**Abstract**

Hydraulic fracturing by shear slippage mechanism (mode II) has been studied in both laboratory and field scales to enhance permeability of geothermal reservoirs by numerous authors and their success stories have been reported. Shear slippage takes place along the planes of pre-existing fractures which causes opening of the fracture planes by the fracture asperities (roughness induced opening). Simplified empirical relationships, which are derived based on simple fracture experiments or best guess, are used to calculate compressive normal surface traction, residual aperture and shear displacement. This introduces ambiguity into the simulation results and often leads to erroneous predictions of reservoir performance.

In this study an innovative analytical approach based on the distributed dislocation technique is developed to simulate the roughness induced opening of fractures in the presence of compressive and shear stresses as well as fluid pressure inside the fracture. This provides fundamental basis for computation of aperture distribution for all parts of the fracture which can then be used in the next step of modeling fluid flow inside the fracture as a function of time. It also allows formulation of change in aperture due to thermal stresses. The stress distribution and the fluid pressure are calculated using the fluid flow modeling inside the fracture in a numerical framework in which thermo-hydro mechanical effects are also consid‐ ered using finite element methods (FEM). In this study, fractures with their characteristic properties are considered to simulate rock deformation.

This new approach is applied to the Soultz-Sous-Forets geothermal reservoir to study changes in permeability and its impact on temperature drawdown. It has been shown that the analytical

© 2013 Doonechaly et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

approach provides a more realistic prediction of residual fracture aperture which agrees well with the experience of existing EGS trials around the world. An average increase in aperture due to fluid induced shear dilation has been found to be lower and time required to obtain a sizeable reservoir volume is greater than those previously estimated.

#### **1. Introduction**

#### **1.1. Reservoir stimulation by induced fluid pressure**

Fractured reservoirs in crystalline rocks are usually stimulated by injected fluid pressure. As the injection of fluid continues the pressure inside the fractures increases gradually. The effective stress due to fluid pressure is expressed as:

$$
\sigma\_{eff} = \sigma\_t - p \tag{1}
$$

can be achieved is called characteristic height of the fracture [6]. Based on an experimental study the characteristic height is measured to be of the order of a fraction of a millimeter [7]. Fracture aperture that can be created by conventional hydraulic fracturing is in the order of tens of millimeters [8]. Reservoir rocks with rough surfaces and high shear strength are highly desirable for stimulation by shear displacement to work. One of the most comprehensive attempts to characterize the shear dilation caused by the fracture surface asperities was developed by [9]. In their model, the rock behavior was studied by considering fracture surface and its aperture, normal and shears closure and shear dilation. In another attempt, [10] proposed a methodology to obtain the mechanical aperture of the fractures. The authors used the methodology proposed by [11] to measure the aperture by a tapered feeler gauge using plane sawn surfaces to gain access to the joints. Mechanical aperture can be calculated using an empirical equation as proposed by [10]. Later [10] used the empirical equation proposed by [11] to model the normal closure of fracture surfaces based on the normal stress. [12] proposed an approach to describe the hydraulic and mechanical properties of the fracture

A New Approach to Hydraulic Stimulation of Geothermal Reservoirs by Roughness Induced Fracture Opening

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573

**Figure 1.** Mohr diagram describing the initiation of the shear dilation and normal fracture surface separation.

Mechanical models for shear displacement include displacement estimation under different stress boundary conditions in which a proper topographical model is used to describe the fracture surface. Also [13] experimentally studied the effect of normal stress and shear dilation on fluid flow properties of a naturally fractured core sample. They have used a servo-controlled axial/torsion load frame to test the fluid flow and mechanical behavior of the fracture surface during normal stress, slip and shear dilation. In another approach [14] proposed a semi-

including the shear dilation by induced fluid pressure.

where *σeff* is the effective stress, *σ<sup>t</sup>* is the total stress and *p* is the pore pressure. With further injection of fluid the effective shear stress, which is a function of effective stress, continuously decreases until it reaches a threshold value at which time it can no longer resist shear dis‐ placement of the fracture surfaces. At this stage the shear dilation will occur. During shear displacement rock fails by the shearing (Mode II) instead of opening (Mode I). In Mode II opening, the surface asperities of the rock slide over each other which cause more separation of the fracture surfaces. Such an interlocking of asperities increases the permeability of the rock. Any further increase in pressure can cause the effective closure stress to decrease to zero at which time the separation and interlocking of the fracture surfaces perpendicular to the fracture walls occur. The amount of pressure required to reach zero effective stress is highly dependent on the rock and fracture properties [1]. If the injection continues at some point it will exceed the tensile strength of the rock, which leads to tensile failure of the rock. This means that a certain level of permeability enhancement by shear displacement can be obtained. Mechanical representation of the shear displacement and the normal separation of the fracture surfaces can be described based on a specific failure criterion, such as Mohr-Coulomb (see Fig. 1). As the pressure inside the fracture increases the effective stress decreases: Mohr's circle moves towards the origin. As shown in Fig.1, when the minimum principal stress (closure stress) reaches zero the normal separation of fracture surfaces (Mode I) occurs. However, the shear dilation happens much earlier: when the Mohr's circle encounters the failure envelope (CD) at E. Shear dilation by induced fluid pressure was first detected in the laboratory experiments in 1970s. One of the earliest attempts by [2] showed a significant permeability increase by shear displacement. This observation was confirmed by [3] and [4]. Since then, shear dilation has been comprehensively studied in geotechnical and mining engineering. However, investigation of permeability enhancement by shear dilation in petroleum reservoirs began much later [5]. Since the shear dilation is caused by slippage of the asperities on top of each other, there is maximum dilation that can be reached. The maximum displacement that can be achieved is called characteristic height of the fracture [6]. Based on an experimental study the characteristic height is measured to be of the order of a fraction of a millimeter [7]. Fracture aperture that can be created by conventional hydraulic fracturing is in the order of tens of millimeters [8]. Reservoir rocks with rough surfaces and high shear strength are highly desirable for stimulation by shear displacement to work. One of the most comprehensive attempts to characterize the shear dilation caused by the fracture surface asperities was developed by [9]. In their model, the rock behavior was studied by considering fracture surface and its aperture, normal and shears closure and shear dilation. In another attempt, [10] proposed a methodology to obtain the mechanical aperture of the fractures. The authors used the methodology proposed by [11] to measure the aperture by a tapered feeler gauge using plane sawn surfaces to gain access to the joints. Mechanical aperture can be calculated using an empirical equation as proposed by [10]. Later [10] used the empirical equation proposed by [11] to model the normal closure of fracture surfaces based on the normal stress. [12] proposed an approach to describe the hydraulic and mechanical properties of the fracture including the shear dilation by induced fluid pressure.

approach provides a more realistic prediction of residual fracture aperture which agrees well with the experience of existing EGS trials around the world. An average increase in aperture due to fluid induced shear dilation has been found to be lower and time required to obtain a

Fractured reservoirs in crystalline rocks are usually stimulated by injected fluid pressure. As the injection of fluid continues the pressure inside the fractures increases gradually. The

where *σeff* is the effective stress, *σ<sup>t</sup>* is the total stress and *p* is the pore pressure. With further injection of fluid the effective shear stress, which is a function of effective stress, continuously decreases until it reaches a threshold value at which time it can no longer resist shear dis‐ placement of the fracture surfaces. At this stage the shear dilation will occur. During shear displacement rock fails by the shearing (Mode II) instead of opening (Mode I). In Mode II opening, the surface asperities of the rock slide over each other which cause more separation of the fracture surfaces. Such an interlocking of asperities increases the permeability of the rock. Any further increase in pressure can cause the effective closure stress to decrease to zero at which time the separation and interlocking of the fracture surfaces perpendicular to the fracture walls occur. The amount of pressure required to reach zero effective stress is highly dependent on the rock and fracture properties [1]. If the injection continues at some point it will exceed the tensile strength of the rock, which leads to tensile failure of the rock. This means that a certain level of permeability enhancement by shear displacement can be obtained. Mechanical representation of the shear displacement and the normal separation of the fracture surfaces can be described based on a specific failure criterion, such as Mohr-Coulomb (see Fig. 1). As the pressure inside the fracture increases the effective stress decreases: Mohr's circle moves towards the origin. As shown in Fig.1, when the minimum principal stress (closure stress) reaches zero the normal separation of fracture surfaces (Mode I) occurs. However, the shear dilation happens much earlier: when the Mohr's circle encounters the failure envelope (CD) at E. Shear dilation by induced fluid pressure was first detected in the laboratory experiments in 1970s. One of the earliest attempts by [2] showed a significant permeability increase by shear displacement. This observation was confirmed by [3] and [4]. Since then, shear dilation has been comprehensively studied in geotechnical and mining engineering. However, investigation of permeability enhancement by shear dilation in petroleum reservoirs began much later [5]. Since the shear dilation is caused by slippage of the asperities on top of each other, there is maximum dilation that can be reached. The maximum displacement that

= - *p* (1)

*eff t* s

 s

sizeable reservoir volume is greater than those previously estimated.

**1.1. Reservoir stimulation by induced fluid pressure**

effective stress due to fluid pressure is expressed as:

**1. Introduction**

572 Effective and Sustainable Hydraulic Fracturing

**Figure 1.** Mohr diagram describing the initiation of the shear dilation and normal fracture surface separation.

Mechanical models for shear displacement include displacement estimation under different stress boundary conditions in which a proper topographical model is used to describe the fracture surface. Also [13] experimentally studied the effect of normal stress and shear dilation on fluid flow properties of a naturally fractured core sample. They have used a servo-controlled axial/torsion load frame to test the fluid flow and mechanical behavior of the fracture surface during normal stress, slip and shear dilation. In another approach [14] proposed a semiempirical correlation to determine the change in fracture aperture based on the amount of shear displacement between the fracture surfaces and the stress boundary condition. Also [15] extended the previous attempt of [14] by considering the effect of fracture propagation in shear dilation.in another attempt [16] used a linear relationship between shear displacement and the dilation of the fracture surfaces.

In this study, an analytical computational methodology based on distributed dislocation technique proposed by [6] is used to estimate the aperture distribution caused by the shear dilation in a fracture subject to different varying stress boundary conditions [6].

Two major assumptions are used in this approach to characterize the shear displacement of the fracture surfaces. The shear slippage between the fracture surfaces is described by using Coulomb friction law which explains the friction stress during the shear slippage based on the normal stress exerted on the fracture planes with a proportionality contact named friction factor as shown in Eq.

$$
\sigma = \mathfrak{c} + f \sigma\_n \tag{2}
$$

*<sup>b</sup> k E L*

**Figure 2.** A schematic representation of a fracture subject to in-situ stress boundary conditions.

s s

s

tt

2 2 for *<sup>y</sup>*

( ) for *y y*

 d

for *<sup>y</sup>* s

> d

<sup>0</sup> ( )for *n y*

0for *<sup>n</sup>* t

contact exists between the fracture asperities.

where, E id the Young modulus of elasticity, L is the fracture length and b is a constant less than unity. Also Eq. (4) implies the complete separation of the fracture surfaces in which no

A New Approach to Hydraulic Stimulation of Geothermal Reservoirs by Roughness Induced Fracture Opening

The complete set of boundary conditions for a fracture as shown in Fig. 2 are listed below [6]:

<sup>D</sup> <sup>=</sup> (5)

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575

*x y* ¥ ¢ = + ®¥ (6)

¢ = D- + < £ *kE p c x a* (7)

= + × D- < £ *f kE c x a* (9)

¢ = £ *p xc* (8)

= £ *x c* (10)

0for *<sup>x</sup> u xa* = ³ (11)

0for *<sup>y</sup> u xa* = ³ (12)

where, *τ*<sup>0</sup> is the threshold shear stress value to initiate the shear slippage between the fracture surfaces. Also the friction factor, *f* , is dependent on the material properties, fracture geometry and surface asperities of the fracture [6]. Because a minor change in the fracture aperture causes a significant alteration of the fracture permeability estimation of the shear slippage of the fracture surfaces is of crucial importance in fluid flow simulation. In this study the coupling between the shear displacement and the change in fracture aperture is described by a step function. Fracture displacement normal to the fracture plane is simulated by using virtual springs distributed along the fracture length. Such springs are characterized by a specific spring constant which can be calculated numerically, experimentally or analytically [6]. Also the spring deformations are modeled in an elastic framework which results in the following system of equations describing the stress between the fracture surfaces:

$$
\sigma\_u = kE(\Delta - \mathcal{S}\_y) \\
\text{for } \mathcal{S}\_y < \Delta \tag{3}
$$

$$
\sigma\_n = 0 \,\text{for} \,\delta\_y > \Delta \tag{4}
$$

where, ∆ is the characteristic height of the fracture as shown in Eq. (3) and k is the spring constant. Equation (3) is associated with the rock compressibility and gives us the normal stress exerted on the fracture surfaces. After calculating the normal stresses on the fracture surfaces, the normal displacement of the surfaces is calculated by the distribution dislocation concept. Also the methodology proposed by [17] is used to calculate the spring constant based on a bed of nails as [17]:

A New Approach to Hydraulic Stimulation of Geothermal Reservoirs by Roughness Induced Fracture Opening http://dx.doi.org/10.5772/56447 575

$$k = E\frac{b\Delta}{L} \tag{5}$$

where, E id the Young modulus of elasticity, L is the fracture length and b is a constant less than unity. Also Eq. (4) implies the complete separation of the fracture surfaces in which no contact exists between the fracture asperities.

empirical correlation to determine the change in fracture aperture based on the amount of shear displacement between the fracture surfaces and the stress boundary condition. Also [15] extended the previous attempt of [14] by considering the effect of fracture propagation in shear dilation.in another attempt [16] used a linear relationship between shear displacement and the

In this study, an analytical computational methodology based on distributed dislocation technique proposed by [6] is used to estimate the aperture distribution caused by the shear

Two major assumptions are used in this approach to characterize the shear displacement of the fracture surfaces. The shear slippage between the fracture surfaces is described by using Coulomb friction law which explains the friction stress during the shear slippage based on the normal stress exerted on the fracture planes with a proportionality contact named friction

*n*

= +*c f* (2)

= D- <D *kE* (3)

= >D (4)

 s

where, *τ*<sup>0</sup> is the threshold shear stress value to initiate the shear slippage between the fracture surfaces. Also the friction factor, *f* , is dependent on the material properties, fracture geometry and surface asperities of the fracture [6]. Because a minor change in the fracture aperture causes a significant alteration of the fracture permeability estimation of the shear slippage of the fracture surfaces is of crucial importance in fluid flow simulation. In this study the coupling between the shear displacement and the change in fracture aperture is described by a step function. Fracture displacement normal to the fracture plane is simulated by using virtual springs distributed along the fracture length. Such springs are characterized by a specific spring constant which can be calculated numerically, experimentally or analytically [6]. Also the spring deformations are modeled in an elastic framework which results in the following

dilation in a fracture subject to different varying stress boundary conditions [6].

t

system of equations describing the stress between the fracture surfaces:

s

s

( )for *n yy*

0for *n y*

 d

where, ∆ is the characteristic height of the fracture as shown in Eq. (3) and k is the spring constant. Equation (3) is associated with the rock compressibility and gives us the normal stress exerted on the fracture surfaces. After calculating the normal stresses on the fracture surfaces, the normal displacement of the surfaces is calculated by the distribution dislocation concept. Also the methodology proposed by [17] is used to calculate the spring constant based on a bed

 dd

dilation of the fracture surfaces.

574 Effective and Sustainable Hydraulic Fracturing

factor as shown in Eq.

of nails as [17]:

The complete set of boundary conditions for a fracture as shown in Fig. 2 are listed below [6]:

**Figure 2.** A schematic representation of a fracture subject to in-situ stress boundary conditions.

$$
\sigma\_y' = \sigma^{\sigma^\circ} \text{ for } \mathfrak{x}^2 + y^2 \to \infty \tag{6}
$$

$$
\sigma\_y' = kE(\Delta - \delta\_y) + p \text{ for } c < |\mathbf{x}| \le a \tag{7}
$$

$$\sigma\_y' = p \operatorname{for} \|x\| \le c \tag{8}$$

$$
\tau\_n = \tau\_0 + f \cdot kE(\Delta - \delta\_y) \text{ for } c < \left| \mathbf{x} \right| \le a \tag{9}
$$

$$
\pi\_n = 0 \,\text{for} \, |x| \le c \tag{10}
$$

$$
\mu\_x = 0 \text{ for} \left| x \right| \ge a \tag{11}
$$

$$\mu\_y = 0 \text{ for} \left| x \right| \ge a \tag{12}$$

where *σ<sup>y</sup>* ' is the effective normal stress exerted on the fracture surfaces, *k* is the spring constant, *p* is the pore pressure, *Δ* is the characteristic height of the fracture, *δ<sup>y</sup>* is the displacement of the fracture surfaces, *E* is the Modulus of elasticity, *τn* is the shear stress exerted on the fracture surfaces, *τ*0 is the threshold stress requires to start the shear displacement of the fracture surfaces and u is the displacement. *As* mentioned above, the aperture distribution along the fracture surface is calculated based on an analytical methodology in which fracture geometry, stress distribution and fluid pressure inside the fracture are needed to be known as a priori. For this purpose a thermo-poro-elastic model is developed to simulate the fluid flow in the reservoir scale.

Permeability tensor for each block is expressed as:

*xx xy yx yy*

A New Approach to Hydraulic Stimulation of Geothermal Reservoirs by Roughness Induced Fracture Opening

Permeability tensors are calculated by simulating fluid flow in individual fractures in each element. The concept of permeability tensor was first introduced by [18] by considering a set of parallel fractures in a Representative Elementary Volume (REV) with zero matrix permea‐ bility [18]. In another attempt [19] developed a methodology for calculation of permeability

(a) (b)

**Figure 3.** Domain discretization by using the hybrid of the single continuum and discrete fracture approach. (a) frac‐ tures equal to and longer than 50 m are explicitly discretized in the reservoir domain by using the triangular elements. (b) after the discretization of the long fractures, the effect of short fractures (<50m) are taken into account by calcula‐

In this study the authors have considered interconnected fractures with fracture surface as infinite plate without roughness. In another approach [20] estimated permeability tensor by assuming fractures as a planar sink/source term [20]. Also [21] extended the approach and studied the effect of vertical fracture/ matrix permeability ratio on the permeability tensor. In a separate study, [22] used a numerical technique (BEM) to calculate the permeability tensor of the REV containing medium sized fractures considering fractures as a sink/source term [22]. Following this work [23] presented an analytical model to calculate the permeability tensor of the blocks containing infinite parallel fracture sets [23]. Also [24] improved the efficiency of their previous approach by considering the effect of short fractures using the analytical method

tion of the permeability tensor of the corresponding blocks which are cut by the fractures.

(13)

577

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*k k*

*k k* é ù = ê ú ê ú ë û

*K*

tensor for arbitrary oriented fractures using superposition technique [19].
