# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 2 Governing Equations
### Variational Formulation  
To derive the variational formulations of Equations (7) and (14), we introduce a vector-valued test function \(\delta\mathbf{u}\), which vanishes on \(\Gamma_{D}^{u}\), and a scalar test function \(\delta p\), which vanishes on \(\Gamma_{D}^{\text{p}}\).  
We start with the total stress formulation of the linear momentum balance equation, which from Equation (7) is given by  
\[\
abla\cdot\mathbf{\sigma}+\rho\mathbf{b}=0. \tag{17}\]  
Multiplying by the test function \(\delta\mathbf{u}\) and integrating over the domain \(\Omega\) gives  
\[\int_{\Omega}\delta\mathbf{u}^{\intercal}\
abla\cdot\mathbf{\sigma}\text{d}\Omega+ \int_{\Omega}\delta\mathbf{u}^{\intercal}\rho\mathbf{b}\text{d}\Omega=\mathbf{0}. \tag{18}\]  
The first term in the above equation contains a double derivative of the unknown displacement, and is relaxed using a form of [PERSON]'s theorem,  
\[\begin{split}\int_{\Omega}\delta\mathbf{u}^{\intercal}\
abla\cdot \mathbf{\sigma}\text{d}\Omega&=\sum_{i}\int_{\Omega}\delta u_{i} \
abla\cdot\mathbf{\sigma}_{i}\text{d}\Omega\\ &=\sum_{i}\int_{\partial\Omega}\delta u_{i}\sigma_{i}\cdot\mathbf{n} \text{d}\Gamma-\sum_{i}\int_{\Omega}\
abla\delta u_{i}\cdot\sigma_{i}\text{d} \Omega\\ &=\int_{\Gamma_{N}^{u}}\delta\mathbf{u}^{\intercal}\text{d}\Gamma- \int_{\Omega}\
abla\delta\mathbf{u}:\mathbf{\sigma}\text{d}\Omega.\end{split} \tag{19}\]  
Due to the symmetry of the stress tensor, the last term is expressible in Voigt notation,  
\[\
abla\delta\mathbf{u}:\mathbf{\sigma}=(\mathbf{L}\delta\mathbf{u})^{\intercal}\mathbf{\sigma}, \tag{20}\]  
yielding the weak form of (7) as  
\[\int_{\Omega}\left(\mathbf{L}\delta\mathbf{u}\right)^{\intercal}\mathbf{D}(\mathbf{L}\mathbf{u}) \text{d}\Omega-\alpha\int_{\Omega}\left(\mathbf{L}\delta\mathbf{u}\right)^{\intercal} \mathbf{I}p^{\text{f}}\text{d}\Omega=\int_{\Omega}\delta\mathbf{u}^{\intercal}\rho\mathbf{ b}\text{d}\Omega+\int_{\Gamma_{N}^{u}}\delta\mathbf{u}^{\intercal}\text{d}\Gamma \tag{21}\]  
where we have used \(\tilde{\mathbf{I}}\) as the Voigt notation identity operator, which for a general three-dimensional case is given by:  
\[\tilde{\mathbf{I}}=\{1,1,1,0,0,0\}^{\intercal} \tag{22}\]  
For the mass balance equation, multiplying (14) by the scalar test function \(\delta p\) and integrating over the domain \(\Omega\), we get  
\[\alpha\int_{\Omega}\delta p\
abla\cdot\mathbf{\dot{n}}\text{d}\Omega+c\int_{ \Omega}\delta p\frac{\partial p^{\text{f}}}{\partial t}\text{d}\Omega+\int_{ \Omega}\delta p\
abla\cdot\left[-\frac{1}{\gamma_{\text{f}}}\mathbf{k}\cdot\left( \
abla p^{\text{f}}-\rho_{\text{f}}\mathbf{b}\right)\right]\text{d}\Omega=0. \tag{23}\]Again, by applying [PERSON]'s theorem to the last term, we obtain  
\[\begin{split}\int_{\Omega}\delta p\
abla\cdot\left[-\frac{1}{\gamma_{t }}\mathbf{k}\cdot\Big{(}\
abla p^{\mathrm{f}}-\rho_{\mathrm{f}}\mathbf{b}\Big{)}\right] \mathrm{d}\Omega=\int_{\Gamma_{N}^{\mathrm{p}}}\delta p\overline{\eta}\mathrm{d} \Gamma\\ -\int_{\Omega}\
abla\delta p\cdot\left[-\frac{1}{\gamma_{t}}\mathbf{k }\cdot\Big{(}\
abla p^{\mathrm{f}}-\rho_{\mathrm{f}}\mathbf{b}\Big{)}\right] \mathrm{d}\Omega.\end{split} \tag{24}\]  
Thus, the weak form of the mass balance equation, (14), is  
\[\begin{split}\alpha\int_{\Omega}\delta p\
abla\cdot\mathbf{i}\mathrm{ d}\Omega+c\int_{\Omega}\delta p\frac{\partial p^{\mathrm{f}}}{\partial t} \mathrm{d}\Omega+\int_{\Omega}\
abla\delta p^{\mathrm{f}}\frac{1}{\gamma_{t}} \mathbf{k}\
abla p^{\mathrm{f}}\mathrm{d}\Omega\\ =\int_{\Omega}\
abla\delta p^{\mathrm{T}}\frac{1}{\gamma_{t}}\mathbf{ k}\rho_{\mathrm{f}}\mathrm{d}\Omega-\int_{\Gamma_{N}^{\mathrm{p}}}\delta p \overline{\eta}\mathrm{d}\Gamma.\end{split} \tag{25}\]