Whenever such extension, improvement or protection shall make necessary the filling up of lowlands, owned by private persons, to a level established by the city and such filling up is proposed to be accomplished by excavation from the bed of a stream bordering the water front so to be extended, improved or protected under a general plan established by the city, such lowlands may be filled up by the city at the expense of such private owners and the costs thereof shall be a lien upon the lands so filled up. The cost to each private owner shall be determined by the proportion which the cubic contents of the filling in each lot or parcel of land, separately owned, shall bear to the cubic contents of the whole area filled. If the municipality and the owner cannot agree as to the cost of the filling aforesaid to be borne by such private owner, then it shall be determined by a proceeding in the nature of an action against such property, to be instituted by the municipality in the court of common pleas for the county in which such municipality is situated, to recover the amount claimed to be due and the reasonable costs to be paid by such lot owner shall be determined by the jury as in the case of similar actions.