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Multiphase Flow in Porous Media pp 135–168Cite as

Generalized Relative Permeability Coefficients during Steady-State Two-Phase Flow in Porous Media, and Correlation with the Flow Mechanisms

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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, κ, were changed systematically, whereas the wettability (contact angle θ e ), the coalescence factor Co, 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. The coupling 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

Bo:

bond number.

Ca:

capillary number= μ 1 q 1/γ 12 ωl.

Co:

coalescence factor (effective probability of coalescence, given a collision between two ganglia in the porous medium).

C ik :

parameters used in the functional representation of k ri in terms of cubic B-splines, (Equation (l0a)).

C ijk :

parameters used in the functional representation of k rij in terms of cubic B-splines, (Equation (10b)).

eµα :

residual for the μth experiment and the αth equation of the model, (Equation (6)).

e:

vector of residuals, e µα .

k:

absolute permeability.

kri :

(conventional) relative permeability to fluid i.

k ori :

value of k ri free from end and boundary effects.

krij :

generalized relative permeability coefficients.

k orij :

value of k 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.

lα :

number of unknown parameters in the αth equation of the model.

N:

number of cubic B splines used to represent k ri or k rij ((Equation (10a,b)).

n:

number of experimental data.

qi :

flowrate of fluid i.

Si :

saturation of fluid i.

υi :

superficial velocity of fluid i.

V:

covariance matrix of the true errors ε μα , for all experiments (μ) and equations (α) of the model.

W:

weighing matrix, (Equation (7)).

ω:

width of the network.

x:

vector of the independent variables, (Equation (5)).

y:

vector of the dependent variables.

γ 12 :

interfacial tension.

εμα :

true error for the µth experiment and the αth equation.

ΔP i :

pressure drop (negative) in fluid i, along a distance L.

θe :

equilibrium contact angle.

θ:

vector of unknown parameters, (Equations (5) and (7)).

θ*:

value of θ that minimizes the objective function Φ, (Equation (7)).

\(\hat \theta\) :

true (but unknown) value of θ.

κ:

µ 2/µ 2 = viscosity ratio.

μi :

viscosity of fluid i.

σ 2 μα :

variance of the error in the µth experiment and in the αth equation.

Φ:

objective function, (Equation (7)).

Φ(1),Φ(2) :

objective function for Model 1 and Model 2, respectively, (Equations (9a)–(b)).

χi :

coupling index for fluid i.

χ oi :

value of χi free from end and boundary effects.

1:

water.

2:

oil.

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Authors and Affiliations

  1. Department of Chemical Engineering, University of Patras, PO Box 1414, GR 26500, Patras, Greece

    D. G. Avraam & A. C. Payatakes

  2. Institute of Chemical Engineering and High Temperature Chemical Processes, PO Box 1414, GR 26500, Patras, Greece

    D. G. Avraam & A. C. Payatakes

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  1. D. G. Avraam
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  1. Institut de Physique du Globe de Paris, France

    Pierre M. Adler

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Avraam, D.G., Payatakes, A.C. (1995). Generalized Relative Permeability Coefficients during Steady-State Two-Phase Flow in Porous Media, and Correlation with the Flow Mechanisms. In: Adler, P.M. (eds) Multiphase Flow in Porous Media. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2372-5_6

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  • DOI: https://doi.org/10.1007/978-94-017-2372-5_6

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