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I am trying to find the effective crack diameter (d) and K-value of the fractures. for the specific case of a rectangular prism with the base area of 20 cm and height of 8 cm.
It looks like the crack depth should be 8 cm, but I am uncertain if there is a crack width.

A:

A general approach to calculating fracture permeability is to assume a constant crack length and a uniform crack spacing in the face plane of the prism. Then, you can calculate the fracture permeability as a function of the fracture length, the depth of the fracture, and the crack angle from the face plane. For a prism, the corresponding expression in terms of the crack length, the crack depth, and the crack angle is
$$\phi=\tan^{ -1}\left(\frac{d}{L}\right)$$
$$\frac{\phi}{2}=\sin\left(\frac{\phi}{2}\right)$$
$$\tan\left(\frac{\phi}{2}\right)=\frac{d}{L}$$
$$\phi=2\sin^{ -1}\left(\frac{d}{L}\right)$$
$$\frac{r_i}{r_o}=\frac{L}{2}$$
$$\frac{r_f}{r_o}=\frac{L}{\pi}$$
$$L=r_f\times\pi$$
$$L_{cr}=2\pi r_f$$
$$\text{K}=\frac{\dot Q}{\text{A} \times L_{cr}^\epsilon}$$
$$\text{K}=\frac{\dot Q}{\text{A}\times 2\pi r_f^\epsilon}$$
In the last step, I took the geometrical average of the ratios of crack length and length to height. Since the crack length is the same as the cross-sectional area of the prism, and there are four parallel sides, the area to which the area ratio applies is
$$\text{A}=\frac{4}{\sin(\phi/2)}\text{A}_f$$
$$\text{A}=4\text{A}_f=4\times L_f$$
$$\text{A}=4r_f^2$$
$$\

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