Comparison of Single, Binary and Temperature-Dependent Adsorption Models Based on Error Function Analysis

Abstract

The choice of adsorption model to use when accounting for gas adsorption in shale gas reservoirs is critical especially for Gas in Place (OGIP) calculations since inaccurate predictions can affect reporting of overall gas reserves. To that end, different adsorption models would have to be compared and evaluated in order to select the model that fits experimental data accurately. In examining the effect of using different error criteria for determining parameters for shale gas adsorption models, a statistically robust error analysis has been performed based on the sum of normalised error (SNE). Most shale gas adsorption modelling are conducted without finding out the most appropriate error function to use which introduces adsorption prediction errors in calculations. Five different error analysis were used including Sum of squared error (SSE), average relative error (ARE), the sum of absolute error (SAE), Marquardt’s Percent standard Deviation (MPSD), and Hybrid fractional error (HYBRID). To account for the influence of temperature in adsorption capacities, the study also compares the use of temperature dependent models, such as Exponential and Bi-Langmuir models for gas adsorption. These models can be conducted at multiple temperatures and ensure adsorption data can be obtained at any temperature beyond laboratory conditions. This is particularly useful when conducting thermal stimulation as an enhanced gas recovery in both coal/shale gas reservoirs. Journal of Oil, Gas and Petrochemical Sciences Submit your Article | www.ologypress.com/submit-article Ology Press Citation: Fianu J, Gholinezhad J, Hassan M. Comparison of Single, Binary and Temperature-Dependent Adsorption Models Based on Error Function Analysis. J Oil Gas Petrochem Sci. (2019);2(2):77-91. DOI: 10.30881/jogps.00027 78 function of pressure, but also of temperature. Section 2 of this study is, therefore, focused on describing the various single component systems, multi-component systems and finally temperature-dependent models used in the modelling of shale gas adsorption. Several works have been conducted on adsorption modelling without taking into consideration the choice of error function used in optimising the adsorption model.6,9,20–22 This often results in only one set of adsorption constants for the adsorption models being used without any serious interrogation to how accurately it fits the adsorption model to experimental data. According to Sreńscek-Nazzal et al.,23 very few detailed studies have been conducted on comparing the accuracy of the error functions used in modelling gas adsorption and also the accuracy of the predicted isotherm parameters. No study has however looked at comparing different error functions on modelling gas adsorption in shale gas reservoirs. In minimising the difference between the experimental data and the predicted results from the adsorption models, several error functions have been proposed and applied to predict optimal isotherms including sum of square error (SSE), average relative error (ARE), sum of absolute error (SAE), Marquardt’s percent standard deviation (MPSD) and Hybrid fractional error (HYBRID).23–26 Shale gas adsorption models Single Component Models Langmuir Isotherm One of the most widely used adsorption isotherms is Langmuir isotherm.6 A key assumption of Langmuir isotherm is that there must be a homogeneous surface and no interaction between the adjacent molecules. This is, however, a difficult concept to apply even in coal or shale systems, because their internal organic matter is chemically heterogeneous.6 Langmuir isotherm is given by the formulae below , 1 L V bp V bp = + Equation 1 Where V is the volume of adsorbed gas at pressure P , L V is the Langmuir volume or maximum gas adsorption at infinite pressure and b is the Langmuir constant. BET Model BET isotherm was developed in 1938 by Stephen Brunauer, P.H. Emmet and Edward Teller.27 A key assumption that was used in the derivation of this isotherm is the fact that the adsorption layers on the surface of the organic carbon were infinite. For relatively flat and non-porous surfaces, the use of Langmuir isotherm is often not valid. The BET isotherm is normally considered a better fit in describing the adsorption processes in certain shale gas reservoirs.11 The BET equation is given as