This is what I found for the state Arizona about covenant marriages. I'm assuming it means about the same in other states that have it.
What is a Covenant Marriage?
The State Legislature has created a type of marriage in Arizona called "covenant marriage." It does not replace the kind of marriage already available. Instead it offers an additional option to couples who wish to marry. The covenant marriage differs both in the steps necessary to get married and the reasons why a legal separation or divorce may be granted by the court.
To enter into a covenant marriage, the couple first must have counseling (called "premarital counseling") from a member of the clergy or a marriage counselor. Then, when applying for a license to be married, both persons must show their intention to enter into a covenant marriage by signing a special statement (or "declaration") on the application form. In a covenant marriage, legal separation or divorce (in Arizona, a "dissolution of marriage") may be granted by the court only for specific reasons listed in state law. These are explained in the following pages.
ENTERING INTO A COVENANT MARRIAGE
To be married in Arizona, a woman and man legally qualified to marry must first get a marriage license. (Sections 25-101 and 25-102 of the Arizona Revised Statutes indicate who may legally marry.) To get a license, a written application must be filed with the Clerk of the Superior Court in any county of the state or with some justices of the peace, city clerks or town clerks. Call the Clerk of the Superior Court in your county for information on where to apply for a marriage license.
For a covenant marriage, certain information must be included in the marriage license application. By law (Section 25-901 of the Arizona Revised Statutes) a person must state their intention to enter into a covenant marriage. This statement (or "declaration:) must contain three things:
A written statement, printed exactly as follows:
A Covenant Marriage
We solemnly declare that marriage is a covenant between a man and a woman who agree to live together as husband and wife for as long as they both live. We have chosen each other carefully and have received premarital counseling on the nature, purposes and responsibilities of marriage. We understand that a covenant marriage is for life. If we experience marital difficulties, we commit ourselves to take all reasonable efforts to preserve our marriage, including marital counseling.
With full knowledge of what this commitment means, we do declare that our marriage will be bound by Arizona law on covenant marriages and we promise to love, honor and care for one another as husband and wife for the rest of our lives.
The signed and sworn statement of both people that they have received premarital counseling from a member of the clergy or from a marriage counselor.
In premarital counseling, both people must be advised that a covenant marriage is a commitment for life. Premarital counseling also must include a discussion of the seriousness of covenant marriage, the requirement to seek marriage counseling if marital difficulties develop and the limited legal reasons available for ending the marriage by legal separation or divorce. The couple also must receive a copy of this pamphlet.
The signatures of both parties witnessed by a court clerk.
The parties must submit with the license application a sworn, notarized statement from the member of the clergy or marriage counselor who provided the premarital counseling. This statement must confirm that the parties were advised about the nature and purpose of a covenant marriage and the limited reasons for ending the marriage by legal separation or divorce. The counselor’s statement also must show that a copy of this informational pamphlet was given to each person.
Sounds like that if you enter into a covenant marriage, divorce is extremely difficult.