On improvements

The analytical results obtained here for the dry case could be improved somewhat several different ways. Instead of replacing $\nu^*$ by $\nu_m$, we could have replaced it by the fixed point value $\nu_c$ obtained in Appendix B. Since the fixed point is an attractor and the values rapidly approach $\nu_c$ for small but finite volume fractions, this approximation would guarantee an improved approximation over most of the range of crack volume fraction. However, it will make the approximation a little worse in the very small volume fraction region. It has been and will continue to be a significant advantage for our analysis to have formulas valid in the small $\phi$ limit, so we have chosen not to do this here. Alternatively, instead of choosing either of the extreme values of $\nu^*$,we could use their mean, their harmonic mean, or their geometric mean, etc., with similar benefits and drawbacks. Or, we could make direct use of the results from Appendix B for the decoupled equation for Poisson's ratio. This approach will improve the results over the whole range of volume fractions, but will complicate the formulas considerably. We want to emphasize, however, that our goal here has not been to achieve high accuracy in the analytical approximation, but rather to gain insight into what the equations were computing and why. Having accomplished this even with the simplest approximation $\nu^* \simeq \nu_m$, we do not think it fruitful to dwell on this issue and we will therefore leave this part of the subject for now. For the interested reader, some additional technical justifications of the analytical approximation are provided in Appendix C.

Next we want to do more detailed comparisons between these results and those of Gassmann (1951) and of Mavko and Jizba (1991) in the remainder of the paper.

 

allk1pg
Figure 1 Bulk modulus for dry and liquid saturated cracked porous media with $\alpha = 0.1$.Full DEM calculation is shown as a solid line for the saturated case and as a dot-dash line for the dry case. The analytical approximations in the text are displayed as a dashed line for both dry and saturated cases. Agreement between full DEM and analytical approximation is excellent in both cases. Gassmann's prediction is shown by the dotted line.

 

allmu1pmj
Figure 2 Shear modulus for dry and liquid saturated cracked porous media with $\alpha = 0.1$.Full DEM calculation is shown as a solid line for the saturated case and as a dot-dash line for the dry case. The analytical approximations in the text are displayed as a dashed line for both dry and saturated cases. Agreement between full DEM and analytical approximation is again excellent in both cases. The Mavko-Jizba (1991) prediction is shown by the dotted line.

 

allk01pg
Figure 3 Same as Figure 1 for $\alpha = 0.01$.

 

allmu01pmj_25
Figure 4 Same as Figure 2 for $\alpha = 0.01$.Note that the Mavko-Jizba agreement is poor except at low porosities ($< \sim 2\%$).

 

allk001pg_0.1
Figure 5 Same as Figure 1 for $\alpha = 0.001$.Gassmann (1951) is in very good agreement with DEM for this case.

 

allmu001pmj
Figure 6 Same as Figure 2 for $\alpha = 0.001$.Again, note that the Mavko-Jizba prediction is in poor agreement except at very low porosities ($< \sim 0.2\%$).