[Paper Review] A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Media

V. S. Maduru, K. B. Nakshatrala|arXiv (Cornell University)|Mar 11, 2026

CO2 Sequestration and Geologic Interactions0 citations

TL;DR

This paper proposes a DeepLS framework that uses a Hopf–Cole transformation to linearize Klinkenberg nonlinear gas flow, paired with a shared-trunk neural network and a Deep Least-Squares solver for stable, accurate pressure and velocity predictions and inverse parameter estimation.

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.

Motivation & Objective

• Address convergence and stability challenges in nonlinear gas-flow models with pressure-dependent permeability (Klinkenberg effect).
• Achieve accurate, stable prediction of both pressure and velocity fields in porous media.
• Enable efficient inverse modeling of pressure-dependent permeability and slippage parameters from limited data.

Proposed method

• Apply the Hopf–Cole transformation to convert nonlinear Klinkenberg-based flow into a linear Darcy-type system in transformed pressure.
• Formulate a mixed, pressure–velocity system and enforce it via a Deep Least-Squares (DeepLS) objective built from a weighted residual functional.
• Use a shared-trunk neural network with separate heads for transformed pressure and velocity to enforce physical coupling.
• Employ Fourier-feature lifting for neural inputs to capture heterogeneous fields and compute derivatives with automatic differentiation.
• Approximate integrals in the DeepLS functional with Monte Carlo collocation points over domain and boundaries.
• Adopt an adaptive loss weighting strategy to balance interior and boundary residuals during training.

Experimental results

Research questions

• RQ1 How does the Hopf–Cole transformation affect the solvability and conditioning of nonlinear gas-flow equations with the Klinkenberg effect?
• RQ2 Can a shared-trunk neural network with DeepLS provide accurate, stable predictions for pressure and velocity fields in nonlinear porous-media flow?
• RQ3 How effectively can the framework recover pressure-dependent permeability and slippage parameters from indirect observations?
• RQ4 What are the convergence and stability properties of the proposed solver in theory and numerically?
• RQ5 How do inverse-modeling capabilities perform across different pressure regimes?

Key findings

• The Hopf–Cole transformation yields a linear system in transformed pressure, enabling a more tractable solution of nonlinear gas-flow equations.
• A mixed formulation with a shared-trunk neural network improves velocity-field fidelity by enforcing consistent coupling between pressure and velocity.
• The DeepLS solver produces a nonnegative, symmetric, positive-definite objective, contributing to stable and robust training and solution accuracy.
• The framework supports inverse modeling to estimate pressure-dependent permeability and slippage parameters from limited observations.
• Numerical results demonstrate accurate recovery of flow dynamics and parameters across a wide range of pressure regimes, with robustness and computational efficiency.