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8.2. Theoretical background 115A mixture of TMOS/oil, with a fraction woT = w 0 , is injected with an injection rate Qat the inlet of the core, which is found at z = −L/2, and subsequently fluids are producedat the outlet which is found at z = L/2. As the TMOS comes in contact with water,the TMOS partitions between both phases. In the aqueous phase the TMOS reacts withwater. The reaction products are silicic acid and methanol. The condensation of silicicacid is neglected. The effect of gravity is also neglected.8.2.2 Material balance equationThe reactive transport can be described by the following material balance equation (withrespect to each species X and phase α):ϕ ∂ ( )ρ α S α wαX = ∂ ()∂P αλ α∂t ∂z ∂z ρ αwαX + Uα X + ϕS α Rα X , (8.1)where ρ α , λ α , P α , U X α and R X α are the density, the mobility, the pressure, the flux ofTMOS, and the reaction terms of phase α, respectively. The mobilities λ α are given byλ α = K absk rαµ α, (8.2)where k rα is the relative permeability, and µ α the viscosity of phase α. The relativepermeabilities k rw and k ro are modeled using the Brooks-Corey relations [141], given byk rw = k ′ rw (S we ) 2+3ζζ, (8.3)respectively,[ ]k ro = k ro ′ (1 − S we ) 2 1 − (S we ) 2+ζζ. (8.4)Here ζ is a sorting factor and the effective saturation S we is defined as( )Sw − S wiS we =, (8.5)1 − S wi − S orwhere S or is the residual oil saturation, and S wi is the irreducible (or residual) watersaturation.The viscosity of the oleic phase, µ o , is a function of the TMOS fraction. Here we usethe Grunberg-Nissan relation to calculate the viscosity, which is given by [61]log µ o = x log µ T + (1 − x) log µ O , (8.6)where x is the mole fraction of TMOS in oil, and µ T and µ O are the viscosities of pureTMOS and pure oil, respectively. In absence of a condensation reaction, the viscosity ofthe aqueous phase remains constant.The pressure in the oleic phase, P o , is related to the pressure in the aqueous phase,P w , byP c = P o − P w , (8.7)