Rapoport Revisited: A Review of Scaling Theory of Two Phase Flow in Porous Media and its Application
<p id="p1">This paper presents a detailed analytical exposition of the classic 1955 paper by Rapoport on the <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">rigorous</italic></em> scaling of t...
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| Format: | Book Chapter |
| Language: | English |
| Published: |
IntechOpen
2025
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| Subjects: | |
| Online Access: | https://www.intechopen.com/chapters/1223948 |
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| Summary: | <p id="p1">This paper presents a detailed analytical exposition of the classic 1955 paper by Rapoport on the <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">rigorous</italic></em> scaling of two phase flow in porous media. It extends Rapoport’s work to anisotropic systems and fills in a number of the missing details in the original paper, which has led to an incomplete appreciation of this seminal work. The various forms of the rigorous force balances (capillary, viscous and gravity) which can exist are shown. Within any given force balance, say viscous/capillary, there are three formulae which emerge but these, are related through the “shape groups” and reduce to one. The same is true for the gravity/capillary and the viscous/gravity force balances, only one of which is then required in addition to the viscous/capillary group. In fact, only two force balances and two shape groups are required (in 3D) and these are intimately related in a way that is explained in detail here for the first time, but which is “hidden” in Rapoport’s paper. Most papers which use scaling theory present <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">approximate</italic></em> scaling groups for the various force balances, although the correct <em><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">forms</italic></em> of the shape groups are usually broadly correct. However, Rapoport’s work on rigorous scaling groups and the full findings from it, also teaches us some important lessons about what can be expected when less rigorous (or approximate) scaling groups are developed. Since there are many different equivalent formula even for rigorous scaling, then a common literature worry may be laid to rest: That is, when some papers give a long list of the various force balance (viscous/capillary, viscous/gravity balance etc.), and then go on to worry about the question, “which is correct”? The exact formulae are not vital, although a correct set of force balances which are consistently linked by the shape groups should be used. However, the precise number of groups is fixed for a given system. There are also some dimensionless quantities, which are sometimes wrongly labelled as “scaling groups” or “force balances”. A number of novel applications of Rapoport’s scaling theory are also presented, coupled with numerical simulations, to tackle a range of problems in a way which Rapoport may not have anticipated. These examples illustrate the exact scaling that is possible using the Rapoport approach and show some surprising consequences of his results.</p> |
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| ISBN: | 978-0-85466-866-3 |
| DOI: | 10.5772/intechopen.1010239 |