interaction of diffusion and boundary conditions on considere le systeme d equations paraboliques u t d u f u x d u n e x u 0 x ou u r n r n n 3 est un ouvert borne de frontiere lisse d diag d 1 d n e diag e 1 e n chaque d j 0 est constant e j r est continu e j 0 j 1 2 n et e 0 est constant pour f e fixe on etudie comment le flot defini ci dessus depend des parametres d

generation of coarse scale continuum flow models from detailed fracture characterizations 1 a procedure for developing coarse scale continuum models from detailed fracture descriptions is developed and applied the coarse models are in the form of a generalized dual porosity representation in which matrix rock and fractures exchange fluid locally while large scale flow occurs through the fracture network the methodology developed here introduces local subgrids to resolve dynamics within the matrix and provides appropriate coarse scale parameters describing fracture fracture matrix fracture and matrix matrix flow the geometry of the local subgrids as well as the required parameters for the coarse scale model are determined from local flow solutions using the underlying discrete fracture model the method is applied to two and three dimensional single phase and two phase flow problems and the accuracy of the coarse models is assessed relative to fully resolved discrete fracture simulations for the cases considered it is shown that the technique is capable of generating highly accurate coarse models with many fewer unknowns than the detailed characterizations and speedups of about a factor of 100 are achieved

geometrical percolation threshold of overlapping ellipsoids a recurrent problem in materials science is the prediction of the percolation threshold of suspensions and composites containing complex shaped constituents we consider an idealized material built up from freely overlapping objects randomly placed in a matrix and numerically compute the geometrical percolation threshold mathit p mathit c where the objects first form a continuous phase ellipsoids of revolution ranging from the extreme oblate limit of platelike particles to the extreme prolate limit of needlelike particles are used to study the influence of object shape on the value of mathit p mathit c the reciprocal threshold 1 mathit p mathit c mathit p mathit c equals the critical volume fraction occupied by the overlapping ellipsoids is found to scale linearly with the ratio of the larger ellipsoid dimension to the smaller dimension in both the needle and plate limits ratios of the estimates of mathit p mathit c are taken with other important functionals of object shape surface area mean radius of curvature radius of gyration electrostatic capacity excluded volume and intrinsic conductivity in an attempt to obtain a universal description of mathit p mathit c unfortunately none of the possibilities considered proves to be invariant over the entire shape range so that mathit p mathit c appears to be a rather unique functional of object shape it is conjectured based on the numerical evidence that 1 mathit p mathit c is minimal for a sphere of all objects having a finite volume

on holographic realization of logarithmic galilean conformal algebra we study two dimensional logarithmic galilean conformal algebra lgca by making use of a contraction of topologically massive gravity at critical point we observe that using a naive contraction at the critical point fails to give a well defined theory though contracting the theory while we are approaching the critical point leads to a well behaved expression for two point functions of the energy momentum tensors of lgca

the exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

a model for steady fluid flow in random three dimensional networks of disc shaped fractures a model for steady fluid flow in three dimensional random networks of fractures has been developed in this model the fractures are disc shaped discontinuities in an impermeable matrix the fracture discs can be arbitrarily located within the rock volume and can have any desired distribution of aperture radius orientation and density thus where the disc model is appropriate it is possible to calculate flow through fracture networks which are statistically similar to those that occur in nature after the boundary conditions and the desired fractures are specified the intersections nodes between these discs elements are identified then steady flow through the network is calculated using a mixed analytical numerical technique in each fracture analytic equations for flow into or out of each node as a function of the average head at each node are developed the equations are based on image theory and the assumption that each node is a source or sink of uniform strength a set of mass balance equations is constructed which equate flow into a node from one of its associated fractures to flow out of the node into the other associated fracture these equations are solved for the average head at each node and flux between fractures can then be calculated by substituting the average head values back into the analytical equations the model has been successfully checked against analytical results for several cases of two and three intersecting fractures we plan to use these techniques to measure the permeability of fracture networks

flow control of fluids through porous media this paper discusses flow control of fluids through porous media specifically related to its application in petroleum science the flow of fluids is described by the boussinesq equation with mixed boundary conditions a neumann boundary condition describes no flow at the outer boundary of the porous media and a nonlinear boundary condition describes the interaction between the porous media and the source potential in many ways the source potential can be controlled this leads to a boundary control problem of the boussinesq equation the problem can be analyzed by the lyapunov method numerical examples are performed to provide a clear understanding of the concept of boundary control for these fluid flow systems

limit theorems for coupled continuous time random walks scaling limits of continuous time random walks are used in physics to model anomalous diffusion in which a cloud of particles spreads at a different rate than the classical brownian motion governing equations for these limit processes generalize the classical diffusion equation in this article we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent this leads to an efficient method of computing the limit and a surprising connection to fractional derivatives

on mapping fracture networks onto continuum 1 discrete fracture network dfn and stochastic continuum sc are two common modeling approaches used for simulating fluid flow and solute transport in fractured media fracture continuum approaches combine the merits of each approach details of the fracture network are preserved and a computationally efficient grid is utilized for the solution of fluid flow by assigning a conductivity contrast between the grid cells representing the rock matrix and those representing fractures in this paper we propose a fracture continuum approach for mapping individual fractures onto a finite difference grid as conductivity fields we focus on several issues that are associated with this approach such as enhanced connectivity between fractures that would otherwise not be in connection in a dfn simulation and the influence of grid cell size to addresses these issues both dfn and the proposed approach are used to solve for fluid flow through two dimensional randomly generated fracture networks in a steady state single phase flow system the dfn flow solution is used as a metric to evaluate the robustness of the method in translating discrete fractures onto grid cell conductivities on four different regularly spaced grids 1 1 m 2 2 m 5 5 m and 10 10 m two correction factors are introduced to ensure equivalence between the total flow of the grid and the original fracture network the first is dependent on the fracture alignment with the grid and is set to account for the difference between the length of the flow path on the grid and that of the fracture the other correction is applied for areas in the grid with high fracture density and accounts for the artificial degree of connectivity that exists on the grid but not in the dfn fifteen different cases are studied to evaluate the effect of fracture statistics on the results of the proposed approach and by taking average results of 100 realizations in each case in a stochastic monte carlo framework the flow equation is solved for the dfn and total flow is obtained the flow is also solved separately for the four grid resolution levels and comparisons between the dfn and the grid total flows are made for the different cases and the different grid resolution levels the approach performed relatively well in all cases for the fine grid resolution but an overestimation of grid flow is observed in the coarse grid resolution especially for cases wherein the network connectivity is controlled by small fractures this overestimation shows minor variation from one realization to another within the same case this allowed us to develop an approach that depends on solving limited number of dfn simulations to obtain this overestimation factor results indicate that the proposed approach provides improvements over existing approaches and has a potential to provide a link between dfn and sc models

communicability angle and the spatial efficiency of networks we introduce the concept of communicability angle between a pair of nodes in a graph we provide strong analytical and empirical evidence that the average communicability angle for a given network accounts for its spatial efficiency on the basis of the communications among the nodes in a network we determine characteristics of the spatial efficiency of more than a hundred real world networks that represent complex systems arising in a diverse set of scenarios in particular we find that the communicability angle correlates very well with the experimentally measured value of the relative packing efficiency of proteins that are represented as residue networks we finally show how we can modulate the spatial efficiency of a network by tuning the weights of the edges of the networks this allows us to predict the effects of external stresses on the spatial efficiency of a network as well as to design strategies to improve important parameters in real world systems

transport of conservative solutes in simulated fracture networks 2 ensemble solute transport and the correspondence to operator stable limit distributions 1 in networks where individual fracture lengths follow a fractal distribution ensemble transport of conservative solute particles at the leading plume edge exhibit characteristics of operator stable densities these densities have as their governing equations of transport either fractional order or integer order advection dispersion equations model selection depends on the identification of either multi gaussian or operator stable transport regimes which in turn depends on the power law exponent of the fracture length distribution low to moderately fractured networks with power law fracture length exponents less than or equal to 1 9 produce solute plumes that exhibit power law leading edge concentration profiles and super fickian plume growth rates for these network types a multiscaling fractional advection dispersion equation mfade provides a model of multidimensional solute transport where different rates of power law particle motion are defined along multiple directions the mfade model is parameterized by a scaling matrix to describe the super fickian growth process in which the eigenvectors correspond to primary fracture group orientations and the eigenvalues code fracture length and transmissivity the approximation of particle clouds by a multi gaussian a subset of the operator stable for densely fractured networks with finite variance fracture length distributions can be ascribed to the classical ade where fickian scaling rates pertain along orthogonal plume growth directions fracture networks show long term particle retention in low velocity fractures so that coupling of the equations of motion with retention models such as continuous time random walk or multirate mobile immobile will increase accuracy near the source particle arrival times at exit boundaries for multi gaussian plumes vary with spatial density generally arrival times are faster in sparsely fractured domains where transport is governed by a few very long fractures

transport behavior of a passive solute in continuous time random walks and multirate mass transfer 1 we consider the transport behavior of a passive solute in a heterogeneous medium modeled by continuous time random walks ctrw and linear multirate mass transfer mrmt within the ctrw framework we formulate a transport model which is formally equivalent to mrmt in both approaches the total concentration is divided into mobile and immobile parts the immobile concentration is given by the convolution in time of the mobile concentration and a memory function the memory function is a functional of the distribution of transition and trapping times for the ctrw and the mrmt approach respectively and determines the transport behavior of the solute based on different expressions for the memory function in the two frameworks we derive conditions for which both approaches describe the same transport behavior we focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime using an expansion of the laplace transform of the memory function we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the macro dispersion coefficients of the solute distribution we observe anomalous power law as well as normal fickian transport behavior depending on the exponents that dominate the trapping and transition time distributions respectively the results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions the unified transport framework presented here comprising ctrw and mrmt shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media

logarithmic correlators in nonrelativistic conformal field theory we show how logarithmic terms may arise in the correlators of fields which belong to the representation of the schrodinger virasoro algebra or the affine galilean conformal algebra gca we show that in gca only scaling operator can have a jordan form and rapidity cannot we observe that in both algebras logarithmic dependence appears along the time direction alone

effective dispersivities for a two dimensional periodic fracture network by a continuous time random walk analysis of single intersection simulations fracture networks are of major importance for transport problems in hydrology their inclusion in models requires assumptions about both the single fracture and the network structure this paper focuses on the macroscopic effect of microscopic mixing conditions in intersections within a periodic fracture network mixing is characterized by the transition probabilities for particles to move from one fracture to another or to stay in the same fracture a periodic network is used which excludes stochastic effects and also single fracture dispersion the periodic network can be solved analytically and is applicable to a variety of network structures we find that for this type of network effective dispersivities in the sense of asymptotic values for long times and large scales do usually exist i e the asymptotic dispersion is gaussian for the particular case of a quadratic network particle transition probabilities at an intersection have been obtained by microscopic lattice boltzmann simulations of flow and transport in addition to the assumptions of complete mixing and stream routing the resulting dispersivities show that complete mixing is not a low peclet number limit in the presence of regions of immobile water volumes dispersivities depend strongly on the direction between flow and network and on the mixing model dispersivities are large for high peclet numbers and flow parallel to the network especially for these cases we also find a scale dependence up to some orders of magnitude which has serious implications for the interpretation of simulations and measurements since these results also appear in stochastic network simulations they are therefore not specific to the periodic structure of the fracture network

stochastic solution of space time fractional diffusion equations classical and anomalous diffusion equations employ integer derivatives fractional derivatives and other pseudodifferential operators in space in this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are mittag leffler type this leads to explicit solutions for space time fractional diffusion equations with multiscaling space fractional derivatives and additional insight into the meaning of these equations

the communicability distance in graphs let g be a simple connected graph with adjacency matrix a the communicabilitygpq between two nodes p and q of the graph is defined as the pq entry of g exp a we prove here that p q gpp gqq 2gpq 1 2 is a euclidean distance and give expressions for it in paths cycles stars and complete graphs with n nodes the sum of all communicabilitydistances in a graph is introduced as a new graph invariant g we compare this index with the wiener and kirchhoff indices of graphs and conjecture about the graphs with maximum and minimum values of this index

spectral measures of bipartivity in complex networks we introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity it is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks trophic interactions in food webs construction principles in metabolic networks or communities in social networks

upscaling discrete fracture network simulations an alternative to continuum transport models 1 particle tracking through stochastically generated networks of discrete fractures provides an alternative to the conventional advection dispersion description of transport in fractured rock however discrete fracture network simulations are computationally intensive and usually limited to small scales an approach for direct upscaling of discrete fracture simulations is described trajectories for nonreacting tracer particles are first extracted from relatively small discrete fracture network simulations tracer rock interaction is represented by also calculating a cumulative reactivity parameter along each trajectory the residence time reactivity information is then used in a monte carlo simulation to construct artificial particle trajectories of any length thereby achieving the upscaling objective in its simplest form the procedure has the form of a random walk evolving in a two dimensional space tests using site specific and generic networks show that it is necessary to modify the random walk to produce sequential correlation along the trajectories we achieve this by using a discrete state markov process to direct the random walk the procedure is computationally efficient easily implemented and compares well with the network simulations

on iterative techniques for computing flow in large two dimensional discrete fracture networks computation of flow in discrete fracture networks often involves solving for hydraulic head values at all intersection points of a large number of stochastically generated fractures inside a bounded domain for large systems this approach leads to the generation of problems involving highly sparse matrices which must be solved iteratively distributions of fracture lengths spanning over several orders of magnitude and the randomness of fracture orientations and locations lead to coefficient matrices that are devoid of any regular structure in the sparsity pattern in addition to the rapid increase in computational effort with increase in the size of the fracture network the spread in the distribution of fracture parameters such as length and transmissivity dramatically influences the convergence behavior of the system of linear equations an overview of the discrete fracture network dfn methodology for computation of flow is presented along with a comparative study of various krylov subspace iterative methods for the resulting class of sparse matrices the rate of convergence of the iterative techniques is found to exhibit a systematic pattern with respect to changes in statistical parameters of the stochastically generated fracture networks salient features of the observed trends in the convergence pattern are discussed and guidelines for design of dfn algorithms are provided

path laplacian operators and superdiffusive processes on graphs i one dimensional case we consider a generalization of the diffusion equation on graphs this generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one dimensional graphs the generalization is based on the k path laplacian operators lk which account for the hop of a diffusive particle to non nearest neighbours in a graph we first prove that the k path laplacian operators are self adjoint then we study the transformed k path laplacian operators using laplace factorial and mellin transforms we prove that the generalized diffusion equation using the laplace and factorial transformed operators always produce normal diffusive processes independently of the parameters of the transforms more importantly the generalized diffusion equation using the mellin transformed k path laplacians k 1k slk produces superdiffusive processes when 1 s 3

on the connectivity of three dimensional fault networks natural fault networks involve a very broad range of fault lengths modeled in general by a power law length distribution n l al such a scaling law does not allow to define any a priori pertinent scale of observation for hydraulic field experiments in fractured media to investigate the relative effects of faults depending on their length we undertake in the spirit of percolation theory a theoretical and numerical study of the connectivity of three dimensional fault networks following power law length distributions we first establish the correct analytical expression of a percolation parameter p which describes the connectivity of the system the parameter p is found to be dependent on the third moment of the length distribution for fault planes it allows us to identify different regimes of connectivity depending on a the exponent of the fault length distribution the geometrical properties of the infinite cluster which partly control transport properties are also established at the percolation threshold for natural fault networks our theoretical analysis suggests that faults larger than a critical length scale may form a well connected network while smaller faults may be not connected on average this result which implies an increase of the connectivity with scale is consistent with scaling effects observed on permeability measurements

geometrical and spectral study of beta skeleton graphs we perform an extensive numerical analysis of skeleton graphs a particular type of proximity graphs in a skeleton graph bsg two vertices are connected if a proximity rule that depends of the parameter 0 is satisfied moreover for 1 there exist two different proximity rules leading to lune based and circle based bsgs first by computing the average degree of large ensembles of bsgs we detect differences which increase with the increase of between lune based and circle based bsgs then within a random matrix theory rmt approach we explore spectral and eigenvector properties of random bsgs by the use of the nearest neighbor energy level spacing distribution and the entropic eigenvector localization length respectively the rmt analysis allows us to conclude that a localization transition occurs at 1

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions the motivations to study the problem considered in this paper come from the theory of composite materials where the heat diffusion properties can change from one part of the domain to another mathematically this leads to a nonlinear second order parabolic equation for which the diffusion coefficient becomes large in a subdomain 0 the equation is supplemented by a nonlinear boundary condition on and an initial condition the authors determine the form of the limit problem the so called shadow system which involves an evolution equation for the averages of the density over 0 the main results include global in time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system

flow simulation in three dimensional discrete fracture networks in fractured rocks fluid flows mostly within a complex arrangement of fractures both the fracture network structure and its hydraulic properties are determined at first order by the broad range of fracture lengths and densities to handle the observed wide variety of fracture properties and the lack of direct fracture visualization we develop a general and efficient stochastic numerical model for discrete fracture networks dfns in a three dimensional 3d computational domain we present an original conforming mesh generation method addressing the penalizing configurations stemming from close fractures and acute angles between fracture intersections flows are subsequently computed by using a mixed hybrid finite element mhfe method the lack of direct fracture knowledge is treated by monte carlo simulations requiring simulations with a large number of networks with various characteristics we analyze the complexity in size and in time for the computation of flow in 3d dfns meshed with our method and compare with the complexities for 2d rectangular domains meshed with a regular grid we find out that complexity in size is similar whereas complexity in time is slightly larger for dfns than for 2d regular domains

upper semicontinuity of attractors and synchronization abstract in this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize we apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems the technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem explicitly exhibited in the diagonal

scaling of production data obtained from random walk particle tracking simulations in highly fractured porous media abstract we modeled tracer injection through highly fractured media by random walk particle tracking simulations and observed that production curves for different r values distance between injection and production wells are similar to each other and can be scaled hence if we define p t as a probability of a particle to reach the production well within traveling time t and plot p t r vs t r we observe that production curves for different values of r overlaid we identified the dependencies of and parameters on fracture network parameters such as fracture spacing fracture density and the fractal characteristics of fracture network

affine extension of galilean conformal algebra in 2 1 dimensions we show that a class of nonrelativistic algebras including noncentrally extended schrodinger algebra and galilean conformal algebra gca has an affine extension in 2 1 hitherto unknown this extension arises out of the conformal symmetries of the two dimensional complex plane we suggest that this affine form may be the symmetry that explains the relaxation of some classical phenomena toward their critical point this affine algebra admits a central extension and maybe realized in the bulk the bulk realization suggests that this algebra may be derived by looking at the asymptotic symmetry of an anti de sitter ads theory this suggests that ads cft conformal field theory duality may take on a special form in four dimensions

global attractor for sectorial evolutionary equation an abstract differential equation with sectorial operator is shown to generate a dissipative semigroup whenever an appropriate introductory estimate for solutions is known based on hale s theory of dissipative systems such an a priori estimate is then shown to be sufficient for existence of a global attractor examples contained in the final part of the paper illustrate presented ideas 1996 academic press inc 1 introductory notes

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

percolation threshold of a random array of discs a numerical simulation the authors evaluate numerically the percolation threshold of a 3d assembly of widthless monodisperse discs radius r number per unit volume n in terms of the quasi invariant nr the value obtained is consistent with previous estimates this model should apply to the permeability problem of fractured rocks

theory of anomalous chemical transport in random fracture networks we show that dominant aspects of chemical particle transport in fracture networks char22 non gaussian propagation char22 result from subtle features of the steady flow field distribution through the network this is an outcome of a theory based on a continuous time random walk formalism structured to retain the key space time correlations of particles as they are advected across each fracture segment the approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions monte carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation the geometry and velocity distributions are used in conjunction with particle mixing rules to map the particle movement between fracture intersections onto a joint probability density ensuremath psi mathbf r t the chemical concentration plume and breakthrough curves can then be calculated analytically particle tracking simulations on these networks exhibit the same non gaussian profiles demonstrating quantitative agreement with the theory the analytic plume shapes display the same basic behavior as extensive field observations at the columbus air force base mississippi the quantitative correlation between the time dependence of the mean and standard deviation of the field plumes and their shape is predicted by the theory

conditional simulations of fluid flow in three dimensional networks of discrete fractures how to predict flow through a network of discrete fractures in a three dimensional domain is investigated fractures are modeled as circular discs of arbitrary size orientation transmissivity and location a fracture network is characterized by the statistical distributions of these quantities fracture traces observed on a wall form the basis for estimates of mean fracture radius fracture orientation parameters and fracture density fracture trace lengths are estimated with the scanline method and from areal sampling on circular regions the traces observed on the wall can also be used to condition the network this trace conditioning is achieved by forcing the network generator to always reproduce the observed traces conditioning might be a means of decreasing the variability of the fracture networks a numerical simulation model has been developed which is capable of generating a fracture network of desired statistical properties and solving for the steady state flow on each fracture disc the flow is discretized with the boundary element method a series of hypothetical examples are analyzed these examples consist of sets of monte carlo simulations of flow through a series of networks generated from the same statistical distributions the examples lead to the following conclusions large fractures and high fracture density implies good connectivity in the networks a high fracture density implies a small variance in the flow through the network trace conditioning decreases estimation variance only when the fracture network consists of large fractures fracture statistics can be estimated reasonably well from fracture traces observed on a wall

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

numerical simulation of non fickian transport in geological formations with multiple scale heterogeneities 1 we develop a numerical method to model contaminant transport in heterogeneous geological formations the method is based on a unified framework that takes into account the different levels of uncertainty often associated with characterizing heterogeneities at different spatial scales it treats the unresolved small scale heterogeneities residues probabilistically using a continuous time random walk ctrw formalism and the large scale heterogeneity variations trends deterministically this formulation leads to a fokker planck equation with a memory term fpme and a generalized concentration flux term the former term captures the non fickian behavior arising from the residues and the latter term accounts for the trends which are included with explicit treatment at the heterogeneity interfaces the memory term allows a transition between non fickian and fickian transport while the coupling of these dynamics with the trends quantifies the unique nature of the transport over a broad range of temporal and spatial scales the advection dispersion equation ade is shown to be a special case of our unified framework the solution of the ade is used as a reference for the fpme solutions for the same formation structure numerical treatment of the equations involves solution for the laplace transformed concentration by means of classical finite element methods and subsequent inversion in the time domain we use the numerical method to quantify transport in a two dimensional domain for different expressions for the memory term the parameters defining these expressions are measurable quantities the calculations demonstrate long tailing arising principally from the memory term and the effects on arrival times that are controlled largely by the generalized concentration flux term

a dynamical approximation for stochastic partial differential equations random invariant manifolds provide geometric structures for understanding stochastic dynamics in this paper a dynamical approximation estimate is derived for a class of stochastic partial differential equations by showing that the random invariant manifold is almost surely asymptotically complete the asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold under suitable conditions as an application stationary states invariant measures are considered for a class of stochastic hyperbolic partial differential equations

dispersion in sub representative elementary volume fracture networks percolation theory and random walk approaches a law for dispersion in fracture networks below the representative elementary volume rev is established by analyzing random walks in two dimensional fracture networks in conjunction with percolation theory irregular fracture networks near the percolation threshold were obtained by removing some of the fractures of a regular orthogonal network consisting of fractures of equal length and different apertures drawn randomly from a lognormal distribution the random walk was directed by an exact solution of flow through the network and monte carlo simulations were performed to track particles through the fracture system the percolation theory analysis indicates a proportionality between the mean square displacement and time raised to the power 1 27 in excellent agreement with the simulations in the fracture networks which indicate a proportionality with time raised to the power 1 3

nonrelativistic conformal groups in this work a systematic study of finite dimensional nonrelativistic conformal groups is carried out under two complementary points of view first the conformal killing equation is solved to obtain a whole family of finite dimensional conformal algebras corresponding to each of the galilei and newton hooke kinematical groups some of their algebraic and geometrical properties are studied in a second step among the groups included in these families one can identify for example the contraction of the minkowski conformal group the analog for a nonrelativistic de sitter space or the nonextended schrodinger group

application of fractal geometry to the study of networks of fractures and their pressure transient typical models for the representation of naturally fractured systems generally rely on the double porosity warren root model or on random arrays of fractures however field observations have demonstrated the existence of multiple length scales in a variety of naturally fractured media present models fail to capture this important property of self similarity we first use concepts from the theory of fragmentation and from fractal geometry to construct numerically a network of fractures that exhibits self similar behavior over a range of scales the method is a combination of fragmentation concepts and the iterated function system approach and allows for great flexibility in the development of patterns next numerical simulation of unsteady single phase flow in such networks is described it is found that the pressure transient response of finite fractals behaves according to the analytical predictions of chang and yortsos 1990 provided that there exists a power law in the mass radius relationship around the test well location finite size effects can become significant and interfere with the identification of the fractal structure the paper concludes by providing examples from actual well tests in fractured systems which are analyzed using fractal pressure transient theory

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

transport and intersection mixing in random fracture networks with power law length distributions the importance of fracture intersection mixing rules complete mixing and streamline routing on simulated solute migration patterns in random fracture networks is assessed for this purpose and based on geological evidence two dimensional model networks having power law fracture length distributions and lognormal fracture permeability distributions are considered different fracture network structures are accounted for by the power law length distribution ranging from networks composed of infinite length fractures to percolation networks with constant length fractures comparison of solute particle statistics shows that there is no significant difference between the bulk transport properties calculated with the two mixing rules in fact it is found that the choice of mixing assumptions has a significant influence in less than 5 of the total number of fracture intersections in most fracture networks this result can be attributed to the small mean effective coordination number defined as the number of branches connected to an intersection with nonzero flux it is concluded that solute transport in fracture networks is strongly influenced by variability and uncertainty in parameters defining the geometrical structure of networks and that by comparison the choice of mixing rules at fracture intersections is of little importance

emergence of stable laws for first passage times in three dimensional random fracture networks we study first passage behaviors in the flow through three dimensional random fracture networks network and flow heterogeneity lead to the emergence of heavy tailed first passage time distributions that evolve with increasing distance between the start and target planes and transition toward stable laws analysis of the spatial memory of the first passage process shows that particle motion can be quantified stochastically by a time domain random walk conditioned on the initial velocity data this approach identifies advective tortuosity the velocity point distribution and the average fracture link length as key quantities for the prediction of first passage times using this approach we develop a theory for the evolution of first passage times with increasing distance between the start and target planes and the convergence towards stable laws

time domain particle tracking methods for simulating transport with retention and first order transformation 1 particle tracking in the time domain has received increasing attention as a technique for robustly simulating transport along one dimensional subsurface pathways using a stochastic lagrangian perspective integral representations of transport including the effects of advection longitudinal dispersion and a broad class of retention models are derived monte carlo sampling of that integral leads directly to new time domain particle tracking algorithms that represent a wide range of physical phenomena retention time distributions are compiled for key retention models an extension to accommodate linear transformations such as decay chains is also introduced detailed testing using first order decay chains and four retention models equilibrium sorption limited diffusion unlimited diffusion and first order kinetic sorption demonstrate that the method is highly accurate simulations using flow fields produced by large scale discrete fracture network simulations a transport problem that is difficult for conventional algorithms demonstrate that the new algorithms are robust and highly efficient

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

a stochastic analysis of macroscopic dispersion in fractured media a stochastic modeling technique has been developed to investigate mass transport in a network of discrete fractures the model is based on the repetitive generation of realizations of a fracture network from probability distributions describing the fracture geometry and on a solution for mass transport within each network using a particle tracking technique the system we work with consists of two orthogonal fracture sets of finite length oriented at various angles with respect to the direction of the mean hydraulic gradient emphasis is placed on describing the character of dispersion which develops as a consequence of fracture interconnectivity and on testing the validity of the conventional diffusion based model of dispersion in describing transport in fractured media results show that mass distributions have a complex form marked longitudinal dispersion can develop even a short distance from a source the distribution of mass in the direction of flow has a consistent negative skew this pattern of dispersion arises from the limited number of pathways for mass to migrate through the network controlling factors in the transport process are the orientation of the fracture sets with respect to the mean hydraulic gradient the difference in the mean flow velocity in the two fracture sets and the standard deviation in velocity for fracture set 1 transport patterns can change greatly as the orientation of the hydraulic gradient changes with respect to the two fracture sets a conventional diffusion based model of dispersion cannot characterize transport in these fracture networks a skewed spatial distribution of mass is observed much more frequently than a gaussian distribution when the mean velocities in the two fracture sets are not equal the form of mass spreading is described by a more general skewed distribution that accounts for the bias in the probability of mass moving along one fracture set over another there is a tendency for mass to form a more symmetric distribution as the orientation of the two fracture sets is rotated toward a 45 angle with respect to the direction of the mean hydraulic gradient furthermore constant dispersivity values or simple dispersivity functions are not definable because of the sensitivity of transport to the local velocity field in the fracture network

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

seepage flow with free surface in fracture networks 1 crystalline rocks are often heterogeneous geological materials that contain numerous fractures of various attitudes and scales although considerable advances have been made in simulation of fluid flow through fractured media our knowledge of seepage flow with free surfaces in fracture networks remains to be an outstanding issue in this paper the partial differential equations pdes defined on the whole fracture network domain are formulated for free surface seepage flow problems by an extension of darcy s law a variational inequality vi formulation is then presented and the proof of the equivalence between the pde and vi formulations is given since the boundary conditions involving the flux components in the pde formulation become the natural conditions in the vi formulation the difficulty of choosing the trial functions for numerical solutions is significantly reduced and the locations of seepage points can be easily determined on the basis of the discrete vi the corresponding numerical procedure for unconfined seepage analysis of discrete fracture network has been developed the results from three typical examples demonstrate the validity and capability of the procedure for unconfined seepage problems involving complicated fracture networks

delay partial differential equations with some large diffusion it is often the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v its eigenvalues and eigenfunctions will also depend upon v in many cases see for example l 7 there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies the remaining eigenvalues blow up at a certain rate in this case one expects the dynamics to be dictated by a system of ordinary differential equations this is indeed the case in several applications and its proof strongly uses the invariant manifold theory in the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain whereas the remaining blow up this is the case for example in parabolic equations in thin domains studied by hale and raugel s we will not be able to address the general case of thin domains but only the case of a thin straight channel however the property that allows this theory to work is present in the general case that is there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up see also arrieta 9 the theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation the main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem a physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that in that direction spatial changes will not be observed the same interpretation can be given to a damped wave problem in a thin straight channel the wave speed can be interpreted as being very large in one direction this would intuitively imply that there would be no spatial changes for the amplitude of the wave in that direction we also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances in this

possible central extensions of non relativistic conformal algebras in 1 1 we investigate possibility of central extension for non relativistic conformal algebras in 1 1 dimension three different forms of charges can be suggested a trivial charge for temporal part of the algebra exists for all elements of l galilei algebra class in attempt to find a central extension as of conformal galilean algebra for other elements of the l galilei class possibility for such extension was excluded for integer and half integer elements of the class we can have an infinite extension of the generalized mass charge for the virasoro like extended algebra for finite algebras a regular charge inspired by schrodinger central extension is possible

quantifying network heterogeneity despite degree distributions give some insights about how heterogeneous a network is they fail in giving a unique quantitative characterization of network heterogeneity this is particularly the case when several different distributions fit for the same network when the number of data points is very scarce due to network size or when we have to compare two networks with completely different degree distributions here we propose a unique characterization of network heterogeneity based on the difference of functions of node degrees for all pairs of linked nodes we show that this heterogeneity index can be expressed as a quadratic form of the laplacian matrix of the network which allows a spectral representation of network heterogeneity we give bounds for this index which is equal to zero for any regular network and equal to one only for star graphs using it we study random networks showing that those generated by the erdos renyi algorithm have zero heterogeneity and those generated by the preferential attachment method of barabasi and albert display only 11 of the heterogeneity of a star graph we finally study 52 real world networks and we found that they display a large variety of heterogeneities we also show that a classification system based on degree distributions does not reflect the heterogeneity properties of real world networks

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays

random neighborhood graphs as models of fracture networks on rocks structural and dynamical analysis abstract we propose a new model to account for the main structural characteristics of rock fracture networks rfns the model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces we study a series of 29 real world rfns and find the best fit with the random rectangular neighborhood graphs rrngs proposed here we show that this model captures most of the structural characteristics of the rfns and allows a distinction between small and more spherical rocks and large and more elongated ones we use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the rfns we find a small set of structural parameters that highly correlates with the average diffusion time in the rfns we found analytically some bounds for the diameter and the algebraic connectivity of these graphs that allow to bound the diffusion time in these networks we also show that the rrngs can be used as a suitable model to replace the rfns in the study of diffusion like processes indeed the diffusion time in rfns can be predicted by using structural and dynamical parameters of the rrngs finally we also explore some potential extensions of our model to include variable fracture apertures the possibility of long range hops of the diffusive particles as a way to account for heterogeneities in the medium and possible superdiffusive processes and the extension of the model to 3 dimensional space

scaling fields in the two dimensional abelian sandpile model we consider the unoriented two dimensional abelian sandpile model from a perspective based on two dimensional conformal field theory we compute lattice correlation functions for various cluster variables at and off criticality from which we infer the field theoretic description in the scaling limit we find perfect agreement with the predictions of a c 2 conformal field theory and its massive perturbation thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory the question of the height 2 variable is also addressed with however no definite conclusion yet

fractional nonlinear diffusion equation solutions and anomalous diffusion abstract we devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms the presence of external forces and absorbent terms is also considered the solutions found here can have a compact or long tail behavior and in particular for the last case in the asymptotic limit we relate these solutions to the levy or tsallis distributions in addition from the results presented here a rich class of diffusive processes including normal and anomalous ones can be obtained

geometry and topology of fracture systems random three dimensional fracture networks are generated according to various rules geometrical and topological features such as the number of three dimensional blocks the percolation thresholds and the cyclomatic numbers are studied with respect to fracture shapes and densities all the results could be successfully interpreted by means of the excluded volume

percolation thresholds of a group of anisotropic three dimensional fracture networks abstract percolation thresholds average number of connections per object of two models of anisotropic three dimensional 3d fracture networks made of mono disperse hexagons have been calculated numerically the first model is when the fracture networks are comprised of two groups of fractures that are distributed in an anisotropic manner about two orthogonal mean directions i e z and x directions we call this model bipolar anisotropic fracture network bfn the second model is when three groups of fractures are distributed about three orthogonal mean directions that is z x and y directions in this model three families of fractures about three orthogonal mean directions are oriented in 3d space we call this model tripolar anisotropic fracture network tfn the finite size scaling method is used to predict the infinite percolation thresholds the effect of anisotropicity on percolation thresholds in x y and z directions is investigated we have revealed that as the anisotropicity of networks increases the percolation thresholds in x y and z directions span the range of 2 3 to 2 0 where 2 3 and 2 0 are extremums of percolation thresholds for isotropic and non isotropic orthogonal fracture networks respectively

a crossing probability for critical percolation in two dimensions langlands et al considered two crossing probabilities and in their extensive numerical investigations of critical percolation in two dimensions cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the q state potts model we extend his results to find an analogous formula for which compares very well with the numerical results