# A Numerical Study on the Effect of Anisotropy on Hydraulic Fractures

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## Abstract

In this paper, we present a two-dimensional numerical model for modelling of hydraulic fracturing in anisotropic media. The numerical model is based on extended finite element method. The saturated porous medium is modelled using Biot’s theory of poroelasticity. An enhanced local pressure model is used for modelling the pressure within the fracture, taking into account the external fluid injection and the leak-off. Directional dependence of all the rock properties, both elastic and flow related, is taken into account. A combination of the Tsai–Hill failure criterion and Camacho–Ortiz propagation criterion is proposed to determine the fracture propagation. We study the impact on fracture propagation (in both magnitude and direction) due to anisotropies induced by various parameters, namely ultimate tensile strength, Young’s modulus, permeability and overburden pressure. The influence of several combinations of all these anisotropies along with different grain orientations and initial fracture directions on the fracture propagation direction is studied. Different regimes are identified where the fracture propagation direction is controlled by the degree of material anisotropy instead of the stress anisotropy.

## Keywords

Rock anisotropy Transverse isotropy Hydraulic fracturing Porous media Extended finite element method## List of symbols

*u*Displacement of the solid grains

*p*Fluid pressure in the pores

- \(p_{\mathrm{d}}\)
Fluid pressure in the fracture

- \(\varGamma _{\mathrm{d}}\)
Line of discontinuity

- \(n_{\mathrm{d}}\)
Normal to the discontinuity

- \(\sigma\)
Total stress in the porous media

- \(\sigma _\mathrm{e}\)
Terzaghi's effective stress

- \(\alpha\)
Biot’s poroelastic coefficient

*C*Constitutive relationship matrix

- \(n_{\mathrm{f}}\)
Porosity of the porous media

- \(\mu\)
Dynamic viscosity of the pore fluid

- \(E_\parallel , E_\perp\)
Young’s moduli parallel and perpendicular to the grain direction, respectively.

- \(\tau _{{{\mathrm{ult}}_\parallel }} , \tau _{{{\mathrm{ult}}_\perp }}\)
Ultimate tensile strength parallel and perpendicular to the grain orientation direction, respectively.

- \(\kappa _\parallel , \kappa _\perp\)
Permeability values parallel and perpendicular to the grain orientation direction, respectively

- \(S_{{\mathrm{ult}}}\)
Ultimate shear strength of porous media

- \(\gamma\)
Grain orientation with respect to global horizontal direction

- \(\phi\)
Fracture propagation direction with respect to global horizontal direction

- \(\theta\)
Angle between fracture propagation direction and the grain orientation direction

- DOA
Degree of anisotropy

## 1 Introduction

Hydraulic fracturing is a process of inducing fractures in rock structures by injecting fluid at high pressures. Interest in hydraulic fracturing has been increasing in recent years due to its applications in the oil and gas industry for enhanced oil recovery in conventional reservoirs, for heat recovery from geothermal reservoirs and for the profitable extraction of oil and gas from unconventional reservoirs. A better understanding of the fracture growth phenomena will enhance productivity and also reduce the environmental footprint as less fractures can be created in a much more efficient way.

Several models have been developed for this purpose, the earliest of which was by Khristianovic and Zheltov (1955) and later extended by Geertsma and Klerk (1969) to form the KGD model, which proposed an analytical solution to the problem by assuming plane strain conditions. Although this model has been accepted as a standard case for hydraulic fracturing due to its simplicity, it suffers from the assumption that there is no leak-off from fracture into formation and also the formation is assumed to be solid. Another analytical solution with a different geometrical assumption (fracture length \(\gg\) fracture height) was given by Perkins and Kern (1961) and extended to include fluid loss by Nordgren (1972). Analytical asymptotic solutions were formulated (Adachi and Detournay 2002; Garagash and Detournay 2005; Detournay 2016), based on a parametric space to identify the significant parameters in a given regime. Later, these asymptotic solutions have been modified to take into account the leak-off, fracture toughness and fluid viscosity (Adachi and Detournay 2008; Kovalyshen 2010; Dontsov 2017).

The first numerical models for modelling hydraulic fracturing were developed by Boone and Ingraffea (1990) using finite elements for modelling the formation and using finite volume for modelling flow with cohesive zones along element edges describing the fractures. In recent times there have been several models developed based on different numerical techniques. A linear elastic fracture mechanics (LEFM)-based finite element model (FEM) was proposed by Hossain and Rahman (2008). To avoid the singularity problems at the crack tip in LEFM, FEM models with zero-thickness elements (describing the fractures using cohesive zones) were developed (Carrier and Granet 2012; Chen 2012). The FEM approach requires re-meshing to capture the fracture propagation accurately, whereas extended finite element (XFEM)-based models allows for fracture propagation in arbitrary directions without the need for re-meshing (Mohammadnejad and Khoei 2013; Remij et al. 2015; Meschke and Leonhart 2015). A novel approach in which the asymptotic behaviour near the fracture tip was resolved with extended finite element method (Gordeliy and Peirce 2013a, b, 2015). An alternate approach based on phase field modelling which combines FEM with continuum damage mechanics has been developed (Mikelic et al. 2015; Miehe and Mauthe 2016) which provides a convenient way for modelling complex fracture interactions. But all the above hydraulic fracture models assume the rock formation to be isotropic in nature.

Most rocks (especially shales, which are the most common rock type to be hydraulically fractured) are highly anisotropic in nature (Jaeger et al. 2009; Barton 2007). Kaarsberg (1959) and Sayers (1994) observed that shales have a bedding plane along which the grains are oriented causing the properties along the grain direction to be vastly different from the properties perpendicular to the grain direction. This causes a special type of anisotropy, called transverse isotropy, where the material properties in any direction in the plane can be obtained by using the material properties along any two mutually perpendicular set of directions in that plane. Although there are several studies (Abousleiman et al. 2008; Zhubayev et al. 2015) experimentally obtaining the anisotropic parameters, there are few papers by Cheng (1997) analytically deriving the anisotropic poroelastic coefficients. Abousleiman et al. (1996) modelled the deformation and pressure in a transversely isotropic porous medium without any fracture. Porous material with a stress-driven fracture in an orthotropic medium was modelled by Remij et al. (2015). More recently, the influence of rock anisotropy on tensile fractures was studied experimentally by Mighani et al. (2016).

## 2 Mathematical Formulation

In order to solve a poroelastic problem, we need to solve for the solid deformation (\(\mathbf {u}\)) and fluid pressure (*p*) at all points within the porous media. Biot’s theory of poroelasticity (Biot 1941) is used to describe the porous media, and fluid flow is described using Darcy’s law. The porous medium is assumed to be saturated.

In addition for the hydraulic fracture problem, we make use of an enhanced local pressure (ELP) as proposed by Remij et al. (2015) to model the pressure inside the fracture at all points along the fracture length. This additional degree of freedom (\({p_\mathrm{d}}\)) enables us to model the complicated phenomenon happening within the fracture, namely the injection of external fluid, moving boundaries of fracture surface and the leak-off. Leak-off from fracture to formation is described using the 1-D Terzaghi's consolidation equation.

For solving the unknowns, a set of governing equations along with auxiliary equations are used. The governing equations used to describe the poroelastic problem are of two types: solid deformation-based momentum balance and fluid flow-based mass balance. We consider an additional equation to ensure the mass balance inside the fracture. The auxiliary equations are used for relating these governing equations with the unknowns and also for coupling them. A schematic flow chart of the mathematical formulation is represented in Fig. 1.

## 3 Implementation

### 3.1 Discretisation

*N*,

*L*are two-dimensional interpolation or shape functions for the displacement and pressure fields, whereas

*V*is a one-dimensional shape function for interpolation of the pressure along the fracture length.

### 3.2 Solution

The governing equations are combined with the auxiliary equations as shown in Fig. 1. Weak form of this set of equations is obtained by integrating them along with a test function. By substituting the discretised unknowns given by Eqs. (1), (2) and (3) into the weak form, we convert the set of differential equations into a set of algebraic equations. In order to solve this set of equations simultaneously, we make use of the Newton–Raphson iterative solver in combination with Euler’s forward scheme for obtaining the time derivative, and Euler’s implicit scheme for time-independent parameters. A detailed description of the solution procedure is given in Remij et al. (2015).

The unknown \(\left( X=\left[ \hat{u}\,\,\tilde{u}\,\, \hat{p}\,\, \tilde{p}\,\, \hat{p}_\mathrm{d}\right] ^T \right)\) degrees of freedom are solved at each grid point (nodes) for every time step.

### 3.3 Propagation

The direction of propagation is taken to be the direction in which the equivalent traction is maximum. The fracture is assumed to propagate through the entire element length in a single time step in a straight line. Further details on the implementation of the solution are described by Remmers (2006) and Remmers et al. (2003).

## 4 Anisotropic Parameters

In this section, we highlight the parameters which need to be modified to incorporate the effect of anisotropy.

### 4.1 Constitutive Relation

*T*is the transformation matrix given as a function of the angle (\(\beta\)) between the global direction and the grain orientation direction.

### 4.2 Ultimate Tensile Strength

### 4.3 Poroelastic Coefficients

Cheng (1997) derived the analytical expressions for the transverse isotropic poroelastic coefficients based on the constitutive relationship matrix.

### 4.4 Permeability

*p*refers to the pressure in the porous media.

## 5 Results

### 5.1 Validation

Since there are no studies which exactly deal with hydraulic fracturing in anisotropic media, we divide the validation into two parts: (1) Mandel’s problem which compares the numerical results with an analytical solution for a transverse isotropic porous medium without fractures and (2) the standard KGD problem which compares the numerical results with the ELP model (Remij et al. 2015) for a hydraulic fracture problem in an isotropic medium.

#### 5.1.1 Mandel’s Problem

We compare the pore pressure solution from the numerical model with the analytical solution at different time periods in Fig. 7. The numerical pore pressure decay from the centre of the specimen to the free edges is found to be consistent with the analytical solution with relative errors (< 5%). The displacement in the x-direction along the centre line of the specimen is plotted and compared with the analytical solution.

#### 5.1.2 KGD

The fracture propagates on a non-predefined path. In Fig. 9, we compare the fracture profiles at various time steps from the current numerical model and the ELP model (Remij et al. 2015). As observed the current model accurately reduces to the ELP solution for isotropic values of the parameters. The fracture mouth opening pressure variation with time is also plotted and compared. A mesh of 50 \(\times\) 50 \({\mathrm{mm}}^2\) is used for the purpose.

### 5.2 Vertical Hydraulic Fracture Problem

Isotropic value of parameters

Parameter | Isotropic value |
---|---|

Young’s modulus | 20 GPa |

Poisson’s ratio | 0.2 |

Toughness (\(G_c\)) | 120 N/m |

Ultimate tensile strength | 6 MPa |

Ultimate shear strength | 60 MPa |

Permeability | \(10^{-19}\,\mathrm{m}^2\) |

Porosity | 0.1 |

Viscosity | \(10^{-2}\) Pa s |

Solid bulk modulus | 36 GPa |

Fluid bulk modulus | 3 GPa |

Injection rate | 0.0006 \(\mathrm{m}^2/\hbox {s}\) |

Overburden pressure | 40 MPa |

By using the parameters in Table 1 to obtain the non-dimensional parameter (\(\mathcal {M}_k\)) described in Bunger et al. (2005), we observe that the hydraulic fracture problem described here lies in the viscosity-dominated regime and closer to the storage edge. In the following Sects. 5.3 and 5.6.1 we try to understand the influence of anisotropy in each individual parameter by keeping all other parameters isotropic. We also look at the possible combination of anisotropy in these parameters in Sects. 5.4 and 5.6.2.

### 5.3 Parametric Anisotropy

In this subsection, we vary one parameter at a time to find out the fracture propagation variation with anisotropy in each individual parameter. In all the considered test cases we assume that the grains are oriented along the horizontal direction (\(0^{\circ }\)) and the initial fracture is oriented in the vertical direction (\(90^{\circ }\)).

#### 5.3.1 Anisotropy Due to Young’s Modulus

From the fracture length plot in Fig. 11, we can observe that \(E_{\parallel }\) has a much greater effect on fracture propagation than \(E_{\perp }\). This is due to the fact that the propagation of the initial vertical fracture is dependent on the stresses which are perpendicular to it. Hence, an increase in \(E_{\parallel }\) results in higher stresses perpendicular to the initial fracture, which promotes fracture growth significantly, whereas a decrease in \(E_{\perp }\) only has a smaller Poisson’s effect on the stress. Since \(E_{\parallel }>E_{\perp }\) always, the fracture prefers to orient itself perpendicular to the grain orientation which is observed in all the three scenarios in the fracture orientation plot.

#### 5.3.2 Anisotropy Due to Ultimate Tensile Strength

Similar to the Young’s modulus variation, we consider the same three scenarios for understanding ultimate tensile strength-induced anisotropy. Fracture propagation is resisted by the ultimate tensile strength perpendicular to the fracture orientation. Hence, the fracture tends to propagate along the direction perpendicular to the minimum ultimate tensile strength. Since \(\tau _{{\mathrm{ult}}_\perp }\) is always lower than \(\tau _{{\mathrm{ult}}_\Vert }\), the fracture tends to propagate parallel to the grain orientation. But since the initial fracture is oriented in an unfavourable direction (perpendicular to the grain direction), the fracture continues to propagate in its initial direction until a threshold level where the effect of anisotropy becomes significant to rotate the fracture as observed from the fracture orientation plot in Fig. 12.

Also looking at the fracture length variation we observe that there is a significant increase in the fracture length when the fracture re-orients itself from its initial direction to the favourable direction. Since the fluid inside the fracture has to go through steep rotation (\(\sim \,80^{\circ }\)), much higher pressures are required in order to drive the fracture.

#### 5.3.3 Anisotropy Due to Permeability

Anisotropy in the permeability of the rocks was considered in the formulation. As indicated in Table 1, the isotropic permeability of shales was assumed to be \(10^{-19}\,\mathrm{m}^2\) (100 nd). It was varied within two orders of magnitude, i.e. \(10^{-18}\) to \(10^{-20}\,\mathrm{m}^2\) (1000–10 nd). However, its impact on the fracture growth was found to be very negligible since shales already have very low permeability values (almost impermeable).

### 5.4 Degree of Material Anisotropy

In the following subsections, we focus only on the fracture propagation direction. This is because variation in fracture length or width due to various anisotropies and combinations can be overcome by varying the fluid injection time, but the fracture orientation direction cannot be modified by means of any external influence as it is solely dependent on the field conditions. Hereafter, all the anisotropies considered are by varying both the values parallel and perpendicular to the grain direction as given by Eq. (22).

In Fig. 14, we observe that contour plots are represented for four different grain orientation (\(\gamma\)) directions. \(\gamma\) refers to a grain orientation with respect to the global horizontal direction (*x*-axis). The non-smooth variations in the threshold values for transition between regimes in the contour plots are due to the variation of anisotropies in step sizes of 5%. Looking at the different contour plots we can see that the size and shape of the different regimes vary with varying grain orientation angles.

From the \(\gamma =0^{\circ }\) plot, we observe that beyond a threshold value of 50% DOA in \(\tau _{\mathrm{ult}}\) the fracture moves from Regime B to Regime A (red regions representing \(\phi <20^{\circ }\)). Looking at the plot for \(\gamma =30^{\circ }\), we observe that the fracture tries to align itself perpendicular to the grain direction (blue regions representing \(\phi >100^{\circ }\)) even for relatively low values of E anisotropy, but requires much higher (\(>\,40\%\)) DOA in Young’s modulus along with low (\(<\,10\%\)) DOA in ultimate tensile strength when \(\gamma =60^{\circ }\). Young’s modulus anisotropy cannot influence the fracture at all when the fracture needs to be completely rotated by \(90^{\circ }\).

Looking at all the contour plots we can observe that the area of Regime A increases with grain orientation angle. The influence of \(\tau _{\mathrm{ult}}\) anisotropy is much higher as the grain orientation angle increases due to the fact that the fracture needs to be rotated by a smaller angle from its initial vertical orientation to align itself with the grain orientation direction. The converse is true for the influence of Young’s modulus anisotropy.

### 5.5 Angle of Orientation

#### 5.5.1 Grain Orientation Direction

We observe that along the vertical axis (for a particular grain orientation direction), after a certain degree of combined material anisotropy the crack always tends to align with the grain direction. This is due to the fact that \(\tau _{\mathrm{ult}}\) anisotropy has a greater impact than the *E* anisotropy at larger values of DOA (\(>30\%\)), whereas *E* anisotropy has a much larger impact when the DOA is lower (\(<30\%\)). Along the horizontal axis (for a particular combined material DOA), we observe that the fracture tends to re-orient itself with relative ease as the angle (\(\theta =\gamma -\phi\)) between the initial fracture orientation and the grain orientation reduces.

#### 5.5.2 Initial Fracture Orientation

In all the previous and later cases, the initial fracture is assumed to be along the vertical direction (\(90^{\circ }\)). In this subsection, three initial fractures at varying initial orientation angles of \(90^{\circ }\), \(60^{\circ }\) and \(0^{\circ }\) are included simultaneously. The grain orientation angle is assumed to be \(60^{\circ }\) with a combined material DOA of 15%.

### 5.6 Lithostatic Stresses

#### 5.6.1 Anisotropy Due to Lithostatic Stress

#### 5.6.2 Combined Anisotropy

The plot for \(\gamma =30^{\circ }\) is the most complete plot with all the regimes which are influenced by the various anisotropy parameters. The dark blue regions representing fracture orientations (\(\phi\)) larger than 110 are indicative of Regime B at low material DOA (\(<25\%\)) and low stress DOA (\(<25\%\)). The high material DOA (\(>25\%\)) and low stress DOA (\(<30\%\)) regions represented by varying intensities of red colour (Regime A) are indicative of the ultimate tensile strength influence. For high stress DOA (\(>30\%\)) (Regime C), the fractures are more influenced by the stress-induced anisotropy. But the fracture orientation angles (\(\phi\)) are not exactly \(90^{\circ }\) which it is supposed to be as the vertical overburden pressures are maximum. This is because of the fact that although external stresses are maximum in one direction the local stress state has maximum values in a different direction as a result of the Young’s modulus anisotropy. Therefore, the fracture ends up oriented at angles (\(\sim 100^{\circ }\)) in between perpendicular to the grain orientation (\(120^{\circ }\)) and the vertical direction (\(90^{\circ }\)).

When \(\gamma =60^{\circ }\), the ultimate tensile strength has great influence over most of the regions except the regions with high stress DOA (\(>30\%\)) and high material DOA(\(>40\%\)) where both the stress and Young’s modulus anisotropy combine. When \(\gamma =90^{\circ }\), both the combined material DOA and the stress anisotropy prefer the fracture to propagate along its initial vertical direction causing all the regimes to coincide.

Looking at all the four contour plots together one can observe the reduction in the influence of stress-induced anisotropy as the grain orientation direction increases, or in other words when the angle (\(\theta =\gamma -\phi\)) between the initial fracture orientation and the grain direction decreases, while the converse is true for material-based anisotropy.

## 6 Conclusion

- (a)
Young’s modulus anisotropy promotes fracture growth perpendicular to the grain direction.

- (b)
Ultimate tensile strength anisotropy promotes fracture growth parallel to the grain direction.

- (c)
Stress-induced anisotropy promotes fracture growth parallel to the maximum overburden pressure.

- (d)
At high degrees of material anisotropy, ultimate tensile strength has a greater influence than the Young’s modulus, while the converse is true for low degrees of anisotropy.

- (e)
Most important angle influencing fracture orientation is the angle between grain orientation and the initial orientation. When this angle decreases, the influence of ultimate tensile strength anisotropy increases, while the influence of Young’s modulus anisotropy and stress-based anisotropy decreases.

## Notes

### Acknowledgements

This research was funded by Shell-Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) under the ‘Computational Sciences for Energy Research’ (CSER) programme with project number 13CSER022. The constructive reviews by two anonymous reviewers are greatly appreciated.

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