Dig a drainage trench on each side of the clay driveway six inches from the edges with a trenching hoe and shovel. The ditches should be 10 inches wide and 18 inches deep. Pitch them in the direction of the street or road by digging a smooth slope down 1/8-inch deeper for every foot of length.
Pack the bottoms of the trenches with the flat end of a tamping tool. Unroll strips of synthetic filtration fabric over the bottoms of the ditches to block loose sediment from seeping up into the pipes.
Fill the bottom six inches of each trench with 1/2- to 1-inch angular gravel, and tamp it.
Arrange lengths of 4-inch rigid PVC perforated pipes down the center of each trench. Use a paint brush to cover the inside of PVC pipe couplings and the outside ends of the pipes with primer and PVC glue. Twist the couplings over the pipe to spread the glue, and attach the next pipe in the open end.
Dump another 6-inch layer of gravel over the pipes, and roll more filtration fabric over the top to protect the pipes from clogs.
Fill in the trenches with some of the dirt you removed. Cover the exposed dirt with flood-tolerant ground-cover seedlings.