Analytical approximations for effective relative permeability in the capillary limit

• Published in: Water resources research 2016-10, Vol.52 (10), p.7645-7667
• Main Authors: Rabinovich, Avinoam, Li, Boxiao, Durlofsky, Louis J.
• Format: Article
• Language: English
• Subjects: Accuracy; analytical method; Analytical methods; Anisotropy; Approximation; capillary heterogeneity; capillary limit; Capillary pressure; Carbon dioxide; Carbon sequestration; CO2‐brine flow; Computer simulation; Correlation; Correlation analysis; Derivation; effective relative permeability; Exact solutions; Exponents; Fields; geometric/power law average; Mathematical models; Membrane permeability; Normality; Permeability; Power law; Pressure; Saturation; Series (mathematics); Solutions; Steady state; Structures; Tests; Variance; Water
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1002/2016WR019234

We present an analytical method for calculating two‐phase effective relative permeability, krjeff, where j designates phase (here CO2 and water), under steady state and capillary‐limit assumptions. These effective relative permeabilities may be applied in experimental settings and for upscaling in the context of numerical flow simulations, e.g., for CO2 storage. An exact solution for effective absolute permeability, keff, in two‐dimensional log‐normally distributed isotropic permeability (k) fields is the geometric mean. We show that this does not hold for krjeff since log normality is not maintained in the capillary‐limit phase permeability field ( Kj=k·krj) when capillary pressure, and thus the saturation field, is varied. Nevertheless, the geometric mean is still shown to be suitable for approximating krjeff when the variance of ln⁡k is low. For high‐variance cases, we apply a correction to the geometric average gas effective relative permeability using a Winsorized mean, which neglects large and small Kj values symmetrically. The analytical method is extended to anisotropically correlated log‐normal permeability fields using power law averaging. In these cases, the Winsorized mean treatment is applied to the gas curves for cases described by negative power law exponents (flow across incomplete layers). The accuracy of our analytical expressions for krjeff is demonstrated through extensive numerical tests, using low‐variance and high‐variance permeability realizations with a range of correlation structures. We also present integral expressions for geometric‐mean and power law average krjeff for the systems considered, which enable derivation of closed‐form series solutions for krjeff without generating permeability realizations. Key Points: Method presented for calculating effective relative permeability using geometric and power averaging Closed‐form series expressions are derived Accuracy of methods is assessed through extensive numerical calculations