The primorial ($\#$) function is defined as the product of first $n$ prime numbers. That is, $ n\# = \prod_{i=1}^nP_i $
And there is some effect named Primorial Influence (explained here)which prevents the numbers near to a Primorial to be prime. That is,except than $n\# + 1$ or $n\# - 1$ which are Primorial prime for sum $n$, $n\# + c$ can be prime only if $c \ge P_{n+1}$. As an example the below near Primorial numbers are prime:
$ P_{1000}\# + P_{1087} \implies \text{3393 digits, }P_{1087} = 8719 \\ P_{1001}\# + P_{1100} \implies \text{3397 digits, }P_{1100} = 8831 \\ P_{1002}\# + P_{1068} \implies \text{3401 digits, }P_{1068} = 8573 $
From samples above it can be seen that difference of prime index of $n$ and $c$ is less than $100$. I know it can't be generalized to any primorial, but isn't there any good estimation on this upper bound?