Miscible displacement theory

The description of these sequential and simultaneous reactions during transport through a soil system is accomplished using miscible displacement theory, which can be formulated for an interacting, degrading solute (e.g. an organic molecule) as (2) ... 

The inherently unstable nature of miscible displacements with unfavorable mobility ratios (the viscosity of the displacing fluid is less than the viscosity of the displaced fluid) and unfavorable density ratios (for a downward vertical displacement, the density of the displacing fluid is greater than the density of the displaced fluid) has been well documented (6-18). For a downward vertical displacement with a favorable mobility ratio and an unfavorable density ratio. Hill ( ) proposed an approximate theory that... 

Perrine, R.L. "A Unified Theory for Stable and Unstable Miscible Displacement," Pet. Trans. AIME 22fi (1963), 205. 

More sophisticated one-dimensional models have included deadend pores, ink-bottle pores, pockets or turner structures, and also, periodically constricted tubes. Two-dimensional and three-dimensional network models have also been developed. The first proposal for a two-dimensional network model was given by Fatt (37, 38). Whereas Fatt primarily dealt with immiscible displacement, Simon and Kelsey (39) used a two-dimensional network model for the simulation of miscible displacement. The first three-dimensional network model is due to Irmay (40). Subsequently, the three-dimensional network model or percolation theory for porous medium has received much attention... 

The allure of miscible displacement for complete oil recovery is obvious it is the only method which permits 100% recovery, even in theory. As long as two phases exist in small capillaries, some trapping of the oil phase will normally occur even under the best conditions. However, if the two fluids are truly miscible, and no other fluid phase is present (seldom true in oil reservoirs), there can be no capillary forces to trap off segments of oil. If the "miscible slug" is large enough, all of the oil can be displaced to achieve a theoretical recovery of 100%. However, to date no practical method has been devised which can provide complete miscibility in real reservoirs (where both oil and water are present) so 100% recovery remains a distant goal. 

In 1960, Gatlin and Slobod (130), by utilizing concepts developed by Wilson et al. (131), proposed their "piston-like" displacement theory which explained that the oil and water flowing simultaneously in front of the miscible alcohol piston would adjust their saturations to form stabilized banks whose saturations could be calculated from the relative permeability and viscosity of the fluids. They carried out a number of experiments which initially appeared to agree quite well with this straightforward theory. 

Simple Theory. The concentration C of a miscible agent of unit injected concentration displacing a fluid of zero concentration in a one-dimensional, semi-infinite medium is given by the familiar error function solution ... 

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