Fluid transfer between tubes in interacting capillary bundle models

The interacting capillary bundle model proposed by Dong et al. Dong, M., Dullien, F.A.L., Zhou, J.: Trans. Porous Media 31, 213–237 (1998); Dong, M., Dullien, F.A.L., Dai, L., Li, D.: Trans. Porous Media 59, 1–18 (2005); Dong, M., Dullien, F.A.L., Dai, L., Li, D.: Trans. Porous Media 63, 289–304 (2006)] has simulated correctly various aspects of immiscible displacement in porous media, such as oil production histories at different viscosity ratios, the effects of water injection rate and of the oil–water viscosity ratio on the shape of the displacement front and the independence of relative permeabilities of the viscosity ratio. In the interacting capillary bundle model pressure equilibrium was assumed at any distance x measured along the bundle. Interaction between the capillaries also results in transfer of fluids across the capillaries. In the first part of this paper the process of fluid transfer between two capillaries is analysed and an algebraic expression for this flow is derived. Consistency with the assumption of pressure equilibration requires that all transfer must take place at the positions of the oil/water menisci in the tubes without any pressure drop. It is shown that fluid transfer between the tubes has no effect on the predictions obtained with the model. In the second part of the paper the interacting tube bundle model is made more realistic by assuming fluid transfer between the tubes all along the single phase flow regions across a uniform resistance, resulting in pressure differences throughout the single phase regions between the fluids present in the different tubes. The results of numerical simulations obtained with this improved interacting capillary bundle model show only small differences in the positions of the displacement front as compared with the predictions of the idealized model.