Eustace Couper Black, often known as 'Couper Black', the son of the eminent South Australian botanist, John McConnell Black, was educated in state schools and Prince Alfred College before graduating in medicine at Adelaide University in 1910. He was appointed in 1911 to the position of Resident Medical Officer at the Adelaide Hospital. He resigned early in 1912 and travelled to the UK in the following year where he sought medical work. With the outbreak of World War 1 he enlisted in the Royal Army Medical Corps and served in France from 1914 to 1917. He was promoted to the rank of captain in 1915. In 1917 and 1918 he served in East Africa as commanding officer of a field ambulance unit. He met and married his wife, Julia Dixon, in England during the war and returned with her to Adelaide in 1920. He was subsequently employed as a lecturer at the University of Adelaide and anaesthetist at the Royal Adelaide Hospital and Adelaide Children's Hospital. He was employed as the Medical Officer with the South Australian Railways from 1940 until his retirement in 1951. He and his wife Julia had two sons and a daughter.

After his return to Adelaide, Black developed an interest in Aboriginal archaeology and anthropology. He participated in the Board of Anthropological Research expeditions to The Granites in 1936, Nepabunna in 1937 and Ooldea in 1939, assisting with the recording of anthropometric data and blood testing. Between 1940 and 1970 Black recorded numerous Aboriginal campsites, stone arrangements and other archaeological sites in the Fleurieu Peninsula, Kangaroo Island, River Murray, Coorong, South-East, Mid-North, Yorke Peninsula, Eyre Peninsula, Port Augusta and Flinders Ranges regions. He also investigated sites in the Broken Hill area and others along the upper Murray and Darling Rivers. Some of these site recording trips were in association with the Anthropological Society of South Australia; other sites were recorded during family holidays. During the 1940s Black carried out a detailed survey of Aboriginal scarred trees along the Finniss River and in the late 1960s he carried out surveys of scarred trees and other Aboriginal sites in the Chowilla area, with Bob Edwards (see AA 83). In 1966 he assisted in archaeological excavations at Lake Robe in the South-East and in the following year he assisted with excavations at Koonalda Cave on the Nullarbor Plain.

Black joined the Anthropological Society of South Australia in 1929. He was elected Vice-President in 1939 and 1940 and President in 1941 and served on the Society's committee for many years. He was also a member of the Royal Society of South Australia. His other interests included the early exploration of Australia and medical history, and he compiled notebooks and reference materials relating to these and other subjects. He published several papers on Aboriginal culture and at the time of his death was working on the final draft of a book on Aboriginal material culture, lifestyle and hunting/gathering techniques, based primarily on historical and ethnographic sources.

Black's wife Julia died in 1970. Black himself suffered a stroke early in 1971 and died on 12 November 1971.