  We propose and analyze two novel decoupled numerical schemes for solving the
Cahn-Hilliard-Stokes-Darcy (CHSD) model for two-phase flows in karstic
geometry. In the first numerical scheme, we explore a fractional step method
(operator splitting) to decouple the phase-field (Cahn-Hilliard equation) from
the velocity field (Stokes-Darcy fluid equations). To further decouple the
Stokes-Darcy system, we introduce a first order pressure stabilization term in
the Darcy solver in the second numerical scheme so that the Stokes system is
decoupled from the Darcy system and hence the CHSD system can be solved in a
fully decoupled manner. We show that both decoupled numerical schemes are
uniquely solvable, energy stable, and mass conservative. Ample numerical
results are presented to demonstrate the accuracy and efficiency of our
schemes.
