Miscible density-driven flows in heterogeneous porous media: Influences of correlation length and distribution of permeability

A random process and highly accurate pseudospectral method associated with compact finite differences are incorporated to evaluate effects of permeability heterogeneity to gravity-driven miscible porous media flows. Flows in heterogeneous permeability fields, based on multiple sets of random realizations with identical statistical characteristics, are simulated to elucidate the global trend of mean quantities and local variations. The presence of heterogeneity provokes more prominent fingering competition and greater variation of fingers' widths, hence the mean values of interested quantitative measures, i.e., breakthrough time, amount of fluid volumes transported, and normalized mixing effectiveness, are generally reduced. Resonant effects, whose influences on the flow fields are the most significant, are verified to occur in an intermediate correlation length, in which the widths of fingers are slightly less than the correspondent correlation length. Nevertheless, under the resonant conditions these measures are the most widely scattered between different random realizations associated with the same control parameters. The widest scatterings are mainly because of the enhancing sensitivity of flow path selections under such resonant conditions, especially in a few realizations whose high or low permeability regions align along with or transverse to the main flow direction. The significant scatterings of these quantitative measures, which had drawn little attention previously, suggest more cautious treatments for practical implementations in the regime of resonant effects.

Figure 1

Representative sketch of a square heterogeneous porous medium with dimension $H$, where a clearer (darker) area corresponds to a region of higher (lower) permeability. Fluid 1 occupied on the top layer is heavier and less viscous than the lower fluid 2, i.e., ${\rho }_1>{\rho }_2$ and ${\eta }_1<{\eta }_2$, so that the miscible interface is gravitationally unstable to produce complex fingering pattern.

Figure 2

Reference distributions of heterogeneous permeability fields generated by a representative set of random signals, denoted as the R1 realizations, for various correlation lengths of $l=$ (a) 0.08, (b) 0.2, (c) 0.4, and (d) 0.8. The fields follow the log-Gaussian distributions with vanishing logarithm means, i.e., $\overline{{\text{log}}_{e}k(x,y)}=0$.

Figure 3

Concentration images at the breakthrough time $\left(t_b\right)$ in the R1 permeability realizations. (a) Homogeneous condition $\left(s=0\right)$, (b) $l=0.08$, (c) $l=0.2$, (d) $l=0.4$, (e) $l=0.8$, and (d) $l=2$. Notice that the correspondent permeability fields for conditions of $l=0.08$, 0.2, 0.4, and 0.8 are shown in Fig. 2. The fastest breakthrough occurs in the case of intermediate correlation length $l=0.8$.

Figure 4

Temporal concentration distribution measured along the top cross section at $y=1.97$ through the entire simulated period, i.e., $t=0\sim t_b$, for the cases shown in Fig. 3. (a) Homogeneous, (b) $l=0.08$, (c) $l=0.2$, (d) $l=0.4$, (e) $l=0.8$, and (f) $l=2$.

Figure 5

Average concentration along the $x$ axis $\left(c_a\right)$ at different times in the R1 realizations. (a) Homogeneous condition, (b) $l=0.08$, (c) $l=0.8$, and (d) $l=2$. For a homogeneous condition, the profiles are nearly linear. Two major features can be identified for cases with heterogeneity, i.e., bumps and plateaus marked by ellipses and rectangles in panels (b), (c), and (d).

Figure 6

Widths of fingers for five representative cross sections at $y=-1,-0.5,0,0.5,1$, as shown by the lines in the left concentration images, of (a) homogeneous case and (b) $l=0.8$. Widths of individual fingers at these cross sections are marked by blue boxes, in which the heights of these blue boxes stand for a concentration value of $c=0.8$.

Figure 7

Concentration (left column), vorticity (middle column), and streamlines superimposed on the correspondent permeability field (right column) at $t=30$ in the R1 permeability realizations. Top row: homogeneous condition; middle row: $l=0.08$; bottom row: $l=0.8$. The brightness in the vorticity image indicates the strength of local flow activity.

Figure 8

Concentration contours of $c=0.05,0.25,0.5$, and 0.95 superimposed on the correspondent permeability fields of $l=0.08$ generated by various sets of random signals: (a) R1 (reference case), (b) R2, (c) R3, and (d) R4 realizations. The overall streamline patterns are similar. Fingering competition is not prominent in all the permeability distributions.

Figure 9

Concentration contours superimposed on the correspondent permeability fields of $l=0.8$ generated by various sets of random signals: (a) R1 (reference case), (b) R2, (c) R3, and (d) R4 realizations. Prominent fingering competition results in very distinct patterns in various distributions.

Figure 10

Average concentration $\left(c_a\right)$ at breakthrough time $t_b$ in four permeability realizations: R1 (reference case), R2, R3, and R4 realizations. (a) $l=0.08$ and (b) $l=0.8$.

Figure 11

(a) Mean fingers' widths $\left(l_f\right)$ for various correlation lengths in eight permeability realizations. Mean width in a homogeneous condition is shown by the horizontal orange lines. (b) Normalized width by the corresponding correlation length $\left(l\right)$. The normalized width decreases monotonically for larger correlation length scale.

Figure 12

Measures of interests for various correlation lengths in eight permeability realizations. (a) breakthrough time $\left(t_b\right)$, (b) volume of lighter fluid penetrating into heavier fluid $\left(Q_b\right)$ at $t=t_b$, (c) mixing length $\left(L_b\right)$ at $t=t_b$, and (d) normalized mixing variance $\left({\sigma }_n^2\right)$ at $t=t_b$. Mean of the eight realizations is connected by the black line. Value of the corresponding measure in a homogeneous condition, which is constant, is shown by the horizontal orange line.