The typical fact pattern is where a U.S. citizen and immigration fulfill all the state requirements for a valid marriage. Those could be medical tests, apply and receive a marriage license, and a valid marriage ceremony. However, the U.S. citizen is paid to marry the immigrant so that he or she can get a green card. The couple doesn’t intend to live together as man and wife. Case law has found that despite being legally married (met state requirement) that marriage is irrelevant to their having defrauded USCIS.
Similar situation arises when the couple has a bona fide legal marriage, but over time the relationship is unsuccessful. However, the couple stays in the marriage in order that the immigrant can keep his or her legal status in the U.S. I believe, that if the initial marriage was bona fide the fact that the relationship failed is not a basis to invalidate a green card. With one exception, if the immigrant has a conditional green card. The Immigration Marriage Fraud Amendment modified 8 U.S.C. 1186a by creating a two year conditional status for marriages that were less than two years old when the initial green card petition was filed. For those marriages, Congress has spoken and USCIS can look at the marriage two years later and determine that is or was a sham. Just because the relationship fails, doesn’t mean the the marriage was a sham.