Examine the driveway and mark the location where water accumulates with chalk. Snap two parallel chalk lines eight inches apart across the driveway and through the problem area.
Remove the pavement or gravel within the two chalk lines. Cut into concrete or asphalt with a diamond-tipped masonry saw. Pry out the pieces with a pry bar and dig out the gravel foundation with a shovel until you have a trench through the pavement 12 inches deep. Slope the bottom by digging down 1/8 inch deeper with every foot toward the side of the driveway. Lay synthetic filtration fabric over the bottom of the trench.
Lay 4-inch perforated rigid PVC drain pipes in the trench. Use a tee connector to attach the lengths of pipe at the center of the driveway. Turn the stem of the connector toward the sky, and measure the distance from the connector to the pavement.
Use a hacksaw to cut a length of 2-inch PVC pipe to fit from the connector to the pavement. Push the pipe into the open end of the tee connector. Attach a drain grid on top so it sits level with the driveway, and caulk the joint with PVC cement.
Outline a trench along the lower side of the driveway to connect with the pipes running through the driveway. Use two lines of ropes spaced 10 inches apart. Remove the grass and dig to a depth of 18 inches, dumping the soil on a plastic tarp. Slope the bottom of this trench toward the street by digging 1/8 inch deeper with each foot.
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Fill the bottom of this trench with a 6-inch thick layer of gravel, and tamp it with a hand tamper.
Lay 4-inch PVC drain pipe down the center of the side trench. Connect the beginning end of the pipe to the pipe running perpendicular through the driveway with an elbow or "y" fitting, depending on the angle.
Cover the pipes along the side of the driveway with a 12-inch layer of gravel. Fill the remaining expanse some of the soil you used to dig the trench. Plant flood-tolerant groundcover over the soil.
Cover the pipes running through the driveway with 4 inches of gravel and lay filtration fabric on top. Spread a 1-inch bed of coarse sand over the fabric. Set pavers, bricks or another material that will complement your existing pavement while providing a water-permeable surface. Place them flush against each other and the edges of the pavement and drain grid.
Fill in gaps between the bricks with polymeric sand, and tamp it down with a stiff brush. Polymeric sand contains binders that will cling to the bricks like cement. It also softens in wet weather to allow the bricks flexibility to shift as the ground expands and shrinks.