Using Particle Swarm Optimization (PSO) Algorithm in Nonlinear Regression Well Test Analysis and Its Comparison with Levenberg-Marquardt Algorithm

1. Introduction

Well testing commences by well head flow rate changes and ends up with recorded well pressure response. According to the physical laws governing the reservoir, any disturbance in the well head flow rate will result in changes in the wellbore pressure drop. The way wellbore pressure changes is a function of: the way the flow rate changes (i.e. type of the well test operations), fluid properties, well geometry, and rock petrophysical characteristics. The purpose of well testing is then to find the underlying mathematical model governing the reservoir in order to estimate some unknown reservoir parameters. Therefore, well testing lies among the inverse problems; it means inputs and outputs of the system are known while the purpose is to find a relationship between these input and output data. The procedure of well testing is at best described in Figure 1 in the Appendix.

Figure 1.

A schematic of well test procedure used in the petroleum industry to acquire knowledge about reservoir parameters

Modern well test analysis is mainly composed of three basic stages (Schlumberger, 2002):

• 1.

  Reservoir model recognition: In the first stage, the well bore pressure response is matched with those belonging to previously calculated theoretical reservoir models. Accordingly, the reservoir model with most similar pressure response is selected as the reservoir’s governing model. Variety of techniques such as conventional analysis, type curve matching, and nonlinear regression are proposed for this purpose;

• 2.

  Estimation of petrophysical reservoir properties: Finding the type of reservoir model is followed by parameter estimation stage where the mathematical equations opted in the first stage are used to estimate the reservoir parameters. Though the first stage forms foundation for other two phases, the techniques employed in the parameter estimation step are also crucial to accurately describe the reservoir under study. Among the techniques used in the second stage, nonlinear regression is one of the most reliable methods used to elicit unknown reservoir parameters from well test pressure data. Type and number of the reservoir parameters could be estimated through nonlinear regression analysis is dependent upon type of the well test operation and identified reservoir model in the first stage;

• 3.

  Verifying the obtained results in stages one and two using well test simulation process: Finally, the assessment of the reservoir model and estimate reservoir parameters through stages one and two, respectively, should be verified via well test simulation process in the third stage. This is also called forward modeling and involves simulation of all the test operations accomplished on a certain well; if the simulation is not consistent with real field data, then the type of the reservoir model should be reassessed and stages one and two have to be repeated accordingly (Schlumberger, 2002).