A Novel Method for Calculating Diffusion Coefficient of Shale Gas Reservoirs: A Case Study of Longmaxi Formation in Weiyuan Area, Sichuan Basin, China

Abstract

The entire process of shale gas generation, migration, and accumulation involves the diffusion of shale gas, and it is impossible to disregard the harm that gas diffusion does to gas reservoirs. The research object for this paper is the Longmaxi Formation shale gas reservoir in the Weiyuan area of the Sichuan Basin. Based on Fick’s diffusion law, an innovative mathematical model of shale gas diffusion is established, and it is clarified that the diffusion amount mainly depends on the free gas content and the diffusion coefficient. Based on the theory of fluid dynamics, the calculation equation of formation paleo-pressure is innovatively deduced. The equation fully considers gas migration, temperature and pressure changes, and the pressure control effect of organic matter gas generation, and restores the evolution history of free gas content in the reservoir. The evolution process of temperature, pressure and stratigraphic physical properties in the study area is the first to calculate the diffusion evolution history and cumulative diffusion amount of shale gas reservoirs in the study area, the reliability of the calculation results is verified by geochemical parameters. Studies have revealed that the existing Longmaxi Formation shale in the Weiyuan area of Sichuan Basin varies from 14.10 to 16.50 × 10⁴ m³/m² per unit area, with an average diffusion loss of 0.30 × 10¹² m³ gas in the positive part and 0.30 × 10¹² m³ in the negative part. The total lost gas accounts for around 1.72 times the present recappable reserves and is estimated to be 0.43 × 10¹² m³ in volume. It is clear that the migration and accumulation of natural gas are significantly influenced by the research of diffusion loss.

1. Introduction

Gas production and preservation are critical factors that govern shale gas enrichment [1,2,3]. The exploration practice in the Sichuan Basin revealed significant differences in shale gas production among various structural regions, indicating that preservation plays a vital role in shale gas enrichment [4,5,6].

Molecular diffusion occurs in geological settings when concentration gradients are present, which are crucial for the migration and preservation of natural gas [7,8,9,10]. Losses due to diffusion have a direct impact on the size of the gas accumulation in shale, the long-term accumulation of shale gas, and the cumulative amount of diffusion that can ruin shale gas reservoirs with industrial exploitation value [11,12,13,14]. Shale gas reservoirs, with their unique features, are classified in a way that constantly causes upwardly diffusing gas to lower the thermal fluid activity and mix with the gas reservoir. The generation, storage, and dispersal effects maintain the shale gas reservoir in a stable dynamic equilibrium state when the gas source is continuously supplied [15,16,17,18]. The persistence of this dynamic equilibrium in a region for natural gas migration and accumulation is a long-term evolutionary process therefore, studying the diffusion history of shale gas reservoirs is equally important as studying their hydrocarbon generation and reservoir development history, and it is a crucial component of understanding the matching history of shale gas accumulation [19,20,21,22]. In consideration of regional resources, accurate appraisal and prediction of shale gas reserves are of paramount importance due to the precise calculation of shale gas diffusion losses [23,24,25,26].

The study of natural gas diffusion in conventional gas reservoirs has made significant progress. Most studies focus on the microscopic diffusion mechanism of conventional natural gas and coal measure strata, as well as the experimental methods for measuring the diffusion coefficient, and calculating the diffusion volume in conventional gas reservoirs [27,28,29,30,31] however, there is a shortage of quantitative research on the dynamic changes of shale gas diffusion, and the current studies on shale gas diffusion are mainly qualitative assessments of preservation circumstances.

Although some researchers have quantitatively studied the diffusion loss of natural gas in the hydrocarbon source rock, their assumptions often differ significantly from the actual geological conditions. For example, some scholars believe that shale pores are all filled with water, ignoring the presence of free gas in shale pores, thus taking a small part of dissolved gas concentration dissolved in formation water as the initial concentration of diffusion. They also assume that the concentration of dissolved gas is only a function of depth, ignoring factors such as pyrolysis of organic matter supplying the diffusion source and the diffusion consumption of the diffusion source, and thus they have incorrectly calculated the amount of natural gas diffusion [32,33,34,35].

To address this issue, this paper comprehensively applies Fick’s diffusion equation and fluid dynamics theory, taking into account the conditions of gas migration, temperature and pressure changes, and organic matter gas generation. Using the shale interval of the Longmaxi Formation in the study area as an example, the study recovers the shale gas diffusion history of the Longmaxi Formation and clarifies the plane distribution characteristics of shale gas diffusion. The reliability of the calculation results is verified by geochemical parameters. Thus, the results can provide theoretical guidance and technical support for the exploration and development of marine shale gas in southern China.

2. Geological Settings

The Weiyuan area is situated in the southwestern region of the Sichuan Basin. The major shale gas production formation in Weiyuan is the Longmaxi Formation, which was formed during the Early Silurian period. This formation has a sedimentary thickness ranging from 200 to 500 m. In the study area, the Longmaxi Formation pinches out in the up-dip direction to form a relatively simple monocline structure that covers approximately 2800 km². The estimated amount of recoverable resources in this area is about 0.25 × 10¹² m³ [6].

The L1₁ layer has an average TOC content of 3.5% and is primarily composed of organic-rich black shale. It is considered the primary gas reservoir interval. On the other hand, the L1₂ and L2 layers have an average TOC content of around 0.8%, and the lithology is mainly gray-black sand mudstone. They can be used as the direct cap rock of L1₁ (Figure 1).

3. Method

3.1. Method of Calculating Shale Gas Diffusion

3.1.1. Fick’s First Law

The classic Fick’s first law states that the rate of material flow due to diffusion across unit area perpendicular to the direction of diffusion is proportional to the concentration gradient, the diffusion coefficient, and the diffusion time, given by Formula (1):

$$Q=D·\frac{dC}{dL}·t$$

(1)

In the formula: $Q$—diffused gas per unit area, m³/m²; $C$—free gas content, m³/t; $D$—cap diffusivity, m²/Ma; $L$—cap thickness, m; $t$—diffusion time, and Ma—million years.

After the initiation of hydrocarbon production from source rocks, light hydrocarbons can diffuse through molecular diffusion from source rocks into permeable strata nearby [36,37]. As the gaseous hydrocarbons continue to diffuse through the cap for a certain period, a relatively constant concentration gradient will occur in the longitudinal direction. As shown in Figure 2a, when the “cap” is securely sealed, the gas that is diffusing from the shale will be fully contained within the “cap” and will not diffuse through the permeable layer above it. This implies that $C_t^0>C_t^x>C_t^L=0$. If the sealing capacity of the cap layer is weak, only a fraction of the gas that diffused from the shale will be confined within the shale, while the remaining fraction will diffuse through the cap layer into the underlying permeable layer; therefore, it is expressed as$C_t^0>C_t^x>C_t^L>0$.

In Formula (1), the unit of the gas concentration, C, is expressed in mol/m³. To determine the actual rate of gas diffusion under strata circumstances, the gas concentration should be converted into free gas content (m³/t) by incorporating the rock’s density. The difference in concentration between the roof and the floor helps calculate the vertical gas content gradient. Thus, during the time interval t₁ to t₂, the cumulative amount of shale gas reservoir that diffuses into the cap layer per unit area via the lower cap interface (x = 0) can be determined as:

$$Q={{\displaystyle \int }}_{t_1}^{t_2}{\rho }_{\mathrm{s}}\frac{D}{L}\left(C_t^0-C_t^L\right)\mathrm{d}t$$

(2)

The calculation points of Equation (2) are the values of $\left(C_t^0-C_t^L\right)$ and D.

In the formula: $Q$—diffused gas per unit area, m³/m²; ${\rho }_{\mathrm{S}}$—shale density, kg/m³; $D$—cap diffusivity, m²/Ma; $L$—cap thickness, m; $C_t^0$—free gas content at cap bottom boundary at any time, m³/t; $C_t^L$—free gas content at cap top boundary at any time, m³/t; $t_1$—initiation time of diffusion, Ma; $t_2$—diffusion termination time, Ma; and ${\rho }_{\mathrm{S}}$—rock density, kg/m³.

3.1.2. Fick’s Second Law

Throughout the gas diffusion process, the free gas content in the cap fluctuates at different points and areas, as shown in Figure 2a. Under the assumption that the diffusion coefficient (D) remains constant, the free gas content within the cap is a binary function of both time (t, 0 ≤ t) and position (x, 0 ≤ x ≤ L). This relationship can be explained using Fick’s second law:

$$\frac{\partial C_t^x}{\partial t}=D\frac{{\partial }^2{C}_t^x}{\partial x^2}$$

(3)

Formula (3) represents a typical partial differential equation that demands specific initial conditions to be considered while solving it. The gas diffusion process starts immediately after its generation, and the amount of free gas at any point inside the cap layer is assumed to be zero ($C_0^x$ = 0). In addition, $C_t^0$ denotes the volume of free gas inside the reservoir and is considered as the source of material for gas diffusion [38,39].

After determining $C_t^0$ and D, the free gas content $C_t^x$ at any given position and time can be obtained by solving Equation (3). This would enable us to determine the value of $\left(C_t^0-C_t^L\right)$ in Formula (2).

3.1.3. Diffusivity in Series Formation Mode

One of the critical physical parameters utilized to measure the diffusion capacity of shale gas is its diffusion coefficient. In stratum environments, the diffusion coefficient of rocks is influenced by the stratum lithology and physical property parameters. As a result, changes in stratum burial depth and lithology interfere with the shale gas diffusion coefficient’s precise value [40,41]. For the purpose of examining the subject, let us assume that shale gas diffuses from the bottom to top through L-thick strata, where the shale layer’s thickness is L_M, and the sandstone layer’s thickness is L_S. Firstly, we consider diffusion through the L_M-thick shale layer and subsequently, diffusion through the L_S-thick sandstone layer. Then, the diffusion coefficient D of shale gas traveling through L-thick formation is estimated using the ratio of the thickness of the shale layer to the thickness of the sandstone layer, the diffusion coefficient D_M of the shale layer, the diffusion coefficient D_S of the sandstone layer, and the series model of formation. (Figure 2b):

$$D=\frac{L}{\frac{L_{\mathrm{M}}}{D_{\mathrm{M}}}+\frac{L_S}{D_{\mathrm{S}}}}=\frac{D_{\mathrm{M}}{D}_{\mathrm{S}}}{\frac{L_{\mathrm{M}}}{L}·D_{\mathrm{S}}+\frac{L_{\mathrm{S}}}{L}·D_{\mathrm{M}}}$$

(4)

3.2. Methods for Recapping Shale Free Gas Content

Due to the relatively weak intermolecular forces experienced by free shale gas, it can diffuse in the formation when pore throats are developed and a concentration difference exists. It is known from Equations (2) and (3) that the key to calculating the diffusion quantity of shale gas is how to recover the amount of free gas at the bottom limit of the cap ($C_t^0$). This can be obtained by applying the forward difference method to solve the implicit function partial differential equations.

3.2.1. Differential Enrichment Model of Free Gas

A mathematical model was developed to calculate the differential enrichment of free gas in the Longmaxi shale located in the Weiyuan area. The model allows for the computation of the evolution process of free gas during different geological periods, providing a representation of the migration and enrichment process of free gas under various structural styles (Figure 3).

The diffusion and migration of free gas in shale follow the principle of energy conservation. Free gas tends to move from areas of high fluid potential energy to those of low fluid potential energy, resulting in its accumulation in regions with low fluid potential energy [16,42]; therefore, shale gas enrichment is usually favored by positive structures over negative ones. Not all high points in positive structures, however, will result in shale gas enrichment, and not all negative structures are incapable of enriching shale gas. Positive structures aim to rise despite the vertical and lateral loss caused by free gas diffusion and supply from negative structures, respectively. On the other hand, the lower part of the negative structure is the prominent area for gas enrichment, which also incurs both lateral free gas loss and vertical free gas diffusion loss. The vertical diffusion loss in the positive structure is always more significant than that in the negative structure due to the denudation of the high points of the positive structure before the low points of the negative structure.

3.2.2. The Calculation Method of Free Gas Content

The free gas content was calculated by the free gas ideal gas equation model [36]:

$$\left\{\begin{array}{c}n=\frac{P\mathrm{V}}{ZR T }=\frac{P_{\mathrm{SC} }{V}_{\mathrm{SC}}}{Z·R T_{\mathrm{SC}} }\\ V=S_{\mathrm{g}} \varnothing ·V_{\mathrm{S}}\\ Q_{\mathrm{f}}=\frac{V_{\mathrm{SC}}}{m_{\mathrm{S}}}=\frac{V_{\mathrm{SC}}}{V_{\mathrm{S}}{\rho }_{\mathrm{S}}}\end{array}\right. \Rightarrow Q_{\mathrm{f}}=\frac{P{T}_{\mathrm{SC}} S_{\mathrm{g}} \varnothing }{Z T P_{\mathrm{SC} }{\rho }_{\mathrm{S}}}$$

(5)

In the formula: $Q_{\mathrm{f}}$—free gas content, m³/t; $n$—amount of free gas, mol; $P$—formation pressure, MPa; $V$—volume of free gas in the formation, m³; $Z$—methane compression factor, dimensionless; $R$—ideal gas constant, J/(mol·K);$T$—formation temperature, K;$P_{\mathrm{SC}}$—surface pressure, MPa; $V_{\mathrm{SC}}$—surface free gas volume, m³; $T_{\mathrm{SC}}$—surface temperature, K; $S_{\mathrm{g}}$—gas saturation of the reservoir, dimensionless; $\varnothing$—porosity of the reservoir, dimensionless; $V_{\mathrm{S}}$—rock volume, m³; $m_{\mathrm{S}}$—rock mass, kg; and ${\rho }_{\mathrm{S}}$—rock density, kg/m³.

Formula (5) reflects the combined effects of formation pressure, temperature, and physical properties on gas content. The temperature of the ancient strata can be determined through a burial history map, the development of physical properties can be obtained via thermal simulation experiments of shale hydrocarbon formation [1], and literature sources can provide data on gas saturation [2]. Once the temperature and pressure parameters of the formation have been established, the methane compression factor can be estimated by referring to a chart therefore, accurately determining the pressure of the ancient strata is crucial in order to determine the content of free gas from ancient times.

3.2.3. The Method of Paleo-Pressure Recovery

This paper utilizes a pressure simulation equation to reconstruct paleo-pressure. A new set of pressure development equations specifically suited for the properties of shale gas reservoirs were developed. The mass conservation theorem governs the migration of shale gas, and is expressed through the continuity equation:

$$\left\{\begin{array}{c}\frac{d\left(\rho \varnothing \right)}{dt}=\nabla \left(\rho ·\varnothing v^{\uparrow }\pm \rho ·\varnothing v^{\to }\right)+\rho ·q_{\mathrm{g}}\\ \varnothing v^{\uparrow }=D\frac{\mu }{P}·\frac{1}{\mu }\nabla P^{\uparrow }\\ \varnothing v^{\to }=K·\frac{1}{\mu }\nabla P^{\to }\\ \rho =\frac{PM}{\mathrm{R}T}\end{array}\right.$$

(6)

The relationship between porosity and shale overlying pressure and formation fluid pressure is as follows:

$$\frac{d\varnothing }{dt}=-{\beta }_{\mathrm{s}}\left(1-\varnothing \right)\frac{d\left(S-P\right)}{dt}$$

(7)

According to the basic theory of fluid mechanics, the ideal state equation of compressible gas is:

$$\frac{1}{\rho }\frac{d\rho }{dt}=\frac{1}{P}\frac{dP}{dt}-\frac{1}{T}\frac{dT}{dt}$$

(8)

Substituting Equations (7) and (8) into Equation (6):

$$\left(\frac{\varnothing }{P}+{\beta }_{\mathrm{s}}-\varnothing {\beta }_{\mathrm{s}}\right)\frac{dP}{dt}=\frac{1}{\rho }·\nabla \left(\rho \frac{D}{P}\nabla P^{\uparrow }\pm \rho \frac{K}{\mu }\nabla P^{\to }\right)+\left(1-\varnothing \right){\beta }_{\mathrm{s}}\frac{dS}{dt}+\varnothing \frac{1}{T}\frac{dT}{dt}+q_{\mathrm{g}}$$

(9)

In the formula: $\varnothing$—porosity of the reservoir, dimensionless; $P$—gas reservoir pressure, MPa; $\nabla P^{\uparrow }$—vertical cap pressure gradient, MPa/m; $\nabla P^{\to }$—reservoir lateral pressure gradient, MPa/m; T—formation temperature, K; ${\beta }_{\mathrm{s}}$—isothermal compression coefficient of shale skeleton, MPa⁻¹; $t$—time, Ma; $\rho$—underground density, kg/m³; $D$—cap diffusivity, m²/s; $K$—Reservoir bedding permeability, m²; $\mu$—shale gas viscosity, MPa·s; M—molar mass of methane, g/mol; $S$—overlying formation pressure, MPa; and $q_{\mathrm{g}}$—rate of gas volume formation per unit volume, m³/Ma.

Formula (9) represents the contribution of four factors to pressure formation in shale gas reservoirs: vertical and lateral migration, compaction, temperature, and gas generation rate. The left side of the equation shows the change in pressure over time. Depending on the stage of shale tectonic uplift, there are two alternative versions of Formula (9) for a given computation.

The stratum’s occurrence was mild due to the nearby tectonic units not having formed prior to the most recent uplift. The pressure between the adjacent tectonic units was roughly equal, so the reservoir’s lateral pressure gradient ($\nabla P^{\to }$) can be disregarded. The pressure gradient of the vertical cap ($\nabla P_{\mathrm{p}}^{\uparrow }$) is determined by the gas reservoir pressure, cap thickness, cap bottom, and hydrostatic pressure at the top of the cap. The pressure evolution equation is used to determine the gas reservoir pressure at the base of the cap based on the time-depth relationship derived from the stratum’s burial history data, as well as the pressure gradient, geothermal gradient, and pressure gradient of the static rock in the study area. The gas yield curve with maturity ($q_{\mathrm{g}}$) was obtained from thermal simulation experiments on hydrocarbon generation in shale [2]. The “time-gas generation rate” relationship is then calculated using the shale’s thermal maturity history and other intermediate characteristics, such as organic matter maturity. The formation pressure evolution equation can be expressed as:

$$\left(\frac{\varnothing }{P}+{\beta }_{\mathrm{s}}-\varnothing {\beta }_{\mathrm{s}}\right)\frac{dP}{dt}=\frac{1}{\rho }·\nabla \left(\rho \frac{D}{P}\nabla P^{\uparrow }\right)+\left(1-\varnothing \right){\beta }_{\mathrm{s}}\frac{dS}{dt}+\varnothing \frac{1}{T}\frac{dT}{dt}+q_{\mathrm{g}}$$

(10)

Shale gas migrates from negative to positive parts of the formation following the most recent formation uplift due to differential uplift. The pressure difference between nearby tectonic units and the bedding distance determine the reservoir bedding pressure gradient ($\nabla P^{\to }$), and the pressure of adjacent tectonic units is obtained using the pressure evolution equation. The organic matter stops the gas at the right end of the equation, and the parameter ($q_{\mathrm{g}}$) is then 0. The overlying pressure permeability experiment is used to determine the major negative correlation between the lateral permeability K and the overlying pressure S. The burial history of the stratum is then used to compute the permeability evolution history. The formation pressure evolution equations have the following specific form:

$$\left\{\begin{array}{c}\left(\frac{\varnothing }{P_{\mathrm{p}}}+{\beta }_{\mathrm{s}}-\varnothing {\beta }_{\mathrm{s}}\right)\frac{d{P}_{\mathrm{p}}}{dt}=\frac{1}{{\rho }_{\mathrm{p}}}\nabla \left({\rho }_{\mathrm{p}}\frac{D_{\mathrm{p}}}{P_{\mathrm{p}}}\nabla P_{\mathrm{p}}^{\uparrow }-{\rho }_{\mathrm{p}}\frac{K}{\mu }\nabla P^{\to }\right)+\left(1-\varnothing \right){\beta }_{\mathrm{s}}\frac{d{S}_{\mathrm{p}}}{dt}+\varnothing \frac{1}{T_{\mathrm{p}}}\frac{d{T}_{\mathrm{p}}}{dt}\\ \left(\frac{\varnothing }{P_{\mathrm{n}}}+{\beta }_{\mathrm{s}}-\varnothing {\beta }_{\mathrm{s}}\right)\frac{d{P}_{\mathrm{n}}}{dt}=\frac{1}{{\rho }_{\mathrm{n}}}\nabla \left({\rho }_{\mathrm{n}}\frac{D_{\mathrm{n}}}{P_{\mathrm{n}}}\nabla P_{\mathrm{n}}^{\uparrow }+{\rho }_{\mathrm{n}}\frac{K}{\mu }\nabla P^{\to }\right)+\left(1-\varnothing \right){\beta }_{\mathrm{s}}\frac{d{S}_{\mathrm{n}}}{dt}+\varnothing \frac{1}{T_{\mathrm{n}}}\frac{d{T}_{\mathrm{n}}}{dt}\end{array}\right.$$

(11)

In the formula: the subscript letter “p” stands for positive construction; the subscript letter “n” stands for negative construction.

3.3. Technical Route of the Method

The working procedure for calculating the amount of diffuse gas in a formation proceeds as follows: initially, the formation pressure, P, for different structural styles is computed using Formulas (10) and (11). Next, the amount of free gas content $C_t^0$, at the bottom of the caprock is determined by using Formula (5). Subsequently, Fick’s third law, expressed in Formula (3), is used to calculate the free gas content $C_t^L$, at the top of the caprock. Finally, Formula (2) is utilized to determine the amount of diffuse gas, while taking into account the results of the previous calculations. The overall process is presented and illustrated in Figure 4.

4. Result and Discussion

The diffusion volume of shale gas in the Weiyuan area of the Sichuan Basin is investigated in the positive part of well Wei201 and the negative part of well Wei204 using the calculation method outlined above. The evolution process of shale gas diffusion is characterized, and the cumulative diffusion volume of shale gas is estimated by determining the direct cap thickness of gas source rock, the initial concentration change of gas reservoir, and the diffusion duration of gas reservoir. Among them, the horizontal distance is 19.1 km, the bedding distance is 20.1 km, the average stratum thickness is 45 m, and the height difference of the target layers between well Wei201 and well Wei204 is roughly 2 km (Figure 1).

4.1. Recovery of Ancient Gas Content

4.1.1. Direct Cap Thickness

The thickness of the cap rock has a direct impact on the capacity to capture shale gas. Previous research [2,5] has clearly identified the segmentation of shale gas cap. In the Weiyuan area of Southern Sichuan, the Longmaxi Formation’s top and bottom are uninterrupted deposits with a significant thickness, consistent distribution, impermeable lithology, high breakthrough pressure and provide excellent isolation. The L1₁ well’s principal pay layer is L1₁, overlaying L1₂ and L2, which are 250 m tall cap layers. The first and second members of the Longmaxi Formation show an average porosity and permeability of 2.4% and 0.0016 × 10³ μm², indicating the formation’s strong isolation properties and that it has great potential as the L11 direct cap rock.

4.1.2. Shale Gas Diffusion Time

Continuous diffusion of shale gas occurs due to the gradual increase in temperature and pressure in the underlying formation, which is initiated by organic matter generating shale gas. By analyzing the stratigraphic burial map of the Weiyuan region in southern Sichuan (

Table 1. Division of diffusion stages.

Time from Now (Ma)	Diffusion Time (Ma)	Stage
244–172	0–72	Slow subsidence, slow generation of gas
172–168	72–76	Rapid subsidence, slow generation of gas
168–161	76–83	Rapid subsidence, rapid generation of gas
161–97	83–147	Slow subsidence, rapid generation of gas
97–29	147–215	Slow uplift
29–0	215–244	Rapid uplift

), the time when the shale rock first surpassed the hydrocarbon generation threshold of Ro = 0.5% 244 million years ago was determined as the start of diffusion.

4.1.3. Gas Yield Rate per Unit Volume

To elucidate the correlation between gas volume yield and formation age, we conducted hydrocarbon generation thermal simulation experiments to acquire the yield–maturity curve (Figure 5a) and then determined the corresponding formation maturity values based on the burial history (Figure 5b). As the yield value represents an instantaneous yield, we integrated the yield–maturity curve over time to generate the production time relationship of cumulative volume of gas. By further combining time, we calculated the cumulative yield rate of gaseous hydrocarbons (q_g).

The period from 168 to 161 million years ago marks the peak of both cumulative gas yield and yield rate (Figure 5c). This phase was characterized by rapid subsidence of the formation, which, in turn, accelerated the evolution rate of R_O due to an increase in geothermal gradient. Within a period of 7 million years, the R_O increased from 1.6% to 2.8%, thereby marking the peak period of gas generation. It was within this phase that the cumulative gas production rate and cumulative production rate reached their highest values.

4.1.4. Shale Gas Diffusion Coefficient

The diffusion coefficient of shale gas, which is mainly controlled by the temperature of the formation and the pore structure of the cap layer, determines the diffusive ability of shale gas within the formation. Earlier studies have established a calculation formula for the diffusion coefficient of shale gas under formation conditions, which is based on the Stokes–Einstein equation [16]:

$$D=\frac{\mathrm{k}T\varnothing }{6\mathsf{\pi }r\mu }$$

(12)

In the formula: D—diffusion coefficient, m²/s; k—Boltzmann’s constant, 1.38 × 10⁻²³ J/K; T—absolute formation temperature, K; $\varnothing$—porosity, %; r—methane molecular radius, m; and μ—gas viscosity, Pa·s.

The Stokes–Einstein equation is a microscopic equation that describes the diffusion behavior of small particles in fluids, and it does not emphasize that small particles and fluids must be two different substances. When methane molecules diffuse in formation water, µ represents the viscosity of formation water; when methane molecules diffuse in pores filled with methane of varying concentrations, µ represents the viscosity of methane.

Combining the information on the burial history of the Weiyuan area, the evolutionary process of shale gas diffusion coefficient (Figure 6) has been restored. As shown in the figure, the evolution of diffusion coefficient is relatively less affected by formation temperature and has good consistency with the evolution of porosity. As the Silurian stratum began to deposit, the shale gas diffusion capacity always maintained a relatively high level due to the weakening of shale compaction and pore reduction effect during the middle period of the Permian. As the Permian stratum entered a stage of prolonged deep burial, the diffusion coefficient rapidly decreased with the change in porosity. As the tectonic activity of the formation became stable during the middle and late Jurassic, the pore remained stable, and the gas reservoir entered a low diffusion capacity stage.

4.1.5. The Content of Ancient Free Gas

According to Figure 6, the bottom boundary of the Silurian stratum in Weiyuan area was buried to a depth of 1800 m in the middle period of the Silurian, with the R_O value reaching 0.5%, entering the hydrocarbon generation threshold. By the late Silurian, the stratum was buried to a depth of about 2100 m, and then underwent uplift and erosion, causing the hydrocarbon generation process to stop. At the end of the Permian, the stratum was rapidly buried again, and the Ro value of the stratum reached 0.5% again, and the organic matter officially entered the hydrocarbon generation threshold. In the middle Jurassic, with the rapid increase in geothermal gradient, the thermal cracking intensity of organic matter was significantly increased, and the pore pressure was about 125 MPa, ensuring the enrichment of gas. In terms of evolution of shale gas content, affected by factors such as overpressure and rapid heating of strata, the free gas content increased rapidly to 11.4 m³/t by the middle Jurassic, and reached the historical maximum of 12.5 m³/t at the middle Cretaceous. In the later period of the Cretaceous, the stratum entered the initial stage of the last uplift, and the intensity of stratum deformation slightly increased, forming two adjacent syncline and anticline areas. At this time, due to the cessation of hydrocarbon generation, the gas supply disappeared, and the gas reservoir pressure and gas content decreased rapidly. By the middle and late Neogene, the gas content in the positive position of well Wei201 was always higher than that in the negative position of well Wei204; however, as the stratum continued to uplift, the squeezing stress significantly changed the structural style of the stratum, and the positive position ruptured before the negative position, greatly accelerating the loss of vertical gas.

The numerical simulation results showed that the current free gas volume for well Wei201 is 1.8 m³/t; for well Wei204, it is 3.0 m³/t (Figure 6). Considering that the proportion of free gas in the Longmaxi shale gas reservoir in the southern Sichuan region is around 60% to 80% [41], the simulated total gas content for well Wei201 is between 2.3 m³/t and 3.0 m³/t; for well Wei204, it is between 3.8 m³/t and 5.0 m³/t. The results of exploration practice show that the measured total gas content for well Wei201 ranges between 2.1 m³/t and 4.8 m³/t; for well Wei204, it ranges between 3.8 m³/t and 7.3 m³/t [43]. The lower limit of the simulation results is similar to that of the measured results, while the upper limit of the simulation results is less than that of the measured results. Overall, the simulation results are consistent with the actual exploration results (

Table 2. Comparison of numerical simulation results and actual exploration results.

Well	Numerical Simulation Results of Free Gas Content	Numerical Simulation Results of Total Gas Content	Actual Exploration Results [5,6]	Accuracy
Wei 201	1.8 m3/t	2.3~3.0 m3/t	2.1~4.8 m3/t	76.8%
Wei 204	3.0 m3/t	3.8~5.0 m3/t	3.8~7.3 m3/t	79.3%

).

4.2. Calculation Equation of Diffused Gas Volume

Due to Equation (11) being a typical partial differential equation, only numerical solutions for P-t can be obtained through mathematical calculations, forming a numerical sequence about P-t, and it is impossible to obtain an exact analytical solution f(x) = P(t). Similarly, the free gas concentration calculated based on the P-t sequence is also a numerical sequence about C-t, and the exact analytical solution f(x) = C(t) cannot be obtained; however, geological history is a lengthy process, and geological parameters change extremely slowly during this time. The evolution of free gas concentration is a continuous function rather than independent scattered points that vary over time. When calculating the diffusion gas volume using Equation (2) and the free gas concentration at the top of the cap layer using Equation (5), it is necessary to obtain a continuous function for the free gas concentration of the reservoir over time, i.e., C(t) = $C_t^0$.

Therefore, when setting boundary conditions, it is necessary to approximate the complex sequence of free gas concentration in geological history as a combination of multiple sets of quadratic polynomial changes (Figure 7). The change in diffusion coefficient in geological history is approximated as a combination of multiple sets of constant values, and the weighted average of diffusion coefficient at multiple time points is used to represent the diffusion coefficient in the short period of time.

The mathematical model of shale gas diffusion in well Wei201 and well Wei204 is:

$$\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2};D=\left\{\begin{array}{c}343.4 {\mathrm{m}}^2/\mathrm{M}\mathrm{a} (0<t\le 72)\\ 230.7 {\mathrm{m}}^2/\mathrm{M}\mathrm{a} (72<t\le 83)\\ 120.3 {\mathrm{m}}^2/\mathrm{M}\mathrm{a} \left(83<t\le 147\right)\\ 120.3 {\mathrm{m}}^2/\mathrm{M}\mathrm{a} \left(147<t\le 244\right)\end{array}\right.$$

(13)

The initial condition of time variation of gas content at the top boundary of Wei201 L1₁ in positive construction is:

$$C_0^x=0;C_t^0=\left\{\begin{array}{c}-0.0001t{ }^2+0.0265t+0.1079 \left(0\le t\le 72\right)\\ -0.1091t{ }^2+16.869t-647.37 (72<t\le 83)\\ -0.0001t{ }^2+0.0528t+6.5066 (83<t\le 147)\\ 0.0012t{ }^2-0.5168t+63.651 (147<t\le 215)\\ 0.0003t{ }^2-0.2644t+49.624 (215<t\le 244)\end{array}\right.$$

(14)

The initial condition of time variation of gas content at the top boundary of Wei204 L1₁ in negative construction is:

$$C_0^x=0;C_t^0=\left\{\begin{array}{c}-0.0001t{ }^2+0.0265t+0.1079 \left(0\le t\le 72\right)\\ -0.1091t{ }^2+16.869t-647.37 (72<t\le 83)\\ -0.0001t{ }^2+0.0528t+6.5066 (83<t\le 147)\\ 0.0021t{ }^2-0.8679t+95.694 (147<t\le 215)\\ 0.0004t{ }^2-0.2313t+34.658 (215<t\le 244)\end{array}\right.$$

(15)

The evolution of free gas content in the cover layer was obtained by solving the partial differential equation of the function using the forward difference approach, as shown in Figure 8. Figure 8 shows that prior to the shale gas diffusion time t = 0 (approximately 244 Ma years ago), the organic matter had not yet reached the hydrocarbon generation threshold, resulting in zero gas content in the 250 m thick cover layer that covered the L1₁. The gas in the shale reservoir is gradually escaping upward due to the continuously strengthened ability of the formation subsidence to generate organic gas, causing the diffusion front to approach the interface of the direct cap top. Due to the hysteresis of diffusion, the change in gas content was much more rapid at the bottom boundary of the direct cap layer compared to the top boundary. Although the gas content at the bottom boundary of the direct cap layer reached its historical maximum around 97 Ma years from now, it took another 78 Ma years for the gas content at the top boundary of the cap layer to achieve its historical maximum.

Due to the influence of differential strata uplift during the later stages, adjacent tectonic sites experience different gas reservoir evolution processes, resulting in variations in the gas diffusion process. Specifically, in the forward section, the gas content only declined to zero about 30 Ma before present at the top of the cap layer, while the concentration of the negative cap layer always remains above zero. This shows that the L1₂ and L2 layers, acting as direct cap layers, cannot completely prevent the gas loss of L1₁ and that a considerable amount of gas continues to leak into the permeable layer beneath the cap layer (Figure 8).

According to Figure 5, the gas content at the top boundary of the L2 of Wei201 in positive construction varies with time as follows:

$$C_t^{250}=\left\{\begin{array}{c}-0.0001t{ }^2+0.0043t-0.0129 \left(0\le t\le 72\right)\\ -0.0024t{ }^2-0.3535t+13.202 (72<t\le 83)\\ -0.0003t{ }^2+0.0973t-5.1672 (83<t\le 147)\\ -0.0002t{ }^2+0.0659t-3.3569 (147<t\le 215)\\ -0.0001t{ }^2+0.0436t-3.5435 (215<t\le 244)\end{array}\right.$$

(16)

The gas content at the top limit of the L2 of Wei204 in negative construction varies with time as follows:

$$C_t^{250}=\left\{\begin{array}{c}-0.0001t{ }^2+0.0043t-0.0129 \left(0\le t\le 72\right)\\ -0.0024t{ }^2-0.3535t+13.202 (72<t\le 83)\\ -0.0003t{ }^2+0.0973t-5.1672 (83<t\le 147)\\ -0.0002t{ }^2+0.0672t-3.0889 (147<t\le 215)\\ -0.0001t{ }^2+0.0002t+1.1646 (215<t\le 244)\end{array}\right.$$

(17)

The history of shale free gas diffusion can be retrieved by substituting the diffusion coefficient of Formula (13) with the gas content of the top boundary of L1₁ and the gas content of the top boundary of L2.

4.3. The Calculation Result of Diffused Gas Volume

By arranging the various tectonic stages of the stratum in a time sequence and converting the generation time into geological time, the cumulative history of free gas diffusion per unit area of the Longmaxi shale in the Weiyuan region is determined (Figure 9). The Weiyuan area, located in southern Sichuan, is characterized by shallow burial in the early phases, profound burial in the middle stages, and fast uplift in the late stages. As the Ro value of the Longmaxi Formation reached 0.5% in the Permian period, the organic matter accumulated to levels that may generate hydrocarbons. The pyrolysis and gas capacity of organic matter improved in the middle and late Jurassic, leading to a rapid increase in the gradient of diffusion concentration and a cumulative diffusion of 2.51 × 10⁴ m³/m². By the middle of the Cretaceous period, during the final stage of uplift, the content of free gas reached its maximum level, and the cumulative diffusion increased to 10.89 × 10⁴ m³/m².

During the final stage of formation uplift, the generation of hydrocarbons from organic matter stopped, and the gas content of the formation decreased rapidly, but gas diffusion continued. About 29 Ma years ago, when the formation was affected by differential uplift, the evolution of the gas diffusion history differed between nearby positive and negative construction. When the positive construction reached the depth of the formation rupture about 23 Ma years ago, the gas loss increased significantly. Ultimately, the cumulative diffused gas volume in the forward section was approximately 15.13 × 10⁴ m³/m², while in the negative section, it was about 13.75 × 10⁴ m³/m².

According to the statistics of shale gas diffusion in different geological eras, it can be concluded that the Jurassic and Cretaceous periods were the peaks of shale gas diffusion. The cumulative diffused gas in the positive part accounted for 32.40% and 47.78%, whereas in the negative part, it accounted for 35.84% and 48.05%, respectively (Figure 10). During the early and middle Cretaceous periods, the Longmaxi Formation was deeply buried for a considerable amount of time, resulting in the complete pyrolysis of organic matter, leading to the generation of large quantities of shale gas. During this process, the gas reservoir also underwent significant diffusion loss. Although hydrocarbon production stopped due to stratigraphic uplift during the later part of the Cretaceous period, diffusion remained high due to the high gas content in the reservoir.

Since the Neogene, the significant upward movement in the positive construction has resulted in a rapid decrease in gas content due to the increase in diffusion, which in turn leads to a relatively smaller diffusion concentration gradient; therefore the free gas diffusion in the negative construction slightly exceeded that in the positive construction only during the Neogene and Quaternary period, while the free gas diffusion in the positive construction has always been greater than that in the negative construction in other periods (Figure 10).

The composition and isotopic characteristics of shale gas are important parameters that can be used to characterize gas diffusion and migration in shale formations [43]. As shale gas diffuses and migrates, rock components have a relatively high adsorption and retention capacity for heavy hydrocarbon components, resulting in isotope fractionation. Consequently, the diffusion direction of CH₄ component enhances its concentration, while the heavy hydrocarbon component of ${\mathrm{C}}_2^{+}$ decreases proportionally. Meanwhile, δ¹³C and δ¹³DC₁ values gradually become lighter. Conversely, diffusion of the CH₄ component from the source in the shale reservoir reduces its concentration, leading to an increase in the ${\mathrm{C}}_2^{+}$ component and heavier δ¹³C and δ¹³DC₁ values [44].

The average CH₄ content, δ¹³C₁, δ¹³C₂ and δ¹³DC₁, of two horizontal wells in the Wei201 well area are 96.02%, −35.3%, −39.8% and −144%, respectively. The average CH₄ content, δ¹³C₁, δ¹³C₂ and δ¹³DC₁, of five horizontal wells in the Wei204 well area are 97.6%, −36.5%, −41.0% and −146.6%, respectively. (Figure 11). The geochemical indicators reveal that the levels of light hydrocarbon components and lighter isotopes are lower in the Wei201 well area compared to those in the Wei204 well area. The reason behind this difference is the greater cumulative shale gas diffusion in the former than the latter, which strengthens the diffusion of light hydrocarbon components and results in a more noticeable isotope fractionation effect. The geochemical features of the gas reservoir confirm the validity of the shale gas diffusion calculation.

The diffusion rate per unit area of Longmaxi Formation shale in Weiyuan region ranges from 14.10 × 10⁴ to 16.50 × 10⁴ m³/m². The basic trend is that diffusion is more prominent in the northwest and less significant in the southeast. The high values are mainly distributed near Weiyuan and Ziliujing anticlines, while the low values are primarily found near the Baimazhen syncline (Figure 12). According to exploration results [43], shale gas in the Longmaxi Formation of Weiyuan region is predominantly enriched in the southeast, indicating that the overall loss of gas reservoirs in this direction is minimal, which is consistent with the numerical simulation results.

The Longmaxi Formation’s shale plane area in the Weiyuan area is about 2800 km² with recoverable resources totaling 0.25 × 10¹² m³ [44]. According to the calculations in this paper, an average loss of 0.13 × 10¹² m³ gas in the positive part and 0.30 × 10¹² m³ gas in the negative part is observed, resulting in a total gas loss of approximately 0.43 × 10¹² m³. The total gas loss is about 1.72 times greater than the current recoverable resources, which is obviously very unfavorable for the accumulation of shale gas in the study area; therefore, when evaluating the prospects of shale gas in other areas, it is necessary to consider the diffusion loss of shale gas in order to make an accurate assessment of gas content prospects. Otherwise, prospect gas content will be overestimated.

The shale gas reservoir has lost a significant amount of methane gas due to diffusion over the long geological history. It is extremely difficult to accurately simulate the amount of gas diffusion therefore, this study can only retroactively estimate the validity of the simulation results based on current exploration and development data. In future research, some experimental methods are planned to restore the evolution history of ancient pressure and gas content in strata. Through the comparison of experimental results and numerical simulation results, the accuracy of the results in this article can be better demonstrated.

5. Conclusions

Strata gas content is mainly controlled by strata pressure. The higher the strata pressure, the richer the gas reservoir, and the evolution of strata structure determines the pressure evolution. In the middle period of the Jurassic, the strata pore pressure was approximately 125 MPa, and the free gas content rapidly increased to 11.4 m³/t. By the mid-Cretaceous, the free gas content reached its historical maximum of 12.5 m³/t, ensuring gas enrichment. In the late-middle Cretaceous, the last uplift of the strata caused the reservoir pressure and gas content to rapidly decrease, resulting in numerical simulation showing that the present-day free gas content of Wei201 well is 1.8 m³/t and that of Wei204 well is 3.0 m³/t.

Shale gas reservoirs serve as supply sources of diffusion, and the gas content directly determines the diffusion intensity of the reservoir. The Jurassic and Cretaceous periods were the peaks of strata gas content, and are also the peaks for studying the diffusion of the Longmaxi Formation shale gas in this area. By the mid-Cretaceous, the strata entered the last uplift stage, and the free gas content reached its peak, while the cumulative diffusion further increased to 10.89 × 10⁴ m³/m². In the late Cretaceous uplift stage, the diffusion was still high due to the high gas content in the reservoir. The accumulated diffusion gas in the positive section was approximately 15.13 × 10⁴ m³/m², and approximately 13.75 × 10⁴ m³/m² in the negative section.

A significant amount of shale gas has been lost due to diffusion during a long geological history, which directly affects the accumulation and preservation of shale gas. Areas with high diffusion in history tend to have low gas content today. In the Weiyan area, the range of diffusion per unit area of the Longmaxi Formation shale is between 14.10 × 10⁴ m³/m² and 16.50 × 10⁴ m³/m², resulting in a total loss of approximately 0.43 × 10¹² m³ of gas, which is about 1.72 times the current recoverable resource volume, and the total generated gas is close to three times the current recoverable resource volume.