John W. Rudnicki

• A novel approach is developed for modeling shear and compaction bands in porous rocks; • A micromechanics-inspired constitutive model is proposed by incorporating various microstructural evolution mechanisms; • A novel non-local formulation is proposed by considering the interaction effect of porosity distribution; • The proposed model depicts the main aspects of mechanical responses of porous rocks and related microstructural evolution; • The proposed approach describes well the various scenarios of shear and compaction bands under different confining pressures. This study develops a novel approach for modeling shear and compaction bands in porous rock-like materials. A micromechanics-inspired constitutive model is first formulated to describe the fundamental mechanical response of porous rocks by incorporating the evolution of microstructure during plastic deformation. The novel non-local formulation is based on the interaction effect of spatial porosity distribution. Within this approach, the void and crack nucleation drives the shear deformation and localization, while the pore collapse controls the volumetric compaction and localization. The new model is implemented in the standard ABAQUS platform, providing a robust tool for investigating the formation of shear-compaction bands under various loading conditions. The results demonstrate that the proposed model accurately reproduces the transition of macroscopic pressure-sensitive plastic behavior from being dominated by pore and crack nucleation to pore collapse as confining pressure increases, thereby correctly capturing the brittle-ductile transition in the mechanical behavior of porous rocks. The model successfully captures the transition from shear bands to compaction bands under varying confining pressures, along with the associated microstructural evolution. During shear band formation, softening behavior driven by pore and crack nucleation dominates within the band, while the region outside the band undergoes elastic unloading. In the formation of shear-enhanced compaction bands, neither pore and crack nucleation nor pore collapse prevails due to their competitive interaction, resulting in multiple inclined or wavy bands, with continuous plastic development of porosity both inside and outside the bands. In the formation of pure compaction bands, discrete bands perpendicular to the maximum principal stress direction are formed, with hardening behavior induced by pore collapse dominating within the band.

Abstract Spring block, and sometimes continuum, models of the effects of the coupling of fluid flow and frictional slip often employ the membrane approximation. This approximation assumes that the fluid flux between the slipping layer and a remote pore fluid reservoir is proportional to the difference between the value of pore pressure in the reservoir and in the slipping zone. In contrast, Darcy's law states that the fluid flux is proportional to the gradient of the pore pressure. We analyze and compare these two formulations by asymptotic analysis of the fluid flow equations and numerical simulations of a spring–block model using rate and state friction. This analysis shows that membrane diffusion agrees with homogeneous diffusion in the limit of undrained conditions and for nearly drained conditions. For homogeneous diffusion and essentially undrained conditions, both the asymptotic analysis and numerical simulations indicate the formation of a boundary layer near the shear zone where gradients of pore pressure are large. Outside this layer the pore pressure rapidly approaches the drained solution. A linearized stability analysis derives the dependence of the non‐dimensional critical stiffness () on fluid diffusivity, dilatancy factor (), and shear zone thickness. The homogeneous and membrane diffusion models exhibit nearly identical in the limits of drained and undrained conditions, but differ between the two limits. In this intermediate range, numerical simulations show that the two models produce similar slip behavior for but significant differences in slip velocity and recurrence patterns for .

Abstract Injection of pore fluid can substantially affect fault slip, often resulting in seismic activity. We simulate the axisymmetric compression experiments of Noël, Passelégue, et al. (2019), https://doi.org/10.1029/2019jb018517 on a saw‐cut specimen of Fontainebleau sandstone subjected to periodic oscillations of fluid pore pressure. We adopt a simple spring‐block model obeying rate and state frictional sliding. Results depend strongly on , the ratio of the period of pore pressure oscillation to time scale of rate and state effects. For small pore pressure oscillation dominates and simulations with constant rate and state parameters show that peak velocities align with pore pressure peaks but not for every peak. For large rate and state effects dominate: multiple velocity peaks occur for each pore pressure period and velocity peaks can occur near the trough or the peak of pore pressure oscillation. In the experiments, measured values of the rate and state parameters varied with velocity. When this variation is included in the simulations, results are consistent with observations at higher confining stress imposed in the experiments. However, for the lower confining stress, it is necessary to include a dependence of the parameters on the effective normal stress for the simulations to be consistent with the observations. In both cases, the magnitude of velocity peaks increases with pore pressure amplitude but the magnitude is less for the higher confining stress. This is consistent with the fewer number of stick‐slip and acoustic emission events observed in the experiments at higher confining stress.

Abstract Reaction‐driven cracking has been discussed for decades. One mechanism is the extension of a microcrack resulting from precipitation. This mechanism can create porosity, permeability and reactive surface area in low‐permeability rock. We model this problem as a fracture loaded over a fraction of its length by a vein. The loading causes crack propagation when the stress intensity factor reaches its critical value. We calculate the conditions for the onset of crack growth, the time required, pressure distribution around the vein, and the crack surface displacements. These results are relevant to many problems. One application is to geological storage of by mineralization. Results depend strongly on rock parameters but using representative values from experiments, our calculations suggest an initiation time within tens of years at low temperature and dilute fluid conditions. Lower critical stress intensity factor, higher reaction rate, and greater carbonate filling ratio reduce the time to initiation.

Abstract Field and laboratory observations indicate that the frictional behaviors of faults depend on hydrothermal conditions. We extend the microphysical Chen‐Niemeijer‐Spiers (CNS) model to hydrothermal conditions by using the observed temperature variation of indentation hardness to infer the temperature dependence of a microphysical parameter . This parameter is assumed constant in previous versions of the CNS model. A simple spring‐slider system is used to simulate the fault system and investigate the steady‐state frictional behaviors of wet granite gouges. Our numerical results quantitatively reproduce experimental data showing the frictional‐plastic transition. The results also describe the transition from velocity‐strengthening at low temperatures (<160°C), to velocity‐weakening at intermediate temperatures (160°C–370°C), then back to velocity‐strengthening at high temperatures (>370°C). In our extended CNS model, these results suggest that the dominant shear deformation mechanism does transition from frictional granular flow to fully plastic creep with increasing temperature.

ABSTRACT Injection of fluids for geothermal stimulation has often caused seismic events that have raised concern and resulted in discontinuing the operations. Consequently, a safe and effective geothermal operation requires an assessment of the potential for induced seismicity. Rate and state friction (RSF) is well-established as a robust description of rock friction. It is, however, strictly empirical and extension to hydrothermal conditions relevant to geothermal stimulation is problematic. We extend a microphysical model to hydrothermal conditions by incorporating thermally activated processes in fault gouges. We use a simple spring-slider model to simulate steady-state friction at a typical range of temperatures and load-point velocities. Numerical simulations show that, consistent with experimental observations, the simulated steady-state friction of granite gouges in wet conditions rises slightly with temperature, and the friction rate parameter (a − b) transitions from velocity-strengthening at low temperatures to velocity-weakening behaviors at high temperatures of about 300°C. This transition indicates the possibility of change from stable to unstable (seismic) slip near this temperature. These results also suggest that the dominant deformation mechanisms may evolve from frictional granular flow with intergranular pressure solution at low temperatures to grain boundary sliding (GBS) with solid-state diffusion creep at high temperatures. INTRODUCTION Induced seismicity in enhanced geothermal systems is generally detected in a depth range from a few hundreds of meters to several kilometers below the Earth surface where temperature and pore fluid pressure are elevated (Tomac and Sauter, 2018; Johnson et al., 2016; Atkinson, 2015; Eberhart-Phillips and Oppenheimer, 1984). The correlation between induced seismicity and geothermal production has been proposed based on many field observations, such as the Utah FORGE geothermal system (Moore et al., 2019), the Reykjanes and Svartsengi geothermal areas (Keiding et al., 2010), the Geysers geothermal field (Eberhart-Phillips and Oppenheimer, 1984), and the Geoven deep geothermal site (Schmittbuhl et al., 2021). The injection of cold fluids and extraction of hot fluids from the subsurface may cause the pressure imbalance and trigger the seismic slip in the seismogenic zone (Xing et al., 2020). Extensive studies suggest that the aseismic-seismic transition in the seismogenic zone is thermally controlled (Currie et al., 2002; Tichelaar and Ruff, 1993; Blanpied et al., 1995; Niemeijer et al., 2016; Den Hartog and Spiers, 2013; Toy et al., 2010). In addition, the seismic slip behaviors are probably linked with temperature dependence of effective frictional properties, rock composition, porosity, and plasticity in fault gouges (Tullis and Yund, 1980; Boettcher et al., 2007; Giger et al., 2007; Barbot, 2022; Scholz, 2019). The complex frictional behaviors are often captured by a standard RSF that uses a state variable to represent the structural evolution within the fault zone (Dieterich, 1979; Ruina, 1983).

Abstract This study uses a spring‐block model and rate and state friction to simulate experiments conducted in a double direct‐shear apparatus on carbonate fault gouge (Scuderi et al., 2017, https://doi.org/10.1016/j.epsl.2017.08.009 ) and on shale bearing rock (Scuderi & Collettini, 2018, https://doi.org/10.1029/2018jb016084 ). Both sets of experiments used the same loading protocol and injected pore fluid under creep conditions. When velocity strengthening rate and state friction is used to simulate the experiments on the carbonate fault gouge the results agree well with the observed onset of tertiary creep in the experiment. Thus, the simulation reinforces the observation that pore fluid injection can induce rapid slip even when the friction relation is velocity strengthening. The rate and state framework provides an interpretation alternative to the standard one of the Mohr's circle moving to the left as pressure increases. In the rate and state framework, the friction coefficient must increase with pore pressure increase. The shale has a low nominal friction coefficient (0.28) and is much more velocity strengthening than the carbonate. The simulation agrees with the observations that increases in pore pressure induce an increase in slip velocity but the magnitudes reach only about 100 microns/s by the end of the experiment. The simulation for the shale also agrees well with the magnitude of the observed displacement at the end of the experiment but observed displacement is increasing much more rapidly the calculated. Although the calculations agree well with features of the observations near failure, the overall curves of displacement and velocity are significantly different.

Recent industrial processes that involve injection of fluids, such as geothermal stimulation, disposal of waste water from hydraulic fracturing and carbon sequestration, have induced seismicity that has caused concern and resulted in discontinuation of the activity. Although field observations are the ultimate test of the effects of pore fluid on failure, their interpretation is complicated by heterogeneity of hydrologic and mechanical structure, and pumping and loading history. In particular circumstances, well-designed field tests can overcome some of these limitations. Laboratory experiments, despite their limited size and time scales, provide a more controlled environment that can yield an understanding of fundamental processes. Simple models that simulate the experiments can assess whether the mechanisms included in the models are sufficient to describe well the response or more complex formulations are needed. In addition, simulations can extend results for parameter values and loading programs beyond those achievable in experiments and aid in extrapolation to field applications.This work uses a spring-block model and rate and state friction to simulate experiments conducted in a double direct shear apparatus on simulated carbonate fault gouge (Scuderi et al., EPSL, 2017) and on a shale bearing rock (Scuderi and Collettini, JGR, 2018). Both sets of experiments used the same loading protocol and injected pore fluid under creep conditions. When velocity strengthening rate and state friction is used to simulate the experiments on the simulated carbonate fault gouge the results agree well with the observed onset of tertiary creep in the experiment. Thus, the simulation reinforces the observation that pore fluid injection can induce rapid slip even when the friction relation is velocity strengthening. The rate and state framework provides an interpretation alternative to the standard one of the Mohr's circle moving to the left as pressure increases. In the rate and state framework, the friction coefficient must increase with pore pressure increase. The shale has a very low nominal friction coefficient (0.28) and is much more velocity strengthening than the carbonate. The simulation agrees with the observations that increases in pore pressure induce an increase in slip velocity but the magnitudes reach only about 100 µm/s by the end of the experiment. The simulation predicts reasonably well the times at which representative values of the slip velocity and displacement occur but the overall agreement of simulation and observation is not as good as for the carbonate. Mechanisms other than rate and state friction, for example, direct dependence of the friction coefficient on slip and porosity changes, may be significant.

Faults are complex structures that substantially influence the mechanical behavior and hydraulic connectivity of rock formations. Therefore, studying faults is important for a variety of disciplines such as geoscience, civil, geotechnical, reservoir engineering, and material science among others. Researchers from these disciplines have considered different aspects of faults, namely geometry, petrophysical properties and mechanics. Until now, these studies have evolved separately and at different scales, making it difficult to connect the geometric development of fault structure to its mechanics. The current understanding of fault geometry and growth is based on fracture mechanics and on many qualitative and quantitative studies on outcrop and seismic reflection surveys among other datasets. The application of fracture mechanics theory is mostly confined to simple geometries: elliptical models for a single fault plane and uniform properties. These applications predict the maximum displacement at the center of the fault, which is not in agreement with the new findings from 3D seismic and outcrop studies. These fracture mechanics models emphasize fault propagation along strike (in 2D). Although they can include the presence of a process zone at the fault tip, the models fail to explain the development of cross-fault damage zones and localization within the fault core as well as fault segmentation and displacement partitioning. Therefore, it is timely to revise the existing applications of fracture mechanics to simple fault geometries and to develop a data-driven fault mechanics possessing closer agreement with real, observed subsurface heterogeneity. This would allow better prediction of fault geometry, propagation, and growth in 3D. We suggest recent advances in non-destructive numerical characterization of faults and application of Deep Neural Networks (DNN) to map fault geometry and predict its properties from seismic data enable us for the first time to extract simultaneously faults' geometrical and mechanical properties at an unprecedented speed and accuracy, thus resolving the 3D fault shape and properties in ways that were unthinkable just a decade ago.