Use of Symbolic Regression for Developing Petrophysical Interpretation Models

Abstract

Complex lithology petrophysical interpretation with multiphysics logging tools has been and continues to be a major challenge in formation evaluation. Many currently used data-driven approaches, such as a neural network (NN), deliver predicted results in numerical quantities rather than analytical equations. It is more challenging if multiphysics logging measurements are collectively used to estimate a petrophysical parameter. To overcome these problems, a physics-guided, artificial intelligence (AI) machine-learning (ML) method for petrophysical interpretation model development is described. The workflow consists of the following five constituents: (1) statistical tools such as correlation heatmaps are employed to select the best candidate input variables for the target petrophysical equations; (2) a genetic programming-based symbolic regression approach is used to fuse multiphysics measurements data for training the petrophysical prediction equations; (3) an optional ensemble modeling procedure is applied for maximally utilizing all available training data by integrating multiple instances of prediction equations objectively, which is especially useful for a small training data set; (4) a means of obtaining conditional branching in prediction equations is enabled in symbolic regression to handle certain formation heterogeneity; and (5) a model discrimination framework is introduced to finalize the model selection based on mathematical complexity, physics complexity, and model performance. The efficacy of the five-constituents petrophysical interpretation development process is demonstrated on a data set collected from six wells with the goal of obtaining formation resistivity factor (F) and permeability (k) equations for heterogeneous carbonate reservoirs. We show quantitatively how individual constituents of the workflow improve the model performance with two error metrics. A comparison of NN-method-predicted permeability values vs. SR-based-workflow-predicted permeability equation is included to showcase many advantages of the latter. Beyond the transparency of an analytical form of the prediction equations, the SR method intrinsically has a more relaxed requirement on the training data size, is less prone to overfitting, yet can deliver superior model performance rival to the NN approach.