This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC^1 in dimension one, leaving the general case open. Moreover, we prove that AP is P-complete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC^1 for one-dimensional sandpits and P-complete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak's model of two-dimensional sandpiles.