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A Fickian diffusion model for the spreading of liquid plumes infiltrating in heterogeneous media

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Abstract

Infiltration of water and non-aqueous phase liquids (NAPLs) in the vadose zone gives rise to complex two- and three-phase immiscible displacement processes. Physical and numerical experiments have shown that ever-present small-scale heterogeneities will cause a lateral broadening of the descending liquid plumes. This behavior of liquid plumes infiltrating in the vadose zone may be similar to the familiar transversal dispersion of solute plumes in single-phase flow. Noting this analogy we introduce a mathematical model for ‘phase dispersion’ in multiphase flow as a Fickian diffusion process.

It is shown that the driving force for phase dispersion is the gradient of relative permeability, and that addition of a phase-dispersive term to the governing equations for multiphase flow is equivalent to an effective capillary pressure which is proportional to the logarithm of the relative permeability of the infiltrating liquid phase. The relationship between heterogeneity-induced phase dispersion and capillary and numerical dispersion effects is established. High-resolution numerical simulation experiments in heterogeneous media show that plume spreading tends to be diffusive, supporting the proposed convection-dispersion model. Finite difference discretization of the phase-dispersive flux is discussed, and an illustrative application to NAPL infiltration from a localized source is presented. It is found that a small amount of phase dispersion can completely alter the behavior of an infiltrating NAPL plume, and that neglect of phase-dispersive processes may lead to unrealistic predictions of NAPL behavior in the vadose zone.

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Abbreviations

a :

parameter in capillary pressure function, Equation (41) (m −1)

C :

solute concentration (kg/m3)

d :

molecular diffusivity (m2/s)

D :

dispersion coefficient (m2/s)

D :

dispersion tensor (m2/s)

e :

unit vector

F :

mass flux (kg/m2 s)

g, g :

acceleration of gravity (m/s2)

I :

identity tensor

k, k :

permeability (m2)

P :

pressure (Pa)

S :

saturation

t :

time (s)

u :

Darcy velocity (m/s)

v :

advection velocity (m/s)

x, y :

horizontal coordinates (m)

z :

vertical coordinate (m)

α :

dispersivity (m)

β :

sorptive number (m−1)

φ :

porosity

κ :

anisotropy coefficient

μ :

viscosity (Pa s)

υ :

parameter in capillary pressure function, Equation (41)

ϱ :

density

σ 2 :

variance, or mean square plume size (m2)

c :

convective

cap:

capillary

d :

diffusive

g :

gas

h :

horizontal

l :

liquid

L :

longitudinal

n :

NAPL

r :

relative

T :

transversal

ν:

vertical

w :

water

x, y, z :

components along coordinate axes

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  1. Earth Sciences Division, Lawrence Berkeley National Laboratory, University of California, 94720, Berkeley, CA, U.S.A.

    Karsten Pruess

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Pruess, K. A Fickian diffusion model for the spreading of liquid plumes infiltrating in heterogeneous media. Transp Porous Med 24, 1–33 (1996). https://doi.org/10.1007/BF00175602

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  • DOI: https://doi.org/10.1007/BF00175602

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