APPROXIMATION OF THE GRAVITATIONAL ATTRACTION OF GEOLOGICAL BODIES*

The gravitational attraction produced by a geological body of irregular shape can be easily determined by dividing it into cubes of small size. The exact expression of the effect of a cube is very complicated, but it can be calculated by using an electronic computer. 4851 values of the gravitational attraction were determined for different positions of a cube with the side of 2l and the center in M(x₀, y₀, z₀), for x₀∈[0;20], y₀∈[0;20] and z₀∈[0;10]. Using these values, templates in double logarithmic representation were drawn, with x₀ and z₀ as parameters and y₀ as abscissa, and with x₀ and y₀ as parameters and z₀ in abscissa; this double set of templates permits a good interpolation for all cubes in the considered domain of M. The use of templates was tested to approximate the effect produced by a theoretical model of spherical shape and in a real case of a three-dimensional salt body of known shape based on a large number of boreholes. In both cases very good results were obtained.     

The α-effect in 2D fast dynamos

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t)?=?(? _y ?ψ?(x, y, t), ?? _x ?ψ?(x, y, t), ?ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R _m . We calculate this effect for different motions and investigate how its limit as k?→?0 depends on R _m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence.     

Finite element-fractional steps solution of the 3-D coastal circulation model

The mathematical model for the nearly horizontal circulation due to wind, tides and density gradients in 3-D coastal areas is solved by a combined use of the method of finite elements and the integration in fractional steps. The discretisation of the flow domain is achieved through a system of 1-D finite elements over the depth, z, and 2-D finite elements in x?y space. The differential operators of the momentum equations in x and y, are split and integrated separately in z and x?y dimensions. The method is an extension of a previously presented approach combining finite differences and expansion in series. The application refers to the wind induced circulation in the 3-D coastal basin of Thessaloniki Bay.     

Anisotropic migration velocity analysis: Application to a data set from West Africa

Although it is widely recognized that anisotropy can have a significant influence on the focusing and positioning of migrated reflection events, conventional depth imaging methods still operate with isotropic velocity fields. Here, we present an application of a 2D migration velocity analysis (MVA) algorithm, designed for factorized v(x, z) VTI (transversely isotropic with a vertical symmetry axis) media, to an offshore data set from West Africa. By approximating the subsurface with factorized VTI blocks, it is possible to decouple the spatial variations in the vertical velocity from the anisotropic parameters with minimal a priori information. Since our method accounts for lateral velocity variation, it produces more accurate estimates of the anisotropic parameters than those previously obtained with time‐domain techniques. The values of the anellipticity parameter η found for the massive shales exceed 0.2, which confirms that ignoring anisotropy in the study area can lead to substantial imaging distortions, such as mis‐stacking and mispositioning of dipping events. While some of these distortions can be removed by using anisotropic time processing, further marked improvement in image quality is achieved by prestack depth migration with the estimated factorized VTI model. In particular, many fault planes, including antithetic faults in the shallow part of the section, are better focused by the anisotropic depth‐migration algorithm and appear more continuous. Anisotropic depth migration facilitates structural interpretation by eliminating false dips at the bottom of the section and improving the images of a number of gently dipping features. One of the main difficulties in anisotropic MVA is the need to use a priori information for constraining the vertical velocity. In this case study, we successfully reconstructed the time–depth curve from reflection data by assuming that the vertical velocity is a continuous function of depth and estimating the vertical and lateral velocity gradients in each factorized block. If the subsurface contains strong boundaries with jumps in velocity, knowledge of the vertical velocity at a single point in a layer is sufficient for our algorithm to determine all relevant layer parameters.     

Directions of preferential flow in a hillslope soil, 1. Quasi‐steady flow

Preferred infiltration is mainly perceived as vertically down whereas subsurface storm flow is thought to occur parallel to slopes. The transition from vertical to lateral flow in a layered hillslope soil is the focus of the contribution. Transient flow is assumed to move as a wetting front. Three time‐domain reflectometry (TDR) wave‐guides, each 0·15 m long, were mounted in the shape of a truncated tetrahedron with its peak pointing down. Each wave‐guide focuses the front velocity along its axis. The three front‐velocity vectors are decomposed into their x, y and z components, which are then assembled to the resultant velocity vector. The volume density flux of preferred flow is the product of the front velocity and the mobile water content. The latter is the amplitude of transient soil moisture measured with each wave‐guide. The resultant vector of the volume flux density is computed similarly to the velocity vector. The experimental approach allows for the rapid assessment of transient flows without relying on the variation of water potentials. The experiments indicate that the directions of the resultant vectors of velocity and volume flux density can be estimated if the moisture variations of the three TDR wave‐guides are strongly correlated during the passing of the wetting front. Copyright © 2004 John Wiley & Sons, Ltd.     

Determining the dilation factor in 4D monitoring of compacting reservoirs by rock-physics models

Hydrocarbon depletion and fluid injection cause compaction and stretching of the reservoir and overburden layers. 4D prestack seismic data can be used to detect these changes because compaction/stretching causes changes in traveltimes and seismic velocities. We show that, by using two different petro‐elastic models at varying effective pressures, a good approximation is to assume that the fractional changes in layer thickness, ΔL/L, and seismic velocity, Δv/v, are related by a linear function of ΔL/L. The slope of this function (the dilation factor, α= (Δv/v)/(ΔL/L) ) is negative and its absolute value generally decreases (shale, low porosity) or increases (sandstone, high porosity) with increasing layer thickness and decreasing effective pressure. The analysis is mainly performed for isotropic deformations. The dilation factor for uniaxial deformations is smaller in absolute value. The dilation factor, which can be calculated from time‐lapse data, can be used to predict reservoir compaction/stretching as a function of depth and surface subsidence.     

Soil water content vertical profiles under natural conditions: matching of experiments and simulations by a conceptual model

The prediction of soil moisture content, θ, as a function of depth, z, and time, t, is of fundamental importance for applications in many hydrological processes. The main objective of this paper is to provide an approach to solve this problem at a local scale in soils with vegetation. The matching of soil moisture vertical profiles observed under natural conditions in grassy plots and their simulations by a conceptual model is presented. Experimental measurements were performed in a plot located in Central Italy, complete with hydrometeorological sensors specifically set up and equipped with a time domain reflectometry system providing the water content, θ_e(z, t). A conceptual model framework earlier proposed for two‐layered soil vertical profiles was modified and adopted for simulations. The changes concern the incorporation of evapotranspiration, the reduction of the original model for applications also to homogeneous soil vertical profiles, and a correction for the differences existing between assumed and observed initial moisture contents. In the model calibration, it was found that the effects of vegetation could be represented adequately by a fictitious soil vertical profile with a more permeable upper layer of saturated hydraulic conductivity, K_s, independent of time. Then, for the validation events, the model simulations in the stages of both infiltration and redistribution/evapotranspiration reproduced appropriately θ_e(z, t) with typical values of root mean square error in the range 0.0017–0.0657. Similar results were obtained by applying the modified two‐layered model for simulations of experimental data observed in three other plots located in Northern Italy and Germany. For all four vegetated sites, the two‐layer profile better matched the experimental data than the assumption of a homogeneous profile. Thus, the conceptual approach based on a two‐layered scheme for representing θ(z, t) in soils with vegetation appears to be appropriate for many hydrological applications. Copyright © 2013 John Wiley & Sons, Ltd.     

Gopher bioturbation: field evidence for non‐linear hillslope diffusion

It has generally been assumed that diffusive sediment transport on soil‐mantled hillslopes is linearly dependent on hillslope gradient. Fieldwork was done near Santa Barbara, California, to develop a sediment transport equation for bioturbation by the pocket gopher (Thomomys bottae) and to determine whether it supports linear diffusion. The route taken by the sediment is divided into two parts, a subsurface path followed by a surface path. The first is the transport of soil through the burrow to the burrow opening. The second is the discharge of sediment from the burrow opening onto the hillslope surface. The total volumetric sediment flux, as a function of hillslope gradient, is found to be: q_s (cm³ cm⁻¹ a⁻¹) = 176(dz/dx)³ − 189(dz/dx)² + 68(dz/dx) + 34(dz/dx)^0·4. This result does not support the use of linear diffusion for hillslopes where gopher bioturbation is the dominant mode of sediment transport. A one‐dimensional hillslope evolution program was used to evolve hillslope profiles according to non‐linear and linear diffusion and to compare them to a typical hillslope. The non‐linear case more closely resembles the actual profile with a convex cap at the divide leading into a straight midslope section. Copyright © 2000 John Wiley & Sons, Ltd.     

GIS‐assisted modelling for debris flow hazard assessment based on the events of May 1998 in the area of Sarno,Southern Italy: Part I. Maximum run‐out

Based on the debris flow events that occurred in May 1998 in the area of Sarno, Southern Italy, this paper presents an approach to simulate debris flow maximum run‐out. On the basis of the flow source areas and an average thickness of 1·2 m of the scarps, we estimated debris flow volumes of the order of 10⁴ and 10⁵ m³. Flow mobility ratios (ΔH/L) derived from the x, y, z coordinates of the lower‐most limit of the source areas (i.e. apex of the alluvial fan) and the distal limit of the flows ranged between 0·27 and 0·09. We performed regression analyses that showed a good correlation between the estimated flow volumes and mobility ratios. This paper presents a methodology for predicting maximum run‐out of future debris flow events, based on the developed empirical relationship. We implemented the equation that resulted from the calibration as a set of GIS macros written in Visual Basic for Applications (VBA) and running within ArcGIS. We carried out sensitivity analyses and observed that hazard mapping with this methodology should attempt to delineate hazard zones with a minimum horizontal resolution of 0·4 km. The developed procedure enables the rapid delineation of debris flow maximum extent within reasonable levels of uncertainty, it incorporates sensitivities and it facilitates hazard assessments via graphic user interfaces and with modest computing resources. Copyright © 2007 John Wiley & Sons, Ltd.     

Point-source inertial particle dispersion

The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small but finite inertia. Our focus is on the evolution equation of the particle joint probability density function p(x,?v,?t), x and v being the particle position and velocity, respectively. For arbitrary inertia, position and velocity variables are coupled, with the result that p(x,?v,?t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small (but nevertheless finite) inertia, (x,?v)-variables decouple and the determination of p(x,?v,?t) is reduced to solve a system of two standard forced advection–diffusion equations in the space variables x. The latter equations are derived here from first principles, i.e., from the well-known Lagrangian evolution equations for position and particle velocity.     

Estimating agricultural deep drainage lag times to groundwater: application to Antelope Valley,California, USA

Estimates of groundwater volumes available in semiarid regions that rely on water balance calculations require the determination of both surface to groundwater lag times and volumes from irrigation or rainfall initiated recharge. Subsurface geologic material hydraulic properties (e.g. hydraulic conductivities, water retention functions) necessary for unsaturated flow modelling are rarely available as are the instrumented field tests that might determine such lag times. Here we develop a simple two‐parameter (specific yield, S_y, and pore‐size distribution index, λ), one‐dimensional unsaturated flow model from simplifications of the Richards equation (using the Brooks‐Corey relationships) to determine lag times from agricultural deep drainage associated with the irrigation of alfalfa hay and various row crops in the Antelope Valley of California, USA. Model‐predicted lag times to depths of 85 m bgs (below ground surface) were similar to that measured in a 2‐year ponded recharge field trial, slightly overestimating that measured by approximately 15% (0.51 vs 0.44 years). Lag time estimates were most sensitive to estimated deep percolation rates and roughly equally sensitive to the model hydraulic parameters. Generally, as subsurface material textures coarsen towards larger S_y and λ values for all S_y >10%, lag times progressively increase; however, at S_y <10%, lag times decrease substantially suggesting that particular combinations of S_y and λ values that may be associated with similarly textured materials can result in the prediction of different lag times for S_y approximately 10%. Overall, lag times of 1–3 years to a depth of 69 m bgs were estimated from deep drainage of agricultural irrigation across a variety of irrigation schedules and subsurface materials. Copyright © 2012 John Wiley & Sons, Ltd.     

A low‐dimensional model of bedrock weathering and lateral flow coevolution in hillslopes: 1. Hydraulic theory of reactive transport

This is the first of a two‐part paper exploring the coevolution of bedrock weathering and lateral flow in hillslopes using a simple low‐dimensional model based on hydraulic groundwater theory (also known as Dupuit or Boussinesq theory). Here, we examine the effect of lateral flow on the downward fluxes of water and solutes through perched groundwater at steady state. We derive analytical expressions describing the decline in the downward flux rate with depth. Using these, we obtain analytical expressions for water age in a number of cases. The results show that when the permeability field is homogeneous, the spatial structure of water age depends qualitatively on a single dimensionless number, Hi. This number captures the relative contributions to the lateral hydraulic potential gradient of the relief of the lower‐most impermeable boundary (which may be below the weathering front within permeable or incipiently weathered bedrock) and the water table. A “scaled lateral symmetry” exists when Hi is low: age varies primarily in the vertical dimension, and variations in the horizontal dimension x almost disappear when the vertical dimension z is expressed as a fraction z/H(x) of the laterally flowing system thickness H(x). Taking advantage of this symmetry, we show how the lateral dimension of the advection–diffusion‐reaction equation can be collapsed, yielding a 1‐D vertical equation in which the advective flux downward declines with depth. The equation holds even when the permeability field is not homogeneous, as long as the variations in permeability have the same scaled lateral symmetry structure. This new 1‐D approximation is used in the accompanying paper to extend chemical weathering models derived for 1‐D columns to hillslope domains.     

On the effects of a bumpy core-mantle interface

Motivated by the high degree of correlation between the variable parts of the magnetic and gravitational potentials of the Earth discovered by Hide and Malin (using a harmonic analysis approach and utilizing the geomagnetic data) when one field is suitably displaced relative to the other, Moffatt and Dillon (1976) studied a simple planar model in an attempt to find a quantitative explanation for the suggestion that this high degree of correlation may be due to the influences produced by bumps on the core-mantle interface. Moffatt and Dillon assumed that the core-mantle interface was z = η(x) where |?η/?χ| ? 1 and such that in the core [z < η(x)] a uniform flow (U₀, 0, 0) prevails in the presence of a uniform ‘toroidal’ field (B₀, 0, 0); (here z is the vertical coordinate and x is the eastward distance). The whole system rotates uniformly about the vertical with angular velocity Ω. The present work extends the model discussed by Moffatt and Dillon to include a horizontal component of angular velocity Ω_H and a uniform small poloidal field B_p. In addition, the uniform toroidal field is here replaced by one which vanishes everywhere in the mantle and increases linearly, from zero on the interface, with z. It is shown that the presence of Ω_H and B_p, together with the present choice of toroidal magnetic field, has a profound effect both on the correlation between the variable parts of the magnetic and gravitational fields of the Earth, and on how far the disturbances caused by the topography of the interface [which is necessarily three-dimensional i.e. z = η(x, y) here] can penetrate into the liquid core. In particular it is found that the highest value of the correlation function is +0.79 which corresponds to a situation in which the magnetic potential is displaced both latitudinally and longitudinally relative to the gravitational potential.     

The Displacement and Strain Field of Three-dimensional Rheologic Model of Earthquake Preparation

Based on the three-dimensional elastic inclusion model proposed by Dobrovolskii, we developed a rheological inclusion model to study earthquake preparation processes. By using the Corresponding Principle in the theory of rheologic mechanics, we derived the analytic expressions of viscoelastic displacement U(r, t) , V(r, t) and W(r, t), normal strains ε_xx (r, t), ε_yy (r, t) and ε_zz (r, t) and the bulk strain θ (r, t) at an arbitrary point (x, y, z) in three directions of X axis, Y axis and Z axis produced by a three-dimensional inclusion in the semi-infinite rheologic medium defined by the standard linear rheologic model. Subsequent to the spatial-temporal variation of bulk strain being computed on the ground produced by such a spherical rheologic inclusion, interesting results are obtained, suggesting that the bulk strain produced by a hard inclusion change with time according to three stages (α, β, γ) with different characteristics, similar to that of geodetic deformation observations, but different with the results of a soft inclusion. These theoretical results can be used to explain the characteristics of spatial-temporal evolution, patterns, quadrant-distribution of earthquake precursors, the changeability, spontaneity and complexity of short-term and imminent-term precursors. It offers a theoretical base to build physical models for earthquake precursors and to predict the earthquakes.     

Electromagnetic scattering by a perfectly conducting half-plane in a conductive half-space

FollowingDmitriev (1960) a rigorous theoretical solution for the problem of scattering by a perfectly conducting inclined half-plane buried in a uniform conductive half-space has been obtained for plane wave excitation. The resultant integral equation for the Laplace transform of scattering current in the half-plane is solved numerically by the method of successive approximation. The scattered fields at the surface of the half-space are found by integrating the half-space Green's function over the transform of the scattering current.The effects of depth of burial and inclination, of the half-plane on the scattered fields are studied in detail. An increase in the depth of burial leads to attenuation of the fields. Inclination introduces asymmetry in the field profiles beside affecting its magnitude. Depth of exploration is greater for quadrature component. An interpretation scheme based on a phasor diagram is presented for the VLF-EM method of exploration for rich vein deposits in a conductive terrain.List of symbols x, y, z Space co-ordinates - Half-space conductivity - ₀ Free-space permeability - Excitation frequency (angular) - T Time - h Depth of the half-plane - a Inclination of the half-plane - E _x x-Directed total electric field - E _x ^p x-Directed primary electric field - E _xo ^p x-Directed primary electric field atz=0 directly over the half-plane - H _y y-Component of total magnetic field - H _y ^p y-Component of primary magnetic field - H _y0 ^p y-Component of primary magnetic field atz=0 directly over the half-plane - H _z z-Component of total magnetic field - H _z ^p z-Component of primary magnetic field - J _x Surface density ofx-directed scattering current - G Green's function - k ₀,K Wave numbers - u,u ₀,u ₁,u ₂ Functions - Space co-ordinate - s Variable in transform domain - Variable of integration - Normalized scattering current - Laplace transform of - N Normalized - , ₀, ₁, ₂ Functions - t Variable of integration - Skin depth - H Total magnetic field - H ^p Primary magnetic field - H ₀ ^p Primary magnetic field atz=0 directly over the half-plane - M,Q,R,S,U,V Functions - N ₁,N ₂ Functions     

Seismoelectric measurements in a porous quartz‐sand sample with anisotropic permeability

Seismoelectric coupling coefficients are difficult to predict theoretically because they depend on a large numbers of rock properties, including porosity, permeability, tortuosity, etc. The dependence of the coupling coefficient on rock properties such as permeability requires experimental data. In this study, we carry out a set of laboratory measurements to determine the dependence of seismoelectric coupling coefficient on permeability. We use both an artificial porous “sandstone” sample, with cracks, built using quartz‐sand and Berea sandstone samples. The artificial sample is a cube with 39% porosity. Its permeability levels are anisotropic: 14.7 D, 13.8 D, and 8.3 D in the x‐, y‐, and z‐directions, respectively. Seismoelectric measurements are performed in a water tank in the frequency range of 20 kHz–90 kHz. A piezoelectric P‐wave source is used to generate an acoustic wave that propagates through the sample from the three different (x, y, and z) directions. The amplitudes of the seismoelectric signal induced by the acoustic waves vary with the direction. The highest signal is in the direction of the highest permeability, and the lowest signal is in the direction of the lowest permeability. Since the porosity of the sample is constant, the results directly show the dependence of seismoelectric coefficients on permeability. Seismoelectric measurements with natural rocks are performed using Berea sandstone 500 and 100 samples. Because the Berea samples are nearly isotropic in permeability, the amplitudes of the seismoelectric signals induced in the different directions are the same within the measurement error. Because the permeability of Berea 500 is higher than that of Berea 100, the amplitude of the seismoelectric signals induced in Berea 500 is higher than those in Berea 100. To determine the relative contributions of porosity and permeability on seismoelectric conversion, we carried out an analysis, using Pride (1994) formulation and Kozeny–Carman relationship; the normalized amplitudes of seismoelectric coupling coefficients in three directions are calculated and compared with the experimental results. The results show that the seismoelectric conversion is related to permeability in the frequency range of measurements. This is an encouraging result since it opens the possibility of determining the permeability of a formation from seismoelectric measurements.     

Cross-matching with interpreted warping of 3D streamer and 3D ocean-bottom-cable data at Valhall for time-lapse assessment

Legacy streamer data and newer 3D ocean‐bottom‐cable data are cross‐matched and analysed for time‐lapse analysis of geomechanical changes due to production in the Valhall Field. The issues relating to time‐lapse analysis using two such distinctly different data sets are addressed to provide an optimal cross‐matching workflow that includes 3D warping. Additionally an assessment of the differences between the imaging using single‐azimuth streamer and multi‐azimuth ocean‐bottom‐cable data is provided. The 3D warping utilized in the cross‐matching procedure is sensitive to acquisition and processing differences but is also found to provide valuable insight into the geometrical changes that occur in the subsurface due to production. As such, this work also provides a demonstration of the use of high‐resolution 3D interpreted warping to resolve the 3D heterogeneity of the compaction and subsidence. This is an important tool for Valhall, and possibly other fields, where compaction and subsidence (and monitoring thereof) are key factors in the reservoir management since the predominant observed production‐induced changes are compaction of the soft, high‐porosity chalk reservoir, due to pore‐pressure reduction, and the resultant overburden subsidence. Such reservoir compaction could have significant implications for production by changing permeabilities and production rates. Furthermore the subsidence effects could impact upon subsea installations and well‐bore stability. Geomechanical studies that have previously been used to model such subsidence and compaction are only constrained by observed surface displacements and measured reservoir pressure changes, with the geological overburden being largely neglected. The approaches suggested herein provide the potential for monitoring and assessment in three dimensions, including the probable heterogeneity and shearing, that is needed for full understanding of reservoir compaction and the resultant effects on the overburden to, for example, mitigate well‐bore failures.     

Model of vibrational level populations of Herzberg states of oxygen molecules at heights of the lower thermosphere and mesosphere

The paper presents a model of the kinetics of electronically excited O₂(c¹Σ_u⁻,v), O₂(A′³Δ_u,v), O₂(A³Σ_u⁺,v) molecules at heights of the lower thermosphere and mesosphere with allowance for electronic excitation transfer processes during molecular collisions. The model is used to calculate the relative O₂(A³Σ_u⁺,v) and O₂(A′³Δ_u,v) populations at heights of 80–110 km. The calculated populations are compared with the available literature results on experimental estimates, and good agreement is obtained. It is shown how the increase in the quenching rates of the considered states by oxygen atoms affects the calculation results.