Deep Model Reduction-Model Learning for Reservoir Simulation
Authors: Zhang Jingyan , Cheung Siu Wun , Efendiev Yalchin , Gildin Eduardo , Chung Eric T. 
Published: March 2019

Abstract
The objective of this work is to design novel multi-layer neural network architectures for simulations of multi-phase flow taking into account the observed data (e.g., production data) and physical modeling concepts. Our approaches use deep learning concepts combined with model reduction methodologies to predict multi-phase flow dynamics. The use of reduced-order model concepts is important for constructing robust deep learning architectures. The reduced-order models provide fewer degrees of freedom and allow handling the cases relevant to reservoir engineering that is limited to production and near-well data.
Multi-phase flow dynamics can be thought as multi-layer networks. More precisely, the solution, pressures and saturations, at the time instant n+1 depends on the solution at the time instant n and input parameters, such as permeability, well rates, and so on. Thus, one can regard the solution as a multi-layer network, where each layer is a nonlinear forward map. The number of time steps is user-defined quantity, which will be treated as an unknown within our deep learning algorithms. We will rely on rigorous model reduction concepts to define unknowns and connections for each layer. Novel proper orthogonal basis functions will be constructed such that the degrees of freedom have physical meanings (e.g., represent the solution values at selected locations) and basis functions have limited support, which will allow localizing the forward dynamics. This will allow writing the forward map for the solution values at selected locations with pre-computed neighborhood structure that will be used in deep learning algorithms.
In each layer, our reduced-order models will provide a forward map, which will be modified (trained) using available data. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Because of the lack of available data, the training will be supplemented with computational data as needed and the interpolation between data-rich and data-deficient models. We will also use deep learning algorithms to train the elements of the reduced model discrete system.
In this case, deep learning architectures will be employed to approximate the elements of the discrete system and reduced-order model basis functions.
The numerical results will use deep learning architectures to predict the solution and reduced-order model variables. Trained basis functions will allow interpolating the solution between the observation points. We show how network architecture, which includes the neighborhood connection, number of layers, and neurons, affect the approximation. Our results show that with a fewer number of layers, the multi-phase flow dynamics can be approximated. The proposed approach uses physical model concepts and deep learning methods to design a novel forward map, which combines the available data and physical models. This will benefit to develop a fast and data-based algorithms for reservoir simulations.