We performed 5 experiments of reactive flooding with distilled water at a flow rate of 0.1 ml/min, with various durations in the range of 4 to 41 hours. However, only 3 of the experiments had good positional match between the measured surface profiles hA,bef ore, hA,af ter, hB,bef ore and hB,af ter. In return, these surface profiles are from experiments spread across the range of flow durations; 4, 22.5 and 41 hours. We analyze these three sets of surface profiles.
The surface roughness is characterized by calculating the elevation fluctuations as function of displacement d, i.e. calculating the structure function [87, 88] for the profiles. We find the 2nd order structure function along the x-direction as
C2(d) =h(h(x+d, y)−h(x, y))2i12, (7.4) for d= 1 to W/2, and along the y-direction as
C2(d) =h(h(x, y+d)−h(x, y))2i12, (7.5) ford= 1 toH/2, where the angle brackets denote the spatial average. The roughness exponent ζ of a fracture surface is evaluated by plotting C2 as function of d in a log-log plot. If the structure function follows a power law
C2(d)∼dζ, (7.6)
the surface is self-affine over a range of scales with an associated roughness exponent.
The structure functions were calculated for the fracture surfaces before and after reactive flooding, and for both sidesAand B. Log-log plots of the results are shown in figure 7.6. The structure functions plotted here are the averages of equations (7.4) and (7.5). We see that all the considered fracture surfaces have self-affine properties, with a roughness exponent of ζ ≈ 0.7. This roughness exponent was measured over the range of scales d = 10−2 mm to 1 mm for the samples used in the experiments with 22.5 and 41 hours of reactive flow, while for the sample used in the 4 hour experiment the range is d = 10−2 mm to 0.16 mm. For scales above d = 0.16 mm, the fracture surfaces on this sample are less rough with ζ around 0.3. In addition, the surface roughness was not observed to change after any of the durations of reactive flooding done here.
Figure 7.7 shows estimated fracture apertures a=hB−hA between the aligned profiles before and after flooding, the change in fracture aperture r = aaf ter − abef ore, and the initial mid-fracture elevation h = (hA +hB)/2. Since the surface profiles are locally and individually measured, it is not trivial to determine the real fracture aperture in the assembled samples. As a crude approach here, the fracture surfaces hA and hB are forced to be in contact but not intersect, such that the apertures are adjusted toa′ =a−min(a). By looking at figure 7.7, we observe some features to investigate; There is a slight pattern in the estimated dissolution for the 4 hour experiment, where there are some distinct lines going roughly in the flow direction. This is not seen in the experiments with longer flow durations. However, for the experiments with 22.5 and 41 hours of flow time, it looks like regions with small initial fracture aperture have more dissolution, while regions with larger initial
log10(d), [mm]
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5
Structure function, log10(C2(d)) [mm]
-3 fit = 0.68*log10(d) - 0.98
4 hours of flow 22.5 hours of flow
Side A, before Side A, after Side B, before Side B, after fit = 0.71*log10(d) - 0.88
log10(d), [mm]
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5
Structure function, log10(C2(d)) [mm]
-3 fit = 0.70*log10(d) - 0.84
log10(d), [mm]
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5
Structure function, log10(C2(d)) [mm]
-3
Figure 7.6: Log-log plots of the structure function C2 as function of spacing d for the different fracture surfaces. A roughness exponent ζ ≈0.7 is found to be typical, both before and after the reactive flooding.
fracture opening have more precipitation. We investigate the changes in fracture aperture by spatial autocorrelation, and cross-correlation with the initial aperture and mid-plane.
We calculate the cross-correlation of two data setsX andY for relative displace-ments ∆y in the flow direction as
CXY(∆y) = h(X(x, y)−X)(Y(x, y+ ∆y)−Y)i σXσY
, (7.7)
and relative displacements ∆x perpendicular to the flow direction as CXY(∆x) = h(X(x, y)−X)(Y(x+ ∆x, y)−Y)i
σXσY
, (7.8)
where σX and σY are the standard deviations of X and Y, and the angle brackets denote spatial average. The autocorrelation of a data set is done by correlating it with itself in equations (7.7) and (7.8).
The autocorrelation of the change r in fracture aperture is shown in figure 7.8 A) as function of relative displacements perpendicular to the flow direction (∆x) and relative displacements parallel with the flow direction (∆y). After 4 hours of reactive flow, the changes in fracture aperture correlate only locally; for relative displacements perpendicular to the flow direction, the correlation coefficient drops below 0.5 for displacements larger than ± 0.05 mm, and in the direction parallel with the flow direction, it drops below 0.5 at displacements of ± 0.07 mm. In ad-dition, there are some periodic fluctuations in the correlation function for relative displacements perpendicular to the flow direction, with peaks at relative displace-ments ±0.3, ±0.5, ±1 and ±1.3 mm. These small fluctuations could be a response to the linear shapes observed in figure 7.7. For the experiment with flow duration of 22.5 hours, we see that the correlation length for r is increased both parallel with and perpendicular to the flow direction, with correlation coefficients above 0.5 for all the relative displacements measured in the flow direction ±2 mm, and within ± 1.2 mm perpendicular to the flow direction. For the experiment with 41 hours of reactive flow, the autocorrelation of r in the flow direction remain above 0.5 within the relative displacements of ± 1.5 mm measured. For displacements in the per-pendicular direction, it remains above 0.45 within the measured displacements ±
1 mm
Figure 7.7: Fracture profiles for the different samples analyzed. A) Fracture aper-ture a′ before reactive flooding, B) Fracture aperture a′ after reactive flooding, C) Changerin fracture aperture after reactive flooding, and D) The initial mid-fracture elevation h before reactive flooding. The flow direction is from the top towards the bottom.
1.5 mm. In general, the spatial correlation of the changes in fracture aperture tend to increase with increased flow duration, first along the flow direction, then also perpendicular to it. Figure 7.8 B) shows the cross-correlation between the change r in fracture aperture and the initial fracture aperturea′bef ore, for relative displace-ments along the flow direction and perpendicular to it. There is a general trend of stronger anticorrelation betweenr anda′bef ore with increasing flow duration. For the 4 h experiment, the correlation is low for all relative displacements measured. For the 22.5 hours experiment, the zero-displacement anticorrelation is moderate with a correlation coefficient at -0.5, as well as a weak anticorrelation (between -0.3 and -0.4) with the upstream initial aperture (seen as positive displacement in the flow direction). For the experiment with 41 hours of reactive flow, the zero-displacement anticorrelation is high, with a correlation coefficient at -0.7. The correlation function also shows weak anticorrelation (from -0.2 to -0.4) ofr with the surrounding regions in the initial aperture. Figure 7.8 C) shows the cross-correlations of the change r in fracture opening with the initial mid-plane h. For the 4 hour experiment the correlation is low, with correlation coefficients within ± 0.2. For the 22.5 hour ex-periment, the correlation coefficient is low for perpendicular displacements and the zero-displacement, more or less within ± 0.2. There is an increase from weak to moderate (0.2 to 0.5) positive correlation with the upstream mid-plane (positive displacements in the flow direction). For the 41 hours experiment, there is a weak anticorrelation for displacements perpendicular to the flow direction (-0.3 to -0.4)
and for the zero-displacement (-0.3), and a moderate positive correlation (0.4 - 0.5) with the mid-plane at 1 to 1.5 mm downstream. With these results we do not find a general trend between the mid-plane and the change in fracture aperture.
Figure 7.9 shows the binned average relationship between the change in fracture aperture r and initial fracture aperture a′bef ore, and the binned average of the rela-tionship between the change in fracture aperture r and initial mid-plane h. For all samples, we see an average decrease in dissolution with increasing initial fracture aperture, and a cross-over to precipitation for the largest initial fracture apertures, which is consistent with the anticorrelation found for this relationship. For the average of the relationship between the fracture gap and the initial mid-fracture elevation, we do not see a typical trend. For the 4 hour experiment the relationship looks uncorrelated. For the 22.5 hour experiment there is more precipitation at the lowest point, no correlation between -0.15 and 0.15 mm, and increasing dissolution at the highest points. For the 41 hour experiment, it is more dissolution at the lowest point, and more precipitation at the highest points. As mentioned, the zero-displacement correlation coefficients between r and h are low or weak (0.2 to - 0.3) for all samples.
Relative displacement [mm]
4 hours 22.5 hours 41 hours
A)
B)
C)
Figure 7.8: Correlation functions as function of relative displacement. A) is the autocorrelation of r, the change in fracture aperture, B) is the cross-correlation between r and the initial fracture aperture a′bef ore, and C) is the cross-correlation between r and the initial mid-planeh
Initial aperture [mm]
0.1 4 hours, distilled water 22.5 hours, distilled water 41 hours, distilled water
0.04 0.08 0.12 0.16 0.2
Figure 7.9: Binned average values of the change in fracture opening as function of initial fracture opening (left) and initial mid-fracture elevation (right)