B777-200 DIVERTS TO AN UNAPPROVED ARPT USING CAPT'S AUTH DUE TO PENDING FUEL EXHAUSTION AND CONVECTIVE WX SEPARATING THE FLT FROM ANY B777 APPROVED ARPT.

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WE WERE FLYING FROM ZZZ5 TO ZZZ1 AND TSTMS WERE AFFECTING THE ARRS INTO ZZZ1. WE WERE REROUTED OVER ZZZ2 SO THAT WE COULD SHOOT THE ARR. WE HAD JUST ENOUGH GAS TO GO MISSED APCH AND MAKE OUR ALTERNATE OF ZZZ3 SO WE WERE WATCHING THE WX VERY CLOSELY. IF POSSIBLE WE WANTED TO SWITCH THE ALTERNATE TO ZZZ2. AS WE APCHED; WE CHKED WITH DISPATCH AND ZZZ2 WAS NOT GOING TO BE A SUITABLE ALTERNATE BUT WE WERE TOLD THERE WAS ONLY 1 LITTLE TSTM CELL BTWN US AND ZZZ3. AS WE SWITCHED TO CTR AT THIS POINT THEY INFORMED US ZZZ1 HAD A GROUND STOP DUE TO A TSTM; SO WE IMMEDIATELY ASKED TO PROCEED TO OUR ALTERNATE ZZZ3. AS WE APCHED THE FIRST POINT ON THE ARR INTO ZZZ3; WE SAW A LINE OF TSTMS BTWN US AND THE FIELD SO WE CHKED THE WX AT THE NEAREST SUITABLE ARPT FOR A B777 (ZZZ4) AND DECIDED IF WE COULD NOT GET INTO ZZZ3 WE WOULD HEAD STRAIGHT TO ZZZ4 AS WE ONLY HAD 1 HR 15 MINS OF FUEL ON BOARD. AS WE SWITCHED ATC CTLRS ABOUT 20 MI FROM THE START OF THE ARR; WE WERE TOLD THAT; 'WE DON'T KNOW HOW WE ARE GOING TO GET YOU IN DUE TO THE STORMS.' WE IMMEDIATELY DIVERTED TO ZZZ4 AND COORD WITH DISPATCH. UNFORTUNATELY; WHILE ON ARR TO ZZZ4; A TSTM WENT DIRECTLY ON TOP OF THE FIELD. AT THIS POINT WE HAD ROUGHLY 1 HR OF FUEL ON BOARD. APCH TOLD US NO ONE WAS SHOOTING APCHS AT ZZZ4 SO WE DISCUSSED OUR OPTIONS. WE WERE DIRECTLY OVER ZZZ SO WE DISCUSSED LNDG THERE AS THE FIELD WAS VMC BUT OUR FOM SAID THAT ZZZ WAS NOT A B777 APPROVED ARPT AND DISPATCH SAID THAT WE NEEDED TO PROCEED TO ZZZ4. AT THIS POINT; APCH TOLD US ZZZ4 REFUSED TO ALLOW US TO COME TO THE ARPT AS THEY WERE SATURATED WITH HOLDING ACFT AND THEY HAD A LEVEL 5 TSTM DIRECTLY OVER THE RWYS. SINCE WE HAD ROUGHLY 55 MINS OF FUEL ON BOARD WE DECIDED TO PROCEED TO ZZZ AND USE 'CAPT'S EMER AUTH' TO LAND. BOTH PLTS WERE SOMEWHAT FAMILIAR WITH ZZZ DUE TO FLYING IN AND OUT OF THE ARPT IN OTHER COMMERCIAL ACFT. WE SHOT A NORMAL ILS APCH TO A SAFE LNDG. AT TOUCHDOWN; WE HAD ROUGHLY 40 MINS OF FUEL ON BOARD. WE FELT ZZZ WAS OUR ONLY OPTION DUE TO THE TSTMS WALLING US OFF FROM ANY OTHER ARPTS.