a fractal model for crustal deformation abstract we hypothesize that crustal deformation occurs on a scale invariant matrix of faults for simplicity we consider a two dimensional pattern of hexagons on which strike slip faulting occurs the behavior of the system is controlled by a single parameter the fractal dimension deformation occurs on all scales of faults the fractal dimension determines the fraction of the total displacement that occurs on the first order or primary faults the value of the fractal dimension can be obtained from the frequency magnitude relation for earthquakes our results are applied to the san andreas fault system in central california earthquake studies give d 1 90 we associate the main strand of the san andreas fault with the primary faults of our fractal system we predict that the relative velocity across the main strand is 2 93 cm yr the remainder of the relative velocity of 5 5 cm yr between the pacific and north american plates occurs on higher order faults the predicted value is in reasonably good agreement with the value 3 39 0 29 cm yr obtained from geological studies

a pseudo genetic stochastic model to generate karstic networks summary in this paper we present a methodology for the stochastic simulation of 3d karstic conduits accounting for conceptual knowledge about the speleogenesis processes and accounting for a wide variety of field measurements the methodology consists of four main steps first a 3d geological model of the region is built the second step consists in the stochastic modeling of the internal heterogeneity of the karst formations e g initial fracturation bedding planes inception horizons etc then a study of the regional hydrology hydrogeology is conducted to identify the potential inlets and outlets of the system the base levels and the possibility of having different phases of karstification the last step consists in generating the conduits in an iterative manner using a fast marching algorithm in most of these steps a probabilistic model can be used to represent the degree of knowledge available and the remaining uncertainty depending on the data at hand the conduits are assumed to follow minimum effort paths in a heterogeneous medium from sinkholes or dolines toward springs the search of the shortest path is performed using a fast marching algorithm this process can be iterative allowing to account for the presence of already simulated conduits and to produce a hierarchical network the final result is a stochastic ensemble of 3d karst reservoir models that are all constrained by the regional geology the local heterogeneities and the regional flow conditions these networks can then be used to simulate flow and transport several levels of uncertainty can be considered large scale geological structures local heterogeneity position of possible inlets and outlets phases of karstification compared to other techniques this method is fast to account for the main factors controlling the 3d geometry of the network and to allow conditioning from available field observations

probabilistic modeling of faults below the limit of seismic resolution in pelican field north sea offshore united kingdom faults which can compartmentalize a reservoir and which are too small to be seen on seismic can be predicted using a self similar or fractal model of faulting at various scales a computerized method has been developed for statistical analysis of a fault pattern interpreted from three dimensional 3 d seismic in order to fractally derive the distributions of fault size frequency and spatial variations and to verify their invariance within the seismic resolution range this analysis can be used for quality control of seismic interpretation and as input for extrapolation toward smaller scales using probabilistic techniques the application of this program to pelican field united kingdom north sea validated the interpretation of the complex fault pattern of that field but suggested that only limited extrapolations could be confidently made probabilistic maps showing distribution of faults down to 3 m 10 ft of throw and not resolved by 3 d seismic were made these maps subsequently were used as a qualitative aid in full field reservoir simulations to model differently zones of potential dense faulting from zones of reduced faulting this was done because nearby field behavior had shown that faulting is a significant factor in reservoir behavior and ultimately field deliverability field development planning therefore could be optimized

multiscale fracture network characterization and impact on flow a case study on the latemar carbonate platform a fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow the network is described in terms of fracture density orientation and length distribution parameters of particular interest is the role of fracture cross connections and abutments on the effective permeability hence the flow simulations explicitly account for network topology by adopting discrete fracture and matrix description the interior of the latemar carbonate platform dolomites italy is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build ups that exhibit a fracture dominated permeability new is our dual strategy to describe the fracture network both as deterministic and stochastic based inputs for flow simulations the fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops airborne imagery and lidar the deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement the fracture networks exhibit a variable degree of multitier hierarchies with smaller sized fractures abutting against larger fractures under both right and oblique angles the influence of network topology on connectivity is quantified using discrete fracture single phase fluid flow simulations the simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1 0 10 14 m2 the permeability enhancement is strongly controlled by the connectivity of the fracture network therefore the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue

scaling and connectivity of joint systems in sandstones from western norway abstract the scaling properties of a joint system in devonian sandstones in western norway have been investigated using seven maps covering areas from 18 to 720 m across which were generated by mapping in the field and from low level aerial photography taken from different heights each map represent a scale window on the fracture population bounded by resolution at small scales and the sample size at large scales a power law relationship between fracture trace length and critical observation height maximum height at which a trace can be identified is derived and used to create a statistical model for the resolution effects the model indicates that a continuous smooth curve without a straight line segment on a log log cumulative frequency distribution plot does not necessarily rule out a power law as the underlying population distribution together the maps indicate a powerlaw trace length distribution with an exponent of 2 1 this power law may be valid over four or more orders of magnitude with natural lower cut off of around 1 m the exponent is significantly different from 2 0 strictly selfsimilar case and is reflected in a decrease in the abundance of fractures with length comparable to map size as map scale decreases the fracture trace length distribution results in a decrease in apparent connectivity with decreasing scale high resolution large scale maps are well connected while the lowest resolution map smallest scale is unconnected fractures in the smallest scale map are however connected by small fractures below the limit of resolution represented by the largest scale map the variation in apparent connectivity with scale has implications for fluid flow when fractures are open to fluid flow the scaling properties of apparent connectivity imply that beyond a certain scale the size of fracture controlling flow will be scale independent in this fracture system this appears to occur in sample areas of around 300 m across

some consequences of a proposed fractal nature of continental faulting a key question in the study of continental deformation concerns the preservation of undeformed regions sometimes quite large in the midst of otherwise penetrative deformation some authors1 have interpreted such regions as being relatively strong but we argue here that the existence of large undeformed regions might instead arise naturally from a fractal style of continental faulting a scaling analysis derived from laboratory simulations of the india asia collision2 5 yields a predicted distribution of undeformed regions which can be tested in the field we also show that if continental faulting is fractal in nature this will limit the validity of homogenization approaches to the modelling of continental deformation in which the rheology of the lithosphere is assumed to be homogeneous at scales larger than a threshold value

connectivity permeability and channeling in randomly distributed and kinematically defined discrete fracture network models a major use of dfn models for industrial applications is to evaluate permeability and flow structure in hardrock aquifers from geological observations of fracture networks the relationship between the statistical fracture density distributions and permeability has been extensively studied but there has been little interest in the spatial structure of dfn models which is generally assumed to be spatially random i e poisson in this paper we compare the predictions of poisson dfns to new dfn models where fractures result from a growth process defined by simplified kinematic rules for nucleation growth and fracture arrest davy et al 2010 2013 this so called kinematic fracture model is characterized by a large proportion of t intersections and a smaller number of intersections per fracture several kinematic models were tested and compared with poisson dfn models with the same density length and orientation distributions connectivity permeability and flow distribution were calculated for 3d networks with a self similar power law fracture length distribution for the same statistical properties in orientation and density the permeability is systematically and significantly smaller by a factor of 1 5 to 10 for kinematic than for poisson models in both cases the permeability is well described by a linear relationship with the areal density p32 but the threshold of kinematic models is 50 larger than of poisson models flow channeling is also enhanced in kinematic dfn models this analysis demonstrates the importance of choosing an appropriate dfn organization for predicting flow properties from fracture network parameters

influence of landscape coverage on measuring spatial and length properties of rock fracture networks insights from numerical simulation natural fractures are ubiquitous in the earth s crust and often deeply buried in the subsurface due to the difficulty in accessing to their three dimensional structures the study of fracture network geometry is usually achieved by sampling two dimensional 2d exposures at the earth s surface through outcrop mapping or aerial photograph techniques however the measurement results can be considerably affected by the coverage of forests and other plant species over the exposed fracture patterns we quantitatively study such effects using numerical simulation we consider the scenario of nominally isotropic natural fracture systems and represent them using 2d discrete fracture network models governed by fractal and length scaling parameters the groundcover is modelled as random patches superimposing onto the 2d fracture patterns the effects of localisation and total coverage of landscape patches are further investigated the fractal dimension and length exponent of the covered fracture networks are measured and compared with those of the original non covered patterns the results show that the measured length exponent increases with the reduced localisation and increased coverage of landscape patches which is more evident for networks dominated by very large fractures i e small underlying length exponent however the landscape coverage seems to have a minor impact on the fractal dimension measurement the research findings of this paper have important implications for field survey and statistical analysis of geological systems

fractal analysis of fracture patterns using the standard box counting technique valid and invalid methodologies abstract on the basis of results of simple box counting analysis it has been suggested in several recent publications that natural fracture patterns are fractal fractal patterns are characterized by self similarity of structure on a range of scales and provide straight line distributions on box counting plots new analysis of a fracture pattern that provided the most convincing straight line box counting curve previously published shows that the curve is non linear and therefore that the fracture pattern is not fractal non linear box counting curves are also characteristic of other natural fracture patterns analysed but spurious linear curves can be obtained if the area analysed extends beyond the mapped area

improved statistical multi scale analysis of fractured reservoir analogues abstract this paper includes the results of the application of novel methodologies for the statistical analysis of tensile fracture attributes spacing density and aperture the statistical methods have been applied to a fractured lower cretaceous carbonate succession cropping out in the sorrento peninsula naples area italy representing a geological analogue of buried reservoirs of southern apennines major oil fields as fracture networks observable from outcrop to the crystal scale can significantly affect the hydraulic behavior of a fractured reservoir the definition of appropriate models of fracture spacing and aperture probability distribution at various scales represents an important goal of structural analysis in order to achieve such an objective sampling has been carried out at different scales by means of traditional scan lines at outcrop as well as micro scan lines carried out both by a digital micro camera 50 and by the optical microscope 200 fracture spacing and density analysis are mainly aimed at establishing the characteristics of joint spatial distribution over a range of scales aperture analysis is aimed at verifying that the aperture cumulative frequency is described by a power law and in the latter instance to provide an effective method for a more precise determination of the exponent of the power law finally new expressions are provided in closed form in order to calculate the confidence interval of the power law exponent the proposed criteria provide handy and effective methods for a significantly improved statistical analysis of fracture attributes

a statistical scaling model for fracture network geometry with validation on a multiscale mapping of a joint network hornelen basin norway 1 fracture patterns are characterized by a complex geometry which involves a large length distribution and nonhomogeneous density distributions here we address the issue of the modeling of this complex geometry over scales through a first order model providing the characterization of the number of fractures of a given length and at a given scale we propose that the simplest model is a power law in both space and fracture length with three main parameters the exponent a of a power law length distribution the fractal dimension d which fixes the scale dependence of the number of fractures and the fracture density we verify the applicability of this model on seven fracture patterns mapped from the metric scale up to almost the kilometric scale in the hornelen basin the model efficiently describes fracture network properties at all scales with a single set of parameters for the hornelen fracture networks we found a simple relationship between basic exponents a d 1 implying that the fracture network is self similar through this analysis we present different methodological developments for deriving the basic exponents a and d and for verifying the consistency of the model overall the main methodological development is about the normalization of the measurements made from different scales of observation finally we discuss both the limitations and the uses of such a model for analyzing the hydraulic and mechanical properties of fracture networks

the gilsa excursion and the matuyama brunhes transition recorded in 40ar 39ar dated lavas from lanai and maui hawaiian islands summary directional results thellier type palaeointensity determinations and ar ar ages are presented from volcanic units of pleistocene age from the hawaiian islands of maui and lanai twenty nine non contiguous lava flows were sampled on lanai the lava flows formed during the late matuyama polarity chron about 1 6 ma and recorded the gilsa geomagnetic polarity excursion the obtained ages of 1 6 ma for the lanai lavas are significantly older than ages previously obtained for this island these new ages are however in agreement with the age progression of the hawaiian hotspot the palaeomagnetic record over the sampled succession is characterized by reversed and intermediate directions palaeointensities are generally very low about half the present day field intensity dropping to values of 5 t during transitional field states transitional virtual geomagnetic poles vgps from lanai are situated first near the west coast of south america and then switch to offshore western australia normal polarity is never reached in the sampled succession transitional vgps located close to australia accompanied by relatively low palaeointensities are found in most records of short geomagnetic excursions during the matuyama polarity chron e g the cobb mountain subchron the punaru event and the matuyama brunhes precursor suggesting very similar physical processes likely related to dipolar dominance during the onset of these excursions and influences of lower mantle heterogeneities approximately 800 kyr after the gilsa excursion lava flows on maui recorded the last reversal the matuyama brunhes transition fourteen lava flows were sampled which recorded the pre and post transitional behaviour of the earth s magnetic field the geomagnetic field intensity is low prior to the reversal approximately 8 t and increases strongly afterwards up to 63 t these values of about twice the present field intensity of hawaii after the transition emphasize a strong asymmetry between pre and post transitional fields which is particularly strong in the region of hawaii

volcanic vent self similar clustering and crustal thickness in the northern main ethiopian rift 1 the spatial distribution of volcanic vents in the northern ethiopian rift has been analysed in terms of their self similar clustering the fractal distribution of vents has been computed for the monogenetic vents cones domes and maar occurring along the rift valley and in the rift margin the self similar clustering of vents has been computed through the analysis of vent centres coordinates by means of the two point correlation method the vents of the northern main ethiopian rift display a self similar clustering characterised by a correlation coefficient of 1 41 0 02 with a size range defined by a lower cut off of 1 1 km and an upper cut off of 26 5 km this analysis indicates a probable crustal thickness of 27 km as indicated by the value of the computed upper cut off this value compares well with other estimates for the crustal and lithosphere thickness in the ethiopian rift

scaling relationships in intraplate fracture systems related to red sea rifting abstract the geometry of fracture systems was analysed at various scales in the western arabian sedimentary cover rocks investigations covered a wide range of scales from regional surveys to outcrops in well exposed areas and were based on the field mapping of fractures and the interpretation and digitizing of fracture patterns on aerial and satellite images as a first step a ground structural analysis was carried out which provided a better understanding of the relationships of joint and fault systems related to red sea rifting secondly fracture datasets were subjected to statistical and geostatistical analyses of direction size spacing and density and the variability in these quantities with space and scale the main result of this combined approach is that fracturing is not a self similar process from the centimetre scale up to the 1000 km scale in this case the spatial organization of joints and faults appears to be highly controlled by both the red sea extensional regime and the thickness of the different mechanical layers that constitute the crust at different scales i e sedimentary beds basins and the seismogenic crust

stochastic fracture generation accounting for the stratification orientation in a folded environment based on an implicit geological model article i nfo this paper presents a new approach in generating stochastic discrete fracture networks the particularity of the approach isthatitallows us tosimulate thetheoreticalfamiliesoffractures thatare expected ina foldedenviron ment the approach produces fractures that are consistent with the local stratigraphic orientation the fractures are modeled as simplerectangular planarobjects whenthey are modeled they arerotatedaccordingtothe local stratigraphic orientation as the stratigraphy is modeled using an implicit approach we use the gradient of this geological potential field to retrieve the information about the geological orientation the fracture number and size are following user defined probability density functions 2015 elsevier b v all rights reserved

development and validation of a snow albedo algorithm for the modis instrument a prototype snow albedo algorithm has been developed for the moderate resolution imaging spectroradiometer modis it complements existing modis products by providing albedo measurements for areas mapped as snow on a global daily basis by modis cloud detection and atmospheric correction are accomplished using existing modis products models of the bidirectional reflectance of snow created using a discrete ordinate radiative transfer disort model are used to correct for anisotropic scattering effects over non forested surfaces initial algorithm validation is undertaken through comparisons with broadband albedo measurements made at the u s national oceanic and atmospheric administration noaa surface radiation budget network surfrad site in fort peck mt in situ surfrad albedo measurements are compared to daily modis snow albedo retrievals for the period 21 26 november 2000 created from five narrow to broadband albedo conversion schemes the prototype modis algorithm produces reasonable broadband albedo estimates maximum daily differences between the five modis broadband albedo retrievals and in situ albedo are 15 daily differences between the best modis broadband estimate and the measured surfrad albedo are 1 8 however no single conversion scheme consistently provides the closest albedo estimate further validation and algorithm development using data from north america and greenland is ongoing

fractal dimension of fault systems in japan fractal structure in rock fracture geometry at various scales based on fault maps whether or not the fracture geometry of rocks is self similar was examined by using a box counting algorithm the statistical self similarity fractal structure of the fault fracture systems holds well at the scale of about 2 to 20 km the fractal dimension in japan varied from 1 05 to 1 60 the fractal dimension is about 1 5 1 6 at the central part of the japan arc and decreases with distance from the center at a smaller scale the fractal structure also holds well in the rock fracture geometry the fractal dimension of the north izu peninsula fault system branching faults is 1 49 at the scale of 0 625 to 10 km the fractal dimension of rock fracture geometry at the scale order of 10 1 to 10 2 meters is about 1 49 1 61 the upper limit of the fractal dimension of rock fracture geometry is about 1 6 judging from the estimation of fractal dimension on actual fracture geometry of rocks this value may impose a restraint on modeling of faulting and the fracture process of rocks

spatial relationship between earthquakes and volcanic vents in the central northern main ethiopian rift abstract during the breakup of continents extension is commonly accommodated by connected networks of fluid filled fractures dykes and by faults despite the importance of these two extension mechanisms their spatial relationship in three dimensions is poorly understood primarily because it is difficult to quantify the subsurface distribution of faulting and intrusion in order to address this problem we conduct a quantitative analysis of the spatial distribution and clustering of earthquakes and volcanic vents in the main ethiopian rift in east africa in order to understand how extension by faulting and intrusion is distributed throughout the volcanic rift we use fractal analysis of earthquake epicentres in order to infer the 2d characteristics of the subsurface fault network and directly test our model results against the 3d distribution of earthquake hypocentres our results show that fractal analysis of these features is a reliable means to characterise the 3d properties of the fault network in addition the strong similarity between the properties of the fault network derived from earthquakes and properties of the magma filled fracture network derived from fractal analysis of volcanic vents strongly suggests that these are genetically linked we then explore their spatial link using computation of earthquake and vent density which shows that the zone of seismicity is generally around 20 30 km wide while the zone of vents is narrower and centred within the zone of seismicity this spatial relationship suggests that the faults which form rift axial grabens are induced above a narrower and central zone of diking we also demonstrate significant along rift variation in degree of magmatism and faulting with regions of increased degree of diking inferred from a higher cone density characterised by reduced degree of faulting

self similar clustering of cinder cones and crust thickness in the michoacan guanajuato and sierra de chichinautzin volcanic fields trans mexican volcanic belt abstract the spatial clustering of basaltic vents in monogenetic volcanic fields has been used as a proxy for crustal thickness in extensional and back arc tectonic settings the basaltic vents have a fractal clustered distribution self similar clustering described by a power law the power law is defined over a range the size range of the distribution of values in this case the vents separation delimited by a lower and an upper cut offs here we apply the fractal clustering analysis to the two largest monogenetic volcanic fields of the trans mexican volcanic belt tmvb a continental arc built on different crustal terranes the michoacan guanajuato volcanic field mgvf located in the central western tmvb includes over 1000 vents of late pliocene to quaternary age built on attenuated crust of mesozoic to tertiary age the sierra de chichinautzin volcanic field scvf in the central eastern tmvb is composed of 220 late pleistocene to holocene vents laying above thicker crust of precambrian to tertiary age monogenetic vents in both volcanic fields show self similar clustering with fractal exponent d 1 67 in the range 1 3 38 km mgvf and d 1 56 in the range 1 5 32 km scvf the upper cut off uco for the power law distribution of the mgvf well fits the crustal thickness below the volcanic field as derived from independent geophysical data the uco value of scvf indicates a crust thickness of about 32 km this value is in agreement with new geophysical data that indicate magma underplating the crust beneath the volcanic field area

a note on the angular correction applied to fracture intensity profiles along drill core we derive the angular correction that has to be applied to borehole fracture intensities to recover the actual three dimensional distribution when the fracture networks have a length distribution we assume that the fracture intensity is calculated from the fractures that fully transect the core because of the length distribution the classical terzaghi correction which involves the cosine of fracture dip is no longer valid solutions have been calculated in the specific case of fractal fracture networks with power law length distribution we show that the terzaghi correction may significantly overestimate the frequency of fractures subparallel to the borehole the correction procedure proposed here was tested on a fracture database recorded at a study site for a repository of spent nuclear fuel located in se sweden with three outcrop maps and three boreholes a consistency between the dip distributions of outcrops and boreholes was achieved when applying the angular correcting term calculated with the power law length distribution deduced from outcrop maps

2 d density and directional analysis of fault systems in the zagros region iran on a regional scale in this paper 2 d spatial variation of the frequency and length density and frequency length relation of large scale faults in the zagros region iran as a typical fold and thrust belt were examined moreover the directional analysis of these faults as well as the scale dependence of the orientations was studied for this purpose a number of about 8000 faults with l 1 0 km were extracted from the geological maps covering the region and then the data sets were analyzed the overall pattern of the frequency length distribution of the total faults of the region acceptably fits with a power law relation with exponent 1 40 with an obvious change in the gradient in l 12 0 km in addition maps showing the spatial variation of fault densities over the region indicate that the maximum values of the frequency and length density of the faults are attributed to the northeastern part of the region and parallel to the suture zone respectively and the fault density increases towards the central parts of the belt moreover the directional analysis of the fault trends gives a dominant preferred orientation trend of 300 330 and the assessment of the scale dependence of the fault directions demonstrates that larger faults show higher degrees of preferred orientations as a result it is concluded that the evolutionary path of the faulting process in this region can be explained by increasing the number of faults rather than the growth in the fault lengths and also it seems that the regional scale faults in this region are generated by a nearly steady state tectonic stress regime

fissural volcanism polygenetic volcanic fields and crustal thickness in the payen volcanic complex on the central andes foreland mendoza argentina shield volcanoes caldera bearing stratovolcanoes and monogenetic cones compose the large fissural payen volcanic complex located in the andes foreland between latitude 35 s and 38 s the late pliocene pleistocene and recent volcanic activity along e w trending eruptive fissures produced basaltic lavas showing a within plate geochemical signature the spatial distribution of fractures and monogenetic vents is characterized by self similar clustering with well defined power law distributions vents have average spacing of 1 27 km and fractal exponent d 1 33 defined in the range 0 7 49 3 km the fractal exponent of fractures is 1 62 in the range 1 5 48 1 km the upper cutoffs of fractures and vent fractal distributions about 48 49 km scale to the crustal thickness in the area as derived from geophysical data this analysis determines fractured media crust thickness associated with basaltic retroarc eruptions we propose that the payen volcanic complex was and is still active under an e w crustal shortening regime

scaling of fracture connectivity in geological formations a new method to quantify fracture network connectivity is developed and applied to analyze two classical examples of fault and joint networks in natural geological formations the connectivity measure accounts for the scaling properties of fracture networks which are controlled by the power law length distribution exponent a the fractal dimension d and the fracture density the connectivity behavior of fracture patterns depends on the scale of measurement for a d 1 analysis of the san andreas fault system shows that a d 1 and that the connectivity threshold is reached only at a critical length scale in contrast for a typical sandstone joint pattern a d 1 which is on the cusp where the connectivity threshold is highly sensitive to the minimum fracture length in the system

fault growth in brittle ductile experiments and the mechanics of continental collisions using laboratory models of continental collisions we study the mechanisms responsible for large scale deformation and the nature of the penetrative deformation localized or homogeneous and its characteristic scales and structure in order to focus the study one of the most spectacular cases of continental collisions namely the india asia collision is considered different models with varying rheologies are analyzed which attempt to respect the brittle ductile stratification corresponding to the crust and mantle structure in the earth each experiment is quantified by studying the strain field and the fault pattern as a function of the position p of the indenter within the system the strain field is characterized by 1 the second invariant of the two dimensional strain tensor and its variation with position for a fixed p the evolution of its average as a function of p 2 the evolution with p of the participation ratio s2 which quantifies the fraction of the system surface area which participates in the deformation 3 the evolution of the escape ratio calculated as the surface increase gained by the system during its eastward lateral escape divided by the surface covered by the indenter penetration qualitatively we find that the deformation first spreads out but later on undergoes localization once the fault pattern is mature the cumulative fault length does not increase anymore we observe that the strain field is essentially controlled by the kinematics of the larger faults each fault pattern is quantified by studying 1 the histograms of fault orientations 2 the capacity and barycenter fractal dimensions 3 the multifractal generalized dimensions and 4 the distribution of fault lengths the fractal dimension d is found to almost constant within experimental uncertainty d 1 7 0 1 and thus appears rather insensitive to the particular chosen rheology we find a correlation between the generalized multifractal dimensions and two exponents the barycenter fractal dimension b and the exponent a of the fault length distribution this shows that the scaling properties of fault patterns can be characterized by the knowledge of only two exponents b and a of the spatial and length distributions of faults our main results are 1 observation of the growth of self similar fault patterns 2 complete characterization of the fault patterns with two scaling exponents a and b 3 wide distribution of undeteriorated domains and large heterogeneity of the deformation field 4 maturation of the fault structure corresponding to a localization of the deformation on a few large faults in the late stage of the deformation

percolation conditions in binary fractal fracture networks applications to rock fractures and active and seismogenic faults 1 the fracture network patterns in crystalline rock are treated using a binary fractal model described by three fractal geometric parameters the fractal dimension d of the spatial distribution of fractures the fractal dimension a of the fracture length distribution and the maximum fracture length lmax normalized by the domain length l lmax l we present a percolation threshold relationship describing the boundary between percolating and nonpercolating conditions in random binary fractal fracture networks rbffns and apply this to observations of natural fractures our results indicate that the three fractal parameters identified indeed control the connectivity of fracture networks and hence the migration of fluids through rock fractures and active and seismogenic faults moreover our numerical approach may be applied in the future to investigating fluid migration associated with seismicity based on fractal parameters determined from earthquake catalog data

estimates of strain due to brittle faulting sampling of fault populations abstract the geometry of sampling domains is a first order consideration in the characterization of brittle fault populations in most cases descriptions of fault size distributions based on map cross section traverse or borehole data systematically underestimate the number of small faults present in a volume the geometry of sampling domains may be accounted for using an empirical proportionality between fault displacement and trace length estimates of strain in which the sampling geometry is considered suggest that small faults accommodate a significant portion of the total strain due to the brittle faulting process

hydraulic connection and fluid overpressure in upper crustal rocks evidence from the geometry and spatial distribution of veins at botrona quarry southern tuscany italy abstract veins are the geologic record of fluids that filled fractures at depth in the crust in southern tuscany italy well exposed oligocene early miocene sandstones hosting vein systems provide insight into the role of pore fluid and the stress state at the time of vein formation the stress ratio 2 3 1 3 and driving stress ratio r p f 3 1 3 were determined by analysing the distribution length and aperture of fractures and veins and the magnitude of fluid overpressure the derived fluid overpressure for the whole vein system ranges from 30 mpa to 64 mpa with an average of 43 mpa these values indicate that veins formed under supra hydrostatic pressure conditions despite their spatial contiguity two different vein arrays show very different stress and driving pressure ratios one vein system is characterised by 0 62 and r 0 60 the other by 0 54 and r 0 78 the described vein systems are an example of a close spatial association of two non hydraulically connected vein systems representing fluids focused through the upper crust

a model of fracture nucleation growth and arrest and consequences for fracture density and scaling 1 in order to improve discrete fracture network dfn models which are increasingly required into groundwater and rock mechanics applications we propose a new dfn modeling based on the evolution of fracture network formation nucleation growth and arrest with simplified mechanical rules the central idea of the model relies on the mechanical role played by large fractures in stopping the growth of smaller ones the modeling framework combines in a time wise approach fracture nucleation growth and arrest it yields two main regimes below a certain critical scale the density distribution of fracture sizes is a power law with a scaling exponent directly derived from the growth law and nuclei properties above the critical scale a quasi universal self similar regime establishes with a self similar scaling the density term of the dense regime is related to the details of arrest rule and to the orientation distribution of the fractures the dfn model so defined is fully consistent with field cases former studied unlike more usual stochastic dfn models ours is based on a simplified description of fracture interactions which eventually reproduces the multiscale self similar fracture size distribution often observed and reported in the literature the model is a potential significant step forward for further applications to groundwater flow and rock mechanical issues

fractal geometry in the san andreas fault system it has been noted that the spatial distribution of earthquakes and the mode of strain release in the san andreas fault system is related to the complexity of fault geometry because of their rough appearance over many length scales faults can be regarded as fractal surfaces direct estimates of fractal dimension d of portions of the san andreas fault system between the northern gabilan range and the salton sea including the postulated extent of the great 1857 fort tejon earthquake are obtained from measured fault lengths analogous to the lengths of coastlines as discussed by mandelbrot regions characterized by complicated fault geometry are associated with larger values of d based on fault traces mapped at a scale of 1 750 000 d is 1 3 for this reach of the fault defined as a 30 km wide band about a main fault trace for that part near parkfield which could be associated with the nucleation of the 1857 earthquake d is 1 1 at this same scale d is 1 4 for the san andreas and related faults near san bernardino where the 1857 rupture stopped compared to 1 2 for the san andreas san juan fault segments near the point of arrest of the 1966 parkfield earthquake at finer map scales 1 24 000 and 1 62 500 critical lengths of 500 m and 1 km are identified which might relate to the extent of off san andreas fault offsets the critical lengths also suggest that fault geometry is not self similar if this fractal geometry persists through the seismic cycle it may be possible to use a quantitative measure of complexity to explain the occurrence of great and characteristic earthquakes along a given reach of fault

brittle ice shell thickness of enceladus from fracture distribution analysis abstract we determine the depth of fracture penetration in multiple regions of enceladus by performing self similar clustering and length distribution analysis of fractures the statistical characterization of fault population attribute such as length and clustering provide a productive avenue for exploring deformation rate stress transmission mode rheology of the medium and mechanical stratification of the ice satellite through this analysis we estimate the depth of the mechanical discontinuity of enceladus ice shell that is the depth to which fractures penetrate the brittle ice layer above the ductile one in this work we find that for the south polar terrain spt the brittle ice shell interested by fracture penetration is about 30 km and corresponds to the total depth of the ice shell because the spt has a very high thermal gradient and hence fractures likely reach the ocean ice interface in the other regions analyzed the depth of fracture penetration increases from 31 to 70 km from the south pole to northern regions up to 75

numerical investigation of hydraulic fracture network propagation in naturally fractured shale formations abstract hydraulic fracture network hfn propagation in naturally fractured shale formations is investigated numerically using a 3d complex fracturing model based on the discrete element method to account for the plastic deformation behavior of shales the drucker prager plasticity model is incorporated into the fracturing model parametric studies are then conducted for different young s moduli horizontal differential stresses natural fracture nf properties injection rates and number and spacing of perforation clusters numerical results show that horizontal differential stress primarily determines the generation of a complex hfn the plastic deformation of shale can reduce the stimulated reservoir volume this is more obvious with young s modulus of less than 20 gpa in addition a higher injection rate could largely increase the fracture complexity index fci moreover increasing perforation cluster numbers per fracturing stage is beneficial for increasing the fci but it also increases the potential merging of neighboring fractures which may lead to non uniform development of hfn in far wellbore regions to achieve uniform development of hfn within a fracturing stage the distribution of nfs should be fully considered the results presented here may provide improved understanding of hfn generation and are favorable for optimizing fracturing treatment designs for shale formations

the scaling of hydraulic conductivity in rock fracture zones beyond the scale of a few meters fracture traces as viewed on aerial photographs represent fracture zones thus for bulk fractured rock permeability the hydraulic properties of fracture zones are important here observations on joint zones in sandstones from western norway are used to build a quantitative model relating joint zone length and effective hydraulic conductivity using simple rules for hydraulic conductors in series and parallel applied to joint zone geometry the effective hydraulic aperture the aperture of a single parallel walled channel that conducts the same flow is estimated for 72 joint zones of known length a power law relation between trace length l and effective hydraulic aperture be with an effective exponent of around 0 27 is found which may extend to km scales this exponent contrasts with values of 0 5 to 2 0 reported for veins

the accommodation of large strains in the upper lithosphere of the earth and other solids by self similar fault systems the geometrical origin of b value plate tectonics has provided a method of visualizing the geometry of the deformation of the earth s lithosphere on a large scale the description is so concise that for many purposes it provides an explanation of geological processes that overshadows the need to understand the driving processes the mechanics of the zones between the plates are less well understood particularly in continental regions where large areas are subject to deformation both continuous and discontinuous models have been tried but both have obvious drawbacks

scaling and geometric properties of extensional fracture systems in the proterozoic basement of yemen tectonic interpretation and fluid flow implications abstract multi scale mappings of fracture systems in the crystalline basement of yemen are presented fracture datasets are described through statistical analyses of direction length spacing density and spatial distribution results are combined with field observations and can be directly used to model the geometry of the fracture networks in analog basement rocks from multi kilometric to decametric scales the fractured reservoir analog is defined with a dual porosity model in which tectonic and joint systems correspond to the basement reservoir backbone and matrix respectively these two end members reveal contrasting geometrical reservoir and scaling properties in tectonic systems multi scale geometries are self similar the fracture network shows fractal behavior power law length distribution and clustered spacing and fault zones show hierarchical organization of geometrical parameters such as length thickness and spacing in joint systems the fracture network is scale dependent with exponential length distribution and shows anti clustered spacing however these two end members have both well connected properties with fault zones acting as main drain and joint systems acting as the fluid supply

thickness and spatial distribution of veins in a porphyry copper deposit rosia poieni romania abstract a statistical analysis of the vein thickness distributions of veins in the rosia poieni porphyry copper deposit romania indicates that vein thickness data conform to a power law distribution whereas the spatial distribution of veins follows a negative exponential law a high clustering of the veins is indicated by both the spacing distribution and the cv values the similar vein densities observed at the open pit scale are representative of a saturation level of vein nucleation differences in the spacing distribution and in the mean vein thickness are explained by different initial nucleation rates in the deepest and the lowest parts of the porphyry copper the highly clustered organization of the system is explained by vein linkage processes during their growth and modification of the fluid pressure and local stress during the hydrofracturing

evaluation of sampling methods for fracture network characterization using outcrops outcrops provide valuable information for the characterization of fracture networks sampling methods such as scanline sampling window sampling and circular scanline and window methods are available to measure fracture network characteristics in outcrops and from well cores these methods vary in their application the parameters they provide and therefore have advantages and limitations we provide a critical review on the application of these sampling methods and apply them to evaluate two typical natural examples 1 a large scale satellite image from the oman mountains oman 120 000 m2 1 291 669 ft2 and 2 a small scale outcrop at craghouse park united kingdom 19 m2 205 ft2 the differences in the results emphasize the importance to 1 systematically investigate the required minimum number of measurements for each sampling method and 2 quantify the influence of censored fractures on the estimation of fracture network parameters hence a program was developed to analyze 1300 sampling areas from 9 artificial fracture networks with power law length distributions for the given settings the lowest minimum number of measurements to adequately capture the statistical properties of fracture networks was found to be approximately 110 for the window sampling method followed by the scanline sampling method with approximately 225 these numbers may serve as a guideline for the analyses of fracture populations with similar distributions furthermore the window sampling method proved to be the method that is least sensitive to censoring bias reevaluating our natural examples with the window sampling method showed that the existing percentage of censored fractures significantly influences the accuracy of inferred fracture network parameters

mud volcanoes as potential indicators of regional stress and pressurized layer depth abstract although the use of magmatic volcanoes as indicators of tectonic stress is established the application of the same principles to mud volcanoes has been overlooked here we analyse the onshore azerbaijan mud volcano province in the foreland of the greater caucasus and we present the first use of the statistical regional scale distribution of active mud volcanoes to obtain information about 1 contemporary regional tectonic stress and 2 the average depth of pressurized source layers self similar clustering suggests that mud is mobilized from a ca 5 km deep mud prone layer and mud volcano strain indicators allow the estimation of a robust present day direction of regional maximum horizontal stress s h striking ca 30 e these results are in excellent agreement with geophysical information about upper crustal structure in the region and stress data and deformation vectors we conclude that besides other fold and thrust belts mud volcano features could be used as a proxy for active stress orientation and source layer determination in inaccessible settings such as underwater or even the surface of planets where mud volcanoes are considered to be indicators of mobilized fluids

temporal and spatial scaling of hydraulic response to recharge in fractured aquifers insights from a frequency domain analysis quantification of the recharge in fractured aquifers is particularly challenging because of the multiscale heterogeneity and the range of temporal scales involved here we investigate the hydraulic response to recharge of a fractured aquifer using a frequency domain approach transfer functions are calculated in a range of temporal scales from 1 day up to a few years for a fractured crystalline rock aquifer located in ploemeur s brittany france using recharge and groundwater level fluctuations as input and output respectively the spatial variability of the response to recharge characteristic response time amplitude temporal scaling is analyzed for 10 wells sampling the different compartments of the aquifer some of the transfer functions follow the linear reservoir model behavior on the contrary others display a temporal scaling at high frequency that cannot be represented by classic models large scale hydraulic parameters estimated from the low frequency response are compared with those estimated from hydraulic tests at different scales the variability of transmissivity and storage coefficient tends to decrease with scale and the average estimates converge toward the highest values at large scale the small scale variability of diffusivities which implies the existence of a range of characteristic temporal scales associated with different pathways is suggested to be at the origin of the unconventional temporal scaling of the hydraulic response to recharge at high frequency

geometry formation and development of tectonic fractures on the reykjanes peninsula southwest iceland abstract a study was made of 903 fractures most of which are of postglacial age on the reykjanes peninsula some 120 fractures belonging to the vogar fissure swarm he completely within the same postglacial lava flow and were studied in great detail for the vogar fractures 75 are pure tension fractures and 25 are normal faults the average length of the vogar fractures is 611 m the average width is 0 6 m the average spacing measured perpendicular to the average strike n54e of fractures is 456 m the estimated average depth is of the order of several hundreds of meters and the average throw of the faults is 2 3 m all major fractures on the peninsula appear to have formed by coalescence of smaller fractures as a result of absolute tensile stress the fractures group into four fissure swarms each being a part of a volcanic system which it is proposed is fed by an elongated partially molten magma reservoir located at a depth of about 8 km beneath the peninsula it is suggested that a slight magmatic pressure increase in these reservoirs and resulting uplift and tensile stress at the surface is the main cause of postglacial fracture formation

the role of the earth s mantle in controlling the frequency of geomagnetic reversals a series of computer simulations of the earth s dynamo illustrates how the thermal structure of the lowermost mantle might affect convection and magnetic field generation in the fluid core eight different patterns of heat flux from the core to the mantle are imposed over the core mantle boundary spontaneous magnetic dipole reversals and excursions occur in seven of these cases although sometimes the field only reverses in the outer part of the core and then quickly reverses back the results suggest correlations among the frequency of reversals the duration over which the reversals occur the magnetic field intensity and the secular variation the case with uniform heat flux at the core mantle boundary appears most earth like this result suggests that variations in heat flux at the core mantle boundary of the earth are smaller than previously thought possibly because seismic velocity anomalies in the lowermost mantle might have more of a compositional rather than thermal origin or because of enhanced heat flux in the mantle s zones of ultra low seismic velocity

percolation conditions in fractured hard rocks a numerical approach using the three dimensional binary fractal fracture network 3d bffn model 1 we numerically investigate fracture connectivity and percolation conditions in fractured hard rocks using a three dimensional binary fractal fracture network 3d bffn model based on three fractal geometric parameters the fractal dimensions d2 of the spatial distribution of fractures the exponent of the power law cumulative fracture length distribution a and the maximum fracture length lmax normalized by the domain length l lmax l numerical results clarify that the percolation threshold in 3d bffn models is strongly controlled by fractal geometric parameters and is independent of any anisotropy in the orientations in addition when a 0 55 are percolated domains the zone of percolation within seismogenic fracture networks between mi and ki reveals that the networks formed from seismic swarm related seismogenic fractures over a 7 week period related to the intrusion of a dyke inferred previously from seismicity and deformation data

hydraulic properties of fractured rock masses with correlated fracture length and aperture permeability of fractured rocks is investigated considering the correlation between distributed fracture aperture and trace length based on a newly developed correlation equation the influence of