How do Covenant Theologians explain the Noahic covenant?

In Covenant Theology, the Noahic Covenant is often called the "covenant of common grace" or the "covenant of nature." Still, it is seen as linked to the Covenant of Grace.

This isn't the only place that a "covenant of common grace" or even a "universal covenant" appears in covenant theology. John Frame argues that the "Edenic covenant" (his preferred term for the "Covenant of Works"), particularly the cultural mandate, has some universal application:

So God as Lord defines the role of human beings as God's vassal kings over the world he has made. [...] So the covenant is individual (with Adam and Eve in their home) and universal (extending to the whole world).

He contends that this aspect of the Edenic covenant continues today, and connects it to the Noahic covenant:

Scripture never repeals the mandate defined as the very purpose of our existence in Genesis 1:26–28; indeed, God reiterates that mandate in Genesis 9:1. (Systematic Theology, chapter 4)

Lee Irons, summarizing Meredith Kline's views, makes a similar point:

[Kline] sees the common grace mode of the cultural mandate in covenantal terms. He views the post-flood Noahic Covenant in Genesis 9 as the reiteration in covenantal form of the common grace order established immediately after the Fall. [...] The cultural mandate is covenantal. [...] That was true in the pre-Fall context, when it was part of the covenant of works. It is just as true in the post-Fall context, when the cultural mandate is re-issued in modified, refracted form as part of the covenant of common grace in Genesis 9. (source)

Bavinck is less explicit regarding the cultural mandate, but still sees differences between the covenant with Noah and the Covenant of Grace:

The grace that manifested itself immediately after the fall now [with Noah] exerted itself more forcefully in the restraint of evil. God made a formal covenant with all his creatures. 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. (Reformed Dogmatics, III, chapter 5)

Berkhof notes the two main differences between the covenants – spiritual blessings vs. temporal blessings, and application to believers and their seed vs. universal application. But he contends that the two covenants are nonetheless connected:

Notwithstanding the differences just mentioned, there is a most intimate connection between the two covenants. (1) The covenant of nature also originated in the grace of God. In this covenant, just as in the covenant of grace, God bestows on man not only unmerited favors, but blessings that were forfeited by sin. By nature man has no claim whatsoever on the natural blessings promised in this covenant. (2) This covenant also rests on the covenant of grace. It was established more particularly with Noah and his seed, because there were clear evidences of the realization of the covenant of grace in this family, Gen. 6:9; 7:1; 9:9,26,27. (3) It is also a necessary appendage (Witsius: "aanhangsel") of the covenant of grace. The revelation of the covenant of grace in Gen. 3:16–19 already pointed to earthly and temporal blessings. These were absolutely necessary for the realization of the covenant of grace. In the covenant with Noah the general character of these blessings is clearly brought out, and their continuance is confirmed. (Systematic Theology, 2.3.5.B.2)

Berkhof's treatment of this topic is both helpful and brief, so it's worth examining more fully. But in closing, here is his summary:

[The covenant with Noah] is a covenant conferring only natural blessings, and is therefore often called the covenant of nature or of common grace. There is no objection to this terminology, provided it does not convey the impression that this covenant is dissociated altogether from the covenant of grace. Though the two differ, they are also most intimately connected.