[论文解读] 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 Interactions被引用 0

一句话总结

本论文提出一个 DeepLS 框架，使用 Hopf–Cole 变换将 Klinkenberg 非线性气体流动线性化，配以共享干网络和 Deep Least-Squares 求解器，实现稳健、准确的压力与速度预测以及对参数的反演估计。

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.

研究动机与目标

• 在具有压力相关渗透率（Klinkenberg 效应）的非线性气体流动模型中解决收敛性与稳定性挑战。
• 实现对多孔介质中压力和速度场的准确、稳定预测。
• 在有限数据下实现对压力相关渗透率与滑移参数的高效反问题建模。

提出的方法

• 将 Hopf–Cole 变换应用于将基于 Klinkenberg 的非线性流动转化为变换后压力下的线性达西型系统。
• 将问题表述为混合的压力–速度系统，并通过一个基于加权残差泛函的 Deep Least-Squares（DeepLS）目标来约束。
• 使用共享干神经网络，为变换后压力与速度设置单独头部，以强化物理耦合。
• 对神经输入使用傅里叶特征提升，以捕捉非均匀场并通过自动微分计算导数。
• 在 DeepLS 泛函中用蒙特卡罗取点在域内及边界上的离散化积分近似。
• 采用自适应损失加权策略，在训练时平衡内部残差与边界残差。

实验结果

研究问题

• RQ1 Hopf–Cole 变换如何影响带有 Klinkenberg 效应的非线性气体流动方程的可解性与条件数？
• RQ2 带有 DeepLS 的共享干神经网络是否能为非线性多孔介质流动中的压力与速度场提供高精度、稳定的预测？
• RQ3 该框架在有限观测下从间接观测中有效恢复压力相关渗透率与滑移参数的能力如何？
• RQ4 所提求解器在理论与数值方面的收敛性与稳定性属性是如何的？
• RQ5 在不同压力区间，逆模型能力的表现如何？

主要发现

• Hopf–Cole 变换在变换后压力下得到线性系统，使非线性气体流动方程更易求解。
• 带有共享干神经网络的混合表述通过强制压力与速度之间的一致耦合，提升了速度场的保真度。
• DeepLS 求解器产生非负、对称、正定目标，有助于稳定的训练与解的精度。
• 该框架支持从有限观测中进行压力相关渗透率与滑移参数的反问题建模。
• 数值结果在广泛的压力区间内显示出对流动动力学与参数的高精度恢复，同时具备鲁棒性与计算效率。