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

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. The model lacks interaction between the flow channels, thus fails at predicting complex flow configuration, including those involving two-phase flow. We propose here to predict spontaneous imbibition in quasi-two-dimensional (quasi-2D) porous-media from a model based on a planar bundle of interacting capillaries. The imbibition flow dynamics, particularly, breakthrough time, global wetting fluid saturation at breakthrough, and capillary carrying the leading meniscus are governed by the distribution of the capillaries' radii and their spatial arrangement. For an 20 interacting capillary system, the breakthrough time can be 39% smaller than that predicted by the classic, non-interacting, capillary-bundle-model of identical capillary radii distribution. We propose a stochastic approach to use this model of interacting capillaries for quantitative predictions. Using the 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 visible leading front, that agree well with measurements taken from the literature. This semi-analytical model is quick to run and provides 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.