Reconstruction of shale image based on Wasserstein Generative Adversarial Networks with gradient penalty

Artificial intelligence methods for oil and gas reservoir development: Current progresses and perspectives

Deep learning for solving partial differential equations (PDEs) has been a major research hotspot. Various neural network frameworks have been proposed to solve nonlinear PDEs. However, most deep learning-based methods need labeled data, while traditional numerical solutions do not need any labeled data. Aiming at deep learning-based methods behaving as traditional numerical solutions do, this paper proposed an approximation-correction model to solve unsteady compressible seepage equations with sinks without using any labeled data. The model contains two neural networks, one for approximating the asymptotic solution, which is mathematically correct when time tends to 0 and infinity, and the other for correcting the error of the approximation, where the final solution is physically correct by constructing the loss function based on the boundary conditions, PDE, and mass conservation. Numerical experiments show that the proposed method can solve seepage equations with high accuracy without using any labeled data, as conventional numerical solutions do. This is a significant breakthrough for deep learning-based methods to solve PDE.

Physics-informed neural networks (PINNs) improve the accuracy and generalization ability of prediction by introducing physical constraints in the training process. As a model combining physical laws and deep learning, it has attracted wide attention. However, the training cost of PINNs is high, especially for the simulation of more complex two-phase Darcy flow. In this study, a physics-informed radial basis function neural network (PIRBFNN) is proposed to simulate two-phase Darcy flow of oil and water efficiently. Specifically, in each time step, oil phase and water phase equations are discretized based on the finite volume method, and then, the physics-informed loss is constructed according to the residual of their coupling equations, and the pressure is approximated by RBFNN. Based on the obtained pressure, another physics-informed loss is constructed according to the residual of discrete water phase equation and the water saturation is approximated by another RBFNN. For boundary conditions, we use “hard constraints” to speed up the training of PIRBFNN. The straightforward structure of PIRBFNN also contributes to an efficient training process. In addition, we have simply proved the ability of RBFNN to fit continuous functions. Finally, the experimental results also verify the computational efficiency of PIRBFNN. Compared with physics-informed convolutional neural network, the training time of PIRBFNN is reduced by more than three times.