# A Unified View of Nonlinear Resistance Formulas for Seepage Flow in Coarse Granular Media

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## Abstract

**:**

## 1. Introduction and Objectives

## 2. Review of Resistance Formulas in Nonlinear Porous Media

#### 2.1. Conceptual Approach

- (a)
- the representative size of the particle ($D$)
- (b)
- the square root of the intrinsic permeability (${K}_{0}$), as a macroscopic property of the porous medium. For the laminar regime, it is determined by Equation (6):$${K}_{0}=\frac{v}{g}\xb7\frac{1}{i}\xb7V$$
- (c)
- the hydraulic mean radius ${R}_{h}$ that was first defined by Taylor (1948) [2] and determined by Equation (7):$${R}_{h}=\frac{n}{{S}_{e}\xb7\left(1-n\right)}$$$${S}_{e}=\frac{SP}{VP}$$

_{d}= 10,000) in order to reach the fully developed turbulent flow.

#### 2.2. Resistance Formulas

#### 2.2.1. Resistance Formulas Based on the Representative Diameter of the Particles

_{e}through Equation (20).

_{2}, N

_{2}, CH

_{4}, and H

_{2}. As a result of this, he obtained the universal values of $\alpha $ = 2.08 and $\beta $ = 2.33 applicable to all porous media.

_{0}is the linear dimensionless coefficient of Engelund and ${\beta}_{0}$ is the quadratic dimensionless coefficient of Engelund. According to the author, “${\alpha}_{0}$ and ${\beta}_{0}$ are dimensionless numerical constants depending, for uniform soil, on the structure and the grain shape.”

_{50}for lack of other data in the bibliography. However, for conceptual reasons, we will continue working with the representative diameter of the particles $D$.

_{d}= 1 for smooth spheres, $f$

_{d}= 2 for river gravel, and $f$

_{d}= 4 for crushed aggregate (see Figure 3).

^{−4}.

_{e},$f$] that tend to an asymptotic value for fully developed turbulent regime (see Figure 1 and Figure 3).

#### 2.2.2. Resistance Formulas Based on Intrinsic Permeability

#### 2.2.3. Resistance Formulas Based on the Hydraulic Mean Radius

_{c}and the hydraulic mean radius ${R}_{h}$ defined by Equation (7), proposed the dimensionless groups Equations (37) and (38):

#### 2.2.4. On the Physical Parameters $r$ and $s$ of the Forchheimer Equation

## 3. Analysis of the Relationships among Parameters of the Different Formulas of Resistance

- (a)
- Among the characteristic lengths, R
_{h}, $\sqrt{{K}_{0}}$, and D; - (b)
- Among the Reynolds numbers ${R}_{p},{R}_{k}\mathrm{and}{R}_{d}$;
- (c)
- Among the different laminar dimensionless coefficients $\alpha $, ${\alpha}_{0}$, ${K}_{l}$, and $\alpha $′, and quadratic dimensionless coefficients $\beta $, ${\beta}_{0}$, ${K}_{t},\beta $′, and $C$.

#### 3.1. Equations with Characteristic Length Based on the Hydraulic Diameter

#### 3.2. Equations with Characteristic Length Based on the Intrinsic Permeability

#### 3.3. Equations with Characteristic Length Based on the Representative Size of Particles

#### 3.4. Relationships among Characteristic Lengths

_{h}and the intrinsic permeability ${K}_{0}$:

- (a)
- Round sand, F = 1.10;
- (b)
- Semi angular sand, F = 1.25;
- (c)
- Angular sand, F = 1.40.

- (a)
- Angular Particles $c$’ = 8.5 (F = 1.47).
- (b)
- Round particles $c$’ = 6.3 (F = 1.05).

#### 3.5. Relationships among Reynolds Numbers

#### 3.6. Relationships among the Laminar Dimensionless Coefficients $\alpha ,{\alpha}_{0},{K}_{l}and\alpha \u2019$

_{0}. If we equal the linear components of Engelund Equation (25) considering the representative diameter of the particle $D$ instead of ${D}_{e}$ and Equation (64), we obtain:

#### 3.7. Relationships among Turbulent Dimensionless Coefficients $\beta ,{\beta}_{0},{K}_{t},{\beta}^{\prime}andC$

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$a$ | Coefficient of the exponential equation that depends on the characteristics of the porous medium |

A_{1} | Generalised dimensionless coefficient of the linear expression r |

A_{2} | Generalised dimensionless coefficient of the quadratic expression |

${A}_{1}^{\u2019}$ | $\mathrm{Linear}\text{}\mathrm{dimensionless}\text{}\mathrm{coefficient}\text{}\mathrm{corresponding}\text{}\mathrm{to}\text{}{L}_{c}={D}_{h}$ |

${A}_{2}^{\u2019}$ | $\mathrm{Quadratic}\text{}\mathrm{dimensionless}\text{}\mathrm{coefficient}\text{}\mathrm{corresponding}\text{}\mathrm{to}\text{}{L}_{c}={D}_{h}$ |

${A}_{1}^{\u2033}$ | $\mathrm{Laminar}\text{}\mathrm{dimensionless}\text{}\mathrm{coefficient}\text{}\mathrm{corresponding}\text{}\mathrm{to}\text{}{L}_{c}=\sqrt{{K}_{0}}$ |

${A}_{2}^{\u2033}$ | $\mathrm{Quadratic}\text{}\mathrm{dimensionless}\text{}\mathrm{coefficient}\text{}\mathrm{corresponding}\text{}\mathrm{to}\text{}{L}_{c}=\sqrt{{K}_{0}}$ |

${A}_{1}^{\u2019\u2019\u2019}$ | $\mathrm{Laminar}\text{}\mathrm{dimensionless}\text{}\mathrm{coefficient}\text{}\mathrm{corresponding}\text{}\mathrm{to}\text{}{L}_{c}=D$ |

${A}_{2}^{\u2019\u2019\u2019}$ | $\mathrm{Quadratic}\text{}\mathrm{dimensionless}\text{}\mathrm{coefficient}\text{}\mathrm{corresponding}\text{}\mathrm{to}\text{}{L}_{c}=D$ |

c’ | Coefficient from Martins |

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

C | Quadratic dimensionless coefficient of Ward |

C_{u} | Coefficient of uniformity |

D | Representative size of the particles in uniform materials |

D_{50} | Sieve opening through which 50% of the material passes |

D_{a} | Average size of sieve openings |

D_{e} | Diameter equivalent or diameter of a sphere with the same volume as the particle |

D_{g} | Geometric mean between the two consecutive sieves |

D_{h} | Hydraulic mean diameter |

D_{m} | Particle mean diameter |

D_{p} | Effective diameter or diameter of a sphere with the same specific surface area as the particle |

D_{x} | Diameter of the permeameter |

F | Coefficient of Loudon which considers the shape and angularity of the particles |

$f$ | Generalised friction factor, by Darcy–Weisbach |

${f}_{L}^{\u2019}$ | Function of porosity, by Engelund |

${f}_{d}$ | Particle friction factor |

${f}_{E}$ | Friction factor of Ergun |

${f}_{k}$ | Friction factor of Ward |

${f}_{p}$ | Function of linear porosity |

${f}_{p}$ | Pore friction factor |

${f}_{T}$ | Function of quadratic porosity |

$g$ | Gravitational acceleration |

$i$ | Hydraulic gradient |

K_{0} | Intrinsic permeability of the porous medium |

K_{b} | Coefficient of Blake that considers the shape of the porous material and the symmetry of the packing |

${\left[{K}_{b}\right]}_{L}$ | Linear dimensionless coefficient, S. P. Burke and W. B. Plummer |

${\left[{K}_{b}\right]}_{T}$ | Quadratic dimensionless coefficient, S. P. Burke y W. B. Plummer |

K_{l} | Linear dimensionless coefficient, by Stephenson |

K_{t} | Quadratic dimensionless coefficient, by Stephenson |

L_{c} | Characteristic length |

M_{g} | Geometric mean of the size of the particles that constitute the porous medium |

𝑛 | Porosity |

r | Linear coefficient of the Forchheimer equation of function of the characteristics of the porous medium and fluid. |

R_{e} | Generalised Reynolds number |

R_{d} | Particle Reynolds number |

R_{E} | Reynolds number, by Ergun |

R_{h} | Hydraulic mean radius |

R_{k} | Reynolds number, by Ward |

R_{p} | Pore Reynolds number of Dh |

s | Quadratic coefficient of the Forchheimer equation of function of the characteristics of the porous medium. |

S | Surface area per volume unit of the packed porous medium |

S_{e} | Average specific surface area of solid particles |

S_{P} | Average surface area of the particles |

V | Average fluid velocity based on the transversal section |

$v$ | Kinematic viscosity |

V_{P} | Pore velocity |

${v}_{p}$ | Average particle volume |

α | Linear dimensionless coefficient of the expression r, by Sabri, Ergun, and A. A. Orning |

α’ | Linear dimensionless coefficient of pores |

α_{0} | Linear dimensionless coefficient r, by Engelund |

β | Quadratic dimensionless coefficient of the expression r, by Sabri, Ergun and A. A. Orning |

β’ | Quadratic dimensionless coefficient of pores |

β_{0} | Quadratic dimensionless coefficient of the expression s, by Engelund |

λ | Linearised generalised friction factor |

$\rho $ | Fluid density |

σs | Geometric standard desviation of the size distribution of the porous medium |

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**Figure 2.**Schematic diagram [${R}_{e},\lambda $] Equation (16). Adapted from Fand et al. (1987) [30].

${\mathit{L}}_{\mathit{c}}$ | ${\mathit{A}}_{\mathbf{1}}$ | ${\mathit{A}}_{\mathbf{2}}$ | ${\mathit{R}}_{\mathit{e}}$ | $\mathit{f}$ | ${\mathit{R}}_{\mathit{h}}$ | $\sqrt{{\mathit{K}}_{\mathbf{0}}}$ | $\mathit{D}$ |

D_{h} | 64α | β | $\frac{4\xb7{R}_{h}\xb7{V}_{p}}{v}$ | $8{R}_{h}\xb7g\xb7\frac{i}{{V}_{p}^{2}}$ | 4·R_{h} | $4\xb7\frac{\sqrt{2\alpha}}{n}\xb7\sqrt{{K}_{0}}$ | $4\xb7\frac{\sqrt{2\alpha}}{\sqrt{{K}_{l}}}\xb7D$ |

$\sqrt{{\mathit{K}}_{\mathbf{0}}}$ | 2n | 2·C·n^{2} | $\frac{\sqrt{{K}_{0}}\xb7{V}_{p}}{v}$ | $\sqrt{{K}_{0}}\xb72g\xb7\frac{i}{{V}_{p}^{2}}$ | $\frac{{n}^{0.5}}{\sqrt{2\alpha}}\xb7{R}_{h}$ | 1 | $\frac{{n}^{0.5}}{\sqrt{{K}_{l}}}\xb7D$ |

D | 2K_{l} | 2·K_{t} | $\frac{D\xb7{V}_{p}}{v}$ | $D\xb72g\xb7\frac{i}{{V}_{p}^{2}}$ | $\frac{\sqrt{{K}_{l}}}{\sqrt{2\alpha}}\xb7{R}_{h}$ | $\frac{\sqrt{{K}_{l}}}{{n}^{0.5}}\xb7\sqrt{{K}_{0}}$ | 1 |

${\mathit{L}}_{\mathit{c}}$ | ${\mathit{R}}_{\mathit{e}}$ | ${\mathit{R}}_{\mathit{p}}$ | ${\mathit{R}}_{\mathit{k}}$ | ${\mathit{R}}_{\mathit{d}}$ |

4R_{h} | R_{p} | 1 | $\frac{4\sqrt{2\alpha}}{{n}^{1.5}}\xb7{R}_{k}$ | $\frac{4\sqrt{2\alpha}}{\sqrt{{K}_{l}}}\xb7{R}_{d}$ |

$\sqrt{{\mathit{K}}_{\mathbf{0}}}$ | R_{k} | $\frac{{n}^{1,5}}{4\sqrt{2\alpha}}\xb7{R}_{p}$ | 1 | $\frac{{n}^{1,5}}{\sqrt{{K}_{l}}}\xb7{R}_{d}$ |

D | R_{d} | $\frac{\sqrt{{K}_{l}}}{4\sqrt{2\alpha}}\xb7{R}_{p}$ | $\frac{\sqrt{{K}_{l}}}{{n}^{1.5}}\xb7{R}_{k}$ | 1 |

${\mathit{L}}_{\mathit{c}}$ | Laminar | $\mathit{\alpha}$ | ${\mathit{K}}_{\mathit{l}}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ |

4R_{h} | α | 1 | $\frac{{n}^{2}}{72\xb7{F}^{2}\xb7{\left(1-n\right)}^{2}}\xb7{K}_{l}$ | $\frac{\left(1-n\right)\xb7n}{72\xb7{F}^{2}}\xb7{\alpha}_{0}$ |

$\sqrt{{\mathit{K}}_{\mathbf{0}}}$ | 1 | |||

D | K_{l} | $\frac{72\xb7{F}^{2}\xb7{\left(1-n\right)}^{2}}{{n}^{2}}\xb7\alpha $ | 1 | $\frac{{\left(1-n\right)}^{2}}{n}\xb7{\alpha}_{0}$ |

D | ${\alpha}_{0}$ | $\frac{72\xb7{F}^{2}}{\left(1-n\right)\xb7n}\xb7\alpha $ | $\frac{n}{{\left(1-n\right)}^{3}}\xb7{K}_{l}$ | 1 |

${\mathit{L}}_{\mathit{c}}$ | Turbulent | $\mathit{\beta}$ | $\mathit{C}$ | ${\mathit{K}}_{\mathit{t}}$ | ${\mathit{\beta}}_{\mathbf{0}}$ |

4R_{h} | β | 1 | $8\xb7\sqrt{2\alpha}\xb7{n}^{1.5}\xb7C$ | $\frac{8}{6}\xb7\frac{n}{F\left(1-n\right)}\xb7{K}_{t}$ | $\frac{8}{6}\xb7\frac{1}{F}\xb7{\beta}_{0}$ |

$\sqrt{{\mathit{K}}_{\mathbf{0}}}$ | C | $\frac{1}{8\xb7\sqrt{2\alpha}\xb7{n}^{1.5}}\xb7\beta $ | 1 | $\frac{1}{6\xb7\sqrt{2\alpha}\xb7F{n}^{0.5}\left(1-n\right)}\xb7\beta $ | $\frac{1}{6\xb7\sqrt{2\alpha}\xb7F\xb7{n}^{1.5}}\xb7{\beta}_{0}$ |

D | K_{t} | $\frac{6\xb7F\left(1-n\right)}{8n}\xb7\beta $ | $6\xb7\sqrt{2\alpha}\xb7F\xb7{n}^{0.5}\xb7\left(1-n\right)\xb7C$ | 1 | $\frac{\left(1-n\right)}{n}\xb7{\beta}_{0}$ |

D | ${\beta}_{0}$ | $\frac{6}{8}\xb7F\xb7\beta $ | $6\xb7\sqrt{2\alpha}\xb7F\xb7{n}^{1.5}\xb7C$ | $\frac{n}{\left(1-n\right)}\xb7{K}_{t}$ | 1 |

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**MDPI and ACS Style**

López, J.C.; Toledo, M.Á.; Moran, R.
A Unified View of Nonlinear Resistance Formulas for Seepage Flow in Coarse Granular Media. *Water* **2021**, *13*, 1967.
https://doi.org/10.3390/w13141967

**AMA Style**

López JC, Toledo MÁ, Moran R.
A Unified View of Nonlinear Resistance Formulas for Seepage Flow in Coarse Granular Media. *Water*. 2021; 13(14):1967.
https://doi.org/10.3390/w13141967

**Chicago/Turabian Style**

López, Juan Carlos, Miguel Ángel Toledo, and Rafael Moran.
2021. "A Unified View of Nonlinear Resistance Formulas for Seepage Flow in Coarse Granular Media" *Water* 13, no. 14: 1967.
https://doi.org/10.3390/w13141967