## Abstract

Steady state relative permeability experiments are performed by coinjection of two (or more) immiscible fluids. The relative permeabilities can be calculated directly from the stabilized pressure drop and saturation of the core if capillary end effects and transient effects are negligible. In most cases such conditions are difficult to obtain.

This work presents an analytical solution in form of explicit expressions for the spatial profiles of pressure gradients and saturation, average saturation and pressure drop for a core being injected simultaneously with two phases at steady state when capillary end effects are significant. When arbitrary saturation functions are applied, such parameters and distributions can only by obtained by numerical integration.

By assumption of simplified saturation function correlations the differential equation describing steady state can be integrated. A new dimensionless capillary number is obtained which contains the fluid and rock parameters, but also the saturation function parameters (relative permeability and capillary pressure), fluid viscosities, injected flow fraction, total flow rate and more. It is shown that when this number is of magnitude 1, end effects cover parts of the core, but parts of the core are also unaffected. For the end effects are negligible, while for end effects are dominant.

This paper gives the first formal proof of the intercept method from basic assumptions. It is shown that when the inlet saturation is sufficiently close to that of a no capillary pressure situation; the average saturation changes linearly with the inverse of total rate towards the saturation corresponding to no capillary forces; also, the pressure drop divided by the pressure drop of a no end effect situation goes linearly towards 1 with the inverse of total rate.

This work presents an analytical solution in form of explicit expressions for the spatial profiles of pressure gradients and saturation, average saturation and pressure drop for a core being injected simultaneously with two phases at steady state when capillary end effects are significant. When arbitrary saturation functions are applied, such parameters and distributions can only by obtained by numerical integration.

By assumption of simplified saturation function correlations the differential equation describing steady state can be integrated. A new dimensionless capillary number is obtained which contains the fluid and rock parameters, but also the saturation function parameters (relative permeability and capillary pressure), fluid viscosities, injected flow fraction, total flow rate and more. It is shown that when this number is of magnitude 1, end effects cover parts of the core, but parts of the core are also unaffected. For the end effects are negligible, while for end effects are dominant.

This paper gives the first formal proof of the intercept method from basic assumptions. It is shown that when the inlet saturation is sufficiently close to that of a no capillary pressure situation; the average saturation changes linearly with the inverse of total rate towards the saturation corresponding to no capillary forces; also, the pressure drop divided by the pressure drop of a no end effect situation goes linearly towards 1 with the inverse of total rate.

Original language | English |
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Article number | 107249 |

Number of pages | 9 |

Journal | Journal of Petroleum Science and Engineering |

Volume | 192 |

Early online date | 5 Apr 2020 |

DOIs | |

Publication status | Published - Sept 2020 |

### Bibliographical note

AcknowledgmentsDr. Andersen acknowledges the Research Council of Norway and the industry partners, ConocoPhillips Skandinavia AS, Aker BP ASA, Vår Energi AS, Equinor ASA, Neptune Energy Norge AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, and Wintershall DEA, of The National IOR Centre of Norway for support. Andersen also acknowledges the Faculty of Science and Technology at University of Stavanger for funding mobility for this research and the University of Aberdeen for hosting. Dr. Yingfang Zhou would like to acknowledge the support from State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), PLN201602.

CRediT authorship contribution statement

Pål Ø. Andersen: Conceptualization, Methodology, Validation, Investigation, Writing - original draft. Yingfang Zhou: Conceptualization, Methodology, Writing - review & editing.

## Keywords

- capillary end effects
- SCAL
- Steady state experiment
- generalized capillary number
- analytical solutions
- intercept method
- Intercept method
- Generalized capillary number
- Analytical solutions
- Capillary end effects