Fluid Displacement in a Dual-Permeability Medium with Local Capillary Equilibrium
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Fluid Displacement in a Dual‑Permeability Medium with Local Capillary Equilibrium Andrey Afanasyev1 Received: 7 February 2020 / Accepted: 23 September 2020 / Published online: 3 October 2020 © Springer Nature B.V. 2020
Abstract The solution to the Buckley–Leverett problem in fractured porous media is investigated by applying the dual-porosity dual-permeability model. It is shown that the dual-permeability medium is equivalent to an upscaled homogenized medium at long injection times when local capillary equilibrium is established. Simple analytical relationships for the petrophysical properties and saturation functions of the equivalent medium are derived. The upscaled relative permeability and the fractional flow function can exhibit kinks, resulting in more complicated solutions to the Buckley–Leverett problem as compared to the classical case. It is found that immiscible displacement in fractured porous media leads to the appearance of saturation profiles with several displacement fronts and rarefaction waves. Up to four solution types are possible, which are constrained in a solution map constructed in the space of the dimensionless parameters of the problem. It is shown that, for particular parameters, the displacing fluid moves faster through the matrix than through the fractures. Keywords Fractured porous medium · Dual-porosity · Dual-permeability · Self-similar solution · Waterflooding
1 Introduction Immiscible displacement in a porous medium is complicated by a wide range of profiles that the relative permeability and capillary pressure functions can show. The influence of these saturation functions on the displacement efficiency, e.g. waterflooding of petroleum reservoirs, is often estimated by applying simplified models. The classical theory of twophase displacement in a homogeneous porous medium was developed by Buckley and Leverett (1942). Under the assumption of negligible capillary pressure and gravitational force, they showed that the distribution of saturation in the medium, i.e. the saturation profile, depends only on the fractional flow function F. In order to determine the saturation profile, one needs to find the convex envelope to F (Welge 1952; Fanchi 2006). Straight segments of the envelope correspond to the displacement fronts, and the other segments, in which * Andrey Afanasyev [email protected] 1
Institute of Mechanics, Moscow State University, 1 Michurinskiy Prospekt, Moscow, Russia 119192
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the envelope coincides with F, correspond to the rarefaction waves. For typical shapes of the relative permeability functions and the S-shaped curve of the function F, the saturation profile consists of the leading displacement front (S) followed by the rarefaction (R). Such a solution type can be abbreviated as RS in order to reflect the sequence of waves propagating into the porous medium. The Buckley–Leverett theory is widely used in reservoir modelling and characterization, in particular, for the interpretation of laboratory measurements of relat
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