When a well that has been producing a mixture of oil, water and gas is shut-in for an extended period of time, the fluid columns expand or contract in response to the temperature changes in the well. These changes in fluid levels in turn, affect the heat transfer in and around the wellbore. This coupled problem of determining the fluid levels as well as the wellbore temperature distribution is addressed. An algorithm is proposed, implemented and discussed for a hydrocarbon that can be modeled as a black-oil. Special cases of the algorithm when one of the phases is absent are also included.

This model helps accurate and rigorous prediction of the wellbore temperature distribution and the wellhead pressure during a shut-in. Accurate temperature and pressure estimates in turn lead to better stress calculations to ensure the design of the wellbore with improved integrity of the tubulars.

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