Herman Bavinck is most helpful when it comes to explaining the details of the Noahic covenant. First of all, he puts it in volume III of Dogmatics, under the covenant of grace instead of volume II where he puts the covenant of works. Speaking of the covenant of grace and the post-flood administration, “we must be careful to distinguish between the covenant of grace in a broader and in a more restricted sense” (III.216). In other words, Bavinck followed his Reformed predecessors and affirmed common grace, teaching that common grace is evident from Genesis 3.15 onwards: “common grace and special grace…flow in a single channel” (Ibid.).
Now concerning the Noahic covenant: “The grace that manifested itself immediately after the fall now exerted itself more forcefully in the restraint of evil. God made a formal covenant with all his creatures” (III.218). “This covenant with Noah…though it is rooted in God’s grace and is most intimately bound up with the actual covenant of grace because it sustains and prepares for it, is not identical with it. It is rather a ‘covenant of long-suffering’ made by God with all humans and even with all creatures” (Ibid.). It is “a covenant of grace in the broad sense” (III.225).
What does this “broad” aspect of the covenant of grace do? “It limits the curse on earth; it checks nature and curbs its destructive power; the awesome violence of water is reined in; a regular alteration of seasons is introduced” (Ibid.). “The whole structure of civil justice is the fruit of God’s common grace” (Ibid.). [For more on Bavinck’s discussion of common grace, see Dogmatics I.302-22.] It is not redemptive grace at work in the post-flood administration of the covenant of grace, but common grace; the latter is broader, the former is more restricted, yet both flow in the same channel.
Side note: Vos has a brief but excellent description of this covenant: the rainbow-sign is “produced upon these [clouds] by the rays of the sun which in the symbolism of Scripture represent the divine grace” (Biblical Theology, p. 55).
One more note (bear with me!): Kline also calls the Noahic covenant “a covenant of common grace” (Kingdom Prologue, 245, 249).