Whenever any municipality shall enter into a contract under any law with the board of chosen freeholders of the county wherein such municipality is situated for the permanent improvement and maintenance of any portion of any road within such municipality, by which contract the said municipality undertakes or agrees to pay any portion of the expense of the improvement, the municipality may enact an ordinance, requiring and directing that the portion of the expense of the improvement which it has contracted to pay shall be assessed upon any real estate that may have been benefited by such improvement in the manner provided by this chapter for the assessment, and shall be paid and collected in the manner provided by this chapter for the payment and collection of assessments levied for local improvements.