If I would decide to leave my marriage (which I won't no matter how hard it is) I would do exactly the way your wife does it
I can tell you from my perspective, that maybe has nothing to do with your wife. It is my story.
But maybe can give you a new perspective.
Marriage is too precious, and your children not to try anything to save it! I will pray for you!
She must have been loved you - otherwise she wouldn't give birth your children, as much as 4!
But something along the way made her so alone, bitter and abandoned (or mad, disapointed or so) that she just closed one by one little piece of her heart to you. You hurt her so many times until nothing left to hurt. And she didn't forgive.
Maybe you don't listen to her.
Maybe you are impatient with her needs.
Maybe you are too critical, or not open enough.
Maybe you are not selfless in giving.
Maybe now she is having an affair - which is also a sign of her lonlines, abandonment and quest for support and love which she wasn't getting in your marriage.
I don't know what exactly is going on. But something hurtful for her was going on for years.
Woman's decision to leave with 4 children is always motivated by deep and helpless suffering in bad communication and relationship.
If she came to that conclusion and is so certain and determined - it only means that all the suffering, second thoughts and pain she experienced you didn't realized or recognized, or you refused to listen. Or withdrawn from communication because watching TV, working or fishing was more satisfying for you. How hard is being ignored? How big gap between husband and wife must be that she can undergo such a venture (coming to terms with idea of leaving him, and becoming determined and without regrets) in her mind without him even realize that SOMETHING is going on? Can you imagine how many nights you lied by her not even realized she is agonized, sad or dissapointed?!
If your wife just decided to leave, and you have no idea why, it is just enough of a sign that you didn't payed attention to her. Many man don't pay attention to women physical need for rest while they have babies, don't help them get daily rest... I know women who were hurt and couldn't make it anymore - just because they were too tired, overworked and overstressed. Family broke because man didn't want to let woman sleep for few hours a day, or help with loundry... how sad is that! How little they should do and yet they tried to ignore that and see what happens - if they still enjoy TV time as much as they like.
When child comes from school you KNOW how it was, did he did something stupid, or is he in love. You KNOW your child, you invest time, you care, you ask, watch and dig until you find out.
But with spouse we ask, demand, ignore and try to fine tune him or her so that our life is as good as it gets - not his or her.
We don't observe, we don't try to find out, we find small talk, or big talk drudgery - it is mostly men's guilt... and when zillion little wounds and sufferings, hopes and dreams passes without togetherness, and slips beside supposedly loving person, it is just naturally our heart closes. And without faith and God it is just plain natural that we just find other source of closeness, love and support. We all need it. It is just too hard to bear all life difficulties if our spouse is distant and not open to communication.
If spouses don't deeply follow Christ, if they don't use benefit of confession and repention, if they don't really understand vow of marriage it is hard to find peace and endure.
No one but Christ is able to love unconditionaly when abandoned, ignored, trivialised, criticised, tortured... no one but Christ and people who "die to oneself and live for the God" - but people always suffer more than they can transform into love and I just can not judge anyone who says - I can't handle this anymore.
As much as having an affair is sinful, bad for the family, and plain stupid, I think that it happens all too often because human nature is made for love, but so often love turns into using us for one's own comfort.
I feel deeply for you, and your wife.
And my suggestion is that you try to go to confession to experienced priest so that Holly Spirit shows you what you can do to regain love and trust of your wife. As a woman I can tell you that it is movingly easy to regain a love of a woman who gave birth to your children.
But it asks for sacrifice, humility and selflesness; not in your terms, but in terms of her needs and needs of her soul.
And yes, she needs to repent and go to confession and understand vow of marriage more deeply, but until than if you want to save marriage - do the first step. Give her your sacrifice, love, support, showing her Christ-like love.
I hope you will be happy toghether!