Spontaneous imbibition dynamics in two-dimensional porous media: A generalized interacting multi-capillary model

Abstract

The capillary bundle model, wherein the flow dynamics of a porous medium is predicted from that of a bundle of independent cylindrical tubes/capillaries whose radii are distributed according to the medium's pore size distribution, has been used extensively. However, as it lacks an interaction between the flow channels, this model fails at predicting a complex flow configuration, including those involving a two-phase flow. We propose here to predict spontaneous imbibition in quasi-two-dimensional porous media from a model based on a planar bundle of interacting capillaries. The imbibition flow dynamics, and in particular, the breakthrough time, the global wetting fluid saturation at breakthrough, and which capillary carries the leading meniscus are governed by the distribution of the capillaries' radii and their spatial arrangement. For an interacting capillary system consisting of 20 capillaries, the breakthrough time can be 39% smaller than that predicted by the classic, non-interacting, capillary bundle model of identical capillary radii distribution, depending on the spatial arrangement of the capillaries. We propose a stochastic approach to use this model of interacting capillaries for quantitative predictions. Comparing bundles of interacting capillaries with the same capillary diameter distribution as that of the pore sizes in the target porous medium, and computing the average behavior of a randomly chosen samples of such interacting capillary bundles with different spatial arrangements, we obtain predictions of the position in time of the bulk saturating front and of that of the leading visible leading front, which agree well with measurements taken from the literature. This semi-analytical model is very quick to run and could be useful to provide fast predictions on one-dimensional spontaneous imbibition in porous media whose porosity structure can reasonably be considered two-dimensional, e.g., paper, thin porous media in general, or layered aquifers.