A Deep-Learning-Based Geological Parameterization Method ......SGeMS Random Real. 1 O-PCA Random...
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A Deep-Learning-Based Geological Parameterization Method for History Matching Yimin Liu Wenyue Sun Louis J. Durlofsky SESAAI Meeting March 30, 2018
Transcript of A Deep-Learning-Based Geological Parameterization Method ......SGeMS Random Real. 1 O-PCA Random...
A Deep-Learning-Based Geological Parameterization Method
for History Matching
Yimin Liu Wenyue Sun Louis J. Durlofsky
SESAAI Meeting
March 30, 2018
Outlineq Background
q Deep-learning-based neural style transfer
q Convolutional neural network PCA (CNN-PCA) for geological Parameterization
q Parameterization results
q History matching results
q Conclusions and future work
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History Matching
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History Matching Problem
𝐚𝐫𝐠𝐦𝐢𝐧𝒎
(
𝟏𝟐 𝒅 𝒎 − 𝒅𝐨𝐛𝐬 𝑻𝐶234(𝒅(𝒎) − 𝒅𝐨𝐛𝐬)
+𝟏𝟐 𝒎− 𝒎8 𝑻𝐶934(𝒎 −𝒎8 )
:
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q Decision variable: 𝒎 - model parameters
q Challenges:
§ 𝒎 can be high dimensional
§ 𝒎 should preserve geology
q Solution: map 𝒎 onto lower dimensional space
Log permeability
Reparameterization for History Matching
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q Map 𝒎 to a new variable 𝝃
q Favorable properties:
§ dim 𝝃 ≪ dim(𝒎)
§ 𝝃 is uncorrelated
§ 𝒎@ preserves geological realism
𝒎 ≈ 𝒎@ = 𝒇(𝝃)
𝐚𝐫𝐠𝐦𝐢𝐧𝝃
(
𝟏𝟐 𝒅 𝝃 − 𝒅𝐨𝐛𝐬 𝑻𝐶234(𝒅(𝝃) − 𝒅𝐨𝐛𝐬)
+𝟏𝟐 𝝃 − 𝝃D
𝑻𝐶𝝃34(𝝃 − 𝝃D)
:
q Optimization variable: 𝝃
Principal Component Analysis (PCA)
q PCA: Oliver (1996), Sarma et al. (2006)
Ø Generate 𝑁F realizations using geostatistical algorithm
Ø Perform SVD and reduce dimension
Ø Generate new realization:
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𝒎𝐩𝐜𝐚 = 𝑼𝒍𝜦𝒍𝟏/𝟐𝝃𝒍 +𝒎8
l << N𝑪 (N𝑪:#ofgridblocks)
𝒀 =𝟏
𝑁F − 𝟏[𝒎𝟏 − 𝒎8 ,𝒎𝟐 − 𝒎8 ,… ,𝒎 _̂ − 𝒎8 ]
𝒀 = 𝑼𝜦𝟏/𝟐𝑽𝑻 ≈ 𝑼𝒍𝜦𝒍𝟏/𝟐𝑽𝒍𝑻
𝝃𝒍~𝑵(𝟎, 𝑰)
PCA Representation for New Realizationsq Works well when 𝒎 follows Gaussian distribution
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PCA 𝒍 = 70SGEMS N𝑪 = 3600
𝒎𝐩𝐜𝐚 = 𝑼𝒍𝜦𝒍𝟏/𝟐𝝃𝒍 +𝒎8 𝝃𝒍~𝑵(𝟎, 𝑰)
Non
-Gau
ssia
nG
auss
ian
PCA 𝒍 = 70SGEMS N𝑪 = 3600
Optimization-based PCA (O-PCA)q Optimization-based PCA: Vo and Durlofsky (2014, 2015)
Ø Formulate PCA as an optimization problem with regularizationØ Objective: minimize difference to 𝒎𝐩𝐜𝐚 and original histogram Ø Essentially post-process 𝒎𝐩𝐜𝐚 with point-wise mapping
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O-PCA 𝒍 = 70SGEMS N𝑪 = 3600
𝒎fghi= 𝐚𝐫𝐠𝐦𝐢𝐧𝒙
𝒎ghi(𝝃𝒍) − 𝒙 𝟐𝟐+ γ𝒙𝑻(𝟏 − 𝒙) 𝑥m ∈ [𝑥o, 𝑥p]
(figures from Vo and Durlofsky, 2014)
Limitations of O-PCAq Underlying PCA honors only two-point correlations
q O-PCA point-wise mapping honors single point statistics
q Difficult to preserve multiple point statistics
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SGeMS Random Real. 1 O-PCA Random Real. 1 O-PCA Random Real. 2
Unconditional Realizations
Neural Style Transfer
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Content Image
Style Image
Output Image
q Recent work in deep learning and computer visionq Gatys et al. (2016), Johnson et al. (2016)q Enables transfer of photo into artistic style
(images from Johnson: github.com/jcjohnson/fast-neural-style)
Neural Style Transfer
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q Recent work in deep learning and computer visionq Gatys et al. (2016), Johnson et al. (2016)q Enables transfer of photo into artistic style
Content ImagePCA Model
Style Image - Training Image
Output ImagePost-processed model
Neural Style Transfer Algorithm
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q O: output image, C: content image, S: style image
q 𝐿hfrstrs 𝑂, 𝐶 : difference between output and content image
q 𝐿vswxt 𝑂, 𝑆 : difference between output and style image
q Output image preservesØ Content (objects) in the content image
Ø Style (color, texture) in the style image
Ø Object and texture are characterized by multipoint statistics
𝑂 = argmin|
{𝐿hfrstrs 𝑂, 𝐶 + 𝛾𝐿vswxt 𝑂, 𝑆 }Content Loss Style Loss
Analogy to O-PCAq Neural style transfer algorithm
q O-PCA
𝑂 = argmin|
𝐿hfrstrs 𝑂,𝐶 + 𝛾𝐿vswxt 𝑂, 𝑆Content Loss Style Loss
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𝒎fghi= argmin𝒙
𝒎ghi − 𝒙 𝟐𝟐 + 𝛾𝒙𝑻(𝟏 − 𝒙)
Style LossContent Loss
q O-PCA losses based on point-wise difference
q Neural style transfer based on high-level features extracted from deep convolutional neural network (CNN)
Artificial Neural Networkq Nonlinear function 𝑦 = 𝑔(𝑥) with multiple layers of neurons
q Linear function: 𝑊�ℎo34 + 𝑏q Nonlinear activation function 𝑓 � , e.g., ReLU, sigmoid
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ℎo = 𝑓(𝑊�ℎo34 + 𝑏)
q Convolutional neural networks (CNN):Ø Convolutional layersØ Suitable for image input
Neural Style Transfer Algorithm
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𝐿h������ 𝑂, 𝐶 =�1
𝑁o𝐷o𝐹o 𝑂 − 𝐹o 𝐶 �
�
o
𝐿vswxt 𝑂, 𝑆 = �1𝑁x� 𝐺o 𝑂 − 𝐺o 𝑆 �
�
o
q Limitations of neural style transfer algorithm: Ø Need to solve optimization onlineØ Derivatives of the output image w.r.t input images not clear
Fast Neural Style Transfer Algorithm
q Train a transform net, Johnson et al. (2016)Ø Hour glass shape deep CNN with same input and output size
q Same loss function, optimize parameters in the transform net
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Fast Neural Style Transfer Algorithmq Construct PCA with 1000 SGeMS realizationsq Train the model transform net on 3000 random PCA modelsq Training takes 3 minutes on 1 GPU (NVIDIA Telsa K80)q Final step: threshold at 0.5
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Outlineq Background
q Deep-learning-based neural style transfer
q Convolutional neural network PCA (CNN-PCA) for geological parameterization
q Parameterization results
q History matching results
q Conclusions and future work
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Unconditional Binary System
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q Binary facies model:
q Training image size: 250 x 250
q Model size: 60 x 60, 𝑁� = 3600
q No hard data
q Goal: low-dimensional representation
q Reduced dimension: 𝑙 = 70
Training image
One model realization
Unconditional Binary System
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PCA Real. 1
PCA Real. 2
O-PCA Real. 1
O-PCA Real. 2 CNN-PCA Real. 2
CNN-PCA Real. 1
Conditional Binary System
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q Binary facies model
q Training image size: 250 x 250
q Model size: 60 x 60, 𝑁� = 3600
q Hard data at 16 well locations
q Reduced dimension: 𝑙 = 70
q 200 new random realizations
q Additional hard-data loss
Training image
One model realization
argmin|
{𝐿hfrstrs 𝑂, 𝐶 + 𝛾4𝐿vswxt 𝑂,𝑆
+𝛾�𝐿�isi(𝑂)}
Conditional Binary System
PCA Real. 1
PCA Real. 2
O-PCA Real. 1
O-PCA Real. 2
CNN-PCA Real. 1
CNN-PCA Real. 2
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q All 200 CNN-PCA models match all hard data
Outlineq Background
q Deep-learning-based neural style transfer
q Convolutional neural network PCA (CNN-PCA) for geological parameterization
q Parameterization results
q History matching results
q Conclusions and future work
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History Match Conditional Modelq Oil-water, 60 x 60 grid
q 2 injectors, 2 producers, BHP controlled
q ksand = 2000 md, kmud = 0.02 md
q Data: production and injection rates for 1000 days every 100days
q Number of data: 𝑁� = 80
q Goal: 30 RML posterior models
q Optimizer: PSO-MADS
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Injector Producer
Permeability Estimation
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True model
One O-PCA prior model O-PCA posterior model
One CNN-PCA prior model CNN-PCA posterior model
PROD-1 Water Rate
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O-PCA Prior models O-PCA Posterior Models
CNN-PCA Prior models CNN-PCA Posterior Models
Infill Well Prediction
infill well locations
q Two infill wells P3, P4q Drilled at 1000 daysq Prediction to 2000 days
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Permeability Estimation
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True model
O-PCA #1 O-PCA #2
CNN-PCA #1 CNN-PCA #2
O-PCA #3
CNN-PCA #3
Infill Well PROD-4 Prediction
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O-PCA Water Rate CNN-PCA Water Rate
24 curves 13 curves
Field Prediction
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O-PCA Oil Rate CNN-PCA Oil Rate
15 curves 6 curves
Conclusions
q Developed CNN-PCA by combining deep-learning-based neural style transfer algorithm with PCA
q CNN-PCA better preserves channel geometry compared to O-PCA
q CNN-PCA history matching solutions provide more accurate prediction for infill wells
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Future Work
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q Extend CNN-PCA to bimodal, three-facies and three dimensional reservoir models
q Apply CNN-PCA with gradient-based history matching
q Implement CNN-PCA treatment with ensemble-based history matching methods
Acknowledgmentsq Stanford CS 231N course
Ø Instructors: Justin Johnson, Serena Yeung, Fei-fei LiØ Course project teammate: Yuanlin Wen (Google)
q Open source code:Ø O-PCA: Hai Xuan Vo (Chevron)Ø Neural style transfer (PyTorch): Hang ZhangØ Fast neural style transfer (PyTorch): Abhishek Kadian
q PSO-MADS: Obi Isebor
q Stanford CEES
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Thank You!Q & A
Backup Slides
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Convolutional Layersq Each neuron connects to a local region in previous layerq Convolution: No filters 𝑤o sliding through previous layersq Convolutional layer 𝑁o channels:
Ø Each channel is called a feature map of 𝐷o neuronsØ All channels form a feature matrix 𝐹o ∈ 𝑅^�× �
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Convolutional Neural Networkq Neural network consists of mainly convolutional layersq Nonlinear layers:
Ø Activation layer, e.g., ReLU 𝑓 𝑥 = max(0,𝑥)Ø Pooling layer: down sampling to reduce dimension
q Image classification:Ø Input: image of size 𝐻×𝑊×3Ø Output: score for predefined image classes
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VGG-16 Deep CNN (Simonyan and Zisserman, 2015)
Feature Matrix and Gram Matrix
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q Content representation: feature matrix 𝐹o ∈ 𝑅^�× �
q Style representation: Gram matrix 𝐺o = 𝐹o𝐹o�/(𝑁o𝐷o)�
* Images modified from Gatys et al. (2016)
CNN-PCA for Reparameterizationq Construct PCA with 𝑁F SGeMS realizations
q Post-process 𝒎𝐩𝐜𝐚 with fast neural style transfer algorithm
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𝒎𝐩𝐜𝐚 = 𝑼𝒍𝜦𝒍𝟏/𝟐𝝃𝒍 +𝒎8 𝝃𝒍~𝑵(𝟎, 𝑰)
