It is because the name comes from a Latin word (they are Roman structures originally).
Although we normally see 'aqua' as the Latin for water, aqueduct comes from 'aquaeductus' - see the following from the Online Etymology Dictionary (etymonline.com):
from L. aquaeductus "conveyance of water," from aquae, gen. of aqua "water" (see aqua-), + ductus "a leading, conducting," pp. of ducere "to lead".
So because it had both a and e, the word became aqu-E-duct, probably because that more closely represents the Latin sound than aqu-A-duct.
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The cerebral aqueduct is referred to as the aqueduct of Sylvius
The longest Roman aqueduct was Constantinople.
She made an aqueduct to transport water to her model village.
The farmer used the water from the aqueduct to irrigate his field.
Since aqueduct is a noun meaning water pipe, you would use it as any other noun. Such as: "The aqueduct is clogged and the water only trickles through." or "The Roman aqueduct is still in use for irrigation".