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arxiv: 2606.07215 · v1 · pith:QCXUS5W7new · submitted 2026-06-05 · 💻 cs.CE

A Comparative Study of Deep Learning Models for Geological Carbon Sequestration

Giovanni Zingaro , Robert Gracie , Yuri Leonenko This is my paper

Pith reviewed 2026-06-27 20:29 UTC · model grok-4.3

classification 💻 cs.CE
keywords deep learningsurrogate modelsgeological carbon sequestrationFourier neural operatorssubsurface flowCO2 saturationpressure build-upU-FNO
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The pith

Deep learning surrogate accuracy for carbon sequestration varies strongly with whether the target field follows a hyperbolic or elliptic PDE.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper compares five deep learning architectures as fast surrogates for expensive numerical reservoir simulations of CO2 injection. It runs the models on a 2D single-wellbore problem whose permeability, porosity, injection rate, and reservoir properties are varied across many realizations. Results show that no single architecture wins on both outputs: the U-FNO variant records the lowest error on CO2 saturation fields, while the plain FNO records the lowest error on pressure build-up. The authors attribute the split to the different mathematical character of the saturation transport equation versus the pressure equation. Because real-time optimization and uncertainty quantification for carbon storage require many repeated solves, knowing which architecture matches each PDE type directly affects which surrogate can be deployed at scale.

Core claim

In a controlled benchmark of U-Net, V-Net, temporal convolutional networks, Fourier neural operators, and U-FNO on the 2D CO2 injection problem, surrogate performance is strongly dependent on the underlying PDE type, with U-FNO achieving the highest accuracy for predicting CO2 saturation fields and FNO providing the best performance for pressure build-up prediction.

What carries the argument

Head-to-head comparison of U-Net, V-Net, TCN, FNO and U-FNO trained to map heterogeneous permeability-porosity fields and injection parameters to transient saturation and pressure fields.

If this is right

  • For saturation prediction tasks the U-FNO architecture should be selected over the other tested models.
  • For pressure build-up prediction the plain FNO should be selected over the other tested models.
  • Architecture selection for coupled flow problems must account for the hyperbolic versus elliptic character of each field.
  • Memory and training-time differences among the models become decision criteria once accuracy rankings are known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the 2D ranking holds in 3D, digital-twin workflows for carbon storage could route saturation and pressure queries to different specialized surrogates rather than a single model.
  • The observed PDE-type dependence suggests that future operator-learning work should test whether separate Fourier or convolutional branches for hyperbolic and elliptic components improve accuracy on fully coupled multi-phase problems.
  • History-matching and optimization loops that alternate between saturation and pressure updates could achieve lower overall error by switching architectures mid-loop.

Load-bearing premise

The 2D single-wellbore injection problem with anisotropic heterogeneous permeability and porosity is representative of the high-dimensional transient subsurface flow problems encountered in real geological carbon sequestration.

What would settle it

Repeating the identical architecture comparison on a 3D heterogeneous reservoir model or on a multi-well injection scenario and checking whether U-FNO and FNO retain their respective top rankings for saturation and pressure.

Figures

Figures reproduced from arXiv: 2606.07215 by Giovanni Zingaro, Robert Gracie, Yuri Leonenko.

Figure 1
Figure 1. Figure 1: Schematic of the radially symmetric single-wellbore GCS domain, including domain geometry, initial conditions, and boundary conditions. 2.1. Governing Equations The governing equations for coupled multi-constituent flow of CO2 and brine in the context of geological CO2 storage consist of statements of mass conservation for the brine and CO2 phases, combined with the multi-phase fluxes of each phase. The ma… view at source ↗
Figure 2
Figure 2. Figure 2: Fourier Neural Operator (FNO) architecture schematic. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: U-Net Enhanced Fourier Neural Operator (U-FNO) architecture schematic. 3.5. Loss Function and Training Design Similar to Wen et al. [34], we employ a multi-component loss function for both the CO2 saturation and pressure build-up fields. However, we extend their formulation by replacing the first-derivative term with the full spatial gradient of the predicted field. The original loss function is defined as… view at source ↗
Figure 4
Figure 4. Figure 4: Test dataset root mean square error (RMSE) vs. standard deviation in permeability field for all models for CO2 saturation field. 4.1.2. Representative Test Cases The predictive performance of each surrogate model is further evaluated on two representative test cases. The input fields and ground truth CO2 saturation fields for Test Cases 1 and 2 are shown in Figures 5 and 6, respectively. The predicted and … view at source ↗
Figure 5
Figure 5. Figure 5: Input fields and ground truth CO2 saturation field for Test Case 1. Scalar inputs: Qinj = 1.99 MT/yr, T = 118.9 ◦C, Pinit = 213.3 bar, Swi = 0.24, λ = 0.52. (a) kr (mD) (b) kz (mD) (c) ϕ (-) (d) s (-), t = 30.0 yr [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Input fields and ground truth CO2 saturation field for Test Case 2. Scalar inputs: Qinj = 0.66 MT/yr, T = 148.6 ◦C, Pinit = 279.4 bar, Swi = 0.15, λ = 0.45. All models perform reasonably well with R2 > 0.94 for Test Case 1. The U￾20 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: CO2 saturation field predictions for Test Case 1: (a) predicted field sˆ at t = 30.0 yr, (b) normalized absolute error |s − sˆ|/|s|max at t = 30.0 yr, and (c) parity plot over the full 30-year injection period. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: CO2 saturation field predictions for Test Case 2: (a) predicted field sˆ at t = 30.0 yr, (b) normalized absolute error |s − sˆ|/|s|max at t = 30.0 yr, and (c) parity plot over the full 30-year injection period. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Power spectral density of the CO2 saturation field s at t = 30.0 yr for Test Case 1. (a) r-direction, t = 30.0 yr (b) z-direction, t = 30.0 yr [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Power spectral density of the CO2 saturation field s at t = 30.0 yr for Test Case 2. The PSD curves demonstrate that all models capture the low-frequency content 23 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Test dataset root mean square error (RMSE) vs. standard deviation in permeability field for all models for pressure build-up field. (a) kr (mD) (b) kz (mD) (c) ϕ (-) (d) p (bar) [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Input fields and ground truth pressure build-up field (t = 30.0 yr) for Test Case 1. Scalar inputs: Qinj = 0.49 MT/yr, T = 87.9 ◦C, Pinit = 159.0 bar, Swi = 0.12, λ = 0.41. (a) kr (mD) (b) kz (mD) (c) ϕ (-) (d) p (bar) [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Input fields and ground truth pressure build-up field (t = 30.0 yr) for Test Case 2. Scalar inputs: Qinj = 0.21 MT/yr, T = 57.0 ◦C, Pinit = 156.4 bar, Swi = 0.21, λ = 0.64. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Pressure build-up field predictions for Test Case 1: (a) predicted field pˆ at t = 30.0 yr, (b) normalized absolute error |p − pˆ|/|p|max at t = 30.0 yr, and (c) parity plot over the full 30-year injection period. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Pressure build-up field predictions for Test Case 2: (a) predicted field pˆ at t = 30.0 yr, (b) normalized absolute error |p − pˆ|/|p|max at t = 30.0 yr, and (c) parity plot over the full 30-year injection period. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Power spectral density of the pressure build-up field p at t = 30.0 yr for Test Case 1. (a) r-direction, t = 30.0 yr (b) z-direction, t = 30.0 yr [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Power spectral density of the pressure build-up field p at t = 30.0 yr for Test Case 2. All models capture the dominant low-frequency spatial features of the pressure field. However, FNO-based architectures more accurately reproduce the full spatial 29 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
read the original abstract

Numerical reservoir simulations are extremely computationally expensive, as they require the repeated solution of large nonlinear algebraic systems derived from the discretized governing equations. With growing demand for real-time optimization, uncertainty quantification, and history matching in digital twin applications, reducing computational cost has become essential. Deep learning (DL)--based surrogate models have emerged as an effective approach for accelerating subsurface flow simulations. Here, we seek to determine which DL architectures are best suited for high-dimensional, transient subsurface flow problems. In this study, we examine the advantages and relative costs associated with training such models, including memory requirements, training speed, accuracy, robustness, and generalization. We conduct a comparative study of several DL architectures commonly used as surrogate models for subsurface flow problems, including U-Net, V-Net, Temporal Convolutional Networks, Fourier Neural Operators (FNO), and a U-Net--enhanced FNO (U-FNO). As a benchmark, we compare the performance of the studied models for geological carbon sequestration to predict transient pressure build-up and CO$_2$ saturation fields. We study the problem of CO$_2$ injection into a single wellbore in a two-dimensional domain, which is parameterized by anisotropic, heterogeneous permeability and porosity fields, injection configurations, and reservoir properties. Results demonstrate that surrogate model performance is strongly dependent on the underlying PDE type (i.e., hyperbolic vs. elliptic). The U-FNO achieves the highest accuracy for predicting CO$_2$ saturation fields, while the FNO provides the best performance for pressure build-up prediction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript performs a comparative evaluation of deep learning surrogate models (U-Net, V-Net, Temporal Convolutional Networks, FNO, and U-FNO) for accelerating numerical simulations of geological carbon sequestration. Using a 2D single-wellbore CO₂ injection benchmark with heterogeneous anisotropic permeability/porosity fields, the study reports relative performance in accuracy, training speed, memory use, robustness, and generalization for predicting transient pressure build-up (elliptic) and CO₂ saturation (hyperbolic) fields, concluding that model ranking depends strongly on PDE type with U-FNO best for saturation and FNO best for pressure.

Significance. If the empirical rankings prove robust under proper statistical validation and controlled isolation of factors, the work would offer practical guidance for selecting operator-learning architectures in subsurface flow surrogates, directly supporting real-time optimization and uncertainty quantification in GCS digital twins. The explicit inclusion of computational costs (memory, training time) alongside accuracy is a positive feature that strengthens applicability.

major comments (2)
  1. [Abstract] Abstract: The central claim that 'surrogate model performance is strongly dependent on the underlying PDE type (i.e., hyperbolic vs. elliptic)' is not load-bearing supported by the described experiments. Both fields are generated by the same coupled multiphase system; no ablation holds field statistics fixed while varying only the operator class (e.g., pure Darcy vs. pure advection problems), so the observed U-FNO vs. FNO ranking could equally reflect output regularity, discontinuity handling, or loss weighting rather than the hyperbolic/elliptic distinction.
  2. [Abstract] Abstract / benchmark description: The 2D single-wellbore injection problem is presented as representative of 'high-dimensional transient subsurface flow problems,' yet no scaling studies or comparisons to 3D heterogeneous cases are referenced to substantiate this; this assumption directly underpins the generalization claims for real GCS applications.
minor comments (2)
  1. [Abstract] Abstract: Dataset sizes, number of training realizations, error metrics (e.g., relative L2, MAE), statistical significance tests, and validation procedures are not reported, making it impossible to assess whether the stated performance differences are statistically meaningful.
  2. The manuscript should clarify whether the same loss weighting and hyperparameter tuning protocol was used across all architectures when comparing saturation versus pressure predictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our comparative study of deep learning surrogates for geological carbon sequestration. We address the two major comments below and will make revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'surrogate model performance is strongly dependent on the underlying PDE type (i.e., hyperbolic vs. elliptic)' is not load-bearing supported by the described experiments. Both fields are generated by the same coupled multiphase system; no ablation holds field statistics fixed while varying only the operator class (e.g., pure Darcy vs. pure advection problems), so the observed U-FNO vs. FNO ranking could equally reflect output regularity, discontinuity handling, or loss weighting rather than the hyperbolic/elliptic distinction.

    Authors: We agree that the experiments are performed on the coupled multiphase flow system and do not include controlled ablations that isolate the PDE type while holding other factors fixed. The observed performance differences between pressure and saturation predictions could indeed arise from other characteristics of the output fields. We will revise the abstract and discussion sections to remove the strong causal claim of dependence on PDE type and instead report the empirical finding that U-FNO performed best on saturation while FNO performed best on pressure, with a note that further work would be needed to isolate the underlying cause. revision: yes

  2. Referee: [Abstract] Abstract / benchmark description: The 2D single-wellbore injection problem is presented as representative of 'high-dimensional transient subsurface flow problems,' yet no scaling studies or comparisons to 3D heterogeneous cases are referenced to substantiate this; this assumption directly underpins the generalization claims for real GCS applications.

    Authors: The study is conducted on a 2D benchmark with heterogeneous fields to enable controlled comparison of architectures under varying permeability, porosity, and injection conditions. We acknowledge that no scaling studies or 3D results are presented. We will revise the abstract and introduction to describe the setup explicitly as a 2D heterogeneous reservoir benchmark and qualify any generalization statements accordingly, noting extension to 3D as future work. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical model comparison on benchmark task

full rationale

The paper is a pure empirical comparative study of existing DL architectures (U-Net, FNO, U-FNO, etc.) on a fixed 2D single-wellbore CO2 injection benchmark. No derivations, ansatzes, fitted parameters renamed as predictions, or self-citation chains appear in the central claims. Performance rankings are reported directly from training and testing on the same dataset; the PDE-type dependence statement is an interpretation of observed numerical results rather than a reduction to prior self-referential inputs. The analysis is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Empirical comparative study with no new mathematical constructs, free parameters, or invented entities.

pith-pipeline@v0.9.1-grok · 5801 in / 1000 out tokens · 28632 ms · 2026-06-27T20:29:28.458309+00:00 · methodology

discussion (0)

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