Carbonate reservoirs can be extremely complex when estimating oil and gas production due to their texture heterogeneity, mainly from the structure of the pore space. Understanding the flow mechanism in carbonate reservoirs is essential for successful hydrocarbon development and production. Permeability is an important parameter and plays a key role in enhancing production and recovery. However, usually permeability anisotropy is observed due to the deposition, diagenesis, and/or tectonic movements of the rock formations, which complicates the reservoir characterization and description.The digital rock physics (DRP) following the computed tomography (CT) scan techniques has progressed at a rapid speed since the three dimensional (3D) pore geometry structure of reservoir rocks can be captured and visualized and it is able to compute core sample parameters (such as porosity and absolute permeability) from the digital image (Kalam et al. 2013). To better understand permeability anisotropy in core samples from a carbonate reservoir, some accurate methods in DRP are followed.In this study, five digital rock samples were selected to do the simulation. For each digital rock sample, in order to reduce the computational cost, three cubic subsamples were selected respectively from the upper, middle, and lower part of whole core plug as digital rock models. Then, the porosity and the absolute permeabilities in three perpendicular directions were computed (one in the axial direction, z, and two, x and y, perpendicular to the axis) in each of these cubic subsamples using the parallel lattice Boltzmann method (PLBM) in the high-performance-computing-cluster (HPCC). The results show that: (