What is and how would a procedure called chest tube drainage be handled?

Same as its name : It is a tube inserted into chest cavity around the lung to drain air or fluid/blood/pus. After it is inserted the amount of fluid removed daily along with type of fluid plus almost daily chest x-ray will tell your doctor when it is ready to come out . In case of air leakage they will wait for air to stop before pulling it out as long as your x ray looks good.