Gaussian Process Prediction Model to Estimate Excess Adsorption Capacity of Supercritical CO2

Deep coal beds have been suggested as possible usable underground geological locations for carbon dioxide storage. Furthermore, injecting carbon dioxide into coal beds can improve the methane recovery. Due to importance of this issue, a novel investigation has been done on adsorption of carbon dioxide on various types of coal seam. This study has proposed four types of Gaussian Process Regression (GPR) approaches with different kernel functions to estimate excess adsorption of carbon dioxide in terms of temperature, pressure and composition of coal seams. The comparison of GPR outputs and actual excess adsorption expresses that proposed models have interesting accuracy and also the Exponential GPR approach has better performance than other ones. For this structure, R2=1, MRE=0.01542, MSE=0, RMSE=0.00019 and STD=0.00014 have been determined. Additionally, the impacts of effective parameters on excess adsorption capacity have been studied for the first time in literature. According to these results, the present work has valuable and useful tools for petroleum and chemical engineers who dealing with enhancement of recovery and environment protection.


Introduction
The Global Carbon Project declares that overall emission of CO2 has been grown to 37 billion tones based on the latest annual evaluation [1]. This growth causes significant effects on global warming and rising sea levels which threat the human future. Various actions are required to decrease the CO2 concentrations in the atmosphere such as injection of carbon dioxide into deep coal seams. The geological sequestration of CO2 is not only purpose of this injection but enhancement of the methane recovery from coal seams [2][3][4][5]. After injection of CO2 into the deep coal seems, it can be easily behaviors in flowing through the coal seams including adsorption behavior, seepage behavior, mechanical weakening and microstructure variance effects (see Figure 1). The flow of CO2 through the coal seams will cause some changes in pore pressure and exerted stress on the coal seams. CH4 will be pushed by CO2 in competitive adsorption within the affected zone. During this process, the CO2 pressure will reduce gradually and also desorption will occur. The changes in desorption and adsorption of CH4 and CO2 can affect the volume deformation and mechanical properties of coal [6].
Furthermore, these changes will influence the CO2 seepage properties from the coal seams. Thus, CO2 injection can be considered as a complicated process dealing with various mechanisms combined with each other. The investigation of adsorption of CO2 and CH4 among the aforementioned mechanisms have the great role because the value of greenhouse gas storage of deep coal beds can be determined through this investigation.

Fig. 1:
Multi-factor combining impact of interaction of coal bed and CO2 [5] There are several researches about the CO2 adsorption on coal. For example, Ramasamy and coworkers investigated the CO2 adsorption on various kinds of coal and concluded that the capacity of adsorption is highly function of coal properties [7]. After that, a new research was done on the competitive adsorption of CH4 and CO2 by Zhang to identify the impact of gas composition and depth on adsorption of CO2 and CH4 [8]. De Silva investigated the CO2 behaviors with different coal beds and expressed that the equations of states have acceptable estimation for CO2 adsorption [9]. Then, Mazzotti implemented the adsorption study of N2, CH4 and CO2 on coal and claimed that CO2 has better adsorption than N2 and CH4 [4].
Although there is a wide investigation on the CO2 adsorption on coal seems, the experimental study on CO2 adsorption on coal in the supercritical conditions faces many difficulties and measured adsorption, whereas others show smooth isotherms of CO2 excess adsorption [12][13][14][15][16][17][18][19][20][21]. Thus, considering the growth of injection of CO2 into coals and their heterogeneous properties, the investigation of supercritical CO2 adsorption requires more investigations to clarify the mechanisms of CO2 sequestration in coal bed.
In the recent years, the computational study of CO2 adsorption on coals has attracted attention of many recent scholars [12,20,22]. These studies commonly include monolayer, multilayer and potential models. The typical monolayer models are the Toth, Langmuir, T-P, and Extended-Langmuir models.
The multilayer approach includes different forms of BET models and also the potential model consists of D-R model, its modification and upgraded D-A model. These approaches have been utilized for estimation of adsorption of supercritical CO2. However, they have significant drawbacks including limitations to a specific coal seams or isothermal conditions. Due to this fact, development of a comprehensive adsorption model which overcomes the limitations of different conditions (temperature, pressure and coal type) becomes necessary.
In the current work, the amount of excess CO2 adsorption on various kinds of coal samples has been predicted by proposing Gaussian Process Regression approaches including four different kernel functions. Furthermore, the impacts of pressure, temperature and composition on CO2 adsorption on coal seams have been investigated.

Data gathering
To develop a comprehensive approach for calculation of excess CO2 adsorption on coal seams, an actual CO2 adsorption databank has been collected from various resources [5,11,17,19,20,22]. This dataset includes 394 actual adsorption points for 16 various coalbeds in temperature range of 20.14 to 79.42 o C and pressure range of 0.048 and 22.887 MPa. The excess adsorption CO2 values for these conditions vary between 0.09325 and 2.41588 mmol/g.

Gaussian Process Regression
One of the non-parametric approaches is Gaussian Process Regression (GPR) which has ability of modeling arbitrary complicated systems. In the most of estimation issues, this algorithm is preferred because of its flexibility in providing the uncertainty descriptions [23]. This approach models series of time by using a covariance function (CovF) k(x,x') and mean function (MF) m(x) as shown in following: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 February 2020 doi:10.20944/preprints202002.0069.v1 4 of 19 In which, y and x are output and input of training set and f(x) point to the latent variable of model.
Usually, the mean function of aforementioned equation is selected to zero in the most applications.
CovF which expresses the similarities among inputs, is known as the main parameter in GPR because the data points with same values of x are likely to have same output. In this study, different forms of kernel function have been used as following: • Squared Exponential: where θ1 and θ2 are hyper-parameters which require to be optimized and d shows the Euclidean distance between x and ′ .
• Exponential Where ∝ is a positive parameter of covariance.
• Matern Where ν and l are positive parameters and Kv is known as modified Bessel function.
In the training of model, the hyper-parameters of kernel matrix (K) are determined by minimizing the negative log marginalized likelihood (NLML): The NLML minimization concludes to determination of unknown θ. The optimization problem for estimation of parameter can be written as following: The NLML is optimized by applying off-the-shelf optimization approaches because it is a convex function. Then, the estimation distribution for testing data can be shown as below: * | , , * ~( * � , ( * )) In which, WGPR denotes the weighting matrix of GPR [23][24][25]. In order to better understand of proposing GPR algorithms for our work, a flowchart is depicted in Figure 2.
Root mean square error ( Standard deviations (STD) = ( Mean squared error As shown in On the other hand, for better judgement about discussing models, the simultaneous demonstration of estimated and experimental excess adsorption are depicted in Figure 3 for all four models. The interesting agreement between GPR outputs and actual excess adsorption is observed. Moreover, the regression or cross plot of actual and estimated excess adsorption are illustrated in Figure 4.  According to this analysis, the clouds of data points are located on bisector lines which express quality of GPR outputs. Additionally, the relative deviation between GPR output and actual excess adsorption has been determined and shown in Figure 5.

Preprints
where U is a matrix of i*j dimensional. i and j point to the number of model parameter and training points which are applied for calculation of critical leverage limit as below [26][27][28]: * = 3( + 1)/ William's plot (see Figure 7) has potential to distinguish the suspected data visually which expresses the standardized residuals in terms of hat values. Subsequently, the green line point to the leverage limit and two red lines are standard residue limits. The data points which are located outside of the bounded area by these lines are known as suspected data. As can be seen in Figure 7, among 394 data point for Exponential, Square exponential, Matern and Rational Quadratic GPRs, 6, 4, 3 and 3 data points are considered as suspected data.
The proposed GPR algorithms construct a relationship between inputs and excess adsorption of CO2 on coal seams. Sensitivity analysis is normally used to study how input variables affect output. In order to detect the most effective variable on excess adsorption of CO2, Relevancy factor (r) is used.
The range of this factor is between -1 and 1, and also the more absolute r expresses a more impact on the excess adsorption of CO2. Negative and positive values of r illustrate that the more the input the less and the more in the excess adsorption of CO2 respectively. The formulation of r can be described as following [29][30][31][32][33]:  (20) where and , denote the output and input. � and ��� are known as averages of outputs and inputs. As shown in Figure 8, the higher pressure and Ash content, the lower excess adsorption of CO2. Moreover, temperature, Volatile, Moisture and Fixed carbon have straight relationships with target. Additionally, the most effective parameter on CO2 adsorption is Ash content.