porous medium or a Hele-Shaw cell. The model in dimensionless form is given as follows

We study two novel decoupled energy-law preserving numerical schemes for solving the Cahn-Hilliard-Darcy (CHD) system which models two-phase flow in porous medium or in a Hele-Shaw cell. In the first scheme, the velocity in the Cahn-Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn-Hilliard equation. In the second scheme, an intermediate velocity is employed in the Cahn-Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time-step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the eciency an