​Why Abraham Denied Sarah As His Wife

Genesis 20:9-13

9  Then Abimelech called Abraham, and said unto him, What hast thou done unto us? and what have I offended thee, that thou hast brought on me and on my kingdom a great sin? thou hast done deeds unto me that ought not to be done. 
10  And Abimelech said unto Abraham, What sawest thou, that thou hast done this thing? 
11  And Abraham said, Because I thought, Surely the fear of God is not in this place; and they will slay me for my wife's sake. 
12  And yet indeed she is my sister; she is the daughter of my father, but not the daughter of my mother; and she became my wife. 
13  And it came to pass, when God caused me to wander from my father's house, that I said unto her, This is thy kindness which thou shalt shew unto me; at every place whither we shall come, say of me, He is my brother.