A Model for Relative Permeabilities Under Gas Liberation or Condensate Precipitation in Porous Medium
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A Model for Relative Permeabilities Under Gas Liberation or Condensate Precipitation in Porous Medium Wael Al‑Masri1 · Alexander Shapiro1 Received: 10 March 2020 / Accepted: 21 October 2020 © Springer Nature B.V. 2020
Abstract We derive a model for relative permeabilities under two-phase flows in porous media, where the phases may partly mix with each other, and one of the phases may be immobile. Particular cases are the bubble formation in an oil reservoir when the pressure falls below the bubble point, or, similarly, condensation and droplet precipitation in a gas-condensate reservoir. The dependencies for the relative permeabilities on the saturation are derived based on a pore-level model of the porous medium, represented as a capillary network. Distribution of the bubbles or droplets in the network is computed with the application of a method similar to the fundamental statistical physics. Model formula for the conductivities of single capillaries, depending on the numbers of bubbles or droplets in them, is converted to the relative permeabilities of the whole lattice by application of the effective medium formalism. We establish universal correlations between the micro-characteristics of the porous medium, constituting our model, and the parameters in the Corey–Brooks dependencies for relative permeabilities: exponents and limiting saturations. In this way, it is also possible to extend the standard Corey–Brooks formula onto the saturation ranges where one of the phases is immobile. A comparison with the available experimental data indicates the good performance of the model. The experimental data may be fitted by variation of a single parameter, and the data from similar rocks have similar parameter values, which indicates the physical soundness of the model. Keywords Gas liberation · Condensate precipitation · Relative permeability · Microscopic model List of Symbols b,d The functions of the coordinate number determining the critical saturation ci The number of capillaries containing i bubbles C The number of combinations D The number of arrangements fo , fg The fractional flow threshold for oil and gas phase, respectively f The fraction of bubbles/droplets to the maximum capacity of the capillary * Alexander Shapiro [email protected] 1
Department of Chemical and Biochemical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
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F The conductivity distribution function g0 The maximum conductivity G1,G2 The Lagrange constraints g The conductivity of a single capillary kr The relative permeability L The length of a capillary l,t The functions of the coordinate number determining the Brooks–Corey exponents mo , mg The flow thresholds for oil and gas phases, respectively M The total number of bubbles m The number of bubbles in a capillary N The total number of capillaries n The Brooks–Corey exponent P The exponents defining the shapes of the conductivity dependences r The capillary radius S The saturation v The volume
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