If, upon the filing of the report of the engineer, together with the estimates as provided in section 31-311, it appears that lands, other than those incorporated by the court in the district, will be benefited by the drainage improvements of the district, the chairperson of the board of supervisors shall file a petition in the district court of the county where the district was originally organized, containing a description of the lands and the name or names of the owners as they appear on the tax duplicate of the county in which the lands are situated and their place or places of residence and alleging that such land will be benefited by the improvements and ought in justice bear its proportion of the expense and cost of such improvement and that such land was not incorporated within the limits of the drainage district as originally established by the court. If the names of the owners of any such tract or tracts of land are unknown, this fact shall be stated. The prayer of the petition shall be that such tract or tracts of land may be incorporated and made a part of the district. Upon the filing of such petition, duly verified, the clerk of the district court shall issue summons or notice to the parties interested as provided by section 31-303 with reference to the original petition for the establishment of the district, the same proceedings shall be had upon the petition and in the same court as upon the original petition for the establishment of the district, and the same provisions of law shall apply thereto insofar as the same are applicable. Upon the return day of such notice or summons, or at any other time to which the court shall adjourn the cause, the court shall have jurisdiction to try and determine such matter at chambers and to make all necessary orders, judgments, and decrees. The owners of such lands may by writing, duly verified, waive the issuance and service of all notice or process and consent that the court may at once upon the filing of the petition and waiver enter the necessary decree. Upon filing the petition it shall be the duty of the clerk to record the cause as a proceeding in and part of the original cause for the establishment of the district. After entering of the decree of the court, the land and all of the parties so brought into the district shall be subject to the same provisions of law as would have applied to them had they been incorporated in the original petition and decree entered thereon. No land shall be included in such drainage district or be subject to taxation for the drainage except wet, submerged, and swamp lands or land within a district subject to overflow.