At any time before the completion of excavations required for the construction, deepening, or widening of a waterway under this chapter, when there will be surplus material dug or dredged from such waterway, any owner of land within the district, for the filling of whose land no provision has theretofore been made, may have such surplus material delivered upon his or her land for filling purposes upon paying the cost of such delivery in a sum to be fixed by the board. The sum so fixed shall be paid to the treasurer at such time and in such manner as the board may prescribe, and shall be credited to the district fund.