Non-monotonic relationship between injection rate and earthquake magnitude from rate-and-state simulations of porous faults

Fluid injection–induced seismicity poses a major challenge for subsurface fluid injection activities such as wastewater disposal and hydraulic fracturing for shale gas production. A common industrial assumption is that reducing injection rates lowers seismic risk; however, the relationship between injection rate and earthquake magnitude remains debated. To investigate this, we develop a two-dimensional planar fault dynamic model that integrates rate-and-state friction and poroelastic coupling to characterize how injection rate affects earthquake magnitude under different initial fault stress conditions. Our simulations reveal a non-monotonic, U-shaped relationship between injection rate and earthquake magnitude, with magnitude decreasing from low to moderate injection rates and increasing again at higher rates. The sensitivity of earthquake magnitude to injection rate becomes pronounced under intermediate initial stress conditions ( S R = \tau_0/\tau_f = 0.4−0.7) and peaks at S R = 0.6−0.7. In this regime, the largest earthquakes occur at low injection rates (0.005 kg·m⁻²·s⁻¹) due to widespread pore-pressure diffusion that expands the seismogenic region and promotes large-scale rupture. These results demonstrate that reducing injection rates does not necessarily mitigate seismic hazard and may even promote larger events within this intermediate stress regime. Our findings highlight the limitations of single-parameter mitigation strategies and suggest that injection protocols should account for both injection rate and the initial fault stress state when assessing induced seismic risk.