Issue 
Sci. Tech. Energ. Transition
Volume 79, 2024



Article Number  34  
Number of page(s)  15  
DOI  https://doi.org/10.2516/stet/2024028  
Published online  11 June 2024 
Regular Article
Modeling reactive flow dynamics of gelled acid in fracturevuggy carbonates: a coupled thermalhydrologicalchemical simulation
^{1}
School of Science, Qingdao University of Technology, Qingdao 266520, China
^{2}
School of Civil Engineering, Qingdao University of Technology, Qingdao 266520, China
^{3}
School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China
^{*} Corresponding author: piyang.liu@qut.edu.cn
Received:
21
September
2023
Accepted:
16
April
2024
In engineering, gelled acids play a crucial role in facilitating reactive flow phenomena across diverse mediums. This study undertakes a comprehensive numerical investigation into the reactive flow dynamics within fracturevuggy carbonates, employing gelled acid as the agent. The mathematical model intricately couples thermal, hydrological, and chemical aspects to provide a holistic understanding of the process. Fractures are meticulously modeled using a pseudofracture approach, while vugs are delineated as highly porous matrix clusters. The simulation meticulously examines the influence of vugs and fractures, in conjunction with the rheological behavior of gelled acid, on the dissolution process. Our findings reveal that compared to hydrochloric acid, gelled acid is more effective in treating hightemperature carbonate rocks. A lower power law index induces a more pronounced response in shear stress, resulting in a more uniform dissolution pattern. Moreover, the presence of vugs and fractures exerts a significant impact on both the trajectory of wormhole growth and its penetration depth. As the length of fractures increases and their number multiplies, their dominant influence on the growth of wormholes becomes more pronounced. Furthermore, an abundance of fractures may attenuate the influence of vugs. This study highlights the importance of controlling the powerlaw index and understanding the complex interactions between fractures and vugs for reactive flow.
Key words: Fracturevuggy carbonate / Reactive flow / Gelled acid / ThermalHydrologicalChemical coupled model
Publisher note: Following the publication of the Erratum, the article has been corrected on 27 September 2024.
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Reactive flow within porous media is a critical phenomenon exerting significant influence on a diverse array of geological and chemical processes, as well as engineering applications [1]. It lies at the core of comprehending processes such as diagenesis [2–4], karstification [5–7], geologic sequestration of carbon dioxide [8–11], contaminant transport in groundwater resources [12–14], safe disposal of nuclear waste [15], and the acidizing treatment of reservoir [16–18]. To investigate the reactiondissolution mechanism of acid in porous media, researchers have conducted numerous experimental studies [19–23]. The results of the core acidization experiments show that the dissolution pattern resulting from acidrock dissolution correlates with parameters such as the type of acid, core size, and injection rate. Nonetheless, reproducing the intricate environments and multifaceted influences encountered in actual reservoirs presents challenges in laboratory experiments. Hence, researchers frequently turn to numerical simulations to delve deeper into reactive flow processes.
Numerical simulations encompass various influencing factors and intricate geological conditions, thereby furnishing forecasts and interpretations of observed phenomena and behaviors in actual reservoirs [18, 24–37]. In numerical simulations of various reaction flows, reactions can be classified based on reaction kinetics into irreversible reactions and reversible reactions. For instance, the reaction of hydrochloric acid with carbonate rocks [38, 39] is considered an irreversible reaction, while the reaction of carbon dioxide solution with rocks [40–42] is classified as a reversible reaction. Additionally, there exist multistep reactions, such as the flow reaction of hydrofluoric acid in sandstone [43–45]. Among these, reactive flows in carbonatite media are valued for their extensive applications across multiple fields [10, 46, 47].
To investigate the dynamics of hydrochloric acid dissolution in carbonate rocks, Panga et al. [25] developed a twoscale continuum (TSC) model. This model delineated the flow and transport of acid at the Darcy scale while characterizing the dynamic alterations of porescale parameters through empirical formulations. Subsequently, Kalia et al. [48] examined the influence of media heterogeneity on wormholes utilizing this model. It was observed that heterogeneity within a rock not only influences the structure of patterns formed during reactive dissolution but also affects the quantity of acid required to achieve a specific increase in permeability. Recognizing the significant impact of heterogeneity on the dissolution dynamics of acidic rocks, researchers began exploring the effects of largerscale heterogeneity, such as the presence of fractures and vugs, on the reaction process and dissolution outcomes. Izgec et al. [49] examined the characteristics of vuggy carbonate rocks during reactive flow through experiments and numerical simulations. Chen et al. [50] and Khoei et al. [51], respectively, solved the fractured medium reaction flow model based on the unified pipenetwork method and the finite element method. Huang et al. [52] treated vugs as anomalous matrices with high porosity, employing COMSOL software to solve the reaction flow model and investigate the influence of vug presence on wormhole growth and development structure. Liu et al. [53] tailored for fractured carbonate reservoirs, integrating concepts from the discrete fracture model and the TSC model. Utilizing the finite volume approach, the model effectively examined the impact of fracture characteristic parameters on wormhole formation. Qi et al. [54] regarded fractures as matrices with specific high porosity and utilized COMSOL software to explore the influence of fracture geometry on the initiation and propagation of wormholes. Through parameter sensitivity analysis, Jia et al. [55] examined the reaction flow processes of two typical core scales of vugular carbonate rocks as well as isolated vug carbonate rocks. They found that the vug filling degrees and vug diameter all had a significant effect on the reactive flow of fluids in carbonate rocks. Asl et al. [56] simulated the reaction flow process by combining the pseudofracture approach with the TSC model, wherein Darcy’s law governed the flow within the matrix and vugs, while the cubic law mimicked the flow within the fractures. They elucidated the impacts of individual fractures and vugs and analyzed the effects of various arrangements of fractures and vugs on reactive flow. Wang et al. [57] devised a statistical model of natural fractures employing the Monte Carlo method. This model integrates the matrix porosity model, fracture distribution model, and TSC model to simulate the formation mechanism of wormholes in naturally fractured reservoirs under radial flow conditions. Liu et al. [58] extended the TSC model to encompass fracturevuggy porous media. They solved the model under both linear and radial conditions in three dimensions and conducted sensitivity analysis on the parameters associated with vugs and fractures.
In specific domains of geoengineering, nonNewtonian fluid acids find application in injection operations. For instance, in geothermal reservoir acidization, gelled acid is frequently employed to moderate the reaction rate owing to the reservoir’s elevated temperature. Investigating the reactive flow of such nonNewtonian acid types in carbonatite media, Ratnakar et al. [59] utilized the TSC model in conjunction with an empirical rheological model to elucidate the reactive transport of gelled acids on the Darcy scale. Their study explored the impact of rheological parameters on dissolution phenomena. Similarly, Maheshwari et al. [60] examined the threedimensional reactive dissolution of gelled acid utilizing the twoscale continuum model, with a specific focus on the implications of nonNewtonian behavior on wormhole propagation. Liu et al. [61] introduced a coupled ThermalHydrologicalChemical (THC) model to delineate the acidization dissolution of carbonate rock using gelled acid. This model scrutinized the dynamics of polymer adsorption, temperature variations, and the nonNewtonian behavior of the introduced acid. However, prevailing research on the reactive flow of gelled acid predominantly concentrates on the rock without fieldscale heterogeneities. Given that carbonate reservoirs commonly encompass fractures and vugs, the significance of these geological structures in the reactive flow process cannot be overstated. To enhance the fidelity of reactive flow simulations in real formations, it is imperative to consider the effects of fractures and vugs, thereby facilitating a comprehensive evaluation of the mechanism underlying wormhole formation induced by gelled acid.
This study extends the THC coupled model established by Liu et al. [62] to encompass fracturevuggy porous media. Vugs are characterized as matrix clusters with significantly elevated porosity, as described by Liu et al. [58], while fractures are delineated using the pseudofracture model [63]. For computational simplification, the matrix, vugs, and fractures are all discretized within the same grid system.
The paper is structured as follows: Section 2 presents the development of a comprehensive mathematical model to describe the reaction flow behavior of gelled acid in carbonate fracturevuggy systems. The alterations in the in situ viscosity of the gelled acid are elucidated using a rheological model. In Section 3, the model from Section 2 is rendered in a dimensionless form. Section 4 outlines the numerical algorithm utilized for solving the model. Section 5 conducts pertinent numerical simulations aimed at scrutinizing the influence of rheological behavior, fractures, and vugs on the reaction flow process. Finally, Section 6 summarizes the paramount findings of this study.
2 Mathematical model
2.1 Rheological model
Gelled acids exhibit nonNewtonian fluid behavior, which differs from the flow properties of Newtonian fluids [60]. The whole rheological model, which takes into account temperature, shearthinning behavior, and polymer concentration, shows how the in situ viscosity changes [61]. It is written as(1)where μ_{situ} represents the viscosity of gelled acid; μ_{HCl} indicates the concentration of pure HCl; μ_{0} is the gelled acid’s zero shear viscosity at temperature T_{0} and polymer concentration c_{p0}; and K is the porous media’s effective permeability. The constant ξ is affected by the fluid’s properties.
2.2 Reaction transport model
This subsection mainly describes heat transfer, injected acid flow in fracturevuggy carbonate rocks, changes in rock porosity in porous media, and species transport [64, 65]
The equations are expressed as follows:(2)where μ_{situ} and u, respectively, represent the viscosity and velocity of the gelled acid; t denotes the time; ϕ signifies the porous media’s porosity; K stands for the porous media’s permeability; c_{h} denotes the hydrogen ions cupmixing concentrations; c_{p} denotes the cupmixing concentrations of polymers; a_{v} is the specific surface area; k_{c} defines the local masstransfer coefficient; k_{s} is the constant for the surface reaction rate.; D_{ep} and D_{e} are efficient dispersion tensors for polymers and hydrogen ions, respectively; and stand for the specific heat capacities of the gelled acid and the rock, respectively; and are the thermal conductivities of the gelled acid and the rock, respectively; ρ_{l} and ρ_{r} are the densities of the gelled acid and the rock, respectively; and ΔH_{r}(T) signifies the temperaturedependent exothermic heat of the reaction per mole of hydrogen ions consumed.
2.3 Porescale model
For the reaction transport model, some physical parameters such as porespecific surface area, permeability, the local masstransfer coefficient, and pore radius are required. However, these parameters undergo changes due to the dissolution reaction, which constantly alters the porous media’s pore structure. To determine these parameters, the porescale model [64] can be employed to calculate their values using empirical equations. Subsequently, these parameters are applied in reaction transport model to describe transport and fluid flow behavior in porous media.
The porescale equation is expressed as follows:(3) (4) (5) (6) (7) (8)where ϕ_{0}, K_{0}, r_{0}, and a_{0} denote initial porosity, initial permeability, initial pore radius, and initial specific surface area, respectively; The pore radius is r_{p}; D_{f} refers to the fractal dimensions of pore space; D_{T} refers to the fractal dimensions of tortuosity; d_{E} denotes the Euclidean dimension, which is assigned a value 2 for a twodimensional space. The smallest and greatest pore diameters are represented by λ_{min} and λ_{max}, respectively. Sh_{∞} indicates asymptotic Sherwood number; Re_{p} indicates pore Reynolds number; S_{c} indicates Schmidt number; u indicates the magnitude of Darcy’s velocity; D_{eL} denotes the diffusion coefficient in the acid injection’s direction; D_{eT} denotes the diffusion coefficient in the direction perpendicular to the acid injection direction; α_{os}, λ_{L}, and λ_{T} are pore structurerelated constants.
2.4 Initial and boundary conditions
The gelled acid is introduced continuously at a constant velocity from the inlet, with the outlet pressure being consistently maintained at a predetermined value.
In terms of fluid flow, the subsequent conditions are applicable:(9)
P_{e} is a representation of the outlet boundary’s fixed pressure.
Regarding the transport of polymers and hydrogen ions, the boundary conditions are as follows:(10)where c_{h0} and c_{p0} are the inlet hydrogen ions and polymer concentrations; u_{0} is the constant injection rate.
For heat transfer,(11)where T_{f} is the temperature at which the acid is injected.
For the transverse boundaries in the context of linear flow cases, the imposition of noflux conditions serves to confine fluid, solute, and heat within the domain.(12)
Periodic boundary conditions are applied across transverse boundaries in the case of radial flow in order to regulate solute transport, fluid flow, and heat transfer, mirroring the following:(13)
The initial conditions for the system are defined as follows:(14)
For fracturevuggy carbonate formations, the establishment of the spatial arrangement encompassing pores, fractures, and vugs is mainly divided into three parts:
 1.
Establishing a random porosity field [62].
 2.
For the realization of fracture generation and simulation. First, we simulate the distribution characteristics of porosity in the formation by using the initial porosity field with specific correlation length and heterogeneity constructed in step 1. Then, create a random fracture model in twodimensional space using random numbers to determine the central location, orientation, length, and aperture of fractures [58, 63]. These fractures are represented as line segments. We deal with fractures beyond the physical area using the circular window [58] perimeter clipping approach, which allows simulation to adapt to radial flow conditions. The fracture grid is recognized during the numerical simulation stage by looking to see if the edge of each grid cell intersects the fracture. The adjacent grid crossing that edge is designated as a fractured grid if a grid cell’s boundary crosses a fracture. Finally, high porosity values are assigned to these grids to reflect the permeability of the fractures. The permeability of fracture grid is calculated by the formula K_{fr} = b ^{2}/12. b is the fracture aperture, whose value is 1 mm in simulation. The calculated fracture permeability is 8 × 10^{7} md. The porosity of the fracture grid is calculated by equation (15) where the average porosity is 0.35 and the average permeability is 0.1 md, which means that the fracture porosity is 0.99.
 3.
During the process of constructing a vug distribution model, regions in the porosity field with porosity exceeding the average porosity are first identified and designated as vug units [58]. Subsequently, the index positions of these vug units within the grid are determined, and their porosity values are adjusted accordingly. The porosity of the vug areas is set within the range between the average matrix porosity to 1. In this study, we have set the porosity of the vug units to 0.99 and mapped them onto the porosity field. By adjusting the spatial correlation length and heterogeneity coefficient, we are able to generate random vug models with varying densities, volumes, and connectivity.
3 Dimensionless model
The following dimensionless parameters are defined.where the subscript letter D indicates dimensionless; For radial flow conditions, L symbolizes the characteristic length along the flow direction, corresponding to the core radius under radial flow conditions; U represents the dimensionless velocity; k_{a} refers to the dimensionless effective permeability; is the Thiele modulus at the pore scale; η signifies the poretodomain scale ratio; A_{E} denotes the dimensionless activation energy; and H_{D} stands for the dimensionless reaction exothermic.
The following nondimensional form can be obtained to express the mathematical model mentioned above.(16) (17) (18) (19) (20) (21) (22) (23) (24)
Boundary and initial conditions:(25) (26) (27) (28) (29) (30)
4 Numerical methods
The equations (16)–(21) are decoupled and solved using the sequential method for simulation of the injection of gelled acid through the rock. The following steps outline the simulation process:

Determine the initial values: Use the hydrogen ions, polymer concentration, temperature, porosity, and velocity field at time t_{0} as initial values.

Solve the flow equations (16) and (17) to compute the velocity U ^{n+1} at t + Δt.

Substituting U ^{n+1} into the species equations (18) and (19), along with the heat transfer equation (20), couple them with the dissolution equation (21) to determine the distributions of hydrogen ions, polymer concentration, temperature, and porosity at t + Δt.

Using the obtained solutions, update the porescale parameters.

Shift from time n + 1 (i.e., t + Δt) back to time n (i.e., t).

Return to step 2 and repeat the process until the injected acid breakthrough occurs.
In brief, the process involves solving the species and heat transfer equations from initial values and flow equations. The process of injecting gelled acid through rock is simulated by continuously updating the porescale parameters and repeating the solution.
Since the gelled acid’s in situ viscosity is a nonlinear function of the Darcy velocity, iterations are performed to obtain an accurate solution. In each iteration, the current viscosity value is substituted into equations (16) and (17) to solve for the velocity. The obtained velocity is then used in the viscosity equation (1) to calculate the new viscosity value. This process of iteration persists until convergence is attained.
For the calculation of polymer, hydrogen ions concentration, and temperature distribution, the reactiontransport process is decoupled using the operator splitting method. For a more detailed description of the solution process, interested readers may refer to [61].
5 Simulation results and analysis
In real reservoir acidizing operation, acid flow is radial. Therefore, we adopted the numerical simulation under the condition of twodimensional radial flow. The corresponding simulation results are given in this section. Table 1 provides a list of the simulation’s input parameters. Unless otherwise stated, all parameters remained unchanged throughout the research.
List of parameter values for the simulation
5.1 Dissolution modes by accordance with various injection rates
Wormhole growth and development in hightemperature carbonate rocks are significantly influenced by the rate of acid injection. Therefore, in this study, the dissolution effects of HCl and gelled acid at different injection rates were compared at 375 K. In the simulation, the initial porosity distribution of all cores is consistent. The simulation conditions are set as follows: fractures are 10, the length is 2 R/6, the average orientation is 60°, and the standard deviation of the orientation is 20°. Set the associated length of the vug to l_{x} = l_{y} = 0.09. Other parameters remain unchanged, as shown in Figure 1, where Figure 1a represents the initial fracturevuggy structure, and Figures 1b–1f show the dissolution mode formed under different injection speeds, and each letter corresponds to the same injection speed value.
Fig. 1 The dissolution modes of HCl and gelled acid were compared at different injection rates at 375 K. (A) HC1 (B) Gelled acid (a) initial fracturevuggy structure, (b) 1/Da = 10^{−6}, (c) 1/Da = 10^{−5}, (d) 1/Da = 10^{−4}, (e) 1/Da = 10^{−3}, (f) 1/Da = 1. 
Through comparative analysis, we can see that under a hightemperature environment, compared with HCl, gelled acid can form wormhole dissolution mode at a lower injection rate, and its dissolution range is wider. At the same injection rate, the wormhole formed of gelled acid has stronger breathability, a larger coverage area of the wormhole, and a better dissolution effect. In addition, from the breakthrough volume data required for injected acid to break through the core (Fig. 2), the amount required for gelled acid under the same conditions is less than the amount of HCl. This indicates that gelled acid not only has high dissolution efficiency under hightemperature conditions but also has less dosage.
Fig. 2 The breakthrough volume required for HCl and gelled acid injection under different injection rate conditions. 
In summary, in the acidization treatment of hightemperature carbonate rocks, gelled acid shows more obvious advantages compared with HCl because of its lower amount required for acidization and more uniform dissolution effect.
5.2 Effect of rheological properties on wormhole propagation
The properties of gelled acid are affected by rheological parameters. According to equation (1), rheological property is related to powerlaw index. The characteristics of fluid viscosity with shear rate are described by the powerlaw index. There is a power law relationship between the fluid’s viscosity and shear rate. This section uses the powerlaw index to examine how rheological characteristics affect wormhole propagation.
5.2.1 Powerlaw index
In this subsection, we analyze the powerlaw index’s impact on the dissolution dynamics. The powerlaw index, denoted by n, is varied as n = 0.1, 0.3, 0.5, and 0.7. While holding other variables constant, the gelled acid is injected into the core at various rates. The simulation results are presented in Figure 3, where columns (A) to (D) represent different powerlaw index values and rows (a) to (e) indicate injection rates ranging from low to high.
Fig. 3 Effect of powerlaw index n on the structure of the dissolution. (A) n = 0.1, (B) n = 0.3, (c) n = 0.5, (D) n = 0.7. The rate of gelled acid injection in different rows are: (a) 1/Da = 10^{−9}, (b) 1/Da = 10^{−6}, (c) 1/Da = 10^{−4}, (d) 1/Da = 10^{−3}, (e) 1/Da = 1. 
In row (a) of Figure 3, face dissolution occurs when 1/Da = 10^{−9}. As shown in row (b), when 1/Da = 10^{−6}, conical dissolution patterns are formed. Rows (c) and (d) in Figure 3 depict the formation of wormholes when 1/Da = 10^{−4} and 1/Da = 10^{−3}, respectively. In row (e) of Figure 3, a ramified wormhole is formed when 1/Da = 1. The breakthrough volume (PVbt ) under different injection rates and powerlaw indices is illustrated in Figure 4.
Fig. 4 Effect of powerlaw index n on the breakthrough volume. 
Examining column (A) in Figure 3, we observe that under different injection rates, the branching of wormholes is more prominent for n of 0.1 compared to the other three conditions. This behavior can be attributed to the gelled acid’s viscosity gradually decreasing with rising shear stress (Greater than the critical shear stress) when the powerlaw index is less than 1. Generally speaking, the viscosity is more susceptible to shear stress the smaller the powerlaw index is. As a result, gelled acid with a power law index value of 0.1 has the highest sensitivity. Due to the rock’s heterogeneity, the different regions’ permeability is different. This leads to the gelled acid spreading to highly permeable regions. This high permeability results in greater shear stress in these regions. The viscosity of the gelled acid decreases dramatically and becomes more fluid, allowing more acid to flow into these areas. This uneven acid dissolution process results in the branching phenomenon of wormholes. Therefore, compared to acids with higher powerlaw indices, those with lower powerlaw indices exhibit a more uniform dissolution pattern.
At low injection rates, the breakthrough volumes (PV_{BT}) of acids with different powerlaw indices are relatively similar, resulting in face dissolution and conical dissolution. Under conditions of moderate injection rates, the breakthrough volume (PV_{ bt}) corresponding to n = 0.3 is the smallest. However, as shown in Figure 3 (row (c), column (B)) and (row (d), column (B)), the resulting wormholes are short and exhibit poor branching. Highpermeability channels formed in these cases do not effectively enhance reservoir conditions. Conversely, acids with powerlaw indices of 0.5 and 0.7 exhibit better dissolution effects.
Furthermore, it can be observed from the figures that regardless of the dissolution mode, when the injected acid comes into contact with fractures, it preferentially grows and develops along them. Fractures provide relatively lowresistance channels, enabling the acid to penetrate the rock more rapidly. This growth and development process inhibits the formation of other wormholes because the acid concentration and flow rate in other areas are lower compared to fractures, resulting in relatively weak dissolution. Thus, fractures play a guiding and concentrating acid role in the dissolution process, preventing the formation of wormholes in other locations. Consequently, the injected acid cannot uniformly penetrate all parts of the carbonate reservoir, leading to reduced uniformity in the treatment effect.
5.3 Effect of fracture parameters
5.3.1 Fracture density
This subsection investigates the effect of fracture number on wormhole propagation by varying the number of fractures N_{f}. The fracture’s orientation and length were kept constant, while the number of fractures ranged from 6 to 30. The fracture length was set as 2 R/3, with a standard deviation is 20° and an average orientation is 60°. Each numerical simulation has a different fracture length and orientation because fractures are generated at random.
In Figure 5, Column (A) illustrates the initial location of the vugs and fractures, with the vugs fixed. Column (B) shows a comparison of the dissolution structure of acid breakthrough cores at different fracture numbers. From Figure 5, it can be observed that at a small number of fractures, such as 6, the distribution of fractures appears sparse, with individual fractures separated from each other. During the propagation of the wormhole, the acid must dissolve a portion of the matrix to establish connections between fractures and vugs. The wormhole primarily grows along the vugs, which become part of the wormhole and contribute to the formation of more branches. This is because the vugs have more permeability than the matrix, which causes the acid to flow into the vugs preferentially. Thus, when there are few fractures, the growth of the wormhole is primarily controlled by the vugs. Additionally, the volume of acid breakthrough is the greatest at a fracture number of 6.
Fig. 5 Fracture number’s impact on the structure of the dissolution. (A) Distribution of vugs and fractures during initial dissolution. (B) dissolution structure when acid breakthrough. (a) N_{f} = 6, (b) N_{f} = 12, (c) N_{f} = 20, (d) N_{f} = 30. 
The connectedness between fractures gets better as the number of fractures rises. Consequently, the impact of vugs diminishes. Fractures become more dominant in the growth of the wormhole, and the injected gelled acid flows predominantly along the fractures. When there are 30 fractures, the effect of vugs is almost negligible. The injected acid directly reaches the outlet from the inlet along the fractures, largely bypassing contact with the matrix. The resulting wormhole structure is formed primarily along a few fractures, while other areas receive less treatment, limiting the extent of acid coverage throughout the reservoir. Moreover, Figure 6 demonstrates that the breakthrough volume (PV_{ bt}) decreases with the increasing number of fractures at the same injection rate. When there are 30 fractures, the corresponding breakthrough volume (PV_{ bt}) is smallest. This is because dissolution within the fractures causes the acid to pass through quickly, reducing the amount of acid dissolved into other areas of the rock.
Fig. 6 The injected acid’s breakthrough volume corresponding to different fracture number. 
Furthermore, as depicted in Figure 5 (Column (B)), when the dissolution process is complete, it will be found that the gelled acid does not pass through every fracture. Only a portion of the fractures becomes part of the wormhole. Fractures closer to the core are more likely to be accessible to the acid, while fractures at the edges of the core are rarely affected by the acid.
5.3.2 Fracture orientation
This subsection investigates the fracture orientation’s influence on the wormhole’s structure. In this case, the number and length of fractures were kept constant. Specifically, there were 20 fractures with a length of 2 R/3, a standard deviation of 5°, and the values of θ_{f} were set to 0°, 30°, 90°, and 150°, as shown in Figure 7 (Column (A)). The resulting wormhole structures corresponding to different fracture orientations are presented in Figure 7 (Column (B)).
Fig. 7 Varying fracture orientation’s influences on the structure of the dissolution. (A) Distribution of fractures and vugs during the initial dissolution (B) dissolution structure when gelled acid breakthroughs. (a) θ_{f} = 0°, (b) θ_{f} = 30°, (c) θ_{f} = 90°, (d) θ_{f} = 150°. 
From Figure 7, it can be observed that when the fracture orientation is 0°, as depicted in Figure 7 (row (a), column (B)), the fracture direction runs parallel to the wormhole’s growth direction. Injection’s gelled acid dissolves a portion of the rock and then connects with the fracture. The wormhole primarily grows in the direction of the fracture, resulting in a thin wormhole with fewer branches. The final wormhole generated aligns with the radius of the core.
When the fracture orientation is 30° or 150° (−30°), as shown in Figure 7 (row (b), column (B)) and (row (d), column (B)), respectively, the wormhole still grows in the direction of the fracture, but with more branching. In contrast, when the fracture orientation is perpendicular to the core radius, as shown in Figure 7 (row (c), column (B)), the branching of the wormhole is strongest. The wormhole no longer grows in the same direction as the fracture. After passing through the vertical fracture, the injected acid extends along the matrix, following the path of least resistance for penetration.
In conclusion, for radial flow, the wormhole’s growth direction depends on the core’s radius direction. The orientation of the fracture does not solely determine the development direction of the wormhole; however, it does influence the growth path and the final morphology of the wormhole. The interaction between the fracture orientation and the injection process leads to variations in the branching pattern and the overall structure of the wormhole.
5.3.3 Fracture length
The fracture length’s impact on the structure of the wormhole is examined in this subsection. The study keeps the number of fractures and the fracture orientation constant. Specifically, there are 20 fractures with a standard deviation of 20°, and the lengths are taken as 2R/6, 2R/3, and 2R, as depicted in Figure 8 (Column (A)). Figure 8 (Column (B)) displays generated wormhole structures corresponding to various fracture lengths.
Fig. 8 Analysis of the impact of fracture length of (a) 2 R/6, (b) 2 R/3, (c) 2 R on dissolution structures. (A) Distribution of fractures and vugs during the initial dissolution (B) dissolution structure when acid breakthrough. 
When the fractures are short, as illustrated in Figure 8 (row (a), column (B)), the acid dissolves a portion of the matrix and then connects with the fracture. The resulting wormhole is relatively wide, exhibits branching, and possesses an erratic growth pattern. As the fracture length increases, as shown in Figure 8 (row (b), column (B)) and (row (c), column (B)), the dissolution area of the wormhole decreases, and the branching becomes weaker. The generated wormhole assumes a more regular shape. Because the fractures are more permeable than the matrix, the acid preferentially flows into the fractures during the acidrock reaction. When fracture length reaches the edge of the core, the injected gelled acid can break through the core more rapidly. Consequently, the breakthrough volume gradually decreases as the fracture length grows. The wormhole lines up with the fracture’s direction, which becomes an integral part of the wormhole structure.
5.4 Effect of vug size
This subsection investigates the impact of vug size on dissolution structure. Figure 9 shows the outcomes of the simulations. The corresponding dissolution structures of the vug distributions are illustrated in Figure 9 (column (B)).
Fig. 9 Twodimensional distribution of vug (A) and dissolution structures (B) at (a) l_{x} = 0.05, l_{y} = 0.05; (b) l_{x} = 0.1, l_{y} = 0.1; (c) l_{x} = 0.2, l_{y} = 0.2. 
From Figure 9 (column (B)), it is evident that, regardless of size, the injected acid does not preferentially flow into the vugs. Only a subset of vugs within the dissolution structure participates in the reaction, while the remaining unreacted vugs remain intact. The growth and propagation of wormholes mainly occur in the direction of fractures, rather than along the vug direction. This finding implies that vugs have little to no impact on wormhole propagation. On the contrary, fractures play a more crucial role in the carbonate rocks’s reaction flow process, as they facilitate the propagation and interconnection of wormholes.
As depicted in Figure 9, the injected acid dissolves a portion of the matrix first and then connects the vugs and the fractures. Fractures serve as highly permeable channels for acid injection, enabling the acid to flow preferentially through the channels with the least resistance. This preferential flow inhibits the acid from reaching other areas. Due to their ability to extend over longer distances and occur in greater numbers, fractures have a more pronounced influence on fluid permeability and the propagation of wormholes in rocks. Conversely, the number of vugs is limited, and their sizes are relatively small, resulting in a comparatively minor effect on wormhole propagation.
6 Conclusion
In this paper, a THC coupled model is established to simulate the reaction flow process in deepburied carbonate rocks with vugs and fractures. Analysis of the influence of the rheological properties, fracturevuggy parameters on the propagation of the gelled acid wormhole from a twodimensional radial flow perspective has led to the following conclusions:

Gelled acid with a lower powerlaw index exhibits a more uniform dissolution pattern compared to that with a higher powerlaw index. This uniform reaction flow enhances the efficiency and effectiveness of the entire treatment process.

Carbonate rocks with fractures are more susceptible to acid erosion compared to rocks without fractures.

Carbonate rocks often contain fractures and vugs, and their existence alters the flow dynamics of injected acid. They significantly affect the distribution and transport efficiency of gelled acids in rocks. Among them, fractures have a higher influence on the growth and evolution of wormholes compared to vugs.

In cases with fewer fractures, wormholes predominantly grow along vugs. But as the quantity of fractures rises, the influence of vugs diminishes while fractures become more significant. Therefore, in carbonate reservoirs with welldeveloped vugs, the focus can be placed on utilizing these vugs to promote wormhole propagation. On the other hand, if there are significant fractures concentrated around the wellbore, it is advisable to seal these fractures to achieve a more uniform treatment effect.

When fractures are perpendicular to the radial direction, they hinder wormhole propagation. Conversely, when the angle between fractures and the radial direction is smaller, wormhole propagation is promoted. Hence, in the study of reactive flows, priority should be given to areas where fractures are approximately parallel to the radial direction. Fractures in these areas are more likely to contribute to wormhole propagation and have a greater impact on the results.

Increasing the fracture length leads to a faster growth rate of the wormhole but weaker branching. To ensure better dissolution effects in practical applications, sealing of long fractures can be conducted prior to acid injection.

Vug size has only a small impact on wormhole growth. Regardless of vug size, the dissolution pattern remains unchanged.
Although the presence of fractures can accelerate acid penetration and dissolution, too many fractures may lead to problems of uneven acid distribution and small coverage. In conclusion, since gelled acids have limited effectiveness in deepburied reservoirs with numerous fractures, the focus of our future work has shifted to selfdiverting acid. Exploring the potential of selfdiverting acids in reservoirs with complex fracture networks.
Funding
Thanks to the support of the National Natural Science Foundation of China No.52374036, Natural Science Foundation of Shandong Province No. ZR2023ME207, 111 Project under No. B08028.
Conflicts of interest
The authors declare no competing relevant financial or nonfinancial interests.
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All Tables
All Figures
Fig. 1 The dissolution modes of HCl and gelled acid were compared at different injection rates at 375 K. (A) HC1 (B) Gelled acid (a) initial fracturevuggy structure, (b) 1/Da = 10^{−6}, (c) 1/Da = 10^{−5}, (d) 1/Da = 10^{−4}, (e) 1/Da = 10^{−3}, (f) 1/Da = 1. 

In the text 
Fig. 2 The breakthrough volume required for HCl and gelled acid injection under different injection rate conditions. 

In the text 
Fig. 3 Effect of powerlaw index n on the structure of the dissolution. (A) n = 0.1, (B) n = 0.3, (c) n = 0.5, (D) n = 0.7. The rate of gelled acid injection in different rows are: (a) 1/Da = 10^{−9}, (b) 1/Da = 10^{−6}, (c) 1/Da = 10^{−4}, (d) 1/Da = 10^{−3}, (e) 1/Da = 1. 

In the text 
Fig. 4 Effect of powerlaw index n on the breakthrough volume. 

In the text 
Fig. 5 Fracture number’s impact on the structure of the dissolution. (A) Distribution of vugs and fractures during initial dissolution. (B) dissolution structure when acid breakthrough. (a) N_{f} = 6, (b) N_{f} = 12, (c) N_{f} = 20, (d) N_{f} = 30. 

In the text 
Fig. 6 The injected acid’s breakthrough volume corresponding to different fracture number. 

In the text 
Fig. 7 Varying fracture orientation’s influences on the structure of the dissolution. (A) Distribution of fractures and vugs during the initial dissolution (B) dissolution structure when gelled acid breakthroughs. (a) θ_{f} = 0°, (b) θ_{f} = 30°, (c) θ_{f} = 90°, (d) θ_{f} = 150°. 

In the text 
Fig. 8 Analysis of the impact of fracture length of (a) 2 R/6, (b) 2 R/3, (c) 2 R on dissolution structures. (A) Distribution of fractures and vugs during the initial dissolution (B) dissolution structure when acid breakthrough. 

In the text 
Fig. 9 Twodimensional distribution of vug (A) and dissolution structures (B) at (a) l_{x} = 0.05, l_{y} = 0.05; (b) l_{x} = 0.1, l_{y} = 0.1; (c) l_{x} = 0.2, l_{y} = 0.2. 

In the text 
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