FRACTURED RESERVOIRS AND CRACK-INFLUENCE PARAMETERS

To illustrate the Sayers and Kachanov (1991) crack-influence parameter method, consider the situation in which all the cracks in the system have the same vertical (or z-)axis of symmetry. (We use 1,2,3 and x,y,z notation interchangeably for the axes.) Then, the cracked/fractured system is not isotropic, and we have the first-order compliance correction matrix for horizontal fractures, which is:  

$$\Delta S^{(1)}_{ij} = \rho_c\left(\begin{array} {cccccc} 0 &... ... \cr & & & & 2\eta_2 & \cr & & & & & 0 \cr\end{array}\right),$$ (9)

where i,j = 1,2,3. The two lowest order crack-influence parameters from the Sayers and Kachanov (1991) approach are $\eta_1$ and $\eta_2$.The scalar crack density parameter is defined, for penny-shaped cracks having number density n = N/V and radius in the plane of the crack equal to a, to be $\rho_c = na^3$.The aspect ratio of the cracks is b/a.

Now it is also not difficult to see that, if the cracks were oriented instead so that all their normals were pointed horizontally along the x-axis, then we would have one permutation of this matrix and, if instead they were all pointed horizontally along the y-axis, then we would have a third permutation of the matrix. To obtain an isotropic compliance correction matrix, we can simply average these three permutations: just add the three $\Delta S$'s together and then divide by three. [Note that this method of averaging, although correct for contributions linear in $\rho_c$, does not necessarily work for higher order corrections (Berryman, 2007).] This construction shows in part both the power and the simplicity of the Sayers and Kachanov (1991) approach. The connection to the isotropic case is of great practical importance, because it permits us to estimate the parameters $\eta_1$ and $\eta_2$ by studying isotropic cracked/fractured systems, using well-understood effective medium theories (Zimmerman, 1991; Berryman and Grechka, 2006).