A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a discrete-in-time setting. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.     

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a continuous in-time setting for a strictly positive constrained specific storage coefficient. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation, and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.     

Employment in retailing: A case study of employment in suburban shopping centres

There have been many studies of the retail industry in recent years, and of the impact that retailing has had on the spatial arrangement of urban areas and on consumer behaviour. Little attention has been paid, however, to employment in retailing, which is disturbing in view of its growing importance and its rapidly changing nature and composition. This paper examines the nature of the broad structural changes within the industry, and investigates their spatial consequences through a case study of employment generation within a system of suburban retail centres in Canberra, Australia. It is shown that the evolution of suburban centres, particularly those of a regional variety, has had an important and enriching impact on the suburban labour market. The regional centres are also seen to be the focus of the major structural employment changes currently affecting the retail industry. This has some important implications for urban planning.     

A time-distribution study of doline collapse: Framework for prediction

A temporal analysis of doline collapse on the Western Highland Rim (Tennessee) indicated one approach to estimating the probability of collapse in areas where the geologic and hydrologic criteria associated with collapse have been identified. The distribution of collapse was examined for trend, autocorrelation, and goodness of fit. The distribution of doline collapse during one 12-month period conformed to a Poisson distribution with a mean occurrence rate of λ=0.346 collapses per week and with the interoccurrence times being exponentially distributed (0.01 level). Although the proposed model is spatially and temporally restricted, it may provide an initial framework for estimating the probability of doline collapse in other karst terrains of similar geologic and hydrologic settings.     

Fertile and barren AlCr-spinel harzburgites from the upper mantle: Ion and electron probe analyses of trace elements in olivine and orthopyroxene: Relation to lherzolites

Ion and electron microprobe analyses of twenty-one CrAl-spinel harzburgite xenoliths from southern African kimberlites show two chemical groups. Orthopyroxenes from “fertile” harzburgites have higher CaO (mean of 11, 0.95 wt.%), Al₂O₃ (3.05 wt.%), Cr₂O₃ (0.85 wt.%) and Li (0.8 ppmw) than those from “barren” harzburgites (mean of 10, CaO 0.24 wt.%, Al₂O₃ 1.10 wt.%, Cr₂O₃ 0.35 wt.%, Li 0.3 ppmw). Olivines from all harzburgites have similar chemistry except that mean values of Li and Na are higher for barren than fertile harzburgites (Li 0.9 vs. 0.4 ppmw; Na₂O 16 vs. 7 ppmw). Orthopyroxenes from fertile harzburgites are chemically distinct from those in garne lherzolites from southern Africa and spinel lherzolites from southwest U.S.A., but orthopyroxenes from barren harzburgites are indistinguishable from those in many coarse garnet lherzolites.Chromium, Ca, Ni, Na and Li in coexisting olivines and orthopyroxenes from the above rock types show complex patterns, which for Ca, Cr and Ni can be related to pressure and temperature. Temperatures from an empirically calibrated thermometer based on Ni-Mg exchange between olivine and orthopyroxene, measured modes of harzburgites (fertile, mean of 10: ol 68, opx 31, spinel-silicate intergrowth <0.5; barren, mean of 8: ol 76, opx 23, spinel and spinel-silicate intergrowth 1), and high-pressure experimental studies suggest (a) that harzburgites are residues of partial melting, (b) that barren harzburgites were melted to a greater extent at a higher temperature (though probably at a similar depth) than fertile harzburgites, and (c) that incomplete reequilibration during retrograde metamorphism has led to development of complex inter- and intragranular textures, probably in the range ～700–900°C.     

Chemistry of micas from kimberlites and xenoliths—II. Primary- and secondary-textured micas from peridotite xenoliths

Micas from coarse granular Iherzolites in S. African kimberlites may be separated into two groups; those showing primary textural relationships with coexisting silicates and those with secondary, alteration relationships with other silicates. Primary-textured micas form a tight cluster with a mean composition from 10 coarse garnet lherzolites of: SiO₂ 41.0, TiO₂ 0.18, Al₂O₃ 13.5, Cr₂O₃ 0.82, total Fe as FeO 2.60, MnO 0.02, MgO 26.0, NiO 0.22, CaO 0.01, BaO 0.29, Na₂O 0.31, K₂O 10.0, Rb₂O 0.028, Cl 0.08, F 0.43 wt%. Primary-textured micas in aggregates with clinopyroxene have higher TiO₂ and four specimens which look similar to the primary group but have textural ambiguities have still higher TiO₂Micas with secondary textures have wide ranges of composition which may be correlated with details of the textural parageneses. Micas from kelyphitic rims around garnets tend to be Cr-rich while those from veinlets are Cr-poor. Both groups tend to have higher FeO and TiO₂ than the primary group. Micas produced by alteration of, or filling veinlets through, orthopyroxene have a wide compositional range which overlaps that of the primary-textured micas, especially for harzburgite specimens.The primary-textured micas show a positive correlation with coexisting pyroxenes for MgO/(MgO + FeO) and TiO₂, but not for Cr₂O₃. Secondary-textured micas do not show correlations with coexisting pyroxenes for any elements.The ‘primary-metasomatic’ micas described by Harte and Gurney (1975) and metasomatic and other micas described by Boettcher et al. (1979) and Boettcher and O'neil (1979) are richer in FeO and TiO₂ than the present primary-textured micas, and are attributed to crystallization from fractionated fluids.     

Stresses in column-supported hyperboloidal shells subject to seismic loading

The computation of stresses within a finite element displacement method analysis of a shell of revolution is considered. The common procedure of applying the kinematic and constitutive laws to the displacement functions is examined and justified for models where the displacements are represented by high-order polynomial expansions. Also, two alternative computational formats within this technique are explored. The influence of the column-supported base condition on a hyperboloidal shell of revolution is studied with respect to the stresses calculated from a response spectrum analysis. These studies emphasize the importance of accurately modelling the base region of a column-supported shell such as a hyperbolic cooling tower.