Deep Learning-Based Proxy Models to Simulate Subsurface Flow of Three-Dimensional Reservoir Systems

Abstract

Summary Reduced-order modeling (ROM) has been used to simulate subsurface flow in porous media for decades. With recent advances in machine learning and deep learning methods, new ROMs have been presented in the literature. In this work, we present extensions to the embed to control-based (E2C) models limited to two-dimensional (2D) reservoir models to three-dimensional (3D) reservoir models. E2C-based models are built to mimic the ROM known as proper orthogonal decomposition trajectory piecewise linearization (POD-TPWL) by using blocks of neural networks, where two of the blocks; namely encoder and decoder, are used to transform back and forth from the space of system states to a low-dimensional space and a transition block that predicts the evolution of system states linearly in the null-space. This framework predicts the system state variables such as pressure and saturation across the reservoir, and system outputs such as rate and pressure at production or injection wells are computed by using the predicted state variables in explicit well model equations for the E2C model. The other E2C-based proxy model, which is referred to as embed to control and observe (E2CO), can predict system outputs directly by using another network block called transition output and does not require explicit well-model equations. We upgraded the existing E2C and E2CO models by using 3D convolution layers and modified the loss functions to address the 3D flux conditions. The proposed methods are tested using a small and a large portion of the SPE10 benchmark reservoir model with channelized heterogeneous permeability for a waterflooding scenario. The small-scale model contains 14,400 cells and 8 wells whereas the large-scale model contains 528,000 cells and 53 wells spread across the reservoir in a 5-spot pattern. 300 simulations from a commercial high-fidelity simulator (HFS) are generated to train the proxy models. Both the E2C and E2CO provide accurate estimates of the state variables with acceptable errors when compared with the test data obtained from HFS for both the small-scale and large-scale reservoir models. It is observed that the well output predictions made by the E2CO are more accurate than the predictions of the E2C. Compared with HFS, these proxy models result in several orders of magnitude faster forward predictions.

Topics & Concepts