Physics-informed reservoir characterization from bulk and extreme pressure events with a differentiable simulator
Authors: Harun Ur Rashid, Mingxin Li, Aleksandra Pachalieva, Georg Stadler, Daniel O'Malley
Organizations: Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA · Courant Institute School of Mathematics, Computing and Data Science, New York University, New York City, NY 10012, USA
Accurate characterization of subsurface heterogeneity is challenging but essential for applications such as reservoir pressure management, geothermal energy extraction and CO2, H2, and wastewater injection operations. This challenge becomes especially acute in extreme pressure events, which are rarely observed but can strongly affect operational risk. Traditional history matching and inversion techniques rely on expensive full-physics simulations, making it infeasible to handle uncertainty and extreme events at scale. Purely data-driven models often struggle to maintain physics consistency when dealing with sparse observations, complex geology, and extreme events. To overcome these limitations, we introduce a physics-informed machine learning method that embeds a differentiable subsurface flow simulator directly into neural network training. The network infers heterogeneous permeability fields from limited pressure observations, while training minimizes both permeability and pressure losses through the simulator, enforcing physical consistency. Because the simulator is used only during training, inference remains fast once the model is learned. In an initial test, the proposed method reduces the pressure inference error by half compared with a purely data-driven approach. We then extend the test over eight distinct data scenarios, and in every case, our method produces significantly lower pressure inference errors than the purely data-driven model. We also evaluate our method on extreme events, which represent high-consequence data in the tail of the sample distribution. Similar to the bulk distribution, the physics-informed model maintains higher pressure inference accuracy in the extreme event regimes. Overall, the proposed method enables rapid, physics-consistent subsurface inversion for real-time reservoir characterization and risk-aware decision-making.
Figures & tables
Figure 1: Simulation domain used for the reservoir characterization method with monitoring points in red and a heterogeneous permeability field in the background. The trained NN model model infers the permeability of the field from the pressure values at the monitoring points.
Parameter
Value
Batch size
5
Number of batches
100
Samples per iteration
500
Number of iterations
10000
Total number of training samples
[5000, 50000]
Number of validation samples
200
Table 1: Parameters used in training the neural network model.
Figure 2: Training method for the data-driven model, which shows the method for NN model training using only precomputed permeability-pressure pairs without consideration of any physical constraint.
Figure 3: Method for the physics-informed learning, where the NN model learn from both pressure and permeability miss match having a differentiable simulator within the training loop.
Parameters
Value
Unit
Domain size
200 × 200
m
Number of grid blocks
51 × 51
-
Number of observation nodes
200
-
Left-right dirichlet pressure
[10 0]
MPa
Top-bottom dirichlet pressure
[0.5 0]
MPa
Variogram (Covariance) type
Matern, exponential
-
Table 2: Parameters used in the physics-based model.
Figure 4: Training and and validation errors for the physics-informed and data-driven model in base case scenario. (a) Training errors, (b) validation errors. The lower training errors for the data-driven model is the result of having only the permeability error in the loss calculation.
Figure 5: Pressure and permeability errors from the validation data. (a) Permeability errors is approximately the same for the data-driven and physics-informed models towards the end of the training (b) pressure error is lower in the physics-informed approach as it accounts for the pressure mismatch in the loss function.
Figure 6: Evaluation of the trained model. (a) Pressure inference error from the data-driven model; (b) pressure inference error from the physics-informed model. The data-driven model produces significantly higher error value in matching the true pressure field.
Figure 7: Evaluation of the trained model. (a) Permeability inference error from the data-driven model; (b) permeability inference error from the physics-informed model. The error differences are insignificant, however, the physics-informed model provides a slightly better permeability inference.
Figure 8: Statistical distribution of inference error. (a) Empirical cumulative distribution function (ECDF) for pressure inference errors. (b) ECDF for permeability inference errors. (c) Histogram of pressure errors. (d) Histogram of permeability errors. (e) Box plot comparison of model performance (Left axis: Pressure MSE; Right axis: Permeability L2 Error). (f) Relative error scatter between physics-informed and data-driven inference. These plots confirm that the physics-informed model performs more accurately in pressure inference while maintaining comparable accuracy in permeability inference.
Training
Number of
Correlation
Case
data size
observations
length
name
Large (50 k)
Large (200)
Large (100 m)
LLL
Large (50 k)
Large (200)
Small (10 m)
LLS
Large (50 k)
Small (50)
Large (100 m)
LSL
Large (50 k)
Small (50)
Small (10 m)
LSS
Small (5 k)
Large (200)
Large (100 m)
SLL
Table 3: Summary of training scenarios defined by dataset size, number of observations, and correlation length of the permeability field. The assigned case names represent combinations of these three factors.
Case
% PI wins perm
% PI wins pres
PI reduces perm error
PI reduces pres error
LLL
59%
100%
-0.8%
59.4%
LLS
97%
100%
2.4%
39.7%
LSL
56%
99%
3.9%
42.3%
LSS
94%
100%
3.8%
63.9%
SLL
59%
99%
3.4%
41.2%
SLS
83%
100%
3.1%
33.1%
Table 4: Comparison of inference accuracy for the physics-informed and data-driven models in variable data conditions.
Figure 9: Comparison of estimated error in inferred pressure between the physics-informed and data-driven model for the eight data scenarios shows the physics-informed model infers pressure with higher accuracy in every scenario.
Figure 10: Comparison of permeability inference errors between the physics-informed and data-driven models for the eight data scenario shows the physics-informed model perform with similar accuracy in permeability inference despite accounting pressure loss in learning.
Figure 11: Generation of rare-event samples. (a) Location of the critical point. (b) Distribution of pressure at the critical location based on 10,000 random permeability samples. Samples which result in pressures above the prescribed threshold value of 1.285 MPa (top 1%) are selected as rare events.
Figure 12: Comparison of pressure inference errors in the rare event regime. Models trained only on bulk data (tBeR) show significantly higher error, while incorporating rare event samples into training reduces error for both approaches. The physics-informed model achieves the largest improvement and maintains strong accuracy even when trained solely on rare event data (tReR).
Reservoir simulation workflows face a fundamental data asymmetry: input parameter fields (geostatistical permeability realizations, porosity distributions) are free to generate in arbitrary quantities, yet existing neural operator surrogates require large corpora of expensive labeled simulation trajectories and cannot exploit this unlabeled structure. We introduce PI-JEPA (Physics-Informed Joint Embedding Predictive Architecture), a surrogate pretraining framework that trains without any completed PDE solves, using masked latent prediction on unlabeled parameter fields under per-sub-operator PDE residual regularization. The predictor bank is structurally aligned with the Lie--Trotter operator-splitting decomposition of the governing equations, dedicating a separate physics-constrained latent module to each sub-process (pressure, saturation transport, reaction), enabling fine-tuning with as few as 100 labeled simulation runs. On single-phase Darcy flow, PI-JEPA achieves 1.9× lower error than FNO and 2.4× lower error than DeepONet at Nℓ=100, with 24% improvement over supervised-only training at Nℓ=500, demonstrating that label-free surrogate pretraining substantially reduces the simulation budget required for multiphysics surrogate deployment.
Brandon Yee, Pairie Koh
Physics Lab, Yee Collins Research Group · Graduate School of Business, Stanford University
Accurate prediction of permeability tensors from pore-scale microstructure images is essential for subsurface flow modeling, yet direct numerical simulation requires hours per sample, fundamentally limiting large-scale uncertainty quantification and reservoir optimization workflows. A physics-informed deep learning framework is presented that resolves this bottleneck by combining a MaxViT hybrid CNN-Transformer architecture with progressive transfer learning and differentiable physical constraints. MaxViT's multi-axis attention mechanism simultaneously resolves grain-scale pore-throat geometry via block-local operations and REV-scale connectivity statistics through grid-global operations, providing the spatial hierarchy that permeability tensor prediction physically requires. Training on 20000 synthetic porous media samples spanning three orders of magnitude in permeability, a three-phase progressive curriculum advances from an ImageNet-pretrained baseline with D4-equivariant augmentation and tensor transformation, through component-weighted loss prioritizing off-diagonal coupling, to frozen-backbone transfer learning with porosity conditioning via Feature-wise Linear Modulation (FiLM). Onsager reciprocity and positive definiteness are enforced via differentiable penalty terms. On a held-out test set of 4000 samples, the framework achieves variance-weighted R2 = 0.9960 (R2_Kxx = 0.9967, R2_Kxy = 0.9758), a 33% reduction in unexplained variance over the supervised baseline. The results offer three transferable principles for physics-informed scientific machine learning: large-scale visual pretraining transfers effectively across domain boundaries; physical constraints are most robustly integrated as differentiable architectural components; and progressive training guided by diagnostic failure-mode analysis enables unambiguous attribution of performance gains across methodological stages.
Mohammad Nooraiepour
Faculty of Mathematics and Natural Sciences, University of Oslo, PO Box 1047 Blindern, 0316 Oslo, Norway
Full waveform inversion (FWI) is the gold standard for subsurface imaging, with applications from carbon sequestration to energy and mineral exploration to earthquake hazard assessment. Machine learning approaches to FWI need field-scale, geologically diverse, and physically realistic training data, but existing resources such as Marmousi, SEAM, and OpenFWI fall short on spatial extent, temporal extent, geological diversity, and physical realism. We address these limitations with SubsurfaceGen, a GPU-accelerated generator for 3D velocity models and seismic data. Along with SubsurfaceGen, we release a paired dataset of 4,276 2D velocity slices, 5 s wavefields, and 8 s shot gathers drawn from 42 realistic, field-scale 3D velocity models, each spanning 10 km x 10 km laterally and 6.19 km deep at 10 m resolution. The dataset spans six geological settings -- four built with SubsurfaceGen and two drawn from prior sources -- relevant for carbon sequestration and hydrocarbon exploration. We use this dataset to evaluate neural operators on wavefield prediction and encoder-decoders on end-to-end velocity inversion, holding out one geological setting for out-of-distribution testing. These experiments surface failure modes at field-scale and demonstrate how SubsurfaceGen and the associated dataset can impact ML-based FWI.