How Did Sneddon And Elliott Solve The Theory Of Cracks?

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In 1946, Sneddon and Elliott modified the theory of cracks in a two-dimensional elastic medium which was developed by Griffith who succeeded in solving the equations of elastic equilibrium in two dimensions for a space bounded by two concentric coaxial ellipses. (Sneddon and Elliott 1946). They solved the equations for the stress field and pressure associated with static pressurized cracks:
W_average=W ̅=(π/4)W(0,t) (2.2)
W(0,t)=(2P_net h_f (1-υ^2))/E (2.3) where W ̅ is the average value of the fracture width, W(0,t) is the fracture width
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7)
C_l=((KC_t ϕ)/πμ)^0.5 (P_f-P_R) (2. 8) where t is time, τ(x) is the time when the fracture tip gets to point x, C_l is the leak-off coefficient, k and ϕ are the reservoir permeability and porosity, respectively, C_t is the total compressibility, μ is the fluid viscosity, and P_f and P_R are the fracture and reservoir pressures, respectively. u(x,t)=C_l/(t-τ (x))^(1/2) (2.7)
C_l=((kc_t∅)/πμ)^(1/2) (P_f-P_r ) (2.8) where τ (x) is the time the fracture tip arrives at the location x, and C is the fluid loss coefficient, in which k,c_t and ∅ are the permeability, total compressibility and porosity of the reservoir, and P_f and P_r are fracture and reservoir pressure. Figure 2.5 Carter Assumptions for fracture
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Each vertical cross section deforms individually and it is not hindered by its neighbors. The cross sections obtain an elliptical shape with the maximum width in the center. The fluid pressure gradient is determined by the flow resistance in the narrow, elliptical flow channel in the propagation direction. The fluid pressure in the fracture decreases toward the tip of the fracture.

The Perkin and Kern’s original theory together with Nortgren’s modification is now referred to as the Perkins-Kern-Nortgren (PKN) model. In the original model, the influence of the growth rate of the fracture width on the flow rate was neglected. Then in the absence of fluid losses, the term ∂P/∂x is equal to zero. Although this assumption is acceptable if fluid loss dominates the material balance, but causes a significant error in the case of little or no leak-off. Then Nortgren corrected this growth- rate effect and rewrote the continuity equation in the form of (Nordgren 1972):
∂P/∂x=-(πh_f)/4 ∂w/∂t

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