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A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

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Abstract

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.

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Abbreviations

D :

Velocity of the shock front

f :

Fractional flow of water phase

\({{f}_{\rm s}^{{\prime}}}\) :

Derivative of fractional flow w.r.t. saturation

F w_wet :

Fraction of the water-wet rock surface

J :

Leverett function

k :

Permeability

k ro :

Oil relative permeability

k ro_max :

Maximum oil relative permeability

k rw :

Water relative permeability

k rw_max :

Maximum water relative permeability

L :

Core length

n :

Exponent for Corey-type power law

p :

Pressure

P c :

Capillary pressure

s :

Water phase saturation

s i :

Initial water saturation

s ir :

Irreducible water saturation

s 0 :

Maximum water saturation, equals to unity minus residual/critical oil saturation

t :

Time

T :

Dimensionless time, p.v. injection

T p :

Pore volumes produced

T op :

Pore volumes of oil produced

T wp :

Pore volumes of water produced

U :

Velocity

x :

Linear coordinate, from injectors towards producers

X :

Dimensionless linear coordinate

λ :

Total mobility of the oil-water fluid

Ψ :

Potential for capillary forces

\({\varepsilon}\) :

Capillary-viscous ratios

\({\xi}\) :

Self-sharpening large scale (slow) coordinate

\({\omega}\) :

Travelling wave fast coordinate near to the displacement front

\({\zeta}\) :

Fast coordinate near to the core outlet

\({\phi}\) :

Porosity

μ :

Fluid viscosity

σ :

Interfacial tension

θ :

Contact angle

W, O:

Water, oil

i:

Initial (of water saturation)

0:

Boundary value on the injector (saturations, flux)

BL:

Buckley–Leverett

BT:

Breakthrough

SZ:

Stabilized zone

ee:

End effect

min:

Minimum

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Authors and Affiliations

  1. University of New South Wales, Sydney, NSW, 2052, Australia

    F. Hussain & Y. Cinar

  2. University of Adelaide, Adelaide, SA, 5005, Australia

    P. Bedrikovetsky

Authors
  1. F. Hussain
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  2. Y. Cinar
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  3. P. Bedrikovetsky
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Correspondence to Y. Cinar.

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Hussain, F., Cinar, Y. & Bedrikovetsky, P. A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure. Transp Porous Med 92, 187–212 (2012). https://doi.org/10.1007/s11242-011-9897-4

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