[Paper Review] Deep Global Model Reduction Learning in Porous Media Flow Simulation

Siu Wun Cheung, Eric T. Chung+4 more|arXiv (Cornell University)|Jul 24, 2018

Model Reduction and Neural Networks18 references3 citations

TL;DR

This paper proposes a deep global model reduction method for nonlinear flow in heterogeneous porous media by combining proper orthogonal decomposition (POD) with deep learning. It treats time-dependent flow dynamics as a multi-layer neural network mapping initial conditions and inputs (e.g., permeability, well rates) to solutions at future times, with degrees of freedom representing values at observation points. The method achieves accurate forward prediction with low error, even using limited observed data, by integrating real observation data with computational simulation data to improve model fidelity.

ABSTRACT

In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.

Motivation & Objective

• To address the challenge of high computational cost in simulating nonlinear flow in heterogeneous porous media.
• To overcome limitations of traditional model order reduction in handling nonlinear dynamics and observed data.
• To develop a data-driven reduced-order model that honors observed data at specific locations (e.g., wells).
• To integrate observed data with computational simulation data to improve generalization and accuracy in model reduction.
• To enable efficient, accurate forward simulations using deep neural networks trained on reduced-order representations.

Proposed method

• The solution at each time step is modeled as a multi-layer feedforward neural network that maps the initial solution and input parameters (e.g., permeability, well rates) to the current state.
• POD modes are constructed such that degrees of freedom correspond directly to solution values at pre-selected observation points, avoiding explicit basis function learning.
• The network is trained using a combination of observed data (from true model) and simulated data (from reduced-order model) to improve robustness and accuracy.
• A composition of the trained network is used to predict the solution over multiple time steps by stacking one-step predictions.
• The method leverages the universal approximation capability of deep neural networks to represent complex, nonlinear flow dynamics efficiently.
• The training process uses L² percentage error as the loss function to compare predictions against observed solutions at final time.

Experimental results

Research questions

• RQ1 Can deep learning be effectively combined with POD-based model reduction to simulate nonlinear flow in heterogeneous porous media?
• RQ2 How can observed data from wells be incorporated into a reduced-order model to improve accuracy?
• RQ3 What is the impact of combining limited observed data with abundant computational simulation data on model performance?
• RQ4 Can a multi-layer neural network architecture accurately approximate the forward map of time-dependent flow dynamics in porous media?
• RQ5 Does constructing degrees of freedom to represent solution values at observation points eliminate the need for explicit basis function learning?

Key findings

• The method achieves a final-time L² percentage error of 15.9991% when training on 100 observation samples, demonstrating high accuracy in forward prediction.
• Using only 10 observation samples (Case 4) results in a higher error of 21.8747%, showing performance degradation with insufficient data.
• Incorporating 90 simulation samples and 10 observation samples (Case 2) reduces error to 16.6937%, indicating improved performance through data supplementation.
• Training on 100 observation samples (Case 3) outperforms training on 100 simulation samples (Case 1), which yields 31.1781% error, highlighting the value of real data.
• The network architecture effectively captures nonlinear flow dynamics in channelized permeability fields, enabling accurate prediction across various configurations.
• The approach successfully modifies the reduced-order model to honor observed data, improving fidelity without requiring explicit basis function reconstruction.