TY - JOUR
AU1 - Xia,, Caichu
AU2 - Gui,, Yang
AU3 - Wang,, Wei
AU4 - Du,, Shigui
AB - Abstract To determine the void spaces of rock joints under different normal stresses and shear displacements, we mainly introduce a numerical method which was developed based on equivalent void space derived from composite topography. The new method requires the 3D surface data of rock joints, and the normal closure data of the compression test under different shear displacements, while in conventional methods, some disparate materials are inserted between the joint surfaces or special equipments are needed for the measurement of the void space of rock joints without shearing occurs. To apply the technique, a modified 3D box counting method that considers the self-affine fractal property of void spaces was employed to calculate the 3D fractal dimension of the void space. Specially designed experiment was conducted on a cylindrical specimen of artificial joints to explore aperture distribution, and the correlations between void space characteristics, 3D fractal dimension and mean aperture, and normal stress under different shear displacements. The present study focuses on the introduction of the new method for estimating void spaces of rock joints, while the void spaces model obtained contains the combined surfaces roughness and aperture information of rock joints under different normal loads and shear displacements is promising in investigating the mechanical and hydraulic properties during the loading process. rock joint, void space, surface roughness, aperture, fractal dimension 1. Introduction The majority of rocks in the shallow surface of the earth contain joints. In fractured rocks, the coupled hydro-mechanical behavior basically depends on the state of the existing discontinuities. The behavior of these discontinuities is strongly affected by their void geometry resulted from the separation or mismatch of two surfaces. Thus, the investigation and characterization of the void spaces of rock joints are of great significance. Void space is comprised of the separation and contact between two rough joint surfaces. The complexity of the morphology of void space lies in the joint surfaces and stress conditions. Certain parameters, such as joint roughness coefficient (JRC) (Barton and Choubey 1977), slope of asperity (Z2) (Park and Song 2013), joint matching coefficient (JMC) (Zhao 1997), have been proposed for joint surface characterization. These parameters cannot capture the non-stationary property of the void space while the void geometry is only valid at a certain state of stress. As in the case of shear loading conditions, if the stress or position between the two opposing surfaces changes, the void geometry also changes. Experimental studies have been conducted to measure the void spaces of rock joints for hydraulic and mechanical purposes. Injecting disparate materials, such as metals with low melting points, between the joint surfaces is one of the conventional methods used (Gentier et al1989, Yeo et al1998, Kulatilake et al2006, Kulatilake et al2008). However, this method can cause considerable problems, especially when the injected material dose not fill all of the voids or stick to the surfaces of the fracture. After successfully obtaining a metal cast, acquiring a digitized fracture aperture requires superimposing the two surface images of the metal cast correctly. This proves to be very difficult because the smallest error could lead to an entirely false model of the void space. In addition, the method can only be applied to joints under static condition, conducting the method during shearing can be difficult, although certain researchers have performed the method before or after shearing of rock joints. More recently, indirect methods for measuring aperture distribution of void spaces, such as magnetic resonance imaging (MRI), spectrophotometric analysis (SA) of epoxy replicas (Dijk et al1999, Brown et al2000), and x-ray computer tomography (CT) number spatial distribution (Re and Scavia 1999), have been developed. However, these methods can only provide qualitative information on the aperture distribution during shearing. A numerical method to determine void spaces during shearing was reported (Koyama et al2006), but two simplifications of the method result in shear behavior deviates far from that of the natural joints: the initial stable state was decided without normal loading, and the upper and lower surfaces were fixed in the normal direction and only relative lateral displacement occurred during the whole shearing process. In the majority of analyses of the hydro-mechanical behavior of the joints during shearing, aperture is characterized by adding the changes of normal dilation to an initial aperture, based on the assumption that the changes of the aperture during shearing are equal to the dilation of the rock joint (Bandis et al1983, Olsson and Barton 2001, Li et al2008). However, normal dilation only provides partial information about the change in the void space volume that occurs with shear displacement. Thus, the normal dilation cannot reflect the change of the surface topography of the void space. Mandelbrot’s introduction of fractal geometry provides a new approach to describe complex irregular geometrical shapes (Feder 1988). The approach has been used to generate joint profiles for the analytical modeling of the aperture or a few joint surfaces for numerical simulations of the flow in rock joints, which were mostly under normal stress. Most studies focus on the fractal description or creation of 2D profiles (Brown 1985, Brown and Scholz 1985, Jiang et al2006); hardly any literature investigates the accurate quantification of 3D void spaces in fractal, especially when shearing is involved. In the present study, a numerical method to determine the void spaces of rock joints under normal stress and shear displacement was developed. The new method requires the 3D surface data of rock joints, and the normal closure data of compression test under different shear displacements. For the application of the technique, a modified 3D box counting method, which considers the self-affine fractal property of the void space, was employed to calculate the 3D fractal dimension characteristic of the void space. Specially designed experiment was conducted on a cylindrical specimen of artificial rock joints to explore the aperture distribution and the correlations between the void geometry 3D fractal dimension, mean aperture and normal stress under different shear displacements. An empirical model relating to the void geometry 3D fractal dimension and normal stress is improved to consider the influence of shear displacement on the void geometry 3D fractal dimension. Additionally, the proposed method is thought to be useful for further charactering hydraulic properties of rock joints which will be the main topic of our future research. 2. Numerical method for determining void space 2.1. Equivalent void space Brown (Brown 1995) proposed the term composite topography, which is the sum of the heights of two surfaces measured relative to a reference plane for each surface (figure 1), to describe a simple mathematical model of rough-walled rock joints. The contact between two rough surfaces is equivalent to the contact of the composite topography with a reference plane, and the surface roughness of the composite topography provides the mismatching of two rough surfaces. Thus, the composite topography contains the information of the combined roughness of two rock joint surfaces. Considering that the distance Z0 between the two reference planes is fixed and the aperture is the separation between two rough surfaces of rock joints, the aperture A can be obtained simply by subtracting the composite topography from the fixed distance Z0. The aperture A is sufficient to simultaneously characterize the roughness and the aperture distribution of the rock joints because it has the same roughness as composite topography. The distribution of A is actually the void space of the rock joint, but the appearance of aperture A is slightly different from the real void space, thus referred to as the equivalent void space. Figure 1. Open in new tabDownload slide (a) Illustration of schematic cross section of a rough rock joint. The surface heights Z1 and Z2 are measured from two parallel reference planes fixed in each coordinate systems, x1o1z1 and x2o2z2 respectively. Z0 is the fixed separation between two reference planes. (b) The composite topography is defined as the sum of Z1 and Z2, i.e. Zc, in the figure, the aperture A can be obtained by simply subtracting the composite topography Zc from the fixed distance Z0. Figure 1. Open in new tabDownload slide (a) Illustration of schematic cross section of a rough rock joint. The surface heights Z1 and Z2 are measured from two parallel reference planes fixed in each coordinate systems, x1o1z1 and x2o2z2 respectively. Z0 is the fixed separation between two reference planes. (b) The composite topography is defined as the sum of Z1 and Z2, i.e. Zc, in the figure, the aperture A can be obtained by simply subtracting the composite topography Zc from the fixed distance Z0. 2.2. Numerical method for determining void spaces The joint surface is commonly reconstructed from discrete data by digitizing the surface at specified intervals. The 3D surface data used in the present study is the height Z(x,y) of each point on the joint surface, relative to an XOY reference plane, measured by optical measuring equipment. The void spaces can be obtained by Zu(x,y)−Zd(x,y) that the coordinates of the heights of the upper and lower surfaces in the same system, Zu(x,y) and Zd(x,y), are acquired. In practice, the simultaneous direct measurement of two joint surfaces and conducting the shear test is difficult to achieve. In the present study, the initial surface data, and the normal compression data under shear displacement are utilized to estimate the void spaces of the rock joints under different shear displacements. The shear displacement here denotes the change in the relative lateral position effected during periods of separation, the treatment avoids the asperities get worn and damaged in the traditional shear test where shear displacement increases with normal load applied. Moreover, according to Bandis et al (1983), under the normal stress less than 5 MPa, the deformation of the intact rock block occupies less than 10% of the total normal deformation for interlocked joint, and this ratio would be even smaller for mismatched joint. So, the elastic deformations of the intact rock block under normal stress of different shear displacements are assumed to be negligible. Zu_ini(x,y) denotes the height of the upper surface at the initial stage, and Zd(x,y) the height of the lower surface which is fixed during the whole process. At shear displacement of dx and dy, the upper and lower surfaces are superimposed with three stable contact points through changing the normal separation of the two surfaces. The method is described in detail in the author’s another study (Xia et al2010). Thus, the height of the upper surface Zu_ini(x,y) changes to be Zus(x,y): Zus(x,y)=Zu_ini(x−dx,y−dy)+dzs1 where, dzs is the shear caused dilation. In the test conducted for the present study, upper surface shears only in the direction of X. Thus, dy equals zero and equation (1) changes to be: Zus(x,y)=Zu_ini(x−dx,y)+dzs2 Considering that the elastic deformation of the intact rock block is neglected, when the normal loading caused normal closure, dzn, of the upper surface relative to the lower surface occurs, the height of the upper surface Zu(x,y) can be described by: Zu(x,y)=Zu_ini(x−dx,y)+dzs−dzn3 When subtracting Zd(x,y) from Zu(x,y), the void spaces A(x,y) are obtained: A(x,y)=Zu(x,y)-Zd(x,y)                =Zu_ini(x-dx,y)-Zd(x,y)+dz4 where, dz=dzs−dzn is the normal displacement under the combined effects of normal stress and shear displacement. Zu_ini(x−dx,y)+dzs and Zd(x,y) would have been decided after determining the stable contact states at each shear displacement; only dzn needs to be obtained from the rock joint compression test. For any A(x,y)<0, A(x,y) should be set at 0 (grey regions in figure 2). Regions where A(x,y)<0 are actually overlapping areas between the two surfaces. The overlapping areas would, in nature, be crushed, deform elastically or plastically, thus, changing the topography of the joint surfaces. The numerical method in the present study provides a way to explore the nature of the void spaces of rock joints at a small shear displacement and normal loading, where the production of crushed material is minimal and can be neglected (the maximum normal stress was 5 MPa). Then, the overlapping areas represent the areas of initially contacted asperities that would deform elastically or plastically under the further applied stress. Figure 2. Open in new tabDownload slide Sketch for estimating the void space A(x,y), Y direction is neglected for simple because dy equals zero in the whole process. Grey regions in the figure indicate A(x,y)<0 that should be set at 0. Figure 2. Open in new tabDownload slide Sketch for estimating the void space A(x,y), Y direction is neglected for simple because dy equals zero in the whole process. Grey regions in the figure indicate A(x,y)<0 that should be set at 0. In the present study, the mean aperture and 3D fractal dimension were selected to demonstrate the application of the void space model. The mean aperture reflects the volumetric characterization, whereas the 3D fractal dimension captures the roughness of the void space. The mean aperture and the 3D fractal dimension together have a higher chance of characterizing the geometry of the void space. After acquiring the void space A ⁠, the mean aperture b can be calculated from: b=V(A)S XOY5 where, V(A) is the volume of A, S XOY is the area projected on the XOY reference plane. 2.3. The modified 3D box counting method The new approach of fractal geometry provides a mathematical model for many complex objects found in nature (e.g. coastlines, mountains, and the void spaces of rock joints). These objects are usually very complex. Possessing the characteristic sizes of these objects and describing them using the traditional Euclidean geometry is difficult. However, a fractal dimension can quantify these objects easily. Box counting method is one of the most frequently used techniques to estimate the fractal dimension. The original box counting method (Li et al2009) is to record the number Nd of boxes of the size d needed to cover the object of self-similarity property; the number Nd changes along with the different box sizes. If log(Nd) is plotted against log(d), then the fractal dimension D could be estimated from the slope Dc of the linear regression line. Therefore, the estimated fractal dimension Dc can be expressed as follows: Nd=1dDc6 Dc=−log(Nd)log(d)7 The reason for the frequent use of the traditional box counting method lies in its simplicity and automatic computability. However, the traditional box counting method cannot be directly applied to self-affine objects because this leads to a false fractal dimension. Brown (1989) suggests the modification of the conventional box counting method when calculating an object of self-affine fractal properties by: (1) making the ranges for the ordinate of the fractal function equal to the maximum dimension of the object along each axes and (2) keeping the number of intervals along each axes equal while changing in the whole calculation process. The modified method has been proved to produce correct fractal dimension for self-affine profiles. Based on the suggestions of Brown, Kulatilake et al (2008) extended the modified 2D box counting method to the 3D box counting method. The main idea of the modified box counting method is to record the number Nm of boxes needed to fill the void spaces, while changing the element box with dimensions of L/n, W/n, and H/n, with n (figure 3). Following the first suggestion of Brown, L, W, and H are determined from the specimen dimension and are commonly not equal in size. L and W are the length and width of the void spatial domain, which are equivalent to the length and width of the sample. H is the largest aperture value available in the void space. According to the second suggestion, n is the number of segments (intervals) of equal size in length, width, and height. Thus, 1/n is equivalent to the rate that the element box dimensions change in. In this study, n is first assigned 1 and is increased by multiplying 2 each time. Considering that n is inversely proportional to the box dimension, the relation between Nm and n in the modified 3D box counting method is different from the conventional one. The fractal dimension D could be estimated from the slope Db of the linear regression of the plot of log(Nm) versus log(n), written as follows: Nm=nDb8 Db=log(Nm)log(n)9 Figure 3. Open in new tabDownload slide Illustration of the element box dimensions change pattern for the modified 3D box counting method. L, W are the length and width of the sample,respectively, H is the largest aperture value of the void space. Figure 3. Open in new tabDownload slide Illustration of the element box dimensions change pattern for the modified 3D box counting method. L, W are the length and width of the sample,respectively, H is the largest aperture value of the void space. 3. Description of experiment 3.1. Specimen preparation The experiment was performed on a marble joint specimen, which is a cylindrical rock core with a diameter of 50 mm and height of 100 mm. The rock core was drilled from an intact marble rock block taken from the construction site at the Jin-ping II hydropower station in Xichang, Sichuan Province, China. Artificial tensile-fractured rock joints were created using Brazilian test (figure 4), a method suggested by ISRM to determine the splitting tensile strength of brittle materials. Before the test, two 2 mm-wide grooves were cut on the opposing side of the sample along the axial direction. Normal strip loads were then applied along the wedges, the loads increasing gradually until a tensile joint was generated. The joint surface generated was 50 mm × 100 mm (figure 5). The two surfaces are considered interlocked and unweathered because the specimen were prepared artificially in the laboratory and were never destroyed by weathering or erosion. Figure 4. Open in new tabDownload slide Brazilian test used for creating an artificial fracture. Figure 4. Open in new tabDownload slide Brazilian test used for creating an artificial fracture. Figure 5. Open in new tabDownload slide A photograph of the specimen. Figure 5. Open in new tabDownload slide A photograph of the specimen. 3.2. Measurement of joint surfaces A 3D scanner (figure 6) which is based on the triangulation measurement principle was employed to obtain the topographical data of the rock joint surfaces and calculate the void spaces. The basic principle of the measurement is that a digital grating projection device projects a series of fringes of continuous width ranges on the surface of object. The fringes distort with the surface topography of the object, and two cameras can catch this distortion from different visual angles. Then, based on the triangulation principle between the camera and projector, the 3D information is obtained through calculating the offsets (or parallax) of the pixels in one graph. The scanner can sample a 200  ×  150 mm area on a minimum spacing of 140 µm and sample a 400  ×  400 mm area on a minimum spacing of 280 µm, it takes only approximately 5 s to scan a sample. In the present study, the sample spatial was 100  ×  50 mm, and measured with a interval of 280 µm and an accuracy of 20 µm. Xie et al (1997) conducted tests to investigate the influence of sampling intervals on the fractal dimension. Results showed that the difference between the fractal dimension at intervals of 0.125 mm and 1 mm is less than 0.05. Hence, the sampling interval of 280 µm used for the present study is thought to be satisfactory. Figure 7 shows the surfaces of the specimen measured by the scanner. Figure 6. Open in new tabDownload slide A photograph of the 3D scanner, a PC computer performs data collecting and processing in real time. Figure 6. Open in new tabDownload slide A photograph of the 3D scanner, a PC computer performs data collecting and processing in real time. Figure 7. Open in new tabDownload slide 3D model of the specimen based on measured surface data before test. The two figures show the two surfaces of the specimen. Figure 7. Open in new tabDownload slide 3D model of the specimen based on measured surface data before test. The two figures show the two surfaces of the specimen. 3.3. Compression test As mentioned in section 2.2, the algorithm for void spaces requires the displacement dzn of the upper surface relative to the lower surface under normal stress. dzn was determined from the compression test. The specimen used in the present study is the joint created in a cylindrical core by Brazilian test. Normal stress should be the confining pressure around the specimen. A triaxial compression apparatus was employed to perform the test (figure 9(a)). Before the test, the relative lateral position of the joint was offset by sticking two semicircle steel sheets to the opposite ends of the two halves of the specimen (figure 8). The steel sheets have the same diameter of 50 mm and heights equal shear displacement (the height of the steel sheet equals the shear displacement, e.g. 2 mm steel sheet height for 2 mm shear displacement). Five shear displacements of 1, 2, 3, 4, and 5 mm were tested. The assembly was sealed with a thin, deformable jacketing material, and then installed between a steel top-cap and a pedestal. The jacket prevents the confining fluid from penetrating into the sample and allows to control and monitor the confining pressures during testing. Two ends of the sample were fixed by applying a small axial load on the steel top-cap just before the test; the load was kept constant during testing. Two displacement transducers were mounted around the sample to measure the normal closure (figure 9(b)). The specimen was tested with the shear displacements of 1, 2, 3, 4, and 5 mm, each under the increasing normal loads of 1, 2, 3, 4, and 5 MPa, respectively. Figure 10 shows the joint closure test results for all cases. Figure 8. Open in new tabDownload slide Shear displacements were obtained by sticking semicircle steel sheet of different height to the opposite ends of the two halves of the specimen. The height of steel sheet equals the shear displacement needed. (a) Illustrates how this method works; (b) and (c) are photographs of the specimen at initial state and after assembling, respectively. Figure 8. Open in new tabDownload slide Shear displacements were obtained by sticking semicircle steel sheet of different height to the opposite ends of the two halves of the specimen. The height of steel sheet equals the shear displacement needed. (a) Illustrates how this method works; (b) and (c) are photographs of the specimen at initial state and after assembling, respectively. Figure 9. Open in new tabDownload slide Triaxial compression apparatus used to conduct the test: (a) gives a general illustration of the apparatus, shear load is just to apply a small constant axial load to fix the specimen; (b) a photograph of installing the assembly. Figure 9. Open in new tabDownload slide Triaxial compression apparatus used to conduct the test: (a) gives a general illustration of the apparatus, shear load is just to apply a small constant axial load to fix the specimen; (b) a photograph of installing the assembly. Figure 10. Open in new tabDownload slide Test results of normal stress versus joint closure of the specimen under different shear displacements. Figure 10. Open in new tabDownload slide Test results of normal stress versus joint closure of the specimen under different shear displacements. 4. Data analysis and discussion 4.1. Aperture distribution Figure 11 shows the 3D geometry of the void spaces of the specimen under different shear displacements at the state of zero normal stress. The aperture distribution is relatively isotropic at 1 mm shear displacement, and the joint becomes more anisotropic with increasing shear displacement. The spatial distribution of larger apertures seems to align perpendicularly to the shear direction. Figure 11. Open in new tabDownload slide 3D model of the void spaces under shear displacements of (a) 1 mm, (b) 2 mm, (c) 3 mm, (d) 4 mm and (e) 5 mm at the state of zero normal stress. Figure 11. Open in new tabDownload slide 3D model of the void spaces under shear displacements of (a) 1 mm, (b) 2 mm, (c) 3 mm, (d) 4 mm and (e) 5 mm at the state of zero normal stress. Fluid flow through the rock joints has been commonly modeled by applying the cubic law followed from Paul Witherspoon’s original work to determine the applicability of the cubic law to natural rock joints under normal stress. The cubic law is expressed as the flow rate of fluid passing through two smooth parallel plates is proportional to the distance between the two plates (Liu 2005), which can be written as: Q=−wge312v∇H10 where Q is the flow rate of the fluid, w is the width of the plate, g is the acceleration due to gravity, v is the kinematic viscosity of the fluid, e is the distance between the two plates (hydraulic aperture) and ∇H is the dimensionless hydraulic gradient. The cubic law is only valid for the smooth parallel plates and the fluid through which is incompressible, laminar and stationary. When the cubic law is applied to natural rock joints with rough surfaces, a correction factor should be added to account for deviations from the ideal conditions assumed in the parallel smooth plate theory, which means the mean aperture, b (the geometrically measured aperture, is usually called mechanical aperture), is generally larger than e owing to the wall friction and the tortuosity result from the complicated geometry of the void space (Zimmerman and Bodvarsson 1996). From figure 11, we can find that the spatial distribution of connected larger apertures seems to align perpendicularly to the shear direction, and the phenomenon appears more evident with the increase of shear displacement. According to the cubic law, fluid flow rate would be larger in those connected larger apertures perpendicularly to the shear direction, the forming of the connected larger apertures is accepted as channeling effect (Tsang and Tsang 1987, Moreno et al1988). This is in accordance with the phenomena other researchers have observed in laboratory experiments or numerical simulations, where the flow rate perpendicular to the shear direction is larger than those along the shear direction (Archambault et al1997, Gentier et al1997, Gentier et al2000). Although it has been clearly demonstrated by early research that void geometry plays a significant role in fluid flow and the evolution of flow paths with normal stress and shear displacement, much work remains in defining geometrical parameters and determining their roles in hydraulic behavior, and the new technique proposed in the present study provide a method for accurate quantification of parameters account for rock joint hydraulic behavior. All those indicate that the numerical method proposed in the present study is promising in future investigations of hydraulic behavior related parameters and the shear-induced flow anisotropy. 4.2. New equation relating to 3D void geometry fractal dimension, normal stress and shear displacement Table 1 lists the calculation results for the 3D void geometry fractal dimension, Db, which is calculated by the 3D modified box counting method, and the mean aperture, b, both based on the 3D model of the void spaces as in figure 11. Table 1. Calculation results for 3D void geometry fractal dimension, Db, and mean aperture, b ⁠. Normal stress . . Shear displacement . 1 mm . 2 mm . 3 mm . 4 mm . 5 mm . 0 MPa Db 2.51 2.56 2.61 2.65 2.70 b/mm 0.82 1.16 1.46 1.63 1.82 1 MPa Db 2.40 2.47 2.52 2.54 2.57 b/mm 0.44 0.72 0.93 1.09 1.23 2 MPa Db 2.38 2.46 2.50 2.52 2.55 b/mm 0.39 0.62 0.80 0.94 1.08 3 MPa Db 2.37 2.44 2.48 2.50 2.53 b/mm 0.36 0.56 0.73 0.85 0.97 4 MPa Db 2.35 2.43 2.46 2.48 2.50 b/mm 0.33 0.51 0.67 0.79 0.89 5 MPa Db 2.34 2.41 2.44 2.46 2.49 b/mm 0.30 0.48 0.62 0.74 0.83 Normal stress . . Shear displacement . 1 mm . 2 mm . 3 mm . 4 mm . 5 mm . 0 MPa Db 2.51 2.56 2.61 2.65 2.70 b/mm 0.82 1.16 1.46 1.63 1.82 1 MPa Db 2.40 2.47 2.52 2.54 2.57 b/mm 0.44 0.72 0.93 1.09 1.23 2 MPa Db 2.38 2.46 2.50 2.52 2.55 b/mm 0.39 0.62 0.80 0.94 1.08 3 MPa Db 2.37 2.44 2.48 2.50 2.53 b/mm 0.36 0.56 0.73 0.85 0.97 4 MPa Db 2.35 2.43 2.46 2.48 2.50 b/mm 0.33 0.51 0.67 0.79 0.89 5 MPa Db 2.34 2.41 2.44 2.46 2.49 b/mm 0.30 0.48 0.62 0.74 0.83 Open in new tab Table 1. Calculation results for 3D void geometry fractal dimension, Db, and mean aperture, b ⁠. Normal stress . . Shear displacement . 1 mm . 2 mm . 3 mm . 4 mm . 5 mm . 0 MPa Db 2.51 2.56 2.61 2.65 2.70 b/mm 0.82 1.16 1.46 1.63 1.82 1 MPa Db 2.40 2.47 2.52 2.54 2.57 b/mm 0.44 0.72 0.93 1.09 1.23 2 MPa Db 2.38 2.46 2.50 2.52 2.55 b/mm 0.39 0.62 0.80 0.94 1.08 3 MPa Db 2.37 2.44 2.48 2.50 2.53 b/mm 0.36 0.56 0.73 0.85 0.97 4 MPa Db 2.35 2.43 2.46 2.48 2.50 b/mm 0.33 0.51 0.67 0.79 0.89 5 MPa Db 2.34 2.41 2.44 2.46 2.49 b/mm 0.30 0.48 0.62 0.74 0.83 Normal stress . . Shear displacement . 1 mm . 2 mm . 3 mm . 4 mm . 5 mm . 0 MPa Db 2.51 2.56 2.61 2.65 2.70 b/mm 0.82 1.16 1.46 1.63 1.82 1 MPa Db 2.40 2.47 2.52 2.54 2.57 b/mm 0.44 0.72 0.93 1.09 1.23 2 MPa Db 2.38 2.46 2.50 2.52 2.55 b/mm 0.39 0.62 0.80 0.94 1.08 3 MPa Db 2.37 2.44 2.48 2.50 2.53 b/mm 0.36 0.56 0.73 0.85 0.97 4 MPa Db 2.35 2.43 2.46 2.48 2.50 b/mm 0.33 0.51 0.67 0.79 0.89 5 MPa Db 2.34 2.41 2.44 2.46 2.49 b/mm 0.30 0.48 0.62 0.74 0.83 Open in new tab Figure 12(a) shows a plot of the 3D void geometry fractal dimension Db versus the normal stress for five different shear displacements. The changes of Db with normal stress indicate an obvious two-stage behavior. For all cases, Db decreases in a relatively high gradient in the first stage of normal stress (0–1 MPa), and then continues to decrease at a lower gradient that gradually reaching to zero. A similar behavior has also been reported in Kulatilake et al (2008). While there is an important difference between two tests because Kulatilake et al (2008) obtained relations between void geometry fractal dimension and normal stress of three rock fractures at the initial condition (zero shear displacement), and this will be discussed latter. Figure 12. Open in new tabDownload slide The plots of the 3D void geometry fractal dimension Db versus normal stress at different shear displacements (a) and shear displacement under different normal stresses (b). Figure 12. Open in new tabDownload slide The plots of the 3D void geometry fractal dimension Db versus normal stress at different shear displacements (a) and shear displacement under different normal stresses (b). Figure 12(b) shows a plot of the 3D void geometry fractal dimension versus the shear displacement for different normal stresses. Figure 12(b) shows that the relation between the void geometry fractal dimension and shear displacement are different from the obvious two-stage behavior, exhibited by the changes of the void geometry fractal dimension with the increasing of normal stress. Db increases with the shear displacement at a lower gradient, gradually decreasing for all cases except for the zero normal stress case. The increase in the shear displacement causes the matching of two surfaces to decrease, which then causes the void geometry fractal dimension to increase. The zero normal stress case shows a linear relation between the Db and the shear displacement; no sign of changes is noticed until the shear displacement of 5 mm, indicating that normal stress greatly affects the reaction that Db shows to the shear displacement. Kulatilake et al (2008) performed laboratory experiments in which they injected liquefied Wood’s metal into three single fractures at different levels of applied normal stress, without the occurrence of shearing. The fractures were opened up when cooled, and the void spaces were obtained by scanning the solidified filling of Wood’s metal to calculate the 3D fractal dimension. A relation between the 3D void geometry fractal dimension and normal stress was proposed: Db−2(Db)max−2=1AσB+111 Where, 2 in the left denotes the minimum void geometry fractal dimension as normal stress reaches infinity, Db is the void geometry fractal dimension, σ is the normal stress, and A and B are constants determined by a regression analysis. Equation (11) can be rewritten as: Db=(Db)max−2AσB+1+212 Kulatilake et al (2008) applied equation (11) to the relation between Db and σ of three fractures. A regression analysis was performed to estimate the constants A and B ⁠, which lie in the ranges of 0.0455–0.0512 and 0.1804–0.3468, respectively, and the coefficients of standard deviation are 0.063 for A and 0.317 for B ⁠. Thus, equation (12) is an empirical model that effectively describes the relation between Db and σ of the rock joints. Notably, equation (12) does not consider shearing, as the equation was proposed based on normal compression tests where shearing did not occur. In the present study, the numerical calculation results were utilized to improve the model to describe the change of Db under the combined effect of normal stress and shear displacement. Equation (11) is utilized, in conjunction with the shear displacement correction usC ⁠, to describe the relation between Db and σ under different shear displacements: Db−2(Db)max−2=usCAσB+113 where us is the shear displacement values, and C is a constant determined by a regression analysis. Figure 12(b) suggests a positive correlation between Db and the shear displacement us ⁠. Thus, usC is located right up in equation (13). The 3D fractal dimension Db can be represented by the normal stress and shear displacement using equation (14). Db=usC((Db)max−2)AσB+1+214 Regression analyses were performed to estimate the constants A, B, and C; fitting results are shown in figure 13. In figure 13, the tried model fits very well with the relation between Db and normal stress under different shear displacements. The constants A, B, and C lie in the narrow ranges of 0.167–0.259, 0.408–0.517, and -0.938  ×  10-4 to -3.618  ×  10-4. The coefficients of standard deviation are 0.181 for A, 0.103 for B, and -0.396 for C, which confirms the potential promise of the empirical model (14) in describing the relation between Db and σ of the rock joints under the effect of shear displacement. Figure 13. Open in new tabDownload slide The plots of the 3D void geometry fractal dimension Db versus the normal stress and the model fits at different displacements. The fitting functions for each shear displacement are listed to the right of the figure. For the case of 1 mm shear displacement, C is invalid. Figure 13. Open in new tabDownload slide The plots of the 3D void geometry fractal dimension Db versus the normal stress and the model fits at different displacements. The fitting functions for each shear displacement are listed to the right of the figure. For the case of 1 mm shear displacement, C is invalid. 4.3. Relation between mean aperture and normal stress Figure 14(a) shows a plot of the mean aperture b versus the normal stress for five different shear displacements, and figure 14(b) shows a plot of the mean aperture b versus the shear displacements for five different normal stresses. Both figures have similar trends to that of the void geometry fractal dimension, it is unnecessary to go into details here. Figure 14. Open in new tabDownload slide The plots of the mean aperture b versus normal stress at different shear displacements (a) and shear displacement under different normal stress (b). Figure 14. Open in new tabDownload slide The plots of the mean aperture b versus normal stress at different shear displacements (a) and shear displacement under different normal stress (b). 4.4. 3D void geometry fractal dimension with correlation to the mean aperture As discussed in sections 4.2 and 4.3, both the 3D fractal dimension and the mean aperture have similar relations with normal stress and shear displacement. Hence, specified relations between the 3D void geometry fractal dimension and the mean aperture under different normal stresses and shear displacements must exist. Data of the fractal dimension and the mean aperture are plotted in one figure, and then sorted following normal stress and shear displacement or neither. Figure 15(a) shows a plot of the mean aperture versus the 3D void geometry fractal dimension. Spots are sorted based on normal stress; spots in the same normal stress denote the fractal dimension and the mean aperture under different shear displacements. Trials of various functions (linear, power, logarithmic) showed that the logarithmic fit gives the best approximation. As can be seen in figure 15(a), the slope and intercept parameter values of the logarithmic fits lie within a small range, and only one case of zero normal stress shows a little deviation from the others. When the fractal dimension versus the mean aperture is re-plotted as the functions of shear displacement, a different figure is obtained. In figure 15(b), the spots in one shear displacement denote the fractal dimension and the mean aperture under different normal stresses. The figure obviously shows a strong linear relationship. The slope and intercept parameter values of each line are shown in figure 15(b). Figure 15. Open in new tabDownload slide The relation between the 3D void geometry fractal dimension and the mean aperture. The spots in the figures are sorted according to normal stress (a), and shear displacement (b). Linear fit functions are listed under each figure. Figure 15. Open in new tabDownload slide The relation between the 3D void geometry fractal dimension and the mean aperture. The spots in the figures are sorted according to normal stress (a), and shear displacement (b). Linear fit functions are listed under each figure. Figure 16 shows data of the fractal dimension and the mean aperture, all plotted in one figure. In figure 16, the spots distribution clearly shows an ‘S’ shape, which indicates that the changes of the mean aperture with the 3D void geometry fractal dimension show a three-stage behavior. When Db < 2.45 or Db > 2.55, the mean aperture b decreases or increases with Db in a gradient decrease, gradually reaching to zero. In the medium range of 2.45 < Db < 2.55, the mean aperture b shows a linear relationship with Db. Thus, the mean aperture b has upper and lower bounds from two aspects: (1) the mean aperture b tends to be a constant in the low or high Db regions; and (2) Db is bounded between 2 and 3, whereas the bounds of b were practically difficult to determine. Hence, the relation between the mean aperture and aperture fractal dimension provides a way to estimate bounds for the mean aperture. Figure 16. Open in new tabDownload slide The ‘S’ shape spot distribution of the mean aperture versus the 3D aperture fractal dimension. The dotted lines indicate two critical Db values that divide the change of the mean aperture with Db into three stages. Figure 16. Open in new tabDownload slide The ‘S’ shape spot distribution of the mean aperture versus the 3D aperture fractal dimension. The dotted lines indicate two critical Db values that divide the change of the mean aperture with Db into three stages. 5. Conclusion Based on the equivalent void space derived from composite topography, a numerical method was developed to determine the void spaces of rock joints under normal stress and shear displacement. The new method needs the 3D surface data of the rock joints, and the closure data of the compression test of the rock joints under different shear displacements. A modified 3D box counting method was applied to calculate the 3D void space fractal dimension. For the application of the technique, specially designed experiment was conducted on a cylindrical specimen of artificial rock joints to explore the aperture distribution and the correlations between the void geometry 3D fractal dimension, mean aperture and normal stress under different shear displacements: (1) The aperture distribution is relatively isotropic at 1 mm shear displacement, and the fracture becomes more anisotropic with increasing shear displacement. The spatial distribution of the larger aperture seems to align perpendicularly with the shear direction; (2) equation (11) proposed by Kulatilake et al (2008) under initial conditions is improved as equation (14) which considers the effect of shear displacement on the relation between Db and normal stress; (3) The ‘S’ shape relation between b and Db provides a way for the conversion between the two parameters and the estimation bounds for the mean aperture. To better understand the mechanical and hydraulic behavior of rock joints, especially the hydraulic behavior, a good physical model of void space is needed, the present study provides a new method for the estimation of the void space of rock joint, and this permit us to take a first step toward the visualization of void space. The next step is to propose the parameters based on the void space model to better characterize the mechanical and hydraulic behavior of rock joints. Acknowledgments This work was supported by National Natural Science Foundation of China (No.41327001 and No.41472248), the Major State Basic Research Development Program of China (973 Program, No. 2011CB013800), and the program for Changjiang Scholars and Innovative Research Team in University (PCSIRT, IRT1029). References Archambault G , Gentier S , Riss J , Flamand R . , 1997 The evolution of void spaces (permeability) in relation with rock joint shear behaviour , Int. J. Rock Mech. Min. 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TI - Numerical method for estimating void spaces of rock joints and the evolution of void spaces under different contact states
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/11/6/065004
DA - 2014-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/numerical-method-for-estimating-void-spaces-of-rock-joints-and-the-sHuJoJiX1F
VL - 11
IS - 6
DP - DeepDyve
ER -