PSI - Issue 6

Paderin Grigory et al. / Procedia Structural Integrity 6 (2017) 276–282 Paderin G.V./ Structural Integrity Procedia 00 (2017) 000 – 000

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circumstance leads to idea for an averaging approach . Since in a homogeneous medium the network of micro fractures will not have any preferential propagation direction, then their system must have distribution of equal density over the entire ring of the same radius. Because the distribution of microfracture s in size and direct ion directly assigns a fractured permeability, it is logical to assume that the permeability of the system depends only on the radial coordinate. Also, the fracture network isotropy leads to the fact that the fractured permeability tensor in this problem is also isotropic. Since the matrix itself is considered isotropic, the permeability is isotropic everywhere and the equation for the propagation of the liquid is rewritten as: + ∇ ∇ + ∆ = 0 (5) For further consideration, the simple hypothesis is used. Let the permeability of the format ion 0 increases rapidly to when the pressure = + is reached. In all other zo nes the permeability considered constant. Here σ is the horizontal rock stress, and is the critical tensile stress of the rock. Then the problem at each time moment will split into two domains: + ∆ = 0; < < ; > (6) (7) In the first region, the medium has a fractured conductivity, in the second - the usual conductivity of the formation. There are several boundary conditions in the problem. The total flu id flow rate at the we ll is the in jection rate , which is constant in t ime. At infinity, the pressure is equal to the reservoir pressure 0 , and at the interface between the two zones it is always . In addition, the fluid flow of the hydraulic fracturing at the interface is continuous. These conditions are written in the following form: = − 2 ℎ ; = (8) → 0 ; → ∞ (9) = ; ( ) = ( ) (10) ( < ) = 0 ( > ) (11) The two coefficients of the piezoconductivity are defined: = ; 0 = 0 (12) The solution in both regions in the radial case has a follows form: = С 1 (− 2 4 ) + 0 (13) The values of the constants in each of the regions are determined by the boundary conditions. The boundary condition is defined on the surfaces 2 4 0 ⁄ = . Starting from some t ime, 2 4 0 ⁄ = ≪ 1 , and this condition could be replaced by the boundary condition for = 0 . This assumption will work properly on a scale of several seconds orminutes . + 0 ∆ = 0; > ; 0 < <