Permeability, porosity, and percolation properties of two-dimensional disordered fracture networks

Standard and inverse site percolation of triangular tiles on triangular lattices: Isotropic and perfectly oriented deposition and removal

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of triangular tiles of side k (k-tiles) on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, k-tiles are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing k-tiles [composed by k(k+1)/2 monomers] from the lattice. Two schemes are used for the depositing and removing process: the isotropic scheme, where the deposition (removal) of the objects occurs with the same probability in any lattice direction; and the anisotropic (perfectly oriented or nematic) scheme, where one lattice direction is privileged for depositing (removing) the tiles. The study is conducted by following the behavior of four critical concentrations with the size k: (i) [(ii)] standard isotropic (oriented) percolation threshold θ_{c,k} (ϑ_{c,k}), which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k} (ϑ_{c,k}) is reached by isotropic (oriented) deposition of k-tiles on an initially empty lattice; and (iii) [(iv)] inverse isotropic (oriented) percolation threshold θ_{c,k}^{i} (ϑ_{c,k}^{i}), which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i} (ϑ_{c,k}^{i}) is reached after removing isotropic (completely aligned) k-tiles from an initially fully occupied lattice. The obtained results indicate that (1)θ_{c,k} (θ_{c,k}^{i}) is an increasing (decreasing) function of k in the range 1≤k≤6. For k≥7, all jammed configurations are nonpercolating (percolating) states and, consequently, the percolation phase transition disappears. (2)ϑ_{c,k} (ϑ_{c,k}^{i}) show a behavior qualitatively similar to that observed for isotropic deposition. In this case, the minimum value of k at which the phase transition disappears is k=5. (3) For both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(ϑ)=0.5. Thus, a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{c,k}^{i}=1), which has not been observed in other regular lattices. (4) Finally, in all cases, the jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the orientation (isotropic or nematic) or the size k considered. In addition, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and five (three) for isotropic (nematic) deposition scheme, has the same universality class as the standard percolation problem.

Study on Fractal Characteristics of Evolution of Mining-Induced Fissures in Karst Landform

The karst landscape is widespread in the southern region of China. As a result of underground mining activities, the original stress equilibrium is disrupted, causing the redistribution of stress in the overlying rock layer, inducing the longitudinal fracture of mining to expand and penetrate upwards, resulting in the rupture and destabilization of the karst cave roof, thus triggering a series of engineering problems such as karst cave collapse, landslide, the discontinuous deformation of the ground surface, and soil erosion. In order to study the evolutionary characteristics of buried rock fissures in shallow coal seam mining under the karst landform, taking the shallow coal seam with the typical karst cave development landform in Guizhou as the engineering background, based on the similarity simulation experiment and fractal theory, the evolution law of buried rock fissures and network fractal characteristics under the disturbance of the karst landform mining are analyzed. The research shows that the mining-induced fracture reaches the maximum development height of 61 m on the left side of the cave, and the two sides of the cave produce uncoordinated deformation. The separation fracture below the cave is relatively developed, and the overall distribution pattern of the cave rock fracture network presents a “ladder” shape. The correlation coefficient of the fractal dimension of the rock fractures under different advancing distances is more than 0.90, and the rock fracture network under the karst landform has high self-similarity. The variation of fractal dimension with the advancing degree of the working face can be divided into four stages. The first and second stages show an exponential growth trend, and the third and fourth stages show linear changes with slopes of 0.0007 and 0.0014, respectively. The fluctuation of the fractal dimension is small. The periodic weighting of the upper roof in the cave-affected zone is frequent, the fragmentation of the fractured rock mass becomes larger, and the fractures of the upper rock mass are relatively developed. The research results can provide a reference for the study on the evolution law of mining-induced rock fissures under similar karst landforms.

Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and perfectly oriented deposition and removal

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (particles occupying k consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θ_{c,k}[ϑ_{c,k}], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k}[ϑ_{c,k}] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θ_{c,k}^{i}[ϑ_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[ϑ_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, ϑ_{c,k}, θ_{c,k}^{i}, and ϑ_{c,k}^{i} are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θ_{c,k}[θ_{c,k}^{i}] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θ_{c,k→∞}=0.500(2) [θ_{c,k→∞}^{i}=0.500(1)]; (2)ϑ_{c,k}[ϑ_{c,k}^{i}] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑ_{c,k→∞}=0.5334(6) [ϑ_{c,k→∞}^{i}=0.4666(6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(ϑ)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{c,k}^{i}=1) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the θ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.

Design and Simulation of Stormwater Control Measures Using Automated Modeling

Stormwater control measures (SCMs) are decentralized technical elements, which can prevent the negative effects of uncontrolled stormwater flow while providing co-benefits. Optimal SCMs have to be selected and designed to achieve the desired hydrological response of an urban catchment. In this study, automated modeling and domain-specific knowledge in the fields of modeling rainfall-runoff (RR) and SCMs are applied to automate the process of optimal SCM design. A new knowledge library for modeling RR and SCMs, compliant with the equation discovery tool ProBMoT (Process-Based Modeling Tool), was developed. The proposed approach was used to (a) find the optimal RR model that best fits the available pipe flow measurements, and (b) to find the optimal SCMs design that best fits the target catchment outflow. The approach was applied to an urban catchment in the city of Ljubljana, Slovenia. First, nine RR models were created that generally had »very good« performance according to the Nash–Sutcliffe efficiency criteria. Second, six SCM scenarios (i.e., detention pond, storage tank, bio-retention cell, infiltration trench, rain garden, and green roof) were automatically designed and simulated, enabling the assessment of their ability to achieve the target outflow. The proposed approach enables the effective automation of two complex calibration tasks in the field of urban drainage.

Percolation phase transition by removal of k^{2}-mers from fully occupied lattices

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of k×k square tiles (k^{2}-mers) from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by occupied sites. Then the system is diluted by removing k^{2}-mers of occupied sites from the lattice following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be removed due to the absence of occupied sites clusters of appropriate size and shape. The central idea of this paper is based on finding the maximum concentration of occupied sites, p_{c,k}, for which the connectivity disappears. This particular value of the concentration is called the inverse percolation threshold and determines a well-defined geometrical phase transition in the system. The results obtained for p_{c,k} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤4. For k≥5, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the size k considered. In addition, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and four, has the same universality class as the standard percolation problem.

Non-universality of the dynamic exponent in two-dimensional random media

The diffusion of solutes in two-dimensional random media is important in diverse physical situations including the dynamics of proteins in crowded cell membranes and the adsorption on nano-structured substrates. It has generally been thought that the diffusion constant, D, should display universal behavior near the percolation threshold, i.e., D ~ (ϕ − ϕ_c)^μ, where ϕ is the area fraction of the matrix, ϕ_c is the value of ϕ at the percolation threshold, and μ is the dynamic exponent. The universality of μ is important because it implies that very different processes, such as protein diffusion in membranes and the electrical conductivity in two-dimensional networks, obey similar underlying physical principles. In this work we demonstrate, using computer simulations on a model system, that the exponent μ is not universal, but depends on the microscopic nature of the dynamics. We consider a hard disc that moves via random walk in a matrix of fixed hard discs and show that μ depends on the maximum possible displacement Δ of the mobile hard disc, ranging from 1.31 at Δ ≤ 0.1 to 2.06 for relatively large values of Δ. We also show that this behavior arises from a power-law singularity in the distribution of transition rates due to a failure of the local equilibrium approximation. The non-universal value of μ obeys the prediction of the renormalization group theory. Our simulations do not, however, exclude the possibility that the non-universal values of μ might be a crossover between two different limiting values at very large and small values of Δ. The results allow one to rationalize experiments on diffusion in two-dimensional systems.

Acoustic wave propagation in heterogeneous two-dimensional fractured porous media

This paper addresses an important fundamental question: the differences between wave propagation in fractured porous media with a uniform matrix (constant bulk modulus) and those in which the matrix is heterogeneous with its bulk modulus distributed spatially. The analysis of extensive experimental data [Phys. Rev. E 71, 046301 (2005)PLEEE81539-375510.1103/PhysRevE.71.046301] indicated that such distributions are self-affine and induce correlations at all the relevant length scales. The comparison is important from a practical view point because in many of the traditional models of fractured rock, particularly those that are used to study wave propagation or fit some data, the matrix is assumed to be uniform. Using extensive numerical simulation of propagation of acoustic waves, we present strong evidence indicating that the waves' amplitude in a fractured porous medium with a heterogeneous matrix decays exponentially with the distance from the source. This is in sharp contrast with a fractured porous medium with a uniform matrix in which not only the waves' amplitude decays with the distance as a stretched exponential function, but the exponent that characterizes the function is also dependent upon the fracture density. The localization length depends on the correlations in the spatial distribution of the bulk modulus, as well as the fracture density. The mean speed of the waves varies linearly with the fractures' mean orientation.

Electro-osmotic flow in disordered porous and fractured media

Electro-osmosis phenomena are studied in a two-dimensional (2D) model disordered porous medium. The flow passages are represented by a network of spatially distributed rectangular channels with random orientations. The channels may represent microfractures in fractured porous media or in a network of interconnected microfractures, pores in a porous medium, or fibers in a fibrous porous material. The linearized equations of electrokinetics are solved numerically in a single channel, and in the 2D network of the channels. The macroscopic electrical conductivity σ and electro-osmotic coupling coefficient β are computed as functions of the electrical surface potential ζ and such geometrical parameters of the network as the channels' number density and widths, as well as the porosity of the medium. Despite the complexity of the phenomena and the model of porous media that is used, both σ and β appear to depend on the characteristics of the phenomena and porous media through very simple relations.

Wave propagation in disordered fractured porous media

Extensive computer simulations have been carried out to study propagation of acoustic waves in a two-dimensional disordered fractured porous medium, as a prelude to studying elastic wave propagation in such media. The fracture network is represented by randomly distributed channels of finite width and length, the contrast in the properties of the porous matrix and the fractures is taken into account, and the propagation of the waves is studied over broad ranges of the fracture number density ρ and width b. The most significant result of the study is that, at short distances from the wave source, the waves' amplitude, as well as their energy, decays exponentially with the distance from the source, which is similar to the classical problem of electron localization in disordered solids, whereas the amplitude decays as a stretched exponential function of the distance x that corresponds to sublocalization, exp(-x(α)) with α < 1. Moreover, the exponent α depends on both ρ and b. This is analogous to electron localization in percolation systems at the percolation threshold. Similar results are obtained for the decay of the waves' amplitude with the porosity of the fracture network. Moreover, the amplitude decays faster with distance from the source x in a fractured porous medium than in one without fractures. The mean speed of wave propagation decreases linearly with the fractures' number density.

Percolation of aligned rigid rods on two-dimensional square lattices

The percolation behavior of aligned rigid rods of length k (kmers) on two-dimensional square lattices has been studied by numerical simulations and finite-size scaling analysis. The kmers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The process was monitored by following the probability R(L,k)(p) that a lattice composed of L×L sites percolates at a concentration p of sites occupied by particles of size k. The results, obtained for k ranging from 1 to 14, show that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the kmer size; (ii) for any value of k (k>1), the percolation threshold is higher for aligned rods than for rods isotropically deposited; (iii) the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered; and (iv) in the case of aligned kmers, the intersection points of the curves of R(L,k)(p) for different system sizes exhibit nonuniversal critical behavior, varying continuously with changes in the kmer size. This behavior is completely different to that observed for the isotropic case, where the crossing point of the curves of R(L,k)(p) do not modify their numerical value as k is increased.