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| − | ==1 Title, abstract and keywords<!-- Your document should start with a concise and informative title. Titles are often used in information-retrieval systems. Avoid abbreviations and formulae where possible. Capitalize the first word of the title. | + | == Abstract == |
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| − | Provide a maximum of 6 keywords, and avoiding general and plural terms and multiple concepts (avoid, for example, 'and', 'of'). Be sparing with abbreviations: only abbreviations firmly established in the field should be used. These keywords will be used for indexing purposes.
| + | Simulation of flow through heterogeneous media often requires discretizing the flow domain into blocks and assigning an equivalent block conductivity value to each one of them. The process of defining block conductivities from point values is termed upscaling. A number of approaches to upscaling are available, most of which consider the uncertainty associated with any natural property, so that they cast the problem in a stochastic frame. Recently, Indelman and Dagan (1993a, b) provided a general stochastic methodology to upscaling in heterogeneous anisotropic formations by means of the dissipation energy function; unfortunately, they did not provide any “practical” method to compute block values from point ones. The objective of this work is twofold: First, we analyze different practical approaches to compute block conductivities and find that all of them provide very similar results in terms of actual computed values; second, we check that all approaches verify approximately a number of conditions stated by Indelman and Dagan (1993a). Specifically, we show analytically that for regular blocks, the methodologies of both Rubin and Gómez‐Hernández (1990) and Desbarats (1992) (which we call “practical” methodologies) satisfy the condition that the effective conductivity obtained from a field where the elementary conductivities are defined over a certain support (we call this the actual formation) is identical to that obtained from the same field with conductivities defined at a larger support (upscaled formation). The analysis is carried out by working with the logarithm of block conductivities and using a small‐perturbation expansion and thus is strictly valid for small variances. On the other hand, we show numerically that the two methodologies satisfy approximately an important condition stated in terms of the dissipation energy: that block‐averaged dissipation values computed are indeed very close to the true dissipation values in each block. The agreement is even better if we consider statistical moments instead of point values. As an important conclusion we should note that all practical methodologies considered in this work perform equally well and, more important, constitute a simple way to treat an otherwise very complex problem. |
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| − | An abstract is required for every document; it should succinctly summarize the reason for the work, the main findings, and the conclusions of the study. Abstract is often presented separately from the article, so it must be able to stand alone. For this reason, references and hyperlinks should be avoided. If references are essential, then cite the author(s) and year(s). Also, non-standard or uncommon abbreviations should be avoided, but if essential they must be defined at their first mention in the abstract itself. -->==
| + | ==Full Document== |
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| − | | + | <pdf>Media:Draft_Samper_875592705_1026_S-nchez-Vila_et_al-1995-Water_Resources_Research.pdf</pdf> |
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| − | ==2 The main text<!-- You can enter and format the text of this document by selecting the ‘Edit’ option in the menu at the top of this frame or next to the title of every section of the document. This will give access to the visual editor. Alternatively, you can edit the source of this document (Wiki markup format) by selecting the ‘Edit source’ option.
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| − | Most of the documents in Scipedia are written in English (write your manuscript in American or British English, but not a mixture of these). Anyhow, specific publications in other languages can be published in Scipedia. In any case, the documents published in other languages must have an abstract written in English.
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| − | 2.1 Subsections
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| − | Divide your article into clearly defined and numbered sections. Subsections should be numbered 1.1, 1.2, etc. and then 1.1.1, 1.1.2, ... Use this numbering also for internal cross-referencing: do not just refer to 'the text'. Any subsection may be given a brief heading. Capitalize the first word of the headings.
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| − | 2.2 General guidelines
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| − | Some general guidelines that should be followed in your manuscripts are:
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| − | * Use decimal points (not commas); use a space for thousands (10 000 and above).
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| − | * Follow internationally accepted rules and conventions. In particular use the international system of units (SI). If other quantities are mentioned, give their equivalent in SI.
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| − | 2.3 Tables, figures, lists and equations
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| − | Please insert tables as editable text and not as images. Tables should be placed next to the relevant text in the article. Number tables consecutively in accordance with their appearance in the text and place any table notes below the table body. Be sparing in the use of tables and ensure that the data presented in them do not duplicate results described elsewhere in the article.
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| − | 1. The first entry in this list
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| − | 2.1. A subentry
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| − | 3. The last entry
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| − | You may choose to number equations for easy referencing. In that case they must be numbered consecutively with Arabic numerals in parentheses on the right hand side of the page. Below is an example of formulae that should be referenced as eq. (1].
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| − | 2.4 Supplementary material
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| − | Supplementary material can be inserted to support and enhance your article. This includes video material, animation sequences, background datasets, computational models, sound clips and more. In order to ensure that your material is directly usable, please provide the files with a preferred maximum size of 50 MB. Please supply a concise and descriptive caption for each file. -->==
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| − | ==3 Bibliography<!--
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| − | ==4 Acknowledgments<!-- Acknowledgments should be inserted at the end of the document, before the references section. -->==
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| − | ==5 References<!--[1] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Article code. Available: http://www.scipedia.com/ucode.
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| − | [2] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Volume number, first page-last page.
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| − | [3] Author, C. (Year). Title of work: Subtitle (edition.). Volume(s). Place of publication: Publisher.
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| − | [4] Author of Part, D. (Year). Title of chapter or part. In A. Editor & B. Editor (Eds.), Title: Subtitle of book (edition, inclusive page numbers). Place of publication: Publisher.
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| − | [5] Author, E. (Year, Month date). Title of the article. In A. Editor, B. Editor, and C. Editor. Title of published proceedings. Paper presented at title of conference, Volume number, first page-last page. Place of publication.
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| − | [6] Institution or author. Title of the document. Year. [Online] (Date consulted: day, month and year). Available: http://www.scipedia.com/document.pdf.
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Simulation of flow through heterogeneous media often requires discretizing the flow domain into blocks and assigning an equivalent block conductivity value to each one of them. The process of defining block conductivities from point values is termed upscaling. A number of approaches to upscaling are available, most of which consider the uncertainty associated with any natural property, so that they cast the problem in a stochastic frame. Recently, Indelman and Dagan (1993a, b) provided a general stochastic methodology to upscaling in heterogeneous anisotropic formations by means of the dissipation energy function; unfortunately, they did not provide any “practical” method to compute block values from point ones. The objective of this work is twofold: First, we analyze different practical approaches to compute block conductivities and find that all of them provide very similar results in terms of actual computed values; second, we check that all approaches verify approximately a number of conditions stated by Indelman and Dagan (1993a). Specifically, we show analytically that for regular blocks, the methodologies of both Rubin and Gómez‐Hernández (1990) and Desbarats (1992) (which we call “practical” methodologies) satisfy the condition that the effective conductivity obtained from a field where the elementary conductivities are defined over a certain support (we call this the actual formation) is identical to that obtained from the same field with conductivities defined at a larger support (upscaled formation). The analysis is carried out by working with the logarithm of block conductivities and using a small‐perturbation expansion and thus is strictly valid for small variances. On the other hand, we show numerically that the two methodologies satisfy approximately an important condition stated in terms of the dissipation energy: that block‐averaged dissipation values computed are indeed very close to the true dissipation values in each block. The agreement is even better if we consider statistical moments instead of point values. As an important conclusion we should note that all practical methodologies considered in this work perform equally well and, more important, constitute a simple way to treat an otherwise very complex problem.
