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The effect of pore geometry in constitutive hysteretic models for unsaturated water flow

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Abstract

Water flow in porous media is strongly controlled by the microscale structure of the pore space. Therefore, understanding the dynamics at pore scale is fundamental to better estimate and describe the hydraulic properties and phenomena associated to water flow which are observed in a macroscale such as field or laboratory experiments. Pore geometry plays a key role since its variations cause modifications in hydraulic behaviour at the macroscale. In this study, we develop a new analytical model which represents the pore space of a medium as a bundle of tortuous sinusoidal capillary tubes with periodic pore throats and a fractal pore-size distribution. This model is compared with a previous model of straight constrictive capillary tubes in order to analyze the effect of pore geometry on hydraulic properties under partially saturated conditions. The comparison of the constitutive models shows that macroscopic hydraulic properties, porosity and permeability, present the strongest differences due to changes in the pore geometry. Nonetheless, no variations are observed in the relative hydraulic properties, effective saturation and relative permeability. The new model has been tested with experimental data consisting on sets of porosity-permeability, water content-pressure head, conductivity-pressure head, and hysteretic water content-pressure values. In all cases, the model is able to satisfactorily reproduce the data. This new analytical model presents an improvement over the previous model since the smoother variation of the pore radii allows a more realistic representation of the porous medium.

Article highlights

  • New constitutive model to describe hydraulic properties of porous media.

  • Variations in pore geometry significantly influence porosity and permeability estimates.

  • The physically-based model has analytical closed-form expressions whose predictions are consistent with laboratory data.

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Abbreviations

REV:

Representative elementary volume

R :

Radius of a circular tube (m)

a :

Radial factor of the constrictivity

c :

Length factor of the constrictivity

\(\lambda\) :

Wavelength (m)

l :

Pore length (m)

r(x):

Pore radius variation along the longitudinal variable x (m)

M :

Integer number

\(V_p\) :

Pore volume (m\(^3\))

\(f_v\) :

Reduction factor in a pore volume and in the porosity

\(q_p\) :

Pore volumetric water flow (m\(^3\) s\(^{-1}\))

\(\varDelta h\) :

Pressure head drop (m)

\(f_k\) :

Reduction factor in a pore volumetric water flow and in the permeability

\(R_{REV}\) :

REV radius (m)

L :

REV length (m)

\(\tau\) :

Tortuosity

N(R):

Number of pores of radius equal or larger than R

D :

Fractal dimension

\(R_{min}\) :

Minimum pore radius (m)

\(R_{max}\) :

Maximum pore radius (m)

\(\phi\) :

Porosity

q :

Volumetric water flow through the REV (m\(^3\) s\(^{-1}\))

k :

Permeability (m\(^2\))

\(k_{rel}\) :

Relative permeability

h :

Pressure head (m)

\(T_s\) :

Surface tension (N m\(^{-1}\))

\(\beta\) :

Contact angle (degrees)

\(S_e\) :

Effective saturation

\(S_e^d\),\(S_e^w\) :

Main drying and wetting effective saturation, respectively

\(h_{min}\) :

Minimum pressure head (m)

\(h_{max}\) :

Maximum pressure head (m)

\(k_{rel}^d\),\(k_{rel}^w\) :

Main drying and wetting relative permeability, respectively

\(\rho\) :

Water density (kg m\(^{-3}\))

g :

Gravity (m s\(^{-2}\))

\(\mu\) :

Water dynamic viscosity (Pa s)

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Acknowledgements

The authors thank the Editor and two anonymous reviewers for their constructive comments and suggestions which helped to improve the original version of the manuscript. This research is partially supported by Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas (Argentina), Sorbonne Université and Centre National de la Recherche Scientifique (France).

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Authors and Affiliations

  1. Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata, Argentina

    Mariangeles Soldi & Luis Guarracino

  2. Sorbonné Université, CNRS, EPHE, UMR 7619 METIS, 75005, Paris, France

    Damien Jougnot

Authors
  1. Mariangeles Soldi
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  2. Luis Guarracino
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  3. Damien Jougnot
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Contributions

Conceptualization: MS, LG, DJ; Methodology: MS, LG, DJ; Writing-original draft preparation: MS, LG, DJ; Writing-review and editing: MS, LG, DJ; Investigation: MS, LG, DJ; Supervision: LG, DJ.

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Correspondence to Mariangeles Soldi.

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Soldi, M., Guarracino, L. & Jougnot, D. The effect of pore geometry in constitutive hysteretic models for unsaturated water flow. Environ Fluid Mech 22, 1283–1305 (2022). https://doi.org/10.1007/s10652-022-09891-0

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