The general rheological modeling process involves two steps: one is to fit dial reading(or shear stress) and shear rate values to specific rheological equations(e.g., Yield Power Law and Bingham Plastic models); the other is to determine the effect of temperature and pressure on dial readings and define correction factors to correlate rheological parameters in down-hole conditions. The commonly reported correction factor is shown as follows:

(1)

where A, B, and C are characteristic constants. Following Eq.(1), McMordie et al. [6W.C. McMordie, R.B. Bennett, and R.G. Bland, "The effect of temperature and pressure on the viscosity of oil-base muds", J. Pet. Technol., pp. 884-886.
[http://dx.doi.org/10.2118/4974-PA] ] modified the power-law model as,

(2)

where K’ and n correspond to the consistency index and behavior index, respectively. This HTHP rheological model has been successfully used in the oil-based drilling fluid. Similarly, Houwen [7O.H. Houwen, and T. Geehan, "Rheology of oil-based muds", SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, .
[http://dx.doi.org/10.2118/15416-MS] ], Hiller [8K.H. Hiller, "Rheological measurement on clay suspensions and drilling fluids at high temperature and pressures", J. Pet. Technol., vol. 15, no. 7, pp. 779-788.
[http://dx.doi.org/10.2118/489-PA] ], and Alderman [9N.J. Alderman, A. Gavignet, D. Guillot, and G.C. Maitland, "High-temperature, high-pressure rheology of water-based muds", SPE Annual Technical Conference and Exhibition, .Houston, Texas
[http://dx.doi.org/10.2118/18035-MS] ] have applied the MF method to the two-parameter Casson and Bingham plastic models, as well as three-parameter Hurschel-Bulkley and Robertson-Stiff models. On the basis of modeling procedure and results, they concluded that the correlation of parameters in base equations can disturb the final predictive accuracy.

To diminish the correlation of parameters, Rommetveit et al. [10R. Rommetveit, and K.S. Bjorkevoll, Temperature and pressure effects on drilling fluid rheology and ECD in very deep wells., SPE/IADC Middle East Drilling Technology Conference: Bahrain, . [https://doi.org/10.2118/39282-MS]
[http://dx.doi.org/10.2118/39282-MS] ] proposed another way to construct base equation, which is written as,

(3)

where f(P, T, γ) is a factor that multiplies shear stress obtained at considered conditions. The constant functions g₁(P, T) and g₂(P, T) are determined by an optimistic fit to measured data. In this case, the intrinsic defects of the selected base models can be avoided, which makes prediction more accurate.

American Petroleum Institute(API) recommends the HTHP rheological model:

(4)

where τ_o denotes the shear stress at a given rotational speed. This model is extended to viscosity expressions [11J.V. Fisk, and D.E. Jamlson, "Physical properties of drilling fluids at high temperatures and pressures", SPE Drill. Eng., vol. 341, p. 346., 12R.C. Minton, and P.A. Bern, Field measurement and analysis of circulating system pressure drops with low-toxicity oil-based drilling fluids., SPE/IADC Drilling Conference: Dallas, Texas, . [https://doi.org/10.2118/17242-MS]
[http://dx.doi.org/10.2118/17242-MS] ].

It should be noted that upon modeling, general HTHP rheological models depend on the initial base equation and Arrhenius empirical relations, which can cause a large predictive deviation.

To avoid designation of initial rheological model, a specific HTHP rheological modeling process based on the RDR approach was developed by Hemphill [13T. Hemphill, "Prediction of rheological behavior of ester-based drilling fFluids under downhole conditions", International Petroleum Conference and Exhibition, .Villahermosa, Mexico
[http://dx.doi.org/10.2118/35330-MS] ]. Rather than using a designated rheological model, it directly correlated the changes in dial readings with temperature, pressure, and shear rate by the non-linear regression method. All the results were related to a relative parameter measured at T_o and P_o. The RDR are defined as:

(5)

where θ is the dial reading and T_o and P_o are the relative temperature and pressure, respectively. Combining with Arrhenius Relation, the RDR model is rewritten as:

(6)

where C₁ and C₂ are model coefficients that are dependent on the pressure and shear rate.

Using the RDR approach, an accurate HTHP rheological model was developed at eight specific shear rates under HTHP conditions(50-240^oF, 0-8000psig) for an ester-based drilling fluid. Similarly, Demirdal [14D. Barkim, "Study of flow of synthetic drilling fluids under elevated pressure and temperature conditions", Adv Cut Trans Stud., pp. 25-85.] applied the RDR approach to establish an accurate HTHP rheological model for ester-based mud under HTHP conditions(40-280^oF, 500-12000psig). As for the RDR modeling strategy, it is not necessary to use a rheological model as the base equation and, thus, the assumption of base rheological model can be avoided.

Previous RDR modeling studies focused mainly on ester- and synthetic-based drilling fluids and, consequently, the implication of RDR approach on the model applicability and the predictive accuracy in non-aqueous drilling fluids, is increasingly coming to light. In contrast, the RDR approach involved in HTHP rheological modeling for water-based drilling fluid has not been reported so far. Herein, the RDR approach was utilized to reveal its applicability of developing a highly accurate HTHP rheological equation in a weighted sulphonated water-based drilling fluid. This work on the RDR application in the water-based drilling fluid is complementary with respect to non-aqueous drilling fluids, which can not only determine a relation that relates RDRs with temperature, pressure, and shear rate, but also verify the validity of RDR approach as a novel HTHP rheological modeling tool for particular drilling fluids.

The sulphonated water-based drilling fluid, as a common kind of HTHP drilling fluid, was typically employed in the oil and gas industry. Such fluid has a high density of 1.5g/ml, and consists of at least 7 functional additives such as hydrophilic solid phase, pH control agent, loss control agent, viscosifier, inhibitor, antioxidant, and weighting material was employed here, as shown in Table 1. The drilling fluid was heat-aged at 180°C for 16 hours before testing.

Table 1
Components of the sulphonated water-based fluid referred here.

Pressures applied in the tests were 0.1, 15, 25, 55, 85, and 100MPa. Temperatures used during the tests were 30, 60, 90, 120, and 150°C. At each combination of pressure and temperature, the rheological data(dial readings) of the weighted sulphonated drilling fluid were measured at six specific shear rates 600, 300, 200, 100, 6, and 3rpm, using a Fann IX77 HTHP Rheometer.

To establish an accurate equation, both statistics and detailed sample selection are essential. In principle, the more the testing data, the more accurate the final formula. Although the statistic data seem to be not enough, the RDR approach was previously demonstrated to be accurate in defining a HTHP rheological model even with limited parameters [13T. Hemphill, "Prediction of rheological behavior of ester-based drilling fFluids under downhole conditions", International Petroleum Conference and Exhibition, .Villahermosa, Mexico
[http://dx.doi.org/10.2118/35330-MS] , 14D. Barkim, "Study of flow of synthetic drilling fluids under elevated pressure and temperature conditions", Adv Cut Trans Stud., pp. 25-85.].

The RDR approach was employed to determine a single relation that relates dial readings with shear rate, temperature, and pressure. The standard procedure of RDR modeling contains relative dial reading, Arrhenius relation, polynomial of pressure coefficients, and polynomial of shear rate coefficients, as shown in Fig. (1). To determine such an equation, non-linear and linear regression analysis has been used by a commercial software STATISTIC. Note that, on account of the requirements of accuracy, different expressions like logarithmic, exponential, and polynomial model can be obtained upon fitting.

Fig. (1) HTHP rheological Modeling procedure of the RDR approach.

To reduce the effect of T and P, a relative treatment was performed on dial reading data. Herein, RDR is defined as the ratio of the dial reading determined at any temperature and pressure to the dial reading determined at 0.1MPa and 30°C, shown as follows:

(7)

Table 2 collects the dial readings and RDRs under the considered conditions. As might be anticipated, the water-based fluid is rarely pressure-dependent and highly temperature-dependent. Meanwhile, pressure effects on rheology can slightly rise at high temperature and low shear rates.

Table 2
Dial readings measured under the considered HPHT conditions ^{a, b}.

Arrhenius relation is considered to be appropriate to relate RDR to temperature. RDR(T) in terms of the investigated parameters can be shown as follows:

(8)

wherein C₁ and C₂ are coefficients of Arrhenius relation.

Table 3 presents the coefficient analysis of RDR vs. T for Arrhenius relation. It is clear that all of the correlations(R²) are larger than 0.98, and the average value is 0.9886±0.0035, suggesting a high predictive accuracy for Arrhenius relation.

Table 3
Coefficient analysis of RDR vs. T for Arrhenius relation.

Once the relation between RDR and T was determined, the next step was to determine whether coefficients of the model(C₁ and C₂) are dependent on P and γ. Table 4 shows that both C₁ and C₂ are functions of P and γ. If these coefficients are defined as the functions of P and γ, RDR(T, P, γ) can be written as (5).

In order to get a general expression, the regression analysis was carried out to determine the effect of P and γ on these constants(C₁ and C₂). Herein, the effect of P on C₁ and C₂ under constant γ was analysed . Third degree polynomial defines the relation of C₁ and C₂ vs. P as:

(9)

(10)

wherein a, b, c, and d are the model coefficients.

Table 4
Coefficient analysis of C₁ and C₂ vs. P for third degree polynomial relation.

It can be readily found that coefficients of multiple determination(R²) for C₁(P) and C₂(P) are all close to 1.0, which reveals significant correlation between the actual and predictive values. Apparently, the proposed third degree polynomial is quite suitable to fit C(P). Next, C₁(P) and C₂(P) were further used to establish C₁(P, γ) and C₂(P, γ).

In order to determine C₁(P, γ) and C₂(P, γ), the effect of γ on coefficients(a, b, c, and d) shown in C(P) should be determined. Non-linear regression analysis was carried out using either one-parameter or two-parameter curves. The relations between the coefficients of C₁(P) and C₂(P) and γ are presented in Table 5.

Table 5
Relations between coefficients of C(P) and γ.

Listed in Table 5 are the relation that well relate coefficient of C(p) to shear rate. For all relation, R² values are almost equal to 0.9, and average value is as high as 0.971±0.027, suggesting a good relation between the predictive and actual values. Substituting relations listed in Table 5 to C₁(P) and C₂(P), the coefficients C can be shown as a function of P and γ as follows:

(11)

(12)

These two Eq.11 and Eq.12 achieved above can be further substituted into RDR(T). As a consequence, a function relating RDR to T, P, and γ can be determined by using one equation (see Eq. 7).

Analysis on the predictive deviation of HTHP rheological model developed here was further carried out to evaluate its applicability. Table 6 lists main results, including the tested and calculated dial readings, residual, and absolute error. The absolute error percentages vary in the range of 0~12%, and the average error percentage is calculated as 3.3397±2.2006, which reflects high predictive accuracy for the developed HTHP rheological equation within certain temperature and pressure range.

Table 6
Analysis on deviation of the HTHP rheological model developed for the weighted sulphonated water-based drilling fluid.

To gain a further insight into the relation between deviation and variables, deviation distributions with respect to T, P, and γ is investigated. All of the results are presented in Fig. (2).

Fig. (2) Box plots of error percentage vs. P, T, and γ, respectively, for the RDR models, where the 10th, 25th, 50th, 75th, and 90th percentile of deviation dataset are marked.

Fig. (2) compares distribution of deviation data related to P for the RDR models. Here, the key percentiles of deviation dataset are marked in box plots. It can be readily found that the extreme deviation is about 11%, and 99% deviation are less than 11%, reflecting excellent prediction for the RDR model. Besides, the median line at 100MPa is slightly higher than others, which suggests that the predictive accuracy at 100MPa is lower than others. The fact that the deviation area of 85MPa exhibits slightly larger means a larger error distribution.

In Fig. (2), the distribution of error percentage vs. T is offered. It is clear that the extreme deviation appears at 150^oC, and the highest median line at 60^oC. Meanwhile, the deviation area is longer at 60^oC than others. These results show that the predictive accuracy is slightly lower at 60^oC than any others.

In addition, Fig. (2) gives box plots of the distribution of error percentage vs. γ. One readily finds that the extreme deviation appears at 3rpm, and the high median lines appear at 3 and 6rpm. This finding reveals that the predictive accuracy should be worse at low shear rates than those at high shear rates. That is to say, the established model does not always accurately predict the measured shear stresses at low shear rates.

According to deviation analysis given above, the RDR model defined for water-based drilling fluid has a low predictive error less than 11.7%. Although large errors occur at high pressure and low shear rates, they meet requirement of predictive accuracy on-site. It can be reasonably concluded that an accurate HTHP rheological model is established for the sulphonated water-based drilling fluid considered here.