In this study, we model avascular tumour growth in epithelial tissue. This can help us to understand that how an avascular tumour interacts with its microenvironment and what type of physical changes can be observed within the tumour spheroid before angiogenesis. This understanding is likely to assist in the development of better diagnostics, improved therapies, and prognostics. In biological systems, most of the diffusive processes are through cellular membranes which are porous in nature. Due to its porous nature, diffusion in biological systems are heterogeneous. The fractional diffusion equation is well suited to model heterogeneous biological systems, though most of the early studies did not use this fact. They described tumour growth with simple diffusion-based model. We have developed a spherical model based on simple diffusion initially, and then the model is upgraded with fractional diffusion equations to express the anomalous nature of biological system. In this study, two types of fractional models are developed: one of fixed order and the other of variable order. The memory formalism technique is also included in these anomalous diffusion models. These three models are investigated from phenomenological point view by measuring some parameters for characterizing avascular tumour growth over time. Tumour microenvironment is very complex in nature due to several concurrent molecular mechanisms. Diffusion with memory (fixed as well as variable) formation may be an oversimplified technique, and does not reflect the detailed view of the tumour microenvironment. However, it is found that all the models offer realistic and insightful information of the tumour microenvironment at the macroscopic level, and approximate well the physical phenomena. Also, it is observed that the anomalous diffusion based models offer a closer description to clinical facts than the simple model. As the simulation parameters get modified due to different biochemical and biophysical processes, the robustness of the model is determined. It is found that the anomalous diffusion models are moderately sensitive to the parameters.