A Unified General Resistance Formula for Uniform Coarse Porous Media

Abstract

Many authors studied the nonlinear relationship between seepage velocity and hydraulic gradient in coarse granular materials. They managed different approaches and variables to define the resistance formula applicable to that type of granular media. Based on the analysis of the different approaches and experimental data obtained by the corresponding authors, we propose a unified general seepage equation applicable to large-sized granular materials. Such an equation gives unity to the main nonlinear resistance formulas developed to date. Particularly relevant are the conclusions regarding the relationship of the linear (${\alpha }^{*}$) and quadratic (${\beta }^{*}$) dimensionless coefficients of the resistance formula with the representative size of the particle and the geometrical features of the porous materials.

1. Introduction and Objectives

Seepage through coarse porous media, such as gravels or ripraps, with a high pore size creates so-called non-Darcy flows, which can be represented by the quadratic equation of Forchheimer [1], Equation (1), or by the exponential equation presented by S.V. Izbash [2], Equation (2):

$$i=r·V+s·V^2$$

(1)

$$i=a·V^b$$

(2)

where $V$ is the mean seepage velocity defined as the mean fluid velocity over the entire cross section, $i$ is the hydraulic gradient,$r$ represents the coefficient of the linear term, $s$ represents the coefficient of the quadratic term, and $a$ and $b$ are coefficients that can be determined experimentally for each porous media.

The Forchheimer equation, Equation (1), contemplates a wide range of flow regimes: nonlinear laminar, turbulent transition, and fully developed turbulent flow, which make it more suitable than the exponential Equation (2) to represent the physical phenomenon of filtration in porous media.

However, it is necessary to deepen the understanding of this physical phenomenon in these transition regimes by analyzing whether the coefficients $r$ and $s$ remain constant for a wide range of gradients.

On the other hand, based on Equation (1), different formulations have been developed that consider different physical parameters. J.C. López et al. [3] have recently published a comparative analysis of these formulations based on the approach of a generalized quadratic equation based on three physical parameters: the Darcy–Weisbach generalized friction factor $f$, the generalized Reynolds number $Re$, and the generalized characteristic length $L_c$.

From this generalized quadratic equation, this research work aims to develop a general unified equation for coarse porous media of uniform granulometry that can be applied to the so-called non-Darcy flows.

Within the development of the USEC, the obtaining of the dimensionless linear coefficient ${\alpha }^{*}$ included in the coefficient $r$ and the quadratic coefficient ${\beta }^{*}$ included in the coefficient $s$, both related to the particle $D$ representative size and the geometric characteristics of the porous media, is of special importance.

On the other hand, we also highlight the analysis of the application of the USEC to a wide range of gradients where the transition regimes are developed.

The research carried out in this article includes the presentation of a conceptual scheme obtained from the review of the existing formulation and its experimental verification with permeability tests carried out at the Hydraulics Laboratory of the Centro de Estudios Hidrográficos (CEDEX) on porous media composed of crushed aggregates duly selected to obtain a uniform granulometry and coming from the same limestone quarry (i.e., with the same geological origin and similar geometric properties).

2. State of the Art

2.1. Review of the Formulation and Conceptual Scheme

Most of the investigations have based the development of their formulas on the analogy with the flow in pipes by means of the applications of two dimensionless groups that we will call the generalized friction factor of the Darcy–Weisbach $f$ Equation (3) and the generalized Reynolds number $Re$ Equation (4) (J.C. López et al. [3] and J. Bear [4]).

$$f=L_c·2g·\frac{i}{V_p^2}$$

(3)

$$Re=\frac{L_c·V_p}{v}$$

(4)

where $L_c$ is the generalized characteristic length, $g$ is the gravitational acceleration,$i$ is the hydraulic gradient, $v$ is the kinematic viscosity, and $V_p$ is the velocity in the pores, determined by Equation (5).

$$V_p=\frac{V}{n}$$

(5)

where $n$ is the porosity of the porous media.

As indicated by J.C. López et al. [3] and J. Bear [4], most of the researchers agree in pointing out that the experimental data corresponding to a granular porous media of the same geometry and size properly represented in a generalized diagram of the type $\left[Re,f\right]$ fall on a smooth curve for all non-Darcy flow regimes that can be fitted by Equation (6).

$$f=\frac{A_1}{Re}+A_2$$

(6)

where $A_1$ is referred to us as a linear generalized dimensionless coefficient and $A_2$ as a quadratic generalized dimensionless coefficient.

Equation (6) is referred to us as a generalized equation in a $\left[Re,f\right]$ diagram form.

However, the existence of these continuous curves in granular porous media contrast with the physical behavior of turbulent flow in pipes where sharp jumps occur in the Reynolds number of the pipe range ${Re}_t$ between 2000 and 4000 (F. W. White [5]).

Substituting the generalized values$f$ (Equation (3)), $Re$ (Equation (4)), and pore velocity $V_p$ (Equation (5)) into Equation (6), we obtain the generalized quadratic Equation (7):

$$i=\frac{v}{2·g}·A_1·\frac{1}{L_c^2}·\frac{V}{n}+\frac{1}{2·g}·A_2·\frac{1}{L_c}·\frac{V^2}{n^2}$$

(7)

Recently, J.C. López et al. [3] have performed a comparative analysis of different existing formulations based on three characteristics lengths, $L_c$: the representative diameter of the particles $D$ and the square root of the intrinsic permeability $K_o\left(\sqrt{K_o}\right)$, which is a macroscopic property of porous media given by Equation (8):

$$K_0=\frac{v}{g}·\frac{1}{i}·V$$

(8)

and the mean hydraulic radius $R_h$ that was first defined by Taylor [6] and is given by Equation (9):

$$R_h=\frac{n}{S_e(1-n)}$$

(9)

where $S_e$ is the average specific surface area of the solid particles constituting the granular porous media and depends on the size, shape, angularity, and surface roughness of the particles (Loudon, Linford and Sounders, Crawford et al., Martins and Escarameria, Martins and Sabin and Hansen [7,8,9,10,11,12]) and is defined by Equation (10):

$$S_e=\frac{6·F}{D}$$

(10)

where $F$ is a dimensionless coefficient that considers shape, angularity, and surface roughness of the particles, as a whole, to be constituents of the porous granular media and represents the frictional energy loss addition to that produced in granular porous media consisting of smooth spherical particles as a result of having a higher average specific surface $S_e$.

Next, we will analyze each of the parameters involved in Equation (7).

• Characteristic Length L_c

Based on the analogy with the flow in pipes for a noncircular section, as considered by J.C. López et al. [3], we will consider the characteristic length $L_c$ the mean hydraulic diameter $D_h$ (Equation (11)):

$$D_h=4·R_h$$

(11)

The characteristic length $D_h$ represents a physical parameter within a microscale model (Ergun and Orning [13] and Huang and Ayoub [14]) that is related to the size and shape of the interconnected channels of the porous media.

Substituting Equation (10) into Equation (9) and considering Equation (11), we obtain Equation (12):

$$D_h=\frac{2}{3}·\frac{1}{F}·\frac{n}{(1-n)}·D$$

(12)

The mean hydraulic diameter $D_h$ contemplates three physical parameters: the representative diameter of the constituent particles of the porous media $D$, which is related to the pore size, the dimensionless coefficient $F$, and the porosity $n$, both latter parameters related to the pore shape. The porosity parameter n is defined in (Equation (12)) by the porosity function $f\left(n\right)$(Equation (13)):

$$f(n)=\frac{n}{(1-n)}$$

(13)

If we consider as the characteristic length only the representative diameter of the particles $D$, we are actually facing a simpler model within the analogy with the flow in pipes, which does not consider these last two fundamental physical parameters to properly characterize the granular porous media: the dimensionless coefficient $F$and the porosity function $f\left(n\right)$.

On the other hand, the square root of intrinsic permeability, $\sqrt{K_0}$, actually represents a macroscopic property of the granular porous media, as is the case with the parameters $r$ and $s$ considered in the quadratic Equation (1) (H. Huang and J. Ayoub [14]).

• Generalized dimensionless coefficients A₁, linear and A₂, quadratic.

However, in the microscale model of the granular porous media represented by the characteristic length $D_h$, in contrast to the flow in pipes, the channels through which the water flows are tortuous and of a non-constant section with a very complex geometry. Therein, the flow experiences continuous cycles of acceleration and deceleration with the consequential loss of associated energy. At velocities much lower than those necessary to produce the turbulent effects, inertial effects appear because of the sinuous trajectories (Ahmed and Sunada [15]), producing nonlinearity (nonlinear laminar regime) through quadratic Equation (1). When velocities increase, the onset of turbulence is attributed to the phenomena of separation of the boundary layer and the corresponding formation of vortices (Wright, Panfilov et al. and Fourar et al. [16,17,18]), phenomena all related to particle shape and angularity.

On the other hand, McCorquodale et al. [19], using as characteristic length $L_c$ the mean hydraulic radius $R_h$, were the first to introduce the concept of particle surface roughness $\varepsilon$ in a porous media in an analogy with the flow of pipes, indicating that “The effect of surface roughness is usually ignored; however, for the high velocities and large particle sizes, this is probably not justified”.

The conclusion of the experimental results carried out by J. Mulqueen [20] on gravels indicate that the flow velocity was declining substantially with increased surface roughness.

As a consequence, in this microscale model, based on the behavior of flow in pipes, the physical parameters, shape, angularity, and surface roughness of the particles in addition to their size, need to be considered to adequately characterize the geometry of the granular porous media. These physical parameters are contemplated through the $A_1$ linear and $A_2$ quadratic dimensionless coefficients represented in the generalized quadratic Equation (7).

With respect to the linear generalized dimensionless coefficient $A_1$, Terzaghi and Peck, Engelund, and Salahi et al. [21,22,23] consider them to be a function of the shape and angularity of particles. On the other hand, the value of this dimensionless coefficient for these non-Darcy flows is different from that obtained for the laminar regime (Fand et al. [24]; Barree and Conway [19,20,21,22,23,24,25]).

With respect to the quadratic generalized dimensionless coefficient $A_2$, Gupta [26], based on Shergold’s previous research [27], related both concepts, the geometry of the granular porous material and the representative size of particles $D$, by function Equation (14):

$$\eta =f·\left(C^{\prime },D\right)=C^{\prime }·D^{n^{\prime }}$$

(14)

where $\eta$ represents the angularity as a set of aggregates, $C^{\prime }$ is a coefficient of form that depends solely on the shape of the particles as a whole and is constant for each type of porous material and is independent of its size, and $n^{\prime }$ is an exponent.

Gupta used materials of various shapes, with uniform granulometry and a wide range of sizes: rounded gravels (6 m <$D$< 100 mm), subangular crushed quartzite (6 mm < $D$< 50 mm), and angular limestone (6 mm <$D$< 50 mm). He obtained the following values: $C^{\prime }$ = 37.89 for rounded gravels, $C^{\prime }$ = 41.44 for subangular crushed quartzite, and $C^{\prime }$= 43.35 for angular limestone. The value of the adjusted exponent $n^{\prime }$ for these granular porous media was −0.032, so according to Equation (14), this exponent did not vary with the shape of the particles defined through the shape coefficient $C^{\prime }$.

In this same line, Dudgeon [28] stated that “the angularity of the particles increases with the decrease in size”.

As indicated by J.C. López et al. [3], several authors in the development of their formulations (Ergun and Orning [13]; Engelund [22]; Stephenson [29] and Martins [11]) have limited themselves to proposing constant values of these generalized dimensionless coefficients, $A_1$ and $A_2$, to be applied according to the three geometries representative of the granular porous material: smooth spheres, rolled aggregates, and crushed aggregates.

However, in this article, the knowledge of the seepage phenomenon in coarse porous media is further deepened by including in the general unified equation to be developed, based on Equation (7), empirical equations of the types, $A_1=F_1\left(D\right)$ and $A_2=F_2\left(D\right)$, that relate the dimensionless linear $A_1$ and quadratic $A_2$ coefficients with the representative size of the particles $D$. These equations have been grouped in the three representative geometries mentioned above and will be adjusted with the experimental data from previous studies described in the following section and experimentally validated in Section 4.

2.2. Experimental Data from Previous Studies

Experimental data have been collected from several authors in order to properly calibrate the unified overall equation (Sabri Ergun and A.A. Orning [13]; Ward [30]; Dudgeon [28]; Ahmed and Sunada [15]; Arbhabhirama and Dinoy [31]; Martins [11]; Sedghi-Als et al. [32]; M-B Salahi et al. [23]; Ferdos [33]). Such data have been appropriately selected to provide coefficients $r$ and $s$ of the Forchheimer equation (Equation (1)).

These experimental data have been included in this research work shown in Appendix A, and their main characteristics have been described: types of aggregates tested, size ranges $D$ used, diameter of the permeameter $D_x$ of the installation, and hydraulic gradients $i$ applied.

3. The Unified Seepage Equation for Coarse Materials

In this section, the unified seepage equation for coarse materials (USEC) from the generalized quadratic equation (Equation (7)) has been developed using the experimental data included in Appendix A. The USEC is based on the analogy with the flow in pipes and properly calibrated by obtaining empirical equations that relate the generalized dimensionless coefficients, linear $A_1$ and quadratic $A_2$, with the representative size of the particles $D$. Section 3.3 also includes its application to existing experimental data to verify the continuity of the curve given by Equation (6) and has served to obtain preliminary considerations on its applicability. Next, in Section 4, we have proceeded to validate the USEC with the additional experimental data obtained from new seepage tests performed in the Laboratory of Hydraulics at the Centro de Estudios Hidrográficos (CEDEX).

3.1. Methodology

The methodology for properly developing the USEC will consist of:

(a)

Selecting as characteristic length $L_c$ the average hydraulic diameter $D_h$ defined by Equation (11) with the physical parameters $D$, $n$, and $F$. Equation (11), together with Equation (7), will result in a unified general equation that contemplates the unified dimensionless coefficients ${\alpha }^{*}$ (linear) and ${\beta }^{*}$ (quadratic), equivalent to the coefficients $A_1$ and $A_2$ from generalized quadratic Equation (7). The rationale for this new formulation is developed in Section 3.2 below.

(b)

Collecting the experimental data duly selected according to the type of granular porous media, smooth spheres, rolled aggregates, and crushed aggregates, in order to calculate the linear term $r$ and the quadratic term $s$ of the Forchheimer equation (Equation (1)) that appear in Appendix A.

(c)

Analyzing with the available experimental data from Appendix A whether the ${\alpha }^{*}$ linear and ${\beta }^{*}$ quadratic dimensionless coefficients for a granular porous media defined by a given geometry and size are constant for all non-Darcy flow regimes, nonlinear laminar, turbulent transition, and turbulent fully developed, as most research seems to confirm (J.C. López et al. [3]), or conversely, whether they need to be adjusted for each particular non-Darcy flow regime. This analysis is developed in Section 3.3 with the experimental data used, and in the absence of further research in this regard, we can consider as a hypothesis that the USEC can be applied to all non-Darcy flow regimes.

(d)

Obtaining the empirical equations relating the dimensionless USEC coefficients ${\alpha }^{*}$ and ${\beta }^{*}$ to the representative particle size $D$ for granular porous media included in Appendix A. These empirical equations have been grouped into three representative geometries, smooth spheres, rolled aggregates, and crushed aggregates, in order to analyze where there are similarities between them. The values obtained have been represented in the type [$D$, ${\alpha }^{*}$] and [$D$, ${\beta }^{*}$] diagrams in order to analyze their evolution regarding the representative size of particles $D$. Section 3.4 and Section 3.5 develop these aspects.

(e)

Finally, in Section 4, we have analyzed the results obtained by applying the newly developed USEC to the experimental data obtained in the tests carried out at the Hydraulics Laboratory of the Centro de Estudios Hidrográficos (CEDEX). The granular porous media used in these tests consist of crushed aggregates with uniform granulometry and are from the same limestone rock quarry, so the dispersions in the geometric properties of the same are minimal.

3.2. Justification for the New Formulation

According to the analysis carried out in Section 2.1, we will select as characteristic length $L_c$ the mean hydraulic diameter $D_h$. Substituting the value of $D_h$ from Equation (12) into the generalized quadratic Equation (7), we obtain Equation (15):

$$i=\frac{v}{2g}·A_1·\frac{3^2·F^2}{2^2·D^2}·\frac{{(1-n)}^2}{n^2}·\frac{V}{n}+\frac{1}{2g}·A_2·\frac{3·F}{2·D}·\frac{(1-n)}{n}·\frac{V^2}{n^2}$$

(15)

from which, simplifying, we obtain Equation (16):

$$i=\frac{v}{g}·\frac{9}{8}·A_1·F^2·\frac{{(1-n)}^2}{n^3}·\frac{1}{D^2}·V+\frac{1}{g}·\frac{3}{4}·A_2·F·\frac{(1-n)}{n^3}·\frac{1}{D}·V^2$$

(16)

Considering a similar approach to Engelund [22], we will define the unified dimensionless coefficients α* linear and β* quadratic by Equations (17) and (18), respectively:

$${\alpha }^{*}=\frac{9}{8}·A_1·F^2$$

(17)

$${\beta }^{*}=\frac{3}{4}·A_2·F$$

(18)

Substituting these coefficients, ${\alpha }^{*}$ and ${\beta }^{*}$, in Equation (16), we obtain Equation (19):

$$i=\frac{v}{g}·{\alpha }^{*}·\frac{{(1-n)}^2}{n^3}·\frac{1}{D^2}·V+\frac{1}{g}·{\beta }^{*}·\frac{(1-n)}{n^3}·\frac{1}{D}·V^2$$

(19)

Equation (19) includes the porosity function of the linear term (Equation (20)):

$$f_L(n)=\frac{{(1-n)}^2}{n^3}$$

(20)

and the porosity function of the quadratic term (Equation (21)):

$$f_T(n)=\frac{(1-n)}{n^3}$$

(21)

We will refer to Equation (19) as the USEC in quadratic form, applicable for coarse granular porous media with uniform particle size. Each granular porous media will therefore be defined by the physical parameters considered therein, representative size of particles $D$, porosity functions of linear $f_L\left(n\right)$, and quadratic $f_T\left(n\right)$ terms, and finally the unified dimensionless coefficients linear ${\alpha }^{*}$ and quadratic ${\beta }^{*}$ that characterize the geometry of the porous material.

Knowing parameters $r$ and $s$ of Forchheimer’s Equation (1) of a given granular porous media, the unified dimensionless coefficients α* and β* that uniquely characterize granular porous media can be obtained by means of the expressions:

$${\alpha }^{*}=\frac{r·g·n^3·D^2}{v·{(1-n)}^2}$$

(22)

$${\beta }^{*}=\frac{s·g·n^3·D}{(1-n)}$$

(23)

In Appendix A, the unified dimensionless coefficients ${\alpha }^{*}$ and ${\beta }^{*}$ are calculated by applying Equations (22) and (23) based on the experimental data used in this research work.

On the other hand, USEC Equation (19) can be represented dimensionlessly in a [$R_p,f_p$] diagram in order to visualize the continuity of the curves given by Equation (6).

The pore Reynolds number $R_p$ based on $D_h$ will be defined by Equation (24):

$$R_p=\frac{4·R_h·V_p}{v}=\frac{4}{v}·\frac{n}{S_e·(1-n)}·\frac{V}{n}=\frac{2}{3}·\frac{1}{v}·\frac{1}{F}·\frac{1}{(1-n)}·D·V$$

(24)

The pore friction factor $f_p$ based on $D_h$ will be defined by Equation (25):

$$f_p=8·R_h·g·\frac{i}{V_p^2}=8·g·\frac{n}{S_e·\left(1-n\right)}·\frac{n^2}{V^2}·i=\frac{4}{3}·g·\frac{1}{F}·\frac{n^3}{\left(1-n\right)}·D·\frac{i}{V^2}$$

(25)

With it, Equation (6), particularized for the pore friction factor $f_p$ and the pore Reynolds number $R_p$, will remain defined by Equation (26):

$$f_p=\frac{8}{3}·\left[\frac{1}{3}·\frac{1}{R_p}·\frac{{\alpha }^{\mathrm{*}}}{F^2}+\frac{1}{2}·\frac{{\beta }^{\mathrm{*}}}{F}\right]$$

(26)

being $A_1$:

$$A_1=\frac{8}{9}·\frac{{\alpha }^{*}}{F^2}$$

(27)

being $A_2$:

$$A_2=\frac{4}{3}·\frac{{\beta }^{*}}{F}$$

(28)

We will refer to Equation (26) as USEC in a $\left[R_p,f_p\right]$ diagram form.

Finally, we will linearize Equation (26). For this, we will consider the linear Equation (29):

$${\lambda }_p=t+u·R_p$$

(29)

where $t$, y, and $u$ are parameters obtained by fitting the experimental data [$R_p,f_p$], and ${\lambda }_p$ is the linearized pore friction factor given by Equation (30).

$${\lambda }_p=f_p·R_p$$

(30)

To obtain the parameters $t$ and $u$, we will multiply both members of Equation (25) by the pore Reynolds number $R_p$(Equation (31)):

$$f_p·R_p=\frac{8}{9}·\frac{{\alpha }^{\mathrm{*}}}{F^2}+\frac{4}{3}·\frac{{\beta }^{\mathrm{*}}}{F}·R_p$$

(31)

Thus, the unified dimensionless coefficients ${\alpha }^{*}$ and ${\beta }^{*}$ are determined by Equations (32) and (33):

$${\alpha }^{*}=t·\frac{9}{8}·F^2$$

(32)

$${\beta }^{*}=u·\frac{3}{4}·F$$

(33)

We will call Equation (29) USEC in linearized form, which can be represented in a $\left[R_p,{\lambda }_p\right]$ diagram.

3.3. Study of the Continuity of the Equation in Non-Laminar Flow Regimes

In order to verify the continuity of the curve given by Equation (26) for a given porous media in a wide range of gradients that includes all non-Darcy flows, we will use USEC expressed in the [$R_p,f_p$] diagram, Equation (26), and in the [$R_p,{\lambda }_p$] diagram, Equation (29).

• Flow regimes: nonlinear laminar, turbulent transition

In Figure 1 are represented in a $\left[R_p,{\lambda }_p\right]$ diagram the experimental data obtained by Dudgeon [28] corresponding to the granular porous media composed of smooth spheres that have the advantages of a well-defined geometry and a wide range of gradients, 1.80 × 10⁻⁴ < $i$< 8.00 (N = 14). In this case, the representative particle size $D$ is the value $D_{50}$ of the granulometric curve, 15.97 mm.

The porosities $n$of 36.90% (N = 5), 41.50% (N = 5), and 37.20% (N = 4) were used for each of the three tests carried out. It can be seen how these data fall on USEC in the linearized form defined by Equation (29), which implies the existence of a continuous curve as given by Equation (26). A good fit has been obtained for parameters $t$ and$u$ (R² = 0.9972), obtaining the unified dimensionless coefficients α* = 232.01 and β* = 1.26 by applying Equations (32) and (33), respectively, and considering a value of the dimensionless coefficient $F$ = 1.00 for smooth spheres.

Thus, it is observed that each granular porous media with a particle size determined by its representative size $D$ and consisting as a whole of particles with a shape, surface, angularity, and surface roughness determined by the dimensionless coefficient $F$ will be represented by a line whose slope $u$ allows the calculation of the quadratic dimensionless coefficient β* in accordance with Equation (33), and the cut with the ordinate axis$t$ allows the calculation of the linear dimensionless coefficient α* according to Equation (32).

• Fully developed turbulent flow

When the fully developed turbulent flow is reached, exponent $b$ of the exponential Equation (2) is equal to 2 (J.C. López et al. [3]), and as a consequence, parameters $a$of Equation (2) and $s$ of Equation (1) are equalized, and parameter $r$ of the linear component of the quadratic Equation (1) becomes zero. With this, USEC in quadratic form for fully developed turbulent flow will be given by the expression (Equation (34)):

$$i=\frac{1}{g}·{\beta }^{*}·\frac{(1-n)}{n^3}·\frac{1}{D}·V^2$$

(34)

On the other hand, USEC in a [$R_p,f_p$] diagram form will be defined by the horizontal asymptote (Equation (35)):

$$f_p=\frac{4}{3}·\frac{{\beta }^3}{F}$$

(35)

Finally, the USEC in linearized form (Equation (29)) would be represented in a [$R_p,{\lambda }_p$] diagram by a straight line passing through the origin of the ordinates by the expression (Equation (36)):

$${\lambda }_p=u·R_p$$

(36)

where ${\beta }^{*}$ is defined by Equation (33).

Figure 2 shows the experimental data corresponding to two tests of Martins [11] for rolled aggregates ($F$ = 1.05 proposed by Martins) in a diagram [$R_p,f_p$]. In this case, the representative particle size corresponds to the average sieve aperture $D_a$ with a value of 22.00 mm. The porosities $n$ of the two tests were 36.22% and 36.83%, taking place in the zone of the fully developed turbulent regime, and are represented in the diagram [$R_p,f_p$]. Values $R_p$ and $f_p$ have been obtained by applying Equations (24) and (25) from the experimental data [$V,i$] (N = 8).

It is observed for both tests that 7 of the 8 points are aligned on a horizontal asymptote defined by Equation (35) with mean values of $f_p$ = 1.62 for the porous media with a porosity of $n$ = 36.22% and $f_p$ = 1.66 for the porous media with a porosity of $n$ = 36.83. The fully developed turbulent regime has been reached in this case for a pore Reynolds number $R_p$ = 1049 for $n=36.22\%$ and $R_p$ = 1034 for $n=36.83\%$ (see

Table 1. Values of the unified dimensionless quadratic β* coefficient for the fully developed turbulent regime. Rolled aggregates. Data from Martins [11] Source: authors.

D (mm)	Test	n	F	Rh/D	N	${\mathit{R}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{p}}$	β* (Equation (35))	N	$\mathit{u}$ (Equation (36))	β* (Equation (33))
22.00	A	36.22%	1.050	0.090	7	1049	1.615	1.272		-	-
22.00	B	36.83%	1.050	0.093	7	1034	1.660	1.308	14	1.639	1.290
44.00	A	36.18%	1.050	0.090	5	4986	1.168	0.920		-	-
44.00	B	35.53%	1.050	0.087	5	5019	1.110	0.874	10	1.143	0.900
89.00	A	38.23%	1.050	0.098	5	17,033	1.055	0.831		-	-
89.00	B	38.84%	1.050	0.101	5	18,097	1.035	0.815	10	1.052	0.828

Note: $F$ = 1.05 Value adopted by Martins [11] for rolled aggregates. $R_h/D$ = 0.1 Value proposed by Parkin [34]. $N$ = Number of points considered to obtain the asymptote $f_p$ from a total of 8 tested points on each porous media. $R_p$ = Reynolds number of the pores at which the fully developed turbulent flow is reached. $u$ = obtained by fitting Equation (36) with the total number of points of each representative size $D$ tested.

).

Figure 3 shows all the points (N = 16) corresponding to the tests represented in Figure 2. A fit for parameter $u$ = 1.6386 (R² = 0.9974) has been obtained considering 14 of the 16 points. The value $t$ = −0.0656 is close to zero as a consequence of the tests being carried out in the fully developed turbulent regime (Equation (35)). Thus, applying Equation (32), we obtain a value of β* = 1.29 for the set of the two porous media evaluated ($n$ = 36.22% and $n$ = 36.83%), with the representative particle size $D$ = 22.00 mm.

Table 1 shows all the values of the unified quadratic dimensionless coefficient β* corresponding to all the rolled aggregates tested by Martins [11].

For each representative size $D$, three values of the unified quadratic dimensionless coefficient β* have been obtained, two values with different porosities for each size by applying Equation (34) and one value by applying Equation (35), by fitting all the points of each size $D$. The three values obtained are close to each other, which highlights the existence of similar geometries for each size $D$.

3.4. Analysis of the Effect of Representative Size $D$ on the Unified Coefficients α* and β*

In accordance with Section 2.1 on the state of the art, the dimensionless coefficients A₁ and $F$ are a function of the geometrical properties of the porous material. Therefore, the unified dimensionless coefficient α* will also depend on the geometry of the granular porous media, according to Equation (16).

Figure 4 shows the values of α* in a [$D$,α*] diagram obtained from the tests carried out by Sedghi-Als et al. [32] and M-B Salahi et al. [23] in the size range 1.77 mm <$D$< 56.80 mm, which have the advantage of having a higher density of points (N = 18) in this size range. In this figure, an increase in the unified dimensionless coefficient α* with the representative particle size $D$ can be observed for all three porous media.

The values of the unified dimensionless coefficient α* for crushed aggregates are in general slightly above the values for the rolled aggregates, except for in the test for $D$ = 13.08 mm. All this seems to indicate that, with the experimental data used and with a representative particle size $D$, the values of the unified dimensionless coefficient α* for crushed aggregates are higher than the corresponding values for rolled aggregates, as shown in the research work of Terzaghi [21] and Engelund [22]. As a consequence, the linear dimensionless coefficient α* should vary according to the geometry of the porous material and the representative size of the particles $D$.

In the case of the unified quadratic dimensionless β^*, the analysis is based on the analogy with the flow in pipes. According to Colebrook [35], the calculation of the coefficient of turbulent friction $f_t$ in the pipes is given by the expression:

$$\frac{1}{{f_t}^{1/2}}=-2.0log·\left(\frac{\raisebox{1ex}{${\epsilon }_t$}\!\left/ \!\raisebox{-1ex}{$D_t$}\right.}{3.7}\right)$$

(37)

where $D_t$ is the diameter of the pipe, and ${\epsilon }_t$ is the absolute pipe roughness.

Equation (38) allows the obtaining of the asymptotic value of the friction coefficient $f_t$ and is independent of the Reynolds number $Re$ of the pipe. In Figure 5, we have plotted two curves of type $f_t$ = $F$·$D_t$, applying Equation (37). Each curve is represented with a geometry defined by means of its absolute roughness ${\epsilon }_t$. For similarity, we have adopted the absolute particle surface roughness ε proposed by McCorquodale et al. [18] (${\epsilon }_t$ = 0.1 mm for the curve corresponding to crushed aggregates and ${\epsilon }_t$ = 0.02 mm for the curve corresponding to rolled aggregates).

It can be observed from Figure 5 that:

(a)

For the same absolute pipe roughness ${\varepsilon }_t$, the friction coefficient $f_t$ can be fitted by a smooth curve $f_t=F\left(D_t\right)$. The friction coefficient $f_t$ decreases as the diameter of the pipe $D_t$ increases. For high values of $D_t$, the friction coefficient $f_t$ tends to an asymptotic value.

(b)

As the roughness of the pipe ${\epsilon }_t$decreases, curves $f_t=F\left(D_t\right)$ approach the horizontal axis with lower values of the friction coefficients, $f_t$, for the same diameter of the pipe, $D_t$.

Since the dimensionless coefficient $F$, which affects the specific surface area $S_e$, can be considered constant for a given geometry of the porous media according to Equation (9), we can consider by the analogy with the flow in pipes that (a) the parameter $D_t$ is equivalent to the parameter $D$ through Equation (12) and (b) the parameter $f_t$ is equivalent to the parameter $f_p$ and, consequently, with parameter β* through Equation (35). Thus, in flows in porous media with remarkably similar fixed geometries, there should be curves of the type β^*= $F\left(D\right)$, equivalent to curves $f_t=F\left(D_t\right)$.

In Table 1 for the rolled aggregates tested by Martins [11], a decrease in the quadratic dimensionless coefficient β* with the representative particle size $D$ is observed

3.5. Analysis of the Effect of the Aggregate Type on the Unified Coefficients α* and β^*

In line with the investigations of Engelund [22], Stephenson [29], Gupta [26], and Martins [10], we group the experimental data into three different types of coarse granular porous media, smooth spheres, rolled aggregates, and crushed aggregates, whose data are shown in Appendix A. To perform this analysis, we have used as a tool diagrams of the [$D$,α*] and [$D$, β*] type.

We have considered a minimum representative size $D$ of 0.50 mm as a coarse porous media, so Darcy’s law is no longer valid (Taylor [6]), placing us in the so-called non-Darcy regimes flow.

3.5.1. Tests with Smooth Spheres

Figure 6 shows the experimental data (N = 24) represented in the [$D$,α^*] diagram for the interval 0.50 mm < $D$< 30.00 mm, where an increase in coefficient α* with the representative particle size $D$ is observed by means of Equation (38):

$${\alpha }^{*}=136.08·e^{0.0371D}$$

(38)

(R² = 0.9689 N = 19)

The curve that best fits, with the experimental data used, for the linear dimensionless coefficient α* is an exponential type.

Data for $D$ = 0.77 mm (Lindquist (38)), $D$ = 3.00 mm (Sunada [39]), $D$ = 5.03 mm (Crawford et al. [9]), and$D$ = 16.00 mm (Kirkham [40]) are close to their theoretical values using Equation (38). $D$ = 3.20 mm (Blake [36]) is higher than its theoretical value.

Figure 7 shows the experimental data (N = 22) plotted on the [$D$, β*] diagram for the unified quadratic dimensionless coefficient β* at the 0.50 mm < $D$< 30.00 mm interval. A decrease in coefficient β* with the representative particle size $D$ is observed by means of Equation (39):

$${\beta }^{*}=2.087·D^{-0.159}$$

(39)

(R² = 0.8698 N = 17)

As can be seen in Figure 7, the data corresponding to the porous media tested by Blake [36] and Sunada [39] are the worst fit to the curve obtained.

3.5.2. Tests with Rolled Aggregates

In granular porous media composed of rolling and crushed aggregates, unlike smooth spheres, the geometrical properties can vary considerably, which makes it difficult to ensure that the experimental data used can be adequately grouped into porous media with similar geometrical properties. However, we have grouped the data by authors on the assumption that they obtained the materials from the same geological origin, since in this case there should be less dispersion in the parameters defining the geometry of the porous media.

In the following, we will obtain the [$D$, α*] diagrams for different geometries of the porous media composed of aggregates from different origins.

Figure 8 shows the [$D$, α*] diagram with the experimental data (N = 28) within the range 0.50 mm < $D$ < 60.00 mm. The theoretical curve defined by Equation (38) for smooth spheres is represented by a yellow dotted line. An increase in coefficient α* is observed with the representative particle size $D$. Over half of the experimental data could be grouped in the size range 0.50 mm < $D$< 30.00 mm, obtaining a fit given by Equation (40).

$${\alpha }^{*}=183.89·e^{0.523D}$$

(40)

(R² = 0.950 N = 15)

The data corresponding to M-B Salahi et al. [23] with a smaller interval could be fitted by Equation (41).

$${\alpha }^{*}=158.1·e^{0.1246D}$$

(41)

(R² = 0.9982 N = 6)

Both Equations (40) and (41) present a greater slope in the variation of the coefficient α* than Equation (38), with corresponding smooth spheres.

The values of coefficient α* corresponding to Arbhabhirama and Dinoy [31] for $D$ = 12.00 mm are above the curve given by Equation (40). The α* values for $D$ = 2.29 mm and $D$ = 5.97 mm corresponding to Dudgeon are much higher than the other values because of two factors: the difference in particle angularity and shape compared to the rest of the porous media ($D$> 6.00 mm) and the high uniformity coefficient values ($C_u=3.28\mathrm{m}\mathrm{m}$ and $C_u=3.25\mathrm{m}\mathrm{m}$, respectively). The α* values corresponding to Dudgeon [28] for sizes $D$ = 54.86 mm and $D$ = 109.73 mm (the latter not shown in Figure 8) are below Equation (37) for smooth spheres. This may be due to the need to have a higher density of points in the low gradient region to obtain a good fit of parameter $r$ of Equation (1) and, therefore, according to Equation (22), of the linear dimensionless coefficient α^*.

In the case of the dimensionless coefficient β^*, Figure 9 shows the experimental data (N = 25) obtained for different geometries of the porous media in the interval 0.50 mm < $D$< 110.00 mm.

It is striking that the dispersion that appears in the results obtained with the experimental data provided by M-B Salahi et al. [23], of which three are aligned with the Dudgeon data ($D$ = 2.10 mm; $D$ = 10.35 mm $D$ = 17.78 mm), while two are of the intermediate values ($D$ = 4.17 mm; $D$ = 6.53 mm), are considerably separated from this alignment. This greater dispersion in the geometries could be due, among other factors, to the geological origin of the materials tested.

The rest of the experimental data could be grouped together to obtain a fit given by Equation (42):

$${\beta }^{*}=2.5966·D^{-0.258}$$

(42)

(R² = 0.9464 N = 10)

Equation (42) includes the experimental data of Martins [11] for rolled aggregates (see Table 1). This equation is close to the theoretical curve for smooth spheres, Equation (38), which is represented by a yellow dotted line, this result is consistent since the value proposed by Martins for the dimensionless coefficient $F$ is $F$ = 1.05 (López et al. [3]).

Data from Ferdos et al. [33] for $D$ = 130.00 mm and $D$ = 200.00 mm have been obtained in the fully developed turbulent regime with a Reynolds number $R_p$ interval of 3.648 < $R_p$ < 187.448. A decrease in the unified dimensionless coefficient β* with representative particle size is observed. However, the values of the quadratic dimensionless coefficient β* are even higher than the data obtained by Dudgeon [28].

3.5.3. Tests with Crushed Aggregates

Figure 10 shows the data (N = 20) corresponding to crushed aggregate in the interval 1.50 mm < $D$< 40.00 mm.

It has been possible to pool the experimental data corresponding to Dudgeon [28], Arbhabhirama and Dinoy [31] and M-B Salahi et al. [23], obtaining a fit given by Equation (43):

$${\alpha }^{*}=285.683·e^{-0.0426·D}$$

(43)

(R² = 0.890 N = 11)

The data corresponding to $D$ = 16.61 (M-B Salahi et al. [23]) give a higher value of the dimensionless unified coefficient α*.

The experimental data obtained by Arbhabhirama and Dinoy [31] for angular gravels are the ones that give a much higher value of the unified dimensionless coefficient ${\alpha }^{*}$.

Finally, Figure 11 represents experimental data grouped into three different geometries (Equations (44)–(46)):

(a)

Dudgeon’s data [28]:

$${\beta }^{*}=8.7408·D^{-0.252}$$

(44)

(R² = 0.7615; N = 5)

(b)

Arbhabhirama and Dinoy [31] and M-B Salahi et al.’s data [23]:

$${\beta }^{*}=6.7305·D^{-0.245}$$

(45)

(R² = 0.8111; N = 9)

(c)

Martins’ data [11]:

$${\beta }^{*}=5.0693·D^{-0.274}$$

(46)

(R² = 0.9429; N = 5)

Figure 11. Relationship between ${\beta }^{*}$ and $D$ for angular particles; 3.00 mm < $D$ < 130.00 mm. Data from Dudgeon [28]; Arbhabhirama and Dinoy [31]; Martins [11] and Salahi et al. [23].

Figure 11. Relationship between ${\beta }^{*}$ and $D$ for angular particles; 3.00 mm < $D$ < 130.00 mm. Data from Dudgeon [28]; Arbhabhirama and Dinoy [31]; Martins [11] and Salahi et al. [23].

We have not considered the experimental data, $D$ = 37.19 mm, from Dudgeon because of the high dispersion it generates (${\beta }^{*}$ = 9.62).

In the case of Martins [11], the data that produce the greatest dispersion are those corresponding to $D$ = 22.00 mm and $D$ = 44.00 mm with values closer to those obtained for smooth spheres (Equation (38)).

Figure 11 also shows the data corresponding to the tests carried out by Ferdos et al. [33] with properly selected uniform granular materials composed of crushed aggregates with sizes $D$ = 130.00 mm and $D$ = 200.00 mm. As in the case of the rolled aggregates, the values of the unified quadratic dimensionless coefficient ${\beta }^{*}$ are once again higher than the other values obtained, which seems to indicate that there is an additional increase in energy loss in these granular porous media of large sizes and subjected to strong gradients. In these tests, the pore Reynolds number range was 3.250 < $R_p$ < 147.484.

4. Experimental Research

To properly verify the USEC represented by Equations (19), (26), and (29), we have proceeded to:

(a)

Obtain crushed aggregates from the same geological origin (limestone quarry in this case) with the intention of reducing the dispersion in the geometric properties of the selected porous media.

(b)

Select four sizes relatively close to each other ($D$ = 1.00 mm; $D$ = 2.00 mm; $D$ = 3.50 mm; $D$ = 4.00 mm) in order to analyze the relationship of the unified dimensionless coefficients ${\alpha }^{*}$ and ${\beta }^{*}$ with the representative particle size $D$.

(c)

Test at least 10 different pressure gradients for each porous material tested in order to obtain a good fit of the $r$ and $s$ parameters from quadratic Equation (1).

d)

Represent in a [$R_p,f_p$] diagram the results obtained to verify that the experimental data corresponding to each porous material tested fall on the theoretical curve defined by Equation (26) for the whole range of non-Darcy flows tested.

(e)

Apply Equations (29), (32), and (33) to obtain the unified dimensionless coefficients ${\alpha }^{*}$ and ${\beta }^{*}$ for each of the four porous media.

(f)

Represent in the [$D,{\alpha }^{*}$] and [$D,{\beta }^{*}$] diagrams the obtained values of the ${\alpha }^{*}$ and ${\beta }^{*}$ unified dimensionless coefficients for the four granular materials tested with the finality of observing their evolution with the representative size of particles $D$.

4.1. Description of Installation

Figure 12 shows a schematic of the installation consisting of three cylindrical pipes with an inner diameter $D_x$ of 388 mm and a length $L_x$ of 2 m. With it and given that the maximum uniform materials tested have a representative size $D$ of 45 mm, the ratio $D_x/D$ reached a value of 8.6:1. On the other hand, the ratio between the length of the permeameter $L_x$ and its diameter $D_x$ was 5.15:1. The sample was placed to complete fill the intermediate stainless-steel tube. The water, coming from a lower reservoir, is pumped through a perforated conduit that ensures that it reaches the sample without turbulence.

Seven digital pressure readings are available for the average hydraulic pressure: one in the water inlet zone, five along the sample tube, and another one in the water outlet zone, as shown in Figure 13.

The first in-sample pressure reading is at 47.3 cm from the origin of the distance; the last in-sample pressure reading is at 204.3 cm.

To ensure the reliability of the measurements, the pressure recorded is the average of the pressures at three equidistant points of the cross-section where the measurement is taken (see Figure 14).

An overview of the elements that make up the installation is shown in Figure 15 and Figure 16.

4.2. Test Procedure

The tests were carried out at flow rates $Q$ between 1 ls⁻¹ and 10 ls⁻¹. The measurements were taken after steady flow was reached. The experimental results were represented in a [x, y] diagram where the x-axis represents the distance from the “zero” point of the different pressure taps measured in cm, and the y-axis represents the water pressure head also measured in cm (see Figure 17, Figure 18, Figure 19 and Figure 20 (origin of distances in Figure 13)). The hydraulic gradient $i$ is determined by the slope of the regression line duly adjusted for each flow rate $Q$ (see Figure 17, Figure 18, Figure 19 and Figure 20). In some tests, it was not possible to obtain results for the whole range of flow rates.

4.3. Materials

The material used in the tests came from crushing in the same limestone quarry in order to reduce the dispersion in the geometry of these materials with the same geological origin.

• Representative particle size D

The representative particle size was used as the equivalent diameter ($D_e$), which corresponds to the diameter of a sphere having the same volume as the particle. Since the materials came from the same quarry and were all the same type of limestone, we proceed to calculate the specific weight of the material δ by weighing each particle with sufficient accuracy to calculate the volume of the material and, from this, the diameter of the equivalent sphere $D_e$ by means of Equation (47):

$$V_e=\frac{G}{\delta }$$

(47)

where$V_e$ = volume of the equivalent sphere (cm³),$G=$ weight of the particle, and$\delta$ = specific gravity of limestone rock.

The volume of the equivalent sphere $V_e$ is given by Equation (48):

$$V_e=\frac{4}{3}·\pi ·{\left(\frac{D_e}{2}\right)}^3$$

(48)

This way, we get the equivalent diameter of each particle in Equation (49):

$$D_e=\sqrt[3]{6·\frac{V_e}{\pi }}$$

(49)

We finally obtain Equation (50):

$$D_e={\delta }^3·\sqrt[3]{\frac{6·G_{}}{\pi }}$$

(50)

From the different values of the particle weights obtained, the 50% percentile of the equivalent diameter $D_e$ was easily calculated.

Four porous media with uniform particle size and representative diameters $D_e$ of 10.00 mm, 20.00 mm, 35.00 mm, and 45.00 mm have been selected.

In order to obtain experimentally the value of the porosity $n$ in each of the test materials, a tank with a fixed and determined volume ($V_d$), uncovered at the top, was used. First, the tank was filled with the granulometric material, and then, the volume of water entering the tank, which is the total volume of the voids ($V_h$), was measured.

The porosity $n$ was calculated by the expression (Equation (51)):

$$n=\left(\frac{V_h}{V_d}\right)·100$$

(51)

Porosities 40.74%, 39.12%, 41.33%, and 43.40% were obtained for sizes 10.00 mm, 20.00 mm, 35.00 mm, and 45.00 mm, respectively.

4.4. Tests Results and Discussion

Pressure loss graphs based on the [x, y] diagram for each of the four granular porous media tested are shown in Figure 17, Figure 18, Figure 19 and Figure 20. In each of the graphs, a good correlation is observed, except for the lowest flow rates ($Q$ = 2 ls⁻¹; $Q$ = 31 s⁻¹; $Q$ = 4 ls⁻¹) corresponding to the largest granular porous media, $D$ = 45.00 mm (see

Table 2. Results obtained for the hydraulic gradients $i$ from the slopes of the regression lines obtained from the [x, y] diagram (Figure 17, Figure 18, Figure 19 and Figure 20).

Q (ls−1)	$\mathit{V}$ (ms−1)	$$\begin{gathered} {\mathit{D}}_{\mathit{e}}=10\mathbf{mm} \\ \mathit{n}=40.74\mathit{\%} \end{gathered}$$	$$\begin{gathered} {\mathit{D}}_{\mathit{e}}=20\mathbf{mm} \\ \mathit{n}=39.12\mathit{\%} \end{gathered}$$	$$\begin{gathered} {\mathit{D}}_{\mathit{e}}=35\mathbf{mm} \\ \mathit{n}=41.33\mathit{\%} \end{gathered}$$	$$\begin{gathered} {\mathit{D}}_{\mathit{e}}=45\mathbf{mm} \\ \mathit{n}=43.40\mathit{\%} \end{gathered}$$
		Hydraulic Gradient
1	0.0085	0.0200	(4)	0.0019	(4)
2	0.0169	0.0627	0.0301	0.0107	0.0013 (1)
3	0.0254	0.1299	0.0614	0.0227	0.0094 (2)
4	0.0338	0.2142	0.1038	0.0395	0.0180 (3)
5	0.0423	0.3263	0.1610	0.0615	0.0312
6	0.0507	0.4428	0.2182	0.0862	0.0459
7	0.0592	0.5921	0.2800	0.1141	0.0613
8	0.0677	(4)	0.3758	0.1487	0.0767
9	0.0761	(4)	0.4659	0.1880	0.1007
10	0.0846	(4)	0.5659	0.2302	0.1260

Note: ⁽¹⁾ R² = 0.0248. ⁽²⁾ R² = 0.6347. ⁽³⁾ R² = 0.8182. ⁽⁴⁾ No results were obtained.

).

Table 2 shows the results obtained for the mean seepage velocity $V$ and the hydraulic gradient $i$ for each of the materials tested, which correspond to the slopes of the regression lines shown in Figure 17, Figure 18, Figure 19 and Figure 20. High correlation coefficients are observed in all of them except the three lowest gradients $i$ corresponding to a size of 45.00 mm, so they have not been considered for the analysis.

• Representation of the results in a [R_p, f_p] diagram

In order to verify that these experimental results fit the general unified Equation (26) in a [$R_p,f_p$] diagram form, we have plotted the data that have been calculated by applying Equations (24) and (25) in Figure 21. The range of the Reynolds number $R_p$ obtained in the tests was 76.12 <$R_p$< 3586.26.

As we can see in Figure 21, the experimental data corresponding to the granular porous materials $D$ = 10.00 mm, $D$ = 20.00 mm, and $D$ = 35.00 mm conform to a smooth curve in the [$R_p,f_p$] diagram. For size $D$ = 45.00 mm and after discarding the tests with low correlation coefficients (see Table 2), it is observed that all values (N = 6) are in the fully developed turbulent regime. Figure 21 shows that Equation (26) in a [$R_p,f_p$] diagram form and consequently Equation (29) in linearized form can apply to all the non-Darcy flows in which the experiments were carried out (76.12 <$R_p$< 3586.26).

On the other hand, all the points (N = 32) fall on a narrow band, which seems to indicate that there is a similarity in the geometries of the four porous media as a consequence of coming from the same geological origin.

Figure 22 shows the same values represented in a [$R_p,{\lambda }_p$] diagram.

• Obtaining coefficients α ^* and β ^*

Figure 23, Figure 24, Figure 25 and Figure 26 show the experimental data [$V,\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$V$}\right.$] for each of the four granular porous media tested, where a good fit is observed in all of them. With the values $r$ and $s$ I Forchheimer’s equation and applying Equations (22) and (23), we obtain the values of the unified dimensionless coefficients ${\alpha }^{*}$ and ${\beta }^{*}$ that are represented in Figure 27 and Figure 28.

• Unified linear dimensionless coefficient α^*. [D,α^*] diagram

Figure 27 shows the experimental data represented in the [$D$,${\alpha }^{*}$] diagram, where we can observe that:

(a)

There is an increase in the unified dimensionless coefficient ${\alpha }^{*}$ when going from the porous media $D$ = 10.00 mm to $D$ = 20.00 mm.

(b)

For the porous media $D$ = 45.00 mm, the reduced value of ${\alpha }^{*}$ = 9.58 indicates that we are in the fully developed turbulent regime.

(c)

The value of the coefficient ${\alpha }^{*}$ = 304.05 should be a matter for discussion. A possible explanation lies in the fact that in 6 of the 9 points the quadratic component of Equation (1) represents more than 90% of the total energy loss, so these 6 points are close to the fully developed turbulent flow, and consequently, the evolution of the dimensionless coefficient ${\alpha }^{*}$ is decreasing as we approach the fully developed turbulent flow.

We can therefore conclude that no clear conclusions can be drawn regarding the unified dimensionless coefficient ${\alpha }^{*}$.

However, as shown in Section 3.5 for non-Darcy flow regimes not close to the fully developed turbulent regime, the linear dimensionless coefficient ${\alpha }^{*}$ varies with the general expression:

$${\alpha }^{*}=A·e^p$$

(52)

where parameters $A$ and $p$ are related to the geometry of the granular porous media and shall be adjusted for each porous media characterized by a given geometry.

• Unified quadratic dimensionless coefficient β^*. [D, β^*] diagram

Figure 28 shows the data for the quadratic dimensionless coefficient ${\beta }^{*}$. In this case, a good fit was obtained as given by Equation (53):

$${\beta }^{*}=3.0653·D^{-0.258}$$

(53)

(R² = 0.9764)

This equation is of the same type as those obtained in Section 3.5, which shows that the relation of the dimensionless coefficient ${\beta }^{*}$ to the representative particle size $D$is given by the general expression of the potential type:

$${\beta }^{*}=B·D^q$$

(54)

where parameters $\beta$ and q are related to the geometry of the granular porous media and shall be adjusted for each porous media characterized by a given geometry.

5. Conclusions

In this research work, a general unified equation for seepage in porous media has been developed which, based on the adoption of the mean hydraulic diameter $D_h$ as a characteristic length $L_c,$ is applicable to coarse granular media with uniform granulometry for non-Darcy flows and can be represented in a quadratic form (Equation (18)), in a [$R_p,f_p]$ diagram form (Equation (26)), and in a linearized form [$R_p,{\lambda }_p]$ (Equation (28)).

The linearized equation [$R_p,{\lambda }_p]$ (Equation (28)) allows us to check whether, for a given granular porous media, there is a smooth curve in a [$R_p,f_p]$ diagram covering the different non-Darcy flow regimes: nonlinear laminar, turbulent transition, and fully developed turbulent.

Based on experimental data from different authors (Appendix A), it has been verified that the unified dimensionless coefficients linear ${\alpha }^{*}$ and quadratic ${\beta }^{*}$, which are considered by the general unified equation, are a function of the geometry of the porous material through parameter $A$ and exponent $p$ for the linear unified dimensionless coefficient $\alpha$* and parameter $B$ and exponent $q$ for the quadratic unified dimensionless coefficient $\beta$*; and of the representative diameter of the particles $D$ through Equations (52) and (54). The linear $r$ and quadratic $s$ parameters of the Forchheimer equation (Equation (1)) are given by the following expressions:

$$r=\frac{v}{g}·A·e^{p·D}·\frac{{(1-n)}^2}{n^3}·\frac{1}{D}$$

$$s=\frac{1}{g}·B·D^q·\frac{(1-n)}{n^3}·\frac{1}{D^2}$$

where $p$ and $q$ are exponents.

Granular porous media with similar geometrical properties will be defined by parameters $A,$ $B,$ $p$, and $q$. For the case of spherical particles and size range 0.50 mm < $D$ < 30.00 mm, a good fit has been obtained with the values $A$ = 136.08, $p$ = 00371 (R² = 0.9689), $B$ = 2.087, and $q$ = 0.159 (R² = 0.8698). For the rest of the porous media, rolled aggregates and crushed aggregates, parameters $A$ and$B$ have varied according to their geometric properties in accordance with the results obtained in Section 3.5.2 and Section 3.5.3, respectively.

An application of the general unified equation has been carried out by means of tests at CEDEX on porous media duly selected in terms of size, uniformity, and origin from the same geological origin (limestone quarry) under the following principles:

(a)

Verification of the existence of smooth curves in the [$R_p,f_p]$ diagram for all the sizes tested, adjusting to the general unified Equation (26).

(b)

Application of the linearized unified general Equation (29) to obtain the dimensionless linear ${\alpha }^{*}$ and quadratic ${\beta }^{*}$ coefficients.

(c)

Verification of the asymptote for the fully developed turbulent flow reached with the largest size, D = 45.00 mm

(d)

No clear conclusions have been drawn for the linear unified dimensionless coefficient ${\alpha }^{*}$.

(e)

For the quadratic unified dimensionless coefficient ${\beta }^{*}$, a good fit is observed (R² = 0.9764) with values $B$ = 3.0653 and $q$ = 0.258.

Further research work is needed to analyze the influence of the geological origin on the geometrical properties of both parameters $a$ and $b$ and exponents $p$ and $q$, as well as to analyze the evolution of the linear unified coefficient ${\alpha }^{*}$ as we approach the fully developed turbulent regime.

Equation USEC can be applied to various fields in science and engineering where coarse porous media are used, such as rockfill dams, breakwaters in coastal engineering, ripraps, embankments, etc.

Of particular importance is its application in the study of the saturation process of the downstream spall of a rockfill dam subjected to overtopping.