Well placement optimization using an analytical formula-based objective function and cat swarm optimization algorithm

Journal of Petroleum Science and Engineering 157 (2017) 1067–1083

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Well placement optimization using an analytical formula-based objective function and cat swarm optimization algorithm Hongwei Chen a, Qihong Feng a, *, Xianmin Zhang a, Sen Wang a, Wensheng Zhou b, Yanhong Geng b a b

School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong 266580, China CNOOC Research Institute, Beijing 100027, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Well placement Optimization efficiency Analytical formula Objective function Cat swarm optimization algorithm

Well placement optimization is a crucial and complex task in oil field development. Well placement is usually optimized by coupling reservoir numerical simulator with optimization algorithm. This method spends most of computing time in objective function evaluation by reservoir numerical simulator which limits its optimization efficiency. In this work, a well placement optimization method using an analytical formula-based objective function and cat swarm optimization (CSO) algorithm is established. The objective function, derived from fluid flow in porous media and material balance principle, can be calculated by the analytical formula to avoid running reservoir numerical simulator. Then the well placement optimization model is built and solved by CSO algorithm. Three examples are applied to justify the feasibility of the new objective function and the efficiency of this optimization method. Results demonstrate this method can significantly accelerate the speed of well placement optimization process. It can help to determine the optimal well placement more efficiently for actual oilfield development.

Nomenclature a

(continued )

fitting coefficients of oil and water relative r1 permeability curve rw fitting coefficients of oil and water relative rand permeability curve Re

CDC count of dimension to change d fitting coefficients of oil and water relative permeability curve dim dimension index

SD SMP SPC SRD

dp/ dx dSw/ dt DVP FS h

pressure gradient, Pa/m

t0

randomly value in CSO, from 0 to 1 well radius, m random value, from 0 to 1 distance between injector and producer, m water saturation present water saturation average present water saturation standard deviation seeking memory pool self-position considering seeking range of the selected dimension the present time, s

water saturation change rate with time, 1/ s Displacement Vector Parameter fitness values effective reservoir thickness, m

V

pore volume, m3

k

absolutely permeability, m2

b

c c1

fitting coefficients of oil and water relative Sw permeability curve Sw0 acceleration constant Swi

Vi,dim velocity of the ith cat w inertia weight xbest,dim position of cat with the best fitness value xi,dim

* Corresponding author. E-mail address: [email protected] (Q. Feng). http://dx.doi.org/10.1016/j.petrol.2017.08.024 Received 7 January 2017; Received in revised form 18 July 2017; Accepted 8 August 2017 Available online 12 August 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.

position the ith cat in dimth dimension kro krw

relative permeability of oil phase relative permeability of water phase average permeability, m2

Greek symbols α the ratio of injectionproduction region K permeability vector flow rate to the total flow rate min minimum β angle between the direction and the x axis M mutated dimensions θ injection-production region angle MR mixture ratio μo oil viscosity, Pa⋅s N Number of injection-production lines μw water viscosity, Pa⋅s NP number of population in CSO ϕ porosity pe average porosity the bottom hole pressures for injector, Pa ϕ pwf the bottom hole pressures for producer, Pa Pi selecting probability Superscript q flow rate in injection-production region, n iteration step 3 m /s oil production rate, m3/s Subscript qo qw water production rate, m3/s x x direction Q total flow rate, m3/s y y direction r distance to the injector, m z z direction k

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Journal of Petroleum Science and Engineering 157 (2017) 1067–1083

map is built by placing a well in different place to calculate the objective function through running reservoir numerical simulator (Ding et al., 2014). Although these methods can reduce a certain number of reservoir simulations and accelerate the well placement optimization process, considerable number of reservoir simulations and a large amount of CPU computing time during well placement optimization process still trap the optimization efficiency. Optimization algorithm is also the influence factor for well placement optimization efficiency. Some derivate-free optimization algorithms, including Genetic Algorithm (GA) (Emerick et al., 2009; Montes et al., 2001), Simulated Annealing (SA) algorithm (Bangerth et al., 2006; Beckner and Song, 1995), Particles Swarm Optimization (PSO) algorithm (Atashnezhad et al., 2014; Humphries and Haynes, 2015) and Differential Evolution (DE) algorithm (Carosio et al., 2015; Awotunde, 2014), are applied to search optimal well placement. Although GA is one of the most widely used optimization algorithms, it is easily trapped in local optimum when the control parameters is inappropriate (Shieh et al., 2011). SA is a probabilistic method to search the optimal solution imitating the recrystallization process of a heated solid object (Raaymakers and Hoogeveen, 2000). It is difficult to design an optimal cooling schedule and it needs a lot of time in the process of iteration (Raaymakers and Hoogeveen, 2000; Guo et al., 2016). PSO and DE are both population-based stochastic search methods. They can converge to optimal solution with fast speed when algorithm control parameters are set properly (Isebor et al., 2014; Nwankwor et al., 2013; Carosio et al., 2015). In order to improve the well placement optimization efficiency, some hybrid optimization algorithms are generated. One method to construct the hybrid optimization algorithm is combining the derivative-free algorithm, such as GA, and derivate algorithm, such as Conjugate Gradient (CC) (Sampaio et al., 2015). This kind of hybrid optimization algorithm can have a good performance to find the global optimal and local optimal. Another hybrid optimization algorithm construction method is to combine two derivative-free algorithms, such as DE and PSO (Nwankwor et al., 2013), PSO and Mesh Adaptive Direct Search (MADS) (Isebor et al., 2014). This kind of hybrid optimization algorithm can maintain the diversity of the population and improve the ability of producing optimal solution. Cat Swarm Optimization (CSO) is a recently developed heuristic algorithm that mimics the behavior of cats (Chu et al., 2006,2007). For CSO algorithm, there are seven parameters used to control its optimization process. But due to CSO has the advantages of flexibility, fast convergence and producing highly consistent result, it has been applied in many engineering regions and shown more excellent search ability than most derivate-free optimization algorithms, such as GA, PSO, SA and DE, in convergence speed and residual mean square error (Guo et al., 2016; Panda et al., 2011; Pradhan and Panda, 2012). However, to the best of our knowledge, there have been no researches on the application of CSO algorithm to deal with well placement optimization problem. In this paper, we apply CSO to optimize well placement. In order to improve well placement optimization efficiency, this paper proposes a well placement optimization method using an analytical formula-based objective function and CSO algorithm. This new objective function, which represents the displacement balance degree of reservoir, is derived from fluid flow in porous media and material balance principle. It can be calculated by the analytical formula to avoid running reservoir numerical simulator. Then the well placement optimization model is built and solved by CSO algorithm. Three examples are applied to justify the feasibility of the new objective function and the efficiency of this optimization method. The feasibility of the new objective function is validated through the relationship between the new objective function and oil recovery in the first example. A well placement optimization problem is solved by this established method to prove its accuracy and efficiency in the second example. In the third example, this method is applied to optimize well placement in SPE 10 model. Additionally, a hybrid well placement optimization method is developed.

1. Introduction In order to improve oilfield development effect and enhance oil recovery, many technologies including water flooding (Yuan et al., 2016; Yuan and Wang, 2016), CO2 flooding (Yu et al., 2015; Guo et al., 2012) and chemical drive (Bai et al., 2015; Imqam and Bai, 2015) are applied to extract more oil from reservoir. Proper well placement is the essential approach to sweep as much oil as possible and enhance oil recovery. Therefore, well placement optimization is a necessary procedure during oil field development (Wang et al., 2016). In recent years, well placement optimization has been the focus of many researches to deal with the increasing world's energy demand (Dossary and Nasrabadi, 2016; Siavashi et al., 2016; Humphries and Haynes, 2015; Yu and Sepehrnoori, 2014; Ariadji et al., 2014). Optimal well placement is influenced by reservoir geological parameters, fluid properties and development schedule. Well placement optimization problem is usually complex, nonlinear and multimodal (Wang et al., 2016; Dossary and Nasrabadi, 2016). To solve this kind of optimization problem, coupling reservoir numerical simulator with optimization algorithm is the most popular method (Yeten et al., 2002; Güyagüler et al., 2002; Bouzarkouna et al., 2012; Carosio et al., 2015). Reservoir numerical simulator is applied to calculate the objective function such as cumulative oil production and oil recovery by finite difference and finite element solution technology (Sepehrnoori and Yu, 2014; Singh and Srinivasan, 2013,2014). Optimization algorithm is used to control the optimization procedure. Depending on the optimization algorithm and the size of search space, the reservoir numerical simulator will be run many times to find the optimal well placement. And these simulations are time consuming. Nearly 99% total CPU computing time is spent on objective function evaluation by reservoir numerical simulator (Yeten, 2003). In order to accelerate the optimization process, two methods are usually applied. One is to accelerate the speed of objective function evaluation and the other is to improve the efficiency of the optimization algorithm. Streamline-based simulation, proxy model, reduced-order model and quality map are usually applied to reduce the objective function evaluation time (Afshari et al., 2011; Caers, 2003; Cruz et al., 2004; Ding et al., 2014; Jansen and Durlofsky, 2016; Gildin et al., 2006). Streamline-based simulator calculates the development index by implicit pressure and explicit saturation. The oil saturation in streamline-based simulator is calculated explicitly on multiple one dimension streamline (Afshari et al., 2011). It needs less computing time than traditional numerical simulator. Therefore, applying streamline-based simulator to calculate the objective function can accelerate the optimization process (Siavashi et al., 2016). Proxy model is also frequently used to improve the well placement optimization efficiency (Caers, 2003; Pan and Horne, 1998). Based on the proxy model construction method, such as artificial neural network, kriging algorithm and statistical methods, the proxy model for the numerical simulator is built after hundreds of simulating and data training (Güyagüler et al., 2002; Knudsen and Foss, 2015; Onwunalu et al., 2008; AlQahtani et al., 2012). Proxy model is applied to instead of the reservoir numerical simulator to reduce the number of simulations and the optimization time. Another approach to reduce the objective function evaluation time is reduced-order model. Proper orthogonal decomposition (POD)-based reduction and trajectory piecewise linearization (PODTPWL) procedure are two basic methods to represent reservoir state based on a highly reduced set of variables (He and Durlofsky, 2015; Jansen and Durlofsky, 2016). It has been used for many applications including subsurface flow simulation (Gildin et al., 2006). Reduced-order model can speed up the optimization performance at early iterations. Besides, quality map is used to simplify the objective function evaluation process. Net present value, cumulative oil production or production potential is described in quality map. High value regions in quality map are the potential placements to place well (Cruz et al., 2004). The quality 1068

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Results show that the method established in this paper can achieve a good performance in searching optimal well placement with high efficiency and accuracy. 2. Analytical formula-based objective function 2.1. Displacement vector parameter (DVP)

k¼

1 ∫ kðx; yÞds Re Re

(1)

ϕ¼

1 ∫ ϕðx; yÞds Re Re

(2)

Swi ¼

Displacement quality can reflect the development performance. The average water saturation of injection-production region can be employed to investigate the displacement quality of this region. The water saturation of injection-production region is a comprehensive parameter of geological and development parameters. The geological parameters include permeability, porosity, oil and water viscosity, and relative permeability curve. Development parameters contain injectionproduction spacing, production time, pressure difference of injectionproduction, and well control strategy. Considering the anisotropic of reservoir which can be described as a vector of permeability, this comprehensive parameter can be a vector denoting the displacement quality in different directions. Therefore, a new vector parameter is built entitled “displacement vector parameter” briefly called “DVP”, which represents the water saturation of injection-production region taking account of the permeability vector. DVP, as an analytical formula, has two advantages. It can reflect the displacement vector characteristic caused by anisotropic permeability. Besides, it can be calculated only by the analytical formula without running reservoir numerical simulator, which can significantly improve the well placement optimization efficiency. In section 2.2 and 2.3, we derive the formula of DVP and establish the analytical formulabased objective function for well placement optimization problem. And in section 4, we justify the feasibility and merits of this parameter.

1 ∫ Swi ðx; yÞds Re Re

(3)

where k, ϕ, and Swi are the average permeability, porosity and present water saturation between injector and producer, respectively; k(x,y), ϕ(x,y) and Swi(x,y) are the permeability, porosity and present water saturation in position (x,y), respectively; Re is the distance between injector and producer. The element in injection-production region is shown in Fig. 1. Two cases are considered. One case is that the bottom hole pressures (BHPs) of injectors and producers are known, and the other case is that the injecting and producing rates are known. According to Darcy's law, the flow rates of oil phase and water phase for the element are (Zhou et al., 2017):

qo ¼ 

kkro ðSw Þθrh dp μo dr

(4)

kkrw ðSw Þθrh dp μw dr

(5)

qw ¼ 

where qo and qw are oil and water production rate, m3/s; k is the absolute permeability, m2; kro and krw are the relative permeability of oil and water phase; θ is the injection-production region angle; r is the distance to the injector, m; h is the effective reservoir thickness, m; μo and μw are the oil and water viscosity, Pa⋅s; dp/dx is the pressure gradient, Pa/m; Sw is the water saturation. In order to get the analytical formula of DVP, it is necessary to know the relationship between relative permeability and saturation. When the BHPs are known, the relationship of kro(Sw) and Sw should be built firstly. When the control rates are known, the relationship between the ratio of relative permeability of oil phase to water phase kro/krw and Sw should be obtained firstly. There are three types of relationships between kro(Sw) and Sw which are given in Eqs. (6)–(8) (Zhou et al., 1998).

2.2. Analytical formula of DVP 2.2.1. Basic assumption The analytical formula of DVP is derived through fluid flow in porous media and material balance principle. The basic assumptions are made (Feng et al., 2013; El-Khatib, 2012): (1) Each injection-production region is assumed to be homogenous. (2) The reservoir and fluid are incompressible. (3) The fluid only contains oil and water phases, and it obeys Darcy's law. (4) The capillary force and gravity are ignored. (5) The displacement process is non-piston like and isothermal. 2.2.2. The expression of DVP Considering the heterogeneous of reservoir, the average geological parameters of the injection-production region need to be calculated firstly. Because the displacement effect is mainly influenced by the parameters on mainstream line, the average parameters are calculated by Eqs. (1)–(3) (Zhou et al., 2017; Feng et al., 2013).

ðIÞ kro ðSw Þ ¼ a  bSw

(6)

ðIIÞ kro ðSw Þ ¼ að1  Sw Þb

(7)

ðIIIÞ kro ðSw Þ ¼ aexpðbSw Þ

(8)

where a and b are fitting coefficients of oil and water relative permeability curve. The ratio of the relative permeability of oil phase to water phase is

Fig. 1. Schematic of injection-production region. 1069

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represented as tracing mode. In CSO, there are NP cats in a D-dimension search space. The total cats are divided into two groups and they are in two different modes, seeking mode and tracing mode. The ratio of number of cats in tracing mode to the number of cats in seeking mode is decided by a mixture ratio (MR). Each cat has a position and a velocity. And the position of each cat represents a solution to the optimization problem. The position of the cat with best fitness value among all cats is found and signed as the best solution xbest. In order to find a better position, each cat adjusts its own position according to seeking mode and tracing mode. The search mechanism of CSO is similar to that of Particle Swarm Optimization (PSO) algorithm (Guo et al., 2016). In PSO, the position of each particle is updated according to personal best position found by each particle and global best position found by all particles. PSO shows a fast convergence speed (Onwunalu and Durlofsky, 2010; Humphries et al., 2014). The tracing mode in CSO takes the advantage of PSO without considering the personal best position. When there is a good balance between exploration and exploitation of CSO, it can find the optimal solutions with fast convergence and highly consistent result (Guo et al., 2016).

formulated as (Cui et al., 2015):

kro =krw ¼ dexpðcSw Þ

(9)

where c and d are fitting coefficients of oil and water relative permeability curve. Considering the anisotropic of reservoir, permeability can be formulated as (Rasolofosaon and Zinszner, 2002; Feng et al., 2014):



kxx
K ¼

kyx
kzx

kxy kyy kzy


kxz

kyz

kzz


(10)

The DVP is constructed by introducing the vector of permeability into water saturation expression (see Appendix A for further details). DVP expressions are listed in Table 1.

2.3. Analytical formula-based objective function Well placement can be placed in a manner that the value of DVP increases in order to displacement more oil from the reservoir. However, if the well placement is not determined properly, some injectionproduction regions will remain a lot of oil while some regions only remain residual oil. To enhance oil recovery, well placement should be arranged properly such that different injection-production regions uniformly produce oil from the reservoir. Therefore, the more balanced the displacement process is, the better the reservoir development effect will be (Zhou et al., 2017). In order to describe the distribution of DVP values of different injection-production regions, standard deviation (SD) is applied. SD ranges from 0 to 1. If the DVP value is distributed more uniformly between injection-production regions in which case the displacement is more balanced, the value of SDDVP will decrease and trend to zero. If the DVP value is distributed extremely nouniformly, the value of SDDVP will approach to one. Based on the SDDVP values of different injection-production regions, well placement is optimized to reach the minimum SDDVP. Therefore, the analytical formula-based objective function is:

minSDDVP

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # " u N  2 u 1 X jDVPi j  jDVPi j ¼t N  1 i¼1

3.1. Seeking mode The seeking mode represents the process that cat stays in resting state. Cat observes the environment before it finds the next position to move to. The seeking mode is used as a global search technique in the search space. In seeking mode, four important parameters are defined (Panda et al., 2011). Seeking memory pool (SMP) denotes the size of seeking memory for each cat, i.e., the number of candidate positions. Seeking range of the selected dimension (SRD) indicates the mutative ratio for the selected dimension. Count of dimension to change (CDC) defines the number of dimensions to be varied. Self-position considering (SPC) decides whether the present position for the cat will be one of the candidates to move to and it is a Boolen variable. There are five steps involved in seeking mode. Step 1: Generate SMP copies of the ith cat. Step 2: Based on CDC and SRD, change the present position of each copy as Eq. (12) (Guo et al., 2016).

(11)

 xi;dim ¼

3. Cat swarm optimization algorithm CSO, proposed by Chu et al. (2006,2007), is a heuristic and population-based algorithm that retains the natural behavior of cats. Cat has a good hunting skill. There are two sub-modes being mathematically modeled to seek the best solution of complex optimization problem. Cats always remain alert and move very slowly when they are resting (Pradhan and Panda, 2012). This behavior is represented as seeking mode. However, when the presence of pray is sensed, cats chase it very quickly with large amount of energy (Pradhan and Panda, 2012). This mode is

½1 þ ð2  rand  1Þ  SRD  xi;dim xi;dim

; if dim 2 M ; otherwise

(12)

where rand is a random value from 0 to 1; dim represents the dimension index; M means the mutated dimensions which is randomly selected based on CDC; xi,dim is the position of the ith cat in dimth dimension. Step 3: Calculate the fitness values (FS) of all candidates. Step 4: If all fitness values are not exactly equal, calculate the selecting probability Pi for each candidate point according to Eq. (13) (Tsai et al., 2012).

Table 1 DVP expressions of injection-production region. Situation

Type of relative permeability curve

BHP known

ðIÞ kro ðSw Þ ¼ a  bSw

DVP expression 2

0

13 t

2b∫ t ðpe pwf Þdt

DVP ¼ 1b 4a  ða  bSw0 Þexp@ ϕμ

0

2 o Re lnðRe =rw Þ

KA5

2

ðIIÞ kro ðSw Þ ¼ að1  Sw Þb

1 31b t

2aðb1Þ∫ t ðpe pwf Þdt 0 DVP ¼ 1  4ð1  Sw0 Þ1b þ K5 2 ϕμo Re lnðRe =rw Þ

2

ðIIIÞ kro ðSw Þ ¼ aexpðbSw Þ

Rate known

3

DVP ¼ 1b ln4expðbSw0 Þ þ

kro =krw ¼ dexpðcSw Þ

DVP þ

1070

μo 1 μw cd expðcDVPÞ

t

2ab∫ t ðpe pwf Þdt 0 ϕμo R2e

lnðRe =rw Þ

¼ Sw0 þ μμo

w

K5 t

1 cd expðcSw0 Þ

þ

2∫ t αQdt 0

ϕμo R2e h

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Fig. 2. Flow chart of well placement optimization with DVP-based CSO optimization.

Fig. 3. The permeability and porosity map of Example I.

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Fig. 4. Oil and water relative permeability curve.

 Pi ¼

1 ; if FSmax ¼ FSmin jFSi  FSb j=ðFSmax  FSmin Þ ; 0 < i < j

Step 5: Pick the position from N copies randomly, and then replace the position of ith cat.

(13)

3.2. Tracing mode

where FSb¼FSmax when the fitness function is to find the minimum solution, FSb¼FSmin otherwise.

The tracing mode corresponds to the process that cat traces the target with high energy. In this process, the position of cat is changed according to its own velocity for every dimension. The tracing mode is used as a local search technique in the search space. There are three steps involved in tracing mode.

Table 2 Fitting results of oil and water relative permeability curve. Situation

Fitting type

BHP known

kro(Sw) kro(Sw) kro(Sw) kro/krw

Rate Known

¼ a-bSw ¼ a(1-Sw)b ¼ aexp(-bSw) ¼ dexp(-cSw)

Fitting coefficients

Correlation coefficient

a ¼ 1.18 a ¼ 1.99 a ¼ 2.82 c ¼ 17.1

0.9226 0.9973 0.9777 0.9961

b ¼ 1.67 b ¼ 3.01 b ¼ 4.90 d ¼ 5070

Step 1: Update the velocity of ith cat Vi,dim using Eq. (14) (Chu et al., 2006).

Fig. 5. Standard deviation of DVP versus oil recovery under different well placement. 1072

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Fig. 6. SDDVP surface map.

Vi;dim ¼ ω  Vi;dim þ c1  r1  ðxbest;dim  xi;dim Þ

Step 3: The fitness values of all cats are evaluated and the best position among all cats xbest is updated. Step 4: Return to Step 2 until the stop criterion is satisfied.

(14)

where ω is inertia weight; c1 is the acceleration constant; r1 is a random value from 0 to 1; xbest,dim is the position of the cat which has the best fitness value. Step 2: Check whether the velocities are all in the boundary of maximum velocity. If the new velocity is over-range, the new velocity will be bounded to the maximum velocity. Step 3: Update the new position of ith cat following Eq. (15) (Pradhan and Panda, 2012).

xi;dim ¼ xi;dim þ Vi;dim

(15)

3.3. Algorithm description For CSO algorithm, there are general four steps to find the optimal solution. Step 1: NP cats are generated firstly and each cat has the random position and velocity within the boundaries. Step 2: Based on the mixture ratio (MR), cats are randomly categorized into two modes. NP  MR cats perform the process of seeking mode and the other cats perform the process of tracing mode.

Fig. 8. Oil saturation distribution and well placements before optimization.

Fig. 7. Oil recovery map. 1073

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The flow chart of well placement optimization using the analytical formula-based objective function and CSO algorithm is given in Fig. 2. In the next section, we call this method as DVP-based CSO optimization.

4.1. Example I: a 2-D heterogeneous model for analytical formula-based objective function validation This model is a 2-D heterogeneous reservoir with a five-spot well pattern. It is represented by a 60  60  1 uniform grid and the size of this model is 360  360  20 m3. The depth of the top face is 1900 m. This model is a cut-off of SPE10 benchmark model (Christie and Blunt, 2001). Permeability and porosity are presented in Fig. 3. The fluid system is only water-oil two-phase flow without capillary pressure. The oil and water relative permeability curve of SPE 10 benchmark model (Christie and Blunt, 2001) is plotted in Fig. 4. And the fitting results are listed in Table 2. There are four vertical producers and one vertical injector. The BHP for each producer is 18 MPa and the BHP for the injector is 20 MPa. The total production time is 7200 days. These four producers (P1, P2, P3 and P4) have fixed locations in each corner of this model (shown in Fig. 3(b)). According to the production schedule of example I and the correlation coefficient (Table 2), the fitting type of relative permeability for this model is type II and the fitting coefficients are a ¼ 1.99 and b ¼ 3.01. In order to obtain the relationship between SDDVP and oil recovery, we place the one injector in this reservoir randomly except for the corner of the model. SDDVP and oil recovery are calculated respectively at each well location. SDDVP is calculated by Eq. (11) and oil recovery is calculated by reservoir numerical simulator ECLIPSE (GeoQuest, 2010), respectively. The relationship of SDDVP and oil recovery is shown in Fig. 5. There are total 3596 points in this figure. It illustrates that the SDDVP is negatively associated with oil recovery. In other words, the reservoir can reach a higher oil recovery when the SDDVP is less trend to zero, in which case DVP values of injection-production regions are distributed more evenly. Fig. 6 shows the SDDVP surface map and contour map. Fig. 7 shows the oil recovery surface map and contour map. Regions with “low” SDDVP which are symbolled as Region 1 and Region 2 in SDDVP map. Regions with “high” oil recovery which are represented by Region 10 and Region 2’ in oil recovery map. It can be seen that the regions with “low” SDDVP coincide with the regions with “high” oil recovery. During the well placement optimization, the objective function we set is minimum SDDVP which is formulated by Eq. (11). Therefore, if we find the well placement which can achieve minimum SDDVP, the well placement corresponding to maximum oil recovery can also be obtained. It

4. Results and discussion In this section, we validate the feasibility of the analytical formulabased objective function by placing one well in the reservoir randomly to get the relationship between analytical formula-based objective function and oil recovery. Next, the well placement in a 2-D homogeneous reservoir is optimized using DVP-based CSO optimization to validate optimization performance of the new method. Finally, the infilling well placement for a 3-D heterogeneous reservoir is optimized through three optimization strategies and a hybrid well placement optimization method is proposed.

Table 3 CSO algorithm parameters for example II (a). D

NP

MR

SMP

SRD

CDC

SPC

c

Iterations

2

20

0.5

5

0.2

0.8

True

2

20

Fig. 9. Cumulative oil production surface for the exhaustive case.

Fig. 10. Convergence process of DVP-based CSO optimization. 1074

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drilled. This new producer is under BHP control and the bottom hole pressure is fixed to 18 MPa. The reservoir produces 7200 days from now on. This optimal well placement is optimized by DVP-based CSO optimization. Values of CSO algorithm parameters are adjusted according to Table 3. The problem dimension (D) is 2 and the number of population members (NP) is set to 20. The mixture ratio (MR) is equal to 0.5. Seeking memory pool (SMP) is 5. Seeking range of the selected dimension (SRD) is equal to 0.2 and count of dimension to change (CDC) is set to 0.8. Selfposition considering (SPC) is set as true. The acceleration constant (c) is 2. Number of iterations is equal to 20. 10 runs are performed to overcome the stochastic property of optimization algorithm. The optimal well configuration is the well-known five-spot well pattern by infilling this one producer, which can also be seen in Fig. 9 for the resulting cumulative oil production for exhaustive run. As expected, this optimal well configuration can be found by DVP-based CSO optimization. The convergence process is shown in Fig. 10. The CPU computing time of DVP-based CSO optimization is 57 s. The optimal infilling well placement (P5) is in (71, 71) cell and the minimum SDDVP is 0.028864. The optimal well configuration and final oil saturation distribution are shown in Fig. 11. Then this problem is also solved to get the maximum cumulative oil production by coupling reservoir numerical simulator ECLIPSE with CSO algorithm, which is used as a traditional well placement optimization method. This method can also converge at the same optimal well placement and the convergence process is illustrated in Fig. 12. The maximum cumulative oil production is 455.87657  104 m3. However, the CPU computing time of this traditional well placement optimization method is 9217 s which is 161 times higher than the DVPbased CSO optimization. Because DVP-based CSO optimization computes the objective function without running reservoir numerical simulator, it can found the optimal well placement in a very shorter time in comparison with the traditional optimization method. Therefore, DVP-based CSO optimization is an effective and precise method to optimize well placement.

validates the feasibility of the analytical formula-based objective function in well placement optimization problem. 4.2. Example II: a 2-D homogeneous model for DVP-based CSO optimization validation A 2-D homogenous model is applied to confirm that DVP-based CSO optimization method can optimize well placement. This model is divided into 141  141  1 cells. The permeability is 1000 mD and the porosity is 0.3. The depth of top face is 2000 m. The relative permeability of this model is same with example I's. There are four injectors and four producers placed in this reservoir. We assume this field has been developed for 1440 days. The bottom hole pressures of injectors and producers are equal to 22 MPa and 18 MPa, respectively. Positions of the existing wells and present oil saturation distribution can be seen in Fig. 8. (a) Single vertical well placement optimization In order to improve development effect, one producer is needed to be

(b) Two vertical well placements optimization Two wells are planned to be drilled in this reservoir model. One of the infilling wells is injector and the other is producer. The bottom hole pressures for the infilling injector and producer are 22 MPa and 18 MPa

Fig. 11. Optimal well placement and oil saturation distribution after optimization.

Fig. 12. Convergence process of maximum cumulative oil production by coupling reservoir numerical simulator. 1075

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(Nwankwor et al., 2013; Hajizadeh et al., 2010) to find the optimal well locations, respectively. Because CSO and DE are both population-based algorithms and stochastic algorithms, we set the similar parameters to obtain the comparison between the two algorithms. There are 4 parameters used to control the optimization process of DE (Carosio et al., 2015). These parameters include number of particles (N), crossover probability (CR), mutation factor (F) and mutation strategy (Carosio et al., 2015). For this well placement optimization problem, the two algorithms are set the same initial populations, the same number of particles and the same number of iteration steps. The values of parameters for CSO and DE algorithms are listed in Table 4 and Table 5. For the purpose of comparison, each optimization process is performed 10 times to obtain the average performance. Table 6 lists the best objective function, the worst objective function, the average objective function and the relative standard deviation of CSO and DE. Fig. 13 shows the convergence progresses of CSO and DE over 10 runs. It can be observed that the objective function SDDVP obtained by CSO is better than DE algorithm in terms of the best SDDVP, the worst SDDVP and the average SDDVP. And the objective functions of CSO over

respectively. The producing time is 7200 days. We apply CSO algorithm and Differential Evolution (DE) algorithm Table 4 CSO algorithm parameters for example II (b). D

NP

MR

SMP

SRD

CDC

SPC

c

Iterations

4

20

0.3

6

0.2

0.8

True

2

20

Table 5 DE algorithm parameters for example II (b). D

N

CR

F

Mutation

Iterations

4

20

0.9

0.5

current-to-best/1

20

Table 6 Comparison of optimization results for CSO and DE. Algorithm

Best

Worst

Average

Relative Standard Deviation (%)

CSO DE

0.0587 0.0592

0.0591 0.0611

0.0590 0.0598

0.216 1.068

Fig. 13. Convergence progresses of CSO and DE over 10 runs. 1076

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which is set 18 MPa. The optimization production time is 7200 days. We consider three optimization strategies to optimize this problem. Strategy A: optimize this problem only by DVP-based CSO optimization to get the minimum SDDVP. Strategy B: optimize this problem only by coupling reservoir numerical simulator ECLIPSE with CSO algorithm to get maximum cumulative oil production, which is used as a traditional well placement optimization method. Strategy C: optimize this problem by a hybrid method. The optimal solution of DVP-based CSO optimization is used as the initial population of the traditional well placement optimization method to realize the maximum cumulative oil production. It should be noted that this hybrid method combines two kinds of objective function calculation methods, i.e., analytical formula-based method and reservoir numerical simulatorbased method, and it is different with the hybrid optimization algorithms which combine two or more optimization algorithms (Sampaio et al., 2015; Isebor et al., 2014 et al., 2014; Nwankwor et al., 2013). For this well placement optimization problem, simple boundary constraints are taken into account. Other complex constraints such as minimum inter-well distance constraint are not considered during the optimization process. In fact, if two wells are placed too closely, the displacement is not balanced and the cumulative oil production is not high generally (Wang et al., 2016). Complex constraints have been studied by many researchers (Jesmani et al., 2016; Humphries and

the 10 optimization runs are spread out over a less range compared to the results optimized by DE. The cumulative oil productions under the optimal well placements, which are the solutions of best realization of CSO and DE, are 4.56  105 m3 and 4.55  105 m3, respectively. Therefore, the well placements found by CSO algorithm can achieve a more balanced displacement and a more cumulative oil production than DE algorithm. 4.3. Example III: a 3-D heterogeneous model for optimization method The third example is a 3-D heterogeneous model taken from the third to sixth layers of SPE10 benchmark model. We assume the permeability is anisotropic and the permeability in y direction is twice as much as permeability in x direction. This example contains 60  120  4 ¼ 28800 grid cells. The total field size is 600  1200  40 m3. The relative permeability of this model is same with example I's. There are six producers and two injectors. The permeability and well positions of this model are shown in Fig. 14(a). The producers are controlled by a constant BHP of 17 MPa and the BHP for each injector is 21 MPa. We assume this model has been developed for a period of time. The present oil saturation distribution can be seen in Fig. 14(b). In order to improve this reservoir development effect, two producers are needed to be drilled. Therefore, the optimization problem is to place two producers in this reservoir, and both of them are controlled via BHP

Fig. 14. Permeability in x direction and oil saturation of example III. 1077

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initial population of the traditional well placement optimization. Then the final well placement can be found with extra 20 iterations for coupling reservoir numerical simulator ECLIPSE with CSO algorithm. This method totally takes 24321 s to get the optimal solution. The results of optimization process, in terms of convergence processes of these three strategies are illustrated in Fig. 15. The comparison of these three methods results is listed in Table 8. It can be seen that DVP-based CSO optimization can quickly find the best solution, which is 243 times faster than the traditional well placement optimization method and 124 times faster than the hybrid method. The advantage of DVP-based CSO optimization is that the objective function is calculated only by SDDVP formula without running reservoir numerical simulator. The reservoir numerical simulator needs a lot of time to evaluate the objective function as illustrated in strategy B. The final cumulative oil productions for these three strategies are plotted in Fig. 16. And the error values between these three strategies are also shown in this figure. It demonstrates that strategy A can reach 3.75  105 m3 cumulative oil production and the final cumulative oil production resulted by strategy B is 3.82  105 m3 which is same with strategy C. The error values of these three strategies are respectively 1.8%, 0.0% and 0.0%, relative to the maximum final cumulative oil production. Because the SDDVP is calculated based on the analytical formula which is derived from fluid flow in porous media and material balance principle containing some basic assumptions, there is a deviation from the calculating result by reservoir numerical simulator in heterogeneous reservoir. The maximum cumulative oil production corresponding to strategy A is less than strategy B, which differs 1.8%. The results can illustrate that the DVP-based CSO optimization can quickly find the optimal solution and the traditional well placement optimization method can accurately find the optimal solution to get maximum cumulative oil production. Strategy C combines these two methods’ advantages. It can be seen that the computing time of this hybrid method is 1.9 times faster than the traditional well placement optimization. The maximum cumulative oil production of this method is same with strategy B. Therefore, this hybrid method can converge to the optimal solution with high rate and accuracy. The optimization results of three strategies demonstrate that optimization based on reservoir numerical simulator can obtain more oil cumulative production. Therefore, if we do not consider time cost, optimization based on reservoir numerical simulator is a better choice than DVP-based CSO optimization. If we take time cost into account, the hybrid optimization method, i.e., strategy C, is a better choice.

Haynes, 2015) and we do not focus on this topic in this work. The purpose of this case is to illustrate the optimization efficiency and accuracy of proposed method. The total number of function evaluations depends on the complexity of the optimization problem. The probability of falling into local optimum could be limited by the number of populations and iteration steps. For the optimization problem with a rough objective function surface, it needs more number of objective function evaluations to obtain the optimal solution. As previous related researches (Jesmani et al., 2016; Dossary and Nasrabadi, 2016; Nwankwor et al., 2013; Hamida et al., 2017; Carosio et al., 2015; Bouzarkouna et al., 2012), the number of objective function evaluations is from 1250 to 10000 for the optimization problem with less than 10 optimization variables. There are 4 optimization variables in this case. We set the population number is 30 and the maximum number of iterations is equal to 50. The maximum number of objective function evaluations is 1500. Parameter values of CSO algorithm for example III are set according to parameters presented in Table 7. This problem dimension (D) is 4; the number of population members (NP) is 30. The mixture ratio (MR) is equal to 0.1. Seeking memory pool (SMP) is 5. Seeking range of the selected dimension (SRD) is equal to 0.2 and count of dimension to change (CDC) is set to 0.5. Self-position considering (SPC) is set as true. The acceleration constant (c) is 2. Number of iterations is equal to 50. CSO algorithm stops when the velocities of all cats are less than the block size. Each optimization strategy is applied 10 times to get the average performance and we take the average of these 10 times solutions as the optimal well placement. Strategy A which only uses DVP-based CSO optimization meets the convergence condition after 45 maximum iterations. The total CPU-time of this method is 196 s. After 7200 days production, the final SDDVP is 0.12555. Then the two optimal well placements are put into ECLIPSE simulator to calculate the final cumulative oil production. For strategy B, the convergence condition is achieved after 39 maximum iterations. And the CPU-time is 47596 s. The calculation process of objective function is time-consuming which takes most of optimization time. Strategy C is a hybrid method. We use the optimal well placements optimized by DVP-based CSO optimization after 45 iterations as the Table 7 CSO algorithm parameters for example III. D

NP

MR

SMP

SRD

CDC

SPC

c

Iterations

4

30

0.1

5

0.2

0.5

True

2

50

Fig. 15. Convergence of three optimization strategies. 1078

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Table 8 Comparison of three optimization strategies in example III. Strategy Converge step CPU-time, s Maximum cumulative oil production, 105 m3 A B C

45 39 65

196 47596 24321

3.75 3.82 3.82

Fig. 16. Final cumulative oil productions and error values for three strategies.

The computing time of reservoir simulator increases significantly with the grid number increasing while the computing time of analytical formula-based objective function is slightly influenced by the number of grids (Feng et al., 2013). For this reason, the method established in this paper can significantly improve well placement optimization efficiency for large or complicated reservoir models. CSO algorithm is a global optimization algorithm. For this kind of optimization algorithms, they have efficient global optimization ability while the ability of finding the local optimum is relatively weaker. The result found by this kind of algorithms is usually a good solution close to the optimal solution (Sampaio et al., 2015). Therefore, there is also a probability of falling into local optimum when optimization control parameters are set improperly (Panda et al., 2011; Tsai et al., 2012). The optimal well placements (PP1 and PP2) optimized by hybrid method and oil distribution after 7200 days are illustrated in Fig. 17. It can be seen that the oil in the reservoir has been effectively displaced after infilling the two producers.

Fig. 17. Optimal infilling well placements and oil distribution after 7200 days.

optimization model is built and solved by CSO algorithm. Results demonstrate that this method can significantly improve the well placement optimization efficiency compared with the objective function being calculated by reservoir numerical simulator. In addition, a hybrid well placement optimization method, which combines the advantages of objective function evaluated by the analytical formula and reservoir numerical simulator, is also established to realize the maximum cumulative oil production. This hybrid method can converge to the optimal well placement with high rate and accuracy. The method proposed in this paper can help to determine the optimal well placement more efficiently for actual oilfield development.

5. Conclusions In this paper, a well placement optimization method, which uses an analytical formula based-objective function and CSO algorithm, is established to improve optimization efficiency. The objective function is derived from fluid flow in porous media and material balance principle. It can reflect the displacement balance degree in the reservoir and can be calculated by the analytical formula without running reservoir numerical simulator. This new objective function is negatively associated with oil recovery, which validates its feasibility in well placement optimization problem. Based on this objective function, the well placement

Acknowledgment This research is supported by the National Science and Technology Major Project of China (Grant No. 2016ZX05025-001-006).

Appendix A. Analytical formula of DVP The element in injection-production region is shown in Fig. 1. According to Darcy's law, the flow rate of oil phase for the element is (Zhou et al., 2017):

kkro ðSw Þθrh dp qo ¼  μo dr

(A.1)

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where qo is oil production rate, m3/s; k is the absolutely permeability, m2; kro is the relative permeability of oil phase; θ is the injection-production region angle; r is the distance to the injector, m; h is the effective reservoir thickness, m; μo is the oil viscosity, Pa⋅s; dp/dx is the pressure gradient, Pa/m; Sw is the water saturation. When the BHPs of injector and producer are known, integrate Eq. (A.1) between the limits r ¼ rw, p ¼ pw and r ¼ Re, p ¼ pe. It can be written as:

  kkro ðSw Þθh pe  pwf qo ¼  μo lnðRe =rw Þ

(A.2)

where pe and pwf are the bottom hole pressures for injector and producer, respectively, Pa; Re is the distance between injector and producer, m; rw is well radius, m. Based on material balance principle, the flow rate of oil phase can also be formulated as:

qo ¼ V

dSw 1 dSw ¼ ϕθR2e h 2 dt dt

(A.3)

where V is the pore volume, m3; ϕ is the porosity; dSw/dt is the water saturation change rate with time, 1/s. Combine Eqs. (A.2) and (A.3) to get Eq. (A.4):

  k pe  pwf 1 dSw ¼ dt kro ðSw Þ ϕμo R2e lnðRe =rw Þ

(A.4)

Integrate Eq. (A.4) between the limits t ¼ t0, Sw ¼ Sw0 and t ¼ t, Sw ¼ Sw, and it can be written as: S ∫ Sww0

 t  k∫ t0 pe  pwf dt 1 dSw ¼ kro ðSw Þ ϕμo R2e lnðRe =rw Þ

(A.5)

where t0 is the present time, s; Sw0 is the present water saturation. There are three types of relationships between kro(Sw) and Sw which are given in Eqs. (A.6) to (A.8) (Zhou et al., 1998).

ðIÞ kro ðSw Þ ¼ a  bSw

(A.6)

ðIIÞ kro ðSw Þ ¼ að1  Sw Þb

(A.7)

ðIIIÞ kro ðSw Þ ¼ aexpðbSw Þ

(A.8)

where a and b are fitting coefficients of oil and water relative permeability curve. Combining Eqs. (9)–(11) with Eq. (8) respectively, the water saturation expressions under different types of oil and water relative permeability curve are:

"  !# t  2bk∫ t0 pe  pwf dt 1 a  ða  bSw0 Þexp Type ðIÞ Sw ¼ b ϕμo R2e lnðRe =rw Þ " Type ðIIÞ Sw ¼ 1  ð1  Sw0 Þ

1b

þ

(A.9)

 #1 t  2aðb  1ÞK∫ t0 pe  pwf dt 1b

(A.10)

ϕμo R2e lnðRe =rw Þ

"  # t  2abk∫ t0 pe  pwf dt 1 Type ðIIIÞ Sw ¼ ln expðbSw0 Þ þ b ϕμo R2e lnðRe =rw Þ

(A.11)

When the injection and production rates are known, the flow rate of water phase can be formulated as:

qw ¼ 

kkrw ðSw Þθrh dp μw dr

(A.12)

where qw is water production rate, m3/s; krw is the relative permeability of water phase; μw is water viscosity, Pa⋅s. The ratio of the relative permeability of oil to water is formulated as (Cui et al., 2015):

kro =krw ¼ dexpðcSw Þ

(A.13)

where c and d are fitting coefficients of oil and water relative permeability curve. According to Eqs. (A.1), (A.12) and (A.13), the relationship between flow rate of water phase and oil phase is given as: 1080

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Journal of Petroleum Science and Engineering 157 (2017) 1067–1083

μo expðcSw Þ μw

(A.14)

The flow rate q in this injection-production region is the sum of flow rates of water phase and oil phase and it can be formulated as q ¼ qw þ qo. Therefore, the flow rate of oil phase can be calculated by:

qo ¼

q 1 þ μμwo d1 expðcSw Þ

(A.15)

where q is the flow rate in this injection-production region, m3/s. Assuming the ratio of the flow rate of injection-production region to the total flow rate is only depended on the percolation resistance, the ratio can be calculated by (Feng et al., 2013):

1=Ri αi ¼ PN i¼1 ð1=Ri Þ Ri ¼

(A.16)

lnðRei =rwi Þ θi Ki hi ðkroi =μo þ krwi =μw Þ

(A.17)

where αi is the ratio of the ith injection-production region flow rate to the total flow rate; θi is the angle of ith injection-production region. Therefore, the flow rate of the ith injection-production region is:

qi ¼ αi Q

(A.18)

where Q is the total flow rate, m3/s. Based on material balance principle, combine Eqs. (A.3), (A.15) and (A.18) to get Eq. (A.19):

1þ

μo 1 αQ expðcSw Þ dSw ¼ dt ϕAL μw d

(A.19)

Integrate Eq. (A.19) between the limits t ¼ t0, Sw ¼ Sw0 and t ¼ t, Sw ¼ Sw, and it can be written as: t

Sw þ

2∫ t0 αQdt μo 1 μ 1 expðcSw Þ ¼ Sw0 þ o expðcSw0 Þ þ μw cd μw cd ϕμo R2e h

(A.20)

Newton iterative method (Pernice and Walker, 1998) is applied to gain the water saturation when the injection and production rates are known. Firstly, the formula is given as: t

f ðSw Þ ¼ Sw þ

2∫ t0 αQdt μo 1 μ 1 expðcSw Þ  Sw0  o expðcSw0 Þ  μw cd μw cd ϕμo R2e h

(A.21)

Then the derivative function is calculated by:

f ' ðSw Þ ¼ 1 þ

μo 1 expðcSw Þ μw d

(A.22)

The Newton iterative formula is:

Sw;nþ1 ¼ Sw;n 

f ðSw;n Þ f ' ðSw;n Þ

(A.23)

When the absolute difference between Sw,nþ1 and Sw,n is less than 1010, the iterate process stops and Sw,nþ1 is the water saturation of this injectionproduction region. Considering the anisotropic of reservoir, permeability can be formulated as (Rasolofosaon and Zinszner, 2002; Feng et al., 2014):



kxx
K ¼

kyx
kzx

kxy kyy kzy


kxz

kyz

kzz


(A.24)

If the main value of the permeability at x and y direction is known and the vertical flow can be ignored, the permeability can be written as:



k K ¼

x 0


0

ky


(A.25)

According to equivalent displacement principle, the permeability in any direction β in position (x, y) is calculated by (Wang et al., 2005):

kðx; y; βÞ ¼ kx cos2 β þ ky sin2 β

(A.26)

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The DVP is constructed by introducing the vector of permeability into water saturation expression. DVP expressions are listed in Table A.1. And the value of DVP at β direction can be calculated by k(x,y,β) replacing K. Table A.1 DVP expressions of injection-production region. Situation

Type of relative permeability curve

BHP known

ðIÞ kro ðSw Þ ¼ a  bSw

DVP expression 2

0

13 t

2b∫ t ðpe pwf Þdt

DVP ¼ 1b 4a  ða  bSw0 Þexp@ ϕμ

0 2 o Re

ðIIÞ kro ðSw Þ ¼ að1  Sw Þ

2

b

1 31b

DVP ¼ 1  4ð1  Sw0 Þ1b þ

t

2aðb1Þ∫ t ðpe pwf Þdt ϕμo R2e

0

lnðRe =rw Þ

2

ðIIIÞ kro ðSw Þ ¼ aexpðbSw Þ

Rate known

lnðRe =rw Þ

KA5

3

DVP ¼ 1b ln4expðbSw0 Þ þ

kro =krw ¼ dexpðcSw Þ

DVP þ μμo

w

1 cd expðcDVPÞ

K5

t

2ab∫ t ðpe pwf Þdt 0 ϕμo Re2

lnðRe =rw Þ

¼ Sw0 þ μμo

w

K5 t

1 cd expðcSw0 Þ

þ

2∫ t αQdt 0

ϕμo R2e h

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