It is well known from experiments that a sample of thin porous material burning against an oxidizing air under microgravity exhibits various finger-like char patterns. The patterns are classified into three distinct types depending on the oxidizer flow rate. (I) Sparse fingers; (II) tip-splitting fingers; (III) connected front. We presently extend our modeling strategy based on the homogenization approach, which has been applied for the case of isotropic porous media, to analyze the pattern behavior on anisotropic porous media. In order to understand the characteristic features of the patterns based on the influence of the local structure, we simply rely on fixed anisotropic two-dimensional geometries representative of the microstructure of interest. Thus, we illustrate numerically the consequence of the material anisotropy on the fingering patterns based on effective diffusion tensors calculated using the homogenization method and the mechanism of thermal-diffusion instability. Besides revealing new insights on the experimental observations, our numerical results show that material anisotropy can influence the uniformity on the patterns, but the distinct fingering regimes are independent of the local microstructure of materials. This effect is consistent with the qualitative experimental findings from Zik and Moses (1999). (C) 2014 Elsevier Ltd. All rights reserved.