Blood vessels are normal on the white of the eye, and necessary to nourish the tissues. The vessels can become enlarged and more visible due to allergies, irritation, dryness, infection and so on. Mover, the white portion of your eye is covered with a thin membrane. This membrane is where the red veins reside. So surgery is a viable option, although most people find the thought of a membrane being removed from their eyes a little unsettling. A topical and local anesthetic is used, and the surgery is painless. The procedure takes between 15 and 20 minutes to perform per eye. In this procedure, the membrane is removed, and a clear, new membrane grows in its place. An anti-scarring agent is used to help the eye and its new membrane heal.