Numerical Simulation of Acid Leakoff in Fracture Walls Based on an Improved Dual-Scale Continuous Model

Abstract

Controlling fluid loss during acid fracturing remains challenging, as acid may partially or completely leak into reservoir pores and fractures, preventing effective flow within the formation and thereby reducing stimulation effectiveness. The acid leakoff mechanism is fundamentally distinct from that of non-reactive pad fluid (fracturing fluid), with the most critical distinction manifested through wall-confined acid-etched wormholes formed during reactive flow processes, which exert a dominant influence on acid filtration behavior. To address this challenge, a modified dual-scale continuum model based on the Brinkman equation was developed. This model establishes a numerical simulation framework for acid fracturing–etching processes in dolomite reservoirs of the Xi Xiangchi Formation. The study systematically reveals acid leakoff patterns at fracture walls under the influence of operational parameters (injection rate, acid concentration, acid viscosity) and reservoir characteristics (porosity heterogeneity). For field operations, medium-viscosity acid initially enhances distal fracture communication, followed by viscosity reduction to promote non-uniform etching. Prioritizing acid concentration over injection rate optimizes fracture connectivity, while minimizing leakoff. In high-porosity reservoirs, process parameters require optimization through acid retardation and leakoff control strategies.

1. Introduction

The study area is located in the eastern Sichuan Basin, bounded by the Qiyueshan Fault to the east and the Huayingshan Fault to the west, with the Daba Mountain thrust-fold belt to the north and the Gulin–Changning structural zone to the south. The Xixiangchi Formation reservoir in this region predominantly develops in its third member, characterized by a pressure coefficient of 1.0–1.3 and an in situ temperature range of 90–100 °C. Although the reservoir exhibits abundant natural gas reserves, its exploration maturity remains relatively low, primarily due to structural complexity and heterogeneous reservoir properties. The measured average porosity is 2.2%, and the measured average permeability is 0.155 mD, indicating that it is a low-porosity and low-permeability reservoir [1,2]. Currently, acid fracturing is predominantly employed for dolomite reservoirs. However, filtration control is challenging during the acid fracturing process. The objects of filtration control in the acid fracturing process are the filtration of non-reactive pad fluids and the filtration of acid fluids with different viscosities [3,4]. In 2018, Xue Heng et al. [5] established a mathematical model to investigate deep carbonate reservoirs, revealing that a higher calcite content in carbonate rocks significantly reduces the effective acid penetration distance. Their findings demonstrated that increased injection rates extend the acid penetration distance, but they narrow acid-etched fracture widths. In 2019, Ai Kun et al. [6] employed UFD (Unified Fracture Design) theory to optimize fracture geometry and operational parameters, targeting the maximum dimensionless productivity index as the key performance indicator. In 2019, Qi Dan et al. [7] derived a fractal model quantifying the effective propagation distance of acid-etched wormholes in carbonate matrices, enabling geometric characterization of irregular dissolution patterns. In 2021, Liu Pingli et al. [8] developed an innovative network acid fracturing technology that strategically integrates vertical chemical diversion to stimulate multiple acid-etched fracture initiations, planar acid etching for enhanced interconnectivity with natural fracture systems, and synergistic interaction between these mechanisms to establish a three-dimensional fracture network with optimized conductivity pathways. Li Xinyong et al. [9] (2021) implemented Monte Carlo algorithms to construct statistically representative natural fracture models, subsequently analyzing wormhole propagation mechanisms under a dual-scale continuum modeling framework. In 2021, Ren et al. [10] proposed a multi-field coupled acid fracturing model based on the concept of stereoscopic acid fracturing. This model integrates the propagation of hydraulic fractures with the etching effects of acid on both natural and hydraulic fractures, while systematically investigating the fluid loss dynamics and heat transfer mechanisms under these conditions. Numerical simulations were conducted to analyze the complex interactions between acid–rock reactions, fracture network evolution, and thermal effects within deep fractured carbonate reservoirs. In 2022, Li Song et al. [11] demonstrated that the key factors influencing the height extension of acid-fracturing fractures are the inter-zonal stress difference and the pumping rate, with the stress difference being the predominant factor, followed by the thickness of the barrier layer and the fluid viscosity. In 2022, Li Xinyong et al. [12] proposed a compound acid fracturing technology to achieve long-distance connectivity of fault-karst reservoirs and generate stable conductivity. In 2022, Liu et al. [13] proposed a composite acid fracturing process termed “multi-cluster limited-entry perforation + friction-reduced water with proppant + clean acid” to enhance reservoir stimulation volume and strengthen the conductivity of acid-etched fractures.

The analysis above elucidates that the acid leakoff mechanism during acid fracturing operations exhibits distinct differences from those observed in non-reactive pad fluids (fracturing fluids) and conventional matrix acidizing, owing to the inherent coupling between acid injection processes and acid–rock reactions [14,15]. The most critical distinction lies in the formation of acid-etched wormholes along fracture walls during reactive flow, which profoundly influences acid filtration behavior. Consequently, the dual-scale continuum model was improved using the Brinkman equation. A numerical simulation model for acid etching during acid fracturing was established based on geological data from target reservoirs, aiming to reveal the acid leakoff behavior along fracture walls under varying pumping rates, acid concentrations, acid viscosities, and reservoir porosities.

2. Numerical Simulation Algorithms and Mathematical Models for Acid-Etched Fracturing

2.1. Numerical Simulation Algorithms or Acid-Etched Fracturing

The acid etching process in acid fracturing numerical simulation [16] belongs to reactive transport flow. The algorithms for reactive transport flow include the Lattice Boltzmann Method (LBM) [17], Smoothed Particle Hydrodynamics (SPH) [18], and Computational Fluid Dynamics (CFD). The LBM is a specialized discretization of the Boltzmann equation, where the fundamental variable is a statistical function describing the probability of finding particles with specific velocities at given spatial locations. SPH represents fluids using particles endowed with intensive properties; these particles are tracked in real time as they move through pore spaces [19,20]. CFD methods, which are more widely applied, can be categorized into the Finite Difference Method (FDM) [21], Finite Volume Method (FVM) [22,23], and Finite Element Method (FEM) [24]. For the dolomite reservoirs of the Xixiangchi Formation in the eastern Sichuan Basin, the FEM was selected for solving the governing equations. This choice primarily stems from its capability to handle complex geometries with adjustable precision and its flexibility in discretization strategies.

2.2. Mathematical Models for Acid-Etched Fracturing

To effectively leverage the merits of Darcy-scale and pore-scale models, while comprehensively capturing seepage mechanisms and microstructural pore evolution, a dual-scale mathematical framework was developed and refined to incorporate realistic acid-fracturing etching dynamics. This approach synergistically integrates a dual-scale continuum model with a discrete fracture network (DFN) representation, conceptualizing fracture zones as highly porous and permeable media. By bridging Darcy-scale continuum descriptions with pore-scale structural evolution, the model concurrently preserves the advantages of both continuum and discrete fracture methodologies, achieving enhanced numerical accuracy with reduced computational demands. Furthermore, the Brinkman equation was implemented to optimize the governing equations of the dual-scale continuum model, ensuring improved alignment with acid etching physics during hydraulic fracturing processes.

(1) Darcy-scale mathematical model

In the simulation, fractures are treated as a matrix with extremely high permeability, while their porosity is assigned the maximum value. The Brinkman equation, which inherently represents the pore-scale Navier–Stokes equations [25], accounts for fluid–matrix friction, and is applicable to modeling acid flow through fractures and the matrix under high-velocity conditions [26]. Consequently, the Brinkman equation replaces the Darcy equation in the Darcy-scale model.

$${\displaystyle \frac{{\rho }_L}{\varphi }}\left({\displaystyle \frac{\partial U}{\partial t}}+\left(U\cdot \nabla \right){\displaystyle \frac{U}{\varphi }}\right)=-\nabla P+\nabla \cdot \left[{\displaystyle \frac{1}{\varphi }}\left\{\mu \left(\nabla U+{\left(\nabla U\right)}^T\right)-{\displaystyle \frac{2}{3}}\mu \left(\nabla \cdot U\right)I\right\}\right]-\left(K^{-1}\mu +{\displaystyle \frac{Q_m}{{\varphi }^2}}\right)U+F$$

(1)

$${\displaystyle \frac{\partial \varphi }{\partial t}}+\nabla U=0$$

(2)

μ: dynamic viscosity of acid, mPa·s;

ρ_L: fluid density, kg/m³;

ϕ: porosity;

U: velocity vector, m/s;

P: pressure, Pa;

K: permeability, m²;

Q_m: mass source term, kg/(m³·s);

F: gravitational and other volumetric forces, kg/(m²·s²).

The chemical field equations describe chemical reactions, namely the advection–diffusion phenomena, which are constructed from the chemical equilibrium equations and porosity variation equations.

$${\displaystyle \frac{\partial \left(\varphi C_f\right)}{\partial t}}+\nabla \left(U{C}_f\right)-\nabla \left(\varphi D\nabla C_f\right)=-k_c{a}_v\left(C_f-C_s\right)$$

(3)

$${\displaystyle \frac{\partial \varphi }{\partial t}}={\displaystyle \frac{k_c{k}_s\alpha a_v}{\left(k_c+k_s\right){\rho }_S}}{C}_f$$

(4)

C_f: mass concentration of acid in pores, mol/m³;

C_s: mass concentration of acid on pore walls, mol/m³;

D: diffusion coefficient, m²/s;

K_c: mass transfer coefficient, m/s;

K_s: reaction rate constant, m/s;

ρ_s: rock density, kg/m³;

a_v: specific surface area of pores, m²/m³;

α: dissolution capacity coefficient, kg/kmol.

Equations (1)–(4) collectively constitute the Darcy-scale model within the dual-scale continuum model.

(2) Pore-scale mathematical model

The modified Carman–Kozeny semi-empirical formula was employed to derive the relationships between permeability, pore-throat radius, specific surface area, and porosity.

$${\displaystyle \frac{K}{K_0}}={\displaystyle \frac{\varphi }{{\varphi }_0}}\left[{\displaystyle \frac{\varphi \left(1-{\varphi }_0\right)}{{\varphi }_0\left(1-\varphi \right)}}\right]2^{\beta }$$

(5)

$${\displaystyle \frac{r_p}{r_{p0}}}=\sqrt{{\displaystyle \frac{K{\varphi }_0}{K_0\varphi }}}$$

(6)

$${\displaystyle \frac{a_v}{a_0}}={\displaystyle \frac{\varphi r_{p0}}{{\varphi }_0{r}_p}}$$

(7)

K₀: initial permeability, m²;

ϕ₀: initial porosity;

r_p: initial pore-throat radius, m;

β: empirical constant;

r_p0: initial pore radius;

a_v: specific surface area of pores;

a₀: initial specific surface area.

The relationship between the mass transfer coefficient and the effective diffusion coefficient was derived using the Panga empirical formula.

$$Sh={\displaystyle \frac{2k{r}_p}{D_m}}=S{h}_{\infty }+0.7m^{-1/2}R{e}^{1/2}S{c}^{1/3}$$

(8)

$$D_{ei}={\alpha }_{os}{D}_m+{\displaystyle \frac{2{\lambda }_i{v}_i{r}_p}{\varphi }}$$

(9)

Sh: Sherwood number;

D_m: molecular effective diffusion coefficient, m²/s;

Sh_∞: asymptotic Sherwood number;

m: ratio of pore length to pore diameter;

Re: Reynolds number at the pore scale;

Sc: Schmidt number;

D_ei: effective diffusion coefficient of acid in the i-direction;

α_os: constant dependent on pore structure;

λ_i: constant dependent on pore structure.

Equations (5)–(9) collectively constitute the pore-scale model within the dual-scale continuum model.

2.3. Parameter Assignment

As shown in Figure 1, the dolomite reservoir (length L, width W, height H_f) connects with the wellbore through a vertical hydraulic fracture (length L_f, width W_f, height H_f). Taking one fracture as an example, the symmetry of dual-wing vertical fractures is considered. The acid first flows into the wellbore, then enters the fracture, and finally infiltrates the reservoir (acid flow direction is indicated by red arrows in Figure 1). The velocity at the fracture inlet is U₀, while the acid concentration on the fracture surface varies with location. The x-axis aligns with the fracture length direction, the y-axis with the fracture width direction, and the z-axis with the fracture height direction, with the intersection point between the wellbore and fracture center serving as the origin. To reduce the computational load, a two-dimensional model was employed, focusing exclusively on the simulation results in the x–y plane.

The core, with a diameter of 2.5 cm, contains fractures with widths of 0.5 mm and lengths typically ranging from 5 to 6 cm. A rectangular domain with dimensions of 2.5 × 6 cm was established and discretized into 595,052 grids to simulate the acid etching process. To approximate the actual formation porosity, the matrix porosity was generated using normally distributed random numbers with a mean value of 2%. Fractures were modeled based on high-porosity and high-permeability porous media with 5% porosity; the initial porosity distribution is shown in Figure 2.

Table 1. Model-related parameters.

Parameter	Value	Unit	Physical Meaning
L	0.06	m	Core length
W	0.025	m	Core width
Lf	0.05	m	Fracture length
Wf	0.0005	m	Fracture width
${\varphi }_0$	2%		Initial porosity
$K_0$	0.15	mD	Initial permeability
$a_0$	5000	m2/m3	Initial specific surface area
$r_p$	1.0 × 10−6	m	Initial pore-throat radius
${\rho }_S$	2710	kg/m3	Rock density
${\rho }_L$	1080	kg/m3	Fluid density
$D_m$	3.6 × 10−5	m2/s	Molecular effective diffusion coefficient
$\alpha$	1	kg/mol	Dissolution capacity coefficient

lists additional simulation parameters used in the modeling.

The boundary conditions of the model were set as follows: a constant flow rate was maintained at the inlet, a constant static pressure was imposed at the outlet, and no fluid exchange occurred at both lateral boundaries.

3. Model Validation Based on Physical Experiments

The numerical simulation model for acid fracturing and acid etching incorporates parameters referenced from physical experiments on acid-etched fracture wall surfaces. To validate the model reliability, the wormhole morphology on fracture wall surfaces under different experimental parameters was qualitatively characterized through observation. The specific experimental parameters are detailed in

Table 2. Specific experimental parameters.

	Injection Displacement (mL/min)	Acid Concentration	Acid Viscosity (mPa·s)	Acid Dosage (mL)
Scheme 1	12.54	20%	5	120
Scheme 2	12.54	20%	5	240

, and the corresponding results are presented in Figure 3, Figure 4, Figure 5 and Figure 6.

As shown in the experimental results in Figure 3, Figure 4, Figure 5 and Figure 6, both numerical simulations (Scheme 1) and physical experiments revealed acid-etched flow channels in fractures, without wormhole formation along the fracture walls. With increased acid volume (Scheme 2), numerical simulations and experiments consistently demonstrated acid-etched flow channels with wormhole development along the fracture walls, confirming that the acid fracturing and etching numerical model adheres to physical experimental laws. This validates the feasibility of replacing physical experiments with numerical models for parametric sensitivity analyses.

4. Analysis of Influencing Factors on Acid Fluid Filtration Loss at Fracture Walls

The formation of acid-etched wormholes during acid fracturing induces acid fluid filtration loss, thereby shortening the effective acid-etched fracture length and compromising stimulation outcomes. Numerical simulations of acid fracturing–etching were employed to investigate the influence of injection rate, acid viscosity, acid concentration, and reservoir porosity on acid fluid filtration loss, providing a foundation for optimizing acid fracturing parameters. To quantitatively characterize acid fluid filtration loss in the experimental results, the average porosity variation (Δϕ) in the matrix adjacent to the front half of the fracture was adopted as an indicator, reflecting the correlation between porosity–permeability alterations and filtration magnitude.

The field injection rates were set at 4, 6, and 8 m³/min, while the simulated injection rates were 6.27, 9.41, and 12.54 mL/min. Field acid viscosities of 5 and 20 mPa·s were selected, and the simulations incorporated viscosities of 1, 5, and 20 mPa·s to analyze their impacts on the acid fracturing and etching results. Field acid mass concentrations of 10%, 15%, and 20% corresponded to simulated concentrations of 2.872, 4.412, and 6.022 mol/L, respectively, to evaluate concentration-dependent effects. For the Xixiangchi Group dolomite reservoirs in the eastern Sichuan Basin, the measured porosity was 2.2%; thus, porosity values of 2%, 5%, and 10% were simulated to assess their influence. The detailed experimental design is summarized in

Table 3. Acid fracturing–etching numerical simulation experimental design.

	Injection Rate (mL/min)	Acid Viscosity (mPa·s)	Acid Concentration (mol/L)	Porosity
Experiment 1	6.27	5	4.412	2%
Experiment 2	9.41	5	4.412	2%
Experiment 3	12.54	5	4.412	2%
Experiment 4	9.41	1	4.412	2%
Experiment 5	9.41	20	4.412	2%
Experiment 6	9.41	5	2.872	2%
Experiment 7	9.41	5	6.022	2%
Experiment 8	9.41	5	4.412	5%
Experiment 9	9.41	5	4.412	10%

.

4.1. Injection Rate

Numerical simulations of acid fracturing–etching under different injection rates were conducted in Experiment 1, Experiment 2, and Experiment 3. The simulation results are presented in

Table 4. Numerical simulation results of acid fracturing–etching under different injection rates.

	Injection Rate (mL/min)	Acid Viscosity (mPa·s)	Acid Concentration (mol/L)	Porosity	$\Delta \mathit{\varphi }$
Experiment 1	6.27	5	4.412	2%	0.03853
Experiment 2	9.41	5	4.412	2%	0.08949
Experiment 3	12.54	5	4.412	2%	0.19086

and Figure 7, Figure 8 and Figure 9.

The numerical simulation results of acid fracturing–etching from Experiment 1, 2, and 3 demonstrate that, as shown in Experiment 1, compared to a high injection rate, the low injection rate not only failed to induce acid leakoff into the matrix reservoir, but also failed to connect the distal fracture zones. As the injection rate increased, the acid fluid connected the distal fracture zones via wormhole-like patterns; however, acid leakoff into the matrix reservoir significantly intensified. To better compare the effects of increased injection rates on acid leakoff into the matrix reservoir, the simulation duration under the conditions of Experiment 2 and 3 was adjusted to the moment of acid breakthrough at the fracture distal end for comparative analysis. Experiment 2 exhibited a Δϕ of 0.06216, while Experiment 3 showed a Δϕ of 0.07561. This indicates that although a higher injection rate improved connectivity to the distal fracture zones, it also notably exacerbated acid leakoff into the matrix reservoir. The simulation results are presented in Figure 10 and Figure 11.

4.2. Acid Viscosity

Numerical simulations of acid fracturing–etching under varying acid fluid viscosities were conducted in Experiments 2, 4, and 5, with the corresponding results summarized in

Table 5. Numerical simulation results of acid fracturing–etching under varying acid fluid viscosities.

	Injection Rate (mL/min)	Acid Viscosity (mPa·s)	Acid Concentration (mol/L)	Porosity	$\Delta \mathit{\varphi }$
Experiment 2	6	5	4.412	2%	0.08949
Experiment 4	6	1	4.412	2%	0.20672
Experiment 5	6	20	4.412	2%	0.00766

and Figure 12, Figure 13 and Figure 14.

The numerical simulation results of acid fracturing–etching from Experiments 2, 4, and 5 under varying acid fluid viscosities reveal that, as demonstrated in Experiment 5, the high-viscosity acid fluid achieved retardation without significant leakoff into the matrix reservoir. In contrast, the results of Experiment 4 show that low-viscosity acid fluid connected distal fracture zones via approximately uniform dissolution patterns, but acid leakoff into the matrix reservoir intensified markedly. Therefore, in field operations, moderate-viscosity acid can be employed to connect distal fracture zones following pad fluid injection for fracture creation, after which viscosity reduction will enable non-uniform etching of the fracture to form high-conductivity channels.

4.3. Acid Concentration

Numerical simulations of acid fracturing–etching under varying acid fluid concentrations were conducted in Experiments 2, 6, and 7, with the corresponding results summarized in

Table 6. Numerical simulation results of acid fracturing–etching under varying acid fluid concentrations.

	Injection Rate (mL/min)	Acid Viscosity (mPa·s)	Acid Concentration (mol/L)	Porosity	$\Delta \mathit{\varphi }$
Experiment 2	6	5	4.412	2%	0.08949
Experiment 6	6	5	2.872	2%	0.06479
Experiment 7	6	5	6.022	2%	0.16243

and Figure 15, Figure 16 and Figure 17.

The acid fracturing–etching numerical simulation results from Experiments 2, 6, and 7 under varying acid fluid concentrations demonstrate that, as shown in Experiment 6, lower acid fluid concentrations failed to connect the distal fracture zones and induced significant fluid loss into the matrix reservoir. However, with increased acid fluid concentration (Experiment 7), the acid effectively connected the distal fracture zones. Although fluid loss into the matrix reservoir remained evident, it was smaller compared to Experiment 3 (where injection rate was increased). Therefore, optimizing the acid fluid concentration—rather than solely enhancing the pumping rate—is recommended during field operations to achieve effective distal fracture connectivity.

4.4. Porosity

Numerical simulations of acid fracturing–etching under varying reservoir porosity were conducted in Experiments 2, 8, and 79, with the corresponding results summarized in

Table 7. Numerical simulation results of acid fracturing–etching under varying reservoir porosities.

	Injection Rate (mL/min)	Acid Viscosity (mPa·s)	Acid Concentration (mol/L)	Porosity	$\Delta \mathit{\varphi }$
Experiment 2	6	5	4.412	2%	0.08949
Experiment 8	6	5	4.412	5%	0.09476
Experiment 9	6	5	4.412	10%	0.10401

and Figure 18, Figure 19 and Figure 20.

As the reservoir porosity increases, the results of the acid fracturing–etching numerical simulations from Experiments 2, 8, and 9 reveal that enhanced heterogeneity of the core (as demonstrated in Experiment 9) leads to significantly aggravated acid leakoff into the matrix reservoir. Therefore, during field operations, optimizing process parameters through leakoff control and acid retardation strategies is critical to mitigate acid loss and ensure effective access to distal fracture zones.

5. Conclusions

By improving the dual-scale continuum model using the Brinkman equation, a mathematical model for acid fracturing–etching was established and solved via the finite element method (FEM), and validated against physical experiments on acid-etched fracture surfaces. The main conclusions are as follows.

(1) As the acid volume and treatment time increased, the results of both the numerical simulations and physical experiments transitioned from no observable wormholes to distinct wormhole formation on fracture walls, confirming consistency between the numerical predictions and experimental trends.

(2) Increasing the injection rate enhanced acid penetration to distal fracture zones, but intensified acid leakoff into the matrix reservoir. Higher acid viscosity reduced matrix leakoff, but compromised distal fracture connectivity. Elevated acid concentration improved fracture communication, with lower leakoff compared to increased injection rates. Enhanced reservoir porosity significantly exacerbated acid leakoff into the matrix.

(3) For the Xixiangchi Formation reservoirs in the eastern Sichuan Basin, comprehensive analysis of injection rate, acid viscosity, acid concentration, and porosity suggests that the following field operation strategies should be employed: initially, medium-viscosity acid should be used to enhance distal fracture communication, followed by viscosity reduction to promote non-uniform etching. Prioritizing the acid concentration over the injection rate optimizes fracture connectivity while minimizing leakoff. In high-porosity reservoirs, process parameters require optimization through acid retardation and leakoff control strategies.

(4) To reduce computational demands, the numerical model employed a homogeneous porosity distribution, which introduced accuracy deviations. Future work with sufficient computational capacity should adopt the actual reservoir porosity distribution to enhance simulation fidelity.

(5) To enhance engineering applicability, subsequent studies could establish 3D models based on the current 2D framework by incorporating fracture height parameters to investigate acid leakoff patterns along fracture walls.