Empirical Formula

According to the corresponding state principle, gas deviation factor *Z* could be introduced to describe real gas behaviors:

$$pV = ZnRT,\tag{8}$$

where *p* is the pressure, Pa; *V* is the gas volume, m3; *Z* is the gas deviation factor, dimensionless; *n* is the molar mass, kg/mol; *R* is the universal gas constant, J/(mol K); and *T* is temperature, K.

In Equation (8), gas deviation factor *Z* could be obtained by experiments, referring to *<sup>Z</sup>*-plate [44], or calculating from an empirical formula [27]. The *<sup>Z</sup>*-plate and empirical formula were mainly obtained by assuming *Z*c as a constant in the range of 0.23–0.29. That is to say, gas deviation factor *Z* is a function of the reduced pressure *p*r and reduced temperature *T*r. Therefore, we also name the *<sup>Z</sup>*-plate as the two-parameter generalized compressibility chart. The critical gas deviation factor *Z*c of most materials varies in the range of 0.23–0.29. Therefore, a more precise expression of *Z* is expected to be obtained by regarding *Z* as a function of *p*r, *T*r, and another parameter ( *Z*c or acentric factor ω)—a three-parameter relationship [44].



Note: *a*e and *b*e are energy and volume parameters, respectively, and their expressions vary in different EOS; *Z*c is the critical gas deviation factor.

## 2.6.2. Adsorbed Gas

Adsorbed gas is mainly stored on the surface of matrix particles, kerogen, and clay minerals, and can account for 20–85% of total gas reserves [19,45–47]. Gas adsorption on shale matrix particles belongs to physical adsorption [48]. Although adsorbed gas contributes to the total gas production, its exact contribution is not clear. Compared to the contribution of adsorbed gas in total gas production, the percentage of adsorbed gas in original gas in place (OGIP) is much clearer. Recent studies have found that adsorbed gas accounts for 50–80% of OGIP when the pressure is lower than 13.79 MPa, while it accounts for 30–50% when the pressure is higher than 13.79 MPa [49,50].

Organic nanopores of 5–750 nm, are significantly developed in shale gas reservoirs when the maturity of organic matter is larger than 0.6% [28,30]. Due to the small pore radius, organic matter has a large surface area. For example, the specific surface area can be up to 300 m<sup>2</sup>/g in nanoporous kerogen [51]. The enormous surface area provides favorable places for gas adsorption. The gas adsorption capacity is mainly a ffected by pore structures, mineral compositions, metamorphism degree, gas components, pressure, temperature, water vapor content, etc. Since adsorbed gas is mainly stored in organic matter, the TOC content significantly a ffects the adsorbed gas content in shale gas reservoirs. As we can see from Figure 18, there is a positive relationship between the adsorbed gas content and TOC content in di fferent shale gas reservoirs. This is because large TOC content means more organic matter in the shale matrix, which can provide su fficient space for gas storage due to its large surface area.

Taking the Barnett Shale as an example, as shown in Figure 18, we can distinguish di fferent gas types from the relationship between adsorbed gas and the total gas amount in shale matrix. Adsorbed gas and free gas in the organic matrix increase as the TOC content increases, while the amount of free gas in the inorganic matrix is not a ffected by the TOC content. In shale gas plays, adsorbed gas is a non-ignorable component in shale gas reserve calculation [27,39,53–55], and gas ad-/desorption is important in the study of gas flow behaviors [56–59]. If we analyze the case further, we could ask how much adsorbed gas could be produced during shale gas production, and how significantly gas ad-/desorption a ffects gas transient flow behaviors in shale gas reservoirs.

**Figure 18.** Adsorbed gas in different shales and its relationship with TOC content (revised from [50,52]).

The gas transporting process of CH4 and He in organic shale samples was compared by an experimental study at 3.4 MPa and 308 K [60], where CH4 serves as the adsorptive gas and He is non-adoptive. The production dynamics of CH4 and He can be seen in Figure 19a. Assuming the free gas amount of CH4 and He is equal in shale samples, the adsorbed volume of CH4 can be obtained by the difference between the total CH4 production volume and the total He production volume, which are 2.60 cm<sup>3</sup>/g and 1.33 cm<sup>3</sup>/g respectively. Therefore, the produced volume of free gas for unit mass shale particles under standard conditions is 1.33 cm<sup>3</sup>/g, while the adsorbed gas amount is 1.27 cm<sup>3</sup>/g. Similarly, simulation results from dynamic adsorption diffusion model show that the production of free gas dominates at an early production period (before point *A*) and drops very fast, while adsorbed gas dominates the later production after point *A* for a relatively long time. Experimental study and model simulation signified that both free gas and adsorbed gas played an important role in gas production.

The above research is conducted at low pressure (3.4 MPa) and temperature (308 K) compared to practical shale gas reservoir conditions. Gas desorption pressure in shale is usually below 12 MPa, which is close to the abandonment pressure of shale gas reservoirs. Meanwhile, the formation pressure is mainly depleted in a small area near the wellbore, i.e., the average pressure of shale formation is much higher than the abandon pressure. Consequently, gas desorption may only occur in a small area near the wellbore or hydraulic fractures, meaning a limited amount of adsorbed gas is produced during the life cycle of the shale gas reservoir. The significance of adsorbed gas, as well as the corresponding seepage mechanisms, need further investigation.

Assuming adsorbed gas volume is a function of pressure, Tang et al. [61] obtained the absolute adsorbed gas amount from excess adsorption and studied the adsorbed gas proportion to total gas at different shale depths (Figure 20). The conventional absolute adsorption refers to results obtained by fitting low and intermediate pressure sorption data using the Langmuir model of Equation (10), while the new absolute adsorption refers to a dual-site Langmuir model considering the adsorbed layer variation and excess adsorption. The conventional model severely underestimates the absolute adsorption amount when the pressure is higher than 6 MPa, as shown in Figure 20a. The percentage of adsorbed gas to total gas in place (GIP) is a function of shale formation depth, where it increases fast in shallow areas and slows down after 2000 m, as shown in Figure 20b. The adsorbed gas accounts for approximately 40–80% at different depths of formation. Meanwhile, the excess adsorption amount needs to be corrected to the absolute adsorption amount when considering the adsorbed gas percentage

in GIP. Otherwise, it will massively underestimate the adsorbed gas amount and overestimate the free gas amount.

 **Figure 19.** Experimental data (**a**) and mathematical simulation results (**b**) for gas transport process at 3.4 MPa and 308 K [60].

#### 2.6.3. Dissolved Gas

After the equilibrium between adsorption and desorption is found, shale gas could dissolve into the liquid hydrocarbon or formation water during the hydrocarbon accumulation process. Meanwhile, organic kerogen continuously generates shale gas and contains a certain amount of gas molecules [62]. The gas in liquid hydrocarbon, formation water, and kerogen is called the dissolved gas, which has been overlooked, but may play a significant role in shale gas reservoir development [40,59]. Gap-filling and hydration are the two main storage mechanism of dissolved gas, and can be described theoretically by Henry's law [63]:

$$\mathbb{C}\_b = \frac{p}{\mathbb{K}\_c},\tag{9}$$

where *C*b is the mole concentration of dissolved gas, mol/m<sup>3</sup> and *K*c is the Henry constant, m<sup>3</sup> Pa/mol.

Since it is hard to differentiate dissolved gas from adsorbed gas, both gas types are usually attributed to one type, namely adsorbed gas. Moreover, adsorbed gas and dissolved gas can be transformed to the other under proper circumstances. Therefore, it can be roughly seen as one type in some cases [48,64].

 **Figure 20.** Adsorption amount versus pressure (**a**) and adsorbed gas percentage versus shale formation depth (**b**) in different models [61].

#### **3. Gas Adsorption and Desorption**

Shale gas can be stored on pore surfaces of organic matter and clays by gas adsorption. Organic matter in the shale matrix is a key parameter that influences gas adsorption characteristics in shale gas reservoirs. On the one hand, a large amount of nanopores are developed in organic shales, which provide enormous surface area for the gas to be adsorbed on. On the other hand, the adsorption potential is significant in organic nanopores compared to in inorganic nanopores or large organic pores. The adsorption-desorption law in organic shale nanopores is a key scientific problem in the practice of shale gas development, a ffecting the accuracy of evaluating shale adsorption capacity, studying the seepage flow behaviors, and developing transient seepage mathematical models [65].

#### *3.1. Di*ff*erent Sorption Types and Models*

Methane is the main component of shale gas underground, with a critical pressure of 4.59 MPa and a critical temperature of 190.53 K. Therefore, shale gas is in a supercritical state under in situ formation conditions (3000–6000 m, with high pressure up to 60 MPa) [66]. The study of supercritical gas sorption is essential for an accurate understanding of adsorption and desorption mechanisms in shale gas reservoirs. Gas sorption mechanisms are quite confusing, and no unified conclusions have been reached. Monolayer adsorption, multilayer adsorption, and micropore filling are three common assumptions in shale gas sorption research. Based on these assumptions, the Langmuir model, BET model, the Dubinin–Radushkevich (D-R) model, and the Dubinin-Astakhov (D-A) model have been established to fit the sorption data, and have obtained good results. However, good fitting results do not guarantee the validity of the assumption in the adsorption models. For example, even if the Langmuir model fits the experimental data very well, we cannot say that gas adsorption belongs to monolayer adsorption.

#### 3.1.1. Monolayer Adsorption Type

Monolayer adsorption means that gas molecules adsorb on the pore surface in one layer, so the thickness of adsorbed gas equals the molecular diameter. Due to the huge surface area in a shale matrix, a considerable amount of adsorbed gas exists in shale gas reservoirs. Assuming 80% gas saturation in shale samples, the adsorbed gas ratio can be 22.65% of the total gas amount according to the Ono-Kondo lattice model established by Zhou et al. [67].

(1) Langmuir model [68]: The Langmuir model was established by monolayer adsorption assumption. Due to its simplicity and accurate fitting to the experimental data, it is widely used to describe monolayer gas adsorption, which can be written as:

$$G\_a = G\_m \frac{bp}{1 + bp} \,\tag{10}$$

where *G*a is the absolute adsorbed gas amount; *G* m is the maximum adsorbed capacity, *p* is the pressure, and *b* is the Langmuir sorption constant, which can be obtained by fitting the experimental or field test data.

(2) Freundlich model [69]: With pressure decreasing, the Langmuir equation of Equation (10) approaches the Henry's law of Equation (9). Therefore, the Henry's law can describe low-pressure sorption behaviors, since any sorption isotherm satisfies the linear relationship between adsorption amount and pressure at low pressure. To broaden the application range of the Henry's law into high-pressure areas, an exponential empirical formulation, namely the Freundlich model, was used:

$$\mathbf{G}\_{\rm il} = k\_f p^{1/n\_f},\tag{11}$$

where *kf* is related to the adsorption interaction and adsorption amount; *nf* is a constant usually between 2 and 3, reflecting the intensity of adsorption. The values of both *kf* and *nf* depend on the type of adsorbent and adsorbate as well as the temperature.

With temperature increasing, constant *nf* approaches to unity and the Freundlich model of Equation (11) becomes the Henry model of Equation (9). The Freundlich model can properly describe monolayer adsorption, especially for low-concentration gases and in the meso pressure range. However, there is no explicit physical meaning of the constants *kf* and *nf*, and it cannot explain the mechanisms of adsorption.

(3) Langmuir-Freundlich model: To modify the assumption of uniform adsorption sites in the Langmuir model, the Freundlich equation and the Langmuir equation were coupled to form a new adsorption model, namely the Langmuir-Freundlich model, which is:

$$\mathbf{G}\_a = \mathbf{G}\_m \frac{bp^l}{1 + bp^l} \tag{12}$$

where *l* reflects the heterogeneity of adsorbents, *l* ≤ 1. The smaller the value of *l*, the stronger the heterogeneity of the adsorbent. If an adsorbent possesses an ideal surface, *l* tends to be 1 and the Langmuir-Freundlich equation is equivalent to the Langmuir equation.

(4) Toth model [69]: To improve the fitting capacity of the Langmuir model, the Toth model was proposed:

$$G\_{\rm d} = G\_m \frac{bp}{\left[1 + (bp)^t\right]^{1/t'}} \tag{13}$$

where *t* is a constant related to the adsorbent properties.

Note that the Toth model of Equation (13) solves two problems: (1) the Freundlich model of Equation (11) and the DR model of Equation (16) do not satisfy Henry's law at low pressure; and (2) no maximum adsorption amount appears in the Freundlich model of Equation (11) with increasing pressure. Bae et al. [69] found that the Toth equation fitted the experimental data better than the extended three-parameter and Langmuir equation, and yielded realistic values of pore volumes of coal samples and adsorbed gas density.

#### 3.1.2. Multilayer Sorption Type

(1) Two-parameter BET model [32]: Multilayer adsorption can be modeled by the BET sorption theory, which assumes that gas molecules can adsorb on a solid surface by infinite layers and no interaction exists between contiguous layers. In other words, any monolayer obeys the Langmuir adsorption theory in the BET model, which can be expressed as follows:

$$\frac{G\_d}{G\_{\rm m}} = \frac{\mathbb{C}\_b(p/p\_s)}{(1 - p/p\_s)[1 + (\mathbb{C} - 1)(p/p\_s)]},\tag{14}$$

where *G*m is the maximum monolayer adsorption amount, *p*s is the saturated vapor pressure, and *C*b is the dimensionless constant controlling the time of multilayer adsorption.

In applying Equation (14), its equivalent expression needs to be adopted, which is Equation (1). A plot of (*p*/*p*s)/[*G*a(1 − *p*/*p*s)] versus *p*/*p*s is employed to figure out whether the adsorption follows the BET theory. If the plot satisfies the linear relationship in the range of 0.005 < *p*/*p*s < 0.35, we can use the scope and intercept of the straight line to obtain the values of *G*m and *C*.

(2) Three-parameter BET model [44,69]: The above two-parameter BET model of Equation (14) assumes infinite layers of adsorption. If the adsorption layers are finite, the three-parameter BET model can be employed:

$$\frac{G\_{\rm tr}}{G\_{\rm m}} = \frac{\mathbb{C}\_{b}(p/p\_{\rm s})}{(1 - p/p\_{\rm s})} \frac{\left[1 - (n+1)(p/p\_{\rm s})^{n\_{\rm b}} + n\_{b}(p/p\_{\rm s})^{n\_{\rm b} + 1}\right]}{\left[1 + (\mathbb{C}\_{b} - 1)(p/p\_{\rm s}) - \mathbb{C}\_{b}(p/p\_{\rm s})^{n\_{\rm b} + 1}\right]}.\tag{15}$$

If *n*b = 1, Equation (15) simplifies into the Langmuir model of Equation (10); if *n*→∞, Equation (15) transforms into the two-parameter BET model of Equation (14). Note that the three-parameter BET model of Equation (15) is applicable to describe adsorption behaviors for *p*/*p*s in the range of 0.35–0.60.

Note that, although in theory the multilayer adsorption assumption may produce a wider application scope with the BET model than the Langmuir model, it may not be suitable to adopt the

BET theory in a shale gas adsorption study, since sorption behaviors in shale gas reservoirs belong to supercritical adsorption and the saturated vapor pressure *p*s of shale gas (mainly methane) does not exist in practice. Besides, as reported, the BET model may have a poorer performance than the Langmuir model at fitting absolute sorption data, as reported by [70].

#### 3.1.3. Micropore Volume Filling

#### Dubinin–Radushkevich and Dubinin-Astakhov Models

Gas molecule behavior in nanopores is significantly different from that in mesopores or macropores, since there is a superposition of adsorption potential from both pore sides. Consequently, the adsorption force of micropore walls on gas molecules is much greater than in mesopores or macropores, leading to large adsorption. Dubinin named the gas adsorption in these small-scale pores micropore volume filling [71]. Compared to the Langmuir adsorption theory, micropore volume filling is more helpful for understanding the gas adsorption mechanism and gas true storage forms, and for evaluating gas adsorption properties. Meanwhile, it has been reported that the D-A model provides a better fit to sorption data of coal than the Langmuir model [72].

The condensed adsorbate looks like microemulsion droplets when adsorption occurs in micropores, which is greatly affected by interfaces. Based on the Polanyi adsorption potential theory [73], the D-R model and D-A model are commonly used in shale gas adsorption studies, and can be expressed as follows:

$$\mathcal{W} = \mathcal{W}\_0 \exp\{-D[\ln(p\_s/p)]^m\},\tag{16}$$

where *W* is the pore volume filled with gas molecules at relative pressure *p*/*p*s; *W*0 is the total volume of micropores; *D* is a parameter related to the adsorbate-adsorbent system; and *m* is a parameter ranging from 2 to 6, reflecting the heterogeneity of potential energy on adsorbent surfaces. If *m* constantly equals 2, Equation (16) is the D-R model. If *m* is a random parameter between 2 and 6, then Equation (16) is the D-A model.

In applying Equation (16), *W* is equivalent to the absolute adsorbed gas amount *Ga*, and *W*0 is equivalent to the maximum adsorbed gas amount *Gm*, Equation (16) can be re-expressed as follows:

$$G\_{\mathfrak{a}} = G\_{\mathfrak{m}} \exp \left| -D[\ln(p\_{\mathfrak{s}}/p)]^{\mathfrak{m}} \right|. \tag{17}$$

Compared to the D-R model, the D-A model performs better fitting with experimental data, according to the study of Wang et al. [70]. This is because the chosen range of structure heterogeneity parameter *m* in the D-A model is broader than that in the D-R model, which is related to pore size distribution in shale formations. The chosen parameter *m* in the D-A model brings the micropore structure information into adsorption prediction and modeling, while it has a constant value of 2 in the D-R model, without considering the structural heterogeneity in shale samples.

#### Calculation of Virtual Saturated Vapor Pressure

Note that the above D-R and D-A models were not initially proposed for supercritical adsorption, but for subcritical sorption. Therefore, the concept of saturated vapor pressure in the D-R and D-A model was replaced by virtual saturated vapor pressure or supercritical adsorption limited pressure [72]. Generally, the virtual saturated vapor pressure can be calculated by the following empirical formulations or approaches:

(1) The first is the Dubinin method [71,74,75]:

$$p\_{\text{s}} = p\_{\text{c}} \left(\frac{T}{T\_{\text{c}}}\right)^2,\tag{18}$$

where *pc* is the critical pressure and *Tc* is the critical temperature.

(2) The second is the Reid method [72]:

$$p\_s = p\_c \exp\left[\frac{T\_b}{T\_c} \times \frac{\ln p\_c}{1 - \frac{T\_b}{T\_c}} \times (1 - \frac{T\_c}{T})\right],\tag{19}$$

where *Tb* is the boiling point of gas at atmospheric pressure.

(3) The third is the Antoine method [74]:

$$p\_s = 0.1 \times \exp(B\_A - \frac{\mathcal{C}\_A}{D\_A + T}). \tag{20}$$

Equation (20) is a three-parameter vapor pressure equation, where the extrapolation technique and saturated vapor pressure data under subcritical conditions are needed. For methane, three parameters can be obtained: *B*A = 8.784, *C* A = 933.51, and *D* A = −5.37, respectively [74]. Then, saturated vapor pressure could be calculated by Equation (20).

(4) The fourth is the Astakhov method [72], the calculation results of which fit the experimental data well in the interval of *ps* from 0.1 MPa up to critical pressure *pc* [71]. Meanwhile, it should be noted that this method gives satisfactory results for temperatures exceeding the critical temperature by 50–100 K.

$$p\_{\mathbb{S}} = \exp(\frac{c\_A}{T} + d\_A)\_{\prime} \tag{21}$$

 where parameters *cA* and *d*A are determined by the gas critical point ( *Tc*, *pc*) and boiling point (*Tb*, 101,325 Pa). For methane, *cA* = −1032.693 and *dA* = 6.945.

(5) The Amankwah method [76] is an improved calculation method for the Dubinin method, which involves a parameter *kA* to account for interactions in an adsorbate-adsorbent system:

$$p\_s = p\_\varepsilon (\frac{T}{T\_c})^{k\_A} \tag{22}$$

where *k*A is a parameter accounting for interactions in the adsorbate-adsorbent systems.

(6) The linearization of isotherm adsorption data is another processing method [77] to extend the D-R and D-A models into supercritical area. By transforming isotherms from *G*ex versus *p* space to ln[ln *G*ex] − 1 versus ln*p* space, where *Gex* is the excess adsorption amount, a bunch of fitting straight lines could be obtained; they converge to a single point *B*, as shown in Figure 21. This merge point *B* is defined as the limiting state of the adsorbate, corresponding to the extreme condition of the adsorption potential field, where no more adsorptive molecules can enter the adsorbent micropores [77]. Therefore, the limiting pressure and limiting adsorption amount of the merge point correspond to the saturated vapor pressure and the saturated adsorption amount in the D-R and D-A models, respectively.

In order to compare these six di fferent methods, the experimental data in Zhang's study for organic-rich Woodford shale [78] are collected for analysis at temperatures of 308.53 K, 323.53 K, and 338.53 K, respectively, as shown in Figure 22a. The linearization processing method of adsorption data in Zhou's study is adopted, which transforms the sorption data into three straight lines and defines a limiting state at the intersection point *A* in Figure 22b. The other five methods are also employed to calculate the virtual saturated vapor pressure for the same shale sample and sorption data. The calculated results are shown in Figure 23, which exhibits grea<sup>t</sup> di fferences to the results from di fferent methods. The results from Antoine and Astakhov have obviously larger values than the others, while the linearization processing results, which are independent of temperature, have much smaller values than the others. Dubinin and Reid's results are basically the same, with slightly lower values than in the Amankwah results. Dubinin's method only considers the properties of the individual adsorbates, while the Amankwah method also takes adsorbent properties into consideration. Moreover, the value of parameter *k* is obtained by nonlinear fitting for isotherm sorption data, which is more practical than the constant value in the Dubinin method. As a result, the Amankwah method is recommended to calculate virtual saturated vapor pressure.

**Figure 21.** Linearized adsorption isotherms of hydrogen [77], where *p* is in KPa and *Gex* is in mmol/g.

**Figure 22.** The sorption experimental data of Woodford shale (**a**) [78] as well as their linearized isotherms (**b**).

Different models

**Figure 23.** The calculated virtual saturated vapor pressure for Woodford shale at different temperatures.

Note that the linearization processing method was proposed to tackle sorption problems in the ranges of 77–298 K and 0–7 MPa [77]. The pressure and temperatures in shale gas reservoirs are generally beyond this range, so this method is not recommended in shale gas sorption studies.

#### *3.2. Sorption Study in Supercritical Area*

#### 3.2.1. Gibbs Excess Adsorption

For high-pressure and -temperature sorption, Gibbs excess adsorption is adopted to describe its unique behaviors [79]. Generally, adsorbed gas and bulk gas both exist for an adsorbate-adsorbent system, where adsorbed gas is distributed on the pore surface as a layer and bulk gas is far from the surface. Bulk phase gas is also distributed in the adsorption layer, which is irrelevant to gas-solid molecular interactions and can be ignored at low pressure. However, it needs to be considered in shale gas sorption research, since the in situ pressure is high in shale gas reservoirs (>30 MPa) [65]. Therefore, the excess adsorption amount corresponds to the part that is larger than the bulk phase density in the adsorption layer. The difference between the absolute adsorption and the excess adsorption can be seen in Figure 25, where the absolute adsorption (Figure 25e) consists of excess adsorption (Figure 25c) and bulk phase gas in the adsorbed layer (Figure 25d). The relationship can be expressed as follows:

$$\mathbf{G}\_{\rm ex} = \mathbf{G}\_{\rm a} - \rho\_{\mathcal{K}} \upsilon\_{\rm ad} \,\prime \tag{23}$$

where *Gex* is the Gibbs excess adsorption amount, *Ga* is the absolute adsorption amount, and *vad* is the adsorbed gas volume, as can be seen in Figure 25.

The measured adsorption amount in high-pressure sorption experiments is the Gibbs excess adsorption amount *Gex* in Equation (23). The relationship has another explanation: namely, the adsorbed gas is under the effect of bulk phase gas buoyancy. Thus, the measured adsorbed gas weight equals the difference between absolute adsorbed gas weight and the buoyancy it received in bulk-phase gas.

For adsorbed gas, the following relationship exists:

$$v\_{ad} = \frac{G\_d}{\rho\_{ad}}\,\,\,\,\,\tag{24}$$

where ρ*ad* is the adsorbed gas density. Then, incorporating Equations (23) and (24), we can obtain:

$$\mathbf{G}\_{\rm cx} = \mathbf{G}\_{\rm d} (1 - \frac{\rho\_{\mathcal{S}}}{\rho\_{\rm ad}}) . \tag{25}$$

Associating Equation (25) with the calculation methods for bulk and adsorbed gas density in Sections 3.4.1 and 3.4.2, the simulation results of absolute adsorption amount *G*a can be transformed into measured excess adsorption amounts *G*ex. In low-pressure sorption studies, the bulk gas density ρg is much lower than the adsorbed gas density ρad, and the excess adsorption amount *G*ex is approximately the same as the absolute adsorption amount *G*a. However, gas density ρg becomes comparable to adsorbed gas density ρad with pressure increasing, as to be introduced in Section 3.4.2. Thus, the di fference between the absolute adsorption amount *G*a and the excess adsorption amount *G*ex cannot be ignored and there is a maximum on the plot of measured adsorption amount versus pressure or bulk phase gas density, as can be seen from Figure 24a. The location of the maximum depends on the interaction between adsorbate and adsorbent, as well as the thermodynamic state of the adsorptive [80].

**Figure 24.** Fitting of models to CO2 adsorption data (**a**) and residuals of curves from di fferent models to CH4 adsorption data, and the base line means the measured equals to the predicted (**b**) [81].


**Figure 25.** Scheme of absolute adsorption amount and excess adsorption amount.

#### 3.2.2. Supercritical Adsorption Models

The introduction of the virtual saturated vapor pressure in Section 3.1.3 extends the micropore filling models from subcritical area into supercritical range. However, Sakurovs et al. [81] noted that this method cannot easily accommodate adsorption at conditions where both the pressure and temperature are above the critical values. Since the adsorbed gas density is greater than the free gas density in supercritical conditions, another method, which replaced the saturated vapor pressure *ps* by adsorbed phase gas density ρ*ad*, and gas pressure *p* by gas density ρ*g*, was proposed to extend the volume-filling models to a wider pressure and temperature application range [72]. Based on this idea and associating Equation (17) with Equation (25), we obtain:

$$G\_{\rm ex} = G\_{\rm m} (1 - \rho\_{\mathcal{S}} / \rho\_{\rm ad}) \exp \left\{ -\mathcal{D} [\ln (\rho\_{\rm ad} / \rho\_{\mathcal{S}})]^m \right\}. \tag{26}$$

Similarly, other models in Section 3.1 can also be transformed into this form, i.e., replacing gas pressure *p* and saturated vapor pressure *ps* with gas density ρ*g* and adsorbed gas density ρ*ad*, respectively, and employing (1 − <sup>ρ</sup>*g*/ρ*ad*) to correct for the true adsorbed gas amount. For example, the Langmuir equation can be transformed into the following form:

$$G\_{cx} = G\_m(1 - \frac{\rho\_\mathcal{F}}{\rho\_{ad}})\frac{b\_r \rho\_\mathcal{F}}{1 + b\_r \rho\_\mathcal{F}},\tag{27}$$

where *br* is a constant similar to the Langmuir constant, which has the relationship ρ*L* = 1/*br*. Langmuir density ρ*L* refers to the gas density at which the adsorption amount is half of the maximum.

The transformation of other models occurs in the same way. A previous study [82] pointed out that gas density is the most meaningful variable in high-pressure sorption areas, so it is recommended that they be used in high-pressure studies instead of pressure.

## *3.3. Adsorption*/*Absorption Models*

Reucroft [83] reported that CO2 dissolved in coal and caused it to swell in addition to being adsorbed on the coal surface. When studying gas sorption behaviors on polymers, Sato et al. [84] found that gas can not only adsorb on a solid surface, but also can be absorbed into the interior of solid material. Larsen [85] proposed a similar two-component sorption on coal samples. Adsorbed gas and dissolved gas exist on the kerogen surface and in the kerogen interior, respectively, which restrict and connect with each other in shale gas reservoirs [38,41]. To model the two types of sorption (adsorption and absorption), di fferent methods were proposed in previous studies.

The first category is the hybrid type, namely the superposition of gas adsorption law and gas absorption law. Gas adsorption can be described by the abovementioned adsorption models, such as the monolayer adsorption models or the micropore filling models, while gas absorption is described by Henry's law. Meanwhile, supercritical sorption characteristics need to be considered in high-pressure and high-temperature sorption studies. Here, we take the Langmuir adsorption and D-A models as examples to introduce a hybrid method, and other adsorption models, such as the Freundlich, Toth, and D-R models, can be handled by the same procedure.

(1) Langmuir/Henry combination: In this model, gas adsorption is modeled by the Langmuir equation, and gas absorption is described by a term proportional to pressure, following Henry's law. Here, the subcritical adsorption and absorption are described in terms of pressure, as in Equation (28), while supercritical adsorption and absorption are in terms of gas density [81] as in Equation (29):

$$\mathcal{G}\_a = \mathcal{G}\_m \frac{bp}{1+bp} + kp\_\prime \tag{28}$$

$$\mathcal{G}\_{\rm ex} = \mathcal{G}\_m (1 - \frac{\rho\_{\mathcal{S}}}{\rho\_{\rm ad}}) \frac{b\_r \rho\_{\mathcal{S}}}{1 + b\_r \rho\_{\mathcal{S}}} + k \rho\_{\mathcal{S}}.\tag{29}$$

(2) Volume filling/Henry combination: Gas adsorption is described by the D-R or D-A model, and gas absorption is described by Henry's law. For subcritical adsorption and absorption, this can be described in terms of pressure:

$$G\_{\rm tr} = G\_{\rm m} \exp\left\{-D\left[\ln\left(p\_{\rm s}/p\right)\right]^{m}\right\} + kp.\tag{30}$$

For supercritical adsorption and absorption description in terms of gas density [81], this is:

$$\mathcal{G}\_{\rm ex} = \mathcal{G}\_{\rm tr} (1 - \rho\_{\mathcal{S}} / \rho\_{\rm ad}) \exp\left\{-\mathcal{D} [\ln(\rho\_{\rm ad} / \rho\_{\mathcal{S}})]^{\rm m}\right\} + k \rho\_{\mathcal{S}}.\tag{31}$$

If gas pressure is adopted in a supercritical adsorption model [86], then the virtual saturated vapor pressure concept introduced in Section 3.1.3 needs to be employed, i.e.:

$$G\_{ex} = G\_m(1 - \rho\_\mathcal{g}/\rho\_{ad}) \exp\left\{-D[\ln(p\_s/p)]^m\right\} + kp.\tag{32}$$

(3) Swelling contribution: Dissolved gas usually swells the solid materials after absorption [83,85]. If the swelling contribution was equal to the condensed gas volume in adsorbents, Equations (28)–(32) need to be modified, because the swelling occupies space that would otherwise be taken up by gases [81]. The term (1 − <sup>ρ</sup>g/ρad) needs to be multiplied by the absorption term. Taking the supercritical volume filling/Henry combination as an example, the model considering swelling contribution can be expressed as follows:

$$\mathcal{G}\_{\rm rx} = G\_{\rm nr} (1 - \rho\_{\mathcal{S}} / \rho\_{\rm ad}) \exp\left\{-D \left[\ln(\rho\_{\rm ad} / \rho\_{\mathcal{S}})\right]^{\rm nr}\right\} + k \rho\_{\mathcal{S}} (1 - \rho\_{\mathcal{S}} / \rho\_{\rm ad}),\tag{33}$$

$$G\_{\rm ex} = G\_m(1 - \rho\_{\rm g}/\rho\_{\rm ad}) \exp\left\{-D[\ln(p\_s/p)]^m\right\} + kp(1 - \rho\_{\rm g}/\rho\_{\rm ad}).\tag{34}$$

Sakurovs et al. [81] compared the calculation results from the Langmuir model of Equation (27), the D-R model of Equation (26), the Langmuir/Henry combination model of Equation (29), and the D-R/Henry combination model of Equation (31) using CO2 and CH4 adsorption data, as shown in Figure 24a,b. The fitting e ffects of the Langmuir and the D-R models are both poor, while the Langmuir/Henry or D-R/Henry combination model has a much better e ffect. This means that the added term (*kp*) in adsorption models improves the fitting e ffect and reduces the calculated surface sorption capacity. The improvement of this term is greater in the D-R model than in the Langmuir model, since the D-R/Henry combination fits the measured sorption data better than the Langmuir/Henry combination. This sugges<sup>t</sup> that the gas sorption mechanism in coal is more likely the volume filling, rather than monolayer coverage.

(4) Bi-Langmuir adsorption model: Assuming absorption and swelling are related, Pini et al. [82] applied the Bi-Langmuir model [87] to describe the combination of adsorption and absorption, where linear superposition is adopted for each Langmuir adsorption term, i.e.:

$$\mathbf{G}\_{a} = \mathbf{G}\_{m}^{a} \frac{b\_{a\varGamma} \rho\_{\varwp}}{1 + b\_{a\varGamma} \rho\_{\varwp}} + \mathbf{G}\_{m}^{a\varGamma} \frac{b\_{a\varGamma} \rho\_{\varwp}}{1 + b\_{a\varGamma} \rho\_{\varwp}},\tag{35}$$

where the first term on the right side of the equation is the adsorption term, and the second term is the absorption term.

Assuming the Langmuir equilibrium constants for adsorption and absorption are equal [82], i.e., *bad* = *bab*, the excess adsorption amount can be expressed as follows:

$$\mathbf{G}\_{a} = \mathbf{G}\_{t} \frac{b\rho\_{\mathcal{S}}}{1 + b\rho\_{\mathcal{S}}} - \rho\_{\mathcal{S}} v\_{\text{ad}\prime} \tag{36}$$

where *G*t is the sum of maximum adsorption amount and maximum absorption amount.

This Bi-Langmuir model was also compared to the experimental data and the D-R/Henry combination model, as shown in Figure 26. Both models fitted the excess sorption data well, but the fitted curve for absolute sorption data from the D-R/Henry combination model is much higher than the experimental data, while the Bi-Langmuir model fitted the absolute sorption data excellently. This is caused by the neglect of the swelling e ffect and the assumption of unlimited sorption capacity in the D-R/Henry combination model. Therefore, it should be seen as an empirical approach to describe excess sorption isotherms, and cannot be used for gas storage capacity estimation. Contrarily, the Bi-Langmuir model has a solid physical basis from experimental observations of a saturation-limited equilibrium between the gas phase and the condensed phase [82]. From this point, the Bi-Langmuir model is more suitable for adsorption and absorption modeling of shale gas.

#### *3.4. Physical Properties Calculation of Shale Gas*

#### 3.4.1. Bulk Gas Properties

Gas Density

(1) Calculated by EOS: Generally, the bulk phase gas density can be calculated by the real gas EOS, as mentioned in Section 2.6.1. After calculation, we can also obtain the relationship between gas density and pressure by the nonlinear fitting technique, which expresses density in terms of pressure:

$$\rho\_{\mathcal{R}} = c\_0 + c\_1 p + c\_2 p^2 + c\_3 p^3 + \dotsb,\tag{37}$$

where *c*0, *c*1, *c*2, and *c*3 are fitting parameters.

As we can see from Figure 27a, free gas density decreases with increasing temperature when 0.1 MPa < *p* < 30 MPa and increases with increasing pressure when 273.13 K < *T* < 373.13 K. Compared to temperature, the influence of pressure on free gas density is more obvious. Since the change in pressure is more marked than that of temperature during shale gas reservoir development, more attention needs to be paid to the change in free gas density with pressure. The influence of temperature on free gas density is the most severe at a pressure of around 15 MPa, while it is weaker at lower or higher temperatures.

**Figure 26.** The comparison of Bi-Langmuir model with CO2 experimental sorption data on coal and D-R/Henry combination model.

**Figure 27.** Free gas density (**a**) as well as its variance (**b**) in different pressure and temperature conditions.

(2) Measured by experiments: Bulk phase gas density can also by measured by experiments. Since analytical modeling is the main method introduced in this article, experimental measurement apparatus and procedures are not introduced.
