## N

Rc.p

where dr is the diameter of the sphere, v, the terminal settling velocity, pf the density of the fluid, and fx its viscosity. Under laminar flow conditions the terminal How velocity is given by Stokes' law:

2gd2p 36

Pp-li

where pp is the density of the particle. Under turbulent flow conditions the terminal settling velocity is given by Rittinger's formula:28

Predicting the terminal velocity of drill cuttings is much more difficult. For one thing, there is the wide range of particles sizes and the particles have irregular shapes; for another, there is the non-Newtonian nature of most drilling fluids.

Terminal velocities in turbulent fall are somewhat easier to predict because the rate of fall is not affected by the rheological properties. Walker and Mayes*" proposed the following equation for flat particles falling face down, (which is the normal orientation for turbulent fall):

Terminal velocities predicted by this equation correlated well with experimental data obtained with artificial cuttings of uniform size and shape.

However, the simplest procedure is to determine the terminal settling velocity of the drill cuttings of interest by direct experiment. When determining settling velocities in mud, a layer of a transparent liquid of greater density than the mud should be placed at the bottom of the settling column so that the particles may be seen when they reach the bottom. Figure 5-49 shows some settling velocities of shale drill cuttings falling through water.63

In a drilling well, cuttings fall under still settling conditions whenever circulation is stopped. In a Newtonian fluid the settling velocity is finite, no matter how viscous the fluid, but, because of the enormous length of the settling column, only a small proportion of the cuttings reach the bottom unless the viscosity approaches that of water. In a non-Newtonian fluid the settling velocity depends on the difference between the stress (t) created by the difference in gravity (pp-p,) and the gel strength of the mud (5). When x < 5, then v, is zero, and the cutting is suspended. The initial gel strength of most muds is too low to suspend large cuttings, and suspension depends on the increase of gel strength with time.

In a rising column of fluid, a particle will move upward if the velocity of the fluid is greater than the settling velocity of the particle. However, the particle slips in the rising column, so that the upward velocity of the cutting is less than the annular velocity. Sifferman et al04 defined hole-cleaning efficiency in terms of a transport ratio, derived as follows:

where v£. is the net rising velocity of the cutting, va is the annular velocity, and vs is the slip velocity of the cutting. Dividing both sides of the equation by v0 gives

24,4 80

0 0