
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39237641
71171
10.1038/s41598-024-71171-2
Article
Stress sensitivity analysis of vuggy porous media based on two-scale fractal theory
Huang Zhaoqin emc.group.upc@outlook.com

12
Zhou Xu 12
Wang Hao 12
Wang Qi 3
Wu Yu-Shu 4
1 https://ror.org/05gbn2817 grid.497420.c 0000 0004 1798 1132 State Key Laboratory of Deep Oil and Gas, China University of Petroleum (East China), Qingdao, 266580 China
2 https://ror.org/05gbn2817 grid.497420.c 0000 0004 1798 1132 School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, 266580 China
3 https://ror.org/02awe6g05 grid.464414.7 0000 0004 1765 2021 Research Institute of Petroleum Exploration and Development, CNPC, Beijing, 100007 China
4 https://ror.org/04raf6v53 grid.254549.b 0000 0004 1936 8155 Department of Petroleum Engineering, Colorado School of Mines, Golden, CO 80401 USA
5 9 2024
5 9 2024
2024
14 2071023 6 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Interparticle pore space and vugs are two different scales of pore space in vuggy porous media. Vuggy porous media widely exists in carbonate reservoirs, and the permeability of this porous media plays an important role in many engineering fields. It has been shown that the change of effective stress has important effects on the permeability of vuggy porous media. In this work, a fractal permeability model for vuggy porous media is developed based on the fractal theory and elastic mechanics. Besides, a Monte Carlo simulation is also implemented to obtain feasible values of permeability. The proposed model can predict the elastic deformation of the fractal vuggy porous media under loading stress, which plays a crucial role in the variations of permeability. The predicted permeability data based on the present fractal model are compared with experimental data, which verifies the validity of the present fractal permeability model for vuggy porous media. The parameter sensitivity analysis indicates that the permeability of stress-sensitivity vuggy porous media is related to the capillary fractal dimension, capillary fractal tortuosity dimension, Young’s modulus, and Poisson’s ratio.

Keywords

Vuggy porous media
Stress sensitivity analysis
Permeability
Fractal
Monte Carlo simulation
Subject terms

Energy science and technology
Engineering
Mathematics and computing
National Nature Science Foundation of China52074336 CNPC Science and Technology Major ProjectRIPED-2022-JS-1563 issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Carbonate reservoirs hold a significant position among the reservoirs discovered worldwide1,2. The carbonate reservoir deposition process is complex, and at the same time by the stripping effect and other influences, the carbonate reservoir pore space is rich and non-homogeneous, which leads to the carbonate reservoir being very complex3. Permeability can directly reflect the seepage characteristics of a reservoir and is a crucial parameter for predicting its oil and gas production. The decrease in permeability caused by an increase in effective stress significantly impacts reservoir development. Thus, studying the stress sensitivity characteristics of permeability is essential.

During the development of oil and gas reservoirs, with the continuous output of fluids in the reservoir, the formation pressure gradually decreases and the effective stress on the rock skeleton increases. The increasing effective stress compresses the rock, which leads to a decrease in pore connectivity and a decrease in reservoir permeability4. The phenomenon that reservoir permeability decreases with increasing effective stress is called permeability stress sensitivity5–7. This phenomenon is widespread in both natural and man-made materials8,9. The study of stress sensitivity of permeability in porous media spans a wide range of scientific and engineering fields, including hydraulics10, physics11, and petroleum engineering12. There are many factors affecting permeability stress sensitivity, and stress sensitivity has a significant impact on the development of oil and gas reservoirs3. Many scholars have investigated the effect of stress sensitivity on reservoir permeability through experiments, theoretical models, and numerical simulations.

Numerical methods play an important role in the analysis of flow in porous media considering stress sensitivity. Faisal et al.13 numerically predicted the variation of elastic properties of carbonate rocks as a function of stress and improved the accuracy of the numerical prediction using a multiscale imaging method and an "up-scaling" framework. Civan14 established a mathematical model to express the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes, this model can depict the stress dependency of the porosity and permeability of porous rocks, and he introduced the Biot–Willis poroelastic coefficient to construct the equation of net confining pressure and make the analysis results more accurate. Al Balushi et al.15 used micro-computed Tomography images to simulate the stress-induced deformation of rocks and used the lattice Boltzmann method to simulate the fluid flow in the deformed medium. Ahmed et al.16 established a rock mechanics model combined with geomechanical modeling and performed reservoir geomechanical simulations of a carbonate gas reservoir to analyze the change of reservoir permeability with effective stress. Quevedo et al.17 predicted the change of reservoir permeability around a carbonate fault by numerical simulation based on the finite element method, combined with an elastic–plastic model and fault damage data. Fu et al.18 digitally imaged the pressurization and depressurization processes of pore, pore–pore, and pore–pore cavities in carbonate rocks by using X-ray tomography, and simulated and predicted the permeability under different peritectic pressures by using the lattice Boltzmann method and pore network model. However, the actual rock pore structure is complex, and the numerical method requires more cumbersome calculations and accurate modeling, which limits its ability to analyze the permeability of highly inhomogeneous reservoirs.

Numerous scholars have conducted indoor experiments to study the relationship between reservoir permeability and effective stress19–24. However, obtaining permeability data of seam-and-hole carbonate reservoirs through experimental methods has its limitations. The cores used in indoor experiments are usually ordinary core samples with a diameter of 1.5 inches, and the core samples only cover a small portion of the reservoir section, which makes it difficult to restore the actual situation of pore structure distribution in carbonate reservoirs. Therefore, it is of great practical significance to study the stress-related permeability of porous media using theoretical methods. The microstructure of porous media is disordered and extremely complex, and it is more difficult to consider its stress sensitivity, so fractal theory can be introduced to analyze the stress sensitivity of flow and permeability in porous media25–31. Luo et al.32 established a dual permeability calculation model considering stress-induced fracture closure by introducing the fractal dimension of the rock matrix and the curvature of the aperture surface of the fracture network. Miao et al.33 predicted the change of permeability and porosity of fractured rock with stress based on fractal theory and Hooke's model. Ge et al.34 proposed a new permeability model based on micro- and nano-scale discrete pore structures based on fractal theory and analyzed the effect of effective stress permeability. Tian et al.35 proposed a bi-fractal permeability model to quantitatively study the effect of coal internal structure on permeability and considered the effect of effective stress and matrix shrinkage evolution on permeability. Jin et al.36 constructed a bound water saturation model, a permeability model, and a relative permeability model based on the capillary bundle model and the fractal theory while considering the effects of water film and stress sensitivity. Currently, some progress has been made in the study of permeability stress sensitivity based on fractal theory, but most of the studies focus on conventional reservoirs. The permeability stress sensitivity characteristics of vuggy porous media lack comprehensive research and analysis.

In this paper, a two-scale fractal permeability model for vuggy porous media is established, which considers the elastotic deformation of vuggy porous media under stress sensitivity conditions. The proposed fractal model, based on the two-scale fractal theory, takes into account both interparticle pores and vugs. Different matching relations between capillary tubes and vugs are examined to predict the permeability of a vuggy porous medium. Additionally, using a set of random match relations, the most likely predicted permeability of an actual vuggy rock core is calculated based on the Monte Carlo method. The model’s accuracy is verified by real experimental data. And the influence of Young’s modulus, Poisson’s ratio, capillary fractal tortuosity dimension, capillary fractal dimension, and vug fractal dimensions on the models are analyzed.

The conceptual two-scale fractal model

Vuggy carbonate rock is a special type of carbonate rock, and when analyzing this type of carbonate rock, the influence of the vug system and matrix system on their permeability is mainly considered. As shown in Fig. 1a, The core image of this type of carbonate rock reveals vugs of varying sizes caused by dissolution. The matrix is relatively dense, exhibiting certain levels of porosity and permeability.Fig. 1 Conceptual model schematic diagram: (a) Dissolution vuggy reservoirs outcrop (b) Simplified vuggy porous model.

The storage and seepage spaces in vuggy carbonate rocks are highly complex, making it difficult to accurately describe their microstructural details using traditional methods. Based on these characteristics, a two-scale fractal conceptual model of vuggy porous media is established, as shown in Fig. 1b. For matrix pores, there are two main fractal structural features: the fractal distribution of pore diameters and the fractal tortuosity of fluid flow paths within the pores. Both structural features can be described by fractal theory. For vugs, the size distribution follows the fractal scaling law. Although the sizes of capillaries and vugs are not on the same scale, their size distributions both conform to the fractal scaling law.

To consider the effect of stress on the permeability of vuggy porous media, this paper reasonably simplifies the actual problem and makes the following reasonable assumptions on the above conceptual model:

For the capillary bundle: (1) The fluid channels in the matrix of porous medium can be seen as curved capillaries with different radii. (2) When the porous medium sample is under pressure, the internal capillary flow channel is uniformly stressed. (3) The total number of the capillary of the porous media remains the same after deformation. (4) The stress–strain and the fluid flow of the porous medium are steady state. (5) A porous medium is an ideal elastic body. Based on this assumption, the thick-walled cylinder model can be used to analyze the deformation of the single capillary caused by stress. As shown in Fig. 2, where λ is the inner radius of the capillary, and tλ is the outer radius, the pressure of the fluid on the inside of the capillary is referred to as Pi, and the outside pressure is subjected to the Po.Fig. 2 Force on a single capillary under elastic deformation: (a) Single capillary (b) Single capillary cross-section.

In the capillary model, the stress-induced change in the permeability of a porous medium can be characterized by the elastic deformation of the capillary cross-sectional area. According to the theory of mechanics of materials37, for the thick-walled cylinder model, the displacement at any radius can be expressed as:1 ur=1-νEpi-t2pot2-1r+1+νEpi-pot2t2-1λ2r,

where E is the Young’s modulus, ν is the Poisson’s ratio.

Therefore, the deformations of the inner surface of the capillary can be expressed as,2 uλ=1-νEpi-t2pot2-1+1+νEpi-pot2t2-1λ=Ceλλ,

3 Ceλ=piEt2+1t2-1+ν-poE2t2t2-1.

Under stress elastic deformation conditions, the radius of the inner surface of the capillary considering the stress can be expressed as,4 λσ=1+Cλeλ0,

where λ0 is the radius of the inner surface of the capillary when the stress equals 0.

For the vugs: (1) The vugs of the porous medium can be modeled as hollow spheres with varying radii. (2) When the porous medium sample is under pressure, the internal vugs experience uniform stress. (3) The total number of the vugs of the porous media remains unchanged after deformation. (4) The stress–strain relationship and the fluid flow of the porous medium are steady state. (5) The porous medium is assumed to be an ideal elastic body. Based on this assumption, the thick-walled hollow sphere model is employed to analyze the deformation of a single vug under stress. As shown in Fig. 3, where a is the inner radius of the capillary, and ta is the outer radius, the pressure of the fluid inside of the vug is denoted as Pi, and the external pressure is Po.Fig. 3 Force on a single vug under elastic deformation: (a) Single vug (b) Single vug cross-section.

According to material mechanics36, the displacement at hollow sphere radius u(rh) can be expressed as5 u(rh)=12E(t3-1)rh2[2(pi-pot3)(1-2ν)rh3+(pi-po)(1+ν)t3a3],

Therefore, the deformations of the inner surface of the vug can be expressed as6 u(a)=12E(t3-1)[2(pi-pot3)(1-2ν)+(pi-po)(1+ν)t3]a=Caea,

7 Cae=12E(t3-1)[2(pi-pot3)(1-2ν)+(pi-po)(1+ν)t3].

Under stress elastic deformation conditions, the radius of the inner surface of the vug considering the stress can be expressed as,8 aσ=1+Caea0,

where a0 is the radius of the inner surface of the vug when the stress equals 0.

The stress-sensitive permeability model of fractal vuggy porous media

According to the fractal theory38, the fractal scaling law of the capillary can be rewritten as,9 Nλl>λσ=λσmaxλσDf,

where Nλ is the number of capillaries, l is a certain length of the capillary, λ is a determined radius of the capillary, λσ max is the maximum pore radius in the porous media sample, and Df is the number of fractal dimensions. When it is a two-dimensional space, 0 < Df < 2, and when it is a three-dimensional space,0 < Df < 3.

Equation (9) can be regarded as a continuous and differentiable function. The number of capillaries between the λ and dλ can be obtained by differentiating,10 -dNλ=Dfλσmaxfλσ-Df+1dλσ.

Based on the fractal scaling law, the capillary length when considering the tortuosity can be expressed as,11 Lpσ=2λσ1-DTLσT,

where Lpσ is the length of the capillary, DT is the capillary tortuosity fractal dimension, and Lσ is the characteristic length, whose value is the same as the core sample’s length in this study.

The same, the fractal scaling law of the vugs can be rewritten as,12 Nal>aσ=aσmaxaσDa,

where Na is the number of vugs, a is the determined radius of the vug, aσ max is the maximum vug radius in the porous media sample, and Da is the vug fractal dimension.

In the study of fluid flow in vuggy carbonate rocks, the flow of fluids in the matrix system and the vug system is mainly considered. In the fractal conceptual model, the matrix system is regarded as capillary tubes with different radii, and the vug system is simplified as spheres with different radii. Existing research results show that in capillary vuggy porous media, the vugs connected with the capillary can be regarded as an equipotential volume, which means the influence of vugs on the capillary can be seen as the vugs reduce the length of the capillary.

As shown in Fig. 4, in the vuggy carbonate reservoir, one vug may equally affect many capillaries, and each capillary is not only affected by one vug, so it is necessary to add parameters to consider the influence of different numbers of dissolution pores on the capillaries. In addition, when considering the effect of stress on the permeability of the vuggy porous media, to simplify the problem more, this paper considers the vug and capillary separately and studies the deformation of the capillary model and the spherical shell model when they are subjected to stress, respectively, with the outer pressure of the model being the external pressure exerted on the core, and the inner pressure of the model being the fluid pressure inside the core.Fig. 4 The connection between capillaries and vugs: (a) One vug connects with many capillaries (b) One capillary connects with many vugs.

Therefore, the capillary length considering the effect of vugs can be expressed as:13 Lpσ=Lσ-2nβaσDT2λσ1-DT,

where λσ is the radius of the capillary; a is the radius of the vug, Lσ is the length of the capillary, DT is the tortuosity fractal dimension of the capillary; and n is the number of vugs that affect the capillary. β indicates the degree of influence of vugs on capillary length, if the vug is a regular sphere and the diameter of the vug just coincides with the capillary, the impact of the vug on the capillary reaches the maximum value, and at this time, β = 1.

The fluid flow through a curved capillary can be described according to the Hagen-Poisenille equation:14 qσ=π8ΔpμLpσλσ4,

where λσ is the radius of the capillary, Lpσ is the length of the capillary, μ is the viscosity of the fluid, Δp is the pressure difference between the two ends of the capillary.

Therefore, the total flow of the entire fractal porous medium can be obtained through integrating Eq. (14)15 Qσ=-∫λσminσmaxqσdN=∫λσminσmaxπλσ48μΔpLσ-2nβaσDT2λσ1-DTDfλσmaxfλσ-Df+1dλσ.

The permeability of porous media can be expressed by Darcy’s law,16 Kσ=QσμLσAσΔp.

According to Hooke’s Law, the radius of the porous media sample considering the stress can be expressed as:17 Rσ=1+σER0,

where σ is the stress, E is Young’s modulus, and R0 is the radius of the porous media sample when the stress equals 0.

The area of the porous media sample considering the cross-section stress can be expressed as,18 Aσ=πRσ2=π1+σE2R02.

The sample length considering the stress can be expressed as,19 Lσ=1-1νσEL0.

Substituting Eq. (15) into Eq. (16), the expression of permeability of capillary vuggy porous media can be obtained.20 Kσ=DfλσmaxfLσ24-DTR021+σE2∫λσminσmaxλσ4-Df+DTLσ-2nβaσDTdλσ.

Vuggy porous media exhibit a certain degree of heterogeneity, and the spatial relationship between vugs and matrix capillaries is complex, making accurate description challenging. However, through theoretical analysis, the theoretical maximum and minimum permeability of vuggy porous media can be determined by idealizing the relationship between the vugs and matrix capillaries.

The maximum permeability model

Under the condition that the total number of vugs and capillaries in the vuggy porous media is constant, if the vug has the greatest influence on the fluid flow in the porous medium, the vug with the largest radius should have an influence on the n capillaries with the largest radius and the vug with the smallest radius affects the n capillaries with the smallest radius.

Equation (9) also means sorting the capillary from largest to smallest. the capillary whose radius is closest to λ and also is greater than λ ranked nth. The same relationship of vugs can be obtained from Eq. (12).

Therefore, the relationship that the largest vug impacts the n largest aperture capillary and that the smallest vug impacts the n smallest aperture capillary can be expressed as,21 Nal>aσ=nNλl>λσ.

Substituting Eqs. (9) and (12) into Eq. (21), the relationship of capillary aperture λ and vug radius a can be obtained.22 aσ=aσmaxn-1Daλσmax-DfDaλσDfDa.

Substituting Eq. (22) into Eq. (20), we can get the expression of permeability of porous media23 Kσmax=DfλσmaxfLσ24-DTR021+σE2∫λσminσmaxλσ4-Df+DTLσ-2nβaσmaxn-1Daλσmax-DfDaλσDfDaDTdλσ.

The minimum permeability model

Conversely, if the vug has the least effect on the fluid flow in the medium, the cave with the largest radius should have an effect on the n capillaries with the smallest radius, and the vug with the smallest radius should have an effect on the n capillaries with the largest radius.

The relationship between capillary aperture λ and vug radius can be expressed as:24 Nal>aσ=nNλl>λσmin-Nλl>λσ+1.

Substituting Eqs. (9) and (12) into Eq. (24), the relationship of capillary radius λ and vug radius a can be obtained.25 aσ=aσmaxn-1DaλσmaxλσminDf-λσmaxλσDf+1-1Da.

Substituting Eq. (25) into Eq. (20), we can get the expression of permeability of porous media,26 Kσmin=DfλσmaxfLσ24-DTR021+σE2∫λσminσmaxλσ4-Df+DTLσ-2nβaσmaxn-1DaλσmaxλσminDf-λσmaxλσDf+1-1DaDTdλσ.

Monte Carlo simulation based on fractal theory

The Monte Carlo simulation method is a kind of random sampling calculation method with probability theory and statistical theory as the basic theory, between the theoretical method and the real reality, which can simulate the actual physical process more realistically and effectively, and has a wide range of cross-applications in the fields of physics, chemistry, economics, and engineering technology. The basic idea is to establish a stochastic process to describe the randomness of the variables in the actual model and to predict the statistical characteristics of the required parameters by random sampling tests on the model to give an approximate solution of the required parameters. Since the pore distribution in porous media is random and proven to satisfy the fractal scalar law, fractal theory and Monte Carlo simulation methods are naturally combined.

In this paper, the prediction of permeability of vuggy porous media considering the stress case is analyzed by the Monte Carlo simulation method based on the previous studies and combined with the above proposed fractal model.

Monte Carlo characterization of the matrix and vugs

The probability density function of the vug fa satisfies the normalization condition27 ∫aminmaxf(a)da=1-aminamaxDa≡1.

The vug fractal dimension Da is always greater than one: Da > 1, which means the second equal sign in Eq. (27) exists only when the condition amin <  < amax is satisfied. Generally, the actual vug satisfied the condition aminamax<10-2, So Eq. (27) is correct in the actual application.

According to Eq. (27), The cumulative probability of vug size within the range of any vug can be expressed as28 W(a)=∫aminaf(a)da=1-aminaDa.

It can be seen from Eqs. (27) and (28) that when the vug radius a infinitely approaching the amin, W(a)≈0 when the vug radius a infinitely approaching the amax, W(a)≈1, which means the value of the W(a) is a random number from 0 to 1.

The Eq. (28) can be also written as,29 1-Wa=aminaDa.

Therefore, the value of amina is also a random number from 0 to 1. The expression of the vug diameter can be expressed as30 a=amin(1-Wa)(1/Da)=aminamaxamax(1-Wa)(1/Da).

The vug radius a can be replaced by ai in the probabilistic model of random vug radius:31 ai=aminamaxamax1-Wia(1/Da),

where i = (1, 2, 3, …, N), N is the total number of Monte Carlo simulations.

The capillary radius λ in the probabilistic model of random capillary radius can be obtained in the same way:32 λi=λminλmaxλmax1-Wiλ(1/Df),

where Wi(λ) is the cumulative probability of capillary radius within the range of any capillary in the ith Monte Carlo simulation. λmin is the minimum capillary radius, λmax is the maximum capillary radius, Df is the capillary fractal number.

Monte Carlo algorithm workflow

Based on the fractal permeability model illustrated in Sect. "The conceptual two-scale fractal model" the permeability of vuggy porous media with fractal Monte Carlo method can be determined. As shown in Fig. 5, the algorithm for the Fractal-Monte Carlo method is summarized as follows.Fig. 5 Fractal-Monte Carlo algorithm flow chart.

Results and discussions

In this part, the validity of the proposed fractal model is verified by the experimental data. Then, the effect of the key parameters of the vuggy porous media is analyzed.

Experimental validation

The correctness of the fractal model proposed in this paper to predict the elastic deformation and permeability change of vuggy porous media under stress conditions is verified. In this paper, two groups of representative dissolution pore-type carbonate rock cores are selected, they are shown in Figs. 6a and 8a. The cores are scanned by CT, and mechanical experiments are conducted to obtain the relevant parameters required by the model, permeability stress sensitivity experiments are conducted on the two groups of cores respectively, and the experimental results are compared with the model prediction results (Fig. 7 and Fig. 9).Fig. 6 Outcrop core and CT scan image, Example 1: (a) Sample 1 (b) Three-dimensional CT scan image (c) Processed CT image.

Fig. 7 A comparison between model predictions and experimental data, Example 1: (a) Fractal-Monte Carlo simulation results of first data (b) Comparison between model and experimental.

Figures 6b and 8b represent three-dimensional CT scan images. Figures 6c and 8c are corresponding processed CT images of samples. To be more consistent with the actual pore structure of the core, the fractal dimensions in multiple CT scans are averaged. From the CT image of the sample, the number of the vug fractal dimension Da can measured as 1.65 and 1.75, respectively, the measurement method is the box dimension method. From the CT image of the matrix of samples, the number of fractal dimensions Df can measured as 1.35 and 1.3, respectively. Figures 7a and 9a show the simulated results of 1000 times the permeability of the vuggy porous media simulated by the Monte Carlo simulation results of the first data. The final experimental results are shown in Figs. 7b and 9b. The experimental data of the permeability of the carbonate rock medium are represented by the red remark. In Figs. 7c and 9c, the maximum permeability model Kmax is represented by a blue solid line, and the minimum permeability model Kmin is represented by a black solid line. Monte Carlo simulation results based on these experimental data are represented by an orange solid line. It can be seen from these two pictures that our Fractal-Monte Carlo method of vuggy porous media shows a great agreement with the experiment data.Fig. 8 Outcrop core and CT scan image, Example 2: (a) Sample 2 (b) Three-dimensional CT scan image (c) Processed CT image.

Fig. 9 A comparison between model predictions and experimental data, Example 2: (a) Fractal-Monte Carlo simulation results of first data (b) Comparison between model and experimental.

Parameter sensitivity analysis

In the following, the effect of the structural parameters (Young’s modulus E and Poisson’s ratio v) on the normalized permeability is analyzed. The relationships between the permeability and the effective stress are shown in Figs. 10 and 11.Fig. 10 The predicted permeability vs effective stress with the different Young’s modulus: (a) Fractal theory (b) Fractal-Monte Carlo method.

Fig. 11 The predicted permeability vs effective stress with the different Poisson's ratio: (a) Fractal theory (b) Fractal-Monte Carlo method.

It shows from Figs. 10 and 11, that with the effective stress increasing gradually, the permeability decreases rapidly. This is because the increase of the effective stress implies the cross-sectional area of the pore and throat decreasing, leading to the decrease of flowing space and the increase of flowing distance. Figure 10 also shows the permeability varies with the effective stress at different Young’s modulus E. Figure 10 depicts that the permeability increased with the higher value of Young’s modulus E. Figure 10 depicts that the permeability increased with the higher value of Young’s modulus E, and this can be interpreted as that the higher Young’s modulus E brings the higher resistance to the capillary bundles. Figure 11 reveals that the higher Poisson’s ratio ν, the higher permeability at the same effective stress condition. This phenomenon attributed to the higher Poisson’s ratio ν leads to a smaller increase in the flowing distance.

Figures 12 and 13 show the permeability of the porous media is affected by the capillary fractal dimension Df and capillary fractal tortuosity dimension DT. It can be seen from Fig. 13 that both the max permeability model Kmax and the minimum permeability model Kmin increase with the increase of the capillary fractal dimension Df. This can be explained by that the capillary fractal dimension is related to the cross-sectional distribution of the capillary in porous media, the increase of the capillary fractal dimension means an increase in the number of the capillary. Under the condition of constant cross-sectional area, the increase in capillary number will lead to an increase in the conductivity of the capillary network, which increases the permeability of the porous media.Fig. 12 The predicted vs effective stress with the different capillary fractal tortuosity dimensions DT: (a) Fractal theory (b) Fractal-Monte Carlo method.

Fig. 13 The predicted permeability vs effective stress with the different capillary fractal dimensions Df: (a) Fractal theory (b) Fractal-Monte Carlo method.

Figure 12 reveals the effect of the capillary fractal tortuosity dimension DT on the vuggy porous media. It can be seen from Fig. 12 that with the increase of tortuosity fractal dimension of the capillary, the prediction value of the maximum and minimum permeability prediction model decreases rapidly: under the same other conditions, the value of max permeability model Kmax when the tortuosity fractal dimension DT = 1.2 is even lower than the permeability of the minimum permeability model Kmin when the tortuosity fractal dimension DT = 1.1.

It can be seen from Fig. 14 that under the same effective stress conditions, a larger fractal dimension of dissolution pores corresponds to higher permeability. This is because a larger fractal dimension indicates a greater number of dissolved pores, which results in less tortuosity of the capillaries within the porous medium and shorter capillary paths. Consequently, the fluid flow path through the capillaries is reduced, leading to an increase in permeability.Fig. 14 The predicted permeability vs effective stress with the different vug fractal dimensions Da: (a) Fractal theory (b) Fractal Monte Carlo method.

Conclusions

A novel fractal predictive model has been developed for the permeability of vuggy porous media considering the stress. Compared with the other available stress sensitivity permeability models, this work shows better accuracy in highly effective stress conditions and can be able to predict permeability change due to the elastic deformation of the fractal vuggy porous media under loading stress. Each parameter in the models has a clear physical meaning. The models are verified by experimental data. The sensitivity analysis of the influencing factors of the models has also been done and the results show that Young’s modulus E and Poisson’s ratio v play significant roles in stress-depend permeability. The Parameter sensitivity analysis shows that the permeability of vuggy porous media increased with the higher value of Young’s modulus, Poisson’s ratio, capillary fractal tortuosity dimensions, capillary fractal dimensions, and vug fractal dimensions. However, the model also has limitations. For example, it ignores the plastic deformation of the fractal vuggy porous. Besides, the model also neglects the change in the number of the capillary and vug of the vuggy porous media. More work can be done in this aspect in the future.

Acknowledgements

The authors would like to thank the support of the National Nature Science Foundation of China (52074336, 52034010) and CNPC Science and Technology Major Project (RIPED-2022-JS-1563, ZD2019-183-008-001).

Author contributions

All authors made a substantial, direct, and intellectual contribution to the work. Z.H., X.Z. and H.W. wrote the manuscript and designed the figures. Q.W. and Y.W. supervised the manuscript. All authors reviewed the manuscript.

Data availability

The data that support the findings of this study is provided within the manuscript.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Jia AL Yan HJ Guo JL Characteristics and experiences of the development of various giant gas fields all over the world Nat. Gas Ind. 2014 34 10 33 46
Jia, A. L. et al. Characteristics and experiences of the development of various giant gas fields all over the world. Nat. Gas Ind. 34(10), 33–46 (2014).
2. Sun Y Lu J Liu H Study on the development laws of large-scale carbonate gas reservoirs at home and abroad Nat. Gas Explor. Dev. 2017 40 04 59 64
Sun, Y. et al. Study on the development laws of large-scale carbonate gas reservoirs at home and abroad. Nat. Gas Explor. Dev. 40(04), 59–64 (2017).
3. Durrani MZA Talib M Ali A Characterization of carbonate reservoir using post-stack global geostatistical acoustic inversion approach: A case study from a mature gas field, onshore Pakistan J. Appl. Geophys. 2021 188 104313 10.1016/j.jappgeo.2021.104313
Durrani, M. Z. A. et al. Characterization of carbonate reservoir using post-stack global geostatistical acoustic inversion approach: A case study from a mature gas field, onshore Pakistan. J. Appl. Geophys. 188, 104313 (2021).10.1016/j.jappgeo.2021.104313
4. Zhong X Zhu Y Liu L The characteristics and influencing factors of permeability stress sensitivity of tight sandstone reservoirs J. Petrol. Sci. Eng. 2020 191 107221 10.1016/j.petrol.2020.107221
Zhong, X. et al. The characteristics and influencing factors of permeability stress sensitivity of tight sandstone reservoirs. J. Petrol. Sci. Eng. 191, 107221 (2020).10.1016/j.petrol.2020.107221
5. Tan X-H Li X-P Liu J-Y Study of the effects of stress sensitivity on the permeability and porosity of fractal porous media Phys. Lett. A 2015 379 39 2458 2465 10.1016/j.physleta.2015.06.025
Tan, X.-H. et al. Study of the effects of stress sensitivity on the permeability and porosity of fractal porous media. Phys. Lett. A 379(39), 2458–2465 (2015).10.1016/j.physleta.2015.06.025
6. Xiao Y Shengbin F Jiong W Stress sensitivity and its influence factors of tight oil reservoir in Chang Member, Ordos Basin China Petrol. Explor. 2017 22 5 64
Xiao, Y. et al. Stress sensitivity and its influence factors of tight oil reservoir in Chang Member, Ordos Basin. China Petrol. Explor. 22(5), 64 (2017).
7. Yang Y Zhuang D Ding G Observation and description of the shape of water bridge retaining between vertical plane surfaces Int. J. Refrig. 2016 64 20 31 10.1016/j.ijrefrig.2016.01.009
Yang, Y. et al. Observation and description of the shape of water bridge retaining between vertical plane surfaces. Int. J. Refrig. 64, 20–31 (2016).10.1016/j.ijrefrig.2016.01.009
8. Jeong H-Y A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices Int. J. Solids Struct. 2002 39 5 1385 1403 10.1016/S0020-7683(01)00260-8
Jeong, H.-Y. A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices. Int. J. Solids Struct. 39(5), 1385–1403 (2002).10.1016/S0020-7683(01)00260-8
9. Worthington PF A diagnostic approach to quantifying the stress sensitivity of permeability J. Petrol. Sci. Eng. 2008 61 2–4 49 57 10.1016/j.petrol.2008.03.003
Worthington, P. F. A diagnostic approach to quantifying the stress sensitivity of permeability. J. Petrol. Sci. Eng. 61(2–4), 49–57 (2008).10.1016/j.petrol.2008.03.003
10. Zhang HJ Jeng D-S Barry DA Solute transport in nearly saturated porous media under landfill clay liners: A finite deformation approach J. Hydrol. 2013 479 189 199 10.1016/j.jhydrol.2012.11.063
Zhang, H. J. et al. Solute transport in nearly saturated porous media under landfill clay liners: A finite deformation approach. J. Hydrol. 479, 189–199 (2013).10.1016/j.jhydrol.2012.11.063
11. Neto LB Kotousov A Bedrikovetsky P Elastic properties of porous media in the vicinity of the percolation limit J. Petrol. Sci. Eng. 2011 78 2 328 333 10.1016/j.petrol.2011.06.026
Neto, L. B., Kotousov, A. & Bedrikovetsky, P. Elastic properties of porous media in the vicinity of the percolation limit. J. Petrol. Sci. Eng. 78(2), 328–333 (2011).10.1016/j.petrol.2011.06.026
12. Nicolas V Salagnac P Glouannec P Modelling heat and mass transfer in deformable porous media: Application to bread baking J. Food Eng. 2014 130 23 35 10.1016/j.jfoodeng.2014.01.014
Nicolas, V. et al. Modelling heat and mass transfer in deformable porous media: Application to bread baking. J. Food Eng. 130, 23–35 (2014).10.1016/j.jfoodeng.2014.01.014
13. Faisal TF Islam A Jouini MS Numerical prediction of carbonate elastic properties based on multi-scale imaging Geomech. Energy Environ. 2019 20 100125 10.1016/j.gete.2019.100125
Faisal, T. F. et al. Numerical prediction of carbonate elastic properties based on multi-scale imaging. Geomech. Energy Environ. 20, 100125 (2019).10.1016/j.gete.2019.100125
14. Civan F Stress-dependent porosity and permeability of porous rocks represented by a mechanistic elastic cylindrical pore-shell model Transport Porous Media 2019 129 3 885 899 10.1007/s11242-019-01311-0
Civan, F. Stress-dependent porosity and permeability of porous rocks represented by a mechanistic elastic cylindrical pore-shell model. Transport Porous Media 129(3), 885–899 (2019).10.1007/s11242-019-01311-0
15. Al Balushi F Taleghani AD Digital rock analysis to estimate stress-sensitive rock permeabilities Comput. Geotech. 2022 151 104960 10.1016/j.compgeo.2022.104960
Al Balushi, F. & Taleghani, A. D. Digital rock analysis to estimate stress-sensitive rock permeabilities. Comput. Geotech. 151, 104960 (2022).10.1016/j.compgeo.2022.104960
16. Ahmed BI Al-Jawad MS Geomechanical modeling and two-way coupling simulation for carbonate gas reservoir J. Petrol. Explor. Prod. Technol. 2020 10 8 3619 3648 10.1007/s13202-020-00965-7
Ahmed, B. I. & Al-Jawad, M. S. Geomechanical modeling and two-way coupling simulation for carbonate gas reservoir. J. Petrol. Explor. Prod. Technol. 10(8), 3619–3648 (2020).10.1007/s13202-020-00965-7
17. Quevedo R de Andrade TJ Santos L Assessment of fault damage zones in carbonate rocks based on numerical and sensitivity analyses Tectonophysics 2023 864 230023 10.1016/j.tecto.2023.230023
Quevedo, R. et al. Assessment of fault damage zones in carbonate rocks based on numerical and sensitivity analyses. Tectonophysics 864, 230023 (2023).10.1016/j.tecto.2023.230023
18. Fu S Zhang L Li Y Influence of stress sensitivity on water-gas flow in carbonate rocks Geofluids 2020 2020 1 12
Fu, S. et al. Influence of stress sensitivity on water-gas flow in carbonate rocks. Geofluids 2020, 1–12 (2020).
19. Fernandes FB Braga AMB de Souza ALS Mechanical formation damage control in permeability Biot’s effective stress-sensitive oil reservoirs with source/sink term J. Petrol. Sci. Eng. 2023 220 111180 10.1016/j.petrol.2022.111180
Fernandes, F. B. et al. Mechanical formation damage control in permeability Biot’s effective stress-sensitive oil reservoirs with source/sink term. J. Petrol. Sci. Eng. 220, 111180 (2023).10.1016/j.petrol.2022.111180
20. Moradi M Shamloo A Asadbegi M Three dimensional pressure transient behavior study in stress sensitive reservoirs J. Petrol. Sci. Eng. 2017 152 204 211 10.1016/j.petrol.2017.02.017
Moradi, M. et al. Three dimensional pressure transient behavior study in stress sensitive reservoirs. J. Petrol. Sci. Eng. 152, 204–211 (2017).10.1016/j.petrol.2017.02.017
21. Cheng Y Chunqiu GUO Pengyu C Stress sensitivity of carbonate gas reservoirs and its microscopic mechanism Petrol. Explor. Dev. 2023 50 1 166 174 10.1016/S1876-3804(22)60377-X
Cheng, Y. et al. Stress sensitivity of carbonate gas reservoirs and its microscopic mechanism. Petrol. Explor. Dev. 50(1), 166–174 (2023).10.1016/S1876-3804(22)60377-X
22. Shovkun I Espinoza DN Coupled fluid flow-geomechanics simulation in stress-sensitive coal and shale reservoirs: Impact of desorption-induced stresses, shear failure, and fines migration Fuel 2017 195 260 272 10.1016/j.fuel.2017.01.057
Shovkun, I. & Espinoza, D. N. Coupled fluid flow-geomechanics simulation in stress-sensitive coal and shale reservoirs: Impact of desorption-induced stresses, shear failure, and fines migration. Fuel 195, 260–272 (2017).10.1016/j.fuel.2017.01.057
23. Wang H Ji B Lv C The stress sensitivity of permeability in tight oil reservoirs Energy Explor. Exploit. 2019 37 4 1364 1376 10.1177/0144598719855819
Wang, H. et al. The stress sensitivity of permeability in tight oil reservoirs. Energy Explor. Exploit. 37(4), 1364–1376 (2019).10.1177/0144598719855819
24. Wang H Zhou Q Sheng J Effect of long-term infiltration on porosity-permeability evolution in carbonate rocks: An online NMR coupling penetration test J. Hydrol. 2023 617 129029 10.1016/j.jhydrol.2022.129029
Wang, H. et al. Effect of long-term infiltration on porosity-permeability evolution in carbonate rocks: An online NMR coupling penetration test. J. Hydrol. 617, 129029 (2023).10.1016/j.jhydrol.2022.129029
25. Yu BM Li JH Zhang DM A fractal trans-plane permeability model for textile fabrics Int. Commun. Heat Mass Transf. 2003 30 1 127 138 10.1016/S0735-1933(03)00014-9
Yu, B. M., Li, J. H. & Zhang, D. M. A fractal trans-plane permeability model for textile fabrics. Int. Commun. Heat Mass Transf. 30(1), 127–138 (2003).10.1016/S0735-1933(03)00014-9
26. Peitgen H-O Jürgens H Saupe D Chaos and Fractals: New Frontiers of Science 2004 Springer 106
Peitgen, H.-O. et al. Chaos and Fractals: New Frontiers of Science 106 (Springer, 2004).
27. Masters BR Fractal analysis of the vascular tree in the human retina Annu. Rev. Biomed. Eng. 2004 6 1 427 452 10.1146/annurev.bioeng.6.040803.140100 15255776
Masters, B. R. Fractal analysis of the vascular tree in the human retina. Annu. Rev. Biomed. Eng. 6(1), 427–452 (2004).15255776 10.1146/annurev.bioeng.6.040803.140100
28. Edgar G Measure, Topology, and Fractal Geometry 2008 Springer New York
Edgar, G. Measure, Topology, and Fractal Geometry (Springer New York, 2008).
29. Falconer K Fractal Geometry: Mathematical Foundations and Applications 2004 Wiley
Falconer, K. Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2004).
30. Xiao B Yu B Wang Z A fractal model for heat transfer of nanofluids by convection in a pool Phys. Lett. A 2009 373 45 4178 4181 10.1016/j.physleta.2009.09.020
Xiao, B. et al. A fractal model for heat transfer of nanofluids by convection in a pool. Phys. Lett. A 373(45), 4178–4181 (2009).10.1016/j.physleta.2009.09.020
31. Shou D Fan J Ding F A difference-fractal model for the permeability of fibrous porous media Phys. Lett. A 2010 374 10 1201 1204 10.1016/j.physleta.2010.01.002
Shou, D., Fan, J. & Ding, F. A difference-fractal model for the permeability of fibrous porous media. Phys. Lett. A 374(10), 1201–1204 (2010).10.1016/j.physleta.2010.01.002
32. Luo X Cheng Y Tan C Calculation method of equivalent permeability of dual-porosity media considering fractal characteristics and fracture stress sensitivity J. Petrol. Explor. Prod. Technol. 2023 13 8 1691 1701 10.1007/s13202-023-01640-3
Luo, X., Cheng, Y. & Tan, C. Calculation method of equivalent permeability of dual-porosity media considering fractal characteristics and fracture stress sensitivity. J. Petrol. Explor. Prod. Technol. 13(8), 1691–1701 (2023).10.1007/s13202-023-01640-3
33. Miao T Chen A Li Z Stress-dependent models for permeability and porosity of fractured rock based on fractal theory Fractals 2023 31 05 2350093 10.1142/S0218348X23500937
Miao, T. et al. Stress-dependent models for permeability and porosity of fractured rock based on fractal theory. Fractals 31(05), 2350093 (2023).10.1142/S0218348X23500937
34. Ge Z Zhang H Zhou Z Pore permeability model based on fractal geometry theory and effective stress J. Energy Resour. Technol. 2023 145 8 081701 10.1115/1.4056890
Ge, Z. et al. Pore permeability model based on fractal geometry theory and effective stress. J. Energy Resour. Technol. 145(8), 081701 (2023).10.1115/1.4056890
35. Tian J Liu J Elsworth D An effective stress-dependent dual-fractal permeability model for coal considering multiple flow mechanisms Fuel 2023 334 126800 10.1016/j.fuel.2022.126800
Tian, J. et al. An effective stress-dependent dual-fractal permeability model for coal considering multiple flow mechanisms. Fuel 334, 126800 (2023).10.1016/j.fuel.2022.126800
36. Yan J Zheng R Chen P Calculation model of relative permeability in tight sandstone gas reservoir with stress sensitivity Geofluids 2021 2021 1 12
Yan, J. et al. Calculation model of relative permeability in tight sandstone gas reservoir with stress sensitivity. Geofluids 2021, 1–12 (2021).
37. Bower AF Applied Mechanics of Solids 2009 CRC Press
Bower, A. F. Applied Mechanics of Solids (CRC Press, 2009).
38. Huang Z Su X Li Y Stress sensitivity analysis of fractal porous media based on the elasto-plastic thick-walled cylinder model Fractals 2021 29 03 2150162 10.1142/S0218348X21501620
Huang, Z. et al. Stress sensitivity analysis of fractal porous media based on the elasto-plastic thick-walled cylinder model. Fractals 29(03), 2150162 (2021).10.1142/S0218348X21501620
