Modeling relative permeability in multi-scale rocks and fractured networks has broad applications to understanding oil production and recovery in reservoir formations. Natural porous media are typically composed of two domains; one incorporates macropores, while the other contains micropores. In the literature, numerous theoretic models have been developed based on the series-parallel tubes approach (Mualem, 1976; van Genuchten, 1980) to estimate wetting-phase relative permeability (krw) from pore size distribution or capillary pressure curve. In this study, we, however, invoke concepts from critical path analysis (CPA), a theoretical technique from statistical physics. CPA has been successfully used to model flow and transport in porous media (Hunt, 2001; Ghanbarian-Alavijeh and Hunt, 2012; Hunt et al., 2013; 2014; Ghanbarian et al., 2016; Ghanbarian and Hunt, 2017). We estimate the wetting-phase relative permeability from the measured capillary pressure curve using two methods: (1) critical path analysis (CPA), and (2) series-parallel tubes (vG-M). To evaluate these models, we use 26 experiments from the literature for which capillary pressure and wetting-phase relative permeability data were measured at 500 data point over a wide range of wetting-phase saturation (Sw). Results demonstrate that CPA estimates krw more precisely than vG-M. We show that accurate krw estimation by the CPA-based model needs precise characterization of capillary pressure curve and accurate calculation of the crossover point (Swx) separating the two domains.

Fractures play an important role in transport in unconventional formations. Both matrix and fracture network in fractured media contribute to the effective permeability (keff). In this study, we investigate the interaction between matrix and fracture permeabilities (km and kf) via the percolation-based effective-medium approximation (P-EMA). We theoretically determine the keff value at different fracture densities and compare our results with two- and three-dimensional numerical simulations. We found that the P-EMA estimated the effective permeability accurately in the matrix-fracture systems studied here. Further investigations are needed to better understand how the parameters of the P-EMA including scaling exponent and critical fracture density might vary from one fractured reservoir to another.

Scaling has been a long-standing challenge in subsurface hydrology, soil physics, and many other research disciplines. The effect of length scale (or sample dimension) has been known in the literature, and inconsistent results have been reported. For example, experimental measurements typically show that permeability k should increase with increasing scale. However, numerical simulations and some theoretical estimations appear to imply the opposite. In this study, we simulated permeability in twelve synthetic and four Fontainebleau pore networks with different pore-throat size distributions. For each pore network, simulations were carried out for ten pore coordination numbers Z = 1.5, 1.65, 1.75, 2, 3, 3.25, 3.5, 4, 5, and 6. We found a transition in the scale dependence of the permeability in the synthetic pore networks. More specifically, our results showed that the permeability increased with the scale for larger pore coordination numbers, while it decreased with the scale for smaller Z. In Fontainebleau pore networks, however, the trends were decreasing permeabilities regardless of the value of Z. We invoked concept of finite-size scaling analysis, a vigorous theoretical framework from physics, to address the effect of scale on the permeability. Although the plot of the permeability versus the network size for each pore network appeared scattered, the data collapsed together by applying finite-size scaling analysis. Our results demonstrated that finite-size scaling analysis is a promising approach to address the effect of scale on permeability.

Klinkenberg-corrected gas permeability (k) estimation in tight-gas sandstones is essential for gas exploration and production in low-permeability porous rocks. Most models for estimating k are a function of porosity (ϕ), tortuosity (τ), pore shape factor (s) and a characteristic length scale (lc). Estimation of the latter, however, has been the subject of debate in the literature. Here we invoke two different upscaling approaches from statistical physics: (1) the EMA and (2) critical path analysis (CPA) to estimate lc from pore throat-size distribution derived from mercury intrusion capillary pressure (MICP) curve. τ is approximated from: (1) concepts of percolation theory and (2) formation resistivity factor measurements (F = τ/ϕ). We then estimate k of eighteen tight-gas sandstones from lc, τ, and ϕ by assuming two different pore shapes: cylindrical and slit-shaped. Comparison with Klinkenberg-corrected k measurements showed that τ was estimated more accurately from F measurements than from percolation theory. Generally speaking, our results implied that the EMA estimated k within a factor of two of the measurements and more precisely than CPA. We further found that the assumption of cylindrical pores yielded more accurate k estimates when τ was estimated from concepts of percolation theory than the assumption of slit-shaped pores. However, the EMA with slit-shaped pores estimated k more precisely than that with cylindrical pores when τ was estimated from F measurements.

Abstract Gas relative permeability, krg, is a key parameter to determine gas production in unconventional reservoirs. Several theoretical approaches were proposed to study gas relative permeability in tight and ultra-tight porous rocks. Some models are based on a “bundle of capillary tubes” concept. Some others were developed based upon a combination of universal scaling laws from percolation theory and the effective-medium approximation (EMA). Although applications from the EMA have been successfully used to estimate single-phase permeability in permeable media (Ghanbarian et al., 2017; Ghanbarian and Javadpour, 2017), non-universal scaling from the EMA has never been invoked to model gas relative permeability in tight and/or ultra-tight porous rocks. In this study, it was assumed that pore-throat sizes follow the log-normal distribution. It was further assumed that gas transport in shales is mainly controlled by molecular and hydraulic flow, two mechanisms contributing in parallel. Using the EMA, effective pore-throat radii, effective conductances, and gas relative permeabilities were determined at various gas saturations. Comparison with three-dimensional pore-network simulations showed that the proposed krg model estimated gas relative permeability accurately. We also compared our model with experimental data reported in Yassin et al. (2016) including three Montney tight gas siltstone samples from the Western Canadian Sedimentary Basin. Results showed that our model estimated krg reasonably well, although it slightly overestimated krg. This might be because the fitted log-normal probability density function underestimated the probability of small pore-throat sizes. References Ghanbarian, B., & Javadpour, F. (2017). Upscaling pore pressure‐dependent gas permeability in shales. Journal of Geophysical Research: Solid Earth, 122(4), 2541-2552. Ghanbarian, B., Torres-Verdin, C., Lake, L. W., & Marder, M. P. (2017). Upscaling gas permeability in tight-gas sandstones. AGU Fall Meeting Abstracts. New Orleans LA. Yassin, M. R., Dehghanpour, H., Wood, J., & Lan, Q. (2016). A theory for relative permeability of unconventional rocks with dual-wettability pore network. SPE Journal, 21(06), 1970-1980.