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.
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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
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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).
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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
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DOI: https://doi.org/10.1007/BF00611965