Laminar Displacement of Non-Newtonian Fluids in Parallel Plate and Narrow Gap Annular Geometries
Authors: Flumerfelt Raymond W. 
Published: December 2007

Abstract
Based on certain maximum gradient approximations similar to those used in lubrication theory, the miscible displacement of one non-Newtonian fluid by another is analyzed. The fluids are assumed to be of the power-law type and the displacement occurs in a parallel-plate system under conditions where dispesion effects associated with molecular diffusion and convection are negligible. The results provide predictions of displacement efficiency in provide predictions of displacement efficiency in terms of certain characteristic groups, including the density ratio of the phases, an effective viscosity ratio, a flow rate group, and the power-law slope parameters. The results should represent first-order parameters. The results should represent first-order approximations to slow flow displacements occurring in many well completion operations.

Introduction
The displacement of one fluid with another has been a subject of keen interest to the oil industry for many years. Although numerous studies have been undertaken on the mechanics of miscible and immiscible displacement and the associated flow instabilities and dispersion phenomena, there is still a wide range of practical problems that cannot be treated within these previous theoretical and experimental studies. This is particularly true with regard to problems involving rheologically complex, or non-Newtonian fluids. Such fluids are encountered in many facets of petroleum production, and the displacement of (or by) these fluids is crucial to current drilling and well completion practices. Such displacement problems are continually encountered in cementing operations, where one Non-Newtonian fluid is displaced with another. Although various laminar and turbulent flow techniques have evolved over the years, the basic fundamentals underlying these displacement practices go well beyond the current status of displacement theory and experimentation. Under normal circumstances these accepted practices generally lead to satisfactory cementing operations, even though it is doubtful that optimal or near-optimal displacements are ever achieved. However, in more abnormal and demanding operations these practices are often inadequate. In order to achieve optimum results, the basic flow phenomena involved in non-Newtonian displacement must be carefully understood and methods must be developed to predict displacement characteristics in terms of the rheological, kinematic, and geometric conditions involved. Unfortunately, the technology related to non-Newtonian displacement is quite limited. Most of the work to date has been concerned with the specific aspects of displacing a given drilling fluid with another (or with a given cement mixture) for a limited range of flow conditions. Only recently have the effects of rheology, density differences, flow rates, and eccentricity been considered in any quantitative detail. Specifically, Graham has presented a calculation technique for predicting the total volume of cement required to predicting the total volume of cement required to displace completely a drilling mud from an eccentric annulus under conditions of plug-flow displacement. Eccentricity is accounted for by dividing the annulus into segments and treating each segment individually. Unfortunately, displacements under plug-flow conditions are seldom realized in practice and, hence, such calculations represent only crude approximations. In a more recent study, Clark and Carters have presented experimental data on mud displacement efficiency under conditions simulating wellbore environment at 8,000 ft. These investigators indicate that higher displacement efficiencies are obtained under turbulent flow conditions. However, these results were obtained using many displacement-zone volumes of the displacing phase. In practice, because of economic considerations, only a few displacement-zone volumes of the displacing phase can be used. It is still not clear whether turbulent displacement is to be preferred in these latter situations.
SPEJ
P. 169