UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE DYNAMIC OPTIMIZATION OF A WATER FLOOD RESERVOIR A THESIS SUBMITTED TO THE GRADUATE FACULTY

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1 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

2 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 ii

3 Copyright by JUDE NWAOZO 2006 All Rights Reserved. iii

4 Dedication To my dad, mum and siblings iv

5 TABLE OF CONTENTS DEDICATION... IV LIST OF FIGURES...VIII LIST OF TABLES...XV ABSTRACT... XVI 1. INTRODUCTION SMART WELLS PRODUCTION OPTIMIZATION LITERATURE REVIEW OIL AND GAS PRODUCTION OPTIMIZATION HISTORY EOR PROCESS OPTIMIZATION OPTIMIZATION OF WELL PLACEMENT AND TYPE PRODUCTION OPTIMIZATION CONSIDERING UNCERTAINTY THIS OPTIMIZATION APPROACH METHODOLOGY ENSEMBLE KALMAN FILTER (ENKF) OPTIMIZATION PROCEDURE RESERVOIR MODEL DESCRIPTION GRID...30 v

6 4.2 SCHEDULE PVT PROPERTIES OF THE RESERVOIR FLUIDS ECONOMICS NET PRESENT VALUE (NPV) RESULTS AND ANALYSIS CASE CASE CONCLUSIONS...71 NOMENCLATURE...73 REFERENCES...74 A. APPENDIX FORTRAN FLOW CHART...80 B. APPENDIX RESULTS FROM CASE 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 vi

7 D. APPENDIX FORTRAN CODE...93 vii

8 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) Figure 2-2: Schematic of reservoir used by Lorentzen et al Figure 2-3 (a) and (b): Development of optimized value Figure 4-1: Top view of the reservoir model showing permeability field distribution and well placements Figure 4-2: Histogram showing permeability field distribution Figure 4-3: Initial distribution of mean values of bottom hole pressure profile for well P Figure 4-4: Pressure profile generated using correlation range a = Figure 4-5: Histogram for pressure realizations for well P Figure 4-6: Histogram for pressure realizations for well P viii

9 Figure 4-7: Histogram for pressure realizations for well P Figure 4-8: Histogram for pressure realizations for well P Figure 4-9: Ten realizations of BHP profile for production well P Figure 4-10: Ten realizations of BHP profile for well P Figure 4-11: Ten realizations of BHP profile of well P Figure 4-12: Ten realizations of BHP profile of well P Figure 4-13: Relative Permeability Curves Figure 6-1: Permeability field distribution for case Figure 6-2: Case 1 Graph showing NPV for all realizations of pressure profiles before and after optimization Figure 6-3: Case 1 Graph showing cumulative oil production for all realizations of pressure profiles before and after optimization Figure 6-4: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P Figure 6-5: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P ix

10 Figure 6-6: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P Figure 6-7: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P Figure 6-8: Case 1 Water saturation distribution a) before (top) and b) after (bottom) optimization after 913 days 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 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 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 Figure 6-12: Graph showing cumulative oil and water production before and after optimization for realization 1 of BHP profile x

11 Figure 6-13: Graph showing cumulative oil and water production before and after optimization for realization 2 of BHP profile Figure 6-14: Graph showing oil and water production rates before and after optimization for realization 1 of BHP profile Figure 6-15: Graph showing oil and water production rates before and after optimization for realization 2 of BHP profile Figure 6-16: Case 1 - Net Present Value vs. iterations Figure 6-17: Permeability field distribution for case Figure 6-18: Case 2 Graph showing NPV for all realizations of pressure profiles before and after optimization Figure 6-19: Case 2 Graph showing cumulative oil production for all realizations of pressure profiles before and after optimization Figure 6-20: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P Figure 6-21: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P xi

12 Figure 6-22: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P Figure 6-23: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P Figure 6-24: Case 2 Water saturation distribution a) before (top) and b) after (bottom) optimization after 913 days 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 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 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 Figure 6-28: Case 2 Graph showing cumulative oil and water production before and after optimization for realization 1 of BHP profile xii

13 Figure 6-29: Case 2 Graph showing cumulative oil and water production before and after optimization for realization 2 of BHP profile Figure 6-30: Case 2 Graph showing oil and water production rates before and after optimization for realization 1 of BHP profile Figure 6-31: Case 2 Graph showing oil and water production rates before and after optimization for realization 2 of BHP profile Figure 6-32: Case 2 Net Present Value vs. iterations Figure B-1: Permeability distribution for case Figure B-2: Case 3 Graph showing NPV for all realizations of pressure profiles before and after optimization Figure B-3: Case 3 Graph showing cumulative oil production for all realizations of pressure profiles before and after optimization Figure B-4 (a), (b), (c), (d): Ten realizations of BHP for the optimized case for all production wells 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 xiii

14 Figure B-6 (a) and (b): Case 3 Graphs showing cumulative oil and water production before and after optimization for realizations 1 and 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 Figure B-8: Case 3 Water saturation distribution a) before (left) and b) after (right) optimization after 913 days Figure B-9: Case 3 Net Present Value vs. iterations Figure C-1: Graph of NPV as a function of alpha for 6 iterations xiv

15 LIST OF TABLES Table 1: Summary of reservoir properties Table 2: Summary of results xv

16 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 xvi

17 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. xvii

18 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. xviii

19 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 depletion 1. 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 CO 2 flooding and hydrocarbon gas injection, which requires a nearby source of inexpensive gas in sufficient volume. 1

20 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 Canada 3 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 gas 4. 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 recoveries 5. 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 2

21 been suggested. One method of mitigating this problem is by employing smart production and injection wells 6, 7, 8, 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 wells 7. Smart wells are development wells that contain permanent downhole measurement and control equipments that enable significant improvement of oil production 8 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, etc 9. 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 packers 10. ICVs divide the injector into different 3

22 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 zone 8. Smart well technology is presently undergoing the process of value identification and quantification in the exploration and production industry 9. The advantages of this new technology to the industry include the following: 4

23 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 5

24 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 6

25 (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 literature 4 : gradient-based algorithms 11,12,13 and stochastic algorithms. Gradientbased algorithms require an efficient technique of calculating the gradient 7

26 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 algorithms 14,15 and simulated annealing 16,17 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. 8

27 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 distribution 18. 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 Aronofsky 19. 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 9

28 this problem using constant well interference coefficients as a substitute for a real reservoir simulation model. Wattenbarger 10, 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. Wattenbarger 10 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 finitedifference 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. 10

29 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. Asheim 11,20 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 Jansen 7 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 11

30 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 study 8, 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 12

31 ultimate oil recovery from a successive step is lower than that obtained with the preceding flow profile. Brouwer and Jansen 7 investigated the optimization problem under two different scenarios of well operating conditions purely pressureconstrained 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. 13

32 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. 14

33 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 15

34 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 Aziz EOR Process Optimization In 1984, Ramirez and Fathi 12 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 problem 13. The control for the process was the surfactant concentration of the injected fluid. They observed a 16

35 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 Ramirez 22 in 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. 17

36 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. 18

37 2.4 Production Optimization considering uncertainty Naevdal, Brouwer and Jansen 26 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 Aziz 27 in In their approach, a gradient-based optimization algorithm was used to determine optimal control settings, while the parameter gradients are used for model 19

38 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 Virnovsky 18. 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 20

39 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. 21

40 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 time 28. 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 simulation 26, 28. It is a Monte-Carlo method in which an ensemble of initial reservoir state 22

41 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 ensemble 28. 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 [ m ] T d T T y = 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 23

42 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 = {x 1, x 2, x 3 x 80 } 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: 24

43 α S( x) = g( x) 2 T 1 ( x x ) C ( x x ) p X p 3-2 where S(x) = objective function g(x) = Net Present Value α = weighting factor x = new state vector x p = prior state vector C x = covariance matrix of the control vector A local quadratic approximation to S(x) at x = x is given as T 1 T F ( x' + δx) = S ( x' ) + γ δx + δx Hδx whereγ is equal to S(x) T, and the Hessian, H is ( S( x)). The value ofδx that maximizes the quadratic approximation to the objective function is the extremum of this function and it occurs at F = 0or γ + Hδx = The Newton equations for iteratively finding the extremum are H δx = γ 3-5 or H x = S l +1 lδ l 3-6 After computing δ l + 1 x from equation 3-6 above, the controls are updated using the equation below. 25

44 l +1 l l +1 x = x + δ x 3-7 The gradient of the objective function S(x) for the lth iteration is S = G l ( x x ) l 1 ( x ) α C X p 3-8 (assuming thatc is symmetric). In the above equation, G( x l ) denotes the x 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, g i (x) is affected by a change in the controls, x. The individual elements of the sensitivity matrix are given by, G g = i i, j 3-9 x j The approximate Hessian matrix is given by: l H G( x ) α C x Assuming that the second derivative of g(x) is negligible, equation 3-10 above becomes H α C x Therefore, substituting equation 3-8 and 3-11 in equation 3-6 gives l [ G αc ( x x )] α C δx = l X l x After further manipulation, equation 3-12 becomes p 26

45 δx l+ 1 1 = C α x G l l ( x x ) p 3-13 Substituting the incremental controlsδ equation 3-7; l+1 x obtained in equation 3-13 into 1 α l+ 1 l l x = x + CxGl x + x p 3-14 Equation 3-14 reduces to 1 α l+ 1 x = C xgl + x p 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 [ y y, y y Ne ] Y =,..., , 2 3 N e 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; C Y = 1 N 1 e ( Y Y )( Y Y ) T

46 wherey = Mean of state vector calculated across the ensemble. T SinceC G = C M, then equation 3-15 becomes X l Y 1 α l+ 1 T x = CY M + x p 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: M = [ 0 I ] 3-19 Where 0 is an N d x (N y N d ) matrix with all 0s as entries and I is an N d x N d identity matrix. Note that N d is the number of measurements (N d = 1 for this project) and N y is the number of variables in the state vector, y (N y = 81 for this project). 28

47 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 29

48 same time. The depth of the top surface of the reservoir is 10,000 ft with a net pay thickness of 50 ft. P1 P2 INJ P3 P4 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 simulator 29. The 30

49 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. Histogram showing permeability distribution in the reservoir N umber o f g ri d block s HmdL Permeability Figure 4-2: Histogram showing permeability field distribution 31

50 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 32

51 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. 10 Histogram showing Mean BHPs for Well P1 N umber o f realization s Mean BHP Figure 4-3: Initial distribution of mean values of bottom hole pressure profile for well P1. 33

52 Step 2: A gaussian covariance function 30 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. 3( h i h j ) 2 C( hi, j ) = σ exp 4-1 a where C(h i,j ) = Covariance function σ = Standard deviation h i, h j = 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. 34

53 L Hi p s B ottom H ol e P ressur e realizations of BHP profile for Well P1 using a = Timestep Figure 4-4: Pressure profile generated using correlation range a = 5. Histogram of BHP realizations for Well P1 60 Number of realization s Bottom Hole Pressure 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 35

54 pressure profiles of production wells P1 P4 used in the initial simulation run is shown in Figure Figure Histogram of BHP realizations for Well P2 Number of realization s Bottom Hole Pressure Figure 4-6: Histogram for pressure realizations for well P2 Histogram of BHP realizations for Well P3 60 Number o f realization s Bottom Hole Pressure Figure 4-7: Histogram for pressure realizations for well P3 36

55 Histogram of BHP realizations for Well P4 60 N umber o f realization s Bottom Hole Pressure 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. 37

56 Bottom Hole pressure schedule for well P1 - Prior BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 4-9: Ten realizations of BHP profile for production well P1. Bottom Hole pressure schedule for well P2 - Prior BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 4-10: Ten realizations of BHP profile for well P2. 38

57 Bottom Hole pressure schedule for well P3 - Prior BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 4-11: Ten realizations of BHP profile of well P3. Bottom Hole pressure schedule for well P4 - Prior BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 4-12: Ten realizations of BHP profile of well P 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 39

58 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 (B o ) is 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 (B w ) of 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 Relative Permeability Krw Kro Water Saturation, Sw Figure 4-13: Relative Permeability Curves 40

59 Table 1: Summary of reservoir properties Number of grid blocks 25 x 25 x 1 Grid block size Water injection rate Reservoir thickness 118ft x 118ft x 50ft 5000 STB/D 50 ft Porosity 25% Actual reservoir area 200 acres Initial Oil Saturation 0.8 Initial Water Saturation 0.2 Well Depth Initial reservoir pressure 10,000 ft 4500 psi Ave. Reservoir Temperature 284 F Production period Time step 10 years 6 months 41

60 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 formula 31 : 42

61 NPV = C T t t t= 1 (1 + r) C where t = Time step C t = Cash inflow after time t, $ r = Annual (or periodic) discount rate, fraction T = Cumulative investment (or production) period C 0 = 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, 43

62 ( fopt $ / bbl) ( fwpt wat) C = $ 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; wwpr $ / bbl 40 wct = = = 93% 5-3 wwpr + wopr $ / bbl + $ wat 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. 44

63 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. P1 P2 INJ P3 P4 Figure 6-1: Permeability field distribution for case 1. 45

64 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 148,000, ,000, ,000, ,000,000 NPV ($) 140,000, ,000, ,000, ,000, ,000, ,000,000 Prior NPV Optimized NPV 128,000, Realization Number 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). 46

65 Cumulative Oil production for initial and optimized realizations 6,100,000 6,000,000 Cumulative Oil Production (STB) 5,900,000 5,800,000 5,700,000 5,600,000 5,500,000 5,400,000 5,300,000 5,200,000 Initial cumulative production Optimized cumulative production 5,100, Realization number 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 BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-4: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P1. 47

66 Bottom Hole pressure schedule for well P2 - Optimized 6000 BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-5: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P2. Bottom Hole pressure schedule for well P3 - Optimized 6000 BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-6: Case 1 Graph showing 10 realizations of optimized pressure profiles for well P3. 48

67 Bottom Hole pressure schedule for well P4 - Optimized BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days 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. 49

68 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 50

69 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 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days Water cut from producing wells for schedule realization 1 after optimization Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 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

70 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. 1 Water cut from producing wells for schedule realization 2 before optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 1 Water cut from producing wells for schedule realization 2 after optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 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

71 The above trend can also be observed in other realizations of BHP profile (See Figure 6-10 and Figure 6-11). 1 Water cut from producing wells for schedule realization 3 before optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 1 Water cut from producing wells for schedule realization 3 after optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P Time, days 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

72 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 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 9,000,000 Cumulative production, STB 8,000,000 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 Cum Oil prod - prior Cum Oil Prod - Optimized Cum Water prod - prior Cum Oil prod - Optimized Time, days Figure 6-12: Graph showing cumulative oil and water production before and after optimization for realization 1 of BHP profile. 54

73 Field cumulative production for schedule realization 2 before and after optimization 8,000,000 Cumulative production, STB 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 Cum Oil prod - prior Cum Oil Prod - Optimized Cum Water prod - prior Cum Water prod - Optimized Time, days Figure 6-13: Graph showing cumulative oil and water production before and after optimization for realization 2 of BHP profile. Production rates, STB/D Field production rates for schedule realization 1 before and after optimization Oil prod rate - prior Oil prod rate - Optimized Water Prod rate - prior Water prod rate - Optimized Time, days Figure 6-14: Graph showing oil and water production rates before and after optimization for realization 1 of BHP profile. 55

74 Field production rates for schedule realization 2 before and after optimization Oil prod rate - prior Oil prod rate - Optimized Water Prod rate - prior Water prod rate - Optimized Production rates, STB/D Time, days 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 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. 56

75 Mean NPV of ensemble as a function of iteration number 147,000, ,000,000 Optimized Mean NPV 145,000, ,000,000 NPV ($) 143,000, ,000, ,000, % increase 140,000, ,000, ,000,000 Prior Mean NPV for initial BHP realizations 137,000, Iteration Number 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. 57

76 P1 P2 INJ P3 P4 Figure 6-17: Permeability field distribution for case 2. NPV for all realizations before and after optimization 150,000, ,000, ,000, ,000,000 NPV ($) 142,000, ,000, ,000, ,000, ,000, ,000,000 Prior NPV Optimized NPV 130,000, Realization Number Figure 6-18: Case 2 Graph showing NPV for all realizations of pressure profiles before and after optimization. 58

77 Cumulative Oil production for initial and optimized realizations 6,100,000 Cumulative Oil Production (STB) 6,000,000 5,900,000 5,800,000 5,700,000 5,600,000 5,500,000 5,400,000 Initial cumulative production Optimized cumulative production 5,300, Realization number 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. 59

78 Bottom Hole pressure schedule for well P1 after optimization BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-20: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P1. Bottom Hole pressure schedule for well P2 - Optimized BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-21: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P2. 60

79 Bottom Hole pressure schedule for well P3 - Optimized 6000 BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-22: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P3. Bottom Hole pressure schedule for well P4 - Optimized BHP, psi Realization 1 Realization 2 Realization 3 Realization 4 Realization 5 Realization 6 Realization 7 Realization 8 Realization 9 Realization Time, days Figure 6-23: Case 2 Graph showing 10 realizations of optimized pressure profiles for well P4. 61

80 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. 62

81 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 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 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. 63

82 Water cut from producing wells for schedule realization 1 before optimization Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days Water cut from producing wells for schedule realization 1 after optimization Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 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

83 1 Water cut from producing wells for schedule realization 2 before optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 1 Water cut from producing wells for schedule realization 2 after optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 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

84 Water cut from producing wells for schedule realization 3 before optimization Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 1 Water cut from producing wells for schedule realization 3 after optimization 0.8 Water cut, stb/stb Water cut from well P1 Water cut from well P2 Water cut from well P3 Water cut from well P Time, days 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

85 Field cumulative production for schedule realization 1 before and after optimization 9,000,000 Cumulative production, STB 8,000,000 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 Cum Oil prod - prior Cum Oil Prod - Optimized Cum Water prod - prior Cum Oil prod - Optimized Time, days 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 8,000,000 Cumulative production, STB 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 Cum Oil prod - prior Cum Oil Prod - Optimized Cum Water prod - prior Cum Water prod - Optimized Time, days Figure 6-29: Case 2 Graph showing cumulative oil and water production before and after optimization for realization 2 of BHP profile. 67

86 Production rates, STB/D Field production rates for schedule realization 1 before and after optimization Oil prod rate - prior Oil prod rate - Optimized Water Prod rate - prior Water prod rate - Optimized Time, days 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 Oil prod rate - prior Oil prod rate - Optimized Water Prod rate - prior Water prod rate - Optimized Production rates, STB/D Time, days Figure 6-31: Case 2 Graph showing oil and water production rates before and after optimization for realization 2 of BHP profile. 68

87 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 148,000, ,000,000 Optimized Mean NPV 146,000, ,000,000 NPV ($) 144,000, ,000, ,000, % increase 141,000, ,000, ,000,000 Prior Mean NPV for initial BHP realizations 138,000, Iteration Number Figure 6-32: Case 2 Net Present Value vs. iterations A summary of the results obtained from this study is presented in table 2. 69

88 Table 2: Summary of results Case REFERENCE OPTIMIZED Cum. Oil Production NPV Cum. Oil Production Increase in Cum. Oil Prodn NPV Increase in NPV x10 6 STB ($Mil) x 10 6 STB ($Mil) % % % % % % 70

89 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 71

90 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. 72

91 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 73

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98 A. APPENDIX FORTRAN Flow Chart Start Compute Cy.H T Declare variables Optimize alpha Read in Permeability values to reservoir grid file Forward Run Compute new state vector Call Forward Run Subroutine Read in pressures/state vector to schedule file Run Eclipse Open Eclipse output files Read out oil and water production for each BHP realization Is new NPV greater than old NPV? Stop No Output data for results and plotting parameters files Yes Compute NPV 80

99 B. APPENDIX Results from Case 3 P1 P2 INJ P3 P4 Figure B-1: Permeability distribution for case 3. NPV for all realizations before and after optimization 148,000, ,000, ,000, ,000,000 NPV ($) 140,000, ,000, ,000, ,000,000 Prior NPV Optimized NPV 132,000, Realization Number Figure B-2: Case 3 Graph showing NPV for all realizations of pressure profiles before and after optimization. 81