Closed-loop geothermal systems: Modeling and predictions

TL;DR

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Abstract

Geothermal energy is a sustainable baseload source recognized for its ability to provide clean energy on a large scale. Advanced Geothermal Systems (AGS) -- offer promising prototypes -- employ a closed-loop vascular layout that runs deep beneath the Earth's surface. A working fluid (e.g., water or supercritical carbon dioxide (sCO2)) circulates through the vasculature, entering the subsurface at the inlet and exiting at the outlet with an elevated temperature. For designing and performing cost-benefit analysis before deploying large-scale projects and maintaining efficiency while enabling real-time monitoring during the operational phase, modeling offers cost-effective solutions; often, it is the only available option for performance assessment. A knowledge gap exists due to the lack of a fast predictive modeling framework that considers the vascular intricacies, particularly the jumps in the solution fields across the channel. Noting that the channel diameter is considerably smaller in scale compared to the surrounding geological domain, we develop a reduced-order modeling (ROM) framework for closed-loop geothermal systems. This ROM incorporates the jump conditions and provides a quick and accurate prediction of the temperature field, including the outlet temperature, which directly correlates with the power production capacity and thermal draw-down. We demonstrate the predictive capabilities of the framework by establishing the uniqueness of the solutions and reporting representative numerical solutions. The modeling framework and the predictions reported in this paper benefit the closed-loop geothermal community, enabling them to determine the system's performance and optimal capacity.

Figures (10)

• Figure 1: The figure illustrates the average power production and temperature profile of geothermal systems featuring U-shaped and comb-shaped closed loop layouts. The geothermal gradient is approximately 30 K/km. Fluid enters the system at the inlet, travels through the channel, and exits the outlet at an elevated temperature. At lower mass flow rates, such as 5 kg/s and 10 kg/s, the U-shaped layout provides better thermal power production compared to comb-shaped layout. However, as the mass flow rate increases, the average power generated by the comb-shaped layout increases significantly. Therefore, the right combination of vascular layout and mass flow rate helps in achieving an efficient closed-loop geothermal system.
• Figure 1: A geothermal system: The figure illustrates a standard closed-loop geothermal system. A coolant, typically water, circulates through an underground network of channels at depths ranging from 10,000 to 20,000 feet, extracting hot water. This heated water undergoes a heat exchange process, transforming into steam. Utilizing this steam, the geothermal facility generates electricity by propelling a turbine. An electric grid facilitated by a network of transmission towers distributes the produced electricity to urban centers. The process releases warm water, which undergoes additional cooling in a dedicated cooling tower, becoming available for other energy consumption (e.g., heating and cooling, agriculture, etc). Additionally, the figure highlights a collaborative approach involving a solar and wind farm to address the region's energy demand.
• Figure 2: Vascular layouts: This graphic depicts two commonly utilized vascular configurations, both in practical applications and within the context of this paper: A) a U-shaped layout devoid of branching, and B) a comb-shaped arrangement featuring branches. $\Sigma$ denotes the vascular layout, $\widehat{\mathbf{t}}(\mathbf{x})$ the unit tangent vector pointing toward increasing arc-length along the vasculature, and $\widehat{\mathbf{n}}^{\pm}(\mathbf{x})$ the unit outward normal vectors on either side of the vasculature. The fluid (coolant) enters the domain at the inlet and exits at the outlet. As the fluid travels through the length of the vasculature, it extracts heat from the surrounding subsurface, leading to an increase in its thermal energy---either raising the temperature or latent heat.
• Figure 3: Test problem. A) The domain is a cuboid of size $L_x \times L_y \times L_z$, featuring a vascular layout within its subsurface. The top face is free to convect and radiate, while the remaining faces (i.e., lateral and bottom faces) are subject to a Dirichlet boundary condition (BC). B) The prescribed temperature increases linearly with depth at a slope of $m$.
• Figure 4: Domain size: This figure guides the selection of the domain size by observing its influence on the solution field. We consider a domain with depth $d$ and a horizontal dimension of $(1 + 2 \, \alpha)s$, where $s$ represents the spacing of the injection and production channels. For various values of $\alpha$ ranging from 0.5 to 2, we plotted the temperature at a depth of $0.7d$ after 63 years of geothermal system operation and for a mass flow rate of 35 kg/s. The solution profile remains mostly consistent within the chosen range of $\alpha$. Inference: As a result, we recommend setting the domain size to be twice the spacing between the vertical legs in the vascular layout, corresponding to $\alpha = 0.5$.

Theorems & Definitions (4)

• Theorem 4.1: Uniqueness neglecting radiation
• proof
• Theorem 4.2: Uniqueness considering radiation
• proof