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Deep-learning-based surrogate flow modeling and geological parameterization for data assimilation in 3D subsurface flow

Meng Tang, Yimin Liu, Louis J. Durlofsky

arXiv:2007.13326v1physics.comp-ph

TL;DR

Subsurface history matching requires many flow simulations while preserving geological realism, motivating surrogate modeling for challenging three-dimensional systems. The paper develops a recurrent R-U-Net approach and reports close agreement with high-fidelity simulations for 3D channelized models.

  • Problem

    History matching requires substantial flow-simulation effort and must preserve geological realism, including for strongly heterogeneous models and large geomodel ensembles.

  • Method

    The paper extends recurrent R-U-Net procedures for three-dimensional systems, using the R-U-Net output as the target state during upsampling.

  • Results

    The 3D recurrent R-U-Net accurately captured saturation and pressure evolution for 3D channelized models generated using CNN-PCA, agreeing closely with reference high-fidelity simulations across 400 test cases.

  • Takeaways & Limitations

    The reported agreement supports using the surrogate for dynamic prediction in 3D channelized subsurface models.

  • Takeaways & Limitations

    The reported application is constrained to strongly heterogeneous models and large geomodel sets involving many wells or detailed property distributions.

Abstract

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Data assimilation in subsurface flow systems is challenging due to the large number of flow simulations often required, and by the need to preserve geological realism in the calibrated (posterior) models. In this work we present a deep-learning-based surrogate model for two-phase flow in 3D subsurface formations. This surrogate model, a 3D recurrent residual U-Net (referred to as recurrent R-U-Net), consists of 3D convolutional and recurrent (convLSTM) neural networks, designed to capture the spatial-temporal information associated with dynamic subsurface flow systems. A CNN-PCA procedure (convolutional neural network post-processing of principal component analysis) for parameterizing complex 3D geomodels is also described. This approach represents a simplified version of a recently developed supervised-learning-based CNN-PCA framework. The recurrent R-U-Net is trained on the simulated dynamic 3D saturation and pressure fields for a set of random `channelized' geomodels (generated using 3D CNN-PCA). Detailed flow predictions demonstrate that the recurrent R-U-Net surrogate model provides accurate results for dynamic states and well responses for new geological realizations, along with accurate flow statistics for an ensemble of new geomodels. The 3D recurrent R-U-Net and CNN-PCA procedures are then used in combination for a challenging data assimilation problem involving a channelized system. Two different algorithms, namely rejection sampling and an ensemble-based method, are successfully applied. The overall methodology described in this paper may enable the assessment and refinement of data assimilation procedures for a range of realistic and challenging subsurface flow problems.

1. Introduction

The paper addresses data assimilation challenges in realistic subsurface flow by combining a 3D deep-learning surrogate with geological parameterization. It extends recurrent R-U-Net and CNN-PCA methods to channelized 3D systems and applies them to uncertainty quantification and history matching.

  • Motivation: The paper targets data assimilation problems requiring many flow simulations while preserving geological realism in posterior geomodels.Effective parameterization reduces the number of calibrated variables and helps posterior geomodels maintain geological realism.
  • Related work: The work builds on prior 2D combinations of residual U-Net and convLSTM and extends surrogate flow modeling to 3D systems.Earlier work combined residual U-Net with convLSTM for 2D oil-water saturation and pressure evolution, whereas this paper applies 3D architectures.
  • Motivation: Existing physics-informed surrogate procedures may struggle with strongly heterogeneous models, many wells, and ensembles with differing detailed property distributions.The paper identifies these capabilities as necessary for realistic data assimilation, while another physics-constrained approach had only been implemented for steady-state problems.
  • Contributions: The combined surrogate and parameterization are used for uncertainty quantification and data assimilation with rejection sampling and ensemble smoother with multiple data assimilation.The two history-matching algorithms are implemented and assessed for the channelized system.

2. Governing Equations for Two-phase Flow

The paper formulates three-dimensional two-phase oil-water flow through conservation laws for phase mass, pressure, saturation, and well-source terms. Spatially discontinuous geology, saturation shocks, gravity, and well responses make the resulting simulation problem challenging.

  • The model describes immiscible oil-water flow in three-dimensional subsurface formations, including gravitational effects.
  • Phase mass conservation uses density, Darcy velocity, source or sink terms, porosity, and phase saturation.
  • Phase flow depends on absolute and relative permeability, viscosity, pressure, gravity, and depth, with relative permeability generally nonlinear in saturation.
  • Capillary pressure is neglected, so oil and water pressures are treated as equal in this formulation.
  • Finite-volume discretization followed by Newton’s method yields pressure and saturation across grid blocks, but discontinuous geology and saturation shocks complicate the solution.
  • Well responses are computed from well-block pressure, saturation, permeability, wellbore pressure, and well-index relations, and are central data-assimilation quantities.

3. 3D CNN-PCA Low-dimensional Model Representation

The 3D CNN-PCA procedure combines low-dimensional PCA representations with CNN post-processing to preserve complex channelized geological structure while honoring well data. The simplified formulation is effective for the conditioning data and geomodels considered.

  • The simplified procedure uses reconstruction and hard-data losses because the available conditioning data make style loss unnecessary for this system.
  • The method maps high-dimensional geomodels m ∈R^nb to low-dimensional variables ξ ∈R^l, with l ≪ nb, using PCA before CNN transformation.
  • 3D CNN-PCA post-processes PCA models with a convolutional neural network to preserve spatial correlations that standalone PCA can lose in channelized systems.
  • The CNN transform is trained to minimize reconstruction error between post-processed PCA models and the original geomodels.
  • Hard-data loss preserves well-location conditioning, with hard data honored in more than 99.9% of newly generated 3D CNN-PCA models.
  • The generated channelized realizations retain channel features and produce flow variability essentially matching that of 200 reference Petrel geomodels.

4. Recurrent R-U-Net Surrogate Model and Training

The recurrent R-U-Net replaces expensive three-dimensional reservoir simulations with a spatial-temporal surrogate for saturation and pressure fields. It combines a 3D residual U-Net with convLSTM features and is trained on high-fidelity oil-water simulations with well-focused loss weighting.

  • 4.1. Recurrent R-U-Net Architecture: The surrogate predicts either water saturation or pressure at 10 selected time steps for the oil-water system.
  • 4.1. Recurrent R-U-Net Architecture: The recurrent R-U-Net replaces expensive numerical simulation with predictions of dynamic subsurface states for different geomodels.
  • 4.1. Recurrent R-U-Net Architecture: A 3D residual U-Net extracts multiscale geological features, while convLSTM propagates compressed global features through time.
  • 4.1. Recurrent R-U-Net Architecture: The decoder combines local features F1 to F4 with upsampled global features to predict pressure or saturation at each target time.
  • 4.2. Flow Problem Setup and Network Training: Training uses 2500 high-fidelity simulations of 1000-day oil-water flow with prescribed injector and producer bottom-hole pressures.
  • 4.2. Flow Problem Setup and Network Training: The training objective weights well-block quantities because well rates are primary data-assimilation observations, and L2 saturation with L1 pressure produced better global predictions.

5. Evaluation of Recurrent R-U-Net Performance

Across 400 new geomodel test cases, the recurrent R-U-Net closely reproduces saturation and pressure fields, well responses, and ensemble production-rate statistics, with low relative errors.

  • Field predictions: The surrogate captures saturation-front evolution and three-dimensional structure, although localized discrepancies remain near some fronts and wells.Differences are reported in the top layer and near producer P2, while cross-sectional views show accurate 3D prediction.
  • Field predictions: At 400 days, pressure maps for three realizations show significant geological variation while retaining accurate visual 3D agreement with high-fidelity simulations.Pressure changes little over time, so the comparison emphasizes different test cases rather than temporal evolution.
  • Field predictions: δS = 5.7% and δp = 0.7%, indicating accurate global saturation and pressure predictions over new geomodels.The evaluation uses 400 test models and reports relatively low relative errors.
  • Well responses: For one geomodel, predicted oil and water production rates closely match high-fidelity results, including trends and water breakthrough, with some late-time discrepancies.The comparison covers all four production wells.
  • Ensemble well statistics: Across 400 test cases, oil-rate and water-rate errors are 6.1% and 8.8%, respectively, while P10, P50, and P90 statistics remain in close agreement.The statistical comparison spans the full ensemble and maintains accuracy across large rate ranges.

6. Use of Deep-Learning Procedures for History Matching

The study combines 3D CNN-PCA geomodel generation with a recurrent R-U-Net surrogate to perform rejection-sampling and ES-MDA history matching efficiently. Posterior models reduce uncertainty, agree closely with high-fidelity simulations, and support evaluation of ES-MDA.

  • 6.2. Problem Setup and Rejection Sampling Results: 151 posterior models were accepted by rejection sampling, after surrogate evaluation of 10^6 generated models that would require about 170,000 high-fidelity simulation hours.Generating the models took about 3 hours and surrogate predictions about 8 hours, compared with the estimated high-fidelity cost.
  • 6.2. Problem Setup and Rejection Sampling Results: Accepted geomodels reproduced key connectivity patterns: limited I1–P2 sand connectivity and consistent I2–P3 connectivity matched the true model and observed water responses.The inferred structures were consistent with low P2 water production and early, high P3 water production.
  • 6.2. Problem Setup and Rejection Sampling Results: Recurrent R-U-Net predictions for the 151 accepted models closely agreed with high-fidelity simulations across oil and water production rates at all four producers.The comparison supports applying the surrogate to this history-matching problem.
  • 6.3. ES-MDA Setup and Results: ES-MDA produced posterior estimates for many well quantities in close agreement with rejection sampling, although it overestimated some results.Rejection sampling was used as a reference for assessing the more efficient ensemble-based method.

7. Concluding Remarks

The paper extends recurrent R-U-Net flow surrogates to 3D and combines them with CNN-PCA for geological parameterization and data assimilation. The methods accurately model channelized flow, reduce uncertainty, and enable comparison and tuning of history-matching algorithms.

  • Concluding Remarks: CNN-PCA preserves geological consistency with the original object-based Petrel models for posterior models generated from any low-dimensional variable.The implementation used here is a simplified formulation containing reconstruction and hard-data loss terms.
  • Concluding Remarks: The 3D recurrent R-U-Net accurately captured saturation and pressure evolution for channelized CNN-PCA geomodels, including well responses and ensemble flow statistics.Agreement with high-fidelity simulations was assessed visually and through aggregate error statistics over 400 test cases.
  • Concluding Remarks: ES-MDA updated low-dimensional CNN-PCA variables iteratively while using the recurrent R-U-Net for flow evaluations, with rejection sampling providing target posterior predictions for tuning.The paper positions rejection sampling as a reference for evaluating more efficient algorithms.
  • Concluding Remarks: A major future boundary is scaling the methods to practical models with O(10^5–10^6) cells, where training time may require different network architectures.The paper also identifies coupled flow-geomechanics as an extension target.

Appendix A.1. 3D CNN-PCA Transform Net Architecture

The 3D CNN-PCA transform net uses convolutional, deconvolutional, residual, normalization, and activation layers to reconstruct 3D geomodels. Its architecture is summarized in Table 1.

  • Appendix A.1. 3D CNN-PCA Transform Net Architecture: The transform net maps low-dimensional variables to 3D geomodels using 3D convolutional and deconvolutional layers with batch normalization and ReLU activations.The final convolutional layer contains one 3D convolutional layer.
  • Appendix A.1. 3D CNN-PCA Transform Net Architecture: Each residual block contains two 3D convolutional layers with 128 filters of size 3 × 3 × 3 and stride (1, 1, 1).The first convolution is followed by batch normalization and ReLU, while the second is followed only by batch normalization.
  • Appendix A.1. 3D CNN-PCA Transform Net Architecture: Residual-block outputs add the block input to the processed features, providing the architecture’s residual connection.The passage identifies the final residual-block output as the sum of the input and transformed features.

Appendix A.2. 3D R-U-Net Schematic

The 3D R-U-Net schematic represents multichannel feature maps and operations across an encoder–decoder network. Encoder features are copied and concatenated with upsampled decoder features.

  • Appendix A.2. 3D R-U-Net Schematic: The schematic’s boxes represent 3D multichannel feature maps, while arrows denote the network operations between them.Values beside boxes indicate feature-map dimensions, and values above boxes indicate channel counts.
  • Appendix A.2. 3D R-U-Net Schematic: The schematic uses convolution, residual-block, deconvolution, and convolution-layer operations as defined for the network.These operations have filter and stride settings specific to the 3D R-U-Net.
  • Appendix A.2. 3D R-U-Net Schematic: Features extracted by the encoding net are copied and concatenated with upsampled decoding features.This connects encoder representations to the decoder in the illustrated U-Net structure.

Appendix A.3. 3D Recurrent R-U-Net Architecture

The 3D recurrent R-U-Net uses a convLSTM3D block within an encoder–decoder architecture to process multichannel feature maps across time steps. Its architecture is documented schematically and in Table 2.

  • Appendix A.3. 3D Recurrent R-U-Net Architecture: Table 2 provides the detailed 3D recurrent R-U-Net architecture, using a single stride value because strides are identical in all directions.
  • Appendix A.3. 3D Recurrent R-U-Net Architecture: The recurrent R-U-Net architecture combines a convLSTM3D block with an encoder–decoder that processes activation maps across nt time steps.The decoder processes the activation maps separately for each time step.
  • Appendix A.3. 3D Recurrent R-U-Net Architecture: The convLSTM3D block uses 64 filters of size 3 × 3 × 3 to perform the LSTM gate operations.
  • Appendix A.3. 3D Recurrent R-U-Net Architecture: The convLSTM network generates activation maps with dimensions (nx/4, ny/4, nz/4, 64) for all nt time steps.
  • Appendix A.3. 3D Recurrent R-U-Net Architecture: The schematic represents different multichannel feature maps, showing their dimensions and channel counts.
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