Fluid injection into low-permeability rock at great depth is commonly used to develop enhanced geothermal systems. Such injections can induce complex, intermittent slip on pre-existing faults, leading to seismicity that manifests as events of varying magnitude distributed in space and time. However, reproducing the rich statistical response of real faults, in particular the intermittency of slip, generally requires computationally demanding simulations, which motivates reduced-order approaches.

Using the Bueckner–Rice weight-function theory (Rice, 1985; Bueckner, 1987; Rice, 1989), we construct reduced-order frameworks for frictionless heterogeneous and frictional homogeneous cracks. Both problems are studied within the common context of understanding fluid-induced seismicity, with the aim of developing single-fault models that can simulate aseismic and microseismic slip during fluid injection.

To address intermittency, we present a variational model of three-dimensional coplanar frictionless crack propagation in a disordered fracture-energy field under mixed-mode I+II+III loading. The approach bridges the variational formulation of Francfort and Marigo (1998) and the perturbation theory of Rice (1985). Equilibrium crack-front configurations are obtained by minimizing the total energy, defined as the sum of (i) the elastic potential energy, evaluated asymptotically from front deformations, and (ii) the dissipated energy, determined by the fracture-energy field. The numerical framework relies on a matrix-free trust-region Newton–conjugate-gradient algorithm with physics-based preconditioning and reproduces the transition from smooth to intermittent crack growth in multiscale simulations.

Fault slip is, however, governed by friction, which the previous model overlooked. We therefore consider three-dimensional coplanar propagation of a frictional crack along a homogeneous fault. To construct a reduced-order model, we extend the perturbation theory of Rice (1985) to Coulomb friction. The resulting first-order semi-analytical framework is used to explore the influence of Poisson’s ratio and of the injection scenario on the quasi-elliptic shape of the aseismic slip front. Finally, we outline the main components required to recover the first-order slip displacement and stress fields on the frictional interface, which are needed to account for the spatial heterogeneities in fault properties.