A new functional form of the heat transfer coefficient for use in simulating EGS processes at the field scale.
Authors: Detournay christine , Radakovic-Guzina Zorica , Damjanac Branko 
Published: February 2023

Heat extraction by circulating a cold fluid in a hot fractured rock mass at depth is the central topic of study in geothermal engineering. Typical reservoir rocks have a low thermal conductivity, and heat exchange between rock and fluid occurs in a thin region, adjacent to the fracture, where the temperature gradient is very high. Capturing this effect is important for accurate predictions of transient fluid temperatures &#8212; a critical aspect of geothermal power systems.The model assumes a temperature jump at the rock/fracture-fluid contact (collapsed boundary layer), and Newton&#8217;s law of coolingqcv = h (Trock -Tfluid )&#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; (1)is used to express the heat exchanged by forced convection between media. The heat transfer coefficient, h has a significant impact on the results of EGS numerical modeling. A pragmatic expression is proposed whereby h is proportional to the rock thermal conductivity, kr and inversely proportional to the square root of the product of rock diffusivity, &#954;, and fluid injection time, t (Detournay C. et al., 2022):&#160; &#160; &#160;h = kr / (&#946;&#8730;&#954;t)&#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; (2)The novelty is that h is primarily a function of rock thermal properties and only indirectly dependent of fracture fluid velocity. Also, Eq. (1) combined with Eq. (2) is the thermal equivalent of Carter&#8217;s equation for 1D leak-off flow. The logic, combined with heat advection-forced convection, is implemented in the commercial hydraulic fracturing code XSite and coupled with mechanical, fracture flow, and heat conduction.Borehole injection of cold water in a penny-shaped pre-existing fracture with a 100 m diameter is simulated. Fluid extraction occurs at constant downhole pressure. Fluid-thermo-mechanical coupling is considered.&#160;Figure 1. Fluid temperature contour (&#176;C) at 1 year.Fluid temperature contours in Figure 1 show an oval cooled-off region surrounding the injection well and a &#8220;dead zone&#8221; near the producing well where the fluid temperature stays close to the warm original (rock) value. The warm fluid, initially present in the thin fracture, is produced and rapidly replaced by the injected fluid. The fluid temperature gradient between wells is caused by the migration of heat from rock to fluid.REFERENCEDetournay C., Damjanac B., Torres M., Cundall P., Ligocki L., Gil, I., 2022. Heat advection and forced convection in a lattice code &#8211; Implementation and geothermal applications. Rock mechanics Bulletin I (2022) 100004.&#160;