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# V Macroscopic sample scale L

[4.1.2.1] Consider again the stationary limit t of fluid flow in P. [4.1.2.2] 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

 vD=ϕ⁢v=kμ⁢d⁢PL, (20)

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. [4.1.2.3] Here μ is the fluid viscosity, k is the (absolute) permeability and ϕ the porosity of the porous medium.

[4.1.3.1] The generalization of Darcy’s law to two immiscible fluids requires some assumptions. [4.1.3.2] 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. [4.1.3.3] It is observed experimentally and then assumed theoretically that for long times t the total volume fractions

 limt→∞⁡W⁢tP=limt→∞⁡SW⁢t=SW=S (21a) limt→∞⁡O⁢tP=limt→∞⁡SO⁢t=SO=1-S (21b)

approach constant limiting values. [4.2.0.1] The values of SW, SO will depend on the phase pressures PW, PO. [4.2.0.2] Because the injection rates are constant the homogenized phase velocities vWt, vOt approach constant values. [4.2.0.3] Specifically,

 limt→∞⁡vW⁢t=vW, limt→∞⁡vO⁢t=0 (22a) limt→∞⁡vW⁢t=0, limt→∞⁡vO⁢t=vO (22b)

because QW0, QO=0 for water flooding and QW=0, QO0 for oil flooding. [4.2.0.4] 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

 vD=ϕ⁢vi=k⁢kir⁢Sμi⁢d⁢PiL (23)

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. [4.2.0.5] 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].

[4.2.1.1] The asymptotic pressure difference POt-PWtfor t reflects microscopic capillarity on macroscales. [4.2.1.2] The difference is assumed to depend only on saturation

 limt→∞⁡PO⁢t-PW⁢t=Pc⁢S=Pb⁢Pc^⁢S (24)

but not on the phase velocities although dynamic capillary effects have been observed [38, 39]. [4.2.1.3] The function PcS is called capillary pressure and Pc^S is its dimensionless form. [4.2.1.4] The functions PcS and kirS are defined on the interval SWi,1-SOr. [4.2.1.5] The parameters SWi, SOr, defined as solutions of the equations

 kWr⁢SW⁢i=0 (25a) kOr⁢1-SO⁢r=0, (25b)

are the irreducible water saturation SWi and the residual oil saturation SOr. [4.2.1.6] Both parameters SWi and SOr are assumed to be small but nonvanishing, i.e. 0<SWi1 and 0<SOr1. [4.2.1.7] With SWi and SOr the parameter Pb in eq. (24) is defined as

 Pb=Pc⁢SW⁢i+1-SO⁢r2. (26)

[page 5, §0]    [5.1.0.1] If there is hysteresis, so that the values PbimPbdr differ, then Pb=Pbim+Pbdr/2 will be used. [5.1.0.2] The dimensionless capillary pressure function Pc^S can be positive and negative. [5.1.0.3] The relative permeabilities are positive and monotone functions.

[5.1.1.1] 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

 Fi⁢S,vi,L=(viscous pressure drop in phase i)(capillary pressure) =d⁢PiPO-PW (27a) =μi⁢ϕ⁢vi⁢Lk⁢kir⁢S⁢Pc⁢S (27b) =Caikir⁢S⁢Pc^⁢S (27c) =fi⁢S,Cai i=W,O (27d)

or as a function of S and the macroscopic capillary number Cai in the last two equalities. [5.1.1.2] The macroscopic capillary number Cai is defined as in [9] by

 Cai=μi⁢ϕ⁢vi⁢Lk⁢Pb⁢i=W,O (28)

and it depends not only on fluid properties, but also on properties of the porous medium such as porosity and permeablity. [5.1.1.3] Equation (27) seems to be a new result that has been overlooked so far. [5.1.1.4] 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. [5.1.1.5] For a derivation of Cai from the traditional macroscale equations of motion see [32, 9].