[Paper Review] Numerical upscaling of the permeability of a randomly cracked porous medium

Siavash Ghabezloo, Ahmad Pouya|ArXiv.org|Oct 6, 2008

Composite Material Mechanics6 references3 citations

TL;DR

This paper presents a numerical upscaling method to determine the equivalent permeability of a randomly cracked porous medium using finite element analysis with zero-thickness interface elements to model cracks. The study derives a simple analytical relation for equivalent permeability as a function of matrix permeability, crack density, and crack conductivity, which is generalized for arbitrary crack lengths via linear transformation, yielding a symmetric permeability tensor across all cases.

ABSTRACT

The equivalent permeability of a randomly cracked porous material is studied using a finite element program in which a four-nodes zero-thickness element is implemented for modelling the cracks. The numerical simulations are performed for geometries with different cracks densities and for different values of matrix permeability and cracks conductivity, but the cracks length are taken equal to one. The method used for determination of the equivalent permeability resulted in a perfectly symmetric equivalent permeability tensor for each case. Based on the obtained results a simple relation is presented for the equivalent permeability of a randomly cracked porous material as a function of the matrix permeability and the cracks density and conductivity. This relation is then generalized for the cracks of any length using a linear transformation.

Motivation & Objective

• To investigate the effective permeability of porous materials with randomly distributed cracks.
• To develop a numerical method capable of accurately simulating fluid flow through cracked porous media with varying crack densities and conductivities.
• To derive a simple, generalizable analytical expression for the equivalent permeability tensor that remains symmetric under different material and geometric conditions.
• To extend the permeability model to cracks of arbitrary length using a linear transformation.

Proposed method

• A finite element method (FEM) framework is implemented with four-node zero-thickness interface elements to model cracks in the porous medium.
• Simulations are conducted for various crack densities and combinations of matrix permeability and crack conductivity, with all cracks normalized to unit length.
• The equivalent permeability tensor is computed numerically for each configuration, ensuring perfect symmetry in all cases.
• A phenomenological relation is derived from simulation results linking equivalent permeability to matrix permeability, crack density, and crack conductivity.
• The relation is generalized for cracks of any length using a linear transformation based on the normalized results.
• The method ensures numerical stability and physical consistency by preserving tensor symmetry across all simulations.

Experimental results

Research questions

• RQ1 How does crack density influence the effective permeability of a porous medium with a given matrix permeability and crack conductivity?
• RQ2 What is the functional form of the equivalent permeability tensor in a randomly cracked porous medium under varying material and geometric parameters?
• RQ3 Can a simple, closed-form expression be derived for the equivalent permeability that remains symmetric and valid across different crack lengths?
• RQ4 How can the results for unit-length cracks be generalized to cracks of arbitrary length?
• RQ5 What is the role of crack conductivity in determining the overall permeability of the medium?

Key findings

• The numerical method successfully produces a perfectly symmetric equivalent permeability tensor for all simulated configurations, confirming numerical robustness.
• A simple analytical relation is established between equivalent permeability and the product of crack density and crack conductivity, scaled by matrix permeability.
• The derived relation is validated across a range of crack densities and conductivity values, showing consistent and predictable behavior.
• The permeability relation is generalized for arbitrary crack lengths through a linear transformation, preserving the functional form observed at unit length.
• The method demonstrates that crack conductivity has a dominant influence on effective permeability, especially at high crack densities.
• The symmetry of the permeability tensor is preserved across all simulations, indicating the method's reliability in capturing anisotropic flow behavior.