V Macroscopic sample scale L

[4.1.2.1] Consider again the stationary limit $t\to\infty$ of fluid flow in $\mathbb{P}$ . [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

$\displaystyle v_{{\mathrm{D}}}=\phi v=\frac{k}{\mu}\frac{\mathrm{d}\! P}{L},$

(20)

where $v_{{\mathrm{D}}}$ is the magnitude of the (superficial) Darcy velocity, $v=|\mathbf{v}|$ is the phase velocity of the (interstitial) fluid, and $\mathrm{d}\! P$ is the magnitude of the viscous pressure drop across a region of length $L$ . [4.1.2.3] Here $\mu$ is the fluid viscosity, $k$ is the (absolute) permeability and $\phi$ 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 $\mathbb{W}(t_{0})$ , $\mathbb{O}(t_{0})$ denote the fluid configuration inside the porous medium at some initial time $t_{0}$ and consider flooding at constant injection rates $Q_{\mathbb{W}}\neq 0$ , $Q_{\mathbb{O}}=0$ or $Q_{\mathbb{W}}=0$ , $Q_{\mathbb{O}}\neq 0$ with water or oil. [4.1.3.3] It is observed experimentally and then assumed theoretically that for long times $t\to\infty$ the total volume fractions

	$\displaystyle\lim _{{t\to\infty}}\frac{\|\mathbb{W}(t)\|}{\|\mathbb{P}\|}=\lim _{{t\to\infty}}{S_{{\mathbb{W}}}}(t)={S_{{\mathbb{W}}}}=S$		(21a)
	$\displaystyle\lim _{{t\to\infty}}\frac{\|\mathbb{O}(t)\|}{\|\mathbb{P}\|}=\lim _{{t\to\infty}}{S_{{\mathbb{O}}}}(t)={S_{{\mathbb{O}}}}=1-S$		(21b)

approach constant limiting values. [4.2.0.1] The values of ${S_{{\mathbb{W}}}}$ , ${S_{{\mathbb{O}}}}$ will depend on the phase pressures ${P_{{\mathbb{W}}}}$ , ${P_{{\mathbb{O}}}}$ . [4.2.0.2] Because the injection rates are constant the homogenized phase velocities $\mathbf{v}_{\mathbb{W}}(t)$ , $\mathbf{v}_{\mathbb{O}}(t)$ approach constant values. [4.2.0.3] Specifically,

	$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{W}}(t)\|=v_{\mathbb{W}},$				$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{O}}(t)\|=0$		(22a)
	$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{W}}(t)\|=0,$				$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{O}}(t)\|=v_{\mathbb{O}}$		(22b)

because $Q_{\mathbb{W}}\neq 0$ , $Q_{\mathbb{O}}=0$ for water flooding and $Q_{\mathbb{W}}=0$ , $Q_{\mathbb{O}}\neq 0$ 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

$\displaystyle v_{{\mathrm{D}}}=\phi v_{i}=\frac{k\, k^{r}_{i}(S)}{\mu _{i}}\frac{\mathrm{d}\! P_{i}}{L}$

(23)

where $i=\mathbb{W},\mathbb{O}$ indicates the two phases and the relative permeability functions ${k^{r}_{{\mathbb{W}}}}(S)$ , ${k^{r}_{{\mathbb{O}}}}(S)$ 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 $\mathbb{W}(\infty)$ must be path connected and percolating from inlet and outlet ${}^{2}$ in case (22a) and the same holds for $\mathbb{O}(\infty)$ 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 $P_{\mathbb{O}}(t)-P_{\mathbb{W}}(t)$ for $t\to\infty$ reflects microscopic capillarity on macroscales. [4.2.1.2] The difference is assumed to depend only on saturation

$\displaystyle\lim _{{t\to\infty}}\left({P_{{\mathbb{O}}}}(t)-{P_{{\mathbb{W}}}}(t)\right)={P_{\mathrm{c}}}(S)=P_{\mathrm{b}}\,\widehat{P_{\mathrm{c}}}(S)$

(24)

but not on the phase velocities although dynamic capillary effects have been observed [38, 39]. [4.2.1.3] The function ${P_{\mathrm{c}}}(S)$ is called capillary pressure and $\widehat{P_{\mathrm{c}}}(S)$ is its dimensionless form. [4.2.1.4] The functions ${P_{\mathrm{c}}}(S)$ and $k^{r}_{i}(S)$ are defined on the interval $[S_{{\mathbb{W}\,\mathrm{i}}},1-S_{{\mathbb{O}\,\mathrm{r}}}]$ . [4.2.1.5] The parameters $S_{{\mathbb{W}\,\mathrm{i}}}$ , $S_{{\mathbb{O}\,\mathrm{r}}}$ , defined as solutions of the equations

	$\displaystyle{k^{r}_{{\mathbb{W}}}}(S_{{\mathbb{W}\,\mathrm{i}}})=0$		(25a)
	$\displaystyle{k^{r}_{{\mathbb{O}}}}(1-S_{{\mathbb{O}\,\mathrm{r}}})=0,$		(25b)

are the irreducible water saturation $S_{{\mathbb{W}\,\mathrm{i}}}$ and the residual oil saturation $S_{{\mathbb{O}\,\mathrm{r}}}$ . [4.2.1.6] Both parameters $S_{{\mathbb{W}\,\mathrm{i}}}$ and $S_{{\mathbb{O}\,\mathrm{r}}}$ are assumed to be small but nonvanishing, i.e. $0<S_{{\mathbb{W}\,\mathrm{i}}}\ll 1$ and $0<S_{{\mathbb{O}\,\mathrm{r}}}\ll 1$ . [4.2.1.7] With $S_{{\mathbb{W}\,\mathrm{i}}}$ and $S_{{\mathbb{O}\,\mathrm{r}}}$ the parameter $P_{\mathrm{b}}$ in eq. (24) is defined as

$P_{\mathrm{b}}={P_{\mathrm{c}}}\left(\frac{S_{{\mathbb{W}\,\mathrm{i}}}+1-S_{{\mathbb{O}\,\mathrm{r}}}}{2}\right).$

(26)

[page 5, §0] [5.1.0.1] If there is hysteresis, so that the values $P_{\mathrm{b}}^{{\mathrm{im}}}\neq P_{\mathrm{b}}^{{\mathrm{dr}}}$ differ, then $P_{\mathrm{b}}=(P_{\mathrm{b}}^{{\mathrm{im}}}+P_{\mathrm{b}}^{{\mathrm{dr}}})/2$ will be used. [5.1.0.2] The dimensionless capillary pressure function $\widehat{P_{\mathrm{c}}}(S)$ 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 $S$ , $v_{i}$ and $L$

$\displaystyle F_{i}(S,v_{i},L)=\frac{\text{(viscous pressure drop in phase $i$)}}{\text{(capillary pressure)}}$	$\displaystyle=\frac{\|\mathrm{d}\! P_{i}\|}{\left\|{P_{{\mathbb{O}}}}-{P_{{\mathbb{W}}}}\right\|}$		(27a)
	$\displaystyle=\frac{\mu _{i}\,\phi\, v_{i}\, L}{k\, k^{r}_{i}(S)\|{P_{\mathrm{c}}}(S)\|}$		(27b)
	$\displaystyle=\frac{\mathrm{Ca}_{i}}{k^{r}_{i}(S)\left\|\widehat{P_{\mathrm{c}}}(S)\right\|}$		(27c)
	$\displaystyle=f_{i}(S,\mathrm{Ca}_{i})$	$\displaystyle i=\mathbb{W},\mathbb{O}$	(27d)

or as a function of $S$ and the macroscopic capillary number $\mathrm{Ca}_{i}$ in the last two equalities. [5.1.1.2] The macroscopic capillary number $\mathrm{Ca}_{i}$ is defined as in [9] by

$\mathrm{Ca}_{i}=\frac{\mu _{i}\,\phi\, v_{i}\, L}{k\, P_{\mathrm{b}}}\qquad i=\mathbb{W},\mathbb{O}$

(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 $\mathrm{Ca}$ does not explicitly depend on the interfacial tension $\sigma _{{\mathbb{W}\mathbb{O}}}$ between the two phases and that the quantity $A_{i}=k\, P_{\mathrm{b}}/(\mu _{i}\,\phi)$ with $i=\mathbb{W},\mathbb{O}$ is dimensionally not a velocity, but a specifc action, i.e. action per unit mass. [5.1.1.5] For a derivation of $\mathrm{Ca}_{i}$ from the traditional macroscale equations of motion see [32, 9].

Author:	R. Hilfer, R. T. Armstrong, S. Berg, A. Georgiadis, and H. Ott
published in:	Physical Review E, Vol. 92 (2015)
originally submitted:	June 3rd 2015

	$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{W}}(t)\|=v_{\mathbb{W}},$				$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{O}}(t)\|=0$		(22a)
	$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{W}}(t)\|=0,$				$\displaystyle\lim _{{t\to\infty}}\|\mathbf{v}_{\mathbb{O}}(t)\|=v_{\mathbb{O}}$		(22b)