Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone

High-resolution x-ray imaging was used in combination with differential pressure measurements to measure relative permeability and capillary pressure simultaneously during a steady-state waterflood experiment on a sample of Bentheimer sandstone 51.6 mm long and 6.1 mm in diameter. After prolonged contact with crude oil to alter the surface wettability, a refined oil and formation brine were injected through the sample at a fixed total flow rate but in a sequence of increasing brine fractional flows. When the pressure across the system stabilized, x-ray tomographic images were taken. The images were used to compute saturation, interfacial area, curvature, and contact angle. From this information relative permeability and capillary pressure were determined as functions of saturation. We compare our results with a previously published experiment under water-wet conditions. The oil relative permeability was lower than in the water-wet case, although a smaller residual oil saturation, of approximately 0.11, was obtained, since the oil remained connected in layers in the altered wettability rock. The capillary pressure was slightly negative and 10 times smaller in magnitude than for the water-wet rock, and approximately constant over a wide range of intermediate saturation. The oil-brine interfacial area was also largely constant in this saturation range. The measured static contact angles had an average of ${80}^{\circ }$ with a standard deviation of ${17}^{\circ }$. We observed that the oil-brine interfaces were not flat, as may be expected for a very low mean curvature, but had two approximately equal, but opposite, curvatures in orthogonal directions. These interfaces were approximately minimal surfaces, which implies well-connected phases. Saddle-shaped menisci swept through the pore space at a constant capillary pressure and with an almost fixed area, removing most of the oil.

Figure 1

Schematic diagram of the experimental apparatus using a Hassler-type flow cell made of carbon fiber epoxy with low x-ray photon attenuation. Tomographic scans were taken for the middle section of the downstream sample over a vertical distance of 10.7 mm with a voxel size of $3.58\mu \mathrm{m}$.

Figure 2

(a),(b) Gray-scale two-dimensional cross sections of a three-dimensional image at $f_w=0.5$ after applying a nonlocal means edge-preserving filter. The dimension of the image is $1600\times 1600\times 3000$ voxels with $3.58\mu \mathrm{m}$. (c) Three-phase segmentation showing rock grains (red), brine (blue), and oil (yellow).

Figure 3

The saturation averaged in each block along the direction of flow. Each block contains $1600\times 1600\times 100$ voxels (approximately 0.36 mm in length along the sample). The saturation is computed using the total porosity of 0.231 from the differential imaging method.

Figure 4

The measured pressure differential across the sample using a differential pressure transducer at steady state for 3 h before starting the scan at different fractional flows. The mean pressure differential and standard deviation are shown.

Figure 5

Relative permeabilities during waterflood calculated at steady state for brine displacing oil for the mixed-wet Bentheimer system in this study are shown by the red circles. The result is compared to previously published measurements on a water-wet Bentheimer system performed at the same experimental conditions apart from sample wettability, shown by the blue squares [20]. The error bars consider uncertainties in the measurement of the length and diameter of the sample, the absolute permeability, pressure measurements, and the pump rates, as well as the sensitivity from image segmentation.

Figure 6

Normalized histograms showing the number of pores occupied by oil, weighted by the pore volume, as a function of pore radius for different fractional flows. The mixed-wet case (top) is compared with a previous water-wet experiment (bottom) [20].

Figure 7

(a) The voxelized surface interfaces between two phases. (b) The voxelized interfaces were smoothed by applying a boundary-preserving algorithm with 150 iterations, shown in red overlaid on top of the voxelized interface (in blue).

Figure 8

Specific interfacial area, $a$, between different phases calculated at steady state for brine-oil displacement. The specific surface is defined as the surface area per unit volume. $a_{os}, a_{ws}$, and $a_{ow}$ represent specific interfacial area between oil and solid phases (red circle), brine and solid phases (blue square), and oil and brine phases (black triangle), respectively. The error bars consider the sensitivity from image segmentation.

Figure 9

Curvature distributions for the two mixed-wet experiments considered and a water wet case from a previous study [20]. All the curvature distributions were computed with $f_w=0.5$ at steady state.

Figure 10

The mean curvatures ($\kappa$) for different fractional flows for all the subvolume blocks. Mean curvatures are measured using the connected oil phase only (a), disconnected oil phase only (b), and all oil-brine interfaces (c). Note that there is no connected oil phase when $f_w=1$ at steady state.

Figure 11

The average capillary pressure for (a) the connected oil phase only and (b) all of the oil phase against average saturation for each block along the flow direction at different fractional flows. The capillary pressure obtained using a centrifuge method on a larger system is also compared to the measurement based on interfacial curvature. The capillary pressure in this mixed-wet system is also compared with the previous study on a water-wet sample (c, d) [20, 47]. Note that in both mixed-wet and water-wet cases, there is no connected oil phase when $f_w=1$ at steady state.

Figure 12

Interfaces between connected oil phase and brine phase in the water-wet case from a previous study [20] and two mixed-wet experiments. The images shown here are at $f_w=0.5$ with a dimension of $250\times 250\times 250$ voxels (approximately $0.73{\mathrm{mm}}^3$)

Figure 13

Distributions showing different curvature values for ${\kappa }_1<0, {\kappa }_2<0$ (blue squares); ${\kappa }_1>0, {\kappa }_2>0$ (red circles), and ${\kappa }_1{\kappa }_2\le 0$ (green triangles). Approximately 70% of the interfaces in the mixed-wet system have a negative Gaussian curvature. This phenomenon occurs at all fractional flows. The same plot for the water-wet case is shown here for comparison where the majority of the interfaces have a positive Gaussian curvature ${\kappa }_1{\kappa }_2>0$ (${\kappa }_1>0$ and ${\kappa }_2>0$).

Figure 14

Distribution of Gaussian curvature for the water-wet (blue squares) and mixed-wet (red circles) cases.

Figure 15

Contact angle distribution computed from a $500\times 500\times 500$ voxels subvolume at each fractional flow. The mean contact angle for the mixed-wet experiment is $80.3^{\circ }$ with a standard deviation of $17.0^{\circ }$. The contact angle distribution is also compared with the water wet system [20], which has a mean contact angle of $66.4^{\circ }$ with a standard deviation of $15.1^{\circ }$.