Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.