# STOKES LAW AND SOLIDS SEPARATION

Stoke’s Law, which gives the mathematical relationship of the factors governing the settling velocity of spheres in a liquid stated in its simplest form as:

JB = dD² (Db – Dl) G / 18μ

• dB = diameter of the ball
• DB = density of the ball
• DL = density of the liquid
• G = acceleration of gravity
• μ = viscosity of the liquid

Using this expression, or a modification of it, the settling velocity of any ball in any liquid can be calculated, or predicted. Likewise, we can let any object settle in a liquid, measure its rate of fall, and calculate what size ball would settle at that rate. This gives us an “Equivalent Spherical Diameter.” This is a method commonly used for sizing small irregular particles. such as drill solids, barites, etc. that are below a size easily screened. This method is particularly valid for drilling fluids work, because settling not only can occur in the hole or in a mud pit, it is an essential part of separation process in centrifuge and hydrocyclones.

Two solids particles of known different specific gravities , or densities, will settle at the same rate if their size, or Equivalent Spherical Diameter, are of a certain mathematical relationship. This can be shown quite simply for two particles c and b if we assume their settling velocities are the same.

If the normal range of values for barite density, light (or drill) solids density, and the liquid phase of the mud system are substituted in this equation the result will always indicate that the light solids particle will be approximately. 1 ½ times the diameter of the barite solids particle when the
two settle together . A 10 micron barite particle and a 15 micron light solids particle will not be separated by any settling device , including those utilizing centrifugal force.

Although Stokes Law as presented here is for Newtonian fluids, the 1 ½ to 1 relationship is as valid in non-Newtonian muds as any rule-of-thumb yet devised , and predictions using it agree well with field results.

Figure 1 presents unusual data in that high density solids were removed from a weighted oil mud by flotation , and then particle size distributions were run independently on each type solids. 15 The actual sizes are shown in Figure 1. The sample is from mud that has passed through a 12 x 12 shale shaker screen and the graph shows that a finer screen might have removed some more drill solids and perhaps without cutting deeply into the barites (assuming a fine screen could function properly in the situation).