Recent field experiments in Iceland and Washington State (USA) show that basalt formations may be favorable targets for carbon capture and sequestration (CCS) because CO2 mineralization reactions proceed rapidly. These results imply that there is tremendous opportunity for implementing CCS in large igneous provinces. However, the magnitude of this opportunity comprises commensurate levels of uncertainty because basalt reservoirs are characterized by highly heterogeneous, fracture‐controlled hydraulic properties. This geologic uncertainty is propagated as parametric uncertainty in quantitative risk models, thus limiting the efficacy of models to predict CCS performance attributes, such as reservoir integrity and storage potential. To overcome these limitations, this study presents a stochastic approach for quantifying the geomechanical performance attributes of CCS operations in a highly heterogeneous basalt reservoir. We utilize geostatistical reservoir characterization to develop an ensemble of equally probable permeability distributions in a flood basalt reservoir with characteristics of the Wallula Basalt Pilot Project. We then simulate industrial‐scale CO2 injections within the ensemble and calculate the mean and variance of fluid pressure over a 1‐year injection period. These calculations are combined with the state of stress in southeast Washington State to constrain the spatial extent at which shear failure, fracture initiation, and borehole breakdown may occur. Results from this study show that (i) permeability uncertainty alone causes injection pressure to vary over 25 MPa, (ii) shear failure is likely to occur at 7 times greater distances from the injection than the CO2 migrates, and (iii) joint initiation pressures are localized within the volume comprising the CO2 plume. © 2019 Society of Chemical Industry and John Wiley & Sons, Ltd.