Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a 'macroscopic capillary number' <(Ca)over bar> which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number <(Ca)over bar> is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. <(Ca)over bar> can be related to the microscopic capillary number Ca and the Leverett J-function. Previous dimensional analyses contain a tacit assumption which amounts to setting <(Ca)over bar> = 1. This fact has impeded quantitative upscaling in the past. Our definition for <(Ca)over bar>, however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at <(Ca)over bar> approximate to 1. The length scale related difference between the macroscopic capillary number <(Ca)over bar> for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.