Electrical anisotropy and conductivity distribution functions of fractal random networks and of the crust: the scale effect of connectivity

All explanations of the high-conductivity layers (HCL) found by magnetotellurics in the middle or lower crust incorporate a mixture of a low-conductivity rock matrix and a highly conductive phase, for example graphite or saline fluids. In most cases the bulk conductivity of the mixture does not depend on the conductivity of the rock matrix but rather (1) on the amount of high-conductivity material and, in particular, (2) on its geometry. The latter is quantitatively described by the parameter 'electrical connectivity'. Decomposition of the observed bulk conductivity of the mixture into these two parameters results in an ill-posed problem. Even if anisotropy occurs in the HCL, three output parameters (highly conductive phase fraction, connectivity with respect to the X direction, connectivity with respect to the Y direction) have to be estimated from the two bulk conductivities of the anisotropic HCL. The additional information required for solving this problem is provided if instead of single-site data the conductivities from many field sites are evaluated: a sample distribution of the conductivity can then be obtained. Ensembles of random networks are used to create theoretical distribution functions which match the empirical distribution functions to some extent. The use of random resistor networks is discussed in the context of other established techniques for the treatment of two-phase systems, such as percolation theory and the renormalization group approach. Models of embedded networks explain the discrepancy between 'small' anisotropy (2-3) on the laboratory scale and large anisotropy (10-100) found in electromagnetic field surveys encompassing volumes of several cubic kilometres. Strong anisotropy can indicate low electrical connectivity, and a possible explanation is that a network stays close to the percolation threshold.     

Modelling the compliance of crustal rock—I. Response of shear-wave splitting to differential stress

We show that seismic shear waves may be used to monitor the in situ stress state of deep inaccessible rocks in the crust. The most widespread manifestation of the stress-related behaviour of seismic waves is the shear-wave splitting (shear-wave birefringence) observed in almost all rocks, where the polarizations of the leading split shear waves are usually subparallel to the direction of the local maximum horizontal stress. It has been recognized that such shear-wave splitting is typically the result of propagation through distributions of stress-aligned fluid-filled microcracks and pores, known as extensive-dilatancy anisotropy or EDA. This paper provides a quantitative basis for the EDA hypothesis. We model the evolution of anisotropic distributions of microcracks in triaxial differential stress, where the driving mechanism is fluid migration along pressure gradients between neighbouring microcracks and pores at different orientations to the stress field. This leads to a non-linear anisotropic poroelasticity (APE) model for the stress-sensitive behaviour of fluid-saturated microcracked rocks. A companion paper shows that APE modelling matches a range of observed phenomena and is a good approximation to the equation of state of a stressed fluid-saturated rock mass.     

Acceleration of buoyancy-driven fractures and magmatic dikes beneath the free surface

By performing buoyancy-driven fracture experiments in brittle gelatin we observe that the ascent velocity of a fracture containing a finite volume of fluid increases when approaching the free surface. We theoretically describe this free-surface effect and quantify it by introducing an effective depth-dependent fracture toughness and developing an ascent model on the basis of linear fracture mechanics. We develop a successful inversion approach and resolve the actual and critical fracture length and the ascent velocity far away from the free surface from the observation of the fracture tip migration alone. Other parameters, as the fluid volume included in the fracture and the in situ fracture toughness, can be derived. Applying the model and inversion to the 1998 eruption at Piton de la Fournaise, Reunion Island, reveals estimates of the length and critical length of the feeding magma batch, the magma batch volume and the in situ fracture toughness. It further indicates that the ascent velocity of the magma batch was probably much smaller at greater depths and that the batch might have been initiated several months or years before the eruption.