Surprisingly, what international supermajor Statoil ASA needed next in its development of a world-scale unconventional-resource asset is headquartered in Austin, Texas. In contrast to Houston, the world’s energy capital that is just a three-hour drive east, Austin is better known for launching U.S. presidential bids, music and film careers, national-championship-winning college football, bicyclist Lance Armstrong and computer-maker Dell.

But tucked into the rolling hills and woods of the city whose natives urge “Keep Austin Weird” is Brigham Exploration Co. and a staff that is teeming with unconventional oil- and gas-resource ...