## 2 L

where P0 is the pressure required to initiate plastic flow. At pressures greaier than P0, laminar flow progresses towards the axis of the pipe, so that flow consists of a plug in the center of the pipe surrounded by a zone of laminar flow, and the velocity profile is as shown in Figure 5-5b. No matter how great the pressure, this plug can never be entirely eliminated because as r becomes very small, P must become very large, (since rPj2L must equal r0), and infinite when r = 0. Thus the consistency curve for flow of a Bingham plastic in a round pipe is, strictly speaking, always nonlinear no matter how great the shear rate. However, an approximate relationship between pressure and flow rate may be derived from the asymptote to the curve, which, as shown in Figure 5 4b, intercepts the pressure axis at 4/3 P0. Buckingham-' derived this relationship from the intercept and Poiseulle's equation as:

Substituting values for P0 from Equation 5-10, Equation 5-11 may be arranged more conveniently as:

In the last term, -( 4----------- , represents the contribution of the area between

the full curve and the asymptote. At high flow rates it may be omitted, but for exact solutions the abbreviated form should not be used unless exceeds the yield stress, r0, by a factor of at least four.4

At very low flow rates an expression lor plug flow should be included in Equation 5 12. According to Buckingham, plug flow is described by:

nDkP 2 fiL

where k is a constant, and Â¡.i is the viscosity of the lubricating layer at the wall. The effect of plug flow on overall flow rate is ordinarily insignificant in drilling applications.

The hydraulic radius is defined as (Z),~D;)/4. It is usually substituted for DI4 in Equation 5-12 when determining flow relations in the annulus. However, this procedure does not give an exact solution.5 If such is required, it may be obtained from a computer program written by Melrose, Savins and Parish.6

### The Concentric Cylinder Rotary Viscometer.

The plastic viscosity and yield point of a Bingham plastic are best determined in a concentric cylinder rotary viscometer. The outstanding advantage of this instrument is that, above a certain rotor speed, plug flov. is eliminated, and the consistency curve becomes linear.

Tht: essential elements of the viscometer are shown in Figure 5-6. An outer cup rotates concentrically around an inner cylinder or bob, which is suspended from a torsion wire. The annulus between the bob and the cup is narrow, about 1 millimeter. A dial affixed to the wire and a fixed pointer enable the angle through which the wire has turned to be measured.

Figure 5-6. Torsion viscometerâ€”schematic. The cup rotates.

The relationship between torque and rotor speed for any speed great enough to maintain all the fluid in the annulus in laminar flow is linear, and i>

defined by the Reiner-Riwlin7 equation:

where o is the angular velocity (in radians per second), and 7 the corresponding torque. The yield point, t0, in Equation 5 16 is defined by the intercept, 7\, of the extrapolation of the linear portion of the curve on the torque axis. T0 is given by Equation 5 16 when cb = 0, because then:

The slope of the line above the critical angular velocity. <u/. defines the plastic viscosity, /v

The value of the torque in the wire may be obtained from the deflection of the dial and the wire constant, C:

where 7 is the torque in dyne centimeters and 0 is the dial deflection in degrees. Usually the wire constant is supplied by the manufacturer of the viscometer; if not, it may be obtained by calibration with Newtonian fluids. Since such fluids have no yield point, C may be determined from Equations 5 16 and 5 18:

A number of commercially-available concentric cylinder rotary viscometers that are suitable for use with drilling muds are described in Chapter 3. The\ are similar in principle to the viscometer shown in Figure 5-6, but use a spring instead of a torsion wire. All are based on a design by Savins and Roper,8 which enables the plastic viscosity and yield point to be calculated very simply from two dial readings, at 600 rpm and 300 rpm, respectively. They will be referred to in this chapter as the direct-indicating viscometer.

According to Savins and Roper, the underlying theory is as follows: The Reiner-Riwlin Equation 5 16 is simplified to

0 0