what method of using geothermal energy involves moving water between 10 and 20 feet deep in the earth to cool or heat the water?

The method is called a geothermal heat pump (also known as a ground‑source heat pump).

This system works by circulating water or an antifreeze solution through pipes buried a short distance underground, where the temperature stays relatively constant year‑round; the fluid either absorbs heat from the ground (for winter heating) or releases heat into the ground (for summer cooling).

In many educational quizzes and references on geothermal energy, the question “What method of using geothermal energy involves moving water between 10 and 20 feet deep in the Earth to cool or heat the water?” is explicitly answered as “geothermal heat pumps.”

TL;DR: The specific method you’re asking about is the geothermal heat pump / ground‑source heat pump system.

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