The year 1996 was just on the cusp of the CG revolution in visual effects. In this film we see one of the dwindling examples of miniature submarines shot using motion control. While the surface shots use a lot of Navy stock footage and shots of the full size sub being towed, all the underwater sequences use the miniatures.
There were two submarine models built for the film. The main boat, the Stingray, was a modeled after the WW2 era Balao class diesel powered USS Pampanito, preserved as a memorial and museum ship in the San Francisco Maritime National Park Association located at Fisherman's Wharf.
The second model was the fictional Los Angeles class USS Orlando a Nuclear powered submarine. The other model of note built for the production was a section of the underwater hull of a supertanker complete with twin rotating props.
I suspect all the background art and the foggy depth effects are all
digital compositing along with CG particle effects for the prop wash
which despite the use of miniatures gives the underwater shots an overall indistinct CG
look. Later in the film there are some pretty obvious Naval ship digital cutouts added to the surface scenes of the real supertanker.
There is a separate miniature sequence where a dilapidated destroyer is blown up at the end by torpedoes from the Stingray. This was supervised by Jack Sessums and is reported to be a model originally made for The Winds of War TV miniseries. Ian O'connor who did the pyro on the shot said that the director who had no experience of miniatures, flipped out when he saw how quickly the explosion took place in real time and complained to the studio how badly the the effect went. It was not until the rushes were viewed the next day that the director and the studio understood that the shot was taken using a high speed camera and played out longer and much more spectacularly when shown as intended.
The other thing to note was the that the control room live action set of the Los Angeles class USS Orlando was the same set from the fictional Russian vessel for Hunt for Red October.