Relative permeability

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 {k_{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 $i$ .

$k_{i}$ is here the phase permeability (i.e., the effective permeability of phase $i$ ), as observed through the equation above.

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

k_{\mathit {ri}}=k_{i}/k

where $k$ 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, owing to capillary hysteresis one often resorts to a function or curve measured under drainage and another 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). [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 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 saturation (e.g. water saturation) can be constructed. If $S_{\mathit {wir}}$ (also denoted $S_{\mathit {wr}}$ , or $S_{\mathit {wc}}$ ) is the irreducible (or minimal or connate) water saturation, and $S_{\mathit {orw}}$ is the residual (minimal) oil saturation after water flooding (note that it is the oil saturation), 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_{\mathit {wn}}(S_{w})={\frac {S_{w}-S_{\mathit {wir}}}{1-S_{\mathit {wir}}-S_{\mathit {orw}}}}

and a normalized oil saturation value

S_{\mathit {on}}=1-S_{\mathit {wn}}={\frac {S_{o}-S_{\mathit {orw}}}{1-S_{\mathit {wir}}-S_{\mathit {orw}}}}

We cannot normalize both oil relative permeability and water relative permeability, but we can normalize oil relative permeability by selecting oil phase permeability, with $S_{\mathit {wir}}$ present, as absolute (single phase) permeability. This gives the desired properties

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

Corey-type

An often used approximation of relative permeability is the Corey correlation [2] [3] [4] which is a power law in saturation. The Corey correlations of the relative permeability for oil and water are then

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

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

when the permeability basis is oil with irreducible water present.

The empirical parameters $N_{\mathit {o}}$ and $N_{\mathit {w}}$ are called curve shape parameters or simply shape parameters, and they 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-correlation or Corey model has only one degree of freedom for the shape of each relative permeability curve, the shape parameter N. The LET-correlation[5] [6] 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 $S_{\mathit {wir}}$ , $S_{\mathit {orw}}$ 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 the following reasonable ranges for the parameters L, E, and T: L ≥ 0.1, E > 0 and T ≥ 0.1.

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

Evaluations

After Morris Muskat et alios established the concept of relative permeability in late 1930'ies, the number of correlations, i.e. models, for relative permeability has steadily increased. This creates a need for evaluation of the most common correlations at the current time. Two of the latest (per 2019) and most thorough evaluations are done by Moghadasi et alios[7] and by Sakhaei et alios[8]. Moghadasi et alios[7] evaluated Corey, Chierici and LET correlations for oil/water relative permeability using a sophisticated method that takes into account the number of uncertain model parameters. They found that LET, with the largest number (three) of uncertain parameters, was clearly the best one for both oil and water relative permeability. Sakhaei et alios[8] evaluated 10 common and widely used relative permeability correlations for gas/oil and gas/condensate systems, and found that LET showed best agreement with experimental values for both gas and oil/condensate relative permeability.

Relative permeability versus TEM-function

Relative permeability is just one of the factors that affect fluid flow dynamics, and therefore can’t fully capture dynamic flow behavior of porous media.[9][10] A criterion/metric has been established to characterize dynamic characteristics of rocks, known as True Effective Mobility or TEM-function.[10] TEM-function is a function of Relative permeability, Porosity, permeability and fluid Viscosity, and can be determined for each fluid phase separately. TEM-function has been derived from Darcy's law for multiphase flow.[10]

$TEM={\frac {kk_{\mathit {r}}}{\phi \mu }}$

in which k is the permeability, kr is the Relative permeability, φ is the Porosity, and μ is the fluid Viscosity. Rocks with better fluid dynamics (i.e., experiencing a lower pressure drop in conducting a fluid phase) have higher TEM versus saturation curves. Rocks with lower TEM versus saturation curves resemble low quality systems.[10]

While TEM-function controls the dynamic behavior of a system, the Relative permeability alone has conventionally been used to classify different fluid flow systems. Despite Relative permeability is itself a function of several parameters including permeability, Porosity and Viscosity, the dynamic behavior of systems may not necessarily be fully captured by this single source of information and, if used, it can even result in misleading interpretations.[10]

TEM-function in analyzing Relative permeability data is analogous with Leverett J-function in analyzing Capillary pressure data.[10]

Averaging Relative permeability curves

In multiphase systems Relative permeability curves of each fluid phase (i.e., water, oil, gas, CO2) can be averaged using the concept of TEM-function as:[10]

${\text{Average kr}}={\frac {\sum _{i=1}^{n}TEM_{i}}{\sum _{i=1}^{n}({\frac {k}{\phi \mu }})_{i}}}={\frac {\sum _{i=1}^{n}({\frac {kk_{\mathit {r}}}{\phi \mu }})_{i}}{\sum _{i=1}^{n}({\frac {k}{\phi \mu }})_{i}}}$

References

Bravo, M.C.; Araujo, M. (2008). "Analysis of the Unconventional Behavior of Oil Relative Permeability during Depletion Tests of Gas-Saturated Heavy Oils". International Journal of Multiphase Flow. 34 (5): 447–460. doi:10.1016/j.ijmultiphaseflow.2007.11.003.
Goda, H.M.; Behrenbruch, P. (2004). Using a Modified Brooks-Corey Model to Study Oil-Water Relative Permeability for Diverse Pore Structures. Paper SPE-88538-MS Presented at SPE Asia Pacific Oil and Gas Conference and Exhibition, Perth, Australia. doi:10.2118/88538-MS. ISBN 978-1-55563-979-2.
Brooks, R.H.; Corey, A.T. (1964). "Hydraulic properties of porous media". Hydrological Papers. 3.
Corey, A.T. (Nov 1954). "The Interrelation Between Gas and Oil Relative Permeabilities". Prod. Monthly. 19 (1): 38–41.
Lomeland, F.; Ebeltoft, E.; Thomas, W.H. (2005). "A New Versatile Relative Permeability Correlation" (PDF). Reviewed Proceedings of the 2005 International Symposium of the SCA, Abu Dhabi, United Arab Emirates, October 31 - November 2, 2005, Available at WWW.scaweb.org/, SCA2005-32.
Lomeland, F. (2018). "Overview of the LET Family of Versatile Correlations for Flow Functions" (PDF). Reviewed Proceedings of the 2018 International Symposium of the SCA, Trondheim, Norway, 27 - 30 August, 2018, Available at WWW.scaweb.org/, SCA2018-056.
Moghadasi, L.; Guadagnini, A.; Inzoli, F.; Bartosek, M. (2015). "Interpretation of two-phase relative permeability curves through multiple formulations and model quality criteria". Article Published in Journal of Petroleum Science and Engineering - JPSE - by Elsevier B.V.,23 October, 2015.
Sakhaei, Z.; Azin, R.; Osfouri, S. (2016). "Assessment of empirical/theoretical relative permeability correlations for gas-oil/condensate systems". Paper Presented at the 1st Persian Gulf Oil, Gas and Petrochemical Biennal Conference Held at the Persian Gulf University In, Bushehr, Iran, 20 April, 2016.
Mirzaei-Paiaman, A., Ostadhassan, M., Rezaee, R., Saboorian-Jooybari, H., and Z. Chen. 2018. A new approach in petrophysical rock typing. J. Pet. Sci. Eng., 166, 445-464
Mirzaei-Paiaman, A.; Saboorian-Jooybari, H.; Chen, Z.; Ostadhassan, M. (2018). "New technique of True Effective Mobility (TEM-Function) in dynamic rock typing: Reduction of uncertainties in relative permeability data for reservoir simulation". Article Published in Journal of Petroleum Science and Engineering - JPSE - by Elsevier B.V., August, 2019.

External links

Relative Permeability Curves

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[Bravo2008-1] Bravo, M.C.; Araujo, M. (2008). "Analysis of the Unconventional Behavior of Oil Relative Permeability during Depletion Tests of Gas-Saturated Heavy Oils". International Journal of Multiphase Flow. 34 (5): 447–460. doi:10.1016/j.ijmultiphaseflow.2007.11.003.

[Goda2004-2] Goda, H.M.; Behrenbruch, P. (2004). Using a Modified Brooks-Corey Model to Study Oil-Water Relative Permeability for Diverse Pore Structures. Paper SPE-88538-MS Presented at SPE Asia Pacific Oil and Gas Conference and Exhibition, Perth, Australia. doi:10.2118/88538-MS. ISBN 978-1-55563-979-2.

[Brooks1964-3] Brooks, R.H.; Corey, A.T. (1964). "Hydraulic properties of porous media". Hydrological Papers. 3.

[Corey1954-4] Corey, A.T. (Nov 1954). "The Interrelation Between Gas and Oil Relative Permeabilities". Prod. Monthly. 19 (1): 38–41.

[Lomeland2005-5] Lomeland, F.; Ebeltoft, E.; Thomas, W.H. (2005). "A New Versatile Relative Permeability Correlation" (PDF). Reviewed Proceedings of the 2005 International Symposium of the SCA, Abu Dhabi, United Arab Emirates, October 31 - November 2, 2005, Available at WWW.scaweb.org/, SCA2005-32.

[Lomeland2018-6] Lomeland, F. (2018). "Overview of the LET Family of Versatile Correlations for Flow Functions" (PDF). Reviewed Proceedings of the 2018 International Symposium of the SCA, Trondheim, Norway, 27 - 30 August, 2018, Available at WWW.scaweb.org/, SCA2018-056.

[Moghadasi2015-7] Moghadasi, L.; Guadagnini, A.; Inzoli, F.; Bartosek, M. (2015). "Interpretation of two-phase relative permeability curves through multiple formulations and model quality criteria". Article Published in Journal of Petroleum Science and Engineering - JPSE - by Elsevier B.V.,23 October, 2015.

[Sakhaei2016-8] Sakhaei, Z.; Azin, R.; Osfouri, S. (2016). "Assessment of empirical/theoretical relative permeability correlations for gas-oil/condensate systems". Paper Presented at the 1st Persian Gulf Oil, Gas and Petrochemical Biennal Conference Held at the Persian Gulf University In, Bushehr, Iran, 20 April, 2016.

[9] Mirzaei-Paiaman, A., Ostadhassan, M., Rezaee, R., Saboorian-Jooybari, H., and Z. Chen. 2018. A new approach in petrophysical rock typing. J. Pet. Sci. Eng., 166, 445-464

[Mirzaei-Paiaman2019-10] Mirzaei-Paiaman, A.; Saboorian-Jooybari, H.; Chen, Z.; Ostadhassan, M. (2018). "New technique of True Effective Mobility (TEM-Function) in dynamic rock typing: Reduction of uncertainties in relative permeability data for reservoir simulation". Article Published in Journal of Petroleum Science and Engineering - JPSE - by Elsevier B.V., August, 2019.