Aug. 23, 1993
Terry Hemphill Baroid Drilling Fluids Inc. Houston Wellington Campos University of Tulsa Tulsa Ali Pilehvari Texas A & M University-Kingsville Kingsville, Tex. The yield-power law rheological model can calculate yield point much more accurately than that calculated by the Bingham plastic model. The yield-power law (Herschel-Bulkley) model offers many advantages over the Bingham plastic and power law models because it more accurately characterizes mud behavior across the entire shear rate range.
Terry Hemphill
Baroid Drilling Fluids Inc.
Wellington Campos
University of Tulsa
Ali Pilehvari
Texas A & M University-Kingsville
Kingsville, Tex.

The yield-power law rheological model can calculate yield point much more accurately than that calculated by the Bingham plastic model.

The yield-power law (Herschel-Bulkley) model offers many advantages over the Bingham plastic and power law models because it more accurately characterizes mud behavior across the entire shear rate range.

The yield-power law model has not found widespread use in the oil field because of the lack of simple analytical solutions for viscometric and hydraulics calculations. These concerns are no longer pertinent, however, because of the rapid spread of personal computers in the field and recent developments in using this model.

Rheological modeling of drilling fluids in the field is visually described by the Bingham plastic or the power law model. Although these models are fairly easy to solve for their specific descriptive parameters, they do not simulate fluid behavior across the entire rheological spectrum very well, particularly in the low shear rate range.

The following definitions help in understanding the various models:

  • Shear rate (y) in a simple flow is the change in fluid velocity divided by the gap or width of the channel through which fluid is moving.

  • Shear stress (t) is the force per unit area required to move a fluid at a given shear rate.

  • Fluid viscosity (u) is the fluid's shear stress divided by the corresponding shear rate.


Most drilling fluids and cement slurries exhibit non-Newtonian behavior; their laminar flow relation between shear stress and shear rate is nonlinear. These fluids also require a measurable amount of shear stress for flow to begin and thereafter require additional stress as the shear rate level increases. The level of shear stress required to initiate flow is commonly called a fluid's yield point, or more precisely, its yield stress. For years the oil industry has relied on two Theological models to determine fluid characteristics and hydraulics: the Bingham plastic and power law models. 1 2


The Bingham plastic model calculates two parameters: yield point (YP) and plastic viscosity (PV). These values are calculated using the Farm VG meter dial readings measured at 600 and 300 rpm. Fluid behavior under shear can be modeled using Equation 1.

The shear stress needed to move the mud at any desired shear rate can be calculated with the mud's YP and PV values. This model represents fairly well the behavior of bentonite slurries, and for this reason it has received widespread use in the oil field. For fluids more complex than simple slurries, however, the Bingham plastic model falters, as shown in the following example.

Fig. 1 shows an oil-based mud's six-speed viscometer dial readings plotted across the shear rate range. According to the Bingham plastic model, this fluid would have a PV of 25 cp and a YP of 13 lb/100 sq ft. However, the fluid's YP, which is supposed to simulate yield stress at zero shear (O rpm), is substantially larger than both the 6 and 3 rpm dial readings. Calculations of shear stress using the Bingham plastic model's basic equation produce significant errors because most drilling fluids are not fully Bingham plastic fluids.

Calculations of fluid hydraulics using this YP value will result in pressure losses and equivalent circulating densities (ECDs) larger than observed in the field. While the Bingham plastic model simulates fluid behavior in the higher shear rate range (300-600 rpm), it usually fails in the low shear rate range, which is the area of interest for simulating annular flow behavior. Shear stresses measured at high shear rates usually are poor indicators of fluid behavior at low shear rates.


With increased use of polymer-based fluids in the oil field, the power law (PL) Theological model became popular because it fits the behavior of these fluids better than the Bingham plastic model. The model's relationship between shear stress and shear rate is given by Equation 2.

The two key terms in the PL model are the consistency index (K) and the fluid flow index (n).

There is no term for YP, and fluids that follow this model have no shear stress when shear rate is zero. Although the model fairly accurately predicts drilling fluid behavior at the higher shear rates, it fails across the lower shear rate range (0-100 rpm), as shown by the example fluid in Fig. 2.

Theoretically, the major stumbling block to modeling of the power law is the fact that most drilling fluids have a yield stress, something for which this model cannot account. The net result is that in hydraulics equations, PL modeling underpredicts both annular pressure losses and ECDS. Moreover, people who use the PL model recognize that different values of n are possible, depending on which particular shear stress/shear rate pairs are used in the calculation methods.


The yield-power law (YPL) model is being reconsidered by the drilling industry.3-6 this model, developed by Herschel and Bulkley, merges the theoretical and practical aspects of Bingham plastic and PL models. Equation 3 describes YPL fluid behavior.

In this model, the parameters n and K are similar to those of the PL model. For fluids having a yield stress, however, the calculated values of n and K will be different from those calculated using the PL model. The parameter to (or tau zero) is the fluid's yield stress at zero shear rate (O rpm). In theory this yield stress is identical to the Bingham plastic YP, though its calculated value is different. As special cases, the model becomes the Bingham plastic model when n = 1 and becomes the PL model when To = 0.

The YPL model "works well for water-based and oil-based drilling fluids because both exhibit shear-thinning behavior and have a shear stress at zero shear rate. Additionally, fracturing fluid and cement researchers have long been interested in this model because fracturing fluids and cement slurries have measurable yield stress values and shear stress/shear rate profiles that make them well-suited to YPL modeling.1 Two technical problems have impeded application of this model since its inception. Solutions for the model's three parameters n, K, and -,o are complex. And, more importantly, mud hydraulics equations written using this model have not been available until only recently.


Thus, some fluid flow scientists have developed other models (such as the Casson and Robertson-Stiff models) that simulate fluid behavior better than Bin-ham plastic and PL models, but the oil field has not been very eager to use them. 8-10 One problem was that these newer two and three-parameter models used different terminology,


The modeling of fluid flow in concentric/eccentric annuli requires accurate rheological data over the entire spectrum of shear rates. Drill pipe eccentricity can drastically affect localized fluid velocities. 11 12 Many researchers and engineers are now studying the yield-power law (,YPL) Theological model to try to predict better the flow velocities and the carrying capacity under the drill pipe.

This renewed interest in the model has resulted in a solution for the model's three parameters calculated with a personal computer. The current form of the solution algorithm is the result of a joint effort by the Tulsa University Drilling Research Projects group at the University of Tulsa and Baroid Drilling Fluids Inc. In addition, the hydraulics of pipe and annular flow of YPL fluids in laminar, transition, and turbulent flow have recently been presented.


To calculate To, n, and K values, a mud's Farm 35 VG meter dial readings and the corresponding revolutions per minute are required. A minimum of three data pairs are required for a solution: 0600, 03 (or 06), and another data pair in the intermediate range. Model accuracy is improved with increasing numbers of data pairs, especially in the lower shear rate range. At the minimum, a six-speed Farm viscometer is required. Calculations made using the 10-sec gel strength in place of the 06 or 03 reading is not recommended because errors will result.

Because the model is nonlinear, the computer algorithm uses a least squares method to calculate values for n, K, and -,,). The routine is based on three partial derivatives that minimize the absolute errors of prediction (Equations 4-6). For N shear stress/shear rate data pairs, these equations are solved to produce Equations 7-9.

For a value of the flow index n, Equations 7 and 8 produce solutions for To and K. Because an explicit solution for n is not possible, an iterative calculation procedure is involved in solving Equation 9. A higher level iterative loop is also required to determine new values for shear rate at the viscometer bob wall after each new set of values for To, K, and n.13 Calculation methods programmed on common spreadsheet software, such as Lotus 1-2-3, can solve the algorithm in a few seconds.

Once the program has solved for the three parameters, the fluid's shear stress profile across the shear rate range can be illustrated. The fluid's shear stress/shear rate data pairs used in the calculation procedure can be overlaid to compare the predicted to the measured shear stress values." Any problems the YPL model has fitting the data will quickly become evident.

Fig. 3 shows the increased accuracy of the YPL model in fitting the example mud. Table 1 is a statistical comparison of the accuracy of the Bingham plastic, power law, and yield-power law models in predicting shear stresses for the example fluid. Table 2 contains the values of the example fluid's Theological parameters calculated using the three models.

As shown in Fig. 3, the example fluid's or yield stress was only 6.6 lb/160 sq ft, substantially less than the 13 lb/100 sq ft calculated using the Bingham plastic model. Some mental adjusting to the numerically lower values of YP produced by the model algorithm will be required. Fig. 4 shows how the YPL model offers improved prediction of low (0100 rpm) shear rate behavior for the example fluid.

Linear regression methods can also be used to determine the parameters for the Bingham plastic and PL models using all the available shear stress/shear rate data. An improvement would be seen over those values calculated from only two data pairs, yet the fit of the data would not have the accuracy of the YPL model. Moreover, the oil industry has not practiced such methods to solve the two-parameter models.


Because the YPL model is a combination of the Bingham plastic and PL models, fluid behavior predicted by the other models will also be well predicted by this model. Sometimes YPL values for n will be almost equal to the values for n calculated using the PL model. When this happens, very low yield stress (TO) values are expected. When a fluid's to value is very close to its Bingham plastic YP, the fluid's flow index value n should be nearly 1.

Muds having high shear stresses at high shear rates do not necessarily exhibit elevated yield stresses, something for which Bingham plastic model cannot account. Fig. 5 depicts a viscous water-based fluid's dial readings and the corresponding viscometer rpm values. According to the Bingham plastic model, the yield point of this fluid is 37 lb/100 sq ft (a high level), whereas the YPL model calculates the yield stress to be only 4 lb/160 sq ft, a difference of 89%.

Table 3 compares the accuracy of the three models in predicting shear stresses. Table 4 shows solutions for the water-based fluid's various Theological parameters calculated according to the different models. Again, the Bingham plastic and PL models are not nearly as accurate in predicting shear stresses, especially in the low shear rate range.

Mud treatments or the lack thereof based on the Bingham plastic YP calculation can lead to problems in the field. In the laboratory, some viscous muds (such as this one) exhibit settling tendencies and present barite sag problems in the field. Common mud Theological parameters such as the PY YP, and 10-sec/10-min gel strengths do not correlate with measurements of barite sag." Thus, some researchers have recommended special attention be paid to the 6 and 3 rpm viscometer dial reading.16 What they are actually referring to is the fluid's yield stress value, something which was not easily or reliably calculated until only recently.


The YPL model can be used to understand better the behavior of some of the newer drilling fluid types, such as the mixed metal hydroxide (MMH) system. These fluids are characterized by a high yield stress and high 10-sec and 10-min gels. Their Theological performance at low shear rates is not usually evaluated using the Bingham plastic YP, but rather with the viscometer 06 and/or 03 dial readings.

Fig. 6 contains flow profiles of two MMH fluids using YPL modeling." The curves indicate that this new fluid type can be accurately simulated with this model. Tables 5 and 6 contain shear stresses for the two MMH fluids as predicted by the three models' and Table 7 contains the relevant rheological parameters. Table 8 lists some of the fluid properties of the MMH muds, the oil-based mud, and the water-based mud.


  • The high value of n for the YPL indicates the fluid behavior is nearly Bingham plastic, something confirmed in the near-equivalence of the YP and To values.

    This fluid's flow behavior can be fairly accurately evaluated using either the YPL or Bingham plastic model.

  • The PL model does not simulate fluid behavior very well, especially in the low shear rate range.

    There is a significant difference in values of n as calculated by the PL and YPL models.

  • The gelation profile is fairly flat over time (18, 20, and 21 lb/100 sq ft at 0, 10, and 600 sec, respectively).


  • The values of the YPL n and to indicate the fluid behavior of this MMH mud is neither Bingham plastic nor power law, but is somewhere between the two.

  • The gelation profile is more progressive than that of MMH Mud A (28, 40, and 64 lb/100 sq ft at 0, 10, and 600 sec, respectively).


The accuracy of hydraulics calculations for fluids not well characterized by the Bingham plastic or PL models is compromised when the PY, YP, power law n, power law K, or the viscometer 06 or 03 dial reading terms are directly used in the calculations. Mud hydraulics equations using the YPL model have been written and are available in sophisticated computer programs."4 Annular pressure losses can be quickly calculated with the relevant hole geometry, pump rate data, mud density, and the model's three parameters.

The result is improved values for ECDs because the model simulates field drilling mud behavior more accurately than the more common Theological models. Usually, annular pressure drops and ECDs calculated using this model will be intermediate between those calculated using the Bingham plastic and PL models.

Modeling fluid behavior at high temperatures and high pressures according to the YPL model is currently under way.


  • The yield-power law, or Herschel-Bulkley, Theological model better describes the behavior of most water-based and oil-based drilling fluids across the shear rate spectrum than the Bingham plastic or power law models.

  • Using a personal computer algorithm, the yield-power law model's three parameters can be quickly solved with conventional six-speed viscometer dial readings.

  • To evaluate a fluid's carrying capacity, the model's calculated yield stress (to) is more accurate and useful than the Bingham plastic yield point. Evaluations of mud performance based on the Bingham plastic rheological model can lead to problems in the field.

  • Industry personnel are already familiar with the yield-power law model's three parameters on a conceptual basis; only the method of calculation is different.

  • The model can accurately simulate flow behavior of newer fluids, such as MMH muds, where the Bingham plastic and power law models often fail.

  • Application of this model has important implications for calculating mud hydraulics and evaluating hole cleaning efficiency in highly deviated wells. The model's parameters can be used to directly calculate fluid hydraulics without using approximations or correction factors.


  1. Bourgoyne, A., Chenevert, M., Milheim, K., and Young, F., Applied Drilling Engineering, Society of Petroleum Engineers, Richardson, Tex., 1986.

  2. API Bulletin on the Rheology of Oil-well Drilling Fluids (API BUL 13D), American Petroleum Institute, Washington, D.C.

  3. Skelland, A.H.P., Non-Newtonian Flow and Heat Transfer, John Wiley & Sons Inc., New York, 1967.

  4. Zamora, M., and Lord, D.L., "Practical Analysis of Drilling Mud Flow in Pipes and Annuli," SPE paper 4976 presented at the Annual Technical Conference and Exhibition, Houston, Oct. 69, 1974.

  5. Houwen, O.H., and Geehan, T., "Rheology of Oil-Base Muds," SPE paper 15416 presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Oct. 5-8, 1986.

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

  7. Shah, S.N., Lord, D.L., and Tan, H.C., "Recent Advances in the Fluid Mechanics and Rheology of Fracturing Fluids," SPE paper 22931 presented at the SPE International Meeting on Petroleum Engineering, Beijing, Mar. 24-27, 1992.

  8. Alderman, N.J., Gavignet, A., Guillot, D., and Maitland, G.C., "High-Temperature, High-Pressure Rheology of Water-Based Muds," presented at the SPE Annual Technical Conference and Exhibition, Houston, Oct. 2-5, 1988.

  9. Lauzon, R.Y., and Reid, K.I.G., "New Theological model offers field alternative," OGJ, May 21, 1979, p.51.

  10. Okafor, M., and Evers, J., "Experimental Comparison of Rheology Models for Drilling Fluids," SPE paper 24086, presented at the Western Regional Meeting, Bakersfield, Calif., Mar. 30-Apr. 1, 1992.

  11. Haciislamoglu, M., "Non-Newtonian Fluid Flow in Eccentric Annuli and Its Application to Petroleum Engineering Problems," PhD thesis, Louisiana State University, 1989.

  12. Hemphill, T., "Research Solving Problems in Cuttings Transport," American Oil and Gas Reporter, August 1992, pp. 32-37.

  13. Whittaker, A., and Exlog staff (ed), Theory and Application of Drilling Fluid Hydraulics, International Human Resources Development Corp., Boston, 1985.

  14. Baroid Drilling Fluids Graphics (DFG) software, Baroid Drilling Fluids Inc., Houston, 1993.

  15. Jamison, D., and Clements, W., "A New Test Method to Characterize Settling/Sag Tendencies of Drilling Fluids Used in Extended Reach Drilling," presented at the 1990 Drilling Technology Symposium, American Society of Mechanical Engineers, PD - Vol. 27, 1990, pp. 109-113.

  16. Hanson, P., Trigg, T., Rachal, C., and Zamora, M., "Investigation of Barite "Sag" in "sighted Drilling Fluids in Highly Deviated Wells," SPE paper 26423, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Sept. 23-26, 1990.

  17. Fraser, L., "Unique Characteristics of Mixed Metal Hydroxide Fluids Provide Gauge Hole in Diverse Types of Formation," SPE paper 22379 presented at the International Meeting on Petroleum Engineering, Beijing, Mar. 24-27, 1992.

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