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

where vD is the magnitude of the (superficial) Darcy velocity, v=v is the phase velocity of the (interstitial) fluid, and d⁢P 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 W⁢t0, O⁢t0 denote the fluid configuration inside the porous medium at some initial time t0 and consider flooding at constant injection rates QW≠0, QO=0 or QW=0, QO≠0 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

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 vW⁢t, vO⁢t approach constant values. [4.2.0.3] Specifically,

because QW≠0, QO=0 for water flooding and QW=0, QO≠0 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

where i=W,O indicates the two phases and the relative permeability functions kWr⁢S, kOr⁢S 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 PO⁢t-PW⁢tfor t→∞ reflects microscopic capillarity on macroscales. [4.2.1.2] The difference is assumed to depend only on saturation

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

are the irreducible water saturation SW⁢i and the residual oil saturation SO⁢r. [4.2.1.6] Both parameters SW⁢i and SO⁢r are assumed to be small but nonvanishing, i.e. 0<SW⁢i≪1 and 0<SO⁢r≪1. [4.2.1.7] With SW⁢i and SO⁢r the parameter Pb in eq. (24) is defined as

[page 5, §0]    [5.1.0.1] If there is hysteresis, so that the values Pbim≠Pbdr 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

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

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 σW⁢O between the two phases and that the quantity Ai=k⁢Pb/μi⁢ϕ 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].