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UNIVERSITY OF OKLAHOMA
GRADUATE COLLEGE
DYNAMIC OPTIMIZATION OF A WATER FLOOD RESERVOIR
A THESIS
SUBMITTED TO THE GRADUATE FACULTY
in partial fulfillment of the requirements for the
Degree of
MASTER OF SCIENCE
By
JUDE NWAOZO Norman, Oklahoma
2006
ii
DYNAMIC OPTIMIZATION OF A WATER FLOOD RESERVOIR
A THESIS APPROVED FOR THE MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL
ENGINEERING
BY
__________________________ Dr. Dean Oliver
__________________________ Dr. Chandra Rai
__________________________ Dr. Dongxiao Zhang
iii
© Copyright by JUDE NWAOZO 2006 All Rights Reserved.
iv
Dedication
To my dad, mum and siblings
v
TABLE OF CONTENTS
DEDICATION..................................................................................................... IV
LIST OF FIGURES .........................................................................................VIII
LIST OF TABLES .............................................................................................XV
ABSTRACT...................................................................................................... XVI
1. INTRODUCTION..........................................................................................1
1.1 SMART WELLS ..........................................................................................................3
1.2 PRODUCTION OPTIMIZATION ....................................................................................5
2. LITERATURE REVIEW .............................................................................9
2.1 OIL AND GAS PRODUCTION OPTIMIZATION HISTORY ..............................................9
2.2 EOR PROCESS OPTIMIZATION ................................................................................16
2.3 OPTIMIZATION OF WELL PLACEMENT AND TYPE....................................................17
2.4 PRODUCTION OPTIMIZATION CONSIDERING UNCERTAINTY ...................................19
2.5 THIS OPTIMIZATION APPROACH .............................................................................20
3. METHODOLOGY ......................................................................................22
3.1 ENSEMBLE KALMAN FILTER (ENKF) .....................................................................22
3.2 OPTIMIZATION PROCEDURE....................................................................................23
4. RESERVOIR MODEL DESCRIPTION...................................................29
4.1 GRID........................................................................................................................30
vi
4.2 SCHEDULE ...............................................................................................................32
4.3 PVT PROPERTIES OF THE RESERVOIR FLUIDS ........................................................39
5. ECONOMICS ..............................................................................................42
5.1 NET PRESENT VALUE (NPV) ..................................................................................42
6. RESULTS AND ANALYSIS ......................................................................45
6.1 CASE 1.....................................................................................................................45
6.2 CASE 2.....................................................................................................................56
7. CONCLUSIONS ..........................................................................................71
NOMENCLATURE.............................................................................................73
REFERENCES.....................................................................................................74
A. APPENDIX – FORTRAN FLOW CHART ..............................................80
B. APPENDIX – RESULTS FROM CASE 3.................................................81
C. APPENDIX – DESCRIPTION OF FORTRAN CODE ...........................86
C.1 PERMEABILITY VALUES..........................................................................................86
C.2 FORWARD RUN.......................................................................................................87
C.3 REVENUE OPTIMIZATION .......................................................................................88
C.3.1 Optimum value of alpha .................................................................................90
C.4 FUNCTION OF ALPHA .............................................................................................91
C.5 PARAMETERS..........................................................................................................92
vii
D. APPENDIX – FORTRAN CODE...............................................................93
viii
LIST OF FIGURES
Figure 1-1: Top view of horizontal, 2-D reservoir model. The shaded zone
represents high permeability streak that is at right angles with the
injector and the producer.................................................................... 7
Figure 2-1: Shape of the oil-water front before breakthrough for the base
case (left) and for the optimized case (right). ................................... 14
Figure 2-2: Schematic of reservoir used by Lorentzen et al. ................... 15
Figure 2-3 (a) and (b): Development of optimized value.......................... 15
Figure 4-1: Top view of the reservoir model showing permeability field
distribution and well placements....................................................... 30
Figure 4-2: Histogram showing permeability field distribution.................. 31
Figure 4-3: Initial distribution of mean values of bottom hole pressure
profile for well P1.............................................................................. 33
Figure 4-4: Pressure profile generated using correlation range a = 5...... 35
Figure 4-5: Histogram for pressure realizations for well P1 ..................... 35
Figure 4-6: Histogram for pressure realizations for well P2 ..................... 36
ix
Figure 4-7: Histogram for pressure realizations for well P3 ..................... 36
Figure 4-8: Histogram for pressure realizations for well P4 ..................... 37
Figure 4-9: Ten realizations of BHP profile for production well P1........... 38
Figure 4-10: Ten realizations of BHP profile for well P2. ......................... 38
Figure 4-11: Ten realizations of BHP profile of well P3. .......................... 39
Figure 4-12: Ten realizations of BHP profile of well P4. .......................... 39
Figure 4-13: Relative Permeability Curves .............................................. 40
Figure 6-1: Permeability field distribution for case 1. ............................... 45
Figure 6-2: Case 1 – Graph showing NPV for all realizations of pressure
profiles before and after optimization. .............................................. 46
Figure 6-3: Case 1 – Graph showing cumulative oil production for all
realizations of pressure profiles before and after optimization.......... 47
Figure 6-4: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P1. ........................................................................... 47
Figure 6-5: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P2. ........................................................................... 48
x
Figure 6-6: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P3. ........................................................................... 48
Figure 6-7: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P4. ........................................................................... 49
Figure 6-8: Case 1 – Water saturation distribution a) before (top) and b)
after (bottom) optimization after 913 days. ....................................... 50
Figure 6-9 (a) and (b): Case 1 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 1 of BHP
profiles.............................................................................................. 51
Figure 6-10(a) and (b): Case 1 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 2 of BHP
profiles.............................................................................................. 52
Figure 6-11(a) and (b): Case 1 – Graphs showing water cuts from all wells
before (left) and after (right) optimization for realization 2 of BHP
profiles.............................................................................................. 53
Figure 6-12: Graph showing cumulative oil and water production before
and after optimization for realization 1 of BHP profile....................... 54
xi
Figure 6-13: Graph showing cumulative oil and water production before
and after optimization for realization 2 of BHP profile....................... 55
Figure 6-14: Graph showing oil and water production rates before and
after optimization for realization 1 of BHP profile.............................. 55
Figure 6-15: Graph showing oil and water production rates before and
after optimization for realization 2 of BHP profile.............................. 56
Figure 6-16: Case 1 - Net Present Value vs. iterations............................ 57
Figure 6-17: Permeability field distribution for case 2. ............................. 58
Figure 6-18: Case 2 – Graph showing NPV for all realizations of pressure
profiles before and after optimization. .............................................. 58
Figure 6-19: Case 2 – Graph showing cumulative oil production for all
realizations of pressure profiles before and after optimization.......... 59
Figure 6-20: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P1. ............................................................ 60
Figure 6-21: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P2. ............................................................ 60
xii
Figure 6-22: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P3. ............................................................ 61
Figure 6-23: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P4. ............................................................ 61
Figure 6-24: Case 2 – Water saturation distribution a) before (top) and b)
after (bottom) optimization after 913 days. ....................................... 62
Figure 6-25(a) and (b): Case 2 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 1 of BHP
profiles.............................................................................................. 64
Figure 6-26(a) and (b): Case 2 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 2 of BHP
profiles.............................................................................................. 65
Figure 6-27(a) and (b): Case 2 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 3 of BHP
profiles.............................................................................................. 66
Figure 6-28: Case 2 – Graph showing cumulative oil and water production
before and after optimization for realization 1 of BHP profile............ 67
xiii
Figure 6-29: Case 2 – Graph showing cumulative oil and water production
before and after optimization for realization 2 of BHP profile............ 67
Figure 6-30: Case 2 – Graph showing oil and water production rates
before and after optimization for realization 1 of BHP profile............ 68
Figure 6-31: Case 2 – Graph showing oil and water production rates
before and after optimization for realization 2 of BHP profile............ 68
Figure 6-32: Case 2 – Net Present Value vs. iterations........................... 69
Figure B-1: Permeability distribution for case 3. ...................................... 81
Figure B-2: Case 3 – Graph showing NPV for all realizations of pressure
profiles before and after optimization. .............................................. 81
Figure B-3: Case 3 – Graph showing cumulative oil production for all
realizations of pressure profiles before and after optimization.......... 82
Figure B-4 (a), (b), (c), (d): Ten realizations of BHP for the optimized case
for all production wells ...................................................................... 82
Figure B-5: Case 3 – Graphs showing water cuts from all wells before
(left) and after (right) optimization for 3 realizations of BHP profiles. 83
xiv
Figure B-6 (a) and (b): Case 3 – Graphs showing cumulative oil and water
production before and after optimization for realizations 1 and 2. .... 84
Figure B-7 (a) and (b): Case 3 – Graphs showing oil and water production
rates before and after optimization for realizations 1 and 2 of BHP
profile................................................................................................ 84
Figure B-8: Case 3 – Water saturation distribution a) before (left) and b)
after (right) optimization after 913 days. ........................................... 85
Figure B-9: Case 3 – Net Present Value vs. iterations ............................ 85
Figure C-1: Graph of NPV as a function of alpha for 6 iterations............. 92
xv
LIST OF TABLES
Table 1: Summary of reservoir properties ............................................... 41
Table 2: Summary of results.................................................................... 70
xvi
ABSTRACT
It is of increasing necessity to produce oil and gas fields more efficiently
and economically because of the ever-increasing demand for petroleum
worldwide. Since most of the significant oil fields are mature fields and the
number of new discoveries per year is decreasing, the use of secondary
recovery processes is becoming more and more imperative. Waterflooding
is one of the most widely used secondary recovery means of production
after primary depletion energy has been exhausted. The use of smart
wells, which are typically wells that are equipped with downhole chokes
and other measuring instruments, is also gaining popularity in the industry
as a more efficient means of enhancing ultimate recovery. This research
presents a methodology of optimizing production or net present value from
a waterflood reservoir by controlling the bottom hole pressures of the
production wells with the use of smart well technology.
The optimization procedure involves maximizing the objective function
(e.g. cumulative oil produced or net present value) from a waterflood
reservoir by adjusting a set of controls (e.g. production wells’ bottom hole
pressure or flow rates). In this project, the pressure profile of the
production well that gives the maximum NPV is the solution to our
xvii
waterflood optimization problem. The production wells in this reservoir are
smart wells whose downhole chokes are automatically adjusted to meet
certain optimal requirements. The production well completions are also set
in such a way that they automatically shut off when certain economic limits
based on the watercut are reached.
Most efficient methods used in solving optimization problems require the
explicit knowledge of the underlying simulator equations for computation
of the gradient of the objective function with respect to the controls. As a
result of the large and complicated nature of reservoir models with large
number of unknowns and non-linear constraints, the software for gradient
calculations for practical optimization problems are very tedious and time
consuming to create. The approach presented in this study does not
require the solution of the adjoint equations. No knowledge of the
simulator equations is required and the simulator is run as a black box. In
this approach, a variant of the ensemble Kalman filter (EnKF) technique is
used in the optimization process. A relationship between the objective
function and the set of controls is obtained from the ensemble of
realizations of the state vector.
xviii
This research work also provides a validation of the optimization
methodology using various heterogeneous 2-D models with five-spot
pattern waterflood schemes. The forward run was carried out with the
Eclipse 100 (black oil) reservoir simulator. The results of the optimization
methodology presented in this study show an increase in net present
value of up to 9% and an increase in cumulative production of up to 12%
of the base case when the geology is known.
1
1. INTRODUCTION
In the past, a variety of secondary oil recovery methods have been
developed and applied to mature and depleted oil reservoirs. These
methods help to improve oil recovery compared to primary depletion1. The
oldest secondary recovery method is waterflooding since water is usually
readily available and inexpensive.
Fundamentally, waterflood involves pumping water through a well
(injector) into the reservoir. The water is forced through the pore spaces
and sweeps the oil towards the producing wells (producers). The
percentage of water in the produced fluids steadily increases until the cost
of removing and disposing of water exceeds the income from oil
production. After this point, it becomes uneconomical to continue the
operation and the waterflooding is stopped. Some wells remain
economical at a watercut of up to 99%2. On the average, about one-third
of the original oil in place (OOIP) is recovered, leaving two-thirds behind
after secondary recovery. Other secondary recovery methods include CO2
flooding and hydrocarbon gas injection, which requires a nearby source of
inexpensive gas in sufficient volume.
2
Waterflooding is most often used as a secondary recovery method of
increasing oil recovery in reservoirs where primary depletion energy has
been exhausted. It is responsible for the high production rates in the U.S.
and Canada3 where most of the fields are mature.
The number of new discoveries of significant oil fields per year is
decreasing worldwide and most of the existing major oilfields are already
at their mature stages. Consequently, it is becoming increasingly
necessary to produce these fields as efficiently as possible in order to
meet the global increase in demand for oil and gas4. For this reason,
waterflooding projects are very commonly found in most of these mature
fields. In many of these reservoirs however, water cuts from the
production wells are very high and sometimes uneconomical thereby
causing low ultimate recoveries5. This is because the injected water finds
its way through conductive fractures and high permeability zones within
the reservoir. Premature breakthrough mostly occurs in highly
heterogeneous reservoirs. As a result, many water injectors do not usually
achieve improved sweep efficiencies and a lot of the oil is by-passed.
Various methods of solving the problem of poor sweep efficiency have
3
been suggested. One method of mitigating this problem is by employing
smart production and injection wells6, 7, 8, 9.
1.1 Smart Wells
Smart well technology provides the opportunity to counteract the effects of
high permeability zones in a waterflood field by imposing a suitable
pressure or flow rate profile along the injection wells7. Smart wells are
development wells that contain permanent downhole measurement and
control equipments that enable significant improvement of oil production8
and increase the efficiency of injectors. Smart wells are equipped with a
battery of completion equipment designed to
a) Monitor well operating conditions downhole including flow rate,
pressure, temperature, phase composition, etc.
b) Control inflow and outflow rates of segregated segments of the
well.
c) Image the distribution of reservoir attributes away from the well.
These attributes may include resistivity, acoustic impedance, etc9.
A smart injector is an injector that has been divided into several intervals,
each of which can be independently controlled using inflow control valves
(ICVs) and open hole packers10. ICVs divide the injector into different
4
segments thereby making it possible to control water injection into
individual injection zones. With this type of controls in place, the injector
well can be used to selectively flood zones with more homogeneous
matrix or areas that result in less water production at the production wells.
This is achieved by opening and closing of the ICVs to flood desired
intervals. During injection, the water cuts from the production wells are
closely monitored to determine the optimal opening and closing of the
ICVs. When the water cuts from the production wells reach unacceptable
limits, the section of the horizontal injector contributing to high water cut at
the production well is isolated and shut-in using the ICVs. During this
period, the water injection continues via the matrix and other areas of
lower conductivity. This process is repeated over time until the sweep
efficiency of the injector is maximized and the water saturation is more
uniformly distributed across the injected zone8.
Smart well technology is presently undergoing the process of value
identification and quantification in the exploration and production industry9.
The advantages of this new technology to the industry include the
following:
5
1. Any sudden changes in the production or injection performance of
the well can be immediately observed and prompt response can be
carried out.
2. There is minimized down time and well interference leading to cost
savings.
3. More reserves can be drained per well due to improved well
management.
4. Improved well management also brings about increased ultimate
recovery.
1.2 Production Optimization
The best production schemes for oil and gas fields is being continually
sought after in order to maximize the production from these existing fields.
The objective of reservoir simulation is to determine the best production
design for a given field. This goal has been commonly achieved by trial
and error method. The reservoir engineer is left to decide what parameters
to change and how the changes are made to improve the results. This
imposes a high level of subjectivity to the optimization process. In the past
few decades, researches have been made to develop simulators that can
be used to determine the best production schemes. This can be
6
conceptually achieved by combining the existing reservoir simulators with
some numerical search algorithms.
The problem of production optimization requires the maximization or
minimization of some objective function g(x). In this optimization problem,
the objective function to be maximized is the net present value or
cumulative oil production. Here, x is a set of controls, which may include
bottom hole pressures, flow rates, choke size, etc and these controls may
be manipulated in order to achieve an optimum value at which the
objective function is maximized (or minimized). Optimization processes
result in the improvement of future performance of a reservoir and
therefore requires a simulation model of the real reservoir on which the
optimization is carried out. The simulation model is a dynamic model that
relates the objective function to the set of controls.
Consider a water injector and a producer in Figure 1-1. Let the objective
function g(x) be net present value and the total fluid production rates at
each of the production well completions be the set of controls, x. Changing
the production rates at each completion in turn changes the dynamic state
of the system (pressures and saturations). These changes subsequently
impact on the cumulative production and hence the objective function
7
(NPV). The controls, x are also subject to other constraints such as
surface production facilities, choke sizes, fracture limits, minimum
allowable bottom hole pressures, etc and these constraints determine
feasible values of the controls. These additional constraints pose major
problems and further complicate the solution of the optimization process.
Figure 1-1: Top view of horizontal, 2-D reservoir model. The shaded zone
represents high permeability streak that is at right angles with the injector
and the producer.
Two major categories of optimization algorithms exist in literature4:
gradient-based algorithms11,12,13 and stochastic algorithms. Gradient-
based algorithms require an efficient technique of calculating the gradient
8
of the objective function g(x) with respect to the controls x. The optimal
control theory is one of the most popular gradient-based algorithms.
The total number of controls to be adjusted is the product of the number of
controls to be updated in time (control steps) and the total number of wells
in the reservoir model. The number of controls could be very large even
for a simple reservoir model with a reasonable number of wells and control
steps, making the gradient estimation a very tedious process. Also,
another major drawback of the gradient-based method using adjoint
equations is that it requires explicit knowledge of the simulation model
equations used to describe the dynamic system.
On the other hand, the stochastic algorithms such as genetic
algorithms14,15 and simulated annealing16,17 require many forward model
evaluations but are capable of finding a global optimum with a sufficiently
large number of simulation runs. Unlike the gradient-based algorithms,
they do not require gradient estimations since the relationship between the
objective function and the controls can be obtained from several forward
models. However, the methods can be inefficient when the number of
variables is large.
9
2. LITERATURE REVIEW
The need for production optimization of reservoir fields has arisen as a
result of the global increase in demand for oil and gas. Several
applications of optimization algorithms have been developed and these
optimization techniques have proved to be beneficial in the various
problems of reservoir development, well testing and gas resource
distribution18. This chapter reviews the various optimization problems that
have been investigated by researchers and their methodologies to solving
the existing problems.
2.1 Oil and Gas Production Optimization History
Production optimization problems involving reservoir modeling with time
was first attempted by Lee and Aronofsky19. The purpose of their study
was to apply linear programming procedure to oil production scheduling
problems. The problem was to determine an oil production schedule from
5 different wells that will give the maximum profit over an eight-year
period. The constraints placed on the individual reservoir production rates
of the wells included well pressures and pipeline capacity. They solved
10
this problem using constant well interference coefficients as a substitute
for a real reservoir simulation model. Wattenbarger10, along with some
other researchers extended this study further with the use of real reservoir
simulation models for estimating the well interference coefficients.
Wattenbarger developed a method for maximizing withdrawals from a
natural gas storage reservoir.
Natural gas is commonly stored in underground reservoirs during the
summer months and then produced during the winter to meet seasonal
demands. This seasonal production can be maximized through optimal
scheduling of the individual wells. Wattenbarger10 proposed a method for
optimizing the withdrawal schedule problem using the linear programming
format. In his case, the withdrawal schedule was optimized in the sense
that no discretized withdrawal schedule can be specified for the finite-
difference model that will give greater total seasonal production while still
meeting the constraints placed on the problem. One of the constraints of
this problem requires that the wellbore pressure of each well not fall below
a minimum value. Also, the total reservoir withdrawal rate at any time is
limited to the demand rates.
11
All the work previously mentioned have been limited by the number of
phases, the phase behavior or by the geometry and size of the reservoir
model. An approach, which uses only the control variables explicitly for
numerical optimization has been developed. Asheim11,20 was involved in
the study of optimal control in water flood reservoirs using reservoir
simulation models. He developed a method for numerical optimization of
the net present value of a natural water drive and water drive by injection.
The method uses an areal two-phase reservoir simulator to calculate the
net present value (NPV) of a waterflooding scheme. In his study, the
variables subject to control were the well rates. The waterflooding scheme
that maximized the net present value was numerically obtained by
combining reservoir simulation with control theory practices of implicit
differentiation. He was able to achieve improved sweep efficiency and
delayed water breakthrough by dynamic control of the well flow rates. For
the reservoir models he considered, there was a net present value
improvement of up to 11%.
Brouwer and Jansen7 studied the optimization of water flooding with fully
penetrating, smart horizontal wells in 2-dimensional reservoirs with simple,
large-scale heterogeneities (Figure 1-1). They used optimal control theory
as an optimization algorithm for valve settings in smart wells. The
12
objective was to maximize the recovery or net present value of the
waterflooding process over a period of time.
In the study, they investigated the static optimization of waterflooding with
smart wells. Static implies that the injection and production rates in the
wells were kept constant during the displacement process, until water
breakthrough occurred. They observed significant improvements from
simple reservoir models. They however, observed that more
improvements could be achieved by dynamic optimization of the
production and injections. In a later study8, they addressed this same
problem using dynamic optimization in which case, the inflow control
valves in the wells were allowed to vary during the waterflooding process.
Waterflood was improved by changing the well profiles according to some
simple algorithm that move flow paths away from the high permeability
zones in order to delay water break-through. This was achieved by
calculating the productivity index (PI) for each segment. For each well, the
segments with the higher PI are shut-in and the rates are equally
distributed among the other segments that are open in order to maintain
the production rates. They repeated this process until the optimum flow
profile is obtained. This optimum flow profile was found to occur when the
13
ultimate oil recovery from a successive step is lower than that obtained
with the preceding flow profile.
Brouwer and Jansen7 investigated the optimization problem under two
different scenarios of well operating conditions – purely pressure-
constrained and purely rate-constrained operating environments. They
concluded that the benefit of smart wells under pressure-constrained
operating conditions was mainly the reduced amount of water production
rather than increased oil production. On the other hand, wells operating
under rate constraints gave an increased production and ultimate recovery
as well as reduced water production.
Their results show that water breakthrough is delayed from 253 days for
the base case to 658 days for the optimized case. Figure 2-1 shows
Brouwer and Jansen’s results for the oil and water saturation distribution
just before breakthrough for both the base case and the optimized case. It
can be observed that the sweep of the low permeability region is much
better for the optimized case, thereby improving the ultimate recovery.
14
Figure 2-1: Shape of the oil-water front before breakthrough for the base
case (left) and for the optimized case (right).
Lorentzen et al.21 also carried out a study on the dynamic optimization of
waterflooding using a different approach from those described above. He
carried out his optimization by controlling the chokes to maximize
cumulative oil production or net present value. Their new approach uses
the ensemble Kalman filter as an optimization routine. The ensemble
Kalman filter was originally used for estimation of state variables but has
been adapted to optimization in their work. In their optimization study, they
demonstrated the use of the ensemble Kalman filter as an optimization
routine on a simple 5-layer reservoir with different permeabilities. The
schematic of the reservoir used by Lorentzen et al. is shown in Figure 2-2.
The results from this approach are shown in Figure 2-3 a. and b.
15
Figure 2-2: Schematic of reservoir used by Lorentzen et al.
The above methodology provided by Lorentzen et al. avoids the use of the
optimal control theory since no adjoint equations were needed and the
model equations are treated as a “black box”.
Figure 2-3 (a) and (b): Development of optimized value
16
This methodology avoids one obvious disadvantage of the optimal control
approach when used as a solution to optimization problems – it entails the
construction and solution of an adjoint set of equations. These adjoint
equations require an explicit knowledge of the reservoir model equations
and also require extensive programming in order to implement them. This
has been shown by Sarma and Aziz4.
2.2 EOR Process Optimization
In 1984, Ramirez and Fathi12 applied the theory of optimal control to
determine the best possible injection policies for enhanced oil recovery
processes. Their study was motivated by the high operating costs
associated with EOR projects. The commercial application of new EOR
processes depends on whether economic projections indicate a decent
return on investment. The objective of their study was to develop an
optimization method to minimize injection costs while maximizing the
amount of oil recovered. The performance of their algorithm was
subsequently examined for surfactant injection as an EOR process in a
one-dimensional core flooding problem13. The control for the process was
the surfactant concentration of the injected fluid. They observed a
17
significant improvement in the ratio of the value of the oil recovered to the
cost of the surfactant injected from 1.5 to about 3.4. Optimal control was
also applied to steam flooding by Liu and Ramirez22 in 1993. They
developed an approach using optimal control theory to determine
operating strategies to maximize the economic attractiveness of steam
flooding process. Their objective was to maximize a performance index
which is defined as the difference between oil revenue and the cost of
injected steam. Their optimization methodology also obtained significant
improvement under optimal operation.
2.3 Optimization of Well placement and type
A great deal of research work has been carried out at Stanford University
to determine the optimum location, type and trajectory of wells to be drilled
in a field. The determination of a well location is a very complex problem
that depends on several variables which include reservoir and fluid
properties, well and surface equipment specifications, and economic
criteria. In 2002, a hybrid optimization technique based on genetic
algorithm (GA) was proposed by Baris et al.23 at Stanford University to
optimize placement of water-injection wells for an offshore field in the Gulf
of Mexico. The objective function used was NPV while the water injection
rates and well placements of up to four injectors were being optimized.
18
Their results showed an incremental NPV of $154 million with three
injectors after optimum placement has been achieved, compared to the no
injection case. Badru et al.24 also carried out a similar investigation using
the Hybrid Genetic Algorithm (HGA) to determine optimal well locations.
They used this technique to optimize both vertical and horizontal wells for
both gas injection and water injection projects using NPV as the objective
function. They compared the results obtained from the optimization of well
placements proposed by the HGA method with those selected by
engineering judgment. The optimized placement results obtained using
HGA showed a significant increase in cumulative production of about 70%
more than that proposed by engineering judgment. Burak et al.25 also at
Stanford University extended the research on well optimization process by
including well type and trajectory of nonconventional wells. This problem is
more complicated than other well optimization problems because of the
wide variety of possible well types that must be considered, which include
number of wells, location, and orientation of laterals. Their optimization
procedure entailed the use of GA in conjunction with other routines such
as artificial neural network. They observed a general increase in the
objective function relative to the reference case, up to 30% in some cases.
19
2.4 Production Optimization considering uncertainty
Naevdal, Brouwer and Jansen26 in 2005, developed a closed-loop control
approach where measurements from smart wells were used to
continuously update a waterflood reservoir model and an adjoint-based
optimal control strategy was computed based on the most recent update
of the reservoir model. The ensemble Kalman filter was used to obtain
frequent updates of the reservoir model. They demonstrated their
methodology on a simple reservoir model with one smart injector and
producer where the objective function was NPV and the total fluid
production rates were used as the controls. In a nut-shell, their
methodology is a combination of an optimal control for waterflood
optimization with automatic history matching of reservoir models using
ensemble Kalman filter to estimate the final permeability field. Naevdal et
al. observed that the results obtained using a closed-loop control starting
from an unknown permeability field, were almost as good as those
obtained assuming a priori knowledge of the permeability field.
Another closed-loop production optimization approach in a water flood
reservoir was presented by Sarma, Durlofsky and Aziz27 in 2005. In their
approach, a gradient-based optimization algorithm was used to determine
optimal control settings, while the parameter gradients are used for model
20
updating. The model-updating component of the closed loop is a problem
of inversion of production data (well pressures and flow rates) in order to
determine the reliable estimates of uncertain model parameters (porosity
and permeability). Their results showed substantial improvements in NPV
of up to 25% of the base case and very close to those obtained if the a
priori reservoir description was known.
Some other applications of optimization algorithms used in different
problems of reservoir development, oil production and well testing have
been surveyed by Virnovsky18. Asides added benefits in oil production
through the development of new waterflooding strategies, optimization
procedures have been successfully applied to gas distribution among a
group of gas lift wells. Mathematical programming algorithms in
conjunction with numerical simulation of the appropriate processes were
used to obtain the optimal solutions for each of the cases he presented.
2.5 This Optimization Approach
The new approach for production optimization of a waterflood reservoir
presented in this study is a variant of the ensemble Kalman filter
procedure. It does not require the explicit knowledge of the reservoir
21
simulation equations that are used to describe the dynamic state of the
reservoir system. Solutions to the adjoint equations are therefore not
required hence making software development less tedious.
As discussed earlier, Lorentzen et al.21 applied the ensemble Kalman filter
technique directly to his waterflood optimization problem as well. In his
application, he replaced the observed measurements by values
representing an upper limit for the possible NPV. The filter then returns
control settings that result in NPV as close to the predefined value. In this
new approach however, a predefined value of NPV is not required in the
optimization process. It simply optimizes NPV by maximizing an objective
function which includes the NPV and a penalty term that penalizes the
controls that are far away from the prior estimate. The gradient of the NPV
with respect to controls is obtained from the ensemble of control
realizations. A complete description of the methodology is presented in the
following chapter.
22
3. METHODOLOGY
The ensemble Kalman filter (EnKF) technique has been adapted to the
problem of NPV in this optimization study. In this study, an ensemble of 40
realizations of the controls (bottom hole pressure profiles) was generated
and continuously updated after each reservoir simulation run until nearly
optimum pressure profiles were obtained. The optimum pressure profile is
the profile at which the net present value is at its maximum. Since EnKF is
a Monte-Carlo approach, the final results will vary for each member of the
ensemble.
3.1 Ensemble Kalman Filter (EnKF)
The Kalman filter is typically used to estimate states in systems that
change with time28. The procedure consists of a forecast step and an
assimilation step in which variables that describe the state of the system
are corrected to honor the observations using a series of equations.
The ensemble Kalman filter (EnKF) is a modified form of the Kalman filter
that has been adapted to history matching in reservoir simulation26, 28. It is
a Monte-Carlo method in which an ensemble of initial reservoir state
23
vectors are generated by sampling from a probability density function and
kept up-to-date as data are assimilated sequentially. The reservoir state
vectors consist of all the reservoir variables that are uncertain and need to
be specified in order to run the reservoir simulator. The uncertainty of
reservoir state vectors is estimated from the ensemble28. The state vector
consists of two parts: model parameters (porosities, permeabilities,
saturations, and pressures) and the theoretical data (e.g. water-oil-ratios,
production rates, bottom hole pressures, etc.). If the reservoir state vector
is denoted by y, then the state vector for the reservoir model can be
written as
[ ]TTT dmy = 3-1
where m = model parameters
d = theoretical data
3.2 Optimization Procedure
The ensemble optimization process also consists of 2 steps – the forecast
(or forward) step and the update step. A numerical reservoir simulator is
used to perform the forecast step. The reservoir model is run for each
member of the ensemble of state vectors using Eclipse 100 for the forward
simulation. The reservoir state vector consists of all the control variables
24
that are uncertain and need to be optimized. In this project, the state
vector is made up of the bottom hole pressures for all the wells at every
time step, as well as the net present value obtained from running the
reservoir simulator with these controls. Since there are 4 wells in our
model and 20 time steps in total, the reservoir state vector for each
member of the ensemble is made up of (80 + 1) members. From the
forecast step, the net present value based on these controls is calculated
using the cumulative reservoir fluid production and the average estimated
oil price.
Let x be used to denote the number of controls on the wells for the time
periods. Then,
x = {x1, x2, x3… x80}
These controls could be choke settings, flow rates, bottom hole pressures,
etc. Also, let the net present value for the production period using controls
x be g(x). Assume also that we wish to penalize the control settings that
are far from our initial guess or that rapidly change with time.
The best control settings in this case will be the set x that maximizes the
following equation:
25
( ) ( )pX
T
p xxCxxxgxS −−−= −1
2)()(
α 3-2
where S(x) = objective function
g(x) = Net Present Value
α = weighting factor
x = new state vector
xp = prior state vector
Cx = covariance matrix of the control vector
A local quadratic approximation to S(x) at x = x’ is given as
xHxxxSxxF TT δδδγδ2
1)'()'( ++=+ 3-3
whereγ is equal to )(xS∇ , and the Hessian, H isTxS ))((∇∇ .
The value of xδ that maximizes the quadratic approximation to the objective
function is the extremum of this function and it occurs at 0=∇F or
0=+ xHδγ 3-4
The Newton equations for iteratively finding the extremum are
γδ −=xH 3-5
or l
l
l SxH −∇=+1δ 3-6
After computing1+lxδ from equation 3-6 above, the controls are updated
using the equation below.
26
11 ++ += lll xxx δ 3-7
The gradient of the objective function S(x) for the lth iteration is
( )p
l
X
lxxCxGS −−=∇ −1)( α 3-8
(assuming that xC is symmetric). In the above equation, )( lxG denotes the
matrix of the sensitivity coefficients or the derivatives of the objective
function with respect to the controls. The sensitivity coefficient is a
measure of how strongly the objective function, gi(x) is affected by a
change in the controls, x. The individual elements of the sensitivity matrix
are given by,
j
iji
x
gG
∂
∂=, 3-9
The approximate Hessian matrix is given by:
1)( −−∇≈ x
lCxGH α 3-10
Assuming that the second derivative of g(x) is negligible, equation 3-10
above becomes
1−−≈ xCH α 3-11
Therefore, substituting equation 3-8 and 3-11 in equation 3-6 gives
( )[ ]p
l
xl
l
X xxCGxC −−−=− −+− 111 αδα 3-12
After further manipulation, equation 3-12 becomes
27
( )p
l
lx
lxxGCx −−=+
αδ
11 3-13
Substituting the incremental controls 1+lxδ obtained in equation 3-13 into
equation 3-7;
p
l
lx
llxxGCxx +−+=+
α
11 3-14
Equation 3-14 reduces to
plx
lxGCx +=+
α
11 3-15
Equation 3-15 is used to calculate the updated state vector for the next
iteration step.
Let y be used to denote the state vector consisting of the controls (bottom
hole pressures) as well as the net present value obtained from using the
controls. The ensemble of state vectors can be written as
[ ]eNyyyyY ,...,,, 321= 3-16
Ne is used to denote the number of the ensemble members. The
covariance matrix for the state variables at any time can be estimated
from the ensemble using the standard statistical formula;
( )( )Te
Y YYYYN
C −−−
=1
1 3-17
28
whereY = Mean of state vector calculated across the ensemble.
SinceT
YlX MCGC = , then equation 3-15 becomes
p
T
Y
lxMCx +=+
α
11 3-18
The vector M is called the measurement operator and it relates the state
vector to the theoretical observation. Since the theoretical observation is
part of the state vector y, M is a simple matrix with 0 and 1 as its
components. The matrix M can be arranged as follows:
[ ]IM 0= 3-19
Where 0 is an Nd x (Ny – Nd) matrix with all 0s as entries and I is an Nd x
Nd identity matrix. Note that Nd is the number of measurements (Nd = 1 for
this project) and Ny is the number of variables in the state vector, y (Ny =
81 for this project).
29
4. RESERVOIR MODEL DESCRIPTION
As described earlier, the ultimate goal of a reservoir simulator is to
determine the optimum production scheme of an oil and gas field. This
can be achieved by combining a reservoir simulator with a numerical
optimization algorithm. The reservoir simulation phase of this study was
carried out with the use of Eclipse 100 – black oil option. The waterflood
optimization procedure developed in this research was tested on various
2-D Cartesian reservoir models consisting of 25 x 25 x 1 grid lattice. In this
study, a reservoir with no-flow boundaries on all sides was considered.
The phases present in the reservoir were oil and water. No free gas was
present. The model represents a 200-acre field (approximately 2950 ft x
2950 ft) with 1 vertical injector well (INJ) located at the center of the
reservoir (in grid block 13:13:1) and 4 vertical production wells (P1, P2,
P3, and P4) located at the corners of the field. Production well P1 is
located in block 1:1:1; well P2 is located in block 25:1:1; well P3 is located
in block 1:25:1; and well P4 is in block 25:25:1. Note that the well locations
are fixed and therefore not subject to optimization. The top view of one of
the reservoir models used in this study is shown in Figure 4-1. The wells
are drilled with a 40-acre spacing and are all brought to operation at the
30
same time. The depth of the top surface of the reservoir is 10,000 ft with a
net pay thickness of 50 ft.
Figure 4-1: Top view of the reservoir model showing permeability field
distribution and well placements.
4.1 Grid
The basic geometry of the simulation grid and various rock properties
(porosity, absolute permeability, etc) in each grid cell are specified in the
grid section. From these properties, the pore volumes of the grid blocks
and the inter-block transmissibilities are calculated by the simulator29. The
P1 P2
P3 P4
INJ
31
keywords used in this section usually depend on the geometry option
selected in the initialization section. In this case, we used the Cartesian,
block-centered geometry option. The porosity distribution in the reservoir
is assumed to be homogeneous with a porosity of 0.25 while the
permeability is heterogeneous with an average value of 60 md for the
base case. The permeability field was generated using the sequential
Gaussian simulation (SGS) algorithm in the Geostatical Software Library
(GSLIB). The orientation of the permeability correlation was set at an
angle of 45 degrees in GSLIB in order to achieve a diagonal permeability
trend.
50 100 150 200 250
Permeability HmdL0
20
40
60
80
100
120
rebmuN
fo
dirg
skcolb
Histogram showing permeability distribution in the reservoir
Figure 4-2: Histogram showing permeability field distribution
32
It should be noted here that the values obtained from the SGS simulation
are log permeabilities (lnk). They must first be converted to actual
permeability values by taking the exponential of the variable X before
applying them to the reservoir grid model. A histogram of the initial
permeability distribution is shown in Figure 4-2. The distribution appears to
be log normal.
The original fluids in place in the reservoir consist of water at a pore
volume saturation of 20% and undersaturated oil contained in 80% of the
pore volume. The residual oil and connate water saturation are 0.15 and
0.20, respectively. The initial reservoir pressure is 4500 psi.
4.2 Schedule
As said earlier, all the wells were drilled vertically and completed with 0.5
ft wellbore internal diameter to a depth of 10,050 ft and brought into
operation at the same time (1-Jan-1990). The wells were operated for a
10-year period with constant control settings for 6 months. In total, there
were 20 control settings for each well in the production period. The injector
well schedule had a rate controlled mode with an injection rate of 5000
stb/day. Bottom hole pressures in the production wells are the constraints
33
that were used for the optimization process. The minimum allowable
bottom hole pressure was set at 200 psi.
The following procedure was used to generate the 40 initial realizations of
pressure constraints for the production wells. These steps were used for
each well’s BHP profile.
Step 1: The mean value of each pressure profile was randomly selected
from a uniform distribution. This distribution characterizes a random
variable whose value is equally likely everywhere within the interval. The
upper limit of the uniform distribution was 3000 psi and the lower limit was
1000 psi. Figure 4-3 shows the distribution of the means of the pressure
profiles for well P1.
1500 2000 2500 3000
Mean BHP
0
2
4
6
8
10
rebmuN
fo
snoitazilaer
Histogram showing Mean BHPs for Well P1
Figure 4-3: Initial distribution of mean values of bottom hole pressure
profile for well P1.
34
Step 2: A gaussian covariance function30 with a practical range of about
2.5 years (5 time periods) was subsequently used to generate pressure
variations from the mean for the 20 time steps, which describe the bottom
hole pressure profile for each realization. The mean of this distribution is
the randomly selected value from step 1.
−−=
a
hhhC
ji
ji
)(3exp)( 2
, σ 4-1
where C(h i,j) = Covariance function
σ = Standard deviation
hi, hj = Random variables (Pressures)
a = Correlation range
Step 3: The generated pressure profiles were exported from mathematica
to a text file to be read into Eclipse schedule include file during the
simulation runs. Also the pressure profiles and histogram showing all the
realizations were plotted.
Step 4: Steps 1, 2 and 3 were repeated for all the 4 wells. A standard
deviation of 200 psi was used in the gaussian covariance model shown in
equation 4-1. Each of the 4 wells has a data file where pressure
realizations are stored. These files were also used to store updated
pressure profiles during each iteration process.
35
0 5 10 15 20
Timestep
1000
2000
3000
4000
mottoB
eloH
erusserP
HispL5 realizations of BHP profile for Well P1 using a = 5
Figure 4-4: Pressure profile generated using correlation range a = 5.
1000 1500 2000 2500 3000
Bottom Hole Pressure
0
10
20
30
40
50
60
rebmuN
fo
snoitazilaer
Histogram of BHP realizations for Well P1
Figure 4-5: Histogram for pressure realizations for well P1
Figure 4-5 – Figure 4-8 show histogram plots of initial pressure settings for
all 4 producing wells. Graphs of 10 realizations of the bottom hole
36
pressure profiles of production wells P1 – P4 used in the initial simulation
run is shown in Figure 4-9 - Figure 4-12.
1000 1500 2000 2500 3000
Bottom Hole Pressure
0
10
20
30
40
50
60
rebmuN
fo
snoitazilaer
Histogram of BHP realizations for Well P2
Figure 4-6: Histogram for pressure realizations for well P2
1000 1500 2000 2500 3000
Bottom Hole Pressure
0
10
20
30
40
50
60
rebmuN
fo
snoitazilaer
Histogram of BHP realizations for Well P3
Figure 4-7: Histogram for pressure realizations for well P3
37
1000 1500 2000 2500 3000 3500
Bottom Hole Pressure
0
10
20
30
40
50
60
rebmuN
fo
snoitazilaer
Histogram of BHP realizations for Well P4
Figure 4-8: Histogram for pressure realizations for well P4
The economic limits of the production wells were set using the CECON
(Economic limits for production well connections) keyword from the list of
Eclipse keywords. If an individual connection (or group of connections)
violates one of the economic limits that have been set, it automatically
shuts off. In the case of this study, the maximum water cut of 0.93 is the
economic limit that has been set. Therefore, if any of the wells exceed a
water cut of 0.93, that well is automatically shut-off.
38
Bottom Hole pressure schedule for well P1 - Prior
0
500
1000
1500
2000
2500
3000
3500
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Figure 4-9: Ten realizations of BHP profile for production well P1.
Bottom Hole pressure schedule for well P2 - Prior
0
500
1000
1500
2000
2500
3000
3500
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Figure 4-10: Ten realizations of BHP profile for well P2.
39
Bottom Hole pressure schedule for well P3 - Prior
0
500
1000
1500
2000
2500
3000
3500
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Figure 4-11: Ten realizations of BHP profile of well P3.
Bottom Hole pressure schedule for well P4 - Prior
0
500
1000
1500
2000
2500
3000
3500
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Figure 4-12: Ten realizations of BHP profile of well P4.
4.3 PVT Properties of the Reservoir Fluids
The reservoir fluids are oil and water. The oil contains a constant and
uniform concentration of 0.2 Mscf/stb of dissolved gas. The oil bubble
40
point pressure is assumed to be 400 psi. At a reference pressure of 4500
psi, the oil has a viscosity of 2.4 cp. The oil formation volume factor (Bo) is
0.972. At surface conditions, the oil is assumed to have a density of 56
lb/cuft while the density of water is assumed to be 62.4 lb/cuft. Water
compressibility is set at 3 x 10-6 psi-1, water formation volume factor (Bw) of
1.0034 rb/stb and viscosity of 0.96 cp at a reference pressure of 4500 psi.
The bulk compressibility of the rock was set at 4 x 10-6 psi-1.
The relative permeability curve used is shown in Figure 4-13 below. A
summary of the reservoir properties is shown in Table 1.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
Water Saturation, Sw
Rela
tive P
erm
eabili
ty
Krw
Kro
Figure 4-13: Relative Permeability Curves
41
Table 1: Summary of reservoir properties
Number of grid blocks 25 x 25 x 1
Grid block size 118ft x 118ft x 50ft
Water injection rate 5000 STB/D
Reservoir thickness 50 ft
Porosity 25%
Actual reservoir area 200 acres
Initial Oil Saturation 0.8
Initial Water Saturation 0.2
Well Depth 10,000 ft
Initial reservoir pressure 4500 psi
Ave. Reservoir Temperature 284 F
Production period 10 years
Time step 6 months
42
5. ECONOMICS
The objective function used in this project is the net present value of the
waterflood operation for a given production period. The objective is to
maximize the net present value over the life of the reservoir and this is
achieved by adjusting a set of controls (bottom hole pressures or flow
rates). This chapter explains the concept of net present value and how it
can be determined.
5.1 Net Present Value (NPV)
Present value of money compares the value of a certain amount of money
today to the value of that same amount in the future and vice versa, taking
into consideration inflation and returns. Net present value (NPV) is the
difference between the present value of cash inflows and the present
value of cash outflows. Given an investment opportunity, NPV is used by
an organization to analyze the profitability of the project or investment and
to make decisions with regards to capital budgeting. It is sensitive to the
future cash inflows that an investment or project will yield.
NPV can be computed using the following formula31:
43
0
1 )1(C
r
CNPV
T
tt
t −+
=∑=
5-1
where t = Time step
Ct = Cash inflow after time t, $
r = Annual (or periodic) discount rate, fraction
T = Cumulative investment (or production) period
C0 = Initial investment
A conservative annual discount rate of 10% was used in this study in the
estimation of the present value of money and is based on the current rates
at which eligible institutions are charged to borrow short-term funds
directly from a Federal Reserve Bank (approximately 6.5%). Also, most oil
companies use this rate for evaluating the viability of proposed
investments.
Cash inflow is calculated from the oil and water production rates obtained
from each of the production wells or from the cumulative production from
the reservoir. The price of oil is pegged at $40 per barrel for the entire 10-
year production period while the cost of water disposal is $3 per barrel of
produced water. The total cash inflow for the entire production period is
given by,
44
( ) ( )watfwptbblfoptC $/$ ×−×= 5-2
where, C = Total cash inflow, $
$/bbl = Price of Oil per bbl, $
$wat = Cost of water disposal per bbl, $
fopt = Cumulative oil production, stb
fwpt = Cumulative water production, stb
The economic limit is determined by the time at which the cost of handling
the water exactly balances the income from selling the oil. The water cut
at which the economic limit is reached can be calculated thus;
%93340
40
$/$
/$≈
+=
+=
+=
watbbl
bbl
woprwwpr
wwprwct 5-3
where wct = Water cut, stb/stb
Therefore, each production well in the simulator has been set up in such a
way that the connection/perforation is automatically closed as soon as it
reaches an economic limit of 93% water cut.
45
6. RESULTS AND ANALYSIS
This chapter presents and analyzes the results obtained from this
research project. The codes developed were tested with three different
permeability fields which will be denoted as case 1 – case 3. Results from
case 1 and case 2 are presented in this chapter. Results from case 3 are
presented in Appendix B.
6.1 Case 1
The permeability field for case 1 is shown in Figure 6-1.
Figure 6-1: Permeability field distribution for case 1.
P1 P2
P3 P4
INJ
46
The increase in NPV and cumulative production after the optimization
process can be observed from Figure 6-2 and Figure 6-3, consecutively.
The percentage increase in NPV ranged between 2.4% and 8.7% while a
percentage increase of up to 9% was observed in the cumulative oil
production after optimization.
NPV for all realizations before and after optimization
128,000,000
130,000,000
132,000,000
134,000,000
136,000,000
138,000,000
140,000,000
142,000,000
144,000,000
146,000,000
148,000,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Realization Number
NP
V (
$)
Prior NPV Optimized NPV
Figure 6-2: Case 1 – Graph showing NPV for all realizations of pressure
profiles before and after optimization.
The optimized pressure profiles that give the highest net present value for
all four production wells are shown in Figure 6-4 – Figure 6-7 for ten
realizations. This can be compared to the initial BHP realizations shown in
section 4.2 of chapter 4 (see Figure 4-9 – Figure 4-12).
47
Cumulative Oil production for initial and optimized realizations
5,100,000
5,200,000
5,300,000
5,400,000
5,500,000
5,600,000
5,700,000
5,800,000
5,900,000
6,000,000
6,100,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Realization number
Cu
mu
lati
ve O
il P
rod
ucti
on
(S
TB
)
Initial cumulative production
Optimized cumulative production
Figure 6-3: Case 1 – Graph showing cumulative oil production for all
realizations of pressure profiles before and after optimization.
Bottom Hole pressure schedule for well P1 after optimization
0
100
200
300
400
500
600
700
800
900
1000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-4: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P1.
48
Bottom Hole pressure schedule for well P2 - Optimized
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-5: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P2.
Bottom Hole pressure schedule for well P3 - Optimized
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-6: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P3.
49
Bottom Hole pressure schedule for well P4 - Optimized
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-7: Case 1 – Graph showing 10 realizations of optimized pressure
profiles for well P4.
From the optimized pressure profiles shown in the figures above, NPV
optimization of case 1 requires that well P1 be produced at the minimum
bottom hole pressure for the duration of the production period. This is as a
result of the low permeability zone between the injector and the producer.
The profile of well P1 is however, contrary to the optimized pressure
profile of well P4 where pressures are continually increased to delay water
breakthrough. The effect of the optimized pressure profiles on the water
saturation distribution can be observed in Figure 6-9 below.
50
Figure 6-8: Case 1 – Water saturation distribution a) before (top) and b)
after (bottom) optimization after 913 days.
There is a considerable change in the distribution of water saturation
across the field at the time of the earliest water breakthrough. This change
can be observed in Figure 6-8. Figure 6-8a shows the water saturation
51
distribution before optimization while b) shows the distribution after
optimization. It can be observed that b) gives a more evenly distributed
water saturation across the field than a). This means that higher sweep
efficiency was attained after the optimization process.
Water cut from producing wells for schedule realization 1 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 1 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Figure 6-9 (a) and (b): Case 1 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 1 of BHP
profiles
52
The graphs of water cuts from all four production wells using realization 1
of BHP profiles before and after optimization are shown in Figure 6-9. It
can be observed that the breakthrough times as well as the water cut
trends come closer to overlapping after the optimization process.
Water cut from producing wells for schedule realization 2 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 2 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Figure 6-10(a) and (b): Case 1 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 2 of BHP
profiles
53
The above trend can also be observed in other realizations of BHP profile
(See Figure 6-10 and Figure 6-11).
Water cut from producing wells for schedule realization 3 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 3 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from wellP1Water cut from wellP2Water cut from wellP3
Figure 6-11(a) and (b): Case 1 – Graphs showing water cuts from all wells
before (left) and after (right) optimization for realization 2 of BHP profiles
54
Graphs of field cumulative production of oil and water with time before and
after optimization are shown in Figure 6-12 (for realization 1) and Figure
6-13 (for realization 2). The oil and water production rates for both
realizations are also shown in Figure 6-14 and Figure 6-15. It can be
observed that the optimization process sought to maximize the rates at the
early stages of production. Since the NPV is the objective function being
maximized, the early oil production contributes most to the NPV than the
later production.
Field cumulative production for schedule realization 1 before and after
optimization
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
9,000,000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Cu
mu
lati
ve p
rod
ucti
on
, S
TB
Cum Oil prod - priorCum Oil Prod - OptimizedCum Water prod - priorCum Oil prod - Optimized
Figure 6-12: Graph showing cumulative oil and water production before
and after optimization for realization 1 of BHP profile.
55
Field cumulative production for schedule realization 2 before and after
optimization
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Cu
mu
lati
ve p
rod
ucti
on
, S
TB
Cum Oil prod - prior
Cum Oil Prod - Optimized
Cum Water prod - prior
Cum Water prod - Optimized
Figure 6-13: Graph showing cumulative oil and water production before
and after optimization for realization 2 of BHP profile.
Field production rates for schedule realization 1 before and after
optimization
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Pro
du
cti
on
rate
s, S
TB
/D
Oil prod rate - prior
Oil prod rate - Optimized
Water Prod rate - prior
Water prod rate - Optimized
Figure 6-14: Graph showing oil and water production rates before and
after optimization for realization 1 of BHP profile.
56
Field production rates for schedule realization 2 before and after
optimization
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Pro
du
cti
on
rate
s, S
TB
/D
Oil prod rate - prior
Oil prod rate - Optimized
Water Prod rate - prior
Water prod rate - Optimized
Figure 6-15: Graph showing oil and water production rates before and
after optimization for realization 2 of BHP profile.
Finally, Figure 6-16 shows a plot of NPV against iterations. It is observed
that the optimized value of NPV continuously increases as a function of
iterations. The total increase in the mean NPV of the ensemble from the
initial to the optimized case is approximately 5.9%.
6.2 Case 2
The permeability field distribution for case 2 is shown in Figure 6-17. The
average permeability of the field is about 60 md. However, large patches
of very low permeability are observed between well P3 and the injector.
57
Mean NPV of ensemble as a function of iteration number
137,000,000
138,000,000
139,000,000
140,000,000
141,000,000
142,000,000
143,000,000
144,000,000
145,000,000
146,000,000
147,000,000
0 1 2 3 4 5 6 7 8 9 10
Iteration Number
NP
V (
$)
Prior Mean NPV for initial BHP realizations
Optimized Mean NPV
5.9 % increase
Figure 6-16: Case 1 - Net Present Value vs. iterations
After the optimization process, an increase in NPV and cumulative
production can be observed from Figure 6-18 and Figure 6-19,
consecutively. The percentage increase in NPV ranged between 3.7% and
7.6%. A percentage increase of up to 5% was observed in the cumulative
oil production after optimization.
58
Figure 6-17: Permeability field distribution for case 2.
NPV for all realizations before and after optimization
130,000,000
132,000,000
134,000,000
136,000,000
138,000,000
140,000,000
142,000,000
144,000,000
146,000,000
148,000,000
150,000,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Realization Number
NP
V (
$)
Prior NPV Optimized NPV
Figure 6-18: Case 2 – Graph showing NPV for all realizations of pressure
profiles before and after optimization.
P1 P2
P3 P4
INJ
59
Cumulative Oil production for initial and optimized realizations
5,300,000
5,400,000
5,500,000
5,600,000
5,700,000
5,800,000
5,900,000
6,000,000
6,100,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Realization number
Cu
mu
lati
ve O
il P
rod
ucti
on
(S
TB
)
Initial cumulative production
Optimized cumulative production
Figure 6-19: Case 2 – Graph showing cumulative oil production for all
realizations of pressure profiles before and after optimization.
The optimized pressure profiles for all four production wells are shown in
Figure 6-20 – Figure 6-23 for ten realizations. These optimized profiles
can be compared with the initial BHP realizations shown in section 4.2 of
chapter 4 (see Figure 4-9 – Figure 4-12).
From the optimized pressure profiles, NPV optimization of case 2 requires
that wells P2 and P3 be produced close to the minimum bottom hole
pressure for the duration of the production period. The profiles of well P1
and P4 have continually increasing pressure profiles with production time.
60
Bottom Hole pressure schedule for well P1 after optimization
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-20: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P1.
Bottom Hole pressure schedule for well P2 - Optimized
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-21: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P2.
61
Bottom Hole pressure schedule for well P3 - Optimized
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-22: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P3.
Bottom Hole pressure schedule for well P4 - Optimized
0
1000
2000
3000
4000
5000
6000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1Realization 2Realization 3Realization 4Realization 5Realization 6Realization 7Realization 8Realization 9Realization 10
Figure 6-23: Case 2 – Graph showing 10 realizations of optimized
pressure profiles for well P4.
62
The effect of the optimized pressure profiles on the water saturation
distribution can be observed in Figure 6-24 below.
Figure 6-24: Case 2 – Water saturation distribution a) before (top) and b)
after (bottom) optimization after 913 days.
63
Figure 6-24a shows the water saturation distribution before optimization
while b) shows the distribution after optimization both for case 2. As
observed in case 1, Figure 6-24 b) gives a more evenly distributed water
saturation across the field than a). This means that higher sweep
efficiency was attained after the optimization process.
The graphs of water cuts from all four production wells using realization 1
of BHP profiles before and after optimization are shown in Figure 6-25. It
can be observed that the breakthrough times as well as the water cut
trends tend to overlap after the optimization process. This trend can also
be observed in other realizations of BHP profile (See Figure 6-26 and
Figure 6-27).
Graphs of field cumulative production of oil and water with time before and
after optimization are shown in Figure 6-28 (for realization 1) and Figure
6-29 (for realization 2). The oil and water production rates for both
realizations are also shown in Figure 6-30 and Figure 6-31. It can be
observed that the optimization process sought to maximize the rates at the
early stages of production. Since the NPV is the objective function being
maximized, the early oil production contributes most to the NPV than the
later production.
64
Water cut from producing wells for schedule realization 1 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 1 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Figure 6-25(a) and (b): Case 2 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 1 of BHP
profiles
65
Water cut from producing wells for schedule realization 2 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 2 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Figure 6-26(a) and (b): Case 2 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 2 of BHP
profiles
66
Water cut from producing wells for schedule realization 3 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 3 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Figure 6-27(a) and (b): Case 2 – Graphs showing water cuts from all wells
before (top) and after (bottom) optimization for realization 3 of BHP
profiles
67
Field cumulative production for schedule realization 1 before and after
optimization
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
9,000,000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Cu
mu
lati
ve p
rod
ucti
on
, S
TB
Cum Oil prod - priorCum Oil Prod - OptimizedCum Water prod - priorCum Oil prod - Optimized
Figure 6-28: Case 2 – Graph showing cumulative oil and water production
before and after optimization for realization 1 of BHP profile.
Field cumulative production for schedule realization 2 before and after
optimization
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Cu
mu
lati
ve p
rod
ucti
on
, S
TB
Cum Oil prod - prior
Cum Oil Prod - Optimized
Cum Water prod - prior
Cum Water prod - Optimized
Figure 6-29: Case 2 – Graph showing cumulative oil and water production
before and after optimization for realization 2 of BHP profile.
68
Field production rates for schedule realization 1 before and after
optimization
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Pro
du
cti
on
rate
s, S
TB
/D
Oil prod rate - prior
Oil prod rate - Optimized
Water Prod rate - prior
Water prod rate - Optimized
Figure 6-30: Case 2 – Graph showing oil and water production rates
before and after optimization for realization 1 of BHP profile.
Field production rates for schedule realization 2 before and after
optimization
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Pro
du
cti
on
rate
s, S
TB
/D
Oil prod rate - prior
Oil prod rate - Optimized
Water Prod rate - prior
Water prod rate - Optimized
Figure 6-31: Case 2 – Graph showing oil and water production rates
before and after optimization for realization 2 of BHP profile.
69
Figure 6-32 shows a plot of NPV against iterations. It is observed that the
optimized value of NPV continuously increases as a function of iterations.
The total increase in the mean NPV of the ensemble from the initial to the
optimized case is approximately 6.02%. The maximum number of
iterations used in the optimization process was 15. This limit was used to
minimize computational time.
Mean NPV of ensemble as a function of iteration number
138,000,000
139,000,000
140,000,000
141,000,000
142,000,000
143,000,000
144,000,000
145,000,000
146,000,000
147,000,000
148,000,000
0 2 4 6 8 10 12 14 16
Iteration Number
NP
V (
$)
Prior Mean NPV for initial BHP realizations
Optimized Mean NPV
6.02 % increase
Figure 6-32: Case 2 – Net Present Value vs. iterations
A summary of the results obtained from this study is presented in table 2.
70
Table 2: Summary of results
REFERENCE OPTIMIZED
Cum. Oil
Production
NPV Cum. Oil
Production
Increase
in Cum.
Oil Prodn
NPV Increase
in NPV
Case
x106 STB ($Mil) x 106 STB ($Mil)
1 5.44 134.2 6.0 10.3% 146 8.8%
2 5.58 136.4 6.03 5.06% 147.9 8.4%
3 5.49 137.2 6.14 11.8% 147 7.1%
71
7. CONCLUSIONS
A new production optimization algorithm has been presented in this
project. The methodology borrows its concept from the ensemble Kalman
filter for continuous model update and has been successfully applied to
various heterogeneous waterflood reservoir models. The optimization
process showed remarkable improvement in net present value of up to 9%
from the initial base case as well as an improvement of cumulative
production of up to 8% from the base case. Also, the water saturation at
breakthrough was observed to be more uniformly distributed across the
reservoir after the optimization process as compared with the unoptimized
case.
The advantage of this methodology over the adjoint-based method is that
it does not require explicit knowledge of the simulator flow equations
thereby making it computationally less tedious. A commercial simulator
can easily be applied to this optimization technique without tampering with
its source code. Another advantage of this methodology over Lorentzen et
al’s is that it does not require a pre-selected NPV, which he used to
72
optimize controls. Rather, it optimizes the controls to the maximum
possible NPV by maximizing an objective function.
As a recommendation for future work, the optimization methodology
presented in this study can be used to optimize other objective functions
like cumulative oil production. Also, other controls including total fluid
production rates can also be used as constraints. The procedure may also
be applied to other waterflood patterns. This approach has been applied to
a simple 2-D heterogeneous reservoir with known geology. Further work
can also be carried out on reservoir geology while considering
uncertainties in the reservoir model parameters and also on large scale
field examples.
73
NOMENCLATURE
Ct = Cash inflow after time step, t
Cx = Covariance matrix of control vector
Cy = Covariance matrix of state vector
fopt = Cumulative oil production
fwpt = Cumulative water production
G = Sensitivity matrix
H = Hessian matrix
M = Measurement operator
Ne = Number of ensemble members
r = discount rate
S = Objective function
T = Cumulative production period
X = Control vector
$/bbl = Price of Oil per bbl
$wat = Cost of water disposal
α = weighting factor
γ = Gradient of objective function
74
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80
A. APPENDIX – FORTRAN Flow Chart
Start
Declare variables
Run Eclipse
Open Eclipse output files
Read out oil and water production for each
BHP realization
Compute NPV
Is new NPV greater than old
NPV?
Yes
No
Stop
Read in Permeability values to reservoir grid file
Read in pressures/state vector to schedule file
Compute Cy.HT
Optimize alpha
Compute new state vector
Call Forward Run Subroutine
Forward Run
Output data for results and plotting
parameters files
81
B. APPENDIX – Results from Case 3
Figure B-1: Permeability distribution for case 3.
NPV for all realizations before and after optimization
132,000,000
134,000,000
136,000,000
138,000,000
140,000,000
142,000,000
144,000,000
146,000,000
148,000,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Realization Number
NP
V (
$)
Prior NPV Optimized NPV
Figure B-2: Case 3 – Graph showing NPV for all realizations of pressure
profiles before and after optimization.
P1 P2
P3 P4
INJ
82
Cumulative Oil production for initial and optimized realizations
5,000,000
5,200,000
5,400,000
5,600,000
5,800,000
6,000,000
6,200,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Realization number
Cu
mu
lati
ve
Oil
Pro
du
cti
on
(S
TB
)
Initial cumulative production
Optimized cumulative production
Figure B-3: Case 3 – Graph showing cumulative oil production for all
realizations of pressure profiles before and after optimization.
Bottom Hole pressure schedule for well P1 after optimization
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Bottom Hole pressure schedule for well P2 - Optimized
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P,
psi
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Bottom Hole pressure schedule for well P3 - Optimized
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Bottom Hole pressure schedule for well P4 - Optimized
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500 3000 3500
Time, days
BH
P, p
si
Realization 1
Realization 2
Realization 3
Realization 4
Realization 5
Realization 6
Realization 7
Realization 8
Realization 9
Realization 10
Figure B-4 (a), (b), (c), (d): Ten realizations of BHP for the optimized case
for all production wells
83
Water cut from producing wells for schedule realization 1 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r c
ut,
stb
/stb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 1 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wa
ter
cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 2 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wate
r c
ut,
stb
/stb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 2 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wa
ter
cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization 3 before optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wa
ter
cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Water cut from producing wells for schedule realization
3 after optimization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Wa
ter
cu
t, s
tb/s
tb
Water cut from well P1
Water cut from well P2
Water cut from well P3
Water cut from well P4
Figure B-5: Case 3 – Graphs showing water cuts from all wells before
(left) and after (right) optimization for 3 realizations of BHP profiles
84
Field cumulative production for schedule realization 1 before and after
optimization
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
9,000,000
10,000,000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Cu
mu
lati
ve p
rod
uc
tio
n, S
TB
Cum Oil prod - priorCum Oil Prod - OptimizedCum Water prod - priorCum Oil prod - Optimized
Field cumulative production for schedule realization 2 before and after
optimization
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
7,000,000
8,000,000
9,000,000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Cu
mu
lati
ve p
rod
ucti
on
, S
TB
Cum Oil prod - prior
Cum Oil Prod - Optimized
Cum Water prod - prior
Cum Water prod - Optimized
Figure B-6 (a) and (b): Case 3 – Graphs showing cumulative oil and water
production before and after optimization for realizations 1 and 2.
Field production rates for schedule realization 1 before and after optimization
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Pro
du
cti
on
ra
tes
, S
TB
/D
Oil prod rate - prior
Oil prod rate - Optimized
Water Prod rate - prior
Water prod rate - Optimized
Field production rates for schedule realization 2 before and after
optimization
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000 2500 3000 3500 4000
Time, days
Pro
du
cti
on
rate
s, S
TB
/D
Oil prod rate - prior
Oil prod rate - Optimized
Water Prod rate - prior
Water prod rate - Optimized
Figure B-7 (a) and (b): Case 3 – Graphs showing oil and water production
rates before and after optimization for realizations 1 and 2 of BHP profile.
85
Figure B-8: Case 3 – Water saturation distribution a) before (left) and b)
after (right) optimization after 913 days.
Mean NPV of ensemble as a function of iteration number
139,000,000
140,000,000
141,000,000
142,000,000
143,000,000
144,000,000
145,000,000
146,000,000
147,000,000
0 2 4 6 8 10 12 14 16
Iteration Number
NP
V (
$)
Prior Mean NPV for initial BHP realizations
Optimized Mean NPV
4.33 % increase
Figure B-9: Case 3 – Net Present Value vs. iterations
86
C. APPENDIX – DESCRIPTION OF FORTRAN CODE
The FORTRAN code developed in this study for the NPV optimization
process has been divided into 5 different subroutines. The various
sections include:
� Permeability values
� Forward Run
� Revenue Optimization
� Function of Alpha
� Parameters
The flow chart of the FORTRAN code is shown in appendix A. Each of the
sections of the code listed above is described in subsequent sections of
this chapter.
C.1 Permeability values
This section of the FORTRAN code is used to populate the grid blocks of
the reservoir model with the generated permeabilities. As described in
earlier chapters, the permeability values are initially generated using
GSLIB and exported to a data file from which this subroutine reads the
values into Eclipse grid data file. The Eclipse include file which contains
87
the grid properties was named RevOpt_gpro.INC (gpro is short for grid
properties). Figure 4-1 shows the top view of one of the permeability fields
used in this study.
C.2 Forward Run
This subroutine is used to run the reservoir simulator model for all 40
realizations of pressure profiles. It opens all the input and output files
(typically data files) as well as the schedule file. The input files are data
files containing the randomly generated bottom hole pressure realizations
(i.e. initial ensemble generated from mathematica) or the updated
pressures after each iteration process. The output files are the files
containing the following data: pressure, water cut, oil and water production
rates and cumulative production and net present values for the various
wells. These values are later used for plotting graphs.
The forward run routine updates the Eclipse schedule include file with the
pressures for the ensemble and runs the simulator model for each case.
The output data from each case is stored in the output files. This routine
also computes the net present value for each production well and the
cumulative net present values for the entire field. These values are
computed from the oil and water flow rates at the various time steps. The
88
rates and cumulative productions are read from Eclipse summary output
file with the RSM extension name.
C.3 Revenue Optimization
This is the main program in the code. In this section, the actual iteration
process is performed. If Y is used to represent the ensemble of the state
vector, then Y can be represented as follows:
=
)(
.
.
3
2
1
i
iNt
i
i
i
i
xg
x
x
x
x
Y C-1
where i = ith member of the ensemble
x = control variable, or BHP
)(xg = obective function, or NPV
Nt = Total number of controls (In this research, Nt = 80)
The mean of the ensemble of state vectorsY is calculated as follows:
89
=
∑
∑
∑
∑
∑
i
i
i
iNt
i
i
i
i
i
i
e
xg
x
x
x
x
NY
)(
.
.1
3
2
1
C-2
Ne represents the number of members in the ensemble of state vectors.
For the purpose of this study, Ne = 40.
Recall from chapter 3;
( )( )Te
Y YYYYN
C −−−
=1
1 C-3
SinceT
YlX MCGC = , then
( )( ) TT
e
T
YlX MYYYYN
MCGC −−−
==1
1 C-4
The product of the covariance matrix Cx and the sensitivity matrix lG can be
approximated using the above equation. The product ( lX GC ) gives a
column vector which is multiplied by a weighting factor (1/α) and then
added to the ensemble of the prior state vector used in the previous
iteration (see equation 3-15). The value of α is a variable that also has to
90
be optimized. The following section describes how this optimization
process is carried out.
C.3.1 Optimum value of alpha
A section of the revenue optimization subroutine also estimates an
optimum value of alpha to be used as a weighting factor. The alpha value
can be thought of as a way to control step length32.
In our problem, we wish to maximize the net present value from the
waterflood reservoir, while minimizing rapid changes in the controls. Alpha
is the independent variable that regularizes the control settings. The
solution to this problem is to find a value of alpha for which the net present
value is nearly maximized.
The procedure for finding the extremum of a NPV function is as follows:
1) Select a relatively large starting guess for alpha.
2) Solve equation 3-18 to obtain the updated estimate of optimal
control variables and then execute the simulator to compute the
NPV.
3) Reduce alpha and find the new NPV.
4) If the new NPV is greater than the old NPV, then repeat steps 2
and 3.
91
5) On the other hand, if the new NPV is less than the old NPV, then
the alpha with the highest NPV is the optimum alpha to be used for
the next iteration step.
An alpha value less than the optimum alpha obtained from the procedure
above is generally used for the next iteration step )1( +l . This is to reduce
the step size of the optimization process and thereby penalize the control
settings that change rapidly with time.
C.4 Function of Alpha
This subroutine is used to compute the net present value when given a
value of alpha. It first solves equation 5-18 to obtain the updated state
vector and exports the new state vectors to the pressure input files. Then
it calls the forward run subroutine from where NPV is obtained. A typical
graph of NPV as a function of alpha is shown in Figure C-1. The run time
for the optimization process can be reduced by truncating the optimization
process of alpha to fewer NPV values or by selecting a constant value of
alpha for each iteration step. This however, will result in reduce the
precision of the optimization process.
92
Figure C-1: Graph of NPV as a function of alpha for 6 iterations
C.5 Parameters
This section of the code is used to declare all the variables and
parameters that are used in the entire FORTRAN program. It also defines
the dimensions of the variables.
NPV as a function of alpha for all iterations
213,000,000
213,500,000
214,000,000
214,500,000
215,000,000
215,500,000
216,000,000
216,500,000
1.E+05 1.E+06 1.E+07
Alpha
NP
V (
$)
iteration 1
iteration 2
iteration 3
iteration 4
iteration 5
iteration 6
Prior Mean NPV for initial BHP realizations
Optimized Mean NPV
93
D. APPENDIX – FORTRAN Code
94
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