## 3 Biphasic and Linear Fractional Poroelastic Models
### Linear Fractional Poroelastic Model: Fractional Consolidation Problem  
The classical linear model of transient flow and deformation of a homogeneous fully saturated elastic porous medium depends on an appropriate coupling of the fluid pressure and solid stress. A change in applied stress produces a change in the fluid pressure or fluid mass, and a change in fluid pressure or fluid mass is responsible for a change in the volume of the porous material. The coupling term affects only the hydrostatic part of the stress tensor. The stress tensor can be written as follows:  
\[\mathbf{\sigma}=2G\mathbf{\epsilon}+\lambda\text{trace}(\mathbf{\epsilon})\mathbf{I}-\alpha p \mathbf{I} \tag{11}\]  
where \(\lambda=K-\frac{2}{3\sigma_{t}}\) is the Lame constant; \(G,K\) are the shear and bulk modulus, respectively; \(p\) is the pore pressure; and \(\alpha\) is the Biot coefficient. In order to solve this one, we need an additional equation, which is given by the pore pressure diffusion equation derived considering a time fractional [PERSON]'s law in Equation (1) [13]:  
\[\frac{\partial p}{\partial t}=\frac{KB}{\alpha}\lambda_{\beta}D_{0}^{\beta}\bm {\
abla}^{2}p-B\frac{\partial\sigma_{H}}{\partial t} \tag{12}\]where \(D_{0}^{\beta}\) indicates the [PERSON]'s time fractional derivative [27]. It is important to note that the classical (non fractional) pore pressure diffusion equation is recovered in case of \(\beta=0\). In this case \(\lambda_{\beta}=\frac{k}{\mu}\) with dimension of \(\frac{[L]^{4}}{[F][T]}\). In the case of \(\beta\
eq 0\) note that \(\lambda_{\beta}\) has dimension of \(\frac{[L]^{4}}{[F][T]^{1-\beta}}\). The pore pressure diffusion equation can also be written in terms of strains [13]:  
\[\frac{\partial p}{\partial t}=\frac{K_{u}-K}{\alpha^{2}}\lambda_{\beta}D_{0}^ {\beta}\mathbf{\
abla}^{2}p-\frac{K_{u}-K}{\alpha}\frac{\partial e_{d}}{\partial t} \tag{13}\]  
The consolidation problem is modelled through the 1D uniaxial strain poroelastic problem [13]. Equation (11) is adapted for the 1D case in which the only non-zero component of strain is \(\epsilon_{zz}\), we obtain:  
\[\left(K+\frac{4G}{3}\right)\frac{\partial\epsilon_{zz}}{\partial z}-\alpha \frac{\partial p}{\partial z}=0 \tag{14}\]  
Equation (14) is then coupled with the pore pressure diffusion equation in Equation (13) in order to obtain the following pore pressure diffusion equation:  
\[\frac{\partial p}{\partial t}=\bar{\lambda}D_{0}^{\beta}\frac{\partial^{2}p}{ \partial z^{2}} \tag{15}\]  
where \(\bar{\lambda}=\lambda_{\beta}\frac{(4G+3K)(K_{u}-K)}{\alpha^{2}(4G+3K_{u})}\). We indicate with the symbol \(\lambda_{0}=\frac{(4G+3K)(K_{u}-K)}{\alpha^{2}(4G+3K_{u})}\). The boundary value problem is given [13]:  
\[\frac{\partial p}{\partial z}_{z=0} =0, \tag{16}\] \[p(z=h,t) =0,\] (17) \[u(z=h,t) =0. \tag{18}\]  
A constant compressive stress in the \(z\)-direction is applied to the cylinder at \(z=0\):  
\[\sigma_{zz}(0,t)=-P_{A} \tag{19}\]  
where \(-P_{A}\) is the applied compressive stress. The initial pore pressure is derived for undrained conditions, i.e.,  
\[p(z,0)=P_{A}\frac{3(K_{u}-K)}{\alpha(4G+3K_{u})} \tag{20}\]  
The analytical solution in terms of pore pressure reads as follows:  
\[\frac{\partial p}{\partial t}=\frac{KB}{\alpha}\lambda_{\beta}D_{0}^{\beta}\bm {\
abla}^{2}p-B\frac{\partial\sigma_{H}}{\partial t} \tag{12}\]0  
where:  
\[\frac{\partial p}{\partial t}=\frac{KB}{\alpha}\lambda_{\beta}D_{0}^{\beta}\bm {\
abla}^{2}p-B\frac{\partial\sigma_{H}}{\partial t} \tag{12}\]1 \[\frac{\partial p}{\partial t}=\frac{KB}{\alpha}\lambda_{\beta}D_{0}^{\beta}\bm {\
abla}^{2}p-B\frac{\partial\sigma_{H}}{\partial t} \tag{12}\]2  
where \(E_{1-\beta,1}\) is the Mittag-Leffler function. In the case of \(\beta=0\), the solution is identical to the classical [PERSON]'s solution in which \(E_{1-\beta,1}=\exp\).