PSI - Issue 6

Paderin Grigory et al. / Procedia Structural Integrity 6 (2017) 276–282 Paderin G.V./ Structural Integrity Procedia 00 (2017) 000 – 000 It should be noted that ( ) - the fracturing front, moves with time together with the constant pressure contour = . For a parabolic equation, it is logical to assume that the constant pressure circle moves according to the law: ( ) 2 4 0 = 02 = (14) This assumption could be confirmed only by solving the problem and fin ding some 0 satisfying the boundary conditions. Using these assumptions, the pressure in the near wellbore zone is derived: = 0 ( (− 02 ∗ 0 ) − (− 2 4 )) + (15) Here = 4 ℎ 0 ⁄ is the characteristic depression on the formation due to fluid inject ion. Solution in the second area: = − 0 (− 0 2 ) (− 2 4 0 ) + 0 (16) The value 02 is found from the condition of equality of flu id flows at the boundary. Converting, we obtain the transcendent equation: − 0 = − ( − 02 ) (− 02 ∗ 0 + 02 ) (17) The left part, in practica l cases, is essentially positive, since the horizontal rock stre ss is almost always higher than the reservoir pressure. The left-hand side is a lso positive for any 0 , since ( − 02 ) is negative. Hydraulic fracturing applied, usually, to the reservoirs with low permeability. So, it could be assumed that 0 ≫ 1 . Then the equation could be rewritten as: + − 0 = − ( − 02 ) ( 02 ) = ( 0 ) (18) As it could be seen (Fig. 1), in the first approximation, the equation and its root, are almost independent on 0 ⁄ . The function on the right side decreases monotonically on the interval (0, +∞) , and takes values in the interval (0, + ∞). Its graph could be seen below. Thus, the equation always could be solved. We denote the parameter on the right side of the equation ( + − 0 ) ⁄ = . With increasing viscosity of the liqu id or flow rate, ( + − 0 ) ⁄ = → 0 => 0 → +∞ . With the increase in the injection rate of the flu id, the front of the fracture network will spread faster, and vice versa. This result qualitatively does not cause objections. From this it is clear that the crit ical parameter fo r the fracturing in the isotropic case is . Dependence on the properties of the material itself is performed only through . It is worth noting that for any reservoir conditions and isotropic materia l of any toughness, one can choose so that the dimensionless parameters are identical. Thus, the character of crack propagation does not determined on certain strength characteristics, like brittleness. 279 4