What is the ruling on an unseen man marrying a second wife out of a desire for chastity, in light of his first wife's refusal, and his distance from her due to his work in the desert, and his occasional arrival while she is menstruating?

If "secret marriage" means without a guardian (wali) or witnesses, then it is invalid. However, if it means a man concealing his marriage from his first wife, then there is no objection to it, provided the marriage fulfills its conditions and pillars. Announcing the marriage (ishhar) is recommended, not obligatory, according to the majority of scholars. As for not documenting the marriage in courts, although it is not a condition for its validity, it has become an urgent necessity today. Neglecting it leads to great harms and the serious loss of legal rights.