Analytical approximations for effective relative permeability in the capillary limit

We present an analytical method for calculating two-phase effective relative permeability, k^eff _rj, 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, k^eff, in two-dimensional log-normally distributed isotropic permeability (k) fields is the geometric mean. We show that this does not hold for k^eff _rj since log normality is not maintained in the capillary-limit phase permeability field (k_j=k . k_rj) when capillary pressure, and thus the saturation field, is varied. Nevertheless, the geometric mean is still shown to be suitable for approximating k^eff _rj when the variance of 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 K_j 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 k^eff _rj 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 k^eff _rj for the systems considered, which enable derivation of closed-form series solutions for k^eff _rj without generating permeability realizations.