Downhole disconnect tool

A method of disconnecting first and second portions of a tubular string in a subterranean well, the method comprising:

• connecting a downhole disconnect tool between the first and second portions of the tubular string;
• deploying a plug into the tubular string in the well, thereby engaging the plug with an inner sleeve in the downhole disconnect tool, in which the inner sleeve comprises a plug seat and at least one opening formed below the plug seat, the at least one opening extending radially from an inner surface to an outer surface of the inner sleeve;
• applying a differential pressure across the plug while the plug is engaged with the plug seat, thereby displacing the inner sleeve from a first position to a second position, the at least one opening permitting fluid communication between a first flow passage above the inner sleeve and a second flow passage below the inner sleeve when the inner sleeve is in the second position;
• and
• then unthreading a first outer housing section of the downhole disconnect tool from a second outer housing section of the downhole disconnect tool in the well, thereby disconnecting the first and second portions of the tubular string.