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.
Similar content being viewed by others
Abbreviations
- f :
-
fractional flow function varying withs andx
- \(\bar f\) :
-
value off outsideI δ
- \(\hat f\) :
-
value off insideI δ
- \(\tilde f\) :
-
local approximation off around¯x
- f −,f + :
-
values of\(\tilde f\)
- f nj :
-
value off atS nj andx j
- g :
-
acceleration due to gravity [ms−2]
- I δ :
-
interval containing a low permeable rock
- k :
-
dimensionless absolute permeability
- k * :
-
absolute permeability [m2]
- k *c :
-
characteristic absolute permeability [m2]
- k ro :
-
relative oil permeability
- k rw :
-
relative water permeability
- L * :
-
characteristic length [m]
- L 1 :
-
the space of absolutely integrable functions
- L ∞ :
-
the space of bounded functions
- P c :
-
dimensionless capillary pressure function
- P *c :
-
capillary pressure function [Pa]
- P *c :
-
characteristic pressure [Pa]
- S :
-
similarity solution
- S nj :
-
numerical approximation tos(xj, tn)
- S 1, S2,S 3 :
-
constant values ofs
- s :
-
water saturation
- \(\bar s\) :
-
value ofs at\(\bar x\)
- s L :
-
left state ofs (wrt.\(\bar x\))
- s R :
-
right state ofs (wrt.\(\bar x\))
- s δ :
-
s for a fixed value ofδ in Section 3
- T :
-
value oft
- t :
-
dimensionless time coordinate
- t * :
-
time coordinate [s]
- t *c :
-
characteristic time [s]
- t n :
-
temporal grid point,t n=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]
- x j :
-
spatial grid piont,x j=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 ofI δ
- ε :
-
parameter measuring the capillary effects
- ζ :
-
argument ofS
- Μ o :
-
dimensionless dynamic oil viscosity
- Μw :
-
dimensionless dynamic water viscosity
- Μ *c :
-
characteristic viscosity [kg m−1s−1]
- Μ *o :
-
dynamic oil viscosity [kg m−1s−1]
- Μ *w :
-
dynamic water viscosity [k gm−1s−1]
- ϱ o :
-
dimensionless density of oil
- ϱ w :
-
dimensionless density of water
- ϱ * c :
-
characteristic density [kgm−3]
- ϱ * o :
-
density of oil [kgm−3]
- ϱ * w :
-
density of water [kgm−3]
- Φ :
-
porosity
- ψ :
-
dimensionless diffusion function varying withs andx
- ψ * :
-
dimensionless function varying with s andx * [kg−1m3s]
- ψ nj :
-
value ofψ atS n j andx j
References
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.
Aziz and Settari, 1979,Petroleum Reservoir Simulation, Applied Science Publishers, London.
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.
Isaacson, E. and Temple, B., 1991, The structure of asymptotic states in a singular system of conservation laws, preprint.
Kruzkov, S. N., 1970, First-order quasilinear sequations with several space variables,Math. USSR. Sb. 10, 217–243.
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.
Lucier, B., 1985, Error bounds for the methods of Glimm, Godunov and LeVeque,SIAM J. Numer. Anal. 22, 1074–1081.
Marie, C. M., 1981,Multiphase Flow in Porous Media, Editions Technip.
Rose, W., 1990, Lagrangian simulation of coupled two-phase flow,Math. Geol. 22, 641–654.
Smoller, J., 1982,Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York.
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.
Author information
Authors and Affiliations
Additional information
This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).
Rights and permissions
About this article
Cite this article
Langtangen, H.P., Tveito, A. & Winther, R. Instability of Buckley-Leverett flow in a heterogeneous medium. Transp Porous Med 9, 165–185 (1992). https://doi.org/10.1007/BF00611965
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00611965