Analytical solutions for brine invasion across caprock and leakage through an abandoned well in CO2 storage sites

Success in subsurface CO2 storage requires a trapping mechanism that inhibits the flow of storage layer fluid. Due to the wide prevalence of abandoned wells across geological formations, unwanted leakage events of brine and CO2 may occur if old wells near storage sites are not evaluated accurately. The abandoned wells, with time, may become conducive to flow if a sufficient pressure gradient is realized. Classical analytical models theorized abandoned well problems based on Theis's solution and proposed strategies to find radial coordinates of the abandoned wells. However, in those formulations, brine invasion from storage to the surrounding formation is either disregarded or assumed as a vertical-only flow. In this study, we report a novel mathematical model that accounts for both brine invasion and abandoned well flow. The leakage rate across the abandoned well segment is evaluated from three contributing pressure terms: injection- and the associated brine invasion-induced pressure build-up, leakage-induced pressure drawdown in the storage layer, and leakage-induced pressure build-up in the overlying aquifer. We evaluate the average radial pressure from a two-dimensional coupled flow problem by applying Laplace and Hankel transforms. The obtained solutions in the Hankel-Laplace space were inverted back to the radial-time domain by employing the integration-summation-extrapolation and Stehfest method, respectively. We verified the developed solution against the classical solution analytically and graphically. We observe that Theis's solution always overestimates the radial pressure as the pressure is diffusing in the lateral direction only. We identified that the abandoned well leakage rate decreases with the increase of permeability ratio, and the degree of reduction is higher in late times. We measured the degree of overestimation in leakage rates compared to classical solutions. A maximum leakage rate is observed for open flow conditions. The abandoned well leakage rate reaches its maximum early when the permeability ratios are higher. The proposed work can be employed as a base solution in old-well leakage modeling to determine the radial coordinate of the abandoned wells with a higher degree of certainty.