From 078f3951706d10d27a53708a194a055630abcfe3 Mon Sep 17 00:00:00 2001
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Date: Wed, 27 Mar 2024 18:22:08 +0000
Subject: [PATCH] Built site for gh-pages

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 search.json                 | 10 +++++-----
 sitemap.xml                 | 12 ++++++------
 yin2024SEG/paper.html       |  2 +-
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@@ -6,7 +6,7 @@
 
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
-<meta name="author" content="Abhinav Prakash Gahlot1*, Haoyun Li,1*, Ziyi Yin1, Rafael Orozco1, and Felix J. Herrmann1 1 Georgia Institute of Technology, * first two authors contributed equally">
+<meta name="author" content="Abhinav Prakash Gahlot1*, Haoyun Li,1*, Ziyi Yin1, Rafael Orozco1, and Felix J. Herrmann1 1 Georgia Institute of Technology, * equal contribution">
 
 <title>Image2024 - A Digital Twin for Geological Carbon Storage with Controlled Injectivity</title>
 <style>
@@ -255,7 +255,7 @@ <h1 class="title">A Digital Twin for Geological Carbon Storage with Controlled I
     <div class="quarto-title-meta-heading">Author</div>
     <div class="quarto-title-meta-contents">
              <p>Abhinav Prakash Gahlot<sup>1*</sup>, Haoyun Li,<sup>1*</sup>, Ziyi Yin<sup>1</sup>, Rafael Orozco<sup>1</sup>, and Felix J. Herrmann<sup>1</sup><br>
-<sup>1</sup> Georgia Institute of Technology, <sup>*</sup> first two authors contributed equally </p>
+<sup>1</sup> Georgia Institute of Technology, <sup>*</sup> equal contribution </p>
           </div>
   </div>
     
@@ -270,7 +270,7 @@ <h1 class="title">A Digital Twin for Geological Carbon Storage with Controlled I
 
 <section id="introduction" class="level2">
 <h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
-<p>Digital Twins refer to dynamic virtual replicas of subsurface systems, integrating real-time data and employing advanced generative Artificial Intelligence (genAI) methodologies, such as neural posterior density estimation via simulation-based inference and sequential Bayesian inference. Thanks to combination of these advanced Bayesian inference techniques, our approach is capable of addressing challenges of monitoring and controlling CO<sub>2</sub> storage projects. These challenges include dealing with the subsurface’s complexity and heterogeneity (seismic and fluid-flow properties), operations optimization, and risk mitigation, e.g.&nbsp;via injection rate control. Because our Digital Twin is capable of handling diverse monitoring data, consisting of time-lapse seismic and data collected at (monitoring) wells, it entails a technology that serves as a platform to integrate seemingly disparate and siloed fields, e.g.&nbsp;geophysics and reservoir engineering. In addition, recent breakthroughs in genAI, allow Digital Twins to capture uncertainty in a principled way <span class="citation" data-cites="yu2023IMAGEmsc herrmann2023president gahlot2023NIPSWSifp">(<a href="#ref-yu2023IMAGEmsc" role="doc-biblioref">Yu et al. 2023</a>; <a href="#ref-herrmann2023president" role="doc-biblioref">Herrmann 2023</a>; <a href="#ref-gahlot2023NIPSWSifp" role="doc-biblioref">Gahlot et al. 2023</a>)</span>. By employing training and inference recursively, the Digital Twin trains its neural networks on samples of the simulated current state—i.e., the CO<sub>2</sub> saturation/pressure, paired with simulated imaged seismic and/or data collected at (monitoring) wells. These training pairs of the simulated state and simulated observations are obtained by sampling the posterior distribution, <span class="math inline">\(\mathbf{x}_{k-1}\sim p(\mathbf{x}_{k-1}\vert \mathbf{y}^\mathrm{o}_{1:k-1})\)</span>, at the previous timestep, <span class="math inline">\(k-1\)</span>, conditioned on field data, <span class="math inline">\(\mathbf{y}^\mathrm{o}_{1:k-1}\)</span>, collected over all previous timesteps, <span class="math inline">\(1:k-1\)</span>, followed by advancing the state to the current timestep, followed by simulating (seismic/well) observations associated with that state. Given these simulated state-observation pairs, the Digital Twin’s networks are trained, so they are current and ready to produce samples of the posterior when the new field data comes in—i.e.&nbsp;<span class="math inline">\(\mathbf{x}_{k}\sim p(\mathbf{x}_{k}\vert \mathbf{y}^\mathrm{o}_{1:k})\)</span>. While this new neural approach to data assimilation for CO<sub>2</sub> storage projects provides what is called an uncertainty-informed <em>Digital Shadow</em>, it lacks decision making and control, which would make it a Digital Twin <span class="citation" data-cites="thelen2022comprehensivea">(<a href="#ref-thelen2022comprehensivea" role="doc-biblioref">Thelen et al. 2022</a>)</span>, capable of optimizing storage operations while mitigating risks including the risk of fracturing the cap rock by exceeding the fracture pressure. The latter risk is illustrated in Figure 1, where the first row contains simulated samples of the differential pressure at timestep, <span class="math inline">\(k=4\)</span>, without control. These samples for the simulated state exceed the fracture pressure and are denoted by the red areas. During this talk, we will demonstrate how the Digital Twin can make informed decisions to avoid exceeding the fracture pressure.</p>
+<p>Digital Twins refer to dynamic virtual replicas of subsurface systems, integrating real-time data and employing advanced generative Artificial Intelligence (genAI) methodologies, such as neural posterior density estimation via simulation-based inference and sequential Bayesian inference. Thanks to combination of these advanced Bayesian inference techniques, our approach is capable of addressing challenges of monitoring and controlling CO<sub>2</sub> storage projects. These challenges include dealing with the subsurface’s complexity and heterogeneity (seismic and fluid-flow properties), operations optimization, and risk mitigation, e.g.&nbsp;via injection rate control. Because our Digital Twin is capable of handling diverse monitoring data, consisting of time-lapse seismic and data collected at (monitoring) wells, it entails a technology that serves as a platform to integrate seemingly disparate and siloed fields, e.g.&nbsp;geophysics and reservoir engineering. In addition, recent breakthroughs in genAI, allow Digital Twins to capture uncertainty in a principled way <span class="citation" data-cites="yu2023IMAGEmsc herrmann2023president gahlot2023NIPSWSifp">(<a href="#ref-yu2023IMAGEmsc" role="doc-biblioref">Yu et al. 2023</a>; <a href="#ref-herrmann2023president" role="doc-biblioref">Herrmann 2023</a>; <a href="#ref-gahlot2023NIPSWSifp" role="doc-biblioref">Gahlot et al. 2023</a>)</span>. By employing training and inference recursively, the Digital Twin trains its neural networks on samples of the simulated current state—i.e., the CO<sub>2</sub> saturation/pressure, paired with simulated imaged seismic and/or data collected at (monitoring) wells. These training pairs of the simulated state and simulated observations are obtained by sampling the posterior distribution, <span class="math inline">\(\mathbf{x}_{k-1}\sim p(\mathbf{x}_{k-1}\vert \mathbf{y}^\mathrm{o}_{1:k-1})\)</span>, at the previous timestep, <span class="math inline">\(k-1\)</span>, conditioned on field data, <span class="math inline">\(\mathbf{y}^\mathrm{o}_{1:k-1}\)</span>, collected over all previous timesteps, <span class="math inline">\(1:k-1\)</span>, followed by advancing the state to the current timestep, followed by simulating (seismic/well) observations associated with that state. Given these simulated state-observation pairs, the Digital Twin’s networks are trained, so they are current and ready to produce samples of the posterior when the new field data comes in—i.e.&nbsp;<span class="math inline">\(\mathbf{x}_{k}\sim p(\mathbf{x}_{k}\vert \mathbf{y}^\mathrm{o}_{1:k})\)</span>. While this new neural approach to data assimilation for CO<sub>2</sub> storage projects provides what is called an uncertainty-informed <em>Digital Shadow</em>, it lacks decision making and control, which would make it a Digital Twin <span class="citation" data-cites="thelen2022comprehensivea">(<a href="#ref-thelen2022comprehensivea" role="doc-biblioref">Thelen et al. 2022</a>)</span>, capable of optimizing storage operations while mitigating risks including the risk of fracturing the cap rock by exceeding the fracture pressure. The latter risk is illustrated in Figure 1, where the first row contains simulated samples of the pressure difference at timestep <span class="math inline">\(k=4\)</span>, between the reservoir pressure and hydraulic pressure, without control. These samples for the simulated state exceed the fracture pressure and are denoted by the red areas. During this talk, we will demonstrate how the Digital Twin can make informed decisions to avoid exceeding the fracture pressure.</p>
 </section>
 <section id="methodology" class="level2">
 <h2 class="anchored" data-anchor-id="methodology">Methodology</h2>
@@ -278,7 +278,7 @@ <h2 class="anchored" data-anchor-id="methodology">Methodology</h2>
 </section>
 <section id="results" class="level2">
 <h2 class="anchored" data-anchor-id="results">Results</h2>
-<p>To calculate injection rates that mitigate the risk of exceeding the fracture pressure, we proceed as follows. First, because the optimized injection rates are close to the fracture pressure, we consider these optimization as approximations to the injection rates where the fracture pressure are exceeded. Next, Kernel Density Estimation (KDE) is applied to produce the smooth red curve in Figure 2(a). This smoothed probability function is used to calculate the Cumulative Density Function (CDF), plotted in Figure 2(b). Using the fact that non-fracture/fracture occurrence entails a Bernoulli distribution, confidence intervals can be calculated, <span class="math inline">\(\pm Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{128}}\)</span> where <span class="math inline">\(\hat{p}\)</span> represents the CDF (blue line) and <span class="math inline">\(Z_{\frac{\alpha}{2}}=1.96\)</span> with <span class="math inline">\(\alpha=0.05\)</span>. From the CDF and confidence intervals (denoted by the grey areas), the following conclusions can be drawn: First, if the initial injection rate of <span class="math inline">\(q_3 = 0.0500 m^3/s\)</span> is kept, the fracture probability lies between 24.47 – 40.71% (vertical dashed line) and has a maximum likelihood of 32.59%, which are all way too high. Second, if we want to limit the fracture occurrence rate to 1% (red dashed line), then we need to lower the injection rate to <span class="math inline">\(q_3=0.0387\mathrm{m^3/s}\)</span>. To ensure the low fracture occurrence rate of 1%, the reduced injection rate is chosen as the smallest injection rate within the confidence interval. As can be observed from Figure 2(b), lowering the injection rate avoids exceeding the fracture pressure at the expense of injecting less CO<sub>2</sub>. Out of 128 samples, 43 samples are fractured with the initial injection rate, while only one sample is fractured with the controlled injection rate.</p>
+<p>To calculate injection rates that mitigate the risk of exceeding the fracture pressure, we proceed as follows. First, because the optimized injection rates are close to the fracture pressure, we consider these optimization as approximations to the injection rates where the fracture pressure are exceeded. Next, Kernel Density Estimation (KDE) is applied to produce the smooth red curve in Figure 2(a). This smoothed probability function is used to calculate the Cumulative Density Function (CDF), plotted in Figure 2(b). Using the fact that non-fracture/fracture occurrence entails a Bernoulli distribution, confidence intervals can be calculated, <span class="math inline">\(\pm Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{128}}\)</span> where <span class="math inline">\(\hat{p}\)</span> represents the CDF (blue line) and <span class="math inline">\(Z_{\frac{\alpha}{2}}=1.96\)</span> with <span class="math inline">\(\alpha=0.05\)</span>. From the CDF and confidence intervals (denoted by the grey areas), the following conclusions can be drawn: First, if the initial injection rate of <span class="math inline">\(q_3 = 0.0500 m^3/s\)</span> is kept, the fracture probability lies between 24.47 – 40.71% (vertical dashed line) and has a maximum likelihood of 32.59%, which are all way too high. Second, if we want to limit the fracture occurrence rate to 1% (red dashed line), then we need to lower the injection rate to <span class="math inline">\(q_3=0.0387\mathrm{m^3/s}\)</span>. To ensure the low fracture occurrence rate of 1%, the reduced injection rate is chosen as the smallest injection rate within the confidence interval. As can be observed from Figure 1 (third row) and Figure 2(b), lowering the injection rate avoids exceeding the fracture pressure at the expense of injecting less CO<sub>2</sub>. Out of 128 samples, 43 samples are fractured with the initial injection rate, while only one sample is fractured with the controlled injection rate.</p>
 </section>
 <section id="conclusion-and-discussions" class="level2">
 <h2 class="anchored" data-anchor-id="conclusion-and-discussions">Conclusion and discussions</h2>
@@ -286,17 +286,17 @@ <h2 class="anchored" data-anchor-id="conclusion-and-discussions">Conclusion and
 <div id="fig-flow" class="quarto-figure quarto-figure-center quarto-float anchored">
 <figure class="quarto-float quarto-float-fig figure">
 <div aria-describedby="fig-flow-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-<a href="./figs/Figure1.png" class="lightbox" data-glightbox="description: .lightbox-desc-1" data-gallery="quarto-lightbox-gallery-1"><img src="./figs/Figure1.png" class="img-fluid figure-img" style="width:90.0%"></a>
+<a href="./figs/Figure1.png" class="lightbox" data-gallery="quarto-lightbox-gallery-1" data-glightbox="description: .lightbox-desc-1"><img src="./figs/Figure1.png" class="img-fluid figure-img" style="width:90.0%"></a>
 </div>
 <figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-flow-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-Figure&nbsp;1: Pressure state at time-step <span class="math inline">\(k=3\)</span> and a comparative analysis of pressure outputs from the digital twin at time-step <span class="math inline">\(k=4\)</span>
+Figure&nbsp;1: Pressure state at time-step <span class="math inline">\(k=3\)</span> (second row) and a comparative analysis of pressure outputs (first and third) from the digital twin at time-step <span class="math inline">\(k=4\)</span>
 </figcaption>
 </figure>
 </div>
 <div id="fig-fracture" class="quarto-figure quarto-figure-center quarto-float anchored">
 <figure class="quarto-float quarto-float-fig figure">
 <div aria-describedby="fig-fracture-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-<a href="./figs/Figure2.png" class="lightbox" data-glightbox="description: .lightbox-desc-2" data-gallery="quarto-lightbox-gallery-2"><img src="./figs/Figure2.png" class="img-fluid figure-img" style="width:90.0%"></a>
+<a href="./figs/Figure2.png" class="lightbox" data-gallery="quarto-lightbox-gallery-2" data-glightbox="description: .lightbox-desc-2"><img src="./figs/Figure2.png" class="img-fluid figure-img" style="width:90.0%"></a>
 </div>
 <figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-fracture-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 Figure&nbsp;2: Injection rate samples and KDE curve(a), CDF curve of fracture probability versus injection rate with 95% confidence interval(b)
@@ -317,13 +317,13 @@ <h2 class="anchored" data-anchor-id="references">References</h2>
 Thelen, Adam, Xiaoge Zhang, Olga Fink, Yan Lu, Sayan Ghosh, Byeng D Youn, Michael D Todd, Sankaran Mahadevan, Chao Hu, and Zhen Hu. 2022. <span>“A Comprehensive Review of Digital Twin—Part 1: Modeling and Twinning Enabling Technologies.”</span> <em>Structural and Multidisciplinary Optimization</em> 65 (12): 354.
 </div>
 <div id="ref-yu2023IMAGEmsc" class="csl-entry" role="listitem">
-Yu, Ting-ying, Rafael Orozco, Ziyi Yin, Mathias Louboutin, and Felix J. Herrmann. 2023. <span>“Monitoring Subsurface CO2 Plumes with Sequential Bayesian Inference.”</span> <a href="https://slimgroup.github.io/IMAGE2023/SequentialBayes/abstract.html">https://slimgroup.github.io/IMAGE2023/SequentialBayes/abstract.html</a>.
+Yu, Ting-ying, Abhinav Prakash Gahlot, Rafael Orozco, Ziyi Yin, Mathias Louboutin, and Felix J. Herrmann. 2023. <span>“Monitoring Subsurface CO2 Plumes with Sequential Bayesian Inference.”</span> <a href="https://slimgroup.github.io/IMAGE2023/SequentialBayes/abstract.html">https://slimgroup.github.io/IMAGE2023/SequentialBayes/abstract.html</a>.
 </div>
 </div>
 
 
 <div class="hidden" aria-hidden="true">
-<span class="glightbox-desc lightbox-desc-1">Figure&nbsp;1: Pressure state at time-step <span class="math inline">\(k=3\)</span> and a comparative analysis of pressure outputs from the digital twin at time-step <span class="math inline">\(k=4\)</span></span>
+<span class="glightbox-desc lightbox-desc-1">Figure&nbsp;1: Pressure state at time-step <span class="math inline">\(k=3\)</span> (second row) and a comparative analysis of pressure outputs (first and third) from the digital twin at time-step <span class="math inline">\(k=4\)</span></span>
 <span class="glightbox-desc lightbox-desc-2">Figure&nbsp;2: Injection rate samples and KDE curve(a), CDF curve of fracture probability versus injection rate with 95% confidence interval(b)</span>
 </div>
 </section>
@@ -907,7 +907,7 @@ <h2 class="anchored" data-anchor-id="references">References</h2>
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diff --git a/Rex2024SEG/paper.html b/Rex2024SEG/paper.html
index 7392c21..dfc3b9f 100644
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     </div>
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 </footer>
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diff --git a/erdinc2024SEG/abstract.html b/erdinc2024SEG/abstract.html
index 6d5ed59..f1fdb80 100644
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@@ -324,7 +324,7 @@ <h2 class="anchored" data-anchor-id="methodology">Methodology</h2>
 <div id="fig-sup" class="quarto-figure quarto-figure-center quarto-float anchored">
 <figure class="quarto-float quarto-float-fig figure">
 <div aria-describedby="fig-sup-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
-<a href="figures/IMAGE2024_figure1_new.png" class="lightbox" data-glightbox="description: .lightbox-desc-1" data-gallery="quarto-lightbox-gallery-1"><img src="figures/IMAGE2024_figure1_new.png" class="img-fluid figure-img"></a>
+<a href="figures/IMAGE2024_figure1_new.png" class="lightbox" data-gallery="quarto-lightbox-gallery-1" data-glightbox="description: .lightbox-desc-1"><img src="figures/IMAGE2024_figure1_new.png" class="img-fluid figure-img"></a>
 </div>
 <figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-sup-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 Figure&nbsp;1: Training-loss formulation: field observations (RTMs and wells) along with column-wise masks (<span class="math inline">\(\mathbf{A}\)</span>) are sampled initially. Conditioned on these masks, further sub-sampling masks (<span class="math inline">\(\mathbf{\widetilde{A}}\)</span>) are generated. The neural denoising model <span class="math inline">\(h_\theta\)</span>, informed by the sub-sampling masks, noisy sub-sampled well information and RTMs, is trained to reconstruct the complete, noise-free velocity model.
@@ -338,7 +338,7 @@ <h2 class="anchored" data-anchor-id="results">Results</h2>
 <div id="fig-unsup" class="quarto-figure quarto-figure-center quarto-float anchored">
 <figure class="quarto-float quarto-float-fig figure">
 <div aria-describedby="fig-unsup-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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+<a href="figures/IMAGE2024_figure2_new.png" class="lightbox" data-gallery="quarto-lightbox-gallery-2" data-glightbox="description: .lightbox-desc-2"><img src="figures/IMAGE2024_figure2_new.png" class="img-fluid figure-img"></a>
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 <figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-unsup-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
 Figure&nbsp;2: The denoising generative model is tested on two previously unseen seismic images. For each case, we fed the corresponding RTM into the network and generated velocity samples (200 samples for each example). Subsequently, we computed both the posterior mean and variance. To assess the accuracy of our model, we included the SSIM between the ground-truth to the posterior means
@@ -963,7 +963,7 @@ <h2 class="anchored" data-anchor-id="references">References</h2>
     </div>
   </div>
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diff --git a/search.json b/search.json
index 693a625..1ce146f 100644
--- a/search.json
+++ b/search.json
@@ -4,7 +4,7 @@
     "href": "GahlotLi2024SEG/paper.html",
     "title": "A Digital Twin for Geological Carbon Storage with Controlled Injectivity",
     "section": "",
-    "text": "Digital Twins refer to dynamic virtual replicas of subsurface systems, integrating real-time data and employing advanced generative Artificial Intelligence (genAI) methodologies, such as neural posterior density estimation via simulation-based inference and sequential Bayesian inference. Thanks to combination of these advanced Bayesian inference techniques, our approach is capable of addressing challenges of monitoring and controlling CO2 storage projects. These challenges include dealing with the subsurface’s complexity and heterogeneity (seismic and fluid-flow properties), operations optimization, and risk mitigation, e.g. via injection rate control. Because our Digital Twin is capable of handling diverse monitoring data, consisting of time-lapse seismic and data collected at (monitoring) wells, it entails a technology that serves as a platform to integrate seemingly disparate and siloed fields, e.g. geophysics and reservoir engineering. In addition, recent breakthroughs in genAI, allow Digital Twins to capture uncertainty in a principled way (Yu et al. 2023; Herrmann 2023; Gahlot et al. 2023). By employing training and inference recursively, the Digital Twin trains its neural networks on samples of the simulated current state—i.e., the CO2 saturation/pressure, paired with simulated imaged seismic and/or data collected at (monitoring) wells. These training pairs of the simulated state and simulated observations are obtained by sampling the posterior distribution, \\(\\mathbf{x}_{k-1}\\sim p(\\mathbf{x}_{k-1}\\vert \\mathbf{y}^\\mathrm{o}_{1:k-1})\\), at the previous timestep, \\(k-1\\), conditioned on field data, \\(\\mathbf{y}^\\mathrm{o}_{1:k-1}\\), collected over all previous timesteps, \\(1:k-1\\), followed by advancing the state to the current timestep, followed by simulating (seismic/well) observations associated with that state. Given these simulated state-observation pairs, the Digital Twin’s networks are trained, so they are current and ready to produce samples of the posterior when the new field data comes in—i.e. \\(\\mathbf{x}_{k}\\sim p(\\mathbf{x}_{k}\\vert \\mathbf{y}^\\mathrm{o}_{1:k})\\). While this new neural approach to data assimilation for CO2 storage projects provides what is called an uncertainty-informed Digital Shadow, it lacks decision making and control, which would make it a Digital Twin (Thelen et al. 2022), capable of optimizing storage operations while mitigating risks including the risk of fracturing the cap rock by exceeding the fracture pressure. The latter risk is illustrated in Figure 1, where the first row contains simulated samples of the differential pressure at timestep, \\(k=4\\), without control. These samples for the simulated state exceed the fracture pressure and are denoted by the red areas. During this talk, we will demonstrate how the Digital Twin can make informed decisions to avoid exceeding the fracture pressure.",
+    "text": "Digital Twins refer to dynamic virtual replicas of subsurface systems, integrating real-time data and employing advanced generative Artificial Intelligence (genAI) methodologies, such as neural posterior density estimation via simulation-based inference and sequential Bayesian inference. Thanks to combination of these advanced Bayesian inference techniques, our approach is capable of addressing challenges of monitoring and controlling CO2 storage projects. These challenges include dealing with the subsurface’s complexity and heterogeneity (seismic and fluid-flow properties), operations optimization, and risk mitigation, e.g. via injection rate control. Because our Digital Twin is capable of handling diverse monitoring data, consisting of time-lapse seismic and data collected at (monitoring) wells, it entails a technology that serves as a platform to integrate seemingly disparate and siloed fields, e.g. geophysics and reservoir engineering. In addition, recent breakthroughs in genAI, allow Digital Twins to capture uncertainty in a principled way (Yu et al. 2023; Herrmann 2023; Gahlot et al. 2023). By employing training and inference recursively, the Digital Twin trains its neural networks on samples of the simulated current state—i.e., the CO2 saturation/pressure, paired with simulated imaged seismic and/or data collected at (monitoring) wells. These training pairs of the simulated state and simulated observations are obtained by sampling the posterior distribution, \\(\\mathbf{x}_{k-1}\\sim p(\\mathbf{x}_{k-1}\\vert \\mathbf{y}^\\mathrm{o}_{1:k-1})\\), at the previous timestep, \\(k-1\\), conditioned on field data, \\(\\mathbf{y}^\\mathrm{o}_{1:k-1}\\), collected over all previous timesteps, \\(1:k-1\\), followed by advancing the state to the current timestep, followed by simulating (seismic/well) observations associated with that state. Given these simulated state-observation pairs, the Digital Twin’s networks are trained, so they are current and ready to produce samples of the posterior when the new field data comes in—i.e. \\(\\mathbf{x}_{k}\\sim p(\\mathbf{x}_{k}\\vert \\mathbf{y}^\\mathrm{o}_{1:k})\\). While this new neural approach to data assimilation for CO2 storage projects provides what is called an uncertainty-informed Digital Shadow, it lacks decision making and control, which would make it a Digital Twin (Thelen et al. 2022), capable of optimizing storage operations while mitigating risks including the risk of fracturing the cap rock by exceeding the fracture pressure. The latter risk is illustrated in Figure 1, where the first row contains simulated samples of the pressure difference at timestep \\(k=4\\), between the reservoir pressure and hydraulic pressure, without control. These samples for the simulated state exceed the fracture pressure and are denoted by the red areas. During this talk, we will demonstrate how the Digital Twin can make informed decisions to avoid exceeding the fracture pressure.",
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-    "text": "Digital Twins refer to dynamic virtual replicas of subsurface systems, integrating real-time data and employing advanced generative Artificial Intelligence (genAI) methodologies, such as neural posterior density estimation via simulation-based inference and sequential Bayesian inference. Thanks to combination of these advanced Bayesian inference techniques, our approach is capable of addressing challenges of monitoring and controlling CO2 storage projects. These challenges include dealing with the subsurface’s complexity and heterogeneity (seismic and fluid-flow properties), operations optimization, and risk mitigation, e.g. via injection rate control. Because our Digital Twin is capable of handling diverse monitoring data, consisting of time-lapse seismic and data collected at (monitoring) wells, it entails a technology that serves as a platform to integrate seemingly disparate and siloed fields, e.g. geophysics and reservoir engineering. In addition, recent breakthroughs in genAI, allow Digital Twins to capture uncertainty in a principled way (Yu et al. 2023; Herrmann 2023; Gahlot et al. 2023). By employing training and inference recursively, the Digital Twin trains its neural networks on samples of the simulated current state—i.e., the CO2 saturation/pressure, paired with simulated imaged seismic and/or data collected at (monitoring) wells. These training pairs of the simulated state and simulated observations are obtained by sampling the posterior distribution, \\(\\mathbf{x}_{k-1}\\sim p(\\mathbf{x}_{k-1}\\vert \\mathbf{y}^\\mathrm{o}_{1:k-1})\\), at the previous timestep, \\(k-1\\), conditioned on field data, \\(\\mathbf{y}^\\mathrm{o}_{1:k-1}\\), collected over all previous timesteps, \\(1:k-1\\), followed by advancing the state to the current timestep, followed by simulating (seismic/well) observations associated with that state. Given these simulated state-observation pairs, the Digital Twin’s networks are trained, so they are current and ready to produce samples of the posterior when the new field data comes in—i.e. \\(\\mathbf{x}_{k}\\sim p(\\mathbf{x}_{k}\\vert \\mathbf{y}^\\mathrm{o}_{1:k})\\). While this new neural approach to data assimilation for CO2 storage projects provides what is called an uncertainty-informed Digital Shadow, it lacks decision making and control, which would make it a Digital Twin (Thelen et al. 2022), capable of optimizing storage operations while mitigating risks including the risk of fracturing the cap rock by exceeding the fracture pressure. The latter risk is illustrated in Figure 1, where the first row contains simulated samples of the differential pressure at timestep, \\(k=4\\), without control. These samples for the simulated state exceed the fracture pressure and are denoted by the red areas. During this talk, we will demonstrate how the Digital Twin can make informed decisions to avoid exceeding the fracture pressure.",
+    "text": "Digital Twins refer to dynamic virtual replicas of subsurface systems, integrating real-time data and employing advanced generative Artificial Intelligence (genAI) methodologies, such as neural posterior density estimation via simulation-based inference and sequential Bayesian inference. Thanks to combination of these advanced Bayesian inference techniques, our approach is capable of addressing challenges of monitoring and controlling CO2 storage projects. These challenges include dealing with the subsurface’s complexity and heterogeneity (seismic and fluid-flow properties), operations optimization, and risk mitigation, e.g. via injection rate control. Because our Digital Twin is capable of handling diverse monitoring data, consisting of time-lapse seismic and data collected at (monitoring) wells, it entails a technology that serves as a platform to integrate seemingly disparate and siloed fields, e.g. geophysics and reservoir engineering. In addition, recent breakthroughs in genAI, allow Digital Twins to capture uncertainty in a principled way (Yu et al. 2023; Herrmann 2023; Gahlot et al. 2023). By employing training and inference recursively, the Digital Twin trains its neural networks on samples of the simulated current state—i.e., the CO2 saturation/pressure, paired with simulated imaged seismic and/or data collected at (monitoring) wells. These training pairs of the simulated state and simulated observations are obtained by sampling the posterior distribution, \\(\\mathbf{x}_{k-1}\\sim p(\\mathbf{x}_{k-1}\\vert \\mathbf{y}^\\mathrm{o}_{1:k-1})\\), at the previous timestep, \\(k-1\\), conditioned on field data, \\(\\mathbf{y}^\\mathrm{o}_{1:k-1}\\), collected over all previous timesteps, \\(1:k-1\\), followed by advancing the state to the current timestep, followed by simulating (seismic/well) observations associated with that state. Given these simulated state-observation pairs, the Digital Twin’s networks are trained, so they are current and ready to produce samples of the posterior when the new field data comes in—i.e. \\(\\mathbf{x}_{k}\\sim p(\\mathbf{x}_{k}\\vert \\mathbf{y}^\\mathrm{o}_{1:k})\\). While this new neural approach to data assimilation for CO2 storage projects provides what is called an uncertainty-informed Digital Shadow, it lacks decision making and control, which would make it a Digital Twin (Thelen et al. 2022), capable of optimizing storage operations while mitigating risks including the risk of fracturing the cap rock by exceeding the fracture pressure. The latter risk is illustrated in Figure 1, where the first row contains simulated samples of the pressure difference at timestep \\(k=4\\), between the reservoir pressure and hydraulic pressure, without control. These samples for the simulated state exceed the fracture pressure and are denoted by the red areas. During this talk, we will demonstrate how the Digital Twin can make informed decisions to avoid exceeding the fracture pressure.",
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     "title": "A Digital Twin for Geological Carbon Storage with Controlled Injectivity",
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-    "text": "Results\nTo calculate injection rates that mitigate the risk of exceeding the fracture pressure, we proceed as follows. First, because the optimized injection rates are close to the fracture pressure, we consider these optimization as approximations to the injection rates where the fracture pressure are exceeded. Next, Kernel Density Estimation (KDE) is applied to produce the smooth red curve in Figure 2(a). This smoothed probability function is used to calculate the Cumulative Density Function (CDF), plotted in Figure 2(b). Using the fact that non-fracture/fracture occurrence entails a Bernoulli distribution, confidence intervals can be calculated, \\(\\pm Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\hat{p}(1 - \\hat{p})}{128}}\\) where \\(\\hat{p}\\) represents the CDF (blue line) and \\(Z_{\\frac{\\alpha}{2}}=1.96\\) with \\(\\alpha=0.05\\). From the CDF and confidence intervals (denoted by the grey areas), the following conclusions can be drawn: First, if the initial injection rate of \\(q_3 = 0.0500 m^3/s\\) is kept, the fracture probability lies between 24.47 – 40.71% (vertical dashed line) and has a maximum likelihood of 32.59%, which are all way too high. Second, if we want to limit the fracture occurrence rate to 1% (red dashed line), then we need to lower the injection rate to \\(q_3=0.0387\\mathrm{m^3/s}\\). To ensure the low fracture occurrence rate of 1%, the reduced injection rate is chosen as the smallest injection rate within the confidence interval. As can be observed from Figure 2(b), lowering the injection rate avoids exceeding the fracture pressure at the expense of injecting less CO2. Out of 128 samples, 43 samples are fractured with the initial injection rate, while only one sample is fractured with the controlled injection rate.",
+    "text": "Results\nTo calculate injection rates that mitigate the risk of exceeding the fracture pressure, we proceed as follows. First, because the optimized injection rates are close to the fracture pressure, we consider these optimization as approximations to the injection rates where the fracture pressure are exceeded. Next, Kernel Density Estimation (KDE) is applied to produce the smooth red curve in Figure 2(a). This smoothed probability function is used to calculate the Cumulative Density Function (CDF), plotted in Figure 2(b). Using the fact that non-fracture/fracture occurrence entails a Bernoulli distribution, confidence intervals can be calculated, \\(\\pm Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\hat{p}(1 - \\hat{p})}{128}}\\) where \\(\\hat{p}\\) represents the CDF (blue line) and \\(Z_{\\frac{\\alpha}{2}}=1.96\\) with \\(\\alpha=0.05\\). From the CDF and confidence intervals (denoted by the grey areas), the following conclusions can be drawn: First, if the initial injection rate of \\(q_3 = 0.0500 m^3/s\\) is kept, the fracture probability lies between 24.47 – 40.71% (vertical dashed line) and has a maximum likelihood of 32.59%, which are all way too high. Second, if we want to limit the fracture occurrence rate to 1% (red dashed line), then we need to lower the injection rate to \\(q_3=0.0387\\mathrm{m^3/s}\\). To ensure the low fracture occurrence rate of 1%, the reduced injection rate is chosen as the smallest injection rate within the confidence interval. As can be observed from Figure 1 (third row) and Figure 2(b), lowering the injection rate avoids exceeding the fracture pressure at the expense of injecting less CO2. Out of 128 samples, 43 samples are fractured with the initial injection rate, while only one sample is fractured with the controlled injection rate.",
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     "title": "A Digital Twin for Geological Carbon Storage with Controlled Injectivity",
     "section": "Conclusion and discussions",
-    "text": "Conclusion and discussions\nThe above example illustrates how Digital Twins can be used to mitigate risks associated with CO2 storage projects. Specifically, we used the Digital Twin’s capability to produce samples of its state (pressure), conditioned on observed seismic and/or well data. Using these samples, in conjunction with samples from the permeability distribution, we were able to capture statistics on the fracture occurrence frequency as a function of the injection rate. Given these statistics, we were in a position to set a fracture frequency and choose the corresponding injection rate as a function of the confidence interval. By following this procedure, exceeding the fracture pressure was avoided by lowering the injection rate. The decision to lower the injection rate, and by which amount, was informed by the Digital Twin, which uses seismic and/or well data to capture reservoir’s state including its uncertainty.\n\n\n\n\n\n\nFigure 1: Pressure state at time-step \\(k=3\\) and a comparative analysis of pressure outputs from the digital twin at time-step \\(k=4\\)\n\n\n\n\n\n\n\n\n\nFigure 2: Injection rate samples and KDE curve(a), CDF curve of fracture probability versus injection rate with 95% confidence interval(b)",
+    "text": "Conclusion and discussions\nThe above example illustrates how Digital Twins can be used to mitigate risks associated with CO2 storage projects. Specifically, we used the Digital Twin’s capability to produce samples of its state (pressure), conditioned on observed seismic and/or well data. Using these samples, in conjunction with samples from the permeability distribution, we were able to capture statistics on the fracture occurrence frequency as a function of the injection rate. Given these statistics, we were in a position to set a fracture frequency and choose the corresponding injection rate as a function of the confidence interval. By following this procedure, exceeding the fracture pressure was avoided by lowering the injection rate. The decision to lower the injection rate, and by which amount, was informed by the Digital Twin, which uses seismic and/or well data to capture reservoir’s state including its uncertainty.\n\n\n\n\n\n\nFigure 1: Pressure state at time-step \\(k=3\\) (second row) and a comparative analysis of pressure outputs (first and third) from the digital twin at time-step \\(k=4\\)\n\n\n\n\n\n\n\n\n\nFigure 2: Injection rate samples and KDE curve(a), CDF curve of fracture probability versus injection rate with 95% confidence interval(b)",
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     "href": "GahlotLi2024SEG/paper.html#references",
     "title": "A Digital Twin for Geological Carbon Storage with Controlled Injectivity",
     "section": "References",
-    "text": "References\n\n\nGahlot, Abhinav Prakash, Huseyin Tuna Erdinc, Rafael Orozco, Ziyi Yin, and Felix J. Herrmann. 2023. “Inference of CO2 Flow Patterns  a Feasibility Study.” https://doi.org/10.48550/arXiv.2311.00290.\n\n\nHerrmann, Felix J. 2023. “President’s Page: Digital Twins in the Era of Generative AI.” The Leading Edge 42 (11): 730–32.\n\n\nThelen, Adam, Xiaoge Zhang, Olga Fink, Yan Lu, Sayan Ghosh, Byeng D Youn, Michael D Todd, Sankaran Mahadevan, Chao Hu, and Zhen Hu. 2022. “A Comprehensive Review of Digital Twin—Part 1: Modeling and Twinning Enabling Technologies.” Structural and Multidisciplinary Optimization 65 (12): 354.\n\n\nYu, Ting-ying, Rafael Orozco, Ziyi Yin, Mathias Louboutin, and Felix J. Herrmann. 2023. “Monitoring Subsurface CO2 Plumes with Sequential Bayesian Inference.” https://slimgroup.github.io/IMAGE2023/SequentialBayes/abstract.html.\n\n\n\n\n\nFigure 1: Pressure state at time-step \\(k=3\\) and a comparative analysis of pressure outputs from the digital twin at time-step \\(k=4\\)\nFigure 2: Injection rate samples and KDE curve(a), CDF curve of fracture probability versus injection rate with 95% confidence interval(b)",
+    "text": "References\n\n\nGahlot, Abhinav Prakash, Huseyin Tuna Erdinc, Rafael Orozco, Ziyi Yin, and Felix J. Herrmann. 2023. “Inference of CO2 Flow Patterns  a Feasibility Study.” https://doi.org/10.48550/arXiv.2311.00290.\n\n\nHerrmann, Felix J. 2023. “President’s Page: Digital Twins in the Era of Generative AI.” The Leading Edge 42 (11): 730–32.\n\n\nThelen, Adam, Xiaoge Zhang, Olga Fink, Yan Lu, Sayan Ghosh, Byeng D Youn, Michael D Todd, Sankaran Mahadevan, Chao Hu, and Zhen Hu. 2022. “A Comprehensive Review of Digital Twin—Part 1: Modeling and Twinning Enabling Technologies.” Structural and Multidisciplinary Optimization 65 (12): 354.\n\n\nYu, Ting-ying, Abhinav Prakash Gahlot, Rafael Orozco, Ziyi Yin, Mathias Louboutin, and Felix J. Herrmann. 2023. “Monitoring Subsurface CO2 Plumes with Sequential Bayesian Inference.” https://slimgroup.github.io/IMAGE2023/SequentialBayes/abstract.html.\n\n\n\n\n\nFigure 1: Pressure state at time-step \\(k=3\\) (second row) and a comparative analysis of pressure outputs (first and third) from the digital twin at time-step \\(k=4\\)\nFigure 2: Injection rate samples and KDE curve(a), CDF curve of fracture probability versus injection rate with 95% confidence interval(b)",
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