Free convection in fractured porous media: a numerical study

TL;DR

This paper addresses density-driven free convection in fractured porous media by developing a mixed-dimensional OB-based model that couples bulk flow and fracture dynamics through dimensional reduction and interface fluxes. It introduces an eigenvalue-based stability analysis that predicts convection onset without time stepping and demonstrates strong agreement with direct time-dependent simulations while offering substantial computational efficiency. The MPFA-based finite-volume discretization and PorePy implementation enable robust handling of complex fracture networks, including 3D HRL configurations. The findings emphasize the joint influence of fracture circuits and matrix permeability on convection, showing that fractures alone are insufficient predictors and that matrix–fracture coupling critically governs stability and transport in fractured reservoirs.

Abstract

The objective of this study is to better understand the influence of fractures on the possibility of free convection in porous media. To this aim, we introduce a mathematical model for density driven flow in the presence of fractures, and the corresponding numerical approximation. In addition to the direct numerical solution of the problem we propose and implement a novel method for the assessment of convective stability through the eigenvalue analysis of the linearized numerical problem. The new method is shown to be in agreement with existing literature cases both in simple and complex fracture configurations. With respect to direct simulation in time, the results of the eigenvalue method lack information about the strength of convection and the steady state solution, they however provide detailed (quantitative) information about the behavior of the solution near the initial equilibrium condition. Furthermore, not having to solve a time-dependent problem makes the method computationally very efficient. Finally, the question of how the porous matrix interacts with the fracture network to enable free convection is examined: the porous matrix is shown to be of key importance in enabling convection for complex fracture networks, making stability criteria based on the fracture network alone somewhat limited in applicability.

Figures (26)

• Figure 1: Convective motion. On the left, concentration profile. At center, perturbation of concentration and flow velocity. On the right, the diffusive flow trying to restore the concentration imbalance.
• Figure 2: Domain $\Omega$ containing a single planar fracture $\Omega_f$.
• Figure 3: $\Omega_f$ is a rectangular fracture of height $b$. $\Gamma^+$ and $\Gamma^-$ are both subsets of its boundary: $\Gamma^\pm \subset \partial \Omega_f$.
• Figure 4: Multidimensional coupling for a sample fracture network embedded in a three-dimensional domain.
• Figure 5: In this configuration the Rayleigh number computed based on the upscaled permeability is well below the critical Rayleigh number. The presence of the fracture circuit however enables convection regardless.

Theorems & Definitions (1)

• Remark 1