# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 4 Numerical Examples
### Low Permeability Layer  
The next example we consider is the consolidation of a very low permeability layer sandwiched between two high permeability layers, as presented in [28]. A one-dimensional consolidation is assumed by applying the appropriate boundary conditions. The fluid is allowed to dissipate at the top boundary and a no flux condition is defined at the lateral and bottom boundaries. The bottom boundary is fixed from vertical and horizontal displacement and the domain is allowed to deform only in the vertical direction. An external load \(p_{0}\) is applied at the top boundary. The problem setup with the boundary conditions is shown in Figure 7.  
The material parameters for this problem are given in Table 2. Simplified material properties are assumed to focus on the permeability differences of the middle and the bounding layers. Figure 6: Convergence rates in the relative \(L^{2}\) norm of pressure, for three different polynomial degrees. The low permeability layer problem is studied using mixed and equal order simulations. The polynomial degrees for the pressure are increased continuously from linear to quartic i.e., \(p_{\mathrm{P}}=1,2,3,4\). The corresponding polynomial degrees for the displacement in a mixed formulation are \(p_{\mathrm{u}}=2,3,4,5\). The continuities at the boundaries between the layers are also varied. We consider \(C^{0}\) and \(C^{p}{}^{-1}\) continuities at these interfaces. In addition, simulations are performed for uniform and graded meshes. The results are presented for these different combinations. The results from simulations with a uniformly refined mesh are shown in Figure 8 for the mixed and equal order cases. Severe pressure oscillations are observed within the low permeability layer for the equal order simulations. For short and early time steps coupled with a very low permeability, the solution to Equation (14) requires divergence-free displacements close to the dissipation boundary and in the layer with the low permeability. Requiring to satisfy this with equal order elements for displacement and pressure locks out most of the displacement degrees of freedom leading to oscillations in the pressure solution. Due to its high permeability, the fluid in the top layer dissipates very quickly for the time step size considered here i.e., \(\Delta t=1\)s. The pressure oscillations start as soon as the fluid in the low permeability layer starts dissipating. The results improve with increasing polynomial degrees but some oscillations are still seen for a quartic solution space for the pressure, \(p_{\mathrm{P}}=4\). The results with \(C^{0}\) continuities at the material interfaces improve slightly better than with \(C^{p_{\mathrm{P}}-1}\) continuity since a \(C^{0}\) continuity is a more accurate representation of material interfaces. The pressure oscillations in the mixed simulations are less severe and are localized at the boundary between the low permeability and bottom layers. These again decrease with increasing polynomial degrees and a \(C^{0}\) continuity at the material interfaces. \begin{table}
\begin{tabular}{l c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline External load, \(p_{0}\) & 1.0 & Pa \\ Darcy coefficient, \(k_{1}/\gamma_{t}\) & 1.0 & m\({}^{2}\)/Pa s \\ Darcy coefficient, \(k_{2}/\gamma_{t}\) & \(1.0\times 10^{-8}\) & m\({}^{2}\)/Pa s \\ Biot’s coefficient, \(\alpha\) & 1.0 & – \\ Young’s modulus, \(E\) & 0.67 & Pa \\ Poisson’s ratio, \(\
u\) & 0.25 & – \\ Storativitty, \(c\) & 0 & Pa\({}^{-1}\) \\ Body forces, \(\mathbf{b}\) & \(\mathbf{0}\) & N \\ \hline \end{tabular}
\end{table} 
Table 2: The Haga problem: Load and material parameters. Figure 7: The Haga problem: Domain and boundary conditions. Figure 8: Numerical solution to the Haga problem using \(N_{\rm e}=\delta\) uniform elements and \(\Delta t=1\)s. All figures are shown after two time steps. On the left the mixed order method, and on the right the equal order method. The continuity in the boundary layer is \(Or^{-1}\) in the top row, and \(C^{0}\) in the bottom row.