Well placement optimization under time-dependent uncertainty using an ensemble Kalman filter and a genetic algorithm

Journal of Petroleum Science and Engineering 109 (2013) 70–79

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Well placement optimization under time-dependent uncertainty using an ensemble Kalman filter and a genetic algorithm John Lyons 1, Hadi Nasrabadi 1,n Department of Petroleum Engineering, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar

art ic l e i nf o

a b s t r a c t

Article history: Received 18 December 2012 Accepted 29 July 2013 Available online 13 August 2013

Determining the optimal well location in a reservoir is a challenging problem. It involves taking several factors into account, including geological uncertainty, reservoir and fluid properties, economic costs, and technical ability. Most research on well placement optimization under uncertainty has assumed static uncertainty in the reservoir parameters, until the introduction of the pseudohistory concept. The pseudohistory concept incorporates the field's probable history and results in the determination of optimal locations of future wells with greater certainty. This approach, however, requires an excessive number of simulations and may not be practical for optimization of a reservoir model having a large number of geological realizations. In this study, we use an ensemble Kalman filter (EnKF) to perform history matching of the PUNQ-S3 reservoir model using data from six production wells over an eight-year period. This is followed by well placement optimization using a genetic algorithm (GA) combined with pseudohistory matching, carried out over two years, following the placement of the first future well. Thus, this approach not only provides increased certainty in optimal well placement but also, using EnKF as a history matching method, requires only a single “best estimate” realization for objective function evaluation during GA optimization. As a result, the total time taken to find the optimal well locations is significantly reduced. We illustrate this through comparison with the previous research. & 2013 Published by Elsevier B.V.

Keywords: reservoir optimization reservoir simulation geological uncertainty ensemble Kalman filter genetic algorithm well placement

1. Introduction Well placement optimization (determination of the optimal number, type, and location of wells) has recently gained significant attention in the petroleum industry because of the increase in the global energy demand and necessity to obtain maximum hydrocarbon recovery with minimum investment in gas and oil fields. Choosing the optimum location and type of well(s) can lead to significant economic gain in hydrocarbon extraction from underground reservoirs. The well placement problem is a highly nonlinear one that depends on many continuous or discrete decision variables and includes factors such as subsurface geomechanics, well drilling and construction, surface facilities, as well as geological and economic uncertainty. This problem becomes more complex when non-conventional (horizontal, deviated, or multilateral) wells are considered. The number of possible solution combinations for the well placement variables increases exponentially with an increase in decision variables. Finding the optimum solution to this problem by conducting a few case studies is unlikely, and using intuitive engineering judgment alone may not be sufficient. For almost n

Corresponding author. Tel.: +1 979 862 6483. E-mail address: [email protected] (H. Nasrabadi). 1 Currently at: Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843, US 0920-4105/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.petrol.2013.07.012

three decades, attempts have been made to solve this problem through automatic optimization procedures. These approaches have differed mainly in terms of the optimization algorithm, reservoir response modeling technique, and available decision variables and constraints (Nasrabadi et al., 2012). There is a significant amount of uncertainty in oil and gas reservoir development planning, mainly due to the uncertainty of the geological model used for the estimation of the reservoir response. This can affect the economics of a field development plan, and thus, should be accounted for in the optimization process. Many researchers have developed well placement optimization techniques that include the effect of uncertainty. Some included the effect of uncertainty by using multiple geostatistical realizations in the evaluation of reservoir response (e.g., Güyagüler and Horne, 2001; Yeten et al., 2003). Another approach used to account for uncertainty in well placement is Monte Carlo simulation (e.g., Cullick et al., 2003, 2005). Some other techniques include D-optimality experimental design (Aanonsen et al., 1995; Vincent et al., 1999), utility theory (Güyagüler and Horne, 2004), quality map method (Loureiro and Fresky, 2005), statistical proxy method (Artus et al., 2006), and ensemble optimization (Leeuwenburgh et al., 2010). An important contribution to well placement under uncertainty is the inclusion of time dependency on geological uncertainty. Previous studies on well placement optimization under uncertainty assumed static uncertainty, while in reality, if all of the wells are not

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drilled simultaneously, the amount of geological uncertainty should decrease with time as more production data becomes available. Özdogan and Horne (2006) addressed this problem by introducing the concept of pseudohistory and incorporating time-dependent information in future well placement decisions. They included the effect of time on uncertainty by performing a history match of a future well using the probable reservoir response model (P50 model). However, their approach requires a significant number of reservoir simulations and, given a reservoir model with a large number of geological realizations, this approach may not be practical for field application. In this study, we use an EnKF for continuously updating a reservoir model, after which well placement optimization using a GA and pseudohistory matching is carried out. The advantage of this approach is not only the integration of time-dependent information in well placement optimization but also the performance of optimization on a single best estimate realization, as opposed to all of the updated realizations, as done in past approaches (e.g., Özdogan and Horne, 2006). The ensemble mean is considered the best estimate of the permeability and porosity fields that honors the production data from the reference model (and is thus used for future forecasting). The number of objective function evaluations during the optimization process is thus reduced by a factor equal to the number of realizations (which can be several orders of magnitude). This method is applied to the PUNQ-S3 reservoir model in which we carry out history matching using field history and pseudohistory for six production wells over an eight-year period. Well placement optimization of three future wells is then conducted on the best estimate realization following Özdogan and Horne's well placement approach. The rest of this paper is organized as follows. Section 2 describes the EnKF history matching algorithm. Section 3 outlines the pseudohistory concept. Section 4 describes GA as the well placement optimization technique. The combined EnKF-GA algorithm for well placement optimization under time-dependent geological uncertainty is described in Section 5. The results of application of the EnKF-GA algorithm to the PUNQ-S3 reservoir model are presented in Section 6. Section 7 concludes this paper with a brief summary.

2. EnKF history matching The EnKF method was first introduced by Evensen (1994) for updating non-linear ocean models. Subsequent clarification and support for the use of the ensemble mean as the best estimate was then given by Burgers et al. (1998). EnKF has been successfully used to update oceanography, meteorology, and reservoir engineering models. There is extensive literature on the topic (Evensen, 2009), including an overview of the development and applications of EnKF (Evensen, 2003). Its use in reservoir model history matching was first proposed by Nævdal and Vefring (2002) for updating the near-well permeability field, thereby forecasting reservoir production. Nævdal et al. (2003) further applied EnKF to an entire reservoir model. Gu and Oliver (2006) investigated the suitability of EnKF for updating state variables and the performance of a relatively small ensemble. This was followed by a case study on the North Sea conducted by Seiler et al. (2010), who presented a detailed workflow for updating reservoir simulation models using EnKF. EnKF has also performed well on two further North Sea studies (Evensen et al., 2007; Haugen et al., 2008) along with studies carried out on the PUNQ-S3 model (Gao et al., 2005; Gu and Oliver, 2005). Updating an ensemble of reservoir models (realizations) involves modifying the model parameters such that the (oil, gas, water cut, etc.) production profiles of each of the realizations of the ensemble match, more closely, those of the reservoir's production history. The main advantage of using EnKF in a well-placement

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optimization process is that after updating the ensemble, only a single best estimate realization is required to forecast future production. Another advantage is that when new production data is made available, it is necessary to use only that data to update the ensemble, and previous data need not be considered. EnKF is based on a set of linear equations that allows for updating of large non-linear models. The ensemble update equation is given by Evensen (2003): Aa ¼ A þ KD′:

ð1Þ

where the superscript a denotes analysis and A contains both the model paramaters to be updated (e.g., permeability [a measure of the conductivity of a fluid through a porous medium] and porosity [the ratio of connected space in a porous medium to its total volume]) and the predicted data (e.g., gas–oil ratio [the ratio of the volume of gas to the volume of oil produced from a well at standard conditions] and water cut [the ratio of the volume of water produced to the total volume of liquid produced from a well at standard conditions]), for all ensemble members. D′ ¼ DHA represents the difference between the predicted and observed production data. The Kalman gain matrix, K, is given by: K ¼ A′A′T H T ðHA′A′T H T þ ϒ ϒ T Þ1 ;

ð2Þ

where H is an operator, H ¼ ½0jI, that extracts the predicted data from A, and A′ ¼ AA is the ensemble perturbation matrix. ϒ is the ensemble of measurement perturbations, with an ensemble mean of zero. For further details see Gu and Oliver (2006) and Oliver et al. (2008). EnKF forecasting is based on the approach that, after updating the ensemble, the best estimate of the reservoir model is given by the mean of the ensemble. The spread of realizations about the mean then provides an estimate of the error in the ensemble.

3. Pseudohistory concept The pseudohistory (Özdogan and Horne, 2006) is defined as the probable (future) response of the reservoir that is generated by a probabilistic forecasting model. Given an ensemble, the simulated response of the P50 realization (or the “probable history”) is used as the pseudohistory. The ensemble can then be updated based on this response, and by doing so, time-dependent information is used to enhance subsequent decisions in an implicit manner. The pseudohistory approach, with, for example, the placement of two future wells, can be described as follows: 1. History matching is carried out on the ensemble until the current time. 2. Two possible well locations are chosen. 3. The first well is placed at the current time and the ensemble response is measured until the time of the second well placement. 4. The P50 realization is determined based on the ensemble response. The future response of the P50 realization (which has just been measured) will then be used as the pseudohistory. 5. With the first well in place, the ensemble is then updated until the second well is placed, based on the simulated response of the P50 realization. Thus time-dependent information is integrated into the model and can be used to implicitly enhance subsequent decisions (because the location of the second future well has already been proposed.) By carrying out the pseudohistory matching, using the probable history, information already available to us is expected to be used more efficiently. For further details on the pseudohistory approach, the reader is referred to Özdogan and Horne (2006).

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4. Well placement optimization technique Various optimization techniques for well placement exist in the literature. The most common techniques are gradient-based algorithms (e.g., Bangerth et al., 2006; Castineira and Alpak, 2009; Handels et al., 2007; Sarma and Chen, 2008; Vlemmix et al., 2009; Wang and Li, 2007; Zandvliet et al., 2008) and GA (e.g., Bangerth et al., 2006; Bittencourt and Horne, 1997; Larionov et al., 2006; Litvak et al., 2007; Yeten et al., 2003). Other techniques such as simulated annealing (Beckner and Song, 1995; Norrena and Deutsch, 2002; Bangerth et al., 2006) and particle swarm optimization (Onwunalu and Durlofsky, 2010) have also been applied to the well placement problem with promising results. We use GA as the well placement optimization technique. The GA concept was first introduced by Holland (1975). GAs are a partially stochastic group of optimization algorithms that are based on natural evolution, including features such as reproduction, selection, crossover and mutation (Goldberg, 1989). The optimization is carried out by a population of strings (or individuals) that are initially randomly selected. Each string is composed of a collection of 1's and 0's. Each individual (location) in the search domain is also characterized by a binary string. Once the population size, Np, is initialized, a predefined objective function (or fitness) is calculated for each individual. Sets of two individuals are chosen by tournament selection, and crossover occurs at a randomly selected location within each string. The crossover occurs with probability pc. Following crossover, mutation of one of the bits may occur, i.e., a 0 bit may change to a 1 bit or vice versa. The probability of mutation is pm. The process of reproduction is completed by selecting the individual with the highest fitness and saving a copy of its string. This ensures that the individual with the highest fitness value is neither discarded nor altered during the optimization. This indicates the end of a generation. The number of generations, Nmax, required to find the optimum locations is dependent on the complexity of the solution space. Finally, the well locations corresponding to the individual with the highest fitness are chosen as the optimal locations. We calculate individual fitness as the net present value (NPV) of the field. The NPV for realization i is given by: Net Cash Flowi ðtÞ ¼ Oil Productioni ðtÞ  Oil Price þGas Productioni ðtÞ  Gas Price Water Productioni ðtÞ  Water Handling Cost Net Cash Flowi ðtÞ NPVi ¼ ∑ t: t ð1 þ discount rateÞ

ð3Þ

Note that in our application, operating expenses and capital expenditure are not affected by well location and hence are not included in the NPV calculation.

5. for t ¼ 1; Nt 6. for i ¼ 1; Ne 7. Run simulator, recording predicted production history 8. end 9. Update ensemble using EnKF 10. end 11. for i ¼ 1; N max 12. for j ¼ 1; Np 13. Record proposed well locations 14. for k ¼ 1; Ne 15. Run simulator until 2nd future well starts production 16. Calculate NPVk 17. end 18. Sort NPVs and determine the P50 realization 19. Record P50 response over pseudohistory period 20. for t ¼ 1; N pt 21. Repeat steps 6 to 9 22. end 23. Calculate NPV over field lifetime using ensemble mean 24. end 25. Record the highest NPV and corresponding well locations 26. end

The individual with the highest fitness is then chosen as the optimum individual and the well locations that it parameterizes are most likely to give the highest economic return. Özdogan and Horne's algorithm, which uses gradual deformation as a history matching technique and utility as the objective function, can be outlined as follows: Algorithm 2. Özdogan and Horne's algorithm. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

5. EnKF-GA algorithm We present EnKF as a history matching technique during well placement optimization (EnKF-GA) and highlight the benefit of using it over other techniques. The following is a comparison between optimization using EnKF-GA and Özdogan and Horne's (2006) algorithm, and illustrates the speed-up time achieved. We carry out history matching over Nt time intervals, followed by pseudohistory matching over Npt intervals. The EnKF-GA algorithm can then be outlined as follows: Algorithm 1. EnKF-GA algorithm. 1. 2. 3. 4.

Define EnKF variables ðNe ; N t ; N pt Þ Define GA variables ðN p ; N max ; pm ; pc Þ Read field production history Read ensemble of realizations

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Define gradual deformation variables Define GA variables ðN p ; N max ; pm ; pc Þ Read field production history Read ensemble of realizations for t ¼ 1; N t for i ¼ 1; Ne Run simulator, recording predicted production history end Update ensemble using gradual deformation end for i ¼ 1; N max for j ¼ 1; Np Record proposed well locations for k ¼ 1; Ne Run simulator until 2nd future well starts production Calculate NPVk end Sort NPVs and determine the P50 realization Record P50 response over pseudohistory period for t ¼ 1; N pt Repeat steps 6 to 9 end for k ¼ 1; Ne Calculate NPV over field lifetime end Calculate expected utility end Record the highest utility and corresponding well locations end

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To illustrate the difference between the total computation time required by steps (23) to (25) in Özdogan and Horne's algorithm and that required by step (23) in the EnKF-GA algorithm, the following time comparison is made. In well placement optimization, the reservoir simulation time is usually significantly longer than the EnKF updating process, and thus, we do not consider the EnKF update time. It is assumed that this is also the case for the gradual deformation update time. Let Th, Tph, and Tfh be the times taken for the simulator to run over the history, pseudohistory, and forecasted history periods, respectively. Let Tfl be the time taken for the simulator to run over the field lifetime, i.e., T fl ¼ T h þ T ph þ T fh . Both approaches spend t 1 ¼ N e T h during the history update, t 2 ¼ Ne T ph during P50 calculation, and t 3 ¼ N e T ph during the pseudohistory update. However, Özdogan and Horne's algorithm requires a further t 4 ¼ N e T fl , while the EnKF-GA algorithm only requires t′4 ¼ T fl . This results in a significant reduction in total CPU time (see Table 1).

6. Application of EnKF-GA algorithm to PUNQ-S3 reservoir model 6.1. PUNQ-S3 model The PUNQ-S3 reservoir model is based on a real field operated by Elf Exploration Production. It was developed by oil companies Table 1 Number of reservoir simulations and total CPU time. Simulation type

EnKF-GA model

Özdogan and Horne's model

History match P50 calculation Pseudohistory match Field lifetime (GA)

Ne N e N p N max N e N p N max N e N p N max

Ne N e N p N max N e N p N max N p N max

Total CPU time

8 N T > < e h þ2N e N p N max T ph > : þN N T p

max fl

Ne T h þ 2N e N p N max T ph þ N e N p N max T fl

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and research institutes in the European Union as a test to compare methods for quantifying uncertainty assessment in history matching. The following description is a brief overview of this model and further details can be found in Floris et al. (2001). The model contains 19  28  5 grid blocks, of which 1761 are active. The gridblocks are uniform, 180  180 m. The field is bounded by a fault in the south and east and by a strong aquifer in the north and west (see Fig. 1). A gas cap is present in the center of the domeshaped field. The field initially contains six production wells located around the gas–oil contact. Due to the strong aquifer influx no injection wells are present. The production scheduling consists of one year of extensive well testing, followed by a three year shut-in period before field production begins. The well testing year consists of three-monthly production periods, each with its own constrained production rate. During field production, each of the wells is shut-in for a two-week period to collect shut-in pressure data. Field production is to be carried out for a total of 16.5 years. 6.2. Generation of ensemble We use the geostatistical data analysis software S-GeMS (Remy et al., 2008) to generate an ensemble of 50 realizations. We then import the porosity data from all five layers of the six well locations into S-GeMS. Sequential Gaussian simulation is then carried out using a simple Kriging interpolation method. We calculate the vertical and horizontal permeabilities using the following correlation (Boss, 1999): log ðkh Þ ¼ 0:77 þ 9:03ϕ kv ¼ 3:124 þ 0:306kh

ð4Þ

6.3. Parallel processing To evaluate the objective function in well placement optimization, a reservoir response model is required. We use a commercial reservoir simulator to evaluate the reservoir response. Although this is a reliable technique, a large number of simulations is required by the GA. Including uncertainty can further increase

Fig. 1. Left: Top surface map with initial 6 well locations. Right: Reservoir model showing the fault (black line), water–oil contact and gas–oil contact. Reprinted by permission of Hajizadeh et al. (2011).

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Fig. 2. Production profiles for “PRO-1” before (blue) and after (red) EnKF application for (a) oil production rate, (b) bottom-hole pressure (BHP), (c) gas–oil ratio (GOR), and (d) water cut. The black dots indicate the observed production history. Only 25 realization responses are shown for clarity. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 3. Production profiles for “PRO-11” before (blue) and after (red) EnKF application for (a) oil production rate, (b) bottom-hole pressure, (c) gas–oil ratio, and (d) water cut. The black dots indicate the observed production history. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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the number of required simulations. The use of parallel computing to distribute the simulations over a network of CPUs has been suggested previously (e.g., Bangerth et al., 2005; Cullick et al., 2005; Parashar et al., 2005) and we implement this here. During the EnKF update process, data is acquired from the commercial simulator in P parallel streams, where P is the number of required slave processes. This is implemented using the Fortran Message Passing Interface (MPI) on the Suqoor supercomputer (TAMUQ, 2012), which is a 64-node, 512-core computer cluster. The above-mentioned job requires P+1 cores (P slave processes and one master process). The master process carries out the ensemble modification and, when required, each of the slave processes invokes the simulator and records the predicted production data from its output. A similar process governs GA with pseudohistory optimization. The master process carries out selection, reproduction, crossover, and mutation and, when required, each of the slave processes represents an individual, invoking the simulator, performing the pseudohistory update, and calculating the corresponding individual's fitness value.

Table 2 Optimization and economic parameters. Population size, N p Ensemble size, N e History intervals, N t Pseudohistory intervals, N pt Max. number of generations, N max Crossover probability, pc Mutation probability, pm Oil price (USD/bbl) Gas price (USD/bbl) Water-handling cost (USD/bbl)

25 50 40 26 10 0.6 0.05 Adjusted 1967–1983 Prices Adjusted 1967–1983 Prices 3

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6.4. Application results We carry out EnKF history matching over 40 time intervals (between 1967 and 1975) using production history from the PUNQ-S3 model. This is then followed by pseudohistory matching with GA optimization. Three additional wells to be drilled are “PRO-16” in year 8, “PRO-17” in year 10, and “PRO-18” in year 12, however, our approach allows for complete flexibility in the number of wells drilled and the dates at which they are drilled. We choose the horizontal permeability, kh, and porosity, ϕ, as the model parameters to be updated. The oil production rate for all 6 wells remains fixed (e.g., Figs. 2a and 3a) and we use the gas–oil ratio and water cut as the production data to be matched. Table 2 lists the EnKF-GA parameters used. We carry out the optimization of three future well locations following Özdogan and Horne's approach: the locations of all three wells are proposed when making the location decision of the first future well, i.e., the wells are not sequentially placed. Each set of three locations is represented by an individual in the GA. The optimal locations are chosen based on the maximum NPV achieved. If a proposed well location is in an inactive gridblock, we assign a fitness value of zero to that individual. Each of the proposed wells is perforated in layers 3 and 4. We integrate time-dependent information by continuing from step 11 of the EnKF-GA algorithm. For simplicity, the gridblock saturations and pressures are not updated, and instead, the simulator is run from the beginning of field deployment in order to obtain the predicted production data. The production data profiles of “PRO-1” for each of the realizations before and after the EnKF update are shown in Fig. 2. The production data profiles from “PRO-11” for each of the realizations before and after the EnKF update are shown in Fig. 3. As can be observed from both figures, the profiles after the EnKF update resemble the production

Fig. 4. Production profiles for “PRO-1” before (blue) and after (red) EnKF application for (a) oil production rate, (b) bottom-hole pressure, (c) gas–oil ratio, and (d) water cut. The black dots indicate the predicted production pseudohistory. Note that the blue profiles in this figure are the same as the red profiles in Fig. 2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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Fig. 5. Production profiles for “PRO-11” before (blue) and after (red) EnKF application for (a) oil production rate, (b) bottom-hole pressure, (c) gas–oil ratio, and (d) water cut. The black dots indicate the predicted production pseudohistory. Note that the blue profiles in this figure are the same as the (updated) red profiles in Fig. 3. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 6. Production profiles for “PRO-16” before (blue) and after (red) EnKF application for (a) oil production rate, (b) bottom-hole pressure, (c) gas–oil ratio, and (d) water cut. The black dots indicate the predicted production pseudohistory. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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history profiles more closely than those before the update. Note that each of the six wells are shut-in from years 1 to 4. We perform the pseudohistory update from years 8 to 10, the period between “PRO-16” and “PR0-17” beginning production. The ensemble, which already matches the first eight years of production history, is further updated to honor two years of “probable history”

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with “PRO-16” in place. We perform pseudohistory matching using the production response of the P50 realization over 26 time intervals. The production profiles for “PRO-1” and “PRO-11” before and after pseudohistory matching can be seen in Figs. 4 and 5, respectively, and the profiles of the first optimally placed well, “PRO-16,” are shown in Fig. 6. In each case, the profiles after the EnKF

Fig. 7. (a) Layer 3 of the initial horizontal permeability field of realization 3. (b) Layer 3 of the horizontal permeability field of realization 3 after history and pseudohistory matching.

Fig. 8. (a) Layer 3 of the true horizontal permeability field. (b) Layer 3 of the mean initial horizontal permeability field of the ensemble. (c) Layer 3 of the mean horizontal permeability field of the ensemble (best estimate) after history and pseudohistory matching.

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Fig. 9. The original 6 well locations and optimal well locations of “PRO-16,” “PRO-17,” and “PRO-18”.

pseudohistory update resemble the production pseudohistory profile more closely than those before the update. To determine how accurately the EnKF-updated permeability and porosity fields represent the reference model, we calculate the ensemble mean (best estimate) before any update and again after the pseudohistory update. For example, Fig. 7a represents the initial horizontal permeability field for layer 3 of realization 3, while Fig. 7b represents the same field but after the pseudohistory update. We find that with each successive update, the rock characteristics of the best estimate model more closely resemble those of the reference model. Fig. 8 shows layer 3 of the reference model along with the best estimate initially and after the pseudohistory update. Following the well placement optimization, we simulated the reference PUNQ-S3 model using the resulting optimal well locations. The NPV was calculated over the field lifetime and found to be $1:6257  109 . The optimal well locations are illustrated in Fig. 9. Given that we use an ensemble size of 50, our approach reduces the time required to calculate the objective function over the field lifetime, Tfl, by a factor of 50, in comparison with Özdogan and Horne (2006) approach. The magnitude of the speed-up is mainly dependent on the ratio of the pseudohistory time to the field lifetime, T ph =T fl . If, for example, pseudohistory was not incorporated (i.e., T ph =T fl ¼ 0), then the speed-up would occur by a factor of 46. Conversely, if the pseudohistory period was one half of the field lifetime (i.e., T ph =T fl ¼ 0:5), the speed-up would occur by a factor of 2. The length of the pseudohistory period is determined by how soon the second future well is to be drilled. Simulating the PUNQ-S3 model, the CPU times (for an Intel (R) Xeon(R) processor @ 3.60 GHz, 32.0 GB RAM) for the field lifetime and the history matching period are T fl ¼ 6:4 s and T h ¼ 2:85 s, respectively. The time taken during the pseudohistory simulation is T ph ¼ 1:09 s. We can then compare the CPU time for reservoir simulation for each model, and find that a speed-up factor of approximately 4 is achieved.

7. Conclusions We carry out well placement optimization under time-dependent uncertainty on the PUNQ-S3 model. Using EnKF as a history matching method, only a single realization is required to determine the GA

objective function and therefore the overall optimization time can be significantly reduced. Based on the CPU time required to simulate the history, P50, pseudohistory, and field lifetime production data, and using 50 geological realizations, we could reduce the overall time to obtain the optimal well locations by a factor of 4. Furthermore, the speed-up factor is increased in two ways: (i) when using a greater number of realizations and (ii) when using a shorter pseudohistory period.

Acknowledgments This work is supported by the Qatar National Research Fund under Grant NPRP 28-6-7-33. We greatly appreciate this support. We would also like to acknowledge the Fortran-based EnKF code and advice given by Geir Evensen, the modified GA provided by Adrian Morales and the CMG-compatible version of the PUNQ-S3 model provided by Uchenna Odi. References Aanonsen, S.I., Eide, A.L., Holden, L., Aasen, J.O., 1995. Optimizing reservoir performance under uncertainty with application to well location. Paper SPE 30710 Presented at the 1995 SPE Annual Technical Conference and Exhibition, 22–25 October, Dallas, Texas, USA. Artus, V., Durlofsky, L.J., Onwunalu, J., Aziz, K., 2006. Optimization of nonconventional wells under uncertainty using statistical proxies. Comput. Geosci. 10 (4), 389–404. Bangerth, W., Klie, H., Matossian, V., Parashar, M., Wheeler, M.F., 2005. An autonomic reservoir framework for the stochastic optimization of well placement. Cluster Comput. J. Netw. Softw. Tools Appl. 8 (4), 255–269. Bangerth, W., Klie, H., Wheeler, M.F., Stoffa, P.L., Sen, M.K., 2006. On optimization algorithms for the reservoir oil well placement problem. Comput. Geosci. 10 (3), 303–319. Beckner, B.L., Song, X., 1995. Field development planning using simulated annealing— optimal economic well scheduling and placement. Paper SPE 30650 Presented at the 1995 SPE Annual Technical Conference and Exhibition, 22–25 October, Dallas, Texas, USA. Bittencourt, A.C., Horne, R.N., 1997. Reservoir development and design optimization. Paper SPE 38895 Presented at the 1997 SPE Annual Technical Conference and Exhibition, 5–8 October, San Antonio, Texas, USA. Boss, C., 1999. Production forecasting with uncertainty quantification: a comparative study. Netherlands Institute of Applied Geoscience TNO, Technical Report. Burgers, G., van Leeuwen, P.J., Evensen, G., 1998. Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724. Castineira, D., Alpak, F.O., 2009. Automatic well placement optimization in a channelized turbidite reservoir using adjoint based sensitivities. Paper SPE

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