Instability of Buckley-Leverett flow in a heterogeneous medium
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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 wavesSymbols 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
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References
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