Abstract
In constructing simulations, it is commonly assumed that steady-state relative permeabilities can be used to describe displacements, which are unsteady-state flows. In order to test this assumption, a statistical structural model is used in the context of local volume averaging to examine the effects of interfacial tension, the interfacial viscosities, wetting and the viscosity ratio upon the capillary pressure and relative permeability functions in a displacement. The statistical structural model is a highly simplified idealization of the local pore structure, in which pores are randomly oriented in space, but no pore interconnections are recognized. While the statistical structural model is too simplified to give accurate descriptions of the relative permeabilities, it should be sufficient to determine whether significant qualitative differences between steady-state and unsteady-state relative permeabilities might be expected.
Our results suggest that, whether we are concerned with an imbibition or a drainage, it is not important to distinguish between steady-state and upsteady-state relative permeabilities, when the local (not macroscopic) capillary number N ca≪10-2. Unless the interfacial tension is very small everywhere or the overall pressure gradient very large, it may be satisfactory to use steady-state representations for the relative permeabilities. This means that, for a waterflood in a petroleum reservoir, the difference between steady-state and unsteady-state relative permeabilities is almost certainly negligible. For an optimum surfactant flood in a petroleum reservoir, at least within the immediate neighborhood of the displacement front, it may be important to make the distinction.
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Abbreviations
- a :
-
Free parameter introduced in Equation (20)
- a *, b*, c* :
-
Defined by Equations (55), (56), and (57)
- a(i):
-
Defined by Equation (4)
- a(i,j):
-
Defined by Equation (9)
- a τ :
-
Defined by Equation (10)
- A :
-
Defined by Equation (44)
- dA :
-
Denotes area integration is to be performed
- b :
-
Body force (gravity) per unit mass
- B(m, n) :
-
Beta function defined by Equation (18)
- B (i) :
-
Some quantity associated with phase i
- c :
-
Defined by Equation (14)
- C :
-
Three-phase line of contact or closed curve bounding S (nw, w)
- e :
-
Unit vector defined by Equation (32)
- f(R*):
-
Probability density function for pore size distribution. Represented by Equation (17)
- G :
-
Defined by Equation (45)
- H :
-
Mean curvature
- I :
-
Identity tensor that transforms vectors into themselves
- k :
-
Single-phase permeability
- k (i)** :
-
Relative permeability for phase i
- ksk0/(i)* :
-
Normalized steady-state relative permeability for phase i
- k (nw)*, k (w)* :
-
Normalized relative permeabilities defined by Equations (73) and (74).
- L :
-
Length and diameter of cylindrical averaging surface S
- L t :
-
Length of an individual capillary tube
- L t ,min :
-
Length of pore whose radius is R max
- L sup*inft :
-
Defined by Equation (20)
- L (w) :
-
Length of that portion of a pore occupied by the wetting phase
- L (i)* :
-
Defined by Equation (46)
- N :
-
Total number of pores contained within averaging surface S
- N ca :
-
Local capillary number defined by Equation (64). See also Equation (67)
- N κ+∈ :
-
Dimensionless sum of interfacial viscosities, defined by Equation (48)
- N μ :
-
Dimensionless viscosity ratio, defined by Equation (47)
- p :
-
Pressure
- p(θ, φ) :
-
Probability density function for orientation of capillaries. Represented by Equation (15)
- P c :
-
Capillary pressure, defined by Equation (13); see also Equation (21)
- P *infc :
-
Dimensionless capillary pressure defined by Equation (24)
- Δp :
-
Defined by Equation (22)
- {Δp*}:
-
Defined by Equation (43)
- p sup-1infv , p Emphasis>-1 v-1 :
-
Associated Legendre functions of the first kind
- R :
-
Capillary radius
- R * :
-
Defined by Equation (19)
- R max :
-
Maximum pore radius that occurs within S
- R (i) :
-
Region occupied by phase i within S
- R sup*infb :
-
Dimensionless radius characteristic of the majority of the pores
- R sup*infc :
-
Defined by Equation (69)
- R sup*inft :
-
Determined by Equation (68) at any given dimensionless time t * with c * fixed by Equation (57)
- R sup*inftL , R sup*inftU :
-
Two solutions of Equation (68) for R sup*inft
- s (nw,w) :
-
Interface separating nonwetting and wetting phases within S
- s (w) :
-
Normalized saturation defined by Equations (27) and (72)
- s (i)** :
-
Saturation of phase i or volume fraction of pore space occupied by phase i within S
- s sup(w)**infi :
-
Irreducible saturation of wetting phase at conclusion of drainage
- s (w)**infr :
-
Saturation of wetting phase at conclusion of imbibition, corresponding to residual saturation of nonwetting phase
- Sskinfint/sup(i):
-
Portion of closed surface bounding R (i)composed of phase interfaces
- S :
-
Averaging surface
- S :
-
Viscous portion of stress tensor
- S (σ) :
-
Viscous portion of surface stress tensor
- t * :
-
Dimensionless time defined by Equation (54)
- v :
-
Velocity
- ν (i)* :
-
Defined by Equation (42)
- V :
-
Volume of region enclosed by averaging surface S
- V (i) :
-
Volume of phase i contained within S
- d V :
-
Denotes that a volume integration is to be performed
- X :
-
Defined by Equation (28)
- Y :
-
Defined by Equation (A4)
- α, β :
-
Free parameters introduced in Equation (17)
- γ :
-
Interfacial tension
- Γ(x):
-
Gamma function
- ε :
-
Interfacial shear viscosity
- θ :
-
Cylindrical coordinates
- θ max :
-
Maximum value of θ for which flow in a capillary is possible
- Θ:
-
Contact angle measured through displacing phase
- κ :
-
Interfacial dilatational viscosity
- μ :
-
Unit vector tangent to S (nw,w), normal to C, and outwardly directly with respect to C
- μ (i) :
-
Viscosity of phase i
- v :
-
Parameter satisfying Equation (49)
- ξ :
-
Unit normal to phase interface whose direction is compatible with the definition of H (McConnell, 1957)
- ξ (i) :
-
Unit normal to Sskinfint/sup(i) pointing into phase i
- π :
-
3.1416...
- ϱ :
-
Density
- φ :
-
Cylindrical coordinate
- ψ :
-
Porosity
- div(σ) :
-
Surface divergence operation (Wei et al., 1974; Briley et al., 1976)
- ▽:
-
Gradient operator
- ▽(σ) :
-
Surface gradient operator (Wei et al., 1974; Briley et al., 1976)
- ...1:
-
Refers to the inlet
- ...2:
-
Refers to the outlet
- ...(w) :
-
Denotes quantity associated with wetting phase
- ...(nw) :
-
Denotes quantity associated with nonwetting phase
- ...(1):
-
Denotes quantity associated with displacing phase
- ...(2):
-
Denotes quantity associated with displaced phase
- 〈...〉(i) :
-
Intrinsic volume average for phase i; defined by Equation (2)
- ...(i) :
-
Local volume average for phase i; defined by Equation (1)
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Currently at Amoco Production Company, P.O. Box 3385, Tulsa, OK 74102, U.S.A.
Currently at Regional Research Laboratory, Industrial Estate P.O., Trivandrum 695 019, Kerala India
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Alemán, M.A., Ramamohan, T.R. & Slattery, J.C. The difference between steady-state and unsteady-state relative permeabilities. Transp Porous Med 4, 449–493 (1989). https://doi.org/10.1007/BF00179531
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DOI: https://doi.org/10.1007/BF00179531