Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media (1807.00882v1)

Abstract: Surrogate strategies are used widely for uncertainty quantification of groundwater models in order to improve computational efficiency. However, their application to dynamic multiphase flow problems is hindered by the curse of dimensionality, the saturation discontinuity due to capillarity effects, and the time-dependence of the multi-output responses. In this paper, we propose a deep convolutional encoder-decoder neural network methodology to tackle these issues. The surrogate modeling task is transformed to an image-to-image regression strategy. This approach extracts high-level coarse features from the high-dimensional input permeability images using an encoder, and then refines the coarse features to provide the output pressure/saturation images through a decoder. A training strategy combining a regression loss and a segmentation loss is proposed in order to better approximate the discontinuous saturation field. To characterize the high-dimensional time-dependent outputs of the dynamic system, time is treated as an additional input to the network that is trained using pairs of input realizations and of the corresponding system outputs at a limited number of time instances. The proposed method is evaluated using a geological carbon storage process-based multiphase flow model with a 2500-dimensional stochastic permeability field. With a relatively small number of training data, the surrogate model is capable of accurately characterizing the spatio-temporal evolution of the pressure and discontinuous CO2 saturation fields and can be used efficiently to compute the statistics of the system responses.

• The paper introduces a novel two-stage training method combining regression and segmentation to accurately capture saturation discontinuities.
• The methodology leverages fully convolutional dense blocks to manage high-dimensional inputs, enhancing feature propagation in complex subsurface simulations.
• Dynamic system characterization is achieved by treating time as an additional input, enabling robust predictions with fewer simulations.

Deep Convolutional Encoder-Decoder Networks for Uncertainty Quantification of Dynamic Multiphase Flow

The paper by Mo et al. explores the use of deep convolutional encoder-decoder networks to develop surrogate models for uncertainty quantification (UQ) in dynamic multiphase flow in heterogeneous media. The paper addresses significant challenges in modeling these systems, such as high input dimensionality, response discontinuity, and time-dependent outputs.

The authors propose a methodology that transforms the surrogate modeling task into an image-to-image regression problem. This transformation leverages convolutional neural networks (CNNs) to handle the spatial structure inherent in input and output data, especially useful when dealing with large, heterogeneous permeability fields used in subsurface modeling.

Key Contributions and Methodological Innovations

1. Two-Stage Network Training:
   • A novel two-stage training strategy is introduced. This strategy uses both a regression loss and a segmentation loss to better approximate the discontinuous saturation field encountered in multiphase flow systems.
   • The segmentation loss focuses on the regions around the saturation front where discontinuities are most evident, aiming to refine these approximations significantly.
2. Handling High-Dimensional Inputs:
   • The developed network architecture employs fully convolutional dense blocks, which enhance feature propagation and reduce the need for large datasets typically required by deep learning models.
   • This design efficiently handles input dimensionalities as high as 2500, representing complex subsurface heterogeneities without the need for traditional dimensionality reduction methods.
3. Dynamic System Characterization:
   • Time is treated as an additional network input within the model, enabling efficient and accurate predictions across arbitrary time instances beyond those explicitly provided during training.
   • Such flexibility is critical for practical applications where continuous monitoring or prediction of system states is required.

Numerical Results and Implications

The paper presents a comprehensive set of numerical experiments using a geological carbon storage model to validate the proposed approach. Key findings include:

• The network accurately predicts pressure and saturation fields with high $R^2$ values, demonstrating robust performance even when trained on relatively small sample sizes.
• The two-stage training approach, which combines regression and segmentation, provides notable improvement in capturing discontinuous features like saturation fronts compared to using regression alone.
• For uncertainty quantification, the surrogate model replicates outputs produced by the traditional Monte Carlo method using significantly fewer simulations, suggesting improvements in computational efficiency.

The results have significant implications for applying deep learning approaches to UQ in subsurface modeling, particularly in tasks involving complex, multiphase systems with high-dimensional and time-dependent characteristics.

Future Directions

The work opens several avenues for future research and application:

• Generalization to Other Multiphase Systems: The method could be adapted to other geoscientific domains where similar challenges in modeling exist.
• Inverse Problems and Experimental Design: This framework's ability to model dynamic behaviors suggests potential use in tackling inverse problems and optimizing experimental designs.
• Broader Integration and Application: Exploring multi-input scenarios (beyond permeability fields) and integrating with other data sources could broaden the applicability of this approach in coupled subsurface modeling systems.

In conclusion, the authors provide a rigorous methodological advance using convolutional neural networks for UQ in dynamic multiphase flows, addressing key computational and modeling challenges in this field. Such advancements highlight the potential of deep learning to redefine how computational efficiency and accuracy can be achieved in complex subsurface systems modeling.