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

V. S. Maduri; K. B. Nakshatrala

math.NAMar 11, 2026

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

Authors: V. S. Maduri, K. B. Nakshatrala

Organizations: Graduate Student, Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204. · Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204.

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 & tables

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}|_{r=r_{i}}=10

and

p_{p}|_{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.

Rock type	Typical $\beta$
Glass bead packs	$\approx 0$
Metal-ceramic foams	$\approx 0$
Coarse-grained sands	0–0.5
Carbonates	1–10
Tight sandstone	2–20
Shale	10–200

Table 1. Typical values of the Klinkenberg parameter

\beta

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.

Parameter	Value
$\beta$	1
$\mu$	1
$\gamma\,\mathbf{b}(\mathbf{x})$	$\approx\mathbf{0}$
$k_{0}$	1
$r_{i}$	0.3
$r_{0}$	1.0

Table 2. Gas flow through concentric cylinders. Material and geometry parameters.

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.

Figure 5. Gas flow through concentric cylinders . Analytical and DeepLS solution fields for the concentric-cylinder benchmark, comparing the classical Darcy model with the pressure-dependent Klinkenberg formulation. (A) Radial pressure profile as a function of radius: Darcy (constant permeability), analytical Klinkenberg solution obtained via the Hopf–Cole transformation, and the DeepLS prediction. (B) Radial velocity profile showing agreement between DeepLS and the analytical Klinkenberg solution, and deviation from the classical Darcy response due to slip-induced permeability enhancement.

Figure 6. Gas flow through concentric cylinders . Convergence of the DeepLS approximation with increasing network capacity, characterized by the number of hidden layers $L$ and the number of neurons per layer $m$ . (A) $L_{2}(\Omega)$ error between the neural approximation $\widehat{u}_{\theta}$ and the reference solution $u_{\mathrm{ref}}$ as a function of the total number of neurons $N_{\mathrm{tot}}$ for three depths ( $L=2,4,6$ ). (B) $L_{2}(\Omega)$ error as a function of the network width $m$ for multiple depths ( $L=2,4,6$ ).

Figure 7. Footing problem. Schematic of the boundary value problem. The left, right, and bottom boundaries of the domain are impermeable and subject to a no-flux condition (i.e., the normal component of the seepage velocity vanishes, $\mathbf{u}(\mathbf{x})\bullet\widehat{\mathbf{n}}(\mathbf{x})=0$ ). The top boundary ( $y=5$ ) is partitioned into three contiguous segments: (i) $T_{1}$ ( $0\leq x\leq 2.5$ ), where a prescribed pressure $p_{\mathrm{p}}(\mathbf{x})=10$ is applied; (ii) $T_{2}$ ( $2.5<x\leq 7.5$ ), where a no-flux condition $\mathbf{v}(\mathbf{x})\bullet\widehat{\mathbf{n}}(\mathbf{x})=0$ is enforced; and (iii) $T_{3}$ ( $7.5<x\leq 10$ ), where the boundary is vented (i.e., hydraulically connected to an external reservoir and maintained at the reference pressure $p_{\mathrm{p}}(\mathbf{x})=1$ ).

Parameter	Value
$L$	10
$H$	5
$\beta$	1
$\mu$	1
$\phi(\mathbf{x})\,\gamma\,\mathbf{b}(\mathbf{x})$	$\mathbf{0}=[0,0]$
$k_{0}$	1

Table 3. Footing problem. Material and geometry parameters.

Figure 8. Footing problem. This figure illustrates DeepLS predictions computed using a collocation density of 12,000 points: (A) the pressure field, $p(\mathbf{x})$ , and (B) the velocity-magnitude field, $\lVert\mathbf{u}(\mathbf{x})\rVert$ .

Figure 9. Footing problem. Comparison between the DeepLS predictions and the corresponding stabilized mixed finite element solutions: A) absolute error in the pressure field, B) absolute error in the velocity magnitude, and C) relative $L^{2}$ errors in pressure and velocity as functions of collocation density. The DeepLS results are computed using collocation densities ranging from 3000 to 12000 points, while the finite element reference solutions are obtained using the stabilized mixed formulations proposed in Masud and Hughes [2002] ; Nakshatrala et al. [2006] . The relative $L^{2}$ errors decrease systematically with increasing collocation density, indicating improved agreement with the finite element reference solution and demonstrating convergence of the proposed framework.

Figure 10. Footing problem. This figure shows the evolution of the total training loss over epochs for different collocation densities. Each curve corresponds to an independent optimization run initialized with 3,000, 6,000, 9,000, and 12,000 collocation points. Increasing the collocation density yields faster convergence and a lower terminal loss, indicating more stable optimization and improved solution fidelity (i.e., the learned model exhibits reduced residuals with respect to the governing equations).

Figure 11. Flow through a layered porous medium. This figure illustrates gas flow in layered media. A rectangular domain $(L=5,H=4)$ is composed of five horizontal layers, each with a distinct (piecewise constant) intrinsic permeability. A left-to-right pressure drop is enforced by prescribing $(p^{\textrm{left}}=10)$ on the inlet boundary and $(p^{\textrm{right}}=1)$ on the outlet boundary, while the top and bottom boundaries are impermeable $(\big(\mathbf{u}(\mathbf{x})\bullet\widehat{\mathbf{n}}(\mathbf{x})=0\big))$ . This configuration is used to assess how pressure-dependent Klinkenberg slip modifies the pressure and velocity fields across permeability contrasts.

Parameter	Value
$L$	5
$H$	4
$\beta$	1
$\mu$	1
$\phi(\mathbf{x})\,\gamma\,\mathbf{b}(\mathbf{x})$	$\mathbf{0}=[0,0]$
$p^{\textrm{left}}$	10

Table 4. Flow through a layered porous medium. Material and geometry parameters.

Figure 12. Flow through a layered porous medium. This figure showcases the DeepLS results for Klinkenberg gas flow in a layered medium, highlighting interface-resolving behavior. (A) Predicted pressure field exhibits a smooth, essentially one-dimensional left-to-right drop, remaining continuous across material interfaces as required by the governing flow model. (B) Velocity field (magnitude contours with direction vectors) shows strong layer-to-layer contrasts, with flow accelerating in more conductive layers and decelerating in less conductive layers; the vectors remain predominantly aligned with the imposed pressure gradient.

Figure 13. Flow through a layered porous medium. This figure illustrates the depthwise profile of the horizontal velocity,

u_{x}

, obtained from the DeepLS solution for different collocation point densities: 3,000, 6,000, 9,000, and 12,000 points. The results exhibit a distinct step-like structure that closely reflects the underlying layered architecture, indicating that the method captures sharp interfacial variations in flux. Increasing the collocation density improves the resolution of these sharp transitions, particularly near material interfaces. The inset highlights the 12,000-collocation-point solution, which provides the most refined representation of the layered velocity profile. Such features would typically require specialized treatments—such as Discontinuous Galerkin formulations in finite element methods—to properly resolve flux discontinuities across material interfaces [ Joshaghani et al., 2019 ] .

Figure 14. Gas flow through concentric spheres. This figure illustrates the computational geometry and boundary conditions. The domain is defined by an inner sphere of radius

r_{i}

and an outer sphere of radius

r_{o}

. Prescribed pressure boundary conditions are imposed on the inner and outer spherical surfaces, while a no-normal-flux condition is enforced on the planar surface defined by

z=0

Figure 15. Gas flow through concentric spheres. (A) shows the predicted pressure field within the spherical shell, illustrating the smooth pressure decay from the inner to the outer spherical boundary and accurate enforcement of the prescribed pressure conditions. (B) shows the magnitude of the velocity, highlighting the radially varying flow intensity and the expected increase in velocity near the inner sphere. The results demonstrate the ability of the DeepLS framework to resolve both pressure and velocity fields in a fully three-dimensional curved geometry without mesh-based discretization.

Figure 16. Gas flow through concentric spheres. This figure illustrates comparison between analytical solution with the DeepLS prediction. (A) Radial pressure $p(r)$ and (B) radial Darcy velocity $u_{r}(r)$ . The DeepLS prediction overlap the analytical solution profiles confirming accurate recovery of both fields.

Figure 17. Gas flow through concentric spheres. This figure shows $L_{2}(\Omega)$ errors of the DeepLS predictions as the network width is increased for several fixed depths. (A) $L_{2}(\Omega)$ error in the predicted pressure field, $\|p-p_{\mathrm{ref}}\|_{L_{2}(\Omega)}$ . (B) $L_{2}(\Omega)$ error in the predicted velocity field, $\|\mathbf{u}-\mathbf{u}_{\mathrm{ref}}\|_{L_{2}(\Omega)}$ . The reference fields $p_{\mathrm{ref}}$ and $\mathbf{u}_{\mathrm{ref}}$ correspond to the analytical solution. As network width increases, the $L_{2}(\Omega)$ errors in both pressure and velocity decrease and then level off, indicating a diminishing-returns regime once sufficient capacity is reached. For a fixed width, deeper networks consistently yield smaller errors, showing that depth improves approximation quality.

Figure 18. Betti relation. This figure shows normalized reciprocity error against the number of neurons per hidden layer for multiple network depths. All simulations are performed for fixed interior and boundary point sets, ensuring that variations in the reciprocity error are attributable to changes in network representational capacity. Increasing width generally reduces reciprocity violation, while deeper architectures attain smaller errors at comparable widths.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 19. Graph of the Lambert–

W

function showing both branches. In this work, only the principal branch

W_{0}(x)

(branch 0) is relevant, since the pressure field to which the Lambert–

W

transformation is applied is strictly positive. For

x>0

, the mapping is bijective. For

x\in[-e^{-1},0)

, the function admits two real branches,

W_{0}(x)

and

W_{-1}(x)

, and is therefore not single-valued. Of course, the function is undefined for

x<-e^{-1}

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