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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
Article

Abstract

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.

Keywords

Non-local Non-Darcy 

List of Symbols

Variables

f(x′, x)

Material property of locations x′ and x

f = {fji}

Discrete matrix version of f(x′, x)

k

Material constant

L

Length of material

\({{\mathcal L}}\)

Riemann–Liouville fractional derivative defined in Eq. 11a

n

Number of tubes

N

Number of discrete intervals

p

Pressure

q

Flow rate

\({{\mathcal R}}\)

Weyl fractional derivative defined in Eq. 11b

t

Time

x

Spatial coordinate

Greek symbols

Γ

Gamma function

δ

Delta distribution

δ

Derivative of δ

\({\epsilon}\)

Spatial scale of non-locality

\({\delta_\epsilon}\)

Nascent delta distribution

\({\delta'_\epsilon}\)

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

μ, ν

Size of regions

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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

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