The ideal theory of intersections of prime divisors dominating a normal Noetherian local domain of dimension two

Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if a collection of such prime divisors is taken below a certain ``level,'' then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.