My reasoning for the heuristic given above:
A random number would be coprime to 30 for 8/30 of the time. But if you look at the concatenations mod 30 there are only 4 which lead to numbers coprime to 30, so these numbers would be expected to be prime half as often as random numbers. Then all you need is to find the size of the nth term, which is about (n times the average length of a number from 1 to n) decimal digits, or roughly 10^(n*log_{10} n) = exp(n log n), so putting the two together the expected 'chance' that the nth concatenation is prime is 0.5 n log n. Now integrate from n = 1 to x and you get 0.5 log log x.
