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arxiv: 2604.15785 · v1 · submitted 2026-04-17 · ⚛️ physics.comp-ph · physics.flu-dyn

Recognition: unknown

Probabilistic Upscaling of Hydrodynamics in Geological Fractures Under Uncertainty

Sarah Perez , Florian Doster , Hannah Menke , Ahmed ElSheikh , Andreas Busch
Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:55 UTC · model grok-4.3

classification ⚛️ physics.comp-ph physics.flu-dyn
keywords probabilistic upscalingfracture permeabilityuncertainty quantificationdeep learning surrogategeological fracturesDarcy flowBayesian correctionhydrodynamics
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The pith

A probabilistic workflow using Bayesian correction and a Residual U-Net surrogate produces uncertainty-aware permeability estimates for natural geological fractures that remain consistent with physical flow laws.

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

The paper develops a scalable method to connect detailed images of fracture openings in rock to larger-scale predictions of fluid flow while accounting for measurement errors and model limitations. It corrects biases in standard formulas for how aperture affects permeability through Bayesian updating and trains a neural network to map local geometric variations onto statistical permeability distributions. These distributions are then used to run ensembles of flow calculations at the Darcy scale, yielding ranges of possible transmissivity values rather than single numbers. A reader would care because this supplies concrete uncertainty intervals for applications like assessing leakage risk in geological storage without requiring full physics simulations for every possible fracture realization. The results indicate that widely used empirical relations tend to misrepresent flow behavior in real, irregular fractures.

Core claim

The authors show that a hybrid workflow combining Bayesian correction of aperture-permeability model misspecification, a Residual U-Net surrogate trained to capture effects of local heterogeneity and spatial correlation, and subsequent Darcy-scale upscaling generates ensembles of permeability fields from natural shear fracture images. These ensembles propagate uncertainty to effective transmissivity while preserving consistency with Stokes-flow physics and incorporating influences from channelisation, connectivity, and three-dimensional void geometries.

What carries the argument

The hybrid probabilistic upscaling workflow that merges Bayesian aperture-permeability correction, Residual U-Net surrogate for permeability statistics, and Darcy-scale flow simulation.

If this is right

  • Uncertainty bounds on macroscopic transmissivity can be obtained without repeated high-fidelity simulations.
  • The method accounts for channelisation and complex three-dimensional void geometries in the resulting flow statistics.
  • Common empirical aperture-permeability relations exhibit systematic bias when applied to natural fractures.
  • The workflow supports uncertainty propagation across scales while preserving physical consistency.

Where Pith is reading between the lines

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

  • The same hybrid strategy could be tested on fracture networks rather than single fractures to assess connectivity at larger scales.
  • Integration with transport equations would allow probabilistic predictions of solute migration under the same uncertainty framework.
  • Application to additional core datasets from different geological settings would test whether the surrogate generalisation holds beyond the Utah samples used here.

Load-bearing premise

The Residual U-Net surrogate trained on simplified geometries accurately generalizes local heterogeneity and spatial correlation effects to natural shear fractures without overfitting or bias.

What would settle it

High-fidelity Stokes-flow simulations performed on independent natural fracture samples that produce transmissivity values lying consistently outside the workflow's predicted uncertainty intervals or that retain systematic bias relative to the ensemble outputs.

Figures

Figures reproduced from arXiv: 2604.15785 by Ahmed ElSheikh, Andreas Busch, Florian Doster, Hannah Menke, Sarah Perez.

Figure 1
Figure 1. Figure 1: (a) Examples of paired input-target training patches used to learn the probabilistic permeability mapping (4). The first column displays the mechanical aperture 𝑎𝑚(𝑥, 𝑦) in 𝜇𝑚, while the second and third columns show the corresponding log-normal permeability parameters 𝜇(𝑥, 𝑦) and 𝜎(𝑥, 𝑦). Together, these fields encode both the expected hydraulic response and its uncertainty at the patch scale, and serve a… view at source ↗
Figure 2
Figure 2. Figure 2: Residual U-Net architecture used to learn permeability statistics from mechanical aperture patches. The encoder extracts multi-scale geometrical features through residual convolutional blocks with progressively increasing channel depth (32–256), while the decoder reconstructs full-resolution fields using symmetric upsampling and skip connections. Residual shortcuts are applied within each block to improve … view at source ↗
Figure 3
Figure 3. Figure 3: (a) Greyscale 𝜇CT volume rendering of one natural shear fracture, showing physical dimensions and axis orientation (𝑧 denoting the fracture depth). (b) Representative segmented slices extracted along the 𝑦-axis after image processing. These cross-sections highlight the complex aperture structure, including contact zones, locally disconnected void clusters and branching pathways, that influence the hydrauli… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Binary cross-section of a segmented 𝜇CT fracture (zoomed), with red and blue dashed lines marking two columns used to illustrate the computation of the thickness-weighted vertical aperture. (b) Schematic representation of the aperture profiles extracted along these columns. For the red line, two branches coexist with apertures 𝑎1, 𝑎2 and heights ℎ1, ℎ2, so that Eq. (12) gives 𝑎𝑚(𝑥, 𝑦) = (𝑎1ℎ1 + 𝑎2ℎ2)/(… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of mechanical aperture fields of fracture #1: (a) Thickness-weighted mean aperture defined by Eq. (12), which blends coexisting branches according to their open thickness. (b) Maximum vertical aperture extracted from the real fracture, highlighting dominant wide channels. (c) Aperture field of the corresponding synthetic fracture, exhibiting a single connected band. (d) Relative aperture differe… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Mechanical aperture distributions for natural shear fracture #1 and its synthetic counterpart. The mean mechanical aperture, denoted ⟨𝑎𝑚⟩, and the Stokes-based permeability 𝐾𝑁 𝑆 are reported for each configuration. (b) Representative segmented cross-sections extracted along the 𝑦-axis for the synthetic fracture, taken at the same locations as for the natural fracture. These cross-sections highlight the… view at source ↗
Figure 7
Figure 7. Figure 7: Variogram analysis in the main flow direction, taken as the 𝑥-direction, used to determine the correlation lengths of both fractures: (a) Real shear fracture #1 and its synthetic counterpart. (b) Real shear fracture #2 and its synthetic counterpart. Variograms are fitted with an exponential model. The correlation length is the lag distance ℎ at which the variogram reaches 95% of the sill, which has first b… view at source ↗
Figure 8
Figure 8. Figure 8: Training history of the Residual U-Net: (a) Absolute training and validation losses, Ltrain and Lval, as functions of the number of epochs.The epoch corresponding to the minimum validation loss is indicated by the vertical dashed line. (b) Normalised losses, obtained by scaling each curve by its initial value, highlighting the relative convergence rate of the optimisation. (c) Generalisation gap magnitude,… view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the GradNorm task weights during training for the regression objectives associated with the log-normal permeability parameters 𝜇(𝑥, 𝑦) and 𝜎(𝑥, 𝑦). The adaptive weights 𝑤𝜇 and 𝑤 𝜎 are adjusted to balance the gradient magnitudes of the tasks at the U-Net output layer. The gradual decrease of 𝑤𝜇 and increase of 𝑤 𝜎 indicate a dynamic rebalancing of the learning effort between the two outputs, en… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison between target and predicted outputs of the probabilistic permeability mapping for representative samples from the training set. For each sample, the top row shows the target log-normal parameters 𝜇(𝑥, 𝑦) and 𝜎(𝑥, 𝑦) obtained from the Bayesian inference, the middle row shows the corresponding U-Net predictions, and the bottom row reports the relative error fields. descriptors obtained from the … view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between target and predicted outputs of the probabilistic permeability mapping for representative samples from the validation set (same format as [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Distributional accuracy per patch using symmetric KL divergence. For each patch, the pixelwise KL divergence between the predicted and target log-normal distributions is computed and summarized by (a) the median and (b) the 95th percentile across pixels. Training and validation patches are shown separately, with the dashed vertical line indicating the split between training and validation sets. Most patch… view at source ↗
Figure 13
Figure 13. Figure 13: Probabilistic permeability maps for the natural shear fracture #1: (a) Deterministic permeability field obtained from the local Cubic-Law relationship applied to the mechanical aperture, denoted 𝐾 𝑎𝑚 𝐶𝐿 (𝑥, 𝑦) following the notation of Perez et al. (2025) [1]. (b) Permeability fields derived from log-normal distribution predicted by the U-Net, including the lower quantile 𝑄0.025 [𝐾(𝑥, 𝑦)], mode, expected … view at source ↗
Figure 14
Figure 14. Figure 14: Probabilistic permeability maps for the synthetic fracture #1. (a) Deterministic permeability field obtained from the local Cubic-Law, 𝐾 𝑎𝑚 𝐶𝐿 (𝑥, 𝑦). (b) Probabilistic permeability descriptors predicted by the U-Net, including the lower quantile, mode, expected value, and upper quantile. Compared to the natural fracture (see [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Thickness-weighted mechanical aperture fields 𝑎𝑚(𝑥, 𝑦) and corresponding local permeability uncertainty widths, quantified as the difference between the upper and lower quantiles for the natural fracture #1 and its synthetic counterparts. The natural fracture exhibits strongly localised uncertainty patterns associated with abrupt aperture transitions, whereas the synthetic fracture displays a smoother ape… view at source ↗
Figure 16
Figure 16. Figure 16: Relative uncertainty maps of the predicted local permeability distributions for the natural shear fractures #1 and #2 and their synthetic counterparts. The local relative uncertainty is represented by the logarithm of the quantile ratio ln(𝑄0.975/𝑄0.025), which is directly proportional to the predicted log-normal standard deviation 𝜎(𝑥, 𝑦) and therefore provides a scale-independent comparison of uncertain… view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of Darcy velocity magnitude fields for the natural shear fracture #1 obtained using two permeability models: (a) Velocity field computed from the probabilistic permeability predicted by the U-Net, using the expected value E[𝐾(𝑥, 𝑦)] of the local log-normal distribution. (b) Velocity field computed from a deterministic permeability derived from the local Cubic-Law applied to the mechanical apert… view at source ↗
Figure 18
Figure 18. Figure 18: Probability distributions of the upscaled effective permeability 𝐾eff for (a) the natural shear fracture #1 and (b) its synthetic counterpart. Analytical distributions P (in blue) are reconstructed from the Darcy-upscaled permeability moments 𝐾mode and 𝐾mean predicted by the U-Net, and compared with the Monte Carlo distributions PMC (in grey) for 400 permeability realisations (see Eq. (11)). The shaded bl… view at source ↗
Figure 19
Figure 19. Figure 19: Probability distributions of the upscaled effective permeability 𝐾eff for (a) the natural shear fracture #2 and (b) its synthetic counterpart. Analytical distributions reconstructed from the predicted log-normal parameters are compared with Monte Carlo estimates. The shaded blue region represents the 95% confidence interval of the analytical distribution, while vertical lines denote the mean, mode, and re… view at source ↗
Figure 20
Figure 20. Figure 20: Distribution of Stokes permeabilities 𝐾𝑁 𝑆 obtained on the local aperture patches used for U-Net training (blue histogram shown in log10 (𝐾𝑁 𝑆 [𝜇𝑚2 ]) space). The vertical lines indicate the permeability values of the natural shear fractures #1 and #2, with 𝐾𝑁 𝑆 = 4.48 𝜇𝑚2 and 𝐾𝑁 𝑆 = 62.08 𝜇𝑚2 , respectively. The upscaled permeability of fracture #1 lies outside the range spanned by the training patches, … view at source ↗
Figure 21
Figure 21. Figure 21: Representative segmented cross-sections extracted along the 𝑦-axis for (a) the natural shear fracture #2 and (b) its synthetic counterpart. Compared to the natural fracture #1 (see Figure 3b), fracture #2 exhibits fewer branching structures. The synthetic geometry preserves the main aperture envelope while collapsing secondary branches into a single connected channel. (a) (b) [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 22
Figure 22. Figure 22: (a) Mechanical aperture distributions for natural shear fracture #2 and its synthetic counterpart. The mean mechanical aperture, denoted ⟨𝑎𝑚⟩, and the Stokes-based permeability 𝐾𝑁 𝑆 are reported for each configuration. Both distributions exhibit similar trend, reflecting comparable aperture variability. (b) Spatial distribution of the thickness-weighted mechanical aperture for the natural and synthetic fr… view at source ↗
Figure 23
Figure 23. Figure 23: Thickness-weighted mechanical aperture fields 𝑎𝑚(𝑥, 𝑦) and corresponding local permeability uncertainty widths, quantified as the difference between the upper and lower quantiles for the natural fracture #2 and its synthetic counterparts [PITH_FULL_IMAGE:figures/full_fig_p034_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Comparison of Darcy velocity magnitude fields for the natural shear fracture #2 obtained using two permeability models: (a) Velocity field computed from the probabilistic permeability predicted by the U-Net, using the expected value E[𝐾(𝑥, 𝑦)] of the local log-normal distribution. (b) Velocity field computed from a deterministic permeability derived from the local Cubic-Law applied to the mechanical apert… view at source ↗
Figure 25
Figure 25. Figure 25: Probabilistic permeability maps for the natural shear fracture #2 and its synthetic counterpart: (a, c) Deterministic permeability fields obtained from the local Cubic-Law, 𝐾 𝑎𝑚 𝐶𝐿 (𝑥, 𝑦). (b, d) Probabilistic permeability descriptors predicted by the U-Net, including the lower quantiles, modes, expected values, and upper quantiles. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_25.png] view at source ↗
read the original abstract

Flow and transport in fractured geological media are strongly controlled by aperture heterogeneity and uncertainty in subsurface characterisation, yet most upscaling approaches rely on deterministic representations of fracture permeability. This study presents a scalable probabilistic workflow that bridges image-based fracture geometry and uncertainty-aware hydraulic predictions across scales. The approach integrates Bayesian correction of aperture-permeability model misspecification, a deep learning surrogate for predicting spatially distributed permeability statistics, and Darcy-scale flow upscaling to propagate uncertainty to effective transmissivity. The workflow is applied to natural shear fractures from core material in the Little Grand Wash Fault damage zone (Utah) and to simplified geometries derived from the same datasets. The Bayesian component quantifies uncertainty due to measurement errors and imperfect constitutive relations, while a Residual U-Net learns the effects of local heterogeneity and spatial correlation on predicted permeability uncertainty. Together, these components generate ensembles of permeability fields that are subsequently upscaled to probabilistic macroscopic flow responses. Results show that common empirical aperture-permeability relations are systematically biased for natural fractures, whereas the proposed probabilistic workflow yields uncertainty-aware permeability estimates consistent with physics-based behaviour. The method captures the impact of channelisation, connectivity, and complex 3D void geometries on transmissivity while quantifying the resulting uncertainty bounds. Computational efficiency arises from the proposed hybrid strategy for probabilistic upscaling, which combines physics-informed and data-driven approaches, preserves Stokes-flow consistency and supports uncertainty propagation without repeated high-fidelity simulations.

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 proposes a probabilistic workflow for upscaling hydrodynamics in geological fractures under uncertainty. It integrates Bayesian correction of aperture-permeability model misspecification, a Residual U-Net surrogate trained to predict spatially distributed permeability statistics from fracture geometry, and Darcy-scale flow upscaling to obtain probabilistic effective transmissivity. The method is demonstrated on natural shear fractures from Little Grand Wash Fault core samples (Utah) and on simplified geometries derived from the same datasets, with the central claim that empirical aperture-permeability relations are systematically biased while the proposed workflow produces uncertainty-aware estimates consistent with physics-based behaviour, capturing channelisation, connectivity, and 3D void geometry effects.

Significance. If the surrogate generalizes without dataset-specific bias, the hybrid Bayesian–deep-learning–upscaling strategy offers a computationally efficient route to uncertainty propagation in fracture flow that avoids repeated high-fidelity Stokes simulations. This addresses a practical need in subsurface modelling where aperture heterogeneity and characterisation uncertainty dominate macroscopic transport predictions.

major comments (2)
  1. [Abstract] Abstract: the claim that the workflow 'yields uncertainty-aware permeability estimates consistent with physics-based behaviour' for natural fractures rests on the Residual U-Net generalizing the effects of local heterogeneity and spatial correlation. However, the surrogate is trained on simplified geometries derived from the identical Little Grand Wash Fault core datasets later used as the natural-fracture test cases. This overlap creates a risk that the network learns imaging artifacts, segmentation choices, or correlation lengths specific to those cores rather than the underlying Stokes physics; because the Bayesian correction and Darcy upscaling ingest the U-Net output directly, any such bias would propagate into the reported transmissivity uncertainty bounds.
  2. [Abstract] Abstract and methods description: no validation metrics (e.g., R², MAE, or cross-validation scores), error bars on the surrogate predictions, or direct comparison of the upscaled transmissivity ensembles against full Stokes solutions on held-out natural fracture geometries are provided. Without these, it is not possible to confirm that the probabilistic estimates are indeed consistent with physics-based behaviour or that the uncertainty bounds are well-calibrated.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from explicit statements of the training/validation split sizes and the precise definition of 'simplified geometries' versus 'natural shear fractures' to allow readers to assess the degree of data leakage.
  2. Notation for the permeability statistics output by the Residual U-Net (mean, variance, or full distribution) should be defined once and used consistently when describing the Bayesian correction step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects of validation and generalization in our hybrid workflow. We address each major comment below, clarifying the data handling and committing to additions that strengthen the manuscript's claims without overstating current results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the workflow 'yields uncertainty-aware permeability estimates consistent with physics-based behaviour' for natural fractures rests on the Residual U-Net generalizing the effects of local heterogeneity and spatial correlation. However, the surrogate is trained on simplified geometries derived from the identical Little Grand Wash Fault core datasets later used as the natural-fracture test cases. This overlap creates a risk that the network learns imaging artifacts, segmentation choices, or correlation lengths specific to those cores rather than the underlying Stokes physics; because the Bayesian correction and Darcy upscaling ingest the U-Net output directly, any such bias would propagate into the reported transmissivity uncertainty bounds.

    Authors: We agree that training and testing on geometries derived from the same core samples introduces a risk of learning dataset-specific features rather than general Stokes physics. The simplified geometries were created by applying controlled idealizations (e.g., aperture smoothing and removal of minor segmentation noise) to the original LGW images specifically to isolate heterogeneity and connectivity effects for surrogate training, while the natural-fracture cases preserve full imaging artifacts and 3D void complexity for evaluation. However, we acknowledge that this does not constitute fully independent held-out data from different geological settings. In the revision we will (i) explicitly document the train/test split ratios and simplification procedures, (ii) add a sensitivity test using synthetic fractures with randomized correlation lengths generated independently of the LGW dataset, and (iii) qualify the generalization claim in the abstract and discussion to reflect the current data scope. revision: yes

  2. Referee: [Abstract] Abstract and methods description: no validation metrics (e.g., R², MAE, or cross-validation scores), error bars on the surrogate predictions, or direct comparison of the upscaled transmissivity ensembles against full Stokes solutions on held-out natural fracture geometries are provided. Without these, it is not possible to confirm that the probabilistic estimates are indeed consistent with physics-based behaviour or that the uncertainty bounds are well-calibrated.

    Authors: The referee correctly notes the absence of quantitative surrogate validation metrics and direct Stokes-to-upscaled comparisons on held-out natural geometries in the abstract and methods summary. While the full manuscript contains qualitative visualizations of permeability fields and transmissivity ensembles, it does not report R², MAE, cross-validation scores, or error bars on U-Net outputs, nor does it present side-by-side Stokes versus Darcy-upscaled transmissivity statistics for the natural-fracture cases. We will revise the manuscript to include: (a) a dedicated validation subsection with R², MAE, and 5-fold cross-validation results for the Residual U-Net on the simplified training set, (b) error bars or uncertainty maps on predicted permeability statistics, and (c) a new comparison table/figure showing ensemble transmissivity statistics from the full workflow against available full-Stokes reference solutions on additional held-out natural samples (where such simulations exist in our dataset). These additions will allow direct assessment of physics consistency and uncertainty calibration. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's workflow combines Bayesian correction of aperture-permeability model misspecification, a Residual U-Net surrogate trained on simplified geometries, and Darcy-scale upscaling to produce uncertainty-aware transmissivity estimates. No load-bearing step reduces by construction to its inputs: the surrogate learns statistical effects from training data but is not defined in terms of the target natural-fracture outputs, the Bayesian component quantifies separate uncertainty sources, and upscaling applies independent physics. The abstract and described components show no self-definitional relations, fitted inputs renamed as predictions, or self-citation chains that force the central claims. The hybrid strategy is presented as preserving Stokes-flow consistency without tautological equivalence to the input datasets.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; the workflow relies on standard fluid mechanics and machine learning components with limited explicit free parameters or new entities stated.

free parameters (2)
  • Residual U-Net hyperparameters
    Chosen to learn permeability statistics and spatial correlations from fracture image data
  • Bayesian prior distributions
    Used to quantify uncertainty from measurement errors and constitutive model misspecification
axioms (2)
  • domain assumption Stokes flow at the microscale can be upscaled to Darcy flow at the macroscale for effective transmissivity
    Invoked for the final upscaling step to propagate permeability ensembles to macroscopic responses
  • domain assumption Core-derived fracture geometries are representative of natural conditions
    Basis for applying the workflow to Little Grand Wash Fault data

pith-pipeline@v0.9.0 · 5562 in / 1559 out tokens · 44446 ms · 2026-05-10T07:55:17.498515+00:00 · methodology

discussion (0)

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Reference graph

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