Analytical interpretation of hydraulic fracturing initiation pressure and breakdown pressure

Journal of Natural Gas Science and Engineering 76 (2020) 103185

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Analytical interpretation of hydraulic fracturing initiation pressure and breakdown pressure Feipeng Wu a, b, *, De Li a, b, Xianzhang Fan a, b, Jing Liu a, b, Xiaojun Li c a

Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum East China, Ministry of Education, Qingdao, 266580, PR China School of Petroleum Engineering,China University of Petroleum (East China), Qingdao, 266580, PR China c Dongxin Oil Production Plant, Shengli Oilfield, Dongying, Shandong, 257061, PR China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Hydraulic fracturing Fracturing initiation pressure Fracturing breakdown pressure Analytical models Experimental analysis

The rock collapse during Hydraulic Fracturing (HF) is the result of the accumulation of rock damage to a certain extent. That means there are two key pressure values: the fracture initiation pressure (FIP) and the breakdown pressure (FBP), which characterize the micro damage initiation and the macroscopic fracture stable expansion respectively. However, various conventional analytical breakdown pressure models cannot clearly distinguish and quantitatively characterize the difference between FIP and FBP. This paper presents a combined model. The fracture initiation is defined to start when the stress intensity factor at some point inside the rock reaches the fracture toughness. The breakdown is defined as the derivative of stress intensity factor versus radius is positive, not only as the onset of fracture initiation. According to the comparison and analysis of the actual fracturing experiments, this combined model makes it possible to accurate predict the FIP and the FBP respectively.

1. Introduction Hydraulic Fracturing (HF) has been extensively applied in the hy­ drocarbon extraction, geothermal, mining and other related industries applications to increase the permeability of the rock matrix by produc­ ing new fractures and/or stimulating pre-existing discontinuities (Wang, 2018; Richard et al., 2013; Hou et al., 2018). HF is a mechanical process where pressurized fluids are pumped into an isolated segment of a wellbore until a tensile fracture develops. Execution of hydraulic frac­ turing requires a proper understanding of the in-situ stress state, geo-mechanical properties of the formation as well as fracturing fluid dynamics (Huang, 1982; Schmitt and Zoback, 1992). Accurate predic­ tion of breakdown pressure is essential in designing a successful HF operation. Various theoretical models, numerical simulations and experimental investigations have been performed to reveal the mecha­ nism of breakdown process (Fatahi et al., 2016; Brice et al., 2018; Detournay and Carbonell, 1997; Warpinski et al., 1981.). However, the breakdown process is extremely complex, and it is widely accepted that the rock collapse during HF is the result of the accumulation of rock damage to a certain extent. This means that there exist two key pressure values, the fracture initiation pressure (FIP) and the breakdown pressure (FBP).

In field practice, Fracture pressures can be measured directly from LOTs (Leak off test) (Harris et al., 2001; Bishwas et al., 2018), XLOTs (Extended leak-off test) (Fjær et al., 2008; Lavrov et al., 2016), or other similar tests, e.g., minifrac test and diagnostic fracture injection test (DFITs) (Diagnostic Fracture Injection Test) (Cramer and Nguyen, 2013; Naidu and Rylance, 2017; Mohamed et al., 2019). As shown in Fig. 1, the FIP is conventionally considered as the first pressure inflection point where the pressure-ramp-up curve departs from linearity before for­ mation breakdown (Fjær et al., 2008; Feng et al., 2016). Continuing injection results in the bottomhole pressure reaching a peak, which is known as the formation breakdown pressure (FBP). Typically, after the peak point, the bottomhole pressure drops quickly down and stabilizes on a plateau while the fracture propagates. This steady bottomhole pressure equals to the fracture propagation pressure (FPP) (Zhang and Yin, 2017; Lavrov et al., 2016). Numerous factors influence the signature of an LOT, but only the effect of leakoff into the fractures is observably nonlinear (Fu, 2014). It is commonly accepted that the leakoff point should be identical to the FIP. However, the leakoff pressure may be very different from FIP in some certain cases (Ziegler and Jones, 2014). For a fluid injected con­ taining high solids content, the micro fracture hydraulically created may be quickly sealed by the solids. Therefore, the leakoff pressure perhaps

* Corresponding author. Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum East China, Ministry of Education, Qingdao, 266580, PR China. E-mail address: [email protected] (F. Wu). https://doi.org/10.1016/j.jngse.2020.103185 Received 23 August 2019; Received in revised form 4 January 2020; Accepted 26 January 2020 Available online 28 January 2020 1875-5100/© 2020 Elsevier B.V. All rights reserved.

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

occur immediately when the stress in the rock reaches the tensile strength. Instead, the rock starts strain-softening. When the strain soft­ ening zone develops to certain extent, the stress decreases to zero, and then a true crack is induced. Accordingly, the fracture pressure calcu­ lated using tensile strength model is more likely the pressure point where the strain-softening starts. However, one problem is that the fluid pressurization rate is usually high before breakdown. And the rock mechanics experience strongly indicates that a high pressurization rate leads to little strain softening. Therefore, employing Boone’s theory, Ito et al. (1998, 2008) proposed that the peak tangential stress will move toward the formation once the strain softening occurs. And a real fracture could not occur until this peak stress moves to a certain distance. This critical distance was measured experimentally. It should be noted that their experiments did not take the peak pressure as the FBP. Instead, the FBP was determined by a sudden decrease in the constant pressure ascending rate. Therefore, the fracture pressure calculated using point stress model is more likely to be the FIP rather than FBP. The stress intensity factor-based approaches give the most promising explanations for hydraulic break-down of rock masses (Sampath et al., 2018; Zhang et al., 2017). According to the fracturing mechanics, the fracture starts propagation unstably when the stress intensity (KI) up to the value of rock fracture toughness (KIC). During the hydraulic frac­ turing, along with the fluid pressure increasing, the stress distribution around wellbore will change dramatically. A micro fracture occurs and starts propagation unstably once the stress intensity factor at a point away from the wellbore is satisfied KI � KIC. However, the stress near the micro fracture will redistribution after the fracture been generated suddenly, which results in the decrease of KI. Correspondingly, the micro fracture propagation will cease until the stress intensity factor reaches the fracture toughness again. Consequently, the fracture un­ stable propagation (break down) only occurs when the both conditions are satisfied simultaneously. Not only KI � KIC is satisfied, and mean­ while the KI should increase monotonically. Correspondingly, the fracture pressure determined by shear strength model should also be the FIP. Moreover, it may be more reasonable calculated using shear strength model, because the micro cracks were mostly caused by shear failure before fracture initiation according to the experimental results (Lou et al., 2017). Meanwhile, the energy release rate-based approach and stress intensity factor based approach often provide more approximate FBP, which are in line with most of the experimental results pressure (Sampath et al., 2018). Accordingly, these theoretical models fail to predict or distinguish the variations of FIP and FBP, which have been observed in laboratory and in-situ experiments carried out under various conditions. In order to obtain the comprehensive understanding of this complex physical pro­ cess in HF, this paper proposes a combined analytical approach on the

be increased to a higher value than the ideal case in which no fracture exists (Feng et al., 2016). A slope-change point may be undetectable before formation breakdown, or if detected, it may indicate filter-cake breakdown rather than fracture initiation. Additionally, in reality, borehole walls are rarely intact perfectly. The orientation and length of pre-existing fractures (drilling-induced, natural, intersected during drilling) can also affect the results of LOTs, XLOTs and DFIT (Lavrov et al., 2016). The laboratory experiments are also widely employed to study the mechanism of fracture initiation and propagation. To illustrate the concepts of fracture initiation and formation breakdown, the ascending segment of the pressure-time curve is often replotted in the form of dP/dt as a function of P (Song et al., 2001; Guo et al., 2014). Besides, the real-time acoustic emission (AE) monitoring method is also effectively to determine the fracture initiation started (Bunger et al., 2015; Lou et al., 2017). It shows that micro seismic events take place prior to peak pressure, demonstrated by the HF experimental results on crystalline rock (Bunger et al., 2015), permeable sandstone (Stanchits et al., 2014; Lou et al., 2017) as well as granite with little permeability (Lu et al., 2015). Additionally. Lou et al. (2017). The results indicate that under the high confining pressure, micro cracks were mostly generated by shear failure firstly. Subsequently, the fluid penetrates into the micro cracks lead to the tensile failure of the macro fractures. Compared to the field tests, the laboratory experiments are more effective because of the unlimited number of quantities that can be measured. According to the experimental results, the difference between FIP and FBP may be very slight or very obvious. The magnitude of this difference is a multifac­ torial problem, which depends on in-situ stresses, rock mass and frac­ turing fluid properties, wellbore diameters and orientations, pressurization rate and so on (Haimson and Zhao, 1991; Schmitt and Zoback, 1992; Ito and Hayashi, 1991; Song et al., 2001; Guo et al., 2014; Lou et al., 2017). Many theoretical models were developed to predict the breakdown pressure in HF process. These models include the linear elastic model (Hubbert and Willis, 1957), the poro-elastic model (Haimson and Fair­ hurst, 1967), the linear elastic fracture mechanics model (Rummel, 1987), the point stress model (Ito and Hayashi, 1991) and the shear failure based model (Morgenstern, 1962). All these models are derived based on different assumptions involving rock permeability (permeable, impermeable), location of crack initiation point (at wellbore surface, inside the rock), crack initiation criteria (rock tensile strength, stress intensity factor, energy release rate balance, shear strength, etc.). The tensile strength based models assume that the micro fracture initiated from the wellbore wall, once the circumferential effective stress on the borehole wall (HW model, HF model) reaches the rock tensile strength. However, referring to the research conclusions of Boone and Ingraffea (1989), a true crack (no capacity to carry the load) does not

Fig. 1. Schematic pressure-volume/time curves in typical LOTs (Feng et al., 2016). (a) No visible difference between the FIP and formation FBP. (b) A clear FIP point before formation breakdown. (c) Multiple obvious FIP points before formation breakdown. 2

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basis of tensile strength and fracture mechanics theory. This new crite­ rion considers the changes in pore pressure distribution beyond the wellbore wall, as well as the third circumferential stress component caused by this pore pressure distribution. Besides, this new model as­ sumes that the fracture initiation starts when the stress intensity factor at some point inside the rock nearing the wellbore reaches the fracture toughness. As the complete breakdown occurs only when the two con­ ditions are satisfied simultaneously at a certain point inside the rock:(1) the stress intensity factor of this point is greater or equal to the fracture toughness of the rock; (2) at the same time, the derivative of stress in­ tensity factor to fracture length at this point is positive. Furthermore, several hydraulic fracturing experiments were carried out. Then the analysis and the discussion are present, according to the comprehensive comparison of the critical pressures between the values predicted by various models and values obtained from the experimental results.

pressure increases with constant pressurization rate (C), the additional stress component (σ p) and the pore pressure distribution (p(r,t)) can be obtained using following expressions (Haimson, 1968; Haimson and Fairhurst, 1967). � Z r � 1 σp ¼ A 2 pρ d ρ p (3) r a Z

where, r is the horizontal radial distance from the center of the hole; t is the calculation time, ρ is the fracturing fluid density, ν is the rock Poisson’s ratio, KB and KM is the rock frame and the rock matrix modulus, A is the parameter that can be calculated by equation (5), f(r,t) can be calculated by equation (5)(Ito, 2008; Zhang et al., 2017). � � 1 2ν KB 1 (5) A¼ 1 ν KM

The general in-situ stresses (sometimes in-situ principal stresses) in subsurface can be defined in terms of three orthogonal stresses: over­ burden stress σv along the vertical direction, horizontal stresses σ H and σh along two mutually orthogonal horizontal directions (Hossain et al., 2000). The three in-situ stresses in the earth are generally unequal and functions of various parameters including depth, lithology, pore pres­ sure, structure and tectonic setting. When a vertical well is drilled into the formation, the rock mass is replaced with a density controlling fluid exerting a specified pressure on the wellbore wall. The stress distribution around the wellbore can be altered locally, resulting in large stress concentrations (Fig. 2). If it is assumed that the rock formation remains linear elastic and imperme­ able, equations (1) and (2) can be obtained to calculate the circumfer­ ential stress distribution around the wellbore. σinsitu expresses the stress component generated due to the two horizontal principle stress, σpw expresses the stress component induced due to the injection pressure at the borehole bottom pw. � � � � σ þ σh a2 σH σh a4 σinsitu ¼ H 1þ 2 1 þ 3 4 cosð2θÞ (1) 2 r 2 r a2 r

(4)

f ðr; sÞds 0

2. State of stress distribution around wellbore

σpw ¼ 2 pw

t

pðr; tÞ ¼ C

Z

∞

f ðr; tÞ ¼ 1 þ

exp

2

κu t

# Y0 ðurÞJ0 ðuaÞ du u J0 ðuaÞ2 þ Y0 ðuaÞ2

" � J0 Y0 ðuaÞ

0

(6)

where, J0 and Y0 are the Bessel functions of the first and second kind of order zero, κ is a parameter that can be calculated by equation (7). (7)

κ ¼ k=μnβ

where, k is the permeability of the rock, μ is the viscosity of the fluid, n is the porosity of the rock, β is the compressive coefficient of the fluid. Therefore, the complete distribution of the circumferential total stress around the wellbore can be calculated by superposing three different stress components (Fig. 3). According to Terzaghi effective stress rule (Wang, 1982), the effec­ tive circumferential stress can be expressed by Eq. (8).

σ insitu þ σ pw þ σp þ αp ¼ σ θ

(8)

3. Analytical approach for the FIP and FBP of vertical fracture Table 1 presents various models derived based on different as­ sumptions. The basis, evolution as well as limitations of the existing theoretical break-down criterions are comprehensively discussed by Sampath et al. (2018). This study divides the typical breakdown criteria into three categories, according to the location of the crack initiation determining point. For isotropic porous rock formation, two classical criteria are applied to predict the FBP at two extreme cases. If the fracturing fluid does not penetrate the rock through the wellbore wall, the H–W model can solve the FBP via:

(2)

However, the rock formation can be assumed to be a poroelastic system, which has a linear elastic solid skeleton with fluid inside the pores. During the HF process, fracturing fluid penetrates the formation with a high flow rate, leading the dramatic changes in pore pressure around the wellbore. Accordingly, an additional circumferential total stress component (σp) is induced due to the pore pressure distribution (p (r,t)) through the rock mass. If it assumes that the wellbore bottom

σ H p0 ¼ pf (9) If the fracturing fluid completely penetrates the rock around the wellbore, the H–F model is used to calculate the FBP.

σ t þ 3σ h

3σ h

σH 2ð1

2ηp0

ηÞ

¼ pf

(10)

where, pf is the FBP, σt is the tensile strength of the rock, η is an elas­ toplastic parameter defined by equation (11).

η¼

αð1

α¼1

1

2νÞ

ν Cr Cb

0 � η � 0:5

(11) (12)

where,α is Biot pore elasticity coefficient, Cr is rock matrix compression coefficient, Cb is rock bulk compression coefficient.

Fig. 2. Stress concentration around a hole in the absence of wellbore pressure (Thiercelin and Roegiers, 2004). 3

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

Fig. 3. Stress distribution components around the wellbore (The borehole radius is a, the wellbore pressure is pw, the rock pore pressure is p, the rock far-field pore pressure is p0, the far-field maximum horizontal principal stress is σ H, and the far-field minimum horizontal principal stress is σ h. Take the borehole as the center and starting point, and make X-axis along the direction of maximum principal stress, a cylindrical coordinate system is introduced. Fig. 3-(b) shows the circum­ ferential wellbore stress σinsitu due to two horizontal principal stresses; Fig. 3-(c) shows the circumferential wellbore stress σpw due to borehole pressure pw; Fig. 3-(d) shows additional circumferential stress σ p due to pore pressure change. Table 1 The typical breakdown pressure models and the corresponding assumptions. (1) Determining position on the borehole walls Breakdown pressure model

Criteria formula

Tensile strengthbased models

pf ¼ 3σh

H–W model

σH

αp þ σt

H–F model Point stress model Shear failure-based models

σH 2ηp þ σt 2ð1 2ηÞ � σθ ðrw þ dc ; θ ¼ 0 ; pw ¼ pf Þ � �2 1 KIC ¼ σt dc ¼

Fracture mechanic-based models

KI ðrw þ a; σθ ; pa ; pf Þ ¼ KIC

Energy release rate-based approaches

dUγ dUS � dc dc

pf ¼

3σ h

π σt τ ¼ c þ σn tanϕσn ¼ Sn ð1 αa Þp

Rock property hypothesis

Crack criterion

References

intact and impermeable

When the circumferential effective stress on the wall of wellbore reaches the rock tensile strength.

Hubbert and Willis (1957); Bredehoeft et al. (1976); Zhang et al. (2018). Haimson and Fairhurst (1967); Schmitt and Zoback, 1989. Ito and Hayashi (1991); Ito (2008).

intact and permeable

When the circumferential effective stress at a point being constant distance to the wall of wellbore reaches the rock tensile strength. When the effective stress circle first touches the failure envelope (Mohr-Coulomb) in the wall. When the stress intensity for mode I fracture propagation up to the fracture toughness of the rock. The released strain energy of the initial crack propagation should be greater or equal to the required surface energy.

Intact, highly permeable Intact except an initial symmetrical double micro cracks extending from borehole.

Both the conventional breakdown models assume that the break­ down of the rock takes place on the walls of the pressurized borehole, when the circumferential effective stress in the wellbore wall reaches the tensile strength of the rock (Hubbert and Willis, 1957; Haimson and Fairhurst, 1967). These two classical models have been widely used, but the effects of permeability, pressurization rate and fluid viscosity is ignored. However, a large number of experiments indicate that the large wellbore size, high fluid viscosity and low pressurization rate can significantly reduce the FBP (Ito and Hayashi, 1991).

Morgenstern (1962). Hardy (1973); Van Eekelen (1982); Abou-Sayed (1978); Rummel (1987); Zhang (2017). Griffith (1921).

(2) Determining position at the point of critical distance to the wellbore wall Ito and Hayashi (1991) and Ito (2008) introduced a point stress model based on the theory of critical distance. This model assumes the breakdown occurs at wellbore surface, when the maximum effective tensile stress reaches the tensile strength of the rock at a critical distance into the rock. The distance between the point and the well wall is a material constant. It is identified as characteristic length (dc) of tensile 4

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

failure of rock (see Eq. (13)). The expression can be shown in equation (14) (Taylor, 2007). � �2 1 KIC (13) dc ¼

π

σf

where, σ f is the tensile strength of rock, KIC is the type I fracture toughness. � σ θ ¼ σ f r ¼ r w þ d c ; pw ¼ pf ; θ ¼ 0 (14) (3) Modified fracture mechanics based criterion for FIP (Determining position at a certain point inside the rock) Fig. 4. Stress intensity factor-based model for growth of a micro-crack. (a) There is a constant far field force perpendicular to the crack surface. (b) There exist stresses symmetrically distributed on the x, y axis perpendicular to the crack surface.

In the strict sense, the point stress model proposed by ITO should belong to the fracture mechanics based models rather than a tensile strength-based model. The Theory of Critical Distances (TCD) is a wellknown design method allowing the strength of notched/cracked com­ ponents to be estimated accurately by directly post-processing the entire linear-elastic stress fields damaging the material in the vicinity of the stress concentrators being designed (Iason and Luca, 2016). The classical fracture mechanics theory holds that a micro-crack with a length of 2d locates within an infinite plate. A constant far field force σ apply perpendicular to the crack surface. Then this micro crack will extent unstably when the far field stress can satisfy the condition of Eq. (15) K

where KI ðdÞ is the stress intensity factor at the tip of the crack with length d, σθ ðrÞ is the effective circumferential stress calculated by equation (8), rw is the wellbore radius. As is shown in equation (18), when the type I stress intensity factor at the feature point reaches or above the fracture toughness of the wellbore rock, the wellbore formation begins to crack (Detournay and Cheng, 1992). KI ðdÞ � KIC

(15)

ffiffiffiffiffiffi σ ¼ pffiffiIC 2πd

It can be seen that the assumed length of pre-existing micro crack, also the location of the fracture initiation determining point, is controlled by the distribution of effective stress around the wellbore. The determining point is an uncertain random point rather than a spe­ cific fixed position.

From equation (15), the following relation between fracture tough­ ness and characteristic length is obtained. � �2 1 KIC ​ dc ¼ 2d ¼ (16)

π

(18)

σ

(3) Modified fracture mechanics based criterion for FBP

In point stress model, it is assumed that there are two micro cracks of length d symmetrically around the wellbore along the direction of maximum principal stress. Due to the very short crack length, the circumferential tensile stress of the well can be considered as the far field stress. Therefore, the critical length can be determined using equation (13), when the maximum circumferential tensile stress around the wellbore reaches the tensile strength of the rock. Accordingly, the essence of point stress model is a theoretical model based on fracture mechanics. It is similar to the poroelastic model based on the assumption of crack initiation on the wall of wellbore. However, the formation rock around wellbore can be considered as an engineering structure containing cracks. Due to fluid permeation around the borehole, crack initiation is affected by in-situ stress, fluid pressure in the well and pore pressure. During the fracturing process, crack propagation has three steps. Firstly, the crack is almost closed in the initial state. The pore pressure increases as the fracturing fluid penetrates and changes the effective stress around the borehole. Sec­ ondly, the crack is opened and a macroscopic crack is formed. Then, fracturing fluids easily flow into these macroscopic fractures (Sampath et al., 2018). Therefore, during the fracture initiation process, the effective circumferential stress around the wellbore is constantly changing with the radial distance to the wellbore wall. Accordingly, it is more suitable to adopt the assumptions shown in Fig. 4(b). A set of stresses symmetrically distributed over the x and y axis is assumed to apply perpendicularly on the crack face. Thus, according to the classical fracture mechanics theory, the stress intensity factor KI is obtained by equation (17) (Wang, 1982). 2 33 2 rffiffiffiffiffi Z r þd � �12 � �12 w 1 d r 77 6 6 dþr (17) KI ðdÞ ¼ þ 4σθ ðrÞ4 55dr d r dþr π d rw

Although, from the point of view of fracture mechanics theory, a tensile fracture will start to extend when the stress-intensity factor KI reaches fracture toughness KIC. This is the situation for the typical opening mode I under a uniform tensile stress perpendicular to the crack (as Fig. 4(a)). Actually, the total effective stress around wellbore is a complex and multifactorial physical quantity. It changes dramatically with the radial distance, especially in the area extremely near to the wellbore (as shown in Fig. 4(b)). Therefore, the crack expansion can be either stable or unstable, which depends on whether the slight extension of the crack results in a decrease or increase in the stress intensity factor. Accordingly, the stress intensity factor can be introduced to determine the crack initiation stability with respect to the derivative of the radius r. The breakdown criterion can be described using equation (19). It ex­ presses that the complete breakdown is corresponding to the onset of unstable fracture propagation. That is the crack will not propagate continuously, until these two conditions are met together. They are: (1) the stress intensity factor KI(r) at a certain point reaches the fracture toughness KIC; (2) at the same time, the derivative of KI(r) versus radius r is positive. 8 < KI ðrÞ � KIC � (19) p ¼ pf ; θ ¼ 0 : ∂KI ðrÞ > 0 w ∂r Overall, a new model is obtained by combining equation (17)(18) (19). The FIP can be determined by equation (17) and equation (8), while the FBP can be calculated using equation (19).

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

4. Experimental validation and theoretical discussion

the pressure difference if the permeability stays constant. As outflow increases while the inflow is constant, a gradual decline of wellbore pressurization increment presents due to the loss of borehole fluid, which linearly increases with the wellbore pressure. Eventually the in­ jection fluid flow and outgoing flow will be in equilibrium, and then dP/ dt approaches 0 (Song et al., 2001). At the beginning of the third stage, the pressurization rate curve deviates from the linear approximation to a faster drop down trend. That is the point where the total stress near the wellbore starts damage the rock. This will result in the suddenly increasing of permeability near the borehole due to the micro tensile cracks starting initiation and expansion. However, the borehole pressure is still ascending gradually, which indicates that the crack is constantly evolving at this stage. It has not yet reached the level of rapid collapse. After the peak pressure point, the borehole pressure drops down to an extremely low level and stays stable. It is believed that one or multiple macroscopic through fractures have been generated at the peak pressure point. The stable low pressure is mainly because of the friction of fluid flowing through the cracks. Therefore, the FIP can be determined with the deviation point in the pressurization rate curve, while the FBP can be obtained as the peak pressure in the borehole pressure – time curve.

4.1. Physical simulation experiment of HF The breakdown criterion has been tested by experimental methods. The results show that the FBP increases with pressurization rate C. The results approximate this new combined model. In the experiment, the rock samples were prepared with fine quartz sand with a particle diameter 0.253 mm, coarse quartz sand with diameter 0.97 mm and cement. In order to meet the experimental re­ quirements, the rock specimen were determined to be a thick walled, hollow cylinders with a hole diameter of 10 mm, an external diameter of 100 mm and a length of 45 mm. The permeability and porosity of the rock sample are controlled by adjusting the ratio of the two types of quartz sand. The cement used in the experiment was Zhonglian Cement 32.5 ordinary Portland cement. The specific parameters of the rock sample are shown in Table 2. The experimental pressurized fluid is water. The schematic of experimental setup is shown in Fig. 5. The rock samples were settled in a triaxial pressure chamber and subjected to predetermined uniform horizontal stress confining, vertical stress. The peripheral seepage return pressure system is used to set the pore pres­ sure outside the core. The outer surface of the core is evenly cut 18 axial grooves of 2 mm depth to form a radial flow loop of the rock core. A steel sleeve with axial notch and internal groove is set between the rock sample and the rubber jacked, which could help to set up different confining stress and the boundary fluid pressure outside the rock sample separately. Before the fracturing test starting, the rock sample is satu­ rated with pure water at very low flow rate, until the pore pressure reaches up to 5 MPa. During the tests, the fracturing fluid was injected into the borehole at a constant flow rate until the borehole pressure ceased to increase. A commercial software and a personal computer equipped with digital converters were employed. In this way, the entire borehole pressure changes were automated detected and recorded. Total fourteen experiments were carried out. Four of the experiments were jacketed tests, where the central borehole of the rock sample was jacketed by an impermeable thin copper tube. Which allows a pure insight to the effect of fluid pressurization rate on FIP and FBP without the fluid permeation. The other ten experiments were performed under unjacketed condition. In this case, the radial permeation occurs outward from the borehole. Due to the constant confining stress and fluid pres­ sure outside the rock cylinder, it is possible to approximate the actual buildup process of the pore pressure around the wellbore during frac­ turing. In this way, it is more reasonable to consider the effect of the additional stress component (σp) generated by the pore pressure redis­ tribution (P(r,t)). A typical hydraulic fluid pressure – time record curve of unjacketed test is shown in Fig. 6. The fracturing process can be divided into three stages obviously. During the first stage, the fracturing fluid starts to be injected into the borehole with a constant flow rate. As a result, the borehole pressure presents a pressurization process with an increasing loading rate. This is the pressure building up process within the pipe, borehole and the pore near the hole. When the borehole pressure closes to the original pore pressure, the ascending portion of the pressure – time curve is clearly shows deceleration trend. This is because the outflow from the borehole to the outer surface is linearly proportional to

4.2. Validation with experiments In order to validate the accuracy of this combined criterion, the FIP and the FBP is calculated according to various breakdown criterion under the same parameters. The comparison between experimental re­ sults and theoretical predictions is shown in Fig. 7. (1) Analysis of experimental results According to the test results, both FIP and FBP can be determined from each test. Both the FIP and FBP of unjacketed tests increase monotonically with the increasing fluid pressurization rate. This phe­ nomenon has already been tested and analyzed (Ito and Hayashi, 1991; Zhang et al., 2017). This is because the fracturing fluid leaks radially away from the wellbore. With a slow pressurization rate, there is more time for the fluid to fully penetrate the surrounding rock. And the easier permeation of fracturing fluid induces higher pore pressure and stress concentration at the crack tip. This will cause a larger increment of the stress intensity factor and make the micro cracks easier to initiate and propagate. On the contrary, the higher pressurization rate reduces the penetration time, resulting in lower pore pressure and smaller seepage depth under the same injection pressure. The lower pore pressure and smaller seepage area cause smaller stress intensity factor which results in the higher FIP and PBP. This mechanism could also be validated by the jacketed tests. Due to the thin copper tube within the borehole, the fracturing fluid cannot seepage into the rock. The pore pressure remains constant before macro fractures formed, just as fracturing an imper­ meable reservoir. Therefore, no matter how the pressurization rate changes, the FIP and FBP of the jacketed tests are similar and remains stable around the FBP value calculated with HW model。 The unjacketed tests also indicate that the difference between the FIP and FBP values decreases gradually. In addition, at the experimental conditions, there is almost no significant FIP point when the pressuri­ zation rate is up to 15 MPa/s。This can be explained using the fracturing fluid penetration during the fracturing process. After the micro crack been initiated and extended, the stress is released. As a result, the fracture propagation ceases. Therefore, the fluid leakage rate is much less than the injection rate. And the bottom pressure continues to rise rapidly until the fracture is reopened and extended forced by the fluid penetration within the fracture. In the case of a lower pressurization rate, it takes a longer time for the fluid to fully penetrate the crack and re-extend the crack. This process continues repeatedly up to the crack has been extended to a certain length and the stress intensity factor started increasing monotonically. Since then the fracture propagates

Table 2 Experimental parameter table. Parameter

Value

Parameter

KIC σt(MPa) Initial pore pressure(MPa) Wellbore radius(m) σH ¼ σh (MPa)

0.68 3.4 5 0.005 12.6

Young’s modulus(Mpa Poisson’s ratio Permeability(mD) Fluid viscosity(CP) Porosity

Value 1

)

5 � 103 0.32 2.1 78 0.15

6

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

Fig. 5. Schematic illustration of the hydro fracturing test device.

Fig. 6. Typical borehole pressure – time and pressurization rate – time test record.

unstably. Therefore, the difference between FIP and FBP is high. On the contrary, with a higher fluid injection rate, the fluid pressure increases quickly, which can easier to force the fluid penetration. As a result, the unstable propagation is more likely to occur immediately following the crack initiation. This would present a more slight difference between FIP and FBP.

combined model seems more likely to give satisfactory FIP. Besides, it is encouraging to note that the combined model can successfully predict the more accurate FBP, within all the hydro fracturing pressurization rate tested in this paper. 4.3. Validation with oilfield practice data

(2) Models validation with experimental results

Series methods are widely used to determine the FIP and FBP values in oilfield practice, including LOTs (Leak Off Tests), XLOTs (Extended Leak-off Test), DFITs (Diagnostic Fracture Injection Tests), FPIT(For­ mation Pressure Integrity Tests), WFT (Wireline Formation Tests) and so on. Naidu and Rylance (2017) present a new approach to determine the FIP and FBP. This method first calculates the bulk modulus K using equation (20) being directly related to the pressure P, the initial system volume V, the incremental injection volume ΔV and the volumetric strain ΔεV.

The FBP values calculated by the classical HW model and HF model become the upper and lower limits of other models. Because these two models are proposed under extreme cases: either impermeable or completely permeable. The jacketed tests can be taken as the fracturing of impermeable rock, therefore the FIP and FBP obtained from the jac­ keted tests is close to that calculated by the HW model. Meanwhile, the fracture pressure calculated using HF model is slightly lower than the FIP value of unjacketed tests at extremely low pressurization rate. The fracture pressure calculated using point stress model is more likely to be the FIP rather than FBP. In addition, the point stress model seems to be less sensitive to the pressurization rate. And when the pressurization rate is relatively slow, the both models have high accu­ racy to predict the FIP. As the pressurization rate increases, the

K¼

PV P ¼ ΔV ΔεV

(20)

Hence, the FIP is picked as the slop K occurring a marked reduction, and the FBP is picked at the peak pressure point. Applied this methodology, Naidu and Rylance provided nine 7

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

Fig. 7. Validation for the theoretical models using experimental data.

application examples, including one WFT operation, four drilling FPIT measurements and four DFIT injections. All the examples are taken from a suite of wells that have been drilled as part of the onshore, khazzan gas field development in the Sultanate of Oman. The stress conditions, pressurization rates, FIP and FBP values (picked, calculated using HW model, HF model, the new model proposed in this paper) are listed in Table 3 and Table 4. The detail description of the formations, pressure and modulus K curves, and rock characteristics refers to the literature (Naidu and Rylance, 2017; Rylance et al., 2007, 2011; Najwani et al., 2013). Fig. 8 indicates that both the FIP calculated and FBP calculated are reasonably accurate. The FIP calculated using new model and picked using slope K methods are both closer to the FIP values calculated using HW model. Meanwhile, the FBP calculated using new model and picked using slope K methods are right closer to the average value of the FIP values calculated using HW and HF model.

Table 4 The FIP and FBP data picked and calculated.

Table 3 The stress conditions, rock strength and pressurization rate for injection tests. Test name

Depth (m)

σv

σmax

σmin

(MPa)

Ppore (MPa)

σt

WFT1 FPIT1 FPIT2 FPIT3 FPIT4 DFIT1 DFIT2 DFIT3 DFIT4

1732

38

38

30

18

3

dp/dt (MPa/ min) 2

1110

24

24

18

12

3

3

1089

23

23

18

11

3

7

3687

88

88

65

42

14

0

3795

90

90

68

44

14

166

4528

109

109

77

52

8

34

4522

109

109

83

51

8

69

4519

109

109

82

51

8

69

4507

108

108

77

51

8

69

(MPa)

(MPa)

(MPa)

Test name

FIP (MPa)

FBP (MPa)

FIP calculated

Picked

Calculated (New model)

Picked

Calculated (New model)

HW model

HF model

WFT1 FPIT1 FPIT2 FPIT3 FPIT4 DFIT1 DFIT2 DFIT3 DFIT4

32

32

41

34

42

30

18

19

20

20

24

18

19

18

20

20

25

18

67

65

72

72

89

64

70

82

71

90

93

67

63

68

82

79

90

68

79

79

88

90

106

77

76

79

77

90

106

77

65

66

89

79

91

68

4.4. Theoretical discussion (1) The effect of fluid penetration on the effective stress distribution around the wellbore As mentioned above, the existence of pores brings special feature in mechanical behavior of formation rocks. The fluid penetration through the interconnected pores into a rock from borehole wall can cause an additional circumferential effective stress in compression around the borehole (Detournay and Cheng, 1992). We define the compressive stress as negative and tensile stress as positive, respectively. Using the stress calculation model given above (Chapter 2) and the basic data in Table 2, the pore pressure (p) and the stress distribution around the wellbore along the pressurization time were calculated (Fig. 9(a)). The original pore pressure is 5 MPa. Since the borehole pressurization starts (t ¼ 0s) with constant pres­ surization rate C ¼ 1 MPa/s, the borehole pressure raises and generates a stress component σ pw (Fig. 9(c)). Besides, the ascending of the borehole 8

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

compressive stress near the wellbore is gradually eliminated and tends to reverse to tensile stress. Moreover, due to the additional poro-elastic stress, the maximum tensile stress tends to occur at a point not far from the wellbore wall rather than on it (Fig. 9(d)). Accordingly, it is more reasonable to assume the determining point of fracturing initiation is within the formation rock rather than on the borehole, since the fluid penetration to the formation rock is inevitable during HF process. (2) The effective stress distribution at fracturing initiation deter­ mined by different models The radial distribution of stress and pore pressure were estimated when the borehole pressure reached the FIP. These critical pressure points were determined with both the point stress model and combined model respectively. As shown in Fig. 10 and Fig. 11, total five different pressurization rates were assumed to study the models rate sensitivities. As shown in Fig. 10 (a), at the FIP point, all the distribution curves of the total effective stress around the wellbore intersect at a point. The radial distance from this point to borehole wall is the characteristic distance (dc ¼ 3.17 mm, calculated by equation (13) with the basic data listed in Table 2). This is because the point stress model assumes that the fracturing initiation occurs when the maximum effective stress reaches the rock tensile strength at a point being critical distance away from the borehole wall (Ito T., 2008). When the pressurization rate is relatively high, the total effective stress value decreases monotonically with increasing radial distance. By contrast, at a moderate pressurization rate, the total effective stress increases in a very short distance and then decreases gradually. The maximum tensile stress point appears after the characteristic distance point. Before the effective stress been reached the rock tensile strength at the characteristic distance point, the tensile strength may have been reached or exceeded somewhere along the radial direction away from this point. As the combined model (Fig. 10 (b)), the whole change regulation of effective stress around wells is consistent with the results predicted with point stress model. However, the difference of effective stress is more divergent when fracture initiation occurs at different pressurization rates. The stress curves are not intersecting at a certain point. Particu­ larly, with lower pressurization rate (e.g. 0.05 MPa/s), the effective stress in a very small range near the borehole wall is still negative (characterized as compressive stress), while the fracturing initiation has been determined to start. This is because the macroscopic fracturing failure of formation rock is a cumulative process of microscopic damage. As shown in Fig. 8, due to the fluid penetration into the porous rock, the total effective stress will redistributes in a more complicated manner. And the maximum tensile stress is probably to appear inside the rock, rather than on the borehole wall. This means that minor damage may have been occurred inside the rock, while the borehole wall is still in the state of compressive stress. Once these damages occurred, the perme­ ability around the wellbore is significantly improved, although there is no macro fracture propagation yet. Then immediately, the high-pressure fluid will instantaneously leak into the small cracks more easily, resulting in an inflection point deviating from the straight line on the pressure-time curve. Nonetheless, due to the small fracture length and the insufficient effective tensile stress, the fracture initiated will extent and arrest alternatively. As a result, the fracture volume increases with a rate lower than the fracturing fluid penetration rate. Therefore, the borehole pressure continues to rise until the peak FBP reached. During this stage, the fracture length expands intermittently and effective ten­ sile stress around borehole increases continuously. In addition, the “wedge effect” will appear due to the fluid penetrating into the opening cracks, which increases the stress intensity factor of cracks and promotes the continuous expansion of cracks. After a very short time period, the criterion of equation (19) will be satisfied, and the fracture volume ex­ pands at a much greater rate than the fluid injection rate. As a result, the wellbore experiences a sudden pressure drop and will stabilize to the fracture propagation pressure. This peek pressure is determined as the

Fig. 8. Comparison between FIP, FBP picked and the calculated values using new model, HW and HF model. (a) FIP; (b)FBP.

pressure also develops the pore pressure increasing due to the fluid penetration into the interconnected pores. This pore pressure distribu­ tion changes drastically over the injection time and the distance to the borehole wall. As a result, the additional poro-elastic stress caused by the pore pressure are appears in the vicinity of the borehole at early time, and they propagate outward gradually with time (Fig. 9(b)). It’s worth noting that the borehole pressure can develop tensile stress around the wellbore by compressing the wall of the wellbore. Conversely, the pore pressure raising around the wellbore can generate an additional compression stress very near the wellbore. This compression stress will decrease gradually to an external tensile stress far from the wellbore. That is the pore pressure affects to reduce the FBP, and conversely, the poro-elastic stress affects to raise the FBP (Ito T. 2008). As the total effective stress, there exist a strong stress concentration at the near-wellbore before the fluid pressurization starting. And along the borehole pressure increasing linearly, the concentration of 9

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

Fig. 9. The stress distribution around the wellbore along the pressurization time.

Fig. 10. Total effective stress distribution at fracturing initiation.

FBP. The pore pressure and the poro-elastic stress distribution are pre­ dicted by these two models. The results is very consistent at FIP point.

However, the sensitivity of the combined model to pressurization rate is obviously higher than that of the point stress model. The pore pressure presents a monotonic decline trend along the 10

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

Fig. 11. Pore pressure and poro-elastic stress distribution at fracturing initiation.

radial direction (r-rw). The pore pressure distribution versus radial dis­ tance shows a gradually variation trend from moderately to dramati­ cally. It can be interpreted with the time of fracturing initiation. The time from pump injection to fracture initiation is very short at high pressurization, which is not sufficient for the fluid to seepage deep into the formation. So that it will appear a stronger pore pressure gradient in the near wellbore zone. Due to the penetration of high pressure fracturing fluid, the pore pressure near the wellbore rises to an extremely high level rapidly. Correspondingly, the additional compressive poro-elastic stress will be produced, which decreases rapidly and gradually converts to tensile stress along the radial direction away from the borehole. Therefore, the existence of this additional stress component affects to raise the FBP. Besides, the poro-elastic stress effect will be more significant with a higher pressurization. This is mainly because the pore pressure as well as the pressure gradient near the wellbore are extremely strong. It will generate a large additional compression stress also with dramatically gradient. Thus, it is quite probably that the borehole pressure has raised to a an extremely high level, but the total effective stress near the wellbore still is not sufficient to fail the rock due to this strong poroelastic compressive stress. Accordingly, this may be the key reason for the high FIP with high pressurization rate. Comparing Fig. 11(a) and (b), it is obvious that the combined model is much more sensitive than the point stress model to the pressurization rate. This can be elegantly explained by the stress intensity factor, which is employed as the fracturing initiation criterion by combined model. The stress intensity factor is a function of fracture size and the loading conditions perpendicular to the crack surface. Due to the strong addi­ tional compressive stress component produced by dramatically pore pressure, it will need much larger σ pw to counterbalance. Differently, the point stress model criterion only takes concern to the effective stress at the critical distance point. It may be already converted to reach or exceed the rock tensile strength before the high pressure fluid pene­ trating too deep. For example, with the pressurization rate C ¼ 55 MPa/ s, the fracture initiation was successfully started only after less than half a second since the fracturing pump began to work.

fracture is initiated from the wellbore wall, the criteria are related to the stress condition inside the rock. Therefore, it is meaningful to study the distance of fracturing initiation determining point predicted. According to the assumption of point stress model, the fracture initiation point is constant to be the critical distance, which has no relationship with the pressurization. Differently, for the combined model, the radial distance from fracturing initiation point to borehole wall decreases mono­ tonically with increasing the borehole pressurization rate. Fig. 12 presents three distance changes versus pressurization rate, including the critical distance calculated using point stress model, the distance from the fracturing initiation point and the breakdown point to borehole wall separately determined using the combined model pro­ posed in this paper. As shown in Fig. 12, the fracturing initiation points predicted by combined model are all beyond the critical point. Both the fracturing initiation and the breakdown determining points predicted by combined model are very sensitive to the pressurization rate. When the pressuri­ zation rate is large enough, the two points almost coincide (the critical value is above 15 MPa/s determined with parameters listed in this paper.). This indicates that the fluid pressure increases quickly with higher fluid injection rate, which can easier to force the fluid penetra­ tion. As a result, the unstable propagation is more likely to occur immediately after crack initiation. Therefore, it will be possible that the two conditions are given in equation (19) will be satisfied

(3) The distance of fracturing initiation determining point predicted by different models Fig. 12. The distance from the fracturing initiation point to borehole wall versus pressurization rate.

Although the combined model and the point stress model assume the 11

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Journal of Natural Gas Science and Engineering 76 (2020) 103185

simultaneously. As a result, these two pressure points tend to be coin­ cident at high pressurization rates. In such conditions, it is difficult to identify the obvious pressure deviation point on the pressure curve during an actual HF process, and it will linear monotonic increase to the peak value and then drop down directly.

Mechanism of Blasting Induced Hydraulic Volume Fracturing on the Deep Tight Reservoir with High In-situ Stress Difference” (Project Number: 17CX05004)).

5. Conclusion

Supplementary data to this article can be found online at https://doi. org/10.1016/j.jngse.2020.103185.

Appendix A. Supplementary data

FIP and FBP are important considerations for high-pressure water injection, preventing and mitigating lost circulation, HF operation design and so on. Currently, a number of theoretical models have been developed based on different approaches to predict the break-down pressure, while it is not yet possible to accurately predict FIP and FBP respectively. Based on the effective stress distribution analysis around the wellbore, this paper comprehensively reviews several existing breakdown pressure models. Basically, a combined new model is pro­ posed employing the point stress model and the fracture mechanics model, which makes it possible to identify the both important pressure points accurately. Furthermore, a series of hydraulic fracturing experi­ ments are carried out. After that the analysis and the discussion are present, according to the comprehensive comparison of the key pres­ sures between the values calculated and values tested. The main con­ clusions of this study are listed as follows:

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(1) The effect of fluid penetration can generate an additional compression stress very near the wellbore. This poro-elastic stress affects to raise and the sensitivity of FIP and FBP to pressurization rate. Moreover, due to the poro-elastic stress, the maximum tensile stress tends to occur at a point not far to the wellbore wall. (2) The point stress model provides an approach to consider the ef­ fects of permeability, pressurization rate and fluid viscosity. However, the results predicted is closer to FIP values and seems to be less progressively comparing to the experimental results with the increase of pressurization rate. (3) It is encouraging to see that predicted values of FIP and FBP using combined model align very well with experimental results. Accordingly, it is reasonable to define the FIP and FBP using the fracture criterion for stable or unstable propagation of a hypo­ thetical radial crack with uncertain length. (4) Both the calculation and the experimental results indicate that the FIP and FBP are very sensitive to and increases gradually with an increase in pressurization rate. Besides, the difference between the both pressure values decreases dramatically with the increasing pressurization rate. And there is almost no significant crack FIP point when the pressurization rate is up to about 15 MPa/s according to the calculation and tests in this paper。 (5) Both the fracturing initiation and the breakdown determining points predicted by combined model are very sensitive to and decreases dramatically with the pressurization rate. Both the determining points tend to be coincident at high pressurization rates. Declaration of competing interest The authors declare no conflict of interest. Acknowledgments This study was supported by the Natural Science Foundation of China (“Mesoscopic Damage mechanisms and Its Control on Macro Failure in Saturated Brittle Rock during Deflagration Fracturing Process” (Project Number: 51874339)), and the Natural Science Foundation of Shandong (“The Dynamic Mechanism Study Coupling the Rock Damage and Crack Propagation under the High Intensity Pulse of Multi-stage Blasting Fracturing” (Project Number: ZR2016EEQ04)), and the Fundamental Research Funds for the Central Universities (“The 12

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