A permeability evolution model of high-pressure waterflooding with dual-fractal characteristics of mechanical properties and fracture structures

Zheng, Bintao; Zhang, Liaoyuan; Zhao, Jiaqi; Li, Yuan; Xue, Xianjie; Song, Wenhui

doi:10.3389/feart.2025.1720011

ORIGINAL RESEARCH article

Front. Earth Sci., 12 January 2026

Sec. Georeservoirs

Volume 13 - 2025 | https://doi.org/10.3389/feart.2025.1720011

A permeability evolution model of high-pressure waterflooding with dual-fractal characteristics of mechanical properties and fracture structures

Bintao Zheng¹

Liaoyuan Zhang¹

Jiaqi Zhao²

Yuan Li¹

Xianjie Xue²

Wenhui Song ²*

¹ Petroleum Engineering Technology Research Institute of Shengli Oilfield, Sinopec, Dongying, China
² School of Petroleum Engineering, China University of Petroleum (Beijing), Beijing, China

The complexities of evaluating permeability evolution during high-pressure waterflooding and subsequent production are primarily governed by rock mechanical damage and extensive microfracture deformation. In this study, we propose a general permeability evolution model for high-pressure waterflooding that incorporates dual fractal characteristics of mechanical properties and fracture structures. Both fracture aperture distribution and Young’s modulus distribution are described using fractal scaling laws. For a given fractal unit with a specified Young’s modulus, an analytical model is developed to quantify fracture aperture variation during elastic deformation and the failure process based on damage mechanics. The damage-induced reduction in Young’s modulus after failure is further considered in the production stage, and an analytical solution for the fracture-medium permeability is established through double integration within the framework of fractal geometry. A percolation-based effective-medium approximation is employed to calculate matrix–fracture permeability. The effects of fracture-related fractal properties, rock mechanics-related fractal properties, and reservoir properties (porosity, matrix permeability, and minimum Young’s modulus) on matrix–fracture permeability are systematically investigated. The key results demonstrate that matrix–fracture permeability is most strongly influenced by fracture fractal dimension and reservoir properties, particularly porosity, matrix permeability, and minimum Young’s modulus.

1 Introduction

High-pressure waterflooding has gained increasing attention due to the recent development of tight oil/shale oil reservoirs (Li et al., 2024; Li X. et al., 2025; Su et al., 2025; Sun et al., 2024; Wang et al., 2024; Wang et al., 2025). Unlike traditional hydraulic fracturing, which aims to create large main hydraulic fractures with high injection pressure (Wang and Kobina, 2025; Wu et al., 2025), high-pressure waterflooding aims to create large numbers of microfractures with injection pressure slightly higher than fracture pressure (Feng et al., 2021; Li et al., 2025a). Accurate evaluation of permeability evolution during high-pressure waterflooding and subsequent production is essential for designing a reservoir stimulation plan and productivity assessment. Efforts have been made to understand permeability evolution during high-pressure waterflooding and subsequent production by both physical experiments and numerical simulations. Wu et al. (2025) conducted rock core sample permeability tests during high-pressure waterflooding and post-waterflooding stages. They found that permeability first increases slowly and then rapidly when the injection pressure exceeds the fracture pressure. The permeability of the fractured area decreases faster than that of the original rock without high-pressure waterflooding under increasing effective stress conditions. Li et al. (2025a) studied the influence of the displacement rate on high-pressure waterflooding through core experiments and found that microfractures are generated at high displacement rates, the recovery rate increases rapidly at the initial stage, and then no longer increases when it reaches a certain value. Fan et al. (2015) considered the influence of stress paths during high-pressure waterflooding and subsequent production. A model of dynamic fracture permeability was developed using a piecewise function and an effective-stress-based exponential function. Sun et al. (2025) investigated the development characteristics of overburden fractures during repeated mining influenced by multiple key strata, highlighting the significant roles of cumulative overburden damage and the combined action of superimposed mining-induced stress fields in shaping fracture morphology. Li et al. (2025b) applied the threshold pressure gradient to describe permeability changes with displacement pressure gradients and adopted an expansion–recompaction model to consider pressure gradient-dependent permeability variations during high-pressure waterflooding. However, the above studies have not fully accounted for variations in fracture structures and heterogeneous mechanical properties during high-pressure waterflooding and subsequent production. Fractal theory enables an accurate correlation between spatially distributed properties and hydromechanical properties (Song et al., 2019; Song et al., 2022). Wang and Cheng (2020) developed a fractal permeability model for 2D complex tortuous fractures. Yang et al. (2025) further built a fractal-based fracture permeability model that considers the influence of effective stress on the fracture aperture variation. However, there is no such fractal-based model that aims to describe permeability evolution during high-pressure waterflooding and subsequent production. The influence of fracture structure properties and mechanical properties on the permeability evolution has not yet been studied (Hou et al., 2023; Sun et al., 2023). The purpose of this study is to develop a general permeability evolution model for high-pressure waterflooding that incorporates dual fractal characteristics of mechanical properties and fracture structures. Model development is introduced in Section 2. The effects of fracture-related fractal properties, rock mechanics-related fractal properties, and reservoir properties on matrix–fracture permeability are systematically discussed in Section 3. Section 4 concludes the study

2 Model development

The fracture aperture distribution can generally be described by the fractal scaling law (Qi et al., 2018; Vega and Kovscek, 2022; Wang et al., 2017). Because the Young’s modulus notably differs for different minerals, the Young’s modulus of the matrix–fracture medium also exhibits fractal characteristics. The dual-fractal distribution of the matrix–fracture medium is shown in Figure 1.

Figure 1

Diagram showing a cube with alternating blue and yellow layers, marked L₀ for length. An enlarged side view details layer widths labeled w₁ to w₇. Below are three cubes at different elastic moduli: E₀ = 1 GPa, E₀ = 50 GPa, and E₀ = 100 GPa, illustrating material stiffness variations.

Figure 1. Sketch map of the dual-fractal distribution (Young’s modulus and fracture aperture) of the matrix–fracture medium. The value of Young’s modulus E ₀ is provided for reference.

The aperture distribution in a fractal unit can be described according to the fractal scaling law (Yu and Cheng, 2002; Yu and Li, 2001):

N (L \geq w) = {(\frac{w_{\max}}{w})}^{D_{f}}, (1)

where w is the fracture aperture, w _max is the maximum fracture aperture, and D _f is the fractal dimension of the fracture. By differentiating Equation 1 with respect to the fracture aperture w, the number of fractures whose apertures are within the infinitesimal range from w to w + dw can be calculated by

- d N = D_{f} w_{\max}^{D_{f}} w^{- (D_{f} + 1)} d w . (2)

The cumulative fracture aperture in a fractal unit can consequently be expressed as Equation 3

w_{t} = - \int_{w_{\min}}^{w_{\max}} w d N = D_{f} w_{\max}^{D_{f}} \frac{w_{\max}^{1 - D f} - w_{\min}^{1 - D f}}{1 - D_{f}} (3)

where w _min is the minimum fracture aperture.

Therefore, the side length of the fractal unit can be expressed as Equation 4

L_{0} = D_{f} w_{\max}^{D_{f}} \frac{w_{\max}^{1 - D f} - w_{\min}^{1 - D f}}{(1 - D_{f}) ϕ_{f}}, (4)

where $ϕ_{f}$ is the fracture porosity.

The relationship between fracture length along flow direction and fracture aperture can be expressed as follows (Equation 5) (Yu and Cheng, 2002; Yu et al., 2002)

L (w) = w^{1 - D_{t}} L_{0}^{D_{t}}, (5)

where D _t is the fractal dimension of tortuosity of the fracture. The fracture aperture at a given tensile strain can be expressed as

w = w_{0} (1 + ε), (6)

where ε is the tensile strain. In this study, we define tensile strain and tensile stress as positive values, whereas compressive strain and compressive stress are assigned negative values.

Because Young’s modulus varies among different fractal units (Figure 1), the threshold Young’s modulus for a given fractal unit at the elastic limit can be expressed as Equation 7

E_{t} = \frac{σ}{ε_{0}}, (7)

where σ is the tensile stress, and ε ₀ is the strain at the elastic limit. Therefore, the tensile strain of a fractal unit with a Young’s modulus greater than E _t can be expressed as

ε = \frac{σ}{E} . (8)

For a fractal unit with a Young’s modulus smaller than E _t, the rock is damaged, and the extent of the damage is closely related to the original Young’s modulus value. Herein, the tensile strain of a fractal unit with a Young’s modulus smaller than E _t is formulated as

ε = ε_{0} {(\frac{E_{t}}{E})}^{m}, (9)

where m is a damage degree parameter with a constant value. The Young’s modulus value after the rock damage can be expressed as

E^{'} = E {(\frac{E}{E_{t}})}^{m} . (10)

Equations 9, 10 indicate that the damage extent increases in fractal units with smaller initial Young’s moduli. Because the tensile strain at the elastic limit is relatively small, we assume that the fracture porosity of a fractal unit with a Young’s modulus greater than E _t stays constant. The fracture porosity of a fractal unit with a Young’s modulus smaller than E _t is formulated as

ϕ_{f} = ϕ_{f 0} {(\frac{E_{0}}{E_{t}})}^{n}, (11)

where n is a porosity damage degree parameter with a constant value. Equation 11 indicates that the porosity increases more significantly in fractal units with smaller initial Young’s moduli.

Compressive strain shows a linear relationship with compressive stress during the production process, and the compressive strain of the fractal unit can be expressed as

ε = \{\begin{array}{l} \frac{σ}{E}, E \geq E_{t}, \\ \frac{σ}{E {(\frac{E}{E_{t}})}^{m}}, E < E_{t} . \end{array} (12)

Because fracture porosity is proportional to the cube of the fracture strain, the fracture porosity of the fractal unit during the production process can consequently be expressed as

ϕ_{f} = \{\begin{array}{l} ϕ_{f 0} {(1 + ε)}^{3}, E_{0} \geq E_{t}, \\ ϕ_{f 0} {(\frac{E_{0}}{E_{t}})}^{n} {(1 + ε)}^{3}, E_{0} < E_{t} . \end{array} (13)

The water flux in a single fracture at a given pressure drop between the inlet and outlet of the fractal unit can be expressed as Equation 14

q (w) = \frac{w^{3} L_{0}}{12} \frac{Δ P}{μ_{w} L (w)}, (14)

where μ _w is the water viscosity and $Δ P$ is the pressure drop. The fracture aperture is related to Young’s modulus, strain, and stress in Equations 6–9, 10, 12. According to Darcy’s law, the water flux in the fracture medium of a single fractal unit is calculated by integrating the individual water flux q(w) over the entire fracture aperture range, and the water permeability in the fracture system can be expressed as Equations 15, 16

Q = - \int_{w_{\min}}^{w_{\max}} q (w) d N, (15)

k_{f} = \frac{Q u_{w} L_{0}}{L_{0}^{2} Δ P} = \frac{D_{f} w_{\max}^{2 + D_{t}}}{12 L_{0}^{D_{t}} (2 - D_{f} + D_{t})} (1 - {(\frac{w_{\min}}{w_{\max}})}^{2 - D_{f} + D_{t}}) . (16)

The Young’s modulus distribution among different fractal units can also be described by the fractal scaling law (Yu and Cheng, 2002; Yu and Li, 2001):

N (E^{'} \geq E) = {(\frac{E_{\max}}{E})}^{D_{f m}}, (17)

where D _fm is the fractal dimension of Young’s modulus, and E _max is the maximum Young’s modulus value among different fractal units. The total number of fractal units with different Young’s modulus values can be expressed as follows (Equation 18)

N_{t} = {(\frac{E_{\max}}{E_{\min}})}^{D_{f m}}, (18)

where E _min is the maximum Young’s modulus value among different fractal units. By differentiating Equation 17 with respect to Young’s modulus, the number of fractal units whose Young’s modulus is within an infinitesimal range from E to E + dE can be calculated by

- d N = D_{f m} E_{\max}^{D_{f m}} E^{- (D_{f m} + 1)} d E . (19)

Therefore, the proportion of fractal units whose Young’s modulus is within the infinitesimal range from E to E + dE can accordingly be expressed as Equation 20

f (E) = - \frac{d N}{N_{t} d E} = D_{f m} E_{\min}^{D_{f m}} E^{- (D_{f m} + 1)} . (20)

The water permeability in the fracture system, considering dual-fractal distribution (Young’s modulus and fracture aperture), can be expressed as Equation 21

k_{f s} = \int_{E_{\min}}^{E_{t}} k_{f} (σ, E) f (E) d E + \int_{E_{t}}^{E_{\max}} k_{f} (σ, E) f (E) d E . (21)

It should be noted that fracture strain during both high-pressure waterflooding and the subsequent pressure-depleted production process differs notably according to the threshold Young’s modulus (Equations 8, 9,12). Therefore, the water permeability in the fracture system during high-pressure waterflooding can be expressed as

\begin{array}{l} k_{f s} = \int_{E_{\min}}^{E_{t}} k_{f} (σ, E) f (E) d E + \int_{E_{t}}^{E_{\max}} k_{f} (σ, E) f (E) d E \\ = \int_{E_{\min}}^{E_{t}} \frac{D_{f} w_{\max}^{2 + D_{t}} {(1 + ε_{0} {(\frac{E_{t}}{E})}^{m})}^{2 + D_{t}}}{12 L_{0}^{D_{t}} (2 - D_{f} + D_{t})} \\ \times (1 - {(\frac{w_{\min}}{w_{\max}})}^{2 - D_{f} + D_{t}}) D_{f m} E_{\min}^{D_{f m}} E^{- (D_{f m} + 1)} d E \\ + \int_{E_{t}}^{E_{\max}} \frac{D_{f} w_{\max}^{2 + D_{t}} {(1 + \frac{σ}{E})}^{2 + D_{t}}}{12 L_{0}^{D_{t}} (2 - D_{f} + D_{t})} (1 - {(\frac{w_{\min}}{w_{\max}})}^{2 - D_{f} + D_{t}}) \\ \times D_{f m} E_{\min}^{D_{f m}} E^{- (D_{f m} + 1)} d E . \end{array} (22)

The fracture aperture increases during the high-pressure waterflooding process (Equations 8, 9), and the initial fracture aperture differs notably for the production process. Therefore, the water permeability in the fracture system during the production process can be expressed as

\begin{array}{l} k_{f s} = \int_{E_{\min}}^{E_{t}} k_{f} (σ, E) f (E) d E + \int_{E_{t}}^{E_{\max}} k_{f} (σ, E) f (E) d E \\ = \int_{E_{\min}}^{E_{t}} \frac{D_{f} w_{\max}^{2 + D_{t}} {(1 + ε_{0} {(\frac{E_{t}}{E})}^{m})}^{2 + D_{t}} {(1 + \frac{σ}{E {(\frac{E}{E_{t}})}^{m}})}^{2 + D_{t}}}{12 L_{0}^{D_{t}} (2 - D_{f} + D_{t})} \\ \times (1 - {(\frac{w_{\min}}{w_{\max}})}^{2 - D_{f} + D_{t}}) D_{f m} E_{\min}^{D_{f m}} E^{- (D_{f m} + 1)} d E \\ + \int_{E_{t}}^{E_{\max}} \frac{D_{f} w_{\max}^{2 + D_{t}} {(1 + \frac{σ_{\max}}{E})}^{2 + D_{t}} {(1 + \frac{σ}{E})}^{2 + D_{t}}}{12 L_{0}^{D_{t}} (2 - D_{f} + D_{t})} \\ + (1 - {(\frac{w_{\min}}{w_{\max}})}^{2 - D_{f} + D_{t}}) D_{f m} E_{\min}^{D_{f m}} E^{- (D_{f m} + 1)} d E, \end{array} (23)

where σ _max is the maximum tensile stress during the high-pressure waterflooding process. There is no analytical solution to Equations 22, 23, and numerical integration is used to solve Equations 22, 23 in subsequent analyses. Because the fracture porosity of a fractal unit with a Young’s modulus greater than E _t stays constant because of the relatively small tensile strain during elastic deformation, the fracture porosity in the fracture system during the high-pressure waterflooding process can be expressed as Equation 24

\begin{array}{l} ϕ_{f s} = \int_{E_{\min}}^{E_{t}} ϕ_{f} (σ, E) f (E) d E + ϕ_{f} \int_{E_{t}}^{E_{\max}} f (E) d E \\ = ϕ_{f} \frac{D_{f m}}{D_{f m} + n} E_{\min}^{D_{f m}} E_{t}^{n} (E_{t}^{- (D_{f m} + n)} - E_{\min}^{- (D_{f m} + n)}) + ϕ_{f} E_{\min}^{D_{f m}} (E_{t}^{- D_{f m}} - E_{\max}^{- D_{f m}}) . \end{array} (24)

The fracture porosity increases during the high-pressure waterflooding process (Equation 11), and the initial fracture porosity differs notably in terms of the production process. According to Equation 13, the fracture porosity in the fracture system during the production process can be expressed as

\begin{array}{l} ϕ_{f s} = \int_{E_{\min}}^{E_{t}} ϕ_{f} (σ, E) f (E) d E + \int_{E_{t}}^{E_{\max}} ϕ_{f} (σ, E) f (E) d E \\ = ϕ_{f 0} D_{f m} E_{\min}^{D_{f m}} {\int_{E_{t}}^{E_{\max}} (1 + \frac{σ}{E})}^{3} E^{- (D_{f m} + 1)} d E + ϕ_{f 0} D_{f m} E_{\min}^{D_{f m}} \\ \times {\int_{E_{\min}}^{E_{t}} (\frac{E_{t}}{E})}^{n} {(1 + \frac{σ E_{t}^{m}}{E^{1 + m}})}^{3} E^{- (D_{f m} + 1)} d E . \end{array} (25)

There is no analytical solution to Equation 25, and numerical integration is used to solve Equation 25 in the subsequent analysis.

Percolation-based effective-medium approximation (P-EMA) is applied to calculate matrix–fracture permeability. P-EMA is a statistically based upscaling theory that targets media with low- and high-conductivity components (McLachlan, 1987; McLachlan, 1988), and it has been applied to calculate permeability, thermal conductivity, and Young’s modulus in a binary medium (Agbaje et al., 2023; Deprez et al., 1988; Ghanbarian and Daigle, 2016; McLachlan, 2021). Therefore, the relationship between matrix–fracture permeability and the individual permeabilities of the matrix and fractures can be expressed as Equation 26

(1 - ϕ_{f s}) \frac{k_{m}^{\frac{1}{t}} - k_{e f f}^{\frac{1}{t}}}{k_{m}^{\frac{1}{t}} + \frac{1 - ϕ_{f c}}{ϕ_{f c}} k_{e f f}^{\frac{1}{t}}} + ϕ_{f s} \frac{k_{f}^{\frac{1}{t}} - k_{e f f}^{\frac{1}{t}}}{k_{f}^{\frac{1}{t}} + \frac{1 - ϕ_{f c}}{ϕ_{f c}} k_{e f f}^{\frac{1}{t}}} = 0, (26)

where k _eff is the matrix–fracture permeability, k _m is the matrix permeability, $ϕ_{f c}$ is the critical fracture porosity, and t is the scaling exponent. The scaling exponent value typically fluctuates at approximately 2, and we set t = 2 in this study. Detailed input parameters are given in Table 1. The matrix–fracture permeability variation during the high-pressure waterflooding process and the subsequent production process is calculated using the input parameters in Table 1. The related discussion and corresponding sensitivity analysis are given in the next section.

Table 1

Table 1. Reservoir properties of the studied reservoir.

3 Results and discussion

A semi-logarithmic plot in Figure 2a illustrates the variation of matrix–fracture permeability as a function of tensile stress during the high-pressure waterflooding process. In the initial stage, the matrix–fracture permeability value is similar to the matrix permeability value and shows a gradual increase under low tensile stress. With further stress increase, it exhibits a pronounced increase due to extensive rock damage (Equation 9). In particular, the matrix–fracture permeability increases rapidly by at least three orders of magnitude in the range of 20–60 MPa, whereas at stresses exceeding 60 MPa, the growth rate becomes progressively less pronounced. The variation of matrix–fracture permeability exhibits a linear trend on a semi-logarithmic scale with increasing compressive stress. The final matrix–fracture permeability is more than one order of magnitude higher than the matrix permeability, indicating that high-pressure waterflooding significantly increases reservoir permeability. The observed trends are consistent with those reported in the experimental literature (Wu and Ansari, 2025). As the fracture fractal dimension increases from 1.6 to 2.4, the initial value of the fracture aperture distribution (Equation 2) decreases, leading to a reduction in matrix-fracture permeability by one to two orders of magnitude, as shown in Figure 2. As the tortuosity fractal dimension increases from 1 to 1.15, the fracture flow pathways become more tortuous, and the matrix–fracture permeability decreases to approximately 80% of its original value, as shown in Figure 3. An increase in the fractal dimension of Young’s modulus leads to a smaller value in the Young’s modulus distribution (Equation 19). Thus, rock damage is more readily induced under a smaller Young’s modulus at the same tensile stress (Equation 9). With an increase in the fractal dimension of Young’s modulus from 1.6 to 2.8, the matrix–fracture permeability increases significantly, by a factor of approximately two to three, as shown in Figure 4.

Figure 2

Two line graphs depict the relationship between $k_{\text{eff}}/\mu m^2$ and $\sigma/\text{MPa}$ for different fractal dimensions $D_f$. The top graph shows positive $\sigma$ values from 0 to 100, while the bottom graph shows negative values from -40 to 0. Both graphs feature four lines corresponding to $D_f$ values of 1.6, 2.0, 2.4, and 2.8, indicated by different markers and colors. The effective thermal conductivity increases with $\sigma$ and higher $D_f$.

Figure 2. Variation in matrix–fracture permeability as a function of stress for different fracture fractal dimensions: (a) high-pressure waterflooding and (b) production process.

Figure 3

Two line graphs display the relationship between the effective coefficient ($k_{\text{eff}}/\mu \text{m}^2$) and stress ($\sigma/\text{MPa}$). The top graph shows values for $\sigma$ from 0 to 100, and the bottom graph from -40 to 0. Both graphs include four lines representing different values of $D_t$ (1, 1.05, 1.1, 1.15), illustrating an increase in $k_{\text{eff}}$ as $D_t$ rises. A legend differentiates the lines using distinct markers and colors.

Figure 3. Variation in matrix–fracture permeability as a function of stress for different tortuosity fractal dimensions: (a) high-pressure waterflooding process and (b) production process.

Figure 4

Two logarithmic plots showing the relationship between effective thermal conductivity ($k_{\text{eff}}$) and stress ($\sigma$) for different fractal dimensions ($D_{\text{fm}}$). The top graph shows an increasing trend for $D_{\text{fm}} = 1.6, 2, 2.4, 2.8$ as $\sigma$ increases from 0 to 100 MPa. The bottom graph shows a similar trend for the same $D_{\text{fm}}$ values as $\sigma$ ranges from $-40$ to 0 MPa. Each line represents a different $D_{\text{fm}}$ value as indicated in the legends.

Figure 4. Variation in matrix–fracture permeability as a function of stress for different fractal dimensions of Young’s modulus: (a) high-pressure waterflooding process and (b) production process.

Figure 5a shows that the matrix–fracture permeability increases by at least two orders of magnitude in the range of 0–40 MPa when the matrix permeability decreases from 2.5 × 10⁻¹μm² to 2.5 × 10⁻⁴μm². It can be seen that the influence of matrix permeability on matrix–fracture permeability becomes negligible at tensile stresses greater than 60 MPa. In other words, the matrix–fracture permeability after high-pressure waterflooding is primarily governed by fracture properties and mechanical properties. For different matrix permeabilities, the matrix–fracture permeability decreases linearly with increasing compressive stress in the semi-logarithmic plot (Figure 5b). The matrix–fracture permeability decreases by approximately 40% when the matrix permeability decreases from 2.5 × 10⁻¹μm² to 2.5 × 10⁻⁴μm². The enhancement of the final matrix–fracture permeability relative to the original matrix permeability is more significant when the matrix permeability is smaller than 10⁻²μm². The distribution of Young’s modulus depends on the mineral composition (e.g., clay, quartz, and pyrite) and controls the extent of rock damage. During the high-pressure waterflooding process (Figure 6a), the matrix–fracture permeability increases by at least two orders of magnitude when the minimum Young’s modulus is smaller than 15 GPa. For rock with a minimum Young’s modulus greater than 15 GPa, the enhancement of matrix–fracture permeability by high-pressure waterflooding relative to the original matrix permeability is limited, and the matrix–fracture permeability remains nearly constant during the production process (Figure 6b). Figure 7a indicates that the variation of matrix–fracture permeability is highly sensitive to the initial fracture porosity. The enhancement of matrix–fracture permeability by high-pressure waterflooding relative to the original matrix permeability is limited for reservoirs with extremely low fracture porosity (<0.01). The matrix–fracture permeability increases by at least three orders of magnitude when the fracture porosity is greater than 0.04 (Figure 7b), indicating that high-pressure waterflooding is more favorable in reservoirs with high fracture porosity. Figure 8 further illustrates the variation of matrix–fracture permeability under different maximum tensile stresses during the production process. The matrix–fracture permeability increases by at least two orders of magnitude as the maximum tensile stress increases from 40 MPa to 100 MPa. This trend is more pronounced when the maximum tensile stress is below 80 MPa. Furthermore, the variation of matrix–fracture permeability with compressive stress is not affected by the maximum tensile stress. A comparison of the matrix–fracture permeability variations in Figures 2–8 reveals that the matrix–fracture permeability is more influenced by the fracture fractal dimension and reservoir properties (porosity, matrix permeability, and minimum Young’s modulus).

Figure 5

Two logarithmic line graphs display the relationship between effective permeability ($k_{\text{eff}}$) and stress ($\sigma$) in MPa, for different $k_m$ values (2.5×10⁻⁴ to 2.5×10⁻¹ μm²). The top graph shows increases in $k_{\text{eff}}$ with stress ranging from 0 to approximately 100 MPa. The bottom graph covers stress from -40 to 0 MPa. Both graphs use markers for each $k_m$ value: squares, circles, triangles, and inverted triangles. A legend is included.

Figure 5. Variation in matrix–fracture permeability as a function of stress for different matrix permeabilities: (a) high-pressure waterflooding process and (b) production process.

Figure 6

Two line graphs show the effective permeability $ k_{\text{eff}} $ against stress $ \sigma $ in megapascal for different minimum moduli $ E_{\text{min}} $ values (5, 10, 15, 20 GPa). The top graph shows a positive stress range, with curves increasing as $ E_{\text{min}} $ decreases. The bottom graph shows a negative stress range, with similar trends. Legends are consistent in both graphs.

Figure 6. Variation in matrix–fracture permeability as a function of stress for different minimum Young’s moduli: (a) high-pressure waterflooding process and (b) production process.

Figure 7

Two line graphs displaying effective permeability $k_{eff}$ against stress $\sigma$ in MPa. Top graph: stress from 0 to 100 MPa, the lines indicate increasing permeability for fiber volume fractions $\phi_f$ of 0.01, 0.04, 0.07, and 0.1. Bottom graph: stress from -40 to 0 MPa, similar trend observed. Different symbols and colors denote the fiber volume fractions.

Figure 7. Variation in matrix–fracture permeability as a function of stress for different initial fracture porosities: (a) high-pressure waterflooding process and (b) production process.

Figure 8

A graph depicts effective permeability ($k_{\text{eff}}$) in micrometers squared versus stress ($\sigma$) in megapascals. Four lines, using different symbols and colors, represent maximum stress levels: 100 MPa (black squares), 80 MPa (red circles), 60 MPa (blue triangles), and 40 MPa (magenta inverted triangles). The lines show a consistent increase in permeability as the stress increases from -40 to 0 MPa.

Figure 8. Variation in matrix–fracture permeability as a function of stress for different maximum tensile stresses during the production process.

4 Conclusion

This study developed a general permeability evolution model for high-pressure water flooding and subsequent production by incorporating the dual-fractal characteristics of mechanical properties and fracture structures. The model employs fractal scaling laws for fracture aperture and Young’s modulus distributions, integrates damage mechanics to capture fracture aperture changes during elastic deformation and failure, and accounts for post-failure Young’s modulus reduction in the production stage. Analytical solutions for fracture-medium permeability were derived within the framework of fractal geometry and combined with a percolation-based effective-medium approximation to evaluate matrix–fracture permeability. The results indicate that matrix–fracture permeability is most strongly influenced by fracture fractal dimension and reservoir properties such as porosity, matrix permeability, and minimum Young’s modulus, while the impact of rock mechanics-related fractal parameters is less pronounced. These findings highlight the pivotal role of fracture structure and reservoir heterogeneity in governing permeability evolution and provide a theoretical framework for predicting permeability changes and guiding the optimization of high-pressure waterflooding strategies in tight reservoirs.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

BZ: Writing – review and editing, Writing – original draft, Methodology, Validation. LZ: Formal analysis, Methodology, Writing – review and editing. JZ: Investigation, Writing – original draft. YL: Project administration, Writing – original draft. XX: Writing – original draft, Investigation. WS: Writing – review and editing, Investigation, Validation.

Funding

The author(s) declared that financial support was received for this work and/or its publication. Authors from China University of Petroleum (Beijing) acknowledge the support from the Science Foundation of China University of Petroleum, Beijing (No. 2462023QNXZ014).

Conflict of interest

Authors BT, LZ, and YL were employed by the Petroleum Engineering Technology Research Institute of the Shengli Oilfield, Sinopec.

The remaining author(s) declare that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Keywords: permeability evolution, high-pressure waterflooding, fracture, fractal geometry, analytical solution

Citation: Zheng B, Zhang L, Zhao J, Li Y, Xue X and Song W (2026) A permeability evolution model of high-pressure waterflooding with dual-fractal characteristics of mechanical properties and fracture structures. Front. Earth Sci. 13:1720011. doi: 10.3389/feart.2025.1720011

Received: 07 October 2025; Accepted: 17 December 2025;
Published: 12 January 2026.

Edited by:

Qingchao Li, Henan Polytechnic University, China

Reviewed by:

Zhengzheng Cao, Henan Polytechnic University, China
Yanyong Wang, Chengdu University of Technology, China
Xueying Wang, Yangtze University, China
Darya Khan Bhutto, Dawood University of Engineering and Technology, Pakistan
Xu Zhang, China University of Geosciences (Beijing) Energy Institute, China

Copyright © 2026 Zheng, Zhang, Zhao, Li, Xue and Song. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Wenhui Song, c29uZ193ZW5faHVpQGhvdG1haWwuY29t

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