Leakage from gravity currents in a porous medium. Part 1. A localized sink

We consider the buoyancy-driven flow of a fluid injected into a semi-infinite porous medium bounded by a horizontal impermeable barrier through which a single localized sink allows leakage of the injected fluid. Our study is motivated by the geological sequestration of carbon dioxide (CO ₂), which is less dense than the ambient water, and the possibility that fissures in the bounding 'cap' rock may therefore compromise the long-term storage of CO ₂. A theoretical model is presented in which the leakage through the sink, or fissure, is driven by the hydrostatic pressure at the sink of the injected buoyant fluid. We determine numerical solutions for the evolution of the gravity current in the porous medium and for the quantity of fluid that escapes through the sink as a function of time. A quantity of considerable interest is the efficiency of storage, which we define as the flux of fluid that is stably stored relative to the amount injected. At the later stages in the evolution of the current, the region near the source and sink reaches a quasi-steady state. We find analytical solutions to this asymptotic state which show that the efficiency of storage decreases to zero like 1/lnt, where t is the time since initiation of the current, and predict a dependence on the properties of the sink in agreement with our numerical results. The implications of this result for the geological sequestration of CO ₂ are discussed.