THE EXTENDED WASHBURN EQUATION AND ITS APPLICATION TO THE OIL WATER PORE DOUBLET PROBLEM/

The equations of capillary rise or wetting-phase imbibition into a cyl indrical capillary are frequently described by the Washburn equation. From this starting point, the familiar pore doublet model for imbibiti on into a wide/narrow pair of pores has been developed. This simple an alytically soluble model can afford insights into the physics of immis cible water/oil displacements in porous media, which are of importance to waterflooding of oil reservoirs. Here, we reexamine the basis of b oth the Washburn equation and the formulation of the pore doublet mode l by incorporating a more rigorous treatment of the fluid mechanics. W e have extended equations previously proposed by Szekely et al, to des cribe water/oil displacement in a capillary and it transpires that cer tain additional inertial terms, which appear in the full formulation o f the capillary displacement problem, may be particularly important at the pore size and aspect ratio commonly encountered within porous med ia. The main result of this additional physics is that the pore fillin g time and the relative filling order of large and small pores may cha nge since these quantities depend on the pore aspect ratio in tire ext ended 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, r(2)/L, the rati o of capillary radii, r(1)/r(2), and the fluid ''supply,'' i.e., free supply or restricted supply, which is characterized by a velocity, V-2 (0). In the extended pore doublet model there is a fourth governing qu antity, the pore scale Reynolds number, N-Re, which describes the iner tial terms in the extended Washburn equation. The resulting coupled di fferential equations for the extended pore doublet model do not admit an analytical solution and must be solved numerically. Under certain c onditions of very restricted supply of wetting fluid (low V-2(0)), 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 re levance of these findings to the modeling of imbibition processes with in network models of porous media is discussed. (C) 1995 Academic Pres s, Inc.