Optimizing well placement in a coalbed methane reservoir using the particle swarm optimization algorithm

International Journal of Coal Geology 104 (2012) 34–45

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International Journal of Coal Geology journal homepage: www.elsevier.com/locate/ijcoalgeo

Optimizing well placement in a coalbed methane reservoir using the particle swarm optimization algorithm Qihong Feng a, Jiyuan Zhang a,⁎, Xianmin Zhang a, Aimei Hu b a b

School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, China PetroChina Coalbed Methane Company Limited, Beijing, China

a r t i c l e

i n f o

Article history: Received 6 April 2012 Received in revised form 3 September 2012 Accepted 11 September 2012 Available online 19 September 2012 Keywords: Coalbed methane Well placement Particle swarm optimization Reservoir simulation

a b s t r a c t The optimization of well placement in a coalbed methane (CBM) reservoir is challenging and intricate work due to the large number of variables and geological uncertainties. To address these challenges, this paper presents a framework that integrates a reservoir simulator into the particle swarm optimization (PSO) algorithm. The application of the PSO algorithm can greatly reduce optimization time and work volume. In this study, optimizations were conducted of the placement of single and multiple wells by maximizing net present value (NPV) in a synthetic reservoir. The optimizations tracked a field application in a CBM district located in the southeast of Ordos basin. A comparison of optimal well placements, which were determined by the PSO and manual trials respectively, showed complete concurrence for a single well. The effect of swarm size on the convergence speed to an optimal location was analyzed. The results indicated that a minimum swarm size of 10 particles is required to guarantee convergence to the global optima. For the case of multiwell placement, we compared the NPV of the optimized well placement determined by the PSO with 1200 randomly selected well placements and found that none of the randomly set wells surpassed the optimized well placement. The optimization results indicated that higher permeability and well interference have a positive effect on the optimal location. After history matching, the rearrangement of 10 vertical wells and an optimization of infilling scenarios were performed in a 5-year-old CBM district located in the southeast of Ordos basin. The optimization results demonstrated that cumulative gas production (CGP) increased by 22.01%, while cumulative water production (CWP) remained nearly unchanged after optimization. Optimal locations tend to be in regions with higher permeability and/or gas content. The NPVs in six infill scenarios exhibited an initial increase, but later exhibited a decline as the number of infill wells increased from zero to seven. A peak NPV value occurred at one infill well. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Coalbed methane (CBM) has become a substantial supplement to global energy resources. Capturing and utilizing this gas will not only decrease greenhouse gas emissions, alleviate the explosion threat in mining operations, but will also provide an additional energy source that otherwise will be lost (Karacan et al., 2011). The determination of optimal well placement is a crucial task in coalbed methane development. An accurate prediction of the optimal well location is an essential premise for high drainage efficiency, better production rates and satisfactory economics. Optimization work is challenging and intricate because of the large number of variables, especially in the case of multiwell optimization. Ismail (2005) conducted a parametric simulation analysis of well spacing in coalbed methane production and found that closer well spacing can accelerate the dewatering process and enhance recovery. Clarkson and McGovern (2005) developed a new ⁎ Corresponding author. E-mail address: [email protected] (J. Zhang). 0166-5162/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coal.2012.09.004

CBM prospecting tool (CPT) by combining single-well (multilayered) reservoir simulators with a gridded reservoir model, Monte Carlo (MC) simulation, and economic modules. Optimal well spacing was determined by their model. They also used an integrated approach to optimize production of developed reservoirs by including infilling well scenarios. Karacan et al. (2007) investigated the effects of different horizontal methane drainage borehole patterns, borehole lengths and degasification times on reducing methane emissions by using “dynamic” 3D reservoir modeling of a 381-m-wide longwall panel in a Pittsburgh coalbed. The results of their study showed that dual- and tri-lateral boreholes are more effective in decreasing emissions and shielding entries compared with fewer and shorter cross-panel horizontal boreholes set parallel to the longwall face. Chen et al. (2010) investigated the effect of permeability anisotropy on the optimal layout of CBM multi-lateral well through numerical modeling and reservoir simulations. The results demonstrated that the optimal well direction of the quad-lateral well is parallel to the butt cleat direction whereas the optimal main well angle of the pinnate lateral well is significantly affected by the permeability anisotropy ratio. Reservoir modeling conducted by Keim et al.

Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

(2011) concluded that well patterns can be optimized to maintain high methane-production rates in low permeability coals (less than 1.0 mD). In their work, the combined relationships of horizontal lateral spacing and permeability were developed to maximize production. A positive correlation between lateral spacing and permeability was demonstrated. In the majority of the research mentioned above, much emphasis was placed on parametric analysis or optimization for a specific geological case, and optimizations were conducted artificially by a reservoir simulator. Although this method is accurate, it is not feasible due to the number of simulations required (Guyaguler, 2002). To address this problem, various approaches have been proposed for conventional reservoirs. Thus, the application of optimization algorithms has gained popularity over recent years. By optimizing the schedule and location of horizontal wells with fixed orientations in oilfields, Beckner and Song (1995) first applied the simulated annealing (SA) algorithm to maximize the NPV. The genetic algorithm (GA), which is a most popular optimization algorithm, has also been used by researchers for well placement optimizations (Guyaguler, 2002). Bouzarkouna et al. (2011) recently applied an optimization methodology for determining optimal well locations and trajectories based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). In this paper, the PSO algorithm was applied to optimize vertical well placement in coalbed methane reservoirs. PSO algorithm was first used in well placement optimizations by Onwunalu and Durlofsky (2010) for conventional reservoirs with depletion mechanisms that differ greatly from CBM reservoirs. However, the work of Onwunalu omits the optimization of infilling well placement in an already developed reservoir; this issue is addressed in our study. 2. PSO algorithm Since first proposed by Kennedy and Eberhardt (1995), the PSO algorithm has been applied in various fields, such as system design, pattern recognition, and signal processing. The algorithm was derived from research on the social behaviors of bird foraging. The PSO algorithm is similar to the GA in the following ways: (a) the system is initialized with a population of random solutions and (b) both algorithms are global search stochastic computational models of natural or physical processes that do not require the computation of derivatives. However, compared with other evolutionary computational algorithms, the PSO algorithm has its own features: (a) each individual (swarm) is assigned a randomized velocity and flies through the solution space; (b) each swarm is capable of memorization; and (c) the evolution of particles is dependent on the cooperation and competition between different particles. The PSO algorithm can be used to solve nonlinear, nonderivable and multipeak optimization problems for high-efficiency parallel optimizations. Due to its simplicity and requirement of few parameters, the PSO algorithm has experienced increased popularity and rapid development, causing different versions of the algorithm to emerge in recent years. Unlike the genetic algorithm, which uses three operators called selection, crossover and mutation (Goldberg, 1989; Michalewicz, 1996), an individual in the PSO algorithm is treated as a particle (point) without mass or volume in the multidimensional solution space. Each particle “flies” within the solution space to search for a position that maximizes or minimizes an objective function (fitness). The particle changes direction and velocity based on its experiences and the experiences of its neighboring particle(s). Each particle modifies its marching direction and speed to follow its individual optimal fitness and the swarm in the process of iteration. Therefore, a positive feedback mechanism is established in the chasing process for the optimum among the swarm. In the PSO algorithm, the position of each particle represents a solution to the optimization problem. The flying direction and speed are determined by a velocity vector that is updated during each iteration. We denote X as a potential solution in the search space of a d-dimensional

35

k k k k optimization problem, Xik = {xi,1 , xi,2 , xi,3 , …, xi,d } as the position of the k k k k ith particle at kth iteration, and Vik = {vi,1 , vi,2 , vi,3 , …, vi,d } as the corresponding velocity (position change). The velocity and position of the ith particle at the next iteration k + 1 are updated according to the following algorithm:

kþ1

k

kþ1

xi;d ¼ xi;d þ vi;d ⋅Δt kþ1

k

ð1Þ

vi;d ¼ ωvi;d þ c1 rand1

k



 k k k k k pi;d −xi;d þ c2 rand2 gd −xi;d

ð2Þ

where pki,d is the ith particle's best solution (with the optimal fitness) up to kth iteration; g ki,d is the global best solution found by the whole swarm up to kth iteration; ω, c1 and c2 are weights; and rand1k, rand2k are uniformly distributed random variables in the range [0, 1]; Δt is a time increment. To be consistent with stand PSO algorithm, Δt is set to be 1 (Onwunalu and Durlofsky, 2011). To accelerate convergence, Clerc (1999) further proposed the constriction of the PSO algorithm, where a constriction coefficient is applied to terms on the right side of Eq. (2) as follows: h

i kþ1 k k k k k k k vi;d ¼ χ ωvi;d þ c1 rand1 pi;d −xi;d þ c2 rand2 g d −xi;d

ð3Þ

where χ is the constriction coefficient and defined as 2k k k χ ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and φ ¼ c1 rand1 þ c2 rand2 > 4: 2−φ− φ2 −4φ  

ð4Þ

Typically, k is set to 1 and c1 = c2 = 2.05; and the constriction coefficient is 0.7298 (Clerc and Kennedy, 2002). Shi and Eberhart (1998) suggested the use of an inertia weight, which decreases over time, in the range of ωmax = 0.9 to ωmin = 0.4. This methodology has the effect of narrowing the search, and gradually changes the model from an exploratory mode to an exploitative mode (Li and Engelbrecht, 2007). The decrease strategy proposed by Yasuda et al. (2003) is utilized and shown as follows: ω ¼ ω max −

k ðω max −ω min Þ K

ð5Þ

where k is the present iteration and K is the preset maximum number of optimization steps. 3. Implementation of the PSO algorithm for well placement 3.1. Integration For the specific optimization problem in this study, the objective is to find the optimal location of wells that maximize or minimize a given objective function. For a 2D reservoir model, the location of a well is usually expressed by the coordinates (xζ, xη) of the grid block where it is placed. Hence, for a sum of n wells, a total of 2n variables need to be optimized to obtain optima. For a swarm with an amount of Ns particles (referred to as swarm size), each particle is designated to carry 8 columns of variables, including velocity components, position coordinates, optimal positions up to k+ 1th iteration, present fitness (objective function value) and highest fitness up to kth iteration. The optimization control variables are stored as a matrix: 2

kþ1

v1;ς

6 kþ1 6 v2;ς 6 M¼6 6 vkþ1 6 3;ς 4 ⋮ vkþ1 N s ;ς

kþ1

kþ1

kþ1

kþ1

kþ1

kþ1

v1;η

x1;ς

x1;η

p1;ς

p1;η

f it 1

vkþ1 2;η

xkþ1 2;ς

xkþ1 2;η

pkþ1 2;ς

pkþ1 2;η

f it kþ1 2

vkþ1 3;η ⋮ vkþ1 Ns ;η

xkþ1 3;ς ⋮ xkþ1 N s ;ς

xkþ1 3;η ⋮ xkþ1 Ns ;η

pkþ1 3;ς ⋮ pkþ1 N s ;ς

pkþ1 3;η ⋮ pkþ1 Ns ;η

f it kþ1 3 ⋮ f it kþ1 Ns

k

f it 1

3

7 f it k2 7 7 7 f it k3 7 7 ⋮ 5 k f it Ns

ð6Þ

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Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

where fitik represents the highest fitness value of the ith particle up to kth iteration; fitik+1 is the fitness value at k +1th iteration. In the case of the optimization of n wells, each of the velocity or position component is a one-dimension array with n columns that designates the corresponding property to each well. The computational procedure (Fig. 1) can be described as follows. Step 1. Initialization. • Initialize all position vectors (the 3rd and 4th columns in matrix M) randomly in the feasible reservoir space; • Set all velocity vectors (1st and 2nd columns in matrix M) to zero; • Initialize each particle's best solution by assigning the 3rd and 4th column vectors to the 5th and 6th column vectors in matrix M, respectively; • For each particle, run the reservoir simulator and compute the objective function based on the initialized position; the objective function values are assigned to the 8th column vector. Step 2. Update the positions for all particles using Eqs. (1), (3), (4) and (5); Step 3. Run the reservoir simulator and compute the objective function for each particle. New objective function values are stored in the 7th column in matrix M; Step 4. If fitik+1 > fitik, assign the 3rd and 4th column vectors to the 5th and 6th column vectors in matrix M, respectively; otherwise replace the 7th column vector with the 8th column vector; Step 5. Screen out the global optimum by comparing the objective function values (at k + 1th iteration) of all particles;

Step 6. Check if the maximum number of iterations is met. If not, go to Step 2, otherwise exit the optimization loop. 3.2. Objective function The objective functions previously used for optimizing well locations include NPV, cumulative gas (oil) production, and average field pressure drawdown, etc. (Hazlett and Babu, 2005). Assuming that drilling cost is independent of well location and time, the expression for NPV is defined as in Eq. (7) t

NPV ¼ ∑

330n

∑

n¼1 j¼330ðn−1Þþ1


 Q g Pg  C g

 1 1 n −Q w C w n −mC F ð1 þ iÞ ð1 þ iÞ ð7Þ

where Qg and Qw are daily gas and water production rates; Cg and Cw are the processing costs per unit volume of gas and processing costs per unit volume of water per day, respectively; i is the discount rate; Pg is the price per unit volume of gas at wellhead; CF is the total drilling cost for a single well; n and j represent the annual metered drainage time and daily metered drainage time, respectively; and m is the total number of wells. 3.3. Infeasible particles' constraints The determination of well location in a coalbed methane reservoir is a typical boundary-constrained optimization problem. There is the

Fig. 1. Diagram of the computational procedure for the determination of the optimal well position.

Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

37

potential for particles to fly out of the reservoir boundary if not subjected to constraints. The analysis of constrained optimization problems is a rather common challenge to evolutionary algorithms. To address this challenge, techniques such as the preservation of feasible solutions (Hu and Eberhart, 2002) and the use of penalty functions (Michalewicz, 1995) are introduced. In this study, the absorb technique (Onwunalu and Durlofsky, 2010) is applied, which was found to perform the best in Carlisle and Dozier (2001) and Helwig and Wanka (2008). In this technique, once a particle flies out of the feasible region, variables are reset to the nearest boundary. We refer to this constraint as the outer constraint because it is primarily concerned with the outer boundary of the reservoir. In Eqs. (8) and (9), Xmax and Xmin represent the lower and upper boundaries respectively. ( kþ1

xi

¼

kþ1

X min if xi < X min X max if xkþ1 > X max i

ð8Þ

The particle velocity also needs to be constrained because an immense velocity may drive the particle out of the optimal point and a minimal velocity may result in local optima. To prevent these phenomena, the following constraint strategy was utilized: ( kþ1

vi

¼

0 v max

kþ1

kþ1

if xi < X min or xi if vkþ1 > v max i

> X max

ð9Þ

where vmax is set to twenty grid blocks in this study. Another constraint that prevents reduplication of wells in a particular spot is also needed. In optimizing the placement of multiple wells, wells have the chance to overlap one another, which does not occur in actual field applications. In this case, the inner constraint is invoked, where the distance between any two wells is confined to a minimum well spacing once they overlap, i.e., wells will be moved to a spot that has the minimum well spacing once the distance between any two wells is less than the preset minimum. 3.4. Reservoir simulator The development of a numerical simulator for simulating gas and water transportation in coal seams is the premise for optimizing well placement. Though commercial simulation tools such as GEM and Eclipse have been widely used in optimization, the integration of any of these software packages in an auto-optimization program is usually problematic. As a result, a three-dimensional two phase (gas and water) CBM simulator was developed in the presence of the matrix shrinkage effect (Clarkson et al., 2010; Palmer and Mansoori, 1996; Shi and Durucan, 2004) and following the work of Zhang and Tong (2008). The simulator is essentially a dual-porosity, compositional model that is capable of simulating the sorption and diffusion phenomenon in matrix and fluid flow through the cleats. The source code of the simulator is incorporated directly into the PSO algorithm. 4. Optimization runs and results 4.1. Optimization of a single vertical well 4.1.1. Effect of swarm size on convergence An analysis of the effect of swarm size on convergence was performed in a synthetic reservoir model with strong permeability heterogeneity (Fig. 2). Key input parameters for the synthetic reservoir model are listed in Table 1, and economic parameters are listed in the first row of Table 2. Swarms with 5, 10, 20 and 40 particles were utilized in the current analysis. Due to the stochastic nature of the PSO (Onwunalu and Durlofsky, 2010), 15 optimization runs were repeated for each swarm size to draw a general and reliable conclusion.

Fig. 2. Permeability (kx) field for the synthetic model.

In this case, the total number of grid blocks is 1089 and each grid block is a potential solution to the optimal well location issue. By conducting 1089 simulation runs, we've found that the optimal well position that maximizes the NPV is located in grid block (24, 29), with a corresponding NPV of $3.459×105. The swarm with 5 particles got the global optimum in 7 runs and fell in local optima in 8 runs; the other swarms succeeded in finding the global optimal location in all of 15 runs respectively. This indicates that a minimum swarm size of 10 is required to avoid local optima and guarantee convergence to global optimum. Fig. 3 summarizes the maximum, minimum and average number of simulations required to achieve global convergence with various swarm sizes. Swarms with 5, without regard to runs trapping in local optima, and 10 particles achieved the global optima after a minimum number of simulations of 10 and 20, respectively; the global optima for swarm sizes of 20 and 40 were located after a minimum of 20 and 40 simulation runs, respectively. Both the average and maximum simulations increased with swarm size. However, Fig. 4 illustrates that the average number of required simulations and iterations for getting the global optima shows distinct trends with swarm sizes. This discovery leads to the practical conclusion that the more particles there are in the swarm, the fewer iterations are required to achieve convergence at the cost of additional simulation runs in an iteration, which is consistent with the notion by Trelea (2003). In the following study, the swarm size is set to 10 for the optimization of a single well and is set to 20 for multiwell location optimizations.

Table 1 Property parameters of the synthetic model. Parameters

Value

Grid blocks Grid block size (m2) Thickness (m) Matrix porosity (%) Longmuir volume (m3/t) Longmuir pressure (MPa) Gas concentration in cleats (%) Gas content in matrix (m3/t) kx/ky Initial in situ pressure (MPa) Cleat porosity (%) Coefficient of diffusion (m2/d) Bottomhole pressure (MPa) Skin factor Wellbore diameter (m) Reservoir temperature (°C) Gas gravity Simulation time (day)

33 × 33 20 × 20 4 2 21.6 2.0 0 12.6 3 5.1 2 0.002 0.3 −2.5 0.1 32 0.57 4000

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Table 2 Economic parameters used for orthogonal test (after Zhang, 2010). No.

Gas price ($/m3)

Water disposal ($/m3)

Discount rate (%)

Drilling costa ($/well)

Commodity rate

Time (days)

1 2 3 4 5

0.256 0.100 0.413 0.569 0.725

0.98 0.67 0.83 1.14 1.30

15 7 11 19 23

109,375 140,625 78,125 171,875 203,125

0.95 0.86 0.89 0.92 0.98

4000 1000 2000 3000 5000

a

Including drilling, cementation and completion.

4.1.2. Evolution of particles in the optimization process Despite the difference in initialization, results showed that under same swarm size, optimization runs bear similar convergence trend. Hence, we take one of the runs with 10 particles to illustrate how the particles succeed in finding the optimal location. Fig. 5 illustrates the evolution of 10 particles in the process of searching for the optimal well placement. Ten particles were initialized on the plane of the reservoir with a randomized distribution (Fig. 5(a)). After certain iterations of interaction among the swarms, particles tended to gather around the potential optimum location point (Fig. 5(b)). These two stages can be referred to as exploring modes; afterward, the particles switched into an exploiting mode and began searching within a relatively smaller region rather than within the entire reservoir domain. In the exploiting process, particles were observed to gradually overlap one another (Fig. 5(c) and (d)) in a small region around the ultimate optimal well location. This trend predicts that all particles will fly toward the optimum position and will remain there if the iterations are set broadly enough. 4.1.3. Sensitivity analysis of economic parameters To examine the effect of economic parameters on the objective function value and optimal placement, an orthogonal test was conducted. Table 2 summarizes the parameter combination used in the orthogonal test. Six factors, including gas price, water disposal costs, discount rate, well capital, commodity rate and time period, were analyzed; each factor was set with five levels. The orthogonal test table L25 (56) was applied to the test design. Ranges and variance ratios for the six factors were adopted as two evaluation indices. Both indices exhibit similar trends. Gas price exhibited the highest range (2.89) and variance ratio (810,776), followed by time period (2.20 for range and 737,084 for variance ratio). The other four factors, including water disposal cost, drilling cost, commodity rate and discount rate, demonstrated relatively low ranges (from 0.36 to 0.05) and low variance ratios (from 264,075 to

Fig. 4. Average number of simulations/iterations for convergence with different swarm sizes.

111,838). Because a higher range or higher variance ratio indicates a higher degree of influence of the corresponding factor on the objective function, these results show that gas price and time period exert a greater influence on NPV, whereas the other four factors exhibit a smaller influence on NPV. The optimal well locations determined by the PSO algorithm and their corresponding frequencies under 25 combinations of economic parameters were examined. Twenty-two of the 25 experiments resulted in optimal well placement in grid blocks (27±3, 30±2). Optimal well placements in the other three cases were in grid blocks (9, 30), (19, 31) and (30, 23). Quantitative analysis for the purpose of revealing the underlying mechanism seems to be infeasible because the current problem involves a spatial distribution that cannot be described quantitatively. However, it is noted that in 24 experiments (96% of the total), the optimal well placements fell in the top right region of the reservoir with relatively greater permeability (Fig. 2). The simulation by Chen et al. (2010) showed that increasing the absolute permeability could shorten the dewatering stage of the coalbed methane production and locating more wells in butt cleats direction is highly efficient for coalbed methane production. Gu and Chalaturnyk (2005) presented that permeability was found to be one of the most sensitive parameters that influence CBM production. Rozak and Bustin (2001) reported a positive correlation between CBM production and permeability factors using production records and well data from 35 wells in the San Juan Basin. Other researches (Close et al., 1990; Ried et al., 1992; Shi et al., 2002) also confirmed the key control effect of permeability on CBM production. Those previous simulations and field observations indicate that under analogous conditions, the region with higher permeability yields higher gas and dewatering rate, and consequent higher corresponding NPVs in that region. 4.2. Optimization of 4 vertical wells

Fig. 3. Number of simulations for convergence with different swarm sizes.

Optimization of the placement of 4 vertical wells by maximizing the NPV was carried out in the previous synthetic model. Similarly, to achieve reliable conclusions, a total of ten optimization runs were conducted. Fig. 6 showed that the ultimate converged NPVs, after the 40 iterations (800 runs), vary among the ten experiments (3 runs with moderate NPVs were left out for clear display). The highest and lowest NPVs were $1.851 × 10 6 and $1.735 × 10 6, respectively. Moderate NPVs range between those two bounds. Despite the variation in ultimate converged values, the convergence trends appear similar: all runs ended with reaching a convergence after an analogical stage of exploring, i.e., the iterations required to reach a convergence are very close. The maximum and

Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

39

Fig. 5. Evolution of particles in the optimization process.

minimum required iterations among ten runs are 17 and 14, respectively. Onwunalu and Durlofsky (2010) reported a similar phenomenon when conducting an optimization of the placement of 20 wells in a conventional oil reservoir. These results indicated the efficiency of PSO algorithm and highlighted the necessity of conducting multiple optimization runs when facing multiwell optimizations. The creation of a surface map that displays all potential solutions seems to be a logical way to examine whether the PSO algorithm was

successful in determining the global optimal placement of 4 wells. Although the amount of grid blocks is relatively small, the construction of a surface map that shows all potential NPVs of different combinations 4 of the 4 wells requires C1089 ≈ 5.8 × 1010 simulation runs and is likely to be a time-consuming and expensive task. To accomplish the test, 1200 sets of stochastic placements of 4 wells were generated and run. Fig. 7 demonstrates that all the stochastically selected locations generated a lower NPV than the optimized placement, which implies a promising

Fig. 6. NPVs for ten optimization runs.

Fig. 7. Comparison of NPVs by optimal and stochastic placements.

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Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

performance of the PSO algorithm. In spite of a limited number of runs, the results were credible because their placement was randomly chosen from a statistic perspective. The optimal placements of four wells, with the highest and lowest NPVs in the ten optimization runs, are illustrated in Fig. 8. In the two runs, three wells (Well-a, -b, and -c) were located in analogous positions. Only one well (Well-d) was observed to have been moved by the particles to generate a higher NPV. As shown in Fig. 8(a), a greater degree of depressurization occurs within the area controlled by wells compared with Fig. 8(b), which indicates stronger interference between wells. Dunn (1984) confirmed the favorable effect on CBM production of interference after examining well performance in the Mary Lee/Blue Creek coal seam. Gentzis and Bolen (2008) stated that constructive interference can promote depressurizing the coal reservoirs and accelerating gas production over short periods of time. Studies by Chaianansutcharit et al. (2001), Remner et al. (1986), Young et al. (1991), and Zulkarnain (2005) also indicated the positive effect of interference on methane production. Under such basis, more gas is desorbed and produced on the surface with the placement in Fig. 8(a), and subsequently the NPV is higher. Also, it is interesting to point out that, the optimal NPV generated by four wells is greater than four times that by one well. This further confirms the benefits of well interference on gas production improvement.

exinite or semivitrinite. The content of vitrinite in the three seams is generally in the range of 50% to 60%, and the content of inertinite is generally in the range of 40% to 60%. Other properties of the three coal seams are summarized in Table 3.

5. Field application

5.2. History matching of the field data

In this section, a study was conducted of the well rearrangement and infilling in the Hancheng district in the southeast margin of Ordos basin (Fig. 9), using the PSO algorithm.

Impacted by uncertainties, which are derived from complexity, diversity of the coal formation condition, degree of accuracy in exploration and test method limitations, the parameters acquired during the test process may not reflect the true in situ conditions. Once these parameters are employed in the prediction of well performance, reliability and exactitude cannot be guaranteed. To assess the accuracy of the geological model and formation parameters, history matching for 10 vertical wells in the study area was conducted. Because CO2 is not engaged in the project, pure methane was assumed to be adsorbed on the matrix. Langmuir constants were identical within each seam. Parameters including the porosity, permeability, gas content, thickness, top depth and in situ pressure of the three seams were obtained from logging data of the drilled wells. Permeability data with anisotropy were unavailable and are assumed to identify along the butt cleats and face cleats in the history matching. The relative permeability of CBM reservoirs to gas and water had a considerable effect on CBM production characteristics (Clarkson et al., 2011). Because the curve was unavailable, the data was adjusted by

5.1. Geological setting 5.1.1. Coal seams The strata in the study area include the Cambrian, Ordovician, Carbonic, Permian, Triassic and Quaternary systems. The coal-bearing series includes the Taiyuan Formation of the upper Carbonic system and the Shanxi Formation of the lower Permian system. Drilling data and seismic data indicate that the No. 3 coal seam in the Shanxi Formation, and the No. 5 and 10 seams in the Taiyuan Formation are the elite mining formations, with a relatively gentle distribution along the plane. The thicknesses of the No. 3, 5, and 10 seams vary in the range of 1.13 m to 3.8 m, 0 m to 7.7 m and 2.0 m to 10.8 m, respectively. Most of the coals in this study area are classified as semibright durain coals, which mainly bear vitrinite and inertinite and rarely contain

5.1.2. Tectonic The study area is located at the southeast margin of Ordos basin and at the east margin of the Weibei uplift, bounded by the Hancheng normal fault to the south and the Weihe normal fault to the east. Under the impact of these two faults, structure in the middle deep section is relatively simple, and slips and folding rarely occur. The bottom structure of the elite coal seams slopes in the southwest direction. 5.1.3. Well placement Drilling tests conducted in earlier stages have been successfully applied to the degasification of vertical and multilateral wells in the Hancheng district. Thus far, approximately 250 exploitation wells have been drilled, including well groups WL1, WL2 and H-3. Ten vertical producers, deployed in a cuboid area, were selected for our study from group WL1. The regularity in the original well pattern, as illustrated by Fig. 10, demonstrates that the wells are placed approximately parallel to the bottom-right boundary.

Fig. 8. Optimal locations of 4 wells with the highest and lowest NPVs, with a background of pressure field at 4000 days.

Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

41

Fig. 9. Map of the study area (after Yang et al., 2009).

trial and error (Fig. 11) until the simulated gas and water rates agreed with the field data within an acceptable range. To stimulate the formation, hydraulic fracturing was conducted, which was expected to enhance near wellbore permeability. The hydraulic fractures were assumed to extend along the grid blocks parallel to the X-coordinate in the simulation. Half-length and permeability were set to 130 m and 20 mD, respectively. History matching of gas and water rates was conducted for each single well. History error for single well, defined as the ratio of difference between the simulated and field cumulative production to the field cumulative production, was summarized in Table 4. Field CGP and CWP errors were determined to be 5.18% and 8.34%, respectively. History matching results in a field scale were illustrated in Fig. 12.

Table 3 Properties of the three coal seams in the study area. Coal seam no.

3

5

11

Ro,max % Average pressure (MPa) Average gas content (m3/t) Langmuir pressure (MPa) Langmuir volume (m3/t) Yong's modulus (×104 MPa) Poisson ratio

1.85–2.05 2.63 13.16 2.27 22.18 1.245 0.35

1.90–2.12 2.39 8.46 2.04 24.52 0.767 0.28

1.89–2.07 2.65 6.63 1.37 20.02 1.192 0.33

Fig. 10. Original placement of wells in the study area. The gray dots represent well locations.

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Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

a sub-region with higher gas content (Fig. 13(b)). Lv et al. (2012) stated that permeability and gas content are two key factors that affect the productivity of CBM wells. Combined with the previous results, the two observations in our study concluded that higher permeability and gas content are compatible with optimal well placement when gas production is considered a priority.

Fig. 11. Relative permeability to gas and water used in the history matching.

5.3. Development scenario 5.3.1. Comparison of cumulative productions with original and optimized placements Gas and water productions were compared by using the original and optimized well placements. Before conducting the optimization, all original wells were removed from the model and the field was assumed to be a virgin frontier. Due to well schedule management or well workover in the history of production, dynamic level and bottomhole pressure (BHP) vary with time, resulting in fluctuations in fluid rate and uncertainty in the prediction of cumulative production. Although all wells penetrated the three coal seams, they did not produce simultaneously in identical layers, i.e., the schedule and pay sections for each well were distinct and varied with time. To make a comparative evaluation, only the No. 3 seam (Table 5) was involved, and the BHPs of each well were set to a constant 0.3 MPa throughout a production period of 4000 days. The CGP was applied to this section as the objective function. The total CGP and CWP for the wells with the original well placement were determined to be 2.671 × 10 7 m 3 and 2.342 × 10 4 m 3, respectively; while the productions, optimized over 4000 days, were 3.313 × 10 7 m 3 and 2.339 × 10 4 m 3, respectively. An increment of 22.01% of gas production was observed with optimization, while the water production remained nearly unchanged after optimization. Fig. 13 contains the graphical optimization results of well placement in the permeability and gas content fields, which maximizes CGP. No regular discernible well pattern was observed. However, the following should be noted: 1) no well was placed in the lower permeability region (top right section of the model, Fig. 13(a)) and 2) four wells were placed around a sub-region with relatively higher permeability (Fig. 13(a)), whereas four other wells were deployed in

Table 4 History matching errors in the field application. Well ID

Field CGP, m

Well-1 Well-2 Well-3 Well-4 Well-5 Well-6 Well-7 Well-8 Well-9 Well-10

Simulation 3

3,459,556 1,255,606 716,805 559,585 264,232 2,598,134 1,643,854 1,718,520 104,240 302,220

CWP, m 943 772 2152 574 1127 1886 4021 7246 1340 787

3

CGP, m

3

3,178,640 1,157,292 770,996 545,707 243,225 2,493,949 1,822,705 1,679,853 121,065 31,172

5.3.2. Infilling of new vertical well(s) Previously conducted history matching led to the re-distributions of reservoir pressure and residual gas content after about five years of field production. Based on this premise, six infill scenarios involving 0, 1, 2, 3, 5 and 7 infill wells were performed. A zero infill well represents the scenario that applies the original well placement with no infill well. For each infill scenario, ten optimization runs were repeated and the one with highest resulting NPV was selected as the optimal placement. Once the infill wells began operating, all wells, including existing wells and infill wells, operated at a constant BHP of 0.3 MPa. The prediction time period was set at 4000 days. The economic parameters used are listed in the first row of Table 2. Fig. 14 depicts the variation in the highest ultimate NPVs at 4000 days with different numbers of infill wells. NPVs climbed initially but subsequently fell with an increasing number of infill wells. The peak NPV was $1.245 × 10 6 at one infill well. The corresponding average single well controlled area was 0.136 km 2. This is consistent with Dunn (1984) that the ideal pattern, design for proper reservoir interference, early methane removal for mining efficiency, and economic return from gas sales, contemplated 25–40 acre (0.101 km 2 to 0.162 km 2) spacing. Once the number of infill wells exceeded one, the NPV generated by the optimal placement was observed to decline and closely follow a curve of second order. Fig. 15 provides a qualitative interpretation of the decline. The additional CGP per infill well, defined for all infill wells as the average of the difference between the total gas produced by the combination of existing and infilling wells and the total gas produced by existing wells alone, declined with an increase in infill wells. In contrast, the additional CWP per infill well (definition analogous to the additional CGP) increased as infill wells increased. As a result, considering the capital output due to new drillings and higher water disposal cost, the average capital revenue tended to decline with increasing numbers of infill wells, thus causing a drop in NPV. To enhance the economics, an infill of one well is recommended in this case. The optimal location in the scenario of one infill well was located in grid block (7, 23). In other infill scenarios, which involved from 2 to 7 infill wells, at least one well is placed in the grid block or relatively close to the grid block. Such observations indicate that the top left region of the reservoir area is favorable for the optimal placement. This observation can be interpreted by the constructive interference effect and Lv's statement presented previously. As indicated in Fig. 13(b), a region with relatively high gas content occurred in the top left region. A well located in this region could acquire a relative high gas production. Besides, this region bears no well before infilling. A new well incorporated here could help constructing effective interference with the existing wells, thus accelerating dewatering rate and enhancing gas production rate.

Error CWP, m 974 573 1792 656 990 2241 3201 8543 1607 833

3

CGP, %

CWP, %

−8.12 −7.83 7.56 −2.48 −7.95 −4.01 10.88 −2.25 11.14 3.15

3.22 −25.8 −16.7 14.2 −12.2 18.8 −20.4 17.9 19.9 5.87

6. Conclusion and future work Optimization of the placement of vertical wells was conducted by maximizing the NPV and integrating a numerical simulator to the particle swarm optimization algorithm. The optimization results for single and multi-well placements in a synthetic reservoir model indicated a satisfactory performance of the PSO algorithm. A comparison of CGP for the optimized and original placements of wells highlights the necessity of using optimization runs in field applications. Infilling scenarios indicated that the optimal number of infill wells is one.

Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

43

Fig. 12. History matching results for (a) total water rate; (b) total gas rate.

The success of the PSO algorithm in determining optimal well placement in primary recovery of CBM reservoirs inspires the application of the algorithm in ECBM projects, including optimization of well

pattern, numbers of injectors and producers, and operation schedules. Future research on the performance evaluations of different PSO algorithms for these applications is also recommended. To date,

Table 5 Reservoir parameters in coal No. 3 used for interpolation. Well ID

Well-1 Well-2 Well-3 Well-4 Well-5 Well-6 Well-7 Well-8 Well-9 Well-10

Grid coordinate X

Y

30 19 22 13 17 29 40 31 39 7

6 14 7 4 22 15 5 22 15 22

Top depth (m)

Thickness (m)

Gas content (m3/t)

Pressure (MPa)

Porosity

Permeability (mD)

343 359 345 353 372 390 311 420 380 438

1.9 2.3 2.6 1.5 1.5 1.8 1.9 1.7 1.9 2.4

15.5 10.9 11.7 10.4 8.5 11.7 13 10.6 11.8 23.8

2.33 2.50 2.35 2.43 2.64 2.83 1.98 3.15 2.72 3.35

0.022 0.015 0.025 0.03 0.044 0.014 0.024 0.02 0.01 0.035

3.3 3 2.5 2.3 2.5 3.7 2.5 2.0 2.2 2.6

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Q. Feng et al. / International Journal of Coal Geology 104 (2012) 34–45

Fig. 15. Additional CGP and CWP for different numbers of infill wells.

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Fig. 13. Optimal placement of 10 wells in the field application. The gray dots represent well locations.

the parameters in the PSO algorithm have been adapted from previous investigations. Research to identify the best parameter combination for well placement optimization is encouraged.

Acknowledgments This study was supported by the National Natural Science Foundation of China (Grant No. 40974056), the National Basic Research Program of China (Grant No. 2009CB219606), the Fundamental Research Funds for the Central Universities (Grant No. 10CX04011A) and the National Science and Technology Major Project of China (Grant No. 2011ZX05038). Mr. Xianchao Chen and Mr. Hongfu Shi conducted the history matching. Thanks to Dr. C.Ö. Karacan for his patient and careful inputs. Thanks also are due to two anonymous reviewers whose comments improved the manuscript.

Fig. 14. NPVs versus number of infill wells.

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