A challenge when modeling mineral growth inside the pore space of a porous media is to minimize the effect of the computational grid on the shape of the minerals being formed. Pore surface area and volume are important quantities in estimating upscaled permeability and effective rate equations, which emphasize the importance of models that minimize or completely eliminate grid effects. In this paper, we study how the initial orientation of the solid structure on the numerical grid affects the growth pattern due to precipitation in a lattice Boltzmann model. We have implemented a volume of fluid method to represent the solid interface, and we introduce a surface tension term that extensively reduces the dependency on the underlying numerical grid. We study both diffusion-limited and reaction-limited precipitation. In the diffusion-limited case, instabilities will develop on small scales. The surface tension term effectively introduces a short wavelength cut off which limits the unstable precipitation and reduces grid effects. We argue that the surface tension term is needed to obtain a growth pattern independent of the initial orientation on the underlying grid in the diffusion-limited case, and that simpler models can be used in the reaction-limited case.