Who said a nation can only be wealthy with oil?
A nation requires access to energy, but you can access energy using methods other than just having oil. Free markets are useful for that.
Unfortunately, the USA, China, and Japan are also all inconvenient examples of economic success, since they all became "rich" in part by overwhelming deficit spending and central economic planning.
The problem is that eventually the amount of growth you get for the money slows and you're in a situation like Japan -- they're 30 years into a "lost decade", rates stuck at 0%, and the national debt is massive.