Operational calculus

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In multiphase flow in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. 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

$q_{i}=-{\frac {\kappa _{i}}{\mu _{i}}}\nabla P_{i}\qquad {\text{for}}\quad i=1,2$

where $q_{i}$ is the flux, $\nabla P_{i}$ 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 (i.e., the effective permeability of phase $i$), as observed through the equation above.

Relative permeability, $\kappa _{\mathit {ri}}$, for phase $i$ is then defined from $\kappa _{i}=\kappa _{\mathit {ri}}\kappa$ as

$\kappa _{\mathit {ri}}=\kappa _{i}/\kappa$

where $\kappa$ is the permeability of the porous medium in single-phase flow, i.e., the absolute permeability. 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.

Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).

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 through which 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 of relative permeability is the Corey correlation which is power law in the water saturation $S_{w}$. If $S_{\mathit {wi}}$ (also denoted $S_{\mathit {wir}}$, or $S_{\mathit {wr}}$, or $S_{\mathit {wc}}$) is the irreducible (minimal) water saturation, and $S_{\mathit {orw}}$ is the residual (minimal) oil saturation after water flooding, we can define a normalized (or scaled) water saturation value

Normalization of water saturation values

Example of Corey approximation in normalized $S_{w}$ coordinates, here $N_{\mathit {o}}$ = $N_{\mathit {w}}=2$.

$S_{\mathit {wn}}(S_{w})={\frac {S_{w}-S_{\mathit {wi}}}{1-S_{\mathit {wi}}-S_{orw}}}$

The Corey correlations of the relative permeability of oil and water are then

$K_{\mathit {row}}(S_{w})=(1-S_{\mathit {wn}}(S_{w}))^{N_{\mathit {o}}}$ and

$K_{\mathit {rw}}(S_{w})=K{_{\mathit {rw}}^{o}}S{_{\mathit {wn}}(S_{w})}^{N_{\mathit {w}}}$

when the permeability basis is oil with irreducible water present.

We note the desired properties

${\begin{aligned}K_{\mathit {row}}(S_{\mathit {wi}})&=1&K_{\mathit {row}}(1-S_{\mathit {orw}})&=0\\K_{\mathit {rw}}(S_{\mathit {wi}})&=0&K_{\mathit {rw}}(1-S_{\mathit {orw}})&=K_{\mathit {rw}}^{o}\end{aligned}}$

The empirical parameters $N_{\mathit {o}}$ and $N_{\mathit {w}}$ can be obtained from measured data either by optimizing to analytical interpretation of measured data, or by optimizing using a core flow numerical simulator to match the experiment(often called history matching). $N_{\mathit {o}}$ = $N_{\mathit {w}}=2$ is sometimes appropriate.The physical property $K_{\mathit {rw}}^{o}$ is called the end point of the water relative permeability, and it is obtained either before or together with the optimizing of $N_{\mathit {o}}$ and $N_{\mathit {w}}$.

In case of gas-water system or gas-oil system there are Corey correlations similar to the oil-water relative permeabilities correlations shown above.

LET-type

The Corey approximation only has one degree of freedom for the oil relative permeability and two degrees of freedom for the water permeability (in $S_{wn}$). The LET-correlation adds more degrees of freedom in order to accommodate the shape of measured relative permeability curves in SCAL experiments.

Example of LET-correlation with L,E,T all equal to 2, and $K_{\mathit {rw}}^{o}=0.6$. Normalized $S_{w}$ coordinates.

The LET-type approximation is described by 3 parameters L, E, T. The correlation for water and oil relative permeability with water injection is thus

$K_{\mathit {rw}}={\frac {{K_{\mathit {rw}}^{o}}{S_{\mathit {wn}}}^{L_{\mathit {w}}}}{{S_{\mathit {wn}}}^{L_{\mathit {w}}}+{E_{\mathit {w}}}{(1-S_{\mathit {wn}})}^{T_{\mathit {w}}}}}$

and

$K_{\mathit {row}}={\frac {(1-S_{\mathit {wn}})^{L_{o}}}{{(1-S_{\mathit {wn}})^{L_{o}}}+{E_{\mathit {o}}}{S_{\mathit {wn}}}^{T_{\mathit {o}}}}}$

written using the same $S_{w}$ normalization as for Corey.

Only $S_{\mathit {wi}}$ , $S_{\mathit {orw}}$ and $K_{\mathit {rw}}^{o}$ have direct physical meaning, while the parameters L, E and T are empirical. The parameter L describes the lower part of the curve, and by similarity and experience the L-values are comparable to the appropriate Corey parameter. The parameter T describes the upper part (or the top part) of the curve in a similar way that the L-parameter describes the lower part of the curve. The parameter E describes the position of the slope (or the elevation) of the curve. A value of one is a neutral value, and the position of the slope is governed by the L- and T-parameters. Increasing the value of the E-parameter pushes the slope towards the high end of the curve. Decreasing the value of the E-parameter pushes the slope towards the lower end of the curve. Experience using the LET correlation indicates that the parameter, L ≥ 1, E > 0 and T ≥ 0.5.

In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.

See also

• Permeability (earth sciences)
• Capillary pressure
• Imbibition
• Drainage
• Buckley–Leverett equation

References

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