5. Points of intersection

If bilinear flow also takes place, then the point of intersection between the pressure derivatives of the bilinear and biradial flow lines, tBLBRi, given by Eqs. (10) and (18), respectively, allows the development of an equation to find the halffracture as follows:

$$
\left(\frac{\eta t\_{D\text{xf}}}{2\Phi\xi}\right)^{0.36} = \frac{0.6125}{\sqrt{Q\_{fD}}} \left(\frac{t\_{D\text{xf}}}{\xi}\right)^{1/4}.\tag{25}
$$

Simplifying,

$$
\left(\frac{t\not\!\!\_{\text{def}}}{\xi}\right)^{0.11} = \frac{0.2862}{\sqrt{C\_{fD}}}\Big/\tag{26}
$$

Replacing the dimensionless quantities, Eqs. (3) and (7) in Eq. (26) lead to

$$
\left(\frac{0.000263kt}{\phi \mu c\_t \varkappa\_f^2 \xi}\right)^{0.11} = 0.2862 \sqrt{\frac{k}{k\_f} \frac{\varkappa\_f}{\varkappa\_f}} \Big|\tag{27}
$$

Solving for the half-fracture from Eq. (27), we readily obtain

$$k\_f w\_f = 10.5422 \begin{array}{c} \text{)} \frac{\xi \phi \mu c\_t k^{3.5454} \varkappa\_f^{6.5454}}{t\_{BRBLi}}^{6.22} \,. \tag{28}$$

By the same token, the intercept of Eq. (20) with Eq. (18), tLBRi, provides another expression to find the half-fracture length:

$$\mathbf{x}\_f = \sqrt{\frac{k t\_{LBRi}}{39.044 a o \phi \mu c\_t}} \mathbf{(} \tag{29}$$

Bilinear flow regime is absent in the plot of Figure 1. Linear, biradial, and radial flow regimes along with the late pseudosteady-state period are seen. The interception points formed by the possible combinations of such periods can be represented schematically in this plot.

Another way to find the half-fracture length comes from the intersection of Eqs. (2) and (10), tRBRi, and Eq. (10) with Eq. (15), so that

$$\alpha\_f = \frac{1}{4584.16} \sqrt{\frac{k t\_{RBRi}}{\xi \phi \mu c\_t}} \Bigg( \tag{30}$$

and

$$\alpha\_{\hat{f}} = 41.0554 A^{1.3889} \left( \frac{\xi \not\!\!\phi \mu c\_t}{k\_{BRPi}} \right)^{0.8889} \,. \tag{31}$$

The intercept point resulting between linear flow and bilinear flow lines given by the governing pressure derivative solutions, Eqs. (18) and (19), can be used to find either half-fracture length or permeability:

$$k = \left(\frac{k\_f \mu\_f^{\prime}}{\varkappa\_f^2}\right)^2 \frac{16t\_{BLLi}^{\prime}}{13910\xi\phi\mu c\_t}.\tag{32}$$

tBLRi is the intersection of the bilinear pressure derivative line given by Eq. (18) with the radial flow regime line (Eq. (2)). This intersection point serves as the estimation of either permeability or fracture conductivity:

$$\mathbf{t}\_{\rm BLRi} = \mathbf{1} \mathbf{677} \frac{\xi \phi \mu \mathbf{c}\_{\rm f}}{k^3} \left( k\_{\rm f} w\_{\rm f} \right)^2. \tag{33}$$
