Advertisement

Transport in Porous Media

, Volume 9, Issue 3, pp 165–185 | Cite as

Instability of Buckley-Leverett flow in a heterogeneous medium

  • Hans Petter Langtangen
  • Aslak Tveito
  • Ragnar Winther
Article

Abstract

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.

Key words

oil recovery instability heterogeneous medium shock waves 

Symbols and Notation

f

fractional flow function varying withs andx

\(\bar f\)

value off outsideIδ

\(\hat f\)

value off insideIδ

\(\tilde f\)

local approximation off around¯x

f,f+

values of\(\tilde f\)

fjn

value off atS j n andxj

g

acceleration due to gravity [ms−2]

Iδ

interval containing a low permeable rock

k

dimensionless absolute permeability

k*

absolute permeability [m2]

kc*

characteristic absolute permeability [m2]

kro

relative oil permeability

krw

relative water permeability

L*

characteristic length [m]

L1

the space of absolutely integrable functions

L

the space of bounded functions

Pc

dimensionless capillary pressure function

Pc*

capillary pressure function [Pa]

Pc*

characteristic pressure [Pa]

S

similarity solution

Sjn

numerical approximation tos(xj, tn)

S1, S2,S3

constant values ofs

s

water saturation

\(\bar s\)

value ofs at\(\bar x\)

sL

left state ofs (wrt.\(\bar x\))

sR

right state ofs (wrt.\(\bar x\))

sδ

s for a fixed value ofδ in Section 3

T

value oft

t

dimensionless time coordinate

t*

time coordinate [s]

tc*

characteristic time [s]

tn

temporal grid point,tn=n δt

v*

total filtration (Darcy) velocity [ms−1]

W, Β, v

dimensionless numbers defined by Equations (4), (5) and (6)

x

dimensionless spatial coordinate [m]

x*

spatial coordinate [m]

xj

spatial grid piont,xj=j δx

\(\bar x(t)\)

discontinuity curve in (x, t) space

\(\bar x^ + \)

right limiting value of¯x

\(\bar x^ - \)

left limiting value of¯x

α

angle between flow direction and horizontal direction

δt

temporal grid spacing

δx

spatial grid spacing

δ

length ofIδ

ε

parameter measuring the capillary effects

ζ

argument ofS

Μo

dimensionless dynamic oil viscosity

Μw

dimensionless dynamic water viscosity

Μc*

characteristic viscosity [kg m−1s−1]

Μo*

dynamic oil viscosity [kg m−1s−1]

Μw*

dynamic water viscosity [k gm−1s−1]

ϱo

dimensionless density of oil

ϱw

dimensionless density of water

ϱc*

characteristic density [kgm−3]

ϱo*

density of oil [kgm−3]

ϱw*

density of water [kgm−3]

Φ

porosity

ψ

dimensionless diffusion function varying withs andx

ψ*

dimensionless function varying with s andx* [kg−1m3s]

ψjn

value ofψ atS j n andxj

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allen, M. B. III, Behie, G. A., and Trangenstein, J. A., 1988,Multiphase Flow in Porous Media, Lecture Notes in Engineering, Springer-Verlag, New York.Google Scholar
  2. 2.
    Aziz and Settari, 1979,Petroleum Reservoir Simulation, Applied Science Publishers, London.Google Scholar
  3. 3.
    Gimse, T. and Risebro, N. H., 1990, Riemann problems with a discontinuous flux function, in Enquist and Gustafsson (eds).Proc 3rd Internat. Conf. on Hyperbolic Problems, Uppsala.Google Scholar
  4. 4.
    Isaacson, E. and Temple, B., 1991, The structure of asymptotic states in a singular system of conservation laws, preprint.Google Scholar
  5. 5.
    Kruzkov, S. N., 1970, First-order quasilinear sequations with several space variables,Math. USSR. Sb. 10, 217–243.Google Scholar
  6. 6.
    Lax, P. D., 1973, Hyperbolic systems of conservation laws and the mathematical theory of shock waves,Conf. Board Math. Sci. vol. 11, SIAM, Philadelphia, Pa.Google Scholar
  7. 7.
    Lucier, B., 1985, Error bounds for the methods of Glimm, Godunov and LeVeque,SIAM J. Numer. Anal. 22, 1074–1081.Google Scholar
  8. 8.
    Marie, C. M., 1981,Multiphase Flow in Porous Media, Editions Technip.Google Scholar
  9. 9.
    Rose, W., 1990, Lagrangian simulation of coupled two-phase flow,Math. Geol. 22, 641–654.Google Scholar
  10. 10.
    Smoller, J., 1982,Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York.Google Scholar
  11. 11.
    Tveito, A. and Winther, R., 1990, A well posed system of hyperbolic conservation laws, in Enquist and Gustafsson (eds),Proc. 3rd Internat. Conf. on Hyperbolic Problems, Uppsala.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Hans Petter Langtangen
    • 1
  • Aslak Tveito
    • 1
  • Ragnar Winther
    • 2
  1. 1.Center for Industrial ResearchOslo 3Norway
  2. 2.Department of InformaticsUniversity of OsloOslo 3Norway

Personalised recommendations