Porous media apparent viscosity

Hirasaki, G. J. and Lawson, J. B, "Mechanisms of Foam Flow Through Porous Media — Apparent Viscosity in Smooth Capillaries", Soc. Petr. Eng. J. April 1985 pp 176-190. 

Hirasaki G., Lawson J. B., Mechanism of foam flow in porous media apparent Viscosity in smooth Capillaries, Soc. Petr. Engng., 1986, Vol. 25, p. 176-190. 

Apparent viscosity the viscosity of a fluid, or several fluids flowing simultaneously, measured in a porous medium (rock), and subject to both viscosity and permeability effects also called effective viscosity. 

Certain dilute lamellar liquid crystalline phases having relatively low apparent viscosities can propagate through this porous medium micromodel without plugging it. Their behavior followed the trends established with isotropic phases. 

If the foam phase is thought of as a pseudo continuous fluid with an apparent viscosity Vapp = it follows that Papp is greater than that of the aqueous liquid phase. (For the tests here, values of Uapp were on the order of 1 to 50 times that of water). Because of this, when foam and liquid move through a porous medium under an applied pressure drop, the foam, being the most viscous phase, must occupy a larger region of the pore space. Consequently, as observed, the gas saturation is increased over that of non-dispersed phase flow and the liquid permeability is correspondingly decreased. 

Figure 14 shows a very interesting and an important correlation between the rate of coalescence in macroemulsions and the apparent viscosity in the flow through porous media. It was observed that a minimum in apparent viscosity for the flow of macroemulsions in porous media coincides with a minimum in phase separation time at the optimal salinity. This correlation between the phenomena occurring in the porous medium and outside the porous medium allows us to use coalescence measurements as a screening criterion for many oil recovery formulations for their possible behavior in porous media. It is. very likely that a rapidly coalescing macroemulsion may give a lower apparent viscosity for the flow in porous media (53). 

Equation 20 shows that a porous medium is permeative, that is, a shear factor exists to account for the microscopic momentum loss. Our preliminary study recently reveals that, however, a porous medium is not only permeative but dispersive as well. The dispersivity of a porous medium has been traditionally characterized through heat transfer (in a single- or multifluid flow) and mass transfer (in a multifluid flow) studies. For an isothermal single-fluid flow, the dispersivity of a porous medium is characterized by a flow strength and a porous medium property-de-pendent apparent viscosity. For simplicity, we discuss the single-fluid flow behavior in this chapter without considering the dispersivity of the porous medium. 

The emulsion behaviour in porous media is discussed in [235]. O/w emulsions with volume fractions of up to 50% show Newtonian behaviour, whereas those with more than 50% are non-Newtonian liquids, the apparent viscosity of which depends on the shear rate. The viscosities of such emulsions are more than 20 times that of water and sometimes can be even comparable with that of oil. When the emulsion is moving, a temporary permeability reduction of the reservoir may occur due to the capture of small droplets by the surface of the porous medium. In this case, stable o/w emulsions may flow not as a continuous liquid, i.e. the emulsion flow largely depends on the nature of the porous medium. Therefore, it is necessary to know about the structure and physicochemical characteristics of the oil reservoir (porous medium) porosity, the mean pore diameter, the mean pore size and pore size distribution, chemical composition of the minerals ( acidic , basic , neutral ), the nature of the pore surface, first of all wettability, for a successful application of the emulsion flooding method. 

A point of terminology is repeated here to remind the reader. The term apparent viscosity , pp, is used to describe the observed macroscopic rheology of the polymeric fluid in a porous medium. The quantity effective viscosity ,, refers in a rather similar way to the observed effective viscosity in a single capillary. Each quantity is defined phenomenologically—from Darcy s law (Equation 6.4) and rj ff from Poiseuille s law (Equation 3.75). This distinction should be kept clear, especially when considering porous media models based on networks of capillaries, as discussed later in this chapter. The overall viscosity of the non-Newtonian fluid in the network as a whole is whereas the viscosity in each of the capillaries may be different and is In this latter case, will be in some sense an average value of the in the individual capillaries. 

Hirasaki and Pope (1974) took a similar approach in which they described the apparent viscosity in the porous medium, as a power law function of the Darcy velocity, u, as follows ... 

The studies quoted above (Hirasaki and Pope, 1974 Teew and Hesselink, 1980 Greaves and Patel, 1985 Willhite and Uhl, 1986) presented data on biopolymer flow in the shear thinning regime only and, hence, a power law model may be adequate for the purposes of analysis. Also, Willhite and Uhl (1986) contend that, even down to very low flow rates in the porous medium, they see no evidence of a lower Newtonian plateau in the apparent viscosity analogous to that commonly observed in the bulk flow behaviour (see Figures 3.6 and 3.8). 

Cannella et al (1988) use the following expressions for the apparent porous medium shear rate, 7p, and apparent viscosity,... 

Approach (iii) listed above refers to the use of effective medium theory (Kirkpatrick, 1973 Koplik, 1982 Levine and Cuthiell, 1986) for calculating certain average flow properties in idealised porous media models—usually simple networks. Cannella et al (1988) have recently applied this approach to the flow of power law fluids through networks of capillaries. They use this method to derive an expression for the apparent viscosity of the polymer in the porous medium which has the same overall form as the capillary bundle expression (e.g. Equation 6.18). They then adjusted the parameters in the effective medium formula in order to match their particular form of the capillary bundle formula with C = 6 (Equation 6.18). The values of the effective medium parameters are physically interpretable, and Cannella et al (1988) deduced from these that the effective radius for the flow of a power law fluid is larger than that for the flow of a Newtonian fluid. They also... 

Figure 6.15. Apparent viscosity versus corrected porous medium shear rate compared with the bulk fluid rj/y curve for the four networks with bond radius distributions shown inset (Sorbie et ai, 1989c).

The quantity, would then be used to calculate the apparent viscosity of the polymer solution within the porous medium, /(C,y ), which will have been established experiinentally or by using a simple empirical model. 

Fig. 5.80 shows results illustrating behavior in porous medium for a polyacrylamide/Cr(III)/redox system. The data were taken by flowing the gel system at a steady rate through an unconsolidated sandpack. The gel system was mixed at the inlet of the sandpack with an in-line mixer. Pressure drop was measured along the sandpack and converted to apparent viscosity by use of Darcy s law. Apparent viscosity is plotted vs. distance at different times for the gel system up to about 240 hours. 

Another approach to describing foam rheology has been to calculate an apparent foam viscosity from flow-rate and pressure-drop measurements made during flow through a porous medium, Darcy s law is used with the rock permeability, or water relative permeability if an oil phase is present, to calculate apparent foam viscosity. Results show that the apparent viscosity calculated in this manner is a strong function of foam quality, decreasing approximately linearly as foam quality increases. Ifiis approach essentially treats the foam as a single phase. 

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