Fuzzy Simulation of Waterflooding

Abstract

Evaluation of field performance and a long-term production forecast require considerable resources spent on [numerical] reservoir simulation. Uncertainty in reservoir characterization and future prospects concerning oil price, operating cost, etc. advocates for sound sensitivity analysis which requires even more resources.

Most of the problems associated with the uncertainty of evaluation can be handled either by running a sensitivity analysis or by probabilistic methods. The latter being extensively used in the past for resovling numerous engineering problems are often limited by lack of data with statistical properties. Moreover, in many engineering applications amount of accessible information is often not sufficient for its processing by statistical methods. In such cases fuzzy methods seem to be more appropriate technique to solve the problems.

From a mathematical perspective, the difference between probabilistic and fuzzy methods is based on the definition of membership function that does not necessarily rest on probability, but rather on relative preference among the members of the reference set. As a result, probability theory evaluates the likelihood of outcomes, while fuzzy mathematics models the possibility of occurence. Fuzzy methods can handle uncertainty directly, without running the sensitivity analysis. Another advantage of fuzzy technique is that it links uncertainty of input data to the reliability estimation of the final decision.

From a computational point of view, fuzzy methods, being based on rules resembling axioms of deterministic mathematics, are much faster as compared to stochastic methods. However, little effect can be gained when applying those methods to a volumetric reserve estimate, material balance equation, decline curve analysis, etc. Considerable effect can be foreseen in handling problems related to reservoir characterization. In areas of [numerical] reservoir simulation fuzzy technique outperforms probabiliastic methods in the most effective way and seems to have no rivals.

Examples of the application of fuzzy methods to petroleum engineering problems like resources and reserves estimate, reservoir description and characterization, reservoir simulation, optimization and decision making, have been discussed earlier in the literature[16, 11, 5, 19, 20]. However, little attention has been paid to numerical simulation of a multiphase flow in porous media. The emphasis in this paper is given to reservoir simulation problems illustrated by comparizon of classical deterministic and fuzzy solutions to a two-phase flow of incompressible fluids in porous media known as a fractional flow or a Buckley-Leverett problem.