Abstract
A parametric experimental investigation of the coupling effects during steady-state two-phase flow in porous media was carried out using a large model pore network of the chamber-and-throat type, etched in glass. The wetting phase saturation,S 1, the capillary number,Ca, and the viscosity ratio,k, were changed systematically, whereas the wettability (contact angleθ e ), the coalescence factorCo, and the geometrical and topological parameters were kept constant. The fluid flow rate and the pressure drop were measured independently for each fluid. During each experiment, the pore-scale flow mechanisms were observed and videorecorded, and the mean water saturation was determined with image analysis. Conventional relative permeability, as well as generalized relative permeability coefficients (with the viscous coupling terms taken explicitly into account) were determined with a new method that is based on a B-spline functional representation combined with standard constrained optimization techniques. A simple relationship between the conventional relative permeabilities and the generalized relative permeability coefficients is established based on several experimental sets. The viscous coupling (off-diagonal) coefficients are found to be comparable in magnitude to the direct (diagonal) coefficients over board ranges of the flow parameter values. The off-diagonal coefficients (k rij /Μ j ) are found to be unequal, and this is explained by the fact that, in the class of flows under consideration, microscopic reversibility does not hold and thus the Onsager-Casimir reciprocal relation does not apply. Thecoupling indices are introduced here; they are defined so that the magnitude of each coupling index is the measure of the contribution of the coupling effects to the flow rate of the corresponding fluid. A correlation of the coupling indices with the underlying flow mechanisms and the pertinent flow parameters is established.
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Abbreviations
- B o :
-
bond number
- Ca :
-
capillary number=Μ 1 q 1/γ 12 wl
- Co :
-
coalescence factor (effective probability of coalescence, given a collision between two ganglia in the porous medium)
- C i k :
-
parameters used in the functional representation ofk ri in terms of cubic B-splines, (Equation (10a))
- C ij k :
-
parameters used in the functional representation ofk rij in terms of cubic B-splines, (Equation (10b))
- e Μα :
-
residual for theΜth experiment and theαth equation of the model, (Equation (6))
- e :
-
vector of residuals,e Μα
- k :
-
absolute permeability
- k ri :
-
(conventional) relative permeability to fluidi
- k o ri :
-
value ofk ri free from end and boundary effects
- k rij :
-
generalized relative permeability coefficients
- k o rij :
-
value ofk rij free from end and boundary effects
- L :
-
distance along which δP 1 and δP 2 are measured
- l :
-
node-to-node distance of the pore network
- l α :
-
number of unknown parameters in theαth equation of the model
- N :
-
number of cubic B-splines used to representk ri ork rij , ((Equation (10a,b))
- n :
-
number of experimental data
- qi :
-
flowrate of fluidi
- S i :
-
saturation of fluidi
- v i :
-
superficial velocity of fluidi
- V :
-
covariance matrix of the true errorsε Μα , for all experiments (Μ) and equations (α) of the model
- W :
-
weighing matrix, (Equation (7))
- w :
-
width of the network
- x :
-
vector of the independent variables, (Equation (5))
- y :
-
vector of the dependent variables
- γ 12 :
-
interfacial tension
- ε Μα :
-
true error for theΜth experiment and theαth equation
- δP i :
-
pressure drop (negative) in fluidi, along a distanceL
- θ e :
-
equilibrium contact angle
- θ :
-
vector of unknown parameters, (Equations (5) and (7))
- θ * :
-
value ofθ that minimizes the objective function Φ, (Equation (7))
- \(\hat \theta\) :
-
true (but unknown) value ofθ
- κ :
-
Μ 2/Μ 1=viscosity ratio
- Μ i :
-
viscosity of fluid i
- σ 2 Μα :
-
variance of the error in theΜth experiment and in theαth equation
- Φ:
-
objective function, (Equation (7))
- Φ(1), Φ(2) :
-
objective function for Model 1 and Model 2, respectively, (Equations (9a)-(b))
- χ i :
-
coupling index for fluidi
- χ o i :
-
value ofχ i free from end and boundary effects
- 1:
-
water
- 2:
-
oil
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Avraam, D.G., Payatakes, A.C. Generalized relative permeability coefficients during steady-state two-phase flow in porous media, and correlation with the flow mechanisms. Transp Porous Med 20, 135–168 (1995). https://doi.org/10.1007/BF00616928
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DOI: https://doi.org/10.1007/BF00616928