Propagation and arrest of multiple radial thermal fractures transverse to a horizontal well

Transverse thermal fractures in horizontal wells can be induced by cold fluid injection into high-temperature formations during drilling, geothermal production, CO₂ storage, etc. As these fractures propagate, some are arrested due to stress interaction, resulting in a hierarchical fracture pattern. This study investigates the propagation and arrest of radial transverse thermal fractures in a horizontal well driven by 1-D radial heat conduction using an axisymmetric model. We derived a new elasticity equation for multiple radial thermal fractures in a horizontal well, and developed the dimensionless governing equations, in terms of dimensionless fracture penetration depth $L$, spacing $D$, aperture $\Omega$, time $\tau$, and two model parameters (dimensionless net confining stress $T$ and wellbore radius $A$) through parameter scaling. Displacement discontinuity method was employed to discretize the governing equations and the dimensionless solutions [$L\left(\tau ,T,A\right)$, $D\left(\tau ,T,A\right)$, $\Omega \left(\tau ,T,A\right)$] for the critical states at fracture arrests were solved through stability analysis. The fully transient solutions with stepwise fracture spacing were then obtained to predict the hierarchical fracture pattern. Our findings indicate that the solutions for the radial transverse fractures are asymptotic to those for half-plane thermal fractures at early time and the evolution of fracture penetration depth approaches to a scaling law $L=f\left(T,A\right){\tau }^{\left(1-T\right)/2}$ at late time. Application to a real geothermal site showed that thermal fractures reach depths of 0.48, 3.48, and 28.29 m, spacings of 0.46, 2.34, and 13.20 m, and apertures (at the wellbore wall) of 0.37, 1.53, and 6.15 mm at 1, 100 and 10,000 days of cooling, respectively.