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A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

Abstract

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.

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Abbreviations

A :

Cross-sectional area

\(F_{\mathrm{ow}}\) :

Drag force the water phase exerts on the oil phase

\(I_{{i}}\) :

Solid–oil or water interaction friction coefficient

I :

Fluid–fluid interaction drag coefficient

J :

Identity tensor

K :

Absolute or intrinsic permeability

\(K_{\mathrm{eff}}\) :

Effective permeability (= \(k\cdot k_{\mathrm{r}})\)

\(k_{\mathrm{r}}\) :

Relative permeability

\(\hat{{k}}_{\mathrm{rw}} \) :

Generalized relative permeability of water (= \(k_{\mathrm{rww}} + k_{\mathrm{rwo}})\)

\(\hat{{k}}_{\mathrm{ro}} \) :

Generalized relative permeability of oil (= \(k_{\mathrm{row}}\) + \(k_{\mathrm{roo}})\)

\(\hat{{k}}_{\mathrm{rw}\_\mathrm{Coc}} \) :

Generalized relative permeability of water related to co-current flow

\(\hat{{k}}_{\mathrm{rw}\_\mathrm{Cou}} \) :

Generalized relative permeability of water related to counter-current flow

\(M_{\mathrm{im}}\) :

Drag between fluid i and matrix/solid

\(m_{{i}}\) :

Interaction force

\(n_{\mathrm{o}}\) :

Corey exponent to oil

\(n_{\mathrm{w}}\) :

Corey exponent to water

p :

Pressure

\(P_{\mathrm{C}}\) :

Capillary pressure

\((p_{w})_{x}=\partial p_{\mathrm{w}}/\partial x\) :

Water pressure gradient in x-direction

\((p_{\mathrm{o}})_{{x}}=\partial p_{\mathrm{o}}/\partial x\) :

Oil pressure gradient in x-direction

R :

Fluid–fluid interaction term (drag)

\(R_{{i}}\) :

Fluid–solid interaction term for fluid phase i (friction)

s :

Saturation

(\(s_{\mathrm{w}})_{{t}}=\partial s_{\mathrm{w}}/\partial t\) :

Derivative of water saturation with respect to time

\(S_{\mathrm{w}}\) :

Normalized water saturation (to the range where the fluids are mobile)

u :

Interstitial velocity (average)

\(\hat{{u}}_i (x,y,z)\) :

Interstitial velocity

U :

Darcy flux

x :

Coordinate in x-direction parallel to flow direction

y :

Coordinate in y-direction perpendicular to flow direction

\(\alpha \) :

Water–solid interaction saturation exponent

\(\beta \) :

Oil–solid interaction saturation exponent

\(\sigma \) :

Cauchy stress tensor

\(\lambda \) :

Fluid mobility

\(\hat{{\lambda }}_\mathrm{w} \) :

Generalized mobility of water phase (\(= \lambda _{\mathrm{ww}}+\lambda _{\mathrm{wo}})\)

\(\hat{{\lambda }}_\mathrm{o} \) :

Generalized mobility of oil phase (\(= \lambda _{\mathrm{ow}}+\lambda _{\mathrm{oo}})\)

\(\hat{{\lambda }}_{T} \) :

Generalized total mobility (= \(\hat{{\lambda }}_\mathrm{w} +\hat{{\lambda }}_\mathrm{o} )\)

\(\mu \) :

Fluid viscosity

\(\tau _{{i}}\) :

Viscous stress tensor

\(\phi \) :

Porosity

i, j :

Water or oil phase

m :

Matrix or solid

o:

Oil

r:

Relative or residual

t :

Derivative with respect to time

T :

Total

x :

Derivative with respect to x-coordinate

w:

Water

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Acknowledgements

Dag Chun Standnes thanks Statoil ASA for supporting the adjunct prof. position at the University of Stavanger. Steinar Evje and PålØstebø Andersen thank the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS of the National IOR Centre of Norway for support.

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Appendix

Appendix

A1 Positive and negative counter-current relative permeabilities

Equation 45 shows that negative values for counter-current relative permeabilities may occur for certain combination of input parameter values. For both expressions, we note that the denominator is always positive and only the negative sign corresponds to adding negative terms in the nominator. The only way to have negative \(k_{\mathrm{rw}}\) is to have counter-current flow (“-”) and low water saturations (\(s_{\mathrm{w}}- s_{\mathrm{o}} < 0\)). If the fluid–fluid interaction is weak, \(k_{\mathrm{rw}}\) can stay positive for all saturations. In particular, it follows from Eq. 45 that a positive \(k_{\mathrm{rw}}\) for given saturations is equivalent to have

$$\begin{aligned} I<\frac{I_\mathrm{o} }{\mu _\mathrm{w} s_\mathrm{o} S_\mathrm{o}^{-\beta } \left( {s_\mathrm{o} -s_\mathrm{w} } \right) }, \end{aligned}$$
(A1)

which means that if

$$\begin{aligned} I<\frac{I_\mathrm{o} }{\mu _\mathrm{w} (1-s_{\mathrm{wr}} )(S_\mathrm{o}^*)^{-\beta }\left( {1-s_{\mathrm{or}}^ -s_{\mathrm{wr}} } \right) } \end{aligned}$$
(A2)

(where \(S_{\mathrm{o}}^*=\frac{0.5-s_{\mathrm{or}}}{1-s_{\mathrm{or}} -s_{\mathrm{wr}}}\) is the scaled saturation evaluated when the saturations are equal), \(k_{\mathrm{rw}}\) will be positive for all saturations. By similar derivation, we obtain that:

$$\begin{aligned} I<\frac{I_\mathrm{w} }{\mu _\mathrm{o} (1-s_{\mathrm{or}} )(S_\mathrm{w}^*)^{-\alpha }\left( {1-s_{\mathrm{or}} -s_{\mathrm{wr}} } \right) }, \quad S_\mathrm{w}^*=\frac{0.5-s_{\mathrm{wr}}}{1-s_{\mathrm{or}} -s_{\mathrm{wr}}}, \end{aligned}$$
(A3)

produces strictly positive counter-current \(k_{\mathrm{ro}}\).

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Standnes, D.C., Evje, S. & Andersen, P.Ø. A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions. Transp Porous Med 119, 707–738 (2017). https://doi.org/10.1007/s11242-017-0907-z

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Keywords

  • Relative permeability
  • Generalized mobility
  • Interfacial interactions
  • Steady state
  • Coupled multiphase flow