# A Unified General Resistance Formula for Uniform Coarse Porous Media

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction and Objectives

## 2. State of the Art

#### 2.1. Review of the Formulation and Conceptual Scheme

**Characteristic Length L**_{c}

**Generalized dimensionless coefficients A**_{1}, linear and A_{2}, quadratic.

#### 2.2. Experimental Data from Previous Studies

## 3. The Unified Seepage Equation for Coarse Materials

#### 3.1. Methodology

- (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

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

**Flow regimes: nonlinear laminar, turbulent transition**

^{−4}< $i\text{}$< 8.00 (N = 14). In this case, the representative particle size $D$ is the value ${D}_{50}$ of the granulometric curve, 15.97 mm.

^{2}= 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.

**Fully developed turbulent flow**

^{2}= 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.

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

_{1}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).

^{*}, 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:

- (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 ${\epsilon}_{t}\text{}$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, ${D}_{t}$.

^{*}= $F\left(D\right)$, equivalent to curves ${f}_{t}=F\left({D}_{t}\right)$.

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

#### 3.5.1. Tests with Smooth Spheres

^{*}] diagram for the interval 0.50 mm < $D\text{}$< 30.00 mm, where an increase in coefficient α* with the representative particle size $D$ is observed by means of Equation (38):

^{2}= 0.9689 N = 19)

^{2}= 0.8698 N = 17)

#### 3.5.2. Tests with Rolled Aggregates

^{2}= 0.950 N = 15)

^{2}= 0.9982 N = 6)

^{*}.

^{*}, Figure 9 shows the experimental data (N = 25) obtained for different geometries of the porous media in the interval 0.50 mm < $D\text{}$< 110.00 mm.

^{2}= 0.9464 N = 10)

#### 3.5.3. Tests with Crushed Aggregates

^{2}= 0.890 N = 11)

- (a)
- (b)
- (c)

## 4. Experimental Research

- (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

#### 4.2. Test Procedure

^{−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

**Representative particle size D**

^{3}),$\text{}G=$ weight of the particle, and$\text{}\delta $ = specific gravity of limestone rock.

#### 4.4. Tests Results and Discussion

^{−1}; $Q$ = 31 s

^{−1}; $Q$ = 4 ls

^{−1}) corresponding to the largest granular porous media, $D$ = 45.00 mm (see Table 2).

**Representation of the results in a [R**_{p}, f_{p}] diagram

**Obtaining coefficients α**^{*}and β^{*}

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

- (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.

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

^{2}= 0.9764)

## 5. Conclusions

^{2}= 0.9689), $B$ = 2.087, and $q$ = 0.159 (R

^{2}= 0.8698). For the rest of the porous media, rolled aggregates and crushed aggregates, parameters $A$ and$\text{}B$ have varied according to their geometric properties in accordance with the results obtained in Section 3.5.2 and Section 3.5.3, respectively.

- (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 (R
^{2}= 0.9764) with values $B$ = 3.0653 and $q$ = 0.258.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\alpha $ | Coefficient of the exponential equation that depends on the characteristics of the porous media and of the fluid |

$A$ | USEC linear parameter depending on the geometry of the porous material |

${A}_{1}$ | Generalized dimensionless coefficient of the linear expression r |

${A}_{2}$ | Generalized dimensionless coefficient of the quadratic expression s |

$b$ | Exponent of the exponential equation function of the flow conditions |

$B$ | USEC quadratic parameter depending on the geometry of the porous material. |

${C}^{\prime}$ | Particle shape coefficient as set of R. D. Gupta |

$D$ | Representative size of the particles in uniform materials |

${D}_{a}$ | Average consecutive sieve aperture |

${D}_{e}$ | Equivalent diameter or diameter of a sphere with the same volume as the particle |

${D}_{h}$ | Hydraulic mean diameter |

${D}_{t}$ | Pipe diameter |

${D}_{x}$ | Permeameter diameter of an installation |

$F$ | Dimensionless coefficient that considers shape, angularity, and roughness of particles |

$G$ | Relative specific weight of solid particles |

$f$ | Generalized friction factor by Darcy–Weisbach |

$f\left(n\right)$ | Porosity function of hydraulic mean diameter |

${f}_{L}\left(n\right)$ | Porosity function of the linear term of the USEC |

${f}_{T}\left(n\right)$ | Porosity function of the quadratic term of the USEC |

${f}_{p}$ | Pore friction factor based on ${D}_{h}$ |

${f}_{r}$ | Pipe turbulent friction factor |

$g$ | Gravitational acceleration |

$i$ | Hydraulic gradient |

${K}_{0}$ | Intrinsic permeability of the porous media |

${L}_{c}$ | Generalized characteristic length of porous media |

${L}_{x}$ | Length of the permeameter |

$\eta $ | Angularity as a set of aggregates defined by R. D. Gupta |

$N$ | Number of tests performed |

$n$ | Porosity |

${n}^{\prime}$ | R. D. Gupta exponent for the formula that defines the angularity η |

$p$ | USEC linear exponent depending on the geometry of the granular porous media |

$q$ | USEC quadratic exponent depending on the geometry of the granular porous media |

$Q$ | Mean permeameter flow rate (l/s) |

$r$ | Linear coefficient of the Forchheimer equation of function of the characteristic of the porous media and fluid |

$R$ | Correlation coefficient |

${Re}_{}$ | Generalized Reynolds number |

${Re}_{t}$ | Pipe Reynolds number |

${R}_{h}$ | Hydraulic mean radius |

${R}_{p}$ | Pore Reynolds number based on ${D}_{h}$ |

$s$ | Quadratic coefficient of the Forchheimer equation of function of the characteristic of the porous media |

${S}_{e}$ | Average specific surface area of solid particles |

$t$ | Independent term of the USEC linearized equation |

$u$ | Linear term of the USEC linearized equation |

$v$ | Kinematic viscosity |

$V$ | Average seepage velocity |

${V}_{d}$ | Fixed tank volume used for porosity calculation. |

${V}_{e}$ | Volume of equivalent sphere of one particle |

${V}_{h}$ | Total void volume of the reservoir for the porosity calculation |

${V}_{p}$ | Pore velocity |

$x$ | Horizontal distance in cm to obtain pressure loss |

$y$ | Measured pressure in cm to obtain pressure loss |

$\alpha $* | USEC unified linear dimensionless coefficient |

$\beta $* | USEC unified quadratic dimensionless coefficient |

$\varepsilon $ | Particle surface roughness |

${\epsilon}_{t}$ | Pipe absolute roughness |

$\delta $ | Specific weight of limestone rock |

$\lambda $_{p} | Linearized USEC pore friction factor |

## Appendix A

^{−8}·m

^{2}·s

^{−1}< $v\text{}$< 1.39·10

^{−6}·m

^{2}·s

^{−1}. There are no data of this value for each of the 53 trials Table A1).

^{−5}> $i\text{}$< 13.4. The permeameter used had a diameter of ${D}_{x}$ 571 mm and a length of ${L}_{x}$ 1219.20 mm. The author represented the data obtained in the tests in a diagram $\left[Logi-LogV\right]$ to detect the slope changes that define the successive “post-linear regimes” or non-Darcy regime for each porous material. First, it obtained the critical speeds and gradients that correspond to the slope changes in the graphs. Second, for each of these lines, he obtained the values$\text{}a$ and $b$ of the exponential Equation (2). In non-Darcy regimes, the exponent interval $b$ was in the range 1.20 < $b\text{}$< 1.91. Kinematic viscosity $v$ data are available in all tests (Table A1).

_{50}= 130 mm and D

_{50}= 200 mm, respectively. The permeameter is composed of three bodies: the main unit containing the porous material has a diameter of ${D}_{x}\text{}$1000 mm and a length of ${L}_{x}$ 1990 mm, and the input units (${L}_{x}$ = 1500 mm) and output (${L}_{x}$ = 500 mm) decrease their section linearly to a final diameter of ${D}_{x}$ = 276 mm in both cases (Table A2 and Table A3).

Data By | Reference | Medium Type | Particle Size D(m) | Porosity n | ν (m^{2}:s^{−1}) | r (s^{2}.m^{−2}) | s (s^{2}.m^{−2}) | K_{o} (m^{2}) | α | β | F | α* | β* | Rp | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lindquist (1933) | S.Ergun & A.A.Orning [13] | Glass beads | 0.000227 | 0.343 | * | * | 0.00 | * | 1.90 | 3.00 | 1.00 | 136.80 | 2.25 | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000273 | 0.370 | * | * | * | 5.77 × 10^{−11} | * | * | 1.00 | 164.84 | * | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000322 | 0.341 | * | * | * | 6.37 × 10^{−11} | * | * | 1.00 | 148.62 | * | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000322 | 0.355 | * | * | * | 8.36 × 10^{−11} | * | * | 1.00 | 133.37 | * | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000322 | 0.370 | * | * | * | 9.01 × 10^{−11} | * | * | 1.00 | 146.86 | * | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000383 | 0.370 | * | * | * | 1.15 × 10^{−10} | * | * | 1.00 | 162.79 | * | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000458 | 0.370 | * | * | * | 1.87 × 10^{−10} | * | * | 1.00 | 143.16 | * | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Iron shot | 0.000472 | 0.375 | * | * | * | * | 2.00 | 2.50 | 1.00 | 144.00 | 1.88 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Iron shot | 0.000472 | 0.375 | * | * | * | * | 2.00 | 2.60 | 1.00 | 144.00 | 1.95 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Iron shot | 0.000472 | 0.375 | * | * | * | * | 1.80 | 2.50 | 1.00 | 129.60 | 1.88 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Lead shot | 0.000497 | 0.350 | * | * | * | * | 1.90 | 2.70 | 1.00 | 136.80 | 2.03 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Lead shot | 0.000497 | 0.350 | * | * | * | * | 1.80 | 3.10 | 1.00 | 129.60 | 2.33 | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000545 | 0.370 | * | * | * | 2.62 × 10^{−10} | * | * | 1.00 | 144.68 | * | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Lead shot | 0.000562 | 0.350 | * | * | * | * | 1.80 | 2.70 | 1.00 | 129.60 | 2.03 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Lead shot | 0.000562 | 0.352 | * | * | * | * | 1.80 | 3.10 | 1.00 | 129.60 | 2.33 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Glass beads | 0.000570 | 0.330 | * | * | * | * | 1.80 | 3.10 | 1.00 | 129.60 | 2.33 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Glass beads | 0.000570 | 0.330 | * | * | * | * | 1.90 | 2.80 | 1.00 | 136.80 | 2.10 | * | * |

Ward (1964) | Ward [30] | Glass beads | 0.000650 | 0.370 | * | * | * | 3.47 × 10^{−10} | * | * | 1.00 | 155.39 | * | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Iron shot | 0.000769 | 0.371 | * | * | * | * | 2.00 | 2.70 | 1.00 | 144.00 | 2.03 | ||

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Lead shot | 0.000798 | 0.364 | * | * | * | * | 1.90 | 3.30 | 1.00 | 136.80 | 2.48 | * | * |

Lindquist (1933) | S. Ergun & A. A. Orning [13] | Lead shot | 0.000798 | 0.364 | * | * | * | * | 1.90 | 3.30 | 1.00 | 136.80 | 2.48 | * | * |

Lindquist (1933) | S. Ergun & A.A. Orning [13] | Lead shot | 0.001004 | 0.366 | * | * | * | * | 1.80 | 2.90 | 1.00 | 129.60 | 2.18 | * | * |

Burke and Plummer (1928) | S. Ergun & A. A. Orning [13] | Lead shot | 0.001478 | 0.375 | * | * | * | * | 2.00 | 2.60 | 1.00 | 144.00 | 1.95 | * | * |

Sunada (1965) | Ahmed & Sunada [15] | Glass spheres | 0.003000 | 0.360 | 9.17 × 10^{−7} | 14.50 | 648.00 | 6.45 × 10^{−9} | * | * | 1.00 | 158.94 | 1.39 | * | * |

Burke and Plummer (1928) | S. Ergun & A. A. Orning [13] | Lead shot | 0.003077 | 0.383 | * | * | * | * | 2.30 | 2.50 | 1.00 | 165.60 | 1.88 | * | * |

Burke and Plummer (1928) | S. Ergun & A. A. Orning [13] | Lead shot | 0.003077 | 0.390 | * | * | * | * | 2.20 | 2.50 | 1.00 | 158.40 | 1.88 | * | * |

Blake (1922) | Ahmed & Sunada [15] | Glass beads | 0.003200 | 0.360 | 9.79 × 10^{−7} | 14.90 | 623.00 | 6.70 × 10^{−9} | * | * | 1.00 | 174.09 | 1.43 | * | * |

Crawford C.W et al. (1986) | Crawford C.W et al. [9] | Glass sheres | 0.005030 | 0.356 | * | * | * | * | * | * | 1.00 | 166.10 | 1.50 | * | * |

Burke & Plummer (1928) | S. Ergun & A. A. Orning [13] | Lead shot | 0.006270 | 0.421 | * | * | * | * | 2.80 | 2.00 | 1.00 | 201.60 | 1.50 | ||

Burke & Plummer (1928) | S. Ergun & A. A. Orning [13] | Lead shot | 0.006270 | 0.393 | * | * | * | * | 2.80 | 2.10 | 1.00 | 201.60 | 1.58 | ||

Kirkham (1966) | Ahmed & Sunada [15] | Marble | 0.016000 | 0.360 | 1.04 × 10^{−6} | 0.90 | 117.00 | 1.19 × 10^{−7} | * | * | 1.00 | 245.04 | 1.34 | ||

Dudgeon (1966) | McDonald I.F, et al. [44] | Marbles | 0.01597 | 0.369 | 1.28 × 10^{−6} | 1.1000 | 103.0000 | 1.19 × 10^{−7} | * | * | 1.00 | 232.01 | 1.26 | 3.21 | 3590.05 |

Dudgeon (1966) | McDonald I.F, et al. [44] | Marbles | 0.01597 | 0.415 | 1.21 × 10^{−6} | 0.5000 | 63.0000 | 2.46 × 10^{−7} | * | * | 1.00 | 232.01 | 1.26 | 5.33 | 2342.52 |

Dudgeon (1966) | McDonald I.F, et al. [44] | Marbles | 0.01597 | 0.372 | 1.15 × 10^{−6} | 0.7600 | 95.0000 | 1.55 × 10^{−7} | * | * | 1.00 | 232.01 | 1.26 | 29.16 | 2018.50 |

Dudgeon (1966) | McDonald I.F, et al. [44] | Marbles | 0.02487 | 0.369 | 1.27 × 10^{−6} | 0.5800 | 66.0000 | 2.24 × 10^{−7} | * | * | 1.00 | 348.91 | 1.28 | 6.29 | 6669.85 |

Dudgeon (1966) | McDonald I.F, et al. [44] | Marbles | 0.02896 | 0.385 | 1.11 × 10^{−6} | 0.3600 | 49.0000 | 3.16 × 10^{−7} | * | * | 1.00 | 400.87 | 1.29 | 33.47 | 9699.04 |

Data By | Reference | Medium Type | Particle Size D (m) | Porosity n | ν (m^{2}.s^{−1}) | r (s^{2}.m^{−1}) | s (s^{2}.m^{−2}) | K_{o} (m^{2}) | α* | β* | Rp | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Sedghi-Asl M et al. (2013) | Sedghi-Asl M et al. [32] | Rounded alluvial materials | 0.00283 | 0.3200 | 9.12 × 10^{−7} | 16.242 | 887.200 | 5.72 × 10^{−9} | 99.16 | 1.19 | 4.46 | 213.90 |

Sedghi-Asl M et al. (2013) | Sedghi-Asl M et al. [32] | Rounded alluvial materials | 0.00550 | 0.3300 | 8.80 × 10^{−7} | 5.906 | 508.830 | 1.52 × 10^{−8} | 159.51 | 1.47 | 28.04 | 651.29 |

Sedghi-Asl M et al. (2013) | Sedghi-Asl M et al. [32] | Rounded alluvial materials | 0.00870 | 0.3500 | 8.72 × 10^{−7} | 3.622 | 274.880 | 2.46 × 10^{−8} | 312.78 | 1.55 | 55.94 | 1435.90 |

Sedghi-Asl M et al. (2013) | Sedghi-Asl M et al. [32] | Rounded alluvial materials | 0.01560 | 0.3200 | 8.71 × 10^{−7} | 1.692 | 170.590 | 5.25 × 10^{−8} | 328.78 | 1.26 | 134.58 | 3493.76 |

Sedghi-Asl M et al. (2013) | Sedghi-Asl M et al. [32] | Rounded alluvial materials | 0.03110 | 0.3600 | 8.78 × 10^{−7} | 1.301 | 47.046 | 6.88 × 10^{−8} | 1601.53 | 1.05 | 435.61 | 11,837.12 |

Sedghi-Asl M et al. (2013) | Sedghi-Asl M et al. [32] | Rounded alluvial materials | 0.05680 | 0.4000 | 8.77 × 10^{−7} | 0.535 | 22.367 | 1.67*10^{−7} | 3433.40 | 1.33 | 2424.24 | 31,060.61 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Rounded materials | 0.00210 | 0.3750 | 1.32 × 10^{−6} | 42.782 | 3548.800 | 3.15 × 10^{−9} | 189.29 | 6.17 | 12.61 | 34.91 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Rounded materials | 0.00417 | 0.3790 | 1.33 × 10^{−6} | 15.678 | 854.850 | 8.62 × 10^{−9} | 284.72 | 3.07 | 73.20 | 162.01 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Rounded materials | 0.00653 | 0.3830 | 1.32 × 10^{−6} | 7.712 | 501.490 | 1.74 × 10^{−8} | 360.69 | 2.93 | 152.25 | 376.21 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Rounded materials | 0.01035 | 0.3864 | 1.32 × 10^{−6} | 4.924 | 440.700 | 2.73 × 10^{−8} | 602.16 | 4.21 | 286.44 | 703.25 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Rounded materials | 0.01213 | 0.3978 | 1.32 × 10^{−6} | 3.658 | 242.700 | 3.68 × 10^{−8} | 694.30 | 3.02 | 415.65 | 1081.89 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Rounded materials | 0.01778 | 0.4071 | 1.28 × 10^{−6} | 3.161 | 158.580 | 4.14 × 10^{−8} | 1465.41 | 3.15 | 729.85 | 1923.77 |

Dudgeon (1966) | McDonald I.F, et al. [44] | River Gravel | 0.00229 | 0.418 | 1.30 × 10^{−6} | 78.9100 | 2232.0000 | 1.68 × 10^{−9} | 672.79 | 6.29 | 0.07 | 111.74 |

Dudgeon (1966) | McDonald I.F, et al. [44] | River Gravel | 0.00579 | 0.392 | 1.32 × 10^{−6} | 19.0400 | 2174.0000 | 7.06 × 10^{−9} | 773.57 | 12.23 | 0.23 | 278.75 |

Dudgeon (1966) | McDonald I.F, et al. [44] | Rivel Gravel | 0.01585 | 0.367 | 1.30 × 10^{−6} | 1.8900 | 262.0000 | 7.02 × 10^{−8} | 441.68 | 3.18 | 4.27 | 2239.85 |

Dudgeon (1966) | McDonald I.F, et al. [44] | River Gravel | 0.02595 | 0.372 | 1.30 × 10^{−6} | 0.8200 | 145.0000 | 1.62 × 10^{−7} | 543.49 | 3.03 | 8.21 | 5016.46 |

Dudgeon (1966) | McDonald I.F, et al. [44] | River Gravel | 0.05486 | 0.369 | 1.27 × 10^{−6} | 0.2400 | 51.0000 | 5.41 × 10^{−7} | 702.40 | 2.19 | 113.55 | 12,868.54 |

Dudgeon (1966) | McDonald I.F, et al. [44] | River Gravel | 0.10973 | 0.406 | 1.27 × 10^{−6} | 0.0600 | 15.0000 | 2.16 × 10^{−6} | 1055.96 | 1.82 | 156.82 | 36,188.89 |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Sand | 0.001600 | 0.399 | 1.00 × 10^{−6} | 85.23 | 1750.86 | 1.196 × 10^{−6} | 376.43 | 2.90 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Sand | 0.001600 | 0.391 | 1.00 × 10^{−6} | 95.18 | 1819.07 | 1.071 × 10^{−6} | 385.25 | 2.80 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Gravel Round | 0.012000 | 0.373 | 1.00 × 10^{−6} | 6.04 | 207.10 | 1.6885 × 10^{−8} | 1125.78 | 2.02 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama &y Dinoy [31] | Gravel Round | 0.012000 | 0.357 | 1.00 × 10^{−6} | 5.63 | 187.82 | 1.8117× 10^{−8} | 874.70 | 1.56 | * | * |

Ward (1964) | Ward [30] | Sand | 0.000625 | 0.407 | * | * | * | 2.98 × 10^{−10} | 251.31 | * | * | * |

Ward (1964) | Ward [30] | Sand | 0.001260 | 0.400 | * | * | * | 1.36 × 10^{−9} | 207.53 | * | * | * |

Ward (1964) | Ward [30] | Gravel | 0.001800 | 0.410 | * | * | * | 2.98 × 10^{−9} | 215.27 | * | * | * |

Ward (1964) | Ward [30] | Gravel | 0.005040 | 0.389 | * | * | * | 1.69 × 10^{−8} | 237.00 | * | * | * |

Ward (1964) | Ward [30] | Gravel | 0.009210 | 0.417 | * | * | * | 5.26 × 10^{−8} | 344.04 | * | * | * |

Ward (1964) | Ward [30] | Gravel | 0.016100 | 0.422 | * | * | * | 1.80 × 10^{−7} | 323.94 | * | * | * |

Lindquist (1933) | Ahmed & Sunada [15] | Sand | 0.001050 | 0.380 | 9.14 × 10^{−7} | 116.40 | 2920.00 | 8.00 × 10^{−10} | 196.72 | 2.66 | ||

Lindquist (1933) | Ahmed & Sunada [15] | Sand | 0.004920 | 0.380 | 9.12 × 10^{−7} | 6.74 | 368.00 | 1.38 × 10^{−8} | 250.39 | 1.57 | ||

Martins (1990) | Martins [11] | Rounded materials | 0.02200 | 0.3622 | 1.15 × 10^{−6} | * | * | * | * | 1.29 | 752.00 | 2146.00 |

Martins (1990) | Martins [11] | Rounded materials | 0.02200 | 0.3683 | 1.15 × 10^{−6} | * | * | * | * | 1.29 | 751.00 | 2103.00 |

Martins (1990) | Martins [11] | Rounded materials | 0.04400 | 0.3618 | 1.15 × 10^{−6} | * | * | * | * | 0.90 | 2631.00 | 6970.00 |

Martins (1990) | Martins [11] | Rounded materials | 0.04400 | 0.3553 | 1.15 × 10^{−6} | * | * | * | * | 0.90 | 2583.00 | 7027.00 |

Martins (1990) | Martins [11] | Rounded materials | 0.08900 | 0.3823 | 1.15 × 10^{−6} | * | * | * | * | 0.83 | 8881.00 | 23,359.00 |

Martins (1990) | Martins [11] | Rounded materials | 0.08900 | 0.3884 | 1.15 × 10^{−6} | * | * | * | * | 0.83 | 10,765.00 | 25,430.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Cobblestone | 0.13000 | 0.4680 | 1.00 × 10^{−6} | 3.71 × 10^{−13} | 10.079 | 2.74 × 10^{+5} | 0.00 | 2.48 | 9515.00 | 114,176.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Cobblestone | 0.13000 | 0.4680 | 1.00 × 10^{−6} | 2.37 × 10^{−10} | 9.977 | 4.31 × 10^{+2} | 0.00 | 2.45 | 9515.00 | 114,176.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Cobblestone | 0.13000 | 0.4450 | 1.00 × 10^{−6} | 2.00 × 10^{−10} | 11.697 | 5.09 × 10^{+2} | 0.00 | 2.37 | 3648.00 | 115,829.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Cobblestone | 0.20000 | 0.5060 | 1.00 × 10^{−6} | 2.04 × 10^{−11} | 3.947 | 5.00 × 10^{+3} | 0.00 | 2.03 | 15,621.00 | 187,448.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Cobblestone | 0.20000 | 0.5060 | 1.00 × 10^{−6} | 1.49 × 10^{−11} | 4.201 | 6.84 × 10^{+3} | 0.00 | 2.16 | 15,621.00 | 187,448.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Cobblestone | 0.20000 | 0.4590 | 1.00 × 10^{−6} | 1.96 × 10^{−14} | 5.809 | 5.21 × 10^{−6} | 0.00 | 2.04 | 11,411.00 | 184,400.00 |

Data By | Reference | Medium Type | Particle Size D (m) | Porosity n | ν (m^{2}.s^{−1}) | r (s.2m^{−1}) | s (s^{2}.m^{−2}) | K_{o} (m^{2}) | α* | β* | Rp | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Crushed materials | 0.00177 | 0.4200 | 1.30 × 10^{−6} | 45.515 | 2978.800 | 2.91 × 10^{−9} | 236.98 | 6.61 | 7.66 | 25.29 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Crushed materials | 0.00355 | 0.4210 | 1.30 × 10^{−6} | 15.665 | 979.710 | 8.46 × 10^{−9} | 331.59 | 4.40 | 42.22 | 109.77 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Crushed materials | 0.00555 | 0.4225 | 1.30 × 10^{−6} | 4.550 | 514.300 | 2.91 × 10^{−8} | 239.14 | 3.66 | 130.06 | 278.60 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Crushed materials | 0.00869 | 0.4284 | 1.30 × 10^{−6} | 3.508 | 361.700 | 3.78 × 10^{−8} | 481.05 | 4.24 | 178.06 | 471.97 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Crushed materials | 0.01308 | 0.4392 | 1.30 × 10^{−6} | 1.066 | 174.420 | 1.24 × 10^{−7} | 370.88 | 3.38 | 460.45 | 1045.33 |

M-B Salahi et al. (2015) | M-B Salahi et al. [23] | Crushed materials | 0.01661 | 0.4692 | 1.30 × 10^{−6} | 1.789 | 114.040 | 7.41 × 10^{−8} | 1365.18 | 3.62 | 617.94 | 1544.84 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.00320 | 0.477 | 1.31 × 10^{−6} | 16.61 | 959.00 | 8.04 × 10^{−9} | 505.38 | 6.25 | 0.97 | 221.45 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.00640 | 0.458 | 1.31 × 10^{−6} | 7.79 | 573.00 | 1.71 × 10^{−8} | 781.44 | 6.38 | 1.27 | 565.36 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.01402 | 0.428 | 1.29 × 10^{−6} | 1.43 | 220.00 | 9.20 × 10^{−8} | 512.21 | 4.15 | 3.09 | 1776.00 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.01402 | 0.515 | 1.14 × 10^{−6} | 0.51 | 97.00 | 2.28 × 10^{−7} | 500.92 | 3.76 | 6.87 | 1374.02 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.01585 | 0.455 | 1.16 × 10^{−6} | 1.15 | 162.00 | 1.03 × 10^{−7} | 774.83 | 4.35 | 1.83 | 2734.03 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.02499 | 0.438 | 1.31 × 10^{−6} | 0.61 | 117.00 | 2.19 × 10^{−7} | 758.94 | 4.29 | 22.99 | 4368.32 |

Dudgeon (1966) | McDonald I.F. et al. [44] | Blue metal | 0.03719 | 0.483 | 1.29 × 10^{−6} | 0.33 | 121.00 | 3.98 × 10^{−7} | 1463.21 | 9.62 | 12.84 | 7100.78 |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Sand | 0.001600 | 0.399 | 1.00 × 10^{−6} | 85.23 | 1750.86 | 1.196 × 10^{−9} | 376.43 | 2.90 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Sand | 0.001600 | 0.391 | 1.00 × 10^{−6} | 95.18 | 1819.07 | 1.071 × 10^{−9} | 385.25 | 2.80 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Gravel Angular 1 | 0.006400 | 0.467 | 1.00 × 10^{−6} | 8.80 | 359.00 | 1.1581 × 10^{−8} | 1267.97 | 4.31 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Gravel Angular 1 | 0.006400 | 0.470 | 1.00 × 10^{−6} | 11.98 | 390.07 | 8.51 × 10^{−9} | 1778.98 | 4.80 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Gravel Angular 2 | 0.013000 | 0.461 | 1.00 × 10^{−6} | 2.96 | 176.91 | 3.4425 × 10^{−8} | 1655.53 | 4.10 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Gravel Angular 2 | 0.013000 | 0.479 | 1.00 × 10^{−6} | 2.71 | 130.93 | 3.7585 × 10^{−8} | 1820.55 | 3.52 | * | * |

Arbhabhirama & Dinoy (1973) | Arbhabhirama & Dinoy [31] | Gravel Angular 3 | 0.028300 | 0.465 | 1.00 × 10^{−6} | 1.16 | 55.93 | 8.8255 × 10^{−8} | 3187.75 | 2.92 | * | * |

Martins (1990) | Martins [11] | Angular materials | 0.01100 | 0.462 | 1.15 × 10^{−6} | * | * | * | * | 2.74 | 236.00 | 683.00 |

Martins (1990) | Martins [11] | Angular materials | 0.01100 | 0.460 | 1.15 × 10^{−6} | * | * | * | * | 2.74 | 243.00 | 692.00 |

Martins (1990) | Martins [11] | Angular materials | 0.01600 | 0.428 | 1.15 × 10^{−6} | * | * | * | * | 2.19 | 382.00 | 1123.00 |

Martins (1990) | Martins [11] | Angular materials | 0.01600 | 0.427 | 1.15 × 10^{−6} | * | * | * | * | 2.19 | 394.00 | 1101.00 |

Martins (1990) | Martins [11] | Angular materials | 0.02200 | 0.431 | 1.15 × 10^{−6} | * | * | * | * | 1.72 | 647.00 | 2010.00 |

Martins (1990) | Martins [11] | Angular materials | 0.02200 | 0.424 | 1.15 × 10^{−6} | * | * | * | * | 1.72 | 622.00 | 1945.00 |

Martins (1990) | Martins [11] | Angular materials | 0.03200 | 0.429 | 1.15 × 10^{−6} | * | * | * | * | 2.10 | 1150.00 | 3146.00 |

Martins (1990) | Martins [11] | Angular materials | 0.03200 | 0.436 | 1.15 × 10^{−6} | * | * | * | * | 2.10 | 1287.00 | 3405.00 |

Martins (1990) | Martins [11] | Angular materials | 0.04400 | 0.397 | 1.15 × 10^{−6} | * | * | * | * | 1.37 | 1907.00 | 5176.00 |

Martins (1990) | Martins [11] | Angular materials | 0.04400 | 0.395 | 1.15 × 10^{−6} | * | * | * | * | 1.37 | 1860.00 | 5188.00 |

Martins (1990) | Martins [11] | Angular materials | 0.06350 | 0.414 | 1.15 × 10^{−6} | * | * | * | * | 1.53 | 3925.00 | 10,509.00 |

Martins (1990) | Martins [11] | Angular materials | 0.06350 | 0.444 | 1.15 × 10^{−6} | * | * | * | * | 1.53 | 4623.00 | 12,210.00 |

Martins (1990) | Martins [11] | Angular materials | 0.08900 | 0.474 | 1.15 × 10^{−6} | * | * | * | * | 2.04 | 8127.00 | 19,793.00 |

Martins (1990) | Martins [11] | Angular materials | 0.08900 | 0.470 | 1.15 × 10^{−6} | * | * | * | * | 2.04 | 7695.00 | 19,820.00 |

Martins (1990) | Martins [11] | Angular materials | 0.12700 | 0.481 | 1.15 × 10^{−6} | * | * | * | * | 1.38 | 16,581.00 | 43,687.00 |

Martins (1990) | Martins [11] | Angular materials | 0.12700 | 0.475 | 1.15 × 10^{−6} | * | * | * | * | 1.38 | 15,658.00 | 41,238.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Crushed rock | 0.13000 | 0.5060 | 1.00 × 10^{−6} | 8.96 × 10^{−15} | 10.869 | 1.14 × 10^{+7} | 0.00 | 3.64 | 8273.00 | 99,278.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Crushed rock | 0.13000 | 0.5060 | 1.00 × 10^{−6} | 2.08 × 10^{−11} | 10.546 | 4.91 × 10^{+3} | 0.00 | 3.53 | 8273.00 | 99,278.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Crushed rock | 0.13000 | 0.4970 | 1.00 × 10^{−6} | 4.20 × 10^{−10} | 9.860 | 2.43 × 10^{+2} | 0.00 | 3.07 | 3250.00 | 105,626.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Crushed rock | 0.20000 | 0.4900 | 1.00 × 10^{−6} | 2.61 × 10^{−13} | 6.779 | 3.90 × 10^{+5} | 0.00 | 3.07 | 11,478.00 | 137,741.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Crushed rock | 0.20000 | 0.4900 | 1.00 × 10^{−6} | 1.09 × 10^{−11} | 6.651 | 9.40 × 10^{+3} | 0.00 | 3.01 | 11,478.00 | 137,741.00 |

Farzad Ferdos et al. (2015) | Farzad Ferdos et al. [33] | Crushed rock | 0.20000 | 0.4840 | 1.00 × 10^{−6} | 2.53 × 10^{−14} | 5.676 | 4.04 × 10^{+6} | 0.00 | 2.45 | 6807.00 | 147,484.00 |

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**Figure 1.**Relationship between ${\lambda}_{p}$ and ${R}_{p}$ for smooth spheres $D$ = 15.97 mm; n = 36.90%; $n$ = 41.50%; $n$ = 37.20%; 30 < ${R}_{p}\text{}$< 3590. $F$ = 1.0 Data from Dudgeon [28]. Source: authors.

**Figure 2.**Relationship between ${f}_{p\text{}}\mathrm{a}\mathrm{n}\mathrm{d}\text{}{R}_{p}$ for rolled aggregates $D$ = 22.00 mm; 730 < ${\text{}R}_{p}\text{}$< 2150. $F$ = 1.05 Data from Martins [11]. Source: authors.

**Figure 3.**Relationship between$\text{}{\lambda}_{p}$ and ${R}_{p}$ for rolled aggregates $D$ = 22.00 mm; $n$ = 36.22%; $n$ = 36.83%$.\text{}F$ = 1.05. Data from Martins. Source: authors.

**Figure 5.**Relationship between ${f}_{t}\text{}\mathrm{and}\text{}{D}_{t}$ for smooth curves depending on the absolute roughness of the pipe ${\epsilon}_{t}$. Source: authors.

**Figure 16.**Booster pump (

**left**), flowmeter (

**center**), and digital pressure readers of the seven piezometers shown in Figure 13 (

**right**).

**Figure 17.**Relationship between pressure y and distance x in cm for the $D$ = 10 mm material. Source: authors.

**Figure 18.**Relationship between pressure y and distance x in cm for the $D$ = 20 mm material. Source: authors.

**Figure 19.**Relationship between pressure y and distance x in cm for the $D$ = 35 mm material. Source: authors.

**Figure 20.**Relationship between pressure y and distance x in cm for the $D$ = 45 mm material. Source: authors.

**Figure 21.**Relationship between ${f}_{p}$ and ${R}_{p}$ for crushed aggregates; 1.00 mm < $D\text{}$< 45.00 mm from CEDEX tests. Source: authors.

**Figure 22.**Relationship between ${\lambda}_{p}$ and ${R}_{p}$. Equation $\left(29\right)$; 10.00 mm < $D\text{}$< 45.00 mm. CEDEX tests. F = 1.25. Source: authors.

**Figure 27.**Relationship between ${\alpha}^{*}$ and $D$. $D$ = 10.00 mm; $D$ = 20.00 mm; $D$ = 35.00 mm; $D$ = 45.00 mm. Source: authors.

**Figure 28.**Relationship between ${\beta}^{*}$ and $D$. $D$ = 10.00 mm; $D$ = 20.00 mm; $D$ = 35.00 mm; $D$ = 45.00 mm. Source: authors.

**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 | R_{h}/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 |

Q (ls^{−1}) | $\mathit{V}$ (ms^{−1}) | ${\mathit{D}}_{\mathit{e}}=10\text{}\mathbf{mm}\phantom{\rule{0ex}{0ex}}\mathit{n}=40.74\mathit{\%}$ | ${\mathit{D}}_{\mathit{e}}=20\text{}\mathbf{mm}\phantom{\rule{0ex}{0ex}}\mathit{n}=39.12\mathit{\%}$ | ${\mathit{D}}_{\mathit{e}}=35\text{}\mathbf{mm}\phantom{\rule{0ex}{0ex}}\mathit{n}=41.33\mathit{\%}$ | ${\mathit{D}}_{\mathit{e}}=45\text{}\mathbf{mm}\phantom{\rule{0ex}{0ex}}\mathit{n}=43.40\mathit{\%}$ |
---|---|---|---|---|---|

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 |

^{(1)}R

^{2}= 0.0248.

^{(2)}R

^{2}= 0.6347.

^{(3)}R

^{2}= 0.8182.

^{(4)}No results were obtained.

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## Share and Cite

**MDPI and ACS Style**

López, J.C.; Toledo, M.Á.; Moran, R.; Balairón, L.
A Unified General Resistance Formula for Uniform Coarse Porous Media. *Water* **2023**, *15*, 3578.
https://doi.org/10.3390/w15203578

**AMA Style**

López JC, Toledo MÁ, Moran R, Balairón L.
A Unified General Resistance Formula for Uniform Coarse Porous Media. *Water*. 2023; 15(20):3578.
https://doi.org/10.3390/w15203578

**Chicago/Turabian Style**

López, Juan Carlos, Miguel Ángel Toledo, Rafael Moran, and Luis Balairón.
2023. "A Unified General Resistance Formula for Uniform Coarse Porous Media" *Water* 15, no. 20: 3578.
https://doi.org/10.3390/w15203578