# 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 waves## Symbols and Notation

*f*fractional flow function varying with

*s*and*x*- \(\bar f\)
value of

*f*outside*I*_{δ}- \(\hat f\)
value of

*f*inside*I*_{δ}- \(\tilde f\)
local approximation of

*f*around*¯x**f*^{−},*f*^{+}values of\(\tilde f\)

*f*_{j}^{n}value of

*f*at*S*_{j}^{n}and*x*_{j}*g*acceleration due to gravity [ms

^{−2}]*I*_{δ}interval containing a low permeable rock

*k*dimensionless absolute permeability

*k*^{*}absolute permeability [m

^{2}]*k*_{c}^{*}characteristic absolute permeability [m

^{2}]*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*_{j}^{n}numerical approximation to

*s*(x_{j}, t_{n})*S*_{1}, S_{2},*S*_{3}constant values of

*s**s*water saturation

- \(\bar s\)
value of

*s*at\(\bar x\)*s*^{L}left state of

*s*(wrt.\(\bar x\))*s*^{R}right state of

*s*(wrt.\(\bar x\))*s*_{δ}*s*for a fixed value of*δ*in Section 3*T*value of

*t**t*dimensionless time coordinate

*t*^{*}time coordinate [s]

*t*_{c}^{*}characteristic time [s]

*t*_{n}temporal grid point,

*t*_{n}=n δ*t**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 δ*x*- \(\bar x(t)\)
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

- δ
*t* temporal grid spacing

- δ
*x* spatial grid spacing

*δ*length of

*I*_{δ}*ε*parameter measuring the capillary effects

*ζ*argument of

*S**Μ*_{o}dimensionless dynamic oil viscosity

- Μ
_{w} dimensionless dynamic water viscosity

*Μ*_{c}^{*}characteristic viscosity [kg m

^{−1}s^{−1}]*Μ*_{o}^{*}dynamic oil viscosity [kg m

^{−1}s^{−1}]*Μ*_{w}^{*}dynamic water viscosity [k gm

^{−1}s^{−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 with

*s*and*x**ψ*^{*}dimensionless function varying with s and

*x*^{*}[kg^{−1}m^{3}s]*ψ*_{j}^{n}value of

*ψ*at*S*_{ j }^{n}and*x*_{j}

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## References

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