Stochastic analysis of multiphase flow in porous media: II. Numerical simulations

The first paper (Chang et al., 1995b) of this two-part series described the stochastic analysis using spectral/perturbation approach to analyze steady state two-phase (water and oil) flow in a, liquid-unsaturated, three fluid-phase porous medium. In this paper, the results between the numerical simulations and closed-form expressions obtained using the perturbation approach are compared. We present the solution to the one-dimensional, steady-state oil and water flow equations. The stochastic input processes are the spatially correlated logk where k is the intrinsic permeability and the soil retention parameter, . These solutions are subsequently used in the numerical simulations to estimate the statistical properties of the key output processes. The comparison between the results of the perturbation analysis and numerical simulations showed a good agreement between the two methods over a wide range of logk variability with three different combinations of input stochastic processes of logk and soil parameter . The results clearly demonstrated the importance of considering the spatial variability of key subsurface properties under a variety of physical scenarios. The variability of both capillary pressure and saturation is affected by the type of input stochastic process used to represent the spatial variability. The results also demonstrated the applicability of perturbation theory in predicting the system variability and defining effective fluid properties through the ergodic assumption.     

La conducibilità termica di alcune rocce italiane

Riassunto Il lavoro riguarda una serie di misure su campioni di 5 tipi diversi di rocce italiane aventi lo scopo di determinare la rispettiva conducibilità termica. Dopo la descrizione dell'apparecchiatura usata, vengono esposti i risultati ottenuti, analizzando in dettaglio l'influenza della scistosità. In base alle misure fatte su campioni di ardesia questa influenza può esprimersi con la formola:k =k ₉₀+m cos , dove è l'angolo fra la direzione del flusso termico ed il piano di stratificazione, mentrem è una costante caratteristica della scistosità.

---

Summary Experiments made for obtain the thermal conductivity of different kinds of italian rocks are described and discussed. The effect of schistosity was derived for slate rocks and it can be represented with the formula:k =k ₉₀+m cos , where is the angle between the direction of thermal flux and the plain of stratification, whereasm is a constant characterizing the schistosity.

---

---

Lavoro eseguito con un contributo del Consiglio Nazionale delle Ricerche (Comitato per la Geologia).     

Palaeomagnetism of the Börzsöny Mountains (Hungary)

Summary Mean directions of magnetization (29 normal and 31 reversed) were recorded for 60 magmatic localities of middle Miocene age from the Börzsöny Mountains (Hungary). The overall mean direction of RM, irrespective of polarity, isD=0,9°;I=59,8°; withk=8,3 and ₉₅=6,8°. The coordinates of the corresponding geomagnetic north pole are =82,7°, A=193,8 with p=7,7° and m=10,2°.     

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.     

Deformation of a homogeneous gravitating sphere by internal dislocations

Summary An explicit solution is obtained for the system of equations describing the spheroidal motion in a homogeneous, isotropic, gravitating, elastic medium possessing spherical symmetry. This solution is used to derive the Green's dyad for a homogeneous gravitating sphere. The Green's dyad is then employed to obtain the displacement field induced by tangential and tensile dislocations of arbitrary orientation and depth within the sphere.Notation G Gravitational constant - a Radius of the earth - A _o =4/3 G - Perturbation of the gravitational potential - Circular frequency - V _p ,V _s Compressional and shear wave velocities - k _p =/V _p - k _s =/V _s - k _p [(2.8)] - , [(2.17)] - f _l ⁺ Spherical Bessel function of the first kind - f _l ^– Spherical Hankel function of the second kind - x =r - y =r - x _o =r _o - y _o =r_o - x =r k _s - y =r k _p - x _o =r _o k _s - y _o =r _o k _p - =a - =a - [(5.17)] - _{m, l}      

Bottom irregularities in the North Sea

Summary A pressure transducer fixed to a sledge was towed over the bottom of the sea. Pressure variations corresponding to bottom displacement variations were recorded along different profiles in the area west of Sylt. From the time series obtained power spectra of bottom irregularities were computed as a function of wavenumberk or wavelength . Total rms bottom displacement in the wavelength band 50 to 500 m ranges from 3 cm to 36 cm depending on the location of the profile. Presumedly this is due to different types of sediment. Therefore the area cannot be considered as uniform or isotropic concerning the spatial and directional distribution of bottom irregularities. The approximation of spectra byE(k)k– yields -values between 0.4 and 2.9.

---

Bodenunebenheiten in der Nordsee

Zusammenfassung Mit Hilfe einer Druckdose, die von einem Schiff aus über den Meeresboden gezogen wurde, konnten auf ausgewählten Profilen westlich von Sylt Bodenunebenheiten registriert werden. Aus den gewonnenen Zeitserien wurden SpektrenE(k) für die Wellenzahlenk bzw. für die Wellenlänge berechnet. Hieraus lassen sich die Amplituden der Bodenunebenheiten im Wellenlängenbereich von 20 bis 500 m bestimmen. Die effektive Bodenwelligkeit im Bereich von 50 bis 500 m liegt je nach Profil zwischen 3 cm und 36 cm. Die Ursache dafür ist möglicherweise in der unterschiedlichen Sedimentzusammensetzung zu suchen. Eine Isotropie des Bodenspektrums im Meßgebiet ist nicht vorhanden. Die Annäherung der Spektren durchE(k)k– ergibt -Werte zwischen 0,4 und 2,9.

---

Irrégularités du fond en Mer du Nord

Résumé Un manomètre enregistreur fixé à un traîneau a été remorqué sur le fond de la mer. Les variations de pression correspondant aux variations du fond ont été enregistrées le long de différents profils dans la zone Ouest de Sylt. A partir de la série de mesures obtenues, on a calculé le spectre de puissance des irrégularités du fond en fonction du nombre d'ondek ou de la longueur d'onde . Globalement, les variations du fond dans la bande de longueurs d'onde de 50 à 500 m varient de 3 à 36 cm en moyenne quadratique selon la situation du profil. On présume que cela est dû aux différents types de sédiments. Aussi, la zone ne peut pas être considérée comme uniforme ou isotrope en ce qui concerne la distribution dans l'espace et en direction des irrégularités du fond. L'approximation de spectres parE(k)k– donne des valeurs de comprises entre 0,4 et 2,9.

     

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.     

Resuspension of radioactive fall-out from the soil surface and particulate contamination of vegetative cover

A method for the determination of resuspension parameters of radioactive fall-out from nuclear explosions, deposited on soil from the stratospheric reservoir, is presented. The seasonal variation ofk -¹³⁷Cs resuspension parameter, the trend ofk with time in 1963–1978, the trend over this period of time of the wind component fraction of the concentration of global¹³⁷Cs in the surface boundary layer over arable and natural soils are given. An estimate of radioactive contamination of vegetation by the effect considered is given as an example of practical use of this work.     

Primary ionization coefficients for dry and moist air; secondary ionization coefficients for a water surface

Summary A comparison has been made between the Townsend primary ionization coefficient, , for dry air and for air with humidities typical of those in the atmosphere. is defined as the number of new electrons produced by an electron per centimeter of drift in a field. A range of field/pressure ratios,E/p ₀, of 40 to 100 V (cm torr)^–1 was employed. The variation of with humidity is very small.Over the range ofE/p ₀ from 50 to 100 V (cm torr)^–1, the secondary ionization coefficient, , of a water surface has been found from sparking potential data to be typically 2×10^–4. represents the fraction of primary ionizing collisions that ultimately result in the production of additional electrons at the surface.     

Measurement of a freezing nucleus content in a water sample — Evaluation of the method

Summary A drop freezing technique applied in laboratory research to heterogeneous ice nucleation is described. The quality of the obtained experimental data is studied both theoretically and experimentally. Attention is focused on the range of measurable freezing nucleus concentration, the accuracy, the correctness and the reproducibility of the results. Some mechanisms which could influence the correctness of the method are discussed. No mechanism significantly influencing the measurements was found. It is concluded that the present experimental method is an effective method of measuring a freezing nucleus content in a water sample.

---

ama na uu m¶rt;a n¶rt;u mauu ¶rt; ¶rt; a ¶rt;. ¶rt;m ama mam n nuu m¶rt;a. uau ua a ¶rt;uana uum mauu ¶rt; ¶rt;, a mm, naum u nmumm mam. umum m uu a naum m¶rt;a, m u u ma.

     

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.     

Détermination des paramètres d'Ångström par des observations actionmétriques courantes

Summary By comparing the equalities expressing the solar flux within a wide spectral region on the assumption of an extintion within the atmospheric aerosol given by ^– and that of an extintion represented by ₁^–1, a relationship is established by means of which: 1) The error is discussed which is done when the air opacity is expressed by ₁, in the case when 1; 2) A simple and quick method is worked out for determining the parameters and from actinometric observations, carried out within the spectral regions <525 m and 525 m<<625 m.     

The role of wavegroup velocity in the blocking problem

Considering the blocking problem as a baroclinic instability problem in a dispersive wave system with diabatic heating effects, it is of great interest to investigate the role of wavegroup velocityv _gr in blocking processes, becausev _gr controls the energy transfer in the wave field. Using a Newtonian Cooling —type of forcing with a phase differencek to the main field and taking the linearized version of a two-level model, the phase speedc _r, the group velocityv _gr and the growth ratekc _i have been obtained as analytical functions of the mean zonal windU, the thermal windU _T, the coefficient of diabatic heating x, the phase differencek and the wavelengthL. Now the hypothesis is introduced, that a blocking should be expected, ifv _gr has a maximum value in the vicinity ofL _o, for whichc _r vanishes and thee-folding timet=1/kc _i (kc _i>0) is smaller than 6 days (see condition (20) in the text). One finds, that for special parameter combinations (U _T, U, ), where 15 m/secU _T25m/sec,U=10m/sec, 0.8·10^–51.5·10^–5 [sec^–1], certain valuesL _o with an appropriate phase differencek exist, which satisfy the conditions mentioned above (for values see Table 2). TherebyL _o varies within the range 8500 km <L _o<11000 km corresponding to the preferred planetary blocking wavenumber 2 in middle latitudes 50°<<70° N.     

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.     

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).     

Lagrangian Solution of Dynamo Equations and Path Integrals

We discuss an explicit solution of the Cauchy problem for induction equation and suggest its generalization for equations of ²-dynamo. These solutions are based on concepts of multiplicative, Wiener path, and stochastic integrals. Obtained explicit solution can be useful as a tool in investigations of a dynamo with fluctuating helicity.     

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.

---

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.

     

Interpretations and observations of ocean wave spectra

The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of $\left( \mathbf{k},\omega\right) The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of ( k,w)\left( \mathbf{k},\omega\right) -spectra based on analytical expressions and computer simulations of nonlinear waves are presented. Simulations of the dynamic nonlinear evolution of unidirectional free waves using the nonlinear Schr?dinger equation and its generalizations show that components above the spectral peak have larger phase and group velocities than anticipated by linear theory. Moreover, the spectrum does not maintain a thin well-defined dispersion surface, but rather develops into a continuous distribution in ( k,w)\left( \mathbf{k,}\omega\right) -space. The majority of existing measurement systems rely on linear theory for the interpretation of their data, and no measurement systems are currently able to measure the full spectrum in the open ocean with high accuracy. Nevertheless, there exist a few low-resolution systems where data may be interpreted within a minimal assumption of a non-restricted ( k,w)\left( \mathbf{k,}\omega\right) -spectrum. The theory is reviewed, and analyses based on conventional spectral analysis as well as a directional wavelet analysis are carried out on data from a compact laser array at the Ekofisk field in the North Sea. The investigation confirms the strong impact of the second order spectrum below the spectral peak, but is non-conclusive about the off-set in the support of the first order spectrum seen in the dynamical simulations.