Rheological behaviour effects of non-Newtonian fracturing fluids on propagation of a vertical hydraulic fracture

This communication investigates the effects fo the non-Newtonian behaviour of the fracturing fluids in the hydraulic fracturing mechanism. These effects are illustrated on the propagation of a vertical hydraulic fracture into an oil reservoir, in which the fracturing fluid is of power law and of pseudo-plastic type. The analytical solutions for evaluating the fracture lengrhs are presented for the cdases of large fluid loss and for no fiuid loss. The results obtained should be useful in the design of fracture treatments, permitting the finding of the rheological properties of the injected fluid for obtaining the desired fracture length and width.     

Numerical experiment applicable to the latest stage of planet growth

A three-dimensional numerical model was developed with the goal of studying limited dynamical problems relevant to the latest stage of planet growth in the accretion theory. A small number of large protoplanets (～ Moon size) of different masses, moving around the Sun, are considered. The dynamical evolution and growth of the population is studied under mutual gravitational perturbations, accretion, and collisional fragmentation processes. Gravitational encounters are treated exactly by numerical integration of the N-body problem. Outcomes of collisional fragmentation are modeled according to the results of R. Greenberg et al. (1978, Icarus, 35, 1–26). In the present work, we consider 25 protoplanets with uniform mass distribution in the range 2 × 10²⁵?4 × 10²⁶ g on heliocentric orbits in the Earth zone. These bodies are initially confined to a small volume of space to permit gravitational perturbations by close approaches and collisions within a finite length of integration time. The dynamical evolution of the swarm is followed for four different sets of initial ranges in semimajor axis, eccentricity, and inclination: Δa=0.01, 0.02, 0.04, 0.08 AU; Δe= 0.005, 0.01, 0.02, 0.04; Δi=0°3, 0°6, 1°2, 2°4. Among other results, it is found that average eccentricities and inclinations evolve toward a steady state such that $\text{i}\text{?}\text{1}\text{2}\text{,}\text{e}$; it is also found that, whatever the initial conditions, the population evolves toward a quasi-equilibrium relative velocity distribution corresponding to a Safronov parameter value $\text{θ?10}$. Moreover, the growth process of the growing planet presents very similar behavior in the four cases considered, except for the time scale of evolution, which increases with the initial range of orbital elements. Earlier works of this kind have been presented by L.P. Cox and J.S. Lewis (1980, Icarus, 44, 706–721) and by G.N. Wetherill (1980b, In Geol. Soc. Canad. Spec. Publ., p. 20), although a number of differences exist between the three approaches.     

Towards realistic flow modelling. Creation and evaluation of two-dimensional simulated porous media: An image analysis approach

This work is part of an attempt to quantify the relationship between the permeability tensor (K) and the micro-structure of natural porous media. A brief account is first provided of popular theories used to relate the micro-structure toK. Reasons for the lack of predictive power and restricted generality of current models are discussed. An alternative is an empirically based implicit model whereinK is expressed as a consequence of a few pore-types arising from the dynamics of depositional processes. The analytical form of that implicit model arises from evidence of universal association between pore-type and throat size in sandstones and carbonates. An explicit model, relying on the local change of scale technique is then addressed. That explicit model allows, from knowledge of the three-dimensional micro-geometry to calculateK explicitly without having recourse to any constitutive assumptions. The predictive and general character of the explicit model is underlined. The relevance of the change of scale technique is recalled to be contingent on the availability of rock-like three-dimensional synthetic media. A random stationary ergodic process is developed, that allows us to generate three-dimensional synthetic media from a two-dimensional autocorrelation functionr( _x , _y ) and associated probability density function measured on a single binary image. The focus of this work is to ensure the rock-like character of those synthetic media. This is done first through a direct approach:n two-dimensional synthetic media, derived from single set ( ,r( _x , _y )) yieldn permeability tensorsK _i-1,n ⁱ (calculated by the local change of scale) of the same order. This is a necessary condition to ensure thatr( _x , _y ) and carry all structural information relevant toK. The limits of this direct approach, in terms of required Central Process Unit time and Memory is underlined, raising the need for an alternative. This is done by comparing the pore-type content of a sandstone sample andn synthetic media derived fromr( _x , _y ) and measured on that sandstone-sample. Achievement of a good match ensures that the synthetic media comprise the fundamental structural level of all natural sandstones, that is a domainal structure of well-packed clusters of grains bounded by loose-packed pores.     

On a Babcock-Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field, II

In a previous paper (Paper I), we studied a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic field are the source for the poloidal field) that included a thin, deep seated, generating layer (GL) for the toroidal field, B. Meridional motions (of the order of 12 m s^–1 at the surface), rising at the equator and sinking at the poles were essential for the dynamo action. The induction equation was solved by approximating the latitudinal dependence of the fields by Legendre polynomials. No solutions were found with _p = _f where _p and _f are the fluxes for the preceding and following spot, respectively. The solutions presented in Paper I, had _p = –0.5 _f , were oscillatory in time, and large radial fields, B, were present at the surface.Here, we resume the study of Paper I with a different numerical approach allowing for a much higher resolution in , the polar angle. The time dependent partial differential equations for the toroidal and poloidal field are solved with the help of a second order, time and space centered, finite difference scheme. Oscillatory solutions with _p = _f are found for various values of the meridional motions and diffusivity coefficients. The surface values of B, while considerably smaller than those of Paper I, are still unacceptably large, specially at the poles. The reason can be traced to the eruption of toroidal field at high latitudes. It appears that in order to obtain small values for the radial field in the polar regions, high latitude sources ( smaller than /4, say), must reach their maximum below the surface. Weaker meridional motions near the poles than in the equatorial region are also suggested.     

Experimental determination of activities in Fe−Mg olivine at 1400 K

The exchage equilibrium has been used to measure activity-composition relations along the olivine join FeSi_0.5O₂−MgSi_0.5O₂ at 1400 K and 1 atm pressure. Equilibrium Fe−Mg partitioning between the two phases was determined by reversing the compositions of olivine coexisting with oxide and matallic iron over the composition range Fo₂₃ to Fo₉₂. A detailed study of the thermodynamic properties of the oxide phase has recently been made by Srečec et al. and we have confirmed their results in the composition range of interest. Application of the oxide data to the exchange equilibrium enables the properties of olivine to be determined. Within experimental uncertainly (Fe, Mg)Si_0.5O₂ olivine can, at 1400 K, be treated as a symmetric solution with W _Fe-Mg ^o1 of 3.7±0.8 kJ/mol. The data permit the presence of only very slight asymmetry in the series. The data do not support recent assertions that olivine is highly non-ideal (W≈10 kJ/mol) under these conditions.     

Bidirectional Reflectance of Gaussian Random Surfaces and Its Scaling Properties

Using rigorous probabilistic techniques we define and compute the shadowing factor, the conditional mean slope and the conditional standard deviation of the slope for a random rough surface described as an isotropic Gaussian stochastic process with Gaussian autocorrelation function. We use these quantities in order to obtain the bidirectional reflectance distribution function for incoherent light scattering by rough surfaces. The calculated quantities depend on reduced, affine invariant parameters, such that changing roughness is equivalent to rescaling the slopes of incident and emergent direction. We discuss some possible applications of these scaling properties to remote sensing.     

Anatomy of the Indian Summer Monsoon and ENSO relationships in state-of-the-art CGCMs: role of the tropical Indian Ocean

Climate Dynamics - Indian Summer Monsoon (ISM) rainfall and El Niño-Southern Oscillation (ENSO) exhibit an inverse relationship during boreal summer, which is one of the roots of ISM...     

Trace element partitioning and substitution mechanisms in calcium perovskites

We have determined the partition coefficients of a large number of trace elements between CaTiO₃ perovskite and anhydrous silicate melts at atmospheric pressure and 3 GPa. Determination of the concentration limits of Henrys law behaviour in the CaO-Al₂O₃–SiO₂–TiO₂ system reveals that the incorporation of rare earth elements (REE) and tetravalent large ion lithophile elements (LILE⁴⁺ such as U and Th) at the Ca-site of CaTiO₃ perovskite occurs with charge compensation through Ca-vacancy formation rather than by coupled substitution of Al for Ti. When melt composition is varied, we find that partition coefficients for REE and Th are strong functions of the CaO content of the melt. The observed trends are in excellent agreement with those predicted from the Ca-vacancy model. Given that they adopt the same crystal structure and have similar trace element partitioning behaviour, CaTiO₃ perovskite and the deep mantle phase CaSiO₃ perovskite can be considered analogous to one another. When the analogy is pursued in detail, we find that partitioning into both phases follows the composition-dependence predicted by the Ca-vacancy model. Thus, substitution of REE, U⁴⁺ and Th into CaSiO₃ in the lower mantle also occurs with Ca-vacancy formation to balance charge. Furthermore when 2+, 3+ and 4+ partition coefficients for both phases are plotted as functions of CaO melt content, the trends for CaSiO₃ and CaTiO₃ appear to be continuous. This surprising result means that partitioning into Ca-perovskite is independent of pressure and temperature and also of whether or not the host is CaSiO₃ or CaTiO₃. One implication is that CaSiO₃ crystallising from a peridotitic magma ocean may have partition coefficients for Th and U up to about 400. Crystallisation and sequestration of as little as 0.25 volume% of this phase in the lower mantle early in earth history would make a significant contribution to current mantle heat production.Electronic Supplementary Material Supplementary material is available for this article at     

Adaptability of the ocean and earth tidal models based on global observations of the superconducting gravimeters

As the gravity field is the most primary and direct physical quantity reflecting the density variation of the Earth’s interior and its geodynamic properties under various environmental changes and as the knowledge of the fine structure of the Earth’s interior and its geo-dynamics has a strong impact on space research, grav-ity observations become more and more important in Earth sciences. Therefore based on a global network of superconducting gravimeters (SGs), the Global Geo-dynamic Proje…