The Board shall adopt such regulations for the location, size and depth of such excavation and for the work of excavating and removing earth, sand, rock, stone or gravel from any such natural deposit thereof as may be necessary for the public welfare and safety; and the work of excavating and removing said earth, sand, rock, stone or gravel, and supporting the banks of any contiguous public or private property shall be performed according to plans approved by the City Engineer and to the satisfaction of the Board, and under the supervision of an inspector appointed by the Board for that purpose.