This research used a simple centroid mixture design (SCMD) to study the influence of salinity (x2), hardness (x3) and polymer concentration (x1) on the polymer solutions' viscosity used in enhanced oil recovery processes. Components x1, x2 and x3 are in the experimental range from 0 to 1, where one represents that one of the components is 100%. Thus, the SCMD of these components was satisfied with Σ _i x _i = 1. Fig. 1 shows the ten replicated experimental points assessed.

Since the mixture components' sum must be 1, the linear model is redundant for any mixture design with i components. This study was based on a particular quadratic model (Eq. 1) that considers higher-order effects such as ternary interactions, increasing the cubic model with higher-order quadratic interaction terms [22].

Figure 1 Source: The Authors.

Where, у is the response variable (polymer solution viscosity), and the factors x₁, x₂ and x₃ correspond to the variables studied. The parameters ft, 𝛽₁, 𝛽₂, 𝛽₃, 𝛽₁₂, 𝛽𝛽₁₃ 𝛽₂₃ 𝛽₁₂₃, 𝛽₁₁₂₃, 𝛽₁₂₂₃, and ft₂₃₃ are the model interaction components coefficients.

After achieving the empirical model, the model's fit was tested, the three-dimensional response surface was evaluated, and the optimal fit of the proportions of the components was determined. The statistical software R was used to analyze the experimental data and plot the response surface. ANOVA was also used to estimate statistical parameters.

Based on the products results of the design of mixture experiments, with the special quadratic regression model, two (2) prediction models were developed:

A numerical model for determining viscosity estimates the polymer solutions viscosity as a function of salinity, hardness, and polymer concentration, given the regression of Eq. 1.

Source: The Authors.

Table 1

Note* Fractions based on a maximum concentration of (5000 ppm) *.

The R software established that the special quadratic model makes the best fit with a p-value of 0.3868 in the "Lack of Fit" adjustment component. However, to achieve this adjustment and the model's normality, independence, and homoscedasticity statistical assumptions, the model required a natural logarithm transformation (y). Eq. 2 expresses the final model that predicts the polymer solution viscosity (y) as a function of the polymer concentration (x₁), salinity (x₂ ) and hardness (x₃ ) variables.

Figure 2 Source: The Authors.

Fig. 2 shows the viscosity values predicted by the model versus the values obtained from the experiments. It also shows that the model adequately correlates the components of the mixture with an adjusted R ² of 0.9269 and that the terms 𝑥¹ ₂𝑥₂𝑥₃, 𝑥₁𝑥² ₂𝑥₃ and 𝑥₁𝑥₂𝑥₃ ² contribute substantially to the model response variance.

An analysis of variance (ANOVA) was conducted in the R software to assess the suitability and importance of the model. Table 2 lists ANOVA results of the individual parameter effects, and the interaction of the mixture variables on viscosity. The F-value compares the variation of the differences in the average responses at the design points, and the corresponding estimated responses using the linear model with the expected experimental variations calculated from a replicated design point (pure error). The ANOVA results show an F value of 3,101.44, implying that the model is significant; there is only a 0.01% probability that a large F-value will occur due to noise introduced during the experiment. The p-value is the probability of reaching the F value. A p-value lower than 0.05 indicates that there is a statistically significant difference between the means. A p-value greater than 0.10 indicates that there is no difference between the means [22]. Therefore, the p-value of the general model (error probability value) that is less than 0.0001 confirms that the model is highly significant and shows that changes in the independent variables correlate with shifts in the dependent variable. Additionally, the ANOVA results show that all mixture components and their interaction are essential to the model, as the p values are less than 0.05.

Based on the ANOVA results, all the parameters in the equation are significant and can substantially affect the viscosity value. However, among all the parameters, the effect of the linear terms has a more substantial influence due to its higher F value. According to the quartic model (Eq. 2), the linear terms corresponding to salinity (x₂ ) and hardness (x₃) have adverse effects on viscosity as shown by the negative coefficients; increasing these parameters decreases viscosity. Furthermore, the polymer concentration component (x₁) positively affects the increase in viscosity showing a positive coefficient.

Fig. 3 shows the response surface with a ternary contour of ranges in which the viscosity of the polymer solution (y) is optimized as a function of the polymer concentration (x₁), salinity (x₂) and hardness (x₃). The dark grey region represents the polymeric solution's lower viscosity due to high salinity and hardness values. The light grey part represents the area with the highest viscosity of the polymer solution. The results prove that the variables individually have a strong influence on the viscosity, due to their interaction.

Table 2

Source: The Authors.

Figure 3 Source: The Authors.

The response surface shows that if the three variables (33% polymer concentration, 33% salinity, and 33% hardness) are in the same proportion, the viscosity obtained is close to 25 cP. By doubling the concentration of polymer (66%), reducing salinity and hardness (16%), the viscosity is increased by approximately four times, which shows the positive effect of the increase in polymer concentration. However, the hardness has a more significant negative influence on viscosity. As a polymer concentration of 50% and salinity of 50% reaches a viscosity of 100 cP. On the contrary, where the hardness is 50%, it reduces the viscosity of the polymer solution 2.4 times (41 cP).

Cox direction is the line that connects the centroid point with the vertices (see Fig. 4) of the mixture's space. When moving in the Cox direction of say, x ₁ the ratio of the remaining mixture components stays constant, i.e = 1, and this is almost equivalent to an "independent" effect of x _i in the mixture space. The Cox effects are the traces obtained from slicing the response surface along the Cox directions. Fig. 4 shows the Cox direction in a three-dimensional mixture space, and a Cox effect plot of the special quartic model.

According to the Cox direction, which connects the centroid point with the vertices (Fig. 4, left figure), when moving in the polymer concentration directions, the proportion of the remaining components of the mixture remains constant. Fig. 5 (right figure) shows the individual effects obtained for each mix element by cutting the response surface relative to the Cox direction. It proves the significant influence of salinity and hardness on viscosity at low polymer concentrations.

Figure 4 Source: The Authors.

In the experimentation region shown in Figs. 4 (right figure), an increase in the polymer concentration (x_x) predicts an increase in the polymer viscosity (y). On the contrary, an increase in the salinity (x₂) and hardness (x₃) do not show changes in the rise of viscosity (y). The foregoing shows a negative impact of salinity (NaCl) and hardness (CaCl₂) on the polymer viscosity; thus, reservoirs with highly NaCl and CaCl₂ concentrations require a high polymer concentration to reach the desired resulting viscosity.

To validate the quadratic regression model of the ATBS polymer viscosity as a function of the independent variables (polymer concentration, salinity, and hardness) data from the literature [17] was used, considering the type of polymer used in this work. Castro et al. [17] reported experimental evaluations of the viscosity of an ATBS polymer in a polymer concentration ranging from 470 to 2,500 ppm prepared in brines with different salinity and hardness values. Table 3 shows the results and the treatment of data to make the prediction based on the proposed numerical model.

Based on the experimental data taken from Castro et al. [17] and evaluated in algorithm 1, the concentration of polymer required to reach a viscosity was determined using the Newton-Raphsonnumerical method. Eq. 2, which describes the regression model in its canonical form, was transformed into a general equation described in Eq. 3.

The development of the numerical solution of Algorithm 1 for each of the reported experimental results, showed an estimated error of less than 1x10^-10 for each solution. Table 4 compares the data provided by Castro et al. [17] and the data obtained with the mathematical model. The model calculates the polymer concentration necessary to achieve the viscosity of the polymeric solution under the same salinity and hardness conditions used for the experimental tests. According to these results, the model of Eq. 3 can be used with 95% reliability.

Table 3

Note: Percentages based on a maximum concentration of (5,000 ppm) *.

Table 4

Note: Percentages based on a maximum concentration of (5000 ppm) *

Figure 5 Source: The Authors.

Fig. 5 shows the viscosity of the polymer solution as a function of the polymer concentration based on the NaCl concentration (salinity) and CaCl₂ concentration (hardness). The long dash is the literature-reported data measured experimentally (measured values), and the solid line represents the results predicted by the numerical model (fitted values) using the SMCD. The results show a slight difference between the experimental results and the numerical model; thus, a lower polymer concentration is necessary when using the model to achieve the desired viscosity. For example, to reach a viscosity of 67 cP, based on the numerical model, a concentration of 2099 ppm of polymer is needed, although experimentally, 2500 ppm is required.

Lastly, the polymer concentration value predicted with the model and the experimental one is adequate with 5%error (95% reliability). Statistically, the correlation coefficient of 0.945 of the Newton-Raphson numerical model suggests a positive and robust relationship among the variables, making the model reliable and representative.