Fig xA Percentage distribution of fluid into

The fluid flow inside a fracture BIX 02565 modeled by assuming fractures as parallel plates. For a laminar flow the governing equation for a Newtonian fluid between two parallel plates is often called as cubic law (Eq. (6)):equation(6)q=−w3h62μ∂p∂xwhere q is the volumetric flow rate, µ   is dynamic viscosity of the fluid, ∂p/∂x∂p/∂x is pressure gradient in the direction of fluid flow, w and h are fracture width and height, respectively. The continuity Eq. (7) along with cubic law governs the fluid flow inside the fracture:equation(7)∂q∂x+∂A∂t+q=L0where A is the cross-sectional area of the fracture and qL is fluid leak-off volume rate per unit length of the fracture. Because of the ultra-low permeability nature of shale reservoirs, leak-off is assumed to be zero in adaptation calculations ( Rodrigues et al., 2007). Substituting Eq. (6) in Eq. (7) gives us the partial differential Eq. (8) which is called as lubrication equation. The Eq. (8) is solved using a finite difference approximation with the following initial and boundary conditions (Eq (9), (10) and (11)):equation(8)∂w∂t=∂∂x(w312μ∂p∂x)equation(9)w(x,0)=0w(x,0)=0equation(10)q(0,t)=Q0q(0,t)=Q0equation(11)∂p∂x(l,t)=0equation(12)∫02lwdx=Q0t