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Transport in Porous Media

, Volume 6, Issue 2, pp 115–142 | Cite as

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

  • Y.-S. Wu
  • K. Pruess
  • P. A. Witherspoon
Article

Abstract

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.

Key words

Non-Newtonian fluids Buckley-Leverett immiscible displacement power-law fluids rheological models Welge method fractional flow theory enhanced oil recovery 

Nomenclature

Roman Letters

A

cross-sectional area (m2)

fne

fractional flow of Newtonian phase

fnn

fractional flow of non-Newtonian phase

g

gravitational acceleration vector (m/s2)

g

magnitude of the gravitational acceleration (m/s2)

H

power law coefficient (Pa sn)

K

absolute permeability (m2)

krne

relative permeability of non-Newtonian phase

n

power-law exponential index

Np

cumulative displaced Newtonian fluid (m3)

P

pressure (Pa)

Pc

capillary pressure (Pa)

Pne

pressure of Newtonian phase (Pa)

Pnn

pressure of non-Newtonian phase (Pa)

q(t)

injection rate of non-Newtonian fluid (m3/s)

Q(t)

cumulative injection rate (m3)

Sf

saturation at moving front (m)

Sne

Newtonian phase saturation

Sneir

irreducible Newtonian phase saturation

Snn

non-Newtonian saturation

Snnir

connate non-Newtonian saturation

\(\bar S\)nn

average saturation of non-Newtonian phase in swept zone

x

distance from inlet, coordinate (m)

xf

distance to shock saturation front (m)

xsnn

distance of saturation Snn from the inlet (m)

t

time (s)

u

Darcy velocity (m/s)

u(t)

total flux (m/s)

une

Darcy velocity of Newtonian phase (m/s)

unn

Darcy velocity of non-Newtonian phase (m/s)

Greek Letters

α

angle between horizontal plane and flow direction

γ

shear rate (s-1)

μapp

apparent viscosity (Pa s)

μeff

effective viscosity (Pa s n m1 - n)

μne

viscosity of Newtonian fluid (Pa s)

μnn

equivalent viscosity of non-Newtonian fluid (Pa s)

ϱne

density of Newtonian fluid (kg/m3)

ϱnn

density of non-Newtonian fluid (kg/m3)

τ

shear stress (Pa)

φ

porosity of porous media

Φ

flow potential (Pa)

Subscripts

app

apparent

eff

effective

f

front

ne

Newtonian

nn

non-Newtonian

rne

relative to Newtonian phase

rnn

relative to non-Newtonian phase

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Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Y.-S. Wu
    • 1
  • K. Pruess
    • 1
  • P. A. Witherspoon
    • 1
  1. 1.Earth Sciences DivisionLawrence Berkeley LaboratoryBerkeleyUSA

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