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10.1021/acsomega.4c02492
Article
End-Member Mixing Model for Methane Carbon Isotope Fractionation During Shale Gas Desorption and Its Application
Zhai Changbo †‡§
https://orcid.org/0009-0008-5000-9093
Tao Cheng *†‡§
Zou Yu §
Yang Zhenheng §
Ye Xin †‡§
https://orcid.org/0000-0002-3634-1523
Nie Haikuan †‡§
† State Key Laboratory of Shale Oil and Gas Enrichment Mechanisms and Efficient Development, Beijing 102206, China
‡ Sinopec Key Laboratory of Shale Oil/Gas Exploration and Production Technology, Beijing 102206, China
§ Petroleum Exploration and Production Research Institute, SINOPEC, Beijing 102206, China
* E-mail: taocheng.syky@sinopec.com.
21 08 2024
03 09 2024
9 35 3699337001
14 03 2024
09 08 2024
04 08 2024
© 2024 The Authors. Published by American Chemical Society
2024
The Authors
https://creativecommons.org/licenses/by-nc-nd/4.0/ Permits non-commercial access and re-use, provided that author attribution and integrity are maintained; but does not permit creation of adaptations or other derivative works (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Unlike conventional natural gas reservoirs, shale gas development involves systematic changes in methane carbon isotopes that cannot be effectively described by existing isotope fractionation models and mechanisms. Therefore, based on fundamental theories such as Rayleigh fractionation, mass transfer flow, and mass conservation, this study established isotopic fractionation equations for methane in adsorbed and free gas. By considering adsorbed and free gases as two end-members and using an isotope mixing model, a fractionation model for methane carbon isotopes during shale gas desorption was constructed. This model quantifies the isotopic fractionation effects during shale gas desorption and elucidates the mechanism of methane carbon isotope fractionation. Using on-site desorbed gas content and isotope data, parameter fitting and model calculations were conducted to characterize methane carbon isotope variations throughout the process of shale core field desorption. The results show a pattern of “initially negative and then turning positive,” consistent with those of physical simulation experiments. It was clarified that differences in mixing the two end-members and isotopic fractionation play key roles in the variation of methane carbon isotopic composition in shale gas. By applying the methane carbon isotope fractionation model, the contribution of adsorbed gas during shale gas production was explored. It was found that in the early stage of development, the adsorbed gas in Well JY 1 was negligible. After nearly seven years of development, the contribution of adsorbed gas in the later stage has only reached nearly 15%, indicating that the production contribution of adsorbed gas is still less than 0.3 million cubic feet per day. The open flow of Well JY 6–2 is more conducive to the production of adsorbed gas, but the production capacity is still mainly contributed by free gas, indicating that the shale gas production capacity in the later stage in the Jiaoshiba gas field is still primarily dominated by free gas.

National Natural Science Foundation of China 10.13039/501100001809 42130803 China Petrochemical Corporation 10.13039/501100010824 NA National Natural Science Foundation of China 10.13039/501100001809 42202175 document-id-old-9ao4c02492
document-id-new-14ao4c02492
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pmc1 Introduction

Unlike conventional natural gas, methane carbon isotopes exhibit a trend of becoming heavier during shale gas development.1−5 Some scholars hypothesize that carbon isotope fractionation during methane migration is closely related to molecular adsorption–desorption processes and have conducted simulation experiments using materials with different adsorption capacities, suggesting that molecular diffusion has no significant impact.6,7 However, many studies indicate that methane carbon isotope variation is primarily controlled by kinetic fractionation.3,8−12 Owing to the higher diffusivity of 12CH4 compared with that of 13CH4, a mass discrimination effect occurs between the two isotopes during the seepage flow process in micronanopores and low-permeability shale reservoirs, leading to significant changes in methane carbon isotopes.8−12 Molecular dynamics can be used to model and describe these processes. Previous studies suggest that the carbon isotope fractionation of methane during shale gas desorption can be divided into four stages. Initially, the free gas flows out of the shale core sample due to pressure-driven seepage, and the δ13C value of methane remains relatively constant. As the amount of free gas decreases and the proportion of adsorbed gas increases, the δ13C value of methane experiences a slight decrease. Subsequently, the gas stored in matrix pores begins to desorb and diffuse, resulting in a significant increase in the δ13C value. Finally, the diffusion of the residual adsorbed gas results in a lighter δ13C value.3 However, previous studies have shown significant differences in isotopic fractionation characteristics, lacking a clear and unified understanding due to differences in gas migration processes and observation methods. In terms of quantitative characterization, previous studies have established the diffusion fractionation model,12,13 the diffusion-adsorption/desorption coupling model,10,12 and the multiscale coupling model.3,10,14 These models have been systematically analyzed and compared in the review paper.2,15,16

However, through the continuous collection of shale core field desorption data and subsequent gas composition and isotopic analyses, the δ13C value of desorbed methane gradually increases with desorption time.1,13 The amplitude of methane carbon isotope enrichment can exceed 20‰, and there is no continuous decrease in the δ13C value in the later stages,16,17 which may be due to insufficient monitoring duration.16 The process of retrieving shale cores from the drilling site to the wellhead usually takes several hours, during which gas is lost, making it impossible to obtain gas samples from this stage.18 Consequently, isotopic data for this stage are unavailable, and our understanding of the variations in the methane carbon isotope throughout the gas desorption process remains inadequate, leading to discrepancies between the theory of isotope fractionation and shale gas production.11,19,20 Further exploration is needed to discuss the applicability of different models and to strengthen geological applications (e.g., the calculation of reserves and resources) based on isotope fractionation models.

To further clarify the variation patterns of methane carbon isotopes during shale gas production and deepen our understanding of the fractionation mechanism of methane carbon isotopes in shale gas, this study (1) establishes a mathematical model to quantitatively characterize and analyze the carbon isotope fractionation behavior of methane during shale gas desorption and (2) explores a new method for tracing the production of adsorbed gas in shale based on isotopic indicators. These efforts provide theoretical and technical support for shale gas exploration and development. This study has practical significance for shale gas exploration and production, particularly in calculating shale gas reserves.

2 Sample and Experimental Method

Samples were collected from Well JY11-4 (Table 1) in the Fuling shale gas field, located in Chongqing municipality, southwestern China (Figure 1). This is the first large-scale shale gas field in China. Upon retrieval of the shale core at the wellhead, it was promptly placed into a desorption canister and sealed. Subsequently, the shale core was subjected to on-site gas desorption using a gas content measuring instrument, with real-time recording of the desorbed gas volume. At 30 min intervals, gas samples were collected for laboratory analysis of gas composition and alkane carbon isotopes to elucidate the dynamic changes in shale gas during the desorption process.

Figure 1 (a) JY11-4 well location map. Adapted with permission from ref (21). Copyright 2022 American Chemical Society. (b) Lithological columnar section.

Table 1 Sample Information for Well JY11-4

sample ID	Depth/m	Lithology	core extraction duration (h)	period of surface exposure/h	Water saturation/%	Core weight/kg	Porosity/%	Desorption gas content/(m3/t)	Total gas content/(m3/t)	
JY11-4-10	2280.2	Black shale	6.3	16.2	47.8	3.75	4.75	0.55	1.65	
JY11-4-15	2289.7	Black shale	7.1	11.7	38.4	3.68	5.31	0.80	2.24	
JY11-4-24	2308.2	Black shale	6.4	14.9	44.0	3.48	3.94	0.83	2.24	
JY11-4-36	2333.7	Black shale	7.7	11.5	31.8	3.67	6.54	1.49	4.96	
JY11-4-38	2337.4	Black shale	7.5	11.3	39.4	3.31	5.49	1.00	3.22	

The determination of shale porosity employs the helium method. Initially, cylindrical shale samples were dried at 120 °C for 48 h. Subsequently, the skeletal volume was accurately measured using an Ultrapore-300 helium porosity meter, ensuring a balanced time of no less than 30 min. After completing these steps, the porosity of the samples is precisely calculated. The entire measurement process follows the GB/T 34533–2023 standard.

The total organic carbon (TOC) content of the samples was determined using a Leco CS230 carbon/sulfur analyzer. Samples were ground to less than 0.2 mm, pretreated with 1 mol/L HCl to remove carbonates, washed to neutrality with distilled water, and dried at 60–80 °C. The pretreated and dried samples were introduced into the carbon/sulfur analyzer to precisely determine their TOC content.

The mineral content was determined using a Rigaku SmartLab9 X-ray diffractometer. The samples were pulverized to a particle size of less than 40 μm. The testing conditions were set as follows: Cu Kα radiation, operating voltage of 40 kV, scanning step size of 0.02°, and scanning speed of 2°/min. The samples were scanned within the range of 5°–45°. The contents of clay minerals and nonclay minerals were determined using the K-value method and the adiabatic method, respectively.

The gas content measuring instrument for shale cores was developed by the Wuxi Research Institute of Petroleum Geology, Sinope, based on the drainage and gas collection method.22,23 It uses a high-precision pressure sensor to provide real-time tracking through a feedback servo motor and automatic liquid level correction to eliminate interference from water vapor and multicomponent gases on measurement. This instrument enables the automation of measurement, shortening of the desorption time, continuous data recording, and ensuring the accuracy of the results. During the desorption process of the shale cores, the temperature of the first-stage desorption was set at the mud circulation temperature, and the temperature of the second-stage desorption was 110 °C, with desorption data recorded every 30 s. Over a desorption period of approximately 12 h, more than 1,000 data points were collected, providing a comprehensive characterization of the desorption process.

Isotope analysis was conducted using a Delta V Advantage stable isotope mass spectrometer, with the following parameters set: electron impact (EI) ion source, electron energy of 91 eV, emission current of 1.5 mA, and acceleration voltage of 3.0 kV. Its mass range is 1–80 Da at 3 kV. The precision of carbon isotope testing using the continuous flow analysis method is less than 0.06‰. The main external configuration included a Trace GC gas chromatograph and HP Plot Q chromatographic column (30 m × 0.32 mm × 20 μm), with a flow rate of 2.0 mL/min. The analysis method was based on the standard SY/T 5238–2019 for organic compounds and carbonates carbon–oxygen isotope analysis.

3 Model Establishment

Research shows that micro- to mesopores are the primary contributors to the surface area of shale, with pores smaller than 2 nm being the most significant contributors to the specific surface area, accounting for up to 80%.24 The effective molecular diameter of methane is 0.38 nm, and it preferentially adsorbs in small pores. Extensive research indicates that adsorbed methane mainly exists in the state of monolayer adsorption in pores smaller than 2 nm, which can be described by the Langmuir equation.25 In contrast, in mesopores and macropores, methane is primarily present as a free gas, and its quantity is controlled by the density of free-phase methane, pore space volume, and gas saturation.26

Because of the different states of occurrence and spaces they occupy, the state transformation and seepage flow mechanisms during shale desorption processes also differ.27 Free gas mainly migrates into the fracture system through seepage flow under the influence of differential pressure, following Darcy’s law, whereas adsorbed gas molecules must first desorb and then enter the fracture system through diffusion.28−30 Therefore, different carbon isotope fractionation quantification models must be adopted to characterize the two different gas occurrence states.

3.1 Isotopic Fractionation During Adsorbed Gas Desorption

The adsorbed gas is released through desorption-surface diffusion-seepage flow, and the desorption and surface diffusion in nanoscale pores are crucial for gas flow and mass transfer. The pore size, shape, and distribution of shale directly affect the interaction between gas molecules and the pore walls, thus determining the adsorption and desorption behavior of gases.8,20 The more complex the pore morphology, the slower the desorption process. The pore network determines the migration pathways and efficiency of the shale gas. Theoretical calculations suggest that the primary factors contributing to isotope fractionation are the differences in diffusion and adsorption/desorption capabilities between 12CH4 and 13CH4 molecules,10,31,32 which are also closely related to factors such as initial pressure, pore types, pore size, and porosity. In practical production, because of the irreversibility of the adsorbed gas desorption process, the change in the amount of 12CH4 and 13CH4 molecules over time (amount decreased per unit time) can be described by the following equation:1

where Q and Q* are the quantities of 12CH4 and 13CH4 molecules at time t, respectively, Q0 and Q*0 are the quantities of 12CH4 and 13CH4 molecules at the initial time, respectively; K and K* are the analytical constants of the isotopic molecules 12CH4 and 13CH4, respectively; and α1 = K*/K is the fractionation factor of the methane isotope.

We assume that R0 and Rt are the ratios of 13CH4 to 12CH4 molecules at the initial time and time t, respectively; then,2

where f1 is the ratio of methane adsorbed at time t to the initial time (remaining ratio).

The relationship between δ and the isotopic ratio is expressed as follows:3

Assuming that δ13Cad(0) is the carbon isotopic composition of CH4 at the initial time, and δ13Cins-ad(t) is the instantaneous desorption carbon isotopic composition of adsorbed gas at any time t, based on the derivation of the isotope Rayleigh fractionation process,33 we obtain4

Equation 4 shows a practical and direct reflection of the relationship between the remaining ratio f1 of shale adsorbed gas and the isotopic ratio of δ13Cins-ad(t). By fitting the isotope fractionation factor of the adsorbed gas based on extensive isotopic data from shale gas desorption, α1 can be determined.

3.2 Isotopic Fractionation During Free Gas Diffusion

Elucidating the isotopic fractionation characteristics of free gas during the diffusion and seepage flow processes of shale gas and establishing a fractionation model for free gas were the primary objectives of this study. The seepage flow of free gas can be described using a mass transfer equation with pressure as the primary unknown quantity after simplifying it to one-dimensional linear isothermal flow. This can be described as the one-dimensional mass balance equation:105

where ϕ is the porosity,%; P is the pressure, Pa; t is the time, s; r is the distance, m; m is the shape factor (0 for flat plate, 1 for cylinder, and 2 for circle); D is defined as the seepage flow coefficient, a combination of the Knudsen diffusion coefficient (DK), viscous flow (DV), and surface diffusion (DS), where viscous flow follows Darcy’s law DV = Pk/μ (k is the permeability,%; μ is the gas viscosity, Pa·s). Parameters marked with an asterisk represent 13CH4 methane, whereas those without an asterisk represent 12CH4 methane; thus, an isotope fractionation model for free gas is established. For describing the gas diffusion in a material composed of spherical particles, Crank34 constructed a model (eq 6.20, p 91) for describing changes in the amount of diffusion gas, from which we could deduce an analytical solution to Equation 5 as follows:6

By expanding the first-order term of Equation 6 and performing differentiation, we obtain an approximate equation (Equation 7) for the variation in matrix pressure with time:7

Dividing both sides of Equation 7, we obtain8

The left side of Equation 8 corresponds to the instantaneous desorption of different methane isotopic molecular ratios of free gas. Taking the logarithm of both sides of Equation 8 and approximating ln(1 + δ/1000) ≈ δ/1000, -ln(D*/D) ≈ 1- D*/D, we obtain9

where δ13Cfree(0) is the initial carbon isotope composition of CH4 at initial time, δ13Cins-free(t) is the instantaneous diffusion carbon isotope composition of CH4 in free gas at time t, α2 = D*/D is the isotope fractionation factor, and represents the remaining ratio of free gas. Notably, a plane sheet with homologous diversity has a consistent derivation result as a spherical condition, and the difference between the accurate solution (eq 6) and the approximations (eqs 7, 8, and 9) is generally less than 1‰.10

3.3 Fractionation Model for Methane Isotopes During Shale Gas Desorption

The original shale gas reservoirs are in a closed system, and the isotopic differences between adsorbed methane and free methane are mainly influenced by the isotope’s thermodynamic fractionation between different phases.35 Bacsik et al.36 reported a less than 0.8‰ carbon isotopic fractionation of methane between adsorbed and free phases in the temperature range of 0–200 °C. This carbon isotopic fractionation is inversely related to temperature, indicating that the initial carbon isotopic values of adsorbed methane and free methane are essentially consistent under reservoir temperatures and pressures.

Adsorbed and free gases are in different states in shale gas reservoirs and occupy different storage spaces. Assuming that during shale gas development, the two states of gas proceed in parallel, forming two end members. The gas produced is a mixture of adsorbed gas and free gas. With the known isotopic composition of the end members, based on the principle of mass conservation and using the formula for the isotopic composition of a mixture of two end-member substances, we can obtain the instantaneous isotopic composition characteristics of methane during the shale gas desorption process:10

where fA and fB represent the proportions of free and adsorbed gases, respectively, with fA + fB = 1.

Based on the established two-end-member methane carbon isotopic fractionation model, combined with the cumulative desorbed gas volume, instantaneous isotopic ratios, and physical properties of shale obtained through shale desorption, we explore and deduce the variation of methane carbon isotopes during the entire process of shale desorption. This model also attempts to use the methane carbon isotopic ratio of the produced gas to rapidly estimate the proportion of adsorbed gas.

4 Results and Discussion

4.1 Results

It can be seen from Figure 2 that the methane carbon isotope of desorbed gas from Well JY11-4 in Fuling, Sichuan Basin shows a clear fractionation effect of methane carbon isotopes, with the initial average methane carbon isotope value of desorbed gas is approximately −30‰, and then gradually becomes heavier, reaching around −5‰ by the end of desorption. This phenomenon is different from the results of Ma et al.,37 who observed that there is a sudden drop in the methane carbon isotopes with desorption time when the desorption temperature increases to a higher temperature. They ascribed this to the liberation of isotopically lighter methane from small occluded micropores. The discrepancies in methane carbon isotope behavior during shale gas desorption could relate to variations in shale composition, gas content, pore structure, and desorption equipment. Therefore, further measurements and comprehensive analyses are needed to elucidate these observations.

Figure 2 Variation characteristics of methane carbon isotopes in the core gas analysis of well JY11-4

The methane carbon isotope fractionation during this process can be divided into two stages: the slow fractionation stage (at a desorbed temperature of 55 °C), primarily representing the first-stage desorption, and the rapid fractionation stage at 110 °C, representing the second-stage desorption. During the slow fractionation stage, the methane carbon isotopes of most samples slowly increase, with a fractionation range of approximately 5‰. In the second-stage desorption that occurs at 110 °C, the methane carbon isotopes rapidly increase, with a fractionation range exceeding 20‰, corresponding to substantial desorption of adsorbed methane.

4.2 Discussion

4.2.1 Isotopic Fractionation Characteristics

In the second stage, the influence of free gas can be neglected, and the methane isotopic fractionation factor of adsorbed methane can be obtained using second-stage desorption. Following the method introduced by Yu et al.,26 the total amount of adsorbed gas was determined based on the isothermal adsorption experiment using the weight method. This method is combined with the influence of TOC content, temperature, and humidity on the adsorption capacity to obtain the adsorbed gas content under reservoir temperature and pressure.26 Using the isotopic data of this stage, combined with the total adsorbed gas content and the amount of gas desorption at the sampling time, a least-squares fitting is applied using Equation 4 to obtain an approximate value of the isotopic fractionation factor (α1) of adsorbed gas.

Compared with the second stage, the first stage of desorbed gas involves a mixture of adsorbed gas and free gas.38 By subtracting the space occupied by pore water and adsorbed gas from the total pore space, the space occupied by free gas is determined. Based on the methane state formula (e.g., PR formula), the density of free gas is obtained to calculate the total amount of free gas under reservoir conditions:1811

where R is the molar gas constant with a value of 8.314 J/(mol·K); a = 0.45742R2Tc2/pc, α(T) = [1 + k(1 - Tr0.5)]2, k = 0.37464 + 1.5422ω - 0.2692219ω2, b = 0.07780RTc/pc; pc is the critical pressure in Pa; Tc is the critical temperature in K; pr is the reduced pressure; Tr is the reduced temperature; ω is the acentric factor.

Using pressure as the independent variable and combining shale physical properties and adsorption characteristics, changes in adsorbed gas and free gas can be obtained, thereby calculating the proportions of adsorbed gas and free gas at any moment during shale gas desorption, denoted as fA and fB, respectively. Based on the carbon isotopic fractionation factor (α1) of adsorbed gas, fitting of the first-stage desorption data is performed according to Formulas 9 and (10) to obtain the carbon isotopic fractionation factor (α2) of free gas. Based on this approach, we can depict the mixing ratios of the two end members (Figure 3).

Figure 3 Simulation of methane carbon isotopes during shale gas desorption; the red line is calculated based on the α1, α2, fA, and fB factors.

As shown in Table 2, the carbon isotopic fractionation factor for free methane exhibits minimal variation, with values clustering near 1, indicating that the isotopic fractionation during the migration of free gas is smaller than that of adsorbed gas. This suggests that the main transport mechanism for free gas is Darcy flow, with a limited diffusion effect and insignificant isotopic fractionation. Under conditions of pure diffusion, the carbon isotopic fractionation factor of methane is 0.9934 according to one-dimensional seepage flow isotopic fractionation experiments.39 However, most fitted carbon isotopic fractionation factors of adsorbed gas are lower than this value, indicating that the carbon isotopic fractionation effect in adsorbed gas is more significant due to effects such as surface desorption and Knudsen diffusion. Knudsen diffusion is a special mode of diffusion that occurs when the scale of the system is comparable to or smaller than the mean free path of the particles within it. If the size of the pores is smaller than the mean free path of methane molecules, then collisions between methane molecules and the pore walls occur more frequently than those between the molecules themselves. Because of the difference in the molecular masses of different isotopes, their collision frequency with pore walls and diffusion rates also vary, affecting the separation efficiency of isotopes during fractionation.

Table 2 Carbon Isotopic Fractionation Factors of Methane in Adsorbed Gas and Free Gas from Different Depths of the Core in Well JY11-4

Sample ID	TOC/%	Clay/%	Quartz/%	Carbon isotopic fractionation factor (α1) of adsorbed gas	Carbon isotopic fractionation factor (α2) of free gas	
JY11-4-10	1.23	52.8	46.3	0.9941	0.9996	
JY11-4-15	2.78	45.8	43.8	0.9885	0.9996	
JY11-4-24	1.85	39.8	45.2	0.9915	0.9996	
JY11-4-36	3.86	30.4	58.1	0.9852	0.9995	
JY11-4-38	2.62	29.8	48.6	0.9891	0.9995	

After conducting correlation analysis, the adsorbed gas did not show a significant correlation with clay minerals or quartz, with correlation coefficients (R2) of 0.550 and 0.489, respectively. In contrast, there was a significant negative correlation between α1 and TOC content. The linear relationship is expressed as α1= −0.0033 TOC% + 0.9979 (R2 = 0.996). This result demonstrates the correlation between α1 and organic matter. It is precisely because organic matter is enriched in nanopores that significant isotopic fractionation occurs. In addition, the model can be used for rapidly calculating the fractionation parameters, indicating that the conversion of adsorbed gas and its fractionation effect are the main controlling factors causing the regular variations of methane carbon isotopes. Furthermore, α1 is also negatively related to the total gas content in the five samples (see Table 1) because shale gas mainly exists in organic matter.

Shale cores typically require several hours to be brought to the surface from the wellborn, during which gas can escape, making it impossible to obtain samples and consequently lacking isotopic data for this stage. Therefore, the variations in methane carbon isotopes in the entire process of shale desorption remain unknown. Taking Well JY11-4–36 as an example, the carbon isotopic composition characteristics of methane during the entire process of shale desorption were calculated using the above fractionation model. The initial formation pressure was assumed to be 33.6 MPa, and the time correction was set with the moment of core pull-out as zero time. Figure 3 illustrates the variation in methane carbon isotopes during shale desorption. The model-derived variation curve of the later stage of desorbed gas aligns with the measured methane δ13C of the shale desorption, indicating that the desorption process is not monotonically increasing but rather exhibits a curve shape of “initially negative then positive.” These results were consistent with the experimental results obtained from the forward simulation of methane high-pressure saturation in shale cores.40 Overall, the process of shale gas desorption can be divided into three stages:(1) Early Stage: The desorbed gas is mainly in the free state, with minimal isotopic fractionation effects, resulting in a stable methane carbon isotopic composition.

(2) Middle Stage: The proportion of the adsorbed state gradually increases, reflecting a mixture of adsorbed gas and free gas. Changes in the mixing ratio and significant differences in methane carbon isotopic fractionation between adsorbed gas and free gas lead to bidirectional variation during this stage.

(3) Later Stage: The desorbed gas gradually becomes dominated by the adsorbed state, and under the Rayleigh fractionation effect, the methane carbon isotopic composition shows a monotonically increasing trend.

4.2.2 Field Application: Tracing the Proportion of Adsorbed Gas during Shale Gas Development

Determining the proportion of adsorbed gas is crucial for understanding the production patterns of shale gas.40−42 Determining the contribution of adsorbed gas and identifying when it begins to manifest is highly significant for oilfield enterprises in planning shale gas development.43 To address these questions, typical wells JY 1 and JY 6 in the Fuling shale gas field were selected. The established shale gas methane isotopic fractionation model was applied to calculate the carbon isotopic fractionation factor (α1) of adsorbed gas and the carbon isotopic fractionation factor (α2) of free gas (Table 3).

Table 3 Methane Isotopic Fractionation Factor and Shale Physical Propertiesa

 	Parameters	Well JY1	Well JY6	
Shale characteristics	Depth/m	2402	3270	
 	TOC/%	4.08	4.14	
 	Water saturation/%	50	38	
 	Obvious density/(g/cm3)	2.52	2.53	
 	Reservoir temperature/°C	82.6	103	
 	Porosity/%	4.55	4.14	
 	Langmuir volume/(m3/t)	2.64	2.45	
 	Langmuir pressure/MPa	3.45	4.15	
Isotopic fractionation model parameters	Initial isotopic ratio of methane/‰	-29.9	–29.9	
 	Carbon isotopic fractionation factor (α2) of free gas	0.9996	0.9996	
 	Carbon isotopic fractionation factor (α1) of adsorbed gas	0.9844	0.9842	
a Note: the initial δ13C value(δ13C(0)) of methane has been reported by Liu.44

Based on the parameters presented in Table 4, simulations were conducted for wells JY1 and JY6, where the pressure continuously varied from the initial values of 34.6 and 43.2 MPa to atmospheric pressure. The curves showing the variation in the isotopic ratios of adsorbed methane and free methane with changing pressure were obtained. The methane carbon isotopic values of adsorbed gas and free gas correspond to their respective pressure points, representing the characteristics of end-members A and B, respectively (Figure 4). Based on the mixing ratio of the isotopic compositions, the proportion of adsorbed gas is calculated using the following formula:12

where A% represents the proportion of adsorbed gas. δ13C(A) and δ13C(B) are the methane carbon isotopes of adsorbed gas and free gas, respectively. δ13C(Sam) denotes the methane carbon isotopic composition of the gas produced at the wellhead.

Table 4 Proportion of Adsorbed Gas in the Produced Shale Gas Calculated Based on the Isotopic Fractionation Model

Sampling time	Well	Case pressure/MPa	δ13C1 of adsorbed methane/‰	δ13C1 of free methane/‰	δ13C1 of wellhead gas/‰	The proportion of adsorbed methane/%	
2015.5.12	JY1HF	13.5	–43.80	–30.20	–30.70	3.90	
JY6-2HF	5.6	–38.50	–29.77	–30.50	8.50	
2016.6.23	JY1HF	6.37	–40.50	–29.90	–30.60	6.80	
JY6-2HF	6.67	–39.63	–29.82	–31.00	12.00	
2017.01.25	JY1HF	7.42	–41.30	–29.90	–30.50	5.00	
JY6-2HF	6.44	–39.41	–29.81	–30.50	7.20	
2017.07.26	JY1HF	6.75	–40.80	–29.90	–30.90	9.20	
JY6-2HF	5.7	–38.62	–29.76	–30.50	8.30	
2018.01.25	JY1HF	7.53	–41.40	–29.90	–30.50	4.90	
JY6-2HF	6.66	–39.62	–29.82	–30.20	3.80	
2019.03.26	JY1HF	2.39	–33.20	–29.50	–30.00	13.70	
JY6-2HF	3.18	–34.01	–29.53	–30.70	26.00	

Figure 4 Isotopic variation characteristics of adsorbed gas and free gas in Well JY1.

By regularly monitoring the methane carbon isotopic ratio of gas produced from shale gas wells and using the calculation formula for the mixing of isotopes from two end-members, the proportion of adsorbed gas can be rapidly estimated. Well JY 1HF is the first exploration well in the Fuling shale gas field and has the longest development period. Well JY 6–2 HF is a representative well with a typical open flow method. Shale gas samples were collected annually from 2015 to 2019, and their isotopic compositions were analyzed to estimate the proportion of adsorbed gas. During the early stages of development (2015.5 to 2017.1), the contribution of adsorbed gas was minimal. However, with the rapid pressure decline associated with open flow, the proportion of adsorbed gas production increased accordingly (Table 4). In the middle and later stages of development (2017.1 to 2019.3), after a period of relative stability in the proportion of adsorbed gas, a rapid increase was observed. By early 2019, the estimated proportion of adsorbed gas in Well JY 1HF approached 15%, whereas in Well JY 6–2HF, it reached 26%. The production of Well JY 1HF decreased from a steady level of approximately 60,000 m3 per day to approximately 20,000 m3 per day. Over nearly 7 years, the contribution of adsorbed gas production was still less than 3,000 m3 per day, whereas in Well JY 6–2HF, it was approximately 18,000 m3 per day. Overall, it can be seen that in the later stages of development, the produced gas is still mainly dominated by free gas.

5 Conclusion

(1) A fractionation model for methane isotopes was constructed using a mixture of isotopes from the two end-member gases to quantify the isotope fractionation effect during shale gas desorption and elucidate the carbon isotope fractionation mechanism.

(2) The proposed fractionation model accurately predicts the variation in carbon isotope during shale gas desorption, suggesting that the disparity between dual-source mixing and isotope fractionation effects is the primary controlling factor in the carbon isotope composition variation of methane in shale gas.

(3) The fractionation model was employed to calculate the proportion of adsorbed gas in the Jiaoshiba shale gas field. Results show that the contribution from the adsorbed state is negligible during the early stages of production; however, its share progressively increases in later stages, occasionally surpassing 20%. While open flow enhances the production of the adsorbed gas, the free gas predominates the gas production.

The authors declare no competing financial interest.

Acknowledgments

This project was funded by the National Natural Science Foundation of China (Grant No. 42130803, No. 42202175) and Several Sinopec in-house Projects. We thank Sinopec Petroleum Exploration and Production Research Institute, Sinopec Exploration Company, Sinopec Jianghan Oilfield, and Sinopec East China Petroleum Company for providing their valuable data and information. We also thank Sinopec management for permission to publish this work.
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