A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Media

2026-03-11

AI Summary

This work addresses the strong nonlinearity in gas flow through porous media induced by the Klinkenberg effect, which renders the classical Darcy model invalid and complicates the determination of key parameters. To overcome these challenges, the study introduces the Hopf–Cole transformation into a deep learning framework, integrating a mixed-form linearized governing equation, a shared backbone neural network, and a DeepLS solver. This approach enables high-fidelity joint prediction of pressure and velocity fields while simultaneously inverting pressure-dependent permeability and slip coefficients from sparse observational data. The method effectively circumvents the limitations of conventional numerical techniques under strong nonlinearity and uncertainty, demonstrating superior accuracy, robustness, and computational efficiency across a wide pressure range.

Technology Category

Machine Learning: Deep Learning TheorySearch and Optimization: Learning to SearchNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Accurate modeling of gas flow through porous media is critical for many technological applications, including reservoir performance prediction, carbon capture and sequestration, and fuel cells and batteries. However, such modeling remains challenging due to strong nonlinear behavior and uncertainty in model parameters. In particular, gas slippage effects described by the Klinkenberg model introduce pressure-dependent permeability, which complicates numerical simulation and obscures deviations from classical Darcy flow behavior. To address these challenges, we present an integrated modeling framework for gas transport in porous media that combines a Klinkenberg-enhanced constitutive relation, Hopf-Cole-transformed mixed-form linear governing equations, a shared-trunk neural network architecture, and a Deep Least-Squares (DeepLS) solver. The Hopf-Cole transformation reformulates the original nonlinear flow equations into an equivalent linear system closely related to the Darcy model, while the mixed formulation, together with a shared-trunk neural architecture, enables simultaneous and accurate prediction of both pressure and velocity fields. A rigorous convergence analysis is performed both theoretically and numerically, establishing the stability and convergence properties of the proposed solver. Importantly, the proposed framework also naturally facilitates inverse modeling of pressure-dependent permeability and slippage parameters from limited or indirect observations, enabling efficient estimation of flow properties that are difficult to measure experimentally. Numerical results demonstrate accurate recovery of flow dynamics and parameters across a wide range of pressure regimes, highlighting the framework's robustness, accuracy, and computational efficiency for gas transport modeling and inversion in tight formations.

Problem

• nonlinear gas flow
• porous media
• Klinkenberg effect
• pressure-dependent permeability
• Darcy flow deviation

Innovation

• Hopf-Cole transformation
• Klinkenberg effect
• deep least-squares
• shared-trunk neural network
• inverse modeling