1. Introduction
2. Modeling Trapping Mechanisms and Their Intrinsic Uncertainties
3. Uncertain Geological Scenarios and Results
4. Discussion
5. Conclusions
Nomenclature
Acknowledgments
Author Contributions
Ethics Statement
Informed Consent Statement
Funding
Declaration of Competing Interest
References

Author Information

PETROBRAS, Rio de Janeiro, RJ 20231-030, Brazil

Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712, USA

*

Authors to whom correspondence should be addressed.

Received: 05 December 2023 Accepted: 10 January 2024 Published: 16 January 2024

© 2024 The authors. This is an open access article under the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).

ABSTRACT:
Numerical simulation is a widely used tool for studying CO_{2} storage in porous media. It enables the representation of trapping mechanisms and CO_{2} retention capacity. The complexity of the involved physicochemical phenomena necessitates multiphase flow, accurate fluid and rock property representation, and their interactions. These include CO_{2} solubility, diffusion, relative permeabilities, capillary pressure hysteresis, and mineralization, all crucial in CO_{2} trapping during carbon storage simulations. Experimental data is essential to ensure accurate quantification. However, due to the extensive data required, modeling under uncertainty is often needed to assess parameter impacts on CO_{2} trapping and its interaction with geological properties like porosity and permeability. This work proposes a framework combining laboratory data and stochastic parameter distribution to map uncertainty in CO_{2} retention over time. Published data representing solubility, residual trapping, and mineral trapping are used to calibrate prediction models. Geological property variations, like porosity and permeability, are coupled to quantify uncertainty. Results from a saline sandstone aquifer model demonstrate significant variation in CO_{2} trapping, ranging from 17% (P10 estimate) to 56% (P90), emphasizing the importance of considering uncertainty in CO_{2} storage projects. Quadratic response surfaces and Monte Carlo simulations accurately capture this uncertainty, resulting in calibrated models with an R-squared coefficient above 80%. In summary, this work provides a practical and comprehensive framework for studying CO_{2} retention in porous media, addressing uncertainty through stochastic parameter distributions, and highlighting its importance in CO_{2} storage projects.

Keywords:
CO_{2 }trapping mechanisms; CCS; Uncertainty analysis; Proxy models; Saline aquifer

Scenarios considered by international entities such as the International Agency (IEA) point out the complexity of the actions needed to achieve the goals established for reducing emissions of Greenhouse Gases, in the coming decades, simultaneously with the desired transformation in energy generation sources. The report “World Energy Outlook 2022” [1] emphasizes that multiple technologies and energy sources will play an essential role in providing energy resources for the planet in a more sustainable scenario. Among the various options for reducing CO_{2} emissions, Carbon Capture and Storage (CCS) presents itself as a technology with significant potential to reduce CO_{2} emissions in a probable scenario of continued use of fossil fuels in the coming decades.
A CCS project involves CO_{2} capture from high-emission industries and injecting it into geological formations such as saline aquifers and depleted hydrocarbon reservoirs. One of the most critical barriers to long-term and large-volume CO_{2} storage in geological formations is the proof of safe and reliable storage. For that, CO_{2} can take advantage of different trapping mechanisms in porous media [2,3,4,5,6]; for example, free CO_{2} migration is controlled by the structural and stratigraphic trapping exerted by the caprock during the short-term encompassing the injection time, known as a primary mechanism [7]. Part of that mobile CO_{2} will be dissolved in the water (solubility trapping) as time passes. This CO_{2} solubility trapping is more pronounced in low-salinity brine, high-pressure, and low-temperature conditions [8,9]. The rising plume changes CO_{2} saturation leading to an increase in capillary trapped volume due to hysteresis in relative permeability and capillarity [10]. Some CO_{2} can be trapped as minerals because of brine pH reduction by CO_{2} injection of the brine and the mineralogy of the rock [11], on a timescale of more than 100,000 years [5]. Therefore, those additional mechanisms increase storage security when the buoyant CO_{2} is immobile in the pore space or no longer exists as a free phase (generally, as super-critical CO_{2}).
Modeling CO_{2} retention in porous media depends on the reservoir rock properties, such as porosity and permeability, as well as intrinsic parameters related to each trapping mechanism. These parameters can be measured in lab experiments using reservoir cores and reservoir brine samples. This data helps reduce the uncertainty in the amount of CO_{2} that will be trapped in a CCS project. However, it is important to note that a complete and single suite of lab data is not always available, or it may not fully capture the system’s heterogeneity, resulting in inherent modeling uncertainty [12]. Therefore, evaluating storage uncertainty involves assessing associated risks by considering modeling parameter uncertainties and system variability, enabling better understanding and management of CO_{2} storage risks.
To address the evaluation of uncertainties in the context of CCS, various authors have focused on assessing uncertainties related to geological property distributions [13,14], such as porosity and permeability. Some studies have gone further by incorporating intrinsic parameters of CO_{2} trapping in their evaluations [15,16,17,18]. For instance, Jammoul et al. [17] conducted a history-matching study to reproduce experimental data while considering uncertainties. Similarly, Likanapaisal et al. [18] investigated the influence of rock and fluid properties, as well as grid size, on the shape of the CO_{2} plume. In their work, they specifically examined uncertainties related to the geometry of the CO_{2} plume, simplifying the modeling of trapping mechanisms by adopting a black-oil model.
In this study, a framework for uncertainty analysis in CO_{2} retention in porous media is proposed. It integrates petrophysical properties and comprehensive trapping modeling, utilizing data collected from published laboratory studies. By doing so, it is possible to accurately quantify both trapped and free CO_{2} volumes, enabling a thorough analysis at a field scale with simulations spanning hundreds of years (long-term assessment). This is in contrast to previous studies [17], which focused on short-term redistribution over a few days in a smaller-scale lab model.
Our study also emphasizes the importance of conducting risk assessments in geological carbon storage projects. It takes a comprehensive approach to characterize CO_{2} trapping mechanisms, including solubility, residual, and mineral trapping. This is achieved by leveraging lab data to establish distributions and relationships between parameters and geological properties. This unique approach sets our study apart from other studies conducted by different authors [15,16].

```latex
\text{ln}(H_i)=\text{ln}(H_i^∗)+\frac{\overline{v_i}}{RT}(p-p^∗)
```

```latex
S_{gt}=\frac{S_{g\;max}}{1+CS_{g\;max}}
```

```latex
P_{c,d}=(Max\;P_c -P_c^{S_{g \;crit}})(\frac{1-S_{g\;crit}-S_w}{1-S_{g\;crit}-S_{wr} })^{exp,d}+Min\;P_{c,d}
```

```latex
P_{c,i}=(Max\;P_c)(\frac{1-S_{gt}-S_w}{1-S_{gt}-S_{wr} })^{exp,i}
```

OH^{−} + H^{+} = H_{2}O

CO_{2} + H_{2}O = H^{+} + HCO_{3}^{−}

CO_{3}^{2−} + H^{+} = HCO_{3}^{−}

Calcite [CaCO_{3}] + H^{+} = Ca^{2+} + HCO_{3}^{−}

After presenting the trapping modeling, it is worth stating the main assumptions in this paper.
• No geomechanical modeling;
• Pure CO_{2} stream is injected, e.g., without impurities and free water content;
• No dry-out effect modeling due to water vaporization with CO_{2} injection;
• No rock wettability changes;
• No dependence of the diffusion on brine salinity;
• Simplified geochemical modeling, only considering the calcite dissolution reaction;
• The focus on the technical subsurface phenomena related to CO_{2} retention in porous media. It does not include the assessment of risks and uncertainties associated with surface processes [46], process efficiency [47,48], or economic aspects [49].**Table 3. **Kinetic parameters of TST model for calcite reaction.

The final evaluation of the CO_{2} storage potential in the sandstone aquifer was performed using a model based on actual seismic and well data from three wells in the Campos Basin, located in the state of Rio de Janeiro, Brazil. The aquifer covers both onshore and offshore areas. The sand grains in the aquifer are primarily composed of quartz and feldspars, with a notable presence of granitic lithoclasts, which are mechanically formed and deposited fragments of rock derived from older rocks. The average porosity of the sandstone is approximately 20%, and it is primarily generated through secondary processes such as matrix contraction, grain dissolution, or grain fracturing. The average permeability of the aquifer is around 2 Darcy.
To evaluate the CO_{2} storage potential, 75 different petrophysical properties (some examples are given in Figure 7) are considered. Different realizations of permeability and porosity are generated to capture the variability and uncertainty in these properties (Personal communication with PETROBRAS, 2023) and final CO_{2} retention.
These simulations provide insights into the potential capacity and effectiveness of CO_{2} storage in the sandstone aquifer, considering the range of geological and trapping properties observed in the region. This information is crucial for assessing the feasibility and long-term stability of CCS projects in the Campos Basin, contributing to the overall understanding of CO_{2} storage potential in this area.
The reservoir model has a pore volume of 150 billion m³. However, for the specific analysis in this work, a sector model from the offshore region with a smaller pore volume of approximately 160 million m³ is considered. The gridblock size is 100 m × 100 m horizontally and 5 m thick. The grid refinement technique based on the workflow proposed by Machado et al. [41] is used to refine the grid near the CO_{2} injection well. The refinement is implemented to decrease the gridblock size from 100 m to 25 m near the injector. This adjustment aimed to improve the accuracy of representing the injection process and the trapping of CO_{2}, whether as an aqueous component or a residual phase in porous media, or as a free supercritical fluid.
The injection was evaluated for 20 years, from 2025 to 2045, with a constant injection rate of 1.0 million metric tonnes per year. A vertical injector was located in the center of the model in the bottom layers of the aquifer. It is important to note that in the selected area of the field, there is no evidence of an existing caprock (a layer of impermeable rock that can act as a barrier to prevent CO_{2} leakage to out of the storage zone, such as a shallow freshwater aquifer). As a result, the main risk associated with this project is the buoyancy of CO_{2} and the potential for leakage. Therefore, in this case, without the primary entrapment, it is crucial to address and quantify the uncertainty in the amount of CO_{2} trapped by the secondary mechanisms (mainly, by solubility and hysteresis) in the aquifer to assess the feasibility and safety of the project.
Other properties of this actual reservoir model are summarized in Table 4.
Table 5 provides a summary of the uncertainty range considered for the trapping properties.**Table 4. **Summary of reservoir properties.A numerical experiment was conducted with 400 simulations using Latin Hypercube design [50]. These runs combined 75 heterogeneous realizations of permeability and porosity, the ratio of *K*_{v}/*K*_{h} distribution, and the trapping parameters listed in Table 5. The numerical experiment took approximately 1.5 days to complete the simulations, utilizing 20 parallel jobs on a computer cluster with 40 processors (Intel® Xeon® Gold 6230 CPU @ 2.10GHz, 104 GB RAM). Additionally, another set of 1000 runs was carried out to validate the response surface approach. These simulations required an additional 2 days to complete. The four objective functions (OF) that were evaluated are as follows:
• **II**: the CO_{2} injectivity index after 20 years of injection;
• **Trapped_shutoff**: percentage of the total CO_{2} injected that is trapped due to solubility, residual, and mineralization after the injection stops;
• **Trapped_30yr**: percentage of the total CO_{2} injected that is trapped after 30 years of redistribution;
• **Trapped_100yr**: percentage of the total CO_{2} injected that is trapped after 100 years of redistribution.**Table 5. **Uncertainty range of the trapping properties.**Figure 7. **Cross-sectional view with four examples of x-permeability (**A**) and porosity (**B**) scenarios. The injector's location is indicated by the blue box.

Table 6 summarizes the simulation results (SR) and from 65,000 Monte Carlo simulations (MCS) using quadratic response surfaces [51] as proxy models to represent the SR for each objective function (OF). The last column in the table displays the respective R-squared (*R*^{2}) coefficient, which measures the reduction in the variability of the response achieved by using the regressor variables in the model.
It is worth noting that the maximum amount of CO_{2} retention is reached after 30 years of redistribution. As a result, the uncertainty level calculated based on the ratio of P90 and P10 estimates, remains relatively constant at around 4 (in the 400-run case) and 3.5 (in the 1000-run case). In this context, the P90 represents the optimistic estimate for CO_{2} storage, while the P10 represents the pessimistic estimate.
The errors between the SR and MCS results are found to be lower than 30%, with an *R*^{2} higher than 80%, when considering 400 runs. These error levels are commonly observed in reservoir simulation applications, as mentioned in the study by Cremon et al. [52]. Therefore, they were considered sufficiently accurate for this study, considering the numerous variables involved.
Overall, the results obtained from the SR and MCS analyses, along with the *R*^{2} coefficients, indicate that the proxy models built using quadratic response surfaces are reliable and provide satisfactory estimates for the objectives of this study.**Table 6. **Stochastics estimates for the Objective Functions using SR and MCS, associated errors, P90/P10, and *R*^{2}.The results indicate that the proxy models accurately predict the P50 estimate for the amount of CO_{2} trapped and the injectivity index. However, there are significant discrepancies between the P10 and P90 estimates predicted by MCS and those obtained from the cumulative probability distribution derived from the SR in Figure 8. These differences are particularly notable after 30 and 100 years of CO_{2} redistribution, with 400 simulation runs. This discrepancy leads to a reduction in the *R*^{2} from 95% to 83% for proxy models. To address this issue and better represent cases with lower retention (close to the P10 estimate) and higher retention (close to the P90), additional simulation runs are performed. However, it requires a larger number of simulations in the order of one thousand to achieve satisfactory agreement with the results from other authors [52,53]. Nevertheless, it is worth noting that despite the need for a larger number of simulations, the P50 estimate is well predicted by the proxy model even with the limited number of 400 runs. The P50 estimate is typically considered for project evaluation and approval.**Figure 8. **Cumulative probability functions from SR (solid lines) and MCS (dashed lines) for 400 (in blue) and 1000 runs (in orange).Table 7 presents the distribution of CO_{2} within the porous media for each uncertain scenario after 100 years. These distributions are calculated using MCS with a 1000-run case, and their respective *R*^{2} values are also provided. The table reveals that in most scenarios, the majority of the CO_{2} remains in a free state as a supercritical fluid. This finding highlights the potential risk of CO_{2} leakage to the surface in CCS projects, particularly in geological scenarios where the caprock cannot effectively act as a barrier. In these scenarios, when there is a larger amount of free CO_{2} present (P10), its plume has the potential to rise to shallower depths. This increases the risk associated with operations in areas that lack a caprock. Figure 9 illustrates the vertical distribution of CO_{2} as a free phase in models that represent the three different estimates: P10 (minimum plume depth: 1569 m), P50 (2114 m), and P90 (2320 m). This information obtained from the uncertainty analysis is highly relevant as it highlights the potential risk of the CO_{2} plume (under free CO_{2} form) undesirably coming into contact with shallow aquifers or even the sea floor (or surface in onshore targets). It underscores the importance of considering and managing these risks in CO_{2} storage projects. The P90 case represents the lower-risk scenario, as more than 50% of the injected CO_{2} is trapped, as shown in Table 7. Consequently, the free CO_{2} plume remains at deeper depths (Figure 9). It is important to note that in the timeframe studied here, there is no significant CO_{2} mineralized.**Figure 9. **Depth of CO_{2} plume rise from equivalent simulation models for P10 (in red), P50 (in green), and P90 (in blue) estimates._{2} trapped over a span of 120 years. The green symbols in Figure 10 represent simulation results corresponding to the P50 estimate, while the vertical lines represent the range between the P10 and P90 equivalent models. These equivalent models are simulation models that yield results similar to those obtained using the proxy model estimation. The plot highlights that the uncertainty during the injection period is relatively small since the contributions from secondary trapping mechanisms of solubility, capillarity, and mineralization are insignificant. However, as the injection ceases, the importance of these mechanisms gradually increases as CO_{2} redistributes within the porous media, which is consistent with Bachu [7]. This observation indicates that the uncertainty in CO_{2} trapping becomes more pronounced and influential over time after the injection period concludes. In the P50 estimate, it is observed that a plateau is reached from year 50 onwards, indicating the maximum amount of CO_{2} trapped through solubility and hysteresis. This suggests that 30 years of CO_{2} redistribution was sufficient to achieve the maximum entrapment for the given system. However, it should be noted that this plateau may vary if more optimistic (P90) or pessimistic (P10) trapping properties are assumed.
MCS can also be employed to investigate the most influential parameters for CO_{2} retention. The effect estimated in a Tornado plot quantifies how a parameter (and some modification in its value) can impact the objective function (OF). By definition, the effect of an individual parameter on the OF is referred to as a main (or first-order) effect. However, interaction effects consider the combinations of model parameters that cannot be described by the first-order effects only [54]. In this particular problem, interactions are important and integral aspects of the models, such as those between relative permeability (*k*_{r}) and capillary pressure (*P*_{c}) parameters such as residual saturations and relative permeability endpoint and exponents (*S*_{gt}, *S*_{wr}, *S*_{g crit}, *n*_{g}, and Max CO_{2} *k*_{r}, as well as the petrophysical properties (*K*, *ф*, and *K*_{v}/*K*_{h}) derived from the geological models.
Figure 11 presents the Top-7 most significant parameters and their associated interactions (represented by orange bars), which add up to 100% for the “Trapped_100yr” OF. If a linear response surface is employed as a proxy model, the *R*^{2} value reduces from 86% to 66% (in the case of 1000 runs), indicating the importance of capturing the interaction effects represented by the quadratic surface. This finding aligns with the conclusions of Khanal and Shahriar [16], who employed advanced machine-learning techniques to construct proxy models. Although their models had higher *R*^{2} values than the results of this work, they considered only seven input parameters and made simplified assumptions, i.e., no uncertainty in solubility, no capillarity effects, and the use of a single set of relative permeability curves.**Figure 10. **Evolution of the % CO_{2} trapped over 20 years of injection and 100 years of redistribution. Green dots are the P50 estimate, the vertical black lines represent the variation between P10 and P90.**Figure 11. **Tornado plot for the Objective Function “Trapped_100 yr” generated with MCS calibrated with 1000 runs.In Figure 11, the solubility parameter *H*_{i}^{∗} did not appear as a significant property for the objective function “Trapped_100yr”. This can be attributed to the narrow range (lower uncertainty) of the solubility parameter in the 58,000 ppm case, where the model prediction aligns well with the laboratory data, as shown in Figure 1. However, if a higher salinity case (230,000 ppm) is used, where the difference between the initial prediction and the data is more significant (as depicted in Figure 2), *H*_{i}^{∗} will have a main effect of 3.7% on the “Trapped_100yr” objective function, indicating a significant uncertainty in solubility. These observations emphasize the importance of evaluating the uncertainty in the solubility model calculations.
Figure 12 displays the Tornado Plot for “II” at the time of injection shutoff, specifically regarding CO_{2} injectivity. These results support the findings of Machado et al. [35], which concluded that residual CO_{2} saturation does not significantly impact injectivity. Essentially, the injectivity of CO_{2} is influenced by its effective permeability, which is determined by the product of the maximum CO_{2} relative permeability (Max CO_{2} *k*_{r}) and the absolute permeability in both the horizontal (K) and vertical directions. The vertical permeability plays a critical role in enhancing CO_{2} buoyancy, which in turn facilitates its redistribution within porous media and increases its saturation. This amplification of CO_{2} saturation ultimately leads to an enhanced mobility of CO_{2} within the porous media, positively impacting its injectivity. The significant influence of geological properties on injectivity emphasizes the importance of the proposed methodology, which integrates different geological scenarios. Furthermore, Figure 12 indicates that the reactive calcite area, denoted as A, does not have a substantial effect because of small percentage of calcite of 1.1% present in the reservoir rock.
The steps followed in the proposed workflow can be summarized by referring to the flowchart presented in Figure 13. The flowchart illustrates the integration of geological scenarios and trapping mechanisms, supported by experimental measurements, to map the fate of CO_{2} in porous media. This comprehensive approach enables a more precise assessment of the potential impacts of geological and trapping properties on CO_{2} injectivity and entrapment. Ultimately, it facilitates informed decision-making in CCS projects.**Figure 12. **Tornado plot for the OF “II” generated with MCS calibrated with 1000 runs.**Figure 13. **Flowchart with main steps for uncertainty analysis in CCS projects.

Figure 10 depicts the evolution of the percentage of CO

In conclusion, the uncertainty analysis conducted in this study, despite being based on a specific target with a geological model under different scenarios, allows for the expansion and generalization of the proposed workflow. This analysis provides valuable insights for project evaluation, particularly regarding the main subsurface risks associated with CCS operations. The main conclusions of this study can be summarized as follows:
Regarding the experimental data:
• Laboratory-measured data for parameters that control CO_{2} retention in porous media should be used in order to guide the input distribution for uncertainty assessment. This helps to more accurately map their impacts on the resulting trapped CO_{2} over time.
Regarding the uncertainty assessment on the CO_{2} trapping:
• The workflow employed in this study successfully mapped the uncertainty associated with the amount of CO_{2} trapped. This mapping is crucial for determining appropriate mitigation measures to address the risk of leakage in storage targets where the caprock may not serve as an effective barrier. It enables the evaluation of the feasibility of CCS operations in such targets and aids in making informed decisions.
• The results obtained from a geological model of an actual saline sandstone aquifer demonstrated significant variations in CO_{2} trapping, ranging from 17% for the P10 estimate to 56% for the P90 estimate. In the P50 estimate, the CO_{2} entrapment reached a plateau of approximately 29% after 30 years following the cessation of injection. This highlights the importance of considering uncertainty during the risk assessment and approval phases of CO_{2} storage projects.
• Proxy models constructed using quadratic response surfaces were able to accurately represent the objective functions used to evaluate CO_{2} injectivity and retention over different time periods in the P50 estimate. These models performed well even with a relatively small number of runs (400), making them valuable tools for decision-making.
• Although the proxy models for the P10 and P90 estimates can benefit from further improvements, obtaining a larger number of additional simulations (more than 600 cases) can be challenging, especially for larger models.
• Tornado plots revealed a strong interplay between reservoir properties (such as permeability - *K* -, and the ratio of vertical to horizontal permeability - *K*_{v}/*K*_{h}) and dynamic petrophysical parameters (such as *S*_{gt}, *S*_{wr}, *S*_{g crit}, *n*_{g}, and Max CO_{2} *k*_{r}) in determining the amount of CO_{2} trapped as an objective function. This underscores the importance of an integrated workflow for uncertainty and risk evaluation.
Regarding the uncertainty assessment on the CO_{2} injectivity:
• The low reactivity of minerals in the rock matrix led to a negligible impact of mineral trapping, primarily attributed to calcite dissolution. However, this influence was not significant enough to affect CO_{2} injectivity;
• By coupling the methodology with geological considerations, a more comprehensive understanding of the injectivity process can be achieved. The results highlight the greater influence of geological properties, such as the *K*_{v}/*K*_{h} ratio and absolute permeability, compared to trapping parameters.
For future research, the authors recommend and intend to expand the geochemical modeling in the uncertainty evaluation to apply to targets where the matrix contains more reactive minerals. In this regard, the modeling of water vaporization and halite scale should be considered for a more representative and comprehensive study.

The authors express their gratitude to PETROBRAS for granting permission to use data and models in this study. They would also like to acknowledge Karim Salaheddine for his valuable contributions in collecting geochemical data and engaging in insightful discussions; and Dr. Chowdhury Mamun for thorough English revision.

Conceptualization, M.V.B.M., M.D. and K.S.; methodology, M.V.B.M.; software, M.V.B.M.; validation, M.V.B.M., M.D. and K.S.; formal analysis, M.V.B.M., M.D. and K.S.; investigation, M.V.B.M.; resources, M.V.B.M. and M.D.; data curation, M.V.B.M. and M.D.; writing—original draft preparation, M.V.B.M.; writing—review and editing, M.D. and K.S.; visualization, M.V.B.M.; supervision, M.D. and K.S.; project administration, M.D. and K.S. All authors have read and agreed to the published version of the manuscript.

Not applicable.

Not applicable.

This research received no external funding.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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