Microscopic Flow of CO₂ in Complex Pore Structures: A Recent 10-Year Review

Abstract

To prevent CO₂ leakage and ensure the safety of long-term CO₂ storage, it is essential to investigate the flow mechanism of CO₂ in complex pore structures at the pore scale. This study focused on reviewing the experimental, theoretical, and numerical simulation studies on the microscopic flow of CO₂ in complex pore structures during the last decade. For example, advanced imaging techniques, such as X-ray computed tomography (CT) and nuclear magnetic resonance (NMR), have been used to reconstruct the complex pore structures of rocks. Mathematical methods, such as Darcy’s law, the Young–Laplace law, and the Navier-Stokes equation, have been used to describe the microscopic flow of CO₂. Numerical methods, such as the lattice Boltzmann method (LBM) and pore network (PN) model, have been used for numerical simulations. The application of these experimental and theoretical models and numerical simulation studies is discussed, considering the effect of complex pore structures. Finally, future research is suggested to focus on the following. (1) Conducting real-time CT scanning experiments of CO₂ displacement combined with the developed real-time CT scanning clamping device to achieve real-time visualization and provide a quantitative description of the flow behavior of CO₂ in complex pore structures. (2) The effect of pore structures changes on the CO₂ flow mechanism caused by the chemical reaction between CO₂ and the pore surface, i.e., the flow theory of CO₂ considering wettability and damage theory in a complex pore structures. (3) The flow mechanism of multi-phase CO₂ in complex pore structures. (4) The flow mechanism of CO₂ in pore structures at multiscale and the scale upgrade from microscopic to mesoscopic to macroscopic. Generally, this study focused on reviewing the research progress of CO₂ flow mechanisms in complex pore structures at the pore scale and provides an overview of the potential advanced developments for enhancing the current understanding of CO₂ microscopic flow mechanisms.

1. Introduction

Due to the increasing greenhouse effect as well as acute global climate and environmental problems, countries worldwide have reached a consensus to actively respond to climate change and reduce greenhouse gas emissions [1,2]. Since industrialization has continued since the industrial revolution, the use of fossil fuels has increased, and large amounts of CO₂ gas have been directly discharged into the air, which is one of the main reasons for the intensification of the greenhouse effect [3]. CO₂ emissions have led to a series of environmental problems, such as drought, glacier melt, and sea level rise, which have caused incalculable harm to the environment on which human beings depend [4]. At present, many technologies are available to control CO₂ emissions, such as improving the efficiency of fossil energy combustion, the efficient development and utilization of green and clean energy, and carbon capture and storage (CCS) [5]. CCS refers to the use of separation and purification technology to collect a large amount of CO₂ from industrial waste gas and inject it into appropriate underground reservoirs for permanent storage [6], which is recognized as one of the effective methods to manage CO₂ [7]. At present, the membranes used for the selective transport of CO₂ in the mixture of gases based on hollow alumina fiber-supported silica have been developed [8]. The application of this advanced technology greatly improves the efficiency of CO₂ capture. The main geological layers used for storage include abandoned mines, unrecoverable coal seams, depleted oil and gas fields, deep saline aquifers, and the ocean [9]. The storage depth is generally below 800 m. CO₂ geological storage is the most important part of CCS, and the storage capacity and safety of geological storage bodies determine the effectiveness of CO₂ emission control [10]. Compared to other gases, CO₂ is affected by temperature and pressure in the reservoir, and will produce phase transitions (gaseous, liquid, and supercritical). The microscopic flow of different phase states of CO₂ in porous media is different. In addition, when CO₂ flows in a reservoir, it can undergo physical and chemical reactions with reservoir water and matrix, causing damage to the reservoir. Meanwhile, the complexity, heterogeneity, and wettability [11] of the pores directly affect the flow behavior of CO₂ in reservoirs. Therefore, investigating the CO₂ flow law and revealing the CO₂ microscopic flow mechanism are key for evaluating the storage capacity and safety of geological storage bodies.

Numerous studies have been conducted on the flow of CO₂ in geological storage bodies. Lassen et al. [12] injected gaseous CO₂ into heterogeneous porous media at different rates, and the flow of CO₂ was monitored by sensors. The results showed that large-scale heterogeneity controlled the overall migration of gaseous CO₂ in porous media, whereas a smaller scale was important for gas saturation. The higher the injection rate, the larger the transverse diffusion of the gas phase. Zhang et al. [13] proposed that Darcy’s law could be used to describe a two-phase fluid flow in porous media at the macroscopic scale. Saleem et al. [14] compared and verified the constructed two-phase flow model using field observation data, such as the CO₂ eruption time, changes in the sediment pH, gas leakage rate, flow process, fluid interaction, and CO₂ dissolution in the CO₂ plume. The results showed that the CO₂ plume was formed and developed at a stable rate during the flow process, and the dissolution rate increased with an increase in the injection rate. These studies elucidated the flow of CO₂ from a macroscopic perspective. However, the flow behavior of CO₂ in a storage body is easily affected by its complex pore structures. In macroscale research, the complex pore structures of the storage body have not been accurately characterized. The research results were also based on macroscale flow and often ignored the effect of the pore structure complexity. Therefore, an accurate description of the microscopic flow of CO₂ is crucial for determining storage capacity and long-term safety [15].

X-ray computed tomography (CT) and nuclear magnetic resonance (NMR), as non-invasive ad non-destructive advanced imaging technologies [16], are advantageous for reconstructing the complex pore structures of cores [17] and obtaining the key parameters, such as the porosity, permeability, contact angle, and capillary number. The visualization and quantitative characterization of the complex pore structures at the microscopic scale can be realized [5]. Berg and Dalton et al. [18,19] obtained accurate complex pore structures of sandstone and investigated the effects of the permeability and wettability on the CO₂ flow behavior. Liu et al. [20,21] first used a new reconstruction method to establish the fracture-controlled matrix unit then used the CT scanning technique to obtain the relevant parameters of the proposed fracture control matrix unit mass transfer model. The correctness of the model was verified by a comparison with the experimental results of the microscopic spontaneous imbibition. Liu et al. [22,23] studied the dynamic production process of a fractured reservoir and the influence of a complex pore structure on the imbibition recovery based on a fracture-controlled matrix unit. Liu et al. [24] performed microscopic spontaneous imbibition experiments, CT scans, and NMR tests on two mixed wetted core samples. The impact of mixed wettability on the micro-spontaneous imbibition at the pore scale was investigated. The results showed that an effective imbibition was produced so long as water-wet walls were present in the mixed wettability pores. Furthermore, a core-scale mixed wettability model was established by Liu et al. [25] and, based on the phase-field theory, the influence of wettability on the oil–water two-phase imbibition was studied. In addition, Liu and Song et al. [26,27] proposed two new modeling methods based on CT scanning: the finite volume element modeling method (Figure 1a) and the pore network model based on the maximum sphere algorithm (Figure 1b). The feasibility of the model was verified. The parameters of the models constructed using two different modeling methods are shown in

Table 1. Absolute permeability of the simulation and experiment [26].

No.	Size (10−3 m)	Element Number	Porosity	kx/mD	ky/mD	Kz/mD	Kz/mD(exp.)
a1	1.326	338,899	17.83%	8.26	5.15	3.56	3.35
a2	2.136	561,158	47.62%	1646.92	1843.57	1245.11	-

and

Table 2. Basic parameters of the models [27].

No.	Size (mm)	Number of Pores	Number of Throats	Average Connection Number	Porosity of EPNM (%)	Experimental Porosity (%)	EPNM/mD
b1	2.14	6298	12,558	3.92	19.27	18.73	1210
b2	1.46	810	1563	3.72	13.71	13.11	872

, respectively.

In addition, experts conducted several theoretical and numerical simulations on the flow mechanism of CO₂ at the macroscale. Krause et al. [28] performed a numerical simulation study of the CO₂–brine two-phase flow using the modified Carman–Kozeny equation. The distribution of CO₂–brine in the core was predicted. Theoretical studies and numerical simulations at the macroscale play a crucial role in revealing the flow mechanism of CO₂. However, the complex pore structures have a considerable effect on the CO₂ microscopic flow. The lack of mathematical and numerical models that consider the influence of the pore structure complexity may lead to bias in the conclusions.

The previous studies predominantly focused on the macroscopic flow of CO₂ in porous media, often overlooking the microscopic flow mechanisms. Simultaneously, the research processes frequently oversimplified the pore structures and neglected the influence of their complexity on the microscopic flow of CO₂. At present, there is a lack of comprehensive reviews on the microscopic flow of CO₂ in complex pore structures. In order to fill the knowledge gap in this field, it was necessary to conduct detailed academic research and summary. Therefore, we collated and analyzed relevant literature on the microscopic flow of CO₂. In particular, the complex pore structures substantially affected the CO₂ microscopic flow mechanism. At present, experimental methods, theoretical analyses, and numerical simulations are advanced, owing to the pore structure complexity of geological storage bodies, which is difficult to accurately express in physical model reconstruction, mathematical model characterization, and numerical model construction. Therefore, it is challenging to clearly reveal the microscopic flow mechanism of CO₂ in a complex pore structure. To solve this problem, this study systematically reviewed the recent advances in the microscopic flow of CO₂ in complex pore structures in the last decade to provide some technical and theoretical support for CO₂ geological storage. Overall, this review article focused on the effects of complex pore structures on the microscopic flow behavior of CO₂ through three aspects: experimental studies, theoretical studies, and numerical simulations. It aimed to enhance the current understanding of the CO₂ microscopic flow mechanism to help beginners quickly obtain an in-depth understanding of the current state of research in this field. Finally, the potential challenges and new research directions for future work were identified and elucidated.

2. Flow Experiments of CO₂ in Complex Pore Structures

Numerous experiments have been conducted to study the microscopic flow of CO₂ in pore structures, and fruitful results have been achieved. This section introduces the research progress of the microscopic flow mechanism of CO₂ from four different aspects through an analysis of the relevant literature.

2.1. Traditional Characterization of Pore Structures during CO₂ Microscopic Flow

Mercury intrusion porosimetry (MIP) [29], scanning electron microscopy (SEM) [30], and gas adsorption [31] have been used to characterize the pore structures of the storage body and further explore the flow of CO₂. Du et al. [32] used high-pressure mercury intrusion and permeation experiments to investigate the effect of CO₂ injected into coal. The results showed that the reaction between CO₂–water and coal led to an increase in the pore space and greatly increased the permeability, which increased the permeability area of CO₂ and improved the CO₂ storage capacity. Using SEM, Khather et al. [33] observed that a decrease in the pH during CO₂ injection caused the dissolution and migration of minerals, which resulted in an increase in the rock permeability and flow capacity of CO₂. Pearce et al. [34] studied the physical properties of the reservoir after CO₂ injection using SEM and found that the movement of fine particles could result in opening or blocking the pores during CO₂ injection, increasing or decreasing the permeability and affecting the CO₂ injection capacity. Brattekas and Haugen [35] used high-resolution micro-positron emission tomography (micro-PET) and radiotracers to achieve CO₂ tracing during the flow and capillary capture. The results showed that CO₂ mainly flowed into the outer part of the core with a high permeability.

However, these techniques had numerous limitations for accurately characterizing the pore structure complexity in rocks. For example, MIP only obtained the total porosity of the sample and did not characterize the complexity of the pore distribution within the sample. SEM was an effective technique for generating 2D images of the microstructures. However, it did not provide 3D images, which were important for evaluating the pore structure complexity [5]. The measurement results obtained using gaseous adsorption methods were often constrained by the limited pore-scale range, and accurately characterizing the complexity of pore structures became challenging due to the size and interactions of the gas molecules [36]. More importantly, the traditional characterization methods did not visualize and quantitatively describe the microscopic flow of CO₂. In addition, Tang et al. [37] conducted a comprehensive comparison of the rock pore structure characterization techniques, including experimental analyses, image analyses, and digital core techniques, and emphasized the importance of digital core techniques for solving complex pore structure characterization. Therefore, to accurately reconstruct the 3D complex pore structures and visually and quantitatively study the microscopic flow mechanism of CO₂ in the geological storage body, it was necessary to use advanced techniques, such as CT scanning and NMR, to conduct an in-depth analysis using multidisciplinary intersection research ideas.

It is worth noting that with the continuous improvement of artificial intelligence (AI), some research applied it in the field for studying the microscopic flow of CO₂ and achieved some remarkable results. A hybrid artificial intelligence model integrating a back propagation neural network (BPNN), genetic algorithm (GA), and adaptive boosting algorithm (AdaBoost) was proposed by Yan et al. [38], which was used to evaluate the change in the coal strength caused by the interaction between CO₂ and coal in the flow process. Zhang et al. [39] proposed four advanced machine learning schemes, including RBFNN-MVO, RBFNN-GWO, RBFNN-PSO, and MLPLM, to evaluate the contact angle of various shale systems. Overall, AI plays a key role in predicting the parameters of complex pore structures as well as evaluating safety. In addition, the method of using AI to predict fluid flow is a hot topic for future research.

2.2. Effects of the Porosity and Permeability Characteristics on the Microscopic Flow Mechanism of CO₂

The influence of the key parameters, including the porosity and permeability, on CO₂ flow in geological storage bodies is very important. By simulating 2D fluid flow experiments, Kitamura et al. [40] found that the flow behavior of CO₂ was strongly influenced by the small-scale heterogeneity of the pore structures. However, the simulation results had some errors since the model was 2D and could not fully describe the pore structure complexity. To investigate the effect of the reservoir laminar structures on the flow behavior of CO₂, Krishnamuthy et al. [41] used the CT scanning technique to observe the flow path of CO₂ by measuring the CO₂ saturation variation in the rock. The results showed that CO₂ preferentially flowed through the region with a larger porosity and passed unevenly along the axial direction (Figure 2). However, seepage under in situ conditions was not considered. Thus, the pore distribution in reservoir cores under in situ conditions could not be obtained. To accurately study the CO₂ microcosmic flow, Wang et al. [42] injected liquid CO₂ into a brine-saturated core. Multiscale CT scanning of sandstone was conducted and the distribution of the porosity at different locations in the core under in situ flow conditions was obtained. The phenomenon of CO₂ preferentially passing through the locations with a higher porosity was observed from the images. Al-Bayati et al. [43] conducted a displacement experiment on the stratified core samples using the CT scanning technique. Figure 3 shows the 2D images of the fluid distribution in the XY direction of the stratified samples before and after CO₂ displacement. As illustrated in the Figure 3, CO₂ first bypassed the low-permeability layer, leaving a considerable amount of oil to preferentially enter the high-permeability layer. Owing to the occurrence of the cross-flow phenomenon, CO₂ was transferred from the high- to the low-permeability layer.

A clay interlayer, one of the typical representatives of a low-permeability layer, had an important influence on the CO₂ flow. Using CT scanning, Xu et al. [44] studied the flow characteristics of CO₂ in a special sandstone containing multiple thin clay interlayers. The experiment demonstrated that the flow channel of CO₂ was mainly established in the sandstone, and the clay interlayer hindered the CO₂ flow. However, the description of how the CO₂ flow was hindered by the clay interlayer was too vague. Therefore, Xu et al. [45] further monitored the flow process of CO₂ in the interior of the clay interlayer sample (Figure 4). When the injection direction was parallel to that of the clay interlayer, the clay interlayer separated the CO₂ flow (Figure 4a). When the injection direction was perpendicular to that of the clay interlayer, the clay interlayer hindered the forward CO₂ flow (Figure 4b).

2.3. Two-Phase Flow Law in CO₂ Microscopic Flow

When CO₂ was injected into the deep brine layer, it entered the reservoir pore structures to displace the original fluid from the pore space, and the flow process was an immiscible two-phase flow [16]. However, it was difficult to observe the space–time variation of the two-phase interface between the immiscible two-phase fluids in the porous media. Therefore, the immiscible two-phase flow needed to be understood from a pore-scale perspective, which was very important and extremely complex [46].

Using the CT scanning technique, Kogure et al. [47] injected CO₂ into the Berea sandstone at different times in opposite directions, and the flow behavior of CO₂ in the Berea sandstone was observed. The images showed several narrow pore throats that allowed for the CO₂ flow. Despite injections from opposite directions, the distribution of CO₂ was essentially the same in the final stage of injection (Figure 5). Liu et al. [48] injected CO₂ into a glass bead bed in both the upward and downward directions, and a displacement-saturated water experiment was conducted to investigate the distribution of CO₂ in the core. As shown in Figure 6, the displacement effect of CO₂ injected downward was substantially better than that of CO₂ injected upwards (Figure 6b). This was attributed to the “gas channeling” phenomenon when CO₂ was injected upward, but the cause of the “gas channeling” phenomenon was not discussed in depth. Lv et al. [49] combined the CT scanning technique and a micromodel to study the flow process of CO₂–brine in the pore structures at different injection rates under static and transient conditions. The results showed that a higher injection rate caused a higher displacement efficiency but a lower sweep efficiency. Further, using the NMR technique, Teng et al. [50] found that the different viscosities and densities of CO₂ and water caused the “gas channeling” phenomenon of CO₂, which led to the premature breakthrough of CO₂ and reduced the displacement efficiency. In addition, Zhang et al. [13] obtained the local porosity and saturation of the Berea sandstone. The results showed that the forward movement of CO₂ on the capillary pressure was the main reason for the formation of the pathway of the CO₂ flow. CO₂ preferentially passed through the large-sized pores, and the seepage zone gradually expanded whereas the CO₂ saturation increased.

The Influence of pore geometry on the microscopic flow mechanism of CO₂ was not ignored. Zhang et al. [51] used CT scanning to scan the displacement process of CO₂ and found that CO₂ was unevenly distributed in the sandstone samples. A larger flow patch was formed during the flow process. Herring et al. [52] adopted the Bentheimer sandstone core and used the CT scanning technique to observe the displacement process of CO₂ at the pore scale. The images showed that CO₂ invaded the pore space in a capillary fingering regime (Figure 7). Liu et al. [53] visualized and quantitatively analyzed the dynamic diffusion process of CO₂ in n-decane-saturated porous media. The results showed that the channels of the porous media hindered the diffusion of CO₂. The local diffusion coefficient of CO₂ gradually decreased with time along the diffusion path until it reached a steady state.

Figure 5. X-ray CT images of the distribution of scCO₂ in the Berea sandstone [47].

Figure 5. X-ray CT images of the distribution of scCO₂ in the Berea sandstone [47].

Figure 6. Characterization of the displacement process and complex pore structures based on CT scanning: (a) 3D images of the CO₂ distribution in the displacement process and the analysis of the displacement efficiency [48]; (b) 3D and 2D images of the complex pore structures and grayscale 2D images of the pores [52], * Indicates that capillary pressure is estimated for this data point.

Figure 6. Characterization of the displacement process and complex pore structures based on CT scanning: (a) 3D images of the CO₂ distribution in the displacement process and the analysis of the displacement efficiency [48]; (b) 3D and 2D images of the complex pore structures and grayscale 2D images of the pores [52], * Indicates that capillary pressure is estimated for this data point.

2.4. Microscopic Flow of CO₂ in Different Phase States

In geological storage, appropriate CO₂ phase states, including gaseous, liquid, and supercritical, should be selected according to the specific conditions and requirements of the storage body. CT scanning technology has been used to conduct extensive research on the microscopic flow of CO₂ in different phase states. CO₂ bubbles, a typical representative of the gaseous state, have a stable state in porous media with a high resistance factor (i.e., good plugging capacity), which plays an essential role in the safety of long-term storage of CO₂ [54]. Xue et al. [55] found that microbubble CO₂ injection minimized the content of free CO₂ in the reservoir compared to that of conventional CO₂ injection and also efficiently utilized the pore space of the reservoir, which was conducive to the long-term safety of large-scale CO₂ storage. Patmonoaji et al. [56] and Zhai et al. [57] conducted displacement experiments in sandstone and found that the microbubble flow had a stronger sweeping efficiency than that of the conventional flow, which improved the displacement efficiency and storage capacity of CO₂.

The experimental results proved that CO₂ foam not only improved oil and gas recovery compared to conventional CO₂, but also improved the pore space utilization and increased CO₂ storage in the reservoir. Accordingly, Du et al. [58] further investigated the seepage characteristics of CO₂ bubbles in porous media. However, the experimental object was self-made homogeneous porous media, whereas most of the actual reservoir was composed of heterogeneous rocks. Thus, the experimental results had some limitations. For this reason, McLendon et al. [59] injected CO₂ with and without a surfactant into real Berea sandstone under high-pressure conditions to observe the in situ bubble generation. The results showed that CO₂ tended to pass through the high-permeability reservoir without the addition of a surfactant but resulted in a decrease in the sweep efficiency of CO₂. Du et al. [60] studied the dynamic bubble flow behavior in the entrance region of porous media and obtained dynamic three-phase saturation distributions along the sample core. The results showed that the CO₂ bubble pushed most of the liquid phase into the latter part of the porous media, but the forepart was difficult to push, showing an obvious entrance effect.

The above studies investigated the microscopic flow of single-phase CO₂. However, it was unusual that CO₂ not only existed in the actual storage body, but two or multiphase may be present. Therefore, it was essential to further study the microscopic flow of CO₂ under the condition of polyphase coexistence (including miscible, near-miscible, and immiscible phases). Alhosani et al. [61] conducted an in situ study on immiscible-phase CO₂ displacement in oil-wet reservoirs. The images showed that in strongly oil-wet rocks, the largest pore space was occupied by water, the smallest pore space was occupied by oil, and the medium-sized pore space was occupied by CO₂ (Figure 7a). CO₂ was distributed in a connected layer under near-miscible-phase conditions (Figure 7b) and existed as separated “ganglia” in medium-sized pores under immiscible-phase conditions (Figure 7c). Qin et al. [62] conducted a study on the wettability and spatial distribution of near-miscible-phase CO₂ in oil-wet carbonate rocks under a high temperature and pressure. The results showed that at the initial stage of injection, CO₂ had good connectivity with the oil phase and poor connectivity with the brine phase. With the continuous injection of CO₂, the wettability reversal process was triggered, resulting in a decrease in the oil wettability and an improvement in the CO₂ connectivity with brine. Hao et al. [63] conducted multi-phase and multiple injection experiments. It was found that under the immiscible-phase condition, the porous media showed remarkable gas coverage and flow stratification owing to gas buoyancy. The miscible-phase CO₂ injection eliminated the effect of buoyancy, thus expanding the storage area and flow range of CO₂.

Figure 7. 2D and 3D images of a trapped water ganglion in a single pore: (a) segmented pore-scale images; (b) near-miscible condition; (c) immiscible condition [61].

Figure 7. 2D and 3D images of a trapped water ganglion in a single pore: (a) segmented pore-scale images; (b) near-miscible condition; (c) immiscible condition [61].

The reliability of CO₂ geologic storage depends on the flow mechanism of CO₂ in reservoirs. Due to advanced imaging techniques such as CT scanning and NMR in experimental studies, it was beneficial to study the microscopic flow mechanism of CO₂ in complex pore structures. However, few studies used CT scanning and NMR technology to conduct real-time visualization and quantification research on the microscopic flow process of CO₂. In addition, microscopic flow studies of different phases of CO₂ in complex pore structures were not sufficiently comprehensive, which will be the focus of future research.

3. Theoretical Model of CO₂ Flow in Pore Structures

Theory is the cornerstone of all experiments and numerical simulations, and the continuous improvement of the theory is the premise for studying the microscopic flow mechanism of CO₂. Darcy’s law [64] is one of the classical theories describing the macroscopic flow law of fluid in porous media, which is applicable to single- and multi-phase fluid flows in loose sand columns, consolidated sandstone, and other dense porous media [41,65].

Zhang et al. [13] proposed Darcy’s law to describe the two-phase fluid flow in porous media at the macroscopic scale.

$$q_{\alpha }=-\frac{k{k}_{r\alpha }}{\mu }(\nabla P)$$

(1)

where q is the flow flux, m³/s; k is the absolute permeability, m²; k_r is the relative permeability; μ is the viscosity of a fluid, Pa·s; ∇P is the pressure gradient, Pa; and α = 1 and 2 are the two phases of a fluid.

Berg et al. [18] used Darcy’s law for two-phase flows to establish a mathematical model. The migration and mass transfer behaviors of the saturated and unsaturated CO₂–brine systems in sandstone were studied. The governing equations are as follows.

$$\phi \frac{\partial S_i}{\partial t}+\nabla \cdot \overrightarrow{v_i}=0$$

(2)

$$\overrightarrow{v_i}=-\frac{k_{r,i}}{{\mu }_i}K(\nabla p_i-{\rho }_i\overrightarrow{g})$$

(3)

Equation (2) describes the mass balance of CO₂ and water. The saturation of the wetting phase (water or brine, i = w) and the non-wetting phase (CO₂, i = nw) satisfies S_w + S_nw = 1. The flux of the phases $\overrightarrow{v_i}$ is described by Darcy’s law extending to a two-phase flow, m/s; where φ is the porosity; K is the absolute permeability of the rock, m²; μ_i is the viscosity of the fluids, Pa·s; ρ_i is the density of the fluids, kg/m³; and g is the gravity constant, m/s².

However, in the process of CO₂ displacement, it is difficult to accurately capture the spatiotemporal variation of the interface of an immiscible two-phase fluid in the pore structures. For this reason, Porter et al. [66] redefined the simple expression of pressure in the absence of CO₂ gas, considering the pressure decrease caused by the flow and heterogeneity, which reliably predict the position of the single-phase to multi-phase flow transition.

$$P_{cr}=P_{sat}+\frac{\mu q(L-Z_p)}{k_{eff}}$$

(4)

where P_cr is the critical pressure, Pa; P_sat is the hydrostatic pressure, Pa; μ is the viscosity of the water, Pa·s; q is the injection rate of the CO₂-saturated water, m/s; L is the location of the bottom of the porous media, m; Z_p is the location of the top of the porous media, m; and k_eff is the vertical (harmonic) averaged effective permeability, m².

Darcy’s law has successfully solved many CO₂ geological storage safety problems [65,67,68]. However, it also has some limitations. For example, in the case of a high velocity flow, high Reynolds number (Re), and non-linear flow, a turbulent flow will occur in the complex pore structures, making Darcy’s law no longer applicable. Therefore, many scholars have proposed the use of the extended form or a combination with the actual situation to appropriately modify Darcy’s law to study the single-phase CO₂ flow, CO₂–brine, or other miscible or immiscible flow behaviors at the pore scale.

Kogure et al. [47] proposed the extended Darcy’s law to calculate the relative permeability of porous sandstone in the CO₂–water system at the sub-core scale. The flow behavior of CO₂ was studied.

$$Q_i=k_{ri}{k}_{abs}\frac{A}{{\mu }_i}(\frac{\Delta P}{L}-{\rho }_{i}g)$$

(5)

where Q_i is the flow rate of each fluid, m³/s; k_ri is the relative permeability, k_abs is the absolute permeability, m²; A is the cross-sectional area of the sample, m²; L is the length of the sample, m; ΔP is the pressure difference of the sample, Pa; μ_i is the viscosity of the fluid, Pa·s; ρ_i is the density of the fluid, kg/m³; and g is the acceleration of gravity, m/s².

However, the above scholars first determined the permeability of a complex pore structure based on Darcy’s law. Then, the flow behavior of CO₂ in the complex pore structures was studied, which was a very complicated process. Therefore, Wang et al. [69] used the extended form of Darcy’s law to describe the flow behavior of CO₂–water in the pore structures.

$$v_{\alpha }=-\frac{K{k}_{r\alpha }}{{\mu }_{\alpha }}(\nabla p_{\alpha }-{\rho }_{\alpha }g)$$

(6)

where v denotes the Darcy velocity, m/s; K is the intrinsic permeability tensor, m²; k_r is the relative permeability of the fluid phase, μ denotes the viscosity, Pa·s; ρ represents the density, kg/m³; P represents the pressure of the fluid phase, Pa; and g is the gravity vector, m/s².

Apart from Darcy’s law, the microscopic flow of CO₂ can be effectively calculated based on the Young–Laplace law and the Navier-Stokes equation. Chapman et al. [70] used the Young–Laplace law to calculate the displacement sequence in the node of the pore network model for predicting CO₂ displacement. The governing equation is as follows.

$$P_c=2\gamma \cos \theta (\frac{1}{h}+\frac{1}{w})$$

(7)

where P_c is the capillary pressure, Pa; γ is the interfacial tension, N/m; θ is the contact angle, °; h is the height of the channel, mm; and w is the width of the channel, mm.

Ovaysi and Piri [71] simulated the microscopic flow of CO₂ in a deep saline solution as follows.

$$\frac{Dv}{Dt}=-\frac{1}{\rho }\nabla P+\frac{\mu }{\rho }{\nabla }^{2}v+g$$

(8)

$${\nabla }^2=0$$

(9)

where v is the velocity vector, m/s; μ is the viscosity, Pa·s; ρ is the density, kg/m³; g is the gravity vector, m/s²; and P is the pressure, Pa.

Although numerous theoretical models have been used to reveal the flow mechanism of CO₂ in complex pore structures, some existing mathematical equations have weak universality. In particular, CO₂ reacts with other fluid phases and pore surfaces during the flow process, resulting in pore structure damage and changes in the flow of CO₂. In conclusion, theoretical models considering wettability and damage effects are rare. Therefore, it is necessary to further study the flow theory of CO₂ considering wettability and the damage theory in complex pore structures.

4. Numerical Simulation of CO₂ Flow in Pore Structures

Numerical simulation experiments on complex pore scales have become an effective tool for studying the flow of CO₂ in pore structures, which can impose conditions that cannot be reached by experimental methods, provide insight into the main processes of a multi-phase fluid flow in the storage body, obtain complete parameters of the flow process, and conduct systematic and accurate analyses. By comparing the experimental results with theoretical analyses, more comprehensive and objective conclusions can be obtained. At present, a variety of pore-scale simulation methods have been developed for the CO₂ flow in porous media, including the lattice Boltzmann method (LBM), pore network (PN) model, computational fluid dynamics (CFD) studies, and the smooth particle hydrodynamic method (SPH) [72,73,74]. Among them, the LBM and PN model are widely used due to their simple operation and computational efficiency, respectively. The SPH method is increasingly being utilized due to its relative ease of implementation. However, the primary drawback lies in the requirement for model parameter calibration [74]. CFD simulations of a multi-phase flow can effectively capture phase interfaces. However, interface tension often leads to numerical instability within these regions [75]. Therefore, this section focuses on the numerical simulation studies based on the LBM and PN model.

4.1. Numerical Simulation Based on LBM

Computational fluid dynamics (CFD) [76], a traditional numerical simulation method for simulating fluid flow in porous media, has received considerable attention due to its simple algorithm, good conservation, and ability to handle complex geometric shape problems [77]. However, the traditional CFD method exhibits some problems, including a large computational volume, low computational accuracy, difficulty in dealing with complex equations, and simple boundary conditions. To solve these problems, scholars [78,79,80] proposed adopting the LBM method to conduct CO₂ microscopic flow numerical simulations. Compared to the traditional CFD method, the LBM method can be programmed on a computer system with a parallel processing capability. It has the advantages of an easy handling of complex boundaries, simple description of fluid interactions, high computational efficiency, and high numerical accuracy, making it suitable for simulating the CO₂ flow in porous media at the pore scale. Therefore, gradually replacing the traditional CFD method is a promising research method.

Various types of multi-phase multicomponent LBM models, including color models and phase field models, have been developed based on the description of different component interactions. Liu et al. [73] improved the LBM color fluid model to simulate the displacement of CO₂ to water in a double-permeable pore network. It was found that the degree of the directional flow and the flow behavior of CO₂ depended on the capillary number, as shown in Figure 8, which was consistent with the experimental results of Chang et al. [81]. To improve the accuracy of the simulation results, Jiang and Tsuji [82] introduced color function thresholds, velocity fields, and Neumann boundary conditions to construct a novel LBM numerical simulation method for pore-scale CO₂ displacement simulations. The results showed that reducing the initial CO₂ connectivity and sphericity index reduced the CO₂ mobility rate and improved the capillary trapping capacity of CO₂. Yang et al. [83] proposed a hybrid method for simulating the multi-phase flow in porous media. The method used the LB color gradient polyphase model based on CFS and the geometric-based wetting model to improve the accuracy and stability of the simulation.

However, the disadvantages of the color gradient model were also obvious, as it was not applicable to heterogeneous pore structures with a real density and viscosity ratio of CO₂–brine. Therefore, Fakhari et al. [84] developed a phase field-based LBM and verified that the model accurately simulated the multi-phase flow characteristics of heterogeneous porous media at the pore scale through experimental comparison. Bakhshian et al. [85] used a multi-phase LBM model to simulate the flow of CO₂–brine in Tuscaloosa sandstone rock. The results showed that heterogeneous CO₂ wettability led to a more dispersed fluid distribution and more tortuous CO₂ flow paths (Figure 9). Additionally, the residual trap of CO₂ increased with an increase in CO₂-wet regions. Guo et al. [86] studied the effect of the surface contact angle of Bentheimer sandstone on a two-phase flow in porous media after the completion of CO₂ displacement. The results showed that the heterogeneity of wettability had a substantial impact on the relative permeability of CO₂. An immiscible-phase displacement process in the CO₂–water–rock system was simulated by Guo et al. [87] and Atia et al. [88]. The obvious local CO₂ and water redistribution phenomena were caused by the heterogeneity of the pore-scale surface wettability under low water-wetting conditions and enhanced the aggregation or diffusion behavior of CO₂.

In addition, Wang et al. [89] proposed the multi-relaxation time lattice Boltzmann method (MRT-LBM) to simulate the diffusion and miscible-phase flow of a CO₂–oil system in porous media at the pore scale. The results showed that CO₂ diffused into the oil phase more easily from regions that were relatively attractive to CO₂ and repulsive to oil in porous media. Yin et al. [90] identified the dominant transport mechanism in nanoporous media using the LBM. By developing a pore-scale model based on the LBM, Sha et al. [91] conducted simulations of coupled three-phase flow and reaction transport processes. The findings indicated that an increase in the viscosity of the displacing phase led to a decrease in the recovery efficiency of the displaced phase, highlighting significant implications for understanding coupled three-phase flow and reaction transport processes.

4.2. Numerical Simulation Based on the PNModel

The establishment of the PN model [92] was based on the pores, pore throats size distribution, and pore connectivity. The pore is defined as a lattice with a random shape that is connected through the pore throat to simplify the porous media model, which can improve the computational efficiency. Based on the established PN model established, the PN extracted from the images (Figure 10) by Blunt et al. [93] proved that capillary trapping was beneficial to the storage of CO₂ in the pores. Bensinger et al. [94] found that CO₂ injection created favorable conditions for the dissolution and precipitation of minerals, causing changes in the pore structure, porosity, and permeability. In addition, Campos et al. [95] obtained the same conclusion as the former using the PN model constructed by the test method, i.e., that the CO₂ flow caused changes in the pore structure and reduced the CO₂ storage.

All of these studies were based on small rock samples, but they highlight the importance and difficulty of modeling porous media at different scales (from micro to macro). The modeling was also considered to be divided into pore-to-core prediction and then applied to the field-scale simulation. Therefore, it was necessary to model porous media at different scales. Benali et al. [96] developed a PN model with an extended visual field to study the real-time bubble texture dynamics under high pressure. The results showed that continuous and rapid CO₂ injection bubbled with a low surface viscosity. A large number of bubbles remained on the pore surface of the storage body, which improved the flow of CO₂ in the pore structures. Liu et al. [97] developed a new two-phase steady-state model based on the PN method. The model considered the capillary and viscous forces in the pore structures and proved a stronger seepage effect of CO₂ than that of brine. Cao et al. [98] used a high-pressure microscopic model and a PN model to simulate the displacement phenomenon during CO₂ injection into a saturated brine reservoir. The results showed that CO₂ had a better injection efficiency and capillary trap capability in the regions with a larger porosity.

Wettability, an important factor controlling the flow of multi-phase fluids in porous media, had a remarkable influence on the microscopic flow of CO₂ [74]. The rapid development of the PN model was applied to the wetting layer flow, arbitrary wettability, and any sequence of displacement in two- and three- phase flow [93]. Basirat et al. [99] developed a numerical model based on the phase-field method to study the effect of a different wettability on the two-phase flow of CO₂ and brine at the pore scale. The results showed that the trapped wetting phase saturation and normalized interfacial area increased with a decreasing contact angle. However, the wetting condition had no effect on the CO₂ breakthrough time and saturation. In addition, Hu et al. [74] used CFD to study the effect of wettability on CO₂–brine displacement. The results showed that CO₂ had a higher saturation, wider directivity, and a more compact displacement pattern under medium wetting conditions. The CO₂–brine interface was smaller, which inhibited the mutual mass transfer.

The PN model had the advantages of simplifying the complex pore structures, reducing the complexity of the modeling, and improving the computational efficiency. It well simulated and predicted the flow of fluids in the complex pore structure. It also had a good extensibility, including considering the non-uniformity and multi-level structure, so that the model was closer to the natural core. However, there were also shortcomings, such as ignoring the microscopic details and parameter dependence [74], including ignoring the complexity of the pore structure, which reduced the accuracy of the model and led to possible limitations in the microscopic flow, thus requiring the calibration of the model parameters [75].

The numerical simulations of the microscopic flow of CO₂ in complex pore structures have been extensively investigated. However, numerical simulation methods tend to simplify the pore structure complexity and wettability, which reflect the real in situ conditions of the geological storage body. Therefore, considering the pore structure complexity and complex mixed wettability conditions, the establishment of accurate numerical models for geological storage bodies, as well as the improvement of the numerical experimental computational rate and accuracy, are the frontier issues for conducting CO₂ numerical simulations of microscopic flows in complex pore structures. The established heterogeneous model primarily considered the heterogeneity of the matrix. This model was suitable for numerical simulations of single-phase, two-phase, or multi-component flows, such as oil–water imbibition, oil and gas extraction, and groundwater flow. However, it was necessary to conduct a specific analysis based on the specific problem. For instance, the study of the CO₂ flow required the consideration of the phase behavior and damage effects.

5. Conclusions and Outlook

CO₂ flow is easily affected by reservoir complexity. The microscopic flow mechanism of CO₂ is the key to revealing the flow mechanism of CO₂ in the storage body, which is of great importance for evaluating the safety and effects of CO₂ geological storage. This article thereby provided a systematic and comprehensive review for the last decade on the microscopic flow of CO₂ in complex pore structures using experimental research, theoretical research, and numerical simulations. Moreover, the understanding of the microscopic flow mechanism of CO₂ in the complex pore structures was improved. It can be elucidated that pore structure complexity substantially impacts the microscopic flow process of CO₂, and its influence on wettability and damage needs to be further explored.

By considering the real-time visualization technology; multi-phase, multiscale, damage mechanism; and wettability, future research directions for the microscopic flow mechanism of CO₂ can be anticipated.

• Future studies should conduct real-time CT scanning experiments for CO₂ displacement in combination with the research and development of a CO₂ microscopic flow real-time CT scanning clamping device to achieve the real-time visualization and quantitative description of the CO₂ flow behavior in complex pore structures.
• The flow mechanism of single-phase CO₂ in complex pore structures has been extensively studied, but few studies investigated the micro-flow mechanism of multi-phase CO₂. Therefore, more attention should be paid to the microscopic flow mechanism of multi-phase CO₂ in complex pore structures in the future.
• CO₂ geological storage should meet the needs of long-term storage and wide ranges. Therefore, the flow mechanism of CO₂ in multiscale complex pore structures should be explored to upgrade from micro, mesoscopic to macro scales, which is of great importance for evaluating the long-term safety of CO₂ geological storage.
• In the process of CO₂ flow, chemical reactions occur on the pore surface, and the dissolution or precipitation of minerals changes the pore structures, which has a non-negligible impact on the microscopic flow mechanism of CO₂. In the future, attention should be paid to the influence of changes in the pore structures caused by chemical reactions on the microscopic flow mechanism of CO₂. The effects of wettability and damage are considered in the theoretical model study.
• The pore structures of the natural core are very complex, and the characteristics of the pore structure complexity and wettability are often simplified or ignored in the modeling process. In the future, advanced imaging technology should be combined with the complexity of the pore structure, the accuracy of the numerical model should be improved, and a numerical model that is closer to the natural core should be established for numerical simulations.
• The utilization of artificial intelligence (AI) for characterizing complex pore structures and forecasting fluid flows is also a prospective research focus within this domain.