Any body of land, however large, contiguous or adjacent to a water management district organized under this article may be annexed thereto and made a part thereof, the same as if originally included therein, upon petition of one third or more of the landowners owning 50 percent or more in acreage of the real property to be annexed or upon the petition of one half or more of the owners of the real property to be annexed owning more than one third of the area to be annexed. Such petition and all the proceedings relative thereto shall conform as nearly as may be with the provisions of this article for the filing of the petition for the organization of a water management district and shall be filed with the court of probate having jurisdiction over the district to which annexation is sought. Upon filing of such petition, the court shall direct the board of water management commissioners of said district to cause surveys and a report to be made by the district engineer or other competent authority as to whether or not the purposes of the petition for annexation can be accomplished and in what manner the works and property of the existing district would be affected and to file the report together with the recommendations of the board of water management commissioners thereon with the court of probate. Upon the filing of the report by the board of water management commissioners, notice shall be given by the court of probate as for a hearing of a petition for the organization of a district. Upon such hearing, if the court shall find that it will be for the public health, convenience or welfare to annex said lands and to carry out the purposes of the petition, it shall so order, and thereafter the land so annexed shall be considered and made a part of said district, and the board of water management commissioners shall provide for the treatment of the annexed lands by the improvement contemplated in the plan of water management or amendments thereto; provided, that if at this hearing objections to the proposed annexation shall be made by the owners of the land, either within the original boundaries of the district or within the tract proposed to be annexed, representing, respectively, one third of the landowners owning a majority of the acres or a majority of the landowners owning one third of the acres, it shall be the duty of the court to dismiss the petition for annexation and to levy an acre tax upon the lands described in the petition for annexation to reimburse the board of water management commissioners of said district for all expenses incurred in connection with the proceedings therefor. Such taxes when collected shall be delivered to the board of water management commissioners of said district.