The 3-D, two-phase polymer and heat transport equations

The full polymer transport equation in 3-D is usually taken to be a generalised convection-dispersion equation of the following form  

All symbols have their usual meaning and only more important ones are defined here. Cj is the concentration of component j in the aqueous phase (e.g. polymer, tracer, etc.). The viscosity of the aqueous phase, rj, may depend on polymer or ionic concentrations, temperature, etc. Dj is the dispersion of component j in the aqueous phase Rj and qj are the source/sink terms for component j through chemical reaction and injection/production respectively. Polymer adsorption, as described by the Vj term in Equation 8.34, may feed back onto the mobility term in Equation 8.37 through permeability reduction as discussed above. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (Bondor etai, 1972 Vela etai, 1974 Naiki, 1979 Scott etal, 1987), in order to find the velocity fields for each of the phases present, i.e. aqueous, oleic and micellar (if there is a surfactant present). This pressure equation will be rather more complex than that described earlier in this chapter (Equation 8.12). However, the overall idea is very similar except that when compressibility is included the pressure equation becomes parabolic rather than elliptic (as it is in Equation 8.12). This is discussed in detail elsewhere (Aziz and Settari, 1979 Peaceman, 1977). Various forms of the pressure equation for polymer and more general chemical flood simulators are presented in a number of references (Zeito, 1968 Bondor etal, 1972 Vela etal, 1974 Todd and Chase, 1979 Scott etal, 1987). 

In addition to the mass conservation and pressure equations, a heat balance equation may also be included in the simulator in order to calculate temperature distributions, T(x, y, z), within the reservoir. For a system of NP phases (labelled as subscript m) the heat equation may be expressed as follows  

Note that the above equation does not consider heats of reaction arising from the chemical reactions described by the terms. This is only necessary in the types of reaction that occur in processes such as in-situ combustion, which may be modelled using specially written simulators (for example, Grabowski etal, 1979). However, it is important to describe conductive heat transport to the medium surrounding the reservoir—mainly the over- and unburden—and this is sometimes done using modified aquifer influx/outflow equations such as the Carter-Tracy model (Scott etai, 1987). It will be seen below that the effect of conduction of this type is very important in determining how much cooling occurs in a hot reservoir when cool water is injected. The temperature distribution within the reservoir depends on the ratio of horizontal heat convection to vertical heat conduction from the under- and overburden. The principal terms of Equation 8.35 may be written in the form  

An important point to note when considering the cooling effect in reservoirs is that the thermal properties of the reservoir rocks do not vary substantially between rocks of a wide range of different types. Somerton (1958) has measured heat capacities and conductivities of a wide range of rock types from sandstones to shales, containing air, oil and water. The ranges of values obtained do not in most cases differ by more than 20% from those used in this paper. A similar range of values of conductivity is shown specifically for North Sea rocks by Evans (1977). It is not possible for a reservoir to contain 

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