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A theoretical analysis of vertical flow equilibrium

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Abstract

The assumption of Vertical Equilibrium (VE) and of parallel flow conditions, in general, is often applied to the modeling of flow and displacement in natural porous media. However, the methodology for the development of the various models is rather intuitive, and no rigorous method is currently available. In this paper, we develop an asymptotic theory using as parameter the variable\(R_{{L}} = L/H\sqrt {k_{{V}} /k_{{H}} } \). It is rigorously shown that the VE model is obtained as the leading order term of an asymptotic expansion with respect to 1/R 2L . Although this was numerically suspected, it is the first time that it is theoretically proved. Using this formulation, a series of special cases are subsequently obtained depending on the relative magnitude of gravity and capillary forces. In the absence of strong gravity effects, they generalize previous works by Zapata and Lake (1981), Yokoyama and Lake (1981) and Lake and Hirasaki (1981), on immiscible and miscible displacements. In the limit of gravity-segregated flow, we prove conditions for the fluids to be segregated and derive the Dupuit and Dietz (1953) approximations. Finally, we also discuss effects of capillarity and transverse dispersion.

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Abbreviations

C :

concentration, dimensionless

D :

dispersion tensor [L 2 T −1]

f :

fractional flow

g :

gravity acceleration [LT −2]

H :

reservoir thickness [L]

h :

dimensionless front location

k :

mean permeability [L 2]

K :

permeability [L 2]

L :

reservoir length [L]

M :

viscosity ratio, dimensionless

N CT :

transverse capillary number

N G :

gravity number

N TD :

transverse dispersion number

P :

dimensionless pressure

q :

flow velocity [LT −1]

R L :

VE parameter

S :

saturation

T :

time [T]

t :

dimensionless time

u :

dimensionless horizontal velocity

v :

dimensionless vertical velocity

X :

horizontal coordinate [L]

x :

dimensionless horizontal coordinate

y :

dimensionless vertical coordinate

w :

dimensionless vertical velocity

α :

dispersivity [L]

γ :

interfacial tension [MT −2]

δ :

permeability ratio, dimensionless

ε :

aspect ratio, dimensionless

ΘG :

gravity number

κ :

dimensionless permeability

λ :

dimensionless mobility

μ :

viscosity [ML −1 T −1]

Π:

dimensionless pressure

ρ :

density [ML −3]

φ :

porosity, dimensionless

ψ :

normalized mobility, dimensionless

a:

air

c:

capillary

H:

horizontal

L:

longitudinal

o:

oil

or:

residual oil

r:

relative

T:

total

V:

vertical

w:

water

wr:

residual water

O:

leading order

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Authors and Affiliations

  1. Petroleum Engineering Program, Department of Chemical Engineering, University of Southern California, 90089-1211, Los Angeles, CA, USA

    Y. C. Yortsos

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Yortsos, Y.C. A theoretical analysis of vertical flow equilibrium. Transp Porous Med 18, 107–129 (1995). https://doi.org/10.1007/BF01064674

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