Modeling of thermally driven longitudinal fractures along a vertical well

Fluid injection via a vertical well into a high-temperature formation may induce multiple longitudinal thermal fractures, which may eventually transition to two-wing fractures during fracture propagation, depending on horizontal stress ratio $\kappa$. In this study, we develop a plane strain model with radial heat conduction to investigate either two-wing fractures under highly anisotropic stresses ${\kappa }_{ha}\gg 1$ or multiple fractures under isotropic stresses $\kappa =1$. The coupled dimensionless elasticity equation and criteria of fracture propagation (and arrest) are formulated, discretized, and solved iteratively (with two special algorithms). Two additional critical model parameters are dimensionless effective confining stress $T$ and wellbore radius $A$. The multiple-fracture solution of dimensionless fracture length $L$, angular spacing $D$, and aperture consists of solutions for competitive propagation of fractures with arrests in the near-wellbore region and the subsequent stable propagation of fractures away from the wellbore. Both the multiple-fracture and two-wing-fracture solutions accurately capture the early-time transient and late-time power-law changes with dimensionless time $\tau$, as verified numerically. The late-time fracture propagation follows scaling law $L=f\left(T,A,D\right){\tau }^{\left(1-2T\right)/2}$. These solutions and scaling laws can be used to well bound a general solution with $1<\kappa <{\kappa }_{ha}$ as demonstrated numerically for a geothermal site, for which the maximum fracture length reaches 0.45, 2.71, and 16.42 m in 1, 100 and 10,000 days of cooling, respectively. The applicability of the assumptions used in the theoretical and numerical analysis is discussed.