There were two main reasons why Iraq didn't promptly invade Saudi Arabia. First, there were diplomatic reasons. The Iraqi economy had been severely battered by the eight year war against Iran, with Iraq sustained by grants and loans from friendly Arab states (basically, most of them other than Iraq's fellow Ba'athist state and rival, Syria). Iraq needed friends and money, and an early invasion of Saudi Arabia would have left Iraq with no friends and no support, fighting the rest of the Arab world and the USA in a protracted war they couldn't win, with a hostile Iran possibly ready to join in. On the other side, the USA needed Arab support, and worked hard diplomatically to ensure that Arab states contributed troops to the coalition. The importance of Arab participation to the USA is highlighted by the appointment of Saudi prince and general Khalid bin Sultan as Joint Forces commander. Iraq hoped to break this coalition diplomatically, including by provoking Israeli entry into the war by firing missiles into Israel (which, in the event, failed to bring Israel into the war). Attacking Saudi Arabia would have made Iraq overtly the Bad Guy in Arab eyes, and thoroughly cemented the coalition against Iraq.

Second, as many armies do, the Iraqi army prepared to fight its last war again. The methods that had proved successful against Iran were to be used against the Coalition. Forcing the enemy to attack into strong prepared defences would, hopefully, stop them. Stop them or not, it would, it was planned, inflict unacceptable casualties on the US and other Western forces.

The first sign that this wasn't going to work was the start of the protracted bombing campaign by the Coalition, and the failure or Iraqi air defences to stop it. Iraq tried to provoke a Coalition ground attack, by shelling Saudi Arabia; this was also a goal of firing Scud missiles into Israel. No Coalition attack resulted, so Iraq launched an invasion of Saudi Arabia. This began on the 29th of January, 1991, 12 days after the start of the air attacks. The invasion failed; two of the three columns failed to leave Kuwait, and much the column that did enter Saudi Arabia was destroyed by artillery and air attack while still in Kuwait. Of the three divisions committed to the invasion, a small fraction - about 1000 men - captured the Saudi city of Khafji, leading to its liberation by Arab forces in the three day Battle of Khafji.

An earlier invasion of Saudi Arabia would have been more militarily successful. However, even if Iraq had invaded as soon as the Coalition began taking shape, they would have still been facing at least 20,000 US troops in Saudi Arabia, bringing the USA into the war. Immediate loss of support from the other Arab states, and destruction of Iraqi oil facilities by US forces would have followed, which would have pummeled an Iraqi economy still reeling from the war with Iran into the dust. A limited war with an attempted liberation of Kuwait by Coalition ground forces, without a pulverising bombing of Iraq, with heavy Coalition casualties (assuming that the Coalition fought like Iran) was what Iraq hoped for, with a quick peace allowing the recovery of the Iraqi economy. In the event, the war and the liberation of Kuwait didn't go according to the Iraqi plan, but the hope that it would was not irrational (very unrealistic, but rational enough although based on seriously faulty assumptions).