Transport in Porous Media

, Volume 129, Issue 3, pp 941–953 | Cite as

The Extremum Condition of the Local Volumetric Flux for Compositional Displacements

  • Xiao Luo
  • Quoc Nguyen
  • David DiCarloEmail author


Compositional displacements in porous media, where chemical components partition between phases during a displacement, occur in flow processes such as surfactant flooding and gas injection. We develop a new approach to solving compositional displacements with volume change on mixing. The result is a new condition on the local volumetric flux that is equivalent to the tangent construction (condition). We demonstrate how this works by solving for the shocks for the simplest case of a two-component, two-phase system. We discuss the use of this procedure for the more general case of arbitrary volume change on mixing and arbitrary number of components.


Compositional displacement Coherent wave Extremum condition Volume change on mixing 

List of symbols

\(\alpha _i\)

The molar flux component i normalized by the local volumetric flux (kmol/m\(^3\))

\(\eta \), \(\varLambda \)

The characteristic velocity of the composition wave and shock, respectively (m/s)

\(\lambda ^{I}\)

The mobility of phase I (1/cp)

\(\mu ^I\)

The viscosity of phase I (cp)

\(\phi \)

The porosity of the porous media (m\(^3\)/m\(^3\))


The molar volume of pure component i (m\(^3\)/kmol)


The equilibrium or saturated molar volume of phase I (m\(^3\)/kmol)

\(\xi \)

The independent spatial variable (m)


The fractional flow of phase I (m\(^3\)/m\(^3\))


The molar amount of component i per unit volume (kmol/m\(^3\))


The molar flux of component i per unit volume (kmol/m\(^2\)/s)


The saturation of phase I (m\(^3\)/m\(^3\))


The independent temporal variable (s)


The local volumetric flux (m\(^3\)/m\(^2\)/s)


The total pore volume in a control volume (m\(^3\))


The mole fraction of component i in phase I (kmol/kmol)


The overall mole fraction of component i (kmol/kmol)


The equilibrium or saturated mole fraction of component i in phase I (kmol/kmol)


The relative permeability of phase I (m\(^2\)/m\(^2\))


The pore volume occupied by phase I (m\(^3\))



This research was funded by the Equinor Fellowship Program.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Hildebrand Department of Petroleum and Geosystems EngineeringThe University of Texas at AustinAustinUSA

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