The Government of Venezuela , in the case it made to the Arbitration Tribunal , changed its claim on the district just west of the Essequibo , and claimed that the boundary should run from the mouth of the Moruka River southwards to the Cuyuni River , near where it meets the Mazaruni River , and then along the east bank of the Essequibo to the Brazilian border . The Government of Venezuela , in the case it presented to the Arbitral Tribunal , re-established its claim as regards the district immediately west of the Essequibo , and claimed that the boundary should run from the mouth of the Moruka River southwards to the Cuyuni , near its junction with Mazaruni , and then along the east bank of the Essequibo to the Brazilian frontier .