Constructing chains of primes in power series rings

For an integral domain D of dimension n, the dimension of the polynomial ring D[x] is known to be bounded by n+1 and 2n+1. While n+1 is a lower bound for the dimension of the power series ring D[[x]], it often happens that D[[x]] has infinite chains of primes. For example, such chains exist if D is either an almost Dedekind domain that is not Dedekind or a rank one nondiscrete valuation domain. One concern here is developing schemes by which such chains can be constructed in D[[x]] when D is an almost Dedekind domain. A consequence of these constructions is that there are chains of primes similar to the set of ω1 transfinite sequences of 0ʼs and 1ʼs ordered lexicographically.