3. Biradial flow regime

Biradial or elliptical flow normally results in a hydraulically fractured well when areal anisotropy is present. This is recognized on the pressure derivative versus time log-log plot by a straight line with a slope of 0.36. In hydraulic fractures, the flow from the formation to the fracture is described by parallel flow lines resulting in a linear flow geometry better known as linear flow regime and characterized by a slope of 1/2 on the pressure derivative versus time log-log plot.

Both linear flow and biradial/elliptical flow regimes are seen on the plot of dimensionless pressure and pressure derivative versus dimensionless time based on halffracture length for a naturally fractured formation. New expressions for the elliptical flow regime introduced in [13] excluding reservoir drainage area are given by.

$$P\_D = \frac{25}{9} \left(\frac{\pi t\_{D\text{xf}}}{26\xi}\right)^{0.36} \tag{9}$$

and

$$t\_D \, ^\ast P\_D \, ^\prime = \left(\frac{\pi t\_{D \, \text{xf}}}{2 \text{6\%}}\right)^{0.36},\tag{10}$$

being ξ a dummy variable that defines either a homogeneous or naturally fractured formation. When ξ = 1, a homogeneous reservoir is considered. For the case of naturally fractured formations, ξ = ω, the dimensionless storativity coefficient.

Once dimensionless parameters given by Eqs. (3), (5), and (6) are replaced into Eqs. (9) and (10), respectively, and solve for the half-fracture length, which yields

$$\mathbf{x}\_f = 22.5632 \left( \frac{qB}{h(\Delta P)\_{BR}} \right)^{1.3889} \sqrt{\frac{t\_{BR}}{\xi \phi c\_t} \left( \frac{\mu}{k} \right)^{1.778}} \tag{11}$$

and

$$\mathbf{x}\_f = \mathbf{5.4595} \left( \frac{qB}{h(t\*\Delta P')\_{\partial R}} \right)^{1.3889} \sqrt{\frac{t\_{BR}}{\xi \phi c\_l} \left( \frac{\mu}{k} \right)^{1.778}}.\tag{12}$$

TDS technique is based on drawing a straight line throughout a given flow regime; then, the user is expected to read the pressure, ΔPBR, and pressure derivative, (t\*ΔP')BR, at a given time, tBR. A better way to reduce noise effects consists of extrapolating the mentioned straight line (biradial for this case) to the time of 1 h and read the pressure derivative value, (t\*ΔP')BR1, at 1 h. For this case, the pressure and pressure derivative set in Eqs. (11) and (12) is changed to ΔPBR<sup>1</sup> and (t\*ΔP')BR1, respectively.

When bilinear flow is unseen, fracture conductivity can be found with an expression presented in [27]

$$k\_f w\_f = \frac{3.31739k}{\frac{\varepsilon'}{r\_w} - \frac{1.92173}{x\_f}}.\tag{13}$$

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

[5] also provided an equation for the determination of the skin factor using an arbitrary point read during radial flow regime:

$$s = 0.5 \left\{ \frac{\Delta P\_R}{(t^\* \Delta P')\_R} - \ln \left( \frac{kt\_R}{\phi \mu c\_l r\_w^2} \right) + 7.43 \right\}. \tag{14}$$

The pseudosteady-state regime governing the pressure derivative equation is given by

$$[\mathfrak{t}\_{DA}{}^\* P\_D \, ]\_P = 2\pi (\mathfrak{t}\_{DA})\_P. \tag{15}$$

[7] used the point of intersection, tRPi, of Eqs. (2) and (15) to derive an equation for the estimation of the drainage area:

$$A = \frac{kt\_{R\bar{P}i}}{301.77\phi\mu c\_t}.\tag{16}$$

The derivation of Eq. (16) follows a similar idea as that presented later in Section 4 for the use of the points of intersection.
