Pore-scale velocities in three-dimensional porous materials with trapped immiscible fluid

We study and document the influence of wetting and nonwetting trapped immiscible fluid on the probability distribution of pore-scale velocities of the flowing fluid phase. We focus on drainage and imbibition processes within a three-dimensional microcomputed tomographic image of a real rock sample. The probability distribution of velocity magnitude displays a heavier tail for trapped nonwetting than wetting fluid. This behavior is a signature of marked changes in the distribution and strength of preferential flow paths promoted by the wettability property of the trapped fluid. When the latter is wetting the host solid matrix, high-velocity areas initially present during single-phase flow conditions are mainly characterized by increased or decreased velocity magnitudes, and the velocity field remains correlated with its counterpart associated with the single-phase case. Otherwise, when the trapped fluid is nonwetting, features that are observed to prevail are appearance and disappearance of high-velocity areas and a velocity field that is less correlated to the one obtained under single-phase conditions.

Figure 1

Three-dimensional representation of the pore space showing the reflection of the porous structure along the mean flow direction; dashed lines denote the plane of reflection.

Figure 2

The final fluid distribution obtained after primary drainage is shown in (a) where the invading oil is in red and the trapped residual water is represented in translucent blue; oil-water interfaces are in yellow. Likewise, the final state of the secondary imbibition is depicted in (b) with the invading water in blue and the trapped residual oil in translucent red.

Figure 3

(a) Probability density functions, $f_d$, of scaled pore volume-equivalent diameter, $d/\langle d\rangle$, and (b) probability density functions, $f_u$, of scaled velocity magnitude, $u/\langle u\rangle$, for the three settings analyzed. Symbols correspond to the simulation results and the straight dashed line to the power-law model employed to interpret the low-velocity range. (c) Semilogarithmic plot of the same data as in (b) with the stretched exponential functions considered to interpret the pattern of the decays observed in the distribution of high velocities.

Figure 4

Filled contours of scaled velocity magnitude on two planes normal to the mean flow direction for each case analyzed (top row: $z=0.34L_z$; bottom row: $z=0.97L_z$; $L_z$ being the total length of the computational domain along the mean flow direction). Blue represents trapped water in the setting where oil is flowing (second column). Red represents trapped oil in the setting where water is flowing (third column). Symbols A and B, respectively, indicate higher and lower local flow velocities, as compared to the single-phase case. Symbols C indicate areas where velocity vanishes following trapping and D corresponding to appearance of high-velocity areas.

Figure 5

Scatter-plot of scaled velocities corresponding to single-phase flow and fluid flow in the presence of trapped (a) water or (b) oil, for the cross section at $z=0.34L_z$ (see Fig. 4). Associated contour plots are depicted for (middle column) single-phase flow condition and (right column) settings related to the presence of a trapped phase. Points close to the parity line (in red) correspond to regions where feature A or B occurs. Otherwise, points close to the horizontal or vertical axis correspond to locations associated with features C and D, respectively. These two sets of points and the corresponding spatial pattern are respectively identified through dashed magenta and dash-dotted orange curves (see Appendix pp3 for additional details).

Figure 6

Scatter plot of scaled velocities juxtaposing (a) trapped water and single-phase conditions, (b) trapped oil and single-phase conditions, or (c) trapped oil and trapped water conditions. Points above and below the parity line are signatures of high-velocity channels present in both cases considered and characterized by increased or decreased velocity magnitudes, respectively. Points close to the horizontal or to the vertical axis are signatures of high-velocity channels that are either disappearing or appearing.

Figure 7

(a) Probability density functions, $f_u$, of scaled velocity magnitude, $u/\langle u\rangle$, for the cases with no-slip conditions (open symbols) and free-slip conditions (plain symbols) implemented at fluid-fluid interfaces (logarithmic scale). (b) Semilogarithmic plot of the same data as in (a) to emphasize high-velocity distribution.

Figure 8

(a) Probability density functions, $f_u$, of scaled velocity magnitude, $u/\langle u\rangle$, obtained under single-phase flow conditions for the diverse sample sizes investigated. (b) Semilogarithmic plot of the same data as in (a) to emphasize high-velocity distribution.

Figure 9

Selection of points (in magenta) in the scatter plot (a) above and (b) below the parity line (in red), along with the corresponding spatial location, as depicted in magenta in the contour plot of scaled velocity (panels on the right) for the scenario with trapped water. These selections correspond to the points enclosed in the magenta dashed and orange dash-dotted ovals in Fig. 5, respectively.

Figure 10

Selection of points (in magenta) in the scatter plot close to the (a) horizontal and (b) vertical axis, along with the corresponding spatial location, as depicted in magenta in the contour plot of scaled velocity (panels on the right) for the scenario with trapped oil. These selections correspond to the points enclosed in the orange dash-dotted and magenta dashed ovals in Fig. 5, respectively.