4. Bilinear and linear flow regimes

Bilinear flow regime takes place when two linear flows, normal one flowing into the other, take place simultaneously. This situation occurs in low conductivity fractures where linear flow along the fracture and linear flow from the formation to the fracture are observed. Bilinear flow is recognized in the pressure derivative curve by a slope of 0.25. However, this is not shown in Figure 1 since bilinear flow is absent. The governing expressions for early bilinear and linear flow regimes for vertical fractures in naturally fractured systems were, respectively, presented in [16]

$$P\_D = \frac{2.45}{\sqrt{C\_{\mathcal{DD}}}} \left(\frac{t\_{D\mathcal{xf}}}{\xi}\right)^{1/4},\tag{17}$$

$$t\_D \, ^\ast P\_D \, ^\prime = \frac{0.6125}{\sqrt{\overline{\mathbb{C}\_{fD}}}} \left(\frac{t\_{D \, \circ f}}{\xi}\right)^{1/4},\tag{18}$$

$$P\_D = \left(\frac{\pi t\_{Dxf}}{\xi}\right)^{1/2},\tag{19}$$

and

$$t\_D \, ^\ast P\_D \, ^\prime = \frac{1}{2} \left( \frac{\pi t\_{D \ge f}}{\xi} \right)^{1/2} \, . \tag{20}$$

Linear flow regime can be used to find the half-fracture length, and bilinear flow regime allows finding the fracture conductivity. Once the dimensionless quantities of Eqs. (1) and (3)–(5) are replaced in Eqs. (16)–(19), the fracture conductivity is solved for then

$$k\_f w\_f = \frac{1947.46}{\sqrt{\xi \phi \mu c\_l k}} \left(\frac{q \mu B}{h(\Delta P)\_{BL1}}\right)^2,\tag{21}$$

Figure 1. Dimensionless pressure and pressure derivative behavior for an infinite-conductivity fractured vertical well in a naturally fractured bounded reservoir, <sup>λ</sup> = 1 � <sup>10</sup>�<sup>9</sup> and <sup>ω</sup> <sup>=</sup> 0.1 (taken from [13]).

$$k\_f w\_f = \frac{121.74}{\sqrt{\xi \phi \mu c\_l k}} \left( \frac{q \mu B}{h(t \ast \Delta P')\_{BL1}} \right)^2,\tag{22}$$

Once the fracture conductivity is found, Eq. (7) applies to find the dimensionless fracture conductivity if reservoir permeability and the half-fracture length are known. When bilinear flow is absent, the fracture conductivity may be found from Eq. (13), or the dimensionless fracture conductivity can be read from Figure 2:

$$\alpha\_f = \frac{4.064qB}{h(\Delta P')\_{L1}} \sqrt{\frac{\mu}{\xi \phi c\_l k}} \tag{23}$$

Figure 2. Effect of skin factor on fracture conductivity (taken from [28]).

Well Test Analysis for Hydraulically-Fractured Wells DOI: http://dx.doi.org/10.5772/intechopen.80996

and

$$\mathbf{x}\_{f} = \frac{2.032qB}{h(t\*\Delta P')\_{L1}} \sqrt{\frac{\mu}{\xi\phi c\_{l}k}} \mathbf{(} \tag{24}$$
