Facilities Sand Management: Particulate Solids Transport – Saltation Limit (B-FSM028)
The lower transport boundary for solids in fluid flow is set by the Saltation Limit. This limit is the minimum velocity to carry particles in a liquid without settling.
The minimum transport velocity (VMT) is calculated for both upward vertical (VMT-V) and horizontal (VMT-H) liquid flow rates.
Perry’s Chemical Engineers Handbook (8th Ed. Chapter 6) provides good info for these calculations.
For Vertical Upflow in Pipes (VMT-V)
For transporting solids in upward pipe flow (liquid) the recommendation is to use two (2) times the Stokes Settling Limit (v∞). For these calculations I also use the largest particle size (dp-max) from the distribution, not the mean size. This ensures all particles are carried by the liquid.
Note: Refer to B-FSM-024 posted 23-Jan-18 for details on Stokes settling calculations.
The formulas for laminar-turbulent and turbulent flow (most every calculation will be fully turbulent) are shown in the following graphic. Using the turbulent formula the velocity to lift a 3 mm (3000 micron) sand particle (ρs=2650 kg/m³) in produced water (ρl=1050 kg/m³) is 0.74 m/s (2.4 f/s). Calculation steps are;
- v∞ = 1.74[(2650-1050)*9.81*0.003/1050]^0.5 = 0.37 m/s
- Re = 1050*0.37*0.003/0.001 = 1166 (i.e. turbulent regime)
- VMT-V = 2 * v∞ = 0.74 m/s = 2.4 ft/s
This method has no correction for solids concentration (i.e. hindered settling).
For Horizontal Flow in Pipes (VMT-H)
For transporting solids in horizontal liquid flow in pipes I mostly use the Durand equation modified with the Wasp correction for particle size. The equations are given in the following graphic.
The base Durand equation is given in Perry’s, and uses gravity (g), pipe diameter (D), and solid:liquid density ratio (s). This equation also has an empirical constant (FL) from Figure 6-33 in Perry’s. I’ve made a curve fit equation to estimate FL, but the chart should be consulted when available.
The Wasp modification includes correction for particle size (which I think is very important!). This calculation does not have a modification for liquid viscosity.
Using data from vertical upflow example, the horizontal carrying velocity for the 3 mm particle is 1.15 m/s (3.8 ft/s). This is greater than the vertical velocity value, and both values are well below the normal velocity in produced water piping (e.g. typically 2-3 m/s)
Combining Erosive Limit (previous article) and Saltation Limit give the boundary flow rates for our example;
- Minimum flow rate is 1.2 m/s (VMT-H)
- Maximum flow rate is 4.7 m/s (Ve)
While I use the Durand-Wasp equation almost exclusively, there are a few other methods of calculation as shown in the following graphic. The first equation is the combined Durand-Wasp. The next equation is the Oroskar-Turian from the Davies 1987 reference. This attempts to correct Durand-Wasp with a kinematic viscosity term. Lastly is shown the Wilson Nomograph taken from the Wilson et al. reference. While not useable for spreadsheet calculations, the nomograph is handy when in the field without a calculating device.
In the next article we’ll apply these principles to calculating the amount of solids that settle in your production separator (but I think you should remove them with a wellhead desander first!).
- Davies, J.T. 1987. Calculation of Critical Velocities to Maintain Solids Suspension in Horizontal Pipes. Chemical Engineering Science, Vol. 42, No. 7, pp. 1667-1670.
- Green, D.W. and Perry, R.H. 2008. Perry’s Chemical Engineers’ Handbook, 8th Ed. New York: The McGraw-Hill Companies, Inc.
- Rawlins, C.H. 2016. Design of a Cyclonic-Jetting and Slurry-Transport System for Separators, Oil & Gas Facilities, Vol. 3, No. 1, February, pp. 38-46. https://doi.org/10.2118/166118-PA
- Wilson, K.C., Addie, G.R., Sellgren, A., Clift, R. 2006. Slurry Transport Using Centrifugal Pumps, Third Edition, Springer, New York, NY.