Richeng Liu

Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

Fracture networks play a more significant role in conducting fluid flow and solute transport in fractured rock masses, comparing with that of the rock matrix. Accurate estimation of the permeability of fracture networks would help researchers and engineers better assess the performance of projects associated with fluid flow in fractured rock masses. This study provides a review of previous works that have focused on the estimation of equivalent permeability of two-dimensional (2-D) discrete fracture networks (DFNs) considering the influences of geometric properties of fractured rock masses. Mathematical expressions for the effects of nine important parameters that significantly impact on the equivalent permeability of DFNs are summarized, including (1) fracture-length distribution, (2) aperture distribution, (3) fracture surface roughness, (4) fracture dead-end, (5) number of intersections, (6) hydraulic gradient, (7) boundary stress, (8) anisotropy, and (9) scale. Recent developments of 3-D fracture networks are briefly reviewed to underline the importance of utilizing 3-D models in future research.RésuméLes réseaux de fracture jouent un rôle plus significatif dans l’écoulement d’un fluide et du transport de soluté dans les massifs rocheux fracturés, en comparaison avec celui de la matrice rocheuse. Une estimation précise de la perméabilité des réseaux de fracture aiderait les chercheurs et les ingénieurs à mieux évaluer la performance des projets liés à l’écoulement du fluide dans les massifs rocheux fracturés. Cette étude présente un examen des travaux antérieurs qui ont porté sur l’estimation de la perméabilité équivalente de réseaux de fractures discrétisés (RFDs) en deux dimensions (2-D) compte tenu des influences des propriétés géométriques des massifs rocheux fracturés. Les expressions mathématiques pour les effets de neuf paramètres importants qui ont une incidence significative sur la perméabilité équivalente des RFDs sont résumées, comprenant (1) la distribution des longueurs de fracture, (2) la distribution des ouvertures, (3) la rugosité de la surface des fractures, (4) les fractures sans issues , (5) le nombre d’intersections, (6) le gradient hydraulique, (7) les conditions aux limites, (8) l’anisotropie, et (9) l’échelle. Les développements récents des réseaux de fractures en 3-D sont brièvement revus afin de souligner l’importance de l’utilisation de modèles 3-D dans les recherches futures.ResumenLas redes de fracturas juegan un papel significativo en la conducción del flujo del fluido y en el transporte de solutos en las masas de roca fracturada, en comparación con el de la matriz de la roca. La estimación precisa de la permeabilidad de las redes de fracturas ayudaría a los investigadores e ingenieros a evaluar mejor el rendimiento de los proyectos relacionados con el flujo de fluido en masas de rocas fracturadas. Este estudio proporciona una revisión de trabajos previos que se han centrado en la estimación de la permeabilidad equivalente de redes bidimensionales (2-D) discretas de fractura (dfns) teniendo en cuenta las influencias de las propiedades geométricas de las masas de rocas fracturadas. Se resumen las expresiones matemáticas para los efectos de los nueve parámetros importantes que impactan significativamente sobre la permeabilidad equivalente de dfns, incluyendo (1) la distribución de la longitud de la fractura, (2) la distribución de las aberturas, (3) la rugosidad de la superficie de fractura, (4) las fracturas cerradas, (5) el número de intersecciones, (6) el gradiente hidráulico, (7) la tensión límite, (8) la anisotropía, y (9) la escala. Se revisa brevemente el reciente desarrollo de redes de fracturas 3-D para resaltar la importancia de la utilización de modelos 3-D en futuras investigaciones.摘要裂隙网络相对于岩石基质对裂隙岩体内流体流动和污染物运移等性质有着更为重要的影响。准确预测裂隙网络的渗透系数将有助于研究人员更好的评估与裂隙岩体渗流相关的各项工程的安全稳定性。本综述回顾了前人利用裂隙岩体几何信息计算二维离散裂隙网络渗透系数的相关研究,讨论了裂隙网络模型中9个重要参数对渗透系数数学表达式的影响,这9个参数包括:(1) 裂隙长度分布, (2) 裂隙开度分布, (3) 裂隙表面粗糙度, (4) 裂隙断头, (5) 交点数量, (6) 水力梯度, (7) 边界应力, (8) 各向异性, (9) 模型尺寸。本文也介绍了三维裂隙网络模型的发展现状,并强调了在将来的工作中采用三维模型来评估渗透系数的重要性。ResumoRedes fraturadas desempenham um papel mais significante no fluxo de condução do fluido e no transporte de soluto na massa da rocha fraturada, comparado com o mesmo na rocha matriz. Estimativa precisa da permeabilidade das redes de fratura pode ajudar pesquisadores e engenheiros a melhor avaliar o desempenho dos processos associados com o fluxo de fluido na massa de rocha fraturada. Esse estudo fornece uma revisão de trabalhos anteriores que focaram na estimativa de permeabilidade de redes de fratura discretas (RFDs) bidimensionais (2-D) considerando a influência de propriedades geométricas de massa de rochas fraturadas. As expressões matemáticas para os efeitos de nove parâmetros importantes de impacto significante da permeabilidade equivalente das RFDs foram resumidas, incluindo (1) distribuição no comprimento da fratura, (2) distribuição da abertura, (3) rugosidade da superfície da fratura, (4) final da fratura, (5) número de intersecções, (6) gradiente hidráulico, (7) stress do limite, (8) anisotropia e (9) escala. Desenvolvimentos recentes de redes de fraturas 3-D foram brevemente revisados para destacar a importância da utilização de modelos 3-D em pesquisas futuras.

A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures

Selecting appropriate governing equations for fluid flow in fractured rock masses is of special importance for estimating the permeability of rock fracture networks. When the flow velocity is small, the flow is in the linear regime and obeys the cubic law, whereas when the flow velocity is large, the flow is in the nonlinear regime and should be simulated by solving the complex Navier-Stokes equations. The critical conditions such as critical Reynolds number and critical hydraulic gradient are commonly defined in the previous works to quantify the onset of nonlinear fluid flow. This study reviews the simplifications of governing equations from the Navier-Stokes equations, Stokes equation, and Reynold equation to the cubic law and reviews the evolutions of critical Reynolds number and critical hydraulic gradient for fluid flow in rock fractures and fracture networks, considering the influences of shear displacement, normal stress and/or confining pressure, fracture surface roughness, aperture, and number of intersections. This review provides a reference for the engineers and hydrogeologists especially the beginners to thoroughly understand the nonlinear flow regimes/mechanisms within complex fractured rock masses.

Estimation of the REV Size and Equivalent Permeability Coefficient of Fractured Rock Masses with an Emphasis on Comparing the Radial and Unidirectional Flow Configurations

AbstractA method to estimate the representative elementary volume (REV) size for the permeability and equivalent permeability coefficient of rock mass with a radial flow configuration was developed. The estimations of the REV size and equivalent permeability for the rock mass around an underground oil storage facility using a radial flow configuration were compared with those using a unidirectional flow configuration. The REV sizes estimated using the unidirectional flow configuration are much higher than those estimated using the radial flow configuration. The equivalent permeability coefficient estimated using the radial flow configuration is unique, while those estimated using the unidirectional flow configuration depend on the boundary conditions and flow directions. The influences of the fracture trace length, spacing and gap on the REV size and equivalent permeability coefficient were investigated. The REV size for the permeability of fractured rock mass increases with increasing the mean trace length and fracture spacing. The influence of the fracture gap length on the REV size is insignificant. The equivalent permeability coefficient decreases with the fracture spacing, while the influences of the fracture trace length and gap length are not determinate. The applicability of the proposed method to the prediction of groundwater inflow into rock caverns was verified using the measured groundwater inflow into the facility. The permeability coefficient estimated using the radial flow configuration is more similar to the representative equivalent permeability coefficient than those estimated with different boundary conditions using the unidirectional flow configuration.

A PREDICTIVE MODEL OF PERMEABILITY FOR FRACTAL-BASED ROUGH ROCK FRACTURES DURING SHEAR

This study investigates the roles of fracture roughness, normal stress and shear displacement on the fluid flow characteristics through three-dimensional (3D) self-affine fractal rock fractures, whose surfaces are generated using the modified successive random additions (SRA) algorithm. A series of numerical shear-flow tests under different normal stresses were conducted on rough rock fractures to calculate the evolutions of fracture aperture and permeability. The results show that the rough surfaces of fractal-based fractures can be described using the scaling parameter Hurst exponent (H), in which H = 3 − Df, where Df is the fractal dimension of 3D single fractures. The joint roughness coefficient (JRC) distribution of fracture profiles follows a Gauss function with a negative linear relationship between H and average JRC. The frequency curves of aperture distributions change from sharp to flat with increasing shear displacement, indicating a more anisotropic and heterogeneous flow pattern. Both the mean aperture and permeability of fracture increase with the increment of surface roughness and decrement of normal stress. At the beginning of shear, the permeability increases remarkably and then gradually becomes steady. A predictive model of permeability using the mean mechanical aperture is proposed and the validity is verified by comparisons with the experimental results reported in literature. The proposed model provides a simple method to approximate permeability of fractal-based rough rock fractures during shear using fracture aperture distribution that can be easily obtained from digitized fracture surface information.

A novel cloud model for risk analysis of water inrush in karst tunnels

Water inrush is a serious geological hazard in underground engineering. The prediction of possibility and classification of water inrush risk has long been a global problem for the construction of deep-buried tunnels in karst areas. To solve the randomness and fuzziness in the evaluation process of water inrush risk, a novel comprehensive evaluation model was established based on the normal cloud theory. According to the systematic analysis of the influence factors of water inrush, seven factors were selected as evaluation indices, including formation lithology, unfavourable geological conditions, groundwater level, landform and physiognomy, modified strata inclination, contact zones of dissolvable and insoluble rock, and layer and interlayer fissures. Meanwhile, a hierarchy model of the influence factors was established for water inrush, and the analytic hierarchy process was adopted to determine the weighting coefficients for each evaluation index. The normal cloud theory was used to describe the cloud numerical characteristics for each evaluation index of risk classification for water inrush. Normal cloud droplets were generated to reflect the uncertain transformation between the risk levels of water inrush and the evaluation indices. Then, the synthetic degrees of certainty were calculated, and risk level of water inrush was determined. Finally, the proposed model was applied to two typical deep-buried tunnels in karst areas: Jigongling tunnel and Xiakou tunnel. The obtained results were compared with the relevant analysis results and the practical findings, and reasonable agreements were gained. The normal cloud model was found to be more accurate, feasible and effective for risk classification of water inrush prediction. It can not only meet the requirement of tunnel engineering, but also be extended to various applications.

Investigating the Roles of Included Angle and Loading Condition on the Critical Hydraulic Gradient of Real Rock Fracture Networks

List of symbols Q Volumetric flow rate w Width of a fracture bh Hydraulic aperture J Hydraulic gradient Jc Critical hydraulic gradient P Hydraulic pressure u Flow velocity vector ρ Fluid density E Nonlinear effect factor T Transmissivity T/T0 Normalized transmissivity xi X-coordinate of the fracture surface profile zi Z-coordinate of the fracture surface profile Z2 Dimensionless roughness parameter a Linear coefficient in the Forchheimer’s law b Nonlinear coefficient in the Forchheimer’s law l Fracture spacing Fx X-directional horizontal boundary load Fy Y-directional horizontal boundary load σfne Effective stress

SIZE EFFECT ON THE PERMEABILITY AND SHEAR INDUCED FLOW ANISOTROPY OF FRACTAL ROCK FRACTURES

The effect of model size on fluid flow through fractal rough fractures under shearing is investigated using a numerical simulation method. The shear behavior of rough fractures with self-affine properties was described using the analytical model, and the aperture fields with sizes varying from 25 to 200mm were extracted under shear displacements up to 20mm. Fluid flow through fractures in the directions both parallel and perpendicular to the shear directions was simulated by solving the Reynolds equation using a finite element code. The results show that fluid flow tends to converge into a few main flow channels as shear displacement increases, while the shapes of flow channels change significantly as the fracture size increases. As the model size increases, the permeability in the directions both parallel and perpendicular to the shear direction changes significantly first and then tends to move to a stable state. The size effects on the permeability in the direction parallel to the shear direction are m...

Experimental Study on Stress-Dependent Nonlinear Flow Behavior and Normalized Transmissivity of Real Rock Fracture Networks

The mechanism and quantitative descriptions of nonlinear fluid flow through rock fractures are difficult issues of high concern in underground engineering fields. In order to study the effects of fracture geometry and loading conditions on nonlinear flow properties and normalized transmissivity through fracture networks, stress-dependent fluid flow tests were conducted on real rock fracture networks with different number of intersections (1, 4, 7, and 12) and subjected to various applied boundary loads (7, 14, 21, 28, and 35 kN). For all cases, the inlet hydraulic pressures ranged from 0 to 0.6 MPa. The test results show that Forchheimer’s law provides an excellent description of the nonlinear fluid flow in fracture networks. The linear coefficient and nonlinear coefficient in Forchheimer’s law generally decrease with the number of intersections but increase with the boundary load. The relationships between and can be well fitted with a power function. A nonlinear effect factor was used to quantitatively characterize the nonlinear behaviors of fluid flow through fracture networks. By defining a critical value of = 10%, the critical hydraulic gradient was calculated. The critical hydraulic gradient decreases with the number of intersections due to richer flowing paths but increases with the boundary load due to fracture closure. The transmissivity of fracture networks decreases with the hydraulic gradient, and the variation process can be estimated using an exponential function. A mathematical expression for decreased normalized transmissivity against the hydraulic gradient was established. When the hydraulic gradient is small, holds a constant value of 1.0. With increasing hydraulic gradient, the reduction rate of first increases and then decreases. The equivalent permeability of fracture networks decreases with the applied boundary load, and permeability changes at low load levels are more sensitive.

An experimental study of the effect of fillings on hydraulic properties of single fractures

Fluid flow in single rock fractures considering the influences of fracture surface roughness, shearing process, normal loading, and so on has been extensively studied for several decades, yet the significant influence of fillings has not been systematically investigated due to the numerous difficulties such as determination of the physical parameters of fillings. The present study aims to estimate the hydraulic properties of single fractures filled with different graded and gap-graded fillings, based on a series of flow tests on rock-like samples using the MTS815.02 material testing system. With the increment of fracture aperture, the pressure drops before and after fillings are flowed away decrease, whereas the permeabilities before and after fillings are flowed away increase. When the ratio of mechanical aperture of fractures to maximum diameter of fillings decreases from 4 to 1.33, both pressure drop and permeability change significantly before the fillings are flowed away and then hold constant values after the fillings are flowed away. Due to the effects of fraction force and interlocking force between particles, the ratio of mechanical aperture to maximum diameter of fillings that equals to 2.67 is the inflection point, where the pressure drop has the maximum value and permeability has the minimum value. When the fractures are filled with gap-graded fillings, in which the ratio of mechanical aperture of fractures to mean diameter of fillings decreases from 5.76 to 1.45, the variations of both pressure drop and permeability before fillings are flowed away change more significantly than those after fillings are flowed away. The hydraulic aperture of fractures with fillings is approximately 2–3 orders of magnitude smaller than the mechanical aperture.

Effect of Thermal Treatment on the Dynamic Behaviors at a Fixed Loading Rate of Limestone in Quasi-vacuum and Air-filled Environments

Thermal damage in rock engineering occurs in the air‐filled and quasi‐ vacuum environments of rock mass located near or far from the free sur‐ face. Meanwhile, dynamic loads are encountered frequently in engineering practice. In this study, 39 limestone samples are prepared, and a series of laboratory tests, including split Hopkinson pressure bar SHPB , nuclear magnetic resonance NMR and optical microscopy analyses, are conduct‐ ed to investigate the effects of temperature and the environment on the dynamic mechanical properties of limestone. The results show that the macro‐physical and dynamic mechanical properties of limestone after thermal treatment can be divided into two stages by a critical temperature of 450°C, at which the thermal damage factor is 0.71 and 0.75 in the quasi‐ vacuum and air‐filled environments, respectively. In the first stage, with temperatures varying from 25°C to 450°C, the thermal damage due to ex‐ pansion and fracturing slightly influences the related parameters, except the P‐wave velocity. However, in the second stage, with temperatures rang‐ ing from 450°C to 900°C, the thermal damage caused by mineral decompo‐ sition and hydration leads to a remarkable decrease in the dynamic bearing and anti‐deformation capacities. The environment plays a negligible role in the first stage but an important role in the second stage, and the dynamic compressive strength and modulus of samples after thermal treatment in the air‐filled environment are much lower than those in the quasi‐vacuum environment. Both the temperature and environment of thermal treatment should be considered in engineering practice, especially when the temper‐ ature exceeds 450°C.