Experimental studies of capillary-dominated displacements in variable-aperture fractures have demonstrated the occurrence of a satiated state at the end of invasion, where significant entrapment of the displaced phase occurs. The structure of this entrapped phase controls the behavior of flow and transport processes in the flowing phase. Recent studies have shown that the areal saturation of the flowing phase at satiation $\left(S_f\right)$ is largely controlled by a single parameter $C∕\delta$, where $C$, the curvature number, weighs the mean in-plane interfacial curvature relative to the mean out-of-plane interfacial curvature, and $\delta$, the coefficient of variation of the aperture field, represents the strength of interface roughening induced by aperture variations. Here we consider the satiated relative permeability $\left(k_{\mathit{rs}}\right)$ to the flowing phase, which is defined as the relative permeability when the defending phase is fully entrapped. The satiated relative permeability is shown to be a well-defined function of $S_f$ over a wide range of $C∕\delta$, ranging from capillary fingering with significant entrapment $(C∕\delta \to 0)$ to smooth invasion with very little entrapment $(C∕\delta >1)$. We develop a relationship for $k_{\mathit{rs}}$ as a function of $S_f$, by combining theoretical results for the effective permeability in a spatially correlated random permeability field, with results from continuum percolation theory for quantifying the influence of the entrapped phase. The resulting model for $k_{\mathit{rs}}$ also involves a dependence on $\delta$. The predicted relative permeability values are accurate across the entire range of phase structures representative of capillary-dominated displacements in variable-aperture fractures.

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