Scaling Behavior of Thermally Driven Fractures in Deep Low-Permeability Formations: A Plane Strain Model With 1-D Heat Conduction

Injection of cold fluids through/into deep formations may cause significant cooling, thermal stress, and possible thermal fracturing. In this study, the thermal fracturing of low-permeability formations under one-dimensional heat conduction was investigated using a plane strain model. Dimensionless governing equations, with dimensionless fracture length (Formula presented.), aperture (Formula presented.), spacing (Formula presented.), time (Formula presented.), and effective confining stress (Formula presented.), were derived. Solution of single thermal fracture was derived analytically, while solution of multiple fractures with constant (or dynamic) spacing were obtained using the displacement discontinuity method (and stability analysis). For single fracture, (Formula presented.) increases nonlinearly with (Formula presented.) and then transitions to scaling law (Formula presented.), indicating that late-time fracture length increases linearly with the square root of cooling time. For constantly spaced fractures, (Formula presented.) deviates from the single-fracture solution at a later (Formula presented.) for a larger (Formula presented.), showing slower propagation under inter-fracture stress interaction. For dynamically spaced fractures, fracture arrest induced by stress interaction was determined by the stability analysis; the fully transient solution provides evolution of dimensionless fracture length, spacing, aperture, and pattern; a similar scaling law, (Formula presented.) with (Formula presented.), obtained shows the effect of both stress interaction and fracture arrest. The solution and scaling law provide fast predictions for all reservoir and cooling conditions using (single) model parameter (Formula presented.). Application to a geothermal site with (Formula presented.) demonstrates that thermal fractures reach 0.67, 6.25, and 78.00 m in length, 0.49, 2.30, and 13.00 m in spacing, and 0.43, 2.09, and 12.19 mm in aperture at 1, 100, and 10,000 days