[Paper Review] Scale-dependent permeability and formation factor in porous media: Applications from percolation theory

Misagh Esmaeilpour, Behzad Ghanbarian+2 more|arXiv (Cornell University)|Jan 21, 2021

Hydrocarbon exploration and reservoir analysis63 references4 citations

TL;DR

This study proposes theoretical models based on percolation theory to predict scale-dependent permeability and formation factor in 3D porous media, using pore-throat radius distributions. Simulations across five network sizes (1130–6770 μm) show the models predict permeability and formation factor with relative errors of −3.7% to 3.8% and 0.21% to 4.04%, respectively, demonstrating strong agreement with numerical simulations.

ABSTRACT

Understanding porous media properties and their scale dependence have been an active subject of research in the past several decades in hydrology, geosciences and petroleum engineering. The scale dependence of flow in porous media is attributed to small- and large-scale heterogeneities, such as pore size distribution, pore connectivity, long-range correlations, fractures and faults orientations, and spatial and temporal variations. The main objective of this study was to investigate how permeability (k) and formation factor (F) vary with sample dimension at small scales by means of a combination of pore-network modeling and percolation theory. For this purpose, the permeability and formation factor were simulated in twelve three-dimensional pore networks with different levels of pore-scale heterogeneities. Simulations were carried out at five different network sizes, i.e., 1130, 2250, 3380, 4510 and 6770 microns. Four theoretical models were also developed based on percolation theory to estimate the scale dependence of permeability and formation factor from the pore-throat radius distribution. In addition, two other theoretical scale-dependent permeability models were proposed to estimate permeability at different scales from the pore-throat radius distribution and formation factor. Comparing theoretical estimations with numerical simulations showed that the proposed models estimate the scale dependence of permeability and formation factor reasonably. The calculated relative error (RE) ranged between -3.7 and 3.8% for the permeability and between 0.21 and 4.04% for the formation factor in the studied pore-networks.

Motivation & Objective

• To investigate the scale dependence of permeability and formation factor in porous media at small scales.
• To understand how pore-scale heterogeneities such as pore size distribution and connectivity affect macroscopic transport properties.
• To develop theoretical models based on percolation theory that predict permeability and formation factor across different sample dimensions.
• To validate these models against numerical simulations using 12 three-dimensional pore networks with varying heterogeneity levels.

Proposed method

• Simulated permeability and formation factor in 12 three-dimensional pore networks with distinct pore-scale heterogeneities.
• Used network sizes ranging from 1130 to 6770 microns to assess scale dependence.
• Developed four theoretical models based on percolation theory using pore-throat radius distribution to estimate scale-dependent permeability and formation factor.
• Proposed two additional theoretical models to estimate permeability from pore-throat radius distribution and formation factor.
• Compared theoretical predictions with numerical simulation results to evaluate accuracy.
• Calculated relative error (RE) between theoretical estimates and simulation outputs for both permeability and formation factor.

Experimental results

Research questions

• RQ1 How does permeability vary with sample size in porous media due to pore-scale heterogeneities?
• RQ2 To what extent can percolation theory predict the scale dependence of formation factor in 3D porous networks?
• RQ3 Can theoretical models based on pore-throat radius distribution accurately estimate permeability across different scales?
• RQ4 What is the accuracy of these models when validated against numerical simulations of heterogeneous pore networks?

Key findings

• The proposed percolation-based models accurately predict scale-dependent permeability with relative errors between −3.7% and 3.8% across all network sizes.
• Formation factor estimates from the theoretical models show relative errors ranging from 0.21% to 4.04%, indicating high predictive accuracy.
• The models demonstrate consistent performance across a range of pore-network heterogeneities, confirming their robustness.
• The integration of pore-throat radius distribution with percolation theory enables reliable estimation of macroscopic transport properties at varying scales.
• Numerical simulations confirm that permeability and formation factor are strongly influenced by network size and pore connectivity, validating the need for scale-dependent modeling.
• The study establishes a theoretical framework that links microscale pore structure to macroscale flow behavior in porous media.