PARTICLE SWARM OPTIMIZATION OF SOLUTION GAS OIL RATIO

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INTRODUCTION
Pressure-volume-temperature (PVT) analysis is the study of the behavior of vapor and liquid in petroleum reservoirs in terms of phase behavior and composition. PVT properties are used to estimate reserves, evaluate the performance of oil and gas reservoirs, design production facilities and production operations. Also, to estimate stock tank gas production rate, cumulative gas-oil ratio, stock tank gas-oil ratio is a key input parameter (Hillary and Okotie, 2016).
Determination of PVT properties among other data, for estimation of the stock oil initial in place and the evaluation of the future performance of hydrocarbon reservoir, is usually associated with uncertainties and as such, it is vital to properly characterize these properties which are key input data for most oil and gas calculations. According to Ikiensikimama and Egbe (2006), at the earlier stages of a well, it can be difficult or economically impractical to obtain reliable measurements of PVT data. In a scenario where the fluid samples are available, they can be subjected to PVT analysis to determine their properties, but samples are often suspected and PVT analysis usually applies only at reservoir temperature. In addition, Ikiensikimama (2008) has it that the reserves estimation and the design of the best depletion strategies are only feasible when a realistic and precise values of reservoir fluid properties are available.
As stated by Okotie et al. (2017), to estimate the PVT properties of a reservoir fluid, the fluid is usually sampled and taken to the laboratory for experimental analysis such as saturation pressure (Dew point) at reservoir temperature, constant composition expansion test for black oil and compositional reservoir fluid to determine properties such as relative volume, vapor z-factor and liquid drop out. Differential Liberation/Vaporization test for black oil only to determine vapour Z factor, liquid density, gas-oil ratio, relative volume (formation volume factor), gas gravity, liquid viscosity, vapor viscosity. constant volume depletion test for compositional reservoir fluid only to determine the retrograde liquid drop out, cumulative fluid produced, vapor z factor, specific gravity of produced fluid plus, mole weight of produced fluid plus, final weight of produced fluid plus, produced vapor composition and finally, separator test for black oil and compositional reservoir fluid to estimate the gas-oil ratio and stock tank formation volume factor.
Carrying out this study in the laboratory is usually expensive and time-consuming, thus, Engineers in the field resorted to using existing correlation to estimate these properties. The results from these existing correlations are approximations of the field data which yield a considerable amount of error, in order to minimize error associated in estimating the solution gas-oil ratio from experimental, an optimization algorithm is required. Thus, this study is aimed at minimizing the error between the experimented results and the result from existing correlations for accurate estimation of solution gas-oil ratio from correlations EXISTING PVT CORRELATIONS Standing (1947) correlation for gas-oil ratio (GOR) is given by equation (1): Oloruntoba and Onyekonwu GOR equation is given by equation (3): Glaso (1980) proposed a correlation for estimating the gas solubility as a function of the API gravity, pressure, temperature, and gas specific gravity given by equations (4-5): Al-Marhoun (1985)

THEORECTICAL CONCEPT OF THE PSO ALGORITHM
Step 1: Choose the number of particles Step 2: Initialize the initial positions of the particles Step 3: Evaluate the objective function at the initial positions Step 4: Set the iteration number as t = i+1 Step 5: Find the personal best for each particle Step 6: Find the global best Step 7: find the velocities of the particles Step 8: Find the new values of the particles position Step 9: Find the objective function values of step 6 Step 10: Stopping criterion: If the terminal rule is satisfied, go to step 4, otherwise stop the iteration and output the results.  The results are plotted in Figure 2 to show the disparity in value of the correlations from the experimented value.

ESTIMATED RESULT FROM PSO ALGORITHM
The result of Standing's correlation gave the largest disparity than the other two correlations, hence, it was optimized with the particle swamp optimization algorithm to achieve the objective of this study. The particle swamp optimization algorithm is an iterative process that cannot be done manually, thus, it was programmed in Microsoft Excel to get the global best value for the constant A, B and C respectively as shown in Figures A1, A2 and A3 in the appendix section.
Convergence criteria is a phenomenon of PSO in which all particles tend to converge to a single value as shown by Figures A1, A2 and A3 respectively where all three particles maintain a single value at the 216 th iteration

CONCLUSIONS
Based on the results of this evaluations, the following remarks are worth mentioning:  An empirical correlation for estimating the gas solubility at the bubble point pressure and below for black oils has been developed.
 The result obtained with the data from the differential liberation test, shows that the correlation developed in this study performs better than Standing's, Glaso's and Petrosky's correlations. Hence, engineers can rely on the newly developed correlation to an extent after the have validated it with their field data because it is stated in literatures that correlation performs better in the region it was developed.
 The statistical result indicates a lower value of average relative error and a better coefficient of correlation for this study than for Standing's, Glaso's and Petrosky's correlations at pressures below bubble point.
 The particle swarm optimization tool achieves a better accuracy by minimizing the objective function generated in this study. Above bubble point, solution gas oil ratio is the same with solution gas oil ratio at bubble point.
 Further work is recommended in developing a modification of Particle swarm optimization with faster convergence rate.