# The Relationship between the Darcy and Poiseuille Laws

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}], function of the characteristics of the porous media and of the liquid. The minus sign is needed as the differential term dh is negative; the dimensionless ratio dh/dl is known as hydraulic gradient, i, which controls the flow velocity by a linear law.

## 2. Tank-Reservoir Models

_{0}, at time t

_{0}= 0, and applying the principle of mass and volume conservation. Thus, water has been considered as an incompressible liquid; a compressible case [10] would require a different and complex approach.

_{0}.

_{P}[9]:

^{−1}] and expresses the drainage characteristics of the Poiseuille reservoir; the gravity acceleration, g, and the water density, ρ, appear from the substitution of the term ΔP of Equation (2) by the ρ⋅g⋅h.

_{D}(and diameter d

_{D}) and length L, is filled by sand with hydraulic conductivity K, the Darcy law can be used to simulate the drainage from the tank-reservoir. The following formula describes the discharge, Q, from the tank [9]:

^{−1}] and expresses the drainage characteristics of the Darcy reservoir.

## 3. Comparison between Darcy and Poiseuille Laws

^{−1}).

_{0}; the three different curves have the same recession coefficients (Equations (4) and (8)), and would provide straight lines in the semilogarithmic plot with the same slope, which is precisely the recession coefficient [9].

_{e}, computed for the flow in the tube of the Poiseuille reservoir by:

_{50}is the median grain size of the porous medium as a proxy for the characteristic pore length, and q is the specific discharge (q = Q

_{D}/A

_{D}). In the example of Figure 2, for a value of d

_{50}= 0.02 mm, compatible with the hydraulic conductivity K = 1 × 10

^{−4}m/s of the sand, values of the Reynolds number (Equation (11)) are R

_{e}<< 1.

_{D}and d

_{P}can be obtained:

_{i}is the intrinsic permeability of the material, k

_{i}= (μ·K)/(ρ·g) with dimension [L

^{2}].

_{D}, cannot be minor than diameter of Poiseuille tube, d

_{P}, as the first is filled by sand which reduces the actual section of the tube. Imposing as the limit condition the same diameter for d

_{D}and d

_{P}(d

_{D}≡ d

_{P}), Equations (12) and (13) reduce in:

_{P}, and the intrinsic permeability of a porous medium, k

_{i}.

_{P}and d

_{D}(Equation (12)) for different hydraulic conductivity, K. The grey field characterizes the zones with d

_{P}≥ d

_{D}, and does not have physical meaning, and it has been delimited by Equation (14).

## 4. Discussion

_{e}< 2000 [13]; above this value, the flow progressively changes in the turbulent type.

_{e}< 1 could be required to guaranty the laminar condition of the flow [12].

_{D}, and the area of the tube, A

_{D}, for a fixed hydraulic conductivity, K; that is:

_{P}, and the area of the tube, A

_{P}, is quadratic, that is:

_{Pn}or d

_{Dn}replace the larger tube d

_{P}or d

_{D}, for the Poiseuille or Darcy reservoir, respectively, the following relationships are obtained:

^{4}tubes with diameter d

_{Pn}= 0.0001 m discharge as one tube with diameter d

_{P}= 0.001 m. In the Darcy reservoir, n = 10

^{2}tubes with diameter d

_{Dn}= 0.0001 m discharge as one tube with diameter d

_{P}= 0.001 m.

_{D}, and the intrinsic permeability, k

_{i}. Thus, the reason of why in the Darcy reservoir the discharge is proportional to the area of the tube, A

_{D}(Equation (16)) lies in the fact that the intrinsic permeability, k

_{i}, is considered constant. Instead, the Equation (15) would provide the intimate relationship between the size of A

_{P}and k

_{i}.

_{Pn}; d

_{Dn}) rapidly increase as their diameter decreases. Of course, as a corollary, a single tube can drain the discharge of many smaller tubes. These aspects have an important role in the groundwater flow, both in porous and fractured/conduit systems, with particular emphasis in the karst systems, where a complex conduit network is “immersed” in a low permeability fractured limestone volume [15]. Here the karstification processes lead to a hierarchical conduit network into the aquifer [16], which cause the drainage to be converged to specific points: the karst springs. Therefore, the drainage of a karst spring appears “similar” with that of the venous system that leads the blood flow to the heart from the more peripheral areas. In these systems, a single conduit could require a huge number of smaller conduits to be supplyed.

_{P}, and the conduit number, n, following Equation (19); the total volume occupied by conduits, V, rapidly increases when d

_{P}decreases. That is, a large volume of water into the smallest conduits is needed to maintain the discharge of a few wide conduits. For the smallest values of d

_{P}, the conduits system could be replaced by a porous medium, where the hydraulic conductivity, K, has been estimated by the Equation (14).

_{e}, in relation to d

_{P}, for different rock volumes. The porosity, n

_{e}, increases as the conduits diameter decreases, and it decreases when the rock volume increases. To support the discharge of wide conduits by the smallest conduits or matrix, a large rock volume would be required; furthermore, in the smallest conduits or matrix, the water volume is huge compared to that flowing in the wider conduits. The matrix of karst aquifers occupies a zone characterized by the smallest conduits, approximately by d

_{P}< 0.0001 mm (or K < 2.3 × 10

^{−3}m/s) and range of porosity of 0.3 < n

_{e}< 0.001.

## 5. Conclusions

_{D}for the Darcy reservoir) and by intrinsic permeability, k

_{i}, of the medium. In the Poiseuille law, the flow depends directly on the square of the section area interested by the flow (${A}_{P}^{2}$ for the Poiseuille reservoir). Even if these characteristics were already known individually for each hydraulic law, their analytical relationship has been poorly described.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Darcy, H. Les Fontaines Publiques de la Ville de Dijon; Victor Dalmont: Paris, France, 1856. [Google Scholar]
- Todd, D.K.; Mays, L.W. Groundwater Hydrology, 3rd ed.; Wiley: Chichester, UK, 2005; 636p. [Google Scholar]
- Hubbet, M.K. Darcy’s law and the field equations of the flow of underground fluids. Hydrol. Sci. J.
**1957**, 2, 23–59. [Google Scholar] [CrossRef][Green Version] - Poiseuille, J.L.M. Ecoulement des Liquides: Societe Phi/omatique de Paris. In Extraits des Proces-Verbaux des Seances Pendant I’Annee 1838; Rene et Cie: Paris, France, 1838; pp. 1–3, 77–81. [Google Scholar]
- Poiseuille, J.L.M. Sur Ie mouvement des liquides de nature differente dans les tubes de tres petits diametres. Ann. Chim. Phys.
**1847**, 3, 76–110. [Google Scholar] - Sutera, S.; Skalak, R. The hystort of Poiseuille’s law. Annu. Rev. Fluid Mech.
**1993**, 25, 1–19. [Google Scholar] [CrossRef] - Stokes, G.G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Phil. Soc.
**1845**, 8, 287–341. [Google Scholar] - Vennard, J.K. Elementary Fluid Mechanics; John Wiley and Sons: London, UK, 1940; 372p. [Google Scholar]
- Fiorillo, F. Tank-reservoir drainage as a simulation of the recession limb of karst spring hydrographs. Hydrogeol. J.
**2011**, 19, 1009–1019. [Google Scholar] [CrossRef] - Shepherd, J.; Begeal, D.R. Transient Compressible Flow in Porous Materials; United States Department of Energy: Albuquerque, NM, USA, 1988.
- Maillet, E. Essais d’Hydraulique Souterraine et Fluviale [Underground and River Hydrology]; Hermann: Paris, France, 1905; 218p. [Google Scholar]
- Bear, J. Dynamics of Fluids in Porous Media; Dover Publications: New York, NY, USA, 1972. [Google Scholar]
- Colebrook, C. Turbulent Flow in Pipes, with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws. J. Inst. Civ. Eng.
**1939**, 11, 133–156. [Google Scholar] [CrossRef] - Ford, D.; Williams, P. Karst Hydrogeology and Geomorphology; Wiley: Chichester, UK, 2007; 562p. [Google Scholar]
- Kiraly, L. Karstification and groundwater flow. In Evolution of Karst: From Prekarst to Cessation; Gabrovsek, F., Ed.; Zalozba ZRC: Postojna-Ljubljana, Slovenia, 2002; pp. 155–190. [Google Scholar]
- Kaufmann, G. A model comparison of karst aquifer evolution for different matrix flow formulations. J. Hydrol.
**2003**, 283, 281–289. [Google Scholar] [CrossRef]

**Figure 1.**Cylindrical tank-reservoir, with a base A

_{1}. Water discharge occurs through a bottom tank-tube, with diameter d and length L, starting from an initial water height h

_{0}.

**Figure 2.**Discharge-time plot for different initial water level, h

_{0}, using the Darcy reservoir and the Poiseuille reservoir with the same recession coefficient (${\alpha}_{D}$ = ${\alpha}_{P}$ ). The Reynolds number, R

_{e}, refers to the flow into the tube of the Poiseuille reservoir.

**Figure 3.**Relationship between the Poiseuille and Darcy laws; d

_{P}, tube diameter of Poiseuille reservoir; d

_{D}, tube diameter of Darcy reservoir; K, hydraulic conductivity of sand filling the tube of Darcy reservoir. In the grey field (d

_{P}≥ d

_{D}) no-relationship exists.

**Figure 4.**Number of tubes, n, with the same diameter (d

_{Pn}; d

_{Dn}) required to replace a wider tube (d

_{P}; d

_{D}) for a Poiseuille (

**a**) and Darcy (

**b**) reservoir.

**Figure 5.**Relationship between conduit diameter, d

_{P}, and the porosity, n

_{e}, for different rock volumes (data from Table 1). Hydraulic conductivity, K, has been estimated up to a value of 1 m/s; the zone of karst aquifer matrix is also highlighted.

**Table 1.**Diameter of conduit, d

_{P}and relative number of conduits, n, from Equation (19). The volume of all conduits, V, computed by considering constant conduit length, L = 1 m. Hydraulic conductivity, K, and intrinsic permeability, k

_{i}, have been estimated by Equation

^{1}(14).

d_{P}m | n | V m ^{3} | K m/s | k_{i}cm ^{2} |
---|---|---|---|---|

1 | 1 | 7.85 × 10^{−1} | - | - |

10^{−1} | 10^{4} | 7.85 × 10^{1} | - | - |

10^{−2} | 10^{8} | 7.85 × 10^{3} | - | - |

10^{−3} | 10^{12} | 7.85 × 10^{5} | 2.3 × 10^{−1} | 3.13 × 10^{−4} |

10^{−4} | 10^{16} | 7.85 × 10^{7} | 2.3 × 10^{−3} | 3.13 × 10^{−6} |

10^{−5} | 10^{20} | 7.85 × 10^{9} | 2.3 × 10^{−5} | 3.13 × 10^{−8} |

^{1}Values of K > 1 m/s (and k

_{i}> 10

^{−3}cm

^{2}) are outside the natural range of porous medium.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fiorillo, F.; Esposito, L.; Leone, G.; Pagnozzi, M.
The Relationship between the Darcy and Poiseuille Laws. *Water* **2022**, *14*, 179.
https://doi.org/10.3390/w14020179

**AMA Style**

Fiorillo F, Esposito L, Leone G, Pagnozzi M.
The Relationship between the Darcy and Poiseuille Laws. *Water*. 2022; 14(2):179.
https://doi.org/10.3390/w14020179

**Chicago/Turabian Style**

Fiorillo, Francesco, Libera Esposito, Guido Leone, and Mauro Pagnozzi.
2022. "The Relationship between the Darcy and Poiseuille Laws" *Water* 14, no. 2: 179.
https://doi.org/10.3390/w14020179