Transport in Porous Media

, Volume 92, Issue 1, pp 29–39 | Cite as

A Spatially Non-Local Model for Flow in Porous Media

  • Mihir SenEmail author
  • Eduardo Ramos


A general mathematical model of steady-state transport driven by spatially non-local driving potential differences is proposed. The porous medium is considered to be a network of short-, medium-, and long-range interstitial channels with impermeable walls and at a continuum of length scales, and the flow rate in each channel is assumed to be linear with respect to the pressure difference between its ends. The flow rate in the model is thus a functional of the non-local driving pressure differences. As special cases, the model reduces to familiar forms of transport equations that are commonly used. An important situation arises when the phenomenon is almost, but not quite, locally dependent. The one-dimensional form of the model discussed here can be extended to multiple dimensions, temporal non-locality, and to heat, mass, and momentum transfer.


Non-local Non-Darcy 

List of Symbols


f(x′, x)

Material property of locations x′ and x

f = {fji}

Discrete matrix version of f(x′, x)


Material constant


Length of material

\({{\mathcal L}}\)

Riemann–Liouville fractional derivative defined in Eq. 11a


Number of tubes


Number of discrete intervals




Flow rate

\({{\mathcal R}}\)

Weyl fractional derivative defined in Eq. 11b




Spatial coordinate

Greek symbols


Gamma function


Delta distribution


Derivative of δ


Spatial scale of non-locality


Nascent delta distribution


Derivative of \({\delta_\epsilon}\)

μ, ν

Size of regions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barabási A.-L.: Linked: The New Science of Networks. Perseus, Cambridge, MA (2002)Google Scholar
  2. Berkowitz B., Ewing R.P.: Percolation theory and network modeling applications in soil physics. Surv. Geophys. 19, 2372 (1998)CrossRefGoogle Scholar
  3. Berkowitz B., Cortis A., Dentz M., Scher H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003 (2006)CrossRefGoogle Scholar
  4. Darcy H.: Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris (1856)Google Scholar
  5. Di Paola M., Zingales M.: Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int. J. Solids Struct. 45(21), 5642–5659 (2008)CrossRefGoogle Scholar
  6. Di Paola M., Marino F., Zingales M.: A generalized model of elastic foundation based on long-range interactions: integral and fractional model. Int. J. Solids Struct. 46(17), 3124–3137 (2009)CrossRefGoogle Scholar
  7. Fitzgerald S.D., Woods A.W.: On vapour flow in a hot porous layer. J. Fluid Mech. 292, 1–23 (1995)CrossRefGoogle Scholar
  8. Gorenflo R., Mainardi F.: Random walk models for space-fractional diffusion. Fract. Calc. Appl. Anal. 1, 167–191 (1998)Google Scholar
  9. Logvinova K., Néel M.C.: A fractional equation for anomalous diffusion in a randomly heterogeneous porous medium. Chaos 14(4), 982–987 (2004)CrossRefGoogle Scholar
  10. Mitchell V., Woods A.W.: Self-similar dynamics of liquid injected into partially saturated aquifers. J. Fluid Mech. 566, 345–355 (2006)CrossRefGoogle Scholar
  11. Mityushev V., Adler P.M.: Longitudinal permeability of spatially periodic rectangular arrays of circular cylinders II. An arbitrary distribution of cylinders inside the unit cell. Z. Angew. Math. Phys. 53(3), 486–517 (2002)CrossRefGoogle Scholar
  12. Pachepsky, Y., Timlin, D., Rawls, W.: Generalized Richards’ equation to simulate water transport in unsaturated soils. J. Hydrol. 272, 3–13 (2003). See also Erratum, 279, 290 (2003)Google Scholar
  13. Polizzotto C.: Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38(42–43), 7359–7380 (2001)CrossRefGoogle Scholar
  14. Rayleigh L.: On the influence of obstacles arranged in rectangular order upon the properties of a medium. Philos. Mag. 34, 481–502 (1892)Google Scholar
  15. Richards L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)CrossRefGoogle Scholar
  16. Schumer R., Benson D.A., Meerschaert M.M., Wheatcraft S.W.: Eulerian derivation of the fractional advection–dispersion equation. J. Contam. Hydrol. 48(1–2), 69–88 (2001)CrossRefGoogle Scholar
  17. Schwartz L.: Théorie des Distributions. Hermann, Paris (1966)Google Scholar
  18. Sen M., Yang K.T.: An inflow–outflow characterization of inhomogeneous permeable beds. Transp. Porous Media 4, 97–104 (1989)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre DameUSA
  2. 2.Center for Energy ResearchUniversidad Nacional Autónoma de MéxicoTemixcoMor. Mexico

Personalised recommendations