Transport in Porous Media

, Volume 16, Issue 2, pp 105–123 | Cite as

Steady-state solutions to Bentsen's equation

  • Chonghui Shen
  • Douglas W. Ruth
Article
  • 67 Downloads

Abstract

Based on experimentally observed phenomena and the physical requirement of a unique value of saturation at any location within a porous medium, a restrictive condition for a valid solution to Bentsen's equation is derived: ∂2f/∂S2≤0. The steady-state solution to Bentsen's equation is shown to be identical to the Buckley-Leverett solution to the displacement equation, and the steady-state solution for the fractional flow is shown to be independent of the capillary number. It is proved that under steady-state conditions, the capillary term of the fractional flow equation in the frontal region does not depend on the capillary number. Therefore, the unrealistic triple-valued saturation profile of the original Buckley-Leverett solution resulted because the capillary term was in-appropriately neglected. The break-through recovery efficiency,Τ bt , is shown to be a function of the capillary number. As the capillary number decreases, the break-through recovery efficiency increases and the maximum value ofΤ bt can be obtained asN c → 0. The Buckley-Leverett solution is the limiting solution asN c → 0.

Key words

Bentsen's equation Buckley-Leverett solution capillary number immiscible displacement steady-state solution 

Nomenclature

Roman Letters

A

cross-sectional area

C(Sn)

\( = {\text{ }} - \frac{1}{{M_r }}F_w K_{{\text{r}}nw} {\text{ }}\frac{{{\text{d}}\pi _{\text{c}} }}{{{\text{d}}S_n }}\)

Fw(Sn)

\( = {\text{ }}\frac{{M_r K_{{\text{r}}w} }}{{M_r K_{{\text{r}}w} + K_{{\text{r}}nw} }}\)

G(Sn)

\( = {\text{ }}\left( {1 - \frac{{N_g }}{{M_r }}K_{{\text{r}}w} } \right){\text{ }}F_w\)

K

permeability

L

length of system

M

mobility ratio

N

dimensionless number

P

pressure

S

saturation

X

distance

a

constant in Equation (34)

b

constant in Equation (34)

f

fractional flow

g

acceleration of gravity

n

index of the Corey model

q

volumetric flow rate

s

integral variable

t

time

Greek Letters

α

angle of inclination of system

Δρ

ρ w ρ nw

Μ

viscosity

ξ

normalized distance

π

dimensionless pressure

ρ

density

σ

capillary pressure normalizing parameter

Τ

dimensionless time or pore volume

Φ

saturation front shape function defined by Equation (58)

Φ

porosity

Superscripts

first derivative

second derivative

*

inlet quantity

Subscripts

c

capillary

bt

break-through

d

displacement

g

gravity

i

irreducible

I

inflection

L

flood-front

m

matching

n

normalized

nw

nonwetting

nwr

nonwetting residual

rnw

reduced (normalized) nonwetting

rw

reduced (normalized) wetting

si

singular

w

wetting

wr

wetting at residual (irreducible)

or

oil at residual

r

residual

Accent

~

equivalent quantity

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References

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Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Chonghui Shen
    • 1
  • Douglas W. Ruth
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of ManitobaWinnipegCanada

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