FIGURE 16 IRWIN’S KINEMATIC FRACTURE MODES

Figure 16 identifies the distinct criteria for crack propagation within each of these modes. In mode I, or opening

mode, the crack surfaces are pulled apart by tensile forces creating symmetric deformation on the plane parallel to the tip of fracture advancement or propagation. In mode II, or sliding mode, the crack surfaces are subjected to shear forces parallel to the fracture surface. In mode III, or tearing mode, the crack surface are subjected to shear

41 forces parallel to the fracture front. These idealized opening modes often work in combination of each other creating a mixed mode situation. This requires that the stress superposition to be used in order to determine the individual components and their respective contribution.

𝜎𝑖𝑗 =

1 √2𝜋𝑟[𝐾𝐼𝑓𝑖𝑗

𝐼_{(𝜃) + 𝐾}

𝐼𝐼𝑓𝑖𝑗𝐼𝐼(𝜃) + 𝐾𝐼𝐼𝐼𝑓𝑖𝑗𝐼𝐼𝐼(𝜃)] + 𝜗(𝑟1/2)

Where 𝐾𝐼, 𝐾𝐼𝐼, and 𝐾𝐼𝐼𝐼 are the fracture mode stress intensity factors corresponding to Mode I, II or III.

The stresses induced by hydraulic fracturing dynamically change the in-situ stress distributions through a time dependent, or transient relationship. It is therefore highly important to capture these interactions during the injection period when trying to model the dynamic stress regimes during the injection process.

This correlates to the stress intensity factor governing the fracture opening mode.

𝐾𝐼 = 1 √𝜋𝑎∫ 𝑝𝑅(𝑡) 𝑎 −𝑎 √𝑎 + 𝑡 𝑎 − 𝑡𝑑𝑡

This equation is reduced to the following when subject to a uniform stress field, however this is not the case for hetergenous anisotropic materials.

Through this the fracture toughness is defined in terms of the critical sress intensity factor. ∆𝜎𝑐=

𝐾𝐼𝑐

√𝜋𝑎

Where ∆𝜎𝑐 is the defining critical stress differential, 𝐾𝐼𝑐 is the fracture intensity parameter.

These time-dependent relationships require a complex association between hydraulic and mechanical behaviours. The pressure induced by fluid injection surrounding the borehole controls the opening of new fractures, dilation of pre-existing fractures and their associated propagation. The transient pressure relationship during injection can be referenced from the figures below.

The injection pressure increases until surpassing the fracture initiation or breakdown pressure which is controlled by the material matrix surrounding the point of injection. The pressure reuired to start initiation is greater than that of the pressure required to propagate the fracture and therefore the pressure declines slightly after fracture initiation throughout the period of propagation. As the fracture propagates the pressure plateaus at what is known as the closure pressure until reaching the transient reservoir behavior. This process can be visualized in

Figure 17 below.

FIGURE 17 HYDRAULIC FRACTURE INJECTION PRESSURE WITH TIME

42

3.2.2. S

C

F

I

One of the most important parameters controlling the propagation or initiation of fractures is the fracture toughness parameter derived from the concept of the stress intensity factor. It is assumed through fracture mechanics that preexisting defects will always pervade a continuum. These inclusions provide areas of high stress concentration with respect to the surrounding intact media. The initiation and propagation of fracture is predicted to occur in these areas of localized stress concentration. The inclusion of these defects requires complicated stress interaction to be accounted for either directly or indirectly in analytical and numerical methods.

The stress that is required to initiate a fracture usually differs from the amount of stress required to propagate the fracture. The difference in breakdown to extensional pressure relies in the rock specific properties relating to deformation.

𝜎𝑖𝑗 =

𝐾𝐼

√2𝜋𝑟𝑓𝑖𝑗(𝜃) + ⋯

On a microscopic scale the nucleation of fracture is controlled at the fracture tip by the strength of the atomic bonds throughout the rock matrix. The primary stress component for the crack tip region was summarized by Irwin with the following relation to the three displacement modes of failure.

𝜎𝑦= 𝐾𝐼 (2𝜋𝑟)12 𝑓(𝜃) 𝜎𝑥𝑦= 𝐾𝐼𝐼 (2𝜋𝑟)12 𝑓(𝜃) 𝜎𝑦𝑧= 𝐾𝐼𝐼𝐼 (2𝜋𝑟)12 𝑓(𝜃)

The stress intensity factors are related to the strain energy release rate with the following. 𝐺𝐼 = (𝑘 + 1) 8𝜇 𝐾𝐼 2 𝐺𝐼𝐼= (𝑘 + 1) 8𝜇 𝐾𝐼𝐼 2 𝐺𝐼𝐼𝐼= (𝑘 + 1) 8𝜇 𝐾𝐼𝐼𝐼 2

Secondary permeability lies with the efficiency of stimulated fracture treatments as it is typically inferred as the resistance to driving force between two parallel plates.

−𝜇(ℎ𝑤)𝜕

2_𝜇

𝜕𝑤2=

𝜕𝑝 𝜕𝑥(ℎ𝑤)

3.2.3. F

P

G

The real geometry of fracture is an extremely dynamic process controlled by heterogeneity and anisotropy developed over millions of years. In order to simplify the fracture process, certain assumptions are made to limit the uncertainty of the solution while maintaining a relatively close relationship to field data. The classical method

43 of fault geometry in a homogenous, isotropic medium considers a symmetric wing shaped pattern to the development of fractures from the point of initiation, or the wellbore in this case. This follows the assumption that the fracture initiates from a maximum height at the point of injection and declines as the distance from the injection point increases, to the extents of the fracture or “half-length” (SPE Reprint Series, 1990).

There are multiple theories that follow this approach using slightly differing cross-sectional geometries. These geometries are typically described by the use of three common modeling schemes:

(1) PKN fracture geometry (2) KGD fracture geometry (3) Radial fracture geometry

These types of fracture propagation are governed by three sets of equations that require coupling through each of the models. This is in reference to the continuity, elasticity or linear elastic fracture mechanics (LEFM) and fluid- flow (momentum) equations.

K

-G

K

F

M

The Khristianovic-Geertsma de Klerk (KGD) model follows the assumption that the fracture height is fixed and that fracture width is constant in the vertical direction as it does not depend on the height. The rock stiffness is calculated in the horizontal plane only, or 2D plane strain deformation in this plane. The fluid pressure gradient is determined from the equation below.

𝑤(0, 𝑡) =2(1 − 𝑣)𝐿(𝑝𝑓− 𝜎ℎ) 𝐺 𝑝(0, 𝑡) − 𝑝(𝑥, 𝑡) =12𝜇𝑞𝑖 ℎ𝑓 ∫ 𝑑𝑥 𝑤3_{(𝑥, 𝑡)} 𝑥 0