Some sources are saying that $10 billion of their oil was stolen by foreign invaders in the past couple of years alone.
France is only part of all that. Today US is too busy and scared to get involved specially after their base was targeted by Yemenis forces, UAE is too weak and has its infrastructures damaged, Saudis are involved elsewhere and have their Aramco blown up, so now French are deploying troops there (I believe the French Foreign Legion) hoping it discourages any attacks on a NATO member.
I'd say that is stupid thinking, specially when you consider how they were screwed in Mali and elsewhere from groups a lot weaker than Yemenis resistance and with a lot more to lose!
Yemen does have some LNG infrastructure that was built many years ago before the conflict grow too big and out of hand. For example there is one in Balhaf built by a French company called Total. I'm not sure if there is any other.
Another part of the plan is also to take over the Yemen gas fields and build some infrastructure (like pipelines) and connect it to the existing pipelines to transfer some of Yemen's gas to Saudi Arabia and possibly UAE and convert it to LNG there for exports to EU with the money going in the pockets of the France, UAE and Saudis while people in Yemen are starving!