Roberto MaglioneAgip SpA

Milan, Italy

An innovative calculation method can easily determine the state of fluid flow (plug, laminar, transition, or turbulent) and pressure losses in drilling and cementing operations.

A correct evaluation of the type of flow can improve drilling operations, especially with regard to cuttings transport, penetration rate, and cementing operations. Circulation test data from a deep well were used to validate the theory.

Hydraulics parameters such as pump rate, pressure drop, state of flow, and velocity profile of fluids flowing in the drilling hydraulic circuit are crucial for evaluating the optimum fluid performance during drilling and cementing a well.

Proper hydraulics design can improve the drilling rate, cuttings transport, and mud displacement during cementing, all of which can lower the total well cost. An accurate evaluation of the distribution of the pressure along the depth of a well can optimize the pump rate to obtain the maximum available pump rate without fracturing the crossed formation.

The equations derived in this article describe a new way to consider the hydraulics of drilling fluids with yield pseudoplastic behavior.

A dimensionless number, conceived as the ratio between the magnitude of the dynamic frictional forces (due to the motion) to yield forces, is used to determine both the state of the flow (without taking into account the Reynolds number) and to evaluate the pressure drop gradient of the fluid flowing in a circular and annular section (considered concentric). This number, that takes into account the characteristics of a fluid like the three rheological parameters in the Herschel and Bulkley model, gives an immediate evaluation of the flow regime and easily establishes if the flow is in plug, laminar, transient, or turbulent conditions.

The critical conditions of flow are determined by three precise critical values of the dimensionless number. They are the same for all the fluids having yield pseudoplastic behavior. The pressure drop also is calculated using simple expressions, after the state of flow has been determined, by means of a friction correction coefficient that gives the value for a correct evaluation of the pressure drop gradient of the flowing fluid.

To validate the proposed theory, field and calculated data were compared, with only a very small error found. Field data were obtained from circulating tests in a 1712-in. section of an ultradeep well.

### Background

In the mid-1800s, Hagen conducted the first tests using a Newtonian fluid flowing in brass tubes of various diameters, which indicated laminar flow ceased to exist when the fluid velocity increased beyond a certain limit.

The transition of the laminar into the turbulent state depended on the radius of the tube, the velocity, and the temperature of the water. The turbulent flow became laminar as soon as any one of these three values decreased below a certain limit. Around the same time, Poiseuille experimentally found the same law for the laminar flow of water through very narrow glass capillary tubes.

Later, Reynolds established that a change in the flow occurred and correlated this change with a dimensionless ratio in which size of the conduit, average flow velocity, density, and fluid viscosity were the parameters that determined the flow condition.

For a Reynolds number < 1,000-1,400, the flow is considered laminar; for a Reynolds number > 1,600-2,000, the flow is considered to start into the turbulent conditions. The region between the two is the transition state between laminar and turbulent flow. This transition state strongly depends on the temperature of the tests.

Moody correlated the Reynolds number, which became the most important parameter to define the regimes of water flow, to a dimensionless number called "friction factor" to determine the pressure loss gradient of a fluid flowing in a pipe.1

Bingham and his coworkers, studying the capillary flow of suspensions of clay and paints, and later Buckingham, re-elaborated a rheological model conceived by Schwedoff for flow of gelatinous solutions.2 In this rheological model, once initiation of flow occurred, the fluid behaved as a Newtonian or a viscous inelastic fluid.2 Unfortunately, the calculated values of the shear stresses were overestimated at both low and high shear rates.

Later, Schuh developed a new rheological model, conceived by Ostwald-De Waele in a study of the flow of polymer solutions, for fluids having a decreasing viscosity with shear rate (pseudoplasticity).3-4 Ostwald-De Waele's model corrected the weak points of the previous model, but, the proposed model was unable to calculate exactly the pressure drop of the hydraulic circuit because it did not consider an important parameter of a drilling fluid: the yield point to. This model also underestimated the shear stresses at both low and high shear rates.

The Herschel and Bulkley model, conceived around the beginning of this century, was then used to overcome some of the gaps in the previous rheological models.5 The Herschel and Bulkley model, originally used for the flow of solutions of crude rubber in benzene, considers all the parameters representing a drilling fluid: the yield point to, the consistency index k, and the flow behavior index n. Therefore, the shear stresses at both low and high shear rates can be accurately estimated.

During recent years, this model has been deeply developed for well drilling and cementing operations. Recent drilling literature contains all the expressions necessary to define the type of flow and calculate the relative pressure drop of the circuit.15-20 Other models (such as Casson; Parzonka and Vocadlo; Robertston and Stiff; Gucuyener; Shulman; and Collins and Graves) have been used, but these models have not had a large use because of less accuracy or difficulty in developing a simple analytical solution.6-11

### Reynolds number

Over the years, the most important parameter to define the regime of a drilling fluid flow (laminar, transient, or turbulent) has been the Reynolds number. The Reynolds number has been modified for the rheological model that one wanted to use in the calculations of the pressure drop in the hydraulic circuit. This number reveals the magnitude of inertia to frictional or viscous forces (Equation 1 in the equation box)(61554 bytes).

The Reynolds number does not always give an exact evaluation of the regime of flow, however. And the Reynolds numbers calculated from different rheological models may not agree.

Often, to overcome the difficulty of exactly estimating the ranges of plug and transient flow, some approximations are made. Plug flow is considered as a part of laminar flow, and the transient flow range is neglected. It is also usually assumed that the transition from laminar to turbulent flow occurs at the same critical values of the Reynolds number.

Moreover, very little has been published regarding plug flow, and the question of the transition flow is still open today. Most experimental results show that if the fluid is less Newtonian, the Reynolds number corresponding to the end of the purely laminar flow regime and the Reynolds number corresponding to the beginning of the fully turbulent flow regime will be higher.

Several theories have been developed to account for this variation, but most of them refer to a specific rheological model and do not have a general validity.12-14 17-18

Having a correct evaluation of the type of flow is critical to better planning of the drilling operations. A better understanding of the flow regime can help optimize the cuttings transport, improve the rate of penetration, and improve displacement during cementing to avoid channeling.

Work on hydraulic simulations and calculations indicated the state of flow could be individuated not only using the Reynolds number theory but also by another dimensionless ratio, which takes into account only the inner rheological characteristics of the fluid and without considering the density of the fluid. This second method gave practically the same results and was easier and simpler to use.

The equations derived in this article present a simple, practical alternative method to the Reynolds number for defining the regimes of drilling fluid flow and calculating the relative pressure loss gradients. The Reynolds number is still present in the calculations but only with a meaning of relative expression.

The rheological model used to simulate the behavior of the fluids is the Herschel and Bulkley model. All the considerations regarding the conditions of flow and the equations for the calculation of the pressure loss gradient refer to this model.

### Herschel and Bulkley model

An accurate rheological model relating shear stress to shear rate is an essential starting point to estimate accurately the pressure drop along an hydraulic circuit. Most muds and cement slurries used in drilling and cementing operations behave as yield pseudoplastic fluids.

The Herschel and Bulkley rheological expression is a three-parameter model that describes better than other models the behavior of yield pseudoplastic fluid. Equation 2 (61554 bytes) is the mathematical model, in which to is the yield point, k is the consistency index of the fluid, and n is the flow behavior index.

### Evaluation criteria

The evaluation of flow regimes uses a dimensionless ratio (G) that reveals the magnitude of the dynamic frictional forces (due to the motion) to yield forces, relating the three rheological parameters of the model, the size of the conduit, and the average flow velocity (Equation 3)(61554 bytes).Equation 3 (61554 bytes) reduces to Equation 4 (61554 bytes) for a circular section and to Equation 5 (61554 bytes) for an annular section.

Figs. 1 (40042 bytes)and 2 (42844 bytes) represent the behaviors of this G number versus the flow regimes (plug, laminar, transient, and turbulent) for circular and annular sections, respectively. The G number is correlated to a friction correction coefficient (f) needed to determine the relative pressure loss gradient of the flowing fluid.

The expressions of the correction coefficient f (derived for each state of flow later in this article) were obtained using a nonlinear regression method that better fitted the experimental data on the graph.

Fig. 3 (35707 bytes) shows an example of the relationship between the pressure loss gradient (and the state of flow) and the Gc number for a typical drilling fluid flowing in a 4.27-in. ID pipe.

The accompanying box develops the equations to obtain the pressure loss expressions for plug, laminar, and transient flow for circular and annular sections. The developed theory is based on the following assumptions:

- Isothermal and steady state flow
- One dimensional flow
- Incompressible fluid
- Concentric circular and annular sections
- Annular section considered as a rectangular slot.

Pressure Loss Gradient chart (75443 bytes)

### Circular section

The following are the boundaries for the four flow regimes for a drilling fluid and the relative expressions for calculating the pressure loss gradient.

- Plug flow

The following boundary conditions define plug flow: 0 Equation 6(61554 bytes).

- Laminar flow

- Transient flow

Transient flow is defined for the following: 2.80 Equation 9,(61554 bytes) in which b0 = 50.2775, b1 = -25.1309, b2 = -44.0707, b3 = -26.6213, b4 = 5.7484, b5 = 32.4870, b6 = 37.4958, b7 = 14.3102, b8 = -26.5370, b9 = -47.2891, and b10 = 30.9922.

- Turbulent flow

Turbulent flow is defined for the following: Gc 3.20. The Fanning equation gives the expression to calculate the pressure loss gradient (Equation 10)(61554 bytes). The Fanning friction factor is evaluated using the Schuh expression for smooth pipes (Equations 11-15).4(61554 bytes) In Equation 15, xo = 0.67031498, x1 = -1.6832576, x2 = 1.4891205, x3 = -0.54315691, x4 = 0.07022063, j1 = -2.4506364, j2 = 2.129996, j3 = -0.76926932, and j4 = 0.098792419.

### Annular section

- Plug flow

The following boundary conditions define plug flow: 0 Equation 16 (61554 bytes) gives the expression to calculate the pressure loss gradient.

- Laminar flow

Laminar flow is defined for the following: 0.75 Equation 17 (61554 bytes) gives the expression to calculate the pressure loss gradient. The friction coefficient factor fa is given by Equation 18,(61554 bytes) in which do = 5.9876, d1 = -25.4125, d2 = 68.9826, d3 = -109.5874, d4 = 104.2326, d5 = -59.9213, d6 = 20.3599, d7 = -3.7584, and d8 = 0.2903.

- Transient flow

Transient flow is defined for the following: 17.40 Equation 19,(61554 bytes) in which co = 0.77099795, c1 = -0.54389685, c2 = 0.095927315, z1 = -0.70577824, and z2 = 0.12453472.

- Turbulent flow

Turbulent flow is defined for the following: Ga 26.00. The expression to calculate the pressure loss gradient is given by a version of the Fanning equation (Equation 20)(61554 bytes), in which the expression to calculate the Fanning friction factor is given by the Schuh expression for smooth pipes (Equations 21-23)(61554 bytes). In Equation 23,(61554 bytes) eo = -4.1785098 and e1 = 1.8536188, and the coefficients y and z are the same as those for turbulent flow in the circular section.

Table 1 (10713 bytes) shows the corresponding critical points for different flow regimes.

### Data

To validate this proposed calculation method, measured standpipe pressure data were collected from a series of circulation tests performed in a 1712-in. surface section of an ultradeep well in northern Italy. The tests were carried out under 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 from a manometer located on the rig floor.
- The rheological measurements were performed on a mud sample taken from the aspiration pit while the test was in progress.

Standpipe pressure is given by Equation 24,(61554 bytes) which can also be represented by Equations 25-27.(61554 bytes)

The circulation test was carried out at h = 799 m. Table 2 (11732 bytes) shows the Fann VG 35 coaxial cylinder viscometer readings and the corresponding parameters of the Herschel and Bulkley model, calculated using a nonlinear regression method, and the density for the mud used in the test.

Figs. 4(40855 bytes)and 5 (167877 bytes) show the well schematic and drillstring configuration during the test. Table 3 (10319 bytes) compares the field data and calculated data.

### Results

- A simple, practical method to determine the state of flow of a yield pseudoplastic fluid has been developed. This method does not depend on the fluid density and can be used to calculate the relative pressure loss gradient in both circular and annular sections of a well bore.
- The results of this study indicate that the flow regimes of a yield pseudoplastic fluid depend on a dimensionless ratio, G, given in Equation 3(61554 bytes). The state of the flow depends on the rheological parameters of the fluid in addition to the average flow velocity and the sizes of the conduit.
- There are three critical points that define the transition of the flow state from plug to laminar, laminar to transient, and transient to turbulent flow.
- This calculation method can be used also with plastic-behavior fluids because the Bingham rheological model, with a flow behavior index n = 1, can be considered a particular case of the Herschel and Bulkley rheological model. Considering this case, the dimensionless value G becomes: G = (a/Bi), where a = 8 for a circular section and a = 12 for an annular section.
- The calculated pressure loss data were compared to the measured data (standpipe pressure). They are very accurate, and very small error has been found.

### Acknowledgment

The author would like to thank Professor Giovanni Baldini of the Politecnico di Torino for his suggestions in writing this article. The author also thanks the Saipem rig crew and mud logging unit who helped collect the field data and Agip SpA for permission to publish this article.

### References

1.Moody, F.L., "Friction Factor in Pipe Flow," Transactions of the Association of Mechanical Engineers, Vol. 66, 1944, p. 672.

2.Bingham, E.C., "Fluidity and Plasticity," McGraw-Hill, New York, 1922.

3.Ostwald, W., Kolloid Zeit, Vol. 36, 1925, p. 99.

4.Schuh, F.J., "Computer Makes Surge Pressure Calculations Useful," OGJ, Aug. 3, 1964, pp. 96-104.

5.Herschel, W.H., and Bulkley, R., "Konsistenzmessungen Von Gummi-Benzollosungen," Kolloid Zeit, Vol. 39, 1926, p. 291.

6.Casson, N., "Flow Equation for Pigment Oil Suspensions of the Printing Ink Type," Rheology of Dispersed System, Pergamon Press, Oxford, 1959, pp. 81-104.

7.Parzonka, W., and Vocadlo, J., "Methode de la Caracteristique du Comportement Rheologique des Substances Viscoplastiques d'apres les Mesures au Viscometre de Couette (Model Nouveau Trois Parametres)," Rheologica Acta, Vol. 4, 1968, p. 333.

8.Robertson, R.E., and Stiff, H.A., "An Improved Mathematical Model for Relating Shear Stress to Shear Rate in Drilling Fluids and Cement Slurries," Society of Petroleum Engineers Journal, February 1976, pp. 31-36.

9.Gucuyener, I.H., "A Rheological Model for Drilling Fluids and Cement Slurries," SPE paper 11487, presented at the Middle East Oil Technical Conference, Manama, Bahrain, Mar. 14-17, 1983.

10.Shulman, Z.P., "One Phenomenological Generalisation of Viscoplastic Rheostable Disperse System Flow Curves," Teplo-Massoperenos, Minsk, Vol. 10, 1968, pp. 3-10.

11.Graves, W.G., and Collins, R.E., "A New Rheological Model for Non Newtonian Fluids," SPE paper 7654, June 1978.

12.Hedstrom, B.O.A., "Flow of Plastic Materials in Pipes," Industrial and Engineering Chemistry, Vol. 44, March 1952, pp. 651-56.

13.Ryan, N.W., and Johnson, N.M., "Transition from Laminar to Turbulent Flow in Pipes," American Institute of Chemical Engineers Journal, Vol. 5, No. 4, December 1959, pp. 433-35.

14.Hanks, R.W., "The Laminar-Turbulent Transition for Fluids with a Yield Stress," American Institute of Chemical Engineers Journal, Vol. 9, No. 3, 1963, pp. 306-09.

15.Maglione, R., and Robotti, G., "Identificazione del Fango e delle Malte di Cementazione da Prove di Pozzo nei Cantieri Petroliferi," Bulletin Associazone Mineraria Subalpina, No. 4, December 1989, pp. 551-68.

16.Honghai, F., and Xisheng, L., "Flow Plug and Pressure Drop of Herschel and Bulkley Fluids in Drilling Well Concentric Annuli," (in Chinese), Journal of the University of Petroleum of China, Vol. 17, No. 6, December 1993, pp. 28-34.

17.Reed, T., Pilehvari, A., "New Model for Laminar, Transitional and Turbulent Flow of Drilling Muds," SPE paper 25456, presented at the Production & Operation Symposium, Oklahoma City, Mar. 21-23, 1993, pp. 39-52.

18.Hemphill, T., Campos, W., and Pilehvari, A., "Yield power law model more accurately predicts mud rheology," OGJ, Aug. 23, 1993, pp. 45-50.

19.Merlo, A., Maglione, R., and Piatti, C., "An Innovative Model for Drilling Fluid Hydraulics," SPE paper 29259, presented at Asia Pacific Oil & Gas Conference, Mar. 20-22, 1995, Kuala Lumpur.

20.Maglione, R., and Merlo, A., "Non Newtonian Rheological Parameters Determination Using Field Data," presented at the 9th International Conference on Numerical Methods on Laminar and Turbulent Flow, Atlanta, July 10-14, 1995.

### The Author

Maglione's work includes drilling fluids mechanics, heat transfer, multiphase fluid flow, hydraulic optimization, surge and swab, and well control problems in standard, slim hole, and horizontal wells. He holds an MS in mining engineering from Politecnico di Torino and is a registered professional engineer in Italy.