Permeability Model Research Articles

Effect of gravel pack permeability on horizontal well productivity loss under secondary methane hydrate formation: Experimental optimization of 3D randomly distributed mixed sand pack

Horizontal well productivity is limited by secondary hydrate formation (temperature drop results from endothermic of hydrate dissociation) during production. Seepage capacity as one of the key indexes for evaluating production efficiency, and efficiency is declined by sand production. Sand control is achieved by using gravel pack technology (selected sands for hindering the sediments particles migration), which extends the production time. However, forecasting the productivity decline trend from quantifying the permeability of the gravel pack after secondary formation remain challenges. To address these issues, a randomly distributed 3D normalized permeability (Kr) model was developed to describe the seepage capacity of gravel pack composed by various particle sizes of mixed glass beads. The theoretical modelling was established by observing micro spatial structure of glass beads with hydrate formation. The new model considered three consolidation modes of hydrate in pores (cementing + bearing, pore filling, and grain coating) and linked them with the productivity ratio equation of a horizontal gas well. To confirm the viability of the model, a series of permeability measurement experiments with various size ratios (α) from 1 to 10 were conducted. Finally, errors comparison and all parameters of the model were analyzed. The results showed that the productivity ratio declined clearly after saturation is >0.79, when α = 4. For long-term production, α = 10 would be better. The new model can help hydrate horizontal wells increase production from laboratory to industrial exploitation. KeyPoints•Secondary hydrate formation permeability model in gravel pack was established.•After secondary saturation's effect on horizontal well production was developed.•Gravel pack was simulated by random non-coincident particles with varies size ratios.•When the size ratio is 4, productivity ratio of horizontal well was extended.

A novel prediction model of oil-water relative permeability based on fractal theory in porous media

It is significant to accurately evaluate the relative permeability of oil–water two phase for multiphase seepage in porous media in low permeability and tight oil reservoir. However, stress sensitivity is an important characteristic for low permeability and tight oil reservoir. It is an effective way for fractal theory to describe the complexity and heterogeneity of the microstructure of porous media. To describe the relative permeability of oil–water two phase in porous media with complex and irregularity pores, a new relative permeability model oil–water two phases is proposed by the fractal theory and the stress sensitivity is taken into the established model in this paper. Meanwhile, the effects of effective stress, elastic modulus, porosity, maximum and minimum flow radius on oil–water relative permeability are analyzed. The new model is verified by comparing with the laboratory data and the results demonstrate that irreducible water and residual oil saturation have a negative correlation with effective stress. The relative permeability of the oil–water two-phase will shrink to the middle as the rise of effective stress, and the region of co-infiltration will decrease. The deformation quantity of porous media, irreducible water and residual oil saturation will increase as the elastic modulus decreases. The larger the maximum flow radius is, the lower the irreducible water saturation and residual oil saturation is. Both the porosity and the minimum flow radius have slight influences on the relative permeability of oil–water two-phase. The proposed relative permeability model can effectively predict the relative permeability of oil and water and help to describe and reveal the multiphase flow in porous media.

Experimental investigation on methane hydrate-bearing sediment permeability under a wide range of conditions

Depressurization exploitation of hydrate in methane hydrate-bearing sediments (MHBS) causes changes in sediment effective stress, deformation, and hydrate content, often resulting in decreased permeability which reduces gas production efficiency. To investigate the evolution of MHBS permeability subjected to these changing conditions, permeability tests on MHBS of different methane hydrate saturation level were performed during isotropic compression and drained triaxial shearing under a wide range of effective stress conditions. The results obtained from compression tests indicate that with increasing effective stress, MHBS permeability first decreases rapidly and then tends to stabilize, and the influence of hydrate saturation on permeability gradually reduces. The permeability during the triaxial shear process is closely related to volumetric deformation, with shear-contraction specimen exhibiting continuous decrease in permeability and shear-dilation specimens experiencing an increase in permeability. Unique correlation between permeability with effective void ratio (affected by hydrate saturation) was observed at low confining pressure and early stage of shear. However, this relationship becomes nonunique under increased pressure and shear strain. Thus, a unified MHBS permeability model for a range of conditions is established based on the observed relationship between MHBS permeability and effective void ratio, considering the influence of confining stress, shear strain, and the breakage of sediment particles.

Effect Evaluation of Staged Fracturing and Productivity Prediction of Horizontal Wells in Tight Reservoirs

In this paper, the effect evaluation and production prediction of staged fracturing for horizontal wells in tight reservoirs are studied. Firstly, the basic characteristics and value of horizontal wells in tight reservoirs are introduced, their geological characteristics, flow mechanism and permeability model are analyzed and the application of grey theory in effect analysis is discussed. Considering the problems of staged fracturing effect evaluation and the production prediction of horizontal wells in tight reservoirs, a BP neural network model based on deep learning is proposed. Due to the interference of multiple physical parameters and the complex functional relationship in the development of tight reservoir fracturing, the traditional prediction method has low accuracy and it is difficult to establish an accurate mapping relationship. In this paper, a BP neural network is used to simulate multivariable nonlinear mapping by modifying the model, and its advantages in solving the coupling relationship of complex functions are brought into play. A neural network model with fracturing parameters as input and oil and gas production as output is designed. Through the training and testing of data sets, the accuracy and applicability of the proposed model for effect evaluation and yield prediction are verified. The research results show that the model can fit the complex mapping relationship between fracturing information and production and provide an effective evaluation and prediction tool for the development of the staged fracturing of horizontal wells in tight reservoirs.

Application and Research of Multi-Scale Dynamic Diffusion Permeability Model in Different Coal Ranks

The spatial scale of coal mining and rock layer control is distributed between 10-10 and 107, spanning 17 orders of magnitude. There are multi-scale pore cracks from millimeter to micro and nanometers in the coal, and the pore size can reach one million times, which makes the coal permeability also show a multi-scale characteristics of millions and time. For the permeability of the coal seam, it is necessary to test and analyze the distribution of micro and nano pores and cracks in the coal [1]. CBM mining is accompanied by the dynamic change of adsorption pressure in coal and the complex deformation of multi-scale heterogeneous structure of coal, and the multi-scale deformation feature of pore fissure structure plays an important role in the transport of methane [2]. However, existing studies focus on treating coal as a homogeneous isotropic medium, without considering the influence of coal multiscale pore structure on permeability. The variation of heterogeneous degree of coal structure with structural scale and its influence on dynamic diffusion and permeability are rarely reported. Li Zhiqiang et al. [3] conducted an experimental-model-mechanism study of CBM gas micro-nano series multi-scale dynamic diffusion permeability. On this basis, the dynamic diffusion and permeability of different coal scales.

Optimization of an in vitro method for assessing pulmonary permeability of inhaled drugs using alveolar epithelial cells

IntroductionInhalation of drugs for the treatment of pulmonary diseases has been used since a long time. Due to lungs' larger absorptive surface area, delivery of drugs to the lungs is the method of choice for different disorders. Here we present the establishment of a comprehensive permeability model using Type II alveolar epithelial cells and Beclomethasone Dipropionate (BDP) as a model drug delivered by pressurized metered dose inhaler (pMDI). MethodsUsing Type II alveolar epithelial cells, the method was standardized for parameters viz., cell density, viability, incubation period and membrane integrity. The delivery and deposition of drug were using the pMDI device with a Twin Stage Impinger (TSI) modified to accommodate cell culture insert having monolayer of cells. The analytical method for simultaneous estimation of BDP and Beclomathasone-17-Monopropionate (17-BMP) was validated as per the bioanalytical guidelines. The extent and rate of absorption of BDP was determined by quantifying the amount of drug permeated and the data represented by calculating its apparent permeability. ResultsType II alveolar epithelial cells cultured at 0.55 × 105 cells/cm2 for 8–12 days under air-liquid interface were optimized for conducting permeability studies. The data obtained for absorptive transport showed a linear increase in the drug permeated against time for both BDP and 17-BMP along with proportional permeability profile. DiscussionWe have developed a robust in vitro model to study absorptive rate of drug transport across alveolar layer. Such models would create potential value during formulation development for comparative studies and selection of clinical candidates.

Assessing the impact of pore‐fracture structures on permeability sensitivity in tectonic coal using computerized tomography scanning

AbstractThe heterogeneity in permeability of coal reservoirs is primarily attributed to the considerable variation in the morphologies and structures of microscopic pore‐fractures, shaped by complex geological processes. This study emphasizes the necessity of understanding the impact and governance of these morphological and structural variations in pore‐fractures across different types of coal bodies on their permeability. Utilizing computerized tomography scanning and three‐dimensional imaging, we examined coal samples from the Datong coalfield in the southeastern Qinshui Basin, Shanxi Province, to characterize the pore‐fracture morphologies and structures distinct to various coal‐body types based on tomographic data. This introduces a methodology for assessing the influence of microscopic pore‐fracture parameters, such as porosity, specific surface area, tortuosity and fractal dimension, on permeability sensitivity. This is achieved through the application of the modified Kozeny–Carman equation and a fractal permeability model. Findings indicate a predominance of slab fractures in raw coal, whereas fragmented coal under weak brittle deformation exhibits small, isolated pore‐fractures with minimal diameter and volume and poor connectivity. In contrast, granular coal subjected to strong brittle deformation features extensive clusters of large pore‐fractures with significant diameter and volume, enhancing connectivity. Moreover, permeability predictions are refined by integrating the modified Kozeny–Carman equation with tomographic data, offering a more precise understanding of the permeability across different coal bodies.

A Permeability Model for Fractured Granite and its Numerical Implementation

A Permeability Model for Fractured Granite and its Numerical Implementation

A multifractal model for predicting the permeability of dual-porosity media with rough-walled fractures and variable cross-sectional pore channels

The seepage of rock strata is greatly influenced by the pore network and fracture network; however, the prediction of permeability becomes challenging due to the changes in the cross section of pore channels and the morphology of fractures. In this study, a novel pore-fracture permeability model based on a fractal theory is proposed, and the analytical solutions of the model are given. In contrast to the traditional smooth parallel plate and uniform cross section straight capillary, this model not only considers the roughness of the fracture surface, but also the cross section variation and tortuosity of the pore channel. The comparisons between the calculated results and the experimental data verify the reliability of this model. The quantitative analyses of microscopic parameters indicate a positive correlation between the permeability and the fractal dimension, size, and proportion of pores and fractures. Conversely, there is a negative correlation with roughness, tortuosity, and cross-sectional changes. The range in which the seepage contribution of pores cannot be ignored is determined. Two logarithmic relationship expressions for permeability are presented. This study contributes to explore the effects of the geometry and morphology of the pore-fracture media on seepage and supplements the studies on the permeability models.

Advanced machine learning approaches for predicting permeability in reservoir pay zones based on core analyses

Permeability is the most important petrophysical characteristic for determining how fluids pass through reservoir rocks. This study aims to develop and assess intelligent computer-based models for predicting permeability. The research focuses on three novel models—Decision Tree, Bagging Tree, and Extra Trees—while also investigating previously applied techniques such as random forest, support vector regressor (SVR), and multiple variable regression (MVR). The primary dataset consists of 197 data points from a heterogeneous petroleum reservoir in the Jeanne d'Arc Basin, including laboratory-derived permeability (K), oil saturation (SO), water saturation (SW), grain density (ρgr), porosity (φ), and depth. The most effective machine learning models are identified by a thorough analysis that makes use of a variety of statistical metrics, such as the coefficient of the determinant (R2), mean squared error (MSE), mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), maximum error (maxE), and minimum error (minE). Additionally, core features are ranked based on their importance in permeability modeling. This study deviates from conventional approaches by proposing an efficient means of forecasting permeability, reducing reliance on labor-intensive and time-consuming laboratory work. The findings reveal that MVR is unsuitable for permeability prediction, with all developed models outperforming it. Extra Trees emerges as the most accurate model, with an R2 of 0.976, while random forest and bagging tree exhibit slightly lower R2 values of 0.961 and 0.964, respectively. The ranking of these algorithms based on performance criteria is as follows: extra trees, bagging tree, random forest, SVR, decision tree, and MVR. The study also presents a detailed analysis of the impact of input parameters, highlighting porosity (φ) and water saturation (SW) as the most influential, while grain density (ρgr), oil saturation (SO), and depth are considered less important. This study contributes to the petroleum industry's knowledge by showcasing the inadequacy of MVR and highlighting the superior performance of machine learning models, particularly Extra Trees. The proposed models employed in this study can help engineers and researchers determine reservoir permeability quickly and accurately by using a few core attributes, reducing the dependency on resource-intensive and time-consuming laboratory work.

Fractal analysis of dimensionless permeability and Kozeny–Carman constant of spherical granular porous media with randomly distributed tree-like branching networks

The seepage of porous media has garnered significant interest due to its ubiquitous presence in nature, but most of the research is based on the model of a single dendritic branching network. In this study, we derive a fractal model of the dimensionless permeability and the Kozeny–Carman (KC) constant of porous media consisting of spherical particles and randomly distributed tree-like branching networks based on fractal theory. In addition, three different types of corrugated pipes are considered. Then, the relationships between the KC constant, dimensionless permeability, and other structural parameters were discussed in detail. It is worth noting that the KC constant of the porous media composed of three types of pipes decreases sharply first and then increases with the increase in the internal diameter ratio, while the dimensionless permeability has the opposite trend and conforms to the physical law. In addition, empirical constants are not included in the analytical formulas of the present model, and the physical mechanism of fluid flow in spherical granular porous media with randomly distributed tree-like branching networks is clearly revealed.

Enhancing shale gas recovery: An interdisciplinary power-law model of hydro-mechanical-fracture dynamics

This study explores the efficiency of using carbon dioxide (CO2) to extract shale gas, highlighting its potential to enhance extraction while mitigating environmental CO2 pollution. Given the intricate microstructure of shale, CO2 injection inevitably induces deformation within the shale reservoir's internal microstructure, thereby impacting gas displacement efficiency. The organic matter (kerogen) network and fracture network in shale, serving as primary spaces for gas adsorption and migration, exhibit complex microstructural characteristics. Thus, we developed a dynamic coupled hydro-mechanics permeability model for binary gas displacement, and three novel, interdisciplinary fractal power-law parameters are proposed to represent the distribution of shale fractures, considering the adsorption–desorption strength of the kerogen network. Numerical simulations analyzed the changes in gas seepage, diffusion, shale stress, permeability, and factors influencing displacement efficiency during the CO2–EGR (enhanced gas recovery) projects. Key findings include (1) CO2 injection leads to a nonlinear increase in the number of shale fracture networks, thereby enhancing the CH4 output efficiency. (2) Compared to traditional fractal theory, the proposed power-law model is applicable to a wider range of reservoir fracture distributions and effectively characterizes the density (by α), size (by r), and complexity (by n) of the fracture network during the CO2–EGR process. (3) Changes in the proposed interdisciplinary power-law parameters significantly alter CO2 and CH4 adsorption capacities and, in turn, significantly affects displacement efficiency and shale deformation. According to calculations, these parameters have the greatest impact on the CO2–EGR process, ranging from 16.3% to 68.1%.

Pore-scale investigations of permeability of saturated porous media: Pore structure efficiency

Accurate estimation of permeability and the establishment of correlations with pore space properties have garnered attention due to the increasing demand for oil and gas reservoir development as well as geological carbon sequestration. However, prevailing theoretical and empirical models often rely on porosity and pore sizes as key pore space properties to predict permeability, overlooking the variability of pore structure. To delineate the intrinsic associations between permeability and pore space properties, a self-developed visualized experimental system with controllable pore space properties is used to measure permeability of porous media with varying porosity and particle arrangements. Various parameters, encompassing porosity, pore sizes (maximum pore diameter and pore throat diameter) and fractal parameters (pore fractal dimension, succolarity, and lacunarity) are extracted to quantify pore space properties, and further utilized to develop permeability models. Results demonstrate a positive correlation between permeability and porosity, maximum pore diameter, and average pore throat diameter. Additionally, particle arrangements yield pronounced anisotropy in permeability. Relying solely on porosity and pore sizes to portray pore space properties falls short of comprehensively characterizing permeability. Incorporating fractal characterization through metrics like succolarity and lacunarity emerges as a potent avenue for delineating heterogeneity and pore connectivity. This approach proves useful in characterizing the effect of pore structure variability on permeability. The established permeability model rooted in flow direction succolarity and normalized lacunarity aligns effectively with experimental data, capturing the essential physics of permeability variation.

Analysis and modelling of gas relative permeability in reservoir by hybrid KELM methods

Analysis and modelling of gas relative permeability in reservoir by hybrid KELM methods

Finite element modeling of the magnetic property behaviors of electrical steels in rotating fields considering their crystallographic textures

In magnetic device modeling, the crystallographic textures and microstructures of electrical steels critically influence their magnetic properties. Yet, these aspects are often neglected or inadequately represented in finite element (FE) models, leading to inaccuracies in predicting electrical steel performance, especially in complex electromagnetic environments. This paper presents a multi-scale modeling approach, integrating a macroscopic FE model of a magnetic device with a microstructural scalar permeability model that accounts for the crystallographic textures of electrical steels. This approach demonstrates proficiency in predicting nonlinear anisotropic permeability and capturing magnetic anisotropy of electrical steels within a rotational tester, especially when anisotropy is predominantly governed by crystallographic texture and the angles between the B and H vectors are minimal. Furthermore, the model effectively captures the magnetic anisotropy in complex magnetic fields and geometric configurations of a permanent magnet motor. A comparative analysis of this model is conducted against three other material models: an isotropic model, a two-axis model, and a general-vector model. These models are based on different approaches, ranging from a single non-linear BH curve to comprehensive vector BH curves covering all directions. Among them, the general-vector model, reliant on extensive 2D vector BH measurements, is the most accurate but requires substantial experimental data.

Numerical investigation of heat extraction performance affected by stimulated reservoir volume and permeability evolution in the enhanced geothermal system

The high-permeability stimulated reservoir volume (SRV) containing artificial fractures is the main region for deep geothermal extraction. Evaluating the SRV distribution and the contributions of fluid pressure and thermal stress to permeability evolution is essential to accurately quantify heat extraction. Here, we employ the numerically calculated pore pressure and the field-measured rock failure as evaluation indicators to reasonably identify the SRV and investigate the effect of rock and fracture permeability evolution on heat extraction performance. We find that simply assuming the entire model as the SRV would significantly overestimate the heat extraction, and scientifically defining the SRV based on hydraulic fracturing is crucial. The variable rock permeability model extracts marginally less heat as the working fluid squeezes and cools surrounding rocks causing their lower permeability. In regions around the injection well, both the high fluid pressure and the thermal stress increase the fracture aperture and permeability, whereas the fluid pressure shows the opposite effect around the production well as it is lower than the initial pore pressure. When the main fractures connect injection and production wells, the widened fractures accelerate fluid flow, causing higher heat extraction for a short time (first 4 years); however, after 4 years’ operation, the heat extraction remains at a lower value for a long time as the increased fracture permeability causes more fluids to flow directly from main fractures to the production well without playing participating in heat exchange. When the main fractures do not connect injection and production wells, the increased fracture permeability slightly enhances the heat extraction over the geothermal extraction process with a relative error less than 0.025. This study provides guidance for the appropriate selection of geothermal extraction simulation assumptions.

Investigating the Creep Damage and Permeability Evolution Mechanism of Phyllite Considering Non-Darcy’s Flow

This study investigates the intricate interplay of hydraulic coupling, creep damage, and permeability evolution mechanisms in phyllite, focusing on their relevance to tunnel engineering design and long-term stability in soft rock formations. To achieve this, conventional triaxial tests were conducted on saturated phyllite specimens under creep-seepage acoustic emission conditions. The results were systematically analyzed to unveil the inherent characteristics of creep deformation, seepage rate evolution, and damage progression in phyllite when subjected to stress–seepage coupling. Furthermore, a permeability model for representative volume element (RVE) was developed based on meso-mechanics principles, considering the distinctive attributes of low-permeability non-Darcy’s flow in rock. Consequently, a novel relationship between effective damage and permeability was established. We determined that seepage pressure induces three key effects on the creep damage mechanism of phyllite samples: (1) It adds to the existing stress through the superposition of osmotic pressure and axial load, (2) it induces tensile expansion stress because of pore water pressure, and (3) it softens the fracture surfaces to some extent. More importantly, this study validates the relationship between effective damage and permeability through the fitting of creep and damage parameters, which were obtained from the test results, and it reveals a square relationship between subsequent damage and permeability under stress conditions. The findings of this study provide a robust theoretical foundation for comprehending the evolution of damage and permeability characteristics during the creep process of rock under seepage conditions. We obtained essential insights and quantitative analyses for comprehending the damage mechanisms in rock creep under hydraulic coupling, which has significant implications for tunnel engineering and long-term rock stability assessments in soft rock environments.

Calibration and validation of DRAINMOD to predict long-term permeable pavement hydrology

The hydrologic performance of permeable pavement varies widely due to underlying soil type, drainage configuration, contributing drainage area, surface infiltration rate, and aggregate depth. A long-term hydrologic model is needed to better understand the influence of these design variables on surface runoff, drainage, exfiltration, and evaporation from permeable pavement systems. Most permeable pavement models have not been calibrated with field monitored data, are usually unable to accurately model internal water storage (IWS) zones, and do not account for evaporation from the aggregate profile. Because permeable pavement employs drainage and exfiltration as primary hydrologic mechanisms, it was hypothesized that DRAINMOD, a model shown to accurately simulate bioretention hydrology, could be calibrated to predict the hydrologic response from permeable pavements. Hydrologic data were collected from two permeable pavement applications in North Carolina (Boone and Durham) and two permeable pavement applications in Ohio (Perkins Township and Willoughby Hills) to calibrate and test the model. The specific permeable pavements applications varied widely in aggregate depth, drainage configuration, underlying soil type, and contributing drainage area. Nash-Sutcliffe Efficiencies for drainage ranged from 0.77 to 0.95 during calibration and validation of all sites. Nash-Sutcliffe Efficiencies for exfiltration/evaporation ranged from 0.55 to 0.97 for Boone and Perkins Township, but prediction of exfiltration/evaporation volumes on an event basis was poor for the remaining two sites due to low exfiltration rates (less than 0.20 mm/hr). Despite this, the cumulative volume of exfiltration/evaporation was predicted to within 1–9 % of what was measured during monitoring. Modeled and monitored surface runoff from the Willoughby Hills application (8 % of the water balance) was equivalent. The model predicted the percentage of drainage, surface runoff, and exfiltration/evaporation to within 2 % of what was monitored/estimated at each site, suggesting long-term modeling of permeable pavement hydrology with DRAINMOD is viable.

Simulation Study of Coal Seam Gas Extraction Characteristics Based on Coal Permeability Evolution Model under Thermal Effect.

The permeability evolution law of high temperature and high stress coal seam is determined by the influence of multiphase coexistence and multifield coupling. In an environment greatly affected by disturbance and high temperature, the coal permeability model under the coupling of thermal and mechanical creep is not only a vital framework from which to examine gas migration law in multiphase and multifield coal seams but also an important theoretical foundation for gas control in coal seams. The influence of high-temperature environment on creep deformation and permeability is analyzed by several creep seepage tests under different temperature conditions.A mathematical model for the evolution of coal permeability considering the influence of temperature is established through the theory of matrix-crack interaction based on gas adsorption and desorption and thermal expansion deformation. Based on the permeability model under the coupling of thermal and mechanical creep, the numerical model of gas migration, seepage field, diffusion field, stress field, and temperature field is constructed, and the law of gas migration in coal seam under multifield coupling is explored. The influence law of thermal effect on gas extraction characteristics is analyzed, in which the time-varying mechanism of temperature field, the relationship between creep deformation and temperature and pressure, the influence of creep deformation on permeability, the dynamic distribution of gas pressure, and the change of gas extraction quantity are described in detail. It is concluded that the influence of temperature on permeability is much greater than that of creep deformation and that a high initial coal seam temperature is beneficial to gas extraction. It provides theoretical basis and technical guidance for the study of multifield coupled gas migration and coal seam gas treatment.

Study on Permeability Evolution Law of Rock Mass under Mining Stress

In order to study the stress–strain–permeability coefficient relationship of overlying strata in a fractured zone after coal mining, taking the Changcun coal mine in the Changzhi basin as an example, the permeability evolution law of coarse sandstone, fine sandstone, siltstone and mudstone during a stress–strain process was analyzed through a triaxial compression permeability test. The generalized model of the rock mass permeability evolution process under mining stress was summarized, and then a coupling model of the stress–water pressure–permeability coefficient of fractured rock was established based on the continuum model of rock mass. The results showed that the maximum permeability coefficient of different coal overburden types was quite different, and the peak strength of the rock mass preceded the maximum permeability coefficient during the rock mass failure process; the permeability coefficient first decreased and then increased, reaching its maximum value after the peak stress, which occurred during the strain-softening stage; the generalized model of rock mass permeability included the compaction stage, elasticity stage, stable fracture stage, unstable fracture stage, macroscopic failure stage and residual strength stage.