Role of Shear-Thinning-Induced Viscosity Heterogeneity in Regulating Fingering Transition of CO₂ Flooding Within Porous Media

Abstract

During the process of CO₂ displacing shear-thinning oil, the occurrence of fingering is a key factor contributing to a reduction in both displacement and sequestration efficiency. Existing studies typically use the average viscosity to calculate the viscosity ratio M for shear-thinning oil, overlooking the non-uniform viscosity distribution resulting from uneven shear stress. Consequently, a phase diagram based on M fails to accurately capture the underlying mechanism influencing fingering. We investigate the influence of shear-thinning on fingering patterns by analyzing viscosity heterogeneity during immiscible CO₂ flooding in porous media. The results showed the following: (1) An increase in zero-shear viscosity (μ₀) resulted in a greater viscosity difference between the two phases, which intensified interface instability, and the power-law index (n) diminished the shear-thinning effect, promoted fingering formation, and significantly reduced displacement efficiency, with a maximum reduction of 28.6% observed in this study. (2) Shear-thinning oil was more prone to capillary fingering compared to Newtonian oil under the same capillary number Ca and viscosity ratio M. (3) Intense pressure fluctuations at the displacement front combined with non-uniform viscosity distribution exacerbate interfacial instability and make shear-thinning oil more prone to capillary fingering. This study provides guidance for optimizing displacement strategies for shear-thinning fluids and advancing the practical implementation of CO₂ flooding technology.

1. Introduction

China is a major producer of coal and oil [1]. Carbon capture, utilization, and storage (CCUS) technology has emerged as a critical pathway toward achieving carbon neutrality [2,3,4,5]. Within the CCUS framework, CO₂ flooding has garnered significant attention due to the dual advantages of enhancing oil recovery and enabling carbon sequestration [6,7,8,9,10,11]. However, during the displacement process, the pronounced differences in physical properties between CO₂ and reservoir fluids often trigger interfacial instabilities, resulting in the development of fingering patterns that substantially reduce displacement efficiency [12,13,14,15,16,17]. A major challenge is posed to the effective implementation of CO₂ flooding. Consequently, a thorough understanding of the mechanisms governing fingering formation and evolution is essential for CO₂ flooding.

In existing research on the fingering of CO₂ flooding, the oil phase is typically simplified as a Newtonian oil, and the two-phase flow dynamics of CO₂ displacing Newtonian oil have been extensively investigated [18,19,20]. Early experimental studies employed a Hele-Shaw device composed of two parallel transparent plates, which first revealed the characteristic finger-like displacement pattern [21,22]. Subsequent investigations using transparent etched glass micromodels systematically explored two-phase displacement behaviors [23,24]. These studies demonstrated that displacement patterns are governed by the capillary number (Ca) and viscosity ratio (M) of CO₂ and oil and can be classified into three regimes: capillary fingering, viscous fingering, and stable displacement [25,26,27,28]. A phase diagram constructed in the log Ca − log M parameter space delineates the boundaries among these regimes, and a transitional region exhibiting “crossing” behavior associated with multiple fingering modes is observed in the intermediate zone [29,30]. At high Ca, interfacial instability is reduced, promoting a more stable displacement mode. Conversely, at low Ca and high M, the occurrence of fingering is more likely. These findings comprehensively confirmed that the formation of fingering during the CO₂ displacement of Newtonian oil is governed by the synergistic interaction between capillary and viscous forces.

Under reservoir conditions, underground crude oil contains paraffin, surfactants, suspended particles, and other substances, leading to pronounced non-Newtonian behavior, particularly exhibiting shear-thinning characteristics [31,32,33]. This rheological property causes fluid viscosity to vary dynamically with shear stress, significantly influencing the stability of the two-phase displacement front [34,35]. The characteristics of fluids and reservoir conditions have a profound impact on reservoir modification [36]. Differences in fluid viscosity, wettability, and factors such as the pore structure and permeability of the reservoir significantly affect the flow behavior of fluids in porous media and displacement efficiency [37]. Therefore, a thorough understanding of the interactions among these factors is essential for optimizing CO₂ flooding technology and enhancing recovery rates [38]. Previous studies have shown that during the CO₂ displacement of shear-thinning fluids, the power-law index (n) and zero-shear viscosity (μ₀), as two critical rheological parameters, jointly determine the viscosity response to variations in shear stress and exert effects on interfacial stability. Specifically, as either n or μ₀ increases, the shear-thinning effect becomes weaker, thereby making the occurrence of fingering more likely [39]. In previous studies, the viscosity of shear-thinning fluids was typically simplified as an average value. Through the analysis of the viscosity ratio between CO₂ and shear-thinning fluids, it was found that the shear-thinning behavior results in a higher viscosity ratio M. Based on this, a Ca-M fingering phase diagram can be constructed for shear-thinning fluids, and the resulting pattern transition characteristics are similar to those observed in shear-thinning fluids [40]. Omrani developed a computational approach to evaluate and upscale the dispersion of a solute in the flow of a shear-thinning (ST) fluid in a heterogeneous porous medium [41]. And Al-Qenae created a porous medium using glass micromodels to verify spatially variable viscosity [42]. However, the most commonly adopted approach to calculating M in CO₂–shear-thinning fluid systems relies on the effective viscosity of shear-thinning fluids, which neglects the viscosity heterogeneity arising from the non-uniform shear stress distribution during displacement in porous media. However, the unevenness of the viscosity distribution can further enhance the disturbances and instabilities in the two-phase flow during displacement, which is a key factor affecting the formation and development of fingering [43,44]. Analyzing the viscosity distribution can help address the shortcomings in existing research regarding fingering in shear-thinning fluids. Consequently, the Ca-M phase diagram based on effective viscosity cannot fully capture the influence of shear-thinning behavior on fingering dynamics.

In this study, to investigate the influence of the viscosity heterogeneity of shear-thinning oil on fingering patterns and their transitions, CO₂ flooding processes were simulated with varying power-law indices (n) and zero-shear viscosities (μ₀). The variance in fingering transitions under Newtonian and shear-thinning oil was compared, and corresponding phase diagrams were constructed. Through a detailed analysis of the pressure fields and the viscosity distributions along the displacement direction, the underlying mechanisms driving the formation of distinct fingering modes for shear-thinning fluids were revealed. These findings hold significant implications for optimizing shear-thinning oil flooding strategies and advancing the practical implementation of CO₂ enhanced oil recovery technologies.

2. Numerical Modeling

2.1. Geometric Model

To simulate CO₂ flooding, the model utilized in this study comprises a porous medium measuring 14.6 × 9 mm² (as depicted in Figure 1). Within this model, cylinders represent solids, while purple areas denote flow regions. The centers of adjacent cylinders form an equilateral triangle array. Each cylinder has a diameter of 1 mm, with a spacing of 1.15 mm between adjacent cylinder centers, and the angle between two cylinders in adjacent columns is 60°. To introduce non-uniformity and randomness, 10 cylinders marked in pink are randomly selected, and their diameters are increased by 10%. The resulting model has a porosity of approximately 0.34. Initially, the porous medium is saturated with oil, which presents shear-thinning. The entire computational domain is discretized using a free triangular mesh. CO₂ is then injected through the inlet at a certain pressure P_in, displacing the oil phase towards the outlet. Similar geometric models have been proven in previous experiments and numerical studies (such as Hele-Shaw [21] or microscopic model systems) to effectively reveal the fundamental mechanism of multiphase flow.

2.2. Governing Equations

For two-phase fluids, the velocity and pressure fields are determined using the same set of Navier–Stokes equations [45]:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u\right)+\nabla ·\left(\rho uu\right)=-\nabla p+\nabla [\mu (\nabla u+\nabla u^T)]+F_{st}$$

(1)

$$\nabla ·u=0$$

(2)

where u is velocity, p is pressure, ρ and μ are density and dynamic viscosity, and F_st is the momentum source term generated by surface tension at the interface between two fluids. The phase field method was employed to track the interface, and the phase field variable constraints were satisfied:

$${\displaystyle \frac{\partial \varphi }{\partial t}}+u·\nabla \varphi =\nabla ·{\displaystyle \frac{\gamma \lambda }{{\epsilon }^2}}\nabla \psi$$

(3)

$$\psi =-\nabla ·{\epsilon }^2\nabla \varphi +\left({\varphi }^2-1\right)\varphi +{\displaystyle \frac{{\epsilon }^2}{\lambda }}{\displaystyle \frac{\partial f}{\partial \varphi }}$$

(4)

$$F_{st}=\left(G-{\displaystyle \frac{\partial f}{\partial \varphi }}\right)\nabla \varphi$$

(5)

where γ is mobility, ψ is a phase field auxiliary variable, λ* is mixing energy density, f is the total free energy of the system, and ε is a capillary width that scales with the thickness of the interface. Each phase is represented by a phase field variable ϕ, where ϕ = ±1 denotes the two distinct phases, and −1 < ϕ < 1 denotes the phase interface. The chemical potential G is derived from the total energy equation:

$$G=\lambda \left\{-{\nabla }^2\varphi +{\displaystyle \frac{\varphi ({\varphi }^2-1)}{{\epsilon }^2}}\right\}+{\displaystyle \frac{\partial f}{\partial \varphi }}$$

(6)

For shear-thinning oil, the Carreau model is employed to characterize rheological behavior, effectively capturing the shear-thinning oil properties across a broad spectrum of shear rates [46]:

$$\mu \left(\dot{\gamma }\right)={\mu }_{\infty }+\left({\mu }_0-{\mu }_{\infty }\right){\left[1+{\left(\lambda \dot{\gamma }\right)}^2\right]}^{{\displaystyle \frac{n-1}{2}}}$$

(7)

where $\dot{\gamma }$ is the shear rate of the fluid, n is the power-law index (n < 1), λ is the relaxation time (controls the characteristic shear rate at which the fluid deviates from the zero-shear plateau), μ₀ is zero-shear viscosity, and μ_∞ is infinite-shear viscosity (sets the viscosity limit at extremely high shear rates).

The coupled equations were numerically solved using the proven finite element method in the commercial software COMSOL Multiphysics 6.0.

2.3. Boundary and Initial Conditions

In this model, the left boundary was designated as pressure inlet P_in and the right boundary as pressure outlet P_out. The upper and lower boundaries were defined as symmetric. The capillary number Ca was calculated as Ca = μ_gu_g/σ (where μ_g represents the viscosity of CO₂, u_g represents the average velocity of single-phase CO₂ in porous media, and σ signifies the interface tension). The viscosity ratio M was defined as M = μ_g/μ_oil, where μ_oil represents the viscosity of shear-thinning oil. Since the viscosity of shear-thinning oil varies with shear rate, the viscosity ratio M is defined as the ratio of CO₂ viscosity to the equivalent viscosity of the oil phase at the breakthrough moment. This is because using the initial viscosity as a baseline does not accurately reflect the rheological characteristics of the oil phase during the actual displacement process. By taking the average viscosity of the oil phase at breakthrough as the denominator for calculating the viscosity ratio, we can better characterize the “effective viscosity difference” corresponding to interfacial instability. Here, the equivalent viscosity of the oil phase is defined as the average viscosity of the oil phase within the simulation domain at the breakthrough moment. The surface of cylinders acts as a non-slip wall with a fixed contact angle θ_w = 90°. To ensure that crude oil and CO₂ remain immiscible, the physical properties of both fluids were selected at a temperature of 343 K and a pressure of 7 MPa [7,47,48]. The detailed parameter settings are listed in

Table 1. Fluid parameters used in this paper.

Symbol	ρg	ρoil	μg	μ0	μ∞	n	λ
Unit	kg/m3	kg/m3	Pa·s	Pa·s	Pa·s	—	s
Value	142.7	718.9	2 × 10−5	5~150 [47]	0.01	0.2~0.8	0.64

.

2.4. Model Validation

To confirm the precision and effectiveness of the simulation outcomes, the working conditions of capillary number Ca = 3.16 × 10⁻⁴ and viscosity ratio M = 3.14 × 10⁻⁴ were selected for grid independent verification (see Figure S1 in the Supplementary Materials). We compared the breakthrough time t* (defined as the time when CO₂ reaches the outlet boundary) and the residual oil saturation S_o at the breakthrough moment while also calculating the value of convergence rate R_G to assess the convergence behavior during the mesh refinement process. When R_G < 0, it indicates oscillatory convergence; when 0 ≤ R_G ≤ 1, it indicates monotonic convergence; and when R_G > 1, it indicates divergence [49,50]. We found that during the refinement process, the calculations exhibited monotonic convergence, as both t* and S_o had R_G values between 0 and 1, specifically 0.5 and 0.75, respectively. And when the number of grids increases from 105,242 to 121,098, the number and distribution of fingers at the breakthrough moment remain consistent, with changes in S_o and t* both within 1%. Therefore, the grid number of 105,242 was selected for subsequent calculations.

To validate the accuracy of our numerical model, the shear-thinning oil model proposed by Castro et al. [51] was employed as a benchmark to simulate the displacement of a Newtonian oil by a shear-thinning fluid under varying injection rates (u_in). Figure 2A illustrates the distribution of the two-phase fluids at the breakthrough moment. As u_in increased, the area occupied by the displacing fluid decreased, leaving more of the displaced phase trapped within the pore space and resulting in significantly lower displacement efficiency. This occurs because the viscosity of the shear-thinning fluid decreases with increasing u_in. The similarity in flow patterns at the breakthrough moment under different u_in demonstrated the consistency of the results. Furthermore, a quantitative analysis of residual oil saturation (S_o) at different injection rates (Figure 2B) revealed that S_o consistently increased with increasing u_in. The maximum deviation remains within 3%, confirming the accuracy of the proposed model.

3. Results and Discussion

To study the typical flow modes during the CO₂ displacement of shear-thinning oil, displacement patterns and viscosity distributions under various injection pressure P with n = 0.6 and μ₀ = 10 Pa·s were examined, as shown in Figure 3. When P = 80 kPa, owing to the dominance of capillary forces, the resulting fingers are isolated and slender, a phenomenon referred to as capillary fingering. Capillary force mainly determines the direction of finger penetration. If there is a phenomenon that is opposite to the injection direction, this phenomenon is called “reflux” [8]. When P increases to 3000 kPa, under the dominance of viscous forces, CO₂ quickly invades adjacent pores after partially filling the existing ones, forming an interconnected flow structure, which is termed viscous fingering. When P reaches 7000 kPa, the interface between CO₂ and oil becomes relatively flat, and the displacement front appears stable and uniform, with no distinct fingering patterns in the flow field, and the displacement efficiency reaches the maximum, which is referred to as stable displacement. The corresponding viscosity distributions under these fingering patterns are shown in Figure 3D–F. Across all these displacement modes, the viscosity in the channels where fingering occurs is relatively low, approximately 2 × 10⁻⁵ Pa·s, while the viscosity in the regions not swept by CO₂ varies considerably. Specifically, the average viscosity in the unswept region is relatively high in capillary fingering, approximately 6.2 Pa·s. As the displacement mode gradually transitions to viscous fingering and stable displacement, the average oil viscosity in the unswept region progressively decreases to 1.4 Pa·s, indicating a significant viscosity reduction effect.

To investigate the effect of the power-law index (n) on the CO₂ displacement of shear-thinning oil, Figure 4 illustrates the distributions of fluids and viscosities at Ca = 0.24; μ₀ = 10 Pa·s; and n = 0.6, 0.75, and 0.8. Meanwhile, the variations in residual oil saturation (S_o) and average oil viscosity (${\overline{\mu }}_{\mathrm{o}\mathrm{i}\mathrm{l}}$) at the breakthrough moment were analyzed. When n = 0.6, the fingers are relatively wide and partially interconnected, indicating a typical viscous fingering regime. As n increases, the fingers become slender and isolated, and the displacement mode gradually transitions from viscous fingering to capillary fingering, with a significantly larger area occupied by the oil phase. Correspondingly, Figure 4G shows that S_o gradually increases with n and eventually stabilizes, indicating that a higher n reduces displacement efficiency. Figure 4D–F displays the viscosity distributions at different n. As n increases, the viscosity in the region swept by CO₂ remains consistently low, approximately 2 × 10⁻⁵ Pa·s. In contrast, the viscosity of oil in regions unswept by CO₂ increases progressively owing to the transition to capillary fingering. As illustrated in Figure 4G, ${\overline{\mu }}_{\mathrm{o}\mathrm{i}\mathrm{l}}$ in unswept regions rises from 0.4 Pa·s to 3.1 Pa·s, indicating a significant decline in the viscosity reduction effect with increasing n.

To investigate the influence of n on fingering modes, CO₂ displacement at μ₀ = 10 Pa·s under various Ca and n was simulated. Fingering phase diagrams were constructed, as shown in Figure 5. With variations in Ca and n, two typical displacement patterns were observed: capillary fingering and viscous fingering. The critical state between different fingering modes is determined by selecting the midpoint between two adjacent data points representing distinct modes. Increasing Ca leads to a transition from capillary fingering to viscous fingering. However, as n increases, the critical Ca for this transition also increases. n directly affects the shear-thinning behavior of oil, thereby controlling the transition between fingering modes. Lower n corresponds to stronger shear-thinning effects, which promote the development of viscous fingering. In contrast, higher n weakens shear-thinning, favoring the occurrence of capillary fingering.

To investigate the effect of the zero-shear viscosity μ₀ on the CO₂ displacement of shear-thinning oil, Figure 6 illustrates the distributions of fluids and viscosity fields under the conditions of Ca = 0.24 and n = 0.6 for μ₀ = 5, 40, and 100 Pa·s. The variations in residual oil saturation (S_o) and average oil viscosity (${\overline{\mu }}_{\mathrm{o}\mathrm{i}\mathrm{l}}$) at the breakthrough moment were quantitatively analyzed. When μ₀ = 5 Pa·s, CO₂ occupies most area of the porous media, showing a viscous fingering pattern. As μ₀ increases, the area occupied by CO₂ gradually decreases, and the displacement mode gradually transitions from viscous fingering to capillary fingering, as shown in Figure 6A–C. Correspondingly, Figure 6G shows that S_o increases with increasing μ₀, indicating a decline in displacement efficiency. Figure 6D–F display the viscosity distributions under various μ₀. The viscosity in the region swept by CO₂ remains consistently low, approximately 2 × 10⁻⁵ Pa·s. In contrast, the viscosity of the oil in regions unswept by CO₂ increases progressively, as μ₀ increases. As illustrated in Figure 6G, ${\overline{\mu }}_{\mathrm{o}\mathrm{i}\mathrm{l}}$ in unswept regions rises from 0.1 Pa·s to 33 Pa·s, indicating a significant decline in the viscosity reduction effect.

To investigate the influence of μ₀ on fingering pattern transitions during the CO₂ displacement of shear-thinning oil, fingering phase diagrams were constructed under varying Ca and μ₀, as shown in Figure 7. When Ca = 0.11, capillary fingering is observed regardless of μ₀, which indicates that at low Ca, the displacement front is primarily governed by capillary forces, while viscous effects are relatively weak. As Ca increases and μ₀ decreases, the fingering mode gradually transitions from capillary fingering to viscous fingering and eventually to stable displacement. The critical Ca for the transition from capillary fingering to viscous fingering lies between 0.13 and 0.4. With increasing μ₀, the critical Ca for this transition also increases. Stable displacement occurs only when μ₀ reaches 5 Pa·s and Ca exceeds 0.42.

The capillary number (Ca) and viscosity ratio (M) are key parameters controlling fingering patterns and transitions in Newtonian oil displacement. For comparison, Ca and M were used to characterize shear-thinning oil systems. Considering the viscosity of shear-thinning oil depends on the shear rate. M was defined as the ratio of CO₂ viscosity to the average oil viscosity at the breakthrough time, enabling a comparison of fingering phase diagrams between Newtonian and shear-thinning oil. First, we established fingering phase diagrams under various Ca and M for different n, as shown in Figure 8A. The results indicated that capillary fingering and viscous fingering can be observed by adjusting Ca and M. Specifically, capillary fingering primarily occurs at low M and Ca, whereas viscous fingering emerges at high M and Ca. The influence of μ₀ on the fingering transition was further examined. As illustrated in Figure 8B, three distinct displacement regimes were identified by varying Ca and M, which are capillary fingering, viscous fingering, and stable displacement. The effects of Ca and M on the transition behavior exhibit a trend similar to that observed with variations in n. Notably, stable displacement emerges when Ca and M are sufficiently high, marking a key distinction from the behavior governed by n.

To compare the differences in fingering pattern transitions between Newtonian and shear-thinning oil under the influence of Ca and M, a fingering phase diagram for Newtonian oil was first constructed under various Ca and M, as shown in Figure 9A. Although the effects of Ca and M on the trend in the fingering transition of Newtonian and shear-thinning oil are similar, their quantitative fingering transition boundaries differ significantly. An intuitive comparison of fingering transitions between Newtonian and shear-thinning oil under varying Ca and M is shown in Figure 9B. The solid lines represent the transition boundaries from capillary fingering to viscous fingering. The results indicated that Newtonian oil exhibits the lowest critical Ca and M, making it more prone to presenting capillary fingering. The dotted lines denote the transition boundaries from viscous fingering to stable displacement. In this case, shear-thinning oil requires higher Ca and M to achieve the transition, indicating a more unstable displacement process. It is evident that M calculated based on average oil viscosity cannot fully characterize the properties of shear-thinning oil, as it overlooks the non-uniform viscosity distribution. As a result, Newtonian and shear-thinning oils with varying n or μ₀ exhibit different fingering behaviors under the same M.

To clarify the underlying mechanism behind the fingering differences between Newtonian and shear-thinning oil, Figure 10A–C present the pressure distributions when the displacement front l_f reaches the midpoint of the porous media (l_f = L/2). It should be noted that the pressure distribution curve along the vertical cross-section is not strictly extracted from the midpoint (l_f = L/2) but rather from a vertical line that passes through the location of the most significant pressure fluctuations at the interface. In contrast, under viscous fingering (Figure 10B), a smoother front pressure profile and lower pressure gradients were observed. Figure 10D–F present the apparent viscosity distributions when the displacement front reaches the midpoint of the porous medium (l_f = L/2). Regions with larger pressure gradients are generally accompanied by more pronounced viscosity variations. This correlation arises because higher pressure gradients induce stronger local shear rates, leading to a reduction in apparent viscosity for the shear-thinning fluids. Figure 10G–I display the corresponding fluid saturation distributions at the breakthrough moment (l_f = L), which evolves from the pressure fields shown in Figure 10A–C. Under the same M, Newtonian oil ultimately exhibits viscous fingering, whereas shear-thinning oil tends to develop capillary fingering. This difference arises from the irregular, jagged pressure distribution along the vertical section of the displacement front. Figure 10J,K present the pressure profiles at the displacement front and corresponding gradients along the vertical cross-section for the cases shown in Figure 10A–C. Under the same Ca and M, the pressure variation in the shear-thinning oil exhibits more pronounced fluctuations, whereas the Newtonian oil shows a comparatively smoother profile, as shown in Figure 10J. Figure 10K further illustrates this difference where the pressure gradient of the shear-thinning fluid shows larger oscillations. In contrast, the gradient of Newtonian oil is smaller and remains closer to 0 overall, which reflects a more stable displacement front.

To elucidate the dramatic pressure drop fluctuations at the displacement front of the shear-thinning oil, the viscosity distributions when l_f = L/2 and l_f = L are illustrated in Figure 11A and B, respectively. As shown in Figure 11A, the viscosity at the front of different fingers differs markedly, exhibiting a clear non-uniform distribution. To investigate the correlation between viscosity distribution and finger evolution, three representative fingers were selected to characterize the distinct development behaviors observed under identical flow conditions: one finger that ultimately extended to the outlet (h = 3.5 mm), one that ceased advancing during this stage (h = 4.5 mm), and one that exhibited limited further advancement but did not reach the outlet (h = 8.5 mm). By comparing these with the fingering morphology at the breakthrough moment in Figure 11B, it is evident that fingers with lower viscosity at the front experience reduced flow resistance, which favors continuous advancement. In contrast, fingers with higher viscosity at the displacement front increased flow resistance, thereby inhibiting further advancement. The viscosity variations along the flow paths of these three selected fingers were quantitatively analyzed, using the Newtonian oil with constant viscosity throughout the displacement process as a reference (indicated by the red dashed line), as shown in Figure 11C. In the shear-thinning oil, the fingers that extend toward the outlet consistently exhibit lower viscosity than the Newtonian baseline, whereas the stagnant finger exhibits significantly higher viscosity compared to the surrounding flow channels. This indicated that changes in shear rate result in a non-uniform viscosity distribution, where regions with lower viscosity advance more readily, while those with higher viscosity tend to remain stationary. The resulting local viscosity imbalance enhances interfacial instability and facilitates the occurrence of capillary fingering in shear-thinning oil under the same Ca and M.

4. Conclusions

In this study, numerical simulations were conducted to examine the effect of shear-thinning on fingering patterns during immiscible CO₂ displacement in porous media. The influence of the power-law index (n) and zero-shear viscosity (μ₀) on displacement patterns and residual oil saturation were specifically investigated. Corresponding fingering transition phase diagrams were established and compared with those of Newtonian oil, and notable differences were identified. It was found that increases in n and μ₀ led to a weakened shear-thinning effect, which promoted the development of fingering and resulted in a significant reduction in displacement efficiency. In addition, under the same Ca and M, shear-thinning oil was found to exhibit a greater tendency for capillary fingering compared to Newtonian oil. This phenomenon was further interpreted through an analysis of the pressure and viscosity fields and was mainly attributed to the viscosity heterogeneity induced by shear-thinning behavior.

Previous studies on CO₂ displacement in shear-thinning oil mainly relied on effective viscosity to characterize the flow behavior of shear-thinning oil and to investigate the effect of the viscosity ratio between oil and CO₂ on fingering. However, this approach neglects the spatial viscosity heterogeneity caused by the uneven distribution of shear stress. During the CO₂ displacement of shear-thinning oil, oil viscosity significantly decreases, and representing the average viscosity often fails to accurately reflect the evolution of fingering. In contrast, using the oil viscosity distribution instead of average viscosity can more accurately represent the influence of shear-thinning behavior on fingering development. This result facilitates the construction of more reasonable fingering transition phase diagrams for shear-thinning oil and provides theoretical guidance for advancing the practical application of CO₂ flooding. Although the present study provides valuable insights into the transition mechanisms between capillary and viscous fingering during CO₂ immiscible flooding, it should be noted that the findings are based on pore-scale simulations under idealized conditions. Therefore, direct extrapolation to field-scale CO₂ flooding should be made with caution. Future work will focus on incorporating larger-scale heterogeneity and reservoir-specific parameters to improve the practical applicability of the results. In addition, our future research will focus on the influence of shear-thinning on fingering under different wettability conditions, including oil-wet, water-wet, and wettability-gradient scenarios. We will examine the flow characteristics of fluids and the changes in capillary effects to provide more theoretical support for optimizing CO₂ flooding technologies.