Ensemble phase averaged equations for multiphase flows in porous media. Part 1: The bundle-of-tubes model

A bundle-of-tubes construct is used as a model system to study ensemble averaged equations for multiphase flow in a porous material. Momentum equations for the fluid phases obtained from the method are similar to Darcy’s law, but with additional terms. We study properties of the additional terms, and the conditions under which the averaged equations can be approximated by the diffusion model or the extended Darcy’s law as often used in models for multiphase flows in porous media. Although the bundle-of-tubes model is perhaps the simplest model for a porous material, the ensemble averaged equation technique developed in this paper assumes the very same form in more general treatments described in Part 2 of the present work (Zhang, D.Z., 2009. Ensemble Phase Averaged Equations for Multiphase Flows in Porous Media, Part 2: A General Theory. Int. J. Multiphase Flow 35, 640–649). Any model equation system intended for the more general cases must be understood and tested first using simple models. The concept of ensemble phase averaging is dissected here in physical terms, without involved mathematics through its application to the idealized bundle-of-tubes model for multiphase flow in porous media.