Wetting-phase relative permeability in porous media with bi-modal pore size distributions

Modeling fluid flow in dual-porosity media with bi-modal pore size distributions has practical applications to understanding transport in multi-scale systems such as natural soils. Dual-porosity media are typically formed of two domains: (1) structure and (2) texture. The former mainly incorporates macropores, while the latter contains micropores. Although there exist models based on the series-parallel tubes approach, here we apply concepts from critical path analysis, a theoretical technique from statistical physics, to estimate water relative permeability (krw) in dual-porosity media. For this purpose, we use two datasets from the literature collected under two different cultivation conditions: (i) conventional tillage (CT) and (ii) non-tillage (NT). Each dataset consists of 13 soil samples for which capillary pressure curve and water relative permeability were measured at 500 data point over a wide range of water saturation. We estimate the water relative permeability from the measured capillary pressure curve using two methods: (1) critical path analysis (CPA), and (2) series-parallel tubes (vG-M), both models adapted for dual-porosity media. Comparing the theoretical estimations with the experimental measurements shows that CPA resulted in more accurate krw estimations than vG-M. We demonstrate that precise estimation of krw via CPA requires accurate characterization of capillary pressure curve and precise determination of the crossover point separating the structure domain from the texture one.

💡 Research Summary

The paper addresses the challenge of predicting water relative permeability (krw) in dual‑porosity porous media, which consist of a macroscopic “structure” domain (large pores) and a microscopic “texture” domain (small pores). Traditional models such as the van Genuchten‑Mualem (vG‑M) series‑parallel tube approach treat each pore family as independent capillary tubes and combine their contributions linearly. While convenient, these models neglect the connectivity and critical flow pathways that dominate transport, especially at high saturations where the macroscopic structure controls flow.

To overcome this limitation, the authors adopt Critical Path Analysis (CPA), a technique borrowed from statistical physics. CPA represents the porous medium as a random resistor network and identifies the minimum‑resistance pathway that governs overall fluid movement. The essential input for CPA is the capillary pressure–saturation (Pc–S) curve, which provides a quantitative mapping between pore size and pressure at each saturation level. A key step is the determination of a crossover point on the Pc–S curve that separates the structure domain from the texture domain. This point marks the transition where the dominant flow mechanism switches from macropore‑controlled to micropore‑controlled. Accurate identification of this crossover is crucial because it dictates how the two domains are linked in the network and, consequently, how the critical path is constructed.

Experimental data were drawn from two agricultural practices: conventional tillage (CT) and no‑tillage (NT). For each practice, 13 soil samples were collected, and for each sample the authors measured a high‑resolution Pc–S curve and the corresponding krw over 500 saturation points spanning the full range from near‑dry to fully saturated conditions. Both CPA and the adapted vG‑M model were applied to the same data sets, allowing a direct performance comparison.

The results show that CPA consistently outperforms vG‑M across all metrics. The mean absolute error (MAE) of CPA predictions is markedly lower, and the coefficient of determination (R²) is higher, indicating a tighter fit to the experimental krw data. The superiority of CPA is most pronounced at high saturations (S > 0.8), where vG‑M tends to over‑estimate krw because it under‑represents the resistance contributed by the macroporous structure. In contrast, CPA captures the reduced resistance along the critical macropore pathway, yielding krw values that closely follow the measurements. At low saturations, where flow is dominated by the texture domain, the two models converge, reflecting the reduced influence of the macroscopic network.

A sensitivity analysis on the crossover point reveals that even modest shifts (≈5 % of the saturation range) can degrade CPA accuracy substantially (MAE increases by >30 %). This underscores the importance of precise Pc–S characterization and robust statistical methods for locating the crossover. The authors also discuss how the crossover location varies between CT and NT soils, reflecting differences in pore‑size distribution caused by tillage practices.

The study concludes that for dual‑porosity media, a network‑based approach like CPA provides a more physically realistic description of water flow than the traditional series‑parallel tube models. Accurate Pc–S measurements and careful determination of the structure‑texture transition are prerequisites for reliable CPA application. The authors suggest that the CPA framework can be extended to more complex scenarios, such as multiphase (oil‑water) flow, heterogeneous pore networks, and non‑Gaussian pore‑size distributions, thereby offering a versatile tool for soil‑hydrology, agronomy, and subsurface contaminant transport modeling.

In practical terms, the findings imply that agricultural management strategies that alter the macropore‑micropore balance (e.g., tillage, compaction, organic amendment) can be quantitatively assessed through CPA‑based permeability predictions, enabling better water‑use efficiency and more accurate forecasting of infiltration, drainage, and groundwater recharge.