Geological CO₂ storage must operate within pressure and stress limits that preserve caprock, fault, and well integrity while sustaining climate-relevant injection. Existing reviews often treat multiphysics simulation, machine learning, and physics-informed learning separately, which obscures the different evidence required for stability screening, first slip, aseismic deformation, dynamic rupture, monitoring analytics, and containment consequences. This structured critical review integrates direct CO₂ storage observations, laboratory studies, injection analogues, multiphysics numerical methods, data-driven machine learning, and scientific machine learning within a target-specific evidence framework. We compare continuum, discontinuum, interface, and diffuse fracture formulations; one-way, staggered, and monolithic coupling; field and laboratory validation; seismic, deformation, pressure, and fiber optic monitoring; and physics-informed neural networks, neural operators, and reduced-order models. The synthesis herein identifies the strongest evidence base for pressure and deformation modeling and seismic signal processing, for which repeated field or operational applications and task-specific evaluation are available. By contrast, prospective fault slip and seismicity forecasting remain at a limited evidence level because of uncertain in situ stress, fault connectivity, CO₂-conditioned friction, monitoring detection limits, model discrepancy, and scarce cross-site validation. We propose task-appropriate metrics, explicit method maturity criteria, a validation ladder, and a staged, human-supervised digital twin roadmap. Machine learning and physics-informed methods are most credible as bounded complements to verified simulators and monitoring systems, and operational readiness should be judged by uncertainty-calibrated prospective evidence rather than algorithm novelty.

Geological CO₂ storage must operate within pressure and stress limits that preserve caprock, fault, and well integrity while sustaining climate relevant injection. Existing reviews often treat multiphysics simulation, machine learning, and physics informed learning separately, which obscures the different evidence required for stability screening, first slip, aseismic deformation, dynamic rupture, monitoring analytics, and containment consequences. This structured critical review integrates direct CO₂ storage observations, laboratory studies, injection analogues, multiphysics numerical methods, data driven machine learning, and scientific machine learning within a target specific evidence framework. We compare continuum, discontinuum, interface, and diffuse fracture formulations; one way, staggered, and monolithic coupling; field and laboratory validation; seismic, deformation, pressure, and fiber optic monitoring; and physics informed neural networks, neural operators, and reduced order models. The synthesis shows that pressure and deformation modeling and seismic signal processing are comparatively mature, whereas prospective fault slip and seismicity forecasting remain limited by uncertain in situ stress, fault connectivity, CO₂ conditioned friction, monitoring detection limits, model discrepancy, and scarce cross site validation. We propose task appropriate metrics, an explicit validation ladder, and a staged, human supervised digital twin roadmap. Machine learning and physics informed methods are most credible as bounded complements to verified simulators and monitoring systems, and operational readiness should be judged by uncertainty calibrated prospective evidence rather than algorithm novelty.