Dimensional analysis of pore scale and field scale immiscible displacement

Abstract

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’$\overline {Ca}$ 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$\overline {Ca}$ is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure.$\overline {Ca}$ can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting$\overline {Ca}$ = 1. This fact has impeded quantitative upscaling in the past. Our definition for$\overline {Ca}$, 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$\overline {Ca}$ ≈ 1. The length scale related difference between the macroscopic capillary number$\overline {Ca}$ 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.