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A deep-learning-based surrogate model for data assimilation in dynamic subsurface flow problems
Meng Tang, Yimin Liu, Louis J. Durlofsky
TL;DR
The paper addresses the need for computationally efficient and accurate dynamic-flow surrogates for uncertain, high-dimensional channelized subsurface systems. It combines convolutional and recurrent neural networks in a recurrent R-U-Net and uses it for prediction and history matching. The surrogate produces flow responses in close agreement with the underlying simulator at substantially lower computational cost, while surrogate-based history matching reduces prediction uncertainty.
Problem
Accurate and computationally efficient surrogates are needed for dynamic oil-water flow and inverse modeling in high-dimensional channelized systems.
Method
The paper combines a residual U-Net with an LSTM recurrent architecture to predict time-dependent pressure, saturation, and well-rate responses from geological realizations.
Results
The surrogate provides flow predictions in close agreement with the underlying simulator while substantially reducing computational cost, and its well-rate predictions are sufficiently accurate for history matching.
Takeaways & Limitations
Surrogate-based history matching achieved significant uncertainty reduction, with posterior models generated using RML and CNN-PCA parameterization.
Takeaways & Limitations
Higher prediction accuracy corresponds to higher pre-processing cost, constraining the computational trade-off of the training procedure.
Abstract
from arXiv · showhide
A deep-learning-based surrogate model is developed and applied for predicting dynamic subsurface flow in channelized geological models. The surrogate model is based on deep convolutional and recurrent neural network architectures, specifically a residual U-Net and a convolutional long short term memory recurrent network. Training samples entail global pressure and saturation maps, at a series of time steps, generated by simulating oil-water flow in many (1500 in our case) realizations of a 2D channelized system. After training, the `recurrent R-U-Net' surrogate model is shown to be capable of accurately predicting dynamic pressure and saturation maps and well rates (e.g., time-varying oil and water rates at production wells) for new geological realizations. Assessments demonstrating high surrogate-model accuracy are presented for an individual geological realization and for an ensemble of 500 test geomodels. The surrogate model is then used for the challenging problem of data assimilation (history matching) in a channelized system. For this study, posterior reservoir models are generated using the randomized maximum likelihood method, with the permeability field represented using the recently developed CNN-PCA parameterization. The flow responses required during the data assimilation procedure are provided by the recurrent R-U-Net. The overall approach is shown to lead to substantial reduction in prediction uncertainty. High-fidelity numerical simulation results for the posterior geomodels (generated by the surrogate-based data assimilation procedure) are shown to be in essential agreement with the recurrent R-U-Net predictions. The accuracy and dramatic speedup provided by the surrogate model suggest that it may eventually enable the application of more formal posterior sampling methods in realistic problems.
1. Introduction
The paper introduces a deep-learning surrogate for dynamic oil-water flow in channelized geological systems, targeting accurate pressure, saturation, and well-rate predictions at substantially lower computational cost. It applies the recurrent R-U-Net to history matching under pressure-controlled wells, addressing limitations of existing surrogate approaches for high-dimensional, nonlinear problems.
- Reliable subsurface forecasts are important for managing oil, gas, and groundwater resources because characterization uncertainty can substantially affect flow predictions.
- Proposed surrogate model: The proposed model combines CNNs for nonlinear geological-to-flow relationships with an RNN for temporal evolution.The implementation uses a convolutional U-Net for state responses from permeability and an LSTM architecture for time-dependent pressure and saturation maps.
- Limitations of existing approaches: Existing physics-based and data-driven surrogates have limitations, including dependence on low-dimensional settings or test cases close to training runs.POD-based reduced-order models are generally accurate only when new runs are sufficiently close to training runs, while data-driven surrogates have been limited to relatively low-dimensional problems.
- Target problem: The study focuses on channelized, non-Gaussian permeability fields and pressure-controlled wells, where well rates and saturation evolution can vary substantially between realizations.These variations are larger under pressure control than rate control because the injected-fluid amount is not fixed across realizations.
- Application to inverse modeling: The recurrent R-U-Net is trained on multiple geological realizations and provides rapid predictions of pressure, saturation, and well flow-rate data for inverse modeling.The surrogate is designed to replace the original model for most required function evaluations and supports fast history matching in channelized systems.
- Study design: The paper evaluates surrogate accuracy for global states and well rates in multi-realization channelized flow driven by 25 pressure-controlled wells before applying it to history matching.
2. Methodology
The methodology formulates dynamic two-phase flow and replaces repeated numerical simulation with a recurrent deep-learning surrogate. The recurrent R-U-Net maps permeability fields to time-dependent pressure and saturation states under fixed well controls.
- Governing equations: The study models 2D immiscible oil-water flow using pressure and water saturation as primary variables.Capillary and gravitational effects are neglected, and flow is discretized with a fully implicit finite-volume method.
- Governing equations: Well injection and phase-production rates are computed from dynamic well-block pressure and saturation when wellbore pressure is specified.The well source term uses the Peaceman representation and a well index.
- Data-driven surrogate modeling: The surrogate approximates the simulator’s mapping from permeability fields and fixed well controls to pressure and saturation maps at all simulation time steps.Training learns from paired geological models and simulated state responses.
- Architecture comparison: The study reports superior flow-response performance for U-Net compared with DenseED in its experiments.The authors associate this with multiscale information flow between encoding and decoding paths.
- R-U-Net architecture: The R-U-Net combines contracting and expanding paths with residual modules, propagating multiscale encoding features into the decoding path for state-map prediction.The encoder accepts permeability maps, while the decoder upsamples extracted feature maps.
- Recurrent R-U-Net architecture: The recurrent R-U-Net integrates a convLSTM that processes the global encoding feature map to capture temporal dynamics.The convLSTM generates a sequence of feature maps for dynamic predictions.
- Performance and training: After training, the recurrent R-U-Net predicts states at 10 time steps for a new geomodel in about 0.01 seconds using a GPU.The R-U-Net also accurately maps permeability fields to steady-state pressure fields, while L1 pressure loss slightly improves pressure accuracy.
- Performance and training: The authors report that the recurrent R-U-Net does not appear to suffer from over-fitting, while acknowledging that a rigorous explanation is unavailable.This observation is reported without a definitive theoretical explanation.
3. Surrogate Model Evaluation
The recurrent R-U-Net is evaluated on a challenging 2D channelized oil-water flow system, using simulated pressure and saturation sequences to predict dynamic fields and well responses. Across individual and ensemble tests, the surrogate achieves high accuracy, including for well-rate predictions.
- Training procedure: Training uses 1500 channelized permeability fields and corresponding AD-GPRS pressure and saturation maps collected at 10 time steps over 1000 days.Two recurrent R-U-Nets are trained separately for pressure and saturation, using the same architecture but different training sets.
- Training procedure: The L1 norm loss yields more accurate pressure predictions, whereas the L2 norm loss provides better saturation predictions.The two output types are therefore trained with distinct loss choices.
4. History Matching Using Deep-learning-based Surrogate Model
The recurrent R-U-Net surrogate replaces high-fidelity flow simulations within an RML history-matching workflow for 2D channelized reservoirs. The procedure substantially reduces predictive uncertainty while retaining close agreement with numerical-simulation forecasts.
- History matching procedure: RML repeatedly solves an optimization problem to generate multiple posterior samples, with MADS used here as a derivative-free local-search optimizer.Each run produces one posterior sample, and the minimization includes a regularization term that supports geostatistical consistency through CNN-PCA.
- History matching procedure: The workflow represents geomodels with CNN-PCA parameters and uses recurrent R-U-Net flow predictions instead of high-fidelity simulations.The geomodel is approximated as mcnn(ξ), while the flow response is approximated by f̂(mcnn(ξ)).
- History matching procedure: The history-matching data comprise 215 measurements of oil and water production and water injection rates, with Gaussian noise set to 5% of corresponding true data.The setup uses 18 producers, seven injectors, five time steps, and a 1000-day simulation divided into 400-day history and 600-day forecast periods.
- Computational cost: 4 × 10^6 flow-model evaluations are required, making the surrogate-based workflow computationally advantageous for the specified RML and MADS settings.The calculation uses 100 posterior models, 200 MADS iterations per run, and 200 function evaluations per iteration.
- History matching results: Posterior P10–P90 ranges are clearly smaller than prior ranges and generally capture observed and true data, including cases near prior-range boundaries.Uncertainty reduction is also observed for water rates before breakthrough occurs during the history-matching period.
- History matching results: AD-GPRS and recurrent R-U-Net posterior P10, P50, and P90 forecasts generally correspond closely, with small discrepancies in some outer-percentile curves.The reported discrepancies are very small compared with the uncertainty reduction achieved by history matching.
5. Concluding Remarks
The study develops a recurrent R-U-Net surrogate for dynamic oil-water flow and evaluates it on new channelized geomodels and history matching. It achieves accurate state and well-response prediction, substantial uncertainty reduction, and dramatic computational speedup, while remaining limited to 2D systems in this work.
- Surrogate-model development: The recurrent R-U-Net combines residual U-Net and convolutional LSTM components to capture temporal dynamics in high-dimensional oil-water subsurface flow.It is trained on simulated dynamic pressure and saturation maps from 1500 2D channelized realizations.
- Surrogate-model evaluation: The surrogate accurately predicts dynamic pressure and saturation maps and closely matches time-varying oil, water, and injection-rate responses for new geomodels.Accuracy was demonstrated for an individual geomodel and evaluated through flow statistics on an ensemble of 500 new geomodels.
- Surrogate-model evaluation: The surrogate’s flow statistics were sufficiently accurate to demonstrate applicability for uncertainty quantification across P10, P50, and P90 responses.The evaluation compared recurrent R-U-Net predictions with numerical-simulator results for 500 test geomodels.
- History matching: History matching with RML and a 100-parameter CNN-PCA representation achieved significant uncertainty reduction, with surrogate forecasts reasonably accurate against numerical simulations.The posterior-model flow responses were checked against high-fidelity numerical simulation results.
- Implications: The surrogate produced dramatic speedup relative to high-fidelity simulation, suggesting more rigorous inverse-modeling procedures may become feasible for realistic problems.The paper presents this as a future possibility rather than a completed application of formal posterior sampling.
- Limitations and future work: The surrogate model developed in this work is limited to 2D problems, with extensions to 3D systems and larger, more complicated settings left for future research.The authors also identify multiphysics flow-geomechanics applications as a future direction.
Appendix: Recurrent R-U-Net Architecture
The recurrent R-U-Net architecture combines convolutional, residual, transposed-convolutional, and ConvLSTM2D blocks to process dynamic maps across multiple time steps.
- Layer definitions: A convolutional layer in the architecture is followed by batch normalization and ReLU activation, while transposed convolution performs upsampling with the same subsequent operations.These blocks form the encoder and decoder components of the network.
- Residual block: A residual block stacks two 3 × 3 × 128 convolutional layers and connects the first layer to the second layer’s output through a skip connection.The block uses 128 filters in each convolutional layer.
- ConvLSTM2D block: The ConvLSTM2D block performs the LSTM gate operations using 128 filters of size 3 × 3 × 128.This recurrent block supplies temporal processing within the network.
- Temporal processing: The ConvLSTM network generates (N_x/4, N_y/4, 128) activation maps for all n_t time steps, which the decoder processes separately to produce outputs.The architecture therefore preserves separate time-step representations through decoding.