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118 8. Model of reactive transport of an oil-soluble chemical in porous mediawhere C ′ [mol m −3 ] and C [kg m −3 ] are the molar and mass concentration, respectively.Further, k h ′ and k h are the corresponding hydrolysis rate constants, and β and γ arereaction order coefficients. For a higher-order reaction term (i.e. β +γ > 1) the hydrolysisconstant k h ′ is scaled inconveniently. A scale factor Z is introduced such that k′ h can berelated to a first-order reaction rate k [s −1 ], i.e.k ′ h = Zk, (8.21)where Z [(m 3 mol −1 ) β+γ−1 ] can be expressed in terms of the molar volume of TMOS V Tand the molar volume of water V W . In our model we chooseZ = (V T ) β (V W ) γ−1 . (8.22)8.3 Method of solution8.3.1 General equationThe equations are solved based on the IMPES method [142]. First the mass balanceequations are rewritten in terms of the oil pressure using appropriate assumptions andrelations. Subsequently, the equations are discretized. In the IMPES method first thepressure is solved for implicitly, after which the saturation (e.g. S o ) is solved for explicitly.In our extended model the mass fractions of the components are calculated implicitly, andthe mass transfer and reaction terms are updated each time step [140].Summation of the material balance equation for each component, given by Eq. 8.1,and applying the closure conditions result in the following material balance equations forthe oleic phase and the aqueous phase, respectively:ϕ ∂(ρ oS o )= ∂ ( )∂P oρ o λ o − Uw T , (8.23)∂t ∂z ∂zϕ ∂(ρ wS w )∂t= ∂ ∂zExpanding the left-hand side of Eqs. 8.23 and 8.24 yields∂ρ oϕS o∂t + ϕρ ∂S oo = ∂ ( )∂P oρ o λ o∂t ∂z ∂zand∂ρ wϕS w∂t + ϕρ ∂S ww∂t( )∂P wρ w λ w + Uw T . (8.24)∂z= ∂ ∂z+ U T o , (8.25)( )∂P wρ w λ w + Uw T . (8.26)∂zDividing Eq. 8.25 by ρ o , and dividing Eq. 8.26 by ρ w , and adding both resulting equationsyieldsϕ S o ∂ρ oρ o ∂t + ϕS w ∂ρ w= 1 ( )∂ ∂P oρ o λ o + 1 ( ) (∂ ∂P w 1ρ w λ w + − 1 )Uw T .ρ w ∂t ρ o ∂z ∂z ρ w ∂z ∂z ρ w ρ o(8.27)