2. TDS basis

The pioneer publication on the TDS technique, [5], explains in detail the derivation of the equations. The Laplace space solution of the arithmetic pressure derivative for a homogeneous and infinite reservoir with skin and wellbore storage is also presented in [5] and given by

$$P\_D ' = \frac{4}{\pi^2} \int\_0^\infty \left( \frac{e^{-u^2 t\_D}}{u \left\{ \left[ u C\_D I\_0(u) - (1 - C\_D u u^2) I\_1(u) \right]^2 + \left[ u C\_D Y\_0(u) - (1 - C\_D u u^2) Y\_1(u) \right]^2 \right\}} \right) du \,\tag{1}$$

However, we know that the pressure derivative is a horizontal line during radial flow regime. The dimensionless pressure derivative during radial line is easier represented by

$$t\_D \, ^\*P\_D '= \mathbf{0}.\mathbf{5}.\tag{2}$$

Then, to obtain practical equations, dimensionless parameters must be used. The dimensionless time, based upon half-fracture length and reservoir drainage area, is given below:

$$t\_{D\text{xf}} = \frac{0.000263kt}{\phi \mu c\_t \varkappa\_f^2} \tag{3}$$

and

$$t\_{DA} = \frac{0.000263kt}{\phi \mu c\_t A}.\tag{4}$$

The dimensionless pressure and pressure derivative parameters for oil reservoirs are given by

$$P\_D = \frac{kh\Delta P}{141.2q\mu B} \tag{5}$$

and

$$t\_D \, ^\ast P\_D \, ^\prime = \frac{kh \, (t \, ^\ast \Delta P \, ^\prime)}{141.2 q \mu B} \, . \tag{6}$$

Finally, the dimensionless fracture conductivity introduced in [3] is defined as

$$\mathbf{C}\_{f\mathbf{D}} = \frac{k\_f w\_f}{k\_\cdot \mathbf{x}\_f}.\tag{7}$$

It is observed from Eq. (5) that the two key parameters of a hydraulic fracture are the half-fracture length, xf, and the fracture conductivity, kf wf. The total length of the fracture is given by 2 xf.

The easiest application of TDS technique is given by replacing the dimensionless pressure derivative defined by Eqs. (6) and (2), to provide an expression to readily determine formation permeability:

$$k = \frac{70.6q\mu B}{h(t\*\Delta P')\_R},\tag{8}$$

where (t\*ΔP<sup>0</sup> )<sup>R</sup> is the pressure derivative value during radial flow regime. The equations for the TDS technique are derived in the same manner Eq. (8) was obtained.
