Numerical Homogenization of the Rigidity Tensor in Hooke's Law Using the Node-Based Finite Element Method

Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be “upscaled” to one “equivalent homogeneous” coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented.     

The southern cape conductive belt (South Africa): Its composition, origin and tectonic significance

Magnetometer array studies have led to the discovery and mapping of the Southern Cape Conductive Belt (SCCB) crossing the southern tip of Africa from west to southeast coasts. The SCCB lies just south of the Namaqua-Natal Belt of cratonic rocks remobilized about 1000 m.y. B.P. It is shown that it coincides with a zone of weakness which has been exploited by three major geosynclinal accumulations over some 600 m.y. Relationships between the SCCB and the basement geochronology, geology and tectonics are considered in detail. These relationships support the view that the conductive belt was formed by an accumulation of marine sediments and oceanic lithosphere at the top of a Proterozoic subduction which stopped about 1000 to 800 m.y. B.P. Associated with this subduction we propose a Proterozoic range of Andean mountains, whose roots are now exposed in the Namaqua-Natal Belt. Later subduction further south, near the present south coast, is proposed to account for the intrusion, between the south coast and the SCCB, of the Cape Granites in the time interval 600-500 m.y. B.P. There is some evidence for a third, yet more distant, subduction episode off Permian Gondwanaland. After outlining this tectonic history, the paper turns to a closer examination of the hypothesis that the Southern Cape Conductive Belt consists of partly serpentinized basalt accumulated at the top of a Proterozoic subduction. A large static magnetic anomaly, which correlates with the SCCB over most of its length, is well fitted by a model which strongly supports this hypothesis. Bouguer gravity anomalies along western and central profiles likewise support the hypothesis. A discussion follows of the process of formation of the proposed block of serpentinized marine rocks, beginning with serpentinization of the crust near oceanic ridges by reaction of warm, porous, newly-extruded basalt with seawater convecting through it. The serpentinized basalt is stable at crustal temperatures and pressures and so is transported in the seafloor until it reaches a subduction, where it accumulates because of its low density. Examples of such accumulations are cited. Finally, it is shown that any iron in the olivine and pyroxene in the original basalt precipitates, when these silicates are hydrated to serpentine, as magnetite which is the dominant mineral conferring high electrical conductivity and high magnetic susceptibility on serpentinites. In particular the Beattie static magnetic anomaly requires, in our model calculation, a very high susceptibility readily attainable in basalt at 15–20% serpentinization. The authors know of no other rock able to provide this high susceptibility. A similar percentage of serpentinization provides the density required to model the gravity anomalies.     

Improved water-level forecasting for the Northwest European Shelf and North Sea through direct modelling of tide, surge and non-linear interaction

In real-time operational coastal forecasting systems for the northwest European shelf, the representation accuracy of tide–surge models commonly suffers from insufficiently accurate tidal representation, especially in shallow near-shore areas with complex bathymetry and geometry. Therefore, in conventional operational systems, the surge component from numerical model simulations is used, while the harmonically predicted tide, accurately known from harmonic analysis of tide gauge measurements, is added to forecast the full water-level signal at tide gauge locations. Although there are errors associated with this so-called astronomical correction (e.g. because of the assumption of linearity of tide and surge), for current operational models, astronomical correction has nevertheless been shown to increase the representation accuracy of the full water-level signal. The simulated modulation of the surge through non-linear tide–surge interaction is affected by the poor representation of the tide signal in the tide–surge model, which astronomical correction does not improve. Furthermore, astronomical correction can only be applied to locations where the astronomic tide is known through a harmonic analysis of in situ measurements at tide gauge stations. This provides a strong motivation to improve both tide and surge representation of numerical models used in forecasting. In the present paper, we propose a new generation tide–surge model for the northwest European Shelf (DCSMv6). This is the first application on this scale in which the tidal representation is such that astronomical correction no longer improves the accuracy of the total water-level representation and where, consequently, the straightforward direct model forecasting of total water levels is better. The methodology applied to improve both tide and surge representation of the model is discussed, with emphasis on the use of satellite altimeter data and data assimilation techniques for reducing parameter uncertainty. Historic DCSMv6 model simulations are compared against shelf wide observations for a full calendar year. For a selection of stations, these results are compared to those with astronomical correction, which confirms that the tide representation in coastal regions has sufficient accuracy, and that forecasting total water levels directly yields superior results.     

Altimetry-derived tide model for improved tide and water level forecasting along the European continental shelf

Ocean Dynamics - With the continued rise in global mean sea level, operational predictions of tidal height and total water levels have become crucial for accurate estimations and understanding of...     

Effects of self-attraction and loading at a regional scale: a test case for the Northwest European Shelf

The impact of the self-attraction and loading effect (SAL) in a regional 2D barotropic tidal model has been assessed, a term with acknowledged and well-understood importance for global models but omitted for boundary-forced, regional models, for which the implementation of SAL is non-trivial due to its non-local nature. In order to understand the impact of the lack of SAL effects in a regional scale, we have forced a regional model of the Northwest European Continental Shelf and the North Sea (continental shelf model (CSM)) with the SAL potential field derived from a global model (GTSM), in the form of a pressure field. Impacts have been studied in an uncalibrated setup and with only tidal forcing activated, in order to isolate effects. Additionally, the usually adopted simple SAL parameterization, in which the SAL contribution to the total tide is parameterized as a percentage of the barotropic pressure gradient (typically chosen 10%), is also implemented and compared to the results obtained with a full SAL computation. A significant impact on M2 representation is observed in the English Channel, Irish Sea and the west (UK East coast) and south (Belgian and Dutch Coast) of the North Sea, with an impact of up to 20 cm in vector difference terms. The impact of SAL translates into a consistent M2 amplitude and propagation speeds reduction throughout the domain. Results using the beta approximation, with an optimal domain-wide constant value of 1.5%, show a somewhat comparable impact in phase but opposite direction of the impact in amplitude, increasing amplitudes everywhere. In relative terms, both implementations lead to a reduction of the tidal representation error in comparison with the reference run without SAL, with the full SAL approach showing further impacted, improved results. Although the overprediction of tidal amplitudes and propagation speeds in the reference run might have additional sources like the lack of additional dissipative processes and non-considered bottom friction settings, results show an overall significant impact, most remarkable in tidal phases. After showing evidence of the SAL impact in regional models, the question of how to include this physical process in them in an efficient way arises, since SAL is a non-local effect and depends on the instantaneous water levels over the whole ocean, which is non-trivial to implement.     

Immunological studies on microbial mats from Solar Lake (Sinai)— A contribution to the organic geochemistry of sediments

Immunological and biochemical techniques were used to investigate organic matter in microbial mats from Solar Lake (Sinai, Egypt). Antibodies, elicited in rabbits against samples taken at different depths from a core of these mats, were used for the detection of immunological determinants preserved in these samples. Special attention was given to an antiserum directed against a top mat fraction (3–10 mm depth) and an antiserum against a fraction from a lower section of the core (380–390 mm depth). Both antisera, tested in a dot-blot immunobinding assay, were found to give positive reactions with organic matter from different depths of the sediment. An immunohistochemical experiment with the antisera points to antigenic determinants on Recent and subfossil sheaths of Microcoleus. In addition, extracts of sediment samples were submitted to electrophoresis in polyacrylamide gels in order to characterize the separated macromolecules by immunological means. The results showed a rapid decrease in the number of antigenic determinants from Recent to subfossil samples. However, even in the deepest layers of the sediment (690–700 mm depth) intact biopolymers were detected.     

Magnetometer array studies and deep Schlumberger soundings in the Damara orogenic belt, South West Africa

Summary. A zone of concentrated induced electric currents crossing parts of Zimbabwe, Botswana and South West Africa was discovered during a magnetovariational study conducted in 1972. In 1977, a second study was made with 27 recording magnetometers distributed across the width of South West Africa between latitudes 19 and 22°S. Several geomagnetic disturbances were recorded with high recording efficiencies. Three of these time sequences were digitized for analysis. Magnetograms and Fourier transform amplitude and phase maps in the period range 22–128min were used to delineate the westward continuation of the conductive structure revealed by the earlier investigation. The conductive zone runs approximately east-west from the Botswana border (21°E) to 17°E longitude. From here to the Atlantic coast it trends in a NE—SW direction. Anomalous fields, normalized to the horizontal field at a station recording the normal field, were used to obtain maximum depth estimates of around 45 km for the induced currents. Several deep Schlumberger soundings were done over the anomalous zone and the results showed that the conductive structure is, in places, only 3 km from the surface and that it has a resistivity of less than 20 Ωm. The resistivity of the upper crust outside the structure ranges from 5000 to more than 20000 Ωm. Some 14 post-Karoo alkaline igneous complexes occur along the course of the resistivity anomaly. These intrusive complexes represent the youngest igneous activity in the Damara Orogenic Belt and were most probably emplaced along a line of weakness in the lithosphere. The resistivity anomaly would seem to delineate this line of weakness.     

Numerical Homogenization of Two-Phase Flow in Porous Media

Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds.