## Abstract

In a previous paper (Dehghanpour et al., Phys Rev E 83:065302, 2011a), we showed that relative permeability of mobilized oil, \(k_\mathrm{ro}\), measured during tertiary gravity drainage, is significantly higher than that of the same oil saturation in other tests where oil is initially a continuous phase. We also showed that tertiary \(k_\mathrm{ro}\) strongly correlates to both water saturation, \(S_\mathrm{w}\), water flux (water relative permeability), \(k_\mathrm{rw}\), and the change in water saturation with time, \(\mathrm{d}S_\mathrm{w}/\mathrm{d}t\). To develop a model and understanding of the enhanced oil transport, identifying which of these parameters (\(S_\mathrm{w},\,k_{\mathrm{rw}}\), or \(\mathrm{d}S_\mathrm{w}/\mathrm{d}t\)) plays the controlling role is necessary, but in the previous experiments these could not be deconvolved. To answer the remaining question, we conduct specific three-phase displacement experiments in which \(k_{\mathrm{rw}}\) is controlled by applying a fixed water influx, and \(S_\mathrm{w}\) develops naturally. We obtain \(k_{\mathrm{ro}}\) by using the saturation data measured in time and space. The results suggest that steady-state water influx, in contrast to transient water displacement, does not enhance \(k_{\mathrm{ro}}\). Instead, reducing water influx rate results in excess oil flow. Furthermore, according to our pore scale hydraulic conductivity calculations, viscous coupling and fluid positioning do not sufficiently explain the observed correlation between \(k_{\mathrm{ro}}\) and \(S_{\mathrm{w}}\). We conclude that tertiary \(k_{\mathrm{ro}}\) is controlled by the oil mobilization rate, which in turn is linked to the rate of water saturation decrease with time, \(\mathrm{d}S_\mathrm{w}/\mathrm{d}t\). Finally, we develop a simple model which relates tertiary \(k_{\mathrm{ro}}\) to transient two-phase gas/water relative permeability.

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## Acknowledgments

The authors thank the Industrial Associates Gas Flooding Program at The University of Texas at Austin for the support, and Dr. Ramon G. Bentsen for useful discussions.

## Author information

## Appendices

### Appendix 1

### Viscous Coupling

Although two-phase viscous coupling in porous media has been well detailed in the literature (Avraam and Payatakes 1995; Dullien and Dong 1996; Rose 1988; Kalaydjian 1990; Bentsen 1994; Yazzan et al. 2013), this phenomena during the three-phase flow is poorly understood. Especially for three-phase water wet systems with positive spreading coefficients, the momentum transport between wetting and intermediate phases may be significant. Three-phase viscous coupling in circular and angular capillary tubes has been modeled recently (Dehghanpour et al. 2011a, b). Here, we use the two models to investigate the role of viscous coupling and pore-scale fluid distribution on the observed flow coupling between oil and water.

#### Circular Capillary

To model the three-phase viscous coupling at the pore-scale, one needs to solve the creeping flow of oil, water, and gas in capillary stable flow configurations such as the corners of a triangle. The conductance of the oil layers in the corners has been studied experimentally and numerically (Firincioglu et al. 1999; Al-Futaisi and Patzek 2003; Zhou et al. 1997), although these numeric models explicitly do not conserve the momentum across the water/oil interface. We (Dehghanpour et al. 2011a; Dehghanpour 2011) proposed an idealized circular capillary tube model where the momentum can be calculated analytically. From the simple model, we derived the generalized transport coefficients of the proposed coupled Darcy equation.

Figure 9 shows the calculated transport coefficients for a constant oil saturation of 0.15 as a function of water saturation. The standard description of three-phase flow states that \(k_\mathrm{ro}\) will increase with increasing \(S_\mathrm{w}\) as the oil will move to the center of the pore space reducing the friction. This is observed in this simple model as \(k_{\mathrm{roo}}\) increases with increasing \(S_\mathrm{w}\). Furthermore, with increasing \(S_\mathrm{w}\), the oil film is pushed to the center of the tube and becomes thicker. However, above \(S_\mathrm{w} <0.25\), the primary contribution to the oil flow (\(k_{\mathrm{ro}}^{\mathrm{eff}}\)) is through the pressure gradient in the water (\(k_{\mathrm{row}}\)). Simply with increasing \(S_\mathrm{w}\), the water film become thicker and oil flows on a faster moving boundary; this causes \(k_{\mathrm{ro}}^{\mathrm{eff}}\) to increase significantly at constant \(S_\mathrm{o}\). The full derivation of the transport coefficients and effective relative permeabilities (\(k_{\mathrm{ro}}^{\mathrm{eff}}\) and \(k_{\mathrm{rw}}^{\mathrm{eff}}\)) are given elsewhere (Dehghanpour 2011).

This simple model qualitatively explains how the relative permeability of a constant oil saturation increases by increasing \(S_\mathrm{w}\). However, the observed coupling in experimental data is much stronger than the prediction of this model. In the simple model, \(k_{\mathrm{ro}}\) increases by less than two orders of magnitude when \(S_\mathrm{w}\) increases from 0 to 0.85 (two-phase limit). In the experimental data, \(k_{\mathrm{ro}}\) increases by two orders of magnitude when the water saturation increases from 0.2 (late time) to 0.5 (early time) which is shown in Fig. 1b. Furthermore, considering the pore size distribution of the water wet pack, the water saturation in pores containing oil should be less than the measured average \(S_\mathrm{w}\). Therefore, the comparison between the simple model and the experimental data indicates that the observed coupling between oil and water flow is stronger than that predicted by this model.

#### Angular Capillaries

A more realistic estimation of the viscous coupling between oil and water layers at the pore scale requires the solution of Stokes equation in angular capillaries. Figure 10a shows a water film and an oil layer stabilized by the capillary forces in the corner of an angular capillary. The subscripts 1, 2, and 3 represent water, oil and gas phases, respectively. We (Dehghanpour et al. 2011b) numerically simulated the viscous coupling for this system and presented a heuristic model for the generalized transport coefficients.

In Fig. 10b, we plot the components of the dimensionless mobility matrix (Dehghanpour et al. 2011b) versus \(b_1/b_2 = \bar{b}_1\) which is an indication of the water saturation in the corner. There exists a value of \(\bar{b}_1\) above which the coupling term, \(\bar{\lambda }_{12}\), is significantly higher than the diagonal term of oil \(\bar{\lambda }_{22}\), which means that oil is mainly driven by the flowing water. When \(\bar{b}_1\) is lower than a certain limit, the coupling term is greater than the diagonal term of water which means that water mainly flows through coupling of the oil phase. Now the question is: what is the approximate value of \(\bar{b}_1\) in the gas-filled pores of the sand column when we observe flow coupling in Tests 3 and 4? In the following, we estimate the \(\bar{b}_1\) values corresponding to the saturation data measured during the three-phase drainage tests.

Figure 10b shows that the viscous coupling between the oil layer and water film is significant when \(\bar{b}_1\) is high enough. Here, we investigate the existence of such pore scale fluid configurations during our three-phase experiments. The challenge is to estimate an average value of \(\bar{b}_1\) in gas-filled pores along the column during the experiment. The only available data is the macroscopic fluid saturation measured during the three-phase drainage tests and the capillary pressure curve of the sand. Particularly it is interesting to estimate the range of \(\bar{b}_1\) along the column when \(k_{\mathrm{ro}}\) is coupled to water flow during Tests 3 and 4. In Appendix 2, we show that \(\bar{b}_1\) can be approximated by

where \(S_1\) and \(S_2\) represent the water and oil saturations, respectively. The function \(h(S)\) is the equilibrium head measured after two-phase water/gas gravity drainage in the same column. Figure 11 shows the equilibrium water saturation along the column which can be considered as the characteristic capillary pressure curve of the sand pack. We use this curve and the saturation data measured along the column during the three-phase drainage tests to estimate \({\bar{b}}_1\) in possibly existing three-phase pores.

In Fig. 12a, b, we plot the values of \({\bar{b}}_1\) corresponding to the measured saturation data of Test 4, where we observe flow coupling, and Test 1, where we do not observe flow coupling. In general, with decreasing \(S_\mathrm{w}\) in Test 4, \({\bar{b}}_1\) decreases, while with decreasing \(S_\mathrm{o}\) in Test 1, \({\bar{b}}_1\) increases. For Test 4, the values of \({\bar{b}}_1\) corresponding to high \(k_{\mathrm{ro}}\) data is in the range of 0.85-0.9 (Fig. 12a). We also know from Fig. 10b that this range of \({\bar{b}}_1\) may result in a significant coupling between oil and water. For Test 1, the values of \({\bar{b}}_1\) are initially low (about 0.7) and increase to about 0.78 at later time scales (Fig. 12b).

The above results and analysis show that at early times during Tests 3 and 4, water can drag the oil along with it in gas-filled pores because \(\bar{b}_{1}\) values are high enough. However, further analysis is required before we can conclude that viscous drag is responsible for the observed coupling in the lab data. The first concern is the stability of such layers in angular capillaries at the representative saturation state. When the sum of the oil and water saturation exceeds a certain limit the gas/oil interfaces may touch and not allow oil layer formation (Fenwick and Blunt 1998). However, this saturation limit depends on the size and shape of the pore space.

Even if the intermediate layers are stable, we should investigate the dependence of oil layer hydraulic conductance on water content in the corner. For an example corner geometry and fluid property (\(\beta =30^\circ \) and \(\theta _{12}=\theta _{23}=15^\circ \)), we simulate the hydraulic conductance of an oil layer with fixed cross-sectional area as a function of the water film cross-sectional area. In Fig. 13, we plot the oil layer hydraulic conductance versus water cross-sectional area for the three boundary conditions of continuity, no-flow and, perfect slip at the water/oil interface. As the water content in the corner increases, the oil layer moves toward the center of the corner. For the no-flow boundary condition, the layer conductance substantially decreases by increasing the water content because the layer thickness decreases and the length of the no-flow boundary condition increases. For the continuity boundary condition, we observe the same trend but with a lower rate of the layer conductance decrease. This behavior is in contrast to that of the analytical solution for layer flow in the circular tube shown in Fig. 9. Although for both geometries, with increasing water content, the velocity of the water–oil interface increases, the layer thickness in the corner geometry decreases while in the circular tube the layer thickness increases. Furthermore, with increasing water content, the length of the oil/water interface increases in the corner geometry, while it decreases in the circular tube. However, neither the circular model nor the corner geometry truly represent the complicated three dimensional pores.

### Appendix 2

From the Young Laplace equation, the capillary pressure between oil and gas denoted by subscripts 2 and 3, respectively, is given by

We assume water and oil phases (denoted by subscripts 1 and 2, respectively) lumped together as a wetting phase, gas as the non-wetting phase, and approximate \(P_{\mathrm{c}23}\) by

where \(J\) is the Leverett J-fuction. The curvature of the gas/oil interface (\(r_{23}\)) and its distance to the corner apex (\(b_2\)) are related by

By combining Eqs. 8, 9, and 10, \(b_2\) is given by

\(J(S)\) depends on the pore structure of the porous medium. We use the two-phase gas/water equilibrium drainage curve to estimate \(J(S)\). For each saturation, \(S\), along the column, we relate \(J(S)\) to the equilibrium drainage head, \(h(S)\), by

By combining Eqs. 11 and 12, \(b_2\) is given by

We can estimate \(b_1\) by a similar expression:

Finally, we arrive at the following expression for \(b_1/b_2\):

Therefore, in order to estimate \(\bar{b}_1\) in gas-filled pores, the capillary pressure curve, contact angles, and the pore geometry should be measured. However, by assuming zero or identical contact angles, Eq. 15 will reduce to

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Dehghanpour, H., DiCarlo, D.A. Drainage of Capillary-Trapped Oil by an Immiscible Gas: Impact of Transient and Steady-State Water Displacement on Three-Phase Oil Permeability.
*Transp Porous Med* **100, **297–319 (2013). https://doi.org/10.1007/s11242-013-0217-z

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### Keywords

- Three-phase flow
- Gravity drainage
- Relative permeability
- Viscous coupling
- Tertiary gasflood