I am so desperately sorry to hear this OP. My son did this when he was 16, but to his dads. His stepmother picked him up and they drove off into the sunset together with no warning and me feeling totally and utterly bereft. His dad had only ever been a 'Sunday dad'.
There was no abuse, no unhappiness etc, but his dad represented total freedom to him. He was allowed to do what he wanted, when he wanted, go where he wanted, drink etc etc. I had some totally fair boundaries.
I have 2 other teenagers (who belong to me and my husband) and they've honestly never forgiven my son for what he did. When he left, he went off like an abusive husband does - called me controlling etc etc. He made me question literally everything about myself.
I probably didn't deal with it very well - I sunk into a massive depression, and sent my son too many heartbroken text messages which he screenshot to share with his dad and his dad's family about how awful I was.
Anyway, 3 years on, things have much improved. My son has had a lot of problems with his dad and his very uncaring nature - he's realised all that freedom comes with a price! He still comes to me for any nurture, any advice and all the love. He's about to go off travelling.
To everyone replying, there genuinely isn't always a reason other than a teenager being tempted by more freedom etc etc. When they have an option at that age, life my son did with going to his dads, they'll often take it.
OP, I'm so sorry. I looked into this legally too and sadly at age 16, they can live where they want 
Wishing you lots of healing and love.