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PRESSURE, 1,000 psig b. Variation of plastic viscosity of asphalt in diesel oil with temperature and pressure.

Figure 5-43. Variation of plastic viscosity. (From McMordie.49 Courtesy of Oil and Gas J.)

SHEAR RATE (SEC-1)

Figure 5-44. Comparison of actual with calculated flow properties of a 16 lb/gal asphaltic oil base mud. (From McMordie, etal.50 Copyright 1975 by SPE-AIME.)

SHEAR RATE (SEC-1)

Figure 5-44. Comparison of actual with calculated flow properties of a 16 lb/gal asphaltic oil base mud. (From McMordie, etal.50 Copyright 1975 by SPE-AIME.)

McMordie et al50 showed that the behavior of oil-base muds at temperature and pressure may be described by a modification of the power law, viz;

where A is the pressure constant, and B the temperature constant, both of which must be determined separately for each mud. Figure 5-44 shows the excellent correlation between the experimentally determined relationship of shear stress to shear rate with that calculated from Equation 5-57 by means of a computer program. Table 5-2 lists the effective viscosities at two shear rates, calculated according to equation 5-57, for three different oil mud compositions at increasing temperatures and pressures. Note that a mud with the lowest viscosity at the surface may not have the lowest viscosity downhole.

Application of Flow Equations to Conditions in the Drilling Well

The rigorous flow equations which were given in the first part of this chapter were based on two assumptions: first, that the temperature of the fluid remained constant throughout the system and second, that the rheological

*SPE/AIME From McMordie et al.50 Copyright 1975 by SPE-AIME.

properties were not thixotropic. Both these assumptions are violated in the drilling well, but both may be satisfied if the rheological parameters are determined under the flow conditions prevailing at the point of interest in the well. The difficulty lies in ascertaining the flow conditions. The temperature of the mud is constantly changing, as shown in Figure 5-34, and its precise value at a particular point in the circuit and at a particular time in the drilling cycle depends on a number of variables. Also the rate of shear undergoes drastic changes at several points in the circuit, and there is a considerable time lag before the state of shear reaches even approximate equilibrium; indeed, it may never do so (see "The Effect of Thixotropy on Drilling Mud" earlier in this chapter). Along with these uncertainties, there are several unknown factors, such as the width of the annulus in enlarged sections of the hole, and the effect of rotation of the drill pipe.

Because of these limitations, the flow pressures or velocities in a drilling well can never be predicted with the precision that is possible in the piping systems in an industrial plant. The question therefore arises: Are these rigorous equations and the computer programs that are, in some cases, necessary to solve them justifiable in terms of time and expense, given the uncertainty of the input data? Might not some simpler, less exact equations, give equally good results? In this section, we shall endeavor to show that the answer to these questions depends on the section of the flow circuit under consideration, the purpose of the investigation, and where the investigation is being carried out—in the laboratory or at the drilling well.

### Flow Conditions in the Well

Flow in the drill pipe is usually turbulent, and is therefore only influenced by the viscous properties of the mud to a minor extent. The effective shear rate at the pipe wall, as determined from capillary viscometer data and Equation 5 40, is generally between 200 to 1,000 reciprocal seconds. The conduit dimensions are known accurately, so that the pressure loss can be calculated quite accurately. The only uncertain factor is the roughness of the pipe walls. The pressure loss in the drill pipe is about 20% to 45% of the pressure loss over the whole circuit, i.e., of the standpipe pressure.

Flow velocity through the bit nozzles is extremely high, corresponding to shear rates of the order of 100,000 reciprocal seconds. The pressure loss across the nozzles can be calculated accurately because it depends on the coefficient of discharge, which is essentially independent of the viscous properties of the mud. The pressure loss is about 50% to 75% of the standpipe pressure.

Flow in the annulus is normally laminar, and is therefore a function of the viscous properties of the mud. Shear rates generally lie between 50 and 150 reciprocal seconds. Although the pressure loss from the bit to the surface comprises only 2-5% of the standpipe pressure, a knowledge of the pressure and flow velocity in the various sections of the annulus is very important when dealing with such problems as hole cleaning, induced fracturing, and hole erosion. Unfortunately, accurate prediction of flow relationships is usually difficult and often impossible—owing to various unknown factors and uncertainties. Perhaps the greatest of these unknowns is the diameter of the hole, which may be as much as twice the nominal diameter in enlarged sections of the hole, thereby decreasing the rising velocity by a factor of at least five (see Fig. 5 45).

The influence of the rotation of the drill pipe on velocity profiles is also difliculi to account for. There are equations for helical flow,51 but these were derived for drill pipe rotating concentrically in a vertical hole, whereas, in practice, the drill pipe whips around in a randomly deviated hole. Also, equations for flow in eccentric annuli show that annular velocity is greatly reduced when the drill pipe lies against the low side of the hole, as in directionally drilled wells, and that equations based on concentric annuli would be seriously in er~

HOLE DIAMETER (INCHES 10 12 14 16

HOLE DIAMETER (INCHES 10 12 14 16

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