larry wears his soft contacts tech diving and hasn't had any problems so far.
Your mileage will probably vary and you can't rely 100% on what the consensus may be with other divers who use contacts - but you can and should consider the different ways to deal with a problem if it occurs, rather than just speculating about whether or not it's a problem.
For me, "so far" now includes 10 years of cave diving and 12 years of technical diving prior to that, plus about 9 years of recreational an inland commercial diving before that - a bit over 31 years in total.
That has included cave dives in excess of 3 hours, commercial dives in the 6 hour range, trimix dives below 200 ft, fresh water dives in water from 32 to to 85 degrees F, and salt water dives from 40 degrees to 85 degrees. It's also included all the classes and in water drills a diver completes getting to the trimix, technical wreck, full cave, DPV, technical CCR levels.
How stable your lenses are when you have a mask off depends on the material the sense is made out of, how much correction you have (and thus how much curvature the lenses have), whether you are in salt water or fresh water, how long the contact is exposed to it, and how well the lenses fit in the first place.
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In general, lenses will expand slightly in salt water as it is slightly saltier than the tears your contacts are normally soaking in. In fresh water contacts will tend to shrink slightly - which is why they feel a little gritty in your eye for a minute to so if you've ever been in a pinch and rinsed one off in tap water. I've found that if mine are exposed to fresh water for a few minutes, they'll start to get less stable and the risk of losing them increases - but over the years I've also found that tendency has decreased with modern extended wear contacts. Still, I usually just keep my eyes closed with a mask off drill anyway, as the idea is to see how you function in no viz, and I found during countless hours doing commercial work in zero viz that I concentrate better on a task with my guys closed, probably because I'm not trying to see something,
Salt water has never been a particular problem for me, but again I tend to keep my eyes closed during a mask off drill in salt water, mostly because it stings my eyes.
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LiteHedded is probably technically correct that you could encounter a situation where a lens get displaced up under an eyelid. But in 30 plus years of contact lens diving and upwards of 6000 dives, I've never had that happen. In order for it to happen, due to just blinking, you'd have to have very poor fitting contact lenses in the first place, or actually stick something in your eye to manually displace it. You should not be diving in your contacts if they don't fit well, and you should not be doing technical diving if you're stupid enough to poke things in your eyes.
And as someone noted above, the solution to the problem under water is the same as it is above water - take off the mask, lift the eye lid, rotate your eyeball down then press the lid against the contact while you look straight ahead again to pull the contact back in place. The only differences are that a) it would be easier in the water with the water lubricating the process and b) you'd probably lose the lens (see below).
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If you are near sighted and lose a lens, you'll still have one eye providing distance vision and you'll have one eye capable of providing exceptional near vision. And if you lose both lenses, you'll just be able to put all that low viz/zero viz training to good use - and end your dive like you would with a light failure - but with the advantage of being able to see a whole lot more than you can in a lights out drill.
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If you lose both lenses, you might have issues seeing hand signals at longer distances - but light signals like the round "ok" signal or the rapid back and forth "attention" signal will still be evident. Plus of course, as the diver with a problem, you'll be the diver signaling for attention and then motioning your team mate(s) in closer to inform them of the problem and the need to turn the dive, so you'll be exchanging hand signals at close range anyway.
As noted above, you'll be able to read your SPG and computer just fine even if you lose both lenses, so ending the dive won't be much different than swimming through a halocline all the way out of the cave. You've trained for it, so it's not going to be a big deal.
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I didn't always know how well contacts would work for me in a wide range of situations and I do own a mask with corrective lenses. I have used it from time to time. I did not spend big bucks on a custom made mask with lenses ground to my exact prescription. I just bought a mask that accommodates corrective lenses sold in one the local dive shops I worked for. Mine uses a +4.5 lens in the right side and +6.0 lens in the left side - a lot of correction - and it works just fine for any recreational or technical diving purpose. Above ground I'll notice the quarter diopter or so difference that exists in one eye, but underwater it isn't noticeable at all giving the ranges involved and the naturally occurring visibility limits underwater. Plus, given that i need reading glasses if I under correct my left eye by .25 to .5 diopters, it boosts the near vision and lets me read my computer without the need for the reading lenses I have in my regular mask.
My corrective lens mask got used in two ways. Mostly, it got used back in the day when I had a single pair of extended wear contacts that I wore for an entire year, rather than replacing them monthly. A lost or torn lens on a trip could be addressed above water by wearing glasses, but under water it was a problem unless I had a corrective mask along. So it was trip insurance and it got tested and used extensively to ensure it worked as expected.
The other way it got used was in the old school technical classes Diver42 mentions above where the custom was for the instructor to just rip your mask off - often with the regulator as well. In anticipation of that I used daily wear lenses during my classes, and I carried my corrective lens mask as a backup mask. In the event I ever lost a lens in a drill and wanted to equalize the vision, I could just open my eyes and remove the remaining lens, then switch to the back up mask for the rest of the dive. I never needed to actually do that.
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If losing a contact lens under water worries you, the corrective lens mask as a back up mask approach is an option. This type of non custom drop in corrective lens mask can be purchased for around $150-$200 including the mask and lenses. If you happen to have a spare mask that already accommodates drop in corrective lenses, you can do it for about half that amount.
It also gives you the option to dive if you develop conjunctivitis or another eye issue on a dive - things like a black eye, a scratched cornea, etc. I'm still inclined to take my back up along on week long dive trips, just as trip insurance, but again it never gets used.
I will however carry it as my back up mask on DPV cave dive as it would not be a good idea to try to exit at 150 fpm in a dark cave with poor distance vision. In the event of lost lenses in that situation the options are:
Swim out. I'd still have the reserve gas to exit at a swimming speed, and running my scooter at a swimming speed rather than the planned 150 fpm would reduce the gas used, but it would add lots of decompression for the entire team.
Have a team mate tow me out. I could clip my scooter to my butt D -ring, and have Marci tow us out, which would be 100 ish fpm rather than 150 fpm, but it would be about twice as fast as swimming.
The best option though would be a back up mask with corrective lenses that not only lets us exit normally, but also lets us do the entire dive as it's now not really a "failure". All that is lost is the minute or so to remove the remaining contact (if one still remains) and get squared away in the back up mask.
The comment someone made above that a corrective mask on the team could be a problem is only a problem for another team member if your back up corrective lens mask is also the team back up mask, and obviously that is something you can avoid by just not being the person on the team carrying a team back up mask. No one is going to argue with your reasoning when you have them try your back up mask and they experience an effect similar to diving in a halocline.
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