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Fractional Flow Approach to Saturation Overshoot

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Abstract

Saturation overshoot is observed for 1D vertical infiltrations (liquid replacing gas) in many porous media. Aspects of these infiltrations are often described using the Richards equation, which assumes that the gas viscosity is negligible compared to the liquid viscosity. Here, we develop a multi-phase, fractional flow approach to describe the physics behind the displacement front that includes the viscosity of the gas. We show that an overshoot profile will draw in gas behind the overshoot tip. We compare the fractional flow solution to the Richards equation solution and to experimental data, and show that the air viscosity plays an observable role when the infiltrating flux is greater than 50% of the saturated conductivity.

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Abbreviations

C :

Arbitrary constant (arbitrary)

K :

Permeability (m2)

K h :

Hydraulic conductivity (m/s)

g :

Gravity component in flow direction (m/s2)

k ri :

Relative permeability of phase i (–)

f :

Fractional flow of phase 1 (–)

f c :

Capillary component of fractional flow of phase 1 (–)

f g :

Gravity component of fractional flow of phase 1 (–)

f v :

Viscous component of fractional flow of phase 1 (–)

f vg :

Viscous and gravity component of fractional flow of phase 1 (–)

L t :

Length of finger tip (m)

P atm :

Atmospheric pressure (Pa)

P c :

Capillary pressure (Pa)

P cb :

Air entry capillary pressure (Pa)

P c′:

Derivative of capillary pressure with respect to saturation (Pa)

P i :

Pressure of phase i (Pa)

S i :

Saturation of phase i (–)

S :

Saturation of phase 1 (–)

t :

Time (s)

u i :

Flux of phase i (m/s)

v :

Wave velocity (m/s)

v s :

Shock velocity (m/s)

z :

Vertical distance-positive downward (m)

z b :

Vertical distance at the beginning of the finger tip (m)

z e :

Vertical distance at the end of the finger tip (m)

Δρ :

Density difference of phases 1 and 2 (kg/m3)

η :

Traveling wave (inner) variable (m)

λ i :

Mobility of phase i ((Pa s)−1)

μ i :

Viscosity of phase i (Pa s)

\({\phi}\) :

Porosity (–)

ρ i :

Density of phase i (kg/m3)

−:

Value immediately downstream of shock

+:

Value immediately upstream of shock

C :

Value at point C

i :

Phase i

I :

Initial value

J :

Final value

K :

Overshoot value

T :

Total value

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Authors and Affiliations

  1. Department of Petroleum and Geosystems Engineering, University of Texas at Austin, Austin, TX, USA

    D. A. DiCarlo, M. Mirzaei, B. Aminzadeh & H. Dehghanpour

Authors
  1. D. A. DiCarlo
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  2. M. Mirzaei
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  3. B. Aminzadeh
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  4. H. Dehghanpour
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Correspondence to D. A. DiCarlo.

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DiCarlo, D.A., Mirzaei, M., Aminzadeh, B. et al. Fractional Flow Approach to Saturation Overshoot. Transp Porous Med 91, 955–971 (2012). https://doi.org/10.1007/s11242-011-9885-8

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  • DOI: https://doi.org/10.1007/s11242-011-9885-8

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