At higher altitudes, prior to the deployment of the main parachutes, the Orion spacecraft descent to Earth will rely on deceleration by drogue parachutes. These parachutes have a ribbon construction, and in fluid–structure interaction (FSI) modeling this creates geometric and flow complexities comparable to those encountered in FSI modeling of the main parachutes, which have a ringsail construction. The drogue parachutes to be used with the Orion spacecraft have 24 gores, with 52 ribbons in each gore, resulting in hundreds of gaps that the flow goes through. We address this computational challenge, as was done for the main parachutes, with the Homogenized Modeling of Geometric Porosity (HMGP). Like the main parachutes, the drogue parachutes will be used in multiple stages, starting with a "reefed" stage where a cable along the parachute skirt constrains the diameter to be less than the diameter in the subsequent stage. After a certain period of time during the descent, the cable is cut and the parachute "disreefs" (i.e. expands) to the next stage. Computing the parachute shape at the reefed stage and FSI modeling during the disreefing involve computational challenges beyond those in FSI modeling of fully-open drogue parachutes. Orion spacecraft drogue parachutes will have three stages, with FSI modeling of disreefing from Stage 1 to Stage 2 being somewhat more challenging than disreefing from Stage 2 to Stage 3. We present the special modeling techniques we devised to address the computational challenges and the results from the computations carried out. We also present the methods we devised to calculate for a parachute gore the radius of curvature in the circumferential direction. The curvature values are intended for quick and simple engineering analysis in estimating the structural stresses. The flight envelope of the Orion drogue parachutes includes regions where the Mach number is high enough to require a compressible-flow solver. We present some preliminary computations for such cases.