  Out of the more than 5,000 detected exoplanets a considerable number belongs
to a category called 'mini-Neptunes'. Interior models of these planets suggest
that they have some primordial, H-He dominated atmosphere. As this type of
planet does not occur in the solar system, understanding their formation is a
key challenge in planet formation theory. Unfortunately, quantifying the H-He,
based on their observed mass and radius, is impossible due to the degeneracy of
interior models. We explore the effects that different assumptions on planet
formation have on the nebular gas accretion rate, particularly by exploring the
way in which solid material interacts with the envelope. This allows us to
estimate the range of possible post-formation primordial envelopes. Thereby we
demonstrate the importance of envelope enrichment on the initial primordial
envelope which can be used in evolution models. We apply formation models that
include different solid accretion rate prescriptions. Our assumption is that
mini-Neptunes form beyond the ice-line and migrate inward after formation, thus
we form planets in-situ at 3 and 5 au. We consider that the envelope can be
enriched by the accreted solids in the form of water. We study how different
assumptions and parameters influence the ratio between the planet's total mass
and the fraction of primordial gas. The primordial envelope fractions for
small- and intermediate-mass planets (total mass below 15 M$_{\oplus}$) can
range from 0.1% to 50%. Envelope enrichment can lead to higher primordial mass
fractions. We find that the solid accretion rate timescale has the largest
influence on the primordial envelope size. Primordial gas accretion rates can
span many orders of magnitude. Planet formation models need to use a
self-consistent gas accretion prescription.
