Turbulent Dispersion in Porous Materials as Modeled by a Mixing Cell with Stagnant Zone
- C.R. Kyle (California State College) | R.L. Perrine (U. of California)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- March 1971
- Document Type
- Journal Paper
- 57 - 62
- 1971. Society of Petroleum Engineers
- 5.1 Reservoir Characterisation, 5.3.1 Flow in Porous Media, 1.6.9 Coring, Fishing
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This paper reports on a simple theoretical analysis of dispersion in rapid flow through porous materials, giving a comparison of predicted results with experiments. The analytical model considers a pore structure which acts like a sequence of mixing cells, each coupled with a stagnant zone.
Computed results compare very favorably with experimental observations on flow through a staggered matrix of cylinders. This, in turn, has been shown to behave the packed beds of spheres with corresponding properties. Agreement requires that values for certain theoretical parameters be fitted from the data The values required for these parameters are very reasonable. Development of parameters are very reasonable. Development of this approach could be useful for a number of related problems.
The dispersion of two dynamically similar miscible liquids in laminar or turbulent flow through a porous material is a very complex process. However, it can be broken down into four process. However, it can be broken down into four basic mixing mechanisms:
1. Molecular diffusion. Where the flow velocity is appreciable, or pore size is larger, diffusion is usually negligible. Molecular diffusion will not be discussed in this paper.
2. Uneven fluid movement due to irregular pore geometry and inhomogeneities in the media. Both of these factors are difficult to treat, and are usually neglected in theoretical analysis.
3. Uneven fluid movement due to velocity differences within the pores and passages. The zero-velocity boundary condition on each solid surface assures this type of mixing in both laminar and turbulent flow.
4. Mixing by rotational flow, or by turbulent eddies within the pores or passages.
The last two are both convective mixing processes and depend primarily upon the level of processes and depend primarily upon the level of energy dissipation in the media, as well as on the geometry of the system. In general as the velocity increases and the friction losses rise, so does the efficiency of the mixing process.
Dispersion has been reviewed thoroughly by Perkins and Johnston and has been studied Perkins and Johnston and has been studied extensively by others.
DIFFUSION MODEL OF DISPERSION
The most commonly used mathematical model for dispersion in both laminar and turbulent flow is a diffusion-type equation (Refs. 1 or 5). The solution for a step function input with flow in the x-direction only, and with negligible lateral gradients, shows that an initial sharp interface degenerates into a broad mixing zone which grows approximately as the square root of the distance traveled. The solution also predicts a normal distribution for concentration as a function of distance. However, in most real systems "tailing" occurs, causing a skewed distribution. Usually the deviation is not serious and the diffusion equation may be used as a good approximation for the actual process. process. DISPERSION IN A TUBE
Another simple model for laminar dispersion, neglecting molecular diffusion, is to consider a porous material as a bundle of capillary tubes. porous material as a bundle of capillary tubes. Sir Geoffrey Taylor showed that if one fluid in a capillary tube is displaced by another dynamically similar miscible fluid, the average concentration, C, at the tube exit is given by: 2 C = (V /2V) p
(Valid for V 1/2 V ,........................(1) p
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