Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

/pdf/scaling-of-fracture-systems-in-geological-media-1uq6op1ap8.pdf

Scaling of fracture systems in geological media

SUMMARY

We present geological and morphological observations at different scales to constrain rates of faulting and the distribution of deformation in the seismically active Aegean region. We focus first on the 130 km long Corinth Rift, an asymmetric graben where a flight of terraces of marine origin are uplifted. We show that the edges of the terraces lie in the footwall of the normal fault bounding the Corinth Rift and correspond to sea-level highstands of laic Pleistocene age. Using a detailed analysis of aerial and SPOT imagery supported by field observations, we have mapped 10 terrace platforms and strandlines ranging in elevation from 10 to 400 m over distances of 2 to 20 km from the fault. The elevation of the terraces' inner edges was estimated at 172 sites with an error of ±5m. This data set contains a precise description of the uplift and flexure of 10 different palaeohorizontal lines with respect to the present sea level. To date the deformation, we correlate the Corinth terraces with late Pleistocene oxygen-isotope stages of high sea-level stands and with global sea-level fluctuations. Using a thick elastic plate model consistent with our current understanding of the earthquake cycle and a boundary-element technique we reproduce the geometry of the shorelines to constrain both mechanical parameters and the slip on the fault. We show that the seismogenic layer behaves over the long term as if its elastic modulus were reduced by a factor of about 1000. All the terraces are fitted for fault slip increasing in proportion to terrace age, and the component of regional uplift is found to be less than 0.3 mm yr−1. The best fits give a slip rate of 11±3 mm yr−1 on the main rift-bounding fault over the last 350 kyr. Other geological and morphologic information allows us to estimate the total age of the main fault (∼1 Ma) and to examine the mechanical evolution of the Corinth Rift. The minimum observed sediment thickness in the Gulf places an extreme check on the results of the modelling and a lower bound on slip rate of 6–7 mm yr−1 (40 per cent less than estimated with modelling). Even this slip rate is nearly 10 times higher than for comparable features in most of the Aegean and elsewhere in the world.



At a larger scale, the spacing and asymmetry of the rift systems in the Aegean suggest strain localization in the upper mantle, with slow extension starting 15 Myr ago or earlier. The more recent (1 Myr), rapid phase of rifting in Corinth partly reactivated this earlier phase of extension. The younger faulting in Corinth appears to result from its present location in the inhomogeneous stress field (process zone) of the south-westward propagating tip of the southern branch of the North Anatolian Fault. We extend these relations to propose a mechanical model for the Late Cenozoic evolution of the Aegean. As the Arabia/Europe collision progressed in eastern Turkey it caused Anatolia to move to the west and the North Anatolian Fault to propagate into the Aegean, where the early slow extension started to be modified about 5 Ma ago. The process of propagation dramatically increased the activity of some but not all of the earlier rifts. The model we present is compatible with tectonic observations, as well as with the seismicity, the palaeomagnetic rotations and the displacement field now observed with GPS and SLR.

Quaternary evolution of the Corinth Rift and its implications for the Late Cenozoic evolution of the Aegean

Fault growth by segment linkage: an explanation for scatter in maximum displacement and trace length data from the Canyonlands Grabens of SE Utah

https://works.bepress.com/james_evans/139/download/

Permeability of fault-related rocks, and implications for hydraulic structure of fault zones

Cracking and coalescence behavior has been studied experimentally with prismatic laboratory-molded gypsum and Carrara marble specimens containing two parallel pre-existing open flaws. This was done at both the macroscopic and the microscopic scales, and the results are presented in two separate papers. This paper (the first of two) summarizes the macroscopic experimental results and investigates the influence of the different flaw geometries and material, on the cracking processes. In the companion paper (also in this issue), most of the macroscopic deformation and cracking processes shown in this present paper will be related to the underlying microscopic changes. In the present study, a high speed video system was used, which allowed us to precisely observe the cracking mechanisms. Nine crack coalescence categories with different crack types and trajectories were identified. The flaw inclination angle (β), the ligament length (L), that is, intact rock length between the flaws, and the bridging angle (α), that is, the inclination of a line linking up the inner flaw tips, between two flaws, had different effects on the coalescence patterns. One of the pronounced differences observed between marble and gypsum during the compression loading test was the development of macroscopic white patches prior to the initiation of macroscopic cracks in marble, but not in gypsum. Comparing the cracking and coalescence behaviors in the two tested materials, tensile cracking generally occurred more often in marble than in gypsum for the same flaw pair geometries.

Crack Coalescence in Molded Gypsum and Carrara Marble: Part 1. Macroscopic Observations and Interpretation

We report progress made in the last few years on the general problem of the mechanism of fault growth and the scaling laws that result Results are now conclusive that fault growth is a self-similar process in which fault displacement d scales linearly with fault length L Both this result and the overall nature of along-strike fault displacement profiles are consistent with the Dugdale-Barenblatt elastic-plastic fracture mechanics model In this model there is a region of inelastic deformation near the crack tip in which there is a breakdown from the yield strength of the unfractured rock to the residual frictional strength of the fault over a breakdown length S and displacement d0 Limited data also indicate that S and d0 also scale linearly with L, which implies that fracture energy G increases linearly with L The scaling parameters in these relationships depend on rock properties and are therefore not universal In our prime field locality, the Volcanic Tableland of eastern California, we have collected data over 2 orders of magnitude in scale range that show that faults obey a power law size distribution in which the exponent C in the cumulative distribution is ∼13 If the fault is growing within the brittle field, the zone of inelastic deformation consists of a brittle process zone which leaves a wake of fractured rock adjacent to the fault Preliminary results of modeling the process zone are consistent with observations now in hand both in predicting the preferred orientation of cracks in the process zone wake and the rate of falloff of crack density as a function of distance from the fault The preferred orientation of these cracks may be used to infer the mode and direction of propagation of the fault tip past the point in question According to the model, the width of the process zone wake may be used to infer the length of the fault at the time its tip passed the measurement point, but data have not yet been collected to verify this prediction If the fault displacement has been accumulated by repeated seismic slips, each of these will sweep the fault with a crack tip stress field of a smaller spatial extent than that of the fault tip stress field, producing an inner, more intensely fractured, process zone wake This may be the mechanism that creates the cataclasite zone, rather than simple frictional wear, as has been previously supposed

J.-Z. Yu

Papers

Fault growth and fault scaling laws: Preliminary results