During the period of twenty days, or such further time in special cases as is specified in the notice given pursuant to § 322-23, any owner or occupant of land sought to be improved, the owner’s or occupant’s attorney or agent, may file an appeal from the decision of the department of health condemning the land as deleterious to the public health or from its decision or the order of the comptroller as to the nature and extent of the improvements to be made, including the size, character, and location of any drains or ditches, and within twenty days after written demand by the comptroller on the owner or occupant of any land in a drainage district improved by drainage under this chapter, for the payment of the cost of any such drainage, the owner or occupant, the owner’s or occupant’s attorney or agent, may file an appeal from the order of the comptroller apportioning the cost of the drainage to be borne by the owner’s or occupant’s land. All appeals shall be filed with the comptroller, and the comptroller shall transmit them to the circuit court of the circuit wherein the land is situated.