# A Multiscale CNN-Based Intrinsic Permeability Prediction in Deformable Porous Media
## 3 Database Generation
### Intrinsic Permeability Computation  
The proposed CNN model in this work uses CT images of sandstone as input and outputs the components of the permeability tensor. To consider the intrinsic permeability components in the database, 3D samples of Bentheim sandstone at varying levels of deformation are input into the single-phase LBM solver described in Section 3.2. The primary goal of the LBM simulation is to calculate the average lattice fluid velocity for each prescribed pressure gradient applied across the porous domain along the hydrodynamic axes \(\mathbf{x}_{1}\), \(\mathbf{x}_{2}\) and \(\mathbf{x}_{3}\), i.e., \(\
abla p_{1}\), \(\
abla p_{2}\) and \(\
abla p_{3}\), respectively. The simulation results are used to determine the lattice intrinsic permeability tensor \(\mathbf{K}_{l}^{S}\) in lattice units [I.u.]. This is done under the assumption that the permeability tensor is symmetric and positive definite [61]. As proposed by [PERSON] et al. [61], two fluid flow simulations for each direction with different boundary conditions are to be carried out: one with no-slip boundary conditions and another with natural slip boundary conditions on surfaces parallel to the fluid flow direction (see Figure 3, left, for illustration). The rationale behind this is that the average velocity computed in the pressure direction is lower with no-slip boundary conditions compared to that with natural slip boundary conditions. The difference between these velocities indicates additional fluid flow in the direction orthogonal to the pressure gradient. The average velocities for the no-slip boundary conditions are then used to compute the diagonal components of the permeability tensor using a Darcy filter law:  
\[(K_{l}^{S})_{ii}=-v_{l}\,\frac{(u_{l})_{i,\text{avg}}}{\
abla_{i}\,p_{l}}\quad \left[\text{I.u.}\right],\quad\text{with}\quad i=1,\,2,\,3. \tag{24}\]  
In this, the lattice pressure gradient, presented by \(\
abla_{i}\,p_{l}=\partial p_{l}/\partial x_{i}\) is induced between two opposing surfaces perpendicular to the flow direction to calculate the average lat tice fluid velocity \(\left(u_{l}\right)_{i,\text{avg}}\). As for the latter, the unknown off-diagonal elements of the permeability tensor are computed using the average velocity with natural slip boundary conditions following [61] as  
\[ \begin{bmatrix}\
abla_{2}\,p_{l}&\
abla_{3}\,p_{l}&0\\ \
abla_{1}\,p_{l}&0&\
abla_{3}\,p_{l}\\ 0&\
abla_{1}\,p_{l}&\
abla_{2}\,p_{l}\\ \end{bmatrix}\begin{bmatrix}\left(K_{l}^{S}\right)_{12}\\ \left(K_{l}^{S}\right)_{13}\\ \left(K_{l}^{S}\right)_{23}\\ \end{bmatrix} = \begin{bmatrix}-v_{l}\left(u_{l}\right)_{1,\text{avg}}-\left(K_{ l}^{S}\right)_{11}\
abla_{1}\,p_{l}\\ -v_{l}\left(u_{l}\right)_{2,\text{avg}}-\left(K_{l}^{S}\right)_{22}\
abla_{2} \,p_{l}\\ -v_{l}\left(u_{l}\right)_{3,\text{avg}}-\left(K_{l}^{S}\right)_{33}\
abla_{3} \,p_{l}\\ \end{bmatrix} . \tag{25}\]  
The off-diagonal components we computed for Bentheim sandstone are much smaller than the diagonal components. Thus, we neglect them for simplicity from the ML model. In particular, the symmetric permeability tensor and its simplified diagonal form are expressed as follows:  
\[\mathbf{K}_{l}^{S}= \begin{bmatrix}\left(K_{l}^{S}\right)_{11}&\left(K_{l}^{S} \right)_{12}&\left(K_{l}^{S}\right)_{13}\\ \left(K_{l}^{S}\right)_{21}&\left(K_{l}^{S}\right)_{22}&\left(K_{l}^{S} \right)_{23}\\ \left(K_{l}^{S}\right)_{31}&\left(K_{l}^{S}\right)_{32}&\left(K_{l}^{S} \right)_{33}\\ \end{bmatrix} \left(\mathbf{\bar{e}}_{i}\otimes\mathbf{\bar{e}}_{j}\right). \tag{26}\]  
Here, \(\mathbf{\bar{e}}_{i}\), \(\mathbf{\bar{e}}_{j}\) represent the cartesian basis vectors with \(i,j\in\{1,2,3\}\) and \(\otimes\) is the dyadic product (tensor product). In this connection, Figure 3, right, shows the mean and standard deviation of the diagonal intrinsic permeability components \(\left(K_{l}^{S}\right)_{11}\), \(\left(K_{l}^{S}\right)_{22}\), \(\left(K_{l}^{S}\right)_{33}\) for each strain level \(\varepsilon^{V}\), showing the deformation dependency and anisotropy in the flow.