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

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.

Figures (20)

• Figure 1: A) This figure illustrates the three-dimensional concentric spherical porous domain with inner radius $r_i$ and outer radius $r_o$ with prescribed pressure on the spherical boundaries $p_p\vert_{r=r_i}=10$ and $p_p\vert_{r=r_o}=1$ together with a symmetry condition on the cut plane $z=0$ enforcing zero normal flow. B) Shows the resulting pressure profile across the domain demonstrating that the proposed framework delivers stable high-fidelity predictions.
• Figure 1: Conceptual framework and workflow of the proposed DeepLS framework. The nonlinear governing equations are reformulated via the Hopf–Cole transformation, yielding a linear system expressed in terms of a transformed pressure variable. A least-squares energy functional is then constructed for the resulting linear Darcy problem. The functional is minimized using a neural network–based deep least-squares formulation to compute the transformed pressure and velocity fields. Finally, the physical gas pressure is recovered through the inverse Hopf–Cole transformation, completing the solution procedure.
• Figure 2: The figure shows the neural network architecture used under the DeepLS framework.
• Figure 3: Gas flow through concentric cylinders. Schematic of a pressure-driven gas flow problem in an annular porous domain bounded by two concentric circles with radii $r_i$ and $r_o$. Prescribed pressures at the inner ($p_{\mathrm{p}}(r_i)=10$) and outer ($p_{\mathrm{p}}(r_o)=1$) boundaries induce an axisymmetric pressure field and the corresponding radial flow.
• Figure 4: Gas flow through concentric cylinders. (A) Contours of the pressure field $p(\mathbf{x})$ in the concentric annular domain obtained with the proposed DeepLS framework under the Klinkenberg model, showing radial decay from the inner to the outer boundary. (B) Contours of the corresponding velocity magnitude $\lVert \mathbf{u}(\mathbf{x}) \rVert$, with superimposed velocity vectors indicating a purely radial flow, attaining a maximum near the inner boundary and decreasing monotonically toward the outer boundary.

Theorems & Definitions (24)

• Remark 1
• Remark 2
• Remark 3: Admissibility of prescribed pressure data
• Remark 4: Alternative regularity assumptions for the flux
• Theorem 1: Coercivity
• proof
• Theorem 2: Boundedness
• proof
• Lemma 3: Strong convexity of $\Pi_{\mathrm{LS}}$
• proof