Reflection on most scientific decision problems suggests that multiple objectives or criteria are involved. In the past, the majority of multi-objective optimization problems used to be studied as a single-objective problem due to the lack of efficient solutions [17] [18] . However, it should be noted that fundamental differences exist in the nature of multi-objective and single-objective optimization algorithms. Solving multi-objective optimization problems by changing them into single-objective problems dates back to four decades ago. Since then on many algorithms have been proposed and many researchers have attempted to classify them based on various observations. In weighted sum model, user reduces multiple objectives into a single one by multiplying them by a proposed weight. With respect to multiple objectives, this method is the simplest solution which comes into mind. For example, where target function consists of function 1 (enhanced oil recovery) and function 2 (enhanced gas storage), the conventional method makes optimization by maximizing the weighted sum of the functions. While this is a simple idea, it provokes the question that what values the user should use for the weights. Obviously, there is no general answer to this question. The weight of an objective is typically selected according to relative importance of the objective in the problem. In most nonlinear optimization problems, a uniformly distributed collection of weight vectors does not necessarily lead to a uniformly distributed collection of Pareto optimal solutions. Since this mapping is typically unknown, it is difficult to achieve a Pareto optimal solution in a desired region of the objective space of weight vector. Also, different weight vectors do not necessarily lead to different Pareto optimal solutions. In nonlinear problems, it is unclear which weight vectors lead to an optimal solution [17] [18] .

The majority of objective functions defined and used in separate and simultaneous investigations about enhanced oil recovery and CO₂ storage processes are on single-objective basis. The general formula of these functions is as follows:

The majority of cases have a coefficient of 0.5. For instance, Cosk (2004) defined an objective function as a combination of oil recovery and CO₂ storage. Jahangiri et al. (2010) proposed an objective function for simultaneous process of enhanced oil recovery and optimized gas storage. As with previous studies, they used the coefficient of 0.5.

In this method, we first developed the primary population as usual based on problem scale and constraints and evaluated it from the viewpoint of the defined objective functions. Next, we performed non-dominated sorting. According to the definition, A dominates C if it is not worse than C in any function and is better than it in at least one function. The population members were classified into several groups. The first group consisted of fully non-dominated members. The second group was dominated by the first group members, and this went on the same way in other groups. Every member in each group was allocated a rating according to group number. The aim of multi-objective optimization process was to find Pareto solutions (non-dominated) for the problem. Oil production and gas storage were two objectives in the multi-objective optimization process using non-dominated ranking genetic algorithm. We computed population distance parameter for each member in each group, which represented the distance of each sample to other members of the relevant group. Next, we selected parent population for reproduction and carried out mutation and intersection steps. Finally, we formed Pareto front, in which every answer was optimal [19] [20] .

In this study, we utilized two optimization methods. The following represents single-objective and multi-objective optimization methods:

In this study, we utilized a reservoir in Iran and simulated the model using a combined reservoir simulator (Eclipse-300). The model consisted of 4 production wells and 3 injection wells. The average reservoir depth and initial pressure were 11,000 ft and 5620 psi respectively. The average horizontal permeability of the reservoir was 15 md and its porosity was 0.09. Horizontal-vertical permeability ratio was approximately 3. The empty space volume was 10.5 × 10¹⁰ ft³, while initial oil phase saturation was 0.65. The reservoir included 20 × 20 × 20 grade blocks, out of which 3868 blocks were active. The average dimensions of each block in X, Y and Z directions were 830, 800 and 45 ft respectively.

We analyzed the parameters using Design Expert. The required time for optimizing oil recovery and CO₂ storage using single-objective genetic algorithm in continuous injection mode is 100 hours. The reservoir has been naturally exploited for 10 years. The injection scenarios have begun from the beginning of eleventh year and will be continued for twenty years. In the meantime, influential operating parameters are going to be optimized. The combination of objectives in single-objective genetic algorithm is essential.

Since the mean value of objective function variations was smaller than 10⁻⁶ in last generations, optimization stopped in 64^th generation. The corresponding control parameters are shown in Table 1. Injection wells 2 and 3 are approximately located in oil zone and injection well 1 is fully located in water zone. In the case of variation in parameter values, the entire optimization process has to be repeated.

In order to utilize non-dominated ranking multi-objective optimization method, we defined two objective functions for oil production value. We selected these objectives independent of theorem economy (technical function) and there was no need for scaling in contrast to single-objective method. The objectives were optimized independent of price fluctuations. In other words, the objectives were directly optimized in contrast to single-objective method where they are optimized indirectly (by help of net present value). The corresponding control parameters are shown in Tables 2-7. In this approach, the optimized gas injection

scenarios were classified into three groups. The first group consisted of optimal solutions in terms of maximized gas storage (oil was also optimized as far as possible). The third group consisted of optimal solutions in terms of maximized oil production. The second group related to ideal solutions in terms of maximized objective without operating constraints. It should be noted here that all of these points are optimal from the viewpoint of multi-objective optimization process, but the distinguishing feature is the enhanced decision power under various operating conditions. The considerable point was the superiority of optimal solutions in the third group over single-objective optimization solutions. In the third group, oil production increased by 1,500,000 barrels and CO₂ storage increased by 32 billion ft³. These comparisons were made under equal conditions and without operating and economic constraints. Since multi-objective optimization method directly optimizes oil production and CO₂ storage, one can investigate different economic scenarios and select the best mode from among the existing solutions in each scenario. In practice, single-objective method is unable to achieve optimal solutions obtained from multi-objective optimization method. The first reason is that single-objective optimization method has to be executed for different weights, which would be costly and time-consuming in practice. In multi-objective optimization method, the required time for achieving Pareto optimal solutions is 114 hours. The same takes around 1500 hours in single-objective method with the assumption that the weight of each function is known. The second reason is that the weights leading to optimal solutions are unknown.

In order to determine the impact of continuous injection scenario on oil recovery, we first optimized the model on single-objective and multi-objective bases. Figure 1 and Figure 2 illustrate the results.

In this study, we prepared seven samples of Iranian oil reservoirs using CO₂ injection method. Then we computed production rates of both recovery enhancement methods, involving six parameters of porosity, permeability, gravity,

viscosity, temperature and depth. Production rate is obtained from dividing the volume produced under direct influence of recovery enhancement process by total oil production volume (natural production + production obtained from enhanced recovery). This ratio represents the effectiveness of algorithm in the enhanced oil recovery process. We first had to develop a model with the ability to perceive the impact of reservoir parameters on oil production rate obtained from recovery enhancement methods. Then we selected six parameters as network input, with the output being the production rate obtained from recovery enhancement methods. Table 8 summarizes some of these samples and Figure 3 shows the effectiveness of the algorithm.

After studying the simulation results, we inputted the data of seven oil reservoirs located in southwest of Iran into the algorithm and estimated recovery enhancement share of each. Table 9 summarizes the statistical data of input parameters together with estimated recovery enhancement shares and Figure 4 illustrates the algorithm effectiveness on each reservoirs.

Figure 5 compares the data before and after the algorithm. As you can see, the algorithm is able to precisely predict the production rate obtained from recovery enhancement method. Thus, the proposed algorithm achieves the goal by finding recovery enhancement shares in which error function reaches its minimum.