Analytical solutions for forced and spontaneous imbibition accounting for viscous coupling
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OriginalversjonAndersen, P.Ø., Nesvik, E.K., Standnes, D.C. (2019) Analytical solutions for forced and spontaneous imbibition accounting for viscous coupling. Journal of Petroleum Science and Engineering 10.1016/j.petrol.2019.106717
Fluid-ﬂuid momentum transfer can cause higher ﬂow resistance when ﬂuids ﬂow in opposite directions as compared to the same direction. Conventional modelling of ﬂow in porous media using simple, saturation dependent relative permeabilities does not account for such variations. We consider a generalized theory for multiphase ﬂow in porous media based on mixture theory, where ﬂuid mobilities follow from water-rock, oil-rock and water-oil interaction terms deﬁned in momentum equations. Under strictly co- or counter-current ﬂow modes, the generalized model produces explicit relative permeability expressions dependent on the ﬂow mode, saturations, viscosities and interaction parameters. New expressions for counter-current relative permeabilities are derived assuming zero net ﬂux, representative of counter-current spontaneous imbibition. These functions are compared to previously derived co-current relative permeabilities (assuming equal phase pressure gradients). The functions are incorporated into analytical solutions for forced and spontaneous imbibition (FI and SI) using the theory by Buckley and Leverett (1942) and McWhorter and Sunada (1990), respectively. Our results show that when accounting for viscous coupling; Counter-current relative permeabilities are always lower than co-current ones, including the end points. Both phase curves are reduced by the same saturation dependent coeﬃcient. Increased viscous coupling in the FI case led to a more eﬀective displacement, seen as an increased front saturation and average water saturation behind the front. For counter-current SI, increased viscous coupling resulted in lower imbibition rate. Increased viscosities reduces both oil and water counter-current relative permeabilities, and predict greater reduction in imbibition rate than only modifying the viscosities. The analytical solutions for SI were in agreement with numerical solutions of both a conventional and generalized model. The solutions for SI could be scaled exactly to a square root of time curve for arbitrary input parameters in the generalized model, especially including the strength of viscous coupling.