Arab J Geosci (2017) 10:157 DOI 10.1007/s12517-017-2939-7

ORIGINAL PAPER

A fully coupled three-dimensional hydraulic fracture model to investigate the impact of formation rock mechanical properties and operational parameters on hydraulic fracture opening using cohesive elements method Seyed Erfan Saberhosseini 1 & Reza Keshavarzi 2 & Kaveh Ahangari 3

Received: 27 September 2016 / Accepted: 28 February 2017 # Saudi Society for Geosciences 2017

Abstract Hydraulic fracturing operation success is critically dependent on pre-operation fracture geometry analysis. Meantime, hydraulic fracture opening determination is crucial because it deals with maintaining sufficient aperture and efficient communication pathways to accomplish proppant placement and avoid screen-outs or proppant bridging. It is well understood that, to efficiently estimate the hydraulic fracture opening, a thorough understanding of the impact of rock mechanical properties as well as operational parameters such as injection fluid viscosity and leak-off are essential. In this study, a three-dimensional model has been developed for parametric study of fracture stiffness, formation elastic modulus, formation Poisson’s ratio, tensile strength of the rock, fracturing fluid viscosity and leak-off, and their influences on hydraulic fracture opening. Cohesive elements with traction–separation law have been applied to simulate the fracturing process in a fluid-solid coupling model. The maximum nominal stress criterion has been selected for initiation of damage in the cohesive elements. The results revealed that any change in influential rock mechanical properties as well as operational parameters would significantly lead to increasing or decreasing the hydraulic fracture opening. It was observed from the controllable HF parameters including fracturing fluid viscosity and leak-off, by increasing fracturing fluid viscosity from 10−3 to 10 Pa.s, fracture width has been moderately increased from 8.83

* Seyed Erfan Saberhosseini [email protected]

1

Department of Petroleum Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran

2

Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada

3

Department of Mining Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

to 13.3 mm; on the other hand, by reducing leak-off coefficient as a polymer-dependent HF parameter from 5 × 10 −9 to 5 × 10−11 m/kPa.s, fracture width has been gradually risen from 7.27 to 11.52 mm. In addition, all the uncontrollable rock mechanical parameters such as fracture stiffness, Young’s modulus, Poisson’s ratio, and tensile strength have steeply increased or decreased the hydraulic fracture opening and consequently, all of their analysis will be discussed in detail in this research. The results from this study can be applied to hydraulic fracturing jobs in different conditions of fracture stiffness, formation elastic modulus, formation Poisson’s ratio, tensile strength of the rock, fracturing fluid viscosity, and leak-off. Keywords Hydraulic fracturing . Parametric study . Hydraulic fracture opening . Rock mechanical properties . Cohesive elements . XFEM-based cohesive law

Introduction Hydraulic fracturing is currently the most widespread stimulation technique in use to improve and enhance the commercial production of hydrocarbon reservoirs and increase the heat extraction surface area in geothermal resources (Zoback 2007). The efficiency of hydraulic fracturing job is closely tied to estimation of fracture geometry which influences the communication of the well with the fracture. Meanwhile, hydraulic fracture opening is found to be as one of the critical factors that can influence the success or failure of the hydraulic fracturing job since proppant size is calculated based on minimum fracture opening and closure pressure of the formation (Fjaer 2008). In other words, overestimated fracture opening would clearly restrict the fracture growth and in practice narrower fractures would be formed which would also increase the possibility of proppant bridging (Zoback 2007). On the other hand, underestimating the fracture

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opening would practically results in wider fractures that require more volumes of fluid to continue propagating. In addition, for low-permeability reservoirs, long fractures (with narrower opening) are generally expected; while for high-permeability reservoirs, wider fractures are anticipated (Franquet and Economides 1999). Both experimental and numerical studies have revealed that rock mechanical properties such as elastic modulus, Poisson’s ratio, stiffness, and rock tensile strength are influential factors on fracture geometry and proppant placement (Li et al. 2013; Gu and Siebrits 2008; Franquet and Economides 1999) whereas any change in formation rock mechanical properties may lead to enhance or diminish flow resistance caused by a width change (Fisher and Warpinski 2012). Also, it has been found that operational factors such as fluid viscosity and leak-off would affect fracture geometry in different ways (Todd et al. 2011; Franquet and Economides 1999). Fluid viscosity and leak-off coefficient would severely influence the fracture opening by increasing or declining the internal fluid force between the fracture surfaces into the reservoir rock. (Gu and Siebrits 2008). Therefore, determination of fracture opening in different conditions of rock mechanical properties and operational parameters is of great importance to optimize and successfully perform hydraulic fracturing jobs. The PKN and KGD are two of the most common models to calculate hydraulic fracture opening which have been applied for decades. The PKN model was first developed by Perkins and Kern (1961) and then, the effects of fluid loss to the formation were added to the model by Nordgren (1972). This model is valid when the length of a fracture is much greater than the height. The KGD model was developed by Geertsma and de Klerk (1969) which was extended for the case of power-law fluids and fracture toughness by Daneshy (1973) and Spence and Sharp (1985), respectively. The KGD model assumes that the fracture opening at any distance from the wellbore is independent of vertical position which indirectly means that the fracture opening changes much more slowly vertically along the fracture surface than it does horizontally. In other words, KGD model is valid when the fracture height is much greater than the length. In both PKN and KGD, plane strain condition and one-dimensional fluid flow in the fracture are considered and the equations are based on elastic conditions. Therefore, it is clear that these generalized models applied to estimate the opening of hydraulic fracture are accompanied with some limitations and assumptions which bring uncertainties to the calculated fracture opening. As a result, numerical approaches seem to be more efficient tools to determine hydraulic fracture opening in different conditions. The cohesive elements method has its origin in the concepts of a cohesive zone model for fractures which originally proposed by Dugdale (1960) and Barenblatt (1962) is used to simulate the fracture initiation in the ABAQUS program. As the cohesive elements method is based on the XFEM-based traction–separation law, it would be beneficial to show the validation of the cohesive law in hydraulic fracture modeling (Huang et al.

2003; Bowie 1964); thus, Saberhosseini et al. (2014) have developed a true verification with high accuracy for this purpose. So, based on the effectiveness of XFEM-based approaches to account for fractures propagation and interaction (Keshavarzi and Mohammadi 2012; Dahi-Taleghani and Olson 2011), the cohesive elements with XFEM-based traction–separation law (cohesive law) are used for fracture initiation and propagation into the model. The main objective of this paper is to perform a parametric study through a fully coupled numerical modeling to investigate the role of rock mechanical and operational variables such as fracture stiffness, formation elastic modulus, formation Poisson’s ratio, tensile strength of the rock, fracturing fluid viscosity, and leak-off, which have considerable influence the hydraulic fracture opening.

Traction–separation law The traction–separation or cohesive law defines the relationship between the traction tensor t and the displacement δ across the cohesive surfaces (Tomar et al. 2004). The mentioned traction is given by a cohesive potential function φ and the displacement: t¼

∂φ ∂δ

ð1Þ

The traction versus separation law shows the linear elastic behavior until the traction reaches to the cohesive strength Tmax or equivalently the separation exceeds δ0. This law presumes that the cohesive planes are intact without any relative displacement and exhibit linear elastic behavior until the traction reaches the cohesive strength Tmax or equivalently the separation exceeds δ0 (Chen 2011); the mentioned concepts are shown in Figs. 1 and 2.

Fig. 1 Schematic view of fracture and fluid flow (Chen 2011)

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Fracture initiation criterion Fracture or damage initiation related to the starting of degradation of a rock under certain traction loading. The process of degradation begins when the stresses satisfy certain damage initiation criteria and after this point, the non-linear behavior of the traction–separation will be started as fracture propagates (Fig. 2). Several damage initiation criteria are available but we just consider maximum nominal stress criterion (MAXS). A value of 1 or higher indicates that the initiation criterion has been met (Zhang et al. 2010; ABAQUS documentation 2011). According to Fig. 2, t 0n , t 0s , and t 0t represent the peak values of the nominal stresses; d 0n , d 0s , and d 0t are the displacement peak values at initiation of damage and d nf , d sf , and d tf are the displacement peak values at complete failure in the normal and shear directions (Tada et al. 1973).

Fig. 2 Typical traction–separation law (ABAQUS documentation 2011)

where d nf and d 0n are the opening displacements at complete failure and at the initiation of damage, respectively, d max refers to the maximum value of opening displacement n attained during the loading history (Fig. 2).

Maximum nominal stress criterion Fracture or damage initiation occurs when the maximum nominal stress ratio which is defined in the expression below reaches a value of one or higher. The symbol < > signifies that a pure compressive deformation or stress state does not initiate damage (Zhang et al. 2010; ABAQUS documentation 2011). This criterion can be represented as
 < tn > ts tt Max ; 0; o ¼ 1 ð2Þ t 0n ts tt Fracture propagation criterion The fracture propagation is related to the degradation of material stiffness when the damage initiation criterion has been met. Parameter D is a scalar damage variable and represents the damage in the rock. Its value is initially equal to 0; the variable D gradually increases from 0 to 1 as fracture propagates (Chen 2011; Zhang et al. 2010). The stress component in the traction– separation model is affected by the damage according to the following expression: 8 < ð1−DÞ t n; t n ≥ 0 tn ¼ ð3Þ : t n; otherwise where t n is the stress component predicted by the linear elastic behavior without damage and the current strain value. For linear softening, the scalar damage variable, D is expressed with the following form (Chen 2011; Zhang et al. 2010):   0 d nf d max n −d n D ¼ max  f 0  ð4Þ dn d n −d n

Fluid flow into the fracture Tangential flow into the fracture The fluid flow model in the fracture consists of tangential and normal flow (Fig. 3). The term of tangential flow of fluid is referred as the flowing of fluid just into the fracture gap (Chen 2011). To allow tangential flow, we have to define a gap flow property in conjunction with the pore fluid definition. The fluid is assumed to be incompressible with Newtonian rheology. When pore pressure cohesive elements are used in soil procedures in ABAQUS, the fluid constitutive behavior of the cohesive elements can be defined by considering the tangential fluid flow relationship, and by defining fluid leak-off coefficients (ABAQUS documentation 2011; Settari and Cleary 1984).

Fig. 3 Schematic view of tangential and normal fluid flow into the fracture (Chen 2011)

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Newtonian rheology

Normal fluid flow into the fracture

Tangential flow within the gap is governed by the lubrication equations (Batchelor 1967), which is formulated from Poiseuille’s law.

The normal flow is defined as a flow of fluid from rock matrix into the fracture gap due to pressure difference between the formation pore pressure and the fracture gap (Fig. 3). Normal flow can be defined as fluid leak-off properties in ABAQUS which contains top and bottom constant coefficients (Valco and Economides 1997; Chen et al. 2009). The normal flow or the rate of normal flow at the top and bottom of the cohesive elements are defined as follows:

−ðk t  ∇:p f Þ ð5Þ w where the q is the flow rate of tangential flow into the gap, Kt is the tangential permeability that is defined below, ∇.Pf is the fluid pressure gradient along the cohesive zone and w is the gap (fracture) opening, and Kt is defined in the following equation. q¼

kt ¼

w3 12μ

μ is the fluid viscosity. Fig. 4 Finite element meshed region around vertical borehole with stress contours

ð6Þ

Qt ¼ ct ðp f −pt Þ

ð7Þ

Qb ¼ cb ðp f −pb Þ

ð8Þ

where Qt and Qb are top and bottom flow rates, respectively; Pf is cohesive element middle pressure. The terms Pt and Pb

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Fig. 5 The 3D and upper views of vertical well hydraulic fracture model and the fluid injection pressure contours (KPa)

are top and bottom pressure of the cohesive elements, respectively; Ct and Cb are top and bottom coefficients.

continuity between adjacent porous medium fluid and the fracture fluid flow.

Governing equation of fracture fluid flow

  ∂w 1 þ ct ðp f −pt Þ þ cb ðp f −pb Þ ¼ ∇: w3 ∇:p f þ Qðt Þ:δðx; yÞ ∂t 12μ

The continuity equation of mass conservation is described as the governing equation of fluid flow into the fracture and the surrounding porous material (Peirce and Detournay 2008). The continuity equation of mass conservation is ∂w þ ∇:q þ ðqt þ qb Þ ¼ Qðt Þ:δðx; yÞ ∂t

ð9Þ

where q is the fluid flux of the tangential flow, w is the crack opening, Q(t) is the injection rate, qt and qb are the normal flow rates into the top and bottom surfaces of the cohesive elements respectively which reflect the leak-off through the fracture surfaces into the adjacent material (Sarris and Papanastasiou 2011). By combining Eqs. 5 to 9, the extracted Reynolds lubrication equation Eq. 10 can show the fluid conjunction and

ð10Þ

Reservoir hydraulic fracture model In this study, the reservoir top depth of the vertical well model is at 2100 m from surface. The meshed region around the vertical well with stress contours is shown in Fig. 4. Whole reservoir area is considered as a large semicircular with the radius of 250 m and the height of 150 m for the vertical well model (Fig. 4) and the vertical well hydraulic fracture model is symmetric about Y–Z plane. The whole model and the fluid injection pressure contours are shown in Fig. 5. The fracturing operation is started with flow rate of 10 bbl/ min; the fluid injection operation is maintained for 20 min. The formation and cohesive zone material are defined for the vertical well model in Table 1. Also, the in situ stress default values in a specific depth are presented in Table 2.

Table 1 The quantities of formation void ratio, permeability and the displacement peak value Table 2 Void ratio

0.2

d nf

(m)

0.005

Permeability (md)

1

The in situ stress default values

Depth (m)

σv (MPa)

σH (MPa)

σh (MPa)

2100

42

37.8

33.6

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Fig. 6 Fracture evolution along the cohesive element

Numerical results and discussions

The influences of fracture stiffness and formation tensile strength on hydraulic fracture opening

Formation rock mechanical properties such as fracture stiffness, tensile strength, Elastic or Young’s modulus, and Poisson’s ratio have significant influences on fracture opening in hydraulic fracturing operation. It is important to know that, those aforementioned parameters are indeed uncontrollable and are totally dependent to the formation and rock properties which arise from geological sedimentary conditions. Meantime, other parameters such as fluid viscosity and leak-off coefficients play important roles in designing a hydraulic fracturing fluid which can simply change the hydraulic fracture opening. By knowing the influences of the above addressed parameters on opening, the capability of proppant transport and the probability of bridging and eventually any job failure can be somehow predicted. Figure 6 shows the fracture opening as well as the evolution along the cohesive elements in two views of reservoir domain. Table 3 opening

The results of fracture stiffness changes versus fracture

As can be seen from Tables 3 and 4, the fracture stiffness and formation tensile strength have inversely influenced the fracture opening. It is clearly observed from Table 3 that while fracture stiffness is increasing, the fracture opening is decreasing; in other words, by increasing fracture stiffness from 1 GPa to 20 GPa, the fracture opening has been reduced from 11.48 to 8.83 mm. In addition, by increasing formation tensile strength, the fracture opening has been increased from 7.20 to 9.88 mm as shown in Table 4. In fact, after initiation of damage, higher fracture stiffness can significantly restrict the fracture opening which results in narrower fractures but the tensile strength has almost no resistance against the fracture opening after damage initiation within the reservoir rock. In other words, when the tensile strength is high, the extension (length) of the fracture would be reduced; so, in such Table 4 The results of formation tensile strength changes versus fracture opening

Fracture stiffness (GPa)

Fracture opening (mm)

Formation tensile strength (MPa)

Fracture opening (mm)

1 10 20

11.48 9.56 8.83

3 6 9

7.20 8.83 9.88

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Table 5 The results of formation elastic modulus changes versus fracture opening

Table 7 opening

Formation elastic modulus (GPa)

Fracture opening (mm)

Fracturing fluid viscosity (Pa.s)

Fracture opening (mm)

30 40 50

11.35 8.83 6.05

10−3 1 10

8.83 10.56 13.30

cases and in a constant injection rate, by increasing tensile strength of the rock, fluid forces between fracture plates would also be increased and it results in a wider hydraulic fracture width. From another point of view, by increasing the tensile strength in a constant compressive strength, the brittleness index will decrease which leads to fracture length decreasing and somehow hydraulic fracture width increasing under a constant injection rate. However, in case of natural discontinuities or unconsolidated formation the role of tensile strength would be significantly diminished. The influences of formation elastic modulus and Poisson’s ratio on hydraulic fracture opening In this section, the analysis of formation elastic modulus (Young’s modulus) and Poisson’s ratio effects on opening have been conducted. The results show that, by increasing formation elastic modulus from 30 to 50 GPa, the fracture opening has been steeply declined from 11.35 to 6.05 mm as can be seen at Table 5; but increasing Poisson’s ratio from 0.15 to 0.35, the fracture opening decreases from 9.82 to 6.61 mm (Table 6). In other words, from geomechanical point of view, rocks with higher elastic modulus (with more brittleness) can be fractured easier which means that elastic modulus dominates the lateral extent of a fracture and therefore will somehow restrict the fracture opening. Also, Poisson’s ratio indicates how much a rock that is expanded in one direction, contracts in the plane perpendicular to the expanding direction or how much a rock that is shortened in one direction expand in the other two directions (Fossen 2010). So, based on the definition for Poisson’s ratio, it can be deduced that any increase in this parameter would somehow affect the local stresses as well as the extent of strain levels and deformation near the fracture tip which can steeply influence the fracture width.

The results of fracture fluid viscosity changes versus fracture

The influences of fracture fluid viscosity and leak-off coefficient on hydraulic fracture opening Generally, increasing the fracturing fluid viscosity in injection operation can considerably enlarge the fracture width by further opening (Table 7). Based on Table 7, as viscosity is getting higher from 0.001 to 10 Pa.s, a meaningful increase in fracture opening from 8.83 to 13.30 mm can be easily observed. In fact, higher fluid viscosity leads to increasing net wellbore pressure that act on the fracture surface area which results in further opening of the fracture. In addition, by increasing leak-off coefficient from 5 × 10−11 to 5 × 10−9 m/ kPa.s, the fracture opening has been sharply reduced from 11.52 to 7.27 mm as can be seen at Table 8. Fluid leak-off will significantly reduce the required pressure acting on the fracture wall necessary for its propagation. In other words, increasing the fluid leak-off from the body of the fracture into the rock formation reduces the pressure which is acting against minimum principle stress; hence, it will lead to reduction of hydraulic fracture opening. However, in case of excessive leak-off (like reactive clay content formations), a tensile damage zone near the fracture surface would be created which can affect the hydraulic fracture opening.

Conclusion In this work, the problem of hydraulic fracturing has been investigated numerically through an XFEM-based cohesive law workflow for realistic modeling of fracture opening under different conditions of fracture stiffness, formation elastic modulus, formation Poisson’s ratio, tensile strength of the rock, and fracturing fluid viscosity and leak-off. Also, the model couples the fluid flow with fracture propagation while

Table 6 The results of formation Poisson’s ratio changes versus fracture opening

Table 8 opening

The results of leak-off coefficient changes versus fracture

Formation Poisson’s ratio

Fracture opening (mm)

leak-off coefficient (m/kPa.s)

Fracture opening (mm)

0.15 0.25 0.35

9.82 7.91 6.61

5 × 10−9 5 × 10−10 5 × 10−11

7.27 8.83 11.52

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damage initiation and evaluation criteria have been presented as well. It is clearly observed that mechanical properties of rock formation including formation elastic modulus, formation Poisson’s ratio, tensile strength of the rock would considerably affect hydraulic fracture opening directly or inversely. Results from the model suggest that increasing formation elastic modulus and Poisson’s ratio restrict the fracture opening while increasing the tensile strength of the rock would lead to a wider hydraulic fracture. Also, fracture stiffness decrease results in hydraulic fracture opening increase. From operational point of view the impact of fracturing fluid viscosity and leak-off have been illustrated whereas increasing the fluid viscosity would make a wider fracture while increasing the leakoff would inversely affect the hydraulic fracture opening. The results from this work can be applied in the analysis and optimization of hydraulic fracture opening to avoid any proppant bridging or job failure especially where formation modulus contrast is a challenge such as fracturing in multi-layer reservoirs or shale formations. Acknowledgments The authors would like to thank Fatemeh Ataiyan for her special attempts on properly preparing the illustrative pictures of hydraulic fracturing process, the tables and graphs of this research.

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