Several areas of the geosciences have benefitted from the application of the new mathematics of "fuzzy logic."1,2 The oil industry now routinely uses new interpretation techniques, based on fuzzy logic, to predict permeability and litho-facies in uncored wells. Litho-facies and permeability prediction have presented a challenge to formation evaluation due to the lack of tools that measure them directly. The method described can be used as a simple tool for confirming known correlations or as a powerful predictor in uncored wells.
Fuzzy logic is simply an application of recognized statistical techniques. Whereas conventional techniques deal with absolutes, the new methods carry the inherent error term through the calculation rather than ignoring or minimizing it. This retains the information associated with the error and gives surprisingly better results.
One clear application is to litho-facies determination. Litho-facies typing is used in well correlation and is important for building a three-dimensional model of an oil or gas field. The technique makes no assumptions and retains the possibility that a particular litho-facies type can give any well log reading although some are more likely than others. This error or fuzziness has been measured and used to improve the litho-facies prediction in several North Sea fields. In one study, descriptions from 10 cored wells were used to derive litho-facies descriptions in 30 uncored wells. This technique gave near-perfect differentiation between aeolian, fluvial, and sabkha rock types. In addition, it went some way towards differentiating between sandy, mixed, and muddy sabkhas. Using the fuzzy logic technique gives much better predictions than more complicated methods such as neural networks or cluster analysis.
A second application is permeability calculation. Knowledge of permeability is important in determining the well completion strategy and the resulting productivity. The problem with permeability prediction is derived from the fact that permeability is related more to the aperture of pore throats rather than pore size, which logging tools find difficult to measure. Determining permeability from well logs is further complicated by the problem of scale, well logs having a vertical resolution of, typically, 2 ft compared to the 2 in. of core plugs. The new techniques quantify these errors and use them, together with the measurement, to improve the prediction. This new approach has been used in several fields to obtain better estimates of permeability compared to conventional techniques. In addition, the method uses basic log data sets such as gamma ray and porosity rather than depending on new logging technology.
Fuzzy logic is an extension of conventional Boolean logic (zeros and ones) developed to handle the concept of "partial truth"— truth values between "completely true" and "completely false." Dr. Lotfi Zadeh of UC/Berkeley introduced it in the 1960's as a means to model uncertainty.3
Science is heavily influenced by Aristotle's laws of logic initiated by the ancient Greeks and developed by many scientists and philosophers since.4 Aristotle's laws are based on "X or not-X;" a thing either is, or is not. This has been used as a basis for almost everything that we do. We use it when we classify things and when we judge things. Managers want to know whether it is this or that, and even movies have clear goodies and baddies. Conventional logic is an extension of our subjective desire to categorize things. Life is simplified if we think in terms of black and white. This way of looking at things as true or false was reinforced with the introduction of computers that only use bits 1 or 0. When the early computers arrived with their machine-driven binary system, Boolean logic was adopted as the natural reasoning mechanism for them.
Conventional logic forces the continuous world to be described with a coarse approximation; and in so doing, much of the fine detail is lost. We miss a lot in the simplification. By only accepting the two possibilities, the infinite number of possibilities in between them is lost. Reality does not work in black and white, but in shades of gray. Not only does truth exist on a sliding scale, but also because of the uncertainty in measurements and interpretations, a gray scale can be a more useful explanation than two endpoints. For instance, we can look at a map of the earth and see mountains and valleys, but it is difficult to define where mountains start and the valleys end.
This is the mathematics of fuzzy logic. Once the reality of the gray scale has been accepted, a system is required to cope with the multitude of possibilities. Probability theory helps quantify the grayness of fuzziness. It may not be possible to understand the reason behind random events, but fuzzy logic can help bring meaning to the bigger picture. Take, for instance, a piece of reservoir rock. Aeolian rock generally has good porosity and fluvial rock poorer porosity. If we find a piece of rock with a porosity of 2 porosity units (pu) is it aeolian or fluvial? We could say it is definitely fluvial and get on with more important matters. But let us say it is probably fluvial but there is a slim probability that it could be aeolian. Aeolian rocks are generally clean and fluvial rocks shalier. The same piece of rock contains 30% clay minerals. Is it aeolian or fluvial? We could say it is equally likely to be aeolian or fluvial based on this measurement.
This is how fuzzy logic works. It does not accept it is either this or that. It assigns a grayness, or probability, to the quality of the prediction on each parameter of the rock, whether it is porosity, shaliness, or color. There is also the possibility that there is a measurement error and the porosity is 20 pu not 2 pu. Fuzzy logic combines these probabilities and predicts that based on porosity, shaliness, and other characteristics, the rock is most likely to be aeolian. Fuzzy logic says that there is also the possibility it could be fluvial. In essence, fuzzy logic maintains that any interpretation is possible but some are more probable than others. One advantage of fuzzy logic is that we never need to make a concrete decision. What is more, fuzzy logic can be described by established statistical algorithms; and computers, which themselves work in ones and zeros, can do this effortlessly for us.