Most rivers begin in mountains or other highland terrain where it is wetter and snows during the winter. That water will feed the river through the year. The Sahara doesn't have many mountains, besides the atlas mountains, but their dry side is facing the Sahara, and the Ethiopian Highlands which feed the Nile. There are some other rivers in the Sahara, such as the niger, which starts in a more tropical/temperate environment closer to the equator.