## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

The macroscopic permeability of random lattices has been studied when the permeability of each link is a power law of its length with an exponent . When they are sufficiently long, the link lengths are shown to follow exponential laws which depend on the density. The macroscopic permeability is studied as a function of ; it is compared to a modified effective medium theory (EMT).

To read the full-text of this research,

you can request a copy directly from the authors.

... By varying the capillary number, contact angle and initial wetting phase saturation, they obtained five generic types of displacement pattern, namely flat frontal advance, dendritic fractal advance, bond percolation, compact cluster growth and ramified cluster growth. In addition, some researchers studied theoretically and numerically the evolution process of the shape of fluid flow in porous media [7,8] and flow process in fractured media [9][10][11] using percolation model. These researches involved several concepts of interaction between porous media matrix and liquid, nevertheless, in the case of the real porous media such as rock and the real flow process, the results still somewhat differed from the reality. ...

... The five kinds of liquids employed in the present experimental study are water, kerosene, mixture of kerosene and machine oil No. 7, diesel oil No. 10 3.96 and 4.36 mm 2 /s, respectively (at room temperature 20℃). To get better imaging, all liquids are dyed. ...

Researches on the boundary shape of fluid flow in porous media play an important role in engineering practices, such as petroleum
exploitation, nuclear waste disposal and groundwater contamination. In this paper, six types of artificial porous samples
(emery jade) with different porosities are manufactured. With the background of slow flow in porous media, laboratory experiments
are carried out by observing the movement of five types of fluids with different dynamic viscosities in various types of porous
media. A digital video recorder is employed to record the complete process of the fluid flow in the porous media. Based on
the digital photos of the moving boundaries of fluid flow in porous media, the average displacement and fractal dimension
of the moving boundary are estimated for different combinations of porosity and dynamic viscosity. Moreover, the evolution
behavior of the average velocity and fractal dimension of the moving boundary with time is known. The statistical relations
of the average velocity, the fractal dimension of the moving boundary and the porosity of porous media and the dynamic viscosity
of fluids are proposed in this paper. It is shown that the front shape of the moving boundary of fluid flow in porous media
is an integrated result of the porosity of porous media and the dynamic viscosity of fluids.

... However they need large computing resources and are very time consuming. Using Effective Medium Theory (EMT) for calculation the permeability of a lattice network is also reported in the literature as alternative method for numerical solution (Kirkpatrick 1973, Bernasconi 1974, Harris 1990, Zimmerman &Bodvarsson 1996, Jankovic 2003, Fedrov et al 2005, Baghbanan & Dayani 2009, Dayani & Baghbanan 2010. ...

For many engineering rock works, such as rock slopes, foundations of dam, underground excavations; calculating the permeability and evaluating Representative Elementary Volume is very important. The Effective Medium Theory (EMT) approach is an alternative method for network permeability calculation. In this way a fracture network can be replaced by a regular network of conductors that are connected to each other at nodes. Then by evaluation suitable effective conductance, permea-bility is calculated. Using Monte Carlo technique, stochastic DFN models were generated with a large number of realizations with the same fracture density and different aperture patterns when distributed fracture trace lengths are correlated/uncorrelated with fracture aperture distributions and directional permeability components are calculated with developed new code and compared with numerical results. The results show that the calculated mean values of permeability and approximated permeability tensor with EMT method is well fitted with the numerical simulation results in all fracture patterns. It means that the uncertainty in the results of EMT method is small. Theme: Hydrogeology and Grouting.

Fluid flow through geological formations is often concentrated on distinct preferential flow paths owing to the presence of fractures or large‐scale permeability structures. However, the existence of such structures is not a mandatory condition of preferential paths formation. Pore‐scale spatial fluctuations of pore size and/or pore connectivity in statistically stationary porous media, if sufficiently large, can also lead to the concentration of fluid flow on distinct pathways. In this paper, we attempted to establish the conditions of formation of preferential flow paths in heterogeneous porous media in terms of pore‐size heterogeneity and pore connectivity. We simulated steady‐state flow through stochastically constructed two‐ and three‐dimensional pore networks, in which the width of the pore radius distribution and the pore coordination number (a measure of pore connectivity) were varied. We developed new techniques based on graph theory to identify potential preferential flow paths and characterize them. We observed a gradual transition from approximately uniform flow fields in low heterogeneity/high connectivity networks to flow localization on preferential paths with increasing pore‐size heterogeneity and decreasing connectivity. The transition occurred at lower heterogeneity levels in three‐dimensional than in two‐dimensional simulations and was less influenced by pore connectivity variations. These results were summarized in a phase diagram in pore‐size heterogeneity/pore connectivity parameter space, which we found consistent with relevant real rocks data.

Dewatering in surface and underground mining is highly dependent on the permeability of fractured rocks, which is usually investigated by analytical and numerical methods. Analytical methods are used for simple, regular joint systems, while for complicated discrete fracture networks, numerical methods are more suitable. However, such numerical simulations demand considerable computation resources and are time-consuming for large-scale models. An attractive approach is to perform a rough mapping of the fractures into a lattice to apply effective medium theory. A computational code based on effective medium theory for calculating the overall permeability of fracture networks was developed. Several stochastic discrete fracture network models were generated with the fracture aperture distributed uniformly, correlated and uncorrelated with fracture trace length. The results showed that when the aperture was correlated or uncorrelated with trace length, the calculated mean permeability values by the effective medium theory method agreed well with numerical modelling. Also, approximated representative elementary volume was well fitted by numerical methods in the constant fracture aperture case. However, the calculated representative elementary volume by the effective medium theory method was smaller than the numerical estimation in the other two cases.

1] Markov Chain Monte Carlo (MCMC) methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models. However, MCMC methods require the a priori definition of a proposal or sampling distribution, which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the Markov Chain and its rate of convergence. In this paper we present an MCMC sampler entitled the Shuffled Complex Evolution Metropolis algorithm (SCEM-UA), which is well suited to infer the posterior distribution of hydrologic model parameters. The SCEM-UA algorithm is a modified version of the original SCE-UA global optimization algorithm developed by Duan et al. [1992]. The SCEM-UA algorithm operates by merging the strengths of the Metropolis algorithm, controlled random search, competitive evolution, and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution. Three case studies demonstrate that the adaptive capability of the SCEM-UA algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional Metropolis-Hastings samplers.

The Shear Strength of Rock Joints in Theory and Practice The paper describes an empirical law of friction for rock joints which can be used both for extrapolating and predicting shear strength data. The equation is based on three index parameters; the joint roughness coefficientJRC, the joint wall compressive strengthJCS, and the residual friction angleφ r . All these index values can be measured in the laboratory. They can also be measured in the field. Index tests and subsequent shear box tests on more than 100 joint samples have demonstrated thatφ r can be estimated to within ± 1° for any one of the eight rock types investigated. The mean value of the peak shear strength angle (arctanτ/σ n ) for the same 100 joints was estimated to within 1/2°. The exceptionally close prediction of peak strength is made possible by performing self-weight (low stress) sliding tests on blocks with throughgoing joints. The total friction angle (arctanτ/σ n ) at which sliding occurs provides an estimate of the joint roughness coefficientJRC. The latter is constant over a range of effective normal stress of at least four orders of magnitude. However, it is found that bothJRC andJCS reduce with increasing joint length. Increasing the length of joint therefore reduces not only the peak shear strength, but also the peak dilation angle and the peak shear stiffness. These important scale effects can be predicted at a fraction of the cost of performing large scale in situ direct shear tests.

Using two surface profilers, each sensitive to a particular scale of topographic features, we have studied the topography of various natural rock surfaces from wavelengths less than 20 micrometres to nearly 1m. The surfaces studied included fresh natural joints (mode I cracks) in both crystalline and sedimentary rocks, a frictional wear surface formed by glaciation, and a bedding plane surface. There is remarkable similarity among these surfaces. Each surface has a 'red noise' power spectrum over the entire frequency band studied, with the power falling off on average between 2 and 3 orders of magnitude per decade increase in spatial frequency. This implies a strong increase in rms height with surface size, which has little tendency to level off for wavelengths up to 1 meter. These observations can be interpreted using a fractal model of topography. -from Authors

Miscible fluid displacements are studied experimentally in a radial flow between two complementary replica of a self-affine rough granite fracture surface. The displacement front between a dyed fluid and a transparent (but otherwise identical) one is followed optically through one face of the cell. The evolution of its geometry is studied as a function of time, flow-rate, and normal and lateral relative displacements between the two surfaces. For a purely normal displacement, the front is globally smooth, due to the constant local distance between surfaces. For a finite lateral displacement, the front is rough due to spatial variations of this distance; its geometry is fractal and its dimension is directly related to the Hurst exponent H approximately 0.8 of the surface. The fractal regime is observed only above a lower cut-off scale that depends on the normal spacing of the surfaces and an upper one that increases with the injected volume and with the amplitude of the lateral displacement.

rocesses; 3250 Mathematical Geophysics: Fractals and multifractals; 5104 Physical Properties of Rocks: Fracture and flow; 5139 Physical Properties of Rocks: Transport properties; KEYWORDS: fractional, dispersion, fractal, fracture, anomalous, transport Citation: Schumer, R., D. A. Benson, M. M. Meerschaert, and B. Baeumer, Multiscaling fractional advection-dispersion equations and their solutions, Water Resour. Res., 39(1), 1022, doi:10.1029/2001WR001229, 2003. 1. Introduction [2] Hundreds of studies have proposed modeling techniques to address the super-Fickian transport of solutes in aquifers. Among them are fractional advection-dispersion equations (ADEs), analytical equations that employ fractional derivatives in describing the growth and scaling of diffusion-like plume spreading. Fractional ADEs are the limiting equations governing continuous time random walks (CTRW) with arbitrary particle jump length distribution and finite mean waiting time distribution [Compte, 1996]. Th

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

The percolation and conductivity of self-affine fractures are investigated over the whole range of their mean aperture and roughness exponent H, by direct three-dimensional numerical simulations. A scaling behavior is exhibited for the conductivity of tight fractures in the self-affine scale range, with exponent H. All the data can be summarized by two simple models, valid for small to moderate and for large apertures, respectively.

Extensions of percolation theory to treat transport are described. Resistor networks, from which resistors are removed at random, provide the natural generalization of the lattice models for which percolation thresholds and percolation probabilities have previously been considered. The normalized conductance, G, of such networks proves to be a sharply defined quantity with a characteristic concentration dependence near threshold which appears sensitive only to dimensionality. Numerical results are presented for several families of 3D and 2D network models. Except close to threshold, the models based on bond percolation are accurately described by a simple effective medium theory, which can also treat continuous media or situations less drastic than the percolation models, for example, materials in which local conductivity has a continuous distribution of values. The present calculations provide the first quantitative test of this theory. A "Green's function" derivation of the effective medium theory, which makes contact with similar treatments of disordered alloys, is presented. Finally, a general expression for the conductance of a percolation model is obtained which relates G to the spin-stiffness coefficient, D, of an appropriately defined model dilute ferromagnet. We use this relationship to argue that the "percolation channels" through which the current flows above threshold must be regarded as three dimensional.

Calculations of effective conductivities of generalized random bond lattices representing porous media are compared to approximations using effective medium theory (EMT). We use numerical simulations of flow through 2D and 3D random lattice models, which allow for variable lattice densities and a lognormal distribution of local conductivities, to compare effective conductivities to effective medium approximations. We find that the analytical expressions provide good agreement to the simulations in 2D systems, but are in significant error in 3D systems when the standard deviation of the local conductivities is large.

The permeability of self-affine fractures with various roughness exponents H is investigated by direct three-dimensional numerical simulations. A scaling behavior with an exponent H is exhibited in the self-affine scale range. Permeability can be related to the fractional open area and to the percolation probability by simple models.

The existing theories of the resistivity of mixtures assume regular arrangements of the two components, rather than random mixtures. A theory for a random mixture is given, based on the assumption that each crystal acts as if surrounded by a homogeneous medium whose properties are those of the mixture. Comparisons with experiment are made. The experimental data that have been examined fall roughly into two classes. One class consists of mixtures, where the variation of resistivity with composition disagrees violently with this theory, making it clear that the assumptions made are completely inapplicable. The remaining class consists of mixtures which generally agree well with the theory.

The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines bringing the total to well over 300, plus upgraded versions of the original routines, the new edition remains the most practical, comprehensive handbook of scientific computing available today.

Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

Experimental study of miscible displacement fronts in rough self-affine fractures The shear strength of rock joints in theory and practice, Rock Mech

- H Auradou
- J.-P Hulin
- S And Roux

Auradou, H., Hulin, J.-P. and Roux, S.: 2001, Experimental study of miscible displacement fronts in rough self-affine fractures, Phys. Rev. E. 63, 066306. Barton, N. and Choubey, V.: 1970, The shear strength of rock joints in theory and practice, Rock Mech. 10, 1–34.