Artificial Intelligence-based Modeling of Interfacial Tension for Carbon Dioxide Storage

1. Introduction

Storing in depleted oil and gas reservoirs or deep saline aquifers is an attractive technology with the potential for reducing CO₂ emissions to the atmosphere, which are now established as the main cause of global warming (Abedini & Torabi, 2014). The features of porous media which are suitable for CO₂ storage are known (McGrail, Schaef, Ho, Chien, Dooley, & Davidson, 2006) to include:

1. Capacity to accept high quantities of CO₂;

2. Injectivity to absorb CO₂ at rates generated by large-scale carbon dioxide emitters;

3. Confinement (sealing) sufficient to avoid the dispersal and leakage of buoyant and portable CO₂ from the subsurface reservoirs to other below-ground formations, shallow aquifers and/or ultimately back to the Earth’s surface and atmosphere.

Reservoirs at various depths can effectively store CO₂ in the subsurface and at diverse physical and chemical conditions due to the characteristics of CO₂ at the temperature and pressure states found in Earth’s subsurface (Pranesh, 2018). The distinct mechanisms of entrapment are involved: initially, injected is trapped by primary mechanisms, involving static and hydrodynamic trapping; subsequently, secondary mechanisms, including mineralization and chemical interactions between the CO₂ and the formation minerals, act to influence CO₂ containment, not necessarily to increase storage capacity (Metz, Davidson, De Coninck, Loos, & Meyer, 2005).

Storing carbon dioxide in subsurface reservoirs is vulnerable to potential problems and impacted by various uncertainties, some of which may occur during and/or after the injection stage (Damen, Faaij, & Turkenburg, 2003). The most important potential problem is  leakage. Both reservoir pressure and the buoyancy of carbon dioxide once in the reservoir can be sufficient over time to penetrate the reservoir seals (both cap rock and lateral seals) (Ennis-King & Paterson, 2005). If  is able to breach a seal at any point it will ultimately find a pathway out of the reservoir. At high injection rates and pressures, a capillary breakthrough of  may cause leakage from a reservoir. Interfacial tension (IFT) between  and reservoir fluids is known to play a vital role in the capillary breakthrough (De Lary, Loschetter, Bouc, Rohmer & Oldenburg, 2012). Therefore, detailed investigations of the IFT behavior between  and formation fluids are required for each specific subsurface reservoir considered for long-term CO₂ storage to ensure adequate containment can be sustained over time.

Researches focusing on IFT between  and reservoir fluids have provided in recent years substantial experimental data. Early studies provided IFT measurements for pure water-  systems in underground conditions dating back to 1957 (Hebach, Oberhof, Dahmen, Kögel, Ederer, & Dinjus, 2002). However, there are concerns about the accuracy of these earlier data measurements due to the questionable assumptions and equations applied (Ennis-King & Paterson, 2005). Yan, Zhao, Chen, & Guo (2001) applied linear-gradient theory to determine IFT for pure water-  systems at specific temperatures. However, other researchers found out that method tended to underestimate IFT at high pressures and overestimate IFT at low pressures. For salt-water (brine) systems there are much more limited IFT measurements available. Yang & Gu (2004) determined IFT values for a brine-  system. However, the salinity they studied used was constant and very low, resulting in their observation that salinity did not affect IFT. Previously, Aveyard & Saleem (1975) had reported, however, a linear connection between IFT of brine-  systems and molal salt concentration under ambient conditions. Two frequently used techniques for measuring IFT of -brine solutions at high pressures and high temperatures are pendant drop and capillary rise methods (Georgiadis, Maitland, Trusler, & Bismarck, 2010). However, such experimental measurements are time-consuming and require expensive laboratory equipment and sophisticated interpretation procedures.

Mahbob, & Sultan (2016) studied experimentally the changes of interfacial tension and wettability with dolomite rock in the presence of carbon dioxide due to various parameters, such as temperature, pressure, salinity and surfactant type. It was found that with increasing salinity and temperature IFT of brines increases but it decreases with an increase in pressure. These effects are due to the solubility of carbon dioxide in brine. They also concluded that the use of fluoro-surfactants gives the minimum (less than one) interfacial tension. Zhao, Chang, & Feng, (2016) measured the crude oil-carbon dioxide mixture IFT to assess the minimum miscibility pressure. Their research showed that under immiscible conditions, the oil extraction and storage capacity improve dramatically as the injection pressure increases. On the other hand, while the pressure is higher than the MMP, the rise in the injection pressure can only cause a slight increase in oil recovery and storage capacity.

A series of slim tube experiments were planned and presented to measure the impact of cold CO₂ on recovery factor in high-temperature reservoirs (Hamdi, & Awang, 2016). They found that low-temperature CO₂ injection into the high-temperature reservoir can result in a slightly higher recovery factor than isothermal injection. The reason for the increase in recovery was attributed, in particular to the reduction of the interfacial tension between CO₂ and reservoir fluids at the injection point. Golkari, & Riazi (2017) measured experimentally the equilibrium IFT of live and dead crude oil-CO₂ systems. The results of the experiment showed that IFT reduces with different trends as the equilibrium pressure for the live oil/CO₂ and dead oil/CO₂ systems increases. However, it increases with temperature. Karaei, Honarvar, Azdarpour, & Mohammadian (2017) determined the IFT of CO₂/brine at pressures and temperatures up to 7 MPa and 100°C, respectively. Their study confirmed that interfacial tension of CO₂ and brine can be increased by increasing temperature as well as by decreasing pressure.

The CO₂–brine (NaCl + KCl) IFTs were obtained using the method of pendant drop under the condition of 298–373 K temperature, 3–15 MPa pressure (Mutailipu, Liu, Jiang, & Zhang, 2019). They reported that the CO₂–brine IFTs increase with salinity and temperature and decrease with pressure until they reach a plateau. A linear relationship between the increase in IFT and molality was identified for a CO₂–mixed brine system. Furthermore, all samples became less water-wet just as the state of the CO₂ phase changed from subcritical to supercritical.

An empirical equation with an extended uncertainty of 1.6 mN·m^-1 was developed to describe the interfacial tension between carbon dioxide and aqueous solutions of the mixed salt system (0.864 NaCl + 0.136 KCl) with total salt molalities between (0.98 and 4.95) mol·kg^-1 as a function of temperature, pressure, and molality by Li, Boek, Maitland, & Trusler (2012). The results showed that the IFT increases linearly with the molality of the salt solution. A summary of the most important empirical correlations derived by several researchers was compiled by Zhang, Feng, Wang, Zhang, & Wang (2016). However, many of those empirical correlations are inaccurate at high salt concentrations (Li, Wang, Li, Liu, Li, & Lv, 2013). In order to estimate the CO₂−brine IFT more accurately, a newly developed CO₂−brine IFT prediction model based on the alternating condition expectation (ACE) algorithm was proposed by Li, Wang, Li, Liu, Li, & Lv (2013). The CO₂−brine IFT correlation developed resulted in an accurate prediction of the impure CO₂−brine IFT with values of AARE = 10.19% and SD = 13.16%. For captured CO₂ to be stored reliably in underground reservoirs, an extensive and efficient -brine IFT measuring model has yet to be developed that can concurrently consider both impurities in the CO₂ gas injected and a wide range of water salt concentr at ions. Chen & Yang (2018) generalized an IFT correlation between CO₂ and water. The proposed model was a function of mutual solubility and the reduced pressure of CO₂ in a temperature range of 278.2−469.2 K and a pressure range of 0.10−69.10 MPa. Their model generated good prediction accuracy for IFT between CO₂ and brine, provided the salinity is not too high, as the addition of inorganic salt decreases the solubility of CO₂ in the water while increasing the corresponding IFT.

Artificial neural networks (ANN) are widely used (Toghyani, S., Ahmadi, M. H., Kasaeian, A., & Mohammadi, A. H., 2016; Kahani, M., Ahmadi, M. H., Tatar, A., & Sadeghzadeh, M., 2018) for a variety of applications (Maddah, H., Ghazvini, M., & Ahmadi, M. H., 2019; Ramezanizadeh, M., Ahmadi, M. H., Nazari, M. A., Sadeghzadeh, M., & Chen, L., 2019) and continue to be refined and their prediction accuracy improved (Farzaneh-Gord,
M., Mohseni-Gharyehsafa, B., Arabkoohsar, A., Ahmadi, M. H., & Sheremet, M. A., 2020). ANN’s mathematical algorithms can provide credible prediction models for the complicated and nonlinear system between inputs and outputs by establishing multiple correlations between the input variables in their hidden layers. (Abbasi, Madani, Baghban & Zargar, 2017). This study applies ANN models with various algorithmic constructions to accurately estimate -brine IFT from a published dataset of experimental measurements for a wide range of -brine compositions and pressure and temperature conditions.  The novelty aspects of the study are related to the application and performance evaluation of six distinct ANN structures and tuning algorithms. The study identifies the RBF model tuned with particle swarm optimization (PSO) and the RBF model tuned with the farmland fertility algorithm (FFA) as the best performing ANN structures for IFT prediction from its influencing variables.

2. Data Analysis

107 published laboratory measurements of interfacial tension (24.78 to 47.87 mN/m) covering a broad range of conditions: temperatures from 27 to 100 °C; pressures from 48 to 258 bar; salinities from 0.085 to 2.75 M, brine densities from 0.97 to 1.105 g/ ; and, CO₂ densities from 0.093 to 0.933 g/  were compiled from analysis of a published source (Chalbaud, Robin, Lombard, Martin, Egermann & Bertin, 2009). Table 1 lists the value ranges for these variables displayed by the 107 data records. This dataset is divided randomly into 75 data records used for training the ANNs, 16 data records used for testing the ANNs. The remaining 16 data records were used for model validation. The model learns only from training data. The validation data set is used to evaluate a model while tuning, and it indirectly affects a given model. However, the test data set is held independently and is only evaluated once a model is entirely trained.

Table 1. Summary statistics of the variable values of the 107 data points used in this study (Chalbaud, Robin, Lombard, Martin, Egermann & Bertin, 2009)

Table 2 presents a correlation matrix for the six measured variables in the compiled dataset. It reveals that -brine IFT has relatively high negative correlations with CO₂ density and pressure but has a very low correlation with brine density. As should be expected, salinity is highly correlated with brine density. CO₂ density has quite a high inverse correlation with temperature and quite a high positive correlation with pressure. There is a direct (positive) relationship between IFT and brine density, salinity and temperature, whereas there is an inverse relationship between IFT and CO₂ density and pressure.

3. ANN Model Architectures

Six distinct neural network algorithms are developed to compare their IFT CO₂-brine predictions with the experimentally measured data records. Three multilayer perceptrons (MLPs), with two hidden layers, and three radial basis functions (RBFs) networks represent the ANN models evaluated. The MLPs were trained by Levenberg-Marquardt (LM), Bayesian regularization (BR) and scaled conjugate gradient (SCG) back-propagation algorithms, respectively. The RBF networks were optimized by particle swarm optimization (PSO), differential evolution (DE), farmland fertility algorithm (FFA) algorithms (Karkevandi-Talkhooncheh, Rostami, Hemmati-Sarapardeh, Ahmadi, Husein, & Dabir, 2018; Rostami, Hemmati-Sarapardeh & Shamshirband, 2018; Shayanfar & Gharehchopogh, 2018), respectively.  Figure 1 provides a flow diagram detailing the methodology and structure of the ANN models developed.

In all cases, the mean square error (MSE) was evaluated as the cost function to be minimized. Activation functions applied to the MLPs for the input layer to hidden layer 1, hidden layer 1 to hidden layer 2 and hidden layer 2 to the output layer were tensig, logsig and purelin, respectively. Each hidden later had only seven neurons.

The following control variables were applied to the RBF networks:

For RBF-DE: max neurons (35), spread parameter (1.2111), number of population (30), beta min (0.4), beta max (0.8) and crossover probability (0.2).

Where: beta min = lower bound of scaling factor and beta max = upper bound of scaling factor.

For RBF-FFA: max neurons (35), spread parameter (1.1429), number of population (49), 𝛼 (0.6), 𝛽 (0.4), W (1) and Q (0.5).

Where: 𝛼, 𝛽, and Q are numbers between zero and one. W represents the farmland fertility control variable. The values of these variables used for initiating the algorithm are validated by sensitivity analysis.

For RBF-PSO: max neurons (35), spread parameter (1.2129), number of particles in population (50), W (1), W_damp (0.99), C₁ (2) and C₂ (2).

Where: W is inertia weight, W_damp is inertia weight damping ratio, C₁ is the local learning coefficient, and C₂ is the global learning coefficient.

4. Results and Discussion

Four standard statistical measures of prediction accuracy, average percentage error (APRE), average absolute percentage error (AAPRE), root mean square error (RMSE) and coefficient of determination (R²) are derived to compare the relative prediction accuracy of the six neural network models evaluated (Table 3). These prediction accuracy measures are listed separately for the training, testing, and validation subsets of data records and for the respective ANN solutions are applied to all records in the data set.  High prediction accuracies are achieved for IFT of -brine solutions by the six ANN models.

The AAPRE and RMSE values recorded for the independent testing data records (Table 3) reveal that the RBF neural network achieves higher IFT prediction accuracy than the MLP models. The RBF network optimized by farmland fertility algorithm (RBF-FFA) displays the lowest AAPRE for the testing records (1.04253), whereas the RBF-PSO model displays the lowest RMSE (0.510938 mN/m) for the testing records, and RBF-DE displays the highest R² value (0.994912) for the testing records. For the solutions applied to all data records, RBF-FFA has the lowest AAPRE (1.077149), whereas RBF-PSO has the lowest RMSE (0.50456 mN/m) and the highest R² value (0.99225).

Table 2. Correlation matrix for dataset variables with CO₂ Density and pressure showing the highest correlations with measured IFT

Figure 1. Flow diagram illustrating the ANN methodology adopted.

Table 3. IFT prediction performance of six developed ANN models to predict IFT displaying values for four statistical error metrics. The RBF models show slightly better IFT prediction accuracy than the MLP models.

Considering only the MLP networks, the MLP network optimized by Levenberg-Marquardt (MLP-LM) displays the lowest AAPRE (1.550892) and highest R² value (0.98601) for the testing records, whereas the MLP optimized by Bayesian Regularization (MLP-BR) displays the lowest RMSE (0.679927 mN/m) for the testing records. For the solutions applied to all data records, MLP-BR has the lowest AAPRE (1.321031), the lowest RMSE (0.531575 mN/m) and the highest R² value (0.991341). This suggests the MLP-BR model provides the highest IFT prediction accuracy of the three MLP models but is consistently outperformed by the three RBF models.

Figure 2 illustrates the prediction accuracies achieved for IFT of -brine solutions by the six ANN models in terms of the cumulative frequency of the prediction errors of individual data records when arranged in ascending order for all data records. Whereas all six models display high prediction accuracies for IFT of -brine solutions, the best performing model in terms of absolute relative error is the RBF-FFA model. For that model, Figure 2 identifies that nearly 70% of the data records involve an absolute relative error of less than 1, and only 3% of the total data records exceeds an absolute relative error of 4.

Figure 3 displays measured experimental IFT for -brine solutions versus predicted IFT values for each of the six neural network models evaluated for all data records. All data points for the models straddle a line with a 45-degree slope and passing through the origin. This demonstrates the high prediction accuracy achieved collectively by the models, particularly the RBF-FFA and RBF-PSO models. Figure 2 reveals that for the MLP models and the RBF-DE models the greatest dispersion about the unit slope line between measured and predicted data points occurs in the IFT range 26 to 31 mN/m. The RBF-FFA and RBF-PSO models fit the data in that range with much less error than the other models.

Figure 2. Absolute relative error distribution highlighting that for the RBF-FFA model a higher proportion of the data records have a low absolute relative error compared to the other ANN models

Figure 3. Agreement between predicted and experimental IFT values for the six ANN models. The MLP-LM and MLP-BR models show the greatest scatter. The RBF-PSO and RBF-FFA models show the least scatter

Table 4. Correlation coefficients between predicted IFT for -brine solutions and the input variables for the six neural network models developed in this study. The models all show similar relationships between their predictions and the input variables confirming their veracity