Two-Phase Flow Through a Non-Circular Capillary At Low Reynolds Numbers
Authors: Singhal A.K. , Somerton W.H. 
Published: March 2010

Abstract
The shape and size of pores making up the pore structure of most porous media are generally very irregular and random. Modelling pore structure by circular tubes, which has been done extensively in the past, is certainly an oversimplification. The non-circular nature of actual pore channels permits the annular or pendular flow regime to exist over a much larger range of wetting-phase saturations than is possible in circular capillaries. The pendular flow mechanism in non-circular capillaries, particularly those with triangular shapes, was studied in detail in the present work. The fluid distribution within a non-circular capillary depends on the contact angles between walls of the capillaries and the two fluids. Expressions are derived in the present work for the shapes of fluid-fluid interfaces in terms of contact angles and fluid saturations. Velocity distributions within the channel are then obtained by analytical and numerical variational (finite-element) procedures. Suitable integration of fluid velocities over appropriate geometries then yields Irrelative permeabilities" of fluids for the single pore space as functions of contact angles and viscosity ratios.
For fluid-fluid displacement flow, an equation similar to Washburn's equation was obtained; for the slug flow regime, an empirical relation, similar to those obtained for circular tubes, is suggested. The conditions affecting transition between flow regimes are discussed. It is suggested that thin films of wetting fluid and flow history play important roles in determining transition conditions and flow behaviour. The need for experimental work in this field is pointed out. The implication of this work in the description of two-phase flow in porous media is discussed.

Introduction
TWO-PHASE FLOW IN A POROUS MEDIUM may be studied by first considering flow behaviour in individual pore spaces comprising the medium under different flow regimes. Early work in this area was based on pore- structure models made up of circular capillary tubes. The pore structure of most porous media generally comprises very irregular and random-shaped voids, interconnected in some form of a network. The use of non-circular pore channels in the model permits the annular or pendular flow regime to exist over a much larger range of wetting-phase saturations than is possible in circular capillaries..Unfortunately, due to mathematical difficulties, not enough attention has been given to non-circular capillaries and problems associated with two-phase flow through them. Even the corresponding problems in the case of circular capillaries have not been comprehensively studied and literature is scattered in various disciplines. This paper explores the problem of two-phase flow in noncircular capillaries at low Reynolds numbers. Reference and comparisons are made to the behaviour of circular capillaries whenever possible.
In porous media such as sandstones, the pore spaces are generally the interstitial spaces between three or more sand grains, with some cementing materials at their points of contact. The shape of the space may be simulated by a triangle. Pore spaces will be modelled in the present study by equilateral triangular channels for mathematical simplicity. However, the discussion is general and applies qualitatively to straight channels of any arbitrary shape.