

runs for standalone PSO and the hybrid algorithms instead of three.

For metaoptimization, the number of particles and iterations in the superswarm

are five and three respectively. In the subswarm, the total number of function eval-

uations for the PSO and HJDS algorithms is 400. The number of PSO particles in

the subswarm is 20. The number of subswarm iterations is determined in the super-

swarm. The number of function evaluations for HJDS is calculated by subtracting

the number of PSO function evaluations from 400. As stated above, the PSO family

to be used in the subswarm run is determined in the superswarm. The total number

of function evaluations for the full metaoptimization run is 6000 (3 × 5 × 400).

For the standalone PSO runs and hybrid PSO-HJDS runs, we consider each PSO

family. The total number of function evaluations in all of the runs using standalone

PSO is also 6000 (3 PSO families × 5 runs each × 400 function evaluations per run).

By the same calculation, we also use 6000 total function evaluations for the hybrid


Table 4.1: Standalone and hybrid optimization results for Case 1∗

run1 run 2 run 3 run 4 run 5 averageStandard PSO 2.86 2.67 2.91 2.68 2.91 2.81CP-PSO 2.89 3.03 3.02 2.93 3.04 2.98PP-PSO 2.75 2.80 2.36 2.55 2.71 2.63Standard PSO-HJDS 2.98 2.63 2.95 2.63 2.94 2.83CP-PSO-HJDS 2.92 3.09 2.82 2.98 3.07 2.98PP-PSO-HJDS 2.76 2.89 2.79 2.95 3.08 2.89


Table 4.2: Comparison of all optimization methods for Case 1∗

Best PSO Best hybrid Metaoptimization3.04 3.09 3.11


PSO-HJDS runs. Thus, we can compare all three approaches on the basis of 6000

total function evaluations.

4.1.2 Comparison of Metaoptimization to Standalone and

Hybrid Results

Table 4.1 shows the results for five runs for the standalone PSO and hybrid algorithms.

The first three rows are results for different standalone PSO families and the last three

rows are for hybrid algorithms with different PSO families. The first three columns

of results are the same as in Table 3.3. The last column is the average of all five runs.

We see that the best individual solution is found by CP-PSO-HJDS. The highest

average NPV values are achieved using CP-PSO and CP-PSO-HJDS.

Table 4.2 compares the best NPV values found by any of the standalone PSO


(a) Well locations (b) Areal sweep

Figure 4.1: Well locations and the areal sweep at the end of the simulation for themetaoptimization solution (Case 1)

and hybrid algorithms to that obtained by the single metaoptimization run. We see

that the best solution, $3.11×108, is found by metaoptimization, though all solutions

are close. We reiterate that each entry in Table 4.2 is the result achieved after 6000

function evaluations. The well locations determined by metaoptimization, and the

areal sweep after 1000 days of production, are shown in Figure 4.1.


The simulation and economic parameters used in this case are given in Tables 3.4 and

3.5. The optimization parameters for standalone PSO and the hybrid and metaopti-

mization algorithms are the same as were used in Case 1.

Table 4.3 presents the results for five runs for the various standalone PSO and

hybrid algorithms. The best individual solution is found by hybrid CP-PSO-HJDS.

The best average NPV value is also achieved using CP-PSO-HJDS.

Comparison of the solutions found by standalone PSO, hybrid PSO-HJDS and

metaoptimization are shown in Table 4.4. The best solution, $2.63×108, is obtained

















by metaoptimization. Again, the three optimal solutions are very close. The optimum

well locations and the resulting areal sweep for the wells determined by metaoptimiza-

tion are shown in Figure 4.2. Sweep is improved relative to that observed in Figure

3.16(b).

4.3 Case 3: Ten Wells in a 100 × 100 Model

We use the same reservoir and economic parameters for Case 3 as were used in Chapter

3. The parameters for the optimization algorithms are the same as for the other two

cases.

Optimization results using standalone and hybrid algorithms are given in Table

4.5. The best results are obtained by standard PSO-HJDS. In this case hybridization

leads to significant improvement, especially for CP-PSO. The best NPV values found

4.4. SUMMARY OF RESULTS 47



Table 4.7: Optimum hybrid parameters found by metaoptimization for all three cases

Parameter Case 1 Case 2 Case 3Family CP-PSO CP-PSO Standard-PSO# of PSO function evaluations 380 80 220# of HJDS function evaluations 20 320 180

by the various approaches are shown in Table 4.6. The best results are again obtained

by metaoptimization, though differences between the three approaches are small.

Metaoptimization solutions are shown in Figure 4.3.

4.4 Summary of Results

The optimum parameters found by metaoptimization are shown in Table 4.7 for all

cases. For Case 1, the number of PSO function evaluations is 380, which indicates

we should switch to HJDS later than we did in the hybrid runs (which use 300 PSO

function evaluations). Also, the CP-PSO family is used in combination with HJDS.

This is consistent with the results for the various methods in Table 4.1. For Case 2,


metaoptimization switches from PSO to HJDS after 80 function evaluations, which

is much earlier than in the hybrid method. Again, the CP-PSO family is used for

Case 2, which is consistent with the results in Table 4.3. Case 3 uses standard PSO

(consistent with the results in Table 4.5) and switches to HJDS after 220 function

evaluations. Thus, in all three cases, metaoptimization identifies the PSO family that

is observed to perform the best in the detailed hybrid runs presented in Tables 4.1,

4.3 and 4.5.

We also see that the use of standalone PSO did not provide the best results for any

of the three cases. This is in contrast to the results in Chapter 3, where standalone

CP-PSO gave the best results on average for Case 1 (only three runs were performed

in Chapter 3, in contrast to five runs here). The methods that gave the best individual

results for all hybrid cases in this chapter are the same as those that gave the best

individual results in Chapter 3.

Chapter 5

Summary and Future Work

In this thesis, different optimization algorithms were investigated for well placement

problems. A hybrid algorithm was developed by combining particle swarm optimiza-

tion (PSO) and Hooke-Jeeves direct search (HJDS). Results for different standalone

PSO families were compared to the hybrid method. Also, metaoptimization was ap-

plied to determine the optimum way to combine PSO and HJDS for well placement

optimization.

5.1 Summary and Conclusions

• Different explorative PSO families, specifically CP-PSO and PP-PSO, were in-

vestigated and applied to well placement problems. This enabled comparisons

to standard PSO. In many cases CP-PSO gave better results than standard

PSO, which demonstrates the importance of considering different PSO families.

• In the hybrid method, PSO (a global search method) was run for some number

of iterations. Then, the best particle in the swarm was selected and used as the

initial guess in HJDS (a local search method). The hybrid method was found

49

50 CHAPTER 5. SUMMARY AND FUTURE WORK

to generally outperform both standalone PSO and HJDS.

• Metaoptimization was applied to determine the best way of combining PSO

and HJDS. Two parameters were optimized in metaoptimization: the number

of PSO function evaluations and PSO family type (as a categorical variable).

Metaoptimization results were compared to those from standalone PSO families

and hybrid methods. In all cases metaoptimization gave the best results, though

the improvement over results from the best hybrid method was small.

• According to the metaoptimization results, for some problems switching from

PSO to HJDS earlier was preferable, while in other cases switching later was

better. Also, the optimum PSO family found by metaoptimization was different

for different cases.

• Of the three PSO families studied, PP-PSO did not give the best average NPV

or the best individual run for any of the cases considered. Thus, it may be

better to consider other families in place of PP-PSO.

5.2 Future Work

• The maximum number of optimization variables considered in this study was

40. Also, all cases involved two-dimensional reservoir models with vertical wells.

It will be of interest to evaluate the standalone, hybrid and metaoptimization

procedures for problems with larger numbers of variables and for more complex

reservoirs and wells (deviated and multilateral).

• In this study, a single objective function (NPV) was maximized. However, it is

often useful to have more than one objective function in optimization problems.

Extending our approaches to multiobjective optimization problems will provide

the methods with greater applicability.

5.2. FUTURE WORK 51

• The metaoptimization procedure is computationally expensive. Proxy models

such as kriging, statistical proxies, and neural networks should be tested for use

in metaoptimization to decrease the number of simulations required.

• It will also be of interest to integrate the hybrid and metaoptimization pro-

cedures into the well pattern optimization algorithm [16]. This would enable

efficient optimization of large scale field development.

52 CHAPTER 5. SUMMARY AND FUTURE WORK

Nomenclature

Abbreviations

BHP bottom hole pressure

CP centered-progressive

GA genetic algorithm

GPRS General Purpose Research Simulator

HJDS Hooke-Jeeves direct search

NPV net present value

PP progressive-progressive

PSO particle swarm optimization

Symbols

c1 cognitive weight

c2 social weight

D diagonal matrices of random numbers between 0 and 1

xi position of particle i

xnbesti best position in neighborhood of particle i

xpbesti best position found by particle i up to current iteration

vi velocity of particle i

vci cognitive component of velocity

vsi social component of velocity

J(x) objective function

NPSO number of PSO iterations

NHJ number of HJDS iterations

ω inertia weight

Bibliography

[1] O. J. Isebor. Constrained Production Optimization with an Emphasis on

Derivative-free Methods. Master’s thesis, Department of Energy Resources En-

gineering, Stanford University, 2009.

[2] D. E. Ciaurri, O. J. Isebor, and L. J. Durlofsky. Application of derivative-free

methodologies to generally constrained oil production optimization problems.

Int. J. Mathematical Modelling and Numerical Optimisation, 2(2):134–161, 2011.

[3] M. J. Zandvliet, M. Handels, G. M. van Essen, D. R. Brouwer, and J. D. Jansen.

Adjoint-based well-placement optimization under production constraints. Pa-

per SPE 105797 presented at the SPE Reservoir Simulation Symposium held in

Houston, USA, 26-28 February, 2007.

[4] P. Sarma and W. H. Chen. Efficient well placement optimization with gradient

based algorithms and adjoint models. Paper SPE 112257 presented at the SPE

Intelligent Energy Conference and Exhibition held in Amsterdam, The Nether-

lands, 25-27 February, 2008.

[5] H. Wang, D. E. Ciaurri, L. J. Durlofsky, and A. Cominelli. Optimal well place-

ment under uncertainty using a retrospective optimization framework. Paper

SPE 141950 presented at the SPE Reservoir Simulation Symposium, 21-23 Febru-

ary 2011, The Woodlands, Texas, USA, 2011.

53

54 BIBLIOGRAPHY

[6] B. L. Beckner and X. Song. Field development planning using simulated an-

nealing - optimal economic well scheduling and placement. Paper SPE 30650

presented at the SPE Annual Technical Conference and Exhibition held in Dal-

las, Texas, USA, 22-25 October, 1995.

[7] B. Guyaguler. Optimization of Well Placement and Assessment of Uncertainty.

PhD thesis, Department of Petroleum Engineering, Stanford University, 2002.

[8] B. Yeten, L. J. Durlofsky, and K. Aziz. Optimization of nonconventional well

type, location and trajectory. SPE Journal, 8(13):200–210, 2003.

[9] B. Yeten. Optimum Deployment of Nonconventional Wells. PhD thesis, Depart-

ment of Petroleum Engineering, Stanford University, 2003.

[10] J. E. Onwunalu. Optimization of Field Development Using Particle Swarm Opti-

mization and New Well Pattern Descriptions. PhD thesis, Department of Energy

Resources Engineering, Stanford University, 2010.

[11] J. E. Onwunalu and L. J. Durlofsky. Application of a particle swarm optimiza-

tion algorithm for determining optimum well location and type. Computational

Geosciences, 14(1):183–198, 2010.

[12] R. Hooke and T. A. Jeeves. Direct search solution of numerical and statistical

problems. Journal of the ACM, 8(2):212–229, 1961.

[13] A. Morales, H. Nasrabadi, and D. Zhu. A modified genetic algorithm for hor-

izontal well placement optimization in gas condensate reservoirs. Paper SPE

135182 presented at the SPE Annual Technical Conference and Exhibition held

in Florence, Italy, 19-22 September, 2010.

BIBLIOGRAPHY 55

[14] W. Bangerth, H. Klie, M. F. Wheeler, P. L. Stoffa, and M. K. Sen. On opti-

mization algorithms for the reservoir oil well placement problem. Computational

Geosciences, 10:303–319, 2003.

[15] A. C. Emerick, E. Silva, B. Messer, L. F. Almeida, D. Szwarcman, M. A. C.

Pacheco, and M. Vellasco. Well placement optimization using a genetic algorithm

with nonlinear constraints. Paper SPE 118808 presented at the 2009 SPE Reser-

voir Simulation Symposium held in The Woodlands, Texas, USA, 2-4 February,

2009.

[16] J. E. Onwunalu and L. J. Durlofsky. Development and application of a new well

pattern optimization algorithm for optimizing large-scale field development. To

appear in SPE Journal, 2011.

[17] J. L. F. Martinez and E. G. Gonzalo. The generalized PSO: a new door to PSO

evolution. Artificial Evolution and Applications, 1-15, 2008.

[18] J. L. F. Martinez and E. G. Gonzalo. The PSO family: deduction, stochastic

analysis and comparison. Swarm Intelligence, 3(4):245–273, 2009.

[19] J. L. F. Martinez and E. G. Gonzalo. Design of a simple and powerful particle

swarm optimizer. International Conference on Computational and Mathematical

Methods in Science and Engineering, CMMSE2009, Gijon, Spain, 2009.

[20] Mathworks. Genetic Algorithm and Direct Search Toolbox. Users Guide, 2009.

[21] A. C. Bittencourt and R. N. Horne. Reservoir development and design optimiza-

tion. Paper SPE 38895 presented at the SPE Annual Technical Conference and

Exhibition, San Antonio, Texas, USA, 5-8 October, 1997.

56 BIBLIOGRAPHY

[22] M. Wetter and J. Wright. A comparison of deterministic and probabilistic opti-

mization algorithms for nonsmooth simulation-based optimization. Building and

Environment, 39:989–999, 2004.

[23] M. Meissner, M. Schmuker, and G. Schneider. Optimized particle swarm opti-

mization (OPSO) and its application to artificial neural network training. BMC

Bioinformatics, 7(125):111, 2006.

[24] J. Kennedy and R. C. Eberhardt. Particle swarm optimization. IEEE Interna-

tional Joint Conference on Neural Networks, 1942-1947, 1995.

[25] M. Clerc. Stagnation analysis in particle swarm optimization or what happens

when nothing happens. Technical Report CSM-460, Department of Computer

Science, University of Essex, 2006.

[26] C. T. Kelley. Iterative Methods for Optimization. Frontiers in Applied Mathe-

matics. SIAM, 1999.

[27] H. Cao. Development of Techniques for General Purpose Simulators. PhD thesis,

Department of Petroleum Engineering, Stanford University, 2002.

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