## 6 Discussion
The mean values of \(\lambda_{\beta}\), \(\beta\) in the central body are \(\lambda_{\beta}=5.5443\times 10^{-10}\frac{\text{m}^{4}}{\text{Ns}^{1-\beta}}\), \(\beta=0.0434\), while, in the posterior and anterior regions, are \(\lambda_{\beta}=2.851\times 10^{-10}\frac{\text{m}^{4}}{\text{Ns}^{1-\beta}}\), \(\beta=0.0326\) and \(\lambda_{\beta}=1.2636\times 10^{-10}\frac{\text{m}^{4}}{\text{Ns}^{1-\beta}}\), \(\beta=0.0232\), respectively. Although the values of \(\beta\) in the three regions might not significantly diverge from zero, they indeed affect the evolution of the fluid flow rate. In order to investigate the role of the order of the fractional time derivative \(\beta\), we specifically realized a computational simulation in which we applied a constant gradient of pressure \(\|\mathbf{\
abla}p\|\), and we calculated the fluid flow rate by applying the fractional [PERSON]'s law relation as given in Equation (1). Figure 5(a) shows the normalized flow rate (\(\tilde{f}_{f}\)) calculated considering Equation (1) in the case of a constant value of gradient of pressure \(\|\mathbf{\
abla}p\|=3\times 10^{8}\frac{\text{Pa}}{\text{m}}\), \(\lambda_{\beta}=5.5443\times 10^{-10}\frac{\text{m}^{4}}{\text{N}\times 1^{- \beta}}\) and the three values of \(\beta=0.0434,0.0326,0.0232\) obtained for the central body, posterior and anterior horns, respectively. Even a small value of the time fractional derivative \(\beta\) affects the fluid flow response. \(\beta=0\) is equivalent to considering the classical Darcy law, i.e., the fluid flow rate is constant in time. From Figure 5(a), it is possible to observe that the higher the value of \(\beta\), the faster the decrease in the fluid flow rate is. Furthermore, we analysed the evolution of the fluid flow rate in the central body, posterior and anterior horns using the mean values of the parameters \(\beta,\lambda_{\beta}\) above. Figure 5(b) shows the response of the normalized flux in the three regions. As the value of the anomalous permeability \(\lambda_{\beta}\) in the central body is about four-times higher than in the anterior horn, the value of flux is higher in the central body and lowest in the anterior region. Moreover, as the value of \(\beta\) is larger compared to the anterior and posterior horns, the fluid flux is faster (i.e., decrease in the fluid flow rate) in the central body. We also consider how the fluid flow rate evolves in different directions, i.e., radial, vertical or circumferential. In this regard, given the paucity of data related to anterior and posterior horns, it is only possible to analyse the central body of the meniscus. The mean values of \(\beta,\lambda_{\beta}\) in the circumferential, radial and vertical directions are \(\beta=0.0421,\lambda_{\beta}=6.3924\times 10^{-10}\frac{\text{m}^{4}}{\text{Ns}^{ (1-\beta)}},\ \beta=0.0488,\lambda_{\beta}=5.9196\times 10^{-10}\frac{\text{m}^{4}}{ \text{Ns}^{(1-\beta)}}\) and \(\beta=0.0393,\lambda_{\beta}=4.321\times 10^{-10}\frac{\text{m}^{4}}{\text{Ns}^{ (1-\beta)}}\), respectively. Figure 6c pictures the evolution of the normalized fluid flow rate in the three directions. The values of \(\beta\) are very close for all the three directions, differing within a few percentage points. Therefore, the decrease of flux in time (i.e., the flux velocity) can be considered as almost identical in the three main directions. The value of \(\beta\) rules the time evolution of the pore pressure diffusion in Equation (21). Increasing the value of \(\beta\) implies a faster pore pressure diffusion with time. At the beginning of the test, when the fluid saturates the pores, the pore pressure carries most of the load. As the test continues, the fluid flows out of the specimen. The pore pressure decreases, and hence the solid structure starts deforming. The rate at which the fluid flows and the pore pressure decrease and the solid structure deformation increases is ruled by the value of the fractional derivative \(\beta\).