General approach to in-situ rheology

The various early approaches to the mathematical modelling of non-Newtonian rheology in porous media are reviewed by Savins (1969). One of these is via a capillary bundle view of the porous medium combined with a simple (usually power law) fluid model. From the discussion in Section 6.2, one might not expect this approach to be very fruitful. However, it has been used by a number of workers—in fact, virtually all studies of xanthan flow in porous media present a version (see below)—and results have been sufficiently simple and promising to deserve some further attention. The main objective in these studies is to relate the in-situ rheology of the polymer to 

When using the capillary bundle models, the following two-step approach is taken in formulating a macroscopic description of the flow of non-Newtonian fluids in porous media  

A functional relationship is assumed between shear rate and viscosity. By far the most common choice is the power law relation in Equation 3.45 (e.g. Hirasaki and Pope, 1974 Teew and Hesselink, 1980 Willhite and Uhl, 1986,1988), but some work has been presented on the Carreau model (Vogel and Pusch, 1981). 

the above prescription represents the most common general approach to the problem of relating the bulk viscometric behaviour to that observed in porous media. It is applied by taking Equation 3.73 for the wall shear rate of a power law fluid and then replacing y and R by Equations 6.3 and 6.9 respectively. This leads to an expression for the apparent shear rate in the porous medium, y, of the form  

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