Flow and thixotropy of non-contaminating oil drilling fluids formulated with bentonite and sodium carboxymethyl cellulose

Journal of Petroleum Science and Engineering 57 (2007) 294 – 302 www.elsevier.com/locate/petrol

Flow and thixotropy of non-contaminating oil drilling fluids formulated with bentonite and sodium carboxymethyl cellulose M. Dolz a,⁎, J. Jiménez b , M. Jesús Hernández a , J. Delegido a , A. Casanovas a a

Departamento de Física de la Tierra y Termodinámica, Grupo de Reología, Universidad de Valencia, 46100 Burjassot, Spain b Escuela de Ciencias, Unipaz, C/49 n° 10-22, Barrancabermeja, Colombia Received 13 September 2005; received in revised form 23 October 2006; accepted 24 October 2006

Abstract A study has been made of low-contaminating oil drilling mud in water base, composed of bentonite, at concentrations in the range of 6–12% (w/w), and sodium carboxymethyl cellulose at two different concentrations. Flow analysis yielded an empirical formula indicating shear stress as a function of the formulation concentrations of bentonite and sodium carboxymethyl cellulose, stirring time and shear rate. To study the thixotropy of the different formulations the cycles' method has been used. Calculations were made of the thixotropic areas, obtaining a semi-empirical equation for these areas. The results are analyzed on the base of relative thixotropic area and total relative thixotropy — the latter being defined by the authors in this study. © 2006 Elsevier B.V. All rights reserved. Keywords: Drilling mud; Bentonite; Carboxymethyl cellulose; Thixotropy; Rheology

1. Introduction Drilling muds are often described as thixotropic shear-thinning fluids with a yield stress (Coussot et al., 2004). Water-based drilling muds including bentonite are long known in the petroleum industry. The thixotropic behavior of bentonite muds (Barnes, 1997) is a convenient property in the drilling process: in effect, when drilling mud is at rest, it is required to act as a detritus-suspending solid, while moving drilling mud is required to behave as a viscous fluid capable of trans⁎ Corresponding author. Tel.: +34 963544350; fax: +34 963544813. E-mail addresses: [email protected] (M. Dolz), [email protected] (J. Jiménez), [email protected] (M.J. Hernández), [email protected] (J. Delegido), [email protected] (A. Casanovas). 0920-4105/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2006.10.008

porting such detritus. Drilling muds normally act as nonNewtonian fluid with thixotropic behavior. For industrial application, thixotropy means a reversible change from a high viscosity gel state at rest to a lower viscosity sol state activated by shear stress. During this process, the microstructure of the material is reversibly destroyed, i.e., the microstructure is recovered at rest, though at a much slower rate (Barnes, 1997). At present, the desired rheological properties are obtained with less bentonite, thanks to the addition of polymers. This is because high bentonite concentrations give rise to undesirable effects (Mahto and Sharma, 2004). Extended reach and inclined wells place great demands on drilling fluids (Caenn and Chillingar, 1996). The rheological behavior of drilling muds can be optimized with polymers of different chain lengths and properties.

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There are many kinds of drilling mud, and some are strongly contaminating (e.g., those formulated with hydrocarbons, oils, etc.). The need for sustainable development in oilfields has made it necessary to introduce drilling fluids of very low or negligible contaminating potential. These fluids are generally composed of mud with a water base. Water- and oilbased drilling fluids clean similarly under equivalent rheological and flow-velocity profiles (Hemphill and Larsen, 1996). In the formulation of such mud it is becoming increasingly common to use biopolymers as additives, due to their low cost, rheological properties and scant environmental impact. Some examples of these polymers are guar gum (Pérez et al., 2004), tamarind gum, and polyanionic cellulose (Mahto and Sharma, 2004) or sodium carboxymethyl cellulose (NaCMC) (Caenn and Chillingar, 1996; Zhang et al., 1999a,b with modified NaCMC). In relation to water control in producing wells, water-soluble polymers can reduce water permeability much more than hydrocarbon permeability (Ali and Barrufet, 2001). The present study of the flow characteristics of waterbased, bentonite drilling fluids has made use of a water based preparation of different proportions of bentonite and sodium carboxymethyl cellulose as additive, in view of its easy formulation characteristics, widespread use and non-contaminating nature. Bentonite offer properties that have led to their widespread use for a range of purposes. One of their most important industrial applications since 1929 is in drilling mud as a suspension and gelifying agent. This application is based on the physical-chemical properties of bentonite, particularly their extremely small particle size (b 2 μm), their laminar morphology (phyllosilicates), the generation of laminar charges, and the presence of weakly bound cations within the interlaminar space. As a result of these characteristics, bentonites have an active surface value, with non-saturated bonds. They are therefore able to interact with different substances, particularly with polar compounds – exhibiting plastic behavior in clay–water mixtures with a high solid/liquid proportion – and are able to swell and develop rheological properties, in water suspensions. Thus, when added to drilling mud, bentonites increase the well-cleaning properties, reduce filtrations towards permeable formations, improve circulation and increase the stability of the drill hole. Drilling fluids with high concentrations of bentonite may have negative effects, such as excessive friction and increased torque — as a result of which low concentrations of bentonite must be used in the formulation

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of such fluids. Therefore some polymer acting as a thickener must be added to bentonite (Mahto and Sharma, 2004) in order to achieve the desired drilling effects. A dozen of patents on water-based, bentonite drilling muds include sodium carboxymethyl cellulose (NaCMC) (the most recent are those of Zozulya et al., 2001; Fedoseev et al., 2002a,b; Yoshihisa and Shinya, 2002), which is the sodium salt of partially Ocarboxymethylated cellulose obtained in alkaline medium by adding sodium monochloroacetate to cellulose. HV (high viscosity) and LV (low viscosity) classes are manufactured. The use of one class or the other in turn depends on the desired viscosity and filtration reduction properties (Hughes et al., 1993). At high concentrations, carboxymethyl cellulose (CMC) solutions show viscoelastic in addition to thixotropic behavior (Edali et al., 2001). NaCMC is used in industry as a thickener; for stabilizing suspensions and emulsions; for the formation of gels; and also as a modifier of the flow characteristics of water solutions or suspensions (Feddersen and Thorp, 1993). It is moreover also employed as a disaggregant, binder or agglutinant, and as a coating agent. In the oil industry NaCMC is mainly used in drilling wells. When applied to drilling muds, CMC increases mud viscosity and reduces fluid losses (filtration reduction). Lastly, it should be remembered that an adequately formulated fluid can reduce the total cost of drilling by 5–15% (Bloys et al., 1994). Low-contaminating drilling mud in water base composed of bentonite (100 ASTM sieve grade) at different concentrations, and LV sodium carboxymethyl cellulose at two different concentrations are analyzed in the present study for rheological properties and evaluated for thixotropic behavior. 2. Determining thixotropy The study of the thixotropy of drilling muds comprises determination of the flow curve, which U U relates shear stress, σ, to shear rate, γ, σ = f(γ), with a CS rheometer for the different formulations considered, and evaluation of their thixotropic behavior. CS instruments control the shear, whereas CR instruments control the rate. The reason for using a CS instrument is the great heterogeneity of the shear rate distribution of thixotropic materials (Herzhaft et al., 2003; Roussel et al., 2004). Some discrepancies between the field and laboratory results (Maglione et al., 2000) can be attributed to the use of a CR instrument for obtaining thixotropic parameters.

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The experimental method we used is quite similar to that reported by Wang et al. (2006) for measuring the rheological properties of Zaoyuan heavy fuel. This fuel is shear-thinning with a yield point, and shows a weak thixotropic behavior at room temperature (Wang et al., 2006). Among the different procedures used to evaluate thixotropy from the flow curve, mention should be made of the hysteresis cycles (Cheng and Evans, 1965; Mewis, 1979; Dolz et al., 2000) that consists on determining the areas enclosed between one (or more) upcurve rheogram(s), for increasing shear rates, and different down-curve rheograms, for decreasing shear rates, after several different stirring times. These areas are referred to as thixotropic areas. It is accepted that increasing thixotropic area is associated to increased thixotropy, though this is questionable in certain cases. Therefore, to comparatively evaluate the thixotropy of different gels, the use of a magnitude referred to as “relative thixotropic area” has been proposed (Dolz et al., 1995), relating the thixotropic areas to the total area under the up-curve for each formulation. An additional difficulty in the experimental study of thixotropic formulations is to determine the Smax area, which is defined as the area under the theoretical upU U curve between γmin and γmax, extrapolated to a stirring time t = 0, and of the area Smin of the theoretical downcurve, corresponding to infinite stirring time. The present study proposes a method to obtain Smax and Smin. Firstly, a regression curve is used to determine the function S = S(t), where S is the area enclosed by U each of the curves obtained in a flow diagram σ = f(γ) for the different stirring times. The function S = S(t) must satisfy specific limiting conditions, that is: lim SðtÞ ¼ Smax tY0

lim SðtÞ ¼ Smin

ð1Þ

tYl

In effect, for t= 0 the enclosed area S is the maximum area possible, since it corresponds to a non-stirred formulation. In contrast, for very long stirring times, S would tend towards a limiting minimum value corresponding to a formulation with the maximum structural breakdown possible for the type of stirring involved. The areas under the rheograms after different stirring times are fitted to (Dolz et al., 1997): SðtÞ ¼ Smax  ðSmax  Smin Þð1  ef ðtÞ Þ

ð2Þ

where the function f(t) depends on the type of thixotropic system studied, and should satisfy the limiting conditions specified in Eq. (1) for S(t). Knowing the function S(t), the thixotropic area, ST, is given by: ST ðtÞ ¼ Smax  SðtÞ

ð3Þ

or, taking into account Eq. (2) ST ðtÞ ¼ ðSmax  Smin Þð1  ef ðtÞ Þ

ð4Þ

Eq. (4) provides the thixotropic area of each formulation as a function of the stirring time. Nevertheless, knowledge of the absolute thixotropic areas does not suffice to compare formulations with very different viscosities. In fact, some thixotropic formulations with very high viscosities may present a down-curve that differs relatively little from the upcurve, though the enclosed surface is very large, due to the high shear stresses obtained in these formulations. Likewise, in less viscous formulations, this area may be very small, even though the up- and downrheograms are very different. Structural breakdown of the more viscous formulations may be relatively smaller than in less viscous formulations, even though the absolute thixotropic area is much larger in the former. A proposal to obtain relative values of the thixotropic areas has been made in (Dolz et al., 1995): SR ðtÞ ¼ 100ðST ðtÞ=Smax Þ

ð5Þ

which allows to evaluate the percentage of changed area by stirring, versus the total area under the up-rheogram. Thus, a formulation is considered to be increasingly thixotropic the larger the value of SR, allowing comparison on a more standardized base. Besides, to simplify the understanding of the thixotropic behavior of different formulations, we propose also to obtain a parameter that specifies relative thixotropy independently of t. We refer to this parameter, based on Smax and Smin, as the “total relative thixotropy”, which is defined as: ðTR Þtotal ð%Þ ¼

  Smax  Smin 100 Smax

ð6Þ

calculated as the limit cases of Eq. (2). The advantage of using Eq. (6) is that it simplifies the calculations for obtaining thixotropy and – more importantly – such thixotropy is expressed by a single adimensional numerical value.

M. Dolz et al. / Journal of Petroleum Science and Engineering 57 (2007) 294–302

3. Experimental A total of 10 formulations of water base drilling mud with concentrations of bentonite (Acofarma® batch 010036 R-7, Spain, 100 ASTM sieve grade) of 6, 8, 10, 11 and 12% (w/w) were prepared (henceforth referred to as 6B, 8B, 10B, 11B and 12B respectively). To each concentration of bentonite we added carboxymethyl cellulose (LV — Lab D'Hénio), in the ratio 4.8 g or 7.2 g of NaCMC per 100 g of bentonite (henceforth referred to as 1C and 2C, respectively). These compositions are similar to the oil drilling mud standards of the company Petrobras. A total of 100 g of each mud was prepared as follows: bentonite was slowly added to deionized water under stirring conditions, to avoid the formation of aggregates and ensure homogeneous dispersion. Stirring was carried out with a Heidolph RZR 202® electric stirrer. Following bentonite dispersion, the corresponding amount of NaCMC was slowly added, and the whole system was stirred at 760 rpm for 10 min. The mud thus formulated was left to stand in the dark for three days at a temperature of about 24 °C, followed by rheometer evaluation (RheoStress 1®, Thermo Haake). The sensor system consisted of coaxial cylinders (Haake Z-34 DIN 53019 series 1, cylinder diameter 34 mm, and internal sleeve diameter 36.96 mm). The different tests were performed using Haake RheoWin® software, with its programs Pro Job Manager for measurement and Pro Data Manager for evaluation.

After loading the sample in the receptor cylinder, the formulation was allowed to stand for 2 h, after which 8 up- and down-shear rate cycles were carried out. The complete initial cycle was taken to represent the prestirring cycle. Measurements were made of shear U stresses, σ, at shear rates, γ, between 1 and 100 s− 1 in logarithmic distribution, leaving a 10 s waiting period between each measurement. Therefore the first (pre-stirring) cycle was carried out, obtaining the corresponding up-curve and starting with a shear rate of 1 s− 1, followed by increments up to 100 s− 1. After another 10 s, the rheometer measured the next point, corresponding to the start of down-curve B1 at a rate of 100 s− 1. The rate was then gradually reduced until 1 s− 1 was again reached. Then, after another 10 s, the second up-curve was started, followed by the next down-curve, and so on successively. 4. Results and discussion For the rheological characterization of drilling fluids we have obtained the rheograms of the shear stress as a U function of the shear rate, σ = f(γ), corresponding to all the samples studied. As an example, Fig. 1 shows the rheograms obtained in the analysis of drilling fluid 6B2C, formulated with 6% bentonite and the largest concentration of CMC. The figure presents the uprheogram, S, and all the down-curves corresponding to 7 stirring times of between 3.4 and 51.5 min. The rheogram of the other formulations studied are similar to those shown in Fig. 1, and in all of them we found a yield stress and a distribution of experimental points that agree – with good approximation – to the rheological model proposed by Herschel and Bulkley (1926): r ¼ ro þ K dgn

U

Fig. 1. Shear stress (σ) as a function of shear rate (γ) for sample 6B2C. ● up-curve S, ■ down-curve B1, ♦ down-curve B2, ▲ down-curve B3, ○ down-curve B4, □ down-curve B5, △ down-curve B6, down-curve B7.

×

297

ð7Þ

where σo is the yield stress (or minimum shear stress required for the system to begin to flow), K is known as the consistency of the formulation, and n is the power index that determines the shear-thinning or shearthickening character of the sample. Therefore, the experimental values corresponding to shear stress σ, obtained for all the samples have been U fitted, as a function of shear rate γ, by means of the Herschel–Bulkley model via the least squares method using the KaleidaGraph® program — in all cases yielding correlation coefficients, r2, in excess of 0.997. We found the yield stress σo to be very similar in the up-curve and in all the down-curves in the fitted rheograms of each formulation. Therefore, σo was

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fixed to the mean value of all of them, and the fitting was repeated. The results obtained show that the mean yield stress increases as the concentration C of bentonite increases and is practically independent of the concentration of CMC. Therefore we averaged as well the values of σo for the two concentrations of CMC of each formulation. The plot of the yield stress σo, as a function of the concentration of bentonite, is shown in Fig. 2. The yield stress, fitted by least squares to a fourth power function of the bentonite concentration C, yields the following expression: ro ¼ ð0:0058F0:0001ÞC 4

r ¼ 0:998

ð8Þ

where σo units are Pa, and the concentration of bentonite C in % (w/w). From Eq. (8) it is concluded that in the drilling muds studied, yield stress increases exponentially with the concentration of bentonite and, in practice, is not dependent upon the concentration of CMC, within the concentration range considered. On the other hand, in the Herschel–Bulkley model fittings we found that consistency K is different for the different rheograms of one same formulation — and this applies to all the formulations studied. The physical parameter that changes from one curve to another is the time during which the formulation is subjected to stirring. This stirring time, t, is the time considered to represent the total time of sample exposure to shear stress, which for each of the rheograms is about 6.67 min greater than that corresponding to the previous rheogram.

Fig. 3. Consistency (K ) as a function of stirring time for drilling mud. ▲ 12B2C, △ 12B1C, ● 6B2C and ○ 6B1C.

Fig. 3 shows the dependency of consistency K upon stirring time for the two extreme concentrations of bentonite (6% and 12%) and the two concentrations of CMC as an example of K = f(t). As can be seen in Fig. 3 (and as has been confirmed in all the rest of cases), the consistency K of the formulations increases both with the concentration of bentonite and with the concentration of CMC, and progressively decreases with the stirring time. In all cases this decrease is a logarithmic dependence of the following type: K ¼ D  ElnðtÞ

ð9Þ

where D and E are two parameters that depend on the concentrations of bentonite and CMC. Least squares fitting of K = f (t) by means of Eq. (9), yields the values of D and E shown in Table 1. Table 1 Values of D and E of Eq. (9), for the formulations indicated.

Fig. 2. Yield stress (σo) of the up- and down-curves as a function of the concentration of bentonite C (%).

Concentration of bentonite C% (w/w)

Concentration D (Pa.sn) of CMC

E (Pa.sn)

6 6 8 8 10 10 11 11 12 12

1C 2C 1C 2C 1C 2C 1C 2C 1C 2C

0.076 ± 0.002 0.158 ± 0.004 0.087 ± 0.024 0.255 ± 0.025 0.13 ± 0.04 0.62 ± 0.04 0.21 ± 0.02 0.79 ± 0.05 0.57 ± 0.07 0.92 ± 0.12

0.410 ± 0.006 0.87 ± 0.01 0.81 ± 0.08 1.31 ± 0.08 1.55 ± 0.13 3.27 ± 0.13 2.24 ± 0.07 5.25 ± 0.15 3.2 ± 0.2 6.5 ± 0.4

M. Dolz et al. / Journal of Petroleum Science and Engineering 57 (2007) 294–302 Table 2 Values of exponent n of the up- and down-curves in Eq. (7), for the different drilling mud formulations studied Bentonite (% w/w) Concentration of CMC nup

n̄dw

6 6 8 8 10 10 11 11 12 12

0.87 ± 0.02 0.78 ± 0.02 0.81 ± 0.01 0.85 ± 0.02 0.81 ± 0.01 0.79 ± 0.01 0.84 ± 0.01 0.75 ± 0.01 0.86 ± 0.01 0.74 ± 0.02

1C 2C 1C 2C 1C 2C 1C 2C 1C 2C

0.46 ± 0.01 0.47 ± 0.01 0.40 ± 0.01 0.40 ± 0.01 0.50 ± 0.02 0.48 ± 0.01 0.79 ± 0.02 0.68 ± 0.01 0.83 ± 0.03 0.67 ± 0.01

The table shows that for the same concentration of CMC, the parameter D increases on increasing the concentration of bentonite, and that for the same concentration of bentonite, D increases on increasing the CMC concentration. Likewise, the parameter E, which determines the time decrease of consistency K, and therefore the thixotropic character of the mud, increases with the concentration of bentonite and with the proportion of CMC. This increase in E seems to point out that increased concentration formulations present the greatest thixotropic behavior. This conclusion will be further discussed below, on conducting a detailed study of thixotropy with the cycles' method described in the Introduction. Lastly, regarding the power index n of the fitted Herschel–Bulkley models of all the down-curves corresponding to each of the formulations, we found the value to be similar in all cases. Therefore we averaged all these values (n¯dw) and repeated the fittings of the downrheograms, keeping the former mean value fixed. The power index of the corresponding up-curve, nup, was not included in the averaging process since it was clearly different from that of the corresponding downcurves. The results of nup and n̄dw obtained for all the mud formulations are included in Table 2. The data contained in Table 2 show that the values of n̄dw corresponding to the down-curves are very similar, and practically independent of the concentration of bentonite. A mean of all these values yields the adimensional value n̄dw = 0.81 ± 0.03. However, on comparing the values corresponding to the two CMC concentrations, an appreciable difference in the mean values is observed for 1C and 2C formulations:

299

Since n̄dw 1C is greater than n̄dw 2C, and both are less than one (n = 1 is the value corresponding to a Newtonian fluid), it can be concluded that the rheological behavior of this type of formulations after stirring is shear-thinning, and that the more shearthinning formulations correspond to those with the highest concentration of CMC. Regarding the exponents n of the up-curves, two groups can be determined. For the first group, comprising mud formulated with the lowest bentonite concentrations (6, 8 and 10%), very similar power indices n are obtained that, at same time, are smaller than in the rest of the formulations and practically independent of the concentration of CMC. Thus, it can be concluded that these formulations, prior to stirring, exhibit the most shear-thinning behavior. In the second group of muds formulated with 11% and 12% bentonite, the up-curves show n values similar to those of the corresponding down-curves. In both cases we observe a decrease in the power index n, and thus more accentuated shear-thinning behavior, with increasing CMC concentration. As a summary of the former analysis, it can be concluded that drilling fluids composed of bentonite and CMC in water base show a rheological behavior that is shear-thinning and that is well described by the equation: r ¼ ð0:0058F0:0001ÞC 4 þ ðD  ElntÞdgn

ð11Þ

where C is the concentration of bentonite in % (w/w), D and E are the parameters shown in Table 1, t is the

n dw 1C ¼ 0:84F0:01 ¯ n dw 2C ¼ 0:78F0:01 ¯

ð10Þ

Fig. 4. Area under the curves as a function of stirring time for all the mud formulations. ▲ 12B2C, △ 12B1C, ■ 11B2C, □ 11B1C, ▼ 10B2C, ▽ 10B1C, ♦ 8B2C, 8B1C, ● 6B2C, and ○ 6B1C.

◇

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M. Dolz et al. / Journal of Petroleum Science and Engineering 57 (2007) 294–302 Table 3 Values of parameter P in Eq. (14) for the areas and concentrations of CMC indicated

Smax Smin

1C

2C

11.6 ± 0.3 0.72 ± 0.02

13.7 ± 0.3 0.83 ± 0.03

The fitted values of the parameter M are very similar, as a result of which it can be substituted with their mean value, in s− 1 / 2: M ¼ ð0:56F0:08Þ

Fig. 5. Values of Smax and Smin as a function of the concentration of bentonite, C, for the two concentrations of CMC. ▲ Smax for 2C, ● Smax for 1C, △ Smin for 2C and ○ Smin for 1C.

stirring time in min, and n is the power index, reported in Table 2. The determination and analysis of the thixotropic behavior is the second aspect of interest in the rheological study of drilling muds. To this point we used the cycles' method via numerical integration of the area enclosed U under the up- and down-rheograms between γmin = 1 s− 1 U −1 and γmax = 100 s . These areas have been plotted against the stirring time t in Fig. 4, where it can be seen that as stirring is prolonged, the area under the corresponding rheogram decreases. It is also seen that on increasing the concentration of bentonite, the values of the areas increase considerably — a phenomenon that is also observed on increasing the concentration of CMC. This can be appreciated by comparing the values of any concentration of bentonite formulated with two concentrations of CMC. The values of the area S = S(t) obtained have been fitted for each formulation by means of Eq. (2), where f(t) in Eq. (2) is a function of the stirring time to be determined, and Smax and Smin are the areas corresponding to stirring times ta = 0 and ta = ∞, respectively, as said in the Introduction. The Fig. 4 continuous lines are those obtained by least squares fitting. For the f(t) function in Eq. (2), we have used the relation: f ðtÞ ¼ Mt 1=2

ð13Þ

for practical purposes. On the other hand, Fig. 5 shows Smax and Smin plotted against the bentonite concentration for the two concentrations of CMC and for all the formulations. The plotted lines in Fig. 5 are least squares fittings of equations of the form: Smax ; Smin ¼ PC m

ð14Þ

where P is a fitting parameter and the exponent values are found to be m = 3 in the case of the maximum areas of all the formulations, and m = 4 for the minimum areas. The values of P, obtained from fitting with Eq. (14), are shown in Table 3. From these results we obtain the thixotropic areas, ST, corresponding to all the formulations, as a function of the concentrations of bentonite and CMC, and of the stirring time.

ð12Þ

already proposed earlier by us (Dolz et al., 2000). By using Eq. (12) the correlation coefficients of all the fits were found to be N 0.998.

Fig. 6. Thixotropic area as a function of stirring time for the two extreme concentrations and an intermediate concentration of bentonite, and the two concentrations of CMC. ▲ 12B2C, △ 12B1C, ♦ 10B2C, ◇ 10B1C, ● 6B2C and ○ 6B1C.

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from Eq. (4). However, since the latter did not consider the thixotropic relative values, the conclusions drawn from it are similar to those obtained on comparing the absolute thixotropic areas – i.e., the more viscous – appear to be the most thixotropic formulations. However, in relative values, which we feel are the more representative, the most thixotropic formulation corresponds to the least structured formulation \ i.e., that with the lowest concentration. On the other hand, and to simplify the understanding of the thixotropic behavior of the different formulations, we have proposed a parameter referred to as the “total relative thixotropy”, defined by Eq. (6). The total relative thixotropy as a function of bentonite concentration, C, based on the values of Smax and Smin (Eq. (14) and Table 3), substituted in Eq. (6) yields for the lowest CMC concentration Fig. 7. Relative thixotropic areas as a function of stirring time for all the formulations. ▼ 12B2C, ▽ 12B1C, ♦ 11B2C, 11B1C, ▲ 10B2C, △ 10B1C, ■ 8B2C, □ 8B1C, ● 6B2C and ○ 6B1C.

◇

If these areas are taken to be the areas enclosed by the up-curve (theoretically Smax) and each of the downcurves after a stirring time t, the resulting thixotropic area, ST, can be obtained from Eq. (4). So, Fig. 6 shows the thixotropic areas as a function of the stirring time, ST = f(t), for the formulations with bentonite concentrations of 6, 10 and 12%, and with the two CMC concentrations studied. As can be expected, the larger thixotropic areas correspond to more viscous formulations, i.e., those in which the concentrations of bentonite and CMC are greater. However, we consider that the comparison of the thixotropy of a formulation should not be made based on the absolute thixotropic areas as has been mentioned in the Introduction. Rather, the concept of relative thixotropic area, defined by Eq. (5), will be applied by us. The relative thixotropic areas of all the formulations as a function of the stirring time have been plotted in Fig. 7. As can be seen in the figure, the greatest relative thixotropic behavior corresponds to the least structured formulation, i.e., that with the lowest concentration of bentonite, while the least thixotropic formulation corresponds to the most structured formulation — i.e., that containing 12% bentonite. The increase in CMC concentration produces also an increase in the relative thixotropy of the formulation, as can be seen in Fig. 7, regardless of the bentonite concentration involved. While the conclusions drawn from the analysis of the relative thixotropy based on the cycles' method appear consistent, they are in contrast to the results deduced

 ðTR Þtotal−1C ð%Þ ¼

 11:6C 3  0:72C 4 100 11:6C 3

ð15Þ

¼ 100  6:21C and for the greatest CMC concentration:  ðTR Þtotal−2C ð%Þ ¼

 13:7C 3  0:83C 4 100 13:7C 3

ð16Þ

¼ 100  6:05C Considering the small differences between the two equations, the mean value could be taken in practice for both CMC concentrations. Thus, we propose for the analysis of the total thixotropy of these drilling muds as a function of bentonite concentration the following equation: ðTR Þtotal ð%Þ ¼ 100  6:013C

ð17Þ

Therefore the total relative thixotropy decreases linearly with the concentration of bentonite in the formulation, within the range of concentrations considered in the present study. 5. Conclusions The drilling mud formulations studied, involving bentonite and sodium carboxymethyl cellulose at different concentrations in water, exhibit shear-thinning behavior. An empirical equation has been obtained that allows the determination of shear stress – and thus of mud viscosity – as a function of the concentrations of the thickeners used, the shear rate, and the formulation stirring time. The hysteresis cycles method has been used to

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