To construct or extend an assessment drain in two or more counties, a petition must be presented to the several water resource boards for the area in which the drain will lie for the establishment of the drain under this chapter. The boards shall hold a joint meeting and shall determine the necessity or expediency of the establishment of the drain. To proceed with the drain, the boards shall agree upon the proportion of damages and benefits to accrue to the lands affected in each county, and for this purpose the boards shall consider the entire course of the drain through all the counties as one drain. If the boards fail to agree upon the benefits to accrue to the lands in each county, the boards shall submit the points in controversy to the department of water resources, and the department’s decision is final. The boards may apportion the cost of establishing and constructing the entire drain ratably and equitably upon the lands in each county in proportion to the benefits to accrue to the county’s lands. When the boards have apportioned the costs, the boards shall make written reports of the apportionment to the auditors of the several counties affected. The reports must show the portion of cost of the entire drain to be paid by taxes upon the lands in each of the counties and must be signed by the boards of all counties affected. Upon the filing of the reports, the several boards shall meet and assess against the lands in each of the counties, ratably and equitably as provided by this chapter, an amount sufficient to pay the proportion of the cost of the drain in each county. The provisions of this chapter relating to drains within a single county govern the establishment, construction, maintenance, repair, and cleanout of the drains.