Summary Efficiently determining effective permeability involves substantial computational efforts, even when employing local upscaling techniques. This research utilizes a convolutional neural network (CNN) architecture for the rapid estimation of effective permeability. The CNN takes as input the permeability maps of specific layers from the fine-scale model earmarked for upscaling into layer(s) featuring coarser cells. Treating fine-scale permeability maps as 3D high-resolution images, the CNN produces a three- to six-channel lower-resolution image. Each channel represents the upscaled permeability map in a major direction, along with nondiagonal permeabilities of a symmetric permeability tensor. The simplicity and robustness of the proposed architecture stem from its two 3D convolutional hidden layers, characterized by varying filter numbers and kernel sizes. The effectiveness of the network was evaluated using two distinct data sets: (1) a continuous Gaussian model and (2) the Egg model with 100 permeability realizations. In the first data set, 500 geological realizations were generated and upscaled using a pressure solver with periodic boundary conditions (BCs), employing both local and extended local methods. The same upscaling procedure was applied to 100 realizations of the Egg model. A portion of the realizations was allocated for training, and the remainder was allocated for testing. The CNN exhibited a highly promising capture of nonlinear behavior, displaying no signs of overfitting. The results, visually and quantitatively assessed through an exceptionally high mean R2 score, were further validated through the computation of the pressure fields, affirming the accuracy of the estimated permeabilities. Notably, the training runtime proved significantly shorter than the computation time required for upscaling using the pressure-solver method. This proposed method holds the potential to substantially reduce upscaling computations, marking a stride toward more computationally efficient quasiglobal upscaling methods.
Sayyafzadeh et al. (Thu,) studied this question.