PSI - Issue 2_A

Cherny S.G. et al. / Procedia Structural Integrity 2 (2016) 2479–2486

2481

Cherny S.G., Lapin V.N. / Structural Integrity Procedia 00 (2016) 000–000

3

Fig. 2. Domain of the fluid flow and its piecewise planar representation.

and the Navier-Stokes equations without negligible small terms

W 3 12 µ ∇

q = −

P .

(2)

The Reynolds equation is derived from (2) and (1) (see for ex. Hamrock et al. (2004)) ∇ ( a ∇ P ) = f ,

(3)

where

W 3 12 µ

, f = ∂ W

a =

(4)

∂ t .

This equation is supplemented by the boundary conditions at the fluid front x f and at the inflow boundary x f (see Fig. 2) P | x f = p pore , q | x q = q in · n q , (5) where p pore is the pressure of the porous fluid, n q is the normal vector to the boundary x q that lies in the tangent plane to the fracture surface, q in = Q in / L q is the average inflow rate that is calculated using the given inflow rate Q in and the length L q of the inflow boundary x q . In terms of pressure the second condition (5) is rewritten as ∂ P ∂ n = − 12 µ q in W 3 n q . (6) The equation (3) is solved by finite element method (FEM) as is has been described in Shokin et al. (2015), Kuranakov et al. (2016). This method transforms the di ff erential problem (3-5) into a system of linear algebraic equations that can be written as K P = Q + F , (7) where P = ( P 1 , ... P N ) is the vector of pressure values at all N nodes of the computational mesh, K is a N × N matrix and Q , F are vectors of size N . It should be noted that in case of Newtonian fluid K , Q , F are calculated using formulas (4), (6) and do not depend on the pressure P . The equation (2) for the case of the Newtonian fluid is obtained using the Poiseuille velocity profile (see Fig. 3, left) as the solution of the problem about fluid flow between parallel flat plates (see for example Hamrock et al. (2004)). In case of Herschel-Bulkley fluid the velocity profile is di ff erent (see Fig. 3, right) and the equation (2) can be written as q = − n (4 n + 2)(2 K ) 1 / n W 2 + 1 / n |∇ P | 1 / n 1 − 2 z τ W 1 + 1 / n 1 + 2 z τ W n n + 1 , z τ = τ 0 |∇ P | − 1 . (8) It also can be rewritten in the form of (2) by using the variable apparent viscosity µ app ( P ) instead of the constant viscosity µ . The apparent viscosity can be written as the function of the pressure 2.3. Herschel-Bulkley model

(2 K ) 1 / n (2 n + 1) 6 n

(4 n + 2)2 1 / n τ 0 3 n ( W |∇ P | ) 1 / n .

( W |∇ P | ) ( n − 1) / n +

(9)

µ app =