## TKRISK EXAMPLES

###### COVID-19 VACCINE RESPONSE

# In this tutorial we demonstrate how one can use TKRISK to analyze vaccine response to COVID-19. We create a simple model of contamination and show how vaccine and exposition can jointly affect the probability of being infected.

We review with this example the logic and applicability of Bayesian networks. We are able to closely match the analytical solution thanks to efficient sampling methods including low discrepancy sequences. This is the first of a three parts demo.

In the second part, we demonstrate how to update a model based on acquired knowledge. We refine our initial model (built in Part 1) based on prior knowledge. We show how easily information collected in news publications about cases numbers and vaccine efficacy can be integrated in the model. We show how this added information helps improve our original estimations on the probability of being infected. We also review how we can set specific nodes values and obtain the probability of being infected if vaccinated or not. This option allows to test scenarios.

In the third part, we demonstrate how to refine a model to include more priors and posteriors. We keep enriching the model (from part 1 and 2) by adding more nodes and building a practical case study. We show that prior information on demographics, age, job category or risk level can impact the chances of getting infected and eventually result in more cases. We also add posterior nodes on the severity and contagiousness of an infected person based on these priors and review how these eventually affect the overall cost of care. This simple yet informative scenario can be used by public health policymakers or healthcare insurance to establish the best vaccination strategies. We explore three vaccination strategies and evaluate their respective impact. This is the third of a three parts demo.

###### CARBON SEQUESTRATION AND STORAGE

In this video, we describe how TKRISK can be used to quantify the risk associated with carbon sequestration in geologic formations. The evaluation of underground storage capacity is essential to addressing liability questions such as the purchase of the mineral rights (or pore space) under the land area that may be impacted by sequestration activities. We use the following reference publication: "Reservoir characterization and lithostratigraphic division of the Mount Simon Sandstone (Cambrian): Implications for estimations of geologic sequestration storage capacity" by Cristian R. Medina and John A. Rupp. The paper describes the calculation of such capacity using geological, petrophysical and fluid properties. We model the storage capacity (defined as an explicit formula in the paper) using TKRISK's deterministic node. This example shows how one can build and calibrate a model based on both data and expert knowledge while propagating uncertainty through the nodes of the PGM. This demonstrates the versatility of Bayesian network to model risk for a variety of engineering applications.

This video expands the model built in Part 1 and explores how priors can be added to the model. It makes use of deterministic nodes to express explicit relationships (such as pressure or temperature vs depth) and uses calibration when constitutive laws cannot be explicitly derived. Such is the case of CO2 density as a function of pressure and saturation. The graph is also expanded to model total injection capacity based on a single source, an injection period and a number of wells. It concludes with an overall simulation of the land area under which the CO2 may fill the pore space and that would hence be liable for storage.

We quantify the evolution of the CO2 plume during the injection process. We use TKRISK to model the displacement of original fluids (brine, hydrocarbon) in the geological formation with injected CO2. We are able to quickly build a simple radial flow model of the injection process and propagate key uncertainties (average porosity, injection rate as an example) to model the radius of the CO2 plume as a function of time. This example shows how one can build a graph based model of a transient variable using time as a deterministic input.

In this section, we review how we can model the average saturation behind the front of CO2. In instances where the relationship between input and output is too complex or simply does not exist, we can use calibration methods to "tune" a model. TKRISK continuous nodes allow such approach. In the example of the average saturation in the plume, the theory of hyperbolic systems of conservation laws governs the displacement mechanism and hence the property of interest. We have a complex non-linear relationship between rock/fluid properties and the fractional flow tangent that will give the average saturation of CO2 that will itself drive the speed of expansion of the plume... We can use an existing model to build a multi-linear model of the average saturation using ordinary least square regression. This leads to a graph where the CO2 front speed propagation can be tied to the reservoir properties.