Analytical approximations for effective relative permeability in the capillary limit

Abstract

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.

Characterising Drainage Multiphase Flow in Heterogeneous Sandstones

“…This has the overall effect of boosting the equivalent relative permeability to the non-wetting phase as it flows primarily through regions of locally high N 2 saturation. This effect has also been reported before in previous work (Krause and Benson, 2015;Reynolds and Krevor, 2015;Rabinovich et al, 2016). The increased relative permeability manifests most significantly at lower fractional flows whereby the small amounts of nitrogen injected move preferentially through the low capillary pressure layers, greatly affecting the relative permeability.…”

“…The increase in gas permeability from the viscous limit in the parallel layered Bentheimer in both Figure 14a and c signifies it is a physical effect from the capillary pressure layering, as reported by previous authors (Krause and Benson, 2015;Reynolds and Krevor, 2015;Rabinovich et al, 2016). In the Bunter, we see a different effect, whereby both the gas and water relative permeability curves are lowered below the viscous limit as in Virnovsky et al (2004); Rabinovich et al (2016). With combined boundary effects in Figure 14b, the curves have been more significantly reduced, even at high flow rates.…”

“…A significant reduction in the gas or non-wetting phase relative permeability curves is common for perpendicular layered systems at low flow rate (Rabinovich et al, 2016;Virnovsky et al, 2004), where wetting phase relative permeabilities are also lowered, albeit not for the entire saturation range. The water relative permeability is also reduced, as with the parallel layering.…”

Characterizing Drainage Multiphase Flow in Heterogeneous Sandstones

“…The change in equivalent relative permeability from the viscous or capillary limit manifests itself differently for the Bentheimer and Bunter, depending on the orientation of the capillary pressure heterogeneity. The increase in gas permeability from the viscous limit in the parallel layered Bentheimer in both Figures a and c signifies it is a physical effect from the capillary pressure layering, as reported by previous authors (Krause & Benson, ; Rabinovich et al., ; Reynolds & Krevor, ). In the Bunter, we see a different effect, whereby both the gas and water relative permeability curves are lowered below the viscous limit as in Virnovsky et al.…”

“…This has the overall effect of boosting the equivalent relative permeability to the nonwetting phase as it flows primarily through regions of locally high N 2 saturation. This effect has also been reported before in previous work (Krause & Benson, 2015;Rabinovich et al, 2016;Reynolds & Krevor, 2015).…”