Tens of thousands of Syrian Kurds have fled into Iraq in the last five days as fierce fighting between heavily armed Kurdish groups and -Qa’ida-linked militias rages for control of oil wells. UN officials report “a river of people” crossing the Tigris river by pontoon bridge from Syria into the Kurdish autonomous area in northern Iraq.

Some 30,000 refugees have already crossed the border and today the government of Iraqi Kurdistan put in place an entry quota of 3,000 refugees a day to cope with an influx of Kurds, according to the Reuters news agency.

“The Kurdistan regional government authorities have put a daily quota for those refugees who will be allowed in,” Jumbe Omari Jumbe of the International Organization for Migration (IOM) told reporters in Geneva.

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“Today they will allow 3,000 persons in, but yesterday a similar quota of 3,000 was set but, at the end of the day, 5,000 refugees were allowed to cross.”

“A main reason for the fighting is control of the oil,” says Omar Sheikhmous, a veteran Syrian Kurdish leader, in an interview with The Independent. Syria’s oilfields are concentrated in the north east corner of the country where the two million Syrian Kurds are often the majority population. A further important factor in the escalation of the violence is a struggle for control of border crossings from Turkey.

The little-reported battles in north-east Syria are an important development in determining the outcome of the Syrian civil war. The Syrian army withdrew from many Kurdish towns and villages in the summer of 2012 in what Mr Sheikhmous describes as “a devilish plan” to woo the allegiance of Syria’s Kurdish minority.

As the Syrian army withdrew, its place was mostly taken by the well-organised militiamen of the PYD, the most powerful of the Syrian Kurdish groups, which is strongly linked to the largely Turkish Kurdistan Workers Party (PKK). Mr Sheikhmous says the PYD, whom he describes as “undemocratic and authoritarian” is now dominant in Kurdish areas. As the PYD was getting stronger, two jihadi groups, the al-Nusra Front and the Islamic State of Iraq and the Levant (Isis), were becoming the most effective Syrian opposition military force in Arab areas of northern Syria. The PYD’s dominance is not welcomed by all Syrian Kurds and six people were killed in an anti-PYD riot in Amuda on the border with Turkey.

Both sides control oil wells, though output is 95 per cent down on the 380,000 barrels a day output before the start of the rebellion in 2011.

Some of the oil is crudely refined locally, but Mr Sheikhmous says that the PYD and the al-Qa’ida linked groups maintain covert commercial links with the government in Damascus and sell crude to Syrian government refineries.

The complex struggle for control of north-east Syria has sucked in many players. It has also become increasingly violent as both sides try to consolidate and expand their grip. After months of intermittent fighting and ceasefires the PYD captured on 27 July the important town of Ras al-Ain on the Turkish border which is used as a supply route and is link between Kurdish enclaves north of Aleppo and in the north east of the country. On 30 July a Kurdish leader, Issa Hiso, was assassinated by a bomb in his car in Qamishli, a killing blamed by the PYD on the Isis (though others claim this is Syrian security stirring the pot of Arab-Kurdish animosities).

As clashes multiplied jihadi and secular Syrian opposition groups denounced the PYD in a joint statement saying that it “is without doubt a party belonging to the tyrant criminal Bashar al-Assad.” The denunciation alleges that Damascus is seeking to use Kurdish militias against the rebels in north east Syria just as it had used the Lebanese paramilitary movement Hezbollah to capture the town of Qusayr from the opposition in June.

There is some truth in this. The PKK historically was for many years backed more or less openly by Syria in its guerrilla war in Turkey. The Syrian army withdrawal in 2012 was a clear bid for PYD and Kurdish support against the rebels. The Syrian Kurdish groups had previously either declared themselves neutral or had backed whatever side it was in their tactical advantage to support. But the growing power of the al-Nusra Front and the Isis, which frequently targets Kurds in Iraq, is propelling the Kurds as a whole away from the opposition and towards the government side. But these moves are opportunistic and possibly temporary: the PYD leader Salih Muslim has been visiting Turkey which is seeking conciliate its own Kurds and support the Syrian rebels.

In criticising the PYD for collaborating with the government the Arab rebel groups go out of their way to deny that they have any ethnic hostility towards Kurds. But on the ground Kurds are frightened of the jihadi movements and are likely to flee to the safety of the Kurdistan Regional Government autonomous area in northern. Propaganda exacerbates this fear. YouTube film of a mass killing of 400 Kurds this summer appears to have been concocted out of film of earlier massacres in Syria and Iraq.

There is a further more mundane reason why Syrian Kurds are fleeing after two-and-a-half years of wartime conditions and isolation their resources are exhausted. This may be true of much of Syria but deprivation is worse in the north east because very little outside aid reaches its people whether they are Kurds or Arabs.

“It is a very complicated game,” says Mr Sheikhmous and for the Kurds of northern Syria it has become an ever-more lethal one leading so many of them to leave their homes and seek refuge in northern Iraq.