Stochastic analysis of multiphase flow in porous media: 1. Spectral/perturbation approach

Stochastic analysis of steady-state multiphase (water, oil, and air) flow in heterogeneous porous media was performed using the perturbation theory and spectral representation techniques. The gas phase is assumed to have constant pressure. The governing equations describing the flow of oil and water are coupled and nonlinear. The key stochastic input variables are intrinsic permeability,k, and the soil grain size distribution index, . Three different stochastic combinations of these two input parameters were considered. The perturbation/spectral analysis was used to develop closed-form expressions that describe stochastic variability of key output processes, such as capillary and individual phase pressures and specific discharges. The analysis also included the derivation of the mean flow equations and estimation of the effective flow properties. The impact of the spatial variability ofk and on the effective conductivities and the variances of pressures and specific discharges was examined.     

Power-law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering

We consider a general stochastic branching process,which is relevant to earthquakes as well as to many other systems, and we study the distributions of the total number of offsprings (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We apply our results to a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model. The ETAS model assumes that each earthquake can trigger other earthquakes (aftershocks). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (fertility), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime in which the distribution of fertilities is characterized by a power law ~1/¹⁺. For earthquakes we expect such a power-distribution of fertilities with =b/ based on the Gutenberg-Richter magnitude distribution ~ 10^–bm and on the increase ~ 10^–m of the number of aftershocks with the mainshock magnitude m. We derive the asymptotic distributions p_r(r) and p_g(g) of the total number r of offsprings and of the total number g of generations until extinction following a mainshock. In the regime < 2 for which the distribution of fertilities has an infinite variance, we find This should be compared with the distributions obtained for standard branching processes with finite variance. These predictions are checked by numerical simulations. Our results apply directly to the ETAS model whose preferred values =0.8–1 and b=1 puts it in the regime where the distribution of fertilities has an infinite variance. More generally, our results apply to any stochastic branching process with a power-law distribution of offsprings per mother     

On some sources of uncertainty in the Lyman-α ionization-rate calculation

Summary The effect of the uncertainty and variability in the absorption cross-section, ionization cross-section and the Lyman- line shape on the Lyman- ionization-rate calculation is studied. The effect of the variability of the Lyman- line shape seems to be negligible. The effect of the ionization cross-section is rather small. The greatest and very significant effect, particularly at lower altitudes, is due to the discrepancy in the absorption cross-section data. Some ionospheric and atmospheric consequences are briefly discussed.     

Actinometric determination of turbidity parameters in India

Summary A study of the wavelength exponent of aerosol scattering in the Ångström relation for extinction by aerosol has been made from the ground-based measurments of direct solar radiation using Ångström pyrheliometer with and without Schott filters. It has been observed that in India, mainly for the middle part of the year this exponent is zero or even negative which means that the aerosol scattering is nearly neutral which is in marked contrast with the condition prevailing in middle latitudes. It is evident from the -values that the aerosol size distribution in India is far different from that prevailing in middle latitudes. At four representative stations in India, the values of the wavelength exponent and the atmospheric turbidity coefficient have been determined using the method introduced byÅngström [1,2]) and are discussed here.     

Mixed-Parity Solutions in a Mean-Field Dynamo Model

We consider the effect of including both dipole and quadrupole parities in the previous mean-field model of Hollerbach and Jones (1995), which considered dipole parity only. Allowing for both parities, we find that the onset of dynamo action occurs at ₀ 6, in the form of a purely quadrupolar dynamo wave. A symmetry-breaking bifurcation then occurs at ₀ 11, beyond which the solutions are of mixed parity. The quadrupolar component still oscillates about a zero time-average, but the dipolar component about a non-zero average. For even greater ₀ we obtain an unconnected upper-branch solution. In sharp contrast to the HJ95 pure-parity upper branch, however, this mixed-parity upper branch is steady-state rather than periodic. Although it does not appear to be possible to connect these two upper branches by any simple sequence of bifurcations, we nevertheless suggest how aspects of the mixed-parity branch may help in understanding features of the previous pure-parity branch.     

Galactic Spiral Arms and Dynamo Control Parameters

We discuss the effects of galactic spiral arms on the -coefficient, turbulent diffusivity and turbulent energy density of the interstellar turbulence. We argue that the -coefficient and the dynamo number are larger in the interarm regions, whereas the kinetic energy density of turbulence is larger in the arms; the turbulent magnetic diffusivity can be only weakly affected by the spiral pattern.     

Analytical study of the energy rate balance equation for the magnetospheric storm-ring current

We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, E_y, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of E_y, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter compatible with the observed Dst peak values. To facilitate the analytical integration we assume a constant value for through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed Dst the most appropriate values of are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, _m, are found to be very sensitive to the adopted efficiency factor, , decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of and the resulting values for the decay time parameter, _r, do not vary greatly as varies. As a consequence, the relative values of between the main and the recovery phase, _m/_r, can be greater or smaller than one as varies from 0.1 to 1.     

On the propagation of love waves in a heterogeneous layer lying on a homogeneous medium

Summary The present note is devoted to find out the possibility of propagation of love waves in a heterogeneous layer of finite depth lying on a homogenous semi-infinite elastic medium, the rigidity and density in the upper layer varying as (1–sinz), where is a constant andz is the vertical distance from the interface. The numerical results for the phase velocity for some special Earth models are given.     

Approximating the Distribution of Pareto Sums

Heavy tailed random variables (rvs) have proven to be an essential element in modeling a wide variety of natural and human-induced processes, and the sums of heavy tailed rvs represent a particularly important construction in such models. Oriented toward both geophysical and statistical audiences, this paper discusses the appearance of the Pareto law in seismology and addresses the problem of the statistical approximation for the sums of independent rvs with common Pareto distribution F(x)=1 – x^– for 1/2 < < 2. Such variables have infinite second moment which prevents one from using the Central Limit Theorem to solve the problem. This paper presents five approximation techniques for the Pareto sums and discusses their respective accuracy. The main focus is on the median and the upper and lower quantiles of the sums distribution. Two of the proposed approximations are based on the Generalized Central Limit Theorem, which establishes the general limit for the sums of independent identically distributed rvs in terms of stable distributions; these approximations work well for large numbers of summands. Another approximation, which replaces the sum with its maximal summand, has less than 10% relative error for the upper quantiles when < 1. A more elaborate approach considers the two largest observations separately from the rest of the observations, and yields a relative error under 1% for the upper quantiles and less than 5% for the median. The last approximation is specially tailored for the lower quantiles, and involves reducing the non-Gaussian problem to its Gaussian equivalent; it too yields errors less than 1%. Approximation of the observed cumulative seismic moment in California illustrates developed methods.     

Asymmetry in a nearly symmetric dynamo

mam u¶rt;m uu aumuu a m n uamu ¶rt;u nmu umu ¶rt;ua a maum ¶rt;a amu u. a um naa nu nu naama umuu , au a um aumu m n aau umuu mau aum n. aam, m mum au ¶rt;-amu a ma mu aum u u n a aumu m n.     

Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust

Permeability exerts significant control over the development of pore pressure excess in the crust, and it is a physical quantity sensitively dependent on the pore structure and stress state. In many applications, the relation between permeability and effective mean stress is assumed to be exponential and that between permeability and porosity is assumed to be a power law, so that the pressure sensitivity of permeability is characterized by the coefficient and the porosity sensitivity by the exponent . In this study, we investigate experimentally the dependence of permeability on pressure and porosity in five sandstones with porosities ranging from 14% to 35% and we review published experimental data on intact rocks, unconsolidated materials and rock fractures. The laboratory data show that the pressure and porosity sensitivities differ significantly for different compaction mechanisms, but for a given compaction mechanism, the data can often be approximated by the empirical relations. The permeabilities of tight rocks and rock joints show relatively high pressure sensitivity and low porosity sensitivity. A wide range of values for and have been observed in relation to the mechanical compaction of porous rocks, sand and fault gouge, whereas the porosity sensitivity for chemical compaction processes is often observed to be given by 3. We show that since the ratio / corresponds to the pore compressibility, the different dependences of permeability on porosity and pressure are related to the pore structure and its compressibility. Guided by the laboratory data, we conduct numerical simulations on the development of pore pressure in crustal tectonic settings according to the models ofWalder andNur (1984) andRice (1992). Laboratory data suggest that the pressure sensitivity of fault gouge is relatively low, and to maintain pore pressure at close to the lithostatic value in the Rice model, a relatively high influx of fluid from below the seismogenic layer is necessary. The fluid may be injected as vertically propagating pressure pulses into the seismogenic system, andRice's (1992) critical condition for the existence of solitary wave is shown to be equivalent to >1, which is satisfied by most geologic materials in the laboratory. Laboratory data suggest that the porosity sensitivity is relatively high when the permeability is reduced by a coupled mechanical and chemical compaction process. This implies that in a crustal layer, pore pressure may be generated more efficiently than cases studied byWalder andNur (1984) who assumed a relatively low porosity sensitivity of =2.     

Dynamical stability of convection cells in the upper mantle

Summary A two-dimensional flow model of an incompressible fluid with constant viscosity has been used to study the changes in the large-scale flow pattern (aspect ratio 4). Implications for convection in the Earth's mantle are discussed.

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a ¶rt; mn uuu ua u¶rt;mu nm m unm ¶rt; uu uu mu ama¶rt;a. ¶rt;am mam ¶rt; uu amuu u.

     

Curvature of a group of difeomorphisms and the instability of Kármán's vortex street

a auu uu nmama uau u¶rt;a u¶rt;mu (nn ¶rt;uu au m a) ua na ááa m a muau¶rt;¶rt;uaua uma. mumu mmmm uu aa nu¶rt;umu n au uu ma a u ma mu.     

Forced magnetohydrodynamic oscillations of the earth's core

¶rt;m uu maua anu, ¶rt;mu aau ¶rt;a u amuu a aumu¶rt;¶rt;uau ¶rt;uu ¶rt; u. a auum anu m u ¶rt;am au u mu u u¶rt;ua aum n. a nuu ¶rt; ¶rt;u naa, m u anmam u¶rt; . m uu u u ¶rt;uunauu a anmau mu . uu a u¶rt;, m m¶rt; a mu u m aum mm aumaua maum n ¶rt; u.     

Hydromagnetic spin-up under conditions of the earth's core-mantle boundary

Summary The disturbances of the velocity and magnetic fields close to the Earth's core-mantle boundary, caused by sudden irregular changes in the Earth's rotation, are investigated. The problem leads to the investigation of the structure of the Ekman-Hartman hydromagnetic boundary layer, the magnetic diffusive region and the currentless region. Precise Laplacean inversions of the images of all disturbances in the Earth's core-mantle system are obtained for the limiting case of a zero magnetic Prandtl number, =0. The disturbance of the velocity in the direction of the axis of rotation (Ekman suction) in the currentless region has the nature of inertial oscillations with a frequency of 2. Additional disturbances (with respect to the case of =0) of the velocity in the azimuthal and radial directions, particularly for the EHL and MDR region, are determined for the case when 0< « 1. The disturbance in the velocity again has the character of inertial oscillations with the frequency 2, being exponentially damped in EHL asexp (–2²t) and in MDR asexp (–²t).     

ЧИСЛЕННЫЕ ЭКСПЕРИМЕНТЫ С ИСКУССТВЕННЫМИ ЯДРАМИ СУБЛИМАЦИИ В СТРУЙНОЙ МОДЕЛИ КУЧЕВОГО ОБЛАКА

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Summary A formation of ice particles on artificial deposition INis described. INacting in dependence on the temperature and supersaturation over the ice are considered in the 1 D SSconvection cloud model with detailed microphysics including freezing. The limiting influence of the water vapour pressure upon the concentration of active INis shown resulting from the dependence of water vapour pressure on the ice particle concentration.

     

Multifractal measures,especially for the geophysicist

This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach inMandelbrot (1974). The generalization from fractalsets to multifractalmeasures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function (), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity is called Hölder exponent. In terms of the alternative functionf() used in the approach of Frisch-Parisi and of Halseyet al., one has ()=f()–E for measures supported by the Euclidean space of dimensionE. Whenf()0,f() is a fractal dimension. However, one may havef()<0, in which case is called latent. One may even have <0, in which case is called virtual. These anomalies' implications are explored, and experiments are suggested. Of central concern in this paper is the study of low-dimensional cuts through high-dimensional multifractals. This introduces a quantityD _q, which is shown forq>1 to be a critical dimension for the cuts. An enhanced multifractal diagram is drawn, includingf(), a function called (q) andD _q.This text incorporatesand supersedes Mandelbrot (1988). A more detailed treatment, in preparation, will incorporateMandelbrot (1989).     

Comparison of the effective pressure law for permeability and resistivity formation factor in Chelmsford granite

Permeability, resistivity formation factor, and pore volume change were simultaneously measured on samples of Chelmsford granite subjected to confining pressure and pore pressure cycles. Using a technique described in a previous paper, the tangent coefficients of the effective pressure law for permeability _k and for formation factor _F were determined. _k and _F did not differ significantly from one another. They showed a strong stress history dependence as has already been observed for _k in several crystalline rocks. According to the definition of the effective pressure law used here, two physical properties with identical 's must be related through a one-to-one functional relationship. Hence, the observation above suggests that such a relationship may be empirically found between permeability and formation factor. Indeed, analysis of the data revealed that, to a good approximation, permeability was inversely proportional to the formation factor. The same relation has previously been observed in other crystalline rocks. This relationship was included in a recent version of the so-called equivalent channel model. Using this model, the specific surface area of the cracksA _c/V_S, the standard deviation of the distribution of asperities heightsh and the hydraulic radiusm _o were evaluated. The following values were respectively found: 850 cm^–1, 0.008 m and 0.14 m. The specific surface area of the cracks was independently estimated on micrographs of polished sections using a standard quantitative stereology method. The result was in good agreement with the values estimated from the transport properties data.     

L'ordre du coefficient de récombinaison dans la coucheE

Résumé On propose une méthode de détermination de l'ordre du coefficient équivalent de récombinaison en n'utilisant que les données ionosphériques. A cette fin on évalue l'apport de la vitesse des changements passagersdN _m E/dt dans la densité électroniqueN _m E. L'évaluation se fait en présence d'une série de valeurs choisies de , on observe laquelle des courbes définis _m (t) dans cette étude coïncidera de plus près avec _m (t) mesurée. Les changements de montrés sur la figure 4 sont obtenus de cette manière. La valeur diurne de varie de (0.5÷1) 10^–7 cm³ sec^–1; peu après le lever et peu avant le coucher du soleil devient >10^–7 cm³ sec^–1. Lorsque cos>0, d'après la règle décrcit et devient 10^–8 cm³ sec^–1. Les explications des changements de obtenus de cette manière sont données en rendant compte des changements par rapport aux ions atomiques et moléculaires, à la disparition rapide des ions positifs pendant le coucher du soleil en présence d'un coefficient de récombinaison plus grand et des changements de température eventuels. La symétrie ou l'asymétrie des fréquences critiquesf ₀ E quand cos sont égaux permettent dans la marche diurne de juger de l'ordre de . Toutes ces évaluations indiquent également des valeurs de environ 10^–7 cm³ sec^–1. En précisant des mesures def ₀ E il est possible de définir non seulement l'ordre, mais aussi la valeur de elle-même.

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Summary A method for the determination of the order of the equivalent recombination coefficient is suggested, by using ionospheric data only. The increase in the speed of temporary changes indN _m /dt in the electronic densityN _m E is estimated for this purpose. The estimation is done with series of selected for the sake of expediency values of , following at the same time which curves _m (t), determined in the course of work, will coincide most closely with the _m (t) measured. The changes in , shown in figure 4, have been obtained in this way. The diurnal value of is in the range of (0.5 to 1) 10^–7 cm³ sec, being >10^–7 cm³ sec a little after sunrise and a little prior to sunset. At cos<0, by rule decreases and becomes 10^–8 cm³ sec. Explanations of the changes thus obtained in are indicated, taking into consideration the relation of atomic and molecular ions, the rapid disappearance of positive ions at sunset with a higher recombination coefficient and eventual temperature changes. From the symmetry or asymmetry of the critical frequencies off ₀ E at equal cos in the course of the day it can also be judged for the order of . All those estimates show values of in the range of 10^–7 cm³ sec. In the case of precise measurements off ₀ E, not only the determination of the order but also the real value of is possible.