Abstract
The approach to residual oil saturation during the immiscible displacement of oil as predicted by the multiphase Darcy equations is studied. It is well known that when the capillary pressure term is neglected, one arrives at the Buckley-Leverett formulation according to which the inlet face attains residual oil saturation instantaneously. This result may, however, be strongly influenced by the inclusion of the capillary pressure term. In this paper it is shown that when the relative permeability and capillary pressure functions have power law dependencies on the saturation deviation from residual oil condition, the long time solution exhibits a power law decay toward residual saturation. Moreover, the power law decay solution is found to be unique and independent of the initial condition. The relationship of this solution to the classical Buckley-Leverett result is shown. Finally, generalization to the time varying flow rate case is addressed. As a verification of the theoretical conjectures, the power law solution is compared with direct numerical simulation of the two phase flow equations.
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Abbreviations
- a :
-
defined by Equation (30) or (48)
- a supfinfn :
-
coefficients of expansion given in Equation (B10)
- b :
-
defined by Equation (31) or (49)
- C ij :
-
constants involved in Equation (43)
- f :
-
function defined by Equation (33)
- f 0 :
-
f atξ = 0
- f 1 :
-
f′ atξ = 0
- f 2 :
-
2f′' atξ = 0
- f w :
-
fractional flow of water
- g :
-
defined by Equation (A1)
- k :
-
permeability
- k ri :
-
relative permeability to phasei
- k Oro :
-
coefficient for oil relative permeability
- k Orw :
-
coefficient for water relative permeability
- L :
-
sample length
- M i :
-
Mobility of phasei
- N :
-
arbitrary parameter for solution near infinity
- P :
-
derivative of f with respect to ξ
- P i0 :
-
p on zero derivative isocline
- P c :
-
capillary pressure
- P d0c :
-
coefficient for capillary pressure
- P i :
-
pressure in phasei
- r :
-
defined by Equation (A3)
- r 0 :
-
approximation tor
- r 00 :
-
approximation tor 0
- s :
-
defined by Equation (A6)
- s 00 :
-
approximation tos
- S i :
-
saturation of phasei
- S ro :
-
residual oil saturation
- t :
-
time
- x :
-
distance
- x o :
-
characteristic distance
- v :
-
superficial velocity
- v i :
-
superficial velocity of phasei
- V :
-
constant superficial velocity
- v i :
-
velocity of phasei
- w :
-
= f −p
- α :
-
defined by Equation (26)
- β :
-
defined by Equation (39)
- δ :
-
oil saturation
- δ 0 :
-
approximation to residual oil saturation
- μ i :
-
viscosity of phasei
- v :
-
defined by Equation (27)
- ξ :
-
similarity variable defined by Equation (32)
- σ :
-
exponent for capillary pressure
- τ :
-
oil relative permeability exponent
- γ :
-
defined by Equation (C1)
- φ :
-
porosity
- ψ :
-
time exponent in varying velocity relationship
- o :
-
oil
- w :
-
water
- bl :
-
Buckley-Leverett solution variableAccents corresponding dimensionless variable
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Ramakrishnan, T.S., Wilkinson, D. & Dias, M.M. Effect of capillary pressure on the approach to residual saturation. Transp Porous Med 3, 51–79 (1988). https://doi.org/10.1007/BF00222686
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DOI: https://doi.org/10.1007/BF00222686