Micro- and macro-behaviour of fluid flow through rock fractures: an experimental study

Microscopic and macroscopic behaviour of fluid flow through rough-walled rock fractures was experimentally investigated. Advanced microfluidic technology was introduced to examine the microscopic viscous and inertial effects of water flow through rock fractures in the vicinity of voids under different flow velocities, while the macroscopic behaviour of fracture flow was investigated by carrying out triaxial flow tests through fractured sandstone under confining stresses ranging from 0.5 to 3.0 MPa. The flow tests show that the microscopic inertial forces increase with the flow velocity with significant effects on the local flow pattern near the voids. With the increase in flow velocity, the deviation of the flow trajectories is reduced but small eddies appear inside the cavities. The results of the macroscopic flow tests show that the linear Darcy flow occurs for mated rock fractures due to small aperture, while a nonlinear deviation of the flow occurs at relatively high Reynolds numbers in non-mated rock fracture (Re?>?32). The microscopic experiments suggest that the pressure loss consumed by the eddies inside cavities could contribute to the nonlinear fluid flow behaviour through rock joints. It is found that such nonlinear flow behaviour is best matched with the quadratic-termed Forchheimer equation.     

Effect of roughness on water flow through a synthetic single rough fracture

A single fracture is usually idealized theoretically as smooth parallel plates, but the real fractures are rough-walled with points of contact. Though relative roughness is considered in quantifying the flow through a single rough fracture (SRF) previously, additional factors such as the distribution of rough elements and bending degree of streamlines should be considered in order to obtain more accurate results. Semiempirical friction factor (f) and discharge per unit width (q) equations are first deduced taking relative roughness, roughness elements distribution and streamline reattachment length into consideration. A horizontal SRF model was then set up and a series of experiments and simulations were performed. Main conclusions are drawn: Recirculation of streamlines arises in the rough element and the intensity of the recirculation increases with the angle from which the streamlines enter into the rough elements and Reynolds number (Re); streamlines are discontinuously distributed when asperity height is large and nonlinear flow occurs; the nonlinearity of the flow increases with the increasing the asperity height and Re; the critical value of related roughness used to judge whether the influence of roughness on water flow through a SRF can be ignored or not should be much lower than 0.033; the revised f and q equations under laminar flow through a SRF are proved to be better when calculating the f and q values.     

粗糙岩石裂隙低速非线性渗流模型及试验验证

岩体粗糙裂隙非线性渗透特性是岩体渗流研究的重要课题。针对Javadi所提T模型中亚裂隙中速度恒定这一假设的不足,将其修正为速度与开度成正比的经验关系,考虑黏性压降和局部压降,提出了新的低速下粗糙裂隙非线性渗流模型（MT模型）。为了验证MT模型的正确性,对5种不同粗糙裂隙进行了低流量的饱和渗流试验,将现有模型及MT模型的预测值与试验结果进行对比,分析表明MT模型与试验结果更为吻合,且模型适用于粗糙性系数JRC≤10的低粗糙度裂隙,雷诺数小于1 000的低流速情况。对MT模型进行分析,初步揭露了粗糙裂隙的非线性渗流机制,即小雷诺数下的Darcy流和大雷诺数下的Forchheimer流,并用临界雷诺数区分两种流动行为。分析了裂隙的粗糙度和开度对非线性渗流特性的影响,表明裂隙越粗糙或开度越小,则临界雷诺数越小,非线性作用将越强。提出了临界雷诺数与水力开度和绝对粗糙度的经验关系式,同时指出该关系式适用于JRC≤10的低粗糙度裂隙。     

An experimental study of single and two-phase fluid flow through fractured granite specimens

Triaxial tests on the two-phase flow of air and water through fractured granite specimens were performed to discover whether the two-phase fluid flow within rock fractures was laminar or turbulent. The two-phase flow characterization was carried out based on the macroscopic two-phase steady state flow model and the homogeneous steady state flow model. Rock specimens with a single natural fracture (joint roughness coefficient, JRC < 10) were tested using two-phase, high pressure triaxial rig. Experimental results show that the estimated Reynolds numbers for various inlet fluid pressures are well below 1000. The findings of this study reveal that both single and two-phase flow through rock fractures (JRC < 10) can be characterized as laminar flows at moderate inlet fluid pressures. However, for single-phase air flow, an increase in inlet air pressures may result in the formation of turbulent flow.     

Reynolds控制方程下粗糙节理渗流空腔模型研究

在研究节理的渗流时,渗流控制方程对节理渗流分析结果具有显著影响。首先介绍了节理渗流分析中的控制方程：Navier-Stokes方程、Stokes方程、Reynolds方程和立定定理,并分析了各控制方程在节理渗流分析中的适用性。以Reynolds方程作为渗流分析控制方程,建立了粗糙节理渗流空腔模型。然后以节理试件为研究对象,在实测节理三维表面形貌并计算隙宽分布后,分别进行相同渗流边界条件下的室内渗流试验和空腔模型计算,得到节理在不同接触状态下的渗流量实测值和计算值,并分别将计算结果与立方定理下的空腔模型、将整个节理简化为光滑平行板模型的立方定理以及速宝玉经验公式的计算结果进行比较,结果表明,Reynolds方程下的节理渗流空腔模型计算结果与实测值最为吻合,可以较为准确地反映节理的渗流情况。同时,根据Reynolds方程下空腔模型得到的渗流流量分布可以呈现节理渗流的曲折现象,为从本质上研究节理渗流特性奠定了理论基础。     

Single phase water flow through rock fractures

Flow analysis plays a major role in various geotechnical applications, and the understanding of flow mechanisms is essential for the development of a hydro-mechanical flow model suitable for underground excavations in rock. Discrete flow analysis through discontinuities is reviewed including empirical and analytical flow models. The influence of external loading on joint deformation and single-phase flow show that the surface roughness and aperture size are the prime factors influencing flow rate. Nevertheless, the idealization of natural fractures as smooth parallel plate joints is still followed in many numerical models, because of the simplicity of the cubic law when applied to fracture networks. A numerical study of water flow through a network of joints employing Universal Distinct Element Code (UDEC) is used to quantify the effects of joint orientation and external stress acting on idealized joints.It is found that, for the same joint spacing, the flow rate into an excavation depends on the boundary block size (A_b) relative to the excavation size (A_e). The inflow becomes excessive if A_b/A_e is less than 4, but becomes very small if A_b/A_e exceeds 8.     

岩石节理表面几何特性的三维统计分析

在利用三维非接触式高精度激光表面形状测定仪精确测量岩石节理的表面粗糙形状并进行数值化表达的基础上,应用地理信息系统(GIS)技术,实现了岩石节理粗糙面的三维可视化。再对岩石节理粗糙面的三维几何特性参数（诸如表面粗糙度、裂隙张开度和粗糙高度等）进行统计分析并计算频数分布的特征数字。最后,根据频率直方图的分布趋势和频数分布的特征数字研究各几何特性参数的分布规律。结果表明,花岗岩节理粗糙表面起伏比砂岩大,但二者几何特性参数的分布规律相似,节理粗糙高度和裂隙张开度分布近似于正态分布规律,而节理粗糙面倾斜角分布近似于Γ分布规律。     

Quantification of Aperture and Relations Between Aperture,Normal Stress and Fluid Flow for Natural Single Rock Fractures

Accurate quantification of rock fracture aperture is important in investigating hydro-mechanical properties of rock fractures. Liquefied wood’s metal was used successfully to determine the spatial distribution of aperture with normal stress for natural single rock fractures. A modified 3D box counting method is developed and applied to quantify the spatial variation of rock fracture aperture with normal stress. New functional relations are developed for the following list: (a) Aperture fractal dimension versus effective normal stress; (b) Aperture fractal dimension versus mean aperture; (c) Fluid flow rate per unit hydraulic gradient per unit width versus mean aperture; (d) Fluid flow rate per unit hydraulic gradient per unit width versus aperture fractal dimension. The aperture fractal dimension was found to be a better parameter than mean aperture to correlate to fluid flow rate of natural single rock fractures. A highly refined variogram technique is used to investigate possible existence of aperture anisotropy. It was observed that the scale dependent fractal parameter, K _v, plays a more prominent role than the fractal dimension, D _a1d, on determining the anisotropy pattern of aperture data. A combined factor that represents both D _a1d and K _v, D _a1d × K _v, is suggested to capture the aperture anisotropy.     

Natural rock joint roughness quantification through fractal techniques

Accurate quantification of roughness is important in modeling hydro-mechanical behavior of rock joints. A highly refined variogram technique was used to investigate possible existence of anisotropy in natural rock joint roughness. Investigated natural rock joints showed randomly varying roughness anisotropy with the direction. A scale dependant fractal parameter, K _v, seems to play a prominent role than the fractal dimension, D _r1d, with respect to quantification of roughness of natural rock joints. Because the roughness varies randomly, it is impossible to predict the roughness variation of rock joint surfaces from measurements made in only two perpendicular directions on a particular sample. The parameter D _r1d × K _v seems to capture the overall roughness characteristics of natural rock joints well. The one-dimensional modified divider technique was extended to two dimensions to quantify the two-dimensional roughness of rock joints. The developed technique was validated by applying to a generated fractional Brownian surface with fractal dimension equal to 2.5. It was found that the calculated fractal parameters quantify the rock joint roughness well. A new technique is introduced to study the effect of scale on two-dimensional roughness variability and anisotropy. The roughness anisotropy and variability reduced with increasing scale.     

Numerical Investigations of the Dynamic Shear Behavior of Rough Rock Joints

The dynamic shear behavior of rock joints is significant to both rock engineering and earthquake dynamics. With the discrete element method (DEM), the dynamic direct-shear tests on the rough rock joints with 3D (sinusoidal or random) surface morphologies are simulated and discussed. Evolution of the friction coefficient with the slip displacement shows that the 3D DEM joint model can accurately reproduce the initial strengthening, slip-weakening, and steady-sliding responses of real rock joints. Energy analyses show that the strengthening and weakening behavior of the rock joint are mainly attributed to the rapid accumulation and release of the elastic energy in the joint. Then, effects of the surface roughness and the normal stress on the friction coefficient and the micro shear deformation mechanisms, mainly volume change and asperity damage, of the rock joint are investigated. The results show that the peak friction coefficient increases logarithmically with the increasing surface roughness, but decreases exponentially with the increasing normal stress. In addition, the rougher rock joint exhibits both higher joint dilation and asperity degradation. However, high normal stress constrains the joint dilation, but promotes the degree of asperity degradation significantly. Lastly, the effects of the 3D surface morphology on the shear behavior of the rock joint are investigated with a directional roughness parameter. It is observed that the anisotropy of the surface roughness consequently results in the variation of the peak friction coefficient of the joint corresponding to different shearing directions as well as the micro shear deformation mechanisms, e.g., the extent of joint dilation.     

Two-dimensional patterns of metamorphic fluid flow and isotopic resetting in layered and fractured rocks'pa

One-dimensional advection-dispersion models predict that characteristic δ¹⁸O vs. distance and δ¹⁸O vs. δ¹³C profiles should be produced during isothermal metamorphic fluid flow under equilibrium conditions. However, the patterns of isotopic resetting in rocks that have experienced fluid flow are often different from the predictions. Two-dimensional advection-dispersion simulations in systems with simple geometries suggest that such differences may be as a result of fluid channelling and need not indicate disequilibrium, high dispersivities, or polythermal flow. The patterns of isotopic resetting are a function of: (1) the permeability contrast between more permeable layers ('channels') and less permeable layers ('matrix'); (2) the width and spacing of the channels; (3) the width and spacing of discrete fractures; and (4) the orientation of the pressure gradient with respect to layering. In fractured systems, the efficiency of isotopic transport depends on the fracture aperture and the permeability of the surrounding rock. Resetting initially occurs along and immediately adjacent to the fractures, but with time isotopic resetting because of flow through the rock as a whole increases in importance. Application of the one-dimensional advection-dispersion equations to metamorphic fluid flow systems may yield incorrect estimates of fluid fluxes, intrinsic permeabilities, dispersivities, and permeability contrasts unless fluid flow occurred through zones of high permeability that were separated by relatively impermeable layers.     

Correlations Developed for Estimation of Hydraulic Parameters of Rough Fractures Through the Simulation of JRC Flow Channels

The hydro-mechanical response of fractured rock masses is complex, due partly to the presence of fractures at different scales. Surface morphology has a significant influence on fluid flow behaviour of a fracture. Different empirical correlations and statistical models have been proposed to estimate the equivalent hydraulic aperture and determine the pressure drop along a fracture. However, the existing models suffer from not being adequately generalised to be applicable to a wide range of real fracture surfaces. To incorporate the effect of profile roughness in the hydro-mechanical behaviour of fractured rock masses, the joint roughness coefficient (JRC) is the most widely used empirical approach. However, the average JRC of two fracture walls in fluid flow analysis, as is a common practice, appears to be inappropriate. It will be shown how different combinations of pairs of JRCs could lead to a similar JRC value. Also, changing the position of the top and bottom walls of a fracture can significantly change the hydraulic response of the fracture while the average JRC is identical in both cases. In this paper, correlations are developed which are based on the simulation of JRCs using estimated fluid flow parameters of 2D fractures can be estimated. In order to widen the application range of the correlations, JRC flow channels were generated: these are 2D channels with their top and bottom walls being made from two of the JRC profiles. To estimate the JRC of linear profiles, a correlation developed between JRC and a newly developed Riemannian roughness parameter, D _R1, is proposed. Considering ten JRC profiles, a total of 100 JRC flow channels were generated. In order to only investigate the effect of surface roughness on fluid flow, the minimum closure between the top and bottom walls of JRC flow channels were considered to be constant. Three cases with minimum closures of 0.01, 0.05 and 0.10 cm were considered in this study. All JRC flow channels were subjected to fluid analysis using FLUENT software. Based on these results, correlations were developed between the geometrical and hydraulic properties of flow channels. Analysis of several real fractures demonstrated the applicability of these correlations.     

Modified governing equation and numerical simulation of seepage flow in a single fracture with three-dimensional roughness

Roughness and tortuosity influence groundwater flow through a fracture. Steady flow through a single fracture can be described primitively by the well-known Cubic Law and Reynolds equation with the assumption that the fracture is made of smooth parallel plates. However, ignoring the roughness and tortuosity of the fracture will lead to inaccurate estimations of the flow rate. To obtain a more accurate flow rate through a rough fracture, this paper has derived a modified governing equation, taking into account the three-dimensional effect of the roughness. The equation modifies the Reynolds equation by adding correction coefficients to the terms of the flow rates, which are relative to the roughness angles in both the longitudinal and transverse directions. Experiments of steady seepage flow through sawtooth fractures were conducted. The accuracy of the modified equation has been verified by comparing the experimental data and the theoretical computational data. Furthermore, three-dimensional numerical models were established to simulate the steady flow in rough fractures with the triangular, sinusoidal surfaces and the typical joint roughness coefficient (JRC) profiles. The simulation results were compared with the calculation results of the modified equation and the current equations. The comparison indicates that the flow rate calculated by the modified equation is the closest to the numerical result.     

Settling velocities of particulate systems 15: Velocities in turbulent Newtonian flows

The success in previous papers of this series in obtaining closed form solutions to the sedimentation of individual particles and to the suspensions of particles in fluids motivated the application of the same procedure to the turbulent flow of Newtonian fluids in conduits.The purpose of this paper is to present explicit equations for the Fanning friction factor and of the average velocity of a Newtonian fluid in a cylindrical tube, for any value of the Reynolds number and dimensionless roughness.Previously published equations for the friction coefficient are either implicit or explicit in the friction coefficient but dependent on the Reynolds number. Since the Reynolds number depends on the pipe diameter and fluid velocity, in both cases iterative method must be used to calculate the flowrate.By defining a dimensionless pipe diameter and a dimensionless average fluid velocity, and using conduit-flow material properties to define characteristic size and velocity parameters, explicit equations were obtained for the friction coefficient and for the average flow velocity valid for different roughness and flow regimes.     

岩石节理面形态与水力特性研究

通过节理面粗糙度测量、张开度模拟、节理渗流试验及有限元数值分析,探讨了岩石节理复杂的水力特性。研究表明,节理渗流具有明显的不均匀性、各向异性及与变形耦合等性质,节理的渗透系数与法向应力之间呈负指数关系。     

Effects of linking up of discontinuities on fracture growth and groundwater transport

It is proposed that the growth of fractures is the basic process for generating and maintaining permeability in solid rock (bedrock). Many extension fractures grow as hydrofractures, whereas many shear (and extension) fractures grow through the formation of transverse fractures that connect the adjacent tips of existing fractures. In a boundary-element analysis, the hydrofractures are modeled as being driven open by a fluid overpressure that varies linearly from 10 MPa at the fracture centre to 0 MPa at the fracture tip. The host rock has a uniform Young's modulus of 10 GPa, a Poisson's ratio of 0.25, and is dissected by vertical joints and horizontal contacts, each of which is modeled as an internal spring of stiffness 6 MPa m⁻¹. The number of joints and contacts, and their location with respect to the hydrofracture tip are varied in different model runs. The results of the analyses indicate that the tensile stresses generated by overpressured hydrofractures open up joints and contacts out to considerable distances from the fracture tip, so that they tend to link up to form a hydraulic pathway. Using the same Young's modulus, Poisson's ratio, and internal spring constant for joints as in the hydrofracture models, boundary-element models were made to study the interaction stresses that cause neighbouring joints to become interconnected through the growth of linking transverse fractures that, ultimately, may evolve into shear fractures. The models were subjected to tensile stress of 6 MPa acting normal to the joint planes as the only loading. The offset (horizontal distance) and underlap (vertical distance) between the adjacent tips of the joints were varied between model runs. The results show a concentration of tensile and shear stresses in the regions between the neighbouring tips of the joints, but these regions become smaller when the underlap of the joints decreases and changes to overlap. These stress-concentration regions favour the development of transverse (mostly shear) fractures that link up the nearby tips of the joints, so as to form a segmented shear or extension fracture. Analytical results on aperture variation of a hydrofracture in a homogeneous, isotropic rock are compared with boundary-element results for a hydrofracture dissecting layered rocks. The aperture is larger where the hydrofracture dissects soft (low Young's modulus) layers than where it dissects stiff layers. Aperture variation may encourage subsequent groundwater-flow channeling along a pathway generated by a hydrofracture in layered rocks. Electronic Publication     

Review of a new shear-strength criterion for rock joints

Barton, N., 1973. Review of a new shear-strength criterion for rock joints. Eng. Geol., 7: 287–332.

The surface roughness of rock joints depends on their mode of origin, and on the mineralogy of the rock. Amongst the roughest joints will be those that formed in intrusive rocks in a tensile brittle manner, and amongst the smoothest the planar cleavage surface in slates. The range of friction angles exhibited by this spectrum will vary from about 75° or 80° down to 20° or 25°, the maximum values being very dependent on the normal stress, due to the strongly curved nature of the peak strength envelopes for rough unfilled joints.

Direct shear tests performed on model tension fractures have provided a very realistic picture of the behaviour of unfilled joints at the roughest end of the joint spectrum. The peak shear strength of rough—undulating joints such as tension surfaces can now be predicted with acceptable accuracy from a knowledge of only one parameter, namely the effective joint wall compressive strength or JCS value. For an unweathered joint this will be simply the unconfined compression strength of the unweathered rock. However in most cases joint walls will be weathered to some degree. Methods of estimating the strength of the weathered rock are discussed. The predicted values of shear strength compare favourably with experimental results reported in the literature, both for weathered and unweathered rough joints.

The shear strength of unfilled joints of intermediate roughness presents a problem since at present there is insufficient detailed reporting of test results. In an effort to remedy this situation, a simple roughness classification method has been devised which has a sliding scale of roughness. The curvature of the proposed strength envelopes reduces as the roughness coefficient reduces, and also varies with the strength of the weathered joint wall or unweathered rock, whichever is relevant. Values of the Coulomb parameters c and Φ fitted to the curves between the commonly used normal stress range of 5–20 kg/cm² appear to agree quite closely with experimental results.

The presence of water is found in practice to reduce the shear strength of rough unfilled joints but hardly to affect the strength of planar surfaces. This surprising experimental result is also predicted by the proposed criterion for peak strength. The shear strength depends on the compressive strength which is itself reduced by the presence of water. The sliding scale of roughness incorporates a reduced contribution from the compressive strength as the joint roughness reduces. Based on the same model, it is possible to draw an interesting analogy between the effects of weathering, saturation, time to failure, and scale, on the shear strength of non-planar joints. Increasing these parameters causes a reduction in the compressive strength of the rock, and hence a reduction in the peak shear strength. Rough—undulating joints are most affected and smooth—nearly planar joints least of all.     

岩石结构面粗糙度系数尺寸效应的推拉试验研究

粗糙度系数是结构面抗剪强度的主要影响因素,然而由于结构面表面形态的复杂性,粗糙度系数尺寸效应研究并未获得较大进展。总结了结构面粗糙度系数的3种获取手段：标准剖面对比法、理论公式法、试验反分析法。在此基础上分析了3种方法在研究粗糙度系数尺寸效应方面存在的问题和困难。为了研究结构面粗糙度系数与试样尺寸的相关度,对中砂、硅粉、水泥、非引气型萘系减水剂等原材料的配比进行了研究,获得了与天然钙质板岩物理力学特性相类似的岩石模型材料,然后采用研发的结构面制作模具及其制备工艺制作了8组共176对具有不同尺寸和表面起伏粗糙程度的结构面,并利用改进的高精度岩石结构面推拉仪对结构面粗糙度系数进行了推拉试验研究和数据统计分析,结果表明：模型结构面粗糙度系数的统计均值随试样尺寸的增加而降低,但特定结构面粗糙度系数的尺寸效应规律需要根据结构面的具体表面形貌进行测试;Barton理论公式计算的结构面粗糙度系数尺寸效应变化规律与推拉试验测试规律总体上一致,但试验值与理论值有差异,且结构面试样尺寸越小,二者的差异就越大;具有特定表面形貌的模型结构面粗糙度系数也有差异,工程大尺寸岩体结构面粗糙度系数需要根据表面形貌和分布特征进行综合判定。     

低应力下岩石裂隙水力特性

针对地表附近30～50 m的岩体裂隙处于一种低围压状态（约1 MPa左右）这一情况,提出了低应力状态下裂隙变形的两点假设：（1）裂隙面处于线弹性状态;（2）隙宽减小量相对于初始力学隙宽是一个微小量。结合Barton公式,建立了低法向应力下裂隙渗流的半经验理论公式,并从实验的角度验证了该公式的合理性。实验结果和理论分析表明：低法向应力作用下,裂隙渗透系数随法向应力的增加而线性降低。此外,给出了一种间接确定裂隙初始力学隙宽的方法。