Whenever the cost, or any part thereof, of any water or sewer improvement, whether local or general, is or will be assessed against the owners of real estate and such water or sewer improvement will be connected into or will make use of, contracted water or sewer facilities constructed under the provisions of this chapter and to the cost of which such owners, or any of them, did not contribute, there shall be included in the engineer’s estimate before the hearing on any such improvement, separately itemized, and in such assessments, a sum equal to the amount provided in or computed from such contract as the fair pro rata share due from such owners upon and for such contracted water or sewer facilities.