Human activities associated with the injection of fluids at depth are known to trigger earthquakes. Fluid injection increases the internal pore pressure of the host rock, which in turn reduces the effective stress and frictional resistance of faults that maintain the fractured rock system in a state of mechanical equilibrium. Under certain conditions, sufficiently high pore pressure can lower this frictional resistance below a critical threshold and initiate an earthquake – the relative motion of rock on either side of the fault plane. Many of these earthquakes are small and imperceptible without the aid of specialized instruments, but some may be large enough to pose a significant risk to life and property. Several emerging technologies that have the potential to shape the future of low-carbon energy production, including carbon capture and storage and enhanced geothermal energy production, are inextricably linked to large-scale injection of fluids into the subsurface. The risk of injection-induced earthquakes is a primary concern and potential barrier to widespread adoption of these technologies. New tools are required to help operators manage these risks and meet stakeholder expectations. Current knowledge enables operators to predict the conditions that would trigger such an earthquake, but few or no tools exist to predict the severity of the earthquakes, precluding a complete description of the risk associated with operating a large-scale injection well. This dissertation details the theoretical justification and initial validation of a methodology and software to simulate the motion of an earthquake as it occurs and quantify the severity in terms that are germane to experts in earthquake science. Specifically, this work utilizes the finite element method to solve the equations of motion dictated by the three-dimensional linear elastic constitutive equation. Novel aspects of this research include the treatment of friction at the fault interface as a constraint on the motion of the system, and the numerical methods necessary to solve this problem. This software was created exclusively with free and open source software, so that every aspect of its internal machinery may be scrutinized, replicated, and improved by future workers.