Preconditioning and Reduced-Order Modeling of Navier-Stokes Equations in Complex Porous Microstructures

TL;DR

This work develops robust monolithic preconditioners for solving the incompressible Navier–Stokes equations in complex porous microstructures discretized on a fine MAC grid, addressing the resulting saddle-point systems with Krylov solvers. It introduces algebraic ($aPLMM_{NS}$, $aPNM_{NS}$) and geometric ($gPLMM$, $gPNM$) two-level preconditioners built from pore-scale multiscale concepts and watershed-based domain decompositions, plus a coarse-scale solver based on $gPLMM$ for fast steady-state solutions. Through extensive 2D/3D pore geometries and both steady and unsteady flow regimes, the geometric PLMM preconditioner ($gPLMM$) consistently yields the best convergence and speed, with algebraic Stokes-based variants providing strong alternatives when geometry access is limited. The coarse-scale solver offers substantial speedups (up to an order of magnitude in 2D and several-fold in 3D) while maintaining acceptable accuracy, enabling efficient projections of flow properties like permeability for design and subsurface applications.

Abstract

We aim to solve the incompressible Navier-Stokes equations within the complex microstructure of a porous material. Discretizing the equations on a fine grid using a staggered (e.g., marker-and-cell, mixed FEM) scheme results in a nonlinear residual. Adopting the Newton method, a linear system must be solved at each iteration, which is large, ill-conditioned, and has a saddle-point structure. This demands an iterative (e.g., Krylov) solver, that requires preconditioning to ensure rapid convergence. We propose two monolithic \textit{algebraic} preconditioners, $a\mathrm{PLMM_{NS}}$ and $a\mathrm{PNM_{NS}}$, that are generalizations of previously proposed forms by the authors for the Stokes equations ($a\mathrm{PLMM_{S}}$ and $a\mathrm{PNM_{S}}$). The former is based on the pore-level multiscale method (PLMM) and the latter on the pore network model (PNM), both successful approximate solvers. We also formulate faster-converging geometric preconditioners $g\mathrm{PLMM}$ and $g\mathrm{PNM}$, which impose $\partial_n\boldsymbol{u}\!=\!0$ (zero normal-gradient of velocity) exactly at subdomain interfaces. Finally, we propose an accurate coarse-scale solver for the steady-state Navier-Stokes equations based on $g\mathrm{PLMM}$, capable of computing approximate solutions orders of magnitude faster. We benchmark our preconditioners against state-of-the-art block preconditioners and show $g\mathrm{PLMM}$ is the best-performing one, followed closely by $a\mathrm{PLMM_{S}}$ for steady-state flow and $a\mathrm{PLMM_{NS}}$ for transient flow. All preconditioners can be built and applied on parallel machines.

Figures (22)

• Figure 1: Schematic of a pore-scale image and its decomposition into primary grids and dual grids in the PLMM and PNM preconditioner. (a) The white regions of the image represent the void space and black regions represent the solid phase. The Navier-Stokes equations are solved in the void space. (b, d) Primary grids (or subdomains) obtained from PLMM (b) and PNM (d) decompositions. (c, e) Corresponding dual grids that overlap the contact interfaces shared between primary grids. (f, g) Each basis function is associated with one contact interface (yellow). In PLMM, each contact interface is shared by only two adjacent primary grids. While in PNM, each contact interface is shared among multiple primary grids.
• Figure 2: (a) Two primary grids, ${\Omega^p_1}$ and ${\Omega^p_2}$, separated by a contact interface, ${\Gamma_{12}}$. (b) Partitioning of the cells and faces of the fine grid among the two primary grids and the contact interface is shown. Light/dark blue cells/faces belong to ${\Omega^p_1}$, light/dark green cells/faces belong to ${\Omega^p_2}$, and light/dark red cells/faces belong to ${\Gamma_{12}}$. (c) The cell unknowns correspond to pressure, $p$, and the face unknowns correspond to velocity, $\boldsymbol{u}$.
• Figure 3: The inertia ($\boldsymbol{u}\cdot\nabla\boldsymbol{u}$) term is removed from the modified residual in Eq.\ref{['eq:mod_res']} of the fine-grid faces that lie within a few layers of each contact interface (${\Gamma_{12}}$). These faces are tagged with black dots. The operation is required to impose the closure BC $\partial_n\boldsymbol{u}\!=\!\boldsymbol{0}$ exactly at contact interfaces.
• Figure 4: Basis functions (pressure and velocity magnitude) corresponding to the $\mathrm{M_G}$ of $g$PLMM and $g$PNM. Each basis is a column of the matrix $\mathrm{B}$ in Eq.\ref{['eq:prolong_1']}, comprised of two (in $g$PLMM) or four (in $g$PNM) shape vectors associated with the yellow-highlighted interface. Navier-Stokes bases are built from the Jacobian of the modified residual in Eq.\ref{['eq:mod_res']} at $Re\!=\!1.6\times10^4$. The bases belong to primary grids highlighted in Figs.\ref{['fig:main_sketch']}f-g, with BCs $p\!=\!0$ and $\partial\boldsymbol{u}/\partial n\!=\!0$ on all red lines, and $p\!=\!1$ and $\partial\boldsymbol{u}/\partial n\!=\!0$ on yellow lines. No-slip BCs hold on the void-solid interface (gray lines).
• Figure 5: Schematic of 2D and 3D porous geometries considered for testing the proposed preconditioners and the coarse-scale solver.

Theorems & Definitions (2)

• Proposition 3.1
• proof