As described in Genesis 12, Abram and Sarai traveled to Egypt because of a severe drought in Canaan. There he met the pharaoh of Egypt. The question is, which pharaoh did Abram meet?
When Did Abram Live
Based on a discussion of Genesis 14: Battle of the Nine Kings, Abram left Haran when he was 75 years old and was born in 2166 BC. This means he entered Egypt a little after 2091 BC.
First Intermediate Period of Egypt
Based on what many scholars believe, 2091 BC falls within the First Intermediate Period of Egypt. An intermediate period is like a dark age. It is a time in which little information is available. There would have been limited construction and limited use of correspondence through items like clay tablets. This intermediate period is between about 2181 and 2055 BC.
Starting about 2200 BC, a severe drought occurred that affected the entire region. It probably lasted the entire 22nd Century BC. It has been hypothesized to have caused the collapse of the Old Kingdom in Egypt, the Akkadian Empire in Mesopotamia, the Liangzhu Civilization near the Yangtze River, and the Indus Valley Civilization. The Akkadian Empire ended around 2154 BC. The Gutian people, a nomadic group in the Zagros mountains, were involved militarily and politically in the Mesopotamian region from ~2150 to 2050 BC. Based on all this, a date somewhere between 2200 and 2150 BC for the collapse of the Old Kingdom of Egypt is a reasonable conclusion.
Sixth Dynasty of Egypt
During the Sixth Dynasty of Egypt, pharaoh’s power gradually weakened in favor of powerful regional governors. There were local dynasties largely independent from the central authority of the Pharaoh. Egypt experienced civil wars just after Pepi II’s reign.
Pharaohs of the First Intermediate Period of Egypt
The 7th, 8th, 9th, 10th, and part of the 11th Dynasties of Egypt occurred during the First Intermediate Period of Egypt. It was a chaotic time of drought, starvation, civil wars, and general unrest. Little is preserved from this time period.
Egypt was split between several local dynasties. The Ninth and Tenth Dynasties consolidated Lower Egypt from their capital in Herakleopolis Magna. They were descended from a pharaoh named Akhtoy. The Eleventh Dynasty consolidated its power in Thebes and reunited Upper Egypt. Around 2055 BC, a member of the 11th Dynasty defeated the Heracleopolitan Pharaohs of the 10th Dynasty and reunited Egypt thus ending the First Intermediate Period. He ruled as Mentuhotep II.
Based on the Turin Canon possibly written during the time of Ramses II (~1220 BC), Wahankh Intef II, a member of the Eleventh Dynasty, reigned from Thebes (Upper Egypt) for 49 years (~2112 – 2063 BC).
Intef II was able to unite Upper Egypt. After this he clashed with his main rivals, the kings of Herakleopolis Magna for the possession of Abydos. The city changed hands several times, but Intef II was eventually victorious. When Intef II died, he left behind a strong government in Thebes which controlled the whole of Upper Egypt.
Genesis 12:16 states that Abram was given many possessions and treated well by the Pharaoh. This would suggest a man who is stable, wealthy, and has the ability to properly administrate a nation. It is a reasonable conclusion, based on evidence, that Abram and Sarai met Wahankh Intef II.