Conductivity and Transmissivity of a Single Fracture

Abstract

The effective flow and conduction properties of fractures with Gaussian spatial correlations are investigated by solving the microscale governing equations in three-dimensional samples, along the lines initiated by Mourzenko et al. (J Phys II(5):465–482, 1995), Volik et al. (Trasnp Porous Media 27:305–325, 1997) but in greater details, over a wider range of the parameters, and with greatly improved accuracy. The effective transport coefficients are related to intrinsic geometrical characteristics, quantified by the mean aperture, the surface roughness RMS amplitude, its correlation length, and the intercorrelation coefficient of the roughness on the two surfaces. Extensive results are presented and analyzed. An empirical relationship between the transmissivity and conductivity is formulated, the validity of the Reynolds approximation is assessed, and heuristic expressions are proposed for the direct estimation of the transport coefficients as functions of the fracture geometrical characteristics.

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Acknowledgements

Part of this work was performed when P.M.A. was supported at the Mechanical Engineering Department, Technion, Haifa, Israel, by a fellowship of the Lady Davis Foundation.

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Correspondence to J.-F. Thovert.

Appendix: Statistical Fluctuations

Appendix: Statistical Fluctuations

The statistical fluctuations of the transport coefficients are summarized in Fig. 3 for all the calculations with \(L=16 l_c\). The reduced standard deviations of the hydraulic aperture \(\sigma _{b_S} / \langle b_S \rangle \) and of the conductivity \(\sigma _{b_L} / \langle b_{L} \rangle \) are plotted as functions \(b_m / \sigma _h\) for \(l_c/\sigma _h=1\) to 8 with \(\theta _I=0\), 0.5 and 1.

When \(\theta _I\)=0 and 0.5 , the relative fluctuations are decreasing functions of the aperture for both transport processes, with only a very small influence of the correlation length \(l_c\). They roughly follow the power laws

$$\begin{aligned} \frac{\sigma _{b_S} }{\langle b_S \rangle } \approx 0.13 \left[ \frac{b_m}{\sigma _h}\right] ^{-\frac{3}{2}} \, , \qquad \frac{\sigma _{b_{L}}}{\langle b_{L} \rangle } \approx 0.17 \left[ \frac{b_m}{\sigma _h}\right] ^{-\frac{3}{2}} \end{aligned}$$
(34a)

for uncorrelated surfaces (\(\theta _I\)=0) and

$$\begin{aligned} \frac{\sigma _{b_S} }{\langle b_S \rangle } \approx 0.08 \left[ \frac{b_m}{\sigma _h}\right] ^{-\frac{3}{2}} \, , \qquad \frac{\sigma _{b_{L}}}{\langle b_{L} \rangle } \approx 0.12 \left[ \frac{b_m}{\sigma _h}\right] ^{-\frac{3}{2}} \end{aligned}$$
(34b)

for correlated surfaces with \(\theta _I=0.5\).

A different behavior prevails for \(\theta _I=1\). The fluctuations decrease only slightly with the aperture and much more significantly with the correlation length. However, the magnitude of the fluctuations is smaller than when \(\theta _I < 1\). They are generally a few percents or less, except for very small apertures and short correlation lengths, and never exceed 10%.

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Mourzenko, V.V., Thovert, JF. & Adler, P.M. Conductivity and Transmissivity of a Single Fracture. Transp Porous Med 123, 235–256 (2018). https://doi.org/10.1007/s11242-018-1037-y

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Keywords

  • Fracture
  • Conductivity
  • Permeability
  • Transmissivity