# Steady-state solutions to Bentsen's equation

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## 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: ∂^{2}*f*/∂*S*^{2}≤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 of*Τ*_{ bt } can be obtained as*N*_{ c } → 0. The Buckley-Leverett solution is the limiting solution as*N*_{ 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*(*S*_{n})\( = {\text{ }} - \frac{1}{{M_r }}F_w K_{{\text{r}}nw} {\text{ }}\frac{{{\text{d}}\pi _{\text{c}} }}{{{\text{d}}S_n }}\)

*F*_{w}(*S*_{n})\( = {\text{ }}\frac{{M_r K_{{\text{r}}w} }}{{M_r K_{{\text{r}}w} + K_{{\text{r}}nw} }}\)

*G*(*S*_{n})\( = {\text{ }}\left( {1 - \frac{{N_g }}{{M_r }}K_{{\text{r}}w} } \right){\text{ }}F_w\)

*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

- Δ
*ρ* *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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