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# Relative permeability

In multiphase flow in porous media, relative permeability is a dimensionless measure of the effective permeability of each phase. It can be viewed as an adaptation of Darcy's law to multiphase flow.

For two-phase flow in porous media given steady-state conditions, we can write

:$q_i = -frac\left\{kappa_i\right\}\left\{mu_i\right\} abla P_i qquad ext\left\{for\right\} quad i=1,2$

where $q_i$ is the flux, $abla P_i$ is the pressure drop, $mu_i$ is the viscosity. The subscript $i$ indicates that the parameters are for phase $i$.

$kappa_i$ is here the phase permeability, as observed through the equation above.

Relative permeability, $kappa_\left\{mathit\left\{ri$, for phase $i$ is then defined from $kappa_i = kappa_\left\{mathit\left\{rikappa$ as

:$kappa_\left\{mathit\left\{ri = kappa_i / kappa$

where $kappa$ is the permeability of the porous medium in single-phase flow. Relative permeability must be between zero and one.

In applications, relative permeability is often represented as a function of water saturation, however due to capillary hysteresis, one often resorts to one function or curve measured under drainage and one measured under imbibition.

As the flow of each phase is inhibited by the presence of the other phases, the sum of relative permeabilities over all phases is always less than 1.

Assumptions

The above form for Darcy's law is sometimes also called Darcy's extended law, formulated for horizontal, one-dimensional, immiscible multiphase flow in homogeneous and isotropic porous media. The interactions between the fluids are neglected, so this model assumes that the solid porous media and the other fluids form a new porous matrix throughout a phase can flow, implying that the fluid-fluid interfaces remain static in steady-state flow, which is not true, but this approximation has proven useful anyway.

Each of the phase saturation must be larger than the irreducible saturation, and each phase is assumed continuous within the porous medium.

Approximations

Based on experimental data, simplified models of relative permeability as a function of water saturation can be constructed.

Corey-type

An often used approximation is the so-called Corey type, and is polynomial in the water saturation $S_w$ [cite journal|author=R.H. Brooks and A.T. Corey|title=Hydraulic properties of porous media|journal=Hydrological Papers|volume=3|publisher=Colorado State University|date=1964] [] . If $S_mathit\left\{wc\right\}$ is the irreducible (minimal, critical) water saturation and $S_mathit\left\{or\right\}$ is the residual (minimal, critical) oil saturation, we can define a scaled saturation value:$S^* = frac\left\{S_w - S_mathit\left\{wc\left\{1-S_mathit\left\{wc\right\} - S_\left\{or$and approximations of Corey type of the relative permeabilities of water and oil are then:$kappa_mathit\left\{rw\right\} = \left(S^*\right)^n$ and $kappa_mathit\left\{ro\right\}=\left(1-S^*\right)^m$with the properties
* $kappa_mathit\left\{rw\right\}\left(S_mathit\left\{wc\right\}\right) = 0$ and $kappa_mathit\left\{rw\right\}\left(S_mathit\left\{or\right\}\right) = 1$
* $kappa_mathit\left\{ro\right\}\left(S_mathit\left\{wc\right\}\right) = 1$ and $kappa_mathit\left\{ro\right\}\left(S_mathit\left\{or\right\}\right) = 0$and where $n$ and $m$ can be obtained from measured data. $m=n=2$ is sometimes appropriate.

* Permeability (fluid)
* Capillary pressure
* Imbibition
* Drainage
* Buckley–Leverett equation

References

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