While these two words sound exactly the same, they have different meanings. The best way to use each correctly is to observe the root of the word.
“Areal” is the adjective form of the noun “area,” which means a two-dimensional space.
Example: “The well farthest to the west was drilled to determine the areal extent of the field.”
“Aerial” relates to the air; just think of an “aeroplane,” the British form of “airplane.”
Example: “Google Maps can give you an aerial view of your house.”
Aerial coolers, also known as fin-fan coolers, use air blowing past the tubes to cool whatever is inside the pipes.
Therefore, if you had an “aerial” extent of an oilfield, you would be producing oil out of thin air. (Better hurry up and patent that!)