jeanflop/post_ocr_correction2 huggingface dataset:
Kalahari du Congo belge et les Kalahari beds d’Afrique australe, Mem. Soc. géol. Belg. 4, 1952. E. J. WAYLAND, Outlines of Prehistory and Stone Age Climatology in the Bechuanaland Protectorate, Acad. R. Sci. Col. Mem., 25 (4), 1954, pp. 3-47. H. J. COOICE, The paleoclimatic significance of caves and adjacent landforms in Western Ngamiland, Botswana, Geogr. Jl, 141, (3), 1975, pp. 430-444. H. J. COOICE and B. T. VERHAGEN, The dating of cave development: an example from Botswana, Proc. Ith Int. Speleol. Congr., Sheffield, 1984. R. A. DART, A note on the limestone caverns of Leba, near Humpata, Angola, S. Afr. Archaeol. Bull., 5, (20), 1950, pp. 149-151. M. TELLES ANTUNES, Sur la faune de Vertébrés du Pléistocène de Leba, Humpata (Angola), Actes du 5th Congr. Panafr. Prehist. et de l’Étude du Quaternaire, Ténérife, 1965, pp. 127-128. E. B. WRIGHT, Geological studies in Northern Kalahari, Geogr. Jl., 144, (2), pp. 235-249. E. J. WAYLAND, Drodsky’s Cave, Geogr. Jl, 103, 1944, pp. 230-233. M. R. LEEDER, Pedogenic carbonates and flood sediment accretion rates; a quantitative model for arid zone lithofacies, Geol. Mag, 112, 1975, pp. 257-270. A. C. VAN BRUGGEN, Notes on non-marine molluscs from Mozambique and Bechuanaland, with a checklist of Bechuanaland species, Ann. Transvaal Mus, 25, (6), 1966, pp. 99-111. E. S. VRBA, Biostratigraphy and chronology, based particularly on Bovidae, of Southern hominid-associated assemblages: Makapansgat, Sterkfontein, Taung, Kromdraai, Swartkrans; also Elandsfontein (Saldanha), Broken Hill (now Kabwe) and Cave of Hearths, Proc. 1st Int. Congr. Human Palaeont., Nice, C.N.R.S., 2, 1982, pp. 707-752. M. P.: Institut de Paléontologie, 8, rue Buffon, 75005 Paris; P. M.: Département des Sciences de la Terre, Université Claude-Bernard-Lyon-I, 27-43, boulevard du 11-Novembre-1918, 69622 Villeurbanne Cedex. C. R. Acad. Sci. Paris, t. 307, Série II, p. 1687-1692, 1988 1687 Mécanique des milieux continus/Computational Mechanics Approche spectrale du calcul tridimensionnel de propriétés équivalentes d’un milieu hétérogène Claire GOBBÉ, Jean GOUNOT et Michel QUINTARD Résumé — Un algorithme simple, basé sur une analyse spectrale, est proposé pour étudier les propriétés de transport diffusif dans un milieu hétérogène. À partir de champs de valeurs discrètes 3D du tenseur local
of Bechuanaland species, Ann. Transvaal Mus, 25, (6), 1966, pp. 99-111. E. S. VRBA, Biostratigraphy and chronology, based particularly on Bovidae, of Southern hominid-associated assemblages: Makapansgat, Sterkfontein, Taung, Kromdraai, Swartkrans; also Elandsfontein (Saldanha), Broken Hill (now Kabwe) and Cave of Hearths, Proc. 1st Int. Congr. Human Palaeont., Nice, C.N.R.S., 2, 1982, pp. 707-752. M. P.: Institut de Paléontologie, 8, rue Buffon, 75005 Paris; P. M.: Département des Sciences de la Terre, Université Claude-Bernard-Lyon-I, 27-43, boulevard du 11-Novembre-1918, 69622 Villeurbanne Cedex. C. R. Acad. Sci. Paris, t. 307, Série II, p. 1687-1692, 1988 1687 Mécanique des milieux continus/Computational Mechanics Approche spectrale du calcul tridimensionnel de propriétés équivalentes d’un milieu hétérogène Claire GOBBÉ, Jean GOUNOT et Michel QUINTARD Résumé — Un algorithme simple, basé sur une analyse spectrale, est proposé pour étudier les propriétés de transport diffusif dans un milieu hétérogène. À partir de champs de valeurs discrètes 3D du tenseur local de diffusion, on calcule, par une méthode spectrale, le tenseur de diffusion effectif du milieu homogène équivalent. Calculation by a 3D spectral method of effective transport properties of a heterogeneous medium Abstract — A simple numerical technique, based on spectral analysis, is proposed to determine diffusive transport properties in a heterogeneous medium. From 3D discrete point values of the local diffusion tensor, the effective diffusion tensor for the equivalent homogeneous medium is calculated by a spectral method. Abridged English Version — We study a diffusion process for a physical variable [eq. (1)] in a heterogeneous medium. The transport coefficient K is a periodic function of the space variable. The processes we are dealing with can be: (i) molecular diffusion, (ii) thermal conduction, (iii) or one-phase flow through a porous medium (K is the permeability in this last case). We assume that, for a given length scale r0, the medium can be considered as homogeneous with respect to averaged values over volumes of dimension r0. It can be shown [1] that the volume average value <f> obeys a new diffusion equation with an effective transport coefficient K* which is given by equation (3). In this equation, a vector b must be
heterogeneous medium. From 3D discrete point values of the local diffusion tensor, the effective diffusion tensor for the equivalent homogeneous medium is calculated by a spectral method. Abridged English Version — We study a diffusion process for a physical variable [eq. (1)] in a heterogeneous medium. The transport coefficient K is a periodic function of the space variable. The processes we are dealing with can be: (i) molecular diffusion, (ii) thermal conduction, (iii) or one-phase flow through a porous medium (K is the permeability in this last case). We assume that, for a given length scale r0, the medium can be considered as homogeneous with respect to averaged values over volumes of dimension r0. It can be shown [1] that the volume average value <f> obeys a new diffusion equation with an effective transport coefficient K* which is given by equation (3). In this equation, a vector b must be calculated by solving a closure problem expressed by equations (4) and (5). Various numerical methods have been used to solve the closure problem. We are interested here in K fields obtained from discrete measurements or image analysis. The grid node values are known and we would like to analyze them in a simple way. In order to solve equations (4) and (5), we use Fourier analysis. The idea is to use Galerkin’s technique to solve equations (4) and (5) with Fourier decomposition for K [eq. (6)] and b [eq. (7)]. K is analyzed by FFT. The components of b over the Fourier basis are solutions of a linear system [eq. (8)]. The domain studied is of dimensions 2HX x 2H2 x 2H3. From the solution of this linear system, it is easy to obtain K* [eq. (9)]. This method deals simply with the periodicity boundary conditions in equation (5) and resembles signal analysis method. The major drawback is the matrix in equation (8), a non-sparse matrix, which makes it impossible to deal with 3D expansions of b with more than 15 x 15 x 15 terms. Among the various test problems we have solved, we present a calculation of K*
in a simple way. In order to solve equations (4) and (5), we use Fourier analysis. The idea is to use Galerkin’s technique to solve equations (4) and (5) with Fourier decomposition for K [eq. (6)] and b [eq. (7)]. K is analyzed by FFT. The components of b over the Fourier basis are solutions of a linear system [eq. (8)]. The domain studied is of dimensions 2HX x 2H2 x 2H3. From the solution of this linear system, it is easy to obtain K* [eq. (9)]. This method deals simply with the periodicity boundary conditions in equation (5) and resembles signal analysis method. The major drawback is the matrix in equation (8), a non-sparse matrix, which makes it impossible to deal with 3D expansions of b with more than 15 x 15 x 15 terms. Among the various test problems we have solved, we present a calculation of K* for a network of spheres. We use as a reference solution, KMa, the Maxwell equation [eq. (10)] (see extensive analysis in [2]). Figure 1 shows the evolution of the relative error between K* and KMa versus the ratio Ks/K2 (K2, continuous phase; K1, discontinuous phase). The agreement is fair between the two values provided that Ks/K2 remains smaller than 100. The technique is then applied to the estimation of the large-scale permeability of a sandstone. An introduction to this problem can be found in [3]. We distinguish two aspects: the calculation of the principal direction and the calculation of the associated principal values. In the principal axis system, methods exist [4] which produce upper and lower bounds for principal components of K*, but principal directions cannot be determined easily. In a simple way, our method solves both these problems. Figure 2 shows the measured permeability field [5] on a Vosges sandstone sample. The calculated K* components are also plotted; the principal values are 10. ### CORRECTION ### Balkema, Rotterdam/Boston, 1984, pp. 107-146. A. BROOKS and J. E. YELLEN, Archaeological excavations at Ngami: A preliminary report on the first two field seasons, Botswana Notes and Records, 9, 1977, pp. 21-30. L.
the two values provided that Ks/K2 remains smaller than 100. The technique is then applied to the estimation of the large-scale permeability of a sandstone. An introduction to this problem can be found in [3]. We distinguish two aspects: the calculation of the principal direction and the calculation of the associated principal values. In the principal axis system, methods exist [4] which produce upper and lower bounds for principal components of K*, but principal directions cannot be determined easily. In a simple way, our method solves both these problems. Figure 2 shows the measured permeability field [5] on a Vosges sandstone sample. The calculated K* components are also plotted; the principal values are 10. ### CORRECTION ### Balkema, Rotterdam/Boston, 1984, pp. 107-146. A. BROOKS and J. E. YELLEN, Archaeological excavations at Ngami: A preliminary report on the first two field seasons, Botswana Notes and Records, 9, 1977, pp. 21-30. L. CAHEN and J. LEPERSOHN, Équivalence entre le système du Kalahari du Congo belge et les Kalahari beds d’Afrique australe, Mem. Soc. géol. Belg. 4, 1952. E. J. WAYLAND, Outlines of Prehistory and Stone Age Climatology in the Bechuanaland Protectorate, Acad. R. Sci. Col. Mem., 25 (4), 1954, pp. 3-47. H. J. COOICE, The paleoclimatic significance of caves and adjacent landforms in Western Ngamiland, Botswana, Geogr. Jl, 141, (3), 1975, pp. 430-444. H. J. COOICE and B. T. VERHAGEN, The dating of cave development: an example from Botswana, Proc. Ith Int. Speleol. Congr., Sheffield, 1984. R. A. DART, A note on the limestone caverns of Leba, near Humpata, Angola, S. Afr. Archaeol. Bull., 5, (20), 1950, pp. 149-151. M. TELLES ANTUNES, Sur la faune de Vertébrés du Pléistocène de Leba, Humpata (Angola), Actes du 5th Congr. Panafr. Prehist. et de l’Étude du Quaternaire, Ténérife, 1965, pp. 127-128. E. B. WRIGHT, Geological studies in Northern Kalahari, Geogr. Jl., 144, (2), pp. 235-249. E. J. WAYLAND, Drodsky’s Cave, Geogr. Jl, 103, 1944, pp. 230-233. M. R. LEEDER, Pedogenic carbonates and flood sediment accretion rates; a quantitative model for arid zone lithofacies, Geol. Mag, 112, 1975, pp. 257-270. A. C. VAN BRUGGEN, Notes on
J. COOICE, The paleoclimatic significance of caves and adjacent landforms in Western Ngamiland, Botswana, Geogr. Jl, 141, (3), 1975, pp. 430-444. H. J. COOICE and B. T. VERHAGEN, The dating of cave development: an example from Botswana, Proc. Ith Int. Speleol. Congr., Sheffield, 1984. R. A. DART, A note on the limestone caverns of Leba, near Humpata, Angola, S. Afr. Archaeol. Bull., 5, (20), 1950, pp. 149-151. M. TELLES ANTUNES, Sur la faune de Vertébrés du Pléistocène de Leba, Humpata (Angola), Actes du 5th Congr. Panafr. Prehist. et de l’Étude du Quaternaire, Ténérife, 1965, pp. 127-128. E. B. WRIGHT, Geological studies in Northern Kalahari, Geogr. Jl., 144, (2), pp. 235-249. E. J. WAYLAND, Drodsky’s Cave, Geogr. Jl, 103, 1944, pp. 230-233. M. R. LEEDER, Pedogenic carbonates and flood sediment accretion rates; a quantitative model for arid zone lithofacies, Geol. Mag, 112, 1975, pp. 257-270. A. C. VAN BRUGGEN, Notes on non-marine molluscs from Mozambique and Bechuanaland, with a checklist of Bechuanaland species, Ann. Transvaal Mus, 25, (6), 1966, pp. 99-111. E. S. VRBA, Biostratigraphy and chronology, based particularly on Bovidae, of Southern hominid-associated assemblages: Makapansgat, Sterkfontein, Taung, Kromdraai, Swartkrans 6 and 23.4 mD. If the REV (Representative Elementary Volume) length is known, this method of analyzing K fields is useful and allows us to seek effective transport properties. This also permits a process dependent analysis of the REV determination by varying H15 H2, H3. 1. INTRODUCTION. — On considered a transport process, within a heterogeneous medium, ruled by the equation: where K is a tensor, function of the point. The variations of this function are quasi-periodic at an scale 1. The applications that we envisage can be, among others: (i) K is a diffusion molecular tensor and |/ a concentration; (ii) K is a thermal conductivity tensor and |/ the temperature; (iii) K is the permeability tensor of a heterogeneous porous medium and |/ the average intrinsic pressure of phase. At a scale r0, for which the medium can be considered as homogeneous, several methods, that we do not discuss here [1], permit to produce the equation describing <v|/>, average
is known, this method of analyzing K fields is useful and allows us to seek effective transport properties. This also permits a process dependent analysis of the REV determination by varying H15 H2, H3. 1. INTRODUCTION. — On considered a transport process, within a heterogeneous medium, ruled by the equation: where K is a tensor, function of the point. The variations of this function are quasi-periodic at an scale 1. The applications that we envisage can be, among others: (i) K is a diffusion molecular tensor and |/ a concentration; (ii) K is a thermal conductivity tensor and |/ the temperature; (iii) K is the permeability tensor of a heterogeneous porous medium and |/ the average intrinsic pressure of phase. At a scale r0, for which the medium can be considered as homogeneous, several methods, that we do not discuss here [1], permit to produce the equation describing <v|/>, average of [/] on the volume r0; one has: The coefficient of transfer K* is defined by: where b is, on a representative cell, solution of the following problem: Many techniques of resolution can be developed for solving the problem defining the vector b and calculating the coefficient K*, especially in the case of periodic media. We are interested here in fields K obtained in a discrete way, by measurement on a grid of points covering the domain studied, or by image treatment. Thus, the field K can correspond to the thermal conductivity of a composite medium, in the classical sense of the mechanics of continuous media, obtained from the treatment of micrographs. No simple method exists to complement the usual morphological analysis tools to give indications on the principal directions and the main values of the transfer tensor associated with the macroscopic process involving the entire field studied. It is about here an application making use of the change of scale classical microscopically to macroscopically. Other applications make use of an heterogeneity at the macroscopic level. It is important, in this type of applications, to have also information complementary to the morphological analysis concerning the properties of K*, permeability at
the case of periodic media. We are interested here in fields K obtained in a discrete way, by measurement on a grid of points covering the domain studied, or by image treatment. Thus, the field K can correspond to the thermal conductivity of a composite medium, in the classical sense of the mechanics of continuous media, obtained from the treatment of micrographs. No simple method exists to complement the usual morphological analysis tools to give indications on the principal directions and the main values of the transfer tensor associated with the macroscopic process involving the entire field studied. It is about here an application making use of the change of scale classical microscopically to macroscopically. Other applications make use of an heterogeneity at the macroscopic level. It is important, in this type of applications, to have also information complementary to the morphological analysis concerning the properties of K*, permeability at large scale of the system. The evaluation of these properties makes use of a new change of scale macroscopically to an even larger scale. By example, at the paragraph 4, we consider the field of permeability of a heterogeneous medium presenting variations spatial at a scale well above that of the pore. The local permeability corresponds, in fact, to a property defined by taking the average on an ensemble of pores or measured on a small volume including several pores. In view of these applications, it is interesting to have at our disposal simple logics permitting to exploit immediately the fields of values thus obtained. We propose to apply to the resolution of equations (4) and (5) a method of the type analysis of Fourier which has the advantage of treating simply the conditions of periodicity and which is close to the usual techniques of signal treatment. We present here the big lines of the algorithm and the results of several examples of application. 2. METHOD OF RESOLUTION. — The idea is to use the method of Galerkin with decompositions of the type: where, by definition, and where the domain studied has for dimensions 2H15 2H2, 2H3. By multiplying (4)
the pore. The local permeability corresponds, in fact, to a property defined by taking the average on an ensemble of pores or measured on a small volume including several pores. In view of these applications, it is interesting to have at our disposal simple logics permitting to exploit immediately the fields of values thus obtained. We propose to apply to the resolution of equations (4) and (5) a method of the type analysis of Fourier which has the advantage of treating simply the conditions of periodicity and which is close to the usual techniques of signal treatment. We present here the big lines of the algorithm and the results of several examples of application. 2. METHOD OF RESOLUTION. — The idea is to use the method of Galerkin with decompositions of the type: where, by definition, and where the domain studied has for dimensions 2H15 2H2, 2H3. By multiplying (4) by fmu ra2 m3 and integrating on the domain studied, one obtains finally the linear system following defining the components of the vector b: From the resolution of the linear system and of the equation (3) one obtains: Compte tenu du fait qu'un such algorithm produces full matrices, the possibility of resolution in 3D leads to truncate the series defining b at 15 x 15 x 15. 3. PROBLEM TEST. — Among other geometric configurations simple that we have studied (lattice of spheres, lattice of cylinders, ...), we present here the results obtained in the case of a lattice of spheres. The thermal conductivity of the lattice of spheres has been the object of numerous studies [2]. The existing knowledge on this problem being very developed, it does not s'agit here only to verify the method proposed on a known case. The previous studies all agree to recognize the representativeness of Maxwell's formula for non adhesive lattices of spheres. The equivalent conductivity of the lattice is given by: where Xj and X2 are the conductivities respectively of the spheres and of the continuous phase, and E1 is the volumetric fraction occupied by the spheres. The figure 1 represents the evolution
of resolution in 3D leads to truncate the series defining b at 15 x 15 x 15. 3. PROBLEM TEST. — Among other geometric configurations simple that we have studied (lattice of spheres, lattice of cylinders, ...), we present here the results obtained in the case of a lattice of spheres. The thermal conductivity of the lattice of spheres has been the object of numerous studies [2]. The existing knowledge on this problem being very developed, it does not s'agit here only to verify the method proposed on a known case. The previous studies all agree to recognize the representativeness of Maxwell's formula for non adhesive lattices of spheres. The equivalent conductivity of the lattice is given by: where Xj and X2 are the conductivities respectively of the spheres and of the continuous phase, and E1 is the volumetric fraction occupied by the spheres. The figure 1 represents the evolution of the relative error of our results compared to this formula as a function of the contrast of the conductivities respectively of each phase. The characteristics of the lattice studied are defined on the figure. Obviously, the relative error increases with the contrast of conductivity but in a reasonable manner considering the number of terms in the approximation. 4. EXAMPLE OF APPLICATION: PERMEABILITY OF A GRANULAR MEDIUM. — It is here a real case which illustrates the interest of our approach. Many works have been devoted to the permeability of real media (see references in [3]). One distinguishes two problems: (i) the determination of the principal directions associated with the permeability tensor K, (ii) the determination of the values of its components. In the principal reference, there exist methods allowing to provide bounds for the values of the components [4], but we do not know of simple methods for defining the principal directions. The algorithm proposed allows to solve both these problems. [1] BENSOUSSAN A., LUCIANO, D., STROUD, I., GIRARD, G. and LADOUCE, R., 'Méthodes statistiques en physique : de la théorie aux applications', Masson, 1984. [2] CLERC M. and SADOUN G., 'Effective conductivity of two-component composites with arbitrary shapes',
considering the number of terms in the approximation. 4. EXAMPLE OF APPLICATION: PERMEABILITY OF A GRANULAR MEDIUM. — It is here a real case which illustrates the interest of our approach. Many works have been devoted to the permeability of real media (see references in [3]). One distinguishes two problems: (i) the determination of the principal directions associated with the permeability tensor K, (ii) the determination of the values of its components. In the principal reference, there exist methods allowing to provide bounds for the values of the components [4], but we do not know of simple methods for defining the principal directions. The algorithm proposed allows to solve both these problems. [1] BENSOUSSAN A., LUCIANO, D., STROUD, I., GIRARD, G. and LADOUCE, R., 'Méthodes statistiques en physique : de la théorie aux applications', Masson, 1984. [2] CLERC M. and SADOUN G., 'Effective conductivity of two-component composites with arbitrary shapes', J. Physique, 47, 1986, p. 551-562. [3] SNETHEN R., 'Modelling flow in heterogeneous media with application to aquifers', in D. SCOTT, I. STROUD and F. WOODBINE (eds.), 'Homogenization techniques in the analysis of heterogeneous media', Hemisphere, New-York, 1989, p. 35-55. [4] TOUCHEMAN P., 'Bounds on the effective properties of composites', in D. SCOTT, I. STROUD and F. WOODBINE (eds.), 'Homogenization techniques in the analysis of heterogeneous media', Hemisphere, New-York, 1989, p. 91-97. Sur la figure 2, est représenté le champ de perméabilité local mesuré sur un grès des Vosges [5]. Le milieu est supposé localement isotrope. Le champ K(x) mesuré est analysé par FFT (dans cette application particulière, nous avons disposé de 15x21 mesures), ce qui fournit les coefficients du développement (6). Le vecteur b calculé permet la détermination du tenseur K*; les vecteurs principaux de ce tenseur sont représentés sur la figure 2. Les valeurs calculées des composantes principales sont respectivement 10,6 et 23,4 mD. 5. Un tel résultat illustre une méthode d'analyse qui permet d'associer à un champ de données, des propriétés équivalentes pour un processus de type diffusif en traitant convenablement les effets d'anisotropie. On peut, en principe, en modifiant la taille du champ analysé, étudier les