Introduction

Cementing is the most important and the most expensive exercise during drilling operation. In cementing operation, the annulus between the casing and the adjacent rock formation is filled with a certain compound of cement grout and allowed to set, usually after a few hours or a few days, and solidify strongly to join the casing to the formation. This compound could be made from different ingredients with different percentage of weight with respect to the weight of cement in the grout mixture (Labibzadeh et al. 2010). Compressive strength is one of the properties used to test the reliability of cementing and is the ability of a material to withstand deformation when load is applied. Compressive strength of a cement concrete depends on the type of raw materials including additives used, mixture proportions, concrete structure, method and time of curing, and exposure conditions (Herianto and Fathaddin 2005).

Cement with a good compressive strength should be able to withstand hard and corrosive formations, lost circulation zone, carbon (IV) oxide and other toxic gas intrusion, and extremely high temperature (Benjamin et al. 2010). Problems of poor cementing have led to myriad research in this field using different approaches. Sauki and Irawan (2010) investigated the effect of pressure and temperature on well cement degradation by super-critical CO2 and concluded that compressive strength loss was greater at elevated temperature and pressure due to the formation of alpha-calcium silicate, and in CO2 environment due to formation of carbonation which gives temporary strength to the cement. Labibzadeh et al. (2010) considered the effect of contemporary pressure and temperature changes in the early compressive strength of oil well class G cement and concluded that faster early-age compressive strength could lead to reduction in transition phase time (thickening time). They also observed that cement strength could also experience strength retrogression if crystalline silica was added to the cement slurry (Benjamin et al. (2010)).

Zhou and Jia (2010) developed a low-density and high-compressive strength cement slurry based on the theory of particle grading over numerous experimentations. The performance of the compressive strength of the cement slurry developed has been improved over the existing one. Ordinarily, class G cement is one of the types used for sealing off of formations because of its ability to withstand high resistance of pressure, temperature, and sulphate. Despite these qualities, additives are required to improve the properties of the cement (Xi et al. 2010). Bayu et al. (2010) concluded in their work that the addition of 0.2 % of lignosulfonate to a cement slurry increased compressive strength. Above this value, compressive strength was noticed to have reduced. Performance of other additives to enhance cement operations to optimally increase compressive strength during cement operation depends on the correct proportion of each of the additives. The best way is to subject these additives to a series of experimental runs, which is usually time consuming, tedious, and expensive (Isehunwa and Orji 1995).

Factorial design (FD) is a method that monitors the interactions of multiple factors which accommodate the effect of both main and interaction effects (Cheong and Gupta (2005)). FD has been successfully used in solving engineering problems, some of which are analysis of rheological properties of treated Nigerian clay (Adeleye et al. (2009), indentifying and estimating significant geological parameters White et al. (2001), uncertainty assessment Peake et al. (2005), and much more. Arising from advantages of FD and the importance of additives in the performance of cement, this research aims at development of mathematical model to predict compressive strength using four different additives for improving compressive strength during cementing operation using factorial design.

Methodology

Several experiments were run with the selected slurry systems aiming at the evaluation of the compressive strength of oil well class G cement using factorial design. The experiment was conducted based on the American Petroleum Institute (API) specification (American Petroleum Institute 1997).

Experimental design

The number of experimental runs performed for the model development is full factorial design which is governed by Eq. (1):

N = L k
(1)

where L denotes factors which are four in this case, k is the number of levels, which is two, and N is the total number of experimental runs, which is 16. X1 is extender, X2 is accelerator, X3 is antifoam, and X4 is dispersant. The design of the experiment is tabulated in Table 1.

Table 1 Full factorial design for four variables at two levels
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Response variable

The response variable for this experiment is the compressive strength. Sixteen (16) experimental runs were performed according to a full factorial design of four (4) factors using Table 1 as the guide to different formulations of slurry preparations and combination of factors. We use ‘−’ to indicate low level and ‘ + ’ for high level. The quantity of each of the variables under low and high is presented in Table 2. The response variable for the experiment is the compressive strength.

Table 2 Factor level settings
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Cement slurry preparation, curing time, and compressive strength measurement

For the preparation of cement slurry used for this study, 297 g of class G oil well cement was added to 447 ml of fresh water and blended using Waring Blender set at high speed for 35 s. The four additives identified in this study are: extender, accelerator, antifoam and dispersant. The additives which also serve as the variables are added to the blended cement slurry using Table 1, which gave different combinations of different slurry designs with different compressive strengths. The mixture of the based cement slurry and additives was mixed using a Constant Speed Mixer “Model 30-60” of Chandler Engineering Company at 12,000 rpm ± 500 to achieve the pre-calculated slurry density of 11.5 ppg.

The cement slurry formed from the 16 different experimental runs was poured under ambient pressure and temperature conditions into the curing chamber. In the curing chamber, the cement slurry was moulded into different shapes. The cement slurry was moulded to cubic moulds with diameter of 5.08 cm; thereafter the moulded samples underwent curing for 24 h.

After 24 h, the compressive strengths of the 16 formulations were measured using the Ultrasonic Cement Analyzer (UCA). Ultrasonic Cement Analyzer (UCA) provides a continuous non-destructive method of determining compressive strength as a function of time through the measurement of change in velocity of an acoustic signal according to API 8A (American Petroleum Institute (API) 1997). The UCA measures the delay time of an ultrasonic wave pulse that passed through the moulded cubic cement sample using set equations which converted velocity to uniaxial compressive strength and the values of all the compressive strengths were recorded. The response-dependent variable is the compressive strength which is determined for all the 16 samples. Each experimental run was conducted twice and the result of the response variable was recorded in Table 3.

Table 3 Experimental results
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Model development

Running the full complement of all possible factor combinations means that we can estimate all the main and interaction effects. In this experiment, there are four main effects, six two-factor interactions, three three-factor interactions and one four-factor interaction, all of which appear in the full model as follows:

Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 + β 12 X 1 X 2 + β 13 X 1 X 3 + β 14 X 1 X 4 − β 23 X 2 X 3 + β 24 X 2 X 4 + β 34 X 3 X 4 − β 123 X 1 X 2 X 3 − β 124 X 1 X 2 X 4 − β 134 X 1 X 3 X 4 + β 234 X 2 X 3 X 4 − β 1234 X 1 X 2 X 3 X 4 + e
(2)

A Yates algorithm is used to calculate the main and interaction effect. These values are recorded in Table 4. These values are substituted for a corresponding value in Eq. (3) and fully expressed in equation 4 which is as follows:

Table 4 Level of variables used for the prediction
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R = 584.453125 + 36.171875 X 1 + 1.32815 X 2 + 26.859375 X 3 + 16.984375 X 4 + 6.859375 X 1 X 2 + 40.015625 X 1 X 3 + 33.765625 X 1 X 4 − 3.765625 X 2 X 3 + 13.734375 X 2 X 4 + 13.140625 X 3 X 4 − 16.796875 X 1 X 2 X 3 − 25.671875 X 1 X 2 X 4 − 1.140625 X 1 X 3 X 4 + 37.390625 X 2 X 3 X 4 − 13.078124 X 1 X 2 X 3 X 4 + 0.13258
(3)

Results and discussion

Effect of the variables on compressive strength

The experiment conducted in this research showed that the compressive strength varies according to the proportion of the additives included during the design and formulation of the cement slurries for carrying out specific drilling operations. The influence of the various additives on the compressive strength of the cement has been studied in this work and the results of their sensitivities are as clearly displayed in Figs. 1 and 2 the accuracy of the model in Fig. 3.

Fig. 1
figure 1

Effect of extender on the compressive strength

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Fig. 2
figure 2

Effect of extender on the compressive strength of cement slurry

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Fig. 3
figure 3

Cross plot of experimental against predicted values

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It was evident from Figs. 1 and 2 that both the extender and antifoam impacted the compressive strength of the cement under investigation differently, at varying proportion and time. This behaviour is in agreement with literature. At different proportions, the additives change the velocity of the acoustic signal with time, thereby controlling the magnitude of the response measurable by the Ultrasonic Cement Analyzer. For example, the value of the cement compressive strength remains fairly constant with the extender concentration increase from an initial amount of 5 %. This trend continues up to 10 % of extender being included in the slurry; between the intervals, the value of compressive strength approximately remains at 600 psi. However, an additional 50 psi in compressive strength was observed when the extender increased from 10 to 15 % and this is attributable to increasing delay in the acoustic signal. The effect of the antifoam on the compressive strength of the cement is presented in Fig. 2. Antifoam inclusion in the cement slurry experienced a threshold value approximately at 6 %. Below this amount, the compressive strength remains fairly constant at 600 psi. Above this value, the compressive strength begins to fall gradually and at approximately 8 % proportion a value of 580 psi representing a decrease of 20 psi was observed. To maintain the compressive strength of the cement, say at 600 psi for a particular application, the designing of the slurry must ensure that the threshold values of these additives are not exceeded.

Figure 3 shows the parity plot between the experimental values and the model prediction values. The graph can be used to validate the developed model. The alignment of the values along the 45o line reflected the accuracy and adequacy of the model to navigate the sample space. The straight line gives a correlation coefficient of 99.8 %. This suggests that all the selected model terms are significant and the selected model is sufficient to describe the experimental results perfectly.

Statistical analysis of the model

The summary of the weight of the factors used for the 16 experimental run in real percentage is shown in Table 4. It was a systematic design of variables used from low to high values in real values. The summary of the main and interaction effects between the factors used for the development of the model is shown in Table 5. It also showed the residual, degree of freedom, and mean square and F value for the various combinations of the interaction effects, leading to the development of the model. The deviation of the developed model from the experimental values was obtained by the analysis of variance with the calculation of the sum of contrast and mean square value of 1.125 and 0.070313. The coefficient of each of the main variables and their corresponding interaction effect show the contribution of each effect in the equation ti its contribution to the developed model.

Table 5 Analysis of variance of randomized full factorial design
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Conclusion

A factorial design method has been successfully employed in this study to develop a model to predict the compressive strength of cement. The effects of four different additives were considered individually as well as their interaction with each other during the development of the model. The compressive strength varies according to the proportion of the additives included during the design and formulation of the cement slurries for carrying out specific drilling operations.

The model developed has a correlation coefficient of 99.8, standard error of 0.1325, and accuracy of 99.8 %. The model can be used to determine the behaviour of cement slurry when any of these additives are in underused or added in excess of the required quantity.

Recommendation

This investigation can be extended to other properties that affect cementing operations such as thickening time determination, rheological investigation, and effect of contaminants on cement performance.