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Towards a Robust Parameterization for Conditioning Facies Models Using Deep Variational Autoencoders and Ensemble Smoother

Smith W. A. Canchumuni, Alexandre A. Emerick, Marco Aurélio C. Pacheco

arXiv:1812.06900v1stat.MLcs.LG

TL;DR

Conditioning reservoir models to dynamic data while preserving geological realism remains challenging, and robust facies parameterization is still an open problem. The paper combines a convolutional VAE with ES-MDA and tests it on three synthetic channelized-facies problems. The authors report improved results over earlier methods in the 2D cases, while the 3D case retains visible artifacts.

  • Problem

    Robust facies parameterization remains an open problem because ensemble methods’ Gaussian assumptions can hinder geological realism, especially for complex facies.

    Source: §1 · p. 2

  • Method

    The authors use a convolutional VAE to parameterize facies and combine its decoder with ES-MDA to assimilate reservoir data.

    Source: §1 · p. 3, §3 · p. 7

  • Results

    The procedure outperformed standard ES-MDA, ES-MDA with OPCA, and DBN parameterizations in the two 2D tests, while 3D posterior realizations showed some atypical features.

    Source: §4.3 · p. 20, §6 · p. 24

  • Takeaways & Limitations

    In synthetic channelized-facies tests, the approach produced well-defined channel features and results the authors describe as encouraging, though the 3D realizations were not visually indistinguishable from prior models.

    Source: §4.3 · p. 20, §6 · p. 24

    The latent vector cannot be updated with distance-based localization, and the paper’s test cases used no localization.

    Source: §5 · pp. 23–24

Abstract

from arXiv · show

The literature about history matching is vast and despite the impressive number of methods proposed and the significant progresses reported in the last decade, conditioning reservoir models to dynamic data is still a challenging task. Ensemble-based methods are among the most successful and efficient techniques currently available for history matching. These methods are usually able to achieve reasonable data matches, especially if an iterative formulation is employed. However, they sometimes fail to preserve the geological realism of the model, which is particularly evident in reservoir with complex facies distributions. This occurs mainly because of the Gaussian assumptions inherent in these methods. This fact has encouraged an intense research activity to develop parameterizations for facies history matching. Despite the large number of publications, the development of robust parameterizations for facies remains an open problem. Deep learning techniques have been delivering impressive results in a number of different areas and the first applications in data assimilation in geoscience have started to appear in literature. The present paper reports the current results of our investigations on the use of deep neural networks towards the construction of a continuous parameterization of facies which can be used for data assimilation with ensemble methods. Specifically, we use a convolutional variational autoencoder and the ensemble smoother with multiple data assimilation. We tested the parameterization in three synthetic history-matching problems with channelized facies. We focus on this type of facies because they are among the most challenging to preserve after the assimilation of data. The parameterization showed promising results outperforming previous methods and generating well-defined channelized facies.

1 Introduction

Robust facies parameterization remains an open challenge despite many proposed methods, and most work has considered only small 2D problems. This paper investigates a convolutional VAE with ensemble smoothing to address limitations in earlier autoencoder-based facies history matching.

1 Introduction

  • Facies parameterization for data assimilation remains an open problem, and most publications consider only small 2D problems.Ensemble methods rely on Gaussian assumptions, motivating methods to adapt them to non-Gaussian priors.Source: §1 · p. 2
  • Earlier deep-learning approaches included VAEs, GANs, autoencoders, deep belief networks, and neural style transfer for facies generation or conditioning.The cited work spans low-dimensional parameterization, hard-data conditioning, and history matching.Source: §1 · pp. 2–3
  • This work extends prior autoencoder parameterization by using a convolutional VAE to address training difficulties and insufficient geological realism in earlier results.Unlike Laloy et al. (2017), the authors use the parameterization with an ensemble smoother for hard and dynamic production data.Source: §1 · p. 3
  • The paper reviews generative models, presents ES-MDA-CVAE, evaluates three increasingly complex test problems, discusses potential issues, and concludes with a summary.The authors state that the data and code are available for download.Source: §1 · p. 3

2 Generative Models

The paper introduces generative models and reviews autoencoders, VAEs, and convolutional layers as components for constructing a facies parameterization. These concepts motivate using a convolutional VAE to represent facies in a scalable latent space.

2 Generative Models

  • Generative models learn structure in high-dimensional data to generate new samples, and this paper focuses on convolutional neural networks combined with VAEs.The model maps a random latent vector through a neural network to generate facies realizations.Source: §2 · p. 3

2.1 Autoencoders

  • An autoencoder encodes input data into a feature-space code and decodes it back, training to minimize reconstruction dissimilarity.With linear decoding and mean-square-error loss, it spans the same subspace as PCA; nonlinear encoder and decoder functions generalize PCA.Source: §2.1 · p. 4, Figure 1 · p. 4

2.2 Variational Autoencoders

  • A VAE samples latent vectors and adds a KL-divergence term to encourage their encoded distribution to match a specified prior, usually a standard Gaussian.Its loss combines reconstruction error with this regularization; the encoder produces means and log-variances, and sampling feeds the decoder.Source: §2.2 · pp. 4–5

2.3 Convolutional Layers

  • Convolutional layers reduce training parameters for large facies grids, where fully connected networks scale poorly.Small trainable filters operate on grid-structured data, allowing deeper architectures; Figure 2 depicts a VAE architecture.Source: §2.3 · pp. 5–6, Figure 2 · p. 6

3 ES-MDA-CVAE

ES-MDA-CVAE combines a trained convolutional variational autoencoder (CVAE) decoder with ES-MDA, updating latent vectors to condition reconstructed facies models on reservoir data. Prior latent vectors can be sampled from a normal distribution or encoded from geostatistical realizations; the paper uses the latter approach.

3 ES-MDA-CVAE

  • The CVAE is trained on about O(10^4) facies realizations, after which its decoder generates realizations by sampling z from N(0, I), without reservoir simulations during training.The decoder substitutes for the geostatistical algorithm used to construct initial realizations.Source: §3 · p. 7
  • ES-MDA updates an ensemble of latent vectors to account for reservoir data, and the trained decoder reconstructs the corresponding facies models.The paper calls this procedure ES-MDA-CVAE.Source: §3 · p. 7
  • Prior latent vectors are either sampled from N(0, I) or obtained by encoding geostatistical prior facies realizations; the paper uses encoded realizations.Source: §3 · p. 7
  • Each assimilation iteration decodes latent vectors into facies and runs reservoir simulations to predict data, so the ensemble size is limited to about O(10^2) realizations.The simulations can be time-consuming depending on model size.Source: §3 · p. 7
  • The ES-MDA update uses ensemble-estimated cross-covariances between latent vectors and predicted data, data-error covariance, observations, and perturbed observations.The method applies the update a predefined number of times, Na, with data-inflation factors αk.Source: §3 · pp. 7–8
  • The update is written in terms of the latent vector alone, but the authors say additional reservoir uncertainty parameters can be included through an augmented vector.Source: §3 · p. 8

4 Test Cases

The three synthetic tests evaluate CVAE parameterization with ES-MDA on channelized facies, from conditioning to hard facies data to production data. Results generally preserve channel structure and improve on comparison methods, though the 3D posterior retains visible facies defects and some poor production matches.

4.1.1 CVAE architecture and training

  • For test case 1, the CVAE reconstructed facies with 96.7% validation accuracy after training on snesim realizations.The training set contained 24,000 realizations, with 6,000 additional realizations for validation; training took about 13 minutes on four GPUs.Source: §4.1.1 · pp. 8–9, §4.1 · p. 8, Figure 6 · p. 10

4.1.2 Conditioning to facies data

  • With seven or 20 hard facies points, ES-MDA-CVAE honored every data point and retained well-defined channels, although some channels were broken.The final realizations reasonably preserved the prior’s main geological characteristics.Source: §4.1.2 · p. 11, Figure 7 · p. 11, Figure 9 · p. 12

4.1.3 Conditioning to production data

  • For production-data assimilation in test case 1, ES-MDA-CVAE reproduced the reference model’s main features, matched observed water rates, and outperformed ES-MDA, OPCA, and DBN.The comparison used production data without hard facies data; the reference model and wells are shown in Figure 5.Source: Figure 5 · p. 9, §4.1.3 · p. 14, Figure 12 · p. 14

4.2 Test Case 2

  • The test-case-2 CVAE achieved 93.3% validation reconstruction accuracy and reasonably reconstructed channels using 32,000 training and 8,000 validation realizations.Training took 42 minutes on four GPUs.Source: §4.2 · pp. 14, 16, Figure 16 · p. 17
  • In the 100×100 test case, ES-MDA-CVAE produced a better channel representation than standard ES-MDA, OPCA, or DBN, with reasonable water-cut matches.The authors used 200 realizations and 20 MDA iterations; the parameterization may make the problem more nonlinear.Source: §4.2 · p. 16, Figure 17 · p. 17, Figure 19 · p. 18

4.3 Test Case 3

  • The 3D test used 3D convolutional layers to represent vertical facies geometry; training took 49 hours and achieved 89.1% reconstruction accuracy.Batch normalization was used to improve stability and reduce training time, replacing dropout layers.Source: §4.3 · p. 19

5 Comments

The comments identify localization as a limitation of the CVAE latent space and note that training also requires many prior realizations. These constraints complicate use on larger reservoir models.

5 Comments

  • Distance-based localization cannot be applied to update z because the CVAE latent vector occupies a different space, and grid-sized codes do not guarantee spatial correspondence.Grid-sized codes also increase training parameters through fully connected layers, potentially making larger models unfeasible.Source: §5 · p. 23
  • The tests used no localization; the authors report that available non-distance procedures were less effective than distance-dependent approaches.The localization issue remains for further investigation.Source: §5 · p. 24
  • Training the CVAE required 30,000–50,000 prior realizations in these tests, with larger numbers potentially needed for more complex models.The 100,000-gridblock final test took approximately two days to train on four GPUs.Source: §5 · p. 24

6 Conclusions

The paper tests CVAE facies parameterization with ES-MDA on three synthetic channelized reservoir history-matching problems. The first two 2D cases outperform previous parameterizations, while the 3D case has reconstruction and realism issues despite encouraging overall performance.

6 Conclusions

  • The proposed procedure outperformed standard ES-MDA, ES-MDA with OPCA, and DBN parameterizations in the first two, 2D test problems.The paper evaluates three synthetic history-matching problems with increasing complexity.Source: §6 · p. 24
  • The third test problem used 3D channels and three facies, requiring 3D convolutional layers that significantly increased training time.Reconstruction accuracy decreased noticeably, and conditional realizations included features absent from the prior geological description.Source: §6 · p. 24
  • Despite the 3D case issues, the authors characterize the overall performance as encouraging and worth pursuing.They propose using trained CVAEs as GAN generative models to improve reconstruction accuracy, especially for the third case.Source: §6 · p. 24

Appendix: Architecture of the networks

The appendix captions identify CVAE architecture tables for test cases 1 and 3.

Appendix: Architecture of the networks

  • The appendix includes a CVAE architecture table for test case 1.The caption identifies the table's scope but gives no architecture details.Source: Table 1 · p. 31
  • The appendix includes a CVAE architecture table for test case 3.The caption identifies the table's scope but gives no architecture details.Source: Table 2 · p. 32

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