A Deep Neural Network Algorithm For Solving Fractured Reservoir Problems

The fracture-cavity reservoir model is a common reservoir geological model,and the single-phase seepage equation is the basic mathematical model in the field of numerical reservoir simulation.The strong discontinuity of the coefficients and the multi-scale in space and time bring many challenges to traditional numerical methods.PINNs have the advantages of automatically learning the optimal nu-merical solution of complex partial differential equations from a small amount of sample information,which have become a research hotspot in the field of scientific and engineering computing.Based on PINNs,this paper focuses on the numerical solution of fractured reservoir problem.Firstly,based on the traditional PINN,the number of epochs and penalty pa-rameters are optimized,and a PINN with higher solution accuracy is obtained.Numerical experiments show that the solution accuracy of the new PINN is two orders of magnitude higher than that of the traditional PINN in example 1.Then,a PINN with two DNN subnetworks（abbreviated as DD-DNN-PINN）was designed based on the strong format of the equation and the combination of residual idea and domain decomposition method.Numerical experiments show that DD-DNN-PINN has higher solution accuracy than the traditional single PINN network.For example,in example 2,the relative error of the prediction result of the network is improved from 8.65×10^-1to 1.16×10^-3.Based on the weak form of the equation,a new loss function is designed based on Ritz variational idea,and the structure of DNN subnetwork is improved to ResNet subnetwork which has the idea of domain decomposition method.A PINN suitable for solving complex interface problems is obtained（abbreviated as DD-Resnet-Deepritz）.Numerical experiments show that DD-Resnet-Deepritz has higher solution accuracy than DD-DNN-PINN for complex interface problems.For example,the relative error of example 3 is improved from 3.50×10^-2to 1.15×10^-2.