WELL PLACEMENT OPTIMIZATION FOR WATER INJECTION WELLS · 2012-02-09 · 2nd International Conference on Engineering Optimization September 6 - 9, 2010, Lisbon, Portugal 1 WELL PLACEMENT

2nd International Conference on Engineering Optimization September 6 - 9, 2010, Lisbon, Portugal

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WELL PLACEMENT OPTIMIZATION FOR WATER INJECTION WELLS

Hanieh Rasouli1, Fariborz Rashidi1, Ehsan Khamehchi2

1Faculty of Chemical Engineering Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran 2Faculty of Petroleum Engineering Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

Abstract The determination of optimal well locations is a challenging problem in oil production since it depends on geological and fluid properties as well as on economic parameters. Determining the optimal location of wells with the aid of an automated search method can significantly increase a project’s net present value (NPV) as modeled in a reservoir simulator. This work addresses the efficient solution of this problem by using advanced techniques for coupling two important components of autonomic optimization: Dynamic model using Eclipse simulator software (E100), a optimization technique based on the genetic algorithm (GA). In this study optimal placement of up to four water injection wells was studied for one of south west Iranian heavy oil field. Injection rate was also optimized. The net present value of the water flooding project was used as the objective function. Profits and costs during the time period of the project were taken into consideration. Keywords: Well Placement, Optimization, Economic Evaluation, Meta Heuristic Optimization

1. Introduction One of the most important objectives of petroleum engineering discipline is determination of a comprehensive plan for maximum production of oil and gas under physical and economic constraints and with the least possible costs. Locating the best coordination for drilling future wells is one of the keys in this regard. The optimum well location can be interpreted in reservoir engineering terminology as maximum production rate and/or maximum petroleum ultimate recovery factor. The significance of both of these items can show up as a unified economic concept, like net present value (NPV). NPV or other economic indexes can take into account any kind of capital or operational costs like drilling, pipelines and so on. The overall direction is toward minimum costs and consumes, maximum incomes, improvement of recovery factors, and optimum use of resources and facilities. Optimum well placement is a complicated problem related with reservoir rock and fluid properties, surface facility conditions, wells and economic criteria. Different methods have been proposed for this purpose among which optimization methods that directly use simulators for evaluation of objective functions are state-of-the-art and popular technology. One of the direct optimization methods that are in use in different fields of science and engineering is the genetic algorithm.

A combination of this technique with methods like polytope one can obtain powerful algorithms for optimization problems in engineering, the present case being one in the field of petroleum engineering and more specifically entitled as “well placement optimization problem”.

2. Definition of the “well placement optimization” problem The objective of this problem is to optimize the number of water injection wells, well locations and well injection rates. In this study, optimization of water flooding in a synthetic model using hybrid genetic algorithm is discussed and the input parameters are specific to one of Iranian western oil reservoirs.

3. Reservoir Characterization Some of the most important characteristics of the studied case is summarized in the table 1. A schematic illustration of the reservoir geometry is also depicted in figure one.

Table 1: General data of the simulated reservoir Reservoir Type Naturally Fractured, Saturated Present Phases Water, Oil, Gas

Start of Production 1st October 1999 Datum Depth (ft) 900

Oil/Water Contact (ft) 1950 Gas/Oil Contact (ft) 900

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Initial Pressure (psi) 4100 Number of Production Wells 5

Figure 1: A 3D schematic of the studied case (Permeability Map)

4. Sensitivity Analysis For performing sensitivity analysis on a problem one should first determine the uncertain parameters. Then the sensitivity of the simulation model to the outlined parameters should be evaluated. For the present case the following parameters were selected:

1) fracture permeability

2) fracture porosity

3) shape factor

4) block height

Then one can vary the selected parameters in a given range and evaluate the feedback of the model. Then it can be clarified that how sensitive is the model to a certain parameter. The ranges of the present case are given in table 2.

Table 2: Range of variation for uncertain parameters in the sensitivity analysis

MaximumAdopted Value for Model Minimum

1000 500200Fracture Permeability (mD) 8% 5%10%Fracture Porosity 0.01 0.0080.0001Shape Factor 80 5020Block Height (ft)

Output of the initial model with the first guess parameters in terms of well pressure is presented in comparison with the observed data in figure 2.

Figure 2: Well pressure results of the model with guessed paramers as compared with the field data

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Initially the fracture permeability was set equal to 200 mD and the data was compared with the observed field data. Then it is set to 500 mD and the result as shown is figure 4 is obtained. And finally 1000 mD gives figure 5.

Figure 3: Field presure result for a fracture permeability of 200mD

Figure 4: Field presure result for a fracture permeability of 500mD

Figure 5: Field presure result for a fracture permeability of 1000 mD

Like this should be repeated for other parameters. We finally concluded that the model gives the best matches with fracture permeability of 1000 mD, shape factor of 0.0001 and porosity of 8%. This also was observed that block height is not an influential parameter and is an indicative of gravity drainage insignificancy.

5. An introductory controlled test We first tried the applicability of the genetic algorithm method for optimization of an injection well location with an injection rate of 1500 STB/D. Fixing the injection rate reduces the size of search space and this enables us to run simulation for all possible states. All possible states count up to the number all active blocks in the model available to accommodate an injection well, in this case being: (Number of blocks in the X direction, I) 25* (Number of blocks in the Y direction, J) 30=750 After simulation, corresponding NPV for any of the states is calculated as the value of objective function. The work flow is given in figure 6. as per figure 6, one coordination is selected initially for the injection well. This location plus rock, fluid and other reservoir characteristics are input to a general purpose simulator. The output in

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terms of production profile from the entire field in a reservoir lifelong span should be obtained and translated into economic terms using main expenditure and profit categories, summed up and all be referenced to a common time, like the projects starting point. This would base the calculations on the NPV parameter which is the most popular in economic evaluations of the petroleum industry. The values of the objective function are then given to the optimization algorithm and the next well location is suggested by the optimizer. The process is repeated until there are no significant changes in the NPV. The final location would be the optimum well placement and the NPV would be corresponding to the maximum profits associated with the field under study. The input values for economic factors are given in table 7, as per National Iranian Oil Company's analysts.

Figure 6: The general work flow of optimum injection well placement

Table 7: Economic factor as per NIOC recommendations for 2009

Value Parameter 14 Rate of Interest 50 Oil Price, $/bbl 6Gas Price, $/MSCF 2Waste Water Disposal Cost, $/bbl 40000 OPEX, $/Day 20000000 Well Costs, $/well 50000000 CAPEX, $

After completion of all 750 simulations and having all 750 NPVs it was observed that the objective function is oscillatory. The classical methods based on derivative fail to reach the global maximum in these cases; we should turn to heuristic algorithm to fill up this shortcoming. In the present case the optimum location is the block (16, 11). Some improper blocks may even lead to negative NPVs as occurred in this case too. Having a table of outputs for all states makes it possible to cancel out requirement to run the sensitivity without an in loop simulation by referring to the already prepared table. Though the present model is simple but its complete coverage of search space by direct simulation results provided us the possibility of a good control over the program's performance through parameter adjustments. Different arrays of genetic algorithms were tried for this case and using the tabulated results of the comprehensive simulation step we obtained a fine overview on the optimization method and its parameters. The algorithm was run for 100 times and the average and standard deviation in the number of simulations needed for reaching the optimum state was calculated. The project lifelong was taken 17 years, 9 years covering the years after injection commencement. Some rules of thumb were also used as an aid to GA adjustment. More details are presented in the following sections. The upper limit of simulation runs was set to 200 and the generations to 50. The average and standard deviation for the entire state space is given in table 8. In some states the algorithm has not been able to converge to the optimum point with 200 runs and 50 generations, so the success rate is also given in the table 8.

Table 8: Average, Standard Deviation and success rate for the GA in the studied case

GA Average 103 Standard Deviation82.2 Success Rate 84 %

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6. Optimization of multi-well problem with adaptable injection rates Toward a more realistic problem we now advance to a well placement problem in which there exist up to four injection wells and also the possibility for rate variation in each well. Refer to table 10 for a glance at the huge size of state space emerging in such problems. The problem size is so large that comprehensive coverage by simulation runs is not possible. Hence, we broke the continuous injection rate variable into a discrete variable taking any integer value from 1 to 7, each integer representing an interval of 4000 STB/D covering the span between 0 to 28000 overall. This reduces the accuracy to a tolerance of 4000 STB/D but makes the solution feasible instead. Due to the added complexities the number of runs was increased to 500. The project lifelong was 17 years as before and the results summarized in table 11 were obtained. In table 11, the optimum well locations plus the incremental net present values comparative to the base case (the case with no injection) is given. A graphical depiction is also given in figures 7 to 10. It can be concluded that injection would be profit making in all conditions, the profit is also at its maximum value when 2 injection wells are drilled at determined locations. The well placement can be done manually as well, as is a popular practice currently. No one can reject and cannot also firmly insist on his well placement opinion to be the absolutely optimum claim. This automated search methods can assist in judgments by removing the subjectivity and taking into account the nonlinear unpredictable effects of rock, fluid, economic parameters, physics of fluid flow and other factors for NPV calculations. As the number of wells increases the optimum location for previous wells will be altered too. This prohibits one from utilization of successive optimization methods and dynamic programming.

Table 10: Size of search space

Number of Injection Wells Size 16000236000000

Table 11: Comparative Results

q2(i2,j2)q1(i1,j1)Incremental NPV(MM$) nsimninj٠١Base CAse

١٠٨ ١٠٢)١٤،٨( ٢٠ 1 Well ١٦٣ ١٦٩)١٠،٦(٢٤)١٤،١٧( ٢٤ 2 Wells

Figure 7: Optimum well location for 1 injection well problem

Figure 8: Optimum Well Location for 2 injection well problem

7. Rules of Thumb for Estimation of Genetic Algorithm Parameters Goldberg in 1989 [1] and De Jong in 1975 [2] proposed proper population sizes and proper probabilities for GA operators respectively. We made use of these suggestions.

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� Population Size: is equal to the total length of the binary string for a chromosome (n=1) � Cross Over Probability: Pc=0.6 does well for a wide range of problems � Mutation Probability: Pm=1/n, should be adopted equal to the inverse of population size (in a bit-based census) For the first problem a binary structure was used. For the well placement problem of one well with fixed injection rate since the binary array length was 10 the population size was taken 10 (5 bits for each variable), and based on the De Jongs suggestion the mutation probability was selected as 0.5. [3]

8. Cross-Over and Mutation Probabilities The sensitivity of optimization algorithm to the cross-over and mutation probabilities was evaluated by varying these values between 0 and 1 in 0.01 step. In all states the average and standard deviation of number of needed runs was calculated for the first test problem. For this purpose any doublet combination of Pc and Pm the optimization algorithm was run 100 times. The results shows the number of simulation runs needed for reching the global optimum using optimization algorithm versus cross-over and mutation probability [4]. The total number of runs for optimization algorithm for plotting the sensitivity is:

101 (Pm)*101 (Pc)*100 (Number of runs for the optimization algorithm for any combination of Pm, Pc) = 1020100

In any of these 1020100 times the optimization algorithm was allowed to run utmost 200 simulations for the solution. If the optimum was not reached in 200 runs the procedure was programmed to stop. Such a comprehensive sensitivity analysis made it possible to construct a large set of data for the first test [5]. The first look at Results was as surprising as they show independency of the genetic algorithm from the cross-over probability for the well placement problem. A more in-depth analysis would clarify their reasonableness of this result. There is no inception to justify superiority of fitness for a specific (i,j) over another one. In other words, the fitness depends on the combination of (i,j) and single i and j components offer nothing for fitness improvement [6]. When well location for multiple wells and injection rates are adopted as independent variables for optimization, this can be easily verified that fitness of a well arrangement is dependent on locations and rates of other wells [7].

9. Discussion and Conclusions The optimized genetic algorithm was successfully applied for the injection well placement problem. The algorithm solutions were obvious solutions in this case. The case in which the aim was to find the optimum well location for one well with a fixed injection rate shows that the number of simulation runs can be significantly reduced using a genetic algorithm. Though this was a specific problem, but various petroleum engineering problems can be attacked by combination of an optimizer and a simulator. The only task to be done is to select the correct optimization tools with respect to the physics of the problem. Assigning new objective functions can expand the applicability of this approach.

References

[1] Goldberg, D., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, 1989 [2] De Jong, K. A. ,An Analysis of the Behavior of a Class of Genetic Adaptive Systems, Ph.D. thesis,

University of Michigan. Dissertation Abstracts International 36(10), 514B. , 1975 [3] Moravvej Farshi M. & Aziz Kh., Improving Genetic Algorithms for Optimum Well Placement, a report

submitted to the department of petroleum engineering of Stanford university, US. , 2008 [4] Rogers, L. L. & Dowla, F. U., Optimal Groundwater Remediation Using Artificial Neural Networks with

Parallel Solute Transport, Water Resources Research, 30(2), pp 457–483, 1994 [5] Badru , O.& Horne, R. N., well placement optimization using the quality map approach, a report submitted

to the department of petroleum engineering of stanford university, US. ,2003 [6] G¨uyag¨uler, B. & Horne, R. N. & Rogers, L., & Rosenzweig, J. J., Optimization of Well Placement in a

Gulf of Mexico Water flooding Project, SPE 63221, Texas, 2000 [7] Rian, D. T. & Hage, A., Automatic Optimization of Well Locations in a North Sea Fractured Chalk

Reservoir Using a Front Tracking Reservoir Simulator, paper SPE 28716 presented at the SPE International Petroleum & Exhibition of Mexico, Veracruz, Mexico, October 10-13,1994