There appears to have been extensive research done on this issue. While I do not know the original sources, the information is summarised in the "American Electricians' Handbook," sections 8.146- 8.148.
The data presented show very little change by increasing the diameter of the rod.
The depth of the rod has a tremendous effect up to about a four foot depth (This is where a tangent to the curve is approximately at 45 degrees). By ten foot depth, the curve is essentially horizontal; that is, increasing the depth has very little effect.
Finally, multiple ground rods show very little effect, until the rods are spaced (again) at least four foot apart.
In general, our codes here ask for a 5/8" rod, driven 8 ft. into the earth, and at least six feet between rods.
In summary, yes, the type and condition of the earth matter; it's just that there isn't much you can do to change things.
