A Probabilistic Study of CO₂ Plume Geothermal and Hydrothermal Systems: A Sensitivity Study of Different Reservoir Conditions in Williston Basin, North Dakota

Abstract

The exploration of alternative energy sources has gained significant traction in recent years, driven by the urgent need to mitigate greenhouse gas emissions and transition towards sustainable energy. Among these alternatives, ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems have emerged as promising options due to their potential for providing clean, renewable energy. This study presents a probabilistic investigation into the sensitivity of ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems under various reservoir conditions in the Williston Basin, North Dakota. In addition to elucidating the impact of reservoir conditions on system performance, the study utilizes probabilistic methods to assess energy output of ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems. Insights derived from this probabilistic investigation offer valuable guidance for the working fluid selection, systems design and optimization in the Williston Basin and beyond. Results from the sensitivity analysis reveal the profound influence of reservoir conditions on the behavior and efficiency of ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems. Our case study on Red River Formation and Deadwood Formations shows a potential of 34% increase and 32% decrease in heat extraction based on varying reservoir conditions. Our investigations in the Beaver Lodge field within the Red River Formation yielded arithmetic mean values for ${\mathrm{C}\mathrm{O}}_2$ best case resources, hydrothermal resources and the ${\mathrm{C}\mathrm{O}}_2$ worst case as 6.36 $\times {10}^{18}$ J, 4.75 $\times {10}^{18}$ J and 3.24 $\times {10}^{18}$ J, respectively. Overall, this research contributes to advancing the knowledge and understanding of ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems as viable pathways towards sustainable energy production and carbon sequestration. By highlighting the sensitivity of these systems to reservoir conditions, the study provides valuable insights that can inform decision-making processes and future research endeavours aimed at fostering the transition to a low-carbon energy landscape.

1. Introduction

1.1. Climate Change and Geothermal Energy

The burgeoning global population and its accompanying anthropogenic endeavors have significantly escalated the concentration of greenhouse gases in the atmosphere, with carbon dioxide (CO₂) being the primary culprit. The surge in atmospheric CO₂ concentrations can be predominantly attributed to human activities, notably the combustion of fossil fuels. This uptick in CO₂ levels has catalyzed climate change, which manifests in a plethora of adverse impacts worldwide. From shifting weather patterns to rising sea levels, the repercussions of climate change reverberate across the globe, necessitating urgent and concerted efforts to mitigate its effects. The impacts of global warming on the environment, including phenomena such as sea-level rise and extreme weather conditions, are intensifying due to the anthropogenic release of greenhouse gases, particularly CO₂ [1]. In response to this pressing challenge, the adoption of clean, low-carbon, energy-efficient technologies and renewable energy sources emerges as a critical strategy for mitigating emissions. Among these technologies, carbon capture and storage (CCS) stands out as a promising solution. CCS involves capturing CO₂ emissions from the flue gases of power plants and other large industrial facilities, followed by the storage of this captured CO₂ in carefully selected geological formations. This technology plays a vital role in curbing the environmental impacts of anthropogenic emissions, offering a pathway towards a more sustainable and climate-resilient future [2]. Furthermore, in 2017, CO₂ emissions from electricity generation accounted for a substantial 40% of global fuel combustion CO₂ emissions [3]. Consequently, it is increasingly evident that achieving climate goals necessitates most fossil fuel-based electricity generation to incorporate CO₂ capture and geological storage, utilizing saline formations and depleted oil and gas reservoirs [4]. Rather than solely storing CO₂ in sedimentary basins as part of carbon capture and storage (CCS) initiatives, an alternative approach involves circulating the captured CO₂ back to the land surface. This CO₂ can then be utilized to generate geothermal power within a CO₂ Plume Geothermal (CPG) system. This innovative method not only contributes to mitigating CO₂ emissions but also harnesses the energy potential of CO₂ to generate sustainable geothermal power [5,6,7,8].

1.2. Geothermal Energy Exploration in North Dakota

Nestled beneath the surface of western North Dakota lies a reservoir brimming with untapped potential, the Williston Basin, a rich source of geothermal energy. This abundant resource offers a promising avenue towards sustainable, renewable, and environmentally friendly heat and power solutions. Despite longstanding awareness of its existence, the full development of this resource has been impeded by economic competition from traditional fossil fuels. North Dakota’s strategic geographical location in the heart of the north-central United States, characterized by harsh continental climates, underscores the pressing need for reliable heating solutions throughout the year [9]. Central North Dakota, with its mean annual temperature of 4.3 °C and winter temperatures plummeting to an average of −12.4 °C from December through March, faces a constant demand for heating. Interestingly, even during the summer months from June to September, temperatures often dip below the base level, resulting in negative cooling degree-days. With abundant sedimentary resources lying dormant beneath the earth’s surface, North Dakota stands poised to embrace geothermal energy as a dependable source of energy [10,11,12]. Moreover, the exploration and utilization of this resource hold the promise of reducing reliance on fossil fuels, thereby fostering a more sustainable and environmentally conscious energy landscape in the state. Western North Dakota in particular boasts favorable conditions for the implementation of Enhanced Geothermal Systems [13,14]. Leveraging insights gained from deep oil drilling activities in the region, scientists and researchers have begun to unlock the geothermal potential of the Williston Basin formations [15,16,17,18]. Extensive studies dating back to the 1970s have delved into the basin’s heat flow and temperature distributions, with particular emphasis on the basal clastic Deadwood and Winnipeg [19,20]. The Red River Formation has an estimated thickness of over 214 m, temperatures of over 140 °C and permeability of 0.1 mD—38 mD (Hartig, 2018) [21].

Formed in the Ordovician period, the Red River Formation is the fourth layer within the Big Horn Group. It transitions into the Ordovician Winnipeg Shale, also known as the Roughlock Formation, and is overlain by the Stony Mountain Shale. The Red River Formation primarily comprises limestone, interspersed with lenses of dolomitic mudstone, and features an anhydrite cap at its center. The work of M. Sippel [22] delineated four porosity intervals, A, B, C, and D zones in the Red River’s upper reaches, with zones B and D being the most productive for oil extraction. The Beaver Lodge field within the Red River Formation has an estimated 4.0 $\times {10}^{18}$ J of thermal energy [23]. The Deadwood Formation is segmented into six subgroups identified by variations in Gamma-ray readings from well logs. These changes in Gamma-ray sensitivity are essential for differentiating the subgroups. The formation exhibits various lithological changes, although many are quite subtle. During the deposition of the Deadwood Formation, frequent minor transgressions and regressions occur. Consequently, the lithological features remain consistent throughout the Deadwood Formation’s stratigraphic column. This formation, dating back to the late Cambrian and early Ordovician periods, consists predominantly of dark, organic-rich shale, siltstone, and sandstone (See Figure 1 and Figure 2).

1.3. Geothermal Energy Systems (CO₂ Plume and Hydrothermal Systems)

Geothermal energy stands as a beacon of sustainability in the realm of renewable energy, offering a reliable and eco-friendly alternative to traditional fossil fuels. At its core, geothermal energy harnesses the Earth’s natural heat to generate electricity and provide heating and cooling solutions for a variety of applications. Hydrothermal systems stand as the stalwart foundation of geothermal energy production. These systems tap into naturally occurring reservoirs of hot water and steam beneath the Earth’s surface, harnessing the planet’s internal heat to generate power. The mechanism driving hydrothermal systems involves the circulation of water through fractured rock layers, where it is heated before returning to the surface in the form of hot water or steam. Brown [26] initially suggested CO₂ as a viable working fluid in Enhanced Geothermal Systems (EGSs). CO₂ presents three key advantages over brine: first, its high compressibility allows for the creation of a thermosiphon, potentially eliminating the need for circulation pumps [27,28,29]. Second, CO₂ boasts lower kinematic viscosity, reducing pressure losses through the reservoir rock [30,31]. Lastly, CO₂’s lower mineral solubility mitigates pipe and equipment scaling issues [26]. Studies have shown that CO₂ in EGS systems exhibits higher heat extraction rates compared to brine and can facilitate the development of a thermosiphon [28,29,31]. However, widespread deployment of CO₂-EGS faces challenges, notably limited CO₂ storage potential due to fractured reservoir volumes [32,33]). In contrast, Randolph and Saar [32] introduced a different CO₂-based geothermal system termed CO₂-Plume Geothermal (CPG) which operates in common sedimentary basins with low-permeability caprocks to contain buoyant CO₂ [34,35,36]. These basins offer extensive CO₂ storage potential and are targeted for Carbon Capture and Storage (CCS) efforts [2,37]. The superior heat extraction efficiency of CO₂ compared to water presents significant advantages, especially in regions characterized by low- to medium-grade geothermal heat resources. In such areas where conventional geothermal electricity production is economically unfeasible, the utilization of CO₂ could substantially expand geothermal electricity generation on a global scale. Additionally, this approach offers the added benefit of CO₂ sequestration, providing a solution to the pressing challenge of capturing and storing CO₂ emissions from sources such as fossil fuel power plants [32]. CPG, or CO₂-Plume Geothermal, entails the injection of supercritical CO₂ into deep geologic reservoirs that are naturally porous and permeable and are covered by low-permeability cap rock layers. These geological formations are widespread in the United States [2] and globally with approximately half of North America possessing such formations [38,39]. Economically viable sequestration sites have been identified in these regions, making them prime candidates for CPG deployment [40]. These sedimentary basins, which have been utilized for CO₂ disposal in various regions since the mid-1990s [41], are now central to ongoing efforts in CO₂ capture and sequestration (CCS) aimed at mitigating global climate change [2]. In the CPG process, CO₂ displaces native formation fluids such as brine or hydrocarbons, akin to standard CO₂ sequestration or enhanced oil recovery (EOR) practices. As it permeates the reservoir, the CO₂ is heated by both natural in situ heat and geothermal heat flux. A portion of the heated CO₂ is subsequently extracted to the surface and directed through an expansion device, thereby generating power for electrical generators or providing heat for direct use and/or binary power systems. Following this extraction, the CO₂ is reinjected into the reservoir, ensuring long-term storage of all injected CO₂ [32]. By sequestering CO₂ emissions underground, they directly contribute to reducing greenhouse gas emissions. Moreover, they indirectly facilitate further reductions by synergizing with other low-carbon technologies to meet energy demand (Fleming et al., 2022; Ogland-Hand et al., 2019) [42,43].

1.4. Probabilistic Assessment for Geothermal Studies

Utilizing a probabilistic methodology for geothermal assessment is highly important because it effectively manages the uncertainties associated with underground conditions and reservoir properties. This approach plays a crucial role in reducing risks, offering stakeholders valuable insights into the feasibility of projects and investment opportunities by addressing the intricate uncertainties inherent in geothermal exploration. It provides an energy profile by providing probabilities for a range of outcomes. The range of outcomes is of essence to Value at Risk models for economic reasons. Describing a geothermal system and assessing its potential for generating power or thermal energy is known as resource evaluation, marking the first stage in improving the commercial utilization of geothermal resources [44,45]. The estimation of the resource’s capacity often relies on numerical modeling or the volumetric heat storage method. Numerical modeling, extensively utilized across various sectors, is known for its ability to provide reliable calculations [44,45]. It enables heat transfer and fluid flow simulations within complex geothermal systems more effectively than the heat storage technique [46,47]. Assessing uncertainty in numerical model predictions entails conducting simulation experiments and constructing calibrated reservoir models through reverse modeling. This underscores the importance of probabilistic and uncertainty assessment. In South China, probabilistic geothermal resource assessment was conducted by integrating the Response Surface Methodology (RSM) and Experimental Design (ED) with Monte Carlo Simulation, resulting in the development of proxy numerical model [48]. The study demonstrated the operability, applicability, and reliability of ED-RSM in evaluating geothermal potential. Similarly, in the Dikili-Izmir Region of Western Turkey, a study assessed untapped geothermal potential for direct and indirect utilization. With a 50% probability, the study estimated a net electrical power potential of 75 MWe from Yuntdağ and 17 MWe from Kozak, highlighting Kozak’s substantially higher output potential. Additionally, a sensitivity analysis identified critical reservoir parameters affecting net power output while evaluating sustainability aspects from economic and environmental perspectives [49].

2. Methodology

From the North Dakota Industrial Commission [25], core data were collected for the Beaver lodge field. Five wells from the Beaver lodge field were utilized to generate mean and standard deviation for parametric probabilistic assessment. The parametric approach uses mean ($m_j$) and standard deviation ($s_j$) from the reservoir variables and converts them to lognormal standard deviation (${\beta }_j$) and lognormal mean (${\alpha }_j$), Equations (2) and (4), respectively. The sum of the lognormal mean (${\alpha }_j$) and lognormal variance (${\beta }^2$) equates to output lognormal mean (${\alpha }_{\Omega }$) and output lognormal variance (${{\beta }^2}_{\Omega }$) as shown in Equation (5). The Probabilistic Distribution Function (PDF) equation for a random variable x is shown in Equation (9). The lognormal distribution, PDF shown in Equation (10), is usually asymmetric and the skewness is dependent on standard deviation. Also, from the National Institute of Standards and Technology [50], thermo-physical properties were collected for both ${\mathrm{C}\mathrm{O}}_2$ and water. The thermo-physical properties of ${\mathrm{C}\mathrm{O}}_2$ and water are incorporated into Darcy’s flow equation to determine the thermal flux. The thermal flux of ${\mathrm{C}\mathrm{O}}_2$ and water, Equation (12), is then incorporated into Equation (11) to determine Producible geothermal resource.

2.1. Comparison of Reservoir Conditions for Energy Recovery

Red River Formation Case 1: Decreasing temperature and decreasing pressure, see

Table 1. Thermophysical Properties of water at reservoir conditions, Red River Formation. (Red River Formation Case 1).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
4000	138	941.87	4.2061	0.2066
3500	136	941.85	4.211	0.20892
3000	134	941.81	4.216	0.21133
2500	132	941.76	4.221	0.21384
2000	130	941.71	4.2261	0.21644

and

Table 2. Thermophysical Properties of ${\mathrm{C}\mathrm{O}}_2$ at reservoir conditions, Red River Formation. (Red River Formation Case 1).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
4000	138	489.89	1.863	0.039187
3500	136	437.26	1.8645	0.035302
3000	134	375.77	1.8182	0.031426
2500	132	307.16	1.7049	0.027855
2000	130	235.9	1.5343	0.024895

.

Red River Formation Case 2: Decreasing temperature and increasing pressure, see

Table 3. Thermo-physical Properties of water at reservoir conditions, Red River Formation. (Red River Formation Case 2).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
2000	138	934.97	4.2407	0.20315
2265	137	936.76	4.2341	0.20518
2500	136	938.42	4.2281	0.20719
3000	134	941.81	4.216	0.21133
3500	132	945.14	4.2044	0.21558
3690	130	947.39	4.1982	0.21939
4000	130	948.41	4.1933	0.21993

and

Table 4. Thermo-physical Properties of CO₂ at reservoir conditions, Red River Formation. (Red River Formation Case 2).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
2000	138	225.33	1.4743	0.024906
2265	137	263.78	1.5744	0.02626
2500	136	299.26	1.6624	0.027689
3000	134	375.77	1.8182	0.031426
3500	132	449.12	1.9052	0.035927
3690	130	479.1	1.931	0.037999
4000	130	514.89	1.9248	0.040817

.

Deadwood Formation Case 1: Decreasing temperature and decreasing pressure, see

Table 5. Thermo-physical Properties of water at reservoir conditions, Deadwood Formation. (Deadwood Formation Case 1).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
4000	140	940.2	4.2095	0.20351
4250	145	936.83	4.2142	0.1966
4500	150	933.39	4.2193	0.19019
4750	155	929.88	4.2247	0.18424
5000	160	926.32	4.2305	0.17871

and

Table 6. Thermo-physical Properties of ${\mathrm{C}\mathrm{O}}_2$ at reservoir conditions, Deadwood Formation. (Deadwood Formation Case 1).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
4000	140	484	1.8477	0.038827
4250	145	495.7	1.812	0.039957
4500	150	506.02	1.7792	0.041004
4750	155	515.19	1.7491	0.041978
5000	160	523.4	1.7216	0.042888

.

Deadwood Formation Case 2: Decreasing temperature and increasing pressure, see

Table 7. Thermo-physical Properties of water at reservoir conditions, Deadwood Formation. (Deadwood Formation Case 2).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
4000	160	922.68	4.2495	0.17707
4250	155	928.1	4.2338	0.18342
4500	150	933.39	4.2193	0.19019
4750	145	938.54	4.2058	0.19744
5000	140	943.56	4.1932	0.2052

and

Table 8. Thermo-physical Properties of CO₂ at reservoir conditions, Deadwood Formation. (Deadwood Formation Case 2).

Pressure (psi)	Temperature $(°\mathbf{C})$	Density (kg/m3)	Specific Heat Capacity (J/gK)	Viscosity (cp)
4000	160	432.45	1.7043	0.036069
4250	155	469.33	1.747	0.038411
4500	150	506.02	1.7792	0.041004
4750	145	542.08	1.8005	0.043822
5000	140	577.15	1.8124	0.046836

.

2.2. Probabilistic Approach for Geothermal Studies

$$V_j=\raisebox{1ex}{$s_j$}\!\left/ \!\raisebox{-1ex}{$m_j$}\right.$$

(1)

where $V_j$ is the coefficient of variation, $s_j$ is the standard deviation, and $m_j$ is the mean.

$${\beta }^2=In(1+V_J^2)$$

(2)

where ${\beta }^2$ is the lognormal variance.

$$u_j=\raisebox{1ex}{${\beta }_j^2$}\!\left/ \!\raisebox{-1ex}{$\sum {\beta }_j^2$}\right.$$

(3)

where $u_j$ is the Relative impact.

$${\mathsf{\alpha }}_{\mathrm{j}}=\mathrm{I}\mathrm{n}\left({\mathrm{m}}_{\mathrm{j}}\right)-0.5\times {\mathsf{\beta }}_{\mathrm{j}}^2$$

(4)

$${\mathsf{\alpha }}_{\mathrm{\Omega }}=\sum {\mathsf{\alpha }}_{\mathrm{j}}{{\mathsf{\beta }}^2}_{\mathrm{\Omega }}=\sum {{\mathsf{\beta }}^2}_{\mathrm{j}}$$

(5)

where ${\alpha }_j$ is the lognormal mean.

$$P90=\exp (\alpha -1.281\beta )$$

(6)

where P90 is Probability of at least 90%.

$$P50=\exp \left(\alpha \right)$$

(7)

where P50 is Probability of at least 50%.

$$P10=\exp (\alpha +1.281\beta )$$

(8)

where P10 is Probability of at least 10%.

$$f\left(x\right)=\frac{1}{\sqrt{2\pi s}}\exp (-\frac{1}{2}({\frac{x-m}{s})}^2)$$

(9)

where x is a random variable represented by normal distribution

$$f\left(x\right)=\frac{1}{x\sqrt{2\pi }\beta }\exp (-\frac{1}{2}({\frac{In\left(x\right)-\alpha }{\beta })}^2)$$

(10)

where x is a random variable represented by lognormal distribution

$$E=\rho S_w c_p v q \Delta T·(T_f-20)$$

(11)

where E is the Producible geothermal resource, ρ is density (kg/${\mathrm{m}}^3$), $c_p$ is heat capacity (J/kg/K), v is volume (${\mathrm{m}}^3$), q is flowrate (${\mathrm{m}}^3/\mathrm{h}$), $T_f$ is the reservoir fluid temperature ($℃$), and $T_a$ is annual mean temperature ($℃$).

$$Q_{\mathrm{C}\mathrm{O}2/water}=\frac{{\rho }_{\mathrm{C}\mathrm{O}2}}{{\rho }_w}\times \frac{{Cp}_{\mathrm{C}\mathrm{O}2}}{{Cp}_w}\times \frac{{\mu }_w}{{\mu }_{\mathrm{C}\mathrm{O}2}}$$

(12)

where Q is the Thermal Energy Flux (W/m²), $\mu$ is the viscosity (cp), $\rho$ is the density (kg/m³), $C_p$ is the specific heat capacity at constant pressure (J/gK)

See

Table 9. Reservoir Variables for Geothermal Resource, Beaver Lodge field.

Reservoir Variable	Mean	Standard Deviation
Area (${\mathrm{m}}^2$)	52,609,133	0
Gross thickness (m)	205.435	28.65
Density (kg/${\mathrm{m}}^3$)	2600	100
Temperature (°C)	130	10
Heat capacity (J/kg/K)	950	30
Water saturation (stb/rb)	0.73	0.05
Porosity (frac.)	0.084	0.0086
Flow rate (${\mathrm{m}}^3/\mathrm{h}$)	0.223	0.068

for Reservoir Variables for Geothermal Resource, Beaver Lodge field, and Figure 3 for Flowchart for Probabilistic Parametric Approach.

3. Results and Analysis

3.1. Thermal Heat Flux Results (Red River Formation)

The reservoir pressure and temperature were varied within reasonable ranges of the Red River Formation. The reservoir pressure and temperature were varied inversely to depict different thermal and pressure regimes within the formation. For the Red River Formation, reservoir pressure was the main component with a direct proportion on the thermal flux. Higher reservoir pressures resulted in higher thermal flux regardless of the reservoir temperature. It was observed that at certain temperatures and pressures, water preserves more momentum and has better extraction rates because the high specific heat capacity of water outperforms the low kinematic viscosity of ${\mathrm{C}\mathrm{O}}_2$. For the Deadwood Formation, reservoir temperature was the main component with a direct proportion on the thermal flux. Lower reservoir temperature resulted in higher thermal flux regardless of the reservoir pressure. Although not very often, water has better mobilities and extraction efficiency than ${\mathrm{C}\mathrm{O}}_2$ at specific reservoir conditions due high specific heat capacity. Within the concept of ${\mathrm{C}\mathrm{O}}_2$ sequestration, higher ${\mathrm{C}\mathrm{O}}_2$ thermal flux results in more fluid flow within the reservoir hence better storage. Also, when more ${\mathrm{C}\mathrm{O}}_2$ is stored within the reservoir, it makes the concept of ${\mathrm{C}\mathrm{O}}_2$ Plume Geothermal more feasible and economical especially with favorable reservoir temperatures such as the Red River Formation and Deadwood Formation.

Red River Formation Case 1: Decreasing Temperature and Decreasing Pressure, see

Table 10. Thermal Heat Flux of ${\mathrm{C}\mathrm{O}}_2$/water at reservoir conditions, Red River Formation. (Red River Formation Case 1).

Pressure (psi)	Temperature$(°\mathbf{C})$	Thermal Heat Flux $(\mathbf{W}/{\mathbf{m}}^2$)
4000	138	1.214588519
3500	136	1.216510498
3000	134	1.157102696
2500	132	1.011332025
2000	130	0.79069077

.

Red River Formation Case 2: Decreasing Temperature and Increasing Pressure, see

Table 11. Thermal Heat Flux of ${\mathrm{C}\mathrm{O}}_2$/water at reservoir conditions, Red River Formation. (Red River Formation Case 2).

Pressure (psi)	Temperature $(°\mathbf{C})$	Thermal Heat Flux $(\mathbf{W}/{\mathbf{m}}^2$)
2000	138	0.68341184
2265	137	0.818102651
2500	136	0.938216822
3000	134	1.157102696
3500	132	1.292082697
3690	130	1.342954147
4000	130	1.342738376

.

Deadwood Formation Case 1: Increasing Temperature and Increasing Pressure, see

Table 12. Thermal Heat Flux of ${\mathrm{C}\mathrm{O}}_2$/water at reservoir conditions, Deadwood Formation. (Deadwood Formation Case 1).

Pressure (psi)	Temperature $(°\mathbf{C})$	Thermal Heat Flux $(\mathbf{W}/{\mathbf{m}}^2$)
4000	140	1.184344311
4250	145	1.119416798
4500	150	1.060352724
4750	155	1.006749597
5000	160	0.95813409

.

Deadwood Formation Case 2: Decreasing Temperature and Increasing Pressure, see

Table 13. Thermal Heat Flux of ${\mathrm{C}\mathrm{O}}_2$/water at reservoir conditions, Deadwood Formation. (Deadwood Formation Case 2).

Pressure (psi)	Temperature $(°\mathbf{C})$	Thermal Heat Flux $(\mathbf{W}/{\mathbf{m}}^2$)
4000	160	0.922792148
4250	155	0.996407951
4500	150	1.060352724
4750	145	1.114032997
5000	140	1.158311088

.

3.2. Probabilistic Expectation Curve for Geothermal Resource

The Probabilistic Expectation Curve provides a graphical representation of expected probability distribution of thermal energy. It assumes input variables to be lognormally distributed which can be utilized to assess the feasibility of a project. From the expectation curve, the ${\mathrm{C}\mathrm{O}}_2$ best case generated most energy output followed by the hydrothermal system, followed by the ${\mathrm{C}\mathrm{O}}_2$ worst case. P90 has a 90% probability for the estimate. The P50 is usually the best estimate for parametric probabilistic approach, whereas the P10 is the most unlikely estimate in comparison to P90 and P50. From our studies on the Beaver Lodge field in the Red River Formation, the arithmetic means of the ${\mathrm{C}\mathrm{O}}_2$ best case, the hydrothermal system, followed by the ${\mathrm{C}\mathrm{O}}_2$ worst case were 6.360 $\times {10}^{16}$ J, 4.75 $\times {10}^{16}$ J, and 3.24 $\times {10}^{16}$ J, respectively. It was determined that there is a 34% increase in energy output and also a 32% decrease in energy output with ${\mathrm{C}\mathrm{O}}_2$ depending on the reservoir conditions. This information informs decision-making processes related to geothermal resource development and utilization (See Figure 4 and

Table 14. Summary of comparative statistical measure for geothermal resources, Beaver Lodge field.

Statistical Measure	Reserves	Parametric
	${\mathrm{C}\mathrm{O}}_2$ best case (MMJ)	3.74 × 1010
P90	Hydrothermal system (MMJ)	2.70 × 1010
	${\mathrm{C}\mathrm{O}}_2$ worst case (MMJ)	1.91 × 1010
	${\mathrm{C}\mathrm{O}}_2$ best case (MMJ)	5.96 × 1010
P50	Hydrothermal system (MMJ)	4.44 × 1010
	${\mathrm{C}\mathrm{O}}_2$ worst case (MMJ)	3.03 × 1010
	${\mathrm{C}\mathrm{O}}_2$ best case (MMJ)	9.48 × 1010
P10	Hydrothermal system (MMJ)	7.07 × 1010
	${\mathrm{C}\mathrm{O}}_2$ worst case (MMJ)	4.81 × 1010
	${\mathrm{C}\mathrm{O}}_2$ best case (J^2)	5.68 × 1032
Variance	Hydrothermal system (J^2)	3.17 × 1032
	${\mathrm{C}\mathrm{O}}_2$ worst case (J^2)	1.46 × 1032
	${\mathrm{C}\mathrm{O}}_2$ best case (J^2)	2.53 × 1010
Standard deviation	Hydrothermal system (J^2)	1.78 × 1010
	${\mathrm{C}\mathrm{O}}_2$ worst case (J^2)	1.21 × 1010
	CO2 best case (MMJ)	6.36 × 1010
Mean	Hydrothermal system (MMJ)	4.75 × 1010
	CO2 worst case (MMJ)	3.24 × 1010

).

4. Conclusions

Geothermal energy offers a clean and renewable energy in the Williston basin and the utilization of ${\mathrm{C}\mathrm{O}}_2$ for sequestration and energy extraction benefits both environment and investors. From accurate well testing and downhole hole temperature sensors, precise reservoir conditions could determine the selection of working fluids for geothermal exploration. The probabilistic studies provide an alternative to provide energy solutions due to its data-driven approach, cross-validation nature, accessibility and cost effectiveness. The thermal and pressure regimes vary in reservoirs over time due to geological factors, fluid flow, heat flow, and even seasonal changes, hence understanding the variations within a reservoir is essential for reservoir engineering, production optimization, and resource assessment.

Our case study on Red River and Deadwood Formation shows a potential 34% increase and a 32% decrease in heat extraction based on varying reservoir conditions. Our investigations in the Beaver Lodge field within the Red River Formation yielded arithmetic mean values for ${\mathrm{C}\mathrm{O}}_2$ best case resources, hydrothermal resources, and the ${\mathrm{C}\mathrm{O}}_2$ worst case as 6.36 $\times {10}^{18}$ J, 4.75 $\times {10}^{18}$ J, and 3.24 $\times {10}^{18}$ J, respectively. The probabilistic study of ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems in the Williston basin provided major insights into the sensitivity of reservoir conditions for energy extraction systems. It does not account for other key reservoir parameters due to complexity, and numerical simulation is recommended for future work. Randolph et Saar [32] recommended ${\mathrm{C}\mathrm{O}}_2$ plume geothermal for relatively low geothermal resources, and this study corroborated the results (especially Deadwood Formation). For the reservoir conditions in the Red River Formation, high-pressured regions are recommended for ${\mathrm{C}\mathrm{O}}_2$ plume geothermal exploration In summary, our study contributes to the growing body of knowledge on ${\mathrm{C}\mathrm{O}}_2$ plume geothermal and hydrothermal systems in the Williston basin and beyond.