Since most of the area will have only been surveyed from the air it is not possible to say what the true picture is until more detailed seismic surveys and exploratory drilling have been carried out.  Some of that has has now been started.

So, why is it called the empty quarter? As HO's graphic shows, there aren't many exploration wells in the south and east (the Rub' al Khali).

There may well be SOME oil in the empty quarter.

However, due to the rifting and subsequent formation of the Red Sea between Arabia and Africa (and the Gulf of Aden, too), there was extensive eruption, intrusion, and emplacement of numerous and voluminous flood basalts, basaltic dikes and sills, and many igneous intrusions lower in the crust.

For this reason, it is currently thought that the western reaches of Saudi, like the southern and southwestern regions of Yemen, probably do not have very good potential for large-scale economic hydrocarbon deposits. The massive amount of volcanic and igneous activity in the area is thought to have essentially destroyed any pre-existing oil fields.

There is a good analog for this in the North Atlantic: the North Sea fields that are so productive are, in many cases, conjugate to similar age and similar style geological/tectonic provinces in Greenland. However, the East Greenland margin  was swamped by the eruption/intrusion of massive volumes of basaltic material, essentially cooking off any large-scale hydrocarbon plays.