Download - 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
ii
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)
iii
iv
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.
v
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.
vi
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.
vii
viii
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
ix
3.1 Case 1: Five Wells in a 20 × 20 Model . . . . . . . . . . . . . . . . . 18
3.1.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Case 2: Five Wells in a 60 × 60 Model . . . . . . . . . . . . . . . . . 25
3.2.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Case 3: Ten Wells in a 100 × 100 Model . . . . . . . . . . . . . . . . 33
3.3.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Metaoptimization Results 41
4.1 Case 1: Five Wells in a 20 × 20 Model . . . . . . . . . . . . . . . . . 42
4.1.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Comparison of Metaoptimization to Standalone and Hybrid Re-
sults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Case 2: Five Wells in a 60 × 60 Model . . . . . . . . . . . . . . . . . 44
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
3.4 Simulation parameters for Case 2 . . . . . . . . . . . . . . . . . . . . 26
3.5 Economic parameters for Case 2 . . . . . . . . . . . . . . . . . . . . . 26
3.6 Optimization results for Case 2∗ . . . . . . . . . . . . . . . . . . . . . 33
3.7 Simulation parameters for Case 3 . . . . . . . . . . . . . . . . . . . . 34
3.8 Economic parameters for Case 3 . . . . . . . . . . . . . . . . . . . . . 34
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
4.3 Standalone and hybrid optimization results for Case 2∗ . . . . . . . . 45
4.4 Comparison of all optimization methods for Case 2∗ . . . . . . . . . . 45
4.5 Standalone and hybrid optimization results for Case 3∗ . . . . . . . . 46
4.6 Comparison of all optimization methods for Case 3∗ . . . . . . . . . . 46
4.7 Optimum hybrid parameters found by metaoptimization for all three
cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
xi
xii
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
3.8 Permeability field for Case 2 . . . . . . . . . . . . . . . . . . . . . . . 25
xiii
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
3.17 Permeability field for Case 3 . . . . . . . . . . . . . . . . . . . . . . . 35
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
4.2 Well locations and the areal sweep at the end of the simulation for the
metaoptimization solution (Case 2) . . . . . . . . . . . . . . . . . . . 45
4.3 Well locations and the areal sweep at the end of the simulation for the
metaoptimization solution (Case 3) . . . . . . . . . . . . . . . . . . . 47
xiv
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
1
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
4 CHAPTER 1. INTRODUCTION
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
6 CHAPTER 1. INTRODUCTION
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.
12 CHAPTER 2. OPTIMIZATION METHODS FOR WELL PLACEMENT
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.
14 CHAPTER 2. OPTIMIZATION METHODS FOR WELL PLACEMENT
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
16 CHAPTER 2. OPTIMIZATION METHODS FOR WELL PLACEMENT
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.
20 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
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.
3.1. CASE 1: FIVE WELLS IN A 20 × 20 MODEL 21
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
22 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
(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)
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
Figure 3.5: Areal sweep at the end of the simulation for standard PSO and hybridalgorithms (Case 1)
3.1. CASE 1: FIVE WELLS IN A 20 × 20 MODEL 23
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)
24 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
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.
3.2. CASE 2: FIVE WELLS IN A 60 × 60 MODEL 25
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 $108
3.2 Case 2: Five Wells in a 60 × 60 Model
3.2.1 Problem Specification
Figure 3.8: Permeability field for Case 2
The reservoir model used for Case 2 is shown in Figure 3.8. This problem involves
optimizing the locations of three production and two injection wells. Porosity varies
26 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
Table 3.4: Simulation parameters for Case 2
Grid cell dimensions 50 × 50 × 50 ft3
Initial pressure, Pi 4012 psiAverage porosity 0.22cr at atm pressure 1 × 10-6 psi-1
µo at atm pressure 0.5 cpµw at atm pressure 0.3 cpρo 53.1 lbm/ft3
ρw 62.4 lbm/ft3
Bo at atm pressure 1.00 RB/STBBw at atm pressure 1.00 RB/STBInjector BHP 6000 psiProducer BHP 1000 psiSimulation duration 10 years
Table 3.5: Economic parameters for Case 2
Drilling cost, Cdrill 60 × 106 ($)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
3.2. CASE 2: FIVE WELLS IN A 60 × 60 MODEL 27
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.
3.2.2 Optimization Results
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.
28 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
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.
3.2. CASE 2: FIVE WELLS IN A 60 × 60 MODEL 29
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
Figure 3.11: Best well locations from standard PSO and hybrid algorithms (Case 2)
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
Figure 3.12: Areal sweep at the end of the simulation for standard PSO and hybridalgorithms (Case 2)
30 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
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.
3.2. CASE 2: FIVE WELLS IN A 60 × 60 MODEL 31
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.
32 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
Figure 3.15: Best well locations from PP-PSO and hybrid algorithms (Case 2)
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
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
Table 3.6: Optimization results for Case 2∗
run 1 run 2 run 3 averageStandard PSO 2.28 2.60 2.40 2.43CP-PSO 2.51 2.58 2.53 2.54PP-PSO 1.73 1.95 1.92 1.87Standard PSO-HJDS 2.29 2.61 2.42 2.44CP-PSO-HJDS 2.62 2.59 2.58 2.60PP-PSO-HJDS 2.17 2.53 2.46 2.39
∗all entries should be multiplied by $108
3.3 Case 3: Ten Wells in a 100 × 100 Model
3.3.1 Problem Specification
The reservoir model used for Case 3 is shown in Figure 3.17. The locations of seven
production and three injection wells are optimized. Average porosity is 0.15 and
permeability varies from 1 md to 2000 md. The relative permeability curves are the
same as in Case 1. Simulation and economic parameters are given in Tables 3.7 and
3.8. The algorithmic settings are the same as in previous cases (400 total function
evaluations for both standalone PSO and hybrid algorithms).
34 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
Table 3.7: Simulation parameters for Case 3
Grid cell dimensions 200 × 200 × 10 ft3
Initial pressure, Pi 4800 psiAverage porosity 0.15cr at atm pressure 3 × 10-6 psi-1
µo at atm pressure 1.2 cpµw at atm pressure 0.31 cpρo 49.1 lbm/ft3
ρw 64.79 lbm/ft3
Bo at atm pressure 1.03 RB/STBBw at atm pressure 1.04 RB/STBInjector BHP 6000 psiProducer BHP 1000 psiSimulation duration 1000 days
Table 3.8: Economic parameters for Case 3
Drilling cost, Cdrill 100 × 106 ($)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
3.3. CASE 3: TEN WELLS IN A 100 × 100 MODEL 35
Figure 3.17: Permeability field for Case 3
3.3.2 Optimization Results
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).
36 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
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.
3.3. CASE 3: TEN WELLS IN A 100 × 100 MODEL 37
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
Figure 3.19: Best well locations from PP-PSO and hybrid algorithms (Case 3)
(a) After 300 PSO function evaluations (b) After 300 PSO and 100 HJDS functionevaluations
Figure 3.20: Areal sweep at the end of the simulation for PP-PSO and hybrid algo-rithms (Case 3)
38 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
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.
40 CHAPTER 3. RESULTS FOR STANDALONE AND HYBRID ALGORITHMS
Table 3.9: Optimization results for Case 3∗
run 1 run 2 run 3 averageStandard PSO 15.32 11.90 11.01 12.74CP-PSO 9.94 7.15 14.33 10.47PP-PSO 11.82 9.75 12.01 11.19Standard PSO-HJDS 15.72 16.51 14.03 15.42CP-PSO-HJDS 16.12 14.10 16.33 15.52PP-PSO-HJDS 15.80 12.71 14.81 14.44
∗all entries should be multiplied by $108
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.
4.1 Case 1: Five Wells in a 20 × 20 Model
4.1.1 Problem Specification
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
4.1. CASE 1: FIVE WELLS IN A 20 × 20 MODEL 43
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
∗all entries should be multiplied by $108
Table 4.2: Comparison of all optimization methods for Case 1∗
Best PSO Best hybrid Metaoptimization3.04 3.09 3.11
∗all entries should be multiplied by $108
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
44 CHAPTER 4. METAOPTIMIZATION RESULTS
(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.
4.2 Case 2: Five Wells in a 60 × 60 Model
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
4.2. CASE 2: FIVE WELLS IN A 60 × 60 MODEL 45
Table 4.3: Standalone and hybrid optimization results for Case 2∗
run1 run 2 run 3 run 4 run 5 averageStandard PSO 2.28 2.60 2.40 2.34 2.59 2.44CP-PSO 2.51 2.58 2.53 2.41 2.60 2.53PP-PSO 1.73 1.95 1.92 1.92 2.03 1.90Standard PSO-HJDS 2.29 2.61 2.42 2.36 2.61 2.45CP-PSO-HJDS 2.62 2.59 2.58 2.45 2.62 2.58PP-PSO-HJDS 2.17 2.53 2.46 2.39 2.41 2.39
∗all entries should be multiplied by $108
Table 4.4: Comparison of all optimization methods for Case 2∗
Best PSO Best hybrid Metaoptimization2.60 2.62 2.63
∗all entries should be multiplied by $108
(a) Well locations (b) Areal sweep
Figure 4.2: Well locations and the areal sweep at the end of the simulation for themetaoptimization solution (Case 2)
46 CHAPTER 4. METAOPTIMIZATION RESULTS
Table 4.5: Standalone and hybrid optimization results for Case 3∗
run1 run 2 run 3 run 4 run 5 averageStandard PSO 15.32 11.90 11.01 14.3 16.81 13.87CP-PSO 9.94 7.15 14.33 9.37 12.14 10.59PP-PSO 11.82 9.75 12.01 14.92 14.31 12.56Standard PSO-HJDS 15.72 16.51 14.03 15.75 17.22 15.85CP-PSO-HJDS 16.12 14.10 16.33 14.83 14.01 15.08PP-PSO-HJDS 15.83 12.71 14.82 16.37 16.68 15.28
∗all entries should be multiplied by $108
Table 4.6: Comparison of all optimization methods for Case 3∗
Best PSO Best hybrid Metaoptimization16.81 17.22 17.40
∗all entries should be multiplied by $108
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
(a) Well locations (b) Areal sweep
Figure 4.3: Well locations and the areal sweep at the end of the simulation for themetaoptimization solution (Case 3)
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,
48 CHAPTER 4. METAOPTIMIZATION RESULTS
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.