Conceptual definition of porosity function for coarse granular porous media with fixed texture
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Porous media’s porosity value is commonly taken as a constant for a given granular texture free from any type of imposed loads. Although such definition holds for those media at hydrostatic equilibrium, it might not be hydrodynamically true for media subjected to the flow of fluids. This article casts light on an alternative vision describing porosity as a function of fluid velocity, though the media’s solid skeleton does not undergo any changes and remain essentially intact. Carefully planned laboratory experiments support such as hypothesis and may help reducing reported disagreements between observed and actual behaviors of nonlinear flow regimes. Findings indicate that the so-called Stephenson relationship that enables estimating actual flow velocity is a case that holds true only for the Darcian conditions. In order to investigate the relationship, an accurate permeability should be measured. An alternative relationship, therefore, has been proposed to estimate actual pore flow velocity. On the other hand, with introducing the novel concept of effective porosity, that should be determined not only based on geotechnical parameters, but also it has to be regarded as a function of the flow regime. Such a porosity may be affected by the flow regime through variations in the effective pore volume and effective shape factor. In a numerical justification of findings, it is shown that unsatisfactory results, obtained from nonlinear mathematical models of unsteady flow, may be due to unreliable porosity estimates.
KeywordsPorosity function Granular porous media Non-Darcy flow Boundary layer
Theoretical basis of the porosity function
It seems to be confined to geotechnical applications too.
Numerous investigators have studied this relationship, and several formulae have resulted based on experimental work. Kozeny (1927) proposed a formula which was then modified by Carman (1937, 1956) to become the Kozeny–Carman equation. Other attempts were made by Shepherd (1989), Alyamani and Şen (1993) and Terzaghi (1996).
The applicability of these formulae depends on the type of soil for which hydraulic conductivity is to be estimated. Moreover, few formulas give reliable estimates of results because of the difficulty of including all possible variables in porous media. Vukovic and Soro (1992) noted that the applications of different empirical formulae to the same porous medium material can yield different values of hydraulic conductivity, which may differ by a factor of 10 or even 20. The objective of those researches, therefore, is to evaluate the applicability and reliability of some of the commonly used empirical formulae for the determination of hydraulic conductivity of unconsolidated soil/rock materials.
Experimental setup and methods
A test run was always followed by full saturation of the tested medium to remove air bobbles’ blockage of the pores. Three temperature recording sensors were placed at the entrance, middle and the end of the flume to enable monitoring possible heat buildup due to the recirculation of the liquid.
Test materials and experimental result
General characteristics of the tested granular media
Media (coarse granular materials)
Size distribution (mm)
Coefficient of uniformity Cu
Coefficient of concavity Cc
A crucial point in describing the flow regime was to estimate the thickness of the boundary layer through pore spaces. It might exceed the surface roughness of the grains, making it necessary to consider the boundary-layer dispersion (Koch and Brady 1985).
Although the effect of boundary-layer growth may have limited application in practice, it seems to have a noticeable role in small-scale experimental setups where the effects of viscosity may be overlooked.
As shown in Figs. 7, 8, 9, 10, 11, 12, 13 and 14, with regard to the increasing velocity as a result of increasing turbulence, the predicted equation was near to the observed values of the hydraulic gradient with regard to Ergun’s equation. More adaptation of this subject observed with the unsteady flow, especially. The analysis indicated that the predicted equation agrees with the theory and practical procedures. Thus, with the unsteady flow, the nature of the boundary-layer thickness decreases, and hence, the porous porosity with the mentioned Reynolds number (the rigid structure constant) increases. So, following the theory of porous media flow regime is confirmed.
Based on the analysis and results, methods of estimating the hydraulic conductivity from empirical formulae based on hydraulic porosity have been developed and used to overcome relevant issues and problems.
The determination of the actual flow velocity in frictional soils pores cannot be based on Stephens’ theory, which is widely used in geotechnical engineering. The current research by the author of this paper shows that it is necessary to use a porosity correction coefficient, which is always smaller than one, to be porosity, in order to obtain a more logical estimate than the actual velocity. The analysis of the mathematical model under study shows that using such true corrected speeds, the partial differential equation governing the leakage current in frictional soils yields acceptable solutions.
In order to design granular porous media with fixed texture as rubble-mound breakwaters, the hydraulic gradient should be evaluated reliably. For this purpose, the extended Forchheimer’s equation (EFE) has been analyzed and the equations for coefficients a, b and c have been derived. However, reported experimental results did not agree with that theory, and present study shows that this contradiction stems from a misleading in evaluating hydraulic porosity due to some scale effects in the experiments.
In this paper, emphasizing that the expansion of the boundary layer changes the space available for flow, the porosity and shape of the pores are a function of the hydraulic gradient of flow through the porous medium.
If this theory is valid, the assumption that the aforementioned coefficients are constant in the porous medium is not correct and it is necessary a full scale of future experiments to predict better understanding how they change with the flow regime.
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