Philippe Davy

SCALING OF FRACTURE SYSTEMS IN GEOLOGICAL MEDIA

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.

Lateral extrusion in the eastern Alps, Part 1: Boundary conditions and experiments scaled for gravity

Lateral extrusion encompasses extensional collapse (gravitational spreading away from a topographic high in an orogenic belt) and tectonic escape (plane strain horizontal motion of wedges driven by forces applied to their boundaries). In the Eastern Alps it resulted from (1) an overall northerly compression (Apulia against Eurasia), (2) a strong foreland (Bohemian massif), (3) lack of constraint along a lateral boundary (Carpathian region), and (4) a previously thickened, gravitationally unstable, thermally weakened crust (Eastern Alpine orogenic belt). Six indentation experiments reproduce lateral extrusion at lithospheric scale. The models have two to four lithospheric layers, with a Mohr/Coulomb rheology for the upper and a viscous rheology for the lower crust. The lithosphere rests upon a low-viscosity asthenosphere. A broad indenter, a narrow deformable area, and a weakly constrained eastern margin fullfill as closely as possible conditions in the Eastern Alps. Indentation produces both thickening in front of the indenter and escape of triangular wedges. Lateral variations in crustal thickness become attenuated by gravitational spreading. The overall fault pattern includes domains of reverse, strike-slip, oblique normal, and pure normal faults. Strike-slip faults in conjugate sets develop serially. The narrow width of the deformable area and the strength of the foreland determine the angles between the sets. Gravitational spreading produces a rhombohedral pattern of oblique and pure normal faults along the unconstrained margin. Opposite the unconstrained margin, the indenter front shows thrusts and folds intersecting with the conjugate strike-slip sets. A triangular indenter favors spreading. High velocity of indentation favors escape. High confinement limits lateral motion, inhibits spreading, and favors thickening. Lateral extrusion in the Eastern Alps is best modeled by (1) a weak lateral confinement, (2) a broad and straight indenter, (3) a narrow width of the deformable area, and (4) a rigid foreland. Crustal thickening, lateral escape, and gravitational spreading all contribute to the overall deformation.

Connectivity of random fault networks following a power law fault length distribution

We present a theoretical and numerical study of the connectivity of fault networks following power law fault length distributions, n(l) ∼ αl−a, as expected for natural fault networks. Different regimes of connectivity are identified depending on a. For a > 3, faults smaller than the system size rule the network connectivity and classical laws of percolation theory apply. On the opposite, for a < 1, the connectivity is ruled by the largest fault in the system. For 1 < a < 3, both small and large faults control the connectivity in a ratio which depends on a. The geometrical properties of the fault network and of its connected parts (density, scaling properties) are established at the percolation threshold. Finally, implications are discussed in the case of fault networks with constant density. In particular, we predict the existence of a critical scale at which fault networks are always connected, whatever a smaller than 3, and whatever their fault density.

Discharge, discharge variability, and the bedrock channel profile

Long-term bedrock incision is driven by daily discharge events of variable magnitude and frequency, with ineffective events below an incision threshold. We explore theoretically how this short-term stochastic behavior controls long-term steady state incision rates and bedrock channel profiles, combining a realistic frequency-magnitude distribution of discharge with a deterministic, detachment-limited incision model in which incision rate is a power function of basal shear stress above a critical shear stress. Our model predicts a power law relationship between steady state slope and drainage area consistent with observations. The exponent of this power law is independent of discharge mean and variability, while the amplitude factor, which controls mountain belt relief, is a power law function of mean runoff (with an exponent of -0.5) and a complex function of runoff variability. In accordance with evidence that incision occurs between 6 and 20% of time in rapidly incising rivers (>1 mm/yr) our model predicts that channel steepness is virtually insensitive to runoff variability. Runoff variability can only decrease channel steepness for very slow incision rates and/or weak lithologies. The relationship between channel steepness and incision rate is always a power law whose exponent depends on the channel cross-sectional geometry and runoff variability. This contradicts models neglecting discharge stochasticity in which the steepness-incision scaling is set by the incision law exponent. Our results suggest that changes in climate variability cannot explain an increase in bedrock incision rates during the Late Cenozoic within the context of a detachment limited model.

Physical models of extensional tectonics at various scales

Summary In a preliminary series of experiments, using physical models mechanical processes of extensional tectonics have been investigated at various scales. By a suitable choice of model materials, experiments were performed at low cost in a natural gravity field. Upper layers of the lithosphere were modelled using sand; lower layers, using silicone putties of two different densities; the mantle asthenosphere was modelled using honey. The models deformed under their own weight or under absolute horizontal tension. Rates of extension were controlled using a stepper motor. Surface deformation and faulting were monitored using 35 mm time-lapse photography. Lower lithosphere topography was photographed through the transparent asthenosphere. Fault patterns in models with lithosphere only, were observed by serial sectioning. Otherwise, the brittle-ductile interface was observed after suctioning off the sand. Simple experiments with uniformly extended sand layers only show that; (i) spacing of normal faults is a measure of the layer thickness; (ii) the length of fault trace increases with the amount of downthrow; and (iii) faults tend to form domino domains. Some experiments with a brittle layer on a ductile substrate show a mechanism of passive rifting where; (i) major faults occur in conjugate pairs, defining rift valleys; (ii) minor faults localize additional extension in rift-valley floors; and (iii) isostatic uplift of the viscous substrate causes uplift and tilting of rift rims. In freely floating continents, gravitational spreading leads to: (i) highly localized extension and thinning at continental margins and (ii) internal rifting.

Hydraulic properties of two‐dimensional random fracture networks following a power law length distribution: 1. Effective connectivity

Natural fracture networks involve a very broad range of fractures of variable lengths and apertures, modeled, in general, by a power law length distribution and a lognormal aperture distribution. The objective of this two-part paper is to characterize the permeability variations as well as the relevant flow structure of two-dimensional isotropic models of fracture networks as determined by the fracture length and aperture distributions and by the other parameters of the model (such as density and scale). In this paper we study the sole influence of the fracture length distribution on permeability by assigning the same aperture to all fractures. In the following paper [de Dreuzy et al., this issue] we study the more general case of networks in which fractures have both length and aperture distributions. Theoretical and numerical studies show that the hydraulic properties of power law length fracture networks can be classified into three types of simplified model. If a power law length distribution n (l) ∼ l−a is used in the network design, the classical percolation model based on a population of small fractures is applicable for a power law exponent a higher than 3. For a lower than 2, on the contrary, the applicable model is the one made up of the largest fractures of the network. Between these two limits, i.e., for a in the range 2–3, neither of the previous simplified models can be applied so that a simplified two-scale structure is proposed. For this latter model the crossover scale is the classical correlation length, defined in the percolation theory, above which networks can be homogenized and below which networks have a multipath, multisegment structure. Moreover, the determination of the effective fracture length range, within which fractures significantly contribute to flow, corroborates the relevance of the previous models and clarifies their geometrical characteristics. Finally, whatever the exponent a, the sole significant scale effect is a decrease of the equivalent permeability for networks below or at percolation threshold.

Hydraulic properties of two-dimensional random fracture networks following a power law length distribution: 2. Permeability of networks based on lognormal distribution of apertures

The broad length and aperture distributions are two characteristics of the heterogeneity of fractured media that make difficult, and even theoretically irrelevant, the application of homogenization techniques. We propose a numerical and theoretical study of the consequences of these two properties on the permeability of bidimensional synthetic fracture networks. We use a power law for the model of length distribution and a lognormal model for aperture distribution. We have especially studied the two endmost models for which length and aperture are (1) independent and (2) perfectly positively correlated. For the model without correlation between length and aperture we show that the permeability can be adequately characterized by a power-averaging function whose parameters are detailed in the text. In contrast, for the model with correlation we show that the prevailing parameter is the correlation when the power law length exponent a is lower than 3, whereas the random structure of the network is a second-order parameter. We also determine the permeability scaling and the scale dependence of the flow pattern structure. Three types of scale effects are found, depending exclusively on the geometrical properties of the network, i.e., on the length distribution parameter a. For a larger than 3, permeability decreases for scales below a definite correlation length and becomes constant above. We show in this case that a correlation between length and aperture does not fundamentally change the permeability model. In all other cases the correlation entails much larger-scale effects. For a in the range 1-3 in the case of an absence of correlation and for a in the range 2-3 in the case of correlation, permeability increases and tends to a limit, whereas the flow structure is channeled when permeability increases and tends to homogenize when permeability tends to its limit. We note that this permeability model is consistent with natural observations of permeability scaling. For a in the range 1-2, in the case of correlation, permeability increases with scale with no apparent limit. We characterize the channeled flow pattern, and we show that permeability may increase even when flow is distributed in several independent structures.

On the Frequency-Length Distribution of the San Andreas Fault System

The frequency-length distribution of the San Andreas fault system was analyzed and compared with theoretical distributions. Both density and cumulative distributions were calculated, and errors were estimated. Neither exponential functions nor power laws are consistent with the calculated distributions over the range of studied lengths. The best fit on both density and cumulative distributions was achieved with a gamma function which mixes a power law and an exponential function. At small lengths, the gamma function behaves as a power law with an exponent of −1.3±0.3. At large lengths (above 10km), the distribution is a mixed exponential-power law function with a characteristic length scale of about 23±6 km. The gamma distribution is proposed to result from a length-dependent segmentation of a fractal fault pattern. This study shows the importance of comparing both cumulative and density distributions. It also shows that the studied range of lengths (1–100 km) is not appropriate for measuring power law exponents.

Periodic instabilities during compression or extension of the lithosphere 1. Deformation modes from an analytical perturbation method

We argue that the plastic rheology of the lithosphere, rather than its “recoverable” elastic properties, is responsible for the systematic development of periodic instabilities during compression or extension. We use a linear perturbation model with analytical solutions to calculate the instability modes for various rheologies. The growth of such periodic instabilities is enhanced by the highly nonlinear stress-strain rheologies encountered in the brittle layers of the lithosphere. On the contrary, ductile layers, deforming according to high temperature creep flow laws, tend to inhibit these instabilities. For oceanic domains, we assume that the only brittle layer is the upper part of the lithosphere. In compression, the only valid instability is a buckling whose wavelength is around 4 times the thickness of the brittle layer. Calculated wavelengths and growth rates are consistent with observations available for the Indian Ocean. For continental domains, a reasonable assumption is the existence of two plastic layers, one in the upper crust, the other in the upper mantle, for Moho temperatures between 450°C and 600°C. In compression and extension, two instabilities develop a long wavelength instability involving the whole lithosphere (coupling mode) and a short wavelength instability involving the crust and controlled by the upper brittle layer (intrinsic crustal mode). In compression, the coupling mode is a whole-lithosphere buckling, with a wavelength about 4 times the thickness of the active lithosphere (the two plastic layers plus the intermediate ductile layer). In extension, the coupling mode is a boudinage of opposite phase in the two plastic layers and a folding of the intermediate ductile layer. The intrinsic crustal mode is a crustal buckling in compression, crustal boudinage in extension. Neither deflects the Moho. The intrinsic crustal mode is favored by an increase in thermal gradient and by a decrease in strength of the ductile lower crust.

Sedimentary basins and crustal thickening

Abstract We consider the development of sedimentary basins in a tectonic context dominated by horizontal shortening and vertical thickening of the crust. Well-known examples are foreland basins; others are ramp basins and buckle basins. We have reproduced various styles of compressional basins in experiments, properly scaled for gravity. A multilayered model lithosphere, with brittle and ductile layers, floats on a model asthenosphere. A computer-driven piston provides shortening and thickening, synchronous with erosion and sedimentation. After a first stage of lithospheric buckling, thrust faults appear, mainly at inflection points. Slip on an isolated reverse fault is accompanied by flexure. Footwall flexure results in a foreland basin and becomes accentuated by sedimentation. Hangingwall flexure is less marked, but may become accentuated by erosion. Motion on a fault leads to hangingwall collapse at the surface. Either footwall sedimentation or hangingwall erosion tends to prolong the active life of a reverse fault. Slip on any pair of closely spaced reverse faults of opposite vergence results in a ramp basin. Simultaneous slip produces a symmetric ramp basin, whereas alternating slip results in a butterfly-shaped basin, with superposed foredeeps. Some well-developed ramp basins become pushed down, until bounding faults meet at the surface and the basin disappears from view. At this stage, the basin depth is equivalent to 15 km or more. Slip on any pair of widely spaced reverse faults of opposite vergence results in a pronounced central anticline, between two distinct foredeeps. In Central Asia and in Western Europe, Cenozoic crustal thickening is due to continental collision. For Central Asia (Western China, Kyrgyzstan, Uzbekistan, Tajikistan), we have compiled a regional structure-contour map on the base of the Tertiary, as well as 4 regional sections. Foreland basins and ramp basins are numerous and associated with Cenozoic thrusts. Large basins (Tarim, Junggar, Fergana, Tajik) occur around and between mountain ranges, but smaller basins (Issyk-Kul, Naryn) occur within them. In Western Europe, the Alps and Pyrenees are surrounded by foreland basins, ramp basins or intermediate styles. In the Andes and its foreland, Neogene thrusts and compressional basins are due to subduction of oceanic lithosphere. In Colombia, they account for much of the Cordillera Oriental; in NW Argentina, for the Altiplano; in West-Central Argentina, for the Sierras Pampeanas. Compressional basins are also common in other areas of older crustal thickening.