The City Engineer shall, from time to time, make rules and regulations governing the opening of public streets and public grounds and the materials to be used in relaying or repairing the various kinds of pavements or surfaces so opened. The City Engineer shall also prepare a schedule of fees to be charged for the repaving and repairing of all openings made by applicants under the provisions of this chapter. Such repairing and repaving shall be done under the supervision of the Director of Public Works exclusively, upon due notice given to him by the City Engineer.