# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 2 Governing Equations
### Linear Momentum Balance Equation  
The linear momentum balance equation for a fluid-saturated porous medium is given by:  
\[\
abla\cdot\underbrace{\left(\sigma^{\prime}+\alpha p^{\mathsf{f}}\mathbf{I} \right)}_{=\sigma}+\rho\mathbf{b}=\mathbf{0} \tag{1}\]  
where \(\sigma\) is the total stress, \(\sigma^{\prime}\) is the effective stress, \(\alpha\) is Biot's coefficient, \(p^{\mathsf{f}}\) is the fluid pressure, \(\mathbf{I}\) is an identity matrix, \(\rho\) is the overall density of the porous medium and \(\mathbf{b}\) represents body forces. The Biot coefficient \(\alpha\) can be calculated from:  
\[\alpha=1-\frac{K_{\mathsf{t}}}{K_{\mathsf{s}}} \tag{2}\]  
where \(K_{\mathsf{t}}\) and \(K_{\mathsf{s}}\) are the bulk moduli of the porous medium and solid particles, respectively.  
The constitutive equation for poroeasticity relates stress and strain linearly as:  
\[\sigma^{\prime}=\mathbf{D}:\mathbf{\varepsilon} \tag{3}\]  
where \(\mathbf{D}\) is a fourth-order stiffness tensor. Small deformations are also assumed, so the strain \(\mathbf{\varepsilon}\) satisfies a linear first-order equation with respect to the displacement \(\mathbf{u}\),  
\[\mathbf{\varepsilon}=\frac{1}{2}(\
abla\mathbf{u}+\
abla^{\mathsf{T}}\mathbf{u}) \tag{4}\]  
where \(\
icefrac{{1}}{{2}}(\
abla+\
abla^{\mathsf{T}})\) is the symmetrized gradient operator i.e.,  
\[\varepsilon_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}. \tag{5}\]  
In the following, it will be convenient to lower tensors and higher differential operators to Voigt notation, which represents the symmetric \(d\times d\) tensor \(\sigma^{\prime}\) as a \({}^{d(d+1)}/{2}\)-vector, which we will denote with a tilde:  
\[\underbrace{\begin{bmatrix}\sigma^{\prime}_{xx}&\sigma^{\prime}_{xy}&\sigma^ {\prime}_{xz}\\ \sigma^{\prime}_{xy}&\sigma^{\prime}_{yy}&\sigma^{\prime}_{yz}\\ \sigma^{\prime}_{xz}&\sigma^{\prime}_{yz}&\sigma^{\prime}_{zz}\\ \end{bmatrix}}_{\sigma^{\prime}}\]  
A similar conversion takes place for the strains, where the shear strains are replaced by the _engineering_ shear strains:  
\[\underbrace{\begin{bmatrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz }\\ \varepsilon_{xy}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{xz}&\varepsilon_{yz}&\varepsilon_{zz}\\ \end{bmatrix}}_{\varepsilon}\]  
Voigt notation allows us to express the equilibrium equation and the stress-strain equation using the same differential operator \(\mathbf{L}\),\[\mathbf{L}^{\intercal}=\begin{bmatrix}\frac{\partial}{\partial x}&0&0&\frac{\partial}{ \partial y}&0&\frac{\partial}{\partial z}\\ 0&\frac{\partial}{\partial y}&0&\frac{\partial}{\partial x}&\frac{\partial}{ \partial z}&0\\ 0&0&\frac{\partial}{\partial z}&0&\frac{\partial}{\partial y}&\frac{\partial}{ \partial x}\end{bmatrix} \tag{6}\]  
Using \(\mathbf{L}\) yields the following equilibrium equation in terms of the two primary unknowns \(\mathbf{u}\) and \(p^{\intercal}\),  
\[\mathbf{L}^{\intercal}\tilde{\mathbf{D}}\mathbf{L}\mathbf{u}-\alpha\
abla p^{\intercal}+\rho \mathbf{b}=\mathbf{0}, \tag{7}\]  
where \(\tilde{\mathbf{D}}\) is the Voigt notation equivalent of \(\mathbf{D}\), taking into account the aforementioned engineering shear strains. We will generally assume isotropic materials, where \(\tilde{\mathbf{D}}\) takes the block form (in terms of [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\))  
\[\tilde{\mathbf{D}}=\frac{E}{(1+\
u)(1-2\
u)}\begin{bmatrix}\tilde{\mathbf{D}}_{11}& \mathbf{0}\\ \mathbf{0}&\tilde{\mathbf{D}}_{22}\end{bmatrix} \tag{8}\]  
where the two blocks are given as  
\[\tilde{\mathbf{D}}_{11} =(1-2\
u)\mathbf{I}+\
u\mathbf{I} \tag{9}\] \[\tilde{\mathbf{D}}_{22} =\frac{1-2\
u}{2}\mathbf{I}\]  
and \(\mathbf{1}\) is a matrix of ones.