An Intelligent Identification Method of Fuzzy Models and Its Applications to Inversion of NMR Logging Data

Abstract

Abstract In this paper we describe a novel approach to fuzzy model identification that gives solution to the inverse problem of permeability prediction from NMR data. The fuzzy logic approach uses fuzzy If-Then rules to establish the relationship between permeability (output variable) and the NMR T2 distribution mean values fNMR, fFF, fBF (input variables). We introduce an intelligent data-driven method that generates the fuzzy rules in a two-steps learning algorithm. In the first step, fuzzy clustering is performed on a set of input-output core measurements to obtain an initial approximation of the fuzzy rules in a rapid prototyping approach. This set of observations is the only information assumed about the model behavior. In the second step, the antecedent and consequent parameters of the identified fuzzy rules are fine-tuned by means of a gradient descent method. The identified fuzzy model is subsequently used to estimate permeability in uncored wells in the same field. Computer simulations using data from a complex siliclastic sequence in the Maracaibo Basin (western Venezuela) show the advantages of this methodology over the conventional empirical and statistical inversion methods. Introduction Inversion problems have existed in several areas of petroleum engineering and geosciences for many years. These kind of inverse problems include for example determination of petrophysical rock properties such as porosity, permeability, water saturation and electrofacies. However, it is in the past ten to fifteen years when inverse problems have received far more attention, principally due to the increase in computer power and the development of more sophisticated core analysis methods and wireline logging techniques. These techniques allow more accurate measurements of nuclear, electrical and other physical properties that can be related to the parameters being sought, implying a mathematical inversion problem of some sort. For example, nuclear magnetic resonance (NMR) allows determination of movable and irreducible water saturation, which can be subsequently used in a mathematical inversion formula to compute formation permeability [1]. In the past two decades, fuzzy models have been used in many areas of engineering and science to solve a variety of inverse problems. The variety of applications includes plant process and control, decision-making, risk analysis, image analysis and pattern recognition [2, 3, 4, 5]. Fuzzy models are referred to as universal approximators due to their capability of approximating any given model with any degree of accuracy [6]. There are two main characteristics of these free model approximators that give them a better performance for specific inversion applications. First, fuzzy models are suitable for approximate reasoning, especially for problems in which conventional mathematical models are difficult to derive. Second, fuzzy models allow decision-making with estimated values under incomplete or uncertain information. Fuzzy models provide an excellent basis for developing data-driven identification methods that require little prior knowledge of the problem under investigation. Two things are necessary: a set of experimental observations (data set) from the real behavior of the process and a suitable parameter identification algorithm. In the case of correct and dense data, there are analytical methods based on least-squares techniques to find the parameters of all the rules [7, 8]. Most often however, the data is sparsely distributed or highly noised and the model identification problem does not have a direct solution due to matrix singularities. To cope with this reality, a family of numerical iterative algorithms called learning algorithms for fuzzy model identification has been developed [9, 10].