Simulation of the Crack Propagation in Rocks Using Fracture Mechanics Approach

Abstract

Crack propagation in brittle rock was simulated under different conditions to describe fracturing behavior of rocks due to the applied stress e.g., water pressure. It is assumed that pre-existing cracks initiate and propagate from the edges of the borehole. The two-dimensional finite element fracture code Franc2D with the non-cohesive method was used for computing the stress intensity factor (SIF), energy-release rates (G), and crack propagation and fracturing time. Static tensile and normal-distributed stresses were used within Franc2D to describe the fracture creation and propagation. Therefore, the pressure inside the bore hole was distributed as a tensile load along with the crack faces. Different scenarios were simulated by changing the boundary conditions, crack initiation, and propagation paths. The SIF determines the amount of tensile failure that is required to create a fracture. Then, the injection rate and pressure can be determined. The direction of fracturing is perpendicular to the maximum applied stresses. The crack propagation direction was compared with experimental observations taken from the literature. The predefined SIF solutions were modified according to the Franc2D solutions. Hence, the ability to use Franc2D for fracture simulation in brittle rock was demonstrated.

Introduction

The consideration of the opening stress in determination of the crack path has to be taken into account in rock mechanics. Therefore, it is important to investigate the crack path and its variation by calculating stress intensity factor (SIF).

A network of fractures with a certain length and depths is desirable in rock in order to increase the fluid flow within a rock in a productivity of wells by means of hydraulic fracturing [1]. However, fractures are well known in materials like rocks, the behaviour of cracking remains unclear and need to be studied using numerical simulation [2, 3].

Many rocks such as granite or carbonates have a high modulus of elasticity and brittleness with low permeability. The brittle materials are considered to be linear elastic, and SIF can be used to describe the crack tip stress state in fracture mechanics approach. The investigation of the development of hydraulic fracture is associated with the crack propagation through the solid structure. The fracture growth rate and hydraulic fracture typically are based on the assumptions of linear elastic fracture mechanics (LEFM) [4].

Fracture toughness in terms of SIF is widely used as a propagation criterion in LEFM. Crack propagates if the SIF at the tip of a crack matches the rock toughness. For brittle materials, opening mode SIF reaches opening fracture toughness of materials before sliding mode SIF (e.g., K _I = K _IC), where K _I is the opening SIF calculated due to the tensile opening stresses, and K _IC is the opening fracture toughness [4, 5].

To simulate the fracture propagation using LEFM–FEM, the rock is assumed to be isotropic, homogeneous, and linear elastic [6]. This was done because heterogeneity of properties would make the numerical modeling more complicated. However, it is recommended to consider heterogeneity if cracking due to fluid flow and permeability in porous rock is expected.

In this work, SIF’s were calculated using a Franc2D program from Cornell University Fracture Group [7] which needs less data and requirements than a 3D model. Nevertheless, the fracture propagation in 2-D requires a series of analysis to be calculated with automatic meshes around the fracture. Understanding how theses fractures form and propagate is an important part of this simulation. Calculations of SIF due to the tensile and shear applied load K _I, K _II, respectively, are compared with the available solutions. Results from the computing of SIF’s and strain energy-release rate or the dissipation energy during fracture (G) considering the effect of hydraulic pressure are presented. The propagation time was calculated, however, future work is still needed to evaluate the fracture life parameters.

Numerical Method and Calculation

Because LEFM approach is based on the linear relation between peak load, and crack length with SIF (see Eq 1), an accurate crack path simulation is required. Franc2D has been evaluated in determining the accurate crack path, SIF, and G [8].

Understanding the proper modeling of pre-existing cracks is keys to explain the complexity observed during hydraulic fracturing [1]. The numerical finite element tool Franc2D has an advantage in comparison with other codes because it has ability to simulate the accurate cracks’ propagation.

Fracture prediction triggered by fluid pressure by means of Franc2D was not described so far. Hee et al. [9] showed that Franc2D is able to predict the growth of cracks for dam structures. In that case the dam material was assumed as an isotropic elastic material. The crack was subsequently allowed to form and grow according to a fracture criterion under the effect of water high load. However, no static or hydraulic pressure was taken into account. Simulation of fractures in terms of the stress intensity factor (SIF), K, and the fluid pressure is essential to understand the behavior of the reservoir. Simulating the fracking process using Franc2D was carried out under a given stress and strain environment for some extended application, e.g., Geothermal-Hot Dry Rock, see Fig. 1.

In this work, SIF and G were determined using Franc2D under the opening and shear mode (I, and II, respectively) to simulate whether the crack growth is stable or unstable, see Fig. 2.

Fig. 1

Fractures stress distribution and potential cracking

Fig. 2

(a) opening or tensile mode-I, (b) sliding or shearing mode-II

However, because Franc2D does not simulate fluid flow, static pressure distributions were assumed. The opening mode SIF (K _I), where the tensile stress is normal on the crack faces, has been used in propagation life calculation [10] because the fracture mechanics approach assumes that the stress is always effective and the crack remains open during the propagation period. This assumption presents a good description to consider the effect of pressure inside the borehole. The influence of K _I on crack growth was based on the maximum tangential stress criterion presented by [11]. This criterion assumes that the predicted propagation path of the crack is perpendicular to the maximum principal tensile stress under opening mode where stress ranges were fully effective [10]. The main expression for SIF is

$$K_{\text{I}} = f(b)\sigma \surd \pi a$$

(1)

The applied stress in term of water pressure is Δσ. f (b) is the geometrical factor as a function of the crack length to width ratio f (a/t). The analytical solutions of G according to [12] are

$$G_{\text{I}} = K_{\text{I}}^{2} (1 - \upsilon^{2} )/E$$

(2)

$$G_{\text{II}} = K_{\text{II}}^{2} (1 - \upsilon^{2} )/E,$$

(3)

where υ is Poisson’s ratio and E is Young’s modulus. SIF and G per unit thickness were calculated. For plane stress, (i.e., the default problem type in Franc2D) the factor (1 − υ ²) in Eqs 2 and 3 is replaced by unity. Franc2D computes also G. The analysis of a given piece of rock is described in two parts. The first part is building the FE initial mesh using CASCA [7]. CASCA is a pre-processor for Franc2D. But, meshes can be also created by any other FE mesh generating program. A translator code is available to convert the mesh description to the Franc2D*.inp format [7]. The second part is using a post-processor and the FE-based simulator Franc2D code to assign boundary conditions, perform stress analysis, introduce and propagate the cracks.

Model Input Parameters

Franc2D was used to simulate a hydraulic fracturing experiment which was performed by the Petroleum Engineering Department at the Colorado School of Mines [13, 14]. A block of Colton sandstone was cut to a dimension of 0.76 m × 0.76 m × 0.91 m (high) with a centralized 38.1-mm-diameter hole drilled along the entire height of the block, see Fig. 3.

Fig. 3

Colton sandstone with a borehole: (a) Block geometry; (b) fracture mechanics model of hydraulic fracturing [13, 14]

Material properties are based on a hydraulic fracturing experiment (Table 1). The maximum and minimum horizontal stresses of 28.95 and 16.54 MPa have been assigned in x-, and y-axis, respectively (Table 2). Because of the 2D approach and the cracking from a bore hole, the internal water pressure was considered as an opening stresses.

Table 1 Material properties of Colton sandstone [13]

Table 2 Basic parameters and test results [13, 14]

To control how the mesh is created, it is helpful to divide the model into four quarter subregions before subdividing the edges into elements (Fig. 4a). The choice of mesh size and density depends only on the experience of the user. Otherwise several iterations for the same geometry are required. After creating the subregions, the boundary lines were subdivided and the meshes are created (Fig. 4b, c).

Fig. 4

Model and meshes in Franc2D: (a) subregions of the body, (b) meshes distribution, (c) meshes near the hole

Numerical Simulation of Crack Propagation

Different cracking models can be selected in Franc2D. In this work non-cohesive edge crack models were selected at a location where the effective stresses are the maximum under tension. Otherwise, the user would start a crack at another desired location, or if there are other reasons to believe that a crack is likely at a certain location [15]. The notch was developed to the left and rights of the borehole (see Fig. 3). The crack length and increment length were chosen as a _i = 10 mm, Δa = 5 mm, respectively. The numbers of steps are chosen automatically or could be determined by the user according to body size. The program will automatically delete a number of elements and update the mesh around the crack tip, see Fig. 5.

Fig. 5

Crack initiation and propagation sequences, a _i  = 10 mm, Δa = 5 mm: (a–e) deleted meshes around the right side crack tip; and the new meshes around the crack tip have created; (f) the second crack

The bore pressure was considered as a tensile stresses over the initial crack face. The models of the hydraulic fracturing were compared with the experimental observations. The internal loading on body deformation shows that the opening stresses are the main factors influencing the cracking. Therefore, an opening mode-KI was proposed to be the driving force of cracking. The fracture was propagated by increasing the fluid pressure until the crack is initiated. According to the stress analysis, the new crack propagated normal to the applied stresses. A simple 2D cross-section through an idealized hydraulic fracture will be presented in the following cases.

Case 1

The internal bore pressure of 26 MPa was considered as a tensile stresses in y-direction (Fig. 6a). Two cracks above the midpoint of holes were created. The cracks will turn to be normal to the direction of the load (Fig. 6b).

Fig. 6

Cracking due to borehole pressure 26 MPa, Δa = 5 mm; (a) a tensile load on the block in y-direction; (b) failed block and crack path above the midpoint

Case 2

The normal stress on crack faces is equal to the applied pressure from the observations in the work of Ref. [1]. In other words, the uniform pressure distribution inside fracture was used (Fig. 7a). The internal bore pressure 26 MPa is distributed normally inside the hole. Hence, the crack opening displacement (COD) is increasing, i.e., the crack opening area increases (Fig. 7b). The cracking speed increases with the decreasing of the opening SIF and increasing the crack length. As a consequence, the required pressure will be decreasing.

Fig. 7

Cracking due to borehole pressure 26 MPa, Δa = 5 mm; (a) a tensile load normally distributed inside the hole; (b) failed block and crack from path midpoint

Case 3

The bore pressure of 26 MPa was considered as a tensile stress on the block in y-direction, see Fig. 8a, but the cracks in this case were located at midpoint on left and right the hole. The COD in which a measure of crack faces opening is less than previous cases (Fig. 8b).

Fig. 8

Cracking due to borehole pressure of 26 MPa, Δa = 5 mm; (a) a tensile load on the block in the y-direction: (b) failed block and crack path from midpoint

Fracture mechanics assumes that the SIF’s remain constant during propagation and the cracks remain in the horizontal plane due to the bore internal pressure [16]. Therefore, the load causing the propagation was assumed to be distributed along the crack face during propagation.

Results and Discussion

This work presents some cases showing the influence of boundary conditions and internal pressure on the generated hydraulic fracture in term of SIF and G. Modeling of boundary conditions and fracture mechanics criterion of pre-existing fracture has been implemented. The cracking is based on the tensile mode, while the shear mode remains approximately near zero [17].

SIF and G Calculations

The crack propagation occurred due to the fluid pressure. The critical SIF and critical G approach have successfully been used to predict catastrophic crack propagation [18]. Fracture growing simulation and SIF’s were calculated as follows:

Case 1 Load is distributed over the boundary in y-direction. Figure 9a shows the SIF calculation for case 1. The opening SIF mode-I and shearing SIF mode-II SIF’s are calculated. The conventional G also indicates to the rate of cracking. Figure 9b shows that the G in opening mode-I is released at a higher rate than in shearing mode-II. However, the crack length increment was fixed during the propagation and the released energy G increased rapidly. This is because the cracking speed and SIF are increasing during the propagation.

Fig. 9

SIF and G mode-I and II for case 1

Case 2 In this case SIF, K _I decreases as the crack increases (Fig. 10a). In reality, the pressure decreases from the breakdown values of 26 MPa for each crack step. Therefore, fracture toughness (SIF) decreases consequently. Figure 10b shows the release energy rate which has the same behavior of SIF.

Fig. 10

SIF and G mode-I and II for case 2

Case 3 26 MPa was applied as a tensile load on the block in y-direction. But the cracks are located at the midpoint on the left and right of the hole. SIF and G again increase with increasing the cracking, see Fig. 11.

Fig. 11

SIF and G mode-I and II for case 3

As shown the above calculation, the tensile opening mode-I SIF (K _I) is always larger than shear sliding mode-II SIF (K _II). Therefore, the fracture toughness in tensile direction is smaller than those in shear direction. Moreover, and due to the lower value of opening fracture toughness, the opening mode SIF reaches its critical value before the sliding mode [5].

Critical Strain Energy

Franc2D results agree well with the analytical solutions in case of plane stress (Eqs 2, 3), see Figs. 12, 13 14.

Fig. 12

Comparison between Franc2D and analytical solutions, case 1

Fig. 13

Comparison between Franc2D and analytical solutions, case 2

Fig. 14

Comparison between Franc2D and analytical solutions, case 3

Modified the Model using Franc2D

Zhao’s et al. [19] proposed a failure strength model for brittle materials with a pre-existing open hole. Moreover, Sammis and Ashby’s analyzed the growth of crack from a circle hole [19]. They did not use the geometry factor f(a/t) inside their models. A comparison between Franc2D and Sammis–Ashby theoretical model was carried out. Since the current calculated SIF from Franc2D include already the geometry factor, the Ashby’s model was modified accordingly. The crack direction and propagation are shown in Fig. 15.

Fig. 15

Numerical simulation of fractured rock under compression by Franc2D

In addition, Ashby’s models were modified according to the crack propagation path. Therefore, the explicit crack length (a) is used instead of normalizing crack length L (i.e., L = a/r), where r is the radius of the hole and a the crack length in Ashby’s model. Also, f (a/t) is proposed as follows:

$$K_{\text{I}} = - \sigma_{1} \sqrt {\pi a} \left[ {\frac{1.1\sqrt a }{{\left( {1 + a} \right)^{3.3} }}} \right]/f\left( \frac{a}{t} \right),$$

(4)

where σ ₁ is the uniaxial stresses. By considering the interaction of the short cracks, SIF is determined with current consideration of f(a/t) as follows:

$$K_{\text{I}}^{\text{B}} = - \sqrt {\frac{3}{\pi }\mu } \cdot \frac{{\alpha^{2} }}{1 + a}\sigma_{1} \sqrt {\pi r} \left[ {\frac{1}{{1 + \frac{12}{{\pi^{2} }} \cdot \frac{{\sigma_{1} }}{{E_{\text{o}} }} \cdot \mu^{2} \left( {1 + a} \right)^{2} }}} \right]/f\left( \frac{a}{t} \right)$$

(5)

Then, the fracture toughness of the whole specimen with an open hole is thus given as follows [19]:

$$K_{\text{IC}} = K_{\text{I}} + K_{\text{I}}^{\text{B}}$$

(6)

The comparison between modified Ahby’s model and FE calculations of SIF is shown in Fig. 16.

Fig. 16

Comparison between a modified model and numerical solution

Calculation the Time-Dependent Fracture Mechanics

The fracture mechanics fatigue life approach is used to calculate the propagation time until failure. The assumption of fracture mechanics has been used which based on the SIF calculations, since the cyclic load may occur due to the water pressure. The FORTRAN code was written to calculate the numerical integration of fracture time for the block in Fig. 1. It is based on Paris’ law. The details of the numerical integration which is based on Paris’ Eq 7 were presented in [10]

$$\frac{{{\text{d}}a}}{{{\text{d}}N}} = C\left( {\Delta K_{\text{I}} } \right)^{m} ,$$

(7)

where C and m are fatigue life coefficient. They are the material-dependent constants. It is appropriate to extract SIF versus crack length history computed within Franc2D and use this information in numerical calculation code [10]. The experiment fatigue life test was carried out for the sandstone of the area of reservoir [17]. Hence, the parameters C and m were adopted from the fatigue fracture testing. The extending of the fracture is expressed in term of SIF range. The ability of Franc2D for simulating of the hydraulic fracturing processes is shown in Fig. 17.

Fig. 17

Calculated fracture propagation

Conclusions

Using of fracture mechanics criteria that are based on the pre-existing fracture combining SIF and G during the crack propagation is quite new. This work evaluates different loading and cracking conditions to describe the failure behavior. The fracture mechanism model is essential for understanding the flow behavior of rocks. Different problems have been solved to select the appropriate boundary conditions for hydraulic fracture within Franc2D which is a unique tool in simulation of cracking under the opening mode. The numerical simulation shows that Franc2D is able to compute SIF’s and G. Crack pathways were propagated similar to those from the experimental studies of Colton sandstone at Colorado School of Mines. The consideration of normal-distributed load inside the bore hole gives realistic results as compared with experimental observation. The propagated time until failure was calculated using a FORTRAN code which carries out the numerical integration of the fracture life equation.

No big influence is assumed to be occurring due to the limitation of using a 2-D model because SIF already included the geometric ratio (a/t). The new model was validated against laboratory experimental results and a FE numerical model. The limitation of this simulation is the consideration of thermal and fluid distribution. Also, if there is a multi-scale cracks which are interconnected, the behavior will be different. More work is still required to determine the fatigue life coefficients such as C and by comparing the results from Franc2D with those from experiments.