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A Cell-Centred CVD-MPFA Finite Volume Method for Two-Phase Fluid Flow Problems with Capillary Heterogeneity and Discontinuity

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Abstract

A novel finite volume method is presented that is applicable to discontinuous capillary pressure fields. The method is developed within the control-volume distributed multi-point flux approximation (CVD-MPFA) framework (Edwards and Rogers in Comput Geosci 02(04):259–290, 1998; Friis et al. in SIAM J Sci Comput 31(02):1192–1220, 2008). Results are computed on structured and unstructured grids that demonstrate the ability of the method to resolve flow in the presence of a discontinuous capillary pressure field for diagonal and full-tensor permeability fields. In addition to an upwind approximation for the saturation equation flux, the importance of upwinding on capillary pressure flux via a hybrid formulation is shown.

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Abbreviations

\(\phi \) :

Pressure

\(\varPhi \) :

Vector of pressures

\(\varphi \) :

Porosity

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

Water mobility

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

Oil mobility

\(\varLambda \) :

Total mobility \(\lambda _\mathrm{w} + \lambda _\mathrm{o}\)

\(\mu \) :

Viscosity

CVD-MPFA:

Control-volume distributed multi-point flux approximation

F :

Flux

\(\varvec{k}\) :

Permeability tensor

\(\varvec{K}\) :

\(\frac{\varvec{k}}{\mu }\)

q :

Quadrature of CVD-MPFA schemes

\(q_\mathrm{w}\) :

Known water phase source term

\(q_\mathrm{o}\) :

Known oil phase source term

s :

Saturation

\(\varvec{v}\) :

Velocity

c:

Capillary

o:

Oil

w:

Water

tr:

Transpose

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Authors and Affiliations

  1. Dimue Technology Co, Ltd., Wuhan, 430223, China

    Raheel Ahmed

  2. ZCCE, College of Engineering, Swansea University, Swansea, SA18EN, Wales, UK

    Yawei Xie & Michael G. Edwards

Authors
  1. Raheel Ahmed
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  2. Yawei Xie
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  3. Michael G. Edwards
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Correspondence to Raheel Ahmed.

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Ahmed, R., Xie, Y. & Edwards, M.G. A Cell-Centred CVD-MPFA Finite Volume Method for Two-Phase Fluid Flow Problems with Capillary Heterogeneity and Discontinuity. Transp Porous Med 127, 35–52 (2019). https://doi.org/10.1007/s11242-018-1179-y

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  • DOI: https://doi.org/10.1007/s11242-018-1179-y

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