A new analytical model of wellbore strengthening for fracture network loss of drilling fluid considering fracture roughness

Journal Pre-proof A new analytical model of wellbore strengthening for fracture network loss of drilling fluid considering fracture roughness Lisong Zhang, Zhiyuan Wang, Kai Du, Bo Xiao, Wang Chen PII:

S1875-5100(19)30345-2

DOI:

https://doi.org/10.1016/j.jngse.2019.103093

Reference:

JNGSE 103093

To appear in:

Journal of Natural Gas Science and Engineering

Received Date: 12 April 2019 Revised Date:

26 November 2019

Accepted Date: 27 November 2019

Please cite this article as: Zhang, L., Wang, Z., Du, K., Xiao, B., Chen, W., A new analytical model of wellbore strengthening for fracture network loss of drilling fluid considering fracture roughness, Journal of Natural Gas Science & Engineering, https://doi.org/10.1016/j.jngse.2019.103093. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier B.V.

Author Contribution Statement Lisong Zhang: Methodology, Writing - Original draft preparation. Zhiyuan Wang: Conceptualization, Supervision, Funding acquisition. Kai Du: Software, Data curation. Bo Xiao: Software. Wang Chen: Validation.

A new analytical model of wellbore strengthening for fracture network loss of drilling fluid considering fracture roughness Lisong Zhanga, Zhiyuan Wangb,*, Kai Dub, Bo Xiaob, Wang Chenb1 a

College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, China

b

School of Petroleum Engineering, China University of Petroleum, Qingdao 266580, China

Abstract: Currently, the wellbore strengthening mechanism has not been fully understood, especially for the fracture network loss of the drilling fluid. In view of this, a new analytical model was established to discover the wellbore strengthening mechanism, considering the fracture network loss of the drilling fluid. In this model, the additional stress induced by the natural fracture network was calculated based on the fracture mechanics, and the fluid pressure distribution within the fracture network during the drilling fluid loss was investigated considering the fracture roughness. Especially, the wellbore strengthening mechanism was revealed, by comparing the stress intensity factors before and after the wellbore strengthening. The proposed analytical model was verified by comparing its result to the one of the numerical model. The comparison was focused on four aspects, namely, the fracture pressure, the fracture width, the stress acting on the fracture and the stress intensity factor. The maximum difference between the analytical and numerical results is within 5.45 %. Additionally, the results show that the stress intensity factor obviously decreases after the wellbore strengthening. More importantly, the effect of the fracture roughness (JRC) on the fracture network loss was discussed. The results show that the fracture has a lower fracture pressure, a smaller fracture width and a smaller stress intensity factor with the increasing of JRC, which means that the fracture network with a higher JRC has a lower drilling fluid loss. Meanwhile, the effect of JRC on the wellbore strengthening was investigated, and the results show that the fracture with a higher JRC has the larger reduction for the fracture pressure, the fracture width and the stress intensity factor through operating the wellbore strengthening. This means that the fracture network 1

* Corresponding author. E-mail addresses: [email protected] (Z. Wang). 1

with a larger JRC is better to operate the wellbore strengthening. Keywords: Drilling fluid loss; Fracture network; Wellbore strengthening; Fracture roughness; Analytical model

1. Introduction Drilling fluid loss occurs frequently in the field of the petroleum drilling engineering, greatly increasing the drilling cost and the non-productive time (Feng et al., 2015; Xu et al., 2016; Zhang et al., 2018). It is estimated that the annual cost increases up to 24 billion euros in the global drilling industry (Cook et al., 2011). Fractured loss is the most common type of the drilling fluid loss, especially for the deep well and the complex formation. For the fractured loss, the fracture network is the main channel for the drilling fluid loss. Under the high wellbore pressure, the natural fracture opens and connects each other to form the complex fracture network, causing the drilling fluid loss (i.e., the fracture network loss). In order to reduce the drilling fluid loss, the wellbore strengthening technology was concerned since the DEA-13 experiment of the American Association of Drilling Engineers (Morita et al., 1990; Onyia, 1994). The wellbore strengthening was achieved by depositing the lost circulation materials (LCMs) into the fracture. The main purpose of the wellbore strengthening is to improve the pressure bearing capacity, preventing the fracture to propagate. The whole process is involved in different mechanical problems, including the fluid pressure distribution within the fracture, the stress field around the wellbore as well as the stress intensity factor. Considering the complexity of the wellbore strengthening, the wellbore strengthening needs to be further investigated, especially for the drilling fluid loss due to the complex fracture network. Up to now, some researchers have proposed different viewpoints and concepts to illustrate the wellbore strengthening mechanism. Alberty and McLean MR (2004) introduced the viewpoint of the stress cage to illustrate the wellbore strengthening. Dupriest (2005) described the viewpoint of the fracture closure stress (FCS) for relieving the lost circulation during the drilling. Aston et al (2007) verified the mechanism of shale wellbore strengthening through the experiment and the field

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test. Fett et al. (2010) conducted an offshore borehole test in the deep waters of the Gulf of Mexico, to demonstrate the effect of the wellbore strengthening. Van Oort et al. (2011) analyzed the field data, and illustrated the wellbore strengthening mechanism using the fracture propagation resistance (FPR). According to the above analysis, three main viewpoints were proposed to analyze the wellbore strengthening mechanism, namely, the stress cage, the fracture closure stress (FCS) and the fracture propagation resistance (FPR). Besides, other researchers attempted to illustrate the wellbore strengthening mechanism by the numerical and analytical models. Wang et al. (2009) established a boundary element numerical model to discuss the effects of rock properties, fracture pressure and other factors on the wellbore strengthening. Guo et al. (2014) used the finite element method to analyze the effects of in-situ stress and fracture length on the width of two cracks symmetrically distributed on the borehole wall. Gray et al. (2007) used ABAQUS to perform the linear and non-linear hoop stress analysis for the wellbore strengthening. Shahri et al. (2014) applied the dislocation-based fracture model to establish wellbore strengthening model. Morita and Fuh (2012) established the wellbore strengthening model and derived closed solutions considering two-dimensional boundary. Van Oort and Razavi (2014) used hydrofracture model to obtain the fracture pressure and the fracture width during the drilling fluid loss. Arlanoglu et al. (2015) used the finite element model to explain the wellbore strengthening mechanism, considering the elastic model and the pore elastic model. Mehrabian et al. (2015) developed the analytical model to investigate the wellbore strengthening, and especially emphasized the importance of LCM for the drilling fluid loss. Feng and Gray (2016) established an analytical solution to analyze the wellbore strengthening based on the linear elastic fracture mechanics. Zhong et al. (2017a, 2017b, 2018) proposed a coupled fluid flow model for wellbore strengthening, to explain the behavior of near-well induced fractures and to discuss the effect of controllable parameters on the fracture reopening pressure (FROP). Additionally, Onyia (1994), Benaissa et al. (2005, 2006), Song and Rojas (2006), Salehi and Nygaard (2011) and Savari

3

et al. (2014) also made important contributions to illustrate the wellbore strengthening. Although different models have been developed for the wellbore strengthening, there still are the shortcomings to be improved. These shortcomings are presented in the following aspects: (1) In the actual geological condition, the complex fracture network is the main reason of the drilling fluid loss, which means that the fracture network should be emphatically concerned for the wellbore strengthening. However, only two short fractures distributed symmetrically around the wellbore are considered for the current models, which is not sufficient to analyze the wellbore strengthening. (2) In current models, the fracture pressures are assumed two constants, respectively equaling the wellbore pressure in front of LCMs and a relative lower pressure behind LCMs (obtained by the sealing efficiency of LCMs (Mehrabian et al., 2015)). In such case, the fracture pressure is not a continuous distribution function. This may be reasonable for a piece of natural fracture with limited length (at most several meters), but is not correct for the fracture network, because the fracture network has a longer fracture length. Obviously, the fracture pressure should be a continuous distribution function, but instead of two constants. (3) In current models, the natural fracture was assumed as smooth surface (Morita and Fuh, 2012; Mehrabian et al., 2015; Zhang et al., 2015; Feng and Gray, 2016; Zhong et al., 2017a; Zhong et al., 2018), without the surface roughness. This may be suitable for the hydraulic fracturing, because the width and height of the hydraulic fracturing are larger enough. In such case, the fracture roughness can be neglected. However, this assumption may be not reasonable for the natural fracture. This is because the width of the natural fracture is not larger enough, compared to the fracture roughness. In other words, the fracture roughness would affect the fracture permeability, further changing the fluid pressure distribution within the fracture. As a result, the fracture roughness should be considered for the drilling fluid loss. (4) The current numerical validation is not sufficient for the proposed analytical model. Most

4

of numerical models neglect to judge the natural fracture opening, but directly assuming a geometric defect at the location where the natural fracture exists. In fact, the natural fracture opening under the drilling fluid loss should be judged in the numerical validation. In such case, it is important to establish a reasonable numerical model to verify the analytical model. Zhao et al. (2017) and Li et al. (2017) established the respective numerical models to assess the wellbore strengthening based on the cohesive element. Especially, the cohesive element was introduced to judge the natural fracture opening, which is an important progress for the wellbore strengthening. However, only a piece of fracture was set based on the cohesive element, but instead of the fracture network, which is not sufficient for analyzing the drilling fluid loss through the complex fracture network. To overcome the shortcomings mentioned above, a new analytical model was established to analyze the wellbore strengthening mechanism, considering the fracture network loss. Using this model, the fluid pressure distribution was determined within the fracture network, considering the fracture roughness. Meanwhile, the stress field was calculated considering the multi-fractures. Combining the pressure within the fracture network and the stress field, the stress intensity factor was obtained to discover the wellbore strengthening. In addition, a comprehensive numerical analysis was performed to validate the analytical model. Finally, the effect of the fracture roughness was discussed for the fracture network loss and the wellbore strengthening.

2. Model description Differing from the previous models that only considered two short fractures distributed symmetrically around the wellbore (Morita and Fuh, 2012; Mehrabian et al., 2015; Feng and Gray, 2016), the new wellbore strengthening model considers the fracture network, with more fractures around the wellbore. These fractures intersect each other under the closed state. Once the drilling fluid loss occurs, these fractures would open and connect each other to be a fracture network, causing more drilling fluid to lose, which can be treated as the fracture network loss. Obviously, the

5

fracture network loss is more realistic and has a more complex fluid behavior for the wellbore strengthening. Considering that the total length of all the fractures in the fracture network is far higher than the one of a piece of fracture, the pressure continuous distribution within the fracture was concerned in the new model. Additionally, the new model introduced the fracture roughness to calculate the pressure continuous distribution within the fracture, which is different to the previous models assumed with a smooth fracture surface. Note that, although Zhang et al. (2018) established an analytical model of the wellbore strengthening with multi-cleats around the wellbore, this model did not consider the fracture roughness. In detail, the new model of the wellbore strengthening was shown in Fig. 1, where the blue line represented the natural fracture network.

Fig.1. Wellbore strengthening model considering fracture network and fracture roughness.

Corresponding to the above analysis, three advantages can be seen for the new analytical model, including that: (1) the natural fracture network was considered in the new model of the wellbore strengthening; (2) the pressure continuous distribution within the fracture was adequately analyzed, by introducing the fracture roughness; (3) a reasonable and comprehensive numerical validation was performed, which would be seen in the following section of 6.1. Meanwhile, the following assumptions were made for the new model:

6

(1) LCMs can not perfectly prevent the drilling fluid loss, i.e., there is still a part of drilling fluid to pass through LCMs. (2) The additional stress field induced by the natural fracture network is added to the initial stress field, to calculate the total stress field. (3) The decreasing of the stress intensity factor is treated as the validation of effectiveness for the wellbore strengthening. (4) The decreasing of the fracture width can be also regarded as the positive symbol of the wellbore strengthening.

3 Stress analysis considering fracture network In this section, the total stress field was calculated by adding by the initial stress field and the additional stress field. 3.1 Initial in-situ stress field According to the work of Bradley (1979), the initial in-situ stress field can be expressed as Eq. (1), considering the formation as a continuous body.

 σ H + σ h  R2  σ H − σ h  R4 R4  R2 σ r = 1 − 2  +  1+3 4 − 4 2  cos 2θ + 2 P0 2  r  2  r r  r   R4  R2 σ H + σ h  R2  σ H − σ h  = 1 + − 1 + 3 cos 2 − P0 σ θ  θ   2  4  2 2 r 2 r r       4 2   τ rθ = σ H − σ h 1 − 3 R4 +2 R2  sin 2θ 2  r r  

(1)

where σ H is the horizontal maximum stress, σ h is the horizontal minimum stress, P0 is the drilling fluid pressure, σ r is the radial stress, σ θ is the hoop stress, τ rθ is the shear stress, R is the radius of the wellbore, r, θ are respectively the radial and hoop coordinates. 3.2 Additional stress caused by fracture network Based on the fracture mechanics, the natural fracture induces the additional stress based on Irwin (1957), seeing Eq. (2).

7

 i σ xi′ =    i σ yi′ =   τ xi ′ y′ =  i i

K Ii θ′ θ′ 3θ ′  cos i  1 − sin i ⋅ sin i  2 2 2  2πri′

θ′ θ′ 3θ ′  K Ii cos i 1 + sin i ⋅ sin i  2 2 2  2πri′

(2)

θ′ θ′ 3θ ′ K Ii cos i sin i cos i 2 2 2 2πri′

where i is the ith fracture, xi′ yi′ is the local coordinate system on the ith fracture (seeing Fig. 2), ri′ , θi′ are the coordinates in xi′ yi′ , σ xi i′ is the normal stress along xi′ direction induced by the ith

fracture ( xi′ is parallel to the length direction of the ith fracture), σ iyi′ is the normal stress along yi′ direction induced by the ith fracture ( yi′ is perpendicular to the length direction of the ith

fracture), τ xi i′ yi′ is the shear stress in xi′ yi′ coordinate system, K Ii is the stress intensity factor of the ith fracture and can be calculated based on Tada (1985), seeing Eq. (3). 1 K = i πa i I

∫

ai

−a

P ( xi′) i i net

a i + xi′ dxi′ a i − xi′

(3)

i where a i is the half length of the ith fracture, Pnet ( xi′) is the normal net pressure acting on the ith

fracture surface (see Fig. 3).

Fig. 2. The adopted coordinate systems associated with fractures.

8

Fig. 3. Normal net pressure acting on the fracture surface.

In this analysis, the natural fracture was assumed simultaneous opening under the high wellbore pressure, not considering before and after sequence of the fracture opening. In other words, there was no the interaction between the natural fractures. In such case, the normal net pressure is equal to the difference between the fracture pressure and the normal stress component of the initial in-situ stress. In detail, the fracture pressure was calculated by the following section 5.1, and the normal stress component of the initial in-situ stress was calculated by the section 3.1. To reflect the calculation process more clearly, the flowchart of calculating the additional stress was depicted in Fig. 4.

9

Fig. 4. Calculation flowchart for additional stress field.

More importantly, converting the initial stress field and the additional stress field into the same coordinate system, the total stress field acting on the fracture can be obtained by adding the initial stress field and the additional stress field.

4 Roughness characterization and roughness dependent permeability As seen from Fig. 1, the fracture roughness has an important effect on the fracture permeability, further changing the fracture pressure distribution. In the previous research, the fracture pressure model was based on the smooth parallel plate model, without the surface roughness. In such case, it can be treated as a progress by introducing the surface roughness to calculate the fracture pressure. 4.1 Roughness characterization based on JRC-JCS criterion Barton and Choubey (1973) developed a non-linear strength criterion to evaluate the rock joint strength. In this criterion, the joint roughness was introduced as a key parameter, named as JRC. The detailed strength criterion was expressed as Eq. (4): 

 JCS      σn 

τ p = σ n tan  φr + JRC ⋅ lg  

(4)

where τ p is the peak of the shear strength, σ n is the normal stress on the fracture surface, JRC is the joint roughness coefficient, JCS is the compressive strength on the joint wall, φr is the residual friction angle. Considering the similarity between the rock joint and the natural fracture (Zhang et al., 2015), JRC can be used to evaluate the fracture roughness. Fig. 5 below shows the fracture roughness profiles and JRC values.

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Fig. 5. Roughness profiles and corresponding JRC values (Barton and Choubey, 1977).

4.2 Roughness dependent permeability Corresponding to Fig. 5, the effect of the fracture roughness on the fracture permeability can be schematically shown in Fig. 6.

Fig. 6. Roughness dependent permeability.

Under laminar flow, the relation between the fracture permeability coefficient and the fracture width can be expressed as Eq. (5) (Witherspoon et al., 1980): kf =

gb 2 12u

(5)

where kf is the fracture permeability coefficient, cm/s; g is the gravity acceleration, m/s2; b is the hydraulic width of the fracture, m; u is the fluid viscosity, Pa/s. When considering the fracture roughness, the fracture permeability can be expressed as the function of JRC and the mechanical width of the fracture, seeing Eq. (6) (Barton et al., 1985).

Kf =

bm2 12 ρ JRC 5

11

(6)

where K f is the fracture permeability, ρ is the drilling fluid density, bm is the mechanical width of the fracture and can be calculated by b =

bm . JRC 2.5

Then, substituting Eq. (6) into Eq. (5) and combining the relation between the permeability and the permeability coefficient, as well as the generalized Hooke's law, the mechanical width of the fracture bm can be expressed as Eq. (7) (Zhao., 2015), based on the elastic mechanics model shown in Fig. 7.

bm =

σc JRC   − 0.1 e  0.2 JCS 5  

−σ n         σc JRC  JCS    + 2 JRC −10  0.2 − 0.1 0.02      σ JCR  5  JCS   c − 0.1    0.2   JCS   5   

(7)

Fig. 7. Model of the mechanical width fracture.

Substituting Eq. (16) into Eq (8), the fracture permeability K f can be expressed as the function of the normal stress σ n , where σ n can be solved by section 3.

5 Pressure analysis before and after wellbore strengthening Before and after depositing LCMs, the pressure distribution is different for the fracture network. In view of this, the pressure distribution was analyzed from two aspects, seeing section 5.1 and section 5.2. Especially, the pressure at the intersection of the fracture was emphatically discussed for the fracture network loss.

5.1 Pressure analysis before wellbore strengthening After obtaining the relation of roughness dependent permeability, the pressure within the fracture can be calculated. To obtain the pressure distribution within the fracture, the fracture was 12

divided into n infinitesimal elements, with a length of h for each infinitesimal element. The adjacent two infinitesimal elements were shown in Fig. 8.

Fig. 8. Analysis model for the fracture pressure

For adjacent two infinitesimal elements, the pressure drop can be expressed as: P(i +1) − P(i ) h(i )

=−

12uq(i ) W(3i )

(8)

where P(i ) is the pressure of last element and is known, W(i ) is the fracture width of the ith element, h(i ) is the length of the ith element, u is the fluid viscosity, q(i ) is single width flow and can be expressed as Eq. (9). q(i ) =

Qin(i ) W(i )

(9)

where Qin(i ) is the flow that enters into the ith element. Meanwhile, the outflow that is through the fracture surface can be calculated for the ith element using Eq. (10).

Qout (i ) =

2 K m h(i ) h′ ( P(i ) + P(i +1) − 2 Pp ) uLp

(10)

where Qout ( i ) is the outflow from the ith element, h′ is the fracture height, Lp is the seepage

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distance pointed to the formation, Pp is the formation pressure, K m is the formation permeability. The flow of (i+1)th element was marked as Qin(i +1) , and was calculated by Eq. (11). Qin (i +1) =

W f (i ) K (i ) h( P( i ) − P(i +1) ) uL(i )

Qin ( i ) = Qin ( i +1) +2Qout ( i )

(11)

(12)

where K (i ) is the permeability of the ith element. Using Eqs. (8) ~ (12), the unknown variables W(i +1) , P(i +1) , Qin (i +1) can be calculated. Considering the boundary conditions of P1 = P0 and Qin (1) = Q , the pressure distribution within the fracture can be obtained.

5.2 Fracture pressure after wellbore strengthening The objective of the wellbore strengthening is to reduce the drilling fluid loss. To achieve this objective, it can be treated as a prior condition that the fracture no longer propagates. In such case, the flow that enters into the fracture network is equal to the outflow from the fracture network. This is new additive condition for calculating the pressure within the fracture. In addition to this, the pressure analysis after the wellbore strengthening is similar to the one before the wellbore strengthening. To avoid the repetition, a calculation flowchart was depicted to show the calculation process of the fracture pressure after the wellbore strengthening, seeing Fig. 9.

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Fig. 9. The calculation flowchart for fracture pressure after the wellbore strengthening.

5.3 Pressure analysis at fracture intersection To calculate the pressure within the subsequent fracture, the flow distribution was firstly calculated based on Fig. 10.

Fig. 10. The schematic diagram for flow distribution at fracture intersection

Special analysis was made, when calculating the flow distribution of the intersection of fractures. Taking the fracture OC as an example, the maximum width of the fracture OC was assumed as WO3O4 . Substituting WO3O4 into Eqs. (8) ~ (12), the fracture pressure POC ( x ) within OC and the fracture width WOC ( x ) along OC can be obtained. Then, substituting POC ( x ) into Eq. 15

′ ( x ) was calculated again. Furthermore, the error (13) (Perkins et al. 1961), the fracture width WOC ′ ( x ) . If e ≤ 5% , the calculated fracture analysis was performed by substituting WOC ( x ) and WOC ′ ( x ) . Otherwise, width was considered to be reliable and reasonable, regardless WOC ( x ) or WOC ′ ( x ) was extracted to substitute WO3O4 , starting the new calculation to the maximum value of WOC determine POC ( x ) and QO3O4 until e ≤ 5% .

W ( x) =

e=

8 (1 − v 2 ) ( P ( x ) − σ h ) L

πE

′ ( x ) − WOC ( x ) WOC × 100% ′ ( x) WOC

(13)

(14)

where v is the Poisson's ratio, P ( x ) is the fracture pressure within OC, σ h is the horizontal minimum stress, E is the modulus of elasticity. In this way, the fracture pressure POB ( x ) within OB and the flow QO2 O4 can be determined. Then, the flow QO1O2 that enters the surface O1O2 can be calculated using (15). QO1O2 = QO1O3 − QO2 O4 − QO3O4

(15)

After obtaining QO1O2 , the fracture pressure POD ( x ) within OD can be calculated by Eqs. (8) ~ (12). To describe the calculation process more clearly, a flowchart was shown in Fig. 11.

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Fig. 11. The flowchart to calculate flow distribution at fracture intersection

After obtaining the flow distribution at the intersection of the fractures, the pressure within the subsequence fractures can be solved based on the method in section 5.1 or 5.2.

6 Wellbore strengthening evaluation The effectiveness of the wellbore strengthening can be evaluated by calculating the stress intensity factors before and after the wellbore strengthening. As a result, the stress intensity factors before and after the wellbore strengthening were emphatically discussed in this section. 6.1 Stress intensity factor before wellbore strengthening Using section 3 and section 5, the stress intensity factor for each fracture can be calculated based on Eq. (3). The following analysis would show the detailed calculation process of the stress intensity factor before the wellbore strengthening, taking the fracture network in Fig. 12 as the example. For the fracture network, the intersection mode of the fracture has an important effect on the calculation of the stress intensity factor. Generally speaking, there are three intersection modes of the fracture in the natural fracture network, namely, Mode I, Mode II, Mode III, seeing Fig. 13. Corresponding to Fig. 12, Mode I is the fracture intersection of AB and CD, with the intersection point O. Mode II is the fracture intersection of DE and FG, with the intersection point F. Mode III is the fracture intersection of CD and DE, with the intersection point D. For Mode I, it can be found three fracture tips, according to the flow path of the drilling fluid. In such case, three stress intensity factors need to be calculated. For Mode II, two fracture tips can be found, according to the flow path of the drilling fluid. As a result, two stress intensity factors need to be calculated. For Mode III, there are two fracture tips to be observed, according to the flow path of the drilling fluid. Consequently, two stress intensity factors need to be calculated. According to the above analysis, there are five fracture tips to be found for the fracture network in Fig. 12, respectively corresponding to points B, C, D, E, and G. In this case, the lengths of AB, OC, OD, DE, and FG were used to calculate the stress intensity factors at points B, C, D, E, and G.

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Fig. 12. Stress intensity factors before wellbore strengthening.

Fig. 13. Fracture intersection mode.

In detail, the stress intensity factors of the fracture network can be calculated by the Eq. (16). K IAB =

1 πlAB

∫

lAO

K IOC =

1 πlOC

∫

lOC

K IOD =

1 πlOD

∫

K IDE =

1 πlDE

∫

lDF

K IFG =

1 πlFG

∫

lFG

0

0 lOD

0

0

0

′ ) PnetAO ( xAB

′ lAB + xAB 1 ′ + dxAB ′ lAB − xAB πlAB

OC ′ ) Pnet ( xOC

′ lOC + xOC ′ dxOC ′ lOC − xOC

OD ′ ) ( xOD Pnet

′ lOD + xOD ′ dxOD ′ lOD − xOD

DF ′ ) Pnet ( xDE

′ lDE + xDE 1 ′ + dxDE ′ lDE − xDE πlDE

′ ) PnetFG ( xFG

′ lFG + xFG ′ dxFG ′ lFG − xFG

∫

lAB

lAO

OB ′ ) Pnet ( xAB

′ lAB + xAB ′ dxAB ′ lAB − xAB

(16)

∫

lDE

lDF

′ ) PnetFE ( xDE

′ lDE + xDE ′ dxDE ′ lDE − xDE

where lAO , lOB , lAB , lOC , lOD , lDF , lFE , lDE , lFG are the length of the fracture AO, OB, AB, OC, OD, DF, FE, DE, FG, respectively, K IAB , K IOC , K IOD , K IDE , K IFG are the stress intensity factors at points B, C, D, E, and G. Especially, the net pressure within AO is different to the one within OB, due to the pressure re-distribution at the intersection point O. It is also applicable for net pressures within DF and FE. 18

6.2 Stress intensity factor after wellbore strengthening The wellbore strengthening was achieved by depositing LCMs at a location near the wellbore, seeing Fig. 14. Assuming LCMs was deposited at point A, the stress intensity factors of the fracture network can be calculated by the Eq. (17). K IAB =

1 πlAB

∫

lAO

K IOC =

1 πlOC

∫

lOC

K IOD =

1 πlOD

∫

K IDE =

1 πlDE

∫

lDF

K IFG =

1 πlFG

∫

lFG

′AO ( xAB ′ ) Pnet

′ lAB + xAB 1 ′ + dxAB ′ lAB − xAB πlAB

′OC ( xOC ′ ) Pnet

′ lOC + xOC ′ dxOC ′ lOC − xOC

′OD ( xOD ′ ) Pnet

′ lOD + xOD ′ dxOD ′ lOD − xOD

0

0 lOD

0

0

0

′DF ( xDE ′ ) Pnet

′ lDE + xDE 1 ′ + dxDE ′ lDE − xDE πlDE

′FG ( xFG ′ ) Pnet

′ lFG + xFG ′ dxFG ′ lFG − xFG

∫

lAB

lAO

′OB ( xAB ′ ) Pnet

′ lAB + xAB ′ dxAB ′ lAB − xAB

(17)

∫

lDE

lDF

′FE ( xDE ′ ) Pnet

′ lDE + xDE ′ dxDE ′ lDE − xDE

Fig. 14. Stress intensity factor before wellbore strengthening

More importantly, K IAB was substituted as Eq. (18), considering that LCMs was deposited with a distance of lAH away from point A. K IAB =

1 πlAB

∫

1 + πlAB

lAH

0

∫

′AH ( xAB ′ ) Pnet

lAB

lAO

′ lAB + xAB 1 ′ + dxAB ′ lAB − xAB πlAB

l + x′ ′OB ( xAB ′ ) AB AB dxAB ′ Pnet ′ lAB − xAB

19

∫

lAO

lAH

′HO ( xAB ′ ) Pnet

′ lAB + xAB ′ dxAB ′ lAB − xAB

(18)

′ is the net pressure after the wellbore strengthening. where lAH is the length of AH, Pnet Using Eq. (16), the stress intensity factor of the fracture network before the wellbore strengthening K IBe can be the obtained. Similarly, the stress intensity factor of the fracture network after the wellbore strengthening K IAf can be calculated by Eq. (17) or Eq. (18). Through comparing K IBe and K IAf , the effectiveness of the wellbore strengthening can be verified. Note that, the

notation “Be” means “before”, and the notation “Af” means “after”. 6.3 Potential engineering application The potential engineering application is an objective of the proposed model. The proposed model can be used in the real engineering, as long as the natural fracture network has the known distribution around the wellbore. However, the natural fracture network distribution is difficult to be predicted in real engineering. In view of this, the natural fracture network can be assumed a known specified distribution around the wellbore, when using our analytical model in the real engineering. In such case, the potential engineering application of the proposed model can be concluded as: (1) Specifying a known distribution of natural fracture network around the wellbore; (2) Substituting the known distribution of natural fracture network into Section 4 to calculate the fracture permeability; (3) Substituting the known distribution of natural fracture network into Section 5 to analyze the fracture pressure; (4) Calculating stress intensity factors before and after wellbore based on (3) and (4), and achieving their comparison.

7 Numerical validations The drilling fluid loss occurs frequently in the ChangQing oilfield, China. Recently, a serve drilling fluid loss occurs at WH-X well, with the depth of 3000 m. Through the rock core analysis, the drilling fluid loss was thought as the fracture network loss, seeing Fig. 1. Especially, there is a

20

typical characteristic for the fracture network, i.e., all the natural fractures have a larger length of average exceeding 1.5 m. For simplification, four pieces of natural fractures were taken to participate the analytical calculation, i.e., AB, CD, DE, FG, respectively having the same length of 1.5 m for AB, DE, FG and a length of 0.9 m for CD. Note that, CD is a dip fracture, with a dip angle of 60 °. More importantly, the fracture roughness is assumed to equal 5 based on the rock core analysis. To further prevent the drilling fluid to lose, LCMs were deposited at the fracture mouth of AB. Using the parameters mentioned above, the analytical results can be calculated by the proposed method, including the fracture pressure, the stress field, and the fracture width and the stress intensity factor before and after wellbore strengthening. To validate the analytical model, a comprehensive numerical model was established. The numerical model has a special effect, i.e., the natural fracture opening needs to be judged by the numerical model, but instead of directly presetting a geometric defect at the location where the natural fracture exists in the numerical analysis. In such case, ABAQUS was selected as a tool to establish the numerical model, mainly because the cohesive element in ABAQUS can simulate the natural fracture opening. However, a problem occurs during using the numerical model to simulate the natural fracture opening, i.e., the numerical model is difficult to be solved when having three pieces of natural fractures (or more pieces of fractures) to be intersected. In such case, the related literatures were searched to find whether the pioneer succeeded to use ABAQUS to simulate three pieces of intersected fractures to initiate and propagate (Zhao et al. 2017; Li et al. 2017). However, the searched results were unsatisfactory and disappointed. In view of this, a simplification was made for the numerical model. Only two pieces of intersected fractures were contained for the numerical model, where a longer fracture of 4.5 m was used to represent the fractures AB, DE, FG, and a shorter fracture of 0.9 m was used to represent the fracture CD. Establishing a longer fracture represents the fractures AB, DE, FG, mainly because the pressure drop can not be neglected when the total length of fractures within the fracture network is larger enough. This is a very important

21

viewpoint mentioned in section of introduction. Especially, the injection rate maintains a constant to simulate the fracture network loss, while it decreases linearly to 0 to simulate the wellbore strengthening (Li et al. 2017). Correspondingly, the analytical model was forced to make a change to match the numerical model, only considering a simple fracture network composed by two natural fractures. According to the above analysis, a numerical model was developed based on ABAQUS, where two pieces of the natural fracture were considered to form a fracture network. For this analysis, the basic parameters were listed in Table 1, and the detailed numerical model was shown in Fig. 15. Table 1 The parameters used in numerical model. Parameters

Unit

Values

Parameters

Unit

Values

Cohesive strength

MPa

6

MPa

30.24

Leak-off coefficient

m3/Pa/s

1×10-7

MPa

60.66

Fracture energy

kJ/m2

0.12

Formation pressure Horizontal maximum in-situ stress Horizontal minimum in-situ stress

MPa

55.04

Elastic modulus

GPa

15

Formation permeability

mD

100

Formation porosity

/

0.1

Poisson's ratio

/

0.23

Lost circulation rate

m3/s

1×10-4

Fluid viscosity

cp

200

Drilling fluid pressure

MPa

36.61

Borehole radius

mm

76.20

(a)

(b)

22

Fig. 15. Numerical model; (a) The whole model; and (b) Zoom in of (a).

7.1 Verification of fracture pressure Fig.16 (a) compares the fracture pressures between the analytical and numerical models before the wellbore strengthening, while Fig.16 (b) shows the fracture pressures between the analytical and numerical models after the wellbore strengthening.

(a)

(b)

Fig. 16. Comparison of fracture pressures; (a) Pressure distribution before wellbore strengthening; and (b) Pressure distribution after wellbore strengthening.

Comparing analytical results and numerical results, a good overall agreement can be observed for the fracture pressure in Fig. 16, regardless of A1B1 or C1D1. The analysis shows that the analytical result has a very small deviation to the numerical results for the fracture pressure, not exceeding 2.64 %. Fig. 17 below shows the distribution of the fracture pressure based on the numerical model for before and after the wellbore strengthening.

23

(a) Numerical result before wellbore strengthening

(b) Numerical result after wellbore strengthening

Fig. 17. Numerical results of fracture pressure; (a) Before wellbore strengthening; and (b) After wellbore strengthening.

As seen from Fig. 17, the fracture pressure before the wellbore strengthening is higher, compared to the one after the wellbore strengthening. The maximum fracture pressure is 36.61 MPa before the wellbore strengthening, while it decreases to 33.44 MPa after the wellbore strengthening. From this point of view, the wellbore strengthening achieves a decreasing of the fracture pressure of 3.17 MPa, which can be treated as a positive effect due to the wellbore strengthening. Additionally, the fracture length has a decreasing from 4.50 m to 4.18 m, which can be also treated as an important contribution due to the wellbore strengthening. 7.2 Verification of fracture width Fig.18 (a) compares the fracture widths between the analytical and numerical models before the wellbore strengthening, while Fig.18 (b) compares the fracture widths between the analytical and numerical models after the wellbore strengthening.

24

(a)

(b)

Fig. 18. Comparison of fracture width; (a) Fracture width before wellbore strengthening; and (b) Fracture width after before wellbore strengthening.

As seen from Fig. 18, the fracture width shows a good agreement between the analytical model and the numerical model, regardless of before or after the wellbore strengthening. The maximum difference between two models is less than 5.45 %, which means that the analytical method can be used to illustrate the wellbore strengthening. Additionally, the fracture widths have the obvious decreasing from 2.64 mm to 1.08 mm for A1B1 and from 1.30 mm to 0.18 mm for C1D1, when operating the wellbore strengthening. 7.3 Verification of stress field Fig.19 compares the stresses acting on the fractures A1B1 and C1D1 between the analytical and numerical models before the wellbore strengthening, while Fig.20 compares the stresses acting on the fractures A1B1 and C1D1 between the analytical and numerical models after the wellbore strengthening.

25

(a)

(b)

Fig. 19. Comparison of stresses before wellbore strengthening; (a) Stress results acting on fracture A1B1; and (b) Stress results acting on fracture C1D1.

(a)

(b)

Fig. 20. Comparison of stresses after wellbore strengthening; (a) Stress results acting on fracture A1B1; and (b) Stress results acting on fracture C1D1.

The analytical model coincides well with the numerical model for the stresses acting on the fractures A1B1 and C1D1, with the minor deviations of within 4.82 %, which means that the analytical method used to calculate the stress field is effective and reliable. Additionally, the stress field has a small increasing for the fracture A1B1 and C1D1 after operating plugging, mainly resulting from the decreasing of the fracture pressure after the wellbore strengthening. 7.4 Verification of stress intensity factor 26

Fig. 21 (a) compares the stress intensity factors between the analytical and numerical models before the wellbore strengthening, while Fig. 21 (b) compares the stress intensity factors between the analytical and numerical models after the wellbore strengthening.

(a)

(b)

Fig. 21. Comparison of stress intensity factor; (a) Stress intensity factor before wellbore strengthening; and (b) Stress intensity factor after wellbore strengthening.

Comparing the results of stress intensity factors in Fig. 21, it can be found a difference of less than 4.43 % between the analytical and numerical models. Additionally, the stress intensity factor obviously decreases after the wellbore strengthening, which means that the fracture is difficult to propagate once applying the wellbore strengthening.

8 Discussions In this section, the effectiveness of the wellbore strengthening was evaluated by the proposed analytical model. Meanwhile, the effects of JRC on the fracture network loss and the wellbore strengthening were discussed, focusing on the fracture pressure, the fracture width and the stress intensity factor. Especially, the model in this section was based on Fig. 12, mainly considering that: (1) there has no the calculation limit on the number of the fracture when using the analytical model; (2) more fractures means a more complex fracture network and a more realistic formation condition.

27

8.1 Evaluation of wellbore strengthening In this part, the wellbore strengthening was evaluated by comparing the stress intensity factors and the fracture widths before and after the wellbore strengthening. Fig. 22 (a) shows the results of the stress intensity factors before and after the wellbore strengthening, and Fig. 22 (b) shows the fracture widths before and after the wellbore strengthening.

(a)

(b)

Fig. 22. Evaluation of wellbore strengthening; (a) Comparison of stress intensity factors before and after wellbore strengthening; (b) Comparison of fracture widths before and after wellbore strengthening.

As seen from Fig. 22, the stress intensity factor after the wellbore strengthening is obviously lower than the one before the wellbore strengthening. The fracture width shows a similar trend to the stress intensity factor. This means that plugging LCMs can be treated as an effective measure for the wellbore strengthening, if LCMs can maintain stable within the fracture. In practical engineering, there has the unsuccessful case of the wellbore strengthening to be reported, partly because LCMs are difficult to maintain stable under the complex wellbore pressure circumstance. 8.2 Effect of JRC on fracture network loss In this section, the effect of JRC on the fracture network loss was investigated, focusing on the fracture pressure, the fracture width and the stress intensity factor before plugging. The results were shown in Fig. 23. Note that, only a piece of the fracture of AB was selected to investigate the effects of JRC on the fracture pressure and the fracture width (mainly because other fractures have the 28

similar trend with AB), while all the fractures were selected to investigate the effect of JRC on the stress intensity factor.

(a)

(b)

(c) Fig. 23. Effect of JRC on fracture network loss; (a) Effect of JRC on fracture pressure before plugging; (b) Effect of JRC on fracture width before plugging; (c)Effect of JRC on stress intensity factor before plugging.

As seen from Fig. 23 (a), the fracture pressure has an overall decreasing trend with the increasing of JRC, which is beneficial for reducing the fracture network loss. Additionally, the pressure drop from the fracture mouth to the fracture tip also increases as the JRC increasing. For example, the fracture pressure drop is 0.15 MPa at JRC = 1, while increases to 0.75 MPa at JRC = 5. In other words, the fracture pressure drop is larger for the fracture with a higher JRC. 29

As seen from Fig. 23 (b), the fracture width decreases with the increasing of JRC, which means that a higher JRC is positive for reducing the fracture network loss. At JRC = 1, the width of the fracture mouth exceeds 1.52 mm, but decreases to 0.9 mm at JRC = 5. As seen from Fig. 23 (c), the stress intensity factor shows a decreasing trend for all the fractures as JRC increasing. Compared to other fractures, AB has a slowest decreasing for the stress intensity factor with the increasing of JRC. On the contrary, DE shows a quickest decreasing for the stress intensity factor with the increasing of JRC. From this point of view, the decreasing trend of the stress intensity factor is closely related to the location that the fracture exists. Overall, the fracture has the lower fracture pressure, the smaller fracture width and the smaller stress intensity factor as JRC increasing before plugging, which means that the fracture network with a higher JRC has a lower drilling fluid loss. 8.3 Effect of JRC on wellbore strengthening Similar to section 8.2, the effect of JRC on the wellbore strengthening was investigated, focusing on the fracture pressure difference, the fracture width difference and the stress intensity factor difference before and after wellbore strengthening. The results were shown in Fig. 24. Note that, only a piece of the fracture of AB was selected to investigate the effects of JRC on the fracture pressure difference and the fracture width difference (mainly because other fractures have the similar trend with AB), while all the fractures were selected to investigate the effect of JRC on the stress intensity factor difference.

30

(a)

(b)

(c) Fig. 24. Effect of JRC on wellbore strengthening; (a) The effect of JRC on the fracture pressure difference before and after wellbore strengthening; (b) The effect of JRC on the fracture width difference before and after wellbore strengthening; (c) The effect of JRC on the stress intensity factor difference before and after wellbore strengthening.

As seen from Fig. 24 (a), the difference of the fracture pressures before and after wellbore strengthening shows an increasing trend as JRC increasing. In other words, the fracture with a larger JRC has a larger reduction of the fracture pressure when operating the wellbore strengthening. Therefore, the natural fracture network with a larger JRC is more suitable to apply the wellbore strengthening operation.

31

In addition, the difference of the fracture widths before and after wellbore strengthening increases with the increasing of JRC, which can be seen in Fig. 24 (b). In other words, the fracture with a larger JRC has a larger reduction of the fracture width through operating the wellbore strengthening. From this point of view, the fracture network with a higher JRC has a better application effect for the wellbore strengthening. Furthermore, the difference of the stress intensity factors before and after wellbore strengthening increases, as JRC increasing (seeing Fig. 24 (c)). That is to say, the stress intensity factor has a larger reduction for the fracture with a larger JRC, after operating the wellbore strengthening. Additionally, the results show that AB has the maximum difference of the stress intensity factor, while CD has the minimum difference of the stress intensity factor. This indicates that the fracture that directly connects to the wellbore has the maximum reduction of the stress intensity factor, i.e., the best effect for operating the wellbore strengthening. Overall, the fracture with the larger JRC has the larger reduction for the fracture pressure, the fracture width and the stress intensity factor, when operating the wellbore strengthening. In other words, the natural fracture network with the larger JRC is better to apply the wellbore strengthening operation.

9 Conclusions (1) A new analytical model was established to analyze the drilling fluid and the wellbore strengthening, considering the natural fracture network. For this model, the fracture roughness was introduced to analyze the pressure behavior of the fracture network loss. Especially, the wellbore strengthening mechanism was discovered, by comparing the stress intensity factors before and after the wellbore strengthening. (2) The proposed analytical model was verified by comparing its results to the results of the numerical model, including the fracture pressure, the fracture width, the stress acting on the fracture and the stress intensity factor. The maximum difference between the analytical result and the

32

numerical result was within 5.45 %, which means that the proposed analytical model can analyze the pressure behavior, and can illustrate the wellbore strengthening mechanism. (3) The wellbore strengthening can be evaluated by comparing the stress intensity factor and the fracture width before and after the wellbore strengthening. The results show that the stress intensity factor and the fracture width obviously decrease after the wellbore strengthening. (4) The effect of JRC on the fracture network loss was discussed. As JRC increasing, the fracture has a lower fracture pressure, a smaller fracture width and a smaller stress intensity factor for the fracture network loss. This means that the fracture network with a higher JRC has a lower drilling fluid loss. (5) The effect of JRC on the wellbore strengthening was investigated. The results show that the fracture with a higher JRC has a larger reduction for the fracture pressure, the fracture width and the stress intensity factor when operating the wellbore strengthening. This means that the fracture network with a larger JRC is better to operate the wellbore strengthening.

Acknowledgment The authors are very much indebted to the Projects Supported by PetroChina Innovation Foundation (2018D-5007-0309), Focus on Research and Development Plan in Shandong Province (2019GGX103007), the Fundamental Research Funds for the Central Universities (19CX02034A), the National Natural Science Foundation-Outstanding Youth Foundation (51622405), the Shandong Natural Science funds (JQ201716, ZR2016EEM30), the Changjiang Scholars Program (Q2016135) and the National Key Research and Development Plan (2016YFC0303408) for the financial support.

Nomenclature ai

The half length of the ith fracture

b

The hydraulic width of the fracture 33

bh

The hydraulic width of the fracture

bm

The mechanical width of the fracture

bm0

The initial fracture width

E

The modulus of elasticity

G

The elastic constant of the natural fracture

h

The length of each infinitesimal element

JCS The compressive strength on the joint wall JRC The joint roughness coefficient Kf

The fracture permeability

Km

The formation permeability

Kn

The normal stiffness

K (i ) The permeability of the ith element K Ii The stress intensity factor of the ith fracture kf

The fracture permeability coefficient

Lp The seepage distance pointed to the formation lAO

The length of the fracture AO

POB ( x )

The fracture pressure within OB

P(i )

The pressure of ith element

Pp

The formation pressure

P0

The drilling fluid pressure

i Pnet ( xi′)

The normal net pressure acting on the ith fracture surface.

′ The net pressure after the wellbore strengthening Pnet

34

The lost circulation rate

Q

The flow that enters into the ith element

Qin(i )

Qout ( i ) q(i )

The outflow from the ith element The single width flow

R

The radius of the wellbore

r

The radial coordinate

ri′

The coordinate in xi′ yi′

Un

The normal displacement

W(i )

The fracture width of the ith element

WOC ( x ) xi′ yi′

The fracture OC width

The local coordinate system on the ith fracture

θ

The hoop coordinate

θi′

The coordinates in xi′ yi′

λ

The elastic constant of the natural fracture

u

The fluid viscosity

υ

The Poisson ratio

σr

The radial stress

σθ

The hoop stress

σ xi ′

The normal stress along xi′ direction induced by the ith fracture

σn

The normal stress on the fracture surface

σc

The uniaxial compressive strength of fractured rock mass

i

σ iy′ i

The normal stress along yi′ direction induced by the ith fracture

35

τ rθ

The shear stress

τ xi ′ y′ The shear stress in xi′ yi′ coordinate system i i

τp

The peak of the shear strength

τp

The peak of the shear strength

φr

The residual friction angle

ϕ

The formation porosity

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Highlights:

► A model of wellbore strengthening was developed considering fracture network loss. ► Pressure within the fracture network was analyzed considering the fracture roughness. ► Fracture roughness has an effect on drilling fluid loss and wellbore strengthening. ► The proposed analytical model has an overall agreement to the finite element model.

Declaration of interests


The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: