Summary: We propose and study two second-order in time implicit-explicit methods for the coupled Stokes-Darcy system that governs flows in karst aquifers and other subsurface flow systems. The first method is a combination of a second-order backward differentiation formula and the second-order Gear’s extrapolation approach. The second is a combination of the second-order Adams-Moulton and second-order Adams-Bashforth methods. Both algorithms only require the solution of decoupled Stokes and Darcy problems at each time-step. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. The uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long time scales. Error estimates for fully discretized schemes using finite element spatial discretizations are derived. Numerical examples are provided that illustrate the accuracy, efficiency, and long-time stability of the two schemes.