Multinomial Logistic Regression for Bayesian Estimation of Vertical Facies Modeling in Heterogeneous Sandstone Reservoirs

摘要

Precisely prediction of rock facies leads to adequate reservoir characterization by improving the porosity-permeability relationships to estimate the properties in non-cored intervals.It also helps to accurately identify the spatial facies distribution to perform an accurate reservoir model for optimal future reservoir performance. In this paper,the facies estimation has been done through Multinomial logistic regression(MLR)with respect to the well logs and core data in a well in West Africa Sandstone Reservoir.The entire independent variables are caliper log(CCL), deep induction log,medium induction log,gamma rays,neutron porosity,core porosity,deep resistivity,medium resistivity, spontaneous potential(SP),density&corrected density,in addition to core permeability. The MLR has been chosen to estimate the maximum likelihood and minimize the standard error for the nonlinear relationships between facies&core and log data.The MLR is used to predict the probabilities of the di?erent possible facies given each independent variable by constructing a linear predictor function having a set of weights that are linearly combined with the independent variables by using a dot product.Beta distribution of facies has been considered as prior knowledge and the resulted predicted probability(posterior)has been estimated from MLR based on Baye’s theorem that represents the relationship between predicted probability(posterior)with the conditional probability and the prior knowledge.To assess the statistical accuracy of the model,the bootstrap should be carried out to estimate extra-sample prediction error by randomly drawing datasets with replacement from the training data.Each sample has the same size of the original training set and it can be conducted N times to produce N bootstrap datasets to re-fit the model accordingly to decrease the squared di?erence between the estimated and observed categorical variables(facies)leading to decrease the degree of uncertainty.The Deviance and the probability of Chi-squared distribution have been also adopted to assess the impact of each variable on the response in the logistic regression model.