Heglig oil facilities account for about half of Sudan's production of 115, 000 barrels a day.
The UN Security Council has demanded the withdrawal of South Sudanese troops from Heglig.
It comes after Sudan said Friday its forces drove out South Sudanese troops from Heglig.
Heglig is vital because it accounts for about half of Sudan's 115, 000 barrel-a-day oil output, Reuters reports.
They are not even allowed into the area around Heglig, though it is part of their old homeland.
Just to the east of the town, for example, are the Chinese-run oilfields around the town of Heglig.
Earlier, Sudan said it has restarted pumping oil from Heglig, following the recent withdrawal of Southern Sudanese troops.
Satellite images suggest that infrastructure in the Heglig oil fields may have been severely damaged by the fighting.
He said Sudanese troops were on the outskirts of Heglig, and were advancing, according to the Reuters news agency.
South Sudan said Saturday that its troops have left the region, but maintained its claim that Heglig is its territory.
Sudan's military spokesman, Philip Aguer, said Friday that Sudan had bombed a central processing facility in the Heglig oil fields.
CNN: South Sudan says it is withdrawing from disputed oil region
South Sudanese troops have been deployed in the Heglig field since Tuesday.
During the defiant speech, the Sudanese leader told citizens that "Heglig is in Kordofan, " referring to a border state owned by the nation.
South Sudan has said Sudan bombed Heglig field's central processing facility.
Tensions peaked this month when South Sudan seized the oil-producing region of Heglig, a resource that fuels the economies of both nations, from its northern neighbor.
Tensions between the two nation peaked last month when South Sudan seized the oil-producing region of Heglig, which fuels the economies of both nations, from its northern neighbor.
The African Union Peace and Security Council has called the occupation of Heglig "illegal and unacceptable", and also condemned Sudan for carrying out aerial bombardments of South Sudan.
While both countries claim Heglig, Sudan continued administering the region.
Beijing has urged an end to the recent hostilities, during which Southern forces occupied Sudan's most important oil field, in the Heglig area, saying it belonged to the South.
Sudan claims Heglig's oil fields, which account for about half of the nation's oil production, and Sudanese President Omar al-Bashir had vowed to "never give up" the disputed region.
CNN: South Sudan says it is withdrawing from disputed oil region
Benjamin said bases near the border had been repeatedly attacked by Sudanese troops based in Heglig and those attacks were what prompted South Sudan to take the area April 10.
The latest attacks came after Barnaba Marial Benjamin, South Sudan's minister of information, said his country's army had withdrawn from Heglig to bases near the border and inside South Sudan.
Simmering tensions peaked this month when South Sudan seized the oil-producing region of Heglig from Sudan, raising the stakes by targeting a resource that fuels the economies of both nations.
Sudan's oil industry suffered when fighting in Heglig forced a halt to oil production in those fields, which account for about half of the country's production of 115, 000 barrels a day.
Sudan also claims ownership of Heglig and lodged complaints with the United Nations Security Council and the African Union on Wednesday, urging them to pressure South Sudan to withdraw troops from its territory.
Sudan's oil industry suffered further when fighting in Heglig forced a halt to oil production in those fields, which account for about half of the country's entire production of 115, 000 barrels a day.
Heglig and Bamboo produce low-quality crude.
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