Comput GeosciDOI 10.1007/s10596-013-9375-x

ORIGINAL PAPER

Simultaneous and sequential approaches to jointoptimization of well placement and control

Thomas D. Humphries · Ronald D. Haynes ·Lesley A. James

Received: 10 February 2013 / Accepted: 6 September 2013© Springer Science+Business Media Dordrecht 2013

Abstract Determining optimal well placement and controlis essential to maximizing production from an oil field.Most academic literature to date has treated optimal place-ment and control as two separate problems; well placementproblems, in particular, are often solved assuming somefixed flow rate or bottom-hole pressure at injection and pro-duction wells. Optimal placement of wells, however, doesdepend on the control strategy being employed. Determin-ing a truly optimal configuration of wells thus requires thatthe control parameters be allowed to vary as well. Thispresents a challenging optimization problem, since welllocation and control parameters have different propertiesfrom one another. In this paper, we address the placementand control optimization problem jointly using approachesthat combine a global search strategy (particle swarm opti-mization, or PSO) with a local generalized pattern search(GPS) strategy. Using PSO promotes a full, semi-randomexploration of the search space, while GPS allows us tolocally optimize parameters in a systematic way. We focusprimarily on two approaches combining these two algo-rithms. The first is to hybridize them into a single algorithmthat acts on all variables simultaneously, while the secondis to apply them sequentially to decoupled well placement

T. D. Humphries (�) · R. D. Haynes · L. A. JamesMemorial University of Newfoundland, St. John’s,Newfoundland, Canada, A1C 5S7e-mail: thumphries@mun.ca

R. D. Haynese-mail: rhaynes@mun.ca

L. A. Jamese-mail: ljames@mun.ca

and well control problems. We find that although the bestmethod for a given problem is context-specific, decouplingthe problem may provide benefits over a fully simultaneousapproach.

Keywords Production optimization · Well placement ·Well control · Particle swarm optimization · Patternsearch · Joint optimization · Black-box optimization

1 Introduction

Maximizing production from an oil field is a crucialtask, given the enormous financial investment at stake inany large-scale field development. Decisions regarding theplacement of new wells, and control of injection and pro-duction rates at existing wells, have a significant impacton production. Poor placement and/or control of wells mayresult in premature water breakthrough at the productionwells, or make it difficult to achieve high flow rates whilemaintaining reservoir pressure. The vast number of potentialdevelopment scenarios drives the need for efficient, com-puterized optimization approaches to assist in making thesedecisions.

The problems of finding optimal well locations and deter-mining optimal well control parameters are often treatedseparately [12]. Well placement problems involve optimiz-ing over parameters corresponding to the positions andorientations of the injection and production wells. We limitourselves in this paper to considering vertical wells, mean-ing that each well’s position is parameterized simply byits (x, y)-coordinates. A simple control scheme is typicallyassumed in well placement problems; for instance, injec-tion wells are held at a fixed bottom-hole pressure (BHP),

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while producers are held at a lower BHP in order to gener-ate flow. Well control problems, on the other hand, focus onmanaging the injection and production rates at wells that arealready in place. The optimization variables in this case areusually either the BHP or the flow rate for each well, whichcan be changed at specified time intervals. The objectivefunction that one attempts to maximize in these problemsis typically either the total amount of oil extracted, or thenet present value (NPV) of the extracted oil. The NPV func-tion is correlated with the total amount of oil extracted, butemphasizes producing more oil early in the reservoir’s life-time (due to the time value of money), and also typicallyincludes the costs of water injection and disposal of anywater produced. Heterogeneous properties of the reservoir,such as the permeability field, have a significant impact onoptimal well placement; as a result, the objective functionsurface tends to be much rougher in well placement prob-lems than in well control problems. This difference in theobjective function surface typically leads to the use of differ-ent optimization approaches to address these two problems.Optimization studies on well placement have often focusedon global algorithms with some stochastic element in orderto avoid converging prematurely to local optima; well con-trol problems have tended to make use of deterministicalgorithms, based on local search techniques [12].

A unified approach to optimizing well placement andwell control jointly has the potential to provide benefitsover the treatment of these problems separately. In partic-ular, the best well configuration when producers are heldat some fixed BHP is not necessarily the same as the bestconfiguration if the control can vary with time [40]. Addi-tionally, determining the optimal placement of new wellsmay also require adjusting the control parameters at wellsalready in place. The problem of jointly optimizing wellplacement and control has been largely unexplored in theliterature until recently. Here, we investigate approaches toaddressing this problem by using two optimization algo-rithms: particle swarm optimization (PSO) and generalizedpattern search (GPS). These approaches include hybridiz-ing the algorithms to simultaneously optimize over bothplacement and control parameters, as well as applying themin sequential steps in a decoupled approach to the jointoptimization problem.

2 Existing research

Well placement studies have tended to use stochastic opti-mization approaches aimed at exploring the solution spaceglobally. Genetic algorithms (GAs) have received the widestuse [1, 6, 9, 13, 16, 26, 30, 38]. Other optimization algo-rithms that have been applied to the problem include simul-taneous perturbation stochastic approximation (SPSA) [4],

covariance matrix adaptation [7], and PSO [28, 29]. In addi-tion to studying suitable algorithms for the optimizationproblem, papers on well placement have discussed otherissues such as parametrization and optimal placement ofnonconventional wells [9, 38], consideration of geologi-cal uncertainty when determining optimal positions [1, 16],placement of well patterns rather than individual wells [29,30], and inclusion of nonlinear constraints as part of theoptimization [13, 40].

A popular optimization algorithm for well control prob-lems, on the other hand, has been the adjoint method [8, 14,20, 32, 39]. This method, which is based on approximat-ing the gradient of the objective function, is well suited tothe optimal control problem due to the objective function’ssmoothly varying nature. Implementing the adjoint methodmay be challenging, however, since it requires in-depthknowledge on the workings of the reservoir simulator. Analternative is to use “black-box” optimization algorithms,which optimize based only on the output from the simulator.Examples of black-box algorithms that have been appliedto the well control problem include stochastic methods likeSPSA [36] and GAs [37], as well as deterministic methodssuch as GPS and Hooke–Jeeves directed search [11, 12].

Some recent papers have considered optimization of wellplacement and well control in a more unified manner. Oneproposed approach [15] essentially treats the problem asone of optimizing well control rates, with a large number ofinjection and production wells being placed in the reservoirinitially. As the optimization proceeds, flow rates at somewells are driven to zero, resulting in their removal fromthe simulation. Thus, the procedure determines the optimalnumber of injectors and producers, as well as their loca-tions and optimal flow rates. This approach uses a modifiedversion of the adjoint method, which incorporates a num-ber of inequality and equality constraints to allow for theremoval of wells after every iteration. A second approachuses a combination of GPS and adjoint methods in a nestedoptimization procedure [5]. The outer iteration of this pro-cedure consists of using GPS to determine optimal wellpositions; for each configuration of wells, an inner opti-mization routine uses the adjoint method to determine thebest control strategy. The SPSA algorithm has also beenapplied to problems involving both well placement and wellcontrol in [23, 24], where it was found that optimizingover all variables simultaneously was preferable to applyingSPSA sequentially to subproblems involving placement orcontrol only.

The use of PSO and pattern search in tandem to addressthe well placement and control problem has also been inves-tigated in [19]. There, it was found that hybridizing PSO andmesh adaptive direct search (MADS, a variant of GPS) [3]provided better results than the independent application ofthose algorithms, and that optimizing over all variables

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simultaneously was preferable to sequential optimizations.The approach we use in this paper is similar to theirs,but in preliminary experiments presented in [18], we foundthat sequential optimization was sometimes preferable to afully simultaneous approach. In this paper, we refine thetechnique and perform further experiments, to determineunder which circumstances one approach may be preferableto the other.

3 Optimization approach

Combining global and local optimization techniques shouldbe advantageous when addressing well control and wellplacement simultaneously. We use PSO as a global opti-mizer in this study and GPS for the local search. Our choiceof these algorithms is motivated by the fact that both haveperformed well in previous production optimization stud-ies [11, 28]; both are black-box methods that do not requirein-depth knowledge on the simulator, and both are eas-ily parallelized to help mitigate the high cost of functionevaluations. We now give an overview of PSO and GPS,as well as of the specific optimization approaches usedin this paper.

3.1 Particle swarm optimization

Particle swarm optimization [10, 21] is an optimizationalgorithm based on modeling the behavior of a group ofanimals acting collectively. PSO utilizes a number of par-ticles (typically 20 to 40) to explore solution space in asemi-random way. The position of particle i at iteration k,denoted as x(k)

i , is a vector of size N, where N is the numberof variables in the optimization problem. Every position insolution space is associated with the corresponding objec-tive function value, and every particle remembers the bestposition it has found so far. Particles in the swarm alsocommunicate with one another to share the best positionsthat have been found overall. Each iteration of PSO con-sists of determining a new position for every particle in theswarm and then evaluating the objective function at thatposition. Since the objective function evaluations can beperformed independently of one another, the algorithm ishighly parallelizable.

Given x(k)i , the position of the particle at iteration k + 1

is as follows:

x(k+1)i = x(k)

i + v(k+1)i ,

where the particle’s velocity vector v(k+1)i is given by

v(k+1)i = ιv(k)

i +μr(k)1 ⊗

(p(k)

i − x(k)i

)+νr(k)

2 ⊗(

g(k)i − x(k)

i

).

The velocity is a combination of three terms. The firstterm models the tendency of the particle to continue travel-ing in the direction given by its current velocity. The secondterm represents the tendency of the particle to move towardthe best position it has found so far, denoted by p(k)

i . Finally,the third term represents the tendency of the particle to movetoward the best position found by any other particle withwhich it communicates, denoted by g(k)

i . The constants ι, μ,and ν are parameters whose values are chosen to weightthese three terms appropriately. To inject randomness intothe particle movement, the N-vectors r(k)

1 and r(k)2 are gener-

ated from the uniform distribution on (0,1) at every iterationand then multiplied componentwise with the terms in brack-ets by the ⊗ operator. The algorithm iterates until someconvergence criterion is met; for example, until the veloci-ties of the particles have become sufficiently small, until theparticles are sufficiently close to one another, or simply untilsome maximum number of iterations have been performed.

If every particle communicates with every other particlein the swarm, PSO may quickly converge to a local optimumbefore the solution space is fully explored. Thus, it is usuallyrecommended that each particle communicate only with twoto four other particles at any one time [10]. At every iterationof the algorithm, the neighborhood of particles with whicheach particle communicates can be chosen randomly. Thisrandom neighborhood topology was used for this study, aswell as a swarm size of 40 particles, and parameter values ofι = 0.721 and μ = ν = 1.193. These parameter values havebeen found to provide good convergence results in manynumerical experiments [10]. It should be noted that in gen-eral, one cannot guarantee that PSO converges to a global oreven a local optimum; however, in practice, it has proved tobe effective for a wide variety of optimization problems.

3.2 Generalized pattern search

Generalized pattern search [2, 22] is an optimization algo-rithm that begins from a single incumbent point and consistsof a series of search and poll steps. At every iteration k,a discrete mesh, centered at the current incumbent x(k), isdefined by

M(k) ={

x(k) + �(k)Dz : z ∈ NnD

},

where �(k) is the resolution of the mesh at iteration k, D

is a matrix whose columns form the search directions, N isthe set of natural numbers, and nD is the number of searchdirections. The search directions must form a positive span-ning set in solution space; i.e., one must be able to specifyany point in solution space by adding together only positivescalar multiples of these directions. A common choice ofsearch directions is

D = {e1, e2, . . . , eN, −e1, −e2, . . . , −eN } (1)

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where the en are the canonical basis vectors(1, 0, 0, . . . , 0)T , (0, 1, 0, . . . , 0)T , etc. Here, D refers tothe set of search directions, which form the columns of thematrix D.

The search step consists of selecting a finite number ofpoints on M(k) and evaluating the objective function at eachone. If any of those points improves the objective functionvalue, the point with the best value becomes the new incum-bent. The search step can employ any strategy in selectingpoints and may even be omitted, if desired. If none of thepoints selected in the search step are better than the incum-bent, then the algorithm proceeds to the poll step. The pollstep consists of evaluating the objective function at all thepoints that are immediate neighbors of the incumbent pointon the mesh M(k). These points are given by

{y(k)j

}=

{x(k) + �(k)dj | ∀ dj ∈ D

}.

If the poll step finds one or more points with a betterobjective function value than the incumbent, then the pointwith the best value becomes the new incumbent. Option-ally, �(k) may then be increased for the next iteration. If thepoll step is unsuccessful, then �(k) is reduced, and anotheriteration begins, using the same incumbent point as before.The algorithm is considered to have converged once �(k) isreduced beyond a specified threshold, which indicates thatthe current point is at least close to a local optimum. In fact,provided that the objective function is continuously differ-entiable, GPS is guaranteed to converge to a local optimum,at least to mesh precision [22]. Like PSO, GPS is highlyparallelizable because the function evaluations required bythe search and poll steps can be performed independently ofone another.

3.3 Handling of bound and general constraints

Broadly speaking, there can exist two types of constraints onthe optimization vector x—bound and general constraints.Bound constraints are simple componentwise inequalityconstraints of the form

xl ≤ x ≤ xu,

where xl and xu are the lower and upper bounds on x,respectively. In the context of a reservoir optimization prob-lem, these could be the minimum and maximum grid indices(for well placement) or upper and lower limits on the controlparameters.

Both PSO and GPS can easily incorporate bound con-straints. In PSO, any particles that travel outside of thebounds are projected back onto the boundary of searchspace. For instance, if component d of particle i’s position

exceeds the maximum value xud after being updated, then the

particle’s position and velocity are modified as follows:

xi,d = xud, vi,d = 0

The velocity component is set to zero to ensure that the par-ticle does not continue to travel in the direction that led itout of bounds. Bound constraints are treated similarly inGPS; namely, points which lie outside of search space areprojected back onto the boundary during the poll step [22].

General constraints refer to any constraints on the inputparameters other than simple bound constraints. Input thatviolates general constraints (which we refer to as infeasibleinput) can sometimes be identified prior to evaluating theobjective function, for instance, if the input specifies placingtwo wells at the same location. Other constraints, such asan upper limit on well flow rates for wells being driven byBHP, require running the reservoir simulator to determine ifthey are satisfied.

A simple mechanism for PSO to handle general con-straints is to allow particles to move to infeasible positions,but not store these positions in the particle’s memory [17].Thus, particles can explore search space freely, but are onlyattracted to positions that are feasible, in addition to provid-ing good objective function values. This strategy requiresthat every particle be initialized to a feasible position, so thatthe particle always has at least one feasible position storedin its history. In the experiments considered in this paper,we found that this was not an onerous requirement; how-ever, in cases where it is difficult to generate feasible pointsinitially, an approach that is able to incorporate infeasibleinitial positions would be preferable. We note that one hassome control over the relative sizes of feasible and infea-sible space when formulating the problem; for example,by choosing appropriate bounds on the injection and pro-duction BHPs, one can reduce the probability of violatingmaximum or minimum flow constraints.

To handle general constraints in GPS, one can simplyignore any infeasible points during polling and, thus, onlyaccept feasible points which also reduce the objective func-tion value. This approach is not ideal for general-purposeoptimization, as it may prevent the algorithm from travel-ing through the infeasible region to find the true optimum;alternative approaches such as filtering are recommendedinstead [2, 11]. We found that the first approach was suf-ficient for this study, however, possibly because GPS wasused in conjunction with PSO, rather than as a stand-aloneoptimizer.

3.4 Hybrid algorithm

An optimization algorithm that hybridizes PSO and GPShas previously been proposed in [34, 35]. This algorithm,denoted as PSwarm, is essentially a GPS algorithm that uses

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PSO as the search step. Thus, the algorithm behaves exactlylike PSO for as long as the search step continues to findpoints that improve the objective function value. When thisstep fails to improve the solution, polling takes place aroundthe current best position found. If the poll step finds a bet-ter solution, the current best position is updated, and a newiteration of PSO begins; otherwise, the polling stencil size isreduced. The algorithm proceeds until the convergence cri-teria for both PSO and GPS are met; i.e., the velocity of theparticles is sufficiently small, and the polling stencil size isreduced beyond a specified threshold.

In this paper, we have made the following modificationsto PSwarm in order to adapt it to the simultaneous wellplacement and control problem:

1. We have extended the PSO and GPS components of thealgorithm to handle general constraints, as describedin the previous section. The PSwarm algorithm asdescribed in [34] handles linear constraints, but notgeneral constraints of the type seen in this problem.

2. We have replaced the global communication topologyused by the search step of PSwarm with the randomvariable neighborhood topology. Each particle’s com-munication neighborhood consists of itself and twoother particles, which are selected randomly at eachiteration.

3. We have chosen to skew the sampling of control param-eters when initializing the swarm. PSO is typicallyinitialized by assigning a position sampled uniformlywithin the search space to each particle in the swarm.In the context of this problem, however, we can be rea-sonably certain that for production wells, BHP valuestowards the low end of the range will tend to increaseoil production, while BHP values towards the high endof the range will do the same for injection wells. Wetake advantage of this a priori knowledge in order toaccelerate the convergence of the algorithm.

4. We have investigated the effect of allowing the searchstep (PSO) to fail several times consecutively beforea poll step is performed. As a result of the high costof function evaluations in our problem, the poll stepis more computationally expensive than in many otheroptimization problems. Performing the poll step lessfrequently may allow us to reduce the computationalcost of the algorithm.

5. We have investigated the use of specifically selecteddirection vectors D to use during the poll step. The stan-dard GPS search directions (1) are fairly incremental,particularly with respect to control parameters, sinceeach variable corresponds only to the BHP value at asingle well for a single time interval. We may be ableto achieve a larger improvement in a single step bychoosing search directions that raise or lower BHP in

multiple years. The specific choice of search directionsis discussed in the next section.

4 Experiments

We now describe several experiments that were used to testthe performance of the different optimization approaches.All experiments were performed using the Matlab ReservoirSimulation Toolbox (MRST) [25, 33] as the reservoir sim-ulator. MRST is an open-source simulator implemented inMatlab, which includes routines for processing and visual-izing unstructured grids, as well as several solvers for singleand two-phase flow. The flow and transport equations aresolved in alternating steps in order to determine the phasepressures, flow rates, and saturations at every time point.Modeling of simple vertical and horizontal wells is pro-vided using the Peaceman model [31]. We used MRST’sbasic two-phase fluid model, which is a simplified Coreymodel with relative permeabilities given by krw = (Sw)2,kro = (1 − Sw)2.

The objective function we used in these experimentswas the net present value (NPV) over the entire productionperiod [0, T ]. The NPV was computed as in [4], with

NPV(x) =∫ T

0

⎧⎨⎩

∑n∈prod

[coq

−n,o(t) − cw,dispq

−n,w(t)

]

−∑n∈inj

cw,injq+n,w(t)

⎫⎬⎭ (1 + r)−t dt. (2)

The parameters co, cw,disp, and cw,inj represent the price perbarrel of produced oil, disposal cost per barrel of producedwater, and cost per barrel of injected water, respectively. Thefunctions q−

n,o(t) and q−n,w(t) are the production rates (in

barrels/day) of oil and water, respectively, at well n, whileq+n,w(t) is the water injection rate at well n. These rates

are implicitly functions of the optimization vector x, sincethey depend on the well positions and prescribed BHPs. Theyearly interest rate is specified by r . We used the parametervalues provided in Table 1 for all experiments. This choiceof values meant that production became unprofitable once

Table 1 Economic parameters used in all experiments

Parameter Value

co $80/bbl

cw,disp $12/bbl

cw,inj $8/bbl

r 10 or 0 %

Shut-in water cut 78 %

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the water cut at a well reached roughly 78 %. We there-fore implemented a reactive control strategy under whichproducers were shut in upon reaching this water cut value,for all test cases. This threshold value is often as high as90 or 95 % in practice; we chose a lower value to ensurethat shutting in a well was the optimal choice in someexperiments.

Experiment 1

The first experiment used a simple 2D reservoir model,consisting of 50×60 grid cells measuring 32×32×10 m(total field size, 1,600×1,920×10 m). The permeability andporosity fields (Fig. 1) were taken from the third layer ofthe SPE10 Benchmark model. The reservoir was initiallysaturated uniformly with an 80:20 mix of oil to water. Theoptimization problem was to place two injection and twoproduction wells in the reservoir, all of which were subjectto control via BHP. The production period was 10 years,and the BHP at each well could be altered every 2 years.Thus, there were 28 variables being considered in total (twopositional variables and five control variables per well). Theexperimental parameters are summarized in Table 2.

We considered three variations of the optimizationproblem:

• Case 1A: no constraints on injection and productionrates; discounting rate r of 10 %.

• Case 1B: no constraints on injection and productionrates; no discounting.

• Case 1C: maximum flow constraints on the injectionand liquid production rates as described in Table 2;discounting rate r of 10 %.

The goal of considering these three subproblems wasto see how the different conditions affected the optimalsolutions to the problem, as well as the effectiveness ofdifferent optimization approaches. Three main optimizationapproaches were applied to each problem. The fact thatevery optimization approach that we considered included astochastic component necessitated performing multiple runs

Table 2 Parameters (top) and constraints (bottom) used in experiment 1

Parameter Value

Grid cell dimensions 32×32×10 m

Fluid viscosities μo and μw 2.4 and 1.0 cp

Fluid densities ρo and ρw 835 and 1,000 kg/m3

Initial reservoir pressure 260 bars

Injector BHP range 275–450 bars

Producer BHP range 100–250 bars

Production period 10 years

Control interval 2 years

Maximum water injection rate 1,000 m3/day

Maximum fluid production rate 1,000 m3/day

Minimum distance between wells 250 m

of each approach, in order to assess the average perfor-mance. Each approach was, therefore, applied 20 times tothe appropriate problems.

The first approach was to apply PSO to the vector of alldecision variables (well locations and control parameters)simultaneously. PSO was run for up to 250 iterations, oruntil the average velocity of the particle swarm decreasedbelow a certain threshold. We subsequently used GPS to pollrepeatedly around the best solution found by each run ofPSO, in order to see if it could be improved further. This sec-ond step was not considered to be part of the optimizationapproach, but rather as a test to see how close the solutionsfound by PSO were to being locally optimal.

The second approach was to apply 200 iterations of thehybrid algorithm described in the previous section. The fol-lowing three variants of the hybrid algorithm were applied:

• hybrid-1: Polling was performed every time thePSO step failed to improve the solution. We used thestandard search directions (1).

• hybrid-5: Polling was performed only after PSOfailed five times consecutively. We used the standardsearch directions.

Fig. 1 Permeability andporosity fields used in firstexperiment. Permeability valuesare shown on a logarithmic scale

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• hybrid-5S: Polling was performed only after PSOfailed five times consecutively. We used special searchdirections.

The special search directions used by the hybrid-5Sapproach for this problem are illustrated in Fig. 2. Everyrow of the matrix shown corresponds to one of the 28 vari-ables, and every column represents one search direction.Consider the first 14 directions shown, which act on thefirst seven variables only. These variables correspond to thetwo positional parameters (x- and y-coordinates) and fivecontrol parameters for the first injector. The key differencefrom the standard search directions is that we allow the BHPof the injector to be raised for more than one time periodsimultaneously. The BHP of the injector can only be low-ered, however, for one time period at a time. The rationaleis that raising the BHP in an injector increases flow and isthus more likely to raise oil production than lowering theBHP. For producers, the opposite is true, and so in that case,we allow the BHP to be lowered for multiple time periodssimultaneously. It should be noted that the search directionsused in optimization approaches such as MADS and SPSAwould also permit well control parameters corresponding tomultiple years to be simultaneously altered. These searchdirections would be chosen semi-randomly and without anyparticular knowledge on the problem, however.

In all the hybrid approaches, we scaled the directions cor-responding to positional parameters independently of thecontrol variables, so that the x- or y-coordinates of a wellwere only ever perturbed by one grid space during polling.The idea is that the optimization of the well positions is pri-marily achieved by the PSO step, and that the polling step isaimed primarily at optimizing the controls. Moreover, whenwe perform a poll step during the hybrid approach, we areapplying it to a point that has already been optimized tosome extent by PSO; even early on in the optimization pro-cess, the point is the best out of all the points visited by theswarm. Since well positioning has such a significant impact

Fig. 2 Specialized search directions used by the hybrid-5Sapproach for experiment 1. A red entry corresponds to a value of −1,and a black entry to +1. Dashed horizontal lines separate variablescorresponding to each of the four wells

on the quality of the solution (more so than any single con-trol variable), it seems likely that the well positions at thecurrent best solution are already reasonably good. Thus, asmall perturbation of the well positions may be more likelyto improve the solution than a large one.

The third approach we considered was to decouple theplacement and control components of the problem. Thefirst step of this approach consisted of treating the prob-lem strictly as a well placement problem, by assuming thatthe producers were held at some fixed BHP throughout theproduction period. We used up to 200 iterations of PSO todetermine the optimal well positions under this assumption.Once optimal positions were found, we allowed the controlsto vary and optimized the control using GPS with standardsearch directions. The positions could also be incrementallyadjusted in this second step. This second step ensured thatthe solutions found by the decoupled approach were locallyoptimal. The advantage of the decoupled approach is that itsplits the problem into two smaller problems which are eas-ier to solve than the full problem. A potential disadvantageis that we may find suboptimal solutions by not optimizingover all variables at the same time.

The decoupled approach requires defining fixed BHPvalues to assign to the injection and production wells duringthe first (well placement) phase of the approach. Our defaultchoice was to use the maximum BHP value (450 bars) forinjectors and the minimum BHP value (100 bars) for pro-ducers. For case 1C, where there were maximum constraintson the injection and production rates, we also tried a sec-ond variant where a BHP value of 425 bars was used forinjectors and 125 bars for producers. The rationale for thismodification, which we denote by decoupled-M, is thatusing the maximum and minimum BHP values when plac-ing wells will produce the highest flow rates possible for agiven configuration as wells. If these flow rates exceed themaximum flow constraint, then these configurations will beconsidered as infeasible during the first stage of optimiza-tion, even if they could be made feasible by adjusting theBHP values. The net effect is that the well positions foundin the first phase may tend to place wells farther apart thannecessary, or in regions of lower permeability, in order tosatisfy the flow constraints. Thus, by choosing BHP valuesthat are slightly below or above the maximum and mini-mum values, respectively, we may be able to find betterwell positions.

Experiment 2

The second experiment used a reservoir model providedby the Norwegian University of Science and Technology(NTNU) as part of the Norne benchmark case [27]. The fullmodel of the Norne field is a 46×112×22 grid consisting of44,927 active cells. The reservoir model is subdivided into

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four different formations from top to base, denoted Garn,Ile, Tofte, and Tilje. In order to reduce simulation time, weextracted the seven layers corresponding to the Ile formationto provide a smaller reservoir model, consisting of 15,004active cells. The porosity of the reservoir ranged between 25and 30 % and the permeability from 20 to 2,500 mD. Thereservoir geometry is shown in Fig. 3 (left image). The ini-tial saturation was assumed to be 80 % oil and 20 % water,as in experiment 1.

The reservoir’s irregular shape meant that wells whosepositional coordinates fell within the bounds prescribed bythe grid might not correspond to valid locations in the reser-voir. Thus, any positional coordinates in the (x, y)-planewhich did not correspond to a valid reservoir location wereprojected onto the nearest active cell during the optimiza-tion. This process is illustrated in Fig. 3 (right image). Blackcells indicate grid locations which pass through at least oneactive cell in the z-direction. The three red multiplicationsymbols indicate positions that are invalid, which were pro-jected onto the nearest valid location (indicated by the greenmultiplication symbols).

The goal of experiment 2 was to test the optimizationapproaches on a more complex simulation with a largernumber of variables. The experiment consisted of placing11 wells (four injectors and seven producers) in this fieldand optimizing production over a 15-year time period. Theexperimental parameters are summarized in Table 3. As inexperiment 1, there were two positional and five controlparameters associated with each well, meaning that therewere 77 variables in total. The same economic parameterswere used as in experiment 1 (see Table 1). For this experi-ment, only two cases were considered: case 2A, where therewere no constraints on production, and case 2B, which usedthe constraints given at the bottom of Table 3. A discountingrate of 10 % was used for both cases.

We limited the number of optimization approachesapplied in this experiment to PSO (as a benchmark) alongwith the two that were found to be most effective inexperiment 1—hybrid-5S, and one of the two decou-pled approaches. For case 2A, we used the decoupledapproach, where the well BHPs were held at their max-imum and minimum values; for case 2B, we used the

Table 3 Parameters (top) and constraints (bottom) used in experiment 2

Parameter Value

Fluid viscosities μo and μw 2.4 and 1.0 cp

Fluid densities ρo and ρw 835 and 1,000 kg/m3

Initial reservoir pressure 340 bars

Injector BHP range 350–500 bars

Producer BHP range 150–325 bars

Production period 15 years

Control interval 3 years

Maximum water injection rate 5,000 m3/day

Maximum fluid production rate 3,000 m3/day

decoupled-M modification with injectors held at a BHPof 450 bars and producers held at 200 bars. We performedonly five runs of each optimization approach, due to the highcomputational cost of this experiment. The hybrid algorithmwas run for up to 300 iterations per run, while the decou-pled approach involved up to 250 iterations of PSO duringthe well placement phase, followed by running GPS untilconvergence.

5 Results

The results of both experiments are shown in Table 4. Thesection entitled “NPV” shows the average, best, and worstNPV values over the multiple runs of each approach thatwere performed for each test case (20 runs for experi-ment 1 and five runs for experiment 2). The section entitled“reliability” quantifies the reliability of each approach, byindicating how often the solutions found were within 10 and5 % of the best solution found overall. A reliability value of0.55 in the 10 % column, for instance, indicates that 11 outof the 20 solutions found by that method had an NPV within10 % of the best overall. The section entitled “after GPS”shows the average NPV after the GPS algorithm was appliedto each solution found by PSO and the hybrid algorithm, aswell as the percentage improvement (�%) compared to theoriginal average. These values indicate how close, on aver-age, the solutions found by each algorithm were to being

Fig. 3 Left side, reservoirgeometry of the Ile formationfrom the Norne field. Log ofpermeability (in millidarcy) inthe x–y-directions is shown.Right side, projection of invalidvertical well locations in the(x, y)-plane onto the nearestvalid coordinates

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Table 4 Results of first and second experiments

Case Field Constr. Discount. (%) Algorithm NPV Reliability After GPS

Avg. Best Worst 10 % 5 % Avg. �%

($ × 108) ($ × 108) ($ × 108) ($ × 108)

PSO 5.93 6.38 5.55 0.55 0.20 6.19 4.45

hybrid-1 6.17 6.51 5.77 0.90 0.50 6.18 0.07

1A SPE10 No 10 hybrid-5 6.15 6.44 5.76 0.85 0.50 6.18 0.56

hybrid-5S 6.12 6.46 5.79 0.85 0.30 6.15 0.40

decoupled 6.15 6.30 5.97 1.00 0.20 – –

PSO 8.18 8.57 7.50 0.90 0.50 8.35 2.22

hybrid-1 8.23 8.63 7.49 0.80 0.60 8.25 0.21

1B SPE10 No 0 hybrid-5 8.15 8.59 7.69 0.90 0.45 8.20 0.56

hybrid-5S 8.23 8.59 7.66 0.95 0.60 8.25 0.27

decoupled 8.35 8.64 8.04 1.00 0.65 – –

PSO 5.58 6.06 5.05 0.50 0.15 5.87 5.41

hybrid-1 5.74 6.02 5.35 0.85 0.25 5.75 0.11

hybrid-5 5.73 6.00 5.51 0.95 0.25 5.76 0.45

1C SPE10 Yes 10 hybrid-5S 5.78 6.15 5.44 0.80 0.35 5.88 1.70

decoupled 5.89 6.05 5.63 1.00 0.50 – –

decoupled-M 5.99 6.14 5.80 1.00 0.80 – –

PSO 110 116 101 0.80 0.40 117 5.95

2A Norne No 10 hybrid-5S 112 117 109 1.00 0.40 113 0.22

decoupled 113 117 110 1.00 0.80 – –

PSO 95.9 109 91.3 0.20 0.20 105 9.85

2B Norne Yes 10 hybrid-5S 97.2 98.4 95.9 0.00 0.00 97.6 0.35

decoupled-M 106 112 102 1.00 0.60 – –

The best results for each experiment are highlighted in italic font. Columns headed “Field”, “Constr.” and “Discount.” are included to differentiatethe experiments with respect to the field used, presence or absence of maximum constraints on flow, and discounting rate used to calculate NPV.The constraints for cases 1C and 2B are described in Tables 2 and 3, respectively

locally optimal. These values were not calculated for anyof the decoupled approaches, since the solutions found bythose approaches were guaranteed to be locally optimal.Plots of the mean convergence of the respective algorithmsversus the number of function evaluations (fevals) for bothexperiments are shown in Fig 4.

6 Discussion

Experiment 1

Some general observations can be drawn from the resultsshown in Table 4. First, the decoupled approach was morereliable than the simultaneous approaches (i.e., the PSO andhybrid algorithms). In all three test cases, every solutionfound by the decoupled method was within 10 % of the bestoverall; the hybrid approaches typically scored between 0.8and 0.95, and PSO’s reliability was as low as 0.55 in two out

of the three test cases. The decoupled approach also gavethe most results within 5 % of the best overall for test cases1B and 1C. On average, the NPV of solutions found by thedecoupled approach was better than the other methods forcases 1B and 1C.

This result indicates that reducing the size of the prob-lem by focusing on well placement first, while assumingthat wells are held at or near the extreme BHP values,may help to ensure that one obtains a “good” solution. Thereason may be that this approach allows us to explore anumber of well placement possibilities while holding thecontrols at a configuration that is generally likely to resultin higher production. In the simultaneous approaches, goodwell positions may be missed by the algorithm if the wellcontrols are poorly configured. It is worth noting, however,that for case 1A, the decoupled approach had the poorestperformance of any of the methods, in terms of the bestsolution found.

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1A (No constraint, r = 10%) 1B (No constraint, r = 0%) 1C (Max flow 1000 m 3 /day,r = 10%)

0 2000 4000 6000 8000 10000 120004

4.5

5

5.5

6

x 108 Convergence of NPV (mean)

fevals

NP

V

PSOhybrid−1hybrid−5hybrid−5Sdecoupled

0 2000 4000 6000 8000 10000 120005.5

6

6.5

7

7.5

8

8.5x 10

8 Convergence of NPV (mean)

fevals

NP

V

PSOhybrid−1hybrid−5hybrid−5Sdecoupled

0 2000 4000 6000 8000 10000 120003.5

4

4.5

5

5.5

6x 10

8 Convergence of NPV (mean)

fevals

NP

V

PSOhybrid−1hybrid−5hybrid−5Sdecoupleddecoupled−M

r = 10%)2A (No constraint, 2B (Max flow 5000/300 0 m 3 /day,r = 10%)

0.5 1 1.5 2

x 104

7

8

9

10

11

x 109

NP

V

fevals

Convergence of NPV (mean)

PSOhybrid−5Sdecoupled

0.5 1 1.5 2

x 104

6

7

8

9

10

11x 10

9

NP

V

fevals

Convergence of NPV (mean)

PSOhybrid−5Sdecoupled−M

Fig. 4 Plots showing the convergence of different optimizationapproaches for the three problems of experiment 1 and the two prob-lems considered in experiment 2. The best NPV found is shown as a

function of the number of reservoir simulations (fevals), averaged overall runs for each approach. Note that the scale of the y-axis differsbetween plots

Applying standard PSO to the problem gave the worstresults of any of the methods; it had, by far, the low-est reliability ratings and average NPV for cases 1A and1C. For case 1B, its performance was comparable to thatof the hybrid approaches, but worse than the decoupledapproach. This would seem to indicate that a purely stochas-tic approach is insufficient in addressing the combinedwell placement and well control problem. The “After GPS”column of Table 4 indicates that the solutions found byPSO were usually not even locally optimal and could beimproved significantly (from between 2 and 5 %, on aver-age) by applying GPS subsequently. The solutions found bythe hybrid algorithms, on the other hand, tended to be closeto locally optimal and were typically only improved slightlyby this subsequent application of GPS.

Case 1C was the only test case of experiment 1 to fea-ture constraints on the injection and production rates. Inthis case, we found that the decoupled-M approach out-performed the decoupled approach in every measureof performance. This would seem to validate our hypoth-esis that for constrained problems, choosing BHP values

that are slightly below or above the maximum and mini-mum values when placing wells is preferable to using themaximum and minimum values during the well placementphase. The convergence plot (Fig. 4, rightmost plot) showsthat the NPVs of the solutions found by decoupled-Mare much lower than those found by decoupled duringthe well placement phase. During the control optimizationphase, however, the GPS algorithm was able to significantlyimprove the solutions, to the point that they surpass thosefound by decoupled. Examining the best solutions foundby either approach for this test case indicated that the place-ment of wells was similar, but that the decoupled-Mapproach was able to place wells in regions of higherpermeability, which improved production by reaching themaximum flow rate more quickly. This placement of wellswas not found by the decoupled approach since it causeda constraint violation when BHPs were held fixed at theextreme values.

In terms of the solutions found by each method, the threevariants of the hybrid approach (hybrid-1, hybrid-5,and hybrid-5S) were fairly comparable. Across the three

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Fig. 5 Best solution found forcase 1A. The top left plot showspositions of the four wellsoverlaid on the log-perm field,with a circle denoting aproducer and multiplicationsymbol denoting an injector. Thetwo plots on the bottom showthe control parameters (BHPs)for injectors and producers. Theproduction curves for the twoproducers are shown in the topright plot, with solid linesindicating oil production anddashed lines indicating waterproduction

Case 1A (no constraint, r = 10%)Best NPV = $6.51 × 108

test cases, none of the three were markedly more reli-able or provided better NPVs, on average. The convergenceplots shown in Fig. 4 do indicate that, generally speaking,hybrid-1 required more function evaluations to arriveat a comparable solution to the other two approaches.This indicates that the increased polling frequency of thismethod increased the computational cost of the method

without significantly improving its performance. Betweenhybrid-5 and hybrid-5S, we observed the greatestdifference in case 1C, where the hybrid-5S methodtended to converge more rapidly. This method also hadsomewhat better performance for case 1B, while the per-formance of these two approaches for case 1A was essen-tially the same. We conclude that allowing the pattern

Fig. 6 Best solution found forcase 1B. Symbols used are thesame as for Fig. 5

Case 1B (no constraint, r = 0%)Best NPV = $8.64 × 108

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Fig. 7 Best solution found forcase 1C. Symbols used are thesame as for Fig. 5. The third rowof plots shows the liquidinjection and production rates,along with the maximum valueconstraint (black dashed line)

Case 1C (Max flow 1000 m 3/day, r = 10%)Best NPV = $6.15 × 108

search to raise or lower BHPs over multiple years at atime did accelerate the convergence of the algorithm tosome degree.

The best solutions found overall for each test case areshown in Figs. 5, 6, and 7. Qualitatively, we can see thatwhile the best solutions found for cases 1B and 1C are fairlysimilar to one another, the best solution found for case 1Ais quite different. In case 1A, since there is cash discountingof 10 % and no limit on flow rate, there is a heavy incen-tive to produce large amounts of oil quickly. Furthermore,increased water production in later years is not stronglypenalized. We therefore obtain an optimal solution in whichwells are placed more closely together than in the othertwo cases and where water production rates are quite highat later production times. We note as well that the BHP inone of the injectors is lowered to the minimum level after2 years and held there for much of the production cycle. Thisoccurs because the water front from this injector arrives atthe two producers quickly, and so the BHP is subsequentlylowered to allow the front from the other injector to arrive

before the producers are flooded. One producer is eventu-ally shut in after only 8 years, once the water cut reaches the78 % threshold.

Since case 1B does not include any cash discounting, theobjective is now to maximize the total amount of oil pro-duced over the time period of 10 years, while minimizingthe amount of water produced. As a result, the injectorsare placed farther away from the producers, and the pro-duction rates are slower and steadier than in case 1A. Theoptimal controls in this case are essentially to hold the injec-tor and producer BHPs at their maximum and minimumvalues, respectively.

For case 1C, the positioning of the wells is similar,although the first injector is placed in a region of some-what higher permeability. This allows for a slightly higheroverall initial production rate, which is important since case1C includes discounting. It is clear in Fig. 7 that the max-imum flow constraint of 1,000 m3/day comes into playafter 4 years, when the BHP at the first injector mustbe lowered in order to maintain a flow rate below the

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maximum. The flow constraint is also why the wells areplaced farther apart than they are in case 1A. Althoughboth cases 1A and 1C have the same discounting rate, theflow constraint means that we cannot produce as much oilearly on in case 1C compared to case 1A. Therefore, itis advantageous to place the wells farther apart to delaywater production.

The differences between these solutions give someinsight into the relative performances of our optimizationapproaches. For case 1A, we found that although it wasmore reliable, the decoupled approach was not able to findsolutions that were as good as the best ones found by thehybrid and PSO algorithms. This may be attributable to thefact that the optimal solutions for this test case typicallyinvolved placing the wells close together and in regionsof high permeability, which required varying the BHPs ofinjectors and producers in order to prevent premature waterflooding. Thus, the positions found during the placementphase of the decoupled algorithm, which assumes that wellsare held at the extreme BHP values during the entire pro-duction period, were not the best positions for this particularproblem. In cases 1B and 1C, where the optimal solutionsdid not require varying the control parameters to such agreat degree, the performance of the decoupled approachwas better.

Experiment 2

The results presented in Table 4 indicate that for theunconstrained problem (case 2A), the performance of thehybrid-5S and decoupled algorithms was compara-ble in terms of the solutions to which they converged. Thedecoupled algorithm did score slightly better in termsof reliability, however, and the convergence plot for thistest case (Fig. 4) indicates that its convergence was quickeras well. Both algorithms outperformed PSO by a consid-erable margin. The overall best solution found is shownin Fig. 8. We see again that when a discounting rate of10 % is imposed and there are no constraints on produc-tion, the best solution favors producing large amounts ofoil early on, at the cost of increased water production inlater years.

The results for case 2B, which featured a maximumflow constraint of 5,000 m3/day on the four injectors and3,000 m3/day on the seven producers, clearly indicate thatthe decoupled-M approach used for this test case out-performed the hybrid-5S algorithm. The convergenceplot for this test case indicates that during the well place-ment phase of the decoupled-M algorithm, its perfor-mance lags slightly behind that of hybrid-5S, becausethe BHP values for injectors and producers are heldfixed

Fig. 8 Best solution found forcase 2A. Plot meanings are thesame as for Fig. 5. Note thatseveral lines overlap in the plotsof the BHP values

Case 2A (no constraints, r = 10%)Best NPV = $1.18 × 1010

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at 450 and 200 bars, respectively. The hybrid-5S algo-rithm, meanwhile, is able to explore the full range of BHPvalues. During the control optimization phase, however,the decoupled-M algorithm is able to explore the fullrange of BHP values as well and finds significantly bet-ter solutions than those found by hybrid-5S. Again, theconvergence plot shows that both methods outperform PSO.

The overall best solution found for case 2B is shown inFig. 9. It is clear from the figure that the constraints areactive both with respect to injection and production. In thiscase, the incentive to produce more oil early on is coun-terbalanced by the fact that one can only produce a certainamount of oil per day. Thus, we see that in the best solutionfor case 2B, the producers tend to be placed farther awayfrom injectors than in the best solution for case 2A, to delaywater production. We note as well that the total productionis spread much more evenly among producers than in thebest solution found for case 2A.

It is also apparent that the optimal control scheme in thiscase involves holding injectors fairly close to the maximumBHP value and producers close to the minimum BHP value.This indicates why the decoupled-M approach was moresuccessful than hybrid-5S for this test case; since theoptimal control scheme is fairly simple, maximizing pro-duction for this test case is driven primarily by finding goodwell locations. Thus, the decoupled-M approach bene-fits by reducing the size of the problem initially (from 77to 22 variables) and focusing on well placement. For case2A, we see that the optimal solution involves raising theBHP at several producers to values more towards the mid-dle and even upper values of the permitted range. Thus,well control has more of an impact on the optimal NPV inthis test case, and the benefit that the decoupled approachgains by focusing on well placement is offset somewhatby the fact that the best solutions may not have simplecontrol schemes.

Fig. 9 Best solution found forcase 2B. Plot meanings are thesame as for Fig. 5. The third rowof plots shows the liquidinjection and production rates,along with the maximum valueconstraint (black dashed line).Note that several lines overlap inthe plots of the BHP values

Case 2B (max flow 5000/3000 m3 /day, r = 10%)Best NPV = $1.12 × 1010

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7 Conclusions

We have examined several approaches to simultaneousoptimization of well placement and control, which com-bine PSO with pattern search (GPS). We focused on twogeneral approaches: a hybrid algorithm combining PSOand GPS(based on the previously proposed PSwarm algo-rithm [34]), which we applied to all variables simultane-ously and a decoupled method where PSO was appliedinitially to a well placement problem (assuming a fixedcontrol scheme), and GPS was applied to the controls after-wards. These approaches were applied to a total of five testcases, with some variants of the different approaches beingtested as well.

Overall, we find that there may be benefits to decou-pling the well placement and control aspects of the prob-lem. In three out of five experiments, the decoupled algo-rithm found better solutions, on average, than any of theapproaches that attempted to optimize over all variablessimultaneously. In one case (denoted case 2B), every oneof the five solutions found by the decoupled algorithmwas better than those found by the hybrid algorithm. Wehypothesize that this is due to the fact that during the wellplacement phase, the fixed control scheme assumed by thedecoupled approach is one that is conducive to finding goodsolutions. Thus, by reducing the size of the problem andfocusing on well placement, the size of the solution spaceis reduced, and a more thorough exploration of that space ispossible. Although the solution space of the problem whenall variables are considered simultaneously contains anysolution that our decoupled approach could possibly find,it is harder to find those solutions in the larger space. Thisfinding is similar to results published in some papers onoptimal placement of large numbers of wells, where it waspossible to find better solutions by initially placing wellsaccording to some pattern, rather than allowing all wellpositions to vary freely [29, 30].

One caveat is that the decoupled approach is sensitive tothe control scheme that is assumed during the initial wellplacement phase. In our experiments, we found in one case(1A) that the best solution found by the decoupled approachwas not as good as those found by any of the simultane-ous approaches; the optimal solutions found for this casetended to require raising and lowering BHPs significantly inorder to avoid premature water flooding. In another test case(1C), we compared two variants of the decoupled approachand found that the performance of the algorithm could beimproved by holding BHP values closer to the middle ofthe permitted range, rather than at the extremes. Thus, ifone employs a decoupled approach, some thought should begiven to the assumed control scheme during the well place-ment phase. It is also likely that the effectiveness of thedecoupled approach will suffer for cases where the control

scheme is expected to require varying the control parameterssignificantly.

There are many avenues for further exploration of thejoint placement and control problem. These include mod-eling more complicated well types such as horizontal,deviated, or multilateral wells; incorporating other deci-sion variables in addition to well location and control, suchas the number and type of wells to drill, and schedulingof drilling operations; and finally, modeling of geologi-cal uncertainty. Taking these considerations into accountwill also likely require investigating new optimization tech-niques to account for the increasing complexity of theproblem.

Acknowledgments The authors acknowledge the funding fromthe Natural Sciences and Engineering Research Council of Canada(NSERC), the Atlantic Canada Opportunities Agency (ACOA), andResearch & Development Corporation of Newfoundland and Labrador(RDC). The authors thank the developers of MRST at SINTEF AppliedMathematics for making their software freely available and for pro-viding support. The authors also thank StatoilHydro (operator of theNorne field) and its license partners ENI and Petoro for the releaseof the Norne data used in experiment 2. Further, the authors acknowl-edge the Center for Integrated Operations at NTNU for cooperationand coordination of the Norne cases. The views expressed in this paperare the views of the authors and do not necessarily reflect the views ofStatoilHydro and the Norne license partners. Finally, we thank Obi Ise-bor (Stanford University) for several productive conversations on thistopic.

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