A fuzzy model is applied for permeability estimation in heterogeneous sandstone oil reservoirs using core porosity and gamma ray logs. The basic concepts of a fuzzy model are described, and we explain how to use the constructed model to analyze and interpret the results. The fuzzy-logic approach is used to represent a nonlinear relationship as a smooth concatenation of local linear submodels. The partitioning of the input space into fuzzy regions, represented by the individual rules, is obtained through fuzzy clustering. The results from the fuzzy model show that it is not only accurate but also provides some insight into the nonlinear relationship represented by the model. Furthermore, the results of the blind test developed a good agreement between the measured core permeability and the output of the fuzzy model.
Many oil reservoirs have heterogeneity in rock properties. Understanding the form and spatial distribution of these heterogeneities is fundamental to the successful characterization of these reservoirs. Permeability is one of the fundamental rock properties, which reflects the ability to flow when subjected to applied pressure gradients. While this property is so important in reservoir engineering, there is no well log for permeability, and its determination from conventional log analysis is often unsatisfactory (Mohaghegh et al. 1997; Malki et al. 1996).
Estimation of permeability in a heterogeneous reservoir is a very complex task; a poorly estimated permeability will make the model inaccurate and unreliable, thus affecting the degree of success of many oil and gas operations that are based on such models. Major efforts have been made by many researchers to establish a complex mathematical function that relates permeability to other reservoir characteristics. These studies have helped in understanding the factors controlling permeability but have not provided an accurate estimation of permeability. The internal processes of a reservoir correspond to complex physical phenomena where many factors are interacting. Definition of an exact expression for each of these factors as a function of others is an impossible task. The best that can be done is approximate methods that somehow give a hint about the permeability distribution in the reservoir (Berg 1970; Timur 1968).
One of the first practices was finding correlations between permeability and other reservoir characteristics such as porosity, or well logs. Samples extracted from cored wells were used in the laboratory to find values of permeability and porosity; likewise, logs were taken in the same wells. Correlations were obtained from permeability vs. porosity plots or from functional transformation on the well logs wherever enough information existed. These correlations were extrapolated to wells in which little or no information was available. For this method to work, a high amount of reservoir-representative samples was required, something expensive to achieve. Besides, when heterogeneity of a well is high, these correlations become unreliable (Yao and Holditch 1993).
Statistical multivariate techniques arise as a better choice, providing a potential solution through regression analysis. These techniques offer appealing solutions; however, their main drawback is the need to exhaustively identify all the factors affecting permeability and then establish a linear or nonlinear model that best represents interactions among such factors. Because permeability is controlled by both depositional characteristics (such as grain size and sorting) and digenetic features, a precise model should take into account the fundamentals of geology and physics of flow in porous media (Abbaszadeh et al. 1996). Relationships between core-derived pore-throat parameters and log-derived macroscopic petrophysical attributes can be established (Soto B. et al. 1999).
Fuzzy logic uses the benefits of approximate reasoning. Under this type of reasoning, decisions are made on the basis of fuzzy linguistic variables such as "low," "good," and "high," with fuzzy set operators such as "and" or "or." This process simulates the human expert's reasoning process much more realistically than do conventional expert systems. Fuzzy-set theory is an efficient tool for modeling the kind of uncertainty associated with vagueness, imprecision, and/or a lack of information regarding a particular element of the problem at hand (Soto B. et al. 2001).
In this paper, the fuzzy model was applied for permeability estimation in heterogeneous oil reservoirs using core porosity and gamma ray log. Also, the basic concepts of the fuzzy model are described. Finally, a method is presented for using the constructed models to analyze and interpret the results.