At such time as a petition by one or more owners of any lots or lands within the county is filed with the county board of any county stating: (1) That there are areas within said county which are not within the boundaries of any drainage district organized under Chapter 31, article 3 or 4, or any sanitary drainage district organized under Chapter 31, article 5 or 7, as the same now exist or may hereafter be amended; (2) that said lands are subject to overflow or lack of adequate drainage; (3) that public health, convenience and welfare would be served by the assistance in the control and maintenance of drainage in said areas by the county board; and (4) that the county board should assist and regulate the drainage in said areas under the provisions of sections 31-901 to 31-933, the county board shall investigate the facts set out therein and make determination thereon within thirty days from date of filing.