Application of Deep Learning Reduced-Order Modeling for Single-Phase Flow in Faulted Porous Media

Abstract

We apply reduced-order modeling (ROM) techniques to single-phase flow in faulted porous media, accounting for changing rock properties and fault geometry variations using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework that effectively manages the resulting non-conforming mesh. To streamline complex and repetitive calculations such as sensitivity analysis and solution of inverse problems, we utilize the Deep Learning Reduced Order Model (DL-ROM). This non-intrusive neural network-based technique is evaluated against the traditional Proper Orthogonal Decomposition (POD) method across various scenarios, demonstrating DL-ROM's capacity to expedite complex analyses with promising accuracy and efficiency.

Figures (22)

• Figure 1: Generic domain divided in $\Omega_1$ and $\Omega_2$ by a fault, $\gamma$, with normal $\upsilon_\gamma$. The coupling fluxes, $\lambda^+$ and $\lambda^-$, are defined at the additional interfaces, $\gamma^+$ and $\gamma^-$. The parts of the external boundary with the Neumann conditions are indicated by $\partial_q \Omega$ and $\partial_q \gamma$, while the external boundaries with the Dirichlet conditions are denoted by $\partial_p \Omega$ and $\partial_p \gamma$. The internal boundaries facing the fault are called $\partial_{in}\Omega_1$ and $\partial_{in}\Omega_2$.
• Figure 2: Y-shaped intersection. The three faults, $\gamma_1$, $\gamma_2$, $\gamma_3$, cross each other at the point $\iota$. $\hat{\upsilon}_{\gamma_i}$ are the outward unit vectors aligned with the respective fault.
• Figure 3: DL-ROM scheme. The encoder, $\Psi'$, takes the FOM solution, $u_N$, and returns the reduced basis solution, $u_n$. The decoder $\Psi$ reproduces an approximate solution, called reconstructed solution $\Tilde{u}_N$, with the only knowledge of the reduced solution. The reduced map network, $\varphi$, approximates $u_n$ as a function of the given parameter $\mu$.
• Figure 4: Generic faulted domain with control points, in orange. The dashed lines represent sliding surfaces, one is on $\partial_{ex}\Omega$, and the control points therein belong to $C_{s}$, the other two represent the boundaries $\partial_{in}\Omega$ facing a fault, the control points on those lines are included in the set $C_{sf}$. The gap between the two faces of the fault was added for graphical reasons only, and it is actually absent, so the control points on the fault may be coincident. Other control points in $C_d$ are placed on the fracture where a rigid displacement, $\overline{s}$, is applied. The remaining control points in $C_{df}$ are on the boundary of $\Omega$ where a displacement $\overline{s}$ is enforced.
• Figure 5: Mesh deformation. Zoom of the bottom-left corner of the domain of the third test case, see \ref{['sec:test_3']}. A rigid displacement is imposed to the left side of the fault and to the two small intersecting fractures. The left boundary is a sliding surface where sliding conditions are applied, on the bottom one, instead, a null displacement is enforced.