The Extended Washburn Equation and Its Application to the Oil/Water Pore Doublet Problem

Abstract

The equations of capillary rise or wetting-phase imbibition into a cylindrical capillary are frequently described by the Washburn equation. From this starting point, the familiar pore doublet model for imbibition into a wide/narrow pair of pores has been developed. This simple analytically soluble model can afford insights into the physics of immiscible water/oil displacements in porous media, which are of importance to waterflooding of oil reservoirs. Here, we reexamine the basis of both the Washburn equation and the formulation of the pore doublet model by incorporating a more rigorous treatment of the fluid mechanics. We have extended equations previously proposed by Szekely et al. to describe water/off displacement in a capillary and it transpires that certain additional inertial terms, which appear in the full formulation of the capillary displacement problem, may be particularly important at the pore size and aspect ratio commonly encountered within porous media. The main result of this additional physics is that the pore filling time and the relative filling order of large and small pores may change since these quantities depend on the pore aspect ratio in the extended formulation. When this fuller formulation is embedded within the extended pore doublet model, we find that the modified equations lead to different findings for certain cases, compared with previous well-known (analytical) solutions. The original pore doublet model is fully characterised by three quantities, the aspect ratio, r2/L, the ratio of capillary radii, r1/r2, and the fluid "supply," i.e., free supply or restricted supply, which is characterized by a velocity, V02. In the extended pore doublet model there is a fourth governing quantity, the pore scale Reynolds number, NRe , which describes the inertial terms in the extended Washburn equation. The resulting coupled differential equations for the extended pore doublet model do not admit an analytical solution and must be solved numerically. Under certain conditions of very restricted supply of wetting fluid (low V02), the extended model agrees very closely with the conventional pore doublet results. However, there are cases which are common at the pore scale (and pore aspect ratio) where significant differences are seen. The relevance of these findings to the modeling of imbibition processes within network models of porous media is discussed.