Graded rings associated with contracted ideals

We study the associated graded ring G(I) of an ideal I that is contracted from a quadratic extension in a 2-dimensional polynomial ring. When the ideal is complete (i.e. integrally closed) a result of Lipman and Teissier shows that G(I) is Cohen-Macaulay. Complete ideals are contracted and our main goal is to identify conditions on a contracted ideal I implying that G(I) is Cohen-Macaulay or has at least positive depth