# How do I calculate the Reynolds number in multiphase flows?

I am modeling a gas flowing through a liquid. How do I calculate the Reynolds number in multiphase flows? And, at what Reynolds number should I consider the flow to be turbulent?

The problem is of a typical drilled wellbore in the oil and gas industry. Fresh cement is poured into an annulus between the steel casing and the drilled formation. Before the cement is set it is vulnerable to formation gas over coming its pore pressure (or hydrostatic head) and flowing through the cement, often times forming channels for more gas to flow through in the future, rendering the cement sheath useless.

EDIT:

I am currently using a Bingham fluid to describe the viscosity of the cement slurry. The yield stress will be in the range of 300 to 700 kPa. The plastic viscosity will be in the order of 0.05-0.10 Pa.s.

It is hard for me to estimate gas velocity but in preliminary models I have seen velocities in the range of 5-10 mm/s in preliminary models.

• I have a friend working on Bingham fluids, he's on leave atm. I'll try to ask him. Hang on tight for a couple of days :) though as I said in my answer, it is necessary to quantify the regime of flow! do you have any idea which one is it? that would simplify the problem to some extent. Commented Jun 28, 2014 at 12:43
• @darthcoder I'm not sure how to quantify the regime of the flow. I imagine it will differ based on the slurry design, formation gas pressure, overburden pressure, etc. Do you have a resource that I can use to help myself quantify the regime of the flow? Commented Jun 30, 2014 at 2:02
• ideal way would be observation with your eyes :) check out the diagram in my answer. are the bubbles small or are they large enough to separate the liquid into slugs, are they turbulent and ill-defined, or is there a single core of gas in the flow? Corresponding to bubbly, slug, churn and annular flow regimes respectively. Commented Jun 30, 2014 at 8:39
• Have a look in Pipe Flow 2, see drbratland.com, where there is a free book you may download.
– user56352
Commented Jul 30, 2014 at 14:08

## 2 Answers

There are three ways you can proceed in:

## 1. Homogeneous Flow Model

Herein, you would assume single averaged flow quantities and then solve the Navier-Stokes equations as if it were arising from the flow of an averaged liquid. What I mean is that if you had water and steam flowing together, you would take the average density, viscosity and so on.

Obviously this model isn't very accurate.

## 2. Heterogenous or Separated Flow Model

Here, you would consider a flow of liquid and gas superimposed onto each other. Assume the cross-section area to be divided into two sections, one in which liquid flow and one in which the gas flows, the proportion being the same as the void fraction.

Then you would write the complete Navier Stokes for both the phases separately! And include terms for forces that one phase exerts on the other(and on the walls). These terms usually come from correlations and you can find them in texts on Multiphase Flow or papers.

This model is difficult to solve and accuracy is limited to how good your correlations are.

## 3. Flow Regime Models

Two phase flow is characterised by regimes!
(source: drbratland.com)

It would be best that you use a flow regime map/or observe which flow regime your flow is in and then move accordingly. This is the way I will recommend doing Two Phase flow problems.

You'll have to look for a map that is accurate for your problem and then characterize your flow accordingly.

Collier's book is a good place to start learning about Two phase flow :) I don't have much experience in flow through porous media, so I'll not hazard guesses.

Firstly, some general answer on nondimensional groups such as the Reynolds number e.g.: they do not generally characterise the flow as a whole, but a feature that you choose in the flow. If the flow is not an academic problem, you will have several such features, which have different lengths, velocities...

In the case of multiphase flow, this is obvious since you already have to choose between two different densities and viscosities, and in your case of liquid+gas they are orders of magnitude apart.

Your aim in calculating the Reynolds number is to evaluate the importance of inertial effects vs viscous ones: thus, for inertial effects, you should probably take the highest density, the one of the liquid. But then, of course, the characteristic velocity you take should also be characteristic of the liquid phase. All this to say that there is no definite answer in general, tell us about your setup and then the discussion can take place.

Concerning turbulence, the Reynolds number at transition is well characterized only in a handful of setups, you'll only have some gross estimate and then have to describe your flow in order to assess whether it is turbulent or not.

EDIT CONCERNING THE APPLICATION

In the case you mention, the Reynolds number in cement is probably quite low, as I guess the viscosity of the cement is high (please provide data). The gas flow might reach higher Reynolds, as here you do not have a suspension but rather a system of cracks, I would use the maximum crack width, gas speed and kinematic viscosity. This will probably be much higher than the Reynolds in cement.

For the numbers you give, the Reynolds number will clearly be low both in the cement phase, as the velocities there will be less than gas and viscosity is high, and probably low in the gas phase, but you don't give an order of magnitude for the width of your cracks. Anyway, for centimeter-sized cracks Reynolds would be of order 10 (although if your gas has very low viscosity that could be much more!) So you'll be able to focus on the complicated problem of Bingham flow plus free surface without having to worry about turbulence.