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Deep learning for characterization and prediction of complex fluid flow systems
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Deep learning for characterization and prediction of complex fluid flow systems
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Content
Deep learning for Characterization and Prediction of Complex Fluid Flow
Systems
by
Wei Ling
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMICAL ENGINEERING)
December 2024
Copyright 2024 Wei Ling
Dedication
To my family and friends
ii
Acknowledgments
First and foremost, I would like to express my heartfelt gratitude to Professor S. Joe Qin for
giving me the opportunity to embark on my Ph.D. journey and for his invaluable mentorship
during the early stages of my time at USC. I am deeply thankful to my advisor, Professor
Behnam Jafarpour, for his exceptional guidance and unwavering support throughout my
Ph.D. studies. Both professors have significantly shaped my academic and professional
development, fostering my analytical and technical skills, critical thinking, and innovation.
Their passion for science and engineering, combined with their commitment to rigorous
research, will continue to inspire me in my future endeavors.
I would like to extend my heartfelt gratitude to my Ph.D. qualifying exam committee
members: Professor Pin Wang, Professor Felipe De Barros, Professor Birendra Jha, and
Professor Iraj Ershaghi. I am particularly grateful to Professor Wang and Professor Barros
for attending my dissertation defense and providing thoughtful and insightful feedback. Their
constructive suggestions not only enhanced the quality of my dissertation but also enriched
my understanding of the research field.
I would like to thank Dr. Robert E. Young for mentoring me during my first Department
of Energy project. His exceptional support greatly enhanced my knowledge of thermodynamic simulation and control theory, enabling me to complete the research successfully.
I am also grateful to Dr. Vanessa Feng for mentoring me during my internship at Meta.
She provided me with the opportunity to work on an exciting deep-learning project and was
always ready to cheer me on and celebrate our progress. Without her exceptional mentorship,
I would not have successfully completed a challenging internship in a new area.
Most importantly, I would like to express my deepest gratitude and love to my wife, Jiamei
Zhang. Her unwavering support was instrumental in helping me navigate and overcome the
challenges of my Ph.D. journey. I also wish to acknowledge the members of Professor Qin’s
and Professor Jafarpour’s research groups for their valuable discussions and support. I am
iii
deeply grateful to all my friends at USC, in Seattle, the Bay Area, and in China, whose
companionship and mental support have brought me immense joy. Lastly, I want to express
my profound gratitude to my parents for their endless love, encouragement, and support
throughout this journey.
iv
Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1:
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Model Characterization of Subsurface Fluid Flow System . . . . . . . 2
1.1.2 Prediction and Optimization of Surface Geothermal Fluid Flow System 3
1.1.3 Deep Learning for Model Characterization and Prediction . . . . . . 5
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2:
Characterization of Complex Subsurface Flow Properties . . . . . . . . . . 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Inverse Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Parameterization Methods . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 Deep Learning in Subsurface Model Characterization . . . . . . . . . 11
2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Model Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Deep Learning Models . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Numeric Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Generative Models for Channelized Fluid System Distributions . . . . 22
2.3.2 Pumping Test Example . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Two-phase Fluid System Example . . . . . . . . . . . . . . . . . . . . 29
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Chapter 3:
Deep Learning-Based Pilot Point Method for Parameterization . . . . . . 39
v
3.1 Pilot Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Deep Learning Model - U-Net . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Perceptual Loss Function . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Pilot Point Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Numeric Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Parameterization Results . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 Pumping Test Example . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.3 Two-phase Flow Example . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter 4:
Optimization with Deep Learning Predictive Model in Binary Geothermal
Fluid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Metholodgy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Thermodynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.2 Thermodynamic Model Evaluation . . . . . . . . . . . . . . . . . . . 72
4.4 Numerical Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 Case 1: Synthetic Datasets (Uniform Input) . . . . . . . . . . . . . . 73
4.4.2 Case 2: Synthetic Datasets (Field Pattern Input) . . . . . . . . . . . 78
4.4.3 Case 3: Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter 5:
Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
vi
List of Tables
2.1 Workflow for data assimilation and model parameterization . . . . . . . . . . 16
2.2 Detailed VAE decoder structure . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Detailed strcuture of StyleGAN generator . . . . . . . . . . . . . . . . . . . 37
2.4 Pumping test result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Two-phase flow result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Measurement metrics for model variables. . . . . . . . . . . . . . . . . . . . . 72
4.2 Input variables of the ORC model. . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Summary of prediction results on synthetic datasets. . . . . . . . . . . . . . 78
vii
List of Figures
1.1 Schematic configuration of the binary surface geothermal power plant . . . . 4
2.1 Structures of variational autoencoder and generative adversarial networks . . 18
2.2 Structures of deep convolutional generative adversarial networks and StyleGAN2 19
2.3 The original 250x250 fluvial channel training image (left) and exemplar 64x64
image samples (right) utilized for model training and testing. . . . . . . . . . 22
2.4 Image generation using (a) DCGAN, (b) StyleGAN, and image reconstruction
with (c) VAE (latent variables dimension = 16) and (d) VAE (latent variable
dimension = 32). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Latent space sampling and interpolation in VAE, DCGAN, and StyleGAN . 24
2.6 VAE reconstruction ability on the testing dataset . . . . . . . . . . . . . . . 25
2.7 KID and FID of the stylegan model on the training dataset. . . . . . . . . . 25
2.8 Analysis of PPL and interpolation effects across generative models . . . . . . 26
2.9 Visualization of permeability field for four reference cases . . . . . . . . . . . 27
2.10 Ensemble mean and variance of prior and posterior distributions for generative
models in groundwater parameterization . . . . . . . . . . . . . . . . . . . . 29
2.11 Calibrated sample parameter realizations for cases with 16 monitoring wells
using each of the four generative models. . . . . . . . . . . . . . . . . . . . 30
2.12 Comparative visualization of generated images with perturbed latent variables
across models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.13 Permeability configurations and ensemble results for two-phase flow model
parametrization using generative models . . . . . . . . . . . . . . . . . . . . 32
viii
2.14 Performance metrics of the generative models across the two cases in twophase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.15 Well response data match using StyleGAN parameterization for non-wetting
and wetting phases in case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.16 Initial and calibrated parameter realizations for the two well configurations . 34
2.17 Well response data match in case 2 using VAE, DCGAN, and StyleGAN models 35
2.18 Distributions of latent variables for VAE, DCGAN, and StyleGAN on case 2 35
3.1 Workflow of model calibration using U-Net and pilot points method . . . . . 42
3.2 U-Net architecture for geostatistical simulation using pilot points and a training image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Schematic representation of a neural network incorporating perceptual loss
for style similarity assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Model reconstruction comparisons: (a) Comparison of reconstruction outcomes using L2 loss and combined L2 and perceptual loss across increasing
numbers of input points. (b) Effects of L2 versus combined L2 and perceptual
loss with 25 input points across different realizations. . . . . . . . . . . . . . 53
3.5 sensitivity analysis of pilot points on model accuracy metrics . . . . . . . . . 54
3.6 3D field layouts, permeability maps, and true pressure maps for pumping test
cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 Sensitivity maps from the initial iteration of the ES-MDA algorithm, highlighting regional sensitivity variations and pilot point locations based on maximum values in subregions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.8 Influence of pilot point placement strategies on facies realizations, showcasing
the progression from initial models to refined outcomes using the ES-MDA
method across two cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.9 Data assimilation outcomes for reference facies maps: log-permeability evolution through iterations and sampling strategies . . . . . . . . . . . . . . . . . 55
3.10 Mismatch reduction across iterations for two cases with five pilot point spacing
strategies: evenly spaced, random variations, and sensitivity-based . . . . . . 56
ix
3.11 Comparative visualization of reference and simulation results for two case
studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.12 Two-year production profiles for non-wetted and wetted phases at extraction
wells P1, P2, and P3 in two cases, displaying initial realizations, observed
data, and calibrated predictions. . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.13 Comparative visualization of reference and simulation results for two case
studies, demonstrating the impact of integrating soft points selected based
on uncertainty metrics derived from the ensemble mean and variance of the
previous experiment (first row). . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.14 Two-year production profiles for non-wetted and wetted phases at extraction
wells P1, P2, and P3 in two cases, after incorporating soft point constraints. 59
4.1 Overview of working flow on deep learning-based prediction and optimization. 64
4.2 The schematic diagram of the ANN. . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Schematic diagram of the ORC unit in binary cycle powerplant. . . . . . . . 67
4.4 (a) Daily maximum temperature minimum temperature in 2020; (b) Ambient
temperature vs Net power generation; (c) Ambient temperature vs. turbine
exhaust pressure; (d) Ambient temperature vs. turbine exhaust temperature. 69
4.5 Diagram of the ORC model in AVEVA. . . . . . . . . . . . . . . . . . . . . . 70
4.6 (a) Polynomial regression model to estimate the turbine outlet pressure with
ambient temperature; (b) multi-stage pump curve of discharge pressure. . . . 71
4.7 The featured variables of simulated data (red) compared to the field data (blue). 73
4.8 Figure 8. (a) PCA projection of the synthetic datasets with uniform distributed input; (b) Featured variables in testing dataset simulated with the
field input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.9 Sensitive analysis for finalizing the structure of ANN model. Average RMSE
and MAPE of 10 realizations for each coefficient of (a) Hidden layer (b) Hidden
neural (c) Loss function (d) Learning rate. . . . . . . . . . . . . . . . . . . . 76
4.10 Comparative Predictive Performance on Test Data Sets: (a) Models Applied
to Random Input Scenarios (b) Models Validated Against Field Operation
Data Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
x
4.11 (a) PCA projection of the testing data before and after the optimization (b)
Average power production optimization result of the production; (c) Accumulative net energy improvement on 1500h simulation. . . . . . . . . . . . . . . 79
4.12 (a) Left: ambient temperature change during the operation. Right: original
pump operation vs the optimized pump operation by the numerical simulator
and ANN model; (b) Monitoring the optimization process on highlight samples. 80
4.13 Featured synthetic training and testing data after preprocess from 0 to 1. . . 80
4.14 (a) PCA projection of the testing data before and after the optimization; (b)
Average power production optimization result of the production; (c) Accumulative net energy improvement on 1500h simulation. . . . . . . . . . . . . . . 82
4.15 Normalized historical dataset from the field. . . . . . . . . . . . . . . . . . . 82
4.16 Prediction result of testing datasets: (a) prediction value vs true value in
scatter plot; (b) prediction value vs true value in the sequence. . . . . . . . . 83
4.17 (a) PCA projection of the training dataset and testing data before and after
the optimization of Cyrq dataset; (b) Averaged (predicted) optimization result
of the production by ANN model; (c) Accumulative net energy improvement
on 500h simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xi
Abstract
Accurate characterization and prediction of complex fluid flow systems are essential for
advancements in geosciences and renewable energy. This thesis explores the integration of
deep learning techniques with traditional modeling approaches to enhance subsurface flow
model calibration and optimize geothermal power plant operations.
In subsurface modeling, the accurate representation of non-Gaussian spatial patterns
poses significant challenges for conventional parameterization methods. We introduce stylebased Generative Adversarial Networks (StyleGANs) as a novel parameterization tool, demonstrating their superior performance in reconstruction fidelity, robustness, and calibration
flexibility compared to Variational Autoencoders (VAEs) and traditional GANs. Using the
Ensemble Smoother with Multiple Data Assimilation (ES-MDA), we validate StyleGANs’ effectiveness in single-phase and two-phase flow scenarios, highlighting their ability to improve
the representation and calibration of complex geological patterns. Further, we integrate the
pilot point method with a U-Net model, trained with a perceptual loss function, to conditionally map sparse facies data to detailed distributions. This method enhances geological
continuity and calibration accuracy, with sensitivity-based pilot point selection and soft point
incorporation improving model fidelity while reducing computational overhead.
In geothermal energy optimization, we develop a data-driven artificial neural network
(ANN) to predict power output and operational costs, serving as an efficient alternative to
physics-based approaches. This ANN-based model propagates the influence of control and
disturbance variables and is applied to optimize the working fluid circulation rate, maximizing net power output. The workflow is further demonstrated to model and control ambient temperature effects on air-cooled binary cycle geothermal plants, which are challenging
and computationally expensive to address with physics-based predictive models. Numerical experiments with simulated and field datasets validate the model’s ability to improve
operational efficiency.
xii
By integrating deep learning techniques with geostatistical and optimization workflows,
this research advances subsurface modeling and renewable energy systems. The findings
provide practical, scalable, and efficient solutions for addressing the complexities of fluid
flow systems in geosciences and energy sectors.
xiii
Chapter 1
Introduction
This dissertation focuses on applying deep learning to tackle challenges in characterizing
and predicting complex fluid flow systems, with an emphasis on two complementary areas.
The first part addresses Model Characterization of Subsurface Fluid Flow Systems,
leveraging DL techniques to enhance the representation of spatial heterogeneity and predict
subsurface flow properties. This research seeks to improve the accuracy and reliability of
subsurface models, which are essential for resource management and environmental protection.
The second part focuses on Modeling and Prediction of Surface Geothermal Fluid
Flow Systems, utilizing deep learning to optimize the performance and control of geothermal power plants. By integrating real-time data with advanced deep learning algorithms,
this work generates predictive insights into system behavior, facilitating more efficient and
adaptive operations.
The unifying theme of this dissertation is the application of deep learning to extract
valuable insights from complex datasets, overcoming limitations of traditional methods and
enhancing the accuracy, scalability, and adaptability of fluid flow modeling. Together, these
contributions advance the state-of-the-art in managing subsurface and surface fluid systems,
providing innovative solutions to critical challenges in both domains.
1
1.1 Background
1.1.1 Model Characterization of Subsurface Fluid Flow System
Understanding and predicting fluid flow in complex subsurface formations is essential for the
effective development and management of underground resources. High-fidelity simulation
models play a crucial role in forecasting how these formations respond to various development strategies, providing a framework for strategic planning and efficiency improvement.
However, the heterogeneous and intricate nature of subsurface environments poses significant
challenges. These complexities are exacerbated by the high costs and logistical constraints
of directly observing geological formations and the associated flow and transport processes
[1, 2, 3, 4, 5, 6].
Accurate characterization and calibration of subsurface flow models are further hindered
by uncertainties in key geological properties, such as lithofacies distribution, porosity, and
permeability heterogeneity. Sparse and unevenly distributed observational data often leave
these critical properties poorly constrained. Traditional methods struggle to resolve these
issues, as they rely on costly and logistically challenging data acquisition processes. Consequently, simulation models frequently depend on approximations that introduce significant
uncertainties, reducing their predictive accuracy and limiting their applicability in resource
development and environmental management [7, 8]. Addressing these limitations requires
innovative approaches to improve spatial resolution, enhance data integration, and reduce
uncertainties in subsurface modeling.
To address these challenges, inverse modeling techniques have been developed to reduce
uncertainties by integrating monitoring data and estimating uncertain input parameters.
These methods iteratively align observed data with model predictions, refining subsurface
property estimates and improving reliability. Supported by geostatistical approaches, inverse
modeling has enhanced accuracy and predictive power, making it essential for subsurface
2
characterization and resource management [1, 2, 8].
Geostatistical methods incorporate prior information and efficiently represent spatial
complexity, but they struggle with complex, non-Gaussian patterns found in geological
formations like alluvial systems. Such formations often exhibit spatial distributions that
traditional methods fail to capture during calibration [9, 10, 11, 12, 13]. This highlights
the need for advanced techniques to better represent subsurface heterogeneity and improve
model accuracy.
1.1.2 Prediction and Optimization of Surface Geothermal Fluid
Flow System
Geothermal energy is a renewable resource that strengthens energy security, lowers carbon
emissions, and promotes economic resilience [14, 15]. Systems such as dry steam, flash steam,
and binary cycle (illustrated in Figure 1.1) provide dependable baseload power with minimal
environmental impact, positioning geothermal energy as a critical component in the global
shift toward clean and sustainable energy solutions.
In geothermal power systems, surface units are essential for converting geothermal energy
into electricity. These units must optimize efficiency, minimize environmental impact, and
scale to match resource potential. Accurate modeling and prediction of surface geothermal
fluid flow systems are crucial for automating operations and ensuring reliability. Traditionally, these models are grounded in physics-based principles, using governing equations to
describe system behavior [16, 17]. As systems grow in complexity, enhancing these models
is key to unlocking the full potential of geothermal energy.
To achieve this, predictive models are utilized to estimate system responses based on
inputs from controllers and external disturbances, enabling proactive management and optimization. Traditionally, these models have been rooted in physics-based principles, resulting
in a set of governing equations that describe the behavior of the system. Such models have
provided foundational insights into geothermal fluid dynamics and thermoeconomic perfor3
Figure 1.1: Schematic configuration of the binary surface geothermal power plant
mance [16, 17]. However, as geothermal systems grow in complexity, the need for enhanced
predictive capabilities that incorporate both physical laws and advanced computational techniques becomes increasingly evident.
Physics-based models face inherent challenges. Their development and calibration require
extensive technical expertise, deep knowledge of system dynamics, and detailed data on
component interactions. Collecting the required data for accurate modeling often demands
costly experimental designs and data acquisition efforts [14, 15, 18]. Furthermore, the long
lifecycle of geothermal systems, spanning several decades, complicates maintenance. These
systems require regular updates to account for equipment upgrades and process changes,
adding to the complexity and cost of maintaining accurate models [19, 20].
Addressing these challenges calls for the integration of advanced predictive modeling
techniques that can complement or even surpass traditional physics-based approaches in
efficiency and adaptability, paving the way for improved management of geothermal fluid
flow systems.
4
1.1.3 Deep Learning for Model Characterization and Prediction
Recent advancements in machine learning have introduced deep learning-based techniques
that effectively address many limitations in the characterization and prediction of complex
fluid systems. These techniques have proven transformative in two key areas: subsurface
modeling and surface power plant operations.
For subsurface systems, deep learning has proven to be a transformative tool, addressing
many limitations of traditional parameterization and geostatistical methods. These conventional approaches often struggle to capture the complex, non-Gaussian spatial distributions
and intricate geological patterns inherent in subsurface environments. By leveraging large
datasets and advanced algorithms, deep learning excels at identifying and modeling the
spatial and temporal relationships within these systems, providing improved accuracy and
adaptability [21, 22, 23].
For surface power plant operations, data-driven predictive models have become instrumental in real-time control and optimization. These models are designed to forecast system
responses to controller inputs and external disturbances, enabling more efficient and adaptive operation of geothermal plants. Through the integration of real-time monitoring systems
and advanced machine learning techniques, these models offer efficient, scalable, and costeffective alternatives to traditional physics-based methods. They are particularly adept at
uncovering complex input-output relationships in large datasets, resulting in accurate and
reliable performance predictions [24, 25, 26, 27]. Unlike physics-based models, which often require extensive revisions to accommodate new insights, data-driven models seamlessly
integrate new data, ensuring they remain accurate and relevant over time.
However, while these methods hold great promise for enhancing model characterization,
several challenges persist:
• Black-box nature:
Deep learning models often function as black-box systems, making it difficult to in5
terpret decisions or identify influential features. When use it as the proxy model to
predict the input and output relationship. This lack of transparency can hinder trust
and understanding, particularly the goal is to use the proxy model in automation of
the monitoring and control.
• Nonlinearity and topology challenges:
The highly nonlinear nature of physical processes and the changing topology of solution
spaces can undermine the performance of gradient-based inversion techniques. These
factors frequently lead to poor convergence and susceptibility to local minima traps in
solving the inverse problem[28].
• Trade-off between accuracy and feasibility:
Deep learning methods face a trade-off between achieving high generative accuracy
and maintaining computational feasibility. While high-accuracy models can improve
predictions, they are often resource-intensive and impractical for real-time or largescale applications.
• Limited training data:
Inverse problems often lack sufficient labeled data, increasing the risk of overfitting.
This limitation can reduce the ability of deep learning models to generalize effectively
to new scenarios.
• Integration with existing workflows:
Incorporating deep learning methods into existing workflows and simulation models
presents additional complexity. Ensuring seamless compatibility requires careful calibration and adaptation.
Despite these challenges, the potential of deep learning to advance complex fluid system
characterization and prediction is evident. By addressing these limitations, deep learning
techniques can unlock greater adaptability, accuracy, and scalability in model development
and application.
6
1.2 Outline
The rest of this dissertation is organized as follows:
Part 1: Model Characterization of Subsurface Fluid Flow Systems
Chapter 2 introduces the second generation of style-based Generative Adversarial Networks (StyleGANs) for parameterizing complex subsurface flow properties. This chapter
compares the model calibration capabilities and performance of StyleGANs with those of
convolutional Variational Autoencoders (VAEs) and traditional GAN architectures. The
chapter also explores the use of the Ensemble Smoother with Multiple Data Assimilation
(ES-MDA) for model calibration in single-phase and two-phase fluid flow examples.
Chapter 3 presents a novel hybrid approach that combines the pilot point method with a
deep learning model, specifically a U-Net trained using a perceptual loss function, to enable
conditional calibration of complex facies distributions. This integration maps sparse facies
values to detailed geological distributions, improving calibration accuracy and geological continuity. The chapter further investigates three pilot point selection methods and introduces
the use of soft points based on prior knowledge or analysis to enhance calibration precision
in geologically uncertain areas.
Part 2: Prediction and Optimization of Surface Geothermal Fluid Flow Systems
Chapter 4 focuses on a data-driven prediction and optimization framework as a costeffective and efficient alternative to traditional physics-based approaches. This chapter introduces an artificial neural network (ANN) model to predict power output and operational
costs by capturing the influence of control and disturbance variables. The model is subsequently employed to optimize net power production by adjusting the working fluid circulation
rate. The approach is validated using a thermodynamic flowsheet simulation model and applied to manage the impact of ambient temperature on an air-cooled binary cycle geothermal
power plant.
7
Conclusions and Future Directions
Chapter 5 concludes the dissertation by summarizing the key findings and contributions
from both parts. It reflects on the limitations of the presented methods, discusses their
broader implications, and highlights potential directions for future research in deep learning
applications for subsurface and surface fluid flow systems.
8
Chapter 2
Characterization of Complex Subsurface Flow Properties
2.1 Introduction
2.1.1 Inverse Modeling
Deterministic and stochastic inversion techniques have been instrumental in calibrating subsurface flow systems using dynamic flow response data [4, 29]. Deterministic methods aim
to minimize a data mismatch objective function by updating unknown model parameters
through implicit or explicit regularization constraints. These approaches are valued for their
computational efficiency and rapid convergence. Various regularization and parameterization strategies have been developed to constrain the solution space in underdetermined problems—situations where there are more unknowns than available measurements [30, 31, 32].
However, despite their efficiency and ease of implementation, deterministic methods are limited in their ability to thoroughly explore the solution space, especially in high-dimensional
parameter settings [33, 34]. This limitation underscores the importance of stochastic or
Bayesian methods, which use a probabilistic framework to describe both model parameters
and observational data. Probabilistic inversion methods offer the significant advantage of
incorporating and quantifying uncertainty, a crucial feature for generating reliable forecasts
and performing risk assessments.
Bayesian approaches enhance deterministic inversion frameworks by introducing system9
atic probabilistic analyses of uncertainty, characterizing model parameters through probability density functions and related statistical measures [35, 36, 37]. These methods effectively
combine prior knowledge—represented as initial uncertainties—with empirical data via likelihood functions. While linear-Gaussian problems allow for closed-form solutions, real-world
applications often deviate from these idealized conditions [38, 39, 40, 41, 42, 43, 44, 45].
In complex systems like subsurface flow models, characterized by significant spatial heterogeneity, alternative approaches become necessary. Computationally intensive methods such
as Markov chain Monte Carlo (MCMC) and more streamlined approaches like the ensemble
Kalman filter (EnKF) each present specific challenges. Amid this evolving landscape, Ensemble Smoother Multiple Data Assimilation (ES-MDA) has proven particularly effective by
sequentially integrating diverse observational datasets into computational models [46, 47, 48,
49, 50]. By avoiding the computational intensity associated with methods like MCMC and
EnKF, ES-MDA enables faster model updates within practical computational constraints.
Its success in managing the nonlinear and high-dimensional challenges of complex systems
has bolstered its reputation, especially in geoscientific applications such as reservoir characterization and subsurface flow parameterization [51, 52, 53].
2.1.2 Parameterization Methods
When calibrating subsurface flow models, the scarcity of data necessitates incorporating
prior information to effectively represent and constrain spatial patterns, such as geological continuity, in the distribution of high-dimensional flow properties. To address this,
various parameterization methods have been developed to integrate prior knowledge and
efficiently capture spatial complexity. Traditional methods balance model resolution with
data availability but are generally effective only for relatively simple spatial patterns. These
approaches often struggle to preserve intricate, non-Gaussian spatial patterns during calibration, a challenge particularly evident in complex geological formations like alluvial systems.
Established methods—including Level Set [9, 10, 54], Truncated Pluri-Gaussian [12, 11], Dis10
tance Transform [55], and Normal-Score Transform [13]—frequently rely on multi-Gaussian
assumptions. While they offer advancements over simple dimensionality reduction, these
methods sometimes lack the flexibility to capture the detailed spatial statistics required for
advanced geological models. The continued use of parameterization, particularly in combination with advanced inversion techniques like ES-MDA, remains crucial for accurately
representing geological complexity.
Multiple-Point Geostatistics (MPS) is a prominent parameterization method that diverges from traditional approaches by using detailed training images to guide the simulation
of subsurface features [56, 57]. Through iterative parameter adjustments, MPS ensures that
geological models align with both observational data and known geological information. This
method excels at capturing intricate geological patterns—such as the dendritic structures of
fluvial deposits—more accurately than conventional geostatistical models [58, 59, 60]. Although MPS does not directly address the parameterization challenge, it narrows down the
range of geological features, thereby enhancing parameterization efficacy and facilitating the
representation of complex spatial relationships in geological settings.
2.1.3 Deep Learning in Subsurface Model Characterization
Recent advances in machine learning have led to the development of deep learning techniques
specifically designed for parameterizing subsurface flow models, which are essential for capturing the intricate spatial characteristics of subsurface environments. Generative models
like autoencoders (AE) [22] and variational autoencoders (VAE) [61] have gained prominence
for their ability to reduce complex, non-Gaussian, spatially distributed rock properties into
lower-dimensional latent variables with minimal loss of detail [21]. Besides providing a compact representation of complex geological models, these latent variables typically follow a
Gaussian distribution, greatly facilitating model calibration tasks. The study by Jiang et
al. [32] underscored the challenges in calibrating subsurface flow models, particularly when
traditional methods struggle due to insufficient flow data. They suggested that VAEs and
11
other deep learning techniques can significantly enhance traditional parameterization methods like Principal Component Analysis (PCA). However, they also noted that VAEs may
have limitations in maintaining plausible data representations when dealing with complex
geological data. While VAEs are known for their ability to model complex distributions
due to their probabilistic framework, the assumption of a Gaussian distribution in the latent space can sometimes result in outputs that lack detailed geological features, appearing
blurred or overly smooth [62, 63]. This indicates a need to balance model complexity with
the fidelity of the resulting geological features.
In contrast, Generative Adversarial Networks (GAN) [64] have been recognized for generating high-fidelity images, which is advantageous for visualizing and calibrating geological
models [65]. However, GANs come with challenges such as the risk of mode collapse, leading
to limited diversity in outputs and potentially missing crucial geological variations [66]. The
complexity of GAN training, including issues like non-convergence, requires careful management to achieve stable and high-quality results. A recent study by [67] demonstrated
that GANs can outperform VAEs in generating accurate fluvial channel features similar to
training images, while VAEs tend to excel in data assimilation tasks, providing superior
reconstruction of channel features. Additionally, concerns exist regarding the linearity of
GAN generators, which can produce inaccurate inversion results, especially when used with
stochastic gradient-based inversion methods [28].
The introduction of Style-based Generative Adversarial Networks (StyleGAN) marks a
significant advancement in generative modeling [68, 69, 70, 71]. Building upon the foundational GAN framework, StyleGAN enhances image generation quality and offers greater
control than previous models like Deep Convolutional GAN (DCGAN) and VAE. Notably,
StyleGAN consistently produces images that reflect the complex geological patterns learned
during training, which is crucial for inversion tasks that require preserving prior geological
continuity models. Through successive iterations, the style-based generator in StyleGAN
allows for finer manipulation of image details, improving interpolation capabilities and mit12
igating the excessive smoothing often associated with VAEs. Additionally, StyleGAN’s sophisticated training approach includes regularization of the Jacobian matrix norm to ensure
smoother transitions between generated images [69, 72]. This method effectively addresses
the pronounced nonlinearity and intensive training demands typical of traditional GANs
[73].
In this chapter, the advanced deep learning model StyleGAN with the ES-MDA algorithm
is integrated for model calibration, demonstrating its potential to enhance the parameterization of complex geological models. We provide a comparative evaluation of several generative
models—VAE, DCGAN, and the second generation of StyleGAN (StyleGAN2)—in combination with ES-MDA to parameterize complex non-Gaussian geological formations. The
methodology section 2.2 outlines the principles of these generative models, while the experiment section 2.3 presents two numerical examples—a pumping test case and a two-phase flow
scenario—to assess their performance in model calibration using the ES-MDA framework.
2.2 Methodology
2.2.1 Model Parameterization
In subsurface modeling, parameterization plays a pivotal role in representing geological complexities and calibrating models with the available data. This process involves adjusting
high-resolution, three-dimensional input parameters M = {mi}
N
i=1 to match dynamic monitoring measurements while accounting for the heterogeneity and spatial correlations characteristic of subsurface flow properties. Model calibration seeks to refine these parameters
such that the simulated flow responses di = g(mi) align closely with observed data dobs,
often by minimizing a loss function L(dobs, g(mi)). Given the uncertainties and sparse data
typically encountered, a probabilistic framework based on Bayesian inference is commonly
employed. This framework updates the probability of model parameters conditioned on the
observed data as P(m|dobs) = P(dobs|m)P(m)
P(dobs)
, thereby integrating prior knowledge with the
13
likelihood of the observations under specified parameters.
This section leverages generative deep learning models—specifically VAE, GAN, and
StyleGAN—to generate realizations of model parameters mi
. These parameters are derived
using a generative model G, expressed as mi = G(zi
; ϕ), where zi represents latent variables,
and ϕ denotes the trainable parameters optimized during training on prior realizations M
before calibration. Once trained, the forward model g(·) is employed to simulate synthetic
data di = g(G(zi)). Calibration is then achieved by adjusting the latent variables zi to
improve the agreement between the simulated data Dsim (comprising the di) and the observed
data dobs. This optimization aims to maximize a similarity measure S(dobs, Dsim), such as a
correlation coefficient or a likelihood function.
The mapping g(G(zi)) represents a highly complex transformation, combining the nonlinearities inherent in both the generative model architecture and the governing flow equations.
As a result, the effectiveness of parameterization is closely tied to the calibration process’s
performance, underscoring the importance of robust and flexible approaches in capturing
geological complexity.
2.2.2 Data Assimilation
In the dynamic data integration process, generative models for geological parameter properties provide a robust framework for creating realistic and diverse parameter sets. However,
ensuring that these generated models can accurately reproduce nonlinear dynamic flow measurements (dobs) is a complex challenge. To address this, this chapter employs the ES-MDA,
a variant of the Ensemble Smoother (ES) [74], to iteratively update model parameter realizations using dynamic flow response measurements.
The ES-MDA approach updates an ensemble of prior model parameter realizations through
the following scheme:
ma
j = m
f
j + C
MD
C
DD + CD
−1
h
duc,j − d
f
j
i
(2.1)
1
where j = 1, 2, . . . , Na represents the iteration number in the data assimilation process,
and Na is the total number of iterations. Here:
• ma
j
: Updated model parameters at the j-th iteration.
• m
f
j
: Prior model parameters from the previous iteration.
• CMD: Cross-covariance between model parameters and predicted data, reflecting the
sensitivity of predictions to parameter changes.
• CDD: Auto-covariance matrix of predicted data, quantifying interrelationships among
predictions.
• duc,j : Perturbed observations for the j-th iteration.
• d
f
j
: Predicted data from the current model parameters.
• CD: Covariance of the observation noise.
Perturbed observations duc are generated by adding noise to the observed data, expressed
as:
duc = dobs +
√
αjC
1/2
D ϵd (2.2)
where ϵd is a vector of random noise drawn from an uncorrelated Gaussian distribution,
and αj
is an inflation coefficient for the j-th iteration [46]. This inflation prevents overfitting
to observations and maintains ensemble variability [49].
The ES-MDA workflow involves a two-step iterative procedure, detailed in Table 2.1:
1. Forecast Step: Predict system states and measurable outputs based on the current
parameter ensemble.
2. Update Step: Adjust parameters by assimilating perturbed observations.
15
Through iterative updates governed by inflation coefficients, ES-MDA enhances the integration of dynamic data with generative models, achieving a more accurate representation
of subsurface properties while maintaining the ensemble’s diversity.
Table 2.1: Workflow for data assimilation and model parameterization
Data Preparation and Generative Model Training
1. Construction of Training Dataset: Extract reference scenarios from highresolution training images to compile a comprehensive training and testing dataset.
2. Generative Model Training: Establish the architectural framework of the generative model and train the generative components G(z).
3. Acquisition of Observation Data: The observed data dobs come from field
measurements or by applying the forward model g(·) to a reference model of the property
map (parameters).
Parameterization via ES-MDA with Generative Model
Iterative Procedure (for i = 1 to Na, where Na typically ranges from 2 to 6):
1. Ensemble Execution: Conduct N forward simulations utilizing prior model instances mi=1,2,3,...,N with corresponding inputs zi=1,2,3,...,N to generate simulated data
di=1,2,3,...,N .
2. Covariance Calculation: Compute the cross-covariance matrix between the generative model parameters and the data CZD, as well as the data ensemble covariance
matrix CDD.
3. Observation Perturbation and Parameter Update: Introduce stochastic perturbations into the observation vector for each ensemble member duc = dobs+
√αjC
1/2
D ϵd,
followed by the updating of latent variables z according to Equation 2.2.2.
2.2.3 Deep Learning Models
2.2.3.1 Variational Autoencoder
The VAE excels at learning complex data distributions and generating new samples [61].
Built on the autoencoder framework, it comprises an encoder and a decoder. The encoder
maps input data to a reduced-dimensional latent space, while the decoder reconstructs data
from latent representations. Unlike traditional autoencoders, the VAE introduces a probabilistic perspective, where the encoder outputs parameters (typically Gaussian) defining a
distribution over the latent space. Latent vectors are sampled from this distribution, enabling flexibility in data generation. Figure 2.1(a) illustrates the VAE’s architecture. The
16
input data is encoded into a low-dimensional latent space representation and subsequently
decoded to reconstruct the original data, with an emphasis on minimizing information and
quality loss.
The VAE optimizes a loss function that balances reconstruction accuracy with regularization of the latent space:
LVAE(θ, ϕ; x) = −Eqϕ(z|x)
[log pθ(x|z)] + KL(qϕ(z|x)||p(z)) (2.3)
where x represents input data, z denotes latent variables, qϕ(z|x) is the encoder’s posterior distribution, pθ(x|z) is the decoder’s likelihood of reconstruction, and p(z) is the prior
over latent variables.
The KL divergence term,
KL(qϕ(z|x)||p(z)) = −
Z
qϕ(z|x) log p(z)
qϕ(z|x)
dz (2.4)
regularizes the latent space by minimizing the difference between the encoder’s posterior
distribution qϕ(z|x) and the prior p(z), typically Gaussian. This promotes an organized
latent space conducive to sampling and ensures the latent variables conform to the desired
distribution. The KL divergence is non-negative and achieves zero only when the two distributions are identical, fostering a structured latent representation ideal for generating new
data points.
2.2.3.2 Deep Convolutional Generative Adversarial Networks
Another widely used generative model examined for comparative analysis is the DCGAN. At
its core, DCGAN builds on the GAN framework, which comprises two adversarial networks:
a generator and a discriminator (Figure 2.1(b)). The generator aims to produce data that
mimics the distribution of the training set, while the discriminator seeks to distinguish
between real and synthetic data. This interaction creates a min-max optimization problem,
17
Figure 2.1: Structures of variational autoencoder and generative adversarial networks
expressed as:
min
G
max
D
Ex∼pdata(x)
[log D(x)] + Ez∼pz(z)
[log(1 − D(G(z))] (2.5)
where D(x) is the discriminator’s estimate of the probability that data x is real, G(z)
represents the generator’s output for noise input z, and pdata is the true data distribution.
DCGAN improves upon the original GAN architecture [75] by integrating convolutional
neural networks (CNNs) into both the generator and discriminator. Key innovations include
strided convolutions in the discriminator, fractional-strided convolutions in the generator,
batch normalization for training stability, and the removal of fully connected layers above
convolutional features (Figure 2.2(a)). These enhancements stabilize training and improve
output quality, making DCGAN a benchmark for generating detailed and coherent visual
content. Its ability to learn hierarchical data representations has established DCGAN as a
18
foundational model in generative research and applications, particularly for tasks like image
synthesis.
Figure 2.2: Structures of deep convolutional generative adversarial networks and StyleGAN2
2.2.3.3 Style-based Generative Adversarial Networks
StyleGAN represents a significant advancement in generative network architectures, surpassing traditional DCGAN frameworks with its ability to represent complex subsurface
heterogeneity using low-dimensional latent variables, offering improved fidelity and control
over image attributes. A key feature is the regularity of the latent space, ensuring that
proximity in latent space corresponds to proximity in the original spatial domain—an essential property for model calibration as models are iteratively updated to match observed
data. The StyleGAN architecture, illustrated in Figure 2.2, introduces intermediate latent
variables (w) via a mapping network, replacing the direct mapping of latent variables (z) in
DCGAN. This design allows independent control of stylistic features across scales through
adaptive instance normalization (AdaIN) layers, significantly enhancing image quality and
disentanglement [68]. Building on this, StyleGAN2 [69] incorporates path length regularization to improve consistency in latent space transformations, further refining image fidelity
and resolution. These advancements make StyleGAN2 particularly suited for generating
19
precise synthetic data with high accuracy and detail, which is crucial for modeling tasks
involving complex features.
StyleGAN2’s style-based generator employs AdaIN layers at each convolutional layer to
modulate feature maps using style parameters derived from the latent space. This modulation aligns feature maps with desired style attributes across scales, enhancing image detail
and coherence. The process begins with weight modulation:
w
′
i,j,k = si
· wi,j,k (2.6)
where w and w
′
represent the original and modulated weights, respectively, and si
is
the style-dependent scale factor. Demodulation follows to adjust the standard deviation of
feature maps:
σj =
sX
i,k
w′2
i,j,k (2.7)
and weights are normalized for stability:
w
′′
i,j,k =
w
′
q
i,j,k
P
i,k w′2
i,j,k + ϵ
(2.8)
with ϵ preventing division by zero. This advanced modulation-normalization scheme
enables StyleGAN2 to generate high-quality images with precise stylistic control, essential
for detailed parameterization tasks.
Non-linearity in traditional GANs can lead to inaccuracies during the inversion process
[34, 28]. The smoothness of the latent-to-output mapping, quantified by Perceptual Path
Length (PPL) [69] using the LPIPS metric [76], reflects model consistency. PPL measures
the perceptual distance between images generated from neighboring latent points, defined
as:
PPL = Ez0,z1,t [d (G(slerp(z0, z1, t)), G(slerp(z0, z1, t + ϵ)))] (2.9)
20
where G is the generator, slerp is spherical linear interpolation, t the interpolation ratio, ϵ
a small perturbation, and d the perceptual distance. A lower PPL signifies smoother transitions and reduced nonlinearity, beneficial for parameterization tasks by improving precision
and simplifying generation.
StyleGAN2 addresses this with path length regularization, stabilizing transformations by
maintaining vector lengths in the latent space. The regularization term is expressed as:
Ew,y∼N (0,I)
h
∥J
⊤
wy∥2 − a
2
i
(2.10)
where Jw is the Jacobian matrix, a a dynamic constant, and y random image samples with
pixel values following a normal distribution. This reduces nonlinearity, ensuring proportional
and interpretable changes in generated images [69]. To balance computational efficiency,
StyleGAN2 applies this ”lazy regularization” periodically, optimizing the process for complex
tasks.
In subsurface property parameterization for model calibration, StyleGAN2’s enhancements, particularly path length regularization, provide key advantages. This feature ensures
a predictable relationship between latent space changes and resulting images, crucial for iterative adjustments of latent variables based on flow response feedback to match observed data.
The consistent mapping from latent to image space enables the generation of high-quality,
diverse images, making StyleGAN2 well-suited for addressing model calibration challenges.
Its improved latent-space regularity, predictability, and ability to produce robust spatial domain images underscore its effectiveness in subsurface flow modeling and parameterization
tasks.
2.3 Numeric Examples
In this section, the numerical experiments are presented to evaluate the performance of
generative models in model calibration problems involving complex (non-Gaussian) facies
2
distribution. Our initial evaluation focuses on image quality, dimensionality reduction, and
interpolation property of the model. Subsequently, Section 2.3.2 is dedicated to pumping test
examples, where we update the spatial distribution of hydraulic rock properties, specifically
permeability, to match observed pressure measurements taken from an array of monitoring
wells. Section 2.3.3 explores a more complex two-phase flow example to demonstrate the
parameterization application of these methods. Both examples involve flow simulations that
are performed using the MATLAB MRST package [77].
2.3.1 Generative Models for Channelized Fluid System Distributions
Figure 2.3: The original 250x250 fluvial channel training image (left) and exemplar 64x64
image samples (right) utilized for model training and testing.
The parameterization performance of StyleGAN2 is evaluated using fluvial facies and
non-Gaussian parameter examples. The training image, sourced from [78, 79], is a 250x250
grid (Figure 2.3). Using the SNESIM algorithm [78], 3000 realizations of size 64x64 were
generated, with 2000 samples for training and 1000 for testing. Models including the VAE,
DCGAN, and StyleGAN (using StyleGAN2 structure) were trained using the PyTorch library
[80].
Extensive hyperparameter tuning was performed to optimize each model. For the VAE, a
32-dimensional latent space balanced performance and dimensionality. StyleGAN utilized a
4-dimensional latent vector z and a 32-dimensional intermediate w space, generating 64x64
22
Figure 2.4: Image generation using (a) DCGAN, (b) StyleGAN, and image reconstruction
with (c) VAE (latent variables dimension = 16) and (d) VAE (latent variable dimension =
32).
images through upsampling modulated by style vectors, as described in Section 2.2.3.3. Similarly, DCGAN transformed a 4-dimensional latent vector into 64x64 images via transposed
convolutional layers, matching the parameterization structure of StyleGAN.
The generative models in this study share a common structural design and hyperparameter selection framework. All three models incorporate a downsampling mechanism—either
an encoder or discriminator—to reduce 2D images into latent variables or scores from a postsoftmax layer. This is paired with a decoder structure that transforms Gaussian-distributed
latent variables back into images resembling the training dataset. The VAE model features a
four-layer convolutional encoder, where the number of filters increases from 8 to 64 while the
spatial dimensions progressively decrease. Each layer employs a 4x4 kernel with a stride of
2 and padding of 1, along with batch normalization and LeakyReLU activation for effective
pattern detection. The encoder generates a 64x4x4 feature map, which is further reduced into
23
Figure 2.5: Latent space sampling and interpolation in VAE, DCGAN, and StyleGAN
two 32-dimensional vectors representing the mean and log-variance of the latent space. The
decoder reconstructs the original image by upsampling from the latent vector through four
transposed convolutional layers, decreasing the filter count from 64 to 1, and culminating in
a sigmoid-activated layer that restores the input’s original dimensions. Batch normalization
and LeakyReLU activation are applied throughout, except in the final layer. The specific
structure of the decoder is detailed in Table 2.2. This architecture enables the VAE to balance the trade-off between effective compression and faithful reconstruction, ensuring both
detail retention and generative flexibility in the latent space. The GAN model follows a
24
similar architectural framework, while the detailed structure of the StyleGAN2 is described
in Table 2.3.
The selection of latent variable dimensionality for the VAE is guided by its reconstruction
performance on testing data. The relationship between reconstruction efficiency and latent
variable dimensionality is illustrated in Figure 2.6. For the StyleGAN2, the dimensionality
of latent variables is determined by the quality of generated images, evaluated using Fr´echet
Inception Distance (FID) and Kernel Inception Distance (KID) metrics. These metrics assess
the quality, diversity, and distributional similarity of generated images relative to real data.
The correlation between the latent variable dimension z, the number of training images
(kimg), and the performance of the model as measured by FID and KID scores is depicted
in Figure 2.7.
Figure 2.6: VAE reconstruction ability on the testing dataset
Figure 2.7: KID and FID of the stylegan model on the training dataset.
Figure 2.4 compares outputs from random latent inputs. Both DCGAN and StyleGAN
produce coherent patterns with low-dimensional latent spaces. While the VAE enhances
25
Table 2.2: Detailed VAE decoder structure
Layer Type KSize In Channel Out Channel Out Size
1 Fully Connected (Linear) - LD 1024 Reshape to (64, 4, 4)
2 ConvTranspose2d 4 64 32 8x8
3 Batch Normalization - - - 32 Ch
4 LeakyReLU - - - -
5 ConvTranspose2d 4 32 16 16x16
6 Batch Normalization - - - 16 Ch
7 LeakyReLU - - - -
8 ConvTranspose2d 4 16 8 32x32
9 Batch Normalization - - - 8 Ch
10 LeakyReLU - - - -
11 ConvTranspose2d 4 8 1 64x64
12 Sigmoid - - - -
Figure 2.8: Analysis of PPL and interpolation effects across generative models
image sharpness with a larger 32-dimensional latent space, its random samples often fail to
generate clear fluvial channels (Figure 2.5(a)). Interpolation performance, shown in Figure
26
2.5 (b)-(d), highlights StyleGAN’s smooth transitions, outperforming DCGAN, which introduces artifacts and exhibits less consistent interpolations. The VAE shows more stochastic
transitions due to its probabilistic design.
Figure 2.8(a) correlates Perceptual Path Length (PPL) scores with image quality, consistent with [73]. Lower PPL scores in StyleGAN, attributed to path length regularization,
result in improved image quality and consistency. StyleGAN’s output also aligns better with
the training data’s channel density distribution, confirming its superior ability to capture
and reproduce subsurface heterogeneity.The bar graph (b) and the overlaid histogram illustrate the effects of linear and spherical interpolations on PPL scores; the channel density
distribution (c) comparison signifies a closer approximation to the training data by StyleGAN.
2.3.2 Pumping Test Example
Figure 2.9: Visualization of permeability field for four reference cases
To evaluate the calibration performance of parameterization methods, we use a groundwater pumping test [32]. Pressure measurements from monitoring wells around a central
extraction well are used to infer spatial variability in hydraulic conductivity, represented
logarithmically with channel facies at 0.6 ln(cm/s) and matrix facies at −2.4 ln(cm/s). Four
scenarios, varying in channel complexity and monitoring well configurations (nine wells for
Scenarios 1 and 3, sixteen for Scenarios 2 and 4), are shown in Figure 2.9. Yellow pillars
mark the location where water is pumped in, and green pillars indicate monitoring wells.
Cases 1 and 2 represent simpler channel configurations with nine monitoring wells, while
27
Cases 3 and 4 depict more complex structures with sixteen wells. The constant pressure
boundaries and no-flow zones are duly marked, encapsulating the simulated complex fluid
system.
The ES-MDA method updates generative model latent variables over four iterations
with inflation factors [9.33, 7, 4, 2] [46]. Models include VAE (latent dimensions 16 and
32), DCGAN, and StyleGAN. For DCGAN and StyleGAN, latent variables are sampled
from N (0, σ), while VAE latent variables are encoded from the test dataset for consistent
initialization. Results are presented in Figures 2.10 and 2.11.
The ES-MDA results show that posterior means effectively capture channel features,
particularly in scenarios with 16 monitoring wells. In complex cases with fewer wells (e.g.,
Scenario 3), StyleGAN achieves superior posterior reconstruction and better uncertainty
reduction compared to other models.
Table 2.4 compares model performance metrics across 100 ensemble members. VAE
achieves better flow data matching (lower RMSE/MAPE), while StyleGAN produces higherfidelity spatial maps (higher SSIM), attributed to its advanced architecture and stronger
nonlinearity. The ES-MDA scheme relies on covariance-based updates, which can limit datamatching accuracy, especially with nonlinear parameter-to-response mappings. As shown in
Figure 2.11, StyleGAN’s outputs closely align with expected patterns, outperforming other
models in SSIM scores.
Figure 2.12 demonstrates model robustness under Gaussian perturbations (σ = 0.2) in latent variables. StyleGAN outperforms VAE models, maintaining pattern fidelity with higher
SSIM scores. Its advanced architecture and path length regularization enable superior performance, surpassing both VAE and DCGAN in generating accurate and consistent spatial
maps.
28
Figure 2.10: Ensemble mean and variance of prior and posterior distributions for generative
models in groundwater parameterization
2.3.3 Two-phase Fluid System Example
We conducted a two-phase flow simulation as a complex model calibration case study. The
simulation grid consisted of 64×64 cells in the x and y dimensions, with one z-layer covering
a 640x640 meter area with a 10-meter thickness. Porosity was uniformly set to 0.2, and permeability varied between 1000 millidarcies in channels and 5 millidarcies elsewhere. Wetting
29
Figure 2.11: Calibrated sample parameter realizations for cases with 16 monitoring wells
using each of the four generative models.
phase fluid was injected at 500 cubic meters per day, with production wells regulated at 150
bars of bottom-hole pressure. Over 720 days, extraction data for wetting and non-wetting
phases were recorded bi-monthly.
Two scenarios were considered, mirroring the pumping test example, with one injection
well and three extraction wells placed in the channel area. The first 8 months of dynamic data
were used for calibration. Parameterization methods included VAE, DCGAN, StyleGAN,
30
Figure 2.12: Comparative visualization of generated images with perturbed latent variables
across models
and a pixel-based approach for comparison. Figure 2.13 shows the configurations and prior
and updated mean and variance distributions. Significant differences in updated means
were observed among models, with StyleGAN achieving the best reconstruction and the
pixel-based approach leading to disintegrated channel features, emphasizing the need for
parameterization.
Performance metrics (Table 2.5) and distributions (Figure 2.14) indicate similar model
performance in Case 1, with VAE showing the least variability and pixel-based updates
the most. StyleGAN excelled in reproducing well responses (Figure 2.15), with calibrated
realizations and means closely matching observations.
In Case 2, featuring more complex channel patterns, StyleGAN achieved the best agreement with the reference map (Figure 2.16), outperforming VAE and DCGAN, while pixelbased updates showed poor channel continuity. Flow response matches are shown in Figure
2.17, highlighting the influence of ES-MDA’s linear-Gaussian assumptions on data reproduction quality.
Figure 2.18 illustrates prior and updated latent variable distributions, showing adjustments during data assimilation and their impact on calibration.
31
Figure 2.13: Permeability configurations and ensemble results for two-phase flow model
parametrization using generative models
2.4 Summary
Calibration of subsurface flow models requires geological constraints to ensure plausible
solutions consistent with the geologic continuity of the field. However, constraining complex non-Gaussian property distributions during calibration is challenging. Recent advances
in generative deep learning models offer promising methods for learning and replicating
spatial patterns in complex images, making them suitable for subsurface flow model parameterization. Various generative architectures have been developed for low-dimensional
representation and high-quality image synthesis, each with distinct properties that affect
parameterization performance. A key feature of these models is latent space regularity,
which ensures smooth, organized representations, enabling controllability, continuity, and
32
Figure 2.14: Performance metrics of the generative models across the two cases in two-phase
flow
Figure 2.15: Well response data match using StyleGAN parameterization for non-wetting
and wetting phases in case 1
meaningful interpolation between points. Regularity is critical for effective and interpretable
parameterization, especially in calibration methods requiring automatic perturbations to
latent variables.
Achieving latent space regularity involves techniques such as regularization during train33
Figure 2.16: Initial and calibrated parameter realizations for the two well configurations
ing, architectural design, and optimization strategies. This is particularly important for
calibration methods relying on smoothness assumptions, such as cross-covariance or gradientbased updates. However, the added complexity and nonlinearity of GAN-based architectures,
while improving representation performance, can result in disproportionate changes in data
with small latent space perturbations. This makes such models less suitable for calibration
approaches requiring gradual and predictable updates. Consequently, the choice of generative architecture must balance representation quality with the compatibility of latent space
properties for model calibration tasks.
34
Figure 2.17: Well response data match in case 2 using VAE, DCGAN, and StyleGAN models
Figure 2.18: Distributions of latent variables for VAE, DCGAN, and StyleGAN on case 2
This chapter explored the use of StyleGAN, an advanced generative model, for parameterization and calibration of subsurface flow models. We evaluated its ability to generate high-fidelity images using low-dimensional latent variables while preserving geological
continuity in models with complex, non-Gaussian property distributions. Through several
calibration examples using the ES-MDA algorithm, StyleGAN’s performance was compared
35
with traditional generative models, including DCGAN and VAE. StyleGAN demonstrated
significant advantages in reconstruction quality, control over generated outputs, and training stability due to its disentangled latent space and enhanced architectural design. Unlike
DCGAN’s latent space, where small perturbations can cause significant output variations,
StyleGAN’s disentangled latent space enables independent control of features, improving
robustness and reconstruction accuracy. These properties highlight StyleGAN’s superiority
in parameterizing and calibrating subsurface flow models with diverse spatial patterns.
The increased regularity of StyleGAN’s latent space not only enhances ES-MDA-based
calibration but is also expected to benefit gradient-based calibration algorithms. Effective
application of GAN-based models for subsurface flow calibration requires balancing parameterization complexity and fidelity with latent space regularity and the properties of the
calibration algorithm. Further research is needed to better understand these interactions
and develop application-specific generative models for solving complex subsurface flow model
calibration problems.
36
Table 2.3: Detailed strcuture of StyleGAN generator
Step/Component Operation/Layer Type Input Size Output Size Notes
Overall Network Progression
1 Initial Input [batch size, 512, 4, 4] [batch size, 512, 4, 4] Constant input tensor
2 Style Block 1 [batch size, 512, 4, 4] [batch size, 256, 8, 8] Upsamples to double spatial dimensions
3 Style Block 2 [batch size, 256, 8, 8] [batch size, 128, 16, 16] Further upsampling
4 Style Block 3 [batch size, 128, 16, 16] [batch size, 64, 32, 32] Continues the pattern
5 Style Block 4 [batch size, 64, 32, 32] [batch size, 32, 64, 64] Final upsampling to target size
6 ToRGB Layer [batch size, 32, 64, 64] [batch size, 1, 64, 64] Converts to a 1-channel image
Details of Style Block 1
Latent Vec. to Style Vec. (to style) Equalized Linear Transformation [b s, d l] [b s, 512] Transforms w into style s. Style Modulation - - - Modulates conv. weights.
Convolution (conv) Weight Modulated Convolution with Upsampling [b s, 512, 4, 4] [b s, 256, 8, 8] Upsamples and changes channels.
Noise Addition Scaling and Adding Noise [b s, 256, 8, 8] Same Optional noise scaling.
Bias Addition Add Bias [b s, 256, 8, 8] Same Adds bias.
Activation (LeakyReLU) Apply Activation [b s, 256, 8, 8] Same Applies LeakyReLU.
37
Table 2.4: Pumping test result
ES-MDA results:
Metrics RMSE↓ MAPE(%)↓ SSIM↑
Case1 MEAN MIN MAX MEAN MIN MAX MEAN MIN MAX
VAE(16) 0.0297 0.0059 0.0722 3.2976 0.8473 4.8868 0.4809 0.1958 0.6751
VAE(32) 0.0203 0.0041 0.0673 2.0174 1.1586 3.3366 0.4397 0.2006 0.6903
DCGAN 0.0418 0.0264 0.0826 4.1539 3.6301 5.1573 0.201 0.148 0.2922
StyleGAN 0.0283 0.0023 0.0917 3.0052 0.4374 6.6441 0.6004 0.226 0.9164
Case2
VAE(16) 0.0258 0.0045 0.0584 2.2697 0.6146 3.9709 0.5458 0.3523 0.7385
VAE(32) 0.0186 0.0049 0.0499 1.8319 0.4917 4.8739 0.5089 0.2877 0.6903
DCGAN 0.0567 0.0205 0.1005 5.3934 3.5249 9.3862 0.2527 0.0515 0.3644
StyleGAN 0.0384 0.0031 0.092 3.4691 0.4895 7.8887 0.501 0.1923 0.8087
Case3
VAE(16) 0.0341 0.0107 0.0622 2.6511 1.8222 3.8953 0.3135 0.0674 0.4985
VAE(32) 0.0229 0.0085 0.054 2.4386 1.3177 3.4164 0.2897 0.0883 0.5473
DCGAN 0.0488 0.0268 0.0845 3.9026 2.8578 5.5928 0.2562 0.0637 0.3944
StyleGAN 0.025 0.0051 0.0537 2.0633 0.4841 3.6247 0.5504 0.2975 0.883
Case4
VAE(16) 0.0352 0.0183 0.0707 3.9987 2.87 7.2571 0.2913 0.1612 0.4254
VAE(32) 0.0244 0.0097 0.0552 2.3727 1.2573 4.2246 0.2951 0.0788 0.4559
DCGAN 0.033 0.0175 0.0652 3.1914 2.0542 6.3557 0.164 -0.0037 0.4533
StyleGAN 0.035 0.0118 0.0764 3.1485 1.2747 6.1561 0.4456 0.141 0.7909
Table 2.5: Two-phase flow result
Two-phase flow example result
Metric RMSE↓ SSIM↑
Case1 Mean Min Max Mean Min Max
Pixel 0.0517 0.0123 0.0997 0.1174 0.0644 0.1848
VAE 0.0218 0.0045 0.353 0.6618 0.5373 0.7745
DCGAN 0.0374 0.0181 0.0553 0.4836 0.409 0.5228
StyleGAN 0.0342 0.0022 0.2672 0.7477 0.2422 0.9515
Case2
Pixel 0.0554 0.0151 0.1516 0.1147 0.0401 0.1851
VAE 0.0955 0.0124 0.4205 0.3172 0.1047 0.4735
DCGAN 0.1228 0.0144 0.5203 0.2351 0.0781 0.4376
StyleGAN 0.0445 0.0027 0.4183 0.7594 0.3752 0.9517
38
Chapter 3
Deep Learning-Based Pilot Point Method for Parameterization
3.1 Pilot Point Method
The pilot point method, introduced by de Marsily et al. [81], is a versatile geostatistical
approach for parameterizing subsurface flow inverse problems. This method identifies specific
grid cells, known as pilot points, to serve as key reference locations for interpolation. These
points, treated as hard data, guide the interpolation process and reduce reliance on external
statistical estimates for unmeasured regions. By strategically selecting pilot points, the
method ensures that property values are reconstructed with spatial continuity, often using
variogram models, and are consistent with actual measurements [82, 83].
A notable advantage of the pilot point method is its ability to condition interpolations on
both the pilot points and broader geological continuity models, such as variograms or covariance functions. This dual conditioning enables a more accurate representation of geological
heterogeneity [84, 85]. Furthermore, the method integrates seamlessly with established geostatistical techniques like Kriging and co-Kriging, which are widely used to estimate reservoir
properties from limited data samples [82, 86]. Beyond traditional applications, pilot points
are also effective in advanced calibration techniques, such as multiple-point statistical (MPS)
models, which are particularly useful for addressing complex, non-Gaussian fields [56, 87].
By focusing on critical parameters, this method enhances model efficiency and accuracy,
reducing the number of parameters requiring direct estimation and optimizing those most
39
significant to the model’s performance.
The strategic selection and placement of pilot points are critical for successful model
calibration. The number of pilot points must balance two competing objectives: incorporating sufficient flow data to inform the model while maintaining consistency with the prior
parameterization. Using too many points can lead to over-parameterization and computational inefficiencies, while too few may limit the flow data’s influence, reducing calibration
effectiveness. The optimal number of pilot points depends on the complexity of the model,
the available data, and the specific application requirements.
Placement strategies also vary widely, ranging from uniform distributions to more advanced, strategically designed approaches. De Marsily et al. [81] recommended a ”moreor-less uniform” distribution, focusing on zones of high sensitivity or significant hydraulic
property contrasts to improve calibration performance. For Gaussian-distributed fields,
pseudo-regular grids with specific spacing are often used to balance spatial coverage and
computational simplicity.
Modern approaches to pilot point placement emphasize leveraging sensitivity and covariance information to improve calibration efficiency. High-sensitivity regions are prioritized to
reduce data mismatches and better capture key features of the system. For example, Tonkin
and Doherty [88] proposed constructing a highly parameterized base model and using eigenvector decomposition of the normal matrix to define a subspace for calibration, effectively
reducing the dimensionality of the parameter space. Similarly, Yang et al. [89] used Singular
Value Decomposition to identify high-intensity regions for pilot point placement, selecting
locations based on the leading singular vectors of the sensitivity matrix. Further innovations
include adaptive methods such as traveling pilot points, introduced by Khambhammettu et
al. [90]. These points move dynamically within the model domain to improve efficiency and
accuracy, particularly in categorical inverse problems. This approach allows the model to
adjust pilot point locations as needed, enhancing the representation of geological features
and improving the overall calibration process.
40
Strategic placement generally outperforms uniform distribution by maximizing the information yield from the selected pilot points, reducing uncertainty, and improving the quality
of the solution. Methods that incorporate sensitivity analysis, covariance information, or
adaptive placement techniques ensure that pilot points are optimally positioned to capture
the most critical variations in the model domain. These strategies enable more reliable and
accurate calibration, particularly for complex subsurface systems with heterogeneous and
non-Gaussian properties.
In this chapter, we present a novel geological parameterization method that integrates the
strengths of the U-Net model with the strategic use of pilot points. The U-Net architecture,
originally developed for biomedical image segmentation [91], is renowned for its flexibility
and effectiveness. Its distinctive U-shaped structure, combining contracting and expanding
paths, excels at processing high-resolution data and capturing intricate spatial relationships.
By incorporating localized, hard data points into the broader context of learned geological
patterns, this approach enhances the model’s capability to address inverse problems.
We begin by exploring a modified U-Net architecture designed to integrate positional
and locational information from hard data, enabling the generation of complex geological
property maps. To ensure the preservation of geological patterns, we assess the sensitivity
of the input data and evaluate the impact of different loss functions. The model is then
applied to calibration tasks, demonstrating its effectiveness through numerical examples.
Specifically, we combine the trained U-Net model with the ensemble smoother with multiple
data assimilation (ES-MDA) technique [46].
Three pilot point selection strategies are compared, emphasizing their role in guiding the
U-Net model to reconstruct facies realizations that adhere to hard data constraints while
maintaining spatial continuity within the geological model. This innovative approach leverages the U-Net’s ability to learn complex spatial patterns while integrating specific, localized
data points. The result is a more flexible and robust method for conditional geological parameterization, offering significant advantages for solving inverse problems.
41
3.2 Methodology
The characterization workflow integrating the U-Net model with the pilot point method is
illustrated in Figure 3.1. Hard data and pilot points are input into the U-Net model, which
generates permeability maps. These maps are iteratively evaluated and refined by adjusting
the pilot point values through a numerical simulator and ES-MDA iterations. This process
enhances model accuracy and ensures alignment with observed data.
Figure 3.1: Workflow of model calibration using U-Net and pilot points method
Sparse field data often lack sufficient detail for the U-Net model to fully capture geological
patterns. To address this limitation, pilot points are strategically positioned across the
field as auxiliary variables, enabling localized adjustments to geological properties through
dynamic data assimilation. The U-Net model combines hard data and pilot points into a
domain map, generating an interpolated facies map that is subsequently used in a numerical
simulation model to predict dynamic flow behaviors observed in the field. Through iterative
calibration using the ES-MDA algorithm [46, 49], the pilot points are adjusted to minimize
discrepancies between simulated and observed results. This iterative refinement ensures the
42
facies model increasingly aligns with empirical data, accurately capturing the underlying
geological features and dynamics.
3.2.1 Deep Learning Model - U-Net
The architecture of the U-Net model, depicted in Figure 3.2, adheres closely to the original framework proposed by Ronneberger et al. [91]. It utilizes convolutional layers, max
pooling, upsampling, and skip connections to produce a comprehensive facies distribution
map from scattered grid-block measurements. This design is tailored to handle inputs that
match the dimensions of the reference facies realizations, effectively accommodating varying
combinations of hard data in terms of both location and value.
During training, hard data points are strategically extracted from the reference facies
image using a mask derived from the training data. While these points may appear randomly
distributed within subregions, their selection ensures relevance to the geological scenario,
providing meaningful inputs for training. This approach enhances the model’s ability to
generalize effectively across diverse geological settings.
Figure 3.2: U-Net architecture for geostatistical simulation using pilot points and a training
image.
In the U-Net model’s encoding path, input data passes through convolutional layers that
refine and enhance feature extraction. Each convolution is followed by batch normalization
43
and activation via the Rectified Linear Unit (ReLU), progressively improving the detection
of subtle geological differences. After each convolution, max pooling reduces the spatial
dimensions while increasing the number of feature maps, enabling the model to capture
detailed representations at multiple scales. This process culminates in a bottleneck layer that
performs convolutions without further downsampling, retaining the most critical features for
transitioning to the decoding path.
In the decoding path, the bottleneck features undergo upsampling to restore spatial
dimensions while reducing the number of feature maps. Each upsampling step includes convolutional layers with batch normalization and ReLU activations to maintain data integrity.
A defining feature of the decoding path is the use of skip connections, which link encoding and decoding layers at corresponding levels. These connections reintroduce finer spatial
details lost during downsampling, ensuring accurate and detailed facies distribution maps.
3.2.2 Perceptual Loss Function
During training, the U-Net model utilizes inputs with hard data to replicate the geological
patterns from which the data originates. The primary loss function applied is the Mean
Squared Error (MSE), defined as:
MSE = 1
n
Xn
i=1
(yi − yˆi)
2
(3.1)
where yi
is the true value, ˆyi
is the predicted value, and n is the total number of pixels
in the training batch. The model focuses on filling vacant regions in the input image while
excluding hard data areas. The U-Net architecture excels in capturing local and global contextual information, enhancing its ability to predict and detect geological patterns. However,
applying MSE globally often results in blurry outputs, especially when hard data is sparse.
This occurs because MSE minimizes pixel-level differences uniformly, averaging out details
and diminishing high-frequency components like edges and textures, which are crucial for
44
Figure 3.3: Schematic representation of a neural network incorporating perceptual loss for
style similarity assessment.
clarity. Furthermore, MSE’s sensitivity to noise and pixel intensity variations exacerbates
blurring, as it smooths essential textures to reduce overall error [62, 63].
To address these limitations, perceptual loss, also known as style loss, is introduced
[92]. This loss compares high-level features extracted from pre-trained convolutional neural
networks (CNNs), focusing on textures, patterns, and overall style rather than individual
pixel values [93, 94]. Perceptual loss quantifies style similarity by evaluating feature groupings across different scales, orientations, and appearances, ensuring that generated images
replicate the style of the target image effectively.
To further enhance feature representation in geological classifications, a discriminator
(Figure 3.3) is trained during the initial phase. This training involves a minimax game:
max
D
Ex∼pdata(x)
[log D(x)] + Exmask∼pxmask (xmask)
[log(1 − D(G(xmask)))] (3.2)
where D(x) estimates the likelihood of real data x being authentic, and G(xmask) represents the U-Net’s output for a masked input. The discriminator differentiates between the
U-Net’s predictions and target images. After initial training, the discriminator’s embedding
is incorporated into the loss function to enhance prediction accuracy and prevent overfitting:
L = ∥y − yˆ∥
2
2 + λ · Lfeat(y, yˆ), (3.3)
45
where Lfeat(y, yˆ) introduces perceptual loss based on high-level features, and λ is a hyperparameter controlling the scale of the perceptual component. This augmented loss function
improves predictions by capturing both pixel-level accuracy and feature-level coherence, as
shown in Figure 3.3.
3.2.3 Pilot Point Selection
This chapter evaluates three methods for pilot point selection: fixed position, random selection, and sensitivity mapping. In the fixed position and random selection methods, the
study area is divided into equal subzones, with pilot points placed either at the center or
randomly within each subzone, respectively. Sensitivity mapping, the third method, employs cross-correlation analysis between input and output variables to produce a sensitivity
map matching the input map’s dimensions. This map highlights areas of highest sensitivity,
where pilot points are placed to maximize their influence on model output. As the sensitivity
map is derived during the initial data assimilation step, this approach incurs no additional
computational cost. By strategically positioning pilot points in high-sensitivity regions, this
method ensures the model captures critical system features effectively.
In sensitivity analysis, the relationship between the input matrix M and the observed
output matrix dobs is quantified using a covariance matrix. Both M and dobs are centered
around their means, and the cross-covariance is computed as:
Cov(M, dobs) = M′T d
′
obs
n − 1
(3.4)
where M′ = M − M¯ and d
′
obs = dobs − ¯dobs, with M¯ and ¯dobs representing the means of
M and dobs, respectively.
Weights W are then assigned to reflect the significance or variability of each output
feature, calculated as:
46
Wi =
Var (dobs,i)
Pm
j=1 Var (dobs,j )
(3.5)
These weights prioritize features with higher variability, indicating a stronger impact
on sensitivity analysis. Weights can also be manually adjusted based on expert knowledge
or analytical priorities. To maintain consistency, the weights are normalized so their sum
equals one. The weighted covariance matrix is then computed, focusing on the magnitude
of relationships, independent of direction:
Sensitivity = |Cov(M, dobs)| · W (3.6)
The sensitivity scores are aggregated for each feature in M, resulting in a comprehensive
sensitivity score. These scores form the basis for creating a sensitivity map, which visually
identifies the most influential input features. This map is a valuable tool for guiding pilot
point placement, ensuring efficient and effective calibration by targeting regions with the
greatest sensitivity.
3.3 Numeric Examples
This section 3.3.1 is organized similarly to the previous chapter and presents numerical
experiments evaluating the performance of the pilot points method combined with the UNet model for calibrating models with complex non-Gaussian facies distributions. The initial
evaluation highlights model performance and sensitivity to the number of pilot points.
Section 3.3.2 explores pumping test examples, where the spatial distribution of hydraulic
rock properties, such as permeability, is updated to match observed pressure measurements
from monitoring wells. Section 3.3.3 extends the analysis to a more complex two-phase flow
scenario, showcasing the parameterization capabilities of the methods under investigation.
Both examples utilize flow simulations conducted with the MATLAB MRST package [77].
47
3.3.1 Parameterization Results
The U-Net model’s parameterization performance was evaluated using fluvial facies with
complex non-Gaussian distributions. A dataset of 6,000 realizations (64×64) was generated
from a 250×250 training image by Zahner et al. [79], with 5,000 used for training and 1,000
for testing. Channel regions were encoded as 1 and non-channels as 0, with Gaussian noise
σ ∼ N (0, 0.05) added during training. Data points (9, 16, 25, 36, 49) were masked, with at
least four points placed in channel regions to ensure adequate representation. Masks were
assigned a value of 0.5 to indicate classification uncertainty.
Figure 3.4 compares scenarios with varying input data and loss functions. Prediction
quality improves with more input data, but models using only L2 loss struggle with edges
and patterns, especially with sparse inputs. Adding perceptual loss enhances accuracy and
reduces variability, as shown in results for 25 pilot points.
Sensitivity analysis (Figure 3.5) on 50 realizations with 20 input masks (1,000 inferences) shows improved metrics (MSE, Variance, SSIM) with more pilot points. Perceptual
loss increases MSE and variance slightly but significantly improves SSIM, reflecting better
reconstruction quality. The optimal balance for model calibration occurs around 25 pilot
points, highlighting the role of pilot point density and perceptual loss in achieving highquality, diverse realizations in ESMDA.
3.3.2 Pumping Test Example
To evaluate the pilot point method, we applied it to a groundwater pumping test setup
from Jiang et al. [32], featuring an extraction well at the domain center and nine monitoring wells measuring pressure declines to assess hydraulic conductivity. The domain has
impermeable top and bottom boundaries, with lateral pressure heads of 20 and 10 meters.
Hydraulic conductivity is modeled logarithmically, with channel facies at 0.6 and matrix
facies at −2.4 ln(cm/s). Figure 3.6 shows two scenarios with increasing channel complexity,
48
including 3D layouts, permeability maps, and pressure fields. Injection wells (red crosses)
and monitoring wells (green circles, M1–M9) are highlighted, with true pressure maps displaying post-test distributions recorded only at monitoring wells.
The ES-MDA method uses observed dynamic data to refine pilot point values over four
iterations with diminishing inflation factors [9.33, 7, 4, 2], following the summation criterion
P 1
ai
= 1 [46]. Each iteration processes 200 ensemble realizations. Three pilot point sampling
strategies are evaluated: uniform distribution, random distributions with varying seeds, and
sensitivity map-informed methods (Figure 3.7). Based on Section 3.3.1, 16 pilot points and
9 hard data points are used for facies modeling. Initial pilot point values are sourced from
testing data to preserve channel patterns, with updates restricted to these points during the
ES-MDA cycle. A sigmoid mask is applied post-update to ensure compliance with the U-Net
model’s input constraints.
Figure 3.8 presents realizations from different input configurations. Rows depict pilot
point distributions (uniform, random, and sensitivity-based), while columns show the progression from initial log-permeability maps through ES-MDA updates to final realizations
compared against a reference case. Initial patterns from the U-Net model, using pilot points
sampled from the training dataset, often show mismatches with hard data. Despite this,
all input types align channel positions closely with the reference case post-update. The final column highlights improved pattern clarity with increased realizations and shows that
some pilot points converge near 0.5, reflecting reduced mismatches and uncertainty during
updates.
Figure 3.9 shows the ensemble mean and variance for initial and updated distributions.
The updated mean effectively captures the channel features in the reference case, while
variance maps highlight increased uncertainty along channel edges and regions further from
pilot points. Sensitivity maps outperform random placement by providing better and more
stable improvements in pilot point selection, as evidenced by faster convergence rates. In
contrast, the random strategy in Case 2 results in less confident channel connectivity due to
49
inadequate pilot point support.
Figure 3.10 compares outcomes with additional random placements. Sensitivity-based
strategies significantly enhance pilot point parameterization and improve model calibration
accuracy, corroborating findings from prior studies on pilot point methodologies [87].
3.3.3 Two-phase Flow Example
For a more complex model calibration case study, we conducted a two-phase flow simulation
using a 64 × 64 grid in the x and y dimensions with a single z-layer, representing a 640 m
× 640 m area with a 10 m thickness. Porosity was set uniformly at 0.2, and permeability
ranged from 1,000 millidarcies in channel regions to 5 millidarcies elsewhere. The simulation
involved injecting a wetting-phase fluid at 500 cubic meters per day into injection wells, while
production wells were maintained at a bottom-hole pressure of 150 bars. Fluid volumes for
the wetting and non-wetting phases were recorded bi-monthly over 720 days.
We assessed two scenarios to evaluate the impact of well placement on permeability map
reconstruction [95]. Each scenario included one injection well and three extraction wells
within channel zones, using six months of production data for calibration. Figure 3.11 shows
the configurations, including true log-permeability maps, prior and posterior distributions,
sensitivity maps, and U-Net input-output results. Sensitivity-based pilot point selection
(Section 2.3.2) demonstrated improved reconstruction accuracy.
Updated realizations revealed channelized patterns, with ensemble mean and variance
across 200 realizations showing convergence. In Case 1, channels connecting I1 to P1 and
P3 were identified, while Case 2 revealed connectivity between I1, P1, and P2. Uncertainty
increased with distance from data points. Figure 3.11 compares initial realizations (grey),
calibrated realizations (blue), and actual observations (red), highlighting improved model
performance.
Figure 3.12 shows high uncertainty in areas with sparse hard data. Incorporating soft
points reduced uncertainty, integrating prior knowledge effectively. Results in Figure 3.13
50
demonstrate improved continuity and regularization in permeability maps, confirming the
robustness of the U-Net and pilot point method.
Figure 3.14 presents the dynamic data results, including true log-permeability maps differentiating channel and non-channel regions, as well as the mean and variance of prior
and posterior distributions. Sensitivity maps and U-Net input-output results demonstrate
the improved permeability map reconstruction achieved through the pilot point inversion
method. The integration of the U-Net with the pilot point method enhances model accuracy
and builds confidence in predictions across varying data availability scenarios. This robust
approach effectively manages uncertainty and sparsity, delivering geologically plausible subsurface characterizations.
3.4 Summary
The calibration of subsurface flow models demands geological constraints to ensure plausible
solutions that align with hard data and geological continuity, particularly for complex, nonGaussian property distributions. While traditional methods like MPS and pilot points have
proven effective, they face limitations such as high computational costs, sensitivity to input
parameters, and reliance on high-quality training images.
This chapter addresses these challenges by integrating the pilot point method with a
U-Net deep learning model. The U-Net, trained as a parameterization tool, effectively maps
sparse facies data to high-resolution distributions. Incorporating perceptual loss and input
masks during training enhances image quality and the reconstruction of complex geological
patterns. Sensitivity analysis highlights the role of hard and pilot points in achieving accurate
reconstructions.
Coupled with the ES-MDA algorithm, the U-Net improves parameterization and calibration by preserving geological continuity, honoring hard data, and accurately matching
dynamic data. Sensitivity map-based pilot point selection further optimizes efficiency and
51
precision, reducing the need for additional simulations. Adding soft points as constraints in
uncertain regions enhances calibration by integrating prior knowledge.
Although U-Net offers computational efficiency during inference compared to MPS, it
requires careful tuning of hyperparameters and adequate geological data. Future efforts
will focus on optimizing the U-Net for varied subsurface models, addressing non-stationary
patterns, refining pilot point strategies, and extending the method to three-dimensional
models for broader applications.
52
Figure 3.4: Model reconstruction comparisons: (a) Comparison of reconstruction outcomes
using L2 loss and combined L2 and perceptual loss across increasing numbers of input points.
(b) Effects of L2 versus combined L2 and perceptual loss with 25 input points across different
realizations.
53
Figure 3.5: sensitivity analysis of pilot points on model accuracy metrics
Figure 3.6: 3D field layouts, permeability maps, and true pressure maps for pumping test
cases
Figure 3.7: Sensitivity maps from the initial iteration of the ES-MDA algorithm, highlighting regional sensitivity variations and pilot point locations based on maximum values in
subregions
54
Figure 3.8: Influence of pilot point placement strategies on facies realizations, showcasing
the progression from initial models to refined outcomes using the ES-MDA method across
two cases.
Figure 3.9: Data assimilation outcomes for reference facies maps: log-permeability evolution
through iterations and sampling strategies
55
Figure 3.10: Mismatch reduction across iterations for two cases with five pilot point spacing
strategies: evenly spaced, random variations, and sensitivity-based
Figure 3.11: Comparative visualization of reference and simulation results for two case studies.
56
Figure 3.12: Two-year production profiles for non-wetted and wetted phases at extraction
wells P1, P2, and P3 in two cases, displaying initial realizations, observed data, and calibrated predictions.
57
Figure 3.13: Comparative visualization of reference and simulation results for two case studies, demonstrating the impact of integrating soft points selected based on uncertainty metrics
derived from the ensemble mean and variance of the previous experiment (first row).
58
Figure 3.14: Two-year production profiles for non-wetted and wetted phases at extraction
wells P1, P2, and P3 in two cases, after incorporating soft point constraints.
59
Chapter 4
Optimization with Deep Learning Predictive Model in
Binary Geothermal Fluid System
4.1 Introduction
The increasing urgency to transition toward clean and renewable energy sources has underscored the significant potential of geothermal energy [96]. As Earth’s internal heat is
continuously replenished, geothermal energy is both sustainable and renewable. Between
2015 and 2020, global geothermal capacity grew by an impressive 52% [97]. This growing interest highlights not only the viability of geothermal energy but also the necessity for
more efficient operational workflows, particularly in control and automation systems, to optimize the performance of geothermal power plants. Among these advancements, the Organic
Rankine Cycle (ORC) has emerged as a key technology for efficiently converting low- and
medium-temperature heat. Its integration into geothermal systems is attributed to its high
efficiency, reliability, and relatively low investment costs [98, 99].
Efficient process control and optimization require reliable models to predict ORC performance under varying operational conditions. Traditionally, physics-based simulation models have been employed as comprehensive tools for performance prediction in geothermal
systems [100, 16]. These models have been utilized in various control and optimization
strategies. For instance, Cetin et al. [101] applied a Proportional-Integral-Derivative control
strategy and a Simulated Annealing (SA) algorithm [102] to optimize exergy efficiency in the
Sinem geothermal power plant. Cupeiro et al. [103] implemented Model Predictive Control in
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a hybrid geothermal system, achieving 8.12% cost savings. Similarly, Ghasemian et al. [104]
evaluated subcritical ORC systems using eight working fluids, optimizing performance by
considering thermal efficiency, exergy efficiency, and energy production costs.
Despite their numerous successful applications, physics-based models face inherent challenges. Developing and calibrating these models requires extensive technical expertise and
specialization. Accurately capturing the interactions among various power plant components
and understanding system responses to disturbances and controller settings entail significant
effort and expense [14, 15]. Furthermore, parameterizing these models demands detailed
equipment information and carefully designed experiments to collect essential data [18]. The
costs associated with data acquisition and experimental design further complicate the development of precise physics-based models.
The long lifecycle of geothermal systems, often spanning several decades, introduces additional complexities [19, 20]. As equipment is updated and processes evolve, existing models
must be regularly revised to account for these changes, thereby increasing maintenance efforts. In contrast, advancements in real-time monitoring systems and machine learning
techniques offer a compelling alternative. Data-driven proxy models leverage real-time data
to provide efficient, accurate, and adaptable solutions that mitigate many limitations of
traditional models.
Machine learning is revolutionizing the shift from traditional physics-based methodologies
to data-driven approaches by leveraging expansive datasets to uncover complex input-output
relationships, enabling precise performance predictions [24, 25, 26, 27]. A key strength of
data-driven models is their adaptability; while physics-based models require substantial revisions to integrate new theoretical insights, data-driven models seamlessly incorporate new
data, maintaining their accuracy and relevance over time. Data-driven regression methods
encompass a wide range of techniques that learn from data to model and predict outcomes.
These methods range from linear models, such as simple and multiple regression [105, 106,
107], to more advanced non-linear models like polynomial regression and generalized addi61
tive models (GAMs), which capture complex thermodynamic behaviors. Machine learning
advancements have introduced sophisticated approaches, including Support Vector Regression [108, 109, 110, 111], capable of navigating high-dimensional data, and tree-based methods like Random Forest [106, 112, 113] and Gradient Boosting Regression [114, 115], which
combine multiple models to enhance predictive robustness. Ensemble methods further reduce variance and improve accuracy, while neural networks, including feedforward and recurrent types, excel at deciphering intricate, non-linear patterns. Complementing these are
regularization techniques, such as Ridge [116] and Lasso Regression [117], which maintain
generalizability and prevent overfitting [114, 118, 119]. Together, these data-driven techniques provide a powerful toolkit for researchers and practitioners, enabling precise insights
and trend predictions across diverse domains.
Artificial Neural Networks (ANNs) are at the forefront of computational methodologies,
renowned for their complexity and flexibility [120, 121, 122]. These networks, particularly
those leveraging backpropagation algorithms, have driven significant advancements in forecasting and optimizing geothermal energy operations, enabling more efficient utilization of
this renewable resource. Ping et al. [123] demonstrated the effectiveness of ANNs in refining
the power output of a single screw expander by optimizing operational parameters, such
as pump speed, in response to varying conditions [124, 125]. Similarly, Peng et al. [126]
developed discrete ANN models to predict the performance of equipment within the Organic
Rankine Cycle (ORC) process, utilizing data from the REFPROP database to evaluate a
range of working fluids. Wang et al. [127] further compared ANN and Support Vector Regression (SVR) models across diverse ORC configurations and working fluids, demonstrating
ANNs’ superior ability to capture complex thermodynamic interactions using both empirical and simulated data. However, the application of machine learning has predominantly
been limited to the design phase or specific system components, leaving the impact of operational disturbances on ORC unit performance underexplored. Addressing these gaps,
data-driven models provide a powerful tool to forecast and mitigate such disturbances, en62
suring sustained operational efficiency and advancing the broader adoption of ANN-based
optimization in geothermal systems.
In this chapter, we propose a data-driven prediction and optimization framework to enhance energy generation efficiency in a binary geothermal power plant, where factors such
as ambient temperature and brine supply significantly influence performance. The economic
objective is to maximize net power generation by dynamically adjusting the working fluid
pump speed under varying disturbances. To achieve this, we employ proxy-based optimization using ANN models, which are developed and trained to predict turbine energy
output and multi-stage pump costs. In Section 4.3, we introduce a physics-based thermodynamic model to simulate the power generation process, generate synthetic datasets, perform
simulation-based optimization, and validate the data-driven optimization results. Section
4.4 presents two synthetic scenarios, where the ANN model is trained and tested using the
simulator-generated datasets. Lastly, in Section 4.5, we apply the workflow to a real-world
dataset from Cyrq Energy, demonstrating its potential effectiveness in improving operational
performance within an actual geothermal power plant.
4.2 Metholodgy
The methodology used to assess the feasibility and effectiveness of the proposed workflow
for predicting and optimizing power generation in a geothermal power plant is illustrated
in Figure 4.1. The machine learning model, based on ANNs, is designed and trained to
capture the thermodynamic behavior of the power plant directly from historical measurements, eliminating the need for detailed thermodynamic or engineering specifications of the
equipment. Optimal control inputs are determined to maximize net power generation, with
the data-driven model serving as the predictive engine. To validate the workflow before
applying it to field data, a physics-based thermodynamic model is constructed. While this
model cannot fully replicate the complexities of real-world plant operations, it provides a
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valuable reference for evaluating the solutions derived from the data-driven approach. Finally, the performance of the ANN-based predictive model is benchmarked against results
obtained using the simulation model within the same optimization framework, ensuring a
robust evaluation of the workflow.
Figure 4.1: Overview of working flow on deep learning-based prediction and optimization.
4.2.1 Artificial Neural Networks
The Backpropagation Neural Network (BPNN) is one of the most widely used and mature
types of Artificial Neural Networks (ANNs), featuring a multi-layer structure with nonlinear
activation functions [127]. This architecture allows BPNNs to effectively model and fit
nonlinear regression problems using the provided training data. Its learning capabilities are
further enhanced by the error backpropagation algorithm, which optimizes the network’s
weights through iterative adjustments. Figure 4.2 illustrates the structure of a BPNN,
comprising three main components: an input layer that receives the initial data, one or more
hidden layers that perform intermediate computations, and an output layer that generates
the final predictions.
In this chapter, the input layer of the BPNN accepts disturbance variables (w) and control
input variables (u). The hidden layer computes weighted sums of inputs as:
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Figure 4.2: The schematic diagram of the ANN.
zj =
Xm
i=1
wijxi + b (4.1)
where xi are inputs, wij are weights, and b is the bias term. The outputs (zj ) pass through
the ReLU activation function:
f(z) = max(0, z) (4.2)
This nonlinearity enables the network to model complex patterns. The hidden layer
connects densely to the preceding layer, and its outputs feed into the output layer, predicting
performance variables such as gross power (Pg) and pump cost (Pc).
Training the network minimizes the least-squared error:
L =
1
2
X
N
i=1
(yi − yˆi)
2
(4.3)
where yi are actual outputs, and ˆyi are predictions. This loss guides optimization during
training.
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During training, the neural network’s parameters are optimized to minimize the error
between predicted and actual outputs using the error backpropagation algorithm with the
Adaptive Movement Estimation (ADAM) optimizer [128]. ADAM dynamically adjusts the
learning rate for each parameter by estimating the first and second moments of the gradients,
enabling faster convergence and improved training efficiency.
The predictive performance of the ANN model is evaluated using Root Mean Squared
Error (RMSE) and Mean Absolute Percentage Error (MAPE), defined as:
RMSE =
vuut
1
N
X
N
i=1
(yi − yˆi)
2
(4.4)
MAPE = 1
N
X
N
i=1
yi − yˆi
yi
× 100 (4.5)
where yi and ˆyi are the actual and predicted values, respectively, and N is the total
number of data points. RMSE quantifies the average magnitude of errors, with greater
weight assigned to larger errors, while MAPE provides a relative error metric. Lower RMSE
and MAPE values indicate higher predictive accuracy.
4.2.2 Optimization
Once validated, the ANN model is used to optimize the net power generation of the ORC
unit. The economic objective function maximizes net power generation, as shown in:
argmax
ut
[Pg(wt
, ut) − Pc(wt
, ut)] (4.6)
where Pg is the gross power output, Pc is the pump operating cost, wt represents disturbance variables, and ut represents control inputs. For simplicity, a constant condenser
fan speed is assumed. The control inputs are constrained within physical and training data
limits:
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umin < ut < umax (4.7)
During optimization, the control variable range is restricted to the training dataset’s
domain, avoiding extrapolation and ensuring predictions remain valid. The optimization is
performed using the ”L-BFGS-B” algorithm [129].
4.3 Thermodynamic Simulation
4.3.1 System Description
Figure 4.3: Schematic diagram of the ORC unit in binary cycle powerplant.
This section presents a physics-based thermodynamic model of an air-cooled Organic
Rankine Cycle (ORC) unit, designed to evaluate the proposed data-driven workflow and
explore energy production improvements. As shown in Figure 4.3, the ORC system uses an
organic working fluid to convert low-temperature geothermal energy into electricity. Geothermal brine preheats and vaporizes the working fluid, with its flow rate and pressure controlled
by pump speed. The high-pressure vapor drives a turbine connected to a generator, produc67
ing electricity. The exhaust vapor is then condensed in an air-cooled condenser and pumped
back to the preheater, completing the cycle for continuous operation.
Air cooling is preferred in geothermal power plants located in arid regions, where water cooling is prohibitively expensive [130, 131]. However, this makes ambient temperature
fluctuations the primary disturbance affecting short-term power generation. Figure 4.4(a)
illustrates ambient temperature variations over a year, while Figure 4.4(b) shows that net
power generation efficiency decreases as ambient temperature rises, indicated by a negative
correlation coefficient. In our experiment, simulated power generation outputs did not align
with observed values when assuming constant temperature changes in the air-cooled condenser or setting the condensed working fluid temperature equal to the ambient temperature.
To identify the variable most influenced by ambient temperature, the correlation coefficients
between ambient temperature and turbine exhaust temperature and pressure were analyzed
(Figures 4.4(c)-(d)). Results indicate a strong correlation between ambient temperature and
turbine exhaust pressure, consistent with findings reported in previous studies [16].
Simulating the ORC process using traditional thermodynamic models presents challenges
in aligning with field observations. These include: (1) decreasing heat source temperatures
over time, causing initial design parameters to deviate from actual operations; (2) insufficient plant data requiring inferred parameters, such as the working fluid mass flow rate; (3)
the working fluid, R134a, operating above its critical pressure, invalidating the degree of superheat; and (4) incomplete thermophysical property data and environmental disturbances,
such as ambient temperature fluctuations.
To address these issues, a physics-based thermodynamic model is developed with the
following assumptions:
• Negligible heat loss in heat exchangers and pipelines.
• Parameterized inputs derived from operational data.
• Fluid thermodynamics based on the Peng-Robinson equation.
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• Constant isentropic efficiency for pumps and turbines.
• Constant turbine speed to maintain grid frequency.
• Sequential steady-state calculation approach.
Figure 4.4: (a) Daily maximum temperature minimum temperature in 2020; (b) Ambient
temperature vs Net power generation; (c) Ambient temperature vs. turbine exhaust pressure;
(d) Ambient temperature vs. turbine exhaust temperature.
The steady-state assumption is justified by hourly data collection and minimal inertial effects, supported by Roberto et al. [132], which demonstrate that steady-state models capture
heat source fluctuations without significant dynamic impacts on economic performance.
The model is implemented using AVEVA Process Simulation [133], as shown in Figure 4.5.
The configuration includes a vaporizer, preheater, working fluid pump, air-cooled condenser,
and turbine. Working fluids are labeled f1 to f6, and brine fluids are labeled b1 to b3. Detailed
energy analyses for each component are discussed below.
Heat Exchangers: The preheater and vaporizer are modeled as counter-flow heat exchangers, with heat transfer described by the convection heat balance equation:
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Figure 4.5: Diagram of the ORC model in AVEVA.
Q = UA∆Tlm (4.8)
where Q is the heat transferred, U is the heat transfer coefficient, A is the heat transfer
area, and ∆Tlm is the Logarithmic Mean Temperature Difference (LMTD) between the
countercurrent flows of brine and working fluid.
The air-cooled condenser facilitates condensation of the working fluid from the turbine
exhaust at a pressure in equilibrium with the ambient temperature.
Turbine: The turbine outlet pressure (pf6) is modeled based on the strong correlation
with ambient temperature (Tamb) observed in historical data. For Tamb > 273.15 K, pf6 is
estimated using a polynomial regression:
pf6 = α0 + α1Tamb + α2T
2
amb (4.9)
For Tamb ≤ 273.15 K, pf6 is set to a constant value. Figure 4.6(a) shows the polynomial
regression results (red curve) compared to the scatter of actual data points (blue). Deviations
are attributed to simplified dynamic processes in the model.
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Assuming a turbine efficiency (ηtur) of 70% and an isentropic process, the turbine outlet
temperature is calculated using the turbine inlet pressure, inlet temperature, and outlet
pressure. The turbine power output is given by:
Wtur = ˙mwf(Hf3 − Hf4)ηtur (4.10)
where ˙mwf is the mass flow rate of the working fluid, and Hf3 and Hf4 are the specific
enthalpies at the turbine inlet and outlet, respectively.
Figure 4.6: (a) Polynomial regression model to estimate the turbine outlet pressure with
ambient temperature; (b) multi-stage pump curve of discharge pressure.
Multi-Stage Centrifugal Pump: In the absence of a pump curve, the pump discharge
pressure (ppout ) and volumetric flow rate (qpout ) are modeled using second-order polynomial
regression and linear regression (LR), respectively, with pump speed (n) as the input. Figure 64.6 compares the modeled pump curve with available field data.
The pump work is calculated as:
Wpump =
m˙ wf(Hf6 − Hf5)
ηpump
(4.11)
where ˙mwf is the working fluid mass flow rate, Hf6 and Hf5 are the specific enthalpies at
the pump outlet and inlet, respectively, and ηpump is the pump efficiency.
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4.3.2 Thermodynamic Model Evaluation
The system calculates state variables such as temperature, pressure, and volumetric flow
rate based on disturbance and control variables. The disturbance inputs include ambient
temperature, brine inlet pressure, brine inlet temperature, and brine outlet volumetric flow
rate, while the control variable is pump speed. To evaluate the physics-based simulator, 1000
hours of historical data were simulated, solving steady-state equations sequentially. Given
the stable heat source during the two-month operational period, ambient temperature was
identified as the primary disturbance variable.
Table 4.1 presents the RMSE and MAPE values comparing simulated and historical data,
while Figure 4.7 shows the key measurement locations and corresponding results. The simulated brine outlet temperature and working fluid inlet temperature closely match field data,
indicating the effectiveness of the simplified heat exchanger model for thermal dynamics.
Turbine exhaust pressure, modeled polynomially, aligns well with observed values. However, assuming constant turbine and pump efficiencies, the simulated gross power generation
deviates from historical data, showing smoother turbine inlet temperature trends and overestimating net power at higher ambient temperatures. These discrepancies highlight challenges
in thermodynamic modeling, particularly when incomplete information limits simulation accuracy. Despite this, the simulated results reflect trends consistent with field data, validating
the system’s response to disturbance and control inputs.
Table 4.1: Measurement metrics for model variables.
Variables RMSE MAPE (%)
Brine outlet temperature (Tb3
) 0.05 0.28
R134a inlet temperature (Tf1
) 0.077 19.663
Turbine inlet temperature (Tf3
) 0.121 22.323
Turbine outlet pressure (pf4
) 0.072 6.564
Total power generation (Wtur) 0.094 6.133
Net power generation (Wnet) 0.133 13.97
Despite the steady-state assumptions and minor discrepancies between simulated and
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Figure 4.7: The featured variables of simulated data (red) compared to the field data (blue).
historical data, the reproduced dominant variables and power generation suggest that the
simulator serves as a reliable quasi-dynamic proxy model for generating synthetic datasets
and validating optimization results. This model is applied in two case studies to showcase
the effectiveness of our data-driven approach.
4.4 Numerical Experiments and Results
4.4.1 Case 1: Synthetic Datasets (Uniform Input)
This section demonstrates the ANN model’s prediction capabilities using synthetic datasets
and its application in optimizing well pump operation. Two datasets are generated: one
covering the full operational range for training and testing, and another based on historical
field data to ensure realism. Five input variables—brine temperature, pressure, volume flow
rate, ambient temperature, and pump speed—are used to predict net power production.
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The thermodynamic model simulates the power plant, validates ANN predictions, and evaluates optimization results, with ranges of disturbance and control variables summarized in
Table 4.2.
Table 4.2: Input variables of the ORC model.
Variables Units Values
Brine inlet temperature (Tbin) K 420–422
Brine inlet pressure (pbin) kPa 850–950
Brine volumetric flow rate (qbin) m3/s 0.15–0.27
Ambient temperature (Tamb) K 267–300
Pump speed (n) rpm 840–1200
The first synthetic dataset consists of 3000 hourly samples, each generated from uniform
distributions of five input variables. The data is normalized between 0 and 1 and split into
2000 training and 1000 testing samples using a 2-to-1 ratio. PCA is applied to visualize the
dataset by transforming input and output variables into orthogonal principal components,
highlighting data variance, as shown in Figure 4.8(a).
The second dataset contains 1500 samples generated using measured field inputs and
historical data. This dataset evaluates the predictive model’s performance under realistic
disturbances, with the testing data visualized in Figure 4.8(b).
The process of fine-tuning the ANN structure is illustrated in Figure 4.9. Simplicity is
prioritized, with models containing fewer parameters selected when performance is comparable. Training uses 2000 samples from the first synthetic dataset, divided into 70% for
training and 30% for validation, while the remaining 1000 samples and the second dataset
are reserved for testing. The ANN’s predictive performance is benchmarked against LR
and SVR models, including linear and Radial Basis Function (RBF) kernels, to account for
nonlinearity. Prediction results for both test datasets are shown in Figure 4.10.
To address the impact of weight initialization, the ANN was trained 20 times with different initializations. Performance metrics, including RMSE and MAPE, were statistically
analyzed to determine the mean and variability, as shown in Case 1, Table 4.3. A comparative analysis across 20 iterations also included LR and SVR models trained on a randomly
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Figure 4.8: Figure 8. (a) PCA projection of the synthetic datasets with uniform distributed
input; (b) Featured variables in testing dataset simulated with the field input.
selected 70% data subset.
The ANN model outperformed other approaches, particularly in capturing complex nonlinear thermodynamic interactions within the dataset. Its predictive advantage was most
pronounced under extreme operational conditions, effectively modeling the nonlinearity at
dataset boundaries. This robust performance across the operational range establishes the
ANN as the preferred model for subsequent optimization tasks.
Optimization Results
The trained ANN model is utilized to optimize pump operation under field conditions from
the second testing dataset. Optimal control is determined using ten ANN models trained
with different weight initializations. Figure 11(a) projects the test dataset onto the principal
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Figure 4.9: Sensitive analysis for finalizing the structure of ANN model. Average RMSE
and MAPE of 10 realizations for each coefficient of (a) Hidden layer (b) Hidden neural (c)
Loss function (d) Learning rate.
components (PCs) before (red) and after (orange) optimization by one ANN model. The
first two PCs capture over 50% of the training data variance, with most test samples aligning
well with the training dataset.
Figure 4.11(b) compares the average gross power generation, pump consumption, and
net power output for the original control (green), physics-based optimized control (blue),
and ANN-optimized control (orange). The ANN optimization achieves an average net power
improvement of 60 kW over nearly two months, equivalent to an additional 43 MWh for
one ORC unit. Figure 4.11(c) illustrates the cumulative energy improvement, showing a
101 MWh increase with ANN-based optimization compared to 151 MWh for the simulationbased reference case. All ANN models show consistent net power improvements, closely
following the reference trend.
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Figure 4.10: Comparative Predictive Performance on Test Data Sets: (a) Models Applied to
Random Input Scenarios (b) Models Validated Against Field Operation Data Inputs
To explore optimization steps, Figure 4.12(a) presents the ambient temperature (left)
and pump operation (right) over 1500 hours. Figure 4.12(b) visualizes iterations at samples
1, 500, 1500, and 2000, starting from the initial controller position applied in the field. The
dashed blue line represents ANN-predicted values, the yellow line shows simulated values
for the same pump speed, and the red line depicts optimization results directly from the
simulator.
The optimization process ensures that the blue and red curves monotonically increase,
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Table 4.3: Summary of prediction results on synthetic datasets.
Power generation Pump consumption
Testing data RMSE MAPE (%) RMSE MAPE (%)
Case 1 Training data sampled from uniform input
Input (Uniform)
LR 0.045±0.000 6.6±0.055 0.021±0.000 2.515±0.036
SVM (linear) 0.053±0.001 8.64±0.529 0.062±0.000 11.185±0.073
SVM (RBF kernel) 0.048±0.003 8.46±0.584 0.046±0.003 6.835±1.013
ANN 0.016±0.001 1.44±0.41 0.014±0.001 0.88±0.419
Input (Field Type)
LR 0.042±0.002 9.51±0.522 0.029±0.001 3.685±0.248
SVM (linear) 0.030±0.002 4.77±0.457 0.040±0.001 5.105±0.097
SVM (RBF kernel) 0.057±0.009 11.79±1.550 0.031±0.006 4.16±0.937
ANN 0.008±0.002 1.545±0.488 0.010±0.004 1.265±0.548
Case 2 Training data sampled from field type input
Input (Uniform)
LR 0.045±0.000 6.795±0.038 0.027±0.000 3.655±0.074
SVM (linear) 0.076±0.001 14.645±0.209 0.088±0.003 12.945±0.451
SVM (RBF kernel) 0.061±0.004 10.09±0.804 0.115±0.004 15.13±1.346
ANN 0.026±0.002 3.21±0.301 0.018±0.004 1.855±0.673
Input (Field Type)
LR 0.036±0.001 8.055±0.112 0.014±0.000 1.855±0.050
SVM (linear) 0.033±0.001 5.075±0.077 0.036±0.005 4.535±0.816
SVM (RBF kernel) 0.025±0.004 5.21±0.880 0.038±0.007 4.815±1.025
ANN 0.006±0.001 1.095±0.116 0.004±0.002 0.43±0.193
reflecting the increasing objective function. When the ANN-optimized output aligns with
the simulator’s ideal value (red), the ANN achieves optimal operation. Despite differing
trajectories, this convergence aligns with expectations for subsurface flow applications, where
local optimization methods yield similar values at local optima.
4.4.2 Case 2: Synthetic Datasets (Field Pattern Input)
In the Case 1, the training dataset evenly covered the entire range of normal operating
conditions, which is challenging to achieve in real operations where field data often focus on
narrower ranges for optimal performance. To address this, synthetic datasets are generated
entirely from real field inputs to test the proposed workflow in an emulated scenario. Figure
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Figure 4.11: (a) PCA projection of the testing data before and after the optimization (b)
Average power production optimization result of the production; (c) Accumulative net energy
improvement on 1500h simulation.
4.13 shows a synthetic dataset of 10,000 hours based on field inputs, partitioned into 8,500
hours for training and 1,500 hours for testing, matching the second testing dataset from Case
1. The training data, equivalent to one year of operation, is used to build the ANN model to
predict power production for the subsequent two months. The ANN framework, architecture,
and training process are consistent with those in Case 1. Comparative models, including
LR, SVR, and SVR with an RBF kernel, are also trained using the same methodology.
Predictive results for these models are summarized in Case 2 (Table 4.3). Similar to Case 1,
the ANN outperforms other models, showing superior prediction accuracy on the field test
dataset. The improved performance is attributed to the increased training data volume and
its better coverage of test data. These results confirm the robustness of the ANN model
under real-world operational scenarios.
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Figure 4.12: (a) Left: ambient temperature change during the operation. Right: original
pump operation vs the optimized pump operation by the numerical simulator and ANN
model; (b) Monitoring the optimization process on highlight samples.
Figure 4.13: Featured synthetic training and testing data after preprocess from 0 to 1.
When the trained model is applied to predict outputs for data ranges outside the training
data (e.g., the broader operating conditions in Case 1), prediction accuracy declines. This
reduced performance stems from the extrapolation nature of the task, a known limitation
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of ANN models. Figure 4.14(a) illustrates the PCA projection of the training and test data
on the first three principal components (PCs). The linear pattern in the PCA projection
of the training dataset reflects strong correlations between variables, but the model cannot
generalize well beyond the operational range of the training data.
To address this, the optimization workflow constrains control variables within the historical data range. Figure 4.14(b) shows the optimization results using the ANN model,
demonstrating that net power generation closely matches the reference (simulation) values.
The net power improvement reflects the balance between increased power generation and
reduced pump consumption. Figure 4.14(c) highlights the cumulative net energy improvement, with a mean increase of 138 MWh, nearly matching the reference value of 151 MWh
obtained from the simulation model. These results confirm that constraining optimization
within the training data range maintains high accuracy and performance.
4.4.3 Case 3: Field Data
The proposed workflow is applied to a 5000-hour field dataset from Cyrq Energy Inc., ensuring continuity and excluding periods of equipment replacement, controller changes, or
maintenance shutdowns. The dataset is split into 4500 training and 500 testing samples, as
shown in Figure 4.15.
To better align with field data characteristics for power production and pump cost, the
activation function in the second hidden layer is replaced with a hyperbolic tangent function.
The ReLU activation previously caused predictions to fall outside the normalized range of 0
to 1 due to pump and turbine efficiency curves. The remaining model structure and training
procedure remain consistent with prior cases. Prediction results for the test data achieve
RMSE/MAPE values of 0.041/3.3% for power production and 0.009/0.7% for pump cost, as
shown in Figure 4.16.
Optimization using the trained ANN model maximizes net power generation. Figure 4.17(a) demonstrates that the training data, visualized via PCA, sufficiently covers
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Figure 4.14: (a) PCA projection of the testing data before and after the optimization; (b)
Average power production optimization result of the production; (c) Accumulative net energy
improvement on 1500h simulation.
Figure 4.15: Normalized historical dataset from the field.
the test dataset. Figures 4.17(b)-(c) display predicted power generation results. Since the
optimization cannot be physically validated without applying the controls in the field, performance is reported based on model predictions. The workflow can be deployed in actual
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Figure 4.16: Prediction result of testing datasets: (a) prediction value vs true value in scatter
plot; (b) prediction value vs true value in the sequence.
production units to verify alignment of optimization results with field expectations.
4.5 Summary
This paper presents an efficient data-driven workflow for predicting and optimizing air-cooled
Organic Rankine Cycle (ORC) geothermal operations. Using an ANN, the framework predicts net power generation under varying operating conditions and optimizes production by
adjusting the working fluid circulation rate via pump speed control. The workflow’s construction, training, and validation were demonstrated across three case studies of increasing
complexity, including application to real geothermal field data.
Initial testing with simulated data inspired by a geothermal power plant demonstrated
the ANN’s ability to accurately predict performance variables and optimize operations. Tur83
Figure 4.17: (a) PCA projection of the training dataset and testing data before and after the
optimization of Cyrq dataset; (b) Averaged (predicted) optimization result of the production
by ANN model; (c) Accumulative net energy improvement on 500h simulation.
bine exhaust pressure was modeled using polynomial regression, and the ANN consistently
outperformed LR and SVR, including RBF kernels. Results confirmed that the framework
effectively predicts power generation and optimizes operations when training data sufficiently
cover the operational range.
Application to field data validated the framework’s feasibility and effectiveness, with predictions closely matching historical data. Optimization outcomes demonstrated potential for
improving net power generation while constraining controls within the historical data range.
However, the framework’s dependence on high-quality, representative training data and the
”black box” nature of ANNs limit interpretability and present challenges for diagnosing
issues.
In conclusion, while the ANN-based workflow shows significant potential for improving
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ORC geothermal operations, practitioners must address its reliance on data quality and limited interpretability. Future work should focus on deploying the model online for continuous
updates with new samples, developing metrics for evaluating data quality under limited data
availability [134], and integrating dynamic simulators to train models under transient conditions. Incorporating advanced control methods, such as model predictive control, could
further enhance optimization and enable real-time automation for complex economic objectives [135, 136, 137].
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Chapter 5
Conclusions and Future Directions
5.1 Conclusion
This dissertation has developed and implemented advanced deep learning tools for the characterization and prediction of complex fluid flow systems, contributing significantly to two
primary areas.
First, novel methods for parameterizing and calibrating subsurface fluid flow models were
introduced, with a focus on non-Gaussian property distributions and geological continuity.
These approaches address key challenges in subsurface modeling, including uncertainties in
flow properties and the limitations of traditional methods.
Second, a data-driven framework was developed for the prediction and optimization of
surface geothermal fluid flow systems. This framework provides an efficient alternative to
physics-based approaches, enabling better management of geothermal power plants by predicting power output and optimizing operational efficiency.
Chapter 2 investigates the use of StyleGAN, an advanced generative deep learning model,
for parameterizing and calibrating subsurface flow models with complex, non-Gaussian property distributions. Unlike traditional generative models like DCGAN and VAE, StyleGAN
benefits from a disentangled latent space, offering superior control over generated outputs
and improving training stability. The study demonstrates that StyleGAN’s regular latent
space leads to more stable and consistent transformations, which significantly enhance model
calibration accuracy and robustness. These advancements make StyleGAN a promising tool
86
for parameterizing subsurface flow models, addressing challenges in preserving geological
continuity and improving the calibration of models with complex spatial patterns.
Chapter 3 combines the pilot point method with the U-Net deep learning model to
improve the calibration of complex, non-Gaussian facies distributions in subsurface flow
models. The U-Net model, trained with a perceptual loss function, efficiently maps sparse
facies data to high-fidelity distributions, preserving geological continuity. Unlike traditional
MPS methods, U-Net requires intensive computation only during training, with negligible
costs for generating realizations thereafter. The study demonstrates that the integration of
U-Net with the pilot point method and the ES-MDA algorithm enhances model calibration
accuracy, resulting in models that honor hard data and maintain geological plausibility.
Additionally, the sensitivity map-based pilot point selection method, combined with soft
points, further improves calibration precision, especially in areas with uncertain geological
properties.
Chapter 4 presents a data-driven framework using an ANN to predict and optimize
operations in air-cooled ORC geothermal systems. The model predicts net power generation
and maximizes production by adjusting the working fluid circulation rate. The framework is
tested through three case studies, including real-world geothermal field data, demonstrating
the ANN’s superior predictive accuracy compared to traditional models (LR, SVR, and
SVR-RBF). The results show good agreement with historical data, confirming the model’s
potential for optimizing net power generation. However, the quality of training data and
the ANN’s ”black box” nature pose challenges, suggesting the need for ongoing updates and
further development of dynamic models for improved optimization in operational settings.
5.2 Future Work
Future work will prioritize enhancing the scalability, accuracy, and adaptability of tools for
characterizing and predicting complex fluid flow systems.
87
In subsurface fluid flow modeling, future research will focus on refining generative models
like StyleGAN by improving the interplay between latent space regularity and calibration algorithms. This approach aims to create more efficient, application-specific models tailored to
complex subsurface scenarios, particularly in 3D modeling. Additionally, better integration
of geological and dynamic data will be explored, including extending U-Net architectures to
capture non-stationary geological patterns and optimizing pilot point placement strategies.
These advancements are expected to enhance the calibration of models with non-Gaussian
property distributions, addressing challenges related to flow property uncertainties and the
limitations of traditional geostatistical methods.
For geothermal fluid flow system optimization, the focus will shift toward deploying
ANN-based data-driven frameworks for online applications, enabling continuous updates
and improved predictive accuracy over time. A critical component of this work will involve
developing metrics to evaluate the quality of training data, particularly in scenarios with
limited data availability. Moreover, integrating dynamic simulators to generate training data
under transient conditions will help overcome the steady-state assumptions of current models.
By leveraging this data to train dynamic neural networks, advanced control techniques, such
as model predictive control, can be implemented for real-time optimization, enhancing both
operational efficiency and long-term performance in geothermal power systems.
Addressing these challenges will significantly improve the robustness and applicability
of deep learning models in subsurface flow modeling and geothermal system optimization,
offering scalable and efficient solutions to tackle increasingly complex fluid flow problems.
88
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Abstract (if available)
Abstract
Accurate characterization and prediction of complex fluid flow systems are essential for advancements in geosciences and renewable energy. This thesis explores the integration of deep learning techniques with traditional modeling approaches to enhance subsurface flow model calibration and optimize geothermal power plant operations.
In subsurface modeling, the accurate representation of non-Gaussian spatial patterns poses significant challenges for conventional parameterization methods. We introduce style-based Generative Adversarial Networks (StyleGANs) as a novel parameterization tool, demonstrating their superior performance in reconstruction fidelity, robustness, and calibration flexibility compared to Variational Autoencoders (VAEs) and traditional GANs. Using the Ensemble Smoother with Multiple Data Assimilation (ES-MDA), we validate StyleGANs’ effectiveness in single-phase and two-phase flow scenarios, highlighting their ability to improve the representation and calibration of complex geological patterns. Further, we integrate the pilot point method with a U-Net model, trained with a perceptual loss function, to conditionally map sparse facies data to detailed distributions. This method enhances geological continuity and calibration accuracy, with sensitivity-based pilot point selection and soft point incorporation improving model fidelity while reducing computational overhead.
In geothermal energy optimization, we develop a data-driven artificial neural network (ANN) to predict power output and operational costs, serving as an efficient alternative to physics-based approaches. This ANN-based model propagates the influence of control and disturbance variables and is applied to optimize the working fluid circulation rate, maximizing net power output. The workflow is further demonstrated to model and control ambient temperature effects on air-cooled binary cycle geothermal plants, which are challenging and computationally expensive to address with physics-based predictive models. Numerical experiments with simulated and field datasets validate the model’s ability to improve operational efficiency.
By integrating deep learning techniques with geostatistical and optimization workflows, this research advances subsurface modeling and renewable energy systems. The findings provide practical, scalable, and efficient solutions for addressing the complexities of fluid flow systems in geosciences and energy sectors.
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Ling, Wei
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Deep learning for characterization and prediction of complex fluid flow systems
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Viterbi School of Engineering
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Doctor of Philosophy
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Chemical Engineering
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2024-12
Publication Date
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