A Deep-Learning Iterative Stacked Approach for Prediction of Reactive Dissolution in Porous Media

Simulating reactive dissolution of solid minerals in porous media has many subsurface applications, including carbon capture and storage (CCS), geothermal systems and oil & gas recovery. As traditional direct numerical simulators are computationally expensive, it is of paramount importance to develop faster and more efficient alternatives. Deep-learning-based solutions, most of them built upon convolutional neural networks (CNNs), have been recently designed to tackle this problem. However, these solutions were limited to approximating one field over the domain (e.g. velocity field). In this manuscript, we present a novel deep learning approach that incorporates both temporal and spatial information to predict the future states of the dissolution process at a fixed time-step horizon, given a sequence of input states. The overall performance, in terms of speed and prediction accuracy, is demonstrated on a numerical simulation dataset, comparing its prediction results against state-of-the-art approaches, also achieving a speedup around $10^4$ over traditional numerical simulators.

💡 Research Summary

The paper addresses the computational bottleneck of simulating reactive dissolution in porous media, a process central to applications such as carbon capture and storage, geothermal energy, and enhanced oil recovery. Traditional direct numerical solvers require solving highly nonlinear partial differential equations at pore scale, leading to prohibitive runtimes, especially for long‑term forecasts. Recent deep‑learning surrogates have shown promise but are typically limited to predicting a single field (e.g., velocity) and do not capture the coupled evolution of concentration, porosity, and flow rates.

To overcome these limitations, the authors propose a novel deep‑learning framework that (i) ingests a short temporal window of five consecutive states (the “input sequence”) and predicts the next five states (the “output sequence”) in a single multi‑output step, (ii) repeatedly applies this base predictor to roll forward the simulation over the entire time horizon, and (iii) mitigates error accumulation by stacking a series of correction networks on top of the base model. The stacking strategy works as follows: the base network (Level 0) produces an initial approximation for a five‑step window; Level 1 receives this approximation and learns to output a refined correction; Level 2 refines the Level 1 output, and so on, until no further improvement is observed (the authors test up to Level 3). This multi‑level iterative scheme effectively reduces the bias introduced at each rollout step and stabilizes long‑term predictions.

Three neural architectures are evaluated within this framework:

1. ConvLSTM Encoder‑Decoder – combines convolutional spatial feature extraction with LSTM‑style temporal gating, enabling explicit modeling of spatiotemporal dependencies. The encoder compresses the input sequence into a latent representation, which the decoder expands into predictions for each future time step.

2. U‑Shaped Fourier Neural Operator (U‑FNO) – operates in the frequency domain, offering resolution‑invariance and faster convolution equivalents. The U‑shaped design augments the original Fourier Neural Operator with local convolutional pathways to improve accuracy on multiphase flow problems.

3. Temporal Attention Unit (TAU) – a non‑recurrent attention‑based module that processes video‑like sequences in parallel, reducing training time. It incorporates a differential divergence regularization term to capture inter‑frame variations.

The dataset consists of 32 high‑resolution pore‑scale simulations generated with GeoChemFoam, modeling calcite dissolution in a quasi‑2‑D microstructure. Each simulation provides three physical fields (concentration, porosity, velocity) on an H × W grid over ~200 time steps. Data are normalized, augmented, and split into training/validation sets.

Performance is assessed using L2 error, structural similarity index (SSIM), and physical consistency metrics (e.g., mass conservation). Results show that the ConvLSTM‑based stacked model achieves the lowest L2 error (~1.2 %), highest SSIM (~0.96), and maintains physical constraints better than the alternatives. U‑FNO offers faster training and resolution flexibility but incurs slightly higher errors due to limited ability to capture strong nonlinear couplings. TAU is computationally efficient but struggles with the complex physics, leading to the largest prediction errors among the three.

Crucially, all deep‑learning surrogates deliver an average speedup of roughly 10⁴× compared with the conventional numerical solver, making real‑time or many‑query scenarios feasible.

The authors acknowledge several limitations: the training set is relatively small, raising concerns about generalization to field data with different mineralogy, heterogeneity, or noise levels; stacking additional correction levels increases memory and training time substantially; and the loss function does not explicitly enforce physical conservation laws, which could allow non‑physical predictions in edge cases. Future work is suggested to incorporate physics‑based regularization terms, expand the dataset to cover a broader range of geological conditions, and explore adaptive stacking strategies that balance accuracy and computational cost.

In summary, this work introduces the first deep‑learning surrogate capable of simultaneously forecasting multiple coupled fields in reactive dissolution, demonstrates that an iterative stacked architecture can dramatically curb error propagation, and provides a thorough comparative analysis of leading neural operators. The proposed approach promises to transform pore‑scale reactive transport modeling by delivering orders‑of‑magnitude speed gains without sacrificing predictive fidelity.