Impact of water, nitrogen and CO2 fracturing fluids on fracturing initiation pressure and flow pattern in anisotropic shale reservoirs

Accepted Manuscript Impact of water, nitrogen and CO2 fracturing fluids on fracturing initiation pressure and flow pattern in anisotropic shale reservoirs Xiangxiang Zhang, J.G. Wang, Feng Gao, Yang Ju PII:

S1875-5100(17)30248-2

DOI:

10.1016/j.jngse.2017.06.002

Reference:

JNGSE 2200

To appear in:

Journal of Natural Gas Science and Engineering

Received Date: 10 July 2016 Revised Date:

2 June 2017

Accepted Date: 2 June 2017

Please cite this article as: Zhang, X., Wang, J.G., Gao, F., Ju, Y., Impact of water, nitrogen and CO2 fracturing fluids on fracturing initiation pressure and flow pattern in anisotropic shale reservoirs, Journal of Natural Gas Science & Engineering (2017), doi: 10.1016/j.jngse.2017.06.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Impact of water, nitrogen and CO2 fracturing fluids on fracturing initiation pressure and flow pattern in anisotropic shale reservoirs

Xiangxiang Zhang1, J.G. Wang1,2∗, Feng Gao1,2, Yang Ju1, 3

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1, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

2, State Key Laboratory for Geomechanics and Deep Underground Engineering, China

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University of Mining and Technology, Xuzhou 221116, China

3, State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and

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Technology at Beijing, Beijing 100083, China

Abstract Water-based fluids are currently main fracturing fluids but have some unavoidable limitation. Alternative options include nitrogen and CO2 in some new fracturing methods.

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However, the fracturing mechanism with water, CO2 and nitrogen has not been clear so far. This paper proposes a numerical model to investigate the impact of water, CO2 and nitrogen fluids on fracturing initiation pressure and seepage area of a shale reservoir. First, the

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governing equations for anisotropic deformation and anisotropic seepage are further extended

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for bedding shale. Second, a moving seepage boundary is used for fracturing fluid permeation. Third, a fracture initiation criterion is established based on the stress intensity factor of mode I crack. Fourth, a complex parameter is suggested to express the combined effect of shale permeability, fluid viscosity and pressurization rate. This numerical model is then verified with two sets of experimental data and numerical simulations available in literature. Finally, the impacts of the complex parameter, Biot’s coefficient, confining pressure, and critical state ∗

Corresponding author, School of Mechanics and Civil Engineering, China University of Mining and

Technology, email: [email protected]; Tel: +86 151 5210 5228 1

ACCEPTED MANUSCRIPT on fracturing initiation pressure and seepage area are investigated. The permeability evolution during fracturing initiation process is discussed. It is found that this numerical model has the capability to simulate the hydraulic fracturing process with water, CO2 and nitrogen.

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Complex parameter, Biot’s coefficient and confining pressure have significant impacts on fracturing initiation pressure and seepage area. The fracturing initiation pressure decreases with complex parameter and Biot’s coefficient but increases with confining pressure

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regardless of fracturing fluid type. Water has highest fracturing initiation pressure and

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smallest seepage area. These differences are from the change of effective stress in seepage area, particularly nearby the permeation front. CO2 and nitrogen are good alternative fluids for hydraulic fracturing under the same condition of complex parameter, Biot’s coefficient and confining pressure because they have lower fracturing initiation pressure and larger

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seepage area compared to water.

Keywords: anisotropic shale, fracturing fluid, moving mesh, fracturing initiation pressure,

1. Introduction

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seepage area

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Hydraulic fracturing is an effective technology by fluid injection into fracture geologic

formations for the enhancement of formation permeability (Britt, 2012; Stringfellow et al., 2014). Water-based fluids are the main fracturing fluids but water source is usually scarce in the area of shale gas development. Hence, some new fracturing methods have been proposed by using waterless fracturing fluids such as nitrogen, CO2, or liquefied petroleum gas (LPG) (Grundmann et al., 1998; Sinal and Lancaster, 1987; Soni et al., 2014; Zhang et al., 2017). The supercritical CO2 (SC-CO2) has been used in oil and gas explorations due to its superior 2

ACCEPTED MANUSCRIPT properties of high density, strong diffusion capacity, low viscosity, and no pollution to reservoirs (Wang et al., 2015a). It has been demonstrated that using nitrogen and CO2 can reduce the usage of water. A CO2-based stimulation can reduce formation damage and

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residual fracturing fluid (Liao et al., 2009; Wang et al., 2016b). However, the permeation mechanisms of water-based fluids and waterless fluids into the formation rock and their impacts on fracture initiation pressure and seepage area are not clear so far.

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Fracturing initiation pressure is one of the most important parameters to hydraulic

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fracturing. It is impacted by some key factors including Biot’s coefficient, pressurization rate, viscosity, confining pressure, and rock permeability (Hubbert and Willis, 1957; Haimson and Fairhurst, 1967; Zoback et al., 1977; Bruno and Nakagawa, 1991; Ito and Hayashi, 1991; Matsunaga et al., 1993; Detournay and Carbonell, 1997; Lu et al., 2013; Gan et al., 2015). In

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these studies, water-based fluids are the main focus. The impacts of each single factor on fracture initiation pressure were investigated. The evolution of fracturing initiation pressure has not been studied if different fracturing fluids such as waterless fluids are used. Further,

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the impacts of each factor may not be independent. Their combined effect has not been

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explored. Therefore, it is necessary to investigate the impacts of fracturing fluid type on fracture initiation pressure.

The fracturing behaviors induced by different fracturing fluids have been experimentally investigated (Ishida et al., 2012; Gomaa et al., 2014; Chen et al., 2015b; Li et al., 2016; Zhang et al., 2017). These studies observed that fracturing initiation pressure varies significantly with the type of fracturing fluids. Based on experimental results, some criteria have also been proposed to predict the fracturing initiation pressure in porous rocks

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ACCEPTED MANUSCRIPT (Detournay and Carbonell, 1997). These criteria partially consider the above-mentioned influence factors through a theoretical equation. Fracturing fluid type is too complex to be expressed by a theoretical equation or criterion. It is necessary to propose a new criterion

waterless fluids on fracturing initiation pressure.

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which can consider most of the factors for the impacts of both water-based fluids and

The fracturing fluid permeation has an important effect on the fracturing mechanism of

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the rock around the borehole (Solberg et al., 1980; Schmitt and Zoback, 1992; Matsunaga et

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al., 1993; Chen et al., 2015b; Zhang et al., 2017). Based on the experimental study of hydraulic fracturing and SC-CO2 fracturing, Zhang et al. (2017) found that the fracturing initiation pressure is significantly reduced by using SC-CO2. This is due to larger permeation depth and bigger seepage area. After comparing the fracturing experiments using water, oil

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and SC-CO2, Chen et al. (2015b) observed that fracture pattern and branch vary with the permeation ability of fracturing fluid. However, a quantitative analysis of fluid permeation has not been conducted.

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Water-based fluids and waterless fluids have main differences in their compressibility,

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viscosity and density, if the chemical effect is ignored. For instances, the property difference may change the magnitude and distribution of pore pressure in rock and then impact the fracturing mechanism in the hydraulic fracturing process. In addition, these properties of a waterless fluid usually vary with physical states (temperature and pressure), while the properties of a water-based fluid have almost no change with physical state. For example, CO2 has the critical temperature of 30.1oC and the critical pressure of 7.38MPa. This critical state is easily reached in operations. Its density and viscosity change significantly nearby the

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ACCEPTED MANUSCRIPT critical state (Fenghour et al., 1998; Kestin et al., 1980), which may significantly affect the fluid permeation during hydraulic fracturing. Therefore, it is important to include the property difference when seepage and fracture patterns between water-based fluids and waterless

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fluids are investigated in the hydraulic fracturing process. The anisotropy is one of the most distinct features of shale. This feature generally originates from the mineral foliation, stratification, and discontinuities in the micro-scale rock

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mass (Cho et al., 2012; Zhang, 2016). In the bedding shale, the anisotropy of deformation

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properties (Niandou et al., 1997) and fracture toughness (Chen et al., 1998) are significantly impacted by the weak cementation strength and high compressibility of bedding planes (Chong et al., 1987; Chen et al., 1998; Shen et al., 2015; Zhong et al., 2015). The shale permeability is of direction-dependence because the bedding planes have much higher

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permeability and compressibility than the orthogonal planes (Chen et al., 2015a). The change of effective stress induced by the fracturing fluid permeation causes the variation of permeability. Several models have recently proposed to describe the evolution of induced

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anisotropic permeability (Liu et al., 2010; Wu et al., 2010; Chen et al., 2012; Wang et al.,

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2013; Wang et al., 2015b). These previous models partially considered the anisotropy of shale deformation and permeability. Hence, it is essential to take the effect of anisotropy on hydraulic fracturing into full accounts. In this study, a numerical model is proposed to comparatively investigate the impacts of water, nitrogen and CO2 on fracturing initiation pressure and seepage area. This numerical model considers the anisotropy of shale deformation and permeability, the property of fracturing fluid and the moving boundary in the hydraulic fracturing process. A fracture

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ACCEPTED MANUSCRIPT initiation criterion is established based on the stress intensity factor of mode I crack and incorporated into this numerical model. This numerical model is then verified through the comparison with experimental data and numerical simulation results available in literature.

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Finally, this numerical model is used for parametric study to simulate the fracturing processes by injecting water, nitrogen and CO2. A complex parameter is proposed to comprehensively consider the combined effect of permeability, viscosity and pressurization rate. The impacts

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of this complex parameter, Biot’s coefficient, confining pressure and fracturing fluid type on

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the fracturing initiation pressure and seepage area are analyzed. The effects of critical state of fracturing fluids on permeability evolution, compressibility coefficient and viscosity are observed around the borehole.

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2. Governing equations for each physical process

Before any development of governing equations, the following assumptions are made for a shale formation: (1) Shale is continuous and anisotropic. (2) The deformation and strain

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of shale formation are infinitesimal. (3) The sorption of shale matrix is not considered due to

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the short time of injection process. (4) The fluid seepage in the shale satisfies the Darcy’s law. (5) The chemical effect of methane and water within the shale is ignored. (6) All processes involved are isothermal.

2.1. Governing equation for the deformation of orthotropic porous media For orthotropic linear elastic rocks, the constitutive model is expressed as

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ACCEPTED MANUSCRIPT 0

0

0

1 Ez

0

0

0

0

1 2G yz

0

0

0

0

1 2Gzx

0

0

0

0

Ey

−

ν zy Ey

in which

ν ij Ej

=

ν xz

0

1 Ey −

−

ν ji Ei

Ez

ν yz Ez

 0    0  e  σ x   σ e  y 0   e   σz    τ yz  0   τ zx    τ xy  0   1   2Gxy 

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ν xy

(1)

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−

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 1   Ex  ν yx −  ε x   Ex ε    y   − ν xz  ε z   Ex  = ε yz   ε zx   0    ε xy    0    0 

i , j = x , y , z; i ≠ j

(2)

The compliance matrix has nine independent elastic constants. The term Ei is the

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elastic modulus in the ith direction. The term ν ij is the Poisson’s ratio that characterizes the strain response in the ith direction to a stress acting to the jth direction. The term Gij is the shear modulus in the jth direction whose normal direction is in the ith direction. σ ije is the

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effective stress component and ε ij is the strain component.

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The effective stress is expressed as

σ ij e = σ ij − α pδ ij

(3)

where σ ij is the total stress component of a shale element; p is the pore pressure; the Biot’s coefficient is α = 1 - K / K s ; K is the bulk modulus of the shale; K s is the bulk modulus of grains and δ ij is Kronecker delta. For a pseudo-static deformation process, the equation of motion is

σ ije , j + α p,i + fi = 0

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(4)

ACCEPTED MANUSCRIPT where f i is the body force per unit volume in the ith direction. 2.2. Porosity and permeability of anisotropic porous medium 2.2.1 Porosity model for porous medium

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A general porosity model for porous medium is (Zhang et al., 2008; Wang et al., 2012)

φ = 1 + (1 − Rm ) ∆ε e φ0

(5)

S0 − S 1+ S

∆ε e =

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where Rm = α / φ0 and the change of effective volumetric strain is

(6)

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Without considering the sorption of matrix, the current and initial effective volumetric strains are defined by

S = εv +

p , Ks

S0 = ε 0 +

p0 Ks

(7)

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where ε v is the current volumetric strain; ε 0 is the initial volumetric strain. φ is the current porosity; φ0 is the initial porosity; p0 is the initial pore pressure. The change of φ with respect to time t is

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∂φ α − φ  ∂ε v 1 ∂p  = +   ∂t 1 + S  ∂t K s ∂t 

(8)

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2.2.2 Directional permeability model for anisotropic porous medium The shale consists of fracture and matrix. It is a typical heterogeneous porous medium in the micro-scale. Fig. 1 presents a typical one-dimensional deformation and its fracture deformation is expressed by ∆boj = ( s j + boj ) ∆ε ej − s j ∆ε emj

This equation is reformulated as

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(9)

ACCEPTED MANUSCRIPT  ∆ε emj 1 − ∆ε ej 

 sj ∆boj = boj 1 +  boj

  1 − Rmj    ∆ε ej = boj 1 + n  ∆ε ej φ    f 0   

(10)

The local fracture strain is then obtained as

boj

 1 − Rmj  = 1 + n  ∆ε ej φ f 0  

where boj

= φf 0 / n

for nD case

(12)

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sj

(11)

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∆boj

∆ε ej is the increment of average effective strain in a representative length; ∆ε em is the

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increment of effective strain of matrix. Rmj = ∆ε emj / ∆ε ej is the strain ratio of matrix to the whole element in the ith direction; boj is the initial aperture of the fractures in the ith direction and s j is the spacing of fractures in the jth direction. The strain ratio is defined as Rmj = ∆ε emj / ∆ε ej by assuming that the matrix elastic modulus is anisotropic, which is

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different from the formulas proposed by Wang et al. (2013).

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The permeability of a fracture is calculated by (Levine, 1996)

k0 i =

boj3

12s j

i≠ j

,

(13)

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If the fracture aperture boj has some change of ∆boj due to swelling or compaction or both, the current permeability ki is then expressed as ki

(b =

oj

+ ∆boj )

3

12 s j

,

i≠ j

(14)

In a two-dimensional domain, the directional permeability of fractures is associated with the effective strain in its perpendicular direction as

 ki   2 (1 − Rmj )   ∆ε ej  , = 1 + 1 +  k0i   φf 0     3

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i≠ j

(15)

ACCEPTED MANUSCRIPT The permeability anisotropy ratio is expressed as

Ar =

ki kj

(16)

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2.3. Governing equation for fluid flow in anisotropic porous medium The fluid flow in the shale satisfies the mass conservation law as

r ∂m + ∇ ⋅ ( ρ v) = Qs ∂t

(17)

r

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where m is the fluid mass content in the shale, ρ is the fluid density under in-situ

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conditions; ρ = ρ w for water and ρ = ρ g for gas; v is the fluid velocity vector; Qs is the fluid source by distributive injection or other methods.

Without considering the sorption of matrix, the fluid mass content m is expressed by m = ρφ

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where φ is the shale porosity.

(18)

This fluid flow in the shale is described by the Darcy's law (without gravity effect)

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(Burton et al., 2008; Lu et al., 2013; Zhu et al., 2016; Wang et al., 2016a) as k r vi = − i ∇i p

µ

(19)

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where µ is the dynamic viscosity of the fracturing fluid; k i is the shale permeability in the ith direction. ∇i p denotes the directional pressure gradient in the ith direction. ∇i p =

∂p ∂xi

(20)

As no fluid source is considered, Eq. (17) becomes

φ

ρk ∂ρ ∂φ +ρ − ∇ ⋅ ( i ∇i p ) = 0 ∂t ∂t µ

The isothermal coefficient of compressibility is defined as 10

(21)

ACCEPTED MANUSCRIPT 1 ∂ρ ρ ∂p

cρ =

(22)

Applying Eqs. (8) and (22) to Eq. (21) yields

For water, the density is usually constant and cρ = 0 .

ρg =

Mg ZRT

p

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For gas, the density varies with temperature and pressure as

(23)

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 ∂p ki  k α - φ ∂p φ − α ∂ε V cρ φ − (∇i p ) 2  + − ∇ ⋅ ( i ∇i p) = 1 + S ∂t µ  ∂t µ  K s (1 + S ) ∂t

(24)

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where Z (dimensionless) is the real gas compressibility factor; M g is the molar mass of gas; R is the universal gas constant; T is the temperature. Therefore, the isothermal coefficient of compressibility is defined as (Civan et al., 2011) 1 ∂ρ g 1 1 ∂Z = − ρ g ∂p p Z ∂p

(25)

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cρ =

From Eq. (23), the salient difference among different fracturing fluids is the isothermal coefficient of compressibility cρ and viscosity µ , which are the functions of temperature

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and pressure. It may exhibit a strong impact on the fluid flow in the porous medium and

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result in a decisive influence on the magnitude and distribution of pore pressure. Under the critical temperature, the CO2 compressibility factor and viscosity change dramatically nearby its critical state as shown in Fig. 2. They are expressed as f = A1e pr / t1 + A2 e pr / t2 + A3 pr + A4

(26)

where f is either Z or viscosity µ . The viscosity is in µPa ; pr is the reduced pressure of CO2 and pr = p / pc ; pc is the critical pressure of CO2; Tc is the critical temperature of CO2. The tuned coefficients used in Eq. (26) are listed in Table 1.

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ACCEPTED MANUSCRIPT 2.4. Moving front of injection fluid During hydraulic fracturing, the boundary or front of fracturing fluid moves forward and the seepage zone expands with the volume of injection fluid. Therefore, the fracturing fluid

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flow is a moving boundary problem and the seepage front moves forward elliptically due to permeability anisotropy. Numerical computations usually use a fixed boundary and domain for fracturing fluid flow as shown in Fig. 3(a). Such a treatment may induce some errors in

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the magnitude and distribution of pore pressure. Fig. 3(b) presents a typical 1D moving

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boundary problem. The seepage front and domain should be determined during the fracturing process. A simple approach to predict the moving speed of seepage front is described as (Wang et al., 2016a)

∂x k ∂p r vbi = i = − i ∂t µ ∂xi

(27)

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Eq. (27) shows that the moving speed of seepage front is controlled by the permeability, viscosity and directional gradient of pore pressure. The seepage domain can be determined by

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the seepage front. In this study, the seepage area is denoted by A , which is the area of the seepage domain at the time of fracture initiation. The seepage area is used to investigate the

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flow pattern for different fracturing fluids.

3. Fracture initiation criterion 3.1. Classical solutions for fracturing initiation pressure The earlier foundational studies on the fracture initiation criterion have been comprehensively summarized by Detournay and Carbonell (1997). For isotropic porous media, two classical criteria are used to calculate the fracturing initiation pressure 12

ACCEPTED MANUSCRIPT (breakdown pressure) in terms of far-field stress at two extreme cases. If the fracturing fluid does not permeate into the rock, the Hubbert–Willis solution (Hubbert and Willis, 1957) gives the breakdown pressure as PHW = σ t − 3σ 3 + σ 1 − p0

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(28)

If the fracturing fluid completely permeates into the rock, the Haimson–Fairhurst solution (Haimson and Fairhurst, 1967) gives the breakdown pressure as

σ t − 3σ 3 + σ 1 − 2 p0 + p0 2 − α (1 − 2ν ) / (1 −ν )

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PHF =

(29)

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where σ 1 and σ 3 are the first and the third principal stresses in the far-field ( σ 1 > σ 3 );

σ t is the tensile strength. ν is the Poisson’s ratio of rock. PHW and PHF are the fracturing initiation pressures obtained under extreme cases: either impermeable or completely permeable. They do not consider the impact of pore pressure distribution, permeability,

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pressurization rate, and viscosity.

Ito and Hayashi (1991) proposed an alternative with a characteristic length d 0 to

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express the impacts of borehole size and pressurization rate during the tensile-failure process. The fracturing initiation pressure has following upper and lower limits:

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2   d0   Pupper = 1 +  (σ t − Sθ − p0 ) + p0   Rb   2(σ t − Sθ − p0 ) P + p0 lower =  1 + 1/ (1 + d 0 / Rb ) 2  [ 2 − α (1 − 2ν ) / (1 −ν ) ] 

(30)

in which

Sθ =

σ1 + σ 3  2

 σ1 − σ 3  Rb2 3Rb4  Rb2 1 + − 1 + + p0    2  2  (d0 + Rb )4  (d0 + Rb )2  (d 0 + Rb ) 

13

(31)

ACCEPTED MANUSCRIPT 1  K IC  d0 =   2π  σ t 

2

(32)

where K IC is the fracture toughness; Rb is the radius of the borehole. It is noted that the type and property of fracturing fluid are still not considered although these are important to

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hydraulic fracturing. 3.2. Our fracture initiation criterion

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The fracture initiation is affected by the type and property of fracturing fluid, rock permeability and pressurization rate. These factors mainly impact the fracturing fluid

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permeation and result in different magnitudes and distributions of pore pressure. The pore pressure in the seepage region modifies the effective stress around the borehole and thus enhances fracture initiation. Therefore, both the magnitude and the distribution of pore pressure are important to fracture initiation but few fracture initiation criteria consider these

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effects. In this study, a fracture initiation criterion is proposed with the consideration of pore pressure during hydraulic fracturing process.

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The cracks at the initial state are almost closed. Both pore pressure and in-situ stress determine the effective stress on the crack surface. Hydraulic fracture initiation is a mode I

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crack problem. Before fracture initiation, the stress distribution along the crack surface is shown in Fig. 4. Its stress intensity factor at the crack tip is solved by using the superposition principle as

KI = KI

σ0

+ KI

σp

where the components of stress intensity factor are K I pressure of fracturing fluid (Fig. 4b); K I

σp

(33) σ0

for the in-situ stress and injection

for the pore pressure (Fig. 4c), respectively.

The normal stress on the crack surface is calculated from the Fairhurst equation 14

ACCEPTED MANUSCRIPT (Fairhurst, 1968). The stress intensity factor under the in-situ stress and the injection pressure is

K Iσ 0 =（F1σ 3 + F2 P） π a

(34)

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where F1 and F2 are the geometric factors which can be determined by β ( a / R ), specimen size and σ 1 / σ 3 (Nowman, 1971); P is the injection pressure in the borehole ; a is the length of the initial crack as shown in Fig. 4.

σ

KΙ p = ∫

a

RB

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stress intensity factor induced by the pore pressure as

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By using the integral method and the principle of effective stress, one can obtain the

α p(r , t ) a + r a−r ( + )dr a−r a+r πa

(35)

where r is the distance away from the crack center. From Eq. (35), this stress intensity factor is determined by the magnitude and distribution of pore pressure. The total stress

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intensity factor at the crack tip can be obtained as K I = ( F1σ 3 + F2 P ) π a + ∫

a

RB

α p(r , t ) a + r a−r ( )dr + a−r a+r πa

(36)

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Due to the anisotropy of shale, the tensile strength and fracture toughness vary with the

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direction of fracture initiation (Cho et al., 2012; Heng et al., 2015). There are a large number of initial cracks distributed around the borehole. When the stress intensity factor of the crack in a potential direction firstly reaches the fracture toughness, this crack will initiate and propagate. Thus, the fracture initiation criterion is expressed as ( K I )ϕ ≥ ( K IC )ϕ

(37)

where ϕ is the potential initiation angle between the potential fracturing crack and bedding plane. Under the same in-situ stress and injection pressure, ( K I )ϕ varies with direction.

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ACCEPTED MANUSCRIPT 3.3. Fracturing initiation pressure in two extreme cases Impermeable and highly-permeable rocks are two extreme cases. For the impermeable rock, its fracturing initiation pressure is obtained as

 2α  π 1  ( K IC )ϕ 1 − F1σ 3  −   − arcsin  p0   F2  π a β  π F2  2

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Pmax =

(38)

For highly-permeable rock, its fracturing initiation pressure is

F2 + α −

− F1σ 3

2α

π

arcsin

1

(39)

β

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Pmin =

πa

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( K IC )ϕ

Eqs. (28), (29), (30), (38) and (39) give the critical pressures for anisotropic porous rock. These equations all consider the in-situ stress, injection pressure and the magnitude of pore pressure in the rock which can be predicted by different methods. Eqs. (38) and (39)

initial crack length.

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calculated by our criterion introduce the fracture toughness, distribution of pore pressure and

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4. Numerical model verification and application

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4.1. Numerical computation procedure The numerical model is composed of the deformation of orthotropic porous media (Eq. (1-4)), the Darcy flow of fracturing fluid (Eq. (23)), the moving boundary (Eq. (27)) and the fracture initiation criterion (Eq. (37)). Fig. 5 illustrates the cross-coupling and interaction in this numerical model. After the incorporation of these constitutive laws in Eq. (5) and (15), this model can be implemented for numerical simulations of the fracturing initiation pressure and fluid flow in the rock. In each time increment, the governing equations for rock

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ACCEPTED MANUSCRIPT deformation, fracturing fluid flow, and moving boundary are simultaneously solved by COMSOL Multiphysics, and the fracturing initiation criterion is checked by MATLAB. Particularly, the Darcy’s flow equation is implemented through the PDE module with the

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moving boundary problem and solved by Deformed Geometry module. The deformation equations for rocks are solved by Solid Mechanics module. The entire numerical iteration is carried out through COMSOL with MATLAB, a commercial software to connect COMSOL

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Multiphysics to the MATLAB scripting environment. The computational schematic process is

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shown in Fig. 6 and has three steps. Firstly, the couplings of rock deformation, fracturing fluid flow and moving boundary are solved for the given boundary conditions and parameters. Secondly, the results from the first step are used in the fracture initiation criterion to judge fracture initiation. Thirdly, the computational procedure is ended if the fracture initiation

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criterion is satisfied. Otherwise, the boundary conditions are updated and the computational procedure returns to the first step.

4.2. Model verification with experimental data

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This numerical model is verified through the comparison of its simulation results with

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those experimental data obtained by Zoback et al. (1977). In the experiments, the Weber sandstone specimens (6cm in length and 3cm in diameter) were loaded by an axial stress σ 1 of about 40MPa and a confining stress of about 10MPa. Oil as fracturing fluid was injected into the axial hole (2-3mm in diameter) through the base-plate of the triaxial pressure vessel. The pressurization rates were between 10-4 and 102 MPa/s. The main rock properties were taken from the publication by Rice and Cleary (1976). The viscosity of fracturing fluid was assumed as 50mPa ⋅ s and the fracture toughness of the sandstone was assumed as

17

ACCEPTED MANUSCRIPT 0.6MPa m1/2/s in a possible range (Van Eekelen, 1982). All of computational parameters for these experiments were listed in Table 2. Fig. 7 compares our simulation results with these experimental data by Zoback et al. (1977). This figure indicates that the evolution of

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fracturing initiation pressure by our numerical simulations is in good agreement with the experimental data of the Weber sandstone. Those critical pressures for the Weber sandstone evaluated by Eqs. (28), (29), (30), (38) and (39) are also plotted in Fig. 7. This figure shows

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model are more consistent with the experimental data.

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that Pmax > Pupper > PHW and Pmin > Plower > PHF . The critical pressures evaluated by our

In order to further verify this numerical model, the simulation results of this model are compared with the experimental data obtained by Song et al. (2001). The experiments were carried out on the Tablerock sandstone. The specimen was a cylinder with an external

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diameter of 10.2 cm, a length of 13 cm, and an injection hole diameter of 1.3 cm. The specimen was placed in a triaxial pressure chamber and the confining pressure was equal to the pore pressure. Borehole fluid (viscosity: 2.5Pa s at 20oC) was injected at a constant flow

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rate until a peak pressure was achieved. The main physical properties of the Tablerock

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sandstone were taken from the publication by Song et al. (2001), but our Biot’s coefficient was assumed as 0.9 due to the extremely large porosity and permeability of this sandstone. The fracture toughness of the sandstone was assumed as 0.55MPa m1/2/s in a possible range (Van Eekelen, 1982). All computational parameters used in our simulations for these experiments were listed in Table 3. Fig. 8 is the comparison of our simulations with their experimental data. The critical pressures evaluated from Eqs. (28), (29), (30), (38) and (39) are also plotted in the figure. The Plower is almost equal to PHW and the critical pressures

18

ACCEPTED MANUSCRIPT predicted by our model are more reasonable. As a whole, both our simulations and experimental data are in good agreement and can express the linear relationship between fracturing initiation pressure and confining pressure. This also shows that our numerical

4.3. Model verification with Lu et al. simulation results

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model has the capacity to investigate the impacts of confining pressure.

Our model also compares with the numerical simulations conducted by Lu et al. (2013).

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They numerically investigated the effect of rock permeability on hydraulic fracturing through

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a damage model. Table 4 lists the computational parameters used in our simulations. The fracturing fluid was water. Fig. 9 is the comparison between our simulation results with their simulations. Both simulations show that the fracturing initiation pressure decreases with rock permeability. When the rock permeability is larger than 1mD, the fracturing initiation

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pressure obtained by Lu et al.’s model almost keeps a constant, but the fracturing initiation pressure obtained by our model still decreases with the rock permeability and tends to be a smaller constant. On the whole, the evolution of fracturing initiation pressure obtained by our

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model was in good agreement with that obtained by Lu et al.’s model. The critical pressures

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evaluated from Eqs. (28), (29), (30), (38) and (39) are shown in Fig. 9, respectively. The simulation results are between Pmax and Pmin predicted by our model. 4.4. Comparison of fracturing initiation pressure induced by oil, water and CO2 In order to investigate the impact of fracturing fluid type on fracture initiation, our numerical model is used to simulate the hydraulic fracturing in the previous two section but the original fracturing fluids are replaced by CO2 in simulations. The simulation results are plotted in Figs. 7-9 respectively for comparison. Fig. 7 shows that the fracturing initiation

19

ACCEPTED MANUSCRIPT pressure for CO2 almost keeps constant with the increase of pressurization rate. When the pressurization rate is larger than 0.1MPa/s, the difference of fracturing initiation pressures between oil and CO2 becomes bigger. This indicates that the effect of pressurization rate on

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fracture initiation varies with the type of fracturing fluids. In comparison with the oil fluid in Fig. 8, the fracturing initiation pressure for CO2 also increases linearly with confining pressure, but its value and slope are smaller. This indicates that the effect of confining

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pressure on fracture initiation varies with the type of fracturing fluids. In Fig. 9, the fracturing

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initiation pressure for CO2 is smaller than that for water. This fracturing initiation pressure decreases with the increase of permeability. In the same range of permeability, the change of fracturing initiation pressure for CO2 is different from that for water. This indicates that the

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effect of permeability on fracture initiation also varies with the type of fracturing fluids.

5. Numerical performances in hydraulic fracturing process 5.1. Computational model

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A computational model is presented in Fig. 10 to simulate the fracturing fluid flow

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before fracture initiation in the injection process. The simulation geometry is a square of 100 × 100mm with a borehole of 5mm in radius. The confining pressure is the same in both

horizontal and vertical directions. The elastic modulus, Poisson’s ratio, fracture toughness and permeability are anisotropic. The fracture toughness is taken as 0.566MPa ⋅ m1/2 when

ϕ = 0o (Heng et al., 2015). The fluid pressurization rate is 0.1MPa/s . Fracturing fluids are water, nitrogen and CO2, respectively. Their computational parameters are listed in Table 5. The CO2 fracturing fluid is taken as a sample to investigate the fracture initiation direction.

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ACCEPTED MANUSCRIPT The confining pressure is taken as 10MPa. Fig. 11 presents the change of stress intensity factor induced by pore pressure with direction at different injection time. The stress intensity factor increases with injection time and potential initiation angles. Under the same injection

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time, the stress intensity factor is the largest in the direction of ϕ = 0o . As the fracture toughness has the minimum value along the direction of bedding plane ( ϕ = 0o ), the stress

fracture initiation direction is along the bedding plane.

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5.2. Impact of a complex parameter

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intensity factor firstly reaches the fracture toughness in this direction. This indicates that the

As mentioned above, the fracture initiation is influenced by fracturing fluid viscosity, pressurization rate and permeability. The impact of each single parameter on fracture initiation has been investigated (Zoback et al., 1977; Bruno and Nakagawa, 1991; Ito and

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Hayashi, 1991; Matsunaga et al., 1993; Detournay and Carbonell, 1997; Lu et al., 2013; Gomaa et al., 2014). It is found that the fracturing initiation pressure decreases with the increase of permeability and the decrease of viscosity and pressurization rate. However, their

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impacts on the fracture initiation are not independent and these parameters influence each

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other. Therefore, the following complex parameter is proposed to investigate the combined effect of permeability, viscosity and pressurization rate:

κ=

ky

µl p

(40)

where k y µ is the mobility of fracturing fluid, which largely controls fracturing fluid permeation and pore pressure in a porous medium (Batzle et al., 2006). The increase of pressurization rate l p decreases the injection time to achieve the same injection pressure, and then reduces the fracturing fluid permeation. Therefore, this complex parameter κ is a 21

ACCEPTED MANUSCRIPT new parameter to represent the mobility of fracturing fluid. As discussed above, the fracturing fluid permeation controls the magnitude and distribution of pore pressure, which is important to fracture initiation. Thus, this complex parameter is a useful parameter to express

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the comprehensive impacts on hydraulic fracturing process. The permeability (ky0), viscosity and pressurization rate (lp) vary in a range as listed in Table 6. The confining pressure is 0.1MPa. The CO2 viscosity varies with pressure so the

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viscosity for κ calculation is under the injection pressure. The changes of fracturing

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initiation pressure and seepage area with this complex parameter are presented in Fig. 12. This figure shows that the same κ induces the same fracturing initiation pressure and seepage area even if permeability, viscosity or pressurization rates are different. This figure also shows that the fracturing initiation pressure decreases and the seepage area increases

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with the increase of κ . This is because larger κ enhances fracturing fluid permeation and increases the pore pressure around the borehole. For the same κ , the fracturing initiation pressure is the largest and the seepage area is the smallest for water. These differences are

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mainly induced by the low compressibility of water. Fig. 13 shows the pore pressure along x

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direction with the similar κ for different fracturing fluids. Due to the low compressibility of water, the pore-pressure distribution is linear and its pore-pressure gradient at the seepage front is lower than that of nitrogen and CO2. It induces the moving distance of water seepage front shorter and the seepage area smaller. Fig. 14 shows the evolution of stress intensity σ

factor induced by the pore pressure ( K I p ) with the complex parameter under the injection σ

pressure of 5MPa. It shows that K I p is the smallest for water and thus the fracturing initiation pressure is the largest for water. When the complex parameter is larger than

22

ACCEPTED MANUSCRIPT 1× 10-19 m 2 /Pa 2 , the fracturing initiation pressure and the seepage area are almost the same for nitrogen and CO2. When the complex parameter is smaller than 1× 10-19 m 2 /Pa 2 , the fracturing initiation pressure becomes smaller and the seepage area becomes larger for CO2

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than for nitrogen. This is because the CO2 viscosity increases significantly with the increase of injection pressure and induces the complex parameter decrease. Therefore, the complex parameter κ can comprehensively consider the combined effect of parameters k y , µ and

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l p when fracturing initiation pressure and seepage area are concerned. CO2 has the smallest

5.3. Effect of the Biot’s coefficient

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fracturing initiation pressure and the largest seepage area for the same complex parameter.

Effect of the Biot’s coefficient varies with shale and fracturing process. A typical range of Biot’s coefficient is taken from 0.4 to 0.9. In this computation, the confining pressure is

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taken as 10MPa. Fig. 15 presents the variations of fracturing initiation pressure and seepage area with the Biot’s coefficient. This figure shows that larger Biot’s coefficient implies larger effective stress, easier fracture initiation, shorter time and less seepage area for the same

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fracturing fluid. The fracturing initiation pressure is the largest for water and the smallest for

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nitrogen. These curves are nonlinear. The nonlinearity is more obvious for nitrogen or CO2 than for water. With the same Biot’s coefficient, the seepage area for water is the smallest. The difference of seepage area for nitrogen and for CO2 is complex. For CO2, the seepage area is smaller than that for nitrogen with the small Biot’s coefficient and becomes larger with the increase of Biot’s coefficient. Fig. 16 shows the pore pressure distribution of nitrogen and CO2 at different injection times. Their differences increase with injection time. The pore pressure near the borehole is higher for CO2 than for nitrogen when the injection pressure is

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ACCEPTED MANUSCRIPT low and becomes lower with the increase of injection pressure. This is because the CO2 viscosity increases significantly with the injection pressure nearby the critical state. The viscosity increment changes the distribution of pore pressure near the borehole. Therefore,

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both fracturing initiation pressure and seepage area are larger for CO2 than for nitrogen. Fig. σ

17 shows the evolution of stress intensity factor induced by the pore pressure ( K I p ) with σ

injection time. Under the same injection pressure, the K I p varies with fracturing fluid. It

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indicates that the pore pressure distribution near the borehole has a significant influence on

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fracture initiation. As a summary, the Biot’s coefficient has significant impacts on fracturing initiation and seepage area. The distribution of pore pressure is the main mechanism to impact the fracture initiation and fracturing initiation pressure. 5.4. Impact of confining pressure

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The impact of confining pressure on fracturing process is investigated. In the computation, confining pressure varies from 0.1 to 20MPa and the Biot’s coefficient is taken as 0.67. Fig. 18 presents the changes of fracturing initiation pressure and seepage area with

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confining pressure. They almost linearly increase with confining pressure regardless of

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fracturing fluids. Further, the slope and absolute value are the largest for water and the smallest for nitrogen. Under low confining pressure, the fracturing initiation pressure is larger for nitrogen than for CO2. When the confining pressure is higher than 2MPa, the fracturing initiation pressure becomes smaller for nitrogen than for CO2. This is because the CO2 viscosity is lower and its compressibility factor is larger than those of nitrogen under low pressure. It induces higher pore pressure around the borehole and larger stress intensity factor. On this sense, shale is more easily fractured by CO2 than by nitrogen. Furthermore, lower

24

ACCEPTED MANUSCRIPT viscosity and higher compressibility factor cause higher flow velocity of fracturing fluid. Therefore, the seepage area is larger for CO2 than for nitrogen when the confining pressure is lower than 12.5MPa. It becomes smaller for CO2 than for nitrogen when the confining

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pressure is higher than 12.5MPa. The seepage area for water is the smallest. These indicate that confining pressure increase fracturing initiation pressure and seepage area. Nitrogen and

5.5. Permeability evolution during fracturing process

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CO2 have smaller fracturing initiation pressure and larger seepage area than water.

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At the beginning of hydraulic fracture, the shale permeability distribution along x direction varies with the increase of confining pressure as shown in Fig. 19 ( kr is the radial permeability; kr 0 is the initial radial permeability; kθ is the hoop permeability; kθ 0 is the initial hoop permeability). With the increase of confining pressure, the radial permeability

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along x direction decreases and the change is significant nearby the borehole. However, the hoop permeability along x direction increases near the borehole but decreases in the far field. Fig. 20 presents the change of permeability along x direction during CO2 fracturing process

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under the confining pressure of 10MPa. With the increase of injection time, the radial

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permeability increases and the hoop permeability decreases. A significant change also occurs near the borehole. The hoop permeability curves are not always smooth. There are some inflection points in the hoop permeability curves, which corresponds to the seepage front. This is because the pore pressure modifies the effective stress, compresses the shale grain and thus changes the permeability. In order to investigate the impact of pore pressure on the permeability, the simulation is conducted on the impermeable rock with zero pore pressure. Fig. 21 shows the difference of

25

ACCEPTED MANUSCRIPT permeability along xth direction between permeable rock (considering fracturing fluid permeation) and impermeable rock (ignoring fracturing fluid permeation). The pore pressure significantly increases the radial permeability near the borehole and slightly decreases the

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radial permeability in the far field. However, the hoop permeability decreases significantly along x direction in permeable rock. The inflection point in the permeability curves is also induced by the seepage front. This indicates that the impacts of pore pressure on the

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permeability evolution vary with direction.

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The permeability evolution by using different fracturing fluids is studied under the confining pressure of 10MPa. Fig. 22 presents the permeability evolution at the point near the borehole. The radial permeability increases and the hoop permeability decreases more rapidly for nitrogen and CO2 than for water. Fig. 23 presents the permeability along x direction at the

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injection time of t=199s when fracturing fluid is water, CO2 and nitrogen, respectively. On the whole, the radial and hoop permeabilities along x direction are lower for water than those for nitrogen and CO2. They are also a little lower for CO2 than for nitrogen, but the difference

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is very small. Therefore, pore pressure has significant impacts on the permeability. Nitrogen

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and CO2 are more effective to increase the permeability than water. 5.6. Compressibility and viscosity evolutions in fracturing process The coefficient of compressibility plays an important role in fracturing fluid flow. Fig. 24 shows the variations of compressibility coefficient and pore pressure along x direction at the injection time of t=199s. The coefficient of compressibility ( cg ) keeps the constant of 0 for water, while it increases along x direction for both nitrogen and CO2. It is larger for CO2 than for nitrogen when the pore pressure is lower than pr = 1 . This is because the compressibility

26

ACCEPTED MANUSCRIPT factor (Z) decreases with pressure for CO2 but keeps constant for nitrogen. The difference of cg between nitrogen and CO2 is the largest when CO2 reaches the critical state. Fig. 25

shows the change of viscosity along the x direction. The viscosities of water and nitrogen

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keep constant, while the CO2 viscosity decreases with the pore pressure in the seepage zone and keeps constant outside. The most salient change of CO2 viscosity is also observed at the point where the CO2 reaches the critical state and CO2 viscosity decreases from 63.5µPa at

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the point (0.013, 0) to 20.3µPa at the point (0.015, 0). The salient changes of viscosity and

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compressibility induce the different pore pressure distributions among water, nitrogen and CO2. Above the critical state of CO2, the pore pressure for CO2 almost linearly distributes, which is similar to that for water. Below the critical state of CO2, the pore-pressure distribution is similar to that for nitrogen but the pore pressure value is lower for CO2 than for

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nitrogen. Therefore, the critical state of fracturing fluid has significant impacts on seepage area, pore pressure, effective stress, and the evolution of permeability nearby the moving front if the reservoir pressure is below the critical pressure. This effect cannot be ignored in

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the simulation for fracture initiation if the critical state is passed through.

6. Conclusions

A numerical model is proposed in this study to comparatively investigate the impact of water, nitrogen and CO2 as fracturing fluids on fracturing initiation pressure and flow pattern. This model considers the anisotropy of shale deformation and permeability, the property of fracturing fluid and the moving boundary in the hydraulic fracturing process. A fracture initiation criterion is established based on the stress intensity factor of mode I crack and

27

ACCEPTED MANUSCRIPT incorporated into the numerical model. The fracturing processes by injecting water, nitrogen and CO2 are numerically studied. A complex parameter is proposed to consider the combined effect of viscosity, permeability and pressurization rate. The impacts of this complex

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parameter, Biot’s coefficient, confining pressure and fracturing fluid type on the fracturing initiation pressure and seepage area are investigated. Further, the permeability evolution,

preliminary studies, following conclusions can be drawn.

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compressibility coefficient, and viscosity around the borehole are explored. Based on these

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First, the numerical model in this study is able to simulate the fracturing process with the fracturing fluid of water, CO2 and nitrogen in anisotropic shale. It can describe the flow pattern of fracturing fluid and the evolution of permeability in the fracturing process. Further, the fracture initiation criterion can predict the fracturing initiation pressure.

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Second, the complex parameter κ can comprehensively consider the combined effect of permeability, viscosity and pressurization rate when the fracturing initiation pressure and seepage area are concerned. The fracturing initiation pressure decreases and the seepage area

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increases with the increase of complex parameter. When the complex parameter is smaller

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than 1× 10-19 m 2 /Pa 2 , the fracturing initiation pressure becomes the lowest for CO2. Further, the pore pressure distribution varies with fluid compressibility. Water has the largest fracturing initiation pressure and smallest seepage area. Third, the Biot’s coefficient and confining pressure have significant impact on the fracturing initiation pressure and seepage area. Both fracturing initiation pressure and seepage area decrease with the Biot’s coefficient but increase with confining pressure even fracturing fluid is water, nitrogen, or CO2. Under the same Biot’s coefficient and confining pressure, the

28

ACCEPTED MANUSCRIPT fracturing initiation pressure is larger and the seepage area is smaller for water than that for nitrogen and CO2. In general, the fracturing initiation pressure is larger for CO2 than that for nitrogen. The seepage area is smaller for CO2 than that for nitrogen only when the fracturing

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pressure is larger than about 25MPa. Fourth, confining pressure reduces the radial permeability and increases the hoop permeability near the borehole. With the injection of fracturing fluid, the radial permeability

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increases and the hoop permeability decreases. The permeability evolution varies with

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fracturing fluids mainly due to different magnitude and distribution of pore pressure nearby the moving front. Under the same injection pressure, the radial and hoop permeabilities along x direction are lower for water than those for nitrogen and CO2.

Fifth, during fracturing process, compressibility coefficient and viscosity for water keep

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constant, while those for CO2 changes significantly in the seepage domain, especially at the point where the pore pressure is approximately equal to the critical pressure. This obvious difference of viscosity and compressibility induces the different distributions of pore pressure

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for water, nitrogen and CO2.

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Finally, under the same condition of complex parameter, Biot’s coefficient and confining pressure, the fracturing initiation pressure is lower and seepage area is larger for CO2 and nitrogen than those for water. On these senses, nitrogen and CO2 are the better choices of fracturing fluid than water. Under the same complex parameter, CO2 more easily fractures the shale than nitrogen. With the same confining pressure or Biot’s coefficient, nitrogen seems better than CO2. Nitrogen and CO2 have small difference of fracturing initiation pressure and seepage area. However, as the critical state of CO2 is easily reached in operations, CO2 is

29

ACCEPTED MANUSCRIPT more easily pressurized than nitrogen when the required pressure is large. Therefore, CO2 is more suitable than nitrogen when large fracturing initiation pressure is required.

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Acknowledgments The authors are grateful to the financial support from National Natural Science Foundation of China (Grant No. 51674246), Creative Research and Development Group Program of Jiangsu Province (2014-27), and the College Graduate Research and Innovation Program of Jiangsu

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(KYZZ160210).

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References

Batzle, M.L., Han, D.H., Hofmann, R., 2006. Fluid mobility and frequency-dependent seismic velocity—Direct measurements. Geophysics 71(1): N1-N9.

Britt, L., 2012. Fracture stimulation fundamentals. Journal of Natural Gas Science and Engineering 8, 34-51.

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Bruno, M.S., Nakagawa, F.M., 1991. Pore pressure influence on tensile fracture propagation in sedimentary rock. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 28(4), 261-273.

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Burton, M., Kumar, N., Bryant, S.L., 2008. Time-dependent injectivity during CO2 storage in aquifers. In SPE Symposium on Improved Oil Recovery. Society of Petroleum Engineers.

AC C

Chen, C.S., Pan, E., Amadei B., 1998. Fracture mechanics analysis of cracked discs of anisotropic rock using the boundary element method. International Journal of Rock Mechanics and Mining Sciences 35(2), 195-218.

Chen, D., Pan, Z., Liu, J., Connell L., 2012. Characteristic of anisotropic coal permeability and its impact on optimal design of multi-lateral well for coalbed methane production. Journal of Petroleum Science and Engineering 88, 13-28. Chen, D., Pan, Z., Ye, Z., 2015a. Dependence of gas shale fracture permeability on effective stress and reservoir pressure: Model match and insights. Fuel 139, 383-392. Chen, Y., Nagaya, Y., & Ishida, T., 2015b. Observations of fractures induced by hydraulic fracturing

30

ACCEPTED MANUSCRIPT in anisotropic granite. Rock Mechanics and Rock Engineering, 48(4), 1455-1461. Cho, J.W., Kim, H., Jeon, S., Min, K., 2012. Deformation and strength anisotropy of Asan gneiss, Boryeong shale, and Yeoncheon schist. International Journal of Rock Mechanics and Mining Sciences 50, 158-169.

materials. Engineering fracture mechanics 28(1), 43-54.

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Chong, K.P., Kuruppu, M.D., Kuszmaul, J. S., 1987. Fracture toughness determination of layered

Civan, F., Rai, C.S., Sondergeld, C.H., 2011. Shale-gas permeability and diffusivity inferred by improved formulation of relevant retention and transport mechanisms. Transport in Porous Media

SC

86(3), 925–44.

Detournay, E., Carbonell, R., 1997. Fracture-mechanics analysis of the breakdown process in

M AN U

minifracture or leakoff test. SPE Production & Facilities 12(03), 195-199.

Van Eekelen, H.A.M., 1982. Hydraulic fracture geometry: fracture containment in layered formations. Society of Petroleum Engineers Journal 22(03), 341-349.

Fairhurst, C., 1968. Methods of determining in-situ rock stresses at great depths: Tech. Rep. TRI-68, Mo. River Div. Corps of Eng., Omaha, Nebr.

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Fenghour, A., Wakeham, W.A., Vesovic V., 1998. The viscosity of carbon dioxide. Journal of Physical and Chemical Reference Data 27(1), 31-44.

Gan, Q., Elsworth, D., Alpern, J.S., Marone, C., Connolly, P., Elsworth, D., 2015. Breakdown

EP

pressures due to infiltration and exclusion in finite length boreholes. Journal of Petroleum Science and Engineering 127, 329-337.

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Gomaa, A.M., Qu, Q., Maharidge, R., Nelson, S., Reed, T., 2014. New insights into hydraulic fracturing of shale formations. International Petroleum Technology Conference. Grundmann, S.R., Rodvelt, G.D., Dials, G.A., Robert, E.A., 1998. Cryogenic nitrogen as a hydraulic fracturing fluid in the Devonian shale. SPE Eastern Regional Meeting. Society of Petroleum Engineers.

Haimson, B., Fairhurst, C., 1967. Initiation and extension of hydraulic fractures in rocks. Society of Petroleum Engineers Journal 7(03), 310-318. Heng, S., Yang, C.H., Guo, Y.T., Wang, C.Y., Wang, L., 2015. Influence of bedding planes on hydraulic fracture propagation in shale formations. Chinese Journal of Rock Mechanics and Engineering (in Chinese) 34(2), 228-237 31

ACCEPTED MANUSCRIPT Hubbert, M.K., Willis, D.G., 1957. Mechanics of hydraulic fracturing. AIME Petrol Trans 210, 153-168. Ishida, T., Aoyagi, K., Niwa, T., Chen, Y.Q., Murata, S., Chen, Q., Nakayama, Y., 2012. Acoustic emission monitoring of hydraulic fracturing laboratory experiment with supercritical and liquid

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CO2. Geophysical Research Letters 39(16), L16309, doi:10.1029/2012GL052788. Ito, T., Hayashi, K., 1991. Physical background to the breakdown pressure in hydraulic fracturing tectonic stress measurements. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 28(4), 285-293.

SC

Kestin, J., Korfali, Ö., Sengers, J.V., 1980. Density expansion of the viscosity of carbon dioxide near the critical temperature. Physica A: Statistical Mechanics and its Applications 100(2), 335-348.

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Levine, J.R., 1996. Model study of the influence of matrix shrinkage on absolute permeability of coalbed reservoirs. Geological Society Special Publication 109, 197–212. Li, X., Feng, Z., Han, G., Elsworth, D., Marone, C., Saffer, D., Cheon, D., 2016. Breakdown pressure and fracture surface morphology of hydraulic fracturing in shale with H2O, CO2 and N2. Geomechanics and Geophysics for Geo-Energy and Geo-Resources, 2016, 1-14.

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Liao, S., Brunner, F., Mattar, L., 2009. Impact of Ignoring CO2 Injection Volumes on Post-Frac PTA. Canadian International Petroleum Conference. Petroleum Society of Canada. Liu, J., Chen, Z., Elsworth, D., Miao, X., Mao, X., 2010. Linking gas-sorption induced changes in

EP

coal permeability to directional strains through a modulus reduction ratio. International Journal of Coal Geology 83(1), 21-30.

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Lu, Y.L., Elsworth, D., Wang, L.G., 2013 Microcrack-based coupled damage and flow modeling of fracturing evolution in permeable brittle rocks. Computers and Geotechnics 49, 226-244. Matsunaga, I., Kobayashi, H., Sasaki, S., Ishida, T., 1993. Studying hydraulic fracturing mechanism by laboratory experiments with acoustic emission monitoring, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 30(7), 909-912. Niandou, H., Shao, J.F., Henry, J.P., Fourmaintraux, D., 1997. Laboratory investigation of the mechanical behaviour of Tournemire shale. International Journal of Rock Mechanics and Mining Sciences 34(1), 3-16. Nowman, J.C., Jr., 1971. An improved method of collection for the stress analysis of cracked plates with various shaped boundaries. NASA, TN D-6376. 32

ACCEPTED MANUSCRIPT Rice, J.R, Cleary, M.P., 1976. Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Reviews of Geophysics 14(2), 227-241. Schmitt, D.R., Zoback, M.D., 1992. Diminished pore pressure in low-porosity crystalline rock under tensional failure: Apparent strengthening by dilatancy. Journal of Geophysical Research: Solid

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Earth (1978–2012) 97(B1), 273-288. Shen, B., Siren, T., Rinne, M., 2015. Modelling fracture propagation in anisotropic rock mass. Rock Mechanics and Rock Engineering 48(3), 1067-1081.

Sinal, M.L., Lancaster, G., 1987. Liquid CO2 fracturing: Advantages and limitations. Journal of

SC

Canadian Petroleum Technology 26(05), DOI: 10.2118/87-05-01

Solberg, P., Lockner, D., Byerlee, J.D., 1980. Hydraulic fracturing in granite under geothermal

M AN U

conditions. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 17(1), 25-33.

Song, I., Suh, M., Won, K.S., Haimson, B., 2001. A laboratory study of hydraulic fracturing breakdown pressure in tablerock sandstone. Geosciences Journal, 5(3): 263-271. Soni, T.M., 2014. LPG-based fracturing: An alternate fracturing technique in shale reservoirs.

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IADC/SPE Asia Pacific Drilling Technology Conference. Society of Petroleum Engineers. Stringfellow, W.T., Domen, J.K., Camarillo, M.K., Sandelin, W., Borglin, S., 2014. Physical, chemical, and biological characteristics of compounds used in hydraulic fracturing. Journal of hazardous

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materials 275, 37-54.

Wang, H., Li, G., Tian, S., Cheng, Y., He, Z., Yu, S., 2015a. Flow field simulation of supercritical

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carbon dioxide jet: Comparison and sensitivity analysis. Journal of Hydrodynamics, Ser. B, 27(2), 210-215

Wang, J.G., Ju, Y., Gao, F., Liu, J., 2016a. A simple approach for the estimation of CO2 penetration depth into a caprock layer. Journal of Rock Mechanics and Geotechnical Engineering 8(1), 75-86. Wang, J.G., Ju, Y., Gao, F., Peng, Y., Gao, Y., 2015b. Effect of CO2 anisotropic sorption and swelling on caprock sealing efficiency. Journal of Cleaner Production 103, 685-695 Wang, L., Yao, B., Cha, M., Alqahtani, N.B., Patterson, T.W., Kneafsey, T.J., Miskimins, J.L., Yin, X., Wu, Y.S., 2016b. Waterless fracturing technologies for unconventional reservoirs - opportunities for liquid nitrogen. Journal of Natural Gas Science and Engineering, 35, 160-174. Wang, J.G., Liu, J., Kabir, A., 2013. Combined effects of directional compaction, non-Darcy flow and 33

ACCEPTED MANUSCRIPT anisotropic swelling on coal seam gas extraction. International Journal of Coal Geology 109, 1-14. Wang, J.G., Kabir, A., Liu, J.S., Chen, Z.W., 2012. Effect of non-Darcy flow on the performance of coal seam gas wells. International Journal for Coal Geology 93(1), 62-74

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Wu, Y., Liu, J., Elsworth, D., Miao, X., Mao, X., 2010. Development of anisotropic permeability during coalbed methane production. Journal of Natural Gas Science and Engineering 2(4), 197-210.

Zhang, C.L., 2016. The stress-strain-permeability behaviour of clay rock during damage and

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recompaction. Journal of Rock Mechanics and Geotechnical Engineering 8(1), 16-26.

Zhang, H., Liu, J., Elsworth, D., 2008. How sorption-induced matrix deformation affects gas flow in

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coal seams: a new FE model. International Journal of Rock Mechanics and Mining Sciences 45(8), 1226-1236

Zhang, X., Lu, Y., Tang, J., Zhou, Z., Liao, Y., 2017. Experimental study on fracture initiation and propagation in shale using supercritical carbon dioxide fracturing. Fuel, 190, 370-378. Zhong, J., Liu, S., Ma, Y., Yin, C., Liu, C., Li, Z., Liu, X., Li, Y., 2015. Macro-fracture mode and

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micro-fracture mechanism of shale. Petroleum Exploration and Development 42(2), 269-276. Zhu, W.C., Gai, D., Wei, C.H., Li, S.G., 2016. High-pressure air blasting experiments on concrete and implications for enhanced coal gas drainage. Journal of Natural Gas Science and Engineering, 36,

EP

1253-1263.

Zoback, M.D., Rummel, F., Jung, R., Raleigh, C.B., 1977. Laboratory hydraulic fracturing

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experiments in intact and pre-fractured rock. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 14(2), 49-58

34

ACCEPTED MANUSCRIPT Figures

Shale matrix

s

Fracture

boj

i

SC

j

RI PT

∆σ ej

Shale matrix

M AN U

s

Fig. 1 Deformation of fractured shale element

1.2

Viscosity Compressibility factor

TE D

0.8

EP

0.6

120

0.4

60

Viscosity (µPa)

T=Tc

AC C

Compressibility factor

1.0

180

0.2

Nearby critical state

0.0

0

1

0 2

3

4 pr

5

6

7

8

Fig. 2 CO2 compressibility factor (Z) and viscosity (µ) with the reduced pressure ( pr )

35

RI PT

ACCEPTED MANUSCRIPT

SC

(a) Fixed seepage boundary (b) Moving seepage boundary Fig. 3 Two treatment methods for seepage front boundary

σ3

a p(r,t) p

σ1

M AN U

p(r,t)

p

+

σ1 =

σ3 (a)

p(r,t)

p(r,t)

+

(b)

(c)

σ

KΙ p = ∫

a

(

a+r a−r + )dr a−r a+r

Fracturing fluid flow, p(r,t)

EP

πa

MATLAB

AC C

In-suit & injection pressure

RB

σp

Shale deformation, ui

COMSOL

vi = − ki µ ∇ i p

Pore pressure gradient

Effects of pore pressure

vi = − ki µ ∇ i p

Fracture initiation criterion, KI

Seepage front & domain

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Fig. 4 Stress intensity factor model of a crack: (a) original problem; (b) induced by in-situ stress and injection pressure; (c) induced by pore pressure

Moving boundary, vi

Fig. 5 Cross-coupling among rock deformation, fracturing fluid flow, moving boundary and fracture initiation criterion in hydraulic fracturing process

36

ACCEPTED MANUSCRIPT Geometry model Boundary condition & parameter Rock deformation

Moving boundary

Coupling solver Fracturing initiation criterion

No

SC

Yes

RI PT

Fracturing fluid flow

end

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Fig. 6 Schematic computational process for cross-coupling

PHW

60

PHF Pupper

TE D

50

Plower Pmax

40

Pmin

20

EP

30

AC C

Fracturing initiation pressure (MPa)

70

10 -4 10

-3

10

This simulation with CO2 This simulation with oil Experimental data with oil -2

10

-1

10

0

10

1

10

2

10

lP (MPa/s)

Fig. 7 Comparison of fracturing initiation pressure simulated by our model with experimental data by Zoback et al. (1977). The CO2 one is here for contrast to oil fluid

37

ACCEPTED MANUSCRIPT

PHW PHF

45

Pupper

RI PT

Plower Pmax

30

Pmin

15

SC

This simulation with CO2

This simulation with oil Experimental data with oil 0

0

10

20

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Fracturing initiation pressure (MPa)

60

30

Confining pressure, σv(MPa)

Fig. 8 Comparison of fracturing initiation pressure simulated by our model with experimental data by Song et al. (2001). The CO2 one is here for contrast to oil fluid

TE D

PHF Pupper Plower

EP

15

PHW

Pmax Pmin

AC C

Fracturing initiation pressure (MPa)

20

10

This simulation with CO2 This Simulation with water Lu et al's simulation with water

5

-19

10

-18

10

-17

-16

10

10

-15

10

-14

10

2

k (m )

Fig. 9 Comparison of simulation results of our model with Lu et al.’s model. The CO2 one is here for contrast to oil fluid

38

ACCEPTED MANUSCRIPT σv

ϕ

R=5mm

σv

100mm

SC

RI PT

x

100mm

y

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Fig. 10 Computational model for the anisotropic shale

1.2

TE D

0.8 0.6 0.4

AC C

0.2

EP

σp

1/2

K I (MPa*m )

1.0

0.0

t=0s t=40s t=80s t=120s t=160s t=200s

-90°

-60°

-30°

0°

30°

60°

90°

Potential initiation angle, ϕ σ

Fig. 11 Chang of pore pressure induced stress intensity factor ( K I p ) with potential initiation angle at different injection time

39

ACCEPTED MANUSCRIPT

Pmax

RI PT

10

8 Water Nitrogen CO2

Pmin 4

-23

10

-22

10

-21

10

-20

-19

10

10

SC

6

σv=0.1MPa

-18

10

M AN U

Fracturing initiation pressure (MPa)

12

-17

10

κ (m /Pa ) 2

2

(a) Fracturing initiation pressure

TE D

25

Water Nitrogen CO2

EP

15

σv=0.1MPa

10

AC C

2

Seepage area (cm )

20

5

0

-23

10

-22

10

-21

10

-20

-19

10

10

-18

10

-17

10

κ (m /Pa ) 2

2

(b) Seepage area Fig. 12 Changes of fracturing initiation pressure and seepage area with the complex parameter ( κ )

40

ACCEPTED MANUSCRIPT

6

RI PT

2

SC

Pore pressure (MPa)

Water Nitrogen CO2

σv=0.1MPa

4

0

0.01

0.02

0.03

0.04

0.05

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0.00

0.06

x (m)

Fig. 13 Pore pressure along x direction (under injection pressure of 5MPa): for water,

κ =1×10-19 m2 /Pa 2 ; for nitrogen, κ =1.11×10-19 m2 /Pa 2 ; for CO2, κ =1.19 ×10-19 m2 /Pa 2

TE D

0.5

Water Nitrogen CO2

σv=0.1MPa

EP

0.3

1/2

K I (MPa*m )

0.4

σp

AC C

0.2

0.1

0.0

-23

10

-22

10

-21

10

-20

-19

10

10

-18

10

-17

10

κ (m /Pa ) 2

2

σ

Fig. 14 Evolution of pore pressure induced stress intensity factor ( K I p ) with the complex parameter ( κ ) (under injection pressure of 5MPa) 41

ACCEPTED MANUSCRIPT

40 35 30

RI PT

 Pf = 2.54α 2 − 14.09α + 46.62; for water  2  Pf = 14.11α − 35.47α + 37.45; for nitrogen  P = 16.79α 2 − 40.31α + 40.80; for CO 2  f

20

Water Nitrogen CO2

15 10 0.3

0.4

0.5

σv=10MPa

0.6

0.7

SC

25

0.8

M AN U

Fracturing initiation pressure (MPa)

45

0.9

1.0

0.9

1.0

Biot's coefficient, α

(a) Fracturing initiation pressure

9

TE D

8

6

4 3

EP

5

AC C

2

Seepage area (cm )

7

Water Nitrogen CO2

σv=10MPa

2 1

0 0.3

0.4

0.5

0.6

0.7

0.8

Biot's coefficient

(b) Seepage area Fig. 15 Effect of Biot’s coefficient on fracturing initiation pressure and seepage area under confining pressure of 10MPa

42

ACCEPTED MANUSCRIPT

24

Nitrogen

16 160s

σv=10MPa

RI PT

CO2

12 120s 8 4

80s

Critical state of CO2 40s

0 0.01

0.02

0.03

0.04

0.05

M AN U

0.00

SC

Pore pressure (MPa)

20 199s

0.06

x (m)

Fig. 16 Pore pressure distribution for nitrogen and CO2 at different injection times (t=40s, 80s, 120s, 160s and 199s)

TE D

1.5

EP

0.9

0.6

Water Nitrogen CO2 σv=10MPa

σp

AC C

1/2

K I (MPa*m )

1.2

0.3

0.0

0

100

200

300

400

Injection time (s)

Fig. 17 Evolution comparison of pore pressure induced stress intensity factor

43

ACCEPTED MANUSCRIPT

60

Water Nitrogen CO2

RI PT

50 40 30 20 10 0

0

5

10

15

SC

 Pf = 2.48σ v + 10.7; for water   Pf = 1.37σ v + 6.99; for nitrogen  P = 1.50σ + 6.74; for CO v 2  f 20

25

20

25

M AN U

Fracturing initiation pressure (MPa)

70

Confining pressure, σv (MPa)

(a) Fracturing initiation pressure

TE D

15

Water Nitrogen CO2

2

Seepage area (cm )

12

EP

9

AC C

6

3

0

0

5

10

15

Confining pressure, σv (MPa)

(b) Seepage area Fig. 18 Changes of (a) fracturing initiation pressure and (b) seepage area with confining pressure

44

ACCEPTED MANUSCRIPT

RI PT

0.9 σv=0.1MPa σv=5MPa

0.8

σv=10MPa σv=15MPa

SC

Permeability ratio (kr/kr0)

1.0

σv=20MPa

0.01

0.02

0.03

0.04

0.05

M AN U

0.7 0.00

0.06

x (m)

(a) Radial permeability

1.00

TE D

σv=0.1MPa σv=5MPa σv=10MPa σv=15MPa σv=20MPa

EP

1.05

AC C

Permeability ratio (kθ/kθ0)

1.10

0.95

0.00

0.01

0.02

0.03

0.04

0.05

0.06

x (m)

(b) Hoop permeability Fig. 19 Permeability distribution along x direction under different confining pressures

45

ACCEPTED MANUSCRIPT

t=0s t=40s t=80s t=120s t=160s

σv=10MPa

RI PT

1.1

0.9

0.00

0.01

0.02

0.03

SC

1.0

0.04

0.05

M AN U

Permeability ratio (kr/kr0)

1.2

0.06

x (m)

(a) Radial permeability

t=0s t=40s t=80s t=120s t=160s

TE D

1.05

0.95

EP

1.00

Seepage front

AC C

Permeability ratio (kθ/kθ0)

σv=10MPa

0.90

0.00

0.01

0.02

0.03

0.04

0.05

0.06

x (m)

(b) Hoop permeability Fig. 20 Change of permeability along x direction during CO2 fracturing process under confining pressure of 10MPa

46

ACCEPTED MANUSCRIPT

1.2 Impermeable rock, t=40s Impermeable rock, t=160s Permeable rock, t=40s Permeable rock, t=160s

RI PT

1.1

0.9 0.00

0.01

0.02

0.03

SC

1.0

0.04

0.05

M AN U

Permeability ratio (kr/kr0)

σv=10MPa

0.06

x (m)

(a) Radial permeability

1.02

0.94

TE D

0.96

EP

0.98

Seepage front

Impermeable rock, t=40s Impermeable rock, t=160s Permeable rock, t=40s Permeable rock, t=160s

σv=10MPa

AC C

Permeability ratio (kθ/kθ0)

1.00

0.92 0.90

0.00

0.01

0.02

0.03

0.04

0.05

x (m)

(b) Hoop permeability Fig. 21 Comparison of permeability along x direction

47

0.06

ACCEPTED MANUSCRIPT

1.6

σv=10MPa

RI PT

Point (0.005,0)

1.2

Water Nitrogen CO2

0.8

0

100

SC

1.0

200

300

M AN U

Permeability ratio (kr/kr0)

1.4

400

Injection time (s)

(a) Radial permeability

TE D

1.05

σv=10MPa

Point (0.005,0)

Water Nitrogen CO2

EP

0.95

0.90

AC C

Permeability ratio (kθ/kθ0)

1.00

0.85

0.80

0

100

200

300

400

Injection time (s)

(b) Hoop permeability Fig. 22 Permeability evolution in the fracturing process at the point near the borehole

48

ACCEPTED MANUSCRIPT

1.3

Water Nitrogen CO2

σv=10MPa

RI PT

kx/kx0

1.2

1.1

0.01

0.02

0.03

0.04

0.05

M AN U

0.9 0.00

SC

1.0

0.06

x (m)

(a) Radial permeability

0.94

EP

0.96

Seepage front

AC C

Permeability ratio (kr/kr0)

0.98

TE D

1.00

Water Nitrogen CO2

σv=10MPa

0.92

0.00

0.01

0.02

0.03

0.04

0.05

0.06

x (m)

(b) Hoop permeability Fig. 23 Permeability distribution along x direction at the injection time of t=199s

49

ACCEPTED MANUSCRIPT

12

-6

Coefficient of compressiblity (e )

10 Seepage front

RI PT

8 6 4

σv=10MPa

Water Nitrogen CO2

Critical state of CO2

0 0.01

0.02

0.03

0.04

0.05

M AN U

0.00

SC

2

0.06

x (m)

(a) Coefficient of compressibility

20

TE D

25

Water Nitrogen CO2

10

Critical state of CO2

EP

15

AC C

Pore pressure (MPa)

σv=10MPa

Seepage front

5

0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

x (m)

(b) Pore pressure Fig. 24 Change of coefficient of compressibility and pore pressure along x direction at the injection time of t=199s

50

ACCEPTED MANUSCRIPT

Water Nitrogen CO2

0.1

RI PT

σv=10MPa

Critical state of CO2

0.01 0.00

0.01

0.02

0.03

SC

Seepage front

0.04

0.05

M AN U

Viscosity (mPa*s)

1

0.06

x (m)

AC C

EP

TE D

Fig. 25 Change of viscosity along x direction at the injection time of t=199s

51

ACCEPTED MANUSCRIPT Tables Table 1 Tuned coefficients in Eq. (26) A4 1.00157 3.66 15.2715 186.689

SC

A3 -0.43915 0 0 0

Parameter

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Table 2 Computational parameters for Weber sandstone value

50

Shear modulus of sandstone, G (GPa)

12.2

Bulk modulus of rock grain, Ks (GPa)

36

Sandstone Poisson’s ratio, ν

0.15

Initial porosity, φ0

TE D

Viscosity of fracturing fluid, µ (mPa s)

0.06

Initial permeability of sandstone, k0 (mD)

1

Pressurization rate, lP (MPa/s)

10-3-102

EP

µ

p r ≦1 pr>1 p r ≦1 pr>1

A2 -2.8701E-48 0 0 -141.52906

Length of initial crack, a (mm)

3

Tensile strength, σt (MPa)

9.5

Fracture toughness, KIC (MPa m1/2/s)

0.6

AC C

Z

A1 -2.61556E-4 -3.606 0.04744 -2.29081E7

52

t1 0.15155 -26.704 0.2003 -0.0717

t2 0.00927 ~ ~ -7.8103

RI PT

f

ACCEPTED MANUSCRIPT

value

Viscosity of fracturing fluid, µ (Pa s)

2.5

Shear modulus of sandstone, G (GPa)

15.3

Biot’s coefficient, α

0.9

Sandstone Poisson’s ratio, ν

0.2

Initial porosity, φ0

0.26

Initial permeability of sandstone, k0 (D)

0.12

Pressurization rate, lP (MPa/s)

15

Length of initial crack, a (mm)

13

Tensile strength, σt (MPa)

4.4

Fracture toughness, KIC (MPa m1/2/s)

0.55

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Parameter

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Table 3 Computational parameters for Tablerock sandstone

Table 4 Computational parameters for the problem in Lu et al. (2013)

value

TE D

Parameter

10

Viscosity of water, µw (mPa s)

1

Young’s modulus of rock, E (GPa)

10

Bulk modulus of rock grain, Ks (GPa)

40

EP

Radius of the borehole, RB (mm)

0.25

Initial porosity, φ0

0.05

AC C

Shale Poisson’s ratio, ν

Initial permeability of shale, k0 (µD)

10-1-104

Pressurization rate, lP (MPa/s)

0.5

Length of initial crack, a (mm)

18.9

Fracture toughness, KIC (MPa m1/2/s)

1.1

53

ACCEPTED MANUSCRIPT Table 5 Computational parameters for the anisotropic shale value

Model size (m)

0.1×0.1

Radius of the borehole, R (mm)

5

Density of shale, ρs (Kg/m3)

2230

Temperature, T (Κ)

304

Viscosity of water, µw (mPa s)

1

Viscosity of nitrogen, µn (mPa s)

0.018

Young’s modulus of shale, E (GPa)

Ex= Ez=25, Ey=14

Shale Poisson’s ratio, ν

νxy=0.2, νyz=0.36, νxz=0.3

Initial porosity of shale, φ0

0.03

Initial permeability of shale, k0 (µD)

kx0=0.6, ky0=0.1

Pressurization rate, lP(MPa/s)

0.1

Tensile strength, σt (MPa)

SC

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Length of initial crack, a (mm)

RI PT

Parameter

10

3.2

Fracture toughness, KIC (MPa m1/2/s)

TE D

0.566

Table 6 Range of computational parameters for parametric study

Water

µ (Pa*s)

0.1~1000

0.001~0.01

0.01~10

1.8×10-5~18×10-5

0.01~10

1.6×10-5~7.0×10-5

AC C

Nitrogen

ky0 (µD)

EP

Frac fluid

CO2

54

lP (MPa/s)

κ (m2/Pa2)

0.1~1

1×10-17~1×10-23

ACCEPTED MANUSCRIPT Highlights The impacts of water, nitrogen and CO2 fracturing fluids on hydro-fracturing are comparatively studied in anisotropic shale.

A moving boundary method is introduced.

RI PT

A fracturing initiation criterion is established by using pore pressure distribution.

A complex parameter is proposed for the combined effect of permeability,

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EP

TE D

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viscosity and pressurization rate.