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Chapter VI Immiscible Displacement

This chapter discusses the transition between microscopic and macroscopic length scales for the flow of two immiscible fluids through a porous medium. Contrary to the previous chapters the macroscopic equations of motion describing the immiscible displacement process are assumed to be known from averaging the microscopic equations. The upscaling problem is addressed by comparing the dimensional analysis of the given microscopic and macroscopic equations. The original dimensional analysis dates back to [49, 329, 330, 331], but continues to attract the attention of recent authors [332, 333, 334, 335, 336]. This fact indicates the presence of unresolved problems, which can be seen most clearly in capillary desaturation experiments where they appear as large unexplained discrepancies in the macroscopic balance of viscous and capillary forces. Recently these problems were traced to a tacit assumption underlying the traditional dimensional analysis [47, 48]. This finding could lead to a resolution of the discrepancies and has additional implications for laboratory measurements of relative permeabilities [337] which will be discussed in section VI.C.3. The dimensional analysis of immiscible displacement will therefore be reviewed in this chapter providing a basis for quantitative estimates of the relative importance of macroscopic viscous, capillary and gravitational forces. Such estimates were distorted in the traditional analysis. The revised analysis allows to predict segregation front widths or gravitational relaxation times for different porous media [47, 48].