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We have verified in Sec. 5, in the case of the parameter Λ, that the electrical properties of fluid-saturated porous media can be used to give a fairly accurate prediction of its permeability. In this section we study the actual electrical fields and fluid velocity flow fields more closely, and compare them qualitatively, with gray scale pictures, and quantitatively, with correlation functions.
Using the magnitude v(r) of the vector solution to the Stokes i equations for a given pore structure, we define the scaled two-point fluid velocity correlation function Sv(r) as the angular average of :
where the quantity is defined by
and Q also denotes the area of the region Q. In Eq. (12), the numerator is an integral over the total sample volume V, with zero velocity in the solid regions, while the denominator is an integral over the pore space volume Vp only. For an isotropic system, . In our case, the applied pressure gradient breaks the isotropy, so we average over all possible directions of the position vector to get back a correlation function that depends only on the magnitude of the position vector.
The normalization in Eq. (12) was chosen so that Sv(0) = Vp/V = φ, the porosity. In the limit r → ∞, the fluid velocities must become completely uncorrelated in a pore structure with only short-range structural correlations, so that Sv(r) asymptotically goes to
and only takes on the equality in the special (unphysical) case when v(r)=constant everywhere in the pore space. In that special case, the correlation function Sv(r) is actually identical to the two- point correlation function S(r) of the pore structure , where S(r) = < ƒ(r') ƒ(r' + r) > V, and the function ƒ(r') is one for pore space and zero for the solid phase. In the more general case of v(r) ≠ a constant, we define Sv(r→ ∞) ≡ φv2, where φv is an effective porosity that measures the part of the total pore space that plays a role in fluid flow. This definition implies that the inequality φv < φ must strictly hold.
We define the correlation function SE(r) similarly to Sv(r) in Eq. (12), but with the magnitude of the fluid velocities replaced by the magnitude of the electric fields. Using the method of Ref. , it is then straightforward to show that dSE(r)/dr = −φ / (π Λ) at r = 0, so that the initial slope of this correlation function gives another way to measure Λ. The r → ∞ limit of SE(r) defines E2, the effective electrical porosity. Also, the initial slope for S(r), the pore space correlation function , is −2 φ / (πh) in 2D. The quantities h and Λ can then be considered to be length scales defined by their respective correlation functions. However, using the method of Ref. , which turns the numerator of Eq. (12) into a surface integral, it can be shown that dSv (r)/dr =0 at r = 0, a result that is a direct consequence of the no-slip boundary conditions that force the velocity to be zero at the fluid-solid interface. We therefore define a length scale Lv from the small-r behavior of this correlation function in a different way, by expanding Sv to second order around r = 0: Sv(r) = φ − φ(r / Lv )2.
Figure 11 plots the three correlation functions S(r), SE(r), and Sv(r), vs. spatial separation r, for an 80% porosity system. The straight dashed lines in Figs. 11a and 11b are drawn using slopes calculated from direct numerical determinations of h and Λ, while the parabolic dashed line in Fig. 11c was fit to the small r data. Good agreement with the simulation data is seen in all three cases, confirming the above inital slope derivations. The square root of the large-r limit of the correlation functions gives φ = 0.8, φE = 0.77, and φv = 0.62. The fact that φE and v are both less than φ indicates that even at this high porosity, there are areas of the pore space that carry relatively little flow. Furthermore, the inequality φv < φE indicates that there are more of these stagnant areas (with little or no flow) for fluid flow than for electrical current flow.
Figure 11: (a) Pore space, (b) electrical field magnitude, and (c) fluid velocity magnitude correlation functions as a function of distance r, for porosity = 0.8. The correlation functions are defined in the text. The dashed lines indicate the exact small r behavior, defined in the text.
Figure 12a shows a gray-scale image of the electric field magnitudes and Fig. 12b shows the fluid velocity magnitudes, for an 80% porosity system, where white is the maximum and black is the minimum magnitude of the field quantity being considered. The solid particles are shown in red. The field was applied from left to right. The fluid-flow paths are clearly more concentrated, and follow a more tortuous path than do the electric current paths. It is plain to see that there are significantly more stagnant areas for fluid flow than for electrical current flows.
Figure 12a: Gray scale image of the electric field magnitudes for a porosity of 0.8. Black is zero, white is maximum. The particles are shown in red.
Figure 12b: Gray scale image of the fluid velocity magnitudes for a porosity of 0.8. White is maximum, black is zero.
Figures 13 and 14 are the equivalent of Figs. 11 and 12, but for 50% porosity, and Figs. 15 and 16 show the same results for 35% porosity. As the porosity decreases, the differences between the spatial arrangement of the fluid and electrical current flow fields appear to become more pronounced. The fluid-flow field for 35% porosity, shown in Fig. 16b, is particularly striking, as one or two main pathways seem to carry almost all the flow. This can be attributed to the flow becoming increasingly controlled by narrow necks, so that the fluid picks out a continuous path connected by the largest neck sizes. The periodic model from Sec. 4 showed how much more sensitive the fluid flow rate is to the width of narrow necks than is the electrical current flow rate, due mainly to the no-slip boundary condition.
Figure 13: (a) Pore space, (b) electrical field magnitude, and (c) fluid velocity magnitude correlation functions as a function of distance r, for porosity = 0.5. The correlation functions are defined in the text. The dashed lines indicate the exact small r behavior, defined in the text.
Figure 14a: Gray scale image of the electric field magnitudes for a porosity of 0.5. White is maximum, black is zero.
Figure 14b: Gray scale image of the fluid velocity magnitudes for a porosity of 0.5. White is maximum, black is zero.
Figure 15: (a) Pore space, (b) electrical field magnitude, and (c) fluid velocity magnitude correlation functions as a function of distance r, for porosity = 0.35. The correlation functions are defined in the text. The dashed lines indicate the exact small r behavior, defined in the text.
Figure 16a: Gray scale image of the electric field magnitudes for a porosity of 0.35. White is maximum, black is zero.
Figure 16b: Gray scale image of the fluid velocity magnitudes for a porosity of 0.35. White is maximum, black is zero.
Figure 17 shows a plot of φE, φv, and φv / φE, vs. φ. While the inequality φv < φE holds for all φ, the ratio φv / φE, surprisingly, remains fairly constant. The sizes of the dynamically connected pore regions are therefore not exactly the same for the electric and fluid flow cases, but they do decrease in a commensurate fashion with porosity for the model pore structures studied.
Figure 17: Electrical and fluid-defined effective porosities and their ratio, plotted against porosity.
Figure 18 shows a plot of Lv vs. the length scale Λ. It should be recalled that Lv is extracted from the small r portion of the fluid velocity correlation function, and Λ is extracted from the small r portion of the electric field correlation. We have found the surprising result, displayed in Fig. 18, that these two length scales are roughly proportional to each other for the model porous media studied. This is direct quantitative support for the result of Fig. 10c, which showed that Λ worked well as a length scale to predict permeability. Figure 18 shows that in some sense, the electric fields sample the local neighborhood of the important pore necks in a way similar to that of a moving fluid, since Lv and Λ are determined by the small r behavior of their respective correlation functions.
Figure 18: Fluid velocity length scale Lv plotted against electrical length scale Λ. The slope of the dashed line is fit to the data, and the line is constrained to go through the origin.