TECHNOLOGY Equations determine flow states for yield-pseudoplastic drilling fluids

Roberto Maglione, Gualtiero Ferrario
Agip SpA
Milan, Italy

A new calculation method can determine the state of flow and pressure losses, for all ranges of flow, for a yield-pseudoplastic drilling or cementing fluid.

This method can evaluate the flow regimes without considering the fluid density as a determinant parameter.

This method uses parameters related to fluid rheology, average flow velocity, and the geometry of the hydraulic circuit for conventional drilling and cementing conditions. Data from circulation tests in a well in Southern Italy have correlated well with calculated data using these equations.

The hydraulic circuit plays an important role for the correct development of drilling and cementing operations. It is also commonly recognized that a proper design of the hydraulic circuit can considerably contribute to reducing operating costs.

To accurately predict the behavior of the drilling fluids, the following should be considered:

• Controlling the bottom hole pressure

• Correctly evaluating the pressure gradient due to the circulation of the fluid in the well

• Removing cuttings from the annulus out to the surface

• Minimizing the erosive effect of the fluids on the borehole wall

• Increasing the rate of penetration

• Controlling the pressure while running the drillstring in the hole (surge) and pulling it out of the hole (swab)

• Improving the mud displacement, by means of an appropriate flow state design, and avoiding the channeling of the cement slurry during cementing

• Adequately designing mud and cement pumps.

To calculate the pressure distribution along the borehole wall, one must solve the equations of mass and momentum conservation simultaneously. The basic usual assumptions are unidimensional flow, steady state flow, incompressible fluid, and isothermal flow.

The flow in the annular section is commonly assumed to be the same as fluid flow between two parallel plates. The introduction of this approximation produces a very low error for the geometries normally adopted in drilling wells.

Finally, of course, the choice of a proper rheological model gives a determinant contribution for a good modeling of the drilling fluids.

Drilling and cementing fluids behave as non-Newtonian fluids, so a Newtonian rheological model is not suitable for this application.

In the past, the use of two constant-parameter constitutive laws (Bingham rheological model and Ostwald & de Waele rheological model) was a very common practice. More recently, the Herschel & Bulkley model has been suggested, and now it is the most used rheological model for drilling and cementing fluids.

The analytical expression of the Herschel & Bulkley model, containing three constant parameters, is given in Equation 1. This model includes, as particular cases, the Newtonian rheological model (to = 0, n = 1), the Bingham rheological model (n = 1), and the Ostwald & de Waele rheological model (to= 0). (The n is the flow behavior index, and to is the yield point.)

A good agreement between viscometer data and Herschel & Bulkley rheological model results has been observed both for water-based and oil-based mud and cement slurries of different compositions.

Yield-pseudoplastic fluid

The following is the analytical solution for the flow of a yield-pseudoplastic fluid in a circular section. Considering the flow of a fluid in a generic circular section having radius r, internal to a conduit of radius R and length L, and applying the law of motion for laminar flow, one obtains Equation 2.1 2

Making explicit the term for the shear stress at the generic radius (tr), using the expression of the Herschel & Bulkley rheological model, one then obtains Equation 3. Making explicit the expression for the shear rate yields Equation 4. By integrating the expression of the shear rate, between the radius R and Rmax, where the shear rate is not zero, one obtains an expression for the local flow velocity (Equation 5).

The expression for the maximum flow velocity, located in the plug flow zone, is given by Equation 6. Equation 7 gives the expression of the total flow rate.

Considering total flow rate as the sum of the flow rate in the laminar flow zone Q1, between the radius R and Rmax, and the flow rate in the plug flow zone Q2, between the radius 0 and Rmax, one then obtains Equations 8 and 9. By making the terms explicit, the average flow velocity can be calculated (Equation 10).

Solving the integral yields Equation 11, an analytical expression for the average flow velocity, valid for a circular section. The expression of the wall shear rate is given by Equation 12. In this equation, fc is given by the following:

fc 5 f (n,k,to, Vm,R)

fc has been called the friction coefficient factor and depends on the three rheological parameters of the fluid, the average flow velocity, and the radius of the conduit and is not dependent on the fluid density.

Flow regimes for
circular section

The behavior of the friction coefficient factor, for the full range of flow (from plug to turbulent flow condition), has been deduced from experimental data, both for flow in circular and annular sections. During the investigation, it was found that the shape and the slope of the representing curve for fc, for each state of the flow, changed vs. the state of flow. Thus, the regime of the flow could be represented quite well only considering the behavior of the coefficient fc.

For easier use, this coefficient was reduced from being dependent on five variable parameters to three (two of which are dimensionless). This reduction in dependent variables makes it easier to locate the state of the flow. Based on this philosophy then, the determination of the state of the flow can be made using three groups of numbers: Gc, (k/to), and n. (Gc is dimensionless and is defined by Equation 13).

Consequently, the function fc can be written in the following form:

fc = f(Gc, k/to, n)

On this hypothesis, the state of the flow of a yield-pseudoplastic fluid can be represented in three dimensions with the three following coordinates: Gc, (k/to), and n.

The case for (k/to) = constant and n = constant (exactly for k/to = 0.13 secn and n = 0.7), both for circular and annular sections, had been deeply examined in a previous article (OGJ, Sept. 4, 1995, pp. 94-101). The expressions needed to calculate the pressure drop for a yield-pseudoplastic fluid flowing both in circular and annular section were also reported.3

This article covers the case, valid for a circular section, for n fi constant (in the range between 0.5 and 1.0) and (k/to) = constant (k/to = 0.13 sn).

Fig. 1 [49761 bytes]shows the behavior of the friction coefficient factor vs. Gc and the flow behavior index of the fluid, n. Table 1 [14899 bytes] shows the values of the critical Gc points for different flow regimes.

Circulation test

To validate the proposed method, measured data of the standpipe pressure were taken from a series of circulation tests performed in a 66-mm surface section of a microannulus well drilled in Southern Italy. The tests were carried out considering the following operational conditions:

• In all the tests, the circulation was carried out with the bit off bottom and without drillstring rotation.

• The pump pressure (standpipe pressure) was read to the manometer located on the rig floor.

• The rheological measurements were performed on a mud sample taken from the aspiration pit during the test.

Circulation tests were conducted at h = 608 m and 743 m. Table 2 [9477 bytes] shows the Fann VG 35 coaxial cylinder viscometer readings and the corresponding three rheological parameters of the Herschel & Bulkley model, calculated using a nonlinear regression method, and the density for the mud used in the test.

Figs. 3 [22862 bytes]and 4 [48997 bytes]show the well schematic and geometric data for the drillstring, respectively, during the test. Table 3 [13084 bytes] compares the field and calculated data.

Results

A new method to determine state of flow and pressure losses in all the ranges of flow for a yield-pseudoplastic drilling and cementing fluid has been determined.

The theory behind this method is that the flow regimes can be evaluated without considering the fluid density as a determinant parameter. Alternatively to the standard procedure, the use of parameters related to fluid rheology, average flow velocity, and geometry of the hydraulic circuit appears possible for conventional drilling and cementing conditions.

Circulation tests in wells having different geometries (microannulus and standard annulus wells) have been considered to check the applicability of the proposed theory to various scenarios.3 In all the tests conducted, the agreement between calculated and measured data has been good.

Acknowledgment

The authors wish to thank the Microdrill rig crew who helped in collecting all the field data needed for this work and Agip SpA for permission to publish the results.

References

1. Ghiringhelli, L., "Reologia dei Fluidi," Agip SpA, San Donato Milanese, Italy, 1981.

2. Zaho, S., "Idraulica dei Fluidi di Perforazione," Agip SpA, San Donato, Milanese, Italy, 1986.

3. Maglione, R., "New method determines flow regimes and pressure losses during drilling and cementing," OGJ, Sept. 4, 1995, pp. 94-101.

The Authors