Upscaling: Effective Medium Theory, Numerical Methods and the Fractal Dream

Upscaling is a major issue regarding mechanical and transport properties of rocks. This paper examines three issues relative to upscaling. The first one is a brief overview of Effective Medium Theory (EMT), which is a key tool to predict average rock properties at a macroscopic scale in the case of a statistically homogeneous medium. EMT is of particular interest in the calculation of elastic properties. As discussed in this paper, EMT can thus provide a possible way to perform upscaling, although it is by no means the only one, and in particular it is irrelevant if the medium does not adhere to statistical homogeneity. This last circumstance is examined in part two of the paper. We focus on the example of constructing a hydrocarbon reservoir model. Such a construction is a required step in the process of making reasonable predictions for oil production. Taking into account rock permeability, lithological units and various structural discontinuities at different scales is part of this construction. The result is that stochastic reservoir models are built that rely on various numerical upscaling methods. These methods are reviewed. They provide techniques which make it possible to deal with upscaling on a general basis. Finally, a last case in which upscaling is trivial is considered in the third part of the paper. This is the fractal case. Fractal models have become popular precisely because they are free of the assumption of statistical homogeneity and yet do not involve numerical methods. It is suggested that using a physical criterion as a means to discriminate whether fractality is a dream or reality would be more satisfactory than relying on a limited data set alone.     

Brest sea level record: a time series construction back to the early eighteenth century

The completeness and the accuracy of the Brest sea level time series dating from 1807 make it suitable for long-term sea level trend studies. New data sets were recently discovered in the form of handwritten tabulations, including several decades of the eighteenth century. Sea level observations have been made in Brest since 1679. This paper presents the historical data sets which have been assembled so far. These data sets span approximately 300 years and together constitute the longest, near-continuous set of sea level information in France. However, an important question arises: Can we relate the past and the present-day records? We partially provide an answer to this question by analysing the documents of several historical libraries with the tidal data using a ‘data archaeology’ approach advocated by Woodworth (Geophys Res Lett 26:1589–1592, 1999b). A second question arises concerning the accuracy of such records. Careful editing was undertaken by examining the residuals between tidal predictions and observations. It proved useful to remove the worst effects of timing errors, in particular the sundial correction to be applied prior to August 1, 1714. A refined correction based on sundial literature [Savoie, La gnomique, Editions Les Belles Lettres, Paris, 2001] is proposed, which eliminates the systematic offsets seen in the discrepancies in timing of the sea level measurements. The tidal analysis has also shown that shallow-water tidal harmonics at Brest causes a systematic difference of 0.023 m between mean sea level (MSL) and mean tide level (MTL). Thus, MTL should not be mixed with the time series of MSL because of this systematic offset. The study of the trends in MTL and MSL however indicates that MTL can be used as a proxy for MSL. Three linear trend periods are distinguished in the Brest MTL time series over the period 1807–2004. Our results support the recent findings of Holgate and Woodworth (Geophys Res Lett) of an enhanced coastal sea level rise during the last decade compared to the global estimations of about 1.8 mm/year over longer periods (Douglas, J Geophys Res 96:6981–6992, 1991). The onset of the relatively large global sea level trends observed in the twentieth century is an important question in the science of climate change. Our findings point out to an ‘inflexion point’ at around 1890, which is remarkably close to that in 1880 found in the Liverpool record by Woodworth (Geophys Res Lett 26:1589–1592, 1999b).     

Bubble-induced porewater mixing: A 3-D model for deep porewater irrigation

Porewater data from vent sites of the northeastern shelf off Sakhalin Island, Sea of Okhotsk, exhibit bottom-water concentrations down to a sediment depth of up to 300 cm. Below this depth, solute concentrations rapidly change due to methanogenesis and anaerobic methane oxidation (AMO). The profile shapes suggest an irrigation-like process that mixes on a meter scale. At these sites active gas emanation into the overlying water column and near-surface gas hydrates are commonly observed. We propose that methane gas bubbles rise through the soft surface sediments and cause mixing of the porewater.Mathematically, the bubble-induced irrigation can be described by eddy diffusion enhancing the diffusive transport of solutes by several orders of magnitude. A 3-D numerical transport-reaction model was developed to investigate the parameters defining the mixing process, such as bubble rise velocity, tube size, tube distribution in the sediment, and ebullition frequency.Model consistency with the field data requires eddy diffusivities ?1 × 10⁵ cm²/a, tube densities of >4 tubes/m² (equivalent to a tube spacing of <40 cm), active gas seepage for more than a few weeks or months, and moderate to low diagenetic reaction rates of solutes. The corresponding methane gas fluxes that are predicted from the results of the model realizations range from 1 × 10³-5 × 10⁵ L/(m² a). Due to bubble mixing, solute fluxes in these sediments are increased by a factor of 3 and the maximum AMO rate by a factor of 7.     

On the Power in the Legendre Modes of the Solar Radial Magnetic Field

The power in the different modes of an expansion of the solar radial magnetic field at the surface in terms of Legendre polynomials,P , is calculated with the help of a solar dynamo model studied earlier. The model is of the Babcock–Leighton type, i.e., the surface eruptions of the toroidal magnetic field – through the tilt angle, , formed by the magnetic axis of a bipolar magnetic region with the east-west line – are the sources for the poloidal field. In this paper it is assumed that the tilt angle is subject to fluctuations of the form, = ()+ <> where <> is the average value and () is a random normal fluctuation with standard deviation which is taken from Howard's observations of the distribution of tilt angles. For numerical considerations, negative values of were not allowed. If this occurred, was recalculated. The numerical integrations were started with a toroidal magnetic field antisymmetric across the equator, large enough to generate eruptions, and a negligible poloidal field. The fluctuations in the tilt angle destroy the antisymmetry as time increases. The power of the antisymmetric modes across the equator (i.e., odd values of ) is concentrated in frequencies, _p, corresponding to the cycle period. The maximum power lies in the =3 mode with considerable power in the =5 mode, in broad agreement with Stenflo's results who finds a maximum power at =5. For the symmetric modes, there is considerable power in frequencies larger than _p, again in broad agreement with Stenflo's power spectrum.     

Thickness estimates of the upper-mantle transition zone from bootstrapped velocity spectrum stacks of receiver functions

We modify the receiver-functions stacking technique known as velocity spectrum stacking (VSS) so as to estimate combinations of velocity model ( V_P and V_S ) and depth that stack the Ps conversion from upper-mantle discontinuities most coherently. We find that by estimating the differences in the depths to the 660 and 410 km discontinuities using velocities that maximize the stacked amplitudes of P410s and P660s phases we can estimate the thickness of the transition zone more accurately than the depths to either of these discontinuities. We present two examples indicating that the transition zone beneath Obninsk, Russia, is 252±6 km thick and that beneath Pasadena, California, is only 220±6 km thick.     

Non-equilibrium alcohol flooding model for immiscible phase remediation: 1. Equation development

The equations for a compositional model for simulation of a two-phase, three-component system with inter-phase mass transfer are developed. Emphasis is placed on development of inter-phase mass transfer equations for incorporation of rate-limited inter-phase mass transfer. Due to the nature of the three-component systems considered, a single-film model may be inadequate and a two-film model must be utilized. A two-film model accounts for the simultaneous transfer of components in both directions across phase interfaces. The effect of interaction between components on diffusion is considered using a general form of Fick's Law. A Hand Plot representation of ternary phase behavior is chosen since it allows for straightforward calculation of miscibility of bulk phases under conditions of local non-equilibrium. The developed set of equations form the basis for a numerical model to simulate the enhanced removal of non-aqueous phase liquids (NAPLs) from porous media using single-component alcohol floods.     

Lattice dynamical system modelling of molecular clouds

Because a comprehensive microscopic treatment of interstellar molecular clouds is out of reach, an alternative approach is proposed in which most of the crucial ingredients of the problem are considered, but at some 'minimal' level of modelling. This leads to the elaboration of a lattice dynamical system , i.e. a time-dependent, spatially extended, deterministic system of macroscopic cells coupled through radiative transfer. Each cell is characterized by a small set of variables and supports a caricatural chemistry possessing the essential dynamical features of more realistic reaction schemes.   This approach naturally precludes quantitative results, but allows heretofore unavailable insights into some of the basic mechanisms at play. We focus on the response of the transfer process and the chemistry to a frozen 'turbulent' velocity field. It is shown that the system settles generically into a state where the effective coupling between cells is neither local nor global, and for which no single length-scale exists.   The spectral lines reconstructed from the spatiotemporal evolution of our model may, depending on the velocity field, exhibit profiles ranging from Gaussian to bimodal with strong realization effects. In the bimodal case, the model intrinsically displays an energy cascade transport mechanism to the cells that cool most efficiently: the feedback of chemistry on radiative transfer cannot be neglected.   Finally, extensions of this work are discussed and future developments are outlined.