Capillary number correlations for two-phase flow in porous media

Relative permeabilities and capillary number correlations are widely used for quantitative estimates of enhanced water flood performance in porous media. They enter as essential parameters into reservoir simulations. Experimental capillary number correlations for seven different reservoir rocks and 21 pairs of wetting and nonwetting fluids are analyzed. The analysis introduces generalized local macroscopic capillary number correlations. It eliminates shortcomings of conventional capillary number correlations. Surprisingly, the use of capillary number correlations on reservoir scales may become inconsistent in the sense that the limits of applicability of the underlying generalized Darcy law are violated. The results show that local macroscopic capillary number correlations can distinguish between rock types. The experimental correlations are ordered systematically using a three-parameter fit function combined with a novel fluid pair based figure of merit.

Figure 1

Capillary pressure for the samples listed in Table 1. Parameters for Eq. (18) from (Ref. [25], Table 3).

Figure 2

Capillary number correlations $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{miCa}\right)$ (filled symbols) and $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{lmCa}\right)$ (open symbols) for Berea sandstone (sample 880) and Indiana limestone (sample 881). The vertical dashed line marks equality of viscous and capillary forces. The two solid lines are fits with Eq. (26) with parameters $S_{\mathbb{o}\mathrm{r}}^{\mathrm{pl}}=40.5$%, $A={10}^4$, $\alpha =1$, $\beta =0.3$ (Berea sandstone) and $S_{\mathbb{o}\mathrm{r}}^{\mathrm{pl}}=50$%, $A=3\times {10}^4$, $\alpha =0.125$, $\beta =1$ (Indiana limestone).

Figure 3

Capillary number correlations $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{miCa}\right)$ (filled symbols) and $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{lmCa}\right)$ (open symbols) for Bandera sandstone (sample 879) and Indiana limestone (sample 881). The vertical dashed line marks equality of viscous and capillary forces.

Figure 4

Capillary number correlations $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{miCa}\right)$ (filled symbols) and $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{lmCa}\right)$ (open symbols) for Dalton sandstone (sample 798) and Paluxy sandstone (sample 878). The vertical dashed line marks equality of viscous and capillary forces.

Figure 5

Comparison of capillary number correlations $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{miCa}\right)$ (square), $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{maCa}\right)$ (triangle), and $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{lmCa}\right)$ (asterisk) for all samples. The solid lines are fits to $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{lmCa}\right)$ using Eq. (26) with fit parameters listed in Table 3. The vertical dashed line marks equality between capillary and viscous forces.

Figure 6

An individual fit of $S_{\mathbb{o}\mathrm{r}}^{\mathrm{wf}}\left(\mathrm{lmCa}\right)$ to those data for sample 799 having figure of merit $H=0.167\mathrm{m}$/s.

Figure 7

Fit parameters $A$, $\alpha$, and $\beta$ from Eq. (26) as a function of $lgH$, where $H=\sigma {\mu }_{\mathbb{o}}/{\mu }_{\mathbb{w}}^2$ is the figure of merit in Eq. (27).

Figure 8

All fits of local macroscopic capillary number correlations at fixed $H$ for the data of sample 799. The plot shows one solid line for each parameter combination $A\left(H\right)$, $\alpha \left(H\right)$, $\beta \left(H\right)$ with fixed $H$. The parameter combinations are shown in Fig. 7. In all cases, $S_{\mathbb{o}\mathrm{r}}^{\mathrm{pl}}=0.295$ is used.