Complex aperture networks

Physica A 392 (2013) 1028–1037

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Complex aperture networks H.O. Ghaffari a,∗ , M. Sharifzadeh b , R. Paul Young a a

Department of Civil Engineering and Lassonde Institute, University of Toronto, Toronto, 170, College Street, M5S 3E3, ON, Canada

b

Department of Mining, and Metallurgical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave. Tehran, 15875-4413, Iran

article

info

Article history: Received 25 May 2011 Received in revised form 1 November 2012 Available online 7 November 2012 Keywords: Friction Contact areas Aperture evolution Rock joint complex network

abstract A complex network approach is proposed for studying the shear behavior of a rough rock joint. Similarities between aperture profiles are established, and a functional complex network—in each shear displacement—is constructed in two directions: parallel and perpendicular to the shear direction. We find that the growth of the clustering coefficient and that of the number of edges are approximately scaled with the development of shear strength and hydraulic conductivity, which could possibly be utilized to estimate and formulate a friction law and the evolution of shear distribution over asperities. Moreover, the frictional interface is mapped in the global–local parameter space of the corresponding functional friction network, showing the evolution path and, eventually, the residual stage. Furthermore, we show that with respect to shear direction, parallel aperture patches are more adaptable to environmental stimuli than perpendicular profiles. We characterize the pure-contact profiles using the same approach. Unlike the first case, the later networks show a growing trend while in the residual stage; a saturation of links is encoded in contact networks. © 2012 Elsevier B.V. All rights reserved.

1. Introduction A thorough understanding of the behavior of rock joints or fault surfaces is paramount for the study of abrupt motion, seismicity or flow patterns in geomaterials. The evolution of macroscopic friction in frictional interfaces originates from a sequence of contact area variations [1–3]. The formation and rupture of new contact areas (∼bonds/junctions) between two surfaces result in stick–slip motion. Classic characteristics of stick–slip motion include fast frictional strength drop and the release of energy as spikes in ultrasonic waveforms. Recent findings suggest that stick–slip motion is related to the collective interactions of contact areas (Refs. [1–10]). Dissecting the aforementioned collective behavior into a list of interacting units, using complex networks, provides a new mathematical–statistical framework for analyzing a wide range of complex systems [11–17]. In the geosciences, complex earthquake networks, climate networks, volcanic networks, and river networks, which require large-scale measurements, have been taken into account. On a smaller scale, topological complexity has been evaluated with regard to the gradation of soil particles, fracture networks, and apertures of fractures, and the mechanical behavior of granular materials [18–36]. When considering the direct relationship between void spaces and contact areas, one may be interested in considering the induced topological complexity of the opening elements (i.e., apertures) in the fracture treatment. Using linear elastic fracture mechanics, we know that aperture patterns generally are the indexes for the available energy in the growth of a rupture and the eventual fracture length. Also, variations of fluid flow features are controlled directly by aperture patterns. Owing to the complexity of aperture patterns, the understanding and characterization of the features of energy release

∗

Corresponding author. Tel.: +1 64750250. E-mail address: [email protected] (H.O. Ghaffari).

0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.11.001

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Fig. 1. Methodology for extracting aperture networks from: (a) the evolution of apertures through 20 mm shear slip of an interface; each pixel corresponds to a relative contact area (i.e., aperture); (b) visualization of the similarity matrix and functional friction networks achieved.

and fluid flow are of primary interest in this study. The current work will elaborate upon the Euclidean measure of an aperture network, addressing new aspects, such as: the resolution effect, the local–global (c–k) space of complex aperture networks, and stair-like profiles (contact profiles). We will show the scaling of the development of the frictional forces with the attributes of the proper networks, which will give the approximate evolution of the shear stresses acting on the profiles. The next section details our methods and covers the general aspects of network measures. Subsequently, the results of experiments on a rock joint and the complex networks arranged are shown. 2. Networks on apertures To set up a network on the apertures of a joint – or, more generally an interface, under a certain amount of normal stress – we consider patches (or profiles) of elements where each profile is a line with an extra dimension corresponding to the aperture magnitude (i.e., 2 + 1D). Each element on the profile is a pixel with a certain dimension, indicating relative contact area or aperture size. For apertures with zero magnitude, the terms contact areas and asperities are used. In the aperture patterns, we consider each aperture profile as a node. We define X-profiles as the aperture profiles perpendicular to the shear direction and Y-profiles as those parallel to the shear direction. To draw an edge between two nodes, a relationship (or, in general, a relationship matrix) should be defined (Fig. 1). By assuming that there are some hidden metrics between two nodes, or similar functionality between two profiles, the similarity between the nodes is captured. As the simplest metric for a certain profile’s state (X or Y), the Euclidean distance, we will have

dEuc .

  np  (p(x1 , x2 , . . . , xn ) − q(x1 , x2 , . . . , xn ))2 , = p,q=1

(1)

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where p, q are the ith profiles, through np profiles. xn is the height (or magnitude of the aperture) of the nth pixel (or cell) in a profile. When d ≤ ξ , an edge is created between two nodes. The sensitivity of the network parameters with regard to aperture networks has been investigated in Ref. [33]. Here, we use ξ as 5–20 per cent of the maximum d. The selection of the threshold value can be related to the current state of the system and constrained by or dependent on other network parameters [20,22,32,37–39]. By pre-setting either a threshold value or a dependence on the networks or system parameters, the backbone and main features of the corresponding system should be obtained. We notice that the proposed value for the truncation of the Euclidean metric is compatible with minimum variation of the density of edges (⟨k⟩) or with relatively stable local information flow (i.e., betweenness centrality) [32]. From another perspective, the collection of patches (profiles) can be assumed to be an information series in a spatial form. This interpretation is useful for comparing the method described with the characterization of non-linear time series analysis using network theory [37–39]. One of the main attributes of networks is the clustering coefficient. The clustering coefficient shows the collaboration among the connected nodes. Assume the ith node to have ki neighboring nodes. There can exist at most ki (ki − 1)/2 edges between the neighbors (the local complete graph). If we define ci as the ratio [12,14] ci =

actual number of edges between the neighbors of the ith node ki (ki − 1)/2

(2)

then the clustering coefficient is given by the average of ci over all the nodes in the network: N 1 

ci . (3) N i=1 For ki ≤ 1, we define C = 0. The closer C is to 1, the larger the interconnectedness of the network is. The clustering coefficient can also give an indication of the number of triangles around each node. Another property which can be used in characterization is the connectivity distribution (or degree distribution), P (k), which is the probability of finding nodes with k edges in a network. In large networks, there will always be some fluctuations in the degree distribution. Large fluctuations from the average value (⟨k⟩) refer to the highly heterogeneous networks, while homogeneous networks display low fluctuations. There are different kinds of network models which have been investigated extensively. For example, the random Erdős–Rényi networks [40], the small-world Watts–Strogatz networks [11,14–16], and the scale-free Albert and Barabási networks [12,13] are the most well-known network models. Regular networks have a high clustering coefficient (C ≈ 3/4) and a long average path length. Random networks (whose construction is based on random connections of nodes) have a low clustering coefficient and the shortest possible average path length. The small-world networks have high clustering coefficients and small (much smaller than the regular ones) average path lengths [16]. C =

3. Results In this part, we focus on the experimental results, analyzing the hydro-mechanical properties of the rock joint using the approach described in the previous section. It is recognized that the network anatomy is very important in the characterization of the system’s output (because the structure always affects the function). Our objective is to determine the possible relations between the network properties constructed and the measured mechanical/hydraulic properties of a rock joint. To proceed, the results of several laboratory experiments were used. In the experiments, the geometry of the joint, consisting of two surfaces and the aperture between these two surfaces, was measured. The shear and flow tests were performed later on. The rock was granite with a unit weight of 25.9 kN/m3 and a uniaxial compressive strength of 172 MPa. An artificial rock joint was made at the specimen’s mid-height by splitting. A special joint-creating apparatus, which has two horizontal jacks and a vertical jack [1,9], is used for this purpose. The sides of the sample are cut down after creating the joint. The sample’s final size was 180 mm in length, 100 mm in width, and 80 mm in height. A virtual mesh with a square element size of 0.2 mm was generated on each surface, and a laser scanner measured the aperture height at each position. The details of this procedure can be found in Refs. [1,9]. Fig. 2 highlights relevant information about the measured parameters. The samples were subjected to a constant normal stress of 1, 3 and 5 MPa (see Fig. 3(a)) in different experiments, while the variation in surface heights was recorded. In this study, we focus on the patterns obtained from the test with a 3 MPa normal stress. In the experiments, a special hydraulic testing unit is used to allow intermittent linear flow with the shear displacements of the rock joint. The relationship between the hydraulic conductivity and the joint aperture is given by Darcy’s law: Q = Kh Ai,

(4)

where Q , A, i, Kh are the amount of volumetric flow, area, hydraulic gradient and hydraulic conductivity respectively. In the use of this equation, it is assumed that the joint consists of two smooth parallel plates. The flow rate and the hydraulic conductivity can be written as follows: Q = Kh =

ge2h 12υ

(w eh )i,

(5)

,

(6)

ge2h 12υ

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2 Mean Aperture 1 (X-Profile) 0 20 10 SD

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Fig. 2. The information gathered from experimental and network analysis (scatter graph of the measured information/data) during shearing of a rough joint: shear displacement (SD in mm); clustering coefficient of each node in the X-profiles (ci (x)); degree of node (ki (x)); shear stress (SS in MPa); normal displacement (dilatancy versus ND in mm); and hydraulic conductivity (Kh in cm/s). The diagonal plots show the distribution of attributes (without frequency values in this case). Each sub-plot shows two-variable states of the recorded data set through successive shear displacements. For example SD against the mean aperture (of each profile) shows the growth of openings with the displacements; ci (x)–SD and ki (x)–SD show abrupt drops around the peak point (see Fig. 3). The classical behavior of the evolution of frictional forces and the dilatancy of the joint are shown in the SS–SD and ND–SD plots. The spectrum of the network obtained from cumulative measured information is shown as ci (x)–ki (x) (see Fig. 6 for more details).

where g , eh , υ and w are the acceleration due to gravity (m/s2 ), hydraulic aperture, kinematic viscosity of the fluid and width of the specimen, respectively. The analysis is carried out for the rock joint which is under a constant normal stress and is subjected to translational shear displacement with a constant rate of movement (Fig. 1). By setting a threshold value of d ≤ 5 in the Euclidean distance (Eq. (1)) and by building a complex network (Fig. 1(a)) on the X-profiles, the gradual changes of the appearing networks’ adjacency matrix form are obtained. The results demonstrate that, after a phase transition step, the patterns of similarities are restricted to the adjacency of each profile [33]. Except for the boundary profiles, the influence distance on each side of a profile changes from 2 to 20 pixels (0.4–4 mm). A similar procedure is followed for the Y-profile network. It is noticed that the influence distance differs from one profile to another. In other words, in Y-profiles, the influence distance is lower than that of the X-profiles. Also, after a transition stage (near 2 mm), on decreasing the mesh resolution some insignificant features of the variation of the clustering coefficient (scaled with the hydro-mechanical properties) are neglected (Fig. 4). A careful consideration of the changes taking place in the number of edges and the local clustering coefficient during shearing reveals some important properties of the network. Gradual changes of connectivity frequency, either in X-profiles or in Y-profiles, display a similar transitional behavior: a transformation from nearly a single-value function to a semi-stable Gaussian distribution [33]. It can be shown that the degree of nodes changes and a transition to a semi-stable stage occurs with a larger convergence rate. In other words, similarities in Y-profiles occur faster than the similarities in X-profiles. The complementary results obtained from the calculations of clustering coefficients emphasize that the inverse values of C – in X-profiles and Y-profiles – produce patterns similar to those observed in the evolution of the frictional forces and hydraulic conductivity (Fig. 3). If we consider the clustering coefficient as an indication of the circulation and longer transformation of information flow, the inverse of that may be assumed as a sign of the flow conductivity or easy-path flow out of the traps. These result from the alignments of the contacting asperities in the direction perpendicular to the shear slip. Furthermore, the analysis of force chain networks in granular materials has revealed a possible scaling of the clustering coefficient with the shear stress, in both experimental and numerical modeling [31,36].

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Fig. 3. (a) Experimental results for the shear stress versus shear displacement (cases 1, 3, 5 and 6 correspond to different situations of the experimental procedure, depending on the positions of upper and lower boxes and the different normal stresses during the test [1,10]—here, we use case 3 with 3 MPa normal stress), (b) (experimental results) hydraulic conductivity versus shear displacement associated with three different cases, ((c) and (d)) (calculated by using the network model) inverse of the clustering coefficients versus shear displacement on the X-profiles and Y-profiles, respectively.

It is noticed that the sensitivity of the clustering coefficients to the scaling is very high. For instance, with an increase to 5–10 times the virtual meshes (1–2 mm), the details of the C fluctuations are omitted, although the general form of the variations is preserved as is seen in the experimental outputs, particularly for shear displacements 0 ≤ SD ≤ 2 (Fig. 4). Anisotropy in the rock joint behavior related to the shear displacement or fluid flow can be noticed on viewing the plots of CY /CX or KY /KX variations as a function of the shear displacements (Fig. 5). At the quasi-residual stage, these rates are inclined at 0.95 and 0.4. At SD = 3 mm, CY /CX takes its highest value, while for SD  = 2, KY /KX reaches its lowest value. It should also be noted that for 3 ≤ SD ≤ 4,

d(CY /CX ) d(SD)

≺ 0 and at the same time

d(KY /KX ) d(SD)

≻ 0. The aforementioned

ratios are indicators of an anisotropic distribution of the contacting asperities, which induces anisotropic flow with higher permeability in the perpendicular direction [5]. In Ref. [35], we have modeled the fluid flow patterns through the evolution path of the interface using the lattice Boltzmann method. Additionally, we compared the anisotropy of the permeability while discussing its possible relevance to the characteristics of friction networks. j

By introducing a new space associated with ki –ci (i.e., global–local) and assuming 1/⟨cxi ⟩ ∼ τ i ; 1/⟨cy ⟩ ∼ Kh (⟨•⟩ stands for the average value over profiles at a certain displacement), the variation of profiles can be mapped to a parameter space, namely, k– 1c (Fig. 6). The patterns of the clusters in this space can be compared with the variations of the mechanical or j

j

hydro-mechanical properties of a given single joint (Fig. 6(a), (b)). A faster adaptation in Y-profiles distinguishes the ky –1/cy parameter space from kix –1/cxi , in which there is a discernible discontinuity between the two large apparent clusters (at j

j

SD = 2 → SD = 4). Using an intelligent clustering method, the parameter space ky –cy is clustered into smaller modules. We use a self-organizing feature map (SOM) [41], which generally reduces the information space on the basis of competition in the hidden layer of a neural network model. With an elementary topology of 20 × 20 (in the second layer of the neural network and over 500 epochs), the dominant structures of the aforementioned space are identified. j

The unraveled clustered space can be discretized into three categories (Fig. 6(c)), as follows: (1) increasing cy –decreasing j ky

(increasing shear slip–decreasing Kh ); (2) increasing

j cy –increasing

j ky

(increasing shear slip–increasing Kh ), and (3)

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Fig. 4. The resolution effect on the clustering coefficient evolution calculated from the network model: when the profiles are selected within intervals of (a) 10 pixels, (b) 5 pixels and (c) 1 pixel. The high resolution view of the measured apertures discloses a much more realistic picture of the structural complexity of the fracture planes.

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Fig. 5. Anisotropy in evolution of parallel and perpendicular networks (calculated on the basis of the network model). (a) Variation of Cy /Cx (clustering coefficients) with the shear displacement SD; (b) variation of Ky /Kx (the node’s degree) with SD during shearing. j

j

increasing cy –slight change of ky (increasing shear slip–slight change of Kh ). In Fig. 6(d) we have shown the evolution path for the profiles over the defined space, which clearly exhibits the transition of the interface to residual stages. The approximate scaling of the inverse clustering coefficient in X-profiles with the shear stress evolution can be used as the criterion for estimating the distribution of frictional forces over the perpendicular profiles throughout fracture evolution (Fig. 7). The evolution of the shear stress distribution demonstrates that, at the initial stages, the frictional forces of profiles cover a

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j

100 -0.17 -0.15 -0.13 -0.11 -0.09 -0.07 -0.05 10 10 10 10 10 10 10

j

j

Fig. 6. (a) Evolving kix –1/cxi state space; (b) changes in the ky –1/cy relation during shear displacements; (c) clustering of ky –cy space found using a SOM (neural network) with initial topology nx = 20, ny = 20 in the competition layer after 500 epochs of training; (d) path evolution of the joint in the KX –CX space. All of the graphs are the results obtained from the calculated network parameters over the observed experimental data set.

wide-ranging Gaussian-like distribution, while at residual stages the concentration of stresses follows a similar distribution with smaller scope (standard deviation). Then, we confirm that the initial stages of the interface’s evolution, in particular before the slip-weakening stage (from 2 to 5 mm shear slip), scale with the spatial diversity of the fraction of loops rather j than the later stages. Similarly, the distribution of cy – which scales with Kh – reveals that the initial distribution of the fraction of the triangles is kept at a nearly constant shape with increasing shear slip (Fig. 7(b)). Thus, we confirm that the study of parallel profiles as regards the shear direction is much more appropriate for the study of fluid flow characteristics in that direction. Obviously, our results can be used to formulate the evolution of the friction with respect to the evolution of networks. This issue will be our future focus in developing a reasonable friction network model for describing the friction patterns. Next, using a different approach, we constructed a network on the basis of pure-contact profiles. In other words, the role of contact zones is separated from that of non-contact zones. Similarly to the previous measure, the Euclidean distance is utilized, while non-contact zones and contact pixels are transformed into 0 and 1, respectively. In other words, we make a stair-like profile and then identify similar profiles on the basis of their functionality. We notice that the amount of pure contacts per profile indicates the potential energy of that patch. With functional networks, we would probably build a general framework to allow us to understand the flow or entrapping of energy or the trajectory of the system in terms of the energy landscape. In particular, construction of the contact networks reveals a new type of contact profile evolution: growing networks either in X-profiles or Y-profiles (Figs. 8 and 9). At the initial stages of evolution, neither active nodes nor edges are evident. This is due to the large fluctuations of contact profiles in stair-like shapes. The variation of the connectivity degree distribution – especially for X-profiles (Figs. 8(a), 9(a)) – shows a transition. For instance, for X-profiles, initial stages are encoded with P (k) ∼ k−α ; k < 100, while later stages are represented by two branches of the distribution, reading for large node degree as follows: P (k) ∼ kα ; k > 100. Increments of displacement induce a lower instability of contact profiles (growing of non-contact areas), forcing an increase in similarity (Figs. 8(c), 9(c)). However, for instance in X-profiles, after a certain step (SD = 6 ; shear displacement), the development of new sites stops while the growth of edges continues. In

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Fig. 7. Evolution of the clustering coefficient frequency (calculated on the basis of the networks obtained, f (cx,y )): (a) distribution of the clustering coefficients over X-profiles

a

cxi

and (b) the Y-profile

j cy

during shear displacements (in mm).

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other words, we observe a clear phase transition from a growing state to a saturation phase of links. The transition from 1 to 6 mm displacement is the slip-weakening stage where fracture energy is released. In addition to this event, with respect to the number of nodes, at ∂∂Nt = 0 (after SD = 5—compare with Fig. 3(a) and (c)), we have (at the same scale) ∂∂Ct < ∂∂Kt which confirms the promotion of connectivity with other nodes. This interval is followed by a reduction in roughness which represents (even though the percentage of contacts goes to a nearly constant value) a continuously increasing similarity of contact profiles (a coherent variation of contact patterns).

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Conclusions We presented a novel approach to the evolution of complex patterns of friction which are formed during the evolution of a rock interface. By considering the complicated behavior of a rough rock joint that is subjected to shear displacements, functional networks were associated with similar functional profiles of the interface. Our solid observations showed that the mechanical or hydro-mechanical responses for the interfaces are scaled with the aperture networks’ characteristics. Within the aperture networks, the observed patterns of clustering coefficient evolution, as a local complexity of networks, indicated the interface’s mechanical or hydraulic answer. Since the global collection of aperture profiles and the local topology (i.e. subgraphs or loops) are interdependent, the interface’s large-scale and global reaction to shear forces can be uncovered by analyzing the local subgraph (here, triangle) structures. In addition, to highlight the functionality of contacts, a separate network based on binary profiles was utilized. Analysis of the results through network parameters showed a transition of the contact clusters’ density related to the shear displacement. The formation of similar contact profiles was significant during initial displacements, but, after a certain amount of displacement, similar contact profiles with three district patterns were observed. Our method can be developed to analyze monitored acoustic signals through the evolution of rock joints or faults. Further studies would be essential to present a proper model of complex aperture networks. We have observed that a possible model for describing the evolution of aperture networks generally involves characterizing the evolution of friction on the interface and it can then be related to the friction models over the interfaces. The latter point of view is completely novel, opening new windows to a better understanding of the friction mechanism in dry and rough interfaces.

Acknowledgments We thank the editor and referees for useful comments and suggestions on the paper. The second author would like to express his sincere thanks to Tetsuro Esaki and Yasuhiro Mitani for giving him the opportunity to do several experimental tests.

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