Investigation of the isosteric heat of adsorption for supercritical methane on shale under high pressure

Abstract

The isosteric heat of adsorption (IHA) is one of the key thermodynamic variables for evaluating the interaction between shale and methane, which is rarely studied especially under high pressure. In this work, we conducted methane adsorption experiments at pressures up to 30 MPa and different temperatures on shale samples collected from Longmaxi formation in Sichuan Basin, China. Based on the definition of IHA and Langmuir adsorption model, we proposed a new method to analyze the IHA of methane on shale under four conditions. The calculated results show that the commonly used Clausius–Clapeyron equation overestimates the true isosteric heat of shale, especially under high pressure. IHA under four conditions yield a fixed order as q_st,i-va > q_st,r-va > q_st,i+va > q_st,r+va, indicating both the real gas behavior and the adsorbed-phase volume have a negative influence on it, and the effect of adsorbed-phase volume is dominant. Moreover, IHA at zero coverage ( $q_{st}^{0}$ ) in Henry region determined by linear fitting can be regarded as a maximum value in the above four cases, which is independent of pressure and temperature. Therefore, $q_{st}^{0}$ can be used as a unique descriptor to evaluate the adsorption affinity of the shale. This work modified the method to obtain the true IHA of supercritical methane on shale more accurately, which lays the foundation for future investigations of the thermodynamics and heat transfer characteristics of the interaction between high pressure methane and shale.

Keywords

Shale gas isosteric heat of adsorption supercritical methane adsorbed-phase volume Langmuir equation

Introduction

Shale gas has received increasing attentions in recent years due to its remarkable success of commercial production in North America (Curtis, 2002; EIA, 2016; Montgomery et al., 2005), which has also promoted the exploration and development of shale gas in many other countries. The exploration of shale gas reservoirs in China, particularly in southern Sichuan Basin, has made significant progress in the past few years (Zou et al., 2015). Shale gas mainly consists of free gas and adsorbed gas, and exists in nanopores of organic matter and clay minerals (Ambrose et al., 2012; Chen et al., 2016; Zhou et al., 2016). Gas-in-place, the sum of free gas and adsorbed gas, is a key parameter for economic evaluation of a shale gas reservoir. Therefore, accurately determining the adsorbed gas capacity is critical for the assessment of shale gas reserves and the design of effective production strategies (Ambrose et al., 2012; Curtis, 2002).

Methane adsorption in gas shales has been extensively studied by isothermal adsorption experiments (Chareonsuppanimit et al., 2012; Clarkson and Haghshenas, 2013; Tian et al., 2016; Zhang et al., 2012; Zhou et al., 2018a ) and molecular dynamics simulations (Mosher et al., 2013; Xiong et al., 2017), mainly focusing on calculating adsorbed gas quantity and evaluating its effect on shale gas transport (Wu et al., 2016). However, its thermodynamic characteristics have been less reported (Chen et al., 2019; Rexer et al., 2013; Tang et al., 2017; Tian et al., 2016; Zhang et al., 2012). Previous studies have shown that shale gas adsorption belongs to physisorption, which is mainly controlled by London dispersion forces, the weakest type of van der Waals forces (Tang et al., 2017). Therefore, its adsorption heat is only 10–22 kJ/mol, which is significantly lower than that for chemical adsorption (Zhang et al., 2012). The isosteric heat of adsorption (IHA) is one of the key thermodynamic variables for evaluating the interaction between adsorbate and adsorbent, serving as an important descriptor of the shale gas physisorption system (Askalany and Saha, 2015; Chakraborty et al., 2006; Pan et al., 1998; Sircar et al., 1999).

In previous literatures, researchers commonly used the Clausius–Clapeyron (C-C) equation to calculate its IHA after conducting adsorption experiments under different temperatures (Gasparik et al., 2014; Ji et al., 2015; Tian et al., 2016; Wu et al., 2016; Zhang et al., 2012). The C-C equation is an oversimplified model that ignores the volume of adsorbed phase and uses the ideal gas law (Pan et al., 1998). However, neither of the two conditions can be satisfied for high pressure methane adsorption in shale because of the thermodynamic properties of methane (Askalany and Saha, 2015; Chakraborty et al., 2006). Methane in shale formations is typically at the supercritical state as the formation temperature and pressure are much higher than its critical temperature and pressure (190.65 K and 4.64 MPa), so the volume of adsorbed phase cannot be assumed to be negligible and the ideal gas law is no longer applicable (Do and Do, 2003; Zhou et al., 2000, 2018b). Therefore, the calculated results by the C-C equation may not reveal the true IHA of methane in shales and the influence of the adsorbed-phase volume and the real gas behavior on the IHA are also needed to be investigated.

Moreover, prior to calculating IHA, we need to transform the excess (observed) adsorption isotherms into the absolute adsorption isotherms for determining the volume of adsorbed phase by an appropriate adsorption model (Tang et al., 2017; Tian et al., 2016; Zhou et al., 2019). Many models have been proposed and applied to study high-pressure methane adsorption in shale, such as the extended Langmuir model, Dubinin-Radushkevich model, Dubinin-Astakhov model, simplified local density model, Ono-Kondo lattice model and so on (Chareonsuppanimit et al., 2012; Tian et al., 2016; Zhou et al., 2019). However, the most commonly used model in shale gas reservoirs is still the classic Langmuir equation, which assumes that the surface of the solid adsorbent is homogeneous with constant adsorption heat and methane is monolayer-adsorbed (Langmuir, 1918; Yu et al., 2014; Zhang et al., 2012). Although this model is very simple, it can accurately evaluate the adsorption capacity of shale (Gasparik et al., 2014; Zhang et al., 2012; Zhou et al., 2019). Thus, in this study, we use the Langmuir-based excess adsorption model to fit the observed isotherms assuming a constant adsorbed-phase density.

In this study, the methane adsorption isotherms of Longmaxi shales are obtained at pressures up to 30.0 MPa and different temperatures based on the gravimetric method. The objectives of this study are to (1) evaluate the applicability of the C-C equation to calculate the IHA of shale and (2) establish a suitable method for determining the true IHA of supercritical methane on shale under high pressure.

IHA calculation method

Supercritical methane adsorption model

The critical temperature of methane is only 190.65 K, so the adsorption of methane in shale is supercritical adsorption under formation temperature and pressure (Do and Do, 2003; Zhou et al., 2000, 2019). Under supercritical conditions, the excess adsorption capacity and absolute adsorption capacity have great differences, and these differences cannot be ignored at high pressures. Their relationship is defined as

n_{ex} = n_{a} (1 - \frac{ρ_{g}}{ρ_{a}}) = n_{a} - ρ_{g} v_{a}

(1)

where n_ex is the excess adsorption, n_a is the absolute adsorption, ρ_g is the density of free methane at a given temperature and pressure, ρ_a is the density of adsorbed phase, and v_a is the volume of adsorbed phase.

Previous studies have shown that the Langmuir-based excess adsorption model can still fit the high-pressure adsorption isotherms very well (Tian et al., 2016; Zhou et al., 2019). The classic Langmuir adsorption model is

n_{a} = n_{\max} \frac{K (T) P}{1 + K (T) P}

(2)

K (T) = A_{0} e^{- \frac{E_{0}}{RT}}

(3)

So the supercritical adsorption model base on Langmuir equation can be derived by equations (1) to (3), as shown by equation (4)

n_{ex} = n_{\max} \frac{A_{0} e^{- \frac{E_{0}}{RT}} P}{1 + A_{0} e^{- \frac{E_{0}}{RT}} P} (1 - \frac{ρ_{g}}{ρ_{a}})

(4)

θ = \frac{n_{a}}{n_{\max}} = \frac{K (T) P}{1 + K (T) P}

(5)

where n_max is the maximum absolute adsorption, K is the Langmuir constant, A₀ is the pre-exponential coefficient, E₀ is the adsorption energy, R is the universal gas constant, 8.314 J/(mol·K), θ is the coverage ratio on the surface of absolute adsorption, P is the equilibrium pressure, and T is the equilibrium temperature.

This model (equation (4)) can be used directly to fit the high-pressure adsorption isotherms measured in the experiment and obtain the unknown parameters n_max, A₀, E₀, and ρ_a. Then the excess adsorption capacity (n_ex) can be converted to absolute adsorption capacity (n_abs) by equation (1), and the volume of adsorbed phase (v_a) can be calculated to determine its IHA.

IHA determination

The IHA is defined as the differential change in energy that occurs when an infinitesimal number of molecules are transferred at constant pressure, temperature, and adsorbent surface area from the bulk gas phase to the adsorbed phase, and it can be expressed as (Chakraborty et al., 2006; Murialdo et al., 2015; Pan et al., 1998).

q_{st} = T {(\frac{dP}{dT})}_{n_{a}} (v_{g} - v_{a})

(6)

where q_st is the IHA, v_g is the volume of bulk gas phase, v_a is the volume of adsorbed phase, P is the equilibrium pressure, and T is the equilibrium temperature.

For real gas, its volume should be calculated by equation (7). If we assume methane is ideal, its volume can be calculated by equation (8).

v_{gr} = \frac{ZRT}{P} = \frac{1}{ρ_{gr}}

(7)

v_{gi} = \frac{RT}{P} = \frac{1}{ρ_{gi}}

(8)

where Z is the gas compression factor, ρ_gr is the density of real gas, and ρ_gi is the density of ideal gas. If we neglect the adsorbed-phase volume, combining equations (6) and (8), IHA can be transformed as

q_{st} = \frac{R T^{2}}{P} {(\frac{dP}{dT})}_{n_{a}} = R T^{2} {(\frac{d (\ln P)}{dT})}_{n_{a}}

(9)

The above equation is the classic C-C equation (Askalany and Saha, 2015; Pan et al., 1998; Whittaker et al., 2013;). It is obvious that two approximations are introduced in deriving this equation: (1) the bulk gas phase is considered ideal and (2) the adsorbed-phase volume is neglected. Integration of equation (9) leads to

\ln P = - \frac{q_{st,C-C}}{RT} + C

(10)

Here, q_{st, C-C} is the IHA determined by the C-C equation, which is commonly used in shale gas reservoirs (Tang et al., 2017; Tian et al., 2016; Zhang et al., 2012). C is a constant after integration. In a certain amount of adsorption, q_{st, C-C} can be calculated by the linear slope of the variation of lnP with T. Due to the high pressure of shale gas reservoir, methane will deviate significantly from the ideal gas law and the volume of methane adsorption phase under supercritical conditions cannot be negligible. Therefore, the IHA calculated by the C-C equation will definitely be deviated from the formation situation and more detailed analysis is needed.

If the absolute methane adsorption can be characterized by the Langmuir model (equation (2)), we can get the following differential formulas by equations (3) and (5).

{(\frac{dP}{dT})}_{n_{a}} = {(\frac{\partial P}{\partial θ})}_{n_{a}} \times {(\frac{\partial θ}{\partial K})}_{n_{a}} \times {(\frac{\partial K}{\partial T})}_{n_{a}}

(11)

{(\frac{\partial P}{\partial θ})}_{n_{a}} = \frac{{(1 + KP)}^{2}}{K}

(12)

{(\frac{\partial θ}{\partial K})}_{n_{a}} = \frac{P}{{(1 + KP)}^{2}}

(13)

{(\frac{\partial K}{\partial T})}_{n_{a}} = \frac{E_{0} K}{R T^{2}}

(14)

Combining the above four equations, we can obtain

{(\frac{dP}{dT})}_{n_{a}} = \frac{E_{0} P}{R T^{2}}

(15)

Substituting equation (15) into equation (6), the IHA can be calculated by the following equation

q_{st} = \frac{E_{0} P}{RT} (v_{g} - v_{a})

(16)

Therefore, for the IHA of methane on shale, the following four conditions can be considered (Table 1):

Table 1.

The isosteric heat of adsorption determination equations and its applicable conditions.

Isosteric heat of adsorption determination equation	Gas behavior	Volume of adsorbed phase
$q_{{st,i-v}_{a}} = E_{0}$ , equation (17)	Ideal gas	Neglecting
$q_{{st,r-v}_{a}} = Z E_{0}$ , equation (18)	Real gas	Neglecting
$q_{st,i+v}_{_{a}} = \frac{E_{0} P}{RT} (v_{gi} - v_{a})$ , equation (19)	Ideal gas	Without neglecting
$q_{st,r+v}_{_{a}} = \frac{E_{0} P}{RT} (v_{gr} - v_{a})$ , equation (20)	Real gas	Without neglecting

Assuming methane is ideal and neglecting the volume of adsorbed phase, q_st can be expressed as

q_{{st,i-v}_{a}} = E_{0}

(17)

Assuming methane is real and neglecting the volume of adsorbed phase, q_st can be expressed as

q_{{st,r-v}_{a}} = Z E_{0}

(18)

Assuming methane is ideal and without neglecting the volume of adsorbed phase, q_st can be expressed as

q_{st,i+v}_{_{a}} = \frac{E_{0} P}{RT} (v_{gi} - v_{a})

(19)

Assuming methane is real and without neglecting the volume of adsorbed phase, q_st can be expressed as

q_{st,r+v}_{_{a}} = \frac{E_{0} P}{RT} (v_{gr} - v_{a})

(20)

Experiments and data analysis

High-pressure methane adsorption experiment

Shale samples were collected from the Lower Silurian Longmaxi Formation in southern Sichuan Basin, China. The high-pressure adsorption isotherm experiments were conducted on Rubotherm gravimetric adsorption instrument (Dreisbach et al., 2002). Its core component is the magnetic suspension balance (MSB) with high precision of 10 μg. The maximum test pressure and the temperature are 35 MPa and 150°C, respectively. The long time fluctuation range of temperature can be controlled within 0.2°C.

The experimental procedure includes three steps: blank test, buoyancy test, and adsorption test. The first step is to test the mass and volume of sample container. After the shale sample is put in the adsorption cell, the second step is to test the mass and volume of sample in the container. And then the third step is to test methane adsorption amount at a constant temperature and increasing pressures. The observed adsorption quantity can be calculated by the following equation (Zhou et al., 2019).

m_{ex} = Δ m - m_{sc} - m_{s} + (V_{sc} + V_{s}) \times ρ_{g}

(21)

Here, Δm is the MSB reading, m_sc is the mass of the sample container, m_s is the mass of the sample, and m_abs is the mass of adsorbed methane. V_sc is the volume of the sample container, V_s is the sample volume, and ρ_g is the density of free methane at a given temperature and pressure. Note that ρ_g can be tested directly by this gravimetric adsorption instrument at different pressure points (Zhou et al., 2019).

Supercritical methane adsorption isotherms

In this study, four high-pressure adsorption isotherms of methane on shale were measured at 303.15 K, 313.15 K, 323.15 K, and 333.15 K, as shown in Figure 1, which shows that the excess adsorption capacity (n_ex) increases to a maximum value at a pressure of approximate 10 MPa, and then n_ex begins to decline with pressure. Although this abnormal phenomenon has also been observed in some previous studies (Bi et al., 2016; Ottiger et al., 2008; Rexer et al., 2013; Tang et al., 2016; Zhou et al., 2019), most researchers have stated that the adsorption isotherm of methane in shale monotonically increased with pressure and reached a constant value at a high pressure (i.e., typeIisotherm) (Ji et al., 2015; Zhang et al., 2012). The main reason for these two different types of adsorption isotherms is whether the volume of the adsorbed phase (v_a) is assumed to be negligible. This assumption is reasonable as long as the density of the free gas is much lower than the density of the adsorbed phase at low pressures (Do and Do, 2003). However, when the pressure becomes high enough, the volume of the adsorbed phase (v_a) cannot be neglected, and excess adsorption capacity will decrease definitely according to equation (1).

Figure 1.

The observed excess adsorption isotherms at 303.15 K, 313.15 K, 323.15 K, and 333.15 K.

Fitted parameters and absolute adsorption

All four isotherms were then fitted simultaneously with the Langmuir-based excess adsorption model (equation (4)) using a least-squares residual minimization algorithm. The four independent fitting parameters were varied to achieve the global minimum of the residual-squares value within the following limits: 0 < n_max < 0.5 mmol/g, A₀ > 0, 0 < E₀ < 40 kJ/mol, 0 < ρ_a < 0.423 g/cm³, where the adsorption energy (E₀) is independent of temperature.

Figure 2 and Table 2 show the fitted results by the Langmuir-based excess adsorption model (equation (4)). It can be seen that this model fits the high-pressure adsorption isotherms quite well. The maximum methane adsorption capacities (n_max) fitted by this model ranges from 0.1511 to 0.1834 mmol/g. And the fitted adsorbed-phase density (ρ_a) ranges from 0.3310 to 0.3893 g/cm³, which is lower than the liquid density of methane (0.423 g/cm³). Detailed analysis of the adsorbed-phase density was described in our previous work, showing the fitted results are reasonable (Zhou et al., 2019).

Figure 2.

The fitting results of the excess adsorption isotherms of shale sample at different temperatures using Langmuir-based excess adsorption model (solid lines). The solid points represent the observed adsorption capacity under different pressures.

Table 2.

Fitted parameters by Langmuir-based excess adsorption model (equation (4)) at four temperatures.

T (K)	n_max (mmol·g^–1)	A₀ (MPa^–1)	E₀ (J·mol^–1)	ρ_a (g·cm^–3)	K (MPa^–1)	R ²
303.15	0.1834	711.420	17,421.43	0.3893	0.7083	0.9965
313.15	0.1686	499.934	17,421.43	0.3480	0.6206	0.9954
323.15	0.1602	365.280	17,421.43	0.3403	0.5578	0.9953
333.15	0.1511	275.145	17,421.43	0.3310	0.5105	0.9959

After the adsorbed-phase density is determined (Table 2), we can transform the excess adsorption to the absolute adsorption by equation (1). And then the volume of adsorbed phase also can be obtained at different pressures and temperatures by this equation, which will be used for analyzing the IHA. Figure 3 shows the absolute adsorption isotherms at different temperatures, indicating the absolute adsorption capacity increases monotonically as pressure increases. When the experimental pressure reaches 25 MPa, the adsorption capacities tend to be saturated and absolute adsorption capacities hold at maximum values. Moreover, it clearly shows that the adsorption amount decreases with increasing temperature, which proves that the adsorption is an exothermic process (Zhang et al., 2012).

Figure 3.

The transformed absolute adsorption isotherms at different temperatures.

Discussion

IHA calculated by C-C equation

Based on the absolute adsorption amount (n_a) of methane calculated by the Langmuir-based excess adsorption model, the IHA can be determined by the C-C equation via the plot of ln (P) versus 1/T under different n_a, as illustrated in Figure 4. The IHA calculated by the C-C equation increases with increasing adsorption capacity, which was also observed by Wu et al. (2016). However, in some other studies, the IHA was considered to be decreased with increasing adsorption amount (Rexer et al., 2013; Tian et al., 2016).

Figure 4.

(a) The linear relationship between ln (p) and 1/T under different adsorption amount. (b) IHA calculated by classical C-C equation (q_st,C-C) using the slope of the fitted lines in left figure. It can be seen clearly that q_st,C-C increases with the increase of n_a.

Generally, there are two reasons to explain the variation trend of IHA with the increase of n_a (Myers, 2002). On the one hand, the mutual attraction between CH₄ molecules on the adsorbent surface will be enhanced with the increase of adsorption amount, which will lead to the increasing IHA with the increase of n_a. On the other hand, the adsorption sites on the adsorbent surface have different potential due to the anisotropy of shale. During the process of adsorption, CH₄ molecules tend to occupy the stronger sorption sites first, which will lead to the decrease of IHA with the increase of n_a. Therefore, in this study, the IHA is mainly controlled by the mutual attraction between CH₄ molecules on the shale surface and the anisotropy of adsorption sites has a minor effect.

Volume of bulk gas and adsorbed phase

Four different conditions have been listed in “IHA determination” section and their corresponding IHA can be calculated by equations (17) to (20), where the key parameters are the volume of bulk gas (v_g) and adsorbed phase (v_a). When we consider methane as ideal gas, its bulk gas volume (v_gi) can be simply determined by equation (8). However, if we consider methane as real gas, its bulk gas volume (v_gr) is associated with its compressibility factor (Z) which should be determined first. In this study, the gravimetric adsorption instrument can output methane density at every equilibrium pressure points, so we can obtain its compressibility factor experimentally, as shown in Figure 5. When pressure ranges from 0 to 30 MPa, we can see that the compressibility factor decreases at first and then increases. The compressibility factor reaches a minimum value at about 15 MPa, indicating a maximum deviation of real methane from the ideal gas law.

Figure 5.

Methane compressibility factor under different pressures and temperatures.

As the compressibility factor is determined, we can calculate its real volume of bulk gas by equation (7), as shown in Figure 6. We can see v_gr is slightly lower than v_gi because the compressibility factor is less than 1.0. In principle, the adsorbed-phase volume (v_a) should increase during the process of adsorption which is different from that shown is Figure 6. The reason is that the unit of v_a used here is cm³/mol for calculating its IHA with the unit of J/mol. As we fitted the isotherms by assuming the density of adsorbed phase (ρ_a) is a constant value, the value of v_a with the unit of cm³/mol will definitely not change. In addition, as the difference between v_g and v_a decreases, it can be predicted that the IHA will also be decreased with the increase of pressure.

Figure 6.

Variation trend of the volume of bulk gas and adsorbed phase during the process of adsorption. v_gi is the bulk gas volume considering methane is ideal and v_gr is the bulk gas volume considering methane is real. It should be noted that the unit is cm³/mol, leading to an invariable volume of adsorbed phase (v_a) due to the assumption of a constant adsorbed-phase density.

Comparison of IHA under four conditions

A comparison of the IHA calculated under four conditions (equations (17) to (20)) is shown in Figure 7. Overall, the different IHA follow a similar behavior at different temperatures, and their relationship can be expressed as q_st,i-va > q_st,r-va > q_st,i+va > q_st,r+va. In order to interpret the effect of real gas behavior and adsorbed-phase volume on IHA, we take the comparison results at 303.15 K as an example (Figure 7(a)). In the first case, q_st,i-va equals to the adsorption energy of shale (E₀), remaining unchanged during the adsorption process. Logically, since both this case and C-C equation have the same assumptions, the calculation results of IHA should be the same. Their difference is due to the use of the idealized and homogenized Langmuir equation in the situation one, which leads to a constant isosteric heat. Under the second condition, q_st,r-va decreases at first and then increases with the increase of adsorption capacity. This phenomenon is caused by the variation trend of compressibility factor of methane shown in Figure 5. Moreover, if we consider the volume of adsorbed phase, IHA decline rapidly during the adsorption process, especially at high pressure.

Figure 7.

Effect of temperature on IHA of methane on shale under four different conditions calculated by equations (17) to (20), respectively. (a) qst,i-va, (b) qst,r-va, (c) qst,i+va and (d)qst,r+va.

In general, regardless of the gas law employed, the adsorbed-phase volume significantly affects the IHA, especially under high pressure conditions: without considering this effect leads to a higher IHA. And for cases considering the finite volume of the adsorbed phase, the difference between the real gas and ideal gas density also affects the IHA significantly: the ideal gas law always corresponds to a higher IHA. Moreover, it can be seen clearly that the commonly used C-C equation caused a significant overestimation of the IHA of methane in shale, proving that this equation is not applicable in shale gas reservoir.

A separate analysis of the change of IHA with temperature under each condition is shown in Figure 8. In all cases, temperature generally has a negative effect on the IHA: the higher the temperature, the lower the IHA. When pressure is low the IHA has not changed much, but there will be a significant change under high pressure conditions. For shale gas reservoirs, its formation pressure is about 35 MPa given a pressure coefficient of 12 MPa/km and a depth of 3000 m in Longmaxi formation in China, where the assumptions of ideal gas and neglecting adsorbed-phase volume are certainly not valid. Therefore, only the IHA calculated under the fourth condition can represent the actual IHA of methane on shale.

Figure 8.

Effect of temperature on methane on shale under four different conditions at different temperatures calculated by equations (17) to (20), respectively. (a) 303.15k, (b) 313.15K, (c) 323.15K and (d) 333.15K.

IHA at zero coverage

As can be seen from the above discussion, IHA is a variable and therefore cannot be used as a uniform evaluation criterion for adsorption capacity evaluation. Steele (1973) proposed an IHA in Henry region at zero coverage ( $q_{st}^{0}$ ) to obtain important information about the mechanism and properties of adsorption, where $q_{st}^{0}$ calculated would be equal to the maximum theoretical value of IHA (Do et al., 2008; Fan and Chakraborty, 2018; Prasetyo et al., 2018). In this region, the experimentally measured data are fitted with Henry's isotherm equation as given by

n_{a} = K_{H} P

(22)

where K_H indicates the Henry's coefficient as a function of temperature (T), which can be determined in a form of virial expansions of adsorption amount and pressure (Tang et al., 2017).

\ln (n_{a} / P) = a_{0} + a_{1} n_{a} + a_{2} n_{a}^{2} + a_{3} n_{a}^{3} + \dots

(23)

where a₀ is the first virial coefficient related to the Henry’s coefficient, as K_H = exp(a₀). When n_a is small, the higher order terms can be neglected, and then equation (23) can be written as

\ln (n_{a} / P) = a_{0} + a_{1} n_{a}

(24)

Therefore, a₀ can then be obtained by fitting the linear region of ln (n_a/P) as a function of n_a. The IHA in the Henry’s region is written as (Fan and Chakraborty, 2018; Steele, 1974; Zhang et al., 2012)

q_{st}^{0} = R \frac{d (\ln K_{H})}{d (1 / T)}

(25)

Employing the isotherms data, $q_{st}^{0}$ can be determined by the slope of the straight line of ln (K_H) versus 1/T. The fitting results are shown in Figure 9 and Table 3. The Henry’s coefficients are determined by fitting the first three points in low-pressure region, ranging from 0.11 to 0.24 mmol·g⁻¹·MPa⁻¹. Then the IHA at zero coverage ( $q_{st}^{0}$ ) over the entire temperature range is calculated as 21.08 kJ/mol, which is bigger than the value determined by the analytical method as described above (Figures 7 and 8). When the adsorption amount is low, the IHA under four conditions discussed above approach a same limit value, which is the $q_{st}^{0}$ calculated in this section. We can conclude that $q_{st}^{0}$ is the maximum value of IHA and is independent of pressure and temperature. Therefore, IHA at zero coverage can be used as a unique index to evaluate the adsorption affinity of the shale.

Figure 9.

Determination of the Henry’s coefficient (K_H) and IHA at zero coverage ( $q_{st}^{0}$ ) by linear fitting, respectively.

Table 3.

Analysis results of the isosteric heat of adsorption at zero coverage.

T (K)	Linear fitting of ln (n_a/P) versus n_a	R ²	a₀ (mmol·g^–1)	K_H (mmol·g^–1·MPa^–1)
303.15	y = –15.118x–1.4064	0.9769	–1.4064	0.2450
313.15	y = –15.206x–1.7306	0.9786	–1.7306	0.1772
323.15	y = –15.143x–1.9626	0.9800	–1.9626	0.1405
333.15	y = –15.362x–2.1640	0.9812	–2.1640	0.1149

Conclusions

The IHA is one of the key thermodynamic variables for evaluating the interaction between shale and methane. In order to analyze its characteristics under high pressure, methane adsorption isotherms at pressures up to 30 MPa and different temperatures were measured on shale samples collected from Longmaxi formation in Sichuan Basin, China. The conclusions drawn from this study can be summarized as follows:

The commonly used C-C equation for IHA determination was established assuming that methane is ideal and neglecting the volume of adsorbed phase. IHA calculated by this equation overestimates the true isosteric heat of shale, especially under high pressure.

IHA under four conditions were analyzed, yielding a fixed order as q_st,i-va > q_st,r-va > q_st,i+va > q_st,r+va. Both the real gas behavior and the adsorbed-phase volume have a negative influence on it, and the effect of adsorbed-phase volume is dominant.

IHA at zero coverage ( $q_{st}^{0}$ ) in Henry region, which is the maximum value in the four cases and independent of pressure and temperature, can be used as a unique index to evaluate the adsorption affinity of the shale.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by National Science and Technology Major Project (No. 2017ZX05035002-002).

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