Study on CO₂/CH₄ displacement process in shale microscale models with adsorption/desorption behavior by lattice Boltzmann method

Abstract

The competitive adsorption of CH₄/CO₂ in shale has consistently garnered significant attention as a means to enhance the recovery efficiency of shale gas reservoirs. A theoretical formula for a binary gas competitive adsorption rate model was formulated to investigate the adsorption and desorption characteristics and the evolution patterns of carbon dioxide (CO₂) displacing methane (CH₄) in shale. This formula was integrated into the lattice Boltzmann method (LBM) to simulate the displacement process of CH₄ by CO₂, addressing competitive adsorption challenges associated with binary gases exhibiting adsorption/desorption behaviors. The research findings reveal that the proposed theoretical model accurately captures the competitive adsorption dynamics between CO₂ and CH₄, elucidating the patterns of adsorption and desorption characteristics during the displacement of CH₄ by CO₂. This insight is pivotal for understanding the microscopic mechanisms underlying CO₂-induced CH₄ displacement. The displacement process is dynamic, marked by concurrent adsorption and desorption of CO₂ and CH₄, ultimately converging to an equilibrium state where CO₂ adsorption and CH₄ desorption coexist. Notably, the time taken for CO₂ to attain equilibrium is marginally delayed compared to CH₄. Moreover, the concentration of injected CO₂ substantially influences the dynamics of CO₂ replacing CH₄, as the CO₂ injection concentration increases, both the adsorption rate of CO₂ and the desorption rate of CH₄ augment, while the time required to reach equilibrium in the adsorption/desorption process diminishes. This implies that CO₂ injection effectively facilitates CH₄ desorption. Additionally, models with higher porosity exhibit enhanced permeability, resulting in accelerated adsorbate diffusion rates and improved displacement efficiency. The heterogeneity of the pore structure exerts a pronounced impact on the velocity distribution within the flow field, which in turn significantly influences the concentration field distribution of CO₂ and CH₄. It is worth noting that the distribution characteristics of the two gases within the flow field and their concentrations within the particles are complementary. The results underscore that carbon dioxide injection can enhance methane desorption, thereby improving shale gas recovery.

Introduction

Nanoscale pores in shale serve dual functions: as storage spaces for shale gas and effective sequestration sites for greenhouse gases^1,2. The CO₂-Enhanced Shale Gas Recovery (CO₂-ESGR) technology is regarded as an auspicious approach, as it simultaneously facilitates shale gas development and CO₂ sequestration, positioning it as a current focal point of research³. Shale gas primarily exists in nanoscale storage spaces in two forms: adsorbed and free states. These states involve various migration mechanisms, including adsorption/desorption, diffusion, and viscous flow⁴. Particularly under primary reservoir conditions, the displacement and competitive adsorption behaviors of multicomponent gases occur within the porous media⁵. The mechanisms underlying these behaviors remain unclear and represent a significant scientific challenge in achieving CO₂-enhanced shale gas recovery.

Shale reservoir spaces, characterized by their highly complex internal structures, are macroscopically represented as a porous media system dominated by heterogeneous pores and fracture networks⁶. In such intricate porous media, the transport processes of multi-component gases exhibit significantly nonlinear dynamic behaviors, presenting substantial challenges in deciphering their core mechanisms at the laboratory scale⁷. From a physical mechanisms perspective, these phenomena fundamentally relate to fluid–solid coupling issues under complex boundary conditions, often accompanied by multi-physics coupling effects, including heat transfer, mass transfer, and multiphase flow⁸. The interaction of multiple physical fields directly results in solutions to their governing equations being constrained by both strong nonlinearity and high computational complexity. The lattice Boltzmann method (LBM) has been extensively utilized by the academic community for the numerical simulation of relevant physical processes, addressing this research challenge and providing a reliable technical approach for elucidating scientific mechanisms and exploring engineering solutions⁹.

Ren et al.¹⁰ combined the real gas state equation with the lattice Boltzmann method to investigate the microflow characteristics of shale gas within a simplified microchannel physical model by further discussing the boundary slip velocity and pressure distribution under different parameters. Luo et al.¹¹ utilized the lattice Boltzmann method to simulate double diffusion mixed convection, fluid–solid conjugate heat transfer, and the adsorption process at the pore scale within a lid-driven composite enclosure filled with a homogeneous medium by discussing the effects of the buoyancy ratio and adsorption rate on heat and mass transfer within the computational model. Yang et al.¹² used LBM to investigate the flow-coupled mass transfer process of CO₂ adsorption in flue gas by ZIF-8 material at REV scale, including the effects of porosity, arrangement, and particle size on CO₂ adsorption capacity.

However, these researches were conducted within artificially constructed, regularly shaped packing channels, without taking into account the randomness and complexity inherent in the internal structure of porous media. In order to express the porous characteristics of shale micropores, the Quart Structure Generation Set (QSGS) method¹³ provides a feasible approach for the generation of random porous media. This method can generate porous media samples with different structures and properties by controlling four parameters: porosity, pore size distribution, pore shape distribution, and pore arrangement distribution. Zhou et al.¹⁴ conducted numerical studies on the gas–solid adsorption process in reconstructed random porous media using the lattice Boltzmann method. Their work provides detailed insights into the effects of internal and external mass transfer resistance, offering theoretical guidance for the design and optimization of adsorption systems. Similarly, Wang et al.¹⁵ developed a numerical model for the flow, heat transfer, and mass transfer of flashing vapor within porous media using LBM. Their study focused on the adsorption/desorption characteristics and thermal effects of LNG flashing vapor under cryogenic storage conditions.

The process of CO₂ displacing CH₄ within the pores of shale matrices serves as the primary mechanism underlying CO₂-Enhanced Shale Gas Recovery (CO₂-ESGR)¹⁶ technology. Consequently, it is imperative to explore the competitive adsorption and miscible displacement processes between CO₂ and CH₄. Most researchers have concentrated on the adsorption characteristics of single-component gases, whereas relatively few studies have investigated the competition and displacement among multiple gas components¹⁷. Wu et al.⁵ utilized the lattice Boltzmann method to simulate the CO₂/CH₄ displacement in a nanoporous shale matrix by integrating the Navier–Stokes equations with multicomponent advection–diffusion equations. Their findings indicated that the global mass transfer process is susceptible to intra-matrix diffusion; selectivity can significantly influence the outflux concentration when the solid diffusion rate is approximately 10⁻⁴ of the bulk diffusion rate. Nevertheless, there is still a lack of sufficient theoretical support for the competitive adsorption and displacement processes involving multi-component gases. Furthermore, our current understanding of the adsorption and desorption characteristics of gases during the CO₂ displacement of CH₄ remains limited and requires further investigation.

Unlike previous studies, this paper establishes a theoretical model for the competitive adsorption behavior of binary gases within porous media, providing sufficient theoretical support for research on the competitive adsorption and displacement characteristics of multi-component gases. Utilizing the LBM method, the adsorption/desorption characteristics during the CO₂/CH₄ displacement process at the microscale in shale are analyzed. The technical requirements for contemporary shale gas extraction and carbon sequestration are explored, offering robust assistance for the study of the mechanism of CO₂-enhanced shale gas recovery.

Two-component gas competitive adsorption model

The process of gas adsorption by solids involves convection, heat transfer, and mass transfer within the pore channels of a mixed fluid. Additionally, this process encompasses gas adsorption on the surface of particles, along with diffusion and heat transfer within the particles. Figure 1 illustrates the overall framework of the adsorption model.

Fig. 1

Schematic of the adsorption/desorption process in a channel filled with adsorbent particles.

The physical model of the simulation consists of a two-dimensional channel filled with a disordered random porous medium. For simplicity, circular adsorbent particles with complex internal pore structures represent the random porous medium. Given the low velocity in the porous model, the fluid adheres to the assumption of incompressible flow¹⁸. This study considers only adsorption, desorption, and diffusion, while neglecting changes in pore and matrix porosity, as well as alterations in the reaction front caused by dissolution reactions. The adsorption behavior of gases on solid surfaces is a dynamic process that involves simultaneous adsorption and desorption¹⁹. Figure 1 presents a detailed schematic diagram illustrating mass transport during adsorption. Gases such as carbon dioxide and methane are transferred to the surface of particulate matter through convection and diffusion. Initially, the adsorbate is adsorbed onto the outer surface of the particles via surface adsorption, and it subsequently diffuses into the micropores of the adsorbent through intraparticle diffusion. It is important to note that, due to the small size of the micropores, the convection effect within them is negligible; therefore, only the diffusion effect of the adsorbate is considered following this simplification.

Based on the assumption of single-layer physical adsorption and the Langmuir Eq. ²⁰, when two adsorbates A and B compete for the same adsorption site, the expressions of the adsorption rate v_a and desorption rate are v_d :

$$v_{a,A} = k_{a,A} p_{A} \left( {1 - \theta_{A} - \theta_{B} } \right),v_{a,B} = k_{a,B} p_{B} \left( {1 - \theta_{A} - \theta_{B} } \right)$$

(1)

$$v_{d,A} = k_{d,A} \theta_{A} ,v_{d,B} = k_{d,B} \theta_{B}$$

(2)

where p is the partial pressure of gas phase adsorbate, θ is the surface coverage, indicating the fraction of solid surface adsorption sites covered, k_a and k_d are the adsorption and desorption rate constants, respectively.

Combining formulas (1) and (2) and denoting the adsorption equilibrium constant b as:

$$b = \frac{{k_{a} }}{{k_{d} }}$$

(3)

the coverage of the two components on the adsorbent surface can be obtained as:

$$\theta_{A} = \frac{{b_{A} p_{A} }}{{1 + b_{A} p_{A} + b_{B} p_{B} }},\,\theta_{B} = \frac{{b_{B} p_{B} }}{{1 + b_{A} p_{A} + b_{B} p_{B} }}$$

(4)

According to the definition of surface coverage, it can be expressed as:

$$\theta_{A} = \frac{{V_{A} }}{{V_{m} }},\;\theta_{B} = \frac{{V_{B} }}{{V_{m} }}$$

(5)

where V_m is the Langmuir volume, which represents the maximum adsorbed volume at infinite pressure.

Substituting Eq. (5) into Eq. (4) and organizing it, the expression for the equilibrium adsorption volume of the adsorbate can be obtained:

$$V_{A} = \frac{{V_{m} p_{A} }}{{\frac{1}{{b_{A} }} + \frac{{b_{B} }}{{b_{A} }}p_{B} + p_{A} }},\;V_{B} = \frac{{V_{m} p_{B} }}{{\frac{1}{{b_{B} }} + \frac{{b_{A} }}{{b_{B} }}p_{A} + p_{B} }}$$

(6)

Assuming ideal gas behavior, the pressure can be expressed as $p = \frac{1}{\gamma }CMc_{s}^{2}$ and denote $b_{A} ^{\prime}$ and $b_{{\text{B}}} ^{\prime}$ as follows:

$$b_{A} ^{\prime} = \frac{{b_{A} }}{\gamma }Mc_{s}^{2} ,b_{B} ^{\prime} = \frac{{b_{B} }}{\gamma }Mc_{s}^{2}$$

(7)

where M is molar mass (g/mol) , C is concentration of free gas (kmol/m³), γ is the adiabatic index and c_s is sound speed (m/s).

The equilibrium adsorption volume of the adsorbate can be expressed as:

$$V_{A} = \frac{{V_{m} C_{A} }}{{\frac{1}{{b_{A} ^{\prime}}} + \frac{{b_{B} ^{\prime}}}{{b_{A} ^{\prime}}}C_{B} + C_{A} }},V_{B} = \frac{{V_{m} C_{B} }}{{\frac{1}{{b_{B} ^{\prime}}} + \frac{{b_{A} ^{\prime}}}{{b_{B} ^{\prime}}}C_{A} + C_{B} }}$$

(8)

Considering the dynamic adsorption/desorption process and the competitive adsorption behavior of the two-component gas, the adsorption rate equations for adsorbates A and B under the monolayer assumption are given by:

$$\frac{{\partial V_{A} }}{\partial t} = k_{a,A} C_{A} \left( {V_{m} - V_{A} } \right) - k_{d,A} V_{A} \left( {1 + \frac{{k_{a,B} }}{{k_{d,B} }}C_{B} } \right)$$

(9-1)

$$\frac{{\partial V_{B} }}{\partial t} = k_{a,B} C_{B} \left( {V_{m} - V_{B} } \right) - k_{d,B} V_{B} \left( {1 + \frac{{k_{a,A} }}{{k_{d,A} }}C_{A} } \right)$$

(9-2)

When adsorption reaches equilibrium, the above two equations can be reduced to the form of the Langmuir adsorption kinetics Eqs. (8) for adsorbates A and B, when two-component gases compete for the same adsorption sites.

LBM model of two-component gas flow and competitive adsorption

LBM model describing gas flow

In this paper, the lattice Boltzmann method (LBM) is employed to simulate the flow of two-component gases in complex structures. The evolution equation for the density distribution function f of CO₂-CH₄ mixture is given by:

$$f_{i} \left( {{\mathbf{r}} + e_{i} {\Delta }t,t + {\Delta }t} \right) = f_{i} \left( {{\mathbf{r}},t} \right) - \frac{{1}}{{\tau_{f} }}\left[ {f_{i} \left( {{\mathbf{r}},t} \right) - f_{i}^{eq} \left( {{\mathbf{r}},t} \right)} \right]$$

(10)

where $f_{i}^{eq} \left( {r,t} \right)$ is the equilibrium distribution function at position r at time t, Δt is the time step, $\tau_{f}$ is the dimensionless relaxation (collision) time and e_i represents the lattice velocity in the i-th direction. $f_{i}^{eq} \left( {r,t} \right)$ is the equilibrium distribution function. The expressions are given as follows:

$$f_{i}^{eq} \left( {{\mathbf{r}},t} \right) = w_{i} \rho \left[ {1 + \frac{{e_{i} \cdot {\mathbf{u}}}}{{c_{s}^{2} }} + \frac{{\left( {e_{i} \cdot {\mathbf{u}}} \right)^{2} }}{{2c_{s}^{4} }} - \frac{{{\mathbf{u}} \cdot {\mathbf{u}}}}{{2c_{s}^{2} }}} \right]$$

(11)

where u is the macroscopic velocity.

Figure 1 depicts a physical model with a relatively large Knudsen number. When Kn > 10, due to the free molecular flow state of shale gas, the pore walls of shale are rough and there is adsorption behavior. Using non-slip boundary conditions is more in line with the real situation. In addition, in free molecular flow, intermolecular collisions can be ignored, and the physical meaning of traditional viscosity disappears. The definition and role of viscosity coefficient need to be re examined. Therefore, the relaxation time needs to be revised through effective viscosity²¹

$$\tau_{f} = \frac{1}{2} + \frac{{\nu_{f} }}{{{\Delta }tc_{s}^{2} }}\frac{1}{{1 + \frac{4}{\pi }Kn\arctan \left( {\sqrt 2 Kn^{ - 3/4} } \right)}}$$

(12)

where ν_f represents the kinematic viscosity of the fluid; Kn is the Knudsen number; c_s is the lattice sound speed.

In the physical modeling of gas flow, the D2Q9 lattice model²², which employs a two-dimensional nine-velocity scheme, is utilized to discretize particle velocities within a two-dimensional space.

$$e_{i} = c\left( {\begin{array}{*{20}c} 0 & 1 & 0 & { - 1} & 0 & 1 & { - 1} & { - 1} & 1 \\ 0 & 0 & 1 & 0 & { - 1} & 1 & 1 & { - 1} & { - 1} \\ \end{array} } \right)$$

(13)

In the given expression, i = 0,1,2,3,…represent the nine discrete velocity directions in the D2Q9 model; c denotes the lattice speed, expressed as c = ∆x/∆t. The weight coefficients w_i are assigned as follows: w₀ = 4/9, w_1-4 = 1/9, w_5-8 = 1/36; The lattice sound speed c_s is $1/\sqrt 3$. The macroscopic velocity u and density ρ can be obtained through the following relationships.

$$\rho = \sum\limits_{i} {f_{i} }$$

(14)

$$\rho u = \sum\limits_{i} {f_{i} e_{i} }$$

(15)

Using the aforementioned relationships, through the Chapman-Enskog expansion^23,24, Eqs. (10) and (11) can be recovered into the mass and momentum conservation equations.

$$\partial_{t} \rho + \nabla \cdot \left( {\rho {\mathbf{u}}} \right) = 0$$

(16)

$$\partial_{t} \left( {\rho {\mathbf{u}}} \right) + \nabla \cdot \left( {\rho {\mathbf{uu}}} \right) = - \nabla p + \nabla \cdot {{\varvec{\upsigma}}}$$

(17)

In these equations, p denotes the fluid pressure and σ represents the viscous stress tensor.

Description of the LBM model for competitive gas adsorption

When gas adsorbates are transported to the surfaces of adsorbent particles through convection–diffusion, they are adsorbed at the contact points with the solid particle surface. The interaction between the gas adsorbates and the adsorbent particles constitutes a dynamic and reversible physical adsorption process, wherein adsorption and desorption behaviors coexist between the bulk gas phase and the pore walls. Therefore, by employing the derivation results under the monolayer assumption, the adsorption rate equation that characterizes the adsorption process on the adsorbent surface is presented as follows.

$$S_{C,A} = \partial N_{A} /\partial t = k_{a,A} C_{A} \left( {V_{m} - V_{A} } \right) - k_{d,A} V_{A} \left( {1 + \frac{{k_{a,B} }}{{k_{d,B} }}C_{B} } \right)$$

(18-1)

$$S_{C,B} = \partial N_{B} /\partial t = k_{a,B} C_{B} \left( {V_{m} - V_{B} } \right) - k_{d,B} V_{B} \left( {1 + \frac{{k_{a,A} }}{{k_{d,A} }}C_{A} } \right)$$

(18-2)

The adsorbates further diffuse from the outer surface of the adsorbent particles into the micropores within the adsorbent. Neglecting the impact of the internal micropore structure of the adsorbent particles, the diffusion process of adsorbates within the pores can be described using the Homogeneous Solid Diffusion Model (HSDM)²⁵. The mass transport equation for this process is shown as:

$$\frac{\partial V}{{\partial t}} = D_{p} \left( {\frac{{\partial^{2} V}}{{\partial x^{2} }} + \frac{{\partial^{2} V}}{{\partial y^{2} }}} \right)$$

(19)

where D_p is the solid diffusion coefficient within the adsorbent particles.

In mass transfer physics problems, the D2Q5 lattice model is employed, utilizing direction vectors 0 through 4. This model effectively captures the essential dynamics of mass transfer processes within the defined lattice structure²⁶. The evolution equations for the concentration of adsorbates in the bulk gas phase and the adsorbate content within the particles are approximated using the LBGK equation:

$$h_{i,n} \left( {{\mathbf{r}} + e_{i} {\Delta }t,t + {\Delta }t} \right) = h_{i,n} \left( {r,t} \right) - \frac{{1}}{{\tau_{h,n} }}\left[ {h_{i,n} \left( {r,t} \right) - h_{i,n}^{eq} \left( {r,t} \right)} \right] + \omega_{i} S_{C,n} {\Delta }t$$

(20)

$$n_{i} \left( {r + e_{i} \Delta t,t + \Delta t} \right) = n_{i} \left( {r,t} \right) - \frac{1}{{\tau_{n} }}\left[ {n_{i} \left( {r,t} \right) - n_{i}^{eq} \left( {r,t} \right)} \right]$$

(21)

In these equations, h_i and n_i are the distribution functions for the component concentration in the bulk gas phase and the adsorbate content within the particles, respectively. τ_h and τ_n are the corresponding dimensionless relaxation (collision) times. The equilibrium distribution functions, denoted as $h_{i}^{eq} \left( {r,t} \right)$ and $n_{i}^{eq} \left( {r,t} \right)$, are expressed as follows:

$$h_{i}^{eq} \left( {r,t} \right) = w_{i} C\left[ {1 + \frac{{e_{i} \cdot u}}{{c_{s}^{2} }}} \right]$$

(21)

$$n_{i}^{eq} \left( {r,t} \right) = w_{i} N$$

(22)

where w_i are the weight coefficients, and in the D2Q5 model, w₀ = 1/3 and w_1-4 = 1/6. Using the Chapman-Enskog expansion, the collision and evolution equations for the component concentration field and the adsorbate content can be recovered into the corresponding macroscopic equations through the following formulae:

$$D_{s} = c_{s}^{2} (\tau_{h} - 0.5)\Delta t,\;C = \sum\limits_{i} {h_{i} }$$

(23)

$$D_{p} = c_{s}^{2} (\tau_{n} - 0.5)\Delta t,\;N = \sum\limits_{i} {n_{i} }$$

(24)

Implementation of boundary conditions

Boundary treatment plays a crucial role in simulating flow and surface mass transfer in porous media. For flow problems, non-equilibrium extrapolation methods²⁷ are used for the inlet and outlet of the channel domain, with solid boundaries for the upper and lower boundaries and half step bounce-back boundary conditions. For mass transfer problems, detailed implementation of the concentration distribution function near the boundary requires careful analysis. Figure 2 illustrates concentration distribution functions on a fluid–solid boundary node under the D2Q5 model. The concentration distribution function on internal nodes of the fluid needs to be derived from the migration process of adjacent fluid nodes. But for fluid nodes near the boundary of particulate matter, the unknown concentration distribution function h₂ needs to be obtained through the following Eq. ²⁸

$$h_{2} = h_{4} - D_{s} \frac{{\partial C_{w} }}{\partial y}$$

(24)

Fig. 2

Illustration of concentration distribution functions on a fluid–solid boundary.

Taking component A as an example, the Langmuir adsorption kinetics equation can be obtained from formulas (9–1) and (18–1)

$$D_{s} \frac{{\partial C_{A} }}{\partial n} = \frac{{\partial N_{A} }}{\partial t} = k_{a,A} C_{A} \left( {N_{m} - N_{A} } \right) - k_{d,A} N_{A} \left( {1 + \frac{{k_{a,B} }}{{k_{d,B} }}C_{B} } \right)$$

(25)

Finally, by using the finite difference scheme to match the concentration gradient along the normal direction of the wall, and substituting it into formula (24), the unknown distribution function h₂ on the boundary can be obtained.

Verification of the adsorption model

To verify the accuracy of the adsorption model under the simultaneous presence of convection, diffusion, and adsorption–desorption phenomena, we use the classical single-channel steady-state convection–diffusion system Lévêsque example to validate the LBM model algorithm²⁹. The physical model is shown in Fig. 3.

Fig. 3

Physical model of the Single-Channel Steady-State Convection–Diffusion system considering wall adsorption.

The model dimensions are 320 × 320. Since steady-state pipe flow conforms to the characteristics of Poiseuille flow, a fully developed parabolic velocity profile is specified throughout the domain.

$$u\left( y \right) = - 4u_{max} y\left( {y - L} \right)/L^{2}$$

(26)

Maximum velocities are set at 0.02, 0.04, and 0.06, with a constant inlet concentration boundary condition C₀ set to 1. The concentration gradient is zero at the upper surface and outlet boundary, while adsorption occurs at the lower surface, which can be described using Henry’s adsorption kinetic boundary condition:

$$D_{s} \frac{\partial C}{{\partial y}} = kC$$

(27)

where k is the Henry adsorption constant, set to 1.0 in this study; the adsorbate diffusion coefficient D_s is set to 1/6. The Lévêsque analytical solution for the steady-state Nondimensional mass at the lower surface is expressed as

$$\mathop {C_{x} }\limits^{\sim } = \frac{L}{{C_{0} }}\frac{\partial C}{{\partial n}} = 0.854\left( {\frac{{u_{max} L^{2} }}{{xD_{s} }}} \right)^{1/3}$$

(28)

After the adsorption process reaches a steady state, the concentration field contour distribution of the LBM simulation results is shown in Fig. 4.

Fig. 4

Contour plot of the concentration field in the Single-Channel Steady-State Convection–Diffusion process considering wall adsorption.

The comparison between the LBM simulation results and the Lévêsque analytical solution and simulation results of Zhou, etc.³⁰ shows that, apart from a slight discrepancy near the inlet at x = 0 due to singularities, the results of both solutions are in close agreement (Fig. 5). This verifies the accuracy of the LBM method in simulating the coexistence of flow, diffusion, and interfacial adsorption effects.

Fig. 5

Comparison of LBM simulated nondimensional mass flux as u_max = 0.06 with Lévêsque analytical solution and simulation results of Zhou, etc.

Program validation and analysis of simulation results

Shale reservoir pressure usually ranges from 5 to 30 MPa and reservoir temperature mainly varies between 20 and 100 ℃³¹. In this study, a numerical simulation study was conducted on the CO₂ and CH₄ replacement processes, considering that the reservoir temperature is 50 ℃ and the reservoir pressure is 10 MPa (where the partial pressures of CO₂ and CH₄ are 5 MPa respectively).

The boundary conditions of the model are set in Table 1.

Table 1 Boundary conditions for simulation.

The physical model is built within a square channel of 100 × 100 lattice units. At the initial moment, the particles are adsorbed by the saturated CH₄ and CO₂ with a certain velocity and concentration is injected from the inlet on the left side of the flow field. The conversion between physical values and lattice values of simulation parameters is detailed in Table 2. Among them, the physical value of sound speed and kinematic viscosity are the average values of the two gases.

Table 2 Conversion between physical values and lattice values of simulation parameters. Symbols with subscripts l are the lattice parameters.

Adsorption/desorption characteristics of CO₂ displacing CH₄ in a single particle model

Considering shale particles with a diameter of 160nm (16 lattice units) in the flow field, the CO₂ injection concentration is 0.56kmol/m³. Utilizing a binary gas competitive adsorption model, which incorporates adsorption and desorption behaviors (Eqs. 9–1 and 9–2), the LBM method is used to simulate these behaviors within the region of interest numerically.

Figure 6 is the concentration distribution in the flow field and content in the particle of CH₄ and CO₂ at 1.123μs. The simulation results show that after CO₂ is injected from the left inlet, a higher CH₄ concentration appears in the flow field near the windward side of the adsorbent particles. This is because the adsorbent particles first come into contact with CO₂ on the windward side surface, and CO₂ begins to adsorb to the surface of the particles, while prompting rapid desorption of CH₄ inside the particles. It also shows that the injection of CO₂ significantly enhances the desorption ability of CH₄. CH₄ desorbed from particulate matter continuously diffuses into the surrounding flow field and migrates downstream through convection.

Fig. 6

Distribution of CH₄ and CO₂ concentration in flow field and CH₄ and CO₂ content in the particle at 1.123μs.

Due to the continuous desorption of methane on the surface of the particles, the methane content inside the particles continues to decrease, which causes the desorption power of CH₄ on the surface of the particles to continue to weaken, and the desorption speed also decreases. Eventually, CH₄ is completely desorbed from the particles, and the CH₄ concentration inside the particles and in the flow field gradually decreases to zero (as shown in Fig. 7).

Fig. 7

Distribution of CH₄ concentration in flow field and CH₄ content in the particle at different time.

As shown in Fig. 8, from the comparison curves of the content changes of CO₂ and CH₄ adsorbates in particulate matter and the adsorption/desorption rates, it can be seen that the process of CO₂ replacing CH₄ includes two different stages: the competitive adsorption stage and adsorption/desorption equilibrium stage. After CO₂ is injected, the adsorption process of CO₂ and the desorption process of CH₄ occur simultaneously. In the initial stage of competitive adsorption, the adsorption rate of CO₂ and the desorption rate of CH₄ both increase sharply, and the desorption rate value of CH₄ is slightly higher than the adsorption rate value of CO₂. It is worth noting that at about 0.558 μs, the curves of the CO₂ adsorption rate and the CH₄ desorption rate reach their extreme values at the same time. After that, the two values began to decrease and gradually decreased to zero, indicating that the two gases reached adsorption/desorption equilibrium. In this process, the desorption process of CH₄ reaches equilibrium before the adsorption process of CO₂.

Fig. 8

Comparison of CO₂ and CH₄ contents and adsorption/desorption rates in particles over time.

In order to analyze the impact of the CO₂ concentration in the injected fluid on the CO₂ and CH₄ content in the particles, when the CO₂ concentration is 0kmol/m³ (no carbon dioxide injection), 0.2kmol/m³, 0.4kmol/m³, and 0.6kmol/m³, the model area Numerical simulations were performed. The study found that (as shown in Fig. 9), when there is no carbon dioxide injection, the CH₄ content in the particles decreases the slowest with time, indicating that the desorption rate of CH₄ is the slowest. The greater the CO₂ concentration of the injected fluid, the faster the CH₄ content in the particles decreases, indicating that the CH₄ desorption rate is faster and the time required to reach desorption equilibrium is shorter. Under corresponding conditions, the greater the CO₂ concentration of the injected fluid, the higher the CO₂ content in the particles. This indicates that the CO₂ adsorption rate is faster and the time required to reach desorption equilibrium is shorter.

Fig. 9

Comparison of changes in the content of CH₄ and CO₂ adsorbates inside the particles as different CO₂ injection concentration.

Therefore, this model can effectively simulate the CO₂/CH₄ competitive adsorption phenomenon. Numerical results show that increasing the CO₂ concentration of the injected fluid can significantly promote the desorption of CH₄ and help improve the displacement efficiency of CO₂ to CH₄.

Adsorption/desorption characteristics of CO₂ displacing CH₄ in a porous media model

The process of CO₂ displacing CH₄ is a key feature of CO₂-ESGR technology. Combined with the binary gas adsorption rate equation, the LBM method was used to numerically simulate the physical process of CO₂ displacing CH₄ accompanied by adsorption and desorption behavior in the porous media model. The physical model is established within a square channel of 100 nm × 100 nm. The initial adsorbed CH₄ content in the particles is set at 0.14kmol/m³, and CO₂ with a concentration of 0.4kmol/m³ is injected at the Re number about 0.25–0.75. Due to the pore structure’s inhomogeneity, significant velocity changes are observed throughout the flow field. Areas characterized by low permeability exhibit lower fluxes and flow rates, while areas with high permeability exhibit higher fluxes and flow rates, as shown in Fig. 10.

Fig. 10

Distribution of vectors and magnitudes of fluid velocity.

From a time evolution perspective, the migration and distribution of CO₂ concentration are significantly affected by the flow velocity. In the high flow rate area, the fluid flux increases and the CO₂ concentration migrates faster. On the contrary, in the low flow rate area, the fluid flux increases and the CO₂ concentration migrates faster. In addition, CO₂ in the flow field also migrates through self-diffusion. Therefore, the concentration gradient near the front of the CO₂ concentration distribution is large, significantly higher than the concentration gradient at the rear (as shown in Fig. 11).

Fig. 11

Contour line of CO₂ concentration in the flow field at 22.66μs.

The spatial and temporal evolution of CH₄ and CO₂ concentrations (as shown in Fig. 12) shows that the concentrations of the two gases are complementary. Regions with high CO₂ concentration have low CH₄ concentration and vice versa. As time goes on, CO₂ gradually occupies the pore space, displacing CH₄.

Fig. 12

Distribution of CH₄ and CO₂ concentration in flow field at different time.

The behavior of CO₂ displacing CH₄ within the pores is more pronounced in the concentration changes within the particles. As shown in Fig. 13, over time, the CO₂ content within the particles increases while the CH₄ content decreases. This complementary distribution further demonstrates that CO₂ injection facilitates CH₄ desorption.

Fig. 13

Distribution of CH₄ and CO₂ content in particles at different time.

Three porosity models (0.64, 0.76, and 0.89) were employed to numerically simulate the CO₂ displacing CH₄ process to investigate further porosity’s impact on the adsorption and desorption characteristics of porous media. By comparing the CO₂ concentration distribution in the flow field with the spatiotemporal distribution of CO₂ content within the porous media matrix (as shown in Fig. 14), it was observed that the CO₂ concentration distribution in the flow field significantly influences the distribution of adsorbate CO₂ content within the pore matrix. The evolution of these distributions occurs almost synchronously in both space and time.

Fig. 14

Distribution of CO₂ concentration in flow field and CO₂ content inside particles at 14.16μs for different porosity.

Related studies have shown that in high porosity materials, the surface diffusion rate of CO₂ may be increased due to enhanced pore connectivity³⁶, while in low porosity media, gas diffusion is limited by narrow pore throat structures, resulting in slow adsorption desorption rates³⁷. From the simulation results in this article, it can be seen that models with higher porosity exhibit better permeability, while the diffusion rate of adsorbates is faster and the displacement efficiency is higher, which is consistent with the conclusions in the literature and further confirms the correctness of the theoretical model derived in this article.

Furthermore, increased porosity correlates with a reduced time required to achieve adsorption/desorption equilibrium for CO₂ and CH₄. Under identical CO₂ injection concentrations, models with higher porosity more readily attain adsorption/desorption equilibrium, and the total adsorption amount at equilibrium is lower (as illustrated in Fig. 15).

Fig. 15

Changes in CH₄ and CO₂ content inside particles with different porosity over time.

Conclusion

In this study, the lattice-Boltzmann method (LBM) was utilized to simulate the flow, diffusion, and related adsorption/desorption behaviors during the displacement process of CO₂/CH₄ gases in a porous adsorption system at the pore scale. The primary research findings and conclusions are as follows:

• (1) The theoretical formula of the binary gas competitive adsorption rate model was derived, and this formula was applied to the lattice Boltzmann method (LBM) to simulate the displacement process of CH₄ by CO₂. The research results show that the obtained theoretical model can truly reflect the competitive adsorption behavior between CO₂ and CH₄, and helps analyze the adsorption and desorption characteristics in the process of CO₂ displacing CH₄.

• (2) The process of CO₂ displacing CH₄ is a dynamic process in which CO₂ adsorption and CH₄ desorption occur simultaneously. The injected CO₂ is gradually adsorbed by the particles, occupying the original CH₄ adsorption sites. CH₄ inside the particles is gradually desorbed and migrates downstream through convection and diffusion in the flow field. It is worth noting that both gases eventually reach the adsorption/desorption equilibrium state, and CO₂ reaches adsorption equilibrium slightly later than CH₄ reaches desorption equilibrium.

• (3) The injection concentration of CO₂ will significantly affect the characteristics of the CO₂/CH₄ displacement process. The higher the concentration of injected CO₂, the shorter the time required for CO₂/CH₄ to reach adsorption and desorption equilibrium, indicating that increasing the CO2 injection concentration can increase the CO₂ adsorption rate and CH₄ desorption rate, thus promoting the adsorption of CO₂ and the desorption of CH₄. Therefore, the replacement efficiency of CH₄ by CO₂ can be effectively improved by increasing the concentration of carbon dioxide in the injection.

• (4) Models with higher porosity exhibit better permeability, resulting in faster diffusion rates of adsorbate and increased displacement efficiency. The heterogeneity of pore structure significantly affects the velocity distribution in the flow field, and then plays a dominant role in the distribution of CO₂ and CH₄ concentration fields. The concentrations of CH₄ and CO₂ in the flow field and the contents of CO₂ and CH₄ in the particles show obvious complementarity and synchrony in the spatial and temporal distribution. This observation further demonstrates that CO₂ injection can promote CH₄ desorption, thereby helping to improve shale gas recovery.