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V Macroscopic sample scale L

[] Consider again the stationary limit t of fluid flow in P. [] On the scale of a macroscopic sample, the viscous forces are dominated by wall friction and quantified by Darcy’s law for single phase flow


where vD is the magnitude of the (superficial) Darcy velocity, v=v is the phase velocity of the (interstitial) fluid, and dP is the magnitude of the viscous pressure drop across a region of length L. [] Here μ is the fluid viscosity, k is the (absolute) permeability and ϕ the porosity of the porous medium.

[] The generalization of Darcy’s law to two immiscible fluids requires some assumptions. [] Let Wt0, Ot0 denote the fluid configuration inside the porous medium at some initial time t0 and consider flooding at constant injection rates QW0, QO=0 or QW=0, QO0 with water or oil. [] It is observed experimentally and then assumed theoretically that for long times t the total volume fractions


approach constant limiting values. [] The values of SW, SO will depend on the phase pressures PW, PO. [] Because the injection rates are constant the homogenized phase velocities vWt, vOt approach constant values. [] Specifically,


because QW0, QO=0 for water flooding and QW=0, QO0 for oil flooding. [] Under these assumptions Darcy’s law is generalized from single phase flow to two-phase flow as discussed in [33, 34, 35, 36, 37] to


where i=W,O indicates the two phases and the relative permeability functions kWrS, kOrS quantify the change in permeability for phase i due to presence of the second phase. [] Note that the asymptotic water configuration W must be path connected and percolating from inlet and outlet2 in case (22a) and the same holds for O in case (22b).
2: Limitations and previously unnoticed implicit assumptions for the validity of (23) were first addressed in [15] and then formulated mathematically and explicitly as the residual decoupling approximation in [18, 19, 20].

[] The asymptotic pressure difference POt-PWtfor t reflects microscopic capillarity on macroscales. [] The difference is assumed to depend only on saturation


but not on the phase velocities although dynamic capillary effects have been observed [38, 39]. [] The function PcS is called capillary pressure and Pc^S is its dimensionless form. [] The functions PcS and kirS are defined on the interval SWi,1-SOr. [] The parameters SWi, SOr, defined as solutions of the equations


are the irreducible water saturation SWi and the residual oil saturation SOr. [] Both parameters SWi and SOr are assumed to be small but nonvanishing, i.e. 0<SWi1 and 0<SOr1. [] With SWi and SOr the parameter Pb in eq. (24) is defined as


[page 5, §0]    [] If there is hysteresis, so that the values PbimPbdr differ, then Pb=Pbim+Pbdr/2 will be used. [] The dimensionless capillary pressure function Pc^S can be positive and negative. [] The relative permeabilities are positive and monotone functions.

[] The force balance between the viscous and capillary forces of macroscale two-phase flow can now be expressed either as a function of S, vi and L

FiS,vi,L=(viscous pressure drop in phase i)(capillary pressure)=dPiPO-PW(27a)

or as a function of S and the macroscopic capillary number Cai in the last two equalities. [] The macroscopic capillary number Cai is defined as in [9] by


and it depends not only on fluid properties, but also on properties of the porous medium such as porosity and permeablity. [] Equation (27) seems to be a new result that has been overlooked so far. [] Note that Ca does not explicitly depend on the interfacial tension σWO between the two phases and that the quantity Ai=kPb/μiϕ with i=W,O is dimensionally not a velocity, but a specifc action, i.e. action per unit mass. [] For a derivation of Cai from the traditional macroscale equations of motion see [32, 9].