Fracture Initiation

Abstract

Field in-situ constant-flow hydraulic fracturing test is an important technique to determine the tectonic stress field. Classical hydraulic fracturing mechanics regards the maximum tensile stress criterion as the critical initiation condition, which assumes that the hydraulic fracture initiates and expands when the maximum effective tangential stress around the wellbore is larger than the tensile strength of the rock, resulting in tensile failure of the rock.

1 Breakdown Process Under Constant Injection Flow

Field in-situ constant-flow hydraulic fracturing test is an important technique to determine the tectonic stress field. Classical hydraulic fracturing mechanics regards the maximum tensile stress criterion as the critical initiation condition, which assumes that the hydraulic fracture initiates and expands when the maximum effective tangential stress around the wellbore is larger than the tensile strength of the rock, resulting in tensile failure of the rock. Based on this condition, Hubbert and Willis [1] put forward a theoretical criterion for rock breakdown, which is also taken as synonymous with hydraulic fracture initiation,

$$P_{{\text{b}}} = 3\sigma_{{\text{h}}} - \sigma_{{\text{H}}} + \sigma_{{\text{t}}} - p$$

(6.1)

where σ_H and σ_h are the maximum and minimum principal stresses, respectively; σ_t is the tensile strength of rock; p virgin pore pressure in the formation. This criterion is suitable for the fast pressurization case where the fluid is not able to penetrate the near-wellbore region prior to breakdown (or impermeable rock medium).

Involving the effect of impermeability of rock matrix, Haimson and Fairhurst [2] refined the classical criterion of Hubbert and Willis, and proposed a linear elastic prediction model for breakdown pressure of porous rock,

$$P_{{\text{b}}} = \frac{{3\sigma_{{\text{h}}} - \sigma_{{\text{H}}} + \sigma_{{\text{t}}} - 2\eta p}}{2(1 - \eta )}$$

(6.2)

with η is a poroelastic constant and related to the Poisson’s ratio v and Biot coefficient b as

$$\eta = \frac{{\alpha \left( {1 - 2v} \right)}}{{2\left( {1 - v} \right)}},\quad {(0} \le \eta \le 0.5{)}$$

(6.3)

The above two criteria indicate that the breakdown pressure mainly depends on the initial stress state (i.e., σ_H and σ_h). However, the above criteria failed to consider the impact of pressurization/injection rate on breakdown pressure. Moreover, current research tends to assume a linear relationship between fluid flow and pressurization rate, which is limited to reflect fluid compressibility and lag zone within the crack. When considering fluid compressibility and hysteresis, the hydraulic fracturing physical process can be schematically described by Fig. 6.1. In a closed space, the continuously injected fluid compresses the air into the defects around the wellbore. Part of the gas may penetrate into the pores of the rock matrix, and other part of the gas accumulates in front of the fracturing fluid, forming a fluid lag zone. With the continuous injection of fluid, the length of the fluid lag zone may decrease until a limit compression distance is reached. The higher injection rate results in higher fluid pressure in the wellbore per unit time, resulting in the corresponding increase amount of compression. When the fluid compression amount is close to the limit value, the pressurization rate begins to decrease. This section will model the relationship between the injection rate and breakdown pressure by using the quadratic nonlinear relation shown in Fig. 6.1. The detailed theoretical analysis process is as follows:

Fig. 6.1

Diagram of compressed air replaced by invaded water that causes pressure build-up during constant flow injection process

In line with Ito [3], three tangential stress components ($\sigma_{\theta }^{1}$, $\sigma_{\theta }^{2}$ and $\sigma_{\theta }^{3}$) induced by far-field in-situ stress, net fluid pressure with the fracture and pore pressure gradient are considered around the wellbore $\sigma_{\theta }^{1}$ is a function of far-field stress and polar coordinates (r, θ), which can be expressed by

$$\sigma_{\theta }^{1} = \frac{{\sigma_{H} + \sigma_{h} }}{2}(1 + \frac{{a^{2} }}{{r^{2} }}) - \frac{{\sigma_{H} - \sigma_{h} }}{2}(1 + 3\frac{{a^{4} }}{{r^{4} }})\cos 2\theta + \frac{{a^{2} }}{{r^{2} }}p$$

(6.4)

$\sigma_{\theta }^{2}$ is a near-wellbore stress component generated by fluid pressure increase

$$\sigma_{\theta }^{2} = \frac{{a^{2} }}{{r^{2} }}(P_{{{\text{inj}}}} - p)$$

(6.5)

and $\sigma_{\theta }^{3}$ is induced by fluid infiltration into rock matrix

$$\sigma_{\theta }^{3} = \frac{1 - 2v}{{1 - v}}\alpha \left\{ {\frac{1}{{r^{2} }}\int\limits_{a}^{r} {(P_{{{\text{inj}}}} - p)\rho d\rho - (P_{{{\text{inj}}}} - p)} } \right\}$$

(6.6)

where the wellbore pressure P_inj is a function of r and T_in, which can be calculated by

$$P_{{{\text{inj}}}} = \frac{{{\text{d}}P_{{{\text{inj}}}} }}{{{\text{d}}T_{{{\text{inj}}}} }}\int\limits_{0}^{Teq} {f(r,s)} {\text{d}}s + p$$

(6.7)

with $f(r,t) = 1 + \frac{2}{\pi }\int\limits_{0}^{\infty } {\exp ( - \frac{k}{{\mu n\beta_{{\text{w}}} }}u^{2} T_{{{\text{inj}}}} )} \times \left[ {\frac{{J_{0} (ur)Y_{0} (ua) - Y_{0} (ur)J_{0} (ua)}}{{J_{0}^{2} (ua) + Y_{0}^{2} (ua)}}} \right]\frac{{{\text{d}}u}}{u}$, k is the rock permeability, n is the rock porosity, μ is the fluid viscosity, β_w is the fluid compression coefficient, J₀ and Y₀ are the zero-order first and second Bessel functions.

The total tangential stress around the wellbore can be derived based on the principle of stress superposition.

$$\sigma_{\theta }^{{{\text{Total}}}} = \sigma_{\theta }^{1} + \sigma_{\theta }^{2} + \sigma_{\theta }^{3}$$

(6.8)

Using the maximum tensile stress principle, we have

$$\sigma_{\theta }^{{{\text{Total}}}} {(}r = a{)} = \sigma_{{\text{t}}} - p$$

(6.9)

Considering the ultra-low permeability of the shale and assuming the initial pore pressure to be zero, the breakdown pressure under different injection rates can be calculated after substituting Eqs. (6.4) ~ (6.8) into Eq. (6.9). The related calculation parameters are shown in Table 6.1.

Table 6.1 Parameters used for calculating breakdown pressure

Figure 6.2 compares the theoretical calculation results of breakdown pressure with the experimental observations. It is easy to find that the theoretical prediction results are generally larger than the test breakdown pressure value. This may result from the preexistence of natural micro-cracks, holes, impurities and other defects near the wellbore, contrary to the isotropic and homogeneous assumption in the theoretical model. When the injection rate is high, the calculation and the test results are in good agreement, and the deviation gradually increases with the decrease of the injection rate. We believe that this phenomenon is caused by the nonlinear relationship between the pressurization rate and the injection rate.

Fig. 6.2

Comparison between theoretical and experimental results of rock breakdown pressure under different flow rates

A larger flow rate leads to a greater pressurization rate and a shorter nonlinear change section of the pump pressure. The injection flow value based on the quadratic fitting can better reflect the change in the actual pressurization rate, resulting in a more consistent fitting effect at a high injection rate. This conclusion is consistent with the results of Shao et al. [5] who investigated the effects of pressurization rate on the breakdown pressure of limestone. Therefore, in summary, the predictions of the current model are consistent with the experimental observations, and the model can better explain the effect of the injection rate on the rock breakdown pressure.

2 Breakdown Process Under Constant Injection Pressure

To characterize the continuous propagation of hydraulic fractures (case (iii)), we additionally presented a conceptual model based on the wave-like theory of hydraulic fracture introduced by Jiang et al. [6]. As shown in Fig. 6.3a, for a wellbore uniformly pressurized by its internal fluid pressure (P_inj), the rock near the wellbore will suffer a gradual diffusion stress zone starting from the highest pressure area (blue area) to the second-highest pressure area (orange area). Due to the static fatigue (caused by fluid diffusion) inside the specimen, the rock in Circle 1 (called the first splitting ring) will first reach the critical splitting condition (tensile failure), then the high-pressure water enters a fractured zone (orange area), where the natural beddings and induced fractures are interconnected. This process synchronously results in a decline in the pump pressure and an increase in the injection rate. Due to this pressure loss, the fluid pressure acting at the interface between the first split ring and the second split ring no longer satisfies the splitting condition. Thus, the rock beyond Circle 1 is temporarily free from hydraulic splitting failure. As the fluid pressure in the wellbore is built up by continuous injected fluid, the fluid pressure acting at the interface (Circle 1 in Fig. 6.3b–i) increases accordingly. Once the increased fluid pressure meets the splitting condition, a new hydraulic split ring (Circle 2) appears, which corresponds to a yellow-green area in Fig. 6.3b–ii. By analogy, it can be inferred that as the fluid pressure in the wellbore is repeatedly released, increased, and released, the third splitting ring (Circle 3) may be formed as outlined by the green area in Fig. 6.3b–iii. It is worth noting that once a new fracture is initiated, there would be very little fluid exchange between the fracture and the rock matrix due to its low permeability, which means that fluid migration will occur mainly within the crack [7]. The spreading of the hydraulic splitting rings is the so-called wave-like theory of hydraulic fracture in line with Jiang et al. [6]. This process is slowly extended, which agrees well with the variations of the pumping parameters. In addition, according to Jiang et al. [6], the most crucial feature of the splitting process is that there will be more than two wave peaks in the injection rate and fluid pressure curves with the continuous propagation of a hydraulic fracture. This feature can be identified by correlating the two P_inj oscillations during the fracturing process of Specimen P-21. The consistency further demonstrates the feasibility of employing this conceptual model to explain the constant pressure injection fracturing process.

Fig. 6.3

Conceptual model of fractured zone propagation during hydraulic fracturing experiments

The circulation of the parametric variations and the split rings can be alternatively repeated until the radial crack reaches the edge of the specimen surface. Then, the hydraulic fracture starts to propagate vertically, and the propagation state will primarily depend on the pump pressure. For a relatively small value of pump pressure (e.g., P_con = 19 MPa), an approximately constant fracturing aperture may be abidingly maintained to release the internal fluid, which corresponds to the stable leakage phase in the pump pressure curves. This inference was proved by the local leakage of the fracturing fluid observed during the experiment in Fig. 5.4b. Whereas in the case of a higher pump pressure (e.g., P_con = 21 MPa), through comparing the fluid leakage location and trajectory at a different time in Fig. 5.7b, we see that the leakage paths are gradually opened with an overall increase of the injection rate (Fig. 5.3a), corresponding to the subsequent local and unstable cracking stages during the fracturing process. Thus, the speculation for the conceptual model is reliable to elucidate the fracturing process under constant pressure injection conditions.

In summary, the pump’s output modes (constant pressure or constant flow) have a pronounced influence on the morphology of the hydraulic fractures. Moreover, different stages of fluid pressure (or injection rate) curves are closely correlated with varying fracture behaviors in the fracture propagation process, which implies a potential for using the pumping parameter curves to predict the initiation and propagation of hydraulic fractures in actual engineering.