## 2 Fluid Flow in Complex Porous Media: Time-Fractional Darcy's Law  
Fluid flow in porous media is commonly modelled by [PERSON]'s law. This relation was observed experimentally by [PERSON] (2018) and derived in 1986 from the Navier-Stokes equations using the homogenisation theory ([PERSON], 1986). [PERSON]'s law states that the instantaneous flow rate through a homogeneously permeable porous medium of permeability \(k\) is proportional to the dynamic viscosity of the fluid and the pressure drop over a given distance. The total discharge, \(J_{f}\) (units \(L^{3}/T\), where \(L\) and \(T\) indicate units of length and time, respectively) equates the product of the intrinsic permeability, \(k\) (\(L^{2}\)) with the cross-sectional area flow, \(A\) (\(L^{2}\)) and the total pressure drop \(p_{\text{out}}-p_{\text{in}}\) (\(F/L^{2}\)), all divided by the dynamic viscosity, \(\mu\) (\(FT/L^{2}\)) and the length over which the pressure drop is taking place \(\ell\):  
\[J_{f}=-\frac{k\cdot A\cdot(p_{\text{out}}-p_{\text{in}})}{\ell\mu}\]  
More generally, [PERSON]'s law states that the discharge per unit area, \(j_{f}=J_{f}/A\) is proportional to the pressure gradient, and the intrinsic permeability and inversely proportional to the dynamic viscosity:  
\[j_{f}=-\frac{k}{\mu}\mathbf{\
abla}p.\]  
The derivation of [PERSON]'s law from Navier-Stokes equations assumes a creeping, laminar, stationary and incompressible flow of density \(\rho\) and velocity \((u_{i})_{1\leq i\leq 3}\). Incompressibility implies: (\(\frac{D\rho u_{i}}{DT}=0\)), which leads to [PERSON] equation in the presence of gravity \(g_{i}\):  
\[\mu\mathbf{\
abla}^{2}u_{i}+\rho g_{i}-p_{i}=0.\]  
Assuming that the viscous resisting force varies linearly with the velocity, introducing the porosity \(\phi\), and the second order permeability tensor \(\mathbf{k}=k_{ij}\), a simple derivation leads to:  
\[-(k_{ij})^{-1}\mu\phi u_{j}+\rho g_{i}-p_{j}=0.\]  
The discharge per unit area in direction \(n\) can then be written as follows:  
\[j_{fn}=-\frac{k_{ni}}{\mu}(p_{j}-\rho g_{i}).\]  
\[j_{f}=-\frac{\mathbf{k}}{\mu}(\mathbf{\
abla}p-\rho g)\,.\]  
If the resistance to fluid flow offered by the pores varies in space, the components of the permeability tensor \(k_{ij}\) vary in space. \(k_{ij}\) is a symmetric (Onsager reciprocal relations), positive definite (because the flow component parallel to the pressure drop occurs in the same direction as the pressure drop) matrix. \(k_{ij}\) may be isotropic, in which case, it is diagonal, and all diagonal entries are identical: \(k_{ij}=k\delta_{ij}\). In general, the permeability tensor is anisotropic, and may also not be diagonal. In all cases, the permeability tensor can be diagonalised, as it is symmetric positive definite. As previously underlined, when dealing with soft tissue and complex porous media, permeability changes during the deformation process; therefore, it is important to develop a theory, such as the CTRW, which incorporates the change in microstructural features during the transport process. A modified version of [PERSON]'s law involving linear fractional operators can be seen as the deterministic version of CTRW [21; 22]:  
\[\mathbf{j_{f}}=-\frac{\mathbf{k}}{\mu}D_{0}^{\beta}(\mathbf{\
abla}p) \tag{1}\]  
In the following, \(\frac{\mathbf{k}}{\mu}\) will be indicated as \(\mathbf{\lambda_{\beta}}\). \(D_{0}^{\beta}\) indicates [PERSON]'s fractional derivative of order \(\beta\) of \(\mathbf{\
abla}p\), which is defined below:  
\[(D_{0}^{\beta}\mathbf{\
abla}p)(t)=\frac{1}{\Gamma(n-\beta)}\int_{a}^{t}\frac\mathbf{{\
abla}p^{n}(\tau)}{(t-\tau)^{\beta+1-n}}d\tau \tag{2}\]  
The expression is valid for \(n-1<\beta<n\), and \(\Gamma\) is the Euler's Gamma function. In the cases considered in this paper, the lower bond of integration \(a=0\). The fractional derivative method offers the possibility to model, with reduced number of parameters, all of the anomalous diffusion behaviours by changing the order of the derivative.