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Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media
Shaoxing Mo, Yinhao Zhu, Nicholas Zabaras, Xiaoqing Shi, Jichun Wu
TL;DR
Dynamic multiphase-flow uncertainty quantification is difficult because high-dimensional inputs, discontinuous saturation, and time-dependent outputs challenge surrogate models. The paper addresses these issues with an image-to-image deep convolutional encoder-decoder, regression-plus-segmentation training, and time as an input. In a geological carbon-storage model with 2500 uncertain permeability parameters, the surrogate accurately characterizes pressure and saturation fields over time and supports efficient response-statistics computation.
Problem
Dynamic multiphase-flow uncertainty quantification is hindered by high-dimensional inputs, capillarity-induced saturation discontinuity, and time-dependent multi-output responses.
Method
A deep convolutional encoder-decoder transforms surrogate modeling into image-to-image regression and combines regression with segmentation loss to approximate discontinuous saturation fields.
Results
The surrogate accurately reproduces pressure and discontinuous saturation fields and characterizes dynamic multi-output responses at arbitrary time instances in a 2500-parameter GCS model.
Takeaways & Limitations
The method provides an efficient surrogate for computing uncertainty-quantification statistics of dynamic geological carbon-storage responses.
Takeaways & Limitations
The method’s potential use beyond groundwater multiphase flows remains to be explored.
Abstract
from arXiv · showhide
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.
1 Introduction
Dynamic multiphase-flow surrogates face high-dimensional inputs, discontinuous saturation responses, and time-dependent outputs. The paper proposes a convolutional encoder-decoder strategy with two-stage training to address these challenges for uncertainty quantification.
- 1 Introduction: Monte Carlo uncertainty quantification is computationally burdensome because accurate statistics require many simulation runs.Surrogate models trained with relatively few model runs can approximate input-output relationships more efficiently.
- 1 Introduction: Transient multiphase-flow surrogate models must handle high-dimensional heterogeneous-media inputs, discontinuous saturation profiles, and outputs that vary over time.These challenges together make traditional surrogate methods difficult to apply to uncertainty quantification.
- 1 Introduction: High-dimensionality can arise because permeability is represented at spatial grid points, making input dimensionality equal to the number of pixels.Existing surrogate methods often fail as input dimensionality increases.
- 1 Introduction: Capillarity creates spatially discontinuous saturation profiles that are difficult for continuous, often differentiable, surrogate models to approximate.The discontinuity also appears in stochastic space, further complicating approximation.
- 1 Introduction: Approximating outputs at only selected time instances does not adequately capture the time dependence of dynamic response fields.Dynamic surrogates therefore need to predict outputs efficiently at arbitrary times.
- 1 Introduction: The proposed method extends a convolutional encoder-decoder architecture and transforms surrogate modeling into image-to-image regression for high-dimensional inputs and outputs.A two-stage regression-and-segmentation strategy is introduced to improve approximation of the discontinuous saturation front.
2 Problem Formulation
The paper formulates uncertainty quantification for geological carbon storage as a stochastic multiphase-flow problem with spatially distributed random inputs and time-dependent pressure and saturation responses. It targets statistical characterization of these responses while replacing expensive Monte Carlo model evaluations with a deep convolutional surrogate.
- 2 Problem Formulation: The study models CO2-gas and water-liquid multiphase flow with multicomponent H2O, NaCl, and CO2 transport in geological carbon storage.The target responses include pressure buildup and CO2 plume evolution over time.
- 2.1 Governing Equations: For each component κ, the governing balance is a mass conservation equation combining mass accumulation, phase fluxes, and sources or sinks.The total mass of component κ is obtained by summing contributions over phases.
- 2.1 Governing Equations: The mass accumulation term depends on porosity, phase saturation, phase density, and component mass fraction.These quantities define the component mass stored across phases.
- 2.1 Governing Equations: Mass transport is modeled as advection only, with advective fluxes summed over phases and phase fluxes given by a multiphase Darcy equation.The flux formulation uses absolute and relative permeability, viscosity, pressure, and gravity.
- 2.2 Uncertainty Quantification: Random PDE coefficients make the solution stochastic, so uncertainty quantification estimates statistical moments and probability densities of the output responses.The random input field induces uncertainty in the model solution.
- 2.2 Uncertainty Quantification: After spatial discretization, a random input field becomes a high-dimensional vector x, while the response is represented as y = η(x, t) across spatial locations and time instances.The formulation supports multiple output variables over the computational grid.
- 2.2 Uncertainty Quantification: Classical uncertainty quantification estimates the mean and variance of η(x, t), often using Monte Carlo methods.The paper develops a deep convolutional encoder-decoder surrogate to reduce the number of expensive model runs needed for these estimates.
3 Methods
The method formulates high-dimensional surrogate modeling as pixel-wise image-to-image regression using a fully convolutional encoder-decoder. Dense blocks, coarse-to-refine processing, and segmentation-based training address feature propagation, spatial outputs, and discontinuous saturation fields.
- Deep convolutional neural networks: Convolutional neural networks reduce learned parameters through sparse connectivity and parameter sharing while exploiting spatially local correlations.These properties make convolutional layers suitable for image-based prediction.
- Surrogate modeling as image-to-image regression: The surrogate task reorganizes grid-based inputs and outputs as image-like tensors, enabling pixel-wise prediction with a convolutional encoder-decoder.The formulation extends to three-dimensional domains by adding a depth axis.
- Encoder-decoder architecture: The encoder extracts high-level coarse features from input images, and the decoder refines them into output images through a coarse-refine process.The network uses a dense fully convolutional encoder-decoder architecture for high-dimensional input-output mappings.
- Dense blocks: Dense blocks connect each layer to all successor layers by concatenating feature maps from preceding layers.Each dense-block layer applies batch normalization, ReLU, and convolution; L and K control block depth and growth rate.
- Encoding and decoding layers: Transition layers alternate with dense blocks to control feature-map growth and change spatial resolution through encoder downsampling and decoder upsampling.Both encoding and decoding layers reduce the number of feature maps; their second convolutions perform downsampling or transposed-convolution upsampling.
- Two-stage regression and segmentation training: The training strategy augments regression with image segmentation that separates positive saturation from zero-saturation regions.Binarizing the saturation field helps represent the discontinuous saturation front as a boundary between zero and one.
4 Numerical Experiments
The experiments evaluate approximation accuracy, time interpolation, saturation-front reconstruction, and uncertainty statistics for the proposed surrogate in a geological carbon storage model. The model performs well with limited training data, while the combined MSE-BCE loss particularly improves discontinuous saturation-front predictions.
- Approximation Accuracy Assessment: The experiments compare MSE and MSE-BCE networks on time-dependent pressure and CO2 saturation fields using randomly selected permeability test samples.Both networks share the same architecture and training data, while uncertainty-quantification results are compared with vanilla MC.
- Approximation Accuracy Assessment: With 400 training samples, R2 is 0.869 for MSE versus 0.913 for MSE-BCE, with the advantage decreasing as training data increase.The results indicate that MSE-BCE improves performance especially when the training sample size is small.
- Approximation Accuracy Assessment: Training with six time instances supports good predictions at 21 instances from 100 to 200 days, including the untrained 150-day interpolation time.Errors at trained times are described as similar to those at the remaining 15 time instances.
- Approximation Accuracy Assessment: MSE-BCE more accurately characterizes saturation-front positions than MSE, while pressure and saturation errors are otherwise visually similar except near the front.The binary-image comparison identifies CO2-plume pixels and uses the boundary between zero and nonzero pixels to locate the front.
- Discussion: The dense convolutional encoder-decoder handles high-dimensional image inputs and outputs, nonlinear discontinuous responses, and time-dependent predictions with limited training samples.The authors associate dense blocks with improved local-information exploitation and feature flow through the network.
- Uncertainty Modeling: Predicted pressure and saturation means, variances, and bimodal saturation PDFs are close to reference solutions, including at time instances without training data.The surrogate construction uses 1600 model runs for an almost identical uncertainty-quantification result, and the bimodal PDFs are reproduced at 150 and 200 days.
5 Conclusions
The proposed deep convolutional encoder-decoder surrogate transforms dynamic multiphase-flow UQ into image-to-image regression, combining dense CNN information propagation, mixed losses, and time as an input. It accurately approximates discontinuous pressure and saturation responses while reducing the model runs required for efficient UQ.
- Method: The method transforms surrogate modeling into image-to-image regression, using an encoder to extract high-dimensional input features and a decoder to reconstruct output fields.The dense fully convolutional structure strengthens information flow through the network.
- Method: A combined regression and segmentation loss improves approximation of the discontinuous saturation field, especially near the saturation front.The segmentation loss is induced mainly by mismatches around discontinuous-front regions.
- Method: Time is treated as an additional network input, enabling characterization of dynamic multi-output responses at arbitrary time instances from outputs sampled at limited times.
- Results: The surrogate accurately reproduces pressure and saturation fields, including saturation discontinuities, for a geological carbon storage model with 2500 uncertain inputs.
- Results: The deep-network surrogate achieves comparable UQ accuracy to traditional Monte Carlo sampling with much higher efficiency and significantly fewer geological carbon storage model runs.
- Implications: The dynamic modeling capability may also support experimental design and inverse problems, while extension to multiple input fields requires no network-architecture modifications.Applications beyond groundwater multiphase flow remain to be explored.