This work is motivated by the growing interest in injecting carbon dioxide into deep geological formations as a means of avoiding atmospheric emissions of carbon dioxide and consequent global warming. One of the key questions regarding the feasibility of this technology is the potential rate of leakage out of the primary storage formation. We seek exact solutions in a model of gas flow driven by a combination of buoyancy, viscous and capillary forces. Different combinations of these forces and characteristic length scales of the processes lead to different time scaling and different types of solutions. In the case of a thin, tight seal, where the impact of gravity is negligible relative to capillary and viscous forces, a Ryzhik-type solution implies square-root of time scaling of plume propagation velocity. In the general case, a gas plume has two stable zones, which can be described by travelling-wave solutions. The theoretical maximum of the velocity of plume migration provides a conservative estimate for the time of vertical migration. Although the top of the plume has low gas saturation, it propagates with a velocity close to the theoretical maximum. The bottom of the plume flows significantly more slowly at a higher gas saturation. Due to local heterogeneities, the plume can break into parts. Individual plumes also can coalesce and from larger plumes. The analytical results are applied to studying carbon dioxide flow caused by leaks from deep geological formations used for CO2 storage. The results are also applicable for modeling flow of natural gas leaking from seasonal gas storage, or for modeling of secondary hydrocarbon migration.

We describe a general, physics-based approach to numerical reconstruction of the geometrical structure and mechanical properties of natural sedimentary rock in 3D. Our procedure consists of three main steps: sedimentation, compaction, and diagenesis, followed by the verification of rock mechanical properties. The dynamic geologic processes of grain sedimentation and compaction are simulated by solving a dimensionless form of Newton's equations of motion for an ensemble of grains. The diagenetic rock transformation is modeled using a cementation algorithm, which accounts for the effect of rock grain size on the relative rate of cement overgrowth. Our emphasis is on unconsolidated sand and sandstone. The main input parameters are the grain size distribution, the final rock porosity, the type and amount of cement and clay minerals, and grain mechanical properties: the inter-grain friction coefficient, the cement strength, and the grain stiffness moduli. We use a simulated 2D Fontainebleau sandstone to obtain the grain mechanical properties. This Fontainebleau sandstone is also used to study the initiation, growth, and coalescence of micro-cracks under increasing vertical stress. The box fractal dimension of the micro-crack distribution, and its variation with the applied stress are estimated.