One issue with an Eastern Arabian revolt, is that without some sort of cap on the US involvement, any rebellion would be snuffed out by the local US military within the area and united Arab support to maintain the oil production in the area. Perhaps it is best to first see a sort of isolationist US policy form regarding these affairs. Then, forms certain anti-Saudi cadres from multiple directions, both internally and externally.
Say with a near ambivalent US and for granted ambivalent major powers in Europe, we may see Russia, Turkey and Iran competing over the Arabian states to some degree and perhaps a cival war can erupt over this if the Saudi monarchy manages the issue very poorly, which the Saudi monarchy has not done generally speaking, in my view. Reforms have occurred at a pace that has acquiesced to more stringent elements of society whilst giving some reform to those of a more progressive outlook. This sort of pragmatism has saved the Saudi kingdom from much discontent at home, whilst more progressive regimes to the north have difficulties. Another point of issue, Saudi Arabian clerics generally have a large following of people who take rulings on major life matters from them; namely this includes rebellion. If one checks 'Alifta . net,' you will find that there are questions from many to the clerics regarding the situations that rebellion is permitted. The Saudi clerics have been known for discouraging rebellion by calling any sort of rebellion as an offense that may send one to hellfire and likewise have associated rebellion with the most radical elements of the Saudi society, that being al-Qaeda. Unless there is more economic discontent in the society, the marriage between the Saudi clerics and monarchy is breached or the Saudi royalty attempt to progress or retroactively move backwards too radically, then civil war in my view is improbable.
Though, if it was to erupt, out of a break down of Saudi legitimacy, especially from a more progressive tone, there is nothing unreasonable regarding the idea of a Libya-type rebellion. Namely, varied Saudi royal house members will take up arms and command sections of support against the ruler, perhaps with different foreign backers. Additionally, there would most certainly be al-Qaeda elements and other such types and tribal-like entities. To be frank, I could see civil war as more likely to develop through a succession crisis of some sorts, or a broken monarch legitimacy which could then lead to a generalized civil war without royal claimants attempting to form an 'Arabian Republic' or other types of changes with support from outside countries. The issue is, as
@Falecius stated, some of this was more likely in the 1960s and 1970s when there was great dispute over certain reforms, discontent and the movement in the Middle East was toward pan-Arabism and generally progressive westernized experiments such as Saddam Hussein's government in Iraq, the government in Syria, Egypt and finally the situation in Iran.