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116 8. Model of reactive transport of an oil-soluble chemical in porous mediawhere P c is the capillary pressure, which can be modeled using a Leverett-J function[118]. For a water-wet medium the function is given byP c = σ √ () 1/ξ ( ) −1/ξ ϕ 0.5 − Swi Sw − S wi, (8.8)2 K abs 1 − S wi − S or 1 − S wi − S orwhere σ is the interfacial tension (IFT), and ξ is a sorting factor.HM 6 5 E M = J A HFig. 8.2: Schematic view of the capillary bundle system which is used as an upscaling modelfor the pore-scale mass transfer of TMOS. The tube radius is defined as r w .8.2.3 Initial and boundary conditionsInitially the core is homogeneously saturated at the irreducible water saturation S wi . Theaqueous phase only consists of water and the oleic phase only consists of oil. The pressurein the oleic phase is atmospheric. Therefore the initial conditions areP o (z, 0) = 1 ; S w (z, 0) = S wi ; w T o (z, 0) = 0 ; w W w (z, 0) = 1. (8.9)For t > 0, the pressures P α are governed by the injection rate Q, the outlet pressureP out and the capillary pressure P c . The outlet pressure P out is equal to the atmosphericpressure, and the capillary pressure at the outlet is zero. The injection of the mixturestops at t = t s (i.e. the shut-in period of the core starts when the injection stops). Theboundary conditions are therefore given by8.2.4 Mass transferwo T (−L/2, t) = w 0 t > 0, (8.10)∂P o (−L/2, t)Q = −Aλ o∂z0 > t > t s , (8.11)∂P o (−L/2, t)= 0∂zt > t s , (8.12)P o (L/2, t) = P w (L/2, t) = P out t > 0. (8.13)The flux term UαX is non-zero only for X = T (i.e. for TMOS). In order to describe themass transfer of TMOS between both phases an up-scaling model is needed which linksthe microscopic mass transfer (on the pore scale) with the macroscopic transport model.The geometry of the pore space in the stone is complex and difficult to include in themodel. Therefore we adopt a simplified geometry in the form of a capillary bundle system