A generalized approach for estimation of in‐plane curvature in invasion percolation models for drainage in fractures

In‐plane interfacial curvature plays an important role in shaping the phase structure during quasi‐static immiscible displacement in horizontal rough‐walled fractures. Existing approaches used in percolation modeling require the use of a predefined globally representative length scale for calculating the in‐plane curvature. This length scale is typically estimated using the geostatistics of fracture aperture variability. However, there has not been any rigorous and general derivation on how to obtain this length scale in the literature. This paper presents a general method referred to as adaptive circle‐fitting (ACF) approach to estimate in‐plane curvature, recognizing that in some fractures the in‐plane curvature along the fluid‐fluid interface may exhibit various length scales due to the variability both in the locally averaged aperture and in the fracture wall roughness. The ACF involves nonlinear fitting of the interface to an osculating circle whose radius geometrically defines the inverse of the local curvature. This approach does not require predefining an empirical representative length scale. The influence length of the interface to which the circle is fitted is determined adaptively based on an acceptable threshold error. We have implemented the ACF approach to an invasion percolation model and have performed numerical simulations against experimental data on drainage processes in two horizontal rough‐walled fractures. The observed invasion phase structures can be well reproduced using this generalized approach. In comparison to the previous approaches, it also demonstrates better performance in matching the experimental results of invasion phase distributions and fractal dimensions of the invasion clusters.