A TRACON Departure Controller reported an aircraft turned the wrong direction on a departure procedure and flew below the Minimum Vectoring Altitude. The reporter stated it is not uncommon for pilots to turn the wrong direction when using this departure procedure.

Narrative

Aircraft X departed on the ZZZZZ from ZZZ. The aircraft checked in; but did not say 'climbing via' the ZZZZZ departure. I told the pilot to 'delete the 5000 ft. altitude restriction at ZZZZZ1 and climb via the ZZZZZ departure'. The pilot questioned the fix after ZZZZZ1. I told the pilot it was ZZZ1 on the ZZZZZ departure. The pilot said roger and told me he was making a left turn to ZZZ1. The ZZZZZ departure is right turns after ZZZZZ1 due to high terrain to the left. I instructed the pilot not to turn left. The aircraft had already started the left turn. I told the pilot to turn right direct ZZZ1 to keep the aircraft in lower terrain and issued the Low Altitude Alert. Aircraft X's lowest altitude was 057 MSL in a 075 MSL MVA. I gave the pilot the brasher report once he was at a safe altitude and course. Turning left on the ZZZZZ departure has been a big problem for a really long time. The departure name was even changed to the 'RIGHT TURN Departure' for a period of time to try and fix the left turn issue. That did not help. Better pilot briefings about the departure prior to takeoff would be my only recommendation.