Melt the butter in a large skillet over medium heat. Add green beans; cook and stir for about 5 minutes. Mix in the walnuts, and pour in water. Cover and cook for about 10 minutes, or until very little water is left in the pan.
Add the salt, white sugar, and brown sugar; stir until beans and walnuts are evenly coated. Cover, and cook over medium heat for 5 more minutes. Remove from the heat and allow to cool slightly before serving.
3 tablespoons butter
1 pound green beans, trimmed and cut into 2 inch pieces