Novel Equation for the Prediction of Rheological ... - ACS Publications

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Ind. Eng. Chem. Res. 2010, 49, 3374–3385

Novel Equation for the Prediction of Rheological Parameters of Drilling Fluids in an Annulus M. Nasiri and S. N. Ashrafizadeh* Research Lab for AdVanced Separation Processes, Department of Chemical Engineering, Iran UniVersity of Science and Technology, Narmak, Tehran 16846, Iran

Several well-known correlations such as the Bingham-plastic, power-law, and Herschel-Bulkley models have been used so far to determine the rheological parameters of drilling fluids. For some particular fluids, however, even a three-parameter model such as Herschel-Bulkley does not exhibit appropriate behavior. On the other hand, determination of the rheological parameters by numerical methods such as nonlinear regression may provide meaningless values, i.e. negative yield stresses. This is particularly notable in determination of yield stress, which identifies the capacity of the drilling fluid to carry the cuttings. In this work, a new equation has been developed which is capable of determining the rheological parameters and, more particularly, the yield stress of drilling fluids. It is demonstrated that the developed correlation improves the prediction of the rheological parameters of the fluids by including a logarithmic term. The velocity profiles and pressure drop values obtained for several drilling fluids in an annulus geometry exhibit the suitability of this novel equation in comparison with the previously mentioned equations. τ ) C'γ˙ n

1. Introduction Drilling fluids typically used in drilling gas/oil wells are emulsion-suspension systems to which various viscosifiers and surface active reagents may be added to enhance the fluid’s performance. The flow characteristics of such a suspension, which is essentially regarded as a non-Newtonian fluid, are largely governed by the chemical properties of the colloidal bentonite clay particles that form a network with certain strength.1 Three major categories of non-Newtonian fluids are basically recognized, namely, time-independent, time-dependent, and viscoelastic.2 The time-independent category has received a substantial degree of attention in comparison with the other two categories. A large majority of drilling fluids falls into this category.3 In time-dependent flow behavior, the apparent viscosity at a fixed shear rate does not remain constant but varies to some maximum or minimum with the duration of shear. If the apparent viscosity decreased with flow time, the fluid is known to be thixotropic.1 It is generally accepted that drilling fluids can be typified by the Bingham-plastic model.4,5 The Bingham-plastic model is defined by the relationship of eq 1: τ ) τ0 + ηγ˙

(1)

The Bingham-plastic model of flow differs most notably from a Newtonian fluid by the presence of a yield stress. A Binghamplastic fluid will not flow until the applied shear exceeds a minimum value that is known as the yield stress. Once the yield stress exceeds the mentioned minimum, changes in shear stress are proportional to changes in shear rate and the constant of proportionality is called the plastic viscosity. As it will be discussed, the Bingham-plastic model usually does not accurately represent drilling fluids at low shear rates. The power-law model is defined by the relationship of eq 2:5-7 * To whom correspondence should be addressed. E-mail: ashrafi@ iust.ac.ir. Tel.: +98 (21) 77240402. Fax: +98 (21) 77240309.

(2)

The power-law model is frequently more convenient than the Bingham-plastic model. The power-law model demonstrates the behavior of a drilling fluid at low shear rates more accurately. However, this model does not include a yield stress and therefore can give poor results at extremely low shear rates. Therefore, either of the two models, i.e. the Bingham-plastic and powerlaw models, is inefficient at low shear rates. A typical drilling fluid exhibits behavior intermediate between the Bingham-plastic and the power-law models. The Herschel-Bulkley model eq 3, which is a hybrid of the Bingham-plastic and power-law models, includes three parameters.8 This model is in fact a power-law model with a yield stress. The model yields mathematical expressions relating flow rate to pressure drop that are not readily solved analytically but can be solved using nonlinear regression methods. τ ) τ0 + kγ˙ n

(3)

The three-parameter Herschel-Bulkley model provides an appropriate relation between the shear stress and shear rate. This model is also in much stronger agreement with the rheological data of the fluid; especially at low shear rates. The latter is particularly important to horizontal directional drilling (HDD) where the flow regime is laminar and thus has a low shear rate. More complex four-parameter or even five-parameter models have also been proposed by Shulman,9 Mnatsakanov et al.,10 and Maglione et al.11 Detailed descriptions of the various rheological models have been proposed, and derivations of the appropriate flow equations have been given by Bird et al.12 and Maglione et al.13 The latter models are more accurate in predicting the behavior of drilling fluids than the two-parameter models; which are widely accepted at present. However, there is not wide acceptance and wide application of the more complex models because of the difficulty in finding analytical solutions for the differential equations of motion and because of the complexity of the calculations for the derivation of the appropriate hydraulic parameters such as Reynolds number, flow velocity profiles, circular and annular pressure drops, and criteria for transition from laminar to turbulent flow.14

10.1021/ie9009233  2010 American Chemical Society Published on Web 02/22/2010

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010

The standard procedure for the estimation of the rheological parameters for the Herschel-Bulkley model based on eq 3, i.e. τ0, k, n, or any arbitrary equation, is through nonlinear regression of the viscometric data. This is usually done using a numerical software (e.g., Matlab R2006), command cftool in command window, minimizing the sum of square errors, comparing the goodness of fit through the value of the correlation coefficient, RC2, and the sum of square errors, SSE, from the fitting of nonlinear equation with Matlab software. It is obvious that the best nonlinear fit must belong to the highest correlation coefficient, RC2, with the lowest sum of square errors. Therefore, the ratio of SSE to correlation coefficient, RC2, is always equal or more than zero, so that zero value is the best for a nonlinear fitting. Whatever the lower mentioned ratio, the better goodness of fitting. Nonlinear fit (NL) to various viscometric data in the Herschel-Bulkley equation with Matlab software has given the highest correlation coefficient with negative value for the yield stress.14 Therefore, the condition of τ0 > 0 must be imposed to get meaningful results, although the obtained yield stress will be necessarily not the optimum value. This method is called nonlinear regression with penalty (NLP). According to the 1960s-1980s literature, particles will not settle if the yield stress of the fluid satisfies a criterion based on a balance of particle weight and failure of the surrounding fluid. This yield-stress criterion stems from plasticity theory and is only valid for a static plastic material. According to plasticity theory, a particle will settle if the yield stress satisfies the following criterion:15 τ0 e a(Fs - F)gd

(4)

Equation 4 reveals that the yield stress is an important criterion in the settling of solid particles; as such it is necessary to be evaluated precisely. The authors of the present study thus believe that determination of the exact value of τ0 and shear stress at low shear rates, due to their importance in determining the capacity of the fluid to carry the cuttings, is very important in defining the rheological properties of drilling fluids. Consequently, a new four-parameter model is introduced which is a combination of the Bingham-plastic and the power-law models but differs from that of Herschel-Bulkley. This model can be simplifiedthoughtoeitheroftheBingham-plasticorHerschel-Bulkley models in particular conditions. To build the new model, the authors used the simple relations of the mathematics and demonstrated that a logarithmic term can more effectively represent the rheological parameters. The results of this work are compared with the results of Kelessidis et al.14 as well as with those of nonlinear regression with penalty (NLP). 2. Development of a New Equation (NE) As mentioned earlier, drilling fluids, i.e. water-based and oil/ synthetic-based drilling fluids, commonly exhibit non-Newtonian rheological behavior which can be stated with the Binghamplastic, power-law, or Herschel-Bulkley models. The threeparameter Herschel-Bulkley model, although extensively used in drilling fluids, cannot be applied as an efficient model in the vicinity of zero or in the very low shear rates. The latter is due to the fact that, in the vicinity of zero, the slope of the obtained curve from the Herschel-Bulkley model approaches infinity. Therefore in the vicinity of zero shear rates, shear stress and yield stress will not be exact. However, this model has good agreement with experimental data at high shear rates. The new equation (NE), i.e. eq 5, developed by the current authors, is a hybrid of the Bingham-plastic, power-law, and Herschel-Bulkley

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models with a logarithmic correction factor. Mathematical proof of the NE is given in Appendix A. ∞

τ ) τ0 + bγ˙ + kγ˙ n ln(1 +

∑ a γ˙ ) i

i

(5)

i)1

The determination of coefficients ai in the logarithm argument via nonlinear regression is a hard task. Therefore, it was attempted to simplify eq 5 to eq 6, with the assumptions a1 ) 1 and a2 ) a3 ) ... ) an ) 0. The resulting equation would be the following: τ ) τ0 + bγ˙ + kγ˙ n-m ln(1 + γ˙ )

(6)

It would be then appropriate to compensate for the errors raised from the implication of the above assumptions, i.e. neglecting the terms in the logarithmic argument of eq 5, by adding the exponent m in eq 6. Data analysis reveals that the value of exponent m is usually between 0 and 1 for the best fit. At the same time, the value of (n - m) may be rarely less than zero for the best fit. Eventually, the NE can be stated as eq 7 with parameters having the following values: τ ) τ0 + bγ˙ + kγ˙ n-m ln(1 + γ˙ )

(7)

0 e m e 1;0 e n e 1 τ0 g 0;b g 0 3. Experimental Data Used in This Study The experimental data used in this study are taken from four different sources: (a) SI Type Data. The data of Mihalakis et al.16 and Kelessidis et al.,17 reproduced by Kelessidis et al.,14 was used through this study. This group of data (S1-S12) has been derived from drilling fluids prepared for laboratory tests. The measurements have been made with a viscometer (Grace, M3500) at two temperatures of 25 and 65 °C and rotational speeds of 3, 6, 10, 20, 30, 60, 100, 200, 300, 400, 500, and 600 rpm. Water-bentonite suspensions either hydrated for 24 h at room temperature or aged statically in an aging cell for 16 h at 177 °C have been used. Lignite has been added to the suspension at either of 0.5 or 3% w/v. (b) SII Type Data. The data of Merlo et al.,18 reproduced by Kelessidis et al.,14 was used in this study. This group of data (S13-S18) has been derived from drilling fluids used in the field operations during drilling circulation tests at various sections of the well. The fluid samples have been taken from the outlet of the drilling circuit. The measurements have been made with a viscometer (Huxley-Bertran HPHT rotational viscometer) for all of the samples, except for sample S18 which has been measured with a Fann VG 35 6-speed rotational viscometer. (c) SIII Type Data. The data of Blick,19 as reported by AlZahrani20 and Kelessidis et al.,14 was used in this study. This group of data (S19-S21) has been generated using a rotary viscometer for different suspensions prepared by adding various quantities of Wyoming bentonite to water. (d) T Type Data. The data of Enilari,21 for water-based drilling fluids containing KCl and KF salts and xanthan, was also used. The data T1, T4, T10, T41, T42, and T43 have been reproduced in Table B1, Appendix B. 4. Determination of the Rheological Parameters The standard procedure for the estimation of the three parameters for the Herschel-Bulkley fluids is usually through

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nonlinear regression of the viscometric data obtained from concentric cylinder geometry. This is generally done using a numerical package such as Matlab software, by minimizing the sum of square errors and judging the goodness of the fit through the evaluation of the correlation coefficient, RC2, and the sum of square errors (SSE). (a) Nonlinear Regression with Penalty (NLP). Sometimes nonlinear regression would fit to a set of data (with highest correlation coefficient, RC2) and gives negative values for the yield stress; which is meaningless. Thus the condition of τ0 g 0 must be imposed to obtain meaningful results. Obviously, that would affect the optimum determination of three parameters, while there is also a possibility of nonunique solutions.14 (b) Golden Section (GS). This methodology is based on an initial optimal determination of τ0, using a near optimum form of the golden section search, sometimes called the Fibonacci search, followed by linear regression of the linearized form of the Herschel-Bulkley equation. The aim of this method is to find the value of a particular stress, i.e. the yield stress, which minimizes the difference among the predicted and measured values of shear.14 The functional relationship for the optimization is given by the correlation coefficient, RC2, for the chosen value of τ0. (c) New Equation (NE). Although one may realize at first glance that eq 7 incorporates five parameters, but this equation has only four independent parameters, i.e. τ0, b, k, n - m. The solution method of eq 7 is as follows: parameters τ0, b, and k and exponents n and m are determined through a numerical package which runs the nonlinear regression method. That includes Matlab R2006, Maple, or Mathematica software. In this work, Matlab R2006 is used. It is notable that in the Herschel-Bulkely model, the second derivative relative to the shear rate is always a negative value. However, in the NE model, the second derivate can be negative or positive, depending on the k value (such as samples T1, T3, T9, T11, T41, etc). Variation in the sign of the second derivative of the rheology curve would reduce the errors raised in the modeling of the experimental data, particularly in the waterbased drilling fluids. Applying such a logic, the value of exponent m was changed from zero to one within a loop, using a certain step, e.g. 0.1, and then for each particular value of m, the parameters τ0, b, k, exponent n, correlation coefficient RC2, and the sum of square errors (SSE) were determined. The functional relationship for the best value of exponent m is prepared by the ratio of sum of square errors (SSE) to correlation coefficient RC2 for the chosen value of exponent m. Figure 1 represents the minimum value of the SSE/RC2 ratio and thus the best value of the exponent m; the latter ratio refers to the lowest sum of square errors (SSE) and the highest correlation coefficient,RC2. 5. Results and Discussions NLP and NE methods have been applied to all of the data used in this study. In Tables 1-3, the rheological parameters that have been calculated using both procedures, i.e., the nonlinear regression with penalty (NLP) method and the new equation (NE), are presented. Some of the rheological parameters have been also compared with the golden section (GS) technique in the same tables.14 The results show sensible differences among the three methods in terms of both rheological parameters, particularly yield stress, τ0, as well as the indices of errors, i.e. SSE. The results also reveal that almost for all fluid samples, S1-S21, the sum of square errors calculated through NE is smaller than those from NLP and/or GS methods.

Figure 1. Ratio of sum of square errors (SSE) to correlation coefficient RC2 with an assumed value of exponent m.

At the same time, the correlation coefficient, RC2, calculated through NE is simultaneously higher than those from NLP and GS methods. The NLP and NE methods have been applied to all of the data reported in ref 21; part of the data are given in Table B1 in Appendix B. The rheological parameters that have been calculated using both procedures are presented in Table 4. As a similar trend for SI, SII, and SIII type data, the results show sensible differences among NLP and NE, in terms of both rheological parameters (particularly yield stress, τ0) and the indices of errors, i.e. SSE. The results presented in Table 4 show that almost for all fluid samples, the sum of square errors (SSE) calculated through NE is smaller than those calculated with the NLP method. The values of correlation coefficient, RC2, are also better in NE than those of NLP method. Rheograms for some of the samples, S9, S11, S13, S19, T1, T4, T10, and T41, derived from NE along with those derived from NLP and GS methods are shown in Figure 2. It is apparent that the fitting of the new equation is as good as the GS technique; hence, what the NE gives is the ability to overcome infinite at zero shear rates. The results also show that for all fluid samples the yield stress derived from NE is different from those derived from NLP and GS methods. For instance, the yield stress is determined as zero for samples S13 and S19 by the NLP method while it has been calculated as 0.2883 and 2.51 Pa through NE, respectively. Results of the NE for SI, SII, and SIII type data, only partially provided in this paper for the sake of briefness, show that the proposed scheme provides a viable solution, as is evident by the sum of square errors and correlation coefficients. In this research, both parameters of correlation coefficient, RC2, and sum of square errors, SSE, have been considered in the fitting of data. Therefore, the ratio of SSE/RC2 is an appropriate factor as a criterion for the selection of rheological parameters. Meanwhile, it should be noted that the coefficients in eq 7, i.e. k and exponent, (n - m), have no physical meaning and thus can get either a positive or a negative value. However, this equation can predict the values of stresses versus shear rates with the lowest possible error. The term γ˙ n ln(1 + γ˙ ) in the NE facilitates fitting the data in the limit of zero, and consequently, the yield stress can be found with a lower error. The latter is due to the fact that the slope of γ˙ n ln(1 + γ˙ ) does not approach infinite in the vicinity of zero while in the power-law or Herschel-Bulkley models the slope


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Table 1. Comparison of Rheological Parameters Computed by the NLP, GS, and NE Models (SI Type) sample no. S1

NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP GS NE NLP GS NE NLP GS NE NLP GS NE NLP GS NE NLP GS NE

S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12

b (Pa · s)

τ0 (Pa) 8.447 9.752 11.28 15.40 7.62 × 10-2 3.861 0 0 0.7166 0.1224 6.394 6.938 2.409 2.414 2.631 1.183 1.3012 1.391 3.474 2.8973 2.497 0.3793 0.2847 0.3144 0 7.18 × 10-2 0.2883 0 0.3976 0.2367

4.505 × 10-3 5.013 × 10-3 0.1827 2.925 × 10-2 0.2148 0.1757 0 5.644 × 10

-2

0.2747 1.938 × 10-3 -2

4.797 × 10

5.458 × 10-2

k (Pa · sn)

n/n - m

RC2

SSE

source

3.421 1.950 5.924 2.314 2.388 -2.365 × 10-2 5.906 × 10-2 -3.24 × 10-3 6.564 × 10-2 -7.808 × 10-2 0.4499 -4.683 × 10-2 0.1252 0.1369 4.915 × 10-2 0.1267 0.1058 -9.399 × 10-3 3.109 × 10-2 0.1566 -0.1062 5.668 × 10-2 8.39 × 10-2 6.479 × 10-2 0.1881 0.1462 -5.647 × 10-3 0.1697 9.40 × 10-2 -6.198 × 10-3

0.2549 9.452 × 10-3 0.2643 8.062 × 10-2 0.3406 1 0.6927 1 0.7154 0.8597 0.5001 1 0.7010 0.6842 0.5543 0.6434 0.6680 0.9500 0.8053 0.5661 0.8527 0.6196 0.5625 0.4342 0.5685 0.6068 1 0.6198 0.7036 1

0.9876 0.9889 0.9885 0.9888 0.9340 0.9744 0.9907 0.9964 0.9927 0.9973 0.9951 0.9964 0.9975 0.9910 0.9977 0.9965 0.9967 0.9976 0.9666 0.9800 0.9792 0.9983 0.9960 0.9986 0.9952 0.9942 0.9990 0.9958 0.9942 0.9977

3.863 3.446 12.68 12.30 44.88 17.37 0.7126 0.2803 0.8914 0.3262 1.289 0.9276 0.9188 1.3124 0.8453 0.5671 0.6013 0.3931 3.321 4.828 2.065 3.954 × 10-2 6.70 × 10-2 3.349 × 10-2 0.6079 0.7865 0.1286 0.8862 1.5646 0.4974

17 17 17 17 17 17 17 14 17 14 17 14 17 14 17 14 17 14

Table 2. Comparison of Rheological Parameters Computed by the NLP, GS, and NE Models (SII Type) sample no. S13

NLP GS NE NLP GS NE NLP GS NE NLP GS NE NLP GS NE

S14 S15 S16 S17

b (Pa · s)

τ0 (Pa) 1.707 0 0 0 0 0 0 0 0 0.1747 3.79 × 10-2 0.2478 0 0.3767 0.4480

-2

1.031 × 10

5.831 × 10-3 4.195 × 10-3 -3

4.37 × 10 0.3944

k (Pa · sn)

n/n - m

RC2

SSE

source

1.260 1.9940 2.082 1.901 1.9050 1.622 1.735 1.7330 1.354 0.9448 1.020 0.7620 0.8491 0.4160 -0.1326

0.4354 0.3704 1.52 × 10-2 0.3526 0.3523 5.140 × 10-2 0.3559 0.3561 7.780 × 10-2 0.4097 0.3993 0.1147 0.4079 0.4083 0.8729

0.9971 0.9952 0.9994 0.9990 0.9990 0.9999 0.9981 0.9981 0.9997 0.9990 0.9995 0.9994 0.9987 0.9987 1

1.085 1.8105 0.2166 0.2610 0.2608 3.573 × 10-2 0.4583 0.4583 7.848 × 10-2 0.1563 0.1636 8.942 × 10-2 0.1619 0.1619 4.962 × 10-3

18 14 18 14 18 14 18 14 18 14

Table 3. Comparison of Rheological Parameters Computed by the NLP, GS, and NE Models (SIII Type) sample no. S18 S19 S20 S21

NLP GS NE NLP GS NE NLP GS NE NLP GS NE

b (Pa · s)

τ0 (Pa) 2.675 1.681 1.139 0 1.470 2.510 0 0 5.546 0 0 3.953

1.581 × 10-2 0.1891 0.7 0

k (Pa · sn)

n/n - m

RC2

SSE

source

0.2492 0.6496 0.9901 1.107 0.6234 -3.881 × 10-2 3.733 3.578 -0.1927 6.465 6.180 2.466

0.6607 0.5173 5.12 × 10-2 0.4427 0.5203 0.9326 0.3666 0.3739 0.8983 0.3633 0.3712 0.2147

0.9982 0.9950 0.9990 0.9965 0.9927 0.9981 0.9974 0.9975 0.9997 0.9981 0.9987 0.9989

0.7375 5.124 0.4223 1.325 2.224 0.7155 3.333 3.758 0.3331 7.022 8.489 4.093

18 14

of γ˙ n approaches infinity in the same region. Hence, the Herschel-Bulkley and power-law models are very sensitive to shear rate in calculating the yield stress and that would increase the errors in data fitting. Appendix A presents the details of this statement. It has been demonstrated that NE can lead to appropriate values for all of the rheological parameters with a high degree of confidence and for a wide range of data, i.e. 0-600 rpm.

19 14 19 14 19 14

6. Comparison of the Velocity Profiles The new equation (NE) is a complex function of shear rate in the form of series and thus its analytical solution is impossible. Hence one may try to solve this equation through numerical methods for velocity profile and pressure drop. The appropriate flow equations and methodology are summarized in Appendixes C and D, respectively; for a concentric annulus, eqs C.5 and C.6.

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Table 4. Comparison of Rheological Parameters Computed by the NLP, GS, and NE Models (T Type)21 sample no. T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T19 T20 T21 T27 T28 T29 T32 T33 T36 T37 T41 T42 T43 T44 T45 T49 T50 T51 T52 T53

NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE NLP NE

τ0 (Pa) 0 0.2512 2.579 × 10-3 0 0.2161 0.245 0 0 0 0 0.6767 8.875 × 10-2 1.86 1.866 0.3644 4.489 × 10-2 0.1663 0.1707 0.6046 0.4377 0.5608 0.6987 0 0 0 0 1.149 0.8708 2.235 1.988 0 0 0 0 2.251 2.327 2.286 × 10-2 0 0 0 0 0 1.996 2.198 2.112 2.43 0.1286 0.519 1.379 1.674 0 0.255 0.2906 0.2007 0.3524 0.141 2.778 × 10-2 0.1434 0.2098 0.1781 4.026 × 10-2 4.561 × 10-2 0.2901 0.3492 0.3313 0.1642 0.2188 0.2033 0.2731 0.2416

b (Pa · s) 9.931 × 10-2 6.199 × 10-4 7.643 × 10-2 8.259 × 10-4 0 1.222 × 10-3 8.831 × 10-4 1.074 × 10-3 1.926 × 10-2 6.86 × 10-3 3.704 × 10-2 5.777 × 10-4 2 × 10-5 1.074 × 10-3 1.9 × 10-3 6.785 × 10-4 1.298 × 10-3 1.551 × 10-3 7.822 × 10-4 1.099 × 10-3 0.1027 1.057 × 10-3 9.281 × 10-4 3.14 × 10-3 3.176 × 10-3 8.812 × 10-2 1.099 × 10-3 3.528 × 10-3 2.794 × 10-4 2.248 × 10-3 5.87 × 10-4 6.551 × 10-2 2.928 × 10-3 1.057 × 10-3 8.084 × 10-2

k (Pa · sn)

n/n - m

RC2

SSE

0.3796 -4.008 × 10-2 4.301 × 10-2 3.903 × 10-3 0.125 -2.971 × 10-2 0.9203 0.8669 0.9226 0.6667 0.2119 0.7524 6.524 × 10-3 2.81 × 10-3 5.398 × 10-3 0.2656 2.041 × 10-2 -3.568 × 10-3 5.389 × 10-2 0.1838 0.1345 -1.201 × 10-2 0.8804 0.7275 1.294 0.9746 5.253 × 10-2 0.2677 4.995 × 10-3 0.2051 0.1022 8.785 × 10-2 0.5628 0.5153 0.1875 0.1239 0.1016 0.1091 0.5508 0.4456 0.1736 -2.748 × 10-2 0.2947 0.147 0.4094 0.1812 0.7241 0.4159 0.5144 0.2943 0.3482 -3.31 × 10-2 5.977 × 10-2 0.111 5.184 × 10-2 0.158 0.1567 6.854 × 10-2 3.804 × 10-2 4.378 × 10-2 5.696 × 10-2 4.746 × 10-2 0.1209 -2.354 × 10-2 5.536 × 10-2 0.1374 0.1207 0.1194 7.715 × 10-2 -2.673 × 10-2

0.1707 0.8499 0.5208 0.185 0.345 0.8542 0.1373 -0.19053 0.228 -1.49 × 10-2 0.2985 -0.2162 0.9032 0.7 0.769 -0.23427 0.7893 0.9233 0.4265 -0.12063 0.3516 0.8777 0.1985 -8.55 × 10-2 0.1954 -5.58 × 10-2 0.4819 -0.13439 0.8619 -0.2256 0.4284 0.1119 0.2727 -5.22 × 10-2 0.4894 0.2202 0.4181 5.53 × 10-2 0.3093 2.07 × 10-2 0.5509 0.9 0.4397 0.228 0.4183 0.2299 0.424 0.182 0.4785 0.2392 0.2263 0.8596 0.5057 3.79 × 10-2 0.6618 6.91 × 10-2 0.3719 0.1814 0.6786 0.2687 0.4854 0.1688 0.4005 0.8641 0.6344 8.68 × 10-2 0.4451 9.68 × 10-2 0.5888 0.8719

0.8627 0.9874 0.9961 0.9965 0.9574 0.9931 0.9216 0.9878 0.9444 0.9915 0.9439 0.9921 0.9888 0.9892 0.9771 0.9935 0.9993 0.9993 0.9782 0.9952 0.9908 0.991 0.9471 0.9906 0.9032 0.9806 0.9776 0.9928 0.989 0.9911 0.9953 0.997 0.9896 0.9978 0.9938 0.9937 0.9944 0.9978 0.9919 0.9993 0.9902 0.9963 0.9991 0.999 0.9978 0.9974 0.999 0.9993 0.9991 0.9992 0.9787 0.9922 0.9931 0.9949 0.9974 0.9997 0.9993 0.9993 0.9983 0.9986 0.9967 0.9974 0.9886 0.9936 0.9973 0.9997 0.9972 0.9988 0.9984 0.9997

0.1276 1.175 × 10-2 1.002 × 10-2 8.999 × 10-3 7.405 × 10-2 1.192 × 10-2 0.1966 3.073 × 10-2 0.8685 0.1332 0.1375 1.944 × 10-2 0.1326 0.1281 2.971 × 10-2 8.476 × 10-3 1.583 × 10-2 1.604 × 10-2 2.318 × 10-2 5.078 × 10-3 1.976 × 10-2 1.924 × 10-2 0.4295 7.645 × 10-2 1.734 0.3467 5.043 × 10-2 1.61 × 10-2 4.305 × 10-2 3.506 × 10-2 1.819 × 10-2 1.179 × 10-2 0.1142 2.405 × 10-2 0.1942 0.1959 1.826 × 10-2 7.335 × 10-3 0.1569 1.302 × 10-2 0.6625 0.2468 3.283 × 10-2 3.675 × 10-2 0.1172 0.1391 0.1839 0.1324 0.1864 0.1696 4.191 × 10-2 1.53 × 10-2 2.772 × 10-2 1.489 × 10-2 6.953 × 10-2 7.59 × 10-3 2.583 × 10-3 2.565 × 10-3 3.048 × 10-2 2.56 × 10-2 9.056 × 10-3 7.157 × 10-3 4.11 × 10-2 2.324 × 10-2 5.617 × 10-2 5.613 × 10-3 1.945 × 10-2 7.982 × 10-3 3.432 × 10-2 5.634 × 10-3

Since equations are functions of pressure drop, one should know the value of pressure drop in the pipe and annulus to figure out the velocity profile. Assuming a laminar flow in the annulus is typically a logical assumption. In the present research, the velocity profile and pressure drop in the annulus have been calculated, and the results are compared with those from the GS technique.

The rheological parameters determined by NE, and by GS technique, have been utilized to determine the velocity profile and pressure drop in typical oil-well drilling situations. Two annulus geometries are used: (a) an annulus with an internal diameter equal to the outer pipe diameter of 0.311 m and an external diameter equal to the inner pipe diameter of 0.127 m


3379

Figure 2. Comparison of rheograms derived from experimental data and from rheological models (NE, GS, and NLP) for several samples. The rheological measurements were taken at 25 °C.

and (b) an annulus with an internal diameter equal to the outer pipe diameter of 0.216 m and an external diameter equal to the inner pipediameter of 0.089 m. As it can be realized, the ratio of R1/R2 > 0.3 is valid for both cases. According to the Enilari statement21 for the cases that the above criterion is hold, the

approximation of a slot for annulus geometry is a valid approximation. Results have been derived for a range of flow rates encountered in oil-well drilling situations, while keeping the flow regime laminar; which is the normal condition encountered in drilling applications, particularly for flow in an annulus.14

3380


Figure 3. Velocity profiles for several samples with rheological parameters by NE, NLP, and GS in the 0.216 m by 0.089 m concentric annulus (laminar flow).

Figure 4. Velocity profiles for several samples with rheological parameters by NE, NLP, and GS in the 0.311 m by 0.127 m concentric annulus (laminar flow).

The computed velocity profiles using NE and GS methods for the concentric annulus of 0.216 by 0.089 m at various flow rates

for samples S19, S20, and T41 are shown in Figure 3. As it is demonstrated in Appendix D, for an imposed pressure drop per


Figure 5. Pressure drop vs flow rate graph for several samples with rheological parameters determined by NE, NLP, and GS models in a 0.216 m by 0.089 m concentric annulus (laminar flow).

3381

Figure 6. Pressure drop vs flow rate graph for several samples with rheological parameters determined by NE, NLP, and GS models in a 0.311 m by 0.127 m concentric annulus (laminar flow).

3382


unit length (DP), values of the velocity profile and flow rate (Q) have been calculated. The variations of pressure drop vs flow rate, calculated through NE and GS methods for the flow in the same concentric annulus for samples S19, S20, and T41, have been presented in Figure 5. By comparing velocity profiles (Figure 3) and pressure drop profiles (Figure 5) for various samples, it was found that the obtained results for samples S19 and S21 from both methods are in a good agreement. Comparison of the results obtained for sample T41 by NE and NLP methods also show that there is relative agreement between both models. The computed velocity profiles using NE and GS methods for the concentric annulus of 0.311 m by 0.127 m at various flow rates for samples S19, S20, and T41 are shown in Figure 4. The variations of pressure drop vs flow rate, calculated through NE and GS methods for the flow in the same annulus for samples S19, S20, and T41 have been represented in Figure 6. By comparing velocity profiles (Figure 4) and pressure drop profiles (Figure 6) for various samples, it was found that the obtained results for samples S19 and S21 from both methods are in a good agreement. As mentioned earlier, sample T41 shows the largest variation in pressure drop values among samples. However, comparison of the results obtained for sample T41 by two models show that there is a relative agreement between both models.

but not necessarily infinity. However, the slope of the Herschel-Bulkley model approaches infinity in the same region. The larger slopes can create larger sum of square errors in the vicinity of zero. The derivative of the shear stress via the shear rate is as follows: dτ ) τ′ ) b + kγ˙ n-1[ng(γ˙ ) + γ˙ g'(γ˙ )] dγ˙

When γ˙ f 0, γ˙ n-1 goes to infinity at the zero point. Thus, to get a finite slope at the zero point

{

[ng(γ˙ ) + γ˙ g'(γ˙ )] ) 0 g(0) ) 0

(A.3)

Equation A.3 differs from an ordinary differential equation (ODE) because the function of g(γ˙ ) must satisfy not only eq A.3 but also eq A.1. Therefore, a specific method is proposed here to solve it. As mentioned earlier, function g(γ˙) should be found somehow to satisfy eqs A.1-A.3. Hence to find the proper function, one has to do a variable transforming by letting h(γ˙ ) ) eg(γ˙ )

(A.4)

∞

7. Conclusions Numerical methods using nonlinear regression which are used to calculate the rheological parameters may lead to nonoptimal solutions, due to the meaningless values sometimes obtained, i.e. negative yield stress in NE method. On the other hand, imposing a restriction for the positive value of the yield stress gives nonoptimal solutions (NLP method). GS technique, developed by Kelessidis et al. estimates better values than NL or NLP methods, and gives positive values for the yield stress, where numerical methods determine negative values. It was shown that the NE method, developed in this research, can describe adequately the rheological data of drilling fluids and overcomes the problem of infinite viscosity at zero shear rates. It was also demonstrated that the rheological parameters obtained through the NLP method and GS technique have a sum of square errors higher than those provided by the NE method. Furthermore, comparison of the obtained pressure drop and velocity profiles for laminar flow in pipes and a concentric annulus with those from the GS technique demonstrate the viability of the proposed model in predicting the flow behavior of drilling fluids.

(A.2)

eg(γ˙ ) ≡

∑ [g(γn!˙ )]

n

(A.5)

0

when γ˙ f 0: eg

≡ 1 + g(γ˙ )

(γ ˙)

(A.6)

After substitution of eqs A.4 and A.6 into eq A.3: lim ng(γ˙ ) + γ˙ g'(γ˙ ) ) n(h(γ˙ ) - 1)h(γ˙ ) + γf0

γ˙

dh(γ˙ ) ) 0 (A.7) dγ˙

Differential equation eq A.7 is solved with substitution h(γ˙ ) ) ∞ aiγ˙ i in eq A.7: ∑i)0 ∞

n[a0 +

∑

∞

aiγ˙ i - 1][a0 +

i)1

∑

∞

aiγ˙ i] +

i)1

∑ ia γ˙

i

i

)0

i)1

(A.8) when γ˙ f 0 ⇒ a0 ) 1 ∞

h(γ˙ ) ) 1 +

Acknowledgment

∑aγ

i

(A.9)

i

i)1

The research council of the Iran University of Science and Technology as well as the Iranian Ministry of Higher Education are highly acknowledged for their financial support.

After substitution of h(γ˙ ) into eq A.4, the function of g(γ˙ ) is obtained: ∞

g(γ˙ ) ) ln(1 +

Appendix A:Proof of the New Equation (NE) Assumptions for improving the Herschel-Bulkley model at very low shear rates are as follows: (1) The new equation is a nonlinear hybrid of the Binghamplastic and power-law models, eq A.1; τ ) [τ0 + bγ˙ ] + [kγ˙ ]g(γ˙ )

i

This function can satisfy eqs A.1-A.3 ∞

τ ) τ0 + bγ˙ + kγ˙ n ln(1 +

(A.1)

g(γ˙ ) is an arbitrary function which depends on shear rate so that g(0) ) 0. (2) In the vicinity of zero, the slope of the obtained curve from the new equation, eq A.2, would be a certain value,

(A.10)

i

i)1

so n

∑ a γ˙ ) ∑ a γ˙ ) i

i

(A.11)

i)1

0 e n e 1;

τ0 g 0;

bg0

Appendix B: Rheological Data Table B1 contains rheological data for the drilling fluids.


3383

21

Table B1. Rheological Data of Drilling Fluids sample number sample source

T1 a

BF

T4

T10

T41

T42

T43

BF + 2% KCl

BF + 0.5% KF

0.5% KCl + 1 g xanthan

0.5% KCl + 1 g xanthan + barite to 9 ppg

0.5% KCl + 1 g xanthan + barite to 14 ppg

shear rate (1/s)

shear stress (Pa)

1.701 3.402 5.11 10.21 17.02 34.05 51.07 102.14 170.23 340.46 510.69 1021.38 a

0.3 0.4 0.5 0.6 0.7 0.8 0.8 0.9 0.9 0.9 0.9 1.4

0.8 1 1.1 1.3 1.5 1.7 1.7 1.8 1.8 1.9 2 2.5

0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1.1 1.2 1.3 1.7

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 1.3 1.3 1.7

0.3 0.4 0.4 0.5 0.6 0.7 0.8 0.9 1 1.4 1.7 2.3

0.3 0.4 0.5 0.6 0.8 1 1.1 1.5 1.9 2.7 3.5 5.5

BF (base fluid): 15 lbm/bbl of bentonite + caustic soda.

Appendix C: Flow Pattern of Drilling Fluid in Concentric Annuli According to the New Equation

Appendix D: Calculation of Velocity Profile in the Annulus (0.089 × 0.216) for Sample S19

There is a central core of the fluid which moves as a rigid plug if the shear stress levels are smaller than the yield stress of the fluid, Figure C1. In this research since the criterion of R1/R2 > 0.3 holds, the annulus geometry was approximated as a slot.21 Integration of the force balance in a slot for the flow performed on a fluid element gives the following:14,22

Given data: External radius of the inner pipe of annulus R1 ) 0.089/2 m Internal radius of the outer pipe of annulus R2 ) 0.216/2 m Gap of the annulus H ) R2 - R1 ) 0.0635 m Drilling fluid density F ) 1500 kg/m3 Pressure drop per unit length (arbitrary assumption) DP ) 250 Pa/m Step 1: Determination of the rheological parameters for sample S19 using eq 6 or existing data from Table 3.

τ ) τ0 + y(DP) ya ) -

yb )

H DP 2 DP

τ0 -

H DP 2 DP du dy

H τ ) -τ0 - bγ˙ - kγ˙ n-m ln(1 + γ˙ ) ) - DP + 2 y(DP) 0 e y e ya τ ) τ0 + bγ˙ + kγ˙

H ln(1 + γ˙ ) ) - DP + 2 y(DP) yb e y e H

u ) umax

(C.3)

τ0 ) 2.510 Pa b ) 0.1891 Pa·s C ) -0.03881 Pa·s0.9326 n - m ) 0.9326

(C.4)

Step 2: Determination of both distances ya and yb according to Figure C1.

(C.2)

τ0 +

γ˙ ) -

n-m

(C.1)

ya e y e yb

ya )

τ0 H ) 0.0217 m 2 DP

yb )

τ0 H + ) 0.0418 m 2 DP

(C.5)

(C.6) (C.7)

Step 3: Determination of the shear rate, γw, and shear stress, τw, at the wall. At the wall, y ) 0; thus substituting in eq C.5 yields -2.510 - 0.1891γw + 0.03881γw0.9326 ln(1 + γw) ) 0.0635 250 + 0(250) 2 By solving the nonlinear equation through Matlab software: γw ) 91.06 1/s. By substituting the above result in eq C.5, the value of τw is achieved: τw ) 2.510 + 0.1891(91.06) - 0.03881(91.06)0.9326 ln(1 + 0.9326) ) 7.9375 Pa Step 4: Equation C.5 must be solved to find velocity (u) versus distance from the pipe wall (y) according to Figure C1.

Figure C1. Velocity profile of laminar flow in an annulus for a fluid with a yield stress.

γ)

du dy

3384


-2.510 - 0.1891

du du + 0.03881 dy dy

0.9326

( )

(

ln 1 +

du + dy

)

0.0635 250 - y(250) ) 0 2 Step 5 : Steps 1-4 are repeated for yb e y e H by using eq Table D.1. Results Achieved from Solving Equation C.5 for 0 e y e ya no.

y (m)

u (m/s)

no.

y (m)

u (m/s)

1 2 3 4 5 6 7 8 9 10 11

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.00000 0.08797 0.16989 0.24591 0.31621 0.38096 0.44035 0.49455 0.54375 0.58811 0.62783

12 13 14 15 16 17 18 19 20 21 22

0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.021 0.0217

0.66309 0.69409 0.72101 0.74403 0.76335 0.77926 0.79196 0.80159 0.80840 0.81277 0.81494

C.6. The obtained results are summarized in Table D.2. Table D.2. Results Achieved from Solving Equation C.5 for yb e y eH no.

y (m)

u (m/s)

no.

y (m)

u (m/s)

1 2 3 4 5 6 7 8 9 10 11

0.0418 0.0425 0.0445 0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0515 0.0525

0.81494 0.81277 0.80840 0.80159 0.79196 0.77926 0.76335 0.74403 0.72101 0.69409 0.66309

12 13 14 15 16 17 18 19 20 21 22

0.0535 0.0545 0.0555 0.0565 0.0575 0.0585 0.0595 0.0605 0.0615 0.0625 0.0635

0.62783 0.58811 0.54375 0.49455 0.44035 0.38096 0.31621 0.24591 0.16989 0.087971 0.00000

Step 6: From Tables D.1 and D.2, u ) umax ) 0.81494. Step 7: Determination of the average velocity in the annulus. j ) U

1

∫

AAnnulus 2 2 (R2 - R12)

A2

∫

R2

R1

∫

R2 1 u(2πr) dr) 2 R1 π(R2 - R1 ) R2-R1 2 ur dr ) u(R2 2 2 0 (R2 - R1 ) y) dy where r ) R2 - y

udA )

A1

2

∫

By calculating the above integral through a numerical method such as that of Simpson, the average velocity is determined. In j ) 0.64235 m/ this example, the Simpson method is applied: U s. Regarding the surface area of the annulus, annulus_area ) 0.030422 m2; the flow rate is obtained as Q ) 1172.5 L/min. Nomenclature AbbreViations HDD ) horizontal directional drilling GS ) golden section technique NE ) new equation NL ) nonlinear regression NLP ) nonlinear regression with penalty ODE ) ordinary differential equation SSE ) sum of square errors a ) coefficient in the range 0.048-0.2 in the critical settling equation ai ) coefficients in logarithmic argument in the NE (Ti) b ) NE model parameter (T) C′ ) consistency index, in the power-law model (M/(L T2-n))

d ) particle diameter (L) dP/dL, DP ) pressure drop per unit length (M/(L2 T2)) g ) gravity (L/T2) H ) gap of annulus (L) k ) parameter in the NE, consistency index in the Herschel-Bulkley model (M/(L T2-n)) m ) exponent in the NE n ) behavior index in power-law model/exponent in the NE P ) pressure (M/(L T2)) Q ) flow rate (L3/T) r ) radius (L) rp ) radius of plug flow in pipe (L) R1 ) external radius of the inner pipe of annulus (L) R2 ) internal radius of the outer pipe of annulus (L) RC2 ) correlation coefficient u ) velocity (L/T) ya ) distance of top of inner layer from bottom plate (L) yb ) distance of bottom of top layer from bottom plate (L) Greek Letters γ˙ ) shear rate (1/T) µ ) liquid viscosity (M/(L T)) µa ) apparent viscosity (M/(L T)) η ) plastic viscosity (M/(L T)) τw ) wall shear stress (M/(L T2)) F ) fluid density (M/L3) Fs ) particle density (M/L3) τ ) shear stress (M/(L T2)) τ0 ) yield stress (M/(L T2))

Literature Cited (1) Theory and Application of Drilling Fluid Hydraulics; Exlog Series of Petroleum Geology and Engineering Handbooks; Exlog Publications, 1985; pp 1-43. (2) Holland, F. A.; Bragg, R. Fluid Flow for Chemical Engineers, 2nd ed.; Arnold, Edward, A Division of Hodder Headline PLC: London, UK, 1995. (3) Tschirley, N. K.; Chilingarian, G. V.; Vorabutr, P. Testing of Drilling Fluids, Developments in Petroleum Science; Elsevier: New York, 1983; Vol. 11. (4) Bingham, E. C. Fluidity and Plasticity; McGraw-Hill: NewYork, 1922. (5) Lauzon, R. V.; Reid, K. I. G. New Rheological Model Offers Field Alternative. Oil Gas J. 1979, (May), 51. (6) Govier, G. W.; Aziz, K. The Flow of Complex Mixtures in Pipes; Krieger: Malabar, FL, 1972. (7) Bourgoyne, A. T.; Chenevert, M. E.; Millheim, K. K.; Young, F. S., Jr. Applied drilling engineering; SPE Textbook Series; SPE: Richardson, TX, 1991; Vol. 2. (8) Herschel, W. H.; Bulkley, R. Konsistenzmessungen von GummiBenzollosungen. Kolloid-Z. 1926, 39, 291–300. (9) Shulman, Z. P. On Phenomenological Generalization of Viscoplastic Rheostable Disperse System Flow CurVes; Teplo-Massoperenos: Minsk, 1968; Vol. 10. (10) Mnatsakanov, A. V.; Litvinov A. I.; Zadvornykh, V. N. Hydrodynamics of the Drilling in Deep, Thick, Abnormal Pressure Reservoirs. Paper SPE/IADC 21919, Presented at the Drilling Conference, Amsterdam, The Netherlands, 1991. (11) Maglione, R.; Ferrario, G.; Rrokaj, K.; Calderoni, A. A New Constitutive Law for the Rheological Behaviour of Non-Newtonian Fluids. Proceedings of the XII International Congress of Rheology, Quebec, Canada, 1996. (12) Bird, R. B.; Dai, G. C.; Yarusso, B. Y. The Rheology and Flow of Viscoplastic Materials. ReV. Chem. Eng. 1982, 1 (1), 1–70. (13) Maglione, R.; Guarneri, A.; Ferrari, G. Rheologic and hydraulic parameter integration improves drilling operations. Oil Gas J. 1999, 97, 44–48. (14) Kelessidis, V. C.; Maglione, R.; Tsamantaki, C.; Aspirtakis, Y. Optimal Determination of Rheological Parameters for Herschel-Bulkley Drilling Fluids and Impact on Pressure Drop, Velocity Profiles and Penetration Rates During Drilling. J. Pet. Sci. Eng. 2006, 53, 203–224.

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010 (15) Talmon, A. M.; Huisman, M. Fall Velocity of Particles in Shear Flow of Drilling Fluids. J. Tunnell. Underground Space Tech. (Incorp. Trenchless Technol. Res.) 2005, 20 (2), 175. (16) Mihalakis, A.; Makri, P.; Kelessidis, V. S.; Christidis, G.; Foscolos, A.; Papanikolaou, K. Improving rheological and filtration properties of drilling muds with addition of Greek lignite. Proceedings of the 7th National Congress on Mechanics, Chania, Greece, 2004. (17) Kelessidis, V. C.; Mihalakis, A.; Tsamantaki, C. Rheology and Rheological Parameter Determination of Bentonite-Water and BentoniteLignite-Water Mixtures at Low and High Temperatures. Proceedings of the 7th World Congress of Chemical Engineering, Glasgow, UK, 2005. (18) Merlo, A.; Maglione, R.; Piatti, C. An Innovative Model for Drilling Fluid Hydraulics. Paper SPE 29259, Presented at the Asian-Pacific Oil and Gas Conference, Kuala Lumpur, Malaysia, 1995.

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(19) Blick, E. F. Non-Newtonian Fluid Mechanics Notes; Engineering Library, University of Oklahoma: Norman, OK, 1992. (20) Al-Zahrani, S. M. A generalized rheological model for shear thinning fluids. J. Pet. Sci. Eng. 1997, 17, 211–215. (21) Enilari, M. G. Development and Evaluation of Various Drilling Fluids for Slim-Hole Wells. M.Sc. Thesis, University of Oklahama, Norman, OK, 2005. (22) Bourgoyne, A. T., Jr.; Millheim, K. K.; Chenevert, M. E.; Young, F. S., Jr. Applied Drilling Engineering. SPE Textbook Ser. 1991, 2, 41– 183.

ReceiVed for reView October 31, 2008 ReVised manuscript receiVed February 1, 2010 Accepted February 05, 2010 IE9009233

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