Quartet structure generation set (QSGS) method is improved to numerically reconstruct three-dimensional heterogeneous porous wicks with uniform pore size distribution. The capillary pumping processes of the reconstructed random porous wicks are simulated at pore scale by using a three-dimensional two-phase lattice Boltzmann model. The evolutions of two-phase interface and the variations of the imbibed liquid volume fraction with time are analyzed under the conditions of different porosity, pore structure, and surface wettability. The comparisons between the LBM results and those predicted by a macroscopic scale homogenous model are also conducted. It is found that due to the pore scale effects, the two-phase interface in a random porous media is very irregular, especially at the earlier stages of capillary pumping process when the liquid penetration is faster, and the imbibed liquid does not increase exponentially with time as predicted by the macroscopic scale model. Meanwhile, the liquid penetration rate does not decrease monotonously, but exhibits different degrees of fluctuations. The pore scale effects are more prominent in the cases of lower porosity, smaller pore size and better surface wettability. In the parametric range of the present study, the capillary performance increases with the decreasing of porosity, average pore radius (when porosity is fixed) and contact angle.