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

Abstract

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 scriptL $\mathcal{L}$, aperture normalΩ ${\Omega}$, spacing scriptD $\mathcal{D}$, time τ $\tau $, and effective confining stress scriptT $\mathcal{T}$, 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, scriptL(τ,T) $\mathcal{L}(\tau ,\mathcal{T})$ increases nonlinearly with τ $\sqrt{\tau }$ and then transitions to scaling law scriptL=f(scriptT)τ $\mathcal{L}=f(\mathcal{T})\sqrt{\tau }$, indicating that late‐time fracture length increases linearly with the square root of cooling time. For constantly spaced fractures, scriptL(τ,T,D) $\mathcal{L}(\tau ,\mathcal{T},\mathcal{D})$ deviates from the single‐fracture solution at a later τ $\tau $ for a larger scriptD $\mathcal{D}$, 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, scriptL=f′(scriptT)τ $\mathcal{L}={f}^{\prime }(\mathcal{T})\sqrt{\tau }$ with f′(T)<f(T) ${f}^{\prime }(\mathcal{T})< f(\mathcal{T})$, 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 scriptT $\mathcal{T}$. Application to a geothermal site with scriptT=0.11 $\mathcal{T}=0.11$ 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.

Coalescence of thermal fractures initiated at parallel cooling surfaces

“…Thermal fractures are initiated at the walls of the hydraulic fractures and filled with the fluid under the same pressure pf. Fluid flow and heat convection in the low-permeability rock and thermal fractures and the associated poroelastic effect are assumed to be negligible (Chen and Zhou, 2022; Enayatpour et al. , 2019).…”

“…The material deformation, heat conduction from the cooling surfaces and mixed-mode fracture propagation need to be modeled with the governing equations listed below. Note that, for the case of low-permeability underground rock cooled by injected fluid, we ignored the heat conduction and convection in the fluid in newly generated thermal fractures and the heat convection in rock matrix (Chen and Zhou, 2022).…”

“…Thermal fractures are initiated at the walls of the hydraulic fractures and filled with the fluid under the same pressure p f . Fluid flow and heat convection in the low-permeability rock and thermal fractures and the associated poroelastic effect are assumed to be negligible (Chen and Zhou, 2022;Enayatpour et al, 2019). With these assumptions, we only need to model rock deformation, heat conduction from the hydraulic fractures and mixed-mode fracture propagation and arrest.…”

“…where σ ec is the effective confining stress σ H − p f ¼ 10 MPa, and ΔT s is the surface cooling T w − T r . The calculation of dimensionless solution for specific T and the conversion from (EGI, 2019;Robertson, 1988) Coalescence of thermal fractures dimensionless solutions to their dimensional counterparts is referred to Chen and Zhou (2022).…”

“…One of the observations was enhanced fluid injectivity that was attributed to injection-induced longitudinal thermal fractures along vertical injection wells or radial thermal fractures around horizontal injection wells under normal fault regime. Numerical modeling has been used to understand the propagation of thermal fractures and the field observed phenomena (Brudy and Zoback, 1999; Chen and Zhou, 2022; Detienne et al ., 1998; Morales et al ., 1986; Raterman et al ., 2018; Siratovich et al ., 2011; Svendson et al ., 1991; Williams et al ., 1987). For thermal fractures in a half space with a cooling surface (e.g.…”