Estimating the Permeability Field Using the Bayesian Inference and Principal Component Analysis

Surrogate modeling is essential in reducing computational costs for history-matching applications. Yet, traditional deep learning-based surrogate models cannot cope with high dimensional input parameters, such as the permeability field. This work introduces a robust method to automate the history matching process utilizing the Bayesian inversion assisted by a hybrid convolutional neural network and long short-term memory (CNN-LSTM) model and principal component analysis (PCA) method. The method includes five main steps. Step 1: Generate a high-spatial permeability field using a geostatistical approach. Step 2: use the PCA to reduce the dimensionality of the permeability fields, followed by using PCA to generate permeability fields and perform simulations. Step 3: construct the CNNLSTM to map the nonlinear relationship between the extracted features from PCA and the sequential outputs, such as the pressure response. Here, Bayesian optimization is employed to automate hyperparameter tuning. Step 4: perform the Bayesian inversion to inverse the high dimensional inputs, e.g., permeability field, in which the CNN-LSTM serves as the forward model to reduce the computational cost. The inversed PCA features are then fed into the PCA to recover the high dimensional inputs. Step 5: check convergence and if the errors are significant between the inversed high dimensional permeability field and the ground truth, revisit the construction of the CNN-BiLSTM and the prior information for the uncertainty parameters. A 2D reservoir model demonstrates the proposed history-matching method. We can inverse the high dimensional inputs (e.g., permeability field) with minor errors between the prediction and ground truth. We propose a Bayesian inversion assisted by a hybrid CNN-LSTM model and PCA method for high-dimensional parameter inversion, which is superior to the traditional models regarding accuracy and efficiency. This method enables us to perform history matching for reservoir simulation with high dimensional inputs and significant uncertainties.