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Ensemble-Based Optimization of the Water-Alternating-Gas-Injection Process

ARTICLE in SPE JOURNAL OCTOBER 2015

Impact Factor: 1.31 DOI: 10.2118/173217-PA

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2 AUTHORS:

Bailian Chen

University of Tulsa

5PUBLICATIONS 20CITATIONS

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Albert Reynolds

University of Tulsa

167PUBLICATIONS 2,468CITATIONS

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Available from: Bailian ChenRetrieved on: 11 January 2016

https://www.researchgate.net/?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_1https://www.researchgate.net/profile/Albert_Reynolds?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Tulsa?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_6https://www.researchgate.net/profile/Albert_Reynolds?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_5https://www.researchgate.net/profile/Albert_Reynolds?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_4https://www.researchgate.net/profile/Bailian_Chen?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_7https://www.researchgate.net/institution/University_of_Tulsa?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_6https://www.researchgate.net/profile/Bailian_Chen?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_5https://www.researchgate.net/profile/Bailian_Chen?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_4https://www.researchgate.net/?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_1https://www.researchgate.net/publication/282649718_Ensemble-Based_Optimization_of_the_Water-Alternating-Gas-Injection_Process?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_3https://www.researchgate.net/publication/282649718_Ensemble-Based_Optimization_of_the_Water-Alternating-Gas-Injection_Process?enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng%3D%3D&el=1_x_2

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Ensemble-Based Optimization of theWater-Alternating-Gas-Injection Process

Bailian ChenandAlbert C. Reynolds,University of Tulsa

Summary

CO2-water-alternating-gas (CO2-WAG) flooding generally leads tohigher recovery than either continuous CO2flooding or waterflood-ing. Although CO2 injection increases microscopic displacementefficiency, unless complete miscibility is achieved, suboptimalsweep efficiency may be obtained because of gravity segregationand the channeling of CO2 through high-permeability zones or byviscous fingering. Alternating water injection with CO2 injectionresults in better mobility control and increases sweep efficiency.Water injection also increases pressure that promotes miscibility.However, poorly designed WAG parameters can result in subopti-mal WAG performance. In this work, given the number of WAGcycles and the duration of each WAG cycle, we apply a modifica-tion of a standard ensemble-based optimization technique to esti-mate the optimal well controls that maximize life-cycle net present

value (NPV). By optimizing the well controls, we implicitly opti-mize the WAG ratio (volume of water injected divided by the vol-ume of gas injected). We apply the optimization methodology to asynthetic, channelized reservoir. The performances of optimizedWAG flooding, optimized waterflooding, and optimized continuousCO2 flooding are compared. Because of the similarity betweenWAG and surfactant alternating gas (SAG foam), we also optimizethe SAG process and provide a more computationally efficient wayto optimize the SAG process with the optimal well controlsobtained from WAG as the initial guesses for the optimal controlsfor SAG.

Introduction

WAG injection is an enhanced-oil-recovery method designed to

improve sweep efficiency during gas injection with the injectedwater to control the mobility of gas and to stabilize the gas front(Christensenet al. 2001). The focus of this paper is CO2-WAGinjection, which is a cyclic method of injecting alternating cyclesof CO2followed by water and then repeating this process during anumber of cycles. Field applications of WAG were successful fora variety of rock types, but were mainly applied to high-perme-ability reservoirs (Christensen et al. 2001) although there existsimulation studies that suggest WAG can yield on the order of 5%greater oil recovery than is obtained with continuous gas injectioneven for low-permeability highly heterogeneous reservoirs (Liaoet al. 2013).

Poor recovery from the WAG technique can be caused byinappropriately designed WAG parameters (e.g., the WAG ratio,

the number of cycles, and gas- or water-injection rate). Thus, theoptimization of WAG injection is widely recognized as a viabletechnique for controlling mobility of gas and the miscible processto achieve a better recovery (Chen et al. 2010a; Bahagio 2013).Life-cycle production optimization is an optimal well-controlproblem in which the objective is to determine well-operatingconditions that maximize NPV or cumulative oil production dur-ing the presumed reservoir life. The previous studies on produc-tion optimization mainly focus on optimizing the reservoirperformance under waterflooding (Brouwer and Jansen 2004;Sarma et al. 2005; Alhuthali et al. 2007; Kraaijevanger et al.

2007; Jansen et al. 2009; Wang et al. 2009; Chen et al. 2010,

2012; Oliveira and Reynolds 2014); there are few studies onWAG optimization. The existing studies on WAG optimizationare based, in most cases, on an experimental approach (Aminet al. 2012; Srivastava and Mahli 2012; Shafian et al. 2013) ortrial-and-error reservoir-simulation work (Pritchard and Nieman1992;Johns et al. 2003; Mirkalaei et al. 2011; Zhou et al. 2012;Ghaderiand Clarkson 2012; Bender and Yilmaz 2014). However,the well-control parameters determined by an experimentalapproach or reservoir-simulation experiments may not be close toan optimal solution. Thus, it is important to adaptively estimatethe optimal WAG ratio, well-injection rates, and optimal wellcontrols for producers in an automatic way to achieve a better re-covery or NPV.

Esmaiel et al. (2005) and Esmaiel (2007) applied experimental

design and a related response-surface model to optimize theWAG-injection process under geological uncertainty. The opti-mized recovery was more than double the base recovery. How-ever, the optimal WAG parameters determined remain constantthroughout the simulation. Chen et al. (2010) presented a hybridtechnique, which integrates the orthogonal array and Tabu tech-nique into the genetic algorithm, to determine the optimum WAGproduction/injection parameters. However, the possible values ofthe control variables are preselected and discrete, and the authorsdid not provide information about how they chose the values ofcontrol variables for each injector and producer or the WAG ratio.It is difficult to know whether the WAG production/injection pa-rameters obtained with their hybrid technique are optimal.

Bahagio (2013) used an ensemble optimization method to opti-mize the injection process of CO2-WAG in which the length of

each injection cycle is fixed to 1 year (i.e., each gas-injection pe-riod lasts 6 months, and each water-injection period lasts 6months). Moreover, the WAG ratio for the case the author studiedwas fixed to a constant value that is not feasible in practice becauseof the reservoir heterogeneity, miscibility conditions, injectionconditions, and well operational parameters. In addition, theauthor did not estimate the optimal well controls for productionwells, only for injection wells. Although the NPV obtained by theoptimized strategy was higher than that obtained by the base-casestrategy in the study of Bahagio (2013), we believe that one canachieve better results if more control variables (e.g., well controlsfor production wells) are taken into consideration. In an appendixto Bahagio (2013), the author considered optimizing bottomholepressures (BHPs) at injectors together with the switching times fora simple five-spot example with four injectors and one producer. Aswitching time refers to a time at which we change from CO 2injection to water injection or vice versa. Although the addition ofswitching times to the set of design variables seems to be appeal-ing, we conjecture that, for the procedure we use here, the incorpo-ration of switching times into the list of optimization variableswould not lead to a significant increase in the optimal NPV. Theissue of switching times will be revisited after we present compu-tational results. We also note that for polymer flooding, Ranioloet al. (2013) considered optimizing the polymer concentration andtime length of injection of three polymer slugs. Only a single injec-tion well was considered and the BHP at the injection well wasspecified and was not an optimization variable. Thus, it is unclearwhether one could have optimized a higher NPV with both BHPsand injection times as optimization variables.

Here, we apply an ensemble-based optimization (EnOpt) tech-nique, which was successfully used in waterflooding optimization

CopyrightVC 2015 Society of Petroleum Engineers

This paper (SPE 173217) was accepted for presentation at the SPE Reservoir SimulationSymposium, Houston, 2325 February 2015, and revised for publication. Original manuscriptreceived for review 25 January 2015. Revised manuscript received for review 25 May 2015.Paper peer approved19 August 2015.

2015 SPE Journal 1

https://www.researchgate.net/publication/254507005_Review_of_WAG_Field_Experience?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/254507005_Review_of_WAG_Field_Experience?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/241791531_Optimal_Parametric_Design_for_Water_Alternating-Gas_WAG_Process_in_a_CO_Miscible_Flooding_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/266662114_Full-Field_Simulation_and_Optimization_Study_of_Mature_IWAG_Injection_in_a_Heavy_Oil_Carbonate_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/266662114_Full-Field_Simulation_and_Optimization_Study_of_Mature_IWAG_Injection_in_a_Heavy_Oil_Carbonate_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/266662114_Full-Field_Simulation_and_Optimization_Study_of_Mature_IWAG_Injection_in_a_Heavy_Oil_Carbonate_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/245418382_Production_Optimization_in_Closed-Loop_Reservoir_Management?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/254507005_Review_of_WAG_Field_Experience?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/254507005_Review_of_WAG_Field_Experience?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/241791531_Optimal_Parametric_Design_for_Water_Alternating-Gas_WAG_Process_in_a_CO_Miscible_Flooding_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/266662114_Full-Field_Simulation_and_Optimization_Study_of_Mature_IWAG_Injection_in_a_Heavy_Oil_Carbonate_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/266662114_Full-Field_Simulation_and_Optimization_Study_of_Mature_IWAG_Injection_in_a_Heavy_Oil_Carbonate_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==https://www.researchgate.net/publication/266662114_Full-Field_Simulation_and_Optimization_Study_of_Mature_IWAG_Injection_in_a_Heavy_Oil_Carbonate_Reservoir?el=1_x_8&enrichId=rgreq-4f34cc57-318e-45f2-a0a0-4ca529944285&enrichSource=Y292ZXJQYWdlOzI4MjY0OTcxODtBUzoyODE5OTc4OTg0NjkzNzhAMTQ0NDI0NDk0NjA4Ng==

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(Lorentzen et al. 2006; Chen et al. 2009; Stordal et al. 2014; Fon-seca et al. 2015a, 2015b), to adaptively determine the optimalwell controls (for both injectors and producers) for each cycleduring the presumed reservoir lifetime, without preselecting anycontrol variables. The EnOpt formula applied is the one suggestedbyDo and Reynolds (2013) which invokes fewer assumptions inthe underlying mathematical development than does the standardChenet al. (2009) formulation. By optimizing the well controls,we implicitly optimize the WAG ratio. Because of the similaritybetween WAG and SAG foam, we also can apply this optimiza-tion technique to the SAG process. However, the computationaltime required to estimate the optimal well controls for SAG injec-tion is computationally expensive if we start the optimizationdirectly from an initial guess that is far from optimal. The compu-tational time required to run a SAG simulation is much longerthan the time required to run a WAG simulation. The longer com-putational time is a result of the incorporation of the chemicalreactions necessary to model the generation of foam as well asfoam collapse. We illustrate that it is possible to radically reducethe computational costs of SAG optimization by setting the initialguess for the optimal well controls for SAG equal to the optimalwell controls generated for the WAG process.

In this work, we first present the WAG optimization problem,which is followed by the description of the optimization method-ology used in this paper. Then, we present the results of CO 2-WAG optimization on synthetic channelized reservoir examples

and make a comparison between WAG, continuous CO2, andwaterflooding techniques. Thereafter, we provide a more-efficientmethod to optimize the SAG process. Finally, comments and con-clusions are provided.

Water-Alternating-Gas-Injection

Optimization Problem

In this work, the optimization problem we consider is how to esti-mate the well controls that maximize the NPV in a WAGenhanced-oil-recovery process. NPV is defined by

J u XNt

n1

Dtn

1 b tn

365

hXNPj1

ro qno;j cw qnw;j

8 J xk

, xk1 is accepted as the new approximation

to anx that maximizes NPV. If not, then the stepsize ak is repeat-edly reduced by a factor of 2, and xk1 is recomputed with thereduced stepsize until Eq. 5 produces an xk1 such thatJ xk1

>

J xk

or the maximum number of stepsize cuts has been reached.If, with the maximum number of stepsize cuts, we cannot find an

xk1 that increases the NPV, then we generate a new set of per-

turbed controls to try to find an uphill search direction and, if nec-essary, repeat the stepsize cuts. If after NRes consecutiveresamplings of the perturbations to obtain a search direction, westill cannot find an xk1 that increases J x after the maximumnumber of allowable stepsize cuts, then the algorithm is termi-nated. Also, if both the following two equations hold,

J xk1

J xk

maxfJ xk ; 1:0g eJ 12

and kxk1 xkk2maxfkxkk2; 1:0g

ex; 13

the algorithm is terminated. Following the discussion of Subsec-tion 8.5.1 of Oliver et al. (2008), we use eJ 10

4 andex 103.

If total number of allowable simulation runs, NR,is exceeded, weterminate the algorithm. Typically, we choose NR between 1,000

and 5,000, depending on the computational resources available.For the examples considered here and similar other computationsnot shown, increasing NR from 1,000 to 2,000 results in only avery small increase in the NPV.

Adaptation of Modified EnOpt to WAG. To simulate WAGwith a standard reservoir simulator, we represent each actual injec-tion well by two injection wells, a gas-injection well and a water-injection well at the same location. The use of two wells is neces-sary because CMGs GEM, which is the simulator used to modelWAG, does not allow us to switch a well back and forth from a

water injector to a CO2 injector. Each WAG-injection cycle con-tains two half-cycles, a first half-cycle and a second half-cycle. Spe-cifically, gas is injected during the first half-cycle and water isinjected during the second half-cycle. However, the water-injectionwell is turned off (rate fixed at zero) during the first half-cycle ofeach cycle, and the gas-injection well is turned off during the sec-ond half-cycle of each cycle. When an injection rate is fixed at zeroduring a half-cycle, we do not want to include the rate during thisperiod as an optimization variable; thus, we fix both the rate and thecorresponding search direction equal to zero. As discussed next, theuse of two injection wells at the same location where each wellalternates from open to closed at the end of each half-cycle requiresonly minor complications to our in-house optimization code.

Here,nwell represents the number of wells where each physicalinjection well is modeled as two wells in the reservoir simulator, a

water-injection well and a gas-injection well. Each of these twowells is represented by its own subvector in the form of Eq. 7, whichalsocan be expressed in terms of the number of WAG cycles as

dmk dmk

T1;a

; dmk T

1;b; dmk T

2;a; dmk T

2;b; ;

dmk T

ncycle;a; dmk T

ncycle;bT; 14

wherencycle denotes the total number of WAG cycles and a and brepresent the first half-cycle and the second half-cycle of eachinjection cycle, respectively. Note that each half-cycle may con-tain more than one control step. Here, for simplicity, we assumeall the half-cycles contain the same number of control steps, ncs,which means each entry ofdmk, given in the form of Eq. 14, hasncssubentries; for example, d

mk

1;a

can be expressed as

dmk

1;a dmk

11;a

; dmk 2

1;a;; dmk

ncs1;a

T: 15

We see that nc,the total number of control steps for each well,is equal to 2ncsncycle. To account for the fact that, for all the firsthalf-cycles, the water-injection rates and their corresponding searchdirections are zero, and for all the second half-cycles, the gas-injec-tion rates and their corresponding search directions are zero, wemake the modifications discussed next in Steps 1, 2, and 3.

Step 1. For gas injectors, we set the even entries ofdmk, that is,dmk

1;b, dmk

2;b,, dmk

ncycle;b

in Eq. 14, equal to zero to eliminatethe entries that do not correspond to the design (optimization) var-iables. Similarly, for water injectors, we have to set the oddentries of dmk given in the form of Eq. 14 equal to zero, that is,

dm

k

1;a dm

k

2;a dm

k

ncycle;a 0.Step 2. Because we smooth the dkof Eq. 9 by multiplying bythe covariance matrix CX which is a block diagonal matrix toobtain the search direction, the search direction we use in ouralgorithm is first computed as

gk CXdk

C1X 0 0

0 C2X 0

.

.

....

..

....

0 0 Cnwell

X

266664

377775

d1k

d2k

.

.

.

dmk

.

.

.

dnwellk

2666666666664

3777777777775

C1Xd1k

C2Xd2k

.

.

.

CmXdmk

.

.

.

CnwellX dnwellk

2666666666664

3777777777775

;

16

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . .

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where CmX, m 1; 2; ; nwell is the covariance matrix used toforce smoothness on the well controls for well m. In this work, weuse a spherical covariance function; that is, denoting the i;j entry ofCmXbyC

mi;j, we have

Cmi;j r2m 1

3

2

ji jj

Nms

1

2

ji jj

Nms

3" #; 17

ifji jj Nms , and Cmi;j 0 if ji jj> N

ms . Here, i and jdenote

the control step i and j, respectively; rm refers to the standarddeviation; and Nms is the number of control steps over which we

wish the control at well m to be correlated. As discussed later, thechoice ofNms at each refinement level of Hi-MO algorithm (Oli-veira and Reynolds 2014) controls the degree of smoothingimposed on the well controls. Also, see Wang et al. (2009) andYan and Reynolds (2014) for the effect that the choice ofNms hason the temporal smoothness of the estimated optimal controls.Then the mth component of gk, that is, g

mk C

mXd

mk can be

expressed as

gmk Cm

Xdmk

C11 C12 C1nc

C21 C22 C2nc

.

.

....

..

....

Cnc 1 Cnc2 Cnc nc

266664

377775

dmk

1

dmk

2

.

.

.

dmk nc

2666664

3777775

C11 dmk

1 C12 dmk

2 C1nc d

mk

nc

C21 dmk

1 C22 dmk

2 C2nc d

mk

nc

.

.

.

Cnc1 dmk

1 Cnc2 d

mk

2 Cnc nc d

mk

nc

26666664

37777775:

gmk

1; gmk

2; ; gmk

nc

T: 18

But gmkalso can be expressed in terms of the number of WAGcycles as

gmk

gmk

T1;a

; gmk

T1;b

; gmk

T2;a

; gmk

T2;b

; ; gmk

Tncycle;a

; gmk

Tncycle;b

T

:

19

Note that dmk and gmk in Eq. 18 are expressed in the form of

Eq. 7, not in the form of Eq. 14, to show the matrix multiplicationbyCmXmore clearly.

Step 3. From Eqs. 16, 18, and 19, we see that a zero entry ofdmk in the form of Eq. 14 does not imply that the correspondingentry ofgk in the form of Eq. 19 is zero. Thus, for gas injectors,we again set the even entries of the subvectorgmk in the form ofEq. 19 equal to zero; for water injectors, we set the odd entries ofgmk in the form of Eq. 19 equal to zero. This gives the final g

mk,

and hence, the gk, which we use as the modified EnOpt searchdirection. After the previous modifications, the EnOpt techniques

that were developed for life-cycle waterflooding optimizationcan be adapted to optimize the design variables for the WAG-injection process.

Hi-MO Method. In this work, the performances of continuousCO2 flooding and waterflooding are studied to compare their per-formance with that of the CO2-WAG technique. The algorithmapplied to optimize the process of continuous CO2 flooding andwaterflooding is the adaptive Hi-MO algorithm (Oliveira andReynolds 2014), which can yield a higher NPV than is obtainedby applying the standard procedure of EnOpt. The objectives ofthe Hi-MO algorithm are to reduce the number of control varia-bles used to generate a set of less-ill-conditioned optimizationproblems, and to improve computational efficiency without com-

promising the optimal value of the life-cycle NPV obtained.The Hi-MO method can be applied with any optimization algo-

rithm, but here, we only consider its application with the modifiedEnOpt method.

In the multiscale approach, we start with a small number ofcontrol steps and find the optimal controls by maximizing J x ;then, we select the new set of control steps for the next level ofparameterization that is based on two main procedures: mergingand splitting. The control steps are selected to be merged if thefollowing control-variation criterion holds:

juwi uwi;refj

uup;wi u

low;wi

nmaxs , then set

xopt xh1 and terminate the optimization algorithm. Here eJandex are the final convergence tolerances for the change of objectivefunction and control variables, respectively; ns is the counter forthe number of reservoir simulation runs.

Step 3. Reparameterization: Redefine the control steps foreach well,w 1; 2; 3;; nwell.

Perform the splitting and merging procedure: for each controlstep i 1; 2; 3;; nwc of each well, determine whether the ithcontrol should be either merged or split with Eq. 20, where nwcdenotes the number of control steps for well w.

Step 4. Update the covariance matrix.Decrease the temporal correlation length used to define CX

between the control variables for the optimization step for thenext level by multiplying the current correlation length,Th, by adecreasing factorb,where 0< b< 1; that is,

Th1 maxfbTh; Tming; 24

whereTmin is the minimum correlation length which may be zero.Then, generate the covariance matrix,Ch1X , for the next level.

Step 5. Update convergence tolerances.Tighten the tolerance for the optimization step for the next

level by applying the following equations:

eh1J maxfcehJ; eJg 25

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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eh1x maxfceh

x ; e

x g; 26

where c is the tolerance-tightening factor. More total simulationruns for the next refinement level can be allowed by multiplying

the totalnmax;hs by a user-chosen factor greater than unity.Then, seth h 1 and go to Step 1.End (For)We wish to start with very few controls and long control steps

because there exist cases in which one can obtain an optimal solu-tion with a small number of control steps [e.g., case in whichbang-bang controls are optimal (Zandvliet et al. 2007)]. Thus, wechoose the two initial control steps of equal length. The choice ofnsplit 2 is based on the computational experiments of Oliveira(2014), but choosing nsplit 4 is not expected to yield radicallydifferent results. Afternsplitis chosen,hmaxis chosen so that

Dtmin T

nhmaxsplit; 27

whereTis the total production lifetime and Dtmin is the minimumallowable length of a control step. For example, ifT 2,048 daysand Dtmin 32 days, then Eq. 27 yields hmax 6. It is, of course,possible for the algorithm to terminate before reaching the sixthrefinement.

Case Study

The reservoir model studied in this work is a three-layer syntheticreservoir with a 25 25 3 grid with the gridblock dimensionsgiven by Dx Dy 100 ft and Dz 20 ft. It contains four injec-tors and nine producers. The horizontal log-permeability distribu-tion of the first layer is shown in Fig. 1 with three high-permeability channels. The other two layers have the same hori-zontal permeability distribution as the first layer. The porosity ishomogeneous with/ 0.2. The compressibility of rock and wateris equal to 6:103 105 and 3:3 106 psi1, respectively. Theinitial reservoir pressure is 4,300 psi, and the minimum miscibil-ity pressure is equal to 4,150 psi. The reservoir lifetime is setequal to 2,048 days. A compositional reservoir simulator, GEM(CMG 2009) from Computer Modelling Group, is used in reser-voir simulation for water-alternating gas (WAG) injection (Chenet al. 2010). There are seven pesudocomponents used to describethe oil and gas: CO2, N2, C1, C2C4, C5C6, C7C18, andC19C43.

The control variables at each gas injector are the injection rateswith an upper bound of 20 MM scf/D and a lower bound of 0 MMscf/D; the control variables at each water injector are the injection

rates with an upper bound of 4,000 STB/D and a lower bound of 0STB/D; each producer operates under BHP control with an upper

bound of 4,500 psi and a lower bound of 1,500 psi. The initial

guess for rate controls of each water injector is set equal to 2,000STB/D; the initial guess for rate controls of each gas injector isequal to 10 MM scf/D, and the initial guess for BHP controls foreach producer is equal to 3,500 psi.

To optimize the NPV, the oil price is set equal to USD 80.0/STB; the water-injection cost is USD 5.0/STB; the gas-injectioncost is USD 1.5/Mscf; the cost of disposing produced water isUSD 5.0/STB; the annual discount rate is 0.1. Here, we neglectthe revenue of hydrocarbon-gas production and the cost of dispos-ing of breakthrough injection gas. For the optimization of theSAG process, the cost of injecting water containing dissolved sur-factant is set equal to USD 8.0/STB.

We apply the log-transformation to handle the bound con-straints for all the optimization methods considered in this paper.For the ensemble-based optimization (EnOpt) method, we set the

number of samples for gradient averaging to Ne 10; the maxi-mum number of stepsize cuts is set equal to 5, and the maximumnumber of resamples is specified as Nres 5; the initial step sizeis 1.0; the time-correlation length is 512 days; the perturbationsize is equal to 1.0; the maximum number of allowable simulationruns is 1,000. When the Hi-MO method is used to optimize theprocesses of continuous CO2flooding and waterflooding, we spec-ify the maximum number of refinement levels as h 6; the mini-mum control-step length is 32 days; we use two initial controlsteps, and the number of splits is given by nsplit 2; the controltolerance for lumping steps is 0.01; the initial control-correlationlength is T1 2; 048 with a decreasing factor of 2 for each opti-mization level; the tolerance-tightening factor is set equal to 0.5;the maximum number of allowable simulation runs of Level 1 isset equal to nmax;1

s

100, and this number increases by a factorof 2 at each level; the total maximum allowable simulation runsis 1,000.

Comparison Among Different WAG Cycles. To compare theperformance of WAG with different numbers of cycles, we set upfour cases with 4 cycles, 8 cycles, 16 cycles, and 32 cycles. Thenumber of control steps for each well, nc, is set equal to 64 regard-less of the number of cycles, and the length of each control step is32 days. When the total number of WAG cycles is equal to 4, weset the length of each of the eight half-cycles equal to 256 days,and each of the half-cycles contains 8 control steps. For 8 cycles,the length of each half-cycle is equal to 128 days, and each of thehalf-cycles contains 4 control steps, and so on. Table 1 presentsthe summary of the initial NPV, final NPV (estimate of optimalNPV), and cumulative oil production for the different cases. Fromthe results in Table 1, the final NPV in all the cases is increasedcompared with the initial NPV. We see that the initial NPV, theestimated optimal (final) NPV, and cumulative oil productionincrease as the number of cycles increases (cycle time decreases),but they are fairly close for the 16-cycle and 32-cycle cases.

Fig. 2shows the remaining oil saturation as a function of thenumber of cycles (or cycle time). The color scale corresponds tothe oil saturation, with the darkest blue representing the lowest oilsaturation and the darkest red color representing the highest oilsaturation. For the same layer (Layer 1 or Layer 3 in Fig. 2), asthe number of cycles increases, the remaining oil saturationdecreases. Note that, regardless of the number of cycles, the upperlayer has less remaining oil than the bottom layer because the

gravity segregation of injected water and gas causes CO2, whichhas a higher displacement efficiency, to move to the upper layer

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

25Pro1 Pro2 Pro3

Pro4

Inj1 Inj2

Inj3 Inj4

Pro5 Pro6

Pro7 Pro8 Pro9

25

20

20

15

15

10

10

5

8

6

4

2

05

Fig. 1Log-permeability distribution, three-channel case.

Table 1Summary, comparison among different WAG cycles.

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whereas water, which has a lower displacement efficiency, movesto the bottom layer.

One can obtain the estimated optimal controls for all the pro-duction wells and injection wells by applying the modified EnOptalgorithm.Fig. 3 presents the well controls of producers for thecase with 32 WAG cycles. As one can see in Fig. 1, ProducerPro1 is separated from injectors by one of the three high-perme-ability channels, which we refer to here as Channel 1. When theinjected water or gas reaches Channel 1, most of the injected fluidwill move toward Pro2 through this channel. Thus, it is reasonableto expect Pro1 to operate at close-to-minimum allowable BHP(1,500 psi) throughout the production lifetime, but Pro2 can oper-ate at close-to-minimum allowable BHP only at the early times

when it is still economic but shut in from 500 to 1,700 days, asshown in Fig. 3a. Pro3, Pro4, Pro5, and Pro6 are either in a chan-nel or close to a channel, and all operate at close-to-minimumallowable BHP at the early times, and thereafter, they are almostshut in, as shown in Figs. 3b and 3c. Pro7 is shut in at the earlytimes and thereafter operates at close-to-minimum allowableBHP. Pro8 is close to the high-permeability channel so that theestimated control for this producer is close-to-minimum allowableBHP at the early control steps, but after injected gas and waterbreakthrough at this producer, its pressure increases (ratedecreases), which occurs at approximately 1,000 days. Pro9 is farfrom the channel and the injectors, so it operates at close-to-mini-mum allowable BHP during the whole production lifetime, as

(a) 4 cycles (b) 8 cycles (c) 16 cycles (d) 32 cycles

Layer 1

Layer 3

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.600.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.600.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.600.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.600.50

0.40

0.30

0.20

0.10

0.00

Fig. 2Remaining oil-saturation distribution of different cycle cases.

(a) Pro1 and Pro2 (b) Pro3 and Pro4

(c) Pro5 and Pro6 (d) Pro7, Pro8, and Pro9

1,500

2,500

3,500

4,500

0 500 1,000 1,500 2,000

BHP(

psi)

Time (days)

Pro1 Pro2

1,500

2,500

3,500

4,500

0 500 1,000 1,500 2,000

BHP(

psi)

Time (days)

Pro3 Pro4

1,500

2,500

3,500

4,500

0 500 1,000 1,500 2,000

BHP(

psi)

Time (days)

Pro5 Pro6

1,500

2,500

3,500

4,500

0 500 1,000 1,500 2,000

BHP(

psi)

Time (days)

Pro7 Pro8 Pro9

Fig. 3Estimated optimal well controls for production wells, 32 WAG cycles.

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shown in Fig. 3d. The estimated optimal well controls for injec-tion wells for the case with 32 WAG cycles are shown in Fig. 4.Note that the rates plotted correspond to the half-cycle rates forthe different cycles. Various factors affect the gas- and water-injection rates, including the miscibility between CO2and oil, thechanneling of injected fluid, and the shut-in of some productionwells during the production lifetime, all which lead to the optimalwell-injection rates, as shown in Fig. 4.

Fig. 5 shows the estimated optimal WAG ratio (volume ofwater injected divided by the volume of gas injected) for eachcycle for the case with 32 cycles. The WAG ratio is affected by

reservoir heterogeneity, miscibility conditions, injection condi-tions, and well operational parameters, so it is hard to explainwhy the estimated WAG ratio changes with the WAG cycle, asshown in Fig. 5. However, one can obtain some general ideasfrom the results of this figure. First of all, the WAG ratio startswith a relatively high value at the early injection times to injectmore water into the reservoir to block the high-permeability zoneand to increase gas miscibility. Thereafter, the WAG ratiodecreases to a lower value to have more gas injected to obtain abetter microscopic displacement of oil. Finally, the WAG ratioincreases at the end of the reservoir life. From the previous analy-sis, we can see that it is inappropriate to fix the WAG ratio forWAG-injection process in practice although, for most cycles, theWAG ratio is less than 0.2. Note that we implicitly optimize theWAG ratio for different cycles by optimizing the well controls.

Comparison of WAG, Continuous CO2 Injection, and

Continuous Waterflooding. To compare the efficiency of CO2-WAG, continuous CO2, and continuous waterflooding, we con-sider the following three subcases, all which pertain to the three-layer reservoir.

1. WAG with 32 cycles by use of modified EnOpt foroptimization

2. Continuous CO2by use of Hi-MO (two initial control steps;nsplit 2) for optimization

3. Waterflooding by use of Hi-MO (two initial control steps;nsplit 2) for optimization

For continuous CO2 and waterflooding, we apply the Hi-MOalgorithm (Oliveira and Reynolds 2013) with two initial controlsteps and two splits to optimize the processes and to obtain thefinal NPV and cumulative oil produced. Because Hi-MO adap-tively selects the duration and length of the timesteps and isdesigned to mitigate the effects of ill-conditioning, Hi-MO is gen-erally able to provide a higher NPV than is obtained by fixing thenumber and length of the control steps a priori. We have notdeveloped a version of the Hi-MO algorithm applicable for theWAG process, but it is likely that such an algorithm would yieldhigher optimized values of NPV than the ones given in this paperfor the WAG process.

Table 2 presents the summary of the initial NPV, final NPV,and cumulative oil produced for the three techniques. From theresults in Table 2, all the final NPVs are improved compared withthe initial NPVs. The optimized WAG process yields significantlymore oil production and a higher NPV than those obtained byoptimizing continuous CO2or waterflooding.

Figs. 6 and 7 display the CO2 concentration in the bottomlayer (Layer 3), and the pressure distribution for the middle layer(Layer 2), respectively. These two figures compare results fromWAG flooding with those from continuous CO2 flooding after

(a) Inj 1 (b) Inj 2

(c) Inj 3 (d) Inj 4

0

1,000

2,000

3,000

4,000

0

5

10

15

20

0 4 8 12 16 20 24 28 32 Water-InjectionRate

(STB/D)

Gas-InjectionRate

(MMscf/D)

Water-InjectionRate

(STB/D)

Gas-InjectionRate

(MMscf/D)

Water-InjectionRate

(STB/D)

Gas-InjectionRate

(MMscf/D)

Water-InjectionRate

(STB/D)

Gas-InjectionRate

(MMscf/D)

Cycle

Gas injection Water injection

0

1,000

2,000

3,000

4,000

0

5

10

15

20

0 4 8 12 16 20 24 28 32

Cycle

Gas injection Water injection

0

1,000

2,000

3,000

4,000

0

5

10

15

20

0 4 8 12 16 20 24 28 32

Cycle

Gas injection Water injection

0

1,000

2,000

3,000

4,000

0

5

10

15

20

0 4 8 12 16 20 24 28 32

Cycle

Gas injection Water injection

Fig. 4Estimated optimal well controls for injection wells, 32 WAG cycles; the rates plotted correspond to the half-cycle rates forthe different cycles.

0

0.2

0.4

0.6

0.8

1

1.2

0 4 8 12 16 20 24 28 32

WAGRatio

Cycle

Inj1 Inj2 Inj3 Inj4

Fig. 5Estimated optimal WAG ratio of each cycle, 32 WAGcycles. Table 2Summary, comparison among three techniques.

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1,440 days of production and 2,048 days of production. In Fig. 6,the color scale corresponds to CO2 concentration with the darkest

red color representing the highest CO2 concentration (1.00) andthe darkest blue color representing the lowest concentration(0.00). In Fig. 7, the color scale corresponds to pressure with thedarkest red color representing the highest pressure (5,500 psi) andthe darkest blue color representing the lowest pressure (1,500psi); the upper two figures show the pressure distribution forLayer 2 for WAG flooding at 1,440 days and 2,048 days and thebottom two figures show the pressure distribution for Layer 2 ofcontinuous CO2flooding at 1,440 days and 2,048 days.

From Fig. 6, we see that WAG flooding yields a higher CO2-concentration distribution in the bottom layer than does continu-ous CO2 flooding. This is because the water injected in the WAGprocess can control the mobility of CO2by blocking the high-per-meability channel or zones and by reducing the effects of gravitysegregation that keeps more injected CO

2 in the bottom layer,

which, in turn, improves the sweep efficiency and results in higheroil production. Fig. 7 shows that WAG flooding yields a higherpressure distribution than continuous CO2 flooding; this isbecause the water injected in the WAG process helps to maintaina higher reservoir pressure because water is immiscible and has arelatively low compressibility. The higher pressure also enhancesthe miscibility of CO2 and oil and, hence, improves the micro-scopic displacement efficiency.

Fig. 8 shows the Layer 1 and Layer 3 oil-saturation distribu-tion at 2,048 days obtained for the three processes. The color scalecorresponds to oil saturation. From Fig. 8, we can conclude that,compared with the other two techniques, WAG flooding yieldsthe lowest remaining oil saturation both in the upper layer and thebottom layer. This statement also holds for the second layer

although the results are not shown. For continuous CO2 flooding,more oil is produced from the upper layer (Layer 1) because of

the segregation of CO2 to the upper layer; however, for water-flooding, the bottom layer achieves a lower oil saturation because

of water segregation to the bottom layer. Comparing the remain-ing oil-saturation distribution for Layer 1 and Layer 3 of theWAG case, we also can conclude that CO2 does have better mi-croscopic-displacement efficiency than water.

Optimization of SAG-Injection Process. The SAG-injectionprocess is very similar to the WAG process. Thus, the modifiedensemble-based algorithm still can be applied to optimize theSAG-injection process. Generally, the SAG model is simulatedwith the STARS simulator (Rossen and Boeije 2013), which isalso from Computer Modelling Group; STARS is CMGsadvanced reservoir simulator that includes options to simulateprocesses which GEM does not consider. These processes includefoam injection and chemical flooding (Computer Modeling Group2009). In this part of the paper, we first compare the performanceof the two simulators (GEM and STARS) and thereafter move for-ward to the optimization of the SAG-injection process with, asinitial guesses, the optimized well controls obtained by the opti-mization of WAG injection.

Fig. 9presents plots of NPV vs. the number of simulation runsfor SAG and WAG injection for the cases with 16 cycles and 32cycles. Both Fig. 9a and Fig. 9b show that WAG injection imple-mented by GEM yields a much higher final NPV than SAG injec-tion implemented by STARS. However, the cases are such thatwe believe that SAG should achieve a better NPV or oil-recoveryfactor than WAG because of additional enhanced-oil-recoverymechanisms generated by surfactant and foam. To understand thisdiscrepancy, we perform optimization of the WAG process with

STARS as well as GEM, in which the initial guess for the vectorof well controls is the same for both simulators. For the case with

(a) 1,440 days (WAG) (b) 2,048 days (WAG)

(c) 1,440 days (CO2) (d) 2,048 days (CO2)

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

Fig. 6CO2concentration in Layer 3 from WAG flooding and continuous CO2flooding.

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16 cycles, the final NPV obtained by optimizing WAG with GEMis 18.28% higher than the NPV obtained by optimizing WAGwith STARS; for the 32-cycles WAG case, the final NPV

obtained by optimizing WAG with GEM is 18.58% greater thanthe NPV obtained by optimizing WAG with STARS. Our expla-

nation is that this is because STARS usesk-values to model com-positional effects, whereas GEM uses an equation-of-state-basedphase-equilibrium package for multiphase pressure/volume/tem-

perature calculations. However, it is well-known that the liquid-gas phase envelope can change dramatically as the CO2

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

(a) WAG (b) Continuous CO2 (c) Waterflooding

Layer 1

Layer 3

Fig. 8Remaining oil-saturation distribution for three different techniques.

(a) 1,440 days (WAG) (b) 2,048 days (WAG)

(c) 1,440 days (CO2) (d) 2,048 days (CO2)

5,500

5,000

4,500

4,000

3,500

3,000

2,500

2,000

1,500

5,500

5,000

4,500

4,000

3,500

3,000

2,500

2,000

1,500

5,500

5,000

4,500

4,000

3,500

3,000

2,500

2,000

1,500

5,500

5,000

4,500

4,000

3,500

3,000

2,500

2,000

1,500

Fig. 7Pressure distribution for Layer 2 of WAG flooding and continuous CO2flooding.

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concentration increases. Thus, it is unfair to compare the perform-ance of the WAG injection implemented by GEM with the per-formance of the SAG injection implemented by STARS. We haveto choose the same simulator to compare the performances of theSAG and WAG for the same reservoir case. As seen in Fig. 9,SAG injection yields a higher final NPV than WAG injectionwhen both WAG and SAG cases are simulated by STARS. As wehave mentioned previously, GEM cannot model the process ofSAG injection, so we cannot compare the performance of WAGand SAG with GEM.

Table 3lists the computational time required for the optimiza-tion of SAG and WAG processes with different simulators for thetwo different cases. Note that, for the same reservoir case, thecomputational time for the WAG optimization with GEM is lessthan that of WAG simulation with STARS and SAG simulationwith STARS. For the case with 32 cycles, SAG optimizationimplemented by STARS uses the most computational time (241hours). WAG optimization with GEM requires the least computa-tional time (80 hours), less than one-third the computational timeof SAG optimization with STARS. Thus, if we optimize the SAGprocess by directly starting with the original initial guess of thevector of optimal well controls (initial guess for rate controls ofeach water injector is 2,000 STB/D, initial guess for rate controlsof each gas injector is 10 MM scf/D and initial guess for BHP

controls for each producer is 3,500 psi), even though we canfinally obtain optimal controls and NPV by directly optimizingthe SAG process, the optimization could be prohibitively compu-tationally expensive, especially for a real-field case. One possibleway to reduce the computational time for the SAG optimizationwould be to optimize the reservoir for WAG and then use theWAG optimal controls as the initial guess for optimization of theSAG process. Hopefully, by generating a better initial guess by

doing the WAG optimization first, we can reduce the overall com-putational time required for the SAG optimization. In a sense, thismeans that the WAG process can serve as a proxy for optimiza-tion of SAG.

Fig. 10 shows the performance of SAG optimization withthree different initial guesses for the case with 16 cycles. The firstinitial guess is the original initial guess that was used in previouscases; the second initial guess is the WAG optimal-control vectorobtained with GEM; the third initial guess is the WAG optimal-control vector obtained with STARS. Table 4 presents the initialNPV, final NPV, simulation runs, and the computational time forSAG optimization with the three different initial guesses. FromFig. 10 and Table 4, we see that much faster convergence isobtained when the initial guess for the SAG optimization corre-sponds to the WAG optimal well controls.

The total computational times in Table 4 are calculated as fol-lows: For the original initial-guess case, we run 1,000 simulationswith SAG, and the total computational time is 232 hours. For thesecond initial-guess case, we first run 1,000 simulations withGEM to obtain the optimal WAG controls as the initial guess for

SAG; the computational time required is 75 hours. Thereafter, werun 411 simulations for SAG optimization requiring a computa-tional time of 80 hours, so the total computational time is 155hours. For the third initial-guess case, we first run 1,000 simula-tions with STARS to obtain the optimal WAG controls (this opti-mization requires 101 hours) and thereafter run 299 SAGsimulations which require a computational time of 48 hours, sothe total computational time is 149 hours.

(a) 16 cycles (b) 32 cycles

3.00E+08

3.50E+08

4.00E+08

4.50E+08

5.00E+08

0 200 400 600 800 1,000

NPV(

$)

Simulation Runs

SAG-STARS

WAG-STARS

WAG-GEM3.00E+08

3.50E+08

4.00E+08

4.50E+08

5.00E+08

0 200 400 600 800 1,000

NPV(

$)

Simulation Runs

SAG-STARS

WAG-STARS

WAG-GEM

Fig. 9NPV values vs. the number of simulation runs for SAG and WAG injection.

Table 3Summary, computational time for SAG and WAG with

different simulators.3.40E+08

3.65E+08

3.90E+08

4.15E+08

4.40E+08

0 200 400 600 800 1,000

NPV($)

Simulation Runs

Original initial guess

WAG controls-GEM

WAG controls-STARS

Fig. 10Performance of SAG optimization with different initialguesses, 16 WAG cycles.

Table 4SAG optimization with different initial guess, 16 WAG cycles.

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From the previous analysis, we can see that optimizing theWAG process to obtain initial guesses for the optimization of theSAG process improves computational efficiency.

Effect of Random Seed and Ensemble Size. Because of thestochastic nature of the gradient estimate, it is important to con-sider the effect of the random seed and ensemble size on the opti-mization results. Note that, for all the cases studied previously,we ran a few random seeds and chose the result that achieves thehighest final NPV. Table 5 lists the final NPV calculated fromfive different random seeds for the 16- and 32-cycle cases. Note

that both the mean NPV and highest NPV achieved from the 32-cycle case are higher than those obtained from a 16-cycle case.

To quantify the impact of ensemble size on the optimizationresults, we compare the final NPVs achieved from an ensemblesize equal to 30 with an ensemble size equal to 10. For both en-semble sizes, we ran with five different seeds. For the ensemblesize equal to 30, the maximum-allowable simulation run is setequal to 1,500, whereas for the ensemble size equal to 10, themaximum-allowable simulation run is equal to 1,000 as before.Table 6presents the final NPVs achieved from the 32-cycle casewith two different ensemble sizes (10 and 30). As we can seefrom Table 6, the mean value of NPV achieved from the ensemblesize equal to 30 is not significantly higher than the NPV achievedwith an ensemble size equal to 10, and the highest value obtainedfrom the ensemble size equal to 30 is even lower than thatobtained with the ensemble size equal to 10 even though we allow50% more simulation runs when the ensemble size is 30. As dem-onstrated by Fonseca et al. (2015b), the quality of gradient can beimproved with a relatively large ensemble size. However, for thecompositional simulation, it is computationaly expensive to use alarge ensemble size with more allowable maximum simulationruns. As seen in Table 6, at least for this problem, an ensemblesize equal to 10 is sufficient to obtain a good gradient and toachieve a reasonable final NPV.

Comments

The study suggests that the variety of EnOpt techniques that weredeveloped for life-cycle waterflooding optimization can beadapted to optimize the design variables for the WAG and SAGprocesses. An EnOpt algorithm can easily be coupled with anyreservoir simulator. Thus, the ideas presented should lead to prac-tical tools that allow a field engineer to decide whether water-flooding, continuous gas injection, WAG, or SAG will lead to thehighest NPV and/or the highest oil recovery and, at the sametime, optimize the recovery process chosen.

Conceptually, it is possible to optimize the length of each half-cycle as well as the well controls. In Bahagio (2013), water andCO2 injection are alternated at each of four injection wells in afive-spot pattern and the times at which each injector switchesfrom gas injector to water injector and vice versa are optimized.These switching times plus the BHPs of injectors are the optimi-zation variables. The single production well is operated at a fixedBHP. Bahagio (2013) found that a much higher NPV was

obtained with both BHP and switching times as optimization vari-ables than was obtained with only the BHPs, in which each half-

cycle length is equal to 180 days and each control step length forBHPs was also equal to 180 days. From the results, it seems clearthat the use of the length of control steps equal to 180 days yieldsa suboptimal result. Thus, although the incorporation of switchingtimes is a very appealing approach, it is possible that the results ofBahagio (2013) overestimate the gain achieved by adding switch-ing time to the set of optimization variables. Unfortunately, GEMdoes not provide a means to incorporate switching times at aninjection well to alternate gas and water injection to further inves-tigate the procedure suggested by Bahagio (2013). Moreover,

because we use injection rates instead of injector BHPs and usecontrol steps within each cycle equal to 32 days, there are two tosixteen control steps within each cycle, depending on the numberof cycles used. If we consider, for example, the 16-cycle case inwhich each half-cycle is 64 days long (each half-cycle has twocontrol steps), it is effectively possible to control the length of theinjection time of CO2 and water. For example, one could injectCO2 at a high rate for the first 32 days and inject nothing for thefollowing 32 days, and then inject water at a low rate for 64 days,and so on. Although this preceding scenario is not identical toswitching to water injection at the end of 32 days, it is similarenough so that we conjecture that the addition of switching timesto the set of optimization variables would not yield a significantincrease in the NPV obtained.

Conclusions

The estimated optimal NPV of CO2-WAG injection is improvedsignificantly by applying production optimization. The initialNPV, final (estimated) NPV, and cumulative oil production tendto increase with decreasing cycle time (increasing the numberof cycles).

For the channelized reservoir example considered, the opti-mized WAG flood results in significantly higher valves of NPVand oil production than the values obtained with the well controlsobtained by optimizing waterflooding or continuous CO2injection.

If we wish to optimize the SAG-injection process, we can firstoptimize the reservoir for WAG injection to obtain the optimalcontrols for WAG, and then treat the optimal controls for WAGas the initial guesses of the optimal well controls for the SAG pro-cess. This initialization procedure can yield significant reductionin computational cost.

Acknowlegments

The support of the member companies of the University of TulsaPetroleum Reservoir Exploitation Projects (TUPREP) is gratefullyacknowledged. Bailian Chen also wishes to acknowledge all hiscolleagues in TUPREP for their help.

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Bailian Chen is a PhD degree candidate in the McDougallSchool of Petroleum Engineering at the University of Tulsa. Hisresearch interests include production optimization, enhancedoil recovery, and uncertainty quantification of data-acquisi-tion programs (pilot and surveillance). Chen holds BS and MSdegrees in petroleum engineering from the China University ofPetroleum (East China). He is a member of SPE.

Albert C. Reynolds is a professor of petroleum engineeringand mathematics, holder of the McMan Chair in PetroleumEngineering, and Director of TUPREP at the University of Tulsa,where he has been a faculty member since 1970. Reynoldsresearch interests include optimization, scientific computa-

tion, assessment of uncertainty, reservoir simulation, historymatching, and well testing; he has coauthored more than100 refereed papers in these areas. Reynolds holds a BAdegree from the University of New Hampshire, an MS degreefrom Case Institute of Technology, and a PhD degree fromCase Western Reserve University, all in mathematics. He hasreceived the SPE Distinguished Achievement Award for Petro-leum Engineering Faculty, Reservoir Description and Dynam-ics Award, Formation Evaluation Award, and John FranklinCarll Award; he became an SPE Distinguished Member in1999 and an SPE Honorary Member in 2014. Reynolds is anassociate editor for SPE Journal and for ComputationalGeosciences.