Mixing Equations in Various Geometries
- W.E. Brigham (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- March 1986
- Document Type
- Journal Paper
- 203 - 208
- 1986. Society of Petroleum Engineers
- 1.6.9 Coring, Fishing, 1.14 Casing and Cementing, 5.6.5 Tracers, 5.4.9 Miscible Methods
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In flow through porous media, the mixing characteristics of the flowing fluids are often defined by the diffusivity equation. Solutions of this equation are available for linear flow, and close approximations are available for radial flow. In this paper, one of the radial flow approximation methods has been generalized to a variety of geometries. The solution is assumed to be in the form of an error function; and equations are derived for the standard deviation - i.e., the argument of the error function. By breaking the flow system into segments and by repeated use of these solutions, mixing can be calculated in a large variety of flow systems. For example, equations can be used to calculate tracer flow behavior in reservoirs and mixing in vertical miscible displacement.
In the absence of viscous fingering, the equations that define mixing in a linear miscible displacement are well known. For example, in a long system, where the mixed zone is small compared with the length of the system, the concentration can be defined as follows1:
C = concentration of displacing phase, ranges from 0 to 1,
x = distance, ft [m],
= average distance of displacement, by material balance, ft [m],
t = time of displacement, days, and
K' = mixing coefficient, sq ft/D [m2/d].
Radial geometry has also been investigated. For instance, Raimondi et al.2 and Bentsen and Nielsen3 used an interesting approximation to change the radial diffusivity equation into linear coordinates. The result, for a constant injection rate, was
r = radial distance of displacement, ft [m], and
r = average radial distance by material balance, ft [m].
In Eq. 2 the mixing coefficient is not considered constant. It is defined as follows:
K = a constant depending on the diffusion coefficient and the cementation factor, sq ft/D [m 2/d],
a = a constant, depending on the porous medium, ft [m], and
v = velocity of flow, ft/D [m/d].
Lau et al.,4 and later Baldwin,5 also investigated radial geometry. In this approach they recognized that the mixing curve in radial flow can be closely approximated by an error function, and they defined an equation for the argument of the error function to fit the radial geometry. Baldwin showed that this approach was nearly identical to the solution of Raimondi et al. and the data of Bentsen for radial flow.
In this paper, I show how the approach of Lau et al. and Baldwin can be generalized to a number of differing geometries with a more generalized defining equation for the mixing coefficient. The resulting equations can be used in mixing calculations for a number of practical laboratory and field situations that have not been solved previously.
|File Size||332 KB||Number of Pages||6|