Consanguinity is one of only two valid reasons for divorce - namely, that the two parties are too closely related to be legally married. Apart from adultery, i.e. evidence that the woman has been unfaithful to the man, a church divorce is all but impossible in this era.
As a rule, consanguinity can be used to annul any wedding between people more closely related than third cousins. This must be specifically proven, so definitive family trees must be presented to prove the relationship is to close - and that neither party was aware of this fact before the wedding.
Consanguinity is astoundingly expensive to prove and as such is usually reserved for the rich to escape from marriages they no longer wish.