# Conceptual definition of porosity function for coarse granular porous media with fixed texture

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## Abstract

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.

## Keywords

Porosity function Granular porous media Non-Darcy flow Boundary layer## Introduction

*K*with a negative sign indicating an occurrence of the fluid flow from a relatively high head to a relatively low head locates, and the gradient in the relevant driving head is \( \nabla h \). In groundwater hydrology, the knowledge of saturated hydraulic conductivity of porous media is necessary for modeling the water flow in the soil, both in the saturated and unsaturated zone, and transportation of water-soluble pollutants in the soil.

*Re*) increases up to a critical value, the relationship will become nonlinear. To provide a universal relation, including this nonlinear effect, Forchheimer proposed an empirical formula (Forchheimer equation) (Mccorquodale et al. 1978; Leu et al. 2009; Chai et al. 2010; Wahyudi et al. 2002; Shokri and Sabour 2014)

*a*and

*b*are nonlinear coefficients depending on fluid properties, the pore size, porosity and shape.

*h*), \( \varphi_{y} \) and \( \varphi_{x} \) are differentiations of \( \varphi \) in the

*x*and

*y*directions, respectively. In coarse granular media, the flow velocity is much higher under the same driving head compared to fine-grained soils due to the higher hydraulic conductivity that in turn makes Eq. (1) invalid. This nonlinear laminar flow regime persists to a Reynolds number = 150 (Burcharth and Andersen 1995).

*m*and \( \lambda \) are empirical values depending on the media/fluid properties. Parkin (1971) could combine (4) with continuity equation to develop a partial differential equation governing non-Darcian flows in porous media as well (Bazargan 2002). The Parkin equation may be written as:

*Q*is the discharge rate,

*A*is the cross-sectional area of the specimen and

*n*is the soil’s porosity. A search in the literature shows that at early days of evaluating the Darcy Law’s validity range using the Reynolds number, the actual velocity \( \left( {V_{\text{a}} } \right) \) was calculated differently. For instance, Pavlovski (1940) reported a special version of the Reynolds number reexamining validity of the Darcy Law (Lu and Likos 2004), in which the actual velocity was defined by:

## Theoretical basis of the porosity function

*x*,

*y*and

*z*directions, respectively, \( K_{x} \,,K_{y} \,{\text{and}}\,K_{z} \) are corresponding hydraulic conductivities and

*h*is the driving head. If the medium is assumed to be homogenous and isotropic (i.e., \( k = k_{x} = k_{y} = k_{z} \)) the transient flow equation may be simplified as:

*n*is porosity and

*ρ*represents the fluid’s density.

*b*):

*g*is the gravitational acceleration.

*K*:

*K*denotes hydraulic conductivity, \( \mu \) is dynamic viscosity,

*C*the dimensionless constant related to the geometry of the soil pores and

*f*(

*n*) represents porosity function. Odong’s literature survey shows different researchers approach in quantifying porosity function

*f*(

*n*) in Eq. (15) none of which addressing the way porosity controls the flow through porous media. Though in an alternative conceptual assessment of

*n*Vukovic and Soro (1992) made use of a uniformity coefficient (\( U = \frac{{D_{60} }}{{D_{10} }} \)) to estimate porosity as:

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.

_{ t }is sorting coefficient with \( 6.1 \times 10^{ - 3} \le C_{t} \le 10.7 \times 10^{ - 3} \). In this study, we used an average value of

*C*

_{ t }(\( C_{t} \cong 8.4 \times 10^{ - 3} \)). Terzaghi formula is most applicable for coarse granular media (Cheng and Chen 2007).

## 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) |
| Temperature (°C) | Porosity ( | Coefficient of uniformity | Coefficient of concavity |
---|---|---|---|---|---|---|

CGM1 | 2.5–28 | | | | | |

CGM2 | 10.5–63 | | | | | |

^{−1}ranging between 0.059 and 0.350 was maintained for CGM1 series. For CGM2 series, a discharge rate of 13.19 L s

^{−1}under hydraulic gradients ranging between and 0.065–0.300 was adapted. To create unsteady flow condition, a flap gate placed at the downstream end of the flume was used that was maneuvering open and close repeatedly following a prescribed-preset period. Once flow velocity was determined, its corresponding Reynolds numbers (

*Re*) were calculated by:

*L*is a characteristic length (assumed to be equal to \( d_{\text{m}} \) in the present study) and \( \upsilon \) represents the fluid kinematic viscosity. The values of \( \upsilon \) for \( {\text{CGM}}1 \), and \( {\text{CGM}}2 \) materials are 0.915 and 1.004 mm

^{2}/s, respectively.

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).

*Re*, or:

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.

## Experimental verifications

*n*may be taken as either measured porosity or effective porosity leading to two sets of output.

*i*) versus flow velocity, i.e., data obtained from observed (

*i*) in deferent types of media (designated with C curves), calculated (

*i*) using common understanding of porosity (designated with A curves) and calculated (

*i*) using the proposed hydraulic porosity concept (designated with B curves in the following plots Figs. 7, 8, 9, 10, 11, 12, 13 and 14.

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.

## Conclusions

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.

## Notes

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