The so-called Fracture Process Zone (FPZ) is well-known as a precursor of macroscopic failure for concrete. Moreover, it is a reasonable assumption that this FPZ extension is partly governed by the underlying random distribution of the material meso-structure. Therefore, meso-scale (millimeter scale) fracture simulations are effective in improving our understanding of this relationship. In this paper, our objective is to simulate the extension of cracks within the FPZ and to investigate the influence of spatial heterogeneity distribution. To this end, we propose to employ the Enhanced Finite Element Method (E-FEM) as our simulation tool. In the proposed simulation method, two types of discontinuities are incorporated within the elements: strong discontinuities serve to simulate actual cracks, while weak discontinuities allow for an explicit representation of heterogeneities. The combination of these two enhancements makes the model well-suited for investigating the FPZ at the meso-scale for being able to simulate complex crack features and interactions between cracks and aggregates. The simulation captures many crack patterns in the FPZ, such as diffused micro-cracks, macro-crack coalescence, multi-branching, and crack closures. The size effect of inclusions in the FPZ, which is widely observed in experiments, is also realized in simulations. The comparison with the experimental observations shows good consistency.