A petition, which may consist of any number of separate instruments, shall be filed with the county elections official of the principal county in which the proposed water replenishment district is located, signed by registered voters residing within the boundaries of the proposed district equal in number to at least 10 percent of the number of the voters residing within the proposed district; provided, that where the proposed district is situated partly in different counties, the petition must be signed by at least 10 percent of the voters of each area situated within each county, and each petition shall clearly designate in which affected county it was circulated, and each petition shall contain names only of the voters of the affected county in which the petition was circulated.