Clean the area with a rag dampened with soapy water, a 3-percent hydrogen peroxide solution or an ammonia-based cleaner. Do not use anything abrasive, as it may leave fine scratches on the surface of the countertop. If the hole is large enough, clean the insides of it with a soft toothbrush.
Rinse the hole with water or rubbing alcohol. Rubbing alcohol will evaporate faster and won't leave a mark. If you use water, do not let it air-dry but instead wipe it dry with a rag. Water left to air-dry may leave a film on the countertop's surface.
File down a piece of matching Corian to create a powdery substance. Use a fine file if the pattern of the countertop is solid. If the pattern is aggregate, use a rasp. File until you have enough dust to fill the hole. Most home improvement stores have small pieces or samples of Corian you can purchase, or they may even scraps you can have for free.
Fill the hole with the ground Corian. Let it mound a bit above the surface of the hole. Slowly add an industrial-strength, instant-bonding adhesive into the hole until the powder congeals. Wipe away any excess glue before it dries.
Let the compound dry for 24 hours. Do not use the area of the countertop during this time. Afterward, sand down the material in the hole so it is level with the surface of the countertop. Add more of the adhesive and let it rest for one hour. Sand and repeat the steps as necessary.