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Fracture plugging optimization for drill-in fluid loss control and formation damage prevention in fractured tight reservoir
Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227
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Fracture plugging optimization for drill-in fluid loss control and formation damage prevention in fractured tight reservoir Chengyuan Xu a, *, Yili Kang a, Fei Chen b, Zhenjiang You c a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China CCDC Changqing Downhole Technology Company, Xi'an, China c Australian School of Petroleum, The University of Adelaide, Adelaide, Australia b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 January 2016 Received in revised form 1 September 2016 Accepted 21 September 2016 Available online 22 September 2016
Well-developed natural fractures are beneficial for the economic and efficient development of tight reservoirs. However, they also lead to drill-in fluid loss and induced severe formation damage. Fracture plugging with loss control material (LCM) is the most common way to control lost circulation. Fracture plugging effect largely depends on the fracture propagation pressure, because plugging failure is mainly caused by fracture propagation in fractured formation. Nevertheless, the effects of the plugging parameters on the fracture propagation pressure are still unclear. The current paper develops a mathematical model for fracture propagation pressure accounting for fracture plugging. Key indexes are proposed for fracture plugging optimization based on parameter analysis. Laboratory experiments are conducted to select reasonable LCM type and concentration. The application procedure of the proposed model to drill-in fluid loss control is presented and successfully applied to field case study. The modelling results show that the plugging zone length, width and permeability are the major plugging parameters that affect the fracture propagation pressure. The larger the plugging zone width and the smaller the plugging zone length and permeability, the higher the fracture propagation pressure. Maximum plugging pressure, total loss volume before sealing and D90 degradation rate are proposed as the three indexes for LCM selection. Experimental results show that the combination of rigid granule, fiber and elastic particle can create a synergistic effect to optimize the fracture plugging effect. For the 500 mm width fracture, the optimal concentrations for rigid granule, fiber and elastic particle are 5.0%, 1.5% and 2.5%, respectively. © 2016 Elsevier B.V. All rights reserved.
Keywords: Fractured tight reservoir Drill-in fluid loss Formation damage Fracture propagation pressure Fracture plugging zone Loss control material
1. Introduction Unconventional tight reservoir has become one of the hotspots of reservoir exploration and development, since an increasing number of companies move to the exploitation of more and more challenging oil and gas reservoirs in tighter, deeper and more complex conditions (Kang and Luo, 2007; Economides and Wood, 2009). Globally, the tight reservoirs are mainly located in North America, Latin America, the former Soviet Union, Central Asia, the Middle East and North Africa. In China, there is a wide range of tight reservoir distribution in Sichuan basin. The typical features of Sichuan tight reservoirs are well-developed natural fractures and ultra-low matrix permeability. Natural fractures are beneficial for the economic and efficient development of tight reservoirs.
* Corresponding author. E-mail address: [email protected] (C. Xu). http://dx.doi.org/10.1016/j.jngse.2016.09.059 1875-5100/© 2016 Elsevier B.V. All rights reserved.
However, natural fractures also lead to drill-in fluid loss and induced formation damage. During drill-in fluid loss, the liquid and suspended particles deeply invade into the reservoir and cause severe formation damage (Bennion, 2002; Oliveira et al., 2014; Huang et al., 2015; Kalantariasl et al., 2015). The corresponding formation damage mechanisms mainly include particle invasion, phase trapping damage and rock-fluid incompatibility (Bennion et al., 1999; Mahadevan et al., 2007; Civan, 2008; MirzaeiPaiaman et al., 2012; Naik et al., 2015; Sacramento et al., 2015). Fig. 1 illustrates the effect of drill-in fluid loss on gas production of the fractured tight gas reservoirs in Sichuan basin. Data of 24 gas wells is shown in Fig. 1. All these wells have the same production layer, the same open hole completion type, and the same stimulation operation history (acidizing with compound mud acid). Normally, higher fracture density level should correspond to higher gas production level. However, actual gas production level of some gas wells is severely reduced due to formation damage induced by drill-in fluid loss. The permeability of natural fractures can be
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Fig. 1. Effect of drill-in fluid loss on production (In the column of fracture density level, red color represents high fracture density level, blue color represents medium level and yellow color represents low level. In the columns of expected and actual production level, red color represents production higher than 5 104 m3/d, blue color represents production of (1e5) 104 m3/d and yellow color represents production of (0e1) 104 m3/d “-” means no drill-in fluid loss or no gas production). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
reduced by the particles in the drill-in fluid. The corresponding particle retention mechanisms include size exclusion, bridging and deposition (Yuan et al., 2013; Sandrina and Sarah, 2014; You et al., 2015). Phase trapping damage and rock-fluid incompatibility caused by the filtrate of drill-in fluid can reduce the matrix permeability and fracture productivity significantly (Bennion, 2002). Therefore, only 41.7% of all the wells with high level of fracture density correspond to high production level after drill-in fluid loss; only 37.5% of all the wells with medium level of fracture density achieve medium production level (Fig. 1). During the development of fractured tight reservoir, formation damage prevention and efficient drilling demand high level of lost circulation pressure, which refers to the maximum pressure a wellbore can withstand before lost circulation occurs (Kang et al., 2014a). Several methods are available to increase lost circulation pressure, including stress cage, fracture-closure stress and fracture propagation resistance methods (Aston et al., 2004; Dupriest, 2005; van Oort et al., 2011). The investigations conducted by Drilling Engineer Association (DEA) conducted in the late 1980s laid the groundwork for lost circulation control and fracture propagation prevention by investigating the difference in fracturing behavior
between oil-based and water-based drilling fluids (Whitfill and Nance, 2008). The stress cage method and fracture-closure stress method are developed to strengthen the wellbore and increase lost circulation pressure. The stress cage method aims to create additional hoop stress in the wellbore neighborhood by propping the near-wellbore fractures open with high strength LCM (Wang et al., 2008). The fracture-closure stress method attempts to generate more closure stress by deliberately widening induced fractures and keeping them propped open with sized LCM (Dupriest, 2005). Stabilizing the existing and induced fractures is essential to these three methods (Xu et al., 2014). The stress-intensity factor that affects the fracture propagation behavior can be obtained using 2D boundary element method (Wang et al., 2009). However, it does not provide equations to illustrate how the methods work and guide the LCM selection. Fracture plugging with LCMs are the most common way to control lost circulation. The main fracture plugging mechanisms for lost circulation control are size exclusion and bridging (Zhang et al., 2012). Size exclusion occurs when particle size is larger than fracture width. Bridging means that one section of fracture is plugged by several particles simultaneously. It occurs when two or more particles with sizes smaller than fracture width
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Fig. 2. Configuration of plugging zone and fracture tip.
width and permeability. Three key indexes are proposed and laboratory experiments are conducted for fracture plugging optimization and selection of LCM type and concentration. The structure of this paper is as follows. Section 2 presents the derivation of fracture propagation pressure model accounting for fracture plugging. Parameter analysis is conducted in Section 3 to investigate the effects of plugging parameters on fracture propagation pressure. In Section 4, key indexes are proposed for fracture plugging optimization and LCM selection. The detailed description of experimental procedures is given and experiment results are discussed. Field case study is presented in Section 5 to illustrate how the developed model can be used for drill-in fluid loss control in fractured tight reservoir. Finally, Section 6 delivers the conclusions of the study. 2. Model for fracture propagation pressure accounting for fracture plugging
Fig. 3. Mechanical model of a fracture plugged with LCM.
arrive at the fracture face at the same time. In fractured formation, fracture plugging effect largely depends on the fracture propagation pressure, because most plugging failure is caused by fracture propagation (Morita and Fuh, 2012). However, the effects of the plugging parameters on the fracture propagation pressure are still unclear. To our best knowledge, few papers have been published on fracture plugging optimization for drill-in loss control and formation damage prevention in fractured tight reservoir. In the present work, a mathematical model for fracture propagation pressure accounting for fracture plugging is derived. The solution exhibits the effects of fracture plugging parameters on fracture propagation pressure, including plugging zone length,
After the fracture is plugged, stress-intensity factor which quantifies the intensity of stress singularity at fracture tip has a decisive impact on fracture propagation pressure. The configuration of plugging zone and fracture tip is shown in Fig. 2. Here, w and a represent the width and length of plugging zone, respectively; DL represents the length of fracture tip. The shape of fracture plugging zone is assumed to be rectangle. It is because the width of plugging zone w, which ranges from dozens of micrometers to several millimeters, is much smaller than the length of plugging zone a, which is at least several centimeters (w≪a). Furthermore, the length of plugging zone a is much smaller than the length of fracture it plugged (a<
Fig. 4. Stress-intensity factor determined by: (a) in-situ stress component, (b) wellbore pressure component, and (c) fracture pressure component.
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Note that the sign of stress is positive for tension and negative for compression. The pressure and stress in this paper are effective pressure and effective stress, respectively, which correspond to the deviations from the original formation pressure. If one assumes a line fracture with half-length L, the stressintensity is given by (Morita and Fuh, 2012)
1 KI ¼ pffiffiffiffiffiffi pL
ZL L
rffiffiffiffiffiffiffiffiffiffiffi Lþx dx sðx; 0Þ Lx
(1)
where KI is the stress-intensity factor of fracture tip, s is the stress acting on the fracture surface, and L is the sum of wellbore radius rw, fracture plugging zone length a and fracture tip length DL (Fig. 3). The stress-intensity factor accounting for fracture plugging consists of three components (Fig. 4). The first one KI(A) is determined by the in-situ horizontal stresses sh and sH, the second one KI(B) corresponds to the wellbore pressure Pw, and the third one KI(C) is determined by the fracture pressure Pfin. After the fracture is plugged in the plugging zone, the wellbore pressure propagates from wellbore to fracture tip through the plugging zone. The fracture pressure is the total of the pressure in the plugging zone Pfz and the pressure in the fracture tip Pft (Fig. 5). According to the superposition principle, the stress-intensity factor for a fracture connected with wellbore and plugged in the fracture plugging zone is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin io h pða þ DLÞ 1 þ ð1 sÞ 0:5 þ 0:743ð1 sÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pft Pw L DL arccos þ sh pða þ DLÞFl ðsÞ þ L pffiffiffiffiffiffi L pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P DL L w ft arccos L2 r 2w þ p þ 2L pffiffiffiffiffiffi L a pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rw L DL DLð2L DLÞ þ rw arcsin arcsin L L
KI ¼ Pw
(2) The derivations of the stress-intensity factor accounting for fracture plugging are provided in Appendix A. If the length of fracture plugging zone is relatively small compared to the fracture length, it can be assumed to have a uniform thickness. The relation between plugging zone width and fracture tip length is expressed as (Morita and Fuh, 2012)
DL ¼
wE
4 1 n2
sh Pft
(3)
where w is the width of fracture plugging zone, E is Young's Modulus of formation rock, and n is Poisson ratio.
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With the rising wellbore pressure Pw, the stress-intensity factor KI increases accordingly. Fracture propagation occurs when KI increases to the critical stress-intensity factor KIC. The wellbore pressure corresponding to the critical stress-intensity factor is taken as the fracture propagation pressure. Eq. (2) is a transcendental equation and the parameters including fracture tip pressure Pft, fracture tip lengthDL and distance ratio s are all the functions of wellbore pressure. Bisection algorithm and secant algorithm are adopted and combined to obtain the numerical solution of fracture propagation pressure accounting for fracture plugging. The accuracy of the numerical solution is within the accuracy of 0.1 MPa. Finally, the fracture propagation pressure accounting for fracture plugging is given by
FPP ¼ f ðKIC ; E; v; a; w; rw ; Id ; sH ; sh ; Kz ; Ki ; Pi Þ
(4)
From the FPP model shown in Eq. (2), it can be seen that the plugging parameters, which affect the fracture propagation pressure accounting for fracture plugging, include plugging zone length a, plugging zone width w and plugging zone permeability Kz. Fracture tip pressure Pft and fracture tip length DL also have an impact on the fracture propagation pressure. However, they are expressed as plugging parameters and can be controlled by plugging optimization. 3. Plugging parameter analysis The results of fracture propagation pressure accounting for fracture plugging are calculated from the model developed in Section 2. Effects of fracture plugging parameters on the fracture propagation pressure are analyzed in this section, including fracture plugging zone length, width and permeability. Table 1 lists the basic parameters in the fracture propagation pressure model used for plugging parameter analysis. The parameter values are obtained from the fractured tight reservoir in Sichuan basin (China). Other parameters, such as fracture tip length, fracture tip pressure and total fracture length, are calculated from the model and therefore, not listed in Table 1 as basic parameters. 3.1. Effect of plugging zone length In Fig. 6, the nonlinear relationship between fracture propagation pressure and fracture plugging zone length is presented for different plugging zone widths. For the same plugging zone width, larger plugging zone length leads to lower fracture propagation pressure. Fracture propagation pressure varies faster with larger plugging zone width. It is because the increase of plugging zone length can lead to the decrease of fracture tip length (Eqs. (3) and (A-9)). With a larger fracture tip length, the stress concentration is lower at the fracture tip and the fracture is more relaxed.
Table 1 Basic parameters.
Fig. 5. Mechanical model of a fracture plugged with LCM.
Parameter
Symbol
Value
Unit
Critical stress-intensity factor Young's Modulus Poisson ratio Fracture plugging zone length Fracture plugging zone width Wellbore radius Fracture tip pressure decay distance Maximum horizontal effective stress Minimum horizontal effective stress Fracture plugging zone permeability Formation matrix permeablity
KIC E
15.4 20.89 0.25 200 0.1 101.6 500 14 10 0.5 0.1
MPa*mm0.5 GPa e mm mm mm mm MPa MPa mD mD
n a w rw Id
sH sh Kz Ki
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more rapidly for larger plugging zone length. From Eq. (3), the larger the plugging zone width, the larger the fracture tip length and fracture length. Therefore, fracture propagation pressure changes faster for larger plugging zone width (Fig. 6). Moreover, the increase of plugging zone length can also increase the difficulty for formation damage removal in acidizing operation. Because larger plugging zone length indicates larger drill-in fluid invasion depth, which requires larger acidizing radius to remove it. The plugging zone length is affected by plugging zone strength and plugging efficiency. Fracture plugging zone with low strength can be easily forced into the fracture deeply under wellbore pressure. With higher plugging efficiency, fracture plugging zone is formed faster; therefore its length is smaller.
3.2. Effect of plugging zone width
Fig. 6. FPP versus length of fracture plugging zone with plugging zone width of 0.05, 0.10, 0.20 and 0.30 mm.
Moreover, fracture tip pressure accumulation is a main cause of fracture propagation. With a larger tip length, more fracture wall area is available to dissipate the fracture tip pressure into the matrix to keep a lower fracture tip pressure. It is evident from Fig. 6 that under the effect of fracture plugging zone, the effective propagation pressure for the fracture plugged with different plugging zone length is in the range of 16e34 MPa, which is always larger than the minimum horizontal effective stress (10 MPa in Table 1). For comparison, in the absence of fracture plugging, the wellbore pressure Pw exerts on the fracture tip directly. The fracture propagation pressure FPP is equal to the minimum horizontal principal stress sh. This indicates the benefit of fracture plugging with LCM in preventing further fracture propagation and repeated lost circulation in the subsequent operation. Fig. 7 shows that larger fracture tip length results in lower fracture tip stress-intensity factor. Stress concentration is lower at the fracture tip with larger tip length, thus, the fracture is more relaxed and more difficult to propagate. Therefore, the increase of plugging zone length yields the reduction of fracture propagation pressure. Fig. 7 also shows that the stress-intensity factor alters
Fig. 7. Stress-intensity factor versus fracture tip length with plugging zone length of 200, 300, 400 and 500 mm.
The relationship between fracture propagation pressure and the width of fracture plugging zone is presented in Fig. 8 for different plugging zone lengths. Larger plugging zone width leads to higher fracture propagation pressure. It is because the larger plugging zone width leads to longer fracture tip length (Eq. (3)), and the longer fracture tip length yields lower stress-intensity factor (Fig. 7). It may cause slight fracture propagation by propping the fracture open during the fracture plugging process. However, once the fracture is plugged with LCM and forming stable fracture plugging zone in the fracture, the pressure required for further fracture propagation and repeated lost circulation will be higher than the propagation pressure of fracture plugged with lower plugging zone width. Therefore, the plugged fracture with larger plugging zone width and fracture tip length is more difficult for further fracture propagation. Fracture is propped open by LCMs after fracture is plugged and fluid in the fracture dissipates into formation through fracture face. In this case, fracture closure stress applies directly on the LCMs. Larger plugging zone width results in higher fracture closure stress, so that LCM may be crushed when fracture closure stress exceeds the compressive strength of LCM. Then fracture plugging zone width can be reduced after LCMs are crushed, which can further decrease the fracture propagation pressure. Therefore, LCM size degradation should be avoided to maintain high fracture propagation pressure. The type and combination of LCMs have an important effect on LCM size degradation.
Fig. 8. FPP versus width of fracture plugging zone with plugging zone length of 200, 300, 400 and 500 mm.
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Fig. 9. FPP versus permeability of fracture plugging zone with length of 200, 300, 400 and 500 mm.
Fig. 11. Fracture tip pressure versus permeability of fracture plugging zone for different formation matrix permeability.
3.3. Effect of plugging zone permeability
Moreover, to achieve the best fracture plugging effect, the plugging zone strength should not be less than the fracture propagation pressure. Because the fracture tip pressure can be similar or equal to the wellbore pressure after the plugging zone failure occurs. The optimal fracture propagation pressure cannot be achieved with plugging zone with low strength.
In Figs. 9 and 10, the nonlinear relationship between fracture propagation pressure and plugging zone permeability is presented for different plugging zone lengths and formation matrix permeability. They both show that the increase of fracture plugging zone permeability results in the decrease the fracture propagation pressure. It is because higher plugging zone permeability causes higher fracture tip pressure (Fig. 11). With the increase of fracture tip pressure, the fracture can become inflated and unstable, and tends to propagate. With the plugging zone permeability fixed, larger plugging zone length leads to lower fracture propagation pressure. It is because the increase of plugging zone length has a negative impact on fracture propagation pressure, which can be seen in Fig. 6. In Fig. 10, for the same value of plugging zone permeability, larger formation matrix permeability results in higher fracture propagation pressure. It is because fracture tip pressure is more easily to dissipate into formation and keep it at a lower level with higher formation matrix permeability (Fig. 11). To form the plugging zone with low permeability, the plugging zone should be as tight as possible. LCM type and combination have an important impact on the tightness of fracture plugging zone.
4. Experiments for fracture plugging optimization and LCM selection In this section, three key indexes are proposed for fracture plugging optimization based on the above parameter analysis results. These are the maximum plugging pressure, total loss volume before sealing and D90 degradation rate. Maximum plugging pressure is the maximum pressure a fracture plugging zone can withstand before its strength failure. It is used to evaluate the plugging zone strength which is of great importance to remain the plugging effectiveness and prevent the further invasion of the drill-in fluid. Repeated formation and failure of the plugging zone can result in the increase of the plugging zone length and the reduction of the fracture tip length. Therefore, the fracture plugging zone should keep stable under the positive differential pressure between the wellbore pressure and formation pressure after it is formed in the fracture. The maximum plugging pressure should not be less than the fracture propagation pressure. Total loss volume before sealing is used to evaluate the fracture plugging efficiency and permeability of the plugging zone. Higher loss volume before sealing means lower fracture plugging efficiency and higher plugging zone permeability. Therefore, the total loss volume before sealing should be as low as possible. D90 degradation rate (DDR) is used to describe the particle size decrease due to the compressive failure of rigid particle under formation stress. It is defined as
DDR ¼
Fig. 10. FPP versus permeability of fracture plugging zone for different formation matrix permeability.
D90i D90d 100% D90i
(5)
where D90 is the value of the particle diameter at 90% in the cumulative size distribution, D90i is the initial D90 value before particle size degradation, D90d is the D90 value after particle size degradation. The D90d was measured after performing the test. During the test, the confining stress of 15 MPa was applied on the plugging
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Fig. 12. Schematic diagram of plugging test apparatus. Fig. 14. Acid dissolution rate of different bridging particles (Kang et al., 2014b).
zone for 30 min. During this process, the rigid particle experienced size degradation due to compressive failure. Higher D90 degradation rate can lead to higher reduction of plugging zone width under fracture closure stress, which has a negative impact on the fracture propagation pressure. Therefore, the D90 degradation rate of LCM should be as low as possible.
4.1. Experiment design The schematic diagram of the experimental apparatus is shown in Fig. 12. This apparatus can be used to measure the fracture plugging zone strength, plugging zone permeability and D90 degradation rate. The plugging zone length and width can be monitored in real time during the fracture plugging process. Fluid pressure up to 30 MPa is achievable with this apparatus. Core sample with the diameter of 110 mm and fracture width of 500 mm is used for experimental evaluation. Both the fluid pressure and confining pressure can be controlled and measured independently. Each test in the laboratory experiments is repeated for reliability. Test methodology of the fracture plugging experiments is shown in Fig. 13. The test procedure is as follows: (1) Introduce the drill-in fluid added with LCMs into the fluid container and start the motor to simulate the fluid flow state in drill-in process.
(2) Simulate the drill-in fluid loss and fracture plugging process by applying the flowing pressure in increment of 1 MPa. (3) Note the pressure at which the plugging zone breaks and take the previous pressure point as the maximum plugging pressure. Measure the drill-in loss volume at each pressure point and take the cumulative loss volume when the maximum plugging pressure is achieved as the total loss volume before sealing. (4) Form the fracture plugging zone with the tested maximum plugging pressure again. Then apply the confining pressure of 15 MPa for 30 min. (5) Test the D90 of LCM and calculate the D90 degradation rate. Rigid granule, fiber and elastic particle are the most commonly used LCMs in drill-in fluid loss control (Kang et al., 2014b). In this paper, a series of tests are performed with different combinations and concentrations of the LCM mixed in the drill-in fluid. In the laboratory experiments, sized calcium carbonate is used as rigid granule because of its high acid solution rate (Fig. 14). Cellulosic fiber is used as fiber, and graphite carbon is used as elastic particle. Different combinations and concentrations of rigid granule, fiber and elastic particle used for optimization are shown in Table 2, in which R&F represents the combination of rigid granule and fiber, R&E represents the combination of rigid granule and elastic particle, F&E represents the combination of fiber and elastic particle, and C represents the combination of all the three types of LCM.
4.2. Experimental results and discussion
Fig. 13. Test methodology of the fracture plugging experiments.
The optimal values of maximum plugging pressure, total loss volume before sealing and D90 degradation rate for the four types of LCM combination in Table 2 are presented in Figs. 15e17, respectively. For the pair-wise combinations of LCM in Table 2, experimental results show that the F&E combination without rigid granule has the minimum value of the maximum plugging pressure (Fig. 15). The R&E combination without fiber achieves the maximum total loss volume before sealing and maximum plugging zone permeability (Fig. 16). The above results imply that the major effect of rigid granule is to increase the maximum plugging pressure. And the fiber plays an important role in improving plugging efficiency and reducing plugging zone permeability. From Fig. 17 we can see that the F&E combination also achieves the minimum D90
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Table 2 Different combinations and concentrations for the three types of LCM. Number
CRa
CFa
Number
CRa
CEa
Number
CFa
CEa
Number
CRa
C Fa
CEa
R&F-1 R&F-2 R&F-3 R&F-4 R&F-5 R&F-6 R&F-7 R&F-8 R&F-9
3.0 3.0 3.0 5.0 5.0 5.0 7.0 7.0 7.0
0.5 1.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5
R&E-1 R&E-2 R&E-3 R&E-4 R&E-5 R&E-6 R&E-7 R&E-8 R&E-9
3.0 3.0 3.0 5.0 5.0 5.0 7.0 7.0 7.0
1.5 2.5 3.5 1.5 2.5 3.5 1.5 2.5 3.5
F&E-1 F&E-2 F&E-3 F&E-4 F&E-5 F&E-6 F&E-7 F&E-8 F&E-9
0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5
1.5 2.5 3.5 1.5 2.5 3.5 1.5 2.5 3.5
C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8 C-9
3.0 3.0 3.0 5.0 5.0 5.0 7.0 7.0 7.0
0.5 1.5 2.5 2.5 1.5 0.5 2.5 0.5 1.5
1.5 2.5 3.5 1.5 2.5 3.5 1.5 2.5 3.5
a CR is the ratio of rigid particle mass to the carrying fluid volume, g/100 mL; CF is the ratio of fiber mass to the carrying fluid volume, g/100 mL; CE is the ratio of elastic particle mass to the carrying fluid volume, g/100 mL.
Fig. 17. D90 degradation rate for different LCM combinations. Fig. 15. Maximum plugging pressure for different LCM combinations.
Fig. 16. Total loss volume before sealing and plugging zone permeability for different LCM combinations.
degradation rate. Fig. 18 shows that the D90 degradation rate decreases quickly as mass proportion of elastic particle (graphite carbon) to rigid granule (calcium carbonate) increases. Therefore, elastic particle plays a major role in preventing particle size degradation and minimizing the D90 degradation rate.
Fig. 18. D90 degradation rates with different mass ratios of rigid granule to elastic particle.
Experimental results in Figs. 15e17 indicate that the combination of rigid granule, fiber and elastic particle can create a synergistic effect to achieve the optimal value of the maximum plugging pressure, total loss volume before sealing, and D90 degradation rate. For the fracture with width 500 mm, the optimal concentrations for rigid
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Fig. 19. Application procedure of fracture propagation model to drill-in fluid loss control (‘R’ represents rigid particle, ‘E’ represents elastic particle and ‘F’ represents fiber).
granule, fiber and elastic particle are 5.0%, 1.5% and 2.5%, respectively, based on the experimental results in this study. The selection of optimal LCM type and concentration for drill-in fluid loss control and formation damage prevention is achieved by coupling the developed model and laboratory experiment. First, the required fracture propagation pressure is determined by the collection and analysis of field data. Then, the optimal values for maximum plugging pressure, total loss volume before sealing, plugging zone permeability and D90 degradation rate are determined from the fracture propagation pressure model and field data analysis. Finally, experiments are conducted to select the optimal type and concentration of LCMs. Model application and filed case study by the combination of the developed model and experimental results are described in detail in the next section.
5. Model application and field case study In this section, the application procedure of fracture propagation model to drill-in fluid loss control is proposed. It is then applied to the field case study of drill-in fluid loss control in Sichuan basin (China). The application procedure of fracture propagation model is presented in Fig. 19. First, the field data collection is conducted to determine the required fracture propagation pressure. Then, the optimal value for plugging parameters, including plugging zone length, width, permeability and strength, are determined based on the fracture propagation pressure model accounting for fracture plugging developed in this paper. The plugging zone strength should not be less than the required fracture propagation pressure. Afterwards, laboratory experiments are conducted to determine the type and concentration of LCMs based on the indexes including the maximum plugging pressure, total loss volume before sealing, plugging zone permeability and D90 degradation rate. Finally, field test is conducted and operation results of lost circulation control are collected for further analysis. Tight gas reservoirs in Sichuan, China are characterized by well-developed natural fractures. For the Well 4 in Puguang structure of Sichuan basin drilled with water-based drill-in fluid, severe lost circulations of drill-in fluid with density of 1.74 g/cm3 were observed on the top of the Feixianguan target layer in which
natural fractures are well-developed. Based on the collection and analysis of in-situ stress, formation pressure, drill-in fluid density, pump pressure and the loss rate, 12 MPa of fracture propagation pressure is required for drill-in fluid loss control. The in-situ fracture widths are 50e500 mm. Therefore, the particle size D90 is selected as 500 mm and the plugging zone width is equal to the particle size D90. The minimum value of plugging zone strength should be 12 MPa, which is the required fracture propagation pressure. According to the fracture propagation model in this paper, 0.5mD of plugging zone permeability is required with plugging zone width of 500 mm and plugging zone length of 100 mm. The experimental indexes are as follows: the maximum plugging pressure larger than 12 MPa, plugging zone permeability less than 0.5mD, D90 degradation rate less than 10%, and total loss volume before sealing less than 400 mL (based on Darcy's law and plugging zone permeability). Then, laboratory experiments were conducted to select the type and concentration of LCMs. 6.5% rigid particle (calcium carbonate), 1.0% fiber (cellulosic fiber) and 2.5% resilient particle (graphite carbon) were selected as LCMs and added into the original drill-in fluid. Operationally, a pill of 8.2 m3 volume was placed through the drill bit to plug and seal the loss layer. Lost circulation of drill-in fluid was stopped after 30 min. A test was then conducted and the test pressure can reach up to 12.8 MPa without pressure drop. The loss control remained effective until the well was successfully cemented off one week later. High level gas production rate of 52.21 104 m3/d is obtained for this test well with high fracture density. The above test results of drill-in loss control and formation damage prevention demonstrate good performance of the LCMs selected based on the proposed model.
6. Conclusions (1) Well-developed natural fractures can lead to drill-in fluid loss and induce severe formation damage in fractured tight reservoir. Fracture plugging with LCM is the most common way to control drill-in fluid loss and prevent formation damage. Fracture plugging effect largely depends on the fracture propagation pressure in fractured formation.
C. Xu et al. / Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227
(2) A new model for fracture propagation pressure accounting for fracture plugging is developed in this paper. Effects of fracture plugging parameters on the fracture propagation pressure are analyzed, including fracture plugging zone length, width and permeability. The larger the plugging zone width and the smaller the plugging zone length and permeability, the higher the fracture propagation pressure. (3) The maximum plugging pressure, total loss volume before sealing and D90 degradation rate are the three indexes for fracture plugging optimization. Experimental results indicate that rigid granule is important to the maximum plugging pressure; fiber plays an important role in improving plugging efficiency and reducing plugging zone permeability; elastic particle is essential for the reduction of D90 degradation rate. The combination of rigid granule, fiber and elastic particle can create a synergistic effect to optimize fracture plugging effect. For the fracture with width 500 mm, the optimal concentrations for rigid granule, fiber and elastic particle are 5.0%, 1.5% and 2.5%, respectively. (4) Application procedure of fracture propagation model to drillin fluid loss control is designed and successfully applied to the field case study in Sichuan basin (China). Effective drill-in fluid loss control is observed.
Pft Pw Qin Qout rw s w W
s v
sH sh x
l m DL DDR FPP LCM
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pressure in the fracture tip, MPa wellbore fluid pressure, MPa inflow rate from wellbore to fracture, mm/s outflow rate from fracture to formation matrix, mm/s wellbore radius, mm the ratio of the distance from wellbore wall to fracture tip to the distance from wellbore center to fracture tip fracture plugging zone width, mm fracture width, mm stress acting on the fracture surface, MPa Poisson's ratio maximum horizontal principal stress, MPa minimum horizontal principal stress, MPa coordinate, mm the ratio of in-situ maximum and minimum horizontal stress fluid viscosity, mPa$s fracture tip length, mm D90 degradation rate fracture propagation pressure loss control material
Appendix A. Derivation of the stress-intensity factor after fracture plugging
Acknowledgments The authors gratefully acknowledge the financial support from the Young Scholars Development Fund of Southwest Petroleum University (Grant No.201599010086), the Open Fund (PLN201614) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), and Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology) (PLC201409, PLC201410).
For the condition in Fig. 4(a), the stress-intensity factor determined by in-situ stress component KI(A) is given by (Tada et al., 2000)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KI ðAÞ ¼ KI sH; h ¼ sh Fl ðsÞ pða þ DLÞ
(A-1)
where
Nomenclature
i h Fl ðsÞ ¼ 0:5ð1 lÞð3 sÞ 1 þ 1:243ð1 sÞ3 i h þ lð1 sÞ 0:5 þ 0:743ð1 sÞ2 þ l
a D90
l¼
sH sh
(A-3)
s¼
a þ DL rw þ a þ DL
(A-4)
D90i D90d E H Id Ki Kz KI KIc KI(A) KI(B) KI(C) KI(C1) KI(C2) L Pi Pfin Pfz
fracture plugging zone length, mm the value of the particle diameter at 90% in the cumulative size distribution, mm the initial D90 before particle size degradation, mm the D90 value after particle size degradation, mm rock Young's Modulus, GPa fracture height, mm distance for fracture tip pressure decays to formation pressure, mm formation matrix permeability, mD fracture plugging zone permeability, mD stress-intensity fractor (I type), MPa*mm0.5 critical stress-intensity factor, MPa*mm0.5 stress-intensity factor determined by in-situ stress, MPa*mm0.5 stress-intensity factor determined by wellbore pressure, MPa*mm0.5 stress-intensity factor determined by fracture pressure, MPa*mm0.5 stress-intensity factor determined by pressure in plugging zone, MPa*mm0.5 stress-intensity factor determined by pressure in fracture tip, MPa*mm0.5 total fracture length, mm initial formation pressure, MPa fracture pressure, MPa pressure in the plugging zone, MPa
(A-2)
where sH is the maximum horizontal effective stress, sh is the minimum horizontal effective stress, rw is wellbore radius, a is fracture plugging zone length, and DL is fracture tip length. For the condition in Fig. 4(b), the stress-intensity factor determined by wellbore pressure component KI(B) is given by (Tada et al., 2000)
KI ðBÞ ¼ KI ðPw Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin io h ¼ Pw pða þ DLÞ 1 þ ð1 sÞ 0:5 þ 0:743ð1 sÞ2 (A-5) where Pw is wellbore pressure. For the condition in Fig. 4(c), fracture pressure tends to increase the stress-intensity factor and results in fracture propagation. As shown in Fig. 5, fracture pressure Pfin includes two parts: one is plugging zone pressure Pfz, which is the pressure in the fracture plugging zone; the other is fracture tip pressure Pft, which is the pressure in the fracture tip (Fig. 5).
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The stress-intensity factor in condition of Fig. 4(c) is expressed as
1 KI ðCÞ ¼ pffiffiffiffiffiffi pL
ZL
Pfin Pw
ffi rLffiffiffiffiffiffiffiffiffiffi þx Lx
L
dx
(A-6)
where KI(C) is stress-intensity factor determined by fracture pressure component, and Pfin is fracture pressure. The fracture tip pressure can increase from the original formation pressure and become constant when the inflow rate Qin is equal to the outflow rate Qout (Fig. 5). According to the principle of mass conservation:
Qin ¼ Qout
(A-7)
Kz WH Pw Pft
ma
¼
2Ki DLH Pft Pi
(A-8)
mId
where Qin is inflow rate from wellbore to fracture, Qout is outflow rate from fracture to formation matrix, W is fracture width, H is fracture height, Pft is fracture tip pressure, Kz is plugging zone permeability, Ki is formation matrix permeability, m is fluid viscosity, Pi is original formation pressure, and Id is the distance for fracture tip pressure decays to formation pressure. According to Eq. (A-8), the fracture tip pressure Pft is given by
Pft ¼
w Kz P þ 2DLP a Ki w Id i w Kz þ 2DL a Ki Id
Pw Pft ðjxj rw Þ a
(A-10)
8 Pw Pft > ðjxj rw Þ Pfz ¼ Pw > > a > > > > > < > > > > > > > > :
w Kz 2DL Pw þ P a Ki Id i Pft ¼ w Kz 2DL þ a Ki Id
rw jxj < a (A-11) a jxj L
Accordingly, the stress-intensity factor determined by fracture pressure component consists of two parts. One is determined by fracture tip pressure:
KI ðC1 Þ ¼
Pft Pw pffiffiffiffiffiffi pL 0 1 Lþ Z DL rffiffiffiffiffiffiffiffiffiffiffi ZL rffiffiffiffiffiffiffiffiffiffiffi L þ x L þ x B C @ dx þ dxA Lx Lx L
Pft Pw B ¼ pffiffiffiffiffiffi @ a pL
Zrw LþDL
LþDL
rw
1 rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi L Z DL Lþx Lþx C dx þ dxA ðx rw Þ ðx rw Þ Lx Lx rw
(A-13) The total stress-intensity factor caused by fracture pressure component in the condition of Fig. 4(c) is given by
KI ðCÞ ¼ KI ðC1 Þ þ KI ðC2 Þ
(A-14)
Pft Pw L DL DL L arccos arccos þp KI ðCÞ ¼ L pffiffiffiffiffiffi L L pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pft Pw qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 r 2w DLð2L DLÞ þ 2L pffiffiffiffiffiffi a pL rw L DL þ rw arcsin arcsin L L (A-15) The total stress-intensity induced by in-situ stress, wellbore pressure and fracture pressure is obtained by means of superposition:
KI ¼ KI ðAÞ þ KI ðBÞ þ KI ðCÞ
(A-16)
References
where Pfz is plugging zone pressure, and x is the distance from wellbore center along the fracture direction. Consequently, the fracture pressure Pfin which includes plugging zone pressure and fracture tip pressure is given by
Pfin ¼
0
1 L Zrw rffiffiffiffiffiffiffiffiffiffi Z DL rffiffiffiffiffiffiffiffiffiffi Lþx Lþx C dx þ dxA Lx Lx
(A-9)
Linear decrease is assumed when the wellbore pressure Pw is reduced to the fracture tip pressure Pft through the fracture plugging zone. Therefore, the pressure in the plugging zone is given by
Pfz ¼ Pw
0 Pfz Pw B KI ðC2 Þ ¼ pffiffiffiffiffiffi @ pL
(A-12)
LDL
The other is determined by fracture plugging zone pressure:
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C. Xu et al. / Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227 media. J. Petroleum Sci. Eng. 126 (1), 201e210. Sandrina, R., Sarah, D., 2014. A phenomenological model for particle retention in single, saturated fractures. Groundwater 52 (2), 277e283. Tada, H., Paris, P.C., Irwin, G.R., 2000. The Stress Analysis of Cracks Handbook, third ed. ASME, New York. van Oort, E., Friedheim, J.E., Pierce, T., Lee, J., 2011. Avoiding losses in depleted and weak zones by constantly strengthening wellbores. SPE Drill. Complet. 26 (4), 519e530. Wang, H., Sweatman, R., Engelman, R., Deeg, W., Whitfill, D., Soliman, M.Y., Towler, B.F., 2008. Best practice in understanding and managing lost circulation challenges. SPE Drill. Complet. 23 (2), 168e175. Wang, H., Soliman, M.Y., Towler, B.F., 2009. Investigation of factors for strengthening a wellbore by propping fractures. SPE Drill. Complet. 24 (3), 441e451. Whitfill, D.L., Nance, 2008. DEA 13-Investigation of Lost Circulation with Oil-based
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Muds. Technical/Project Report. Drilling Engineering Association (DEA), Houston, Texas. Xu, C.Y., Kang, Y.L., Tang, L., Li, D.Q., Chen, F., 2014. Prevention of fracture propagation to control drill-in fluid loss in fractured tight gas reservoir. J. Nat. Gas Sci. Eng. 21 (1), 425e432. You, Z., Bedrikovetsky, P., Badalyan, A., Hand, M., 2015. Particle mobilization in porous media: temperature effects on competing electrostatic and drag forces. Geophys. Res. Lett. 42, 2852e2860. Yuan, H., You, Z., Shapiro, A., Bedrikovetsky, P., 2013. Improved population balance model for straining-dominant deep bed filtration using network calculations. Chem. Eng. J. 226, 227e237. Zhang, W., Tang, X., Weisbrod, N., Guan, Z., 2012. A review of colloid transport in fractured rocks. J. Mt. Sci. 9 (6), 770e787.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Fracture plugging optimization for drill-in fluid loss control and formation damage prevention in fractured tight reservoir Chengyuan Xu a, *, Yili Kang a, Fei Chen b, Zhenjiang You c a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China CCDC Changqing Downhole Technology Company, Xi'an, China c Australian School of Petroleum, The University of Adelaide, Adelaide, Australia b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 January 2016 Received in revised form 1 September 2016 Accepted 21 September 2016 Available online 22 September 2016
Well-developed natural fractures are beneficial for the economic and efficient development of tight reservoirs. However, they also lead to drill-in fluid loss and induced severe formation damage. Fracture plugging with loss control material (LCM) is the most common way to control lost circulation. Fracture plugging effect largely depends on the fracture propagation pressure, because plugging failure is mainly caused by fracture propagation in fractured formation. Nevertheless, the effects of the plugging parameters on the fracture propagation pressure are still unclear. The current paper develops a mathematical model for fracture propagation pressure accounting for fracture plugging. Key indexes are proposed for fracture plugging optimization based on parameter analysis. Laboratory experiments are conducted to select reasonable LCM type and concentration. The application procedure of the proposed model to drill-in fluid loss control is presented and successfully applied to field case study. The modelling results show that the plugging zone length, width and permeability are the major plugging parameters that affect the fracture propagation pressure. The larger the plugging zone width and the smaller the plugging zone length and permeability, the higher the fracture propagation pressure. Maximum plugging pressure, total loss volume before sealing and D90 degradation rate are proposed as the three indexes for LCM selection. Experimental results show that the combination of rigid granule, fiber and elastic particle can create a synergistic effect to optimize the fracture plugging effect. For the 500 mm width fracture, the optimal concentrations for rigid granule, fiber and elastic particle are 5.0%, 1.5% and 2.5%, respectively. © 2016 Elsevier B.V. All rights reserved.
Keywords: Fractured tight reservoir Drill-in fluid loss Formation damage Fracture propagation pressure Fracture plugging zone Loss control material
1. Introduction Unconventional tight reservoir has become one of the hotspots of reservoir exploration and development, since an increasing number of companies move to the exploitation of more and more challenging oil and gas reservoirs in tighter, deeper and more complex conditions (Kang and Luo, 2007; Economides and Wood, 2009). Globally, the tight reservoirs are mainly located in North America, Latin America, the former Soviet Union, Central Asia, the Middle East and North Africa. In China, there is a wide range of tight reservoir distribution in Sichuan basin. The typical features of Sichuan tight reservoirs are well-developed natural fractures and ultra-low matrix permeability. Natural fractures are beneficial for the economic and efficient development of tight reservoirs.
* Corresponding author. E-mail address: [email protected] (C. Xu). http://dx.doi.org/10.1016/j.jngse.2016.09.059 1875-5100/© 2016 Elsevier B.V. All rights reserved.
However, natural fractures also lead to drill-in fluid loss and induced formation damage. During drill-in fluid loss, the liquid and suspended particles deeply invade into the reservoir and cause severe formation damage (Bennion, 2002; Oliveira et al., 2014; Huang et al., 2015; Kalantariasl et al., 2015). The corresponding formation damage mechanisms mainly include particle invasion, phase trapping damage and rock-fluid incompatibility (Bennion et al., 1999; Mahadevan et al., 2007; Civan, 2008; MirzaeiPaiaman et al., 2012; Naik et al., 2015; Sacramento et al., 2015). Fig. 1 illustrates the effect of drill-in fluid loss on gas production of the fractured tight gas reservoirs in Sichuan basin. Data of 24 gas wells is shown in Fig. 1. All these wells have the same production layer, the same open hole completion type, and the same stimulation operation history (acidizing with compound mud acid). Normally, higher fracture density level should correspond to higher gas production level. However, actual gas production level of some gas wells is severely reduced due to formation damage induced by drill-in fluid loss. The permeability of natural fractures can be
C. Xu et al. / Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227
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Fig. 1. Effect of drill-in fluid loss on production (In the column of fracture density level, red color represents high fracture density level, blue color represents medium level and yellow color represents low level. In the columns of expected and actual production level, red color represents production higher than 5 104 m3/d, blue color represents production of (1e5) 104 m3/d and yellow color represents production of (0e1) 104 m3/d “-” means no drill-in fluid loss or no gas production). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
reduced by the particles in the drill-in fluid. The corresponding particle retention mechanisms include size exclusion, bridging and deposition (Yuan et al., 2013; Sandrina and Sarah, 2014; You et al., 2015). Phase trapping damage and rock-fluid incompatibility caused by the filtrate of drill-in fluid can reduce the matrix permeability and fracture productivity significantly (Bennion, 2002). Therefore, only 41.7% of all the wells with high level of fracture density correspond to high production level after drill-in fluid loss; only 37.5% of all the wells with medium level of fracture density achieve medium production level (Fig. 1). During the development of fractured tight reservoir, formation damage prevention and efficient drilling demand high level of lost circulation pressure, which refers to the maximum pressure a wellbore can withstand before lost circulation occurs (Kang et al., 2014a). Several methods are available to increase lost circulation pressure, including stress cage, fracture-closure stress and fracture propagation resistance methods (Aston et al., 2004; Dupriest, 2005; van Oort et al., 2011). The investigations conducted by Drilling Engineer Association (DEA) conducted in the late 1980s laid the groundwork for lost circulation control and fracture propagation prevention by investigating the difference in fracturing behavior
between oil-based and water-based drilling fluids (Whitfill and Nance, 2008). The stress cage method and fracture-closure stress method are developed to strengthen the wellbore and increase lost circulation pressure. The stress cage method aims to create additional hoop stress in the wellbore neighborhood by propping the near-wellbore fractures open with high strength LCM (Wang et al., 2008). The fracture-closure stress method attempts to generate more closure stress by deliberately widening induced fractures and keeping them propped open with sized LCM (Dupriest, 2005). Stabilizing the existing and induced fractures is essential to these three methods (Xu et al., 2014). The stress-intensity factor that affects the fracture propagation behavior can be obtained using 2D boundary element method (Wang et al., 2009). However, it does not provide equations to illustrate how the methods work and guide the LCM selection. Fracture plugging with LCMs are the most common way to control lost circulation. The main fracture plugging mechanisms for lost circulation control are size exclusion and bridging (Zhang et al., 2012). Size exclusion occurs when particle size is larger than fracture width. Bridging means that one section of fracture is plugged by several particles simultaneously. It occurs when two or more particles with sizes smaller than fracture width
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Fig. 2. Configuration of plugging zone and fracture tip.
width and permeability. Three key indexes are proposed and laboratory experiments are conducted for fracture plugging optimization and selection of LCM type and concentration. The structure of this paper is as follows. Section 2 presents the derivation of fracture propagation pressure model accounting for fracture plugging. Parameter analysis is conducted in Section 3 to investigate the effects of plugging parameters on fracture propagation pressure. In Section 4, key indexes are proposed for fracture plugging optimization and LCM selection. The detailed description of experimental procedures is given and experiment results are discussed. Field case study is presented in Section 5 to illustrate how the developed model can be used for drill-in fluid loss control in fractured tight reservoir. Finally, Section 6 delivers the conclusions of the study. 2. Model for fracture propagation pressure accounting for fracture plugging
Fig. 3. Mechanical model of a fracture plugged with LCM.
arrive at the fracture face at the same time. In fractured formation, fracture plugging effect largely depends on the fracture propagation pressure, because most plugging failure is caused by fracture propagation (Morita and Fuh, 2012). However, the effects of the plugging parameters on the fracture propagation pressure are still unclear. To our best knowledge, few papers have been published on fracture plugging optimization for drill-in loss control and formation damage prevention in fractured tight reservoir. In the present work, a mathematical model for fracture propagation pressure accounting for fracture plugging is derived. The solution exhibits the effects of fracture plugging parameters on fracture propagation pressure, including plugging zone length,
After the fracture is plugged, stress-intensity factor which quantifies the intensity of stress singularity at fracture tip has a decisive impact on fracture propagation pressure. The configuration of plugging zone and fracture tip is shown in Fig. 2. Here, w and a represent the width and length of plugging zone, respectively; DL represents the length of fracture tip. The shape of fracture plugging zone is assumed to be rectangle. It is because the width of plugging zone w, which ranges from dozens of micrometers to several millimeters, is much smaller than the length of plugging zone a, which is at least several centimeters (w≪a). Furthermore, the length of plugging zone a is much smaller than the length of fracture it plugged (a<
Fig. 4. Stress-intensity factor determined by: (a) in-situ stress component, (b) wellbore pressure component, and (c) fracture pressure component.
C. Xu et al. / Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227
Note that the sign of stress is positive for tension and negative for compression. The pressure and stress in this paper are effective pressure and effective stress, respectively, which correspond to the deviations from the original formation pressure. If one assumes a line fracture with half-length L, the stressintensity is given by (Morita and Fuh, 2012)
1 KI ¼ pffiffiffiffiffiffi pL
ZL L
rffiffiffiffiffiffiffiffiffiffiffi Lþx dx sðx; 0Þ Lx
(1)
where KI is the stress-intensity factor of fracture tip, s is the stress acting on the fracture surface, and L is the sum of wellbore radius rw, fracture plugging zone length a and fracture tip length DL (Fig. 3). The stress-intensity factor accounting for fracture plugging consists of three components (Fig. 4). The first one KI(A) is determined by the in-situ horizontal stresses sh and sH, the second one KI(B) corresponds to the wellbore pressure Pw, and the third one KI(C) is determined by the fracture pressure Pfin. After the fracture is plugged in the plugging zone, the wellbore pressure propagates from wellbore to fracture tip through the plugging zone. The fracture pressure is the total of the pressure in the plugging zone Pfz and the pressure in the fracture tip Pft (Fig. 5). According to the superposition principle, the stress-intensity factor for a fracture connected with wellbore and plugged in the fracture plugging zone is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin io h pða þ DLÞ 1 þ ð1 sÞ 0:5 þ 0:743ð1 sÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pft Pw L DL arccos þ sh pða þ DLÞFl ðsÞ þ L pffiffiffiffiffiffi L pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P DL L w ft arccos L2 r 2w þ p þ 2L pffiffiffiffiffiffi L a pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rw L DL DLð2L DLÞ þ rw arcsin arcsin L L
KI ¼ Pw
(2) The derivations of the stress-intensity factor accounting for fracture plugging are provided in Appendix A. If the length of fracture plugging zone is relatively small compared to the fracture length, it can be assumed to have a uniform thickness. The relation between plugging zone width and fracture tip length is expressed as (Morita and Fuh, 2012)
DL ¼
wE
4 1 n2
sh Pft
(3)
where w is the width of fracture plugging zone, E is Young's Modulus of formation rock, and n is Poisson ratio.
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With the rising wellbore pressure Pw, the stress-intensity factor KI increases accordingly. Fracture propagation occurs when KI increases to the critical stress-intensity factor KIC. The wellbore pressure corresponding to the critical stress-intensity factor is taken as the fracture propagation pressure. Eq. (2) is a transcendental equation and the parameters including fracture tip pressure Pft, fracture tip lengthDL and distance ratio s are all the functions of wellbore pressure. Bisection algorithm and secant algorithm are adopted and combined to obtain the numerical solution of fracture propagation pressure accounting for fracture plugging. The accuracy of the numerical solution is within the accuracy of 0.1 MPa. Finally, the fracture propagation pressure accounting for fracture plugging is given by
FPP ¼ f ðKIC ; E; v; a; w; rw ; Id ; sH ; sh ; Kz ; Ki ; Pi Þ
(4)
From the FPP model shown in Eq. (2), it can be seen that the plugging parameters, which affect the fracture propagation pressure accounting for fracture plugging, include plugging zone length a, plugging zone width w and plugging zone permeability Kz. Fracture tip pressure Pft and fracture tip length DL also have an impact on the fracture propagation pressure. However, they are expressed as plugging parameters and can be controlled by plugging optimization. 3. Plugging parameter analysis The results of fracture propagation pressure accounting for fracture plugging are calculated from the model developed in Section 2. Effects of fracture plugging parameters on the fracture propagation pressure are analyzed in this section, including fracture plugging zone length, width and permeability. Table 1 lists the basic parameters in the fracture propagation pressure model used for plugging parameter analysis. The parameter values are obtained from the fractured tight reservoir in Sichuan basin (China). Other parameters, such as fracture tip length, fracture tip pressure and total fracture length, are calculated from the model and therefore, not listed in Table 1 as basic parameters. 3.1. Effect of plugging zone length In Fig. 6, the nonlinear relationship between fracture propagation pressure and fracture plugging zone length is presented for different plugging zone widths. For the same plugging zone width, larger plugging zone length leads to lower fracture propagation pressure. Fracture propagation pressure varies faster with larger plugging zone width. It is because the increase of plugging zone length can lead to the decrease of fracture tip length (Eqs. (3) and (A-9)). With a larger fracture tip length, the stress concentration is lower at the fracture tip and the fracture is more relaxed.
Table 1 Basic parameters.
Fig. 5. Mechanical model of a fracture plugged with LCM.
Parameter
Symbol
Value
Unit
Critical stress-intensity factor Young's Modulus Poisson ratio Fracture plugging zone length Fracture plugging zone width Wellbore radius Fracture tip pressure decay distance Maximum horizontal effective stress Minimum horizontal effective stress Fracture plugging zone permeability Formation matrix permeablity
KIC E
15.4 20.89 0.25 200 0.1 101.6 500 14 10 0.5 0.1
MPa*mm0.5 GPa e mm mm mm mm MPa MPa mD mD
n a w rw Id
sH sh Kz Ki
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more rapidly for larger plugging zone length. From Eq. (3), the larger the plugging zone width, the larger the fracture tip length and fracture length. Therefore, fracture propagation pressure changes faster for larger plugging zone width (Fig. 6). Moreover, the increase of plugging zone length can also increase the difficulty for formation damage removal in acidizing operation. Because larger plugging zone length indicates larger drill-in fluid invasion depth, which requires larger acidizing radius to remove it. The plugging zone length is affected by plugging zone strength and plugging efficiency. Fracture plugging zone with low strength can be easily forced into the fracture deeply under wellbore pressure. With higher plugging efficiency, fracture plugging zone is formed faster; therefore its length is smaller.
3.2. Effect of plugging zone width
Fig. 6. FPP versus length of fracture plugging zone with plugging zone width of 0.05, 0.10, 0.20 and 0.30 mm.
Moreover, fracture tip pressure accumulation is a main cause of fracture propagation. With a larger tip length, more fracture wall area is available to dissipate the fracture tip pressure into the matrix to keep a lower fracture tip pressure. It is evident from Fig. 6 that under the effect of fracture plugging zone, the effective propagation pressure for the fracture plugged with different plugging zone length is in the range of 16e34 MPa, which is always larger than the minimum horizontal effective stress (10 MPa in Table 1). For comparison, in the absence of fracture plugging, the wellbore pressure Pw exerts on the fracture tip directly. The fracture propagation pressure FPP is equal to the minimum horizontal principal stress sh. This indicates the benefit of fracture plugging with LCM in preventing further fracture propagation and repeated lost circulation in the subsequent operation. Fig. 7 shows that larger fracture tip length results in lower fracture tip stress-intensity factor. Stress concentration is lower at the fracture tip with larger tip length, thus, the fracture is more relaxed and more difficult to propagate. Therefore, the increase of plugging zone length yields the reduction of fracture propagation pressure. Fig. 7 also shows that the stress-intensity factor alters
Fig. 7. Stress-intensity factor versus fracture tip length with plugging zone length of 200, 300, 400 and 500 mm.
The relationship between fracture propagation pressure and the width of fracture plugging zone is presented in Fig. 8 for different plugging zone lengths. Larger plugging zone width leads to higher fracture propagation pressure. It is because the larger plugging zone width leads to longer fracture tip length (Eq. (3)), and the longer fracture tip length yields lower stress-intensity factor (Fig. 7). It may cause slight fracture propagation by propping the fracture open during the fracture plugging process. However, once the fracture is plugged with LCM and forming stable fracture plugging zone in the fracture, the pressure required for further fracture propagation and repeated lost circulation will be higher than the propagation pressure of fracture plugged with lower plugging zone width. Therefore, the plugged fracture with larger plugging zone width and fracture tip length is more difficult for further fracture propagation. Fracture is propped open by LCMs after fracture is plugged and fluid in the fracture dissipates into formation through fracture face. In this case, fracture closure stress applies directly on the LCMs. Larger plugging zone width results in higher fracture closure stress, so that LCM may be crushed when fracture closure stress exceeds the compressive strength of LCM. Then fracture plugging zone width can be reduced after LCMs are crushed, which can further decrease the fracture propagation pressure. Therefore, LCM size degradation should be avoided to maintain high fracture propagation pressure. The type and combination of LCMs have an important effect on LCM size degradation.
Fig. 8. FPP versus width of fracture plugging zone with plugging zone length of 200, 300, 400 and 500 mm.
C. Xu et al. / Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227
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Fig. 9. FPP versus permeability of fracture plugging zone with length of 200, 300, 400 and 500 mm.
Fig. 11. Fracture tip pressure versus permeability of fracture plugging zone for different formation matrix permeability.
3.3. Effect of plugging zone permeability
Moreover, to achieve the best fracture plugging effect, the plugging zone strength should not be less than the fracture propagation pressure. Because the fracture tip pressure can be similar or equal to the wellbore pressure after the plugging zone failure occurs. The optimal fracture propagation pressure cannot be achieved with plugging zone with low strength.
In Figs. 9 and 10, the nonlinear relationship between fracture propagation pressure and plugging zone permeability is presented for different plugging zone lengths and formation matrix permeability. They both show that the increase of fracture plugging zone permeability results in the decrease the fracture propagation pressure. It is because higher plugging zone permeability causes higher fracture tip pressure (Fig. 11). With the increase of fracture tip pressure, the fracture can become inflated and unstable, and tends to propagate. With the plugging zone permeability fixed, larger plugging zone length leads to lower fracture propagation pressure. It is because the increase of plugging zone length has a negative impact on fracture propagation pressure, which can be seen in Fig. 6. In Fig. 10, for the same value of plugging zone permeability, larger formation matrix permeability results in higher fracture propagation pressure. It is because fracture tip pressure is more easily to dissipate into formation and keep it at a lower level with higher formation matrix permeability (Fig. 11). To form the plugging zone with low permeability, the plugging zone should be as tight as possible. LCM type and combination have an important impact on the tightness of fracture plugging zone.
4. Experiments for fracture plugging optimization and LCM selection In this section, three key indexes are proposed for fracture plugging optimization based on the above parameter analysis results. These are the maximum plugging pressure, total loss volume before sealing and D90 degradation rate. Maximum plugging pressure is the maximum pressure a fracture plugging zone can withstand before its strength failure. It is used to evaluate the plugging zone strength which is of great importance to remain the plugging effectiveness and prevent the further invasion of the drill-in fluid. Repeated formation and failure of the plugging zone can result in the increase of the plugging zone length and the reduction of the fracture tip length. Therefore, the fracture plugging zone should keep stable under the positive differential pressure between the wellbore pressure and formation pressure after it is formed in the fracture. The maximum plugging pressure should not be less than the fracture propagation pressure. Total loss volume before sealing is used to evaluate the fracture plugging efficiency and permeability of the plugging zone. Higher loss volume before sealing means lower fracture plugging efficiency and higher plugging zone permeability. Therefore, the total loss volume before sealing should be as low as possible. D90 degradation rate (DDR) is used to describe the particle size decrease due to the compressive failure of rigid particle under formation stress. It is defined as
DDR ¼
Fig. 10. FPP versus permeability of fracture plugging zone for different formation matrix permeability.
D90i D90d 100% D90i
(5)
where D90 is the value of the particle diameter at 90% in the cumulative size distribution, D90i is the initial D90 value before particle size degradation, D90d is the D90 value after particle size degradation. The D90d was measured after performing the test. During the test, the confining stress of 15 MPa was applied on the plugging
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Fig. 12. Schematic diagram of plugging test apparatus. Fig. 14. Acid dissolution rate of different bridging particles (Kang et al., 2014b).
zone for 30 min. During this process, the rigid particle experienced size degradation due to compressive failure. Higher D90 degradation rate can lead to higher reduction of plugging zone width under fracture closure stress, which has a negative impact on the fracture propagation pressure. Therefore, the D90 degradation rate of LCM should be as low as possible.
4.1. Experiment design The schematic diagram of the experimental apparatus is shown in Fig. 12. This apparatus can be used to measure the fracture plugging zone strength, plugging zone permeability and D90 degradation rate. The plugging zone length and width can be monitored in real time during the fracture plugging process. Fluid pressure up to 30 MPa is achievable with this apparatus. Core sample with the diameter of 110 mm and fracture width of 500 mm is used for experimental evaluation. Both the fluid pressure and confining pressure can be controlled and measured independently. Each test in the laboratory experiments is repeated for reliability. Test methodology of the fracture plugging experiments is shown in Fig. 13. The test procedure is as follows: (1) Introduce the drill-in fluid added with LCMs into the fluid container and start the motor to simulate the fluid flow state in drill-in process.
(2) Simulate the drill-in fluid loss and fracture plugging process by applying the flowing pressure in increment of 1 MPa. (3) Note the pressure at which the plugging zone breaks and take the previous pressure point as the maximum plugging pressure. Measure the drill-in loss volume at each pressure point and take the cumulative loss volume when the maximum plugging pressure is achieved as the total loss volume before sealing. (4) Form the fracture plugging zone with the tested maximum plugging pressure again. Then apply the confining pressure of 15 MPa for 30 min. (5) Test the D90 of LCM and calculate the D90 degradation rate. Rigid granule, fiber and elastic particle are the most commonly used LCMs in drill-in fluid loss control (Kang et al., 2014b). In this paper, a series of tests are performed with different combinations and concentrations of the LCM mixed in the drill-in fluid. In the laboratory experiments, sized calcium carbonate is used as rigid granule because of its high acid solution rate (Fig. 14). Cellulosic fiber is used as fiber, and graphite carbon is used as elastic particle. Different combinations and concentrations of rigid granule, fiber and elastic particle used for optimization are shown in Table 2, in which R&F represents the combination of rigid granule and fiber, R&E represents the combination of rigid granule and elastic particle, F&E represents the combination of fiber and elastic particle, and C represents the combination of all the three types of LCM.
4.2. Experimental results and discussion
Fig. 13. Test methodology of the fracture plugging experiments.
The optimal values of maximum plugging pressure, total loss volume before sealing and D90 degradation rate for the four types of LCM combination in Table 2 are presented in Figs. 15e17, respectively. For the pair-wise combinations of LCM in Table 2, experimental results show that the F&E combination without rigid granule has the minimum value of the maximum plugging pressure (Fig. 15). The R&E combination without fiber achieves the maximum total loss volume before sealing and maximum plugging zone permeability (Fig. 16). The above results imply that the major effect of rigid granule is to increase the maximum plugging pressure. And the fiber plays an important role in improving plugging efficiency and reducing plugging zone permeability. From Fig. 17 we can see that the F&E combination also achieves the minimum D90
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Table 2 Different combinations and concentrations for the three types of LCM. Number
CRa
CFa
Number
CRa
CEa
Number
CFa
CEa
Number
CRa
C Fa
CEa
R&F-1 R&F-2 R&F-3 R&F-4 R&F-5 R&F-6 R&F-7 R&F-8 R&F-9
3.0 3.0 3.0 5.0 5.0 5.0 7.0 7.0 7.0
0.5 1.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5
R&E-1 R&E-2 R&E-3 R&E-4 R&E-5 R&E-6 R&E-7 R&E-8 R&E-9
3.0 3.0 3.0 5.0 5.0 5.0 7.0 7.0 7.0
1.5 2.5 3.5 1.5 2.5 3.5 1.5 2.5 3.5
F&E-1 F&E-2 F&E-3 F&E-4 F&E-5 F&E-6 F&E-7 F&E-8 F&E-9
0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5
1.5 2.5 3.5 1.5 2.5 3.5 1.5 2.5 3.5
C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8 C-9
3.0 3.0 3.0 5.0 5.0 5.0 7.0 7.0 7.0
0.5 1.5 2.5 2.5 1.5 0.5 2.5 0.5 1.5
1.5 2.5 3.5 1.5 2.5 3.5 1.5 2.5 3.5
a CR is the ratio of rigid particle mass to the carrying fluid volume, g/100 mL; CF is the ratio of fiber mass to the carrying fluid volume, g/100 mL; CE is the ratio of elastic particle mass to the carrying fluid volume, g/100 mL.
Fig. 17. D90 degradation rate for different LCM combinations. Fig. 15. Maximum plugging pressure for different LCM combinations.
Fig. 16. Total loss volume before sealing and plugging zone permeability for different LCM combinations.
degradation rate. Fig. 18 shows that the D90 degradation rate decreases quickly as mass proportion of elastic particle (graphite carbon) to rigid granule (calcium carbonate) increases. Therefore, elastic particle plays a major role in preventing particle size degradation and minimizing the D90 degradation rate.
Fig. 18. D90 degradation rates with different mass ratios of rigid granule to elastic particle.
Experimental results in Figs. 15e17 indicate that the combination of rigid granule, fiber and elastic particle can create a synergistic effect to achieve the optimal value of the maximum plugging pressure, total loss volume before sealing, and D90 degradation rate. For the fracture with width 500 mm, the optimal concentrations for rigid
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Fig. 19. Application procedure of fracture propagation model to drill-in fluid loss control (‘R’ represents rigid particle, ‘E’ represents elastic particle and ‘F’ represents fiber).
granule, fiber and elastic particle are 5.0%, 1.5% and 2.5%, respectively, based on the experimental results in this study. The selection of optimal LCM type and concentration for drill-in fluid loss control and formation damage prevention is achieved by coupling the developed model and laboratory experiment. First, the required fracture propagation pressure is determined by the collection and analysis of field data. Then, the optimal values for maximum plugging pressure, total loss volume before sealing, plugging zone permeability and D90 degradation rate are determined from the fracture propagation pressure model and field data analysis. Finally, experiments are conducted to select the optimal type and concentration of LCMs. Model application and filed case study by the combination of the developed model and experimental results are described in detail in the next section.
5. Model application and field case study In this section, the application procedure of fracture propagation model to drill-in fluid loss control is proposed. It is then applied to the field case study of drill-in fluid loss control in Sichuan basin (China). The application procedure of fracture propagation model is presented in Fig. 19. First, the field data collection is conducted to determine the required fracture propagation pressure. Then, the optimal value for plugging parameters, including plugging zone length, width, permeability and strength, are determined based on the fracture propagation pressure model accounting for fracture plugging developed in this paper. The plugging zone strength should not be less than the required fracture propagation pressure. Afterwards, laboratory experiments are conducted to determine the type and concentration of LCMs based on the indexes including the maximum plugging pressure, total loss volume before sealing, plugging zone permeability and D90 degradation rate. Finally, field test is conducted and operation results of lost circulation control are collected for further analysis. Tight gas reservoirs in Sichuan, China are characterized by well-developed natural fractures. For the Well 4 in Puguang structure of Sichuan basin drilled with water-based drill-in fluid, severe lost circulations of drill-in fluid with density of 1.74 g/cm3 were observed on the top of the Feixianguan target layer in which
natural fractures are well-developed. Based on the collection and analysis of in-situ stress, formation pressure, drill-in fluid density, pump pressure and the loss rate, 12 MPa of fracture propagation pressure is required for drill-in fluid loss control. The in-situ fracture widths are 50e500 mm. Therefore, the particle size D90 is selected as 500 mm and the plugging zone width is equal to the particle size D90. The minimum value of plugging zone strength should be 12 MPa, which is the required fracture propagation pressure. According to the fracture propagation model in this paper, 0.5mD of plugging zone permeability is required with plugging zone width of 500 mm and plugging zone length of 100 mm. The experimental indexes are as follows: the maximum plugging pressure larger than 12 MPa, plugging zone permeability less than 0.5mD, D90 degradation rate less than 10%, and total loss volume before sealing less than 400 mL (based on Darcy's law and plugging zone permeability). Then, laboratory experiments were conducted to select the type and concentration of LCMs. 6.5% rigid particle (calcium carbonate), 1.0% fiber (cellulosic fiber) and 2.5% resilient particle (graphite carbon) were selected as LCMs and added into the original drill-in fluid. Operationally, a pill of 8.2 m3 volume was placed through the drill bit to plug and seal the loss layer. Lost circulation of drill-in fluid was stopped after 30 min. A test was then conducted and the test pressure can reach up to 12.8 MPa without pressure drop. The loss control remained effective until the well was successfully cemented off one week later. High level gas production rate of 52.21 104 m3/d is obtained for this test well with high fracture density. The above test results of drill-in loss control and formation damage prevention demonstrate good performance of the LCMs selected based on the proposed model.
6. Conclusions (1) Well-developed natural fractures can lead to drill-in fluid loss and induce severe formation damage in fractured tight reservoir. Fracture plugging with LCM is the most common way to control drill-in fluid loss and prevent formation damage. Fracture plugging effect largely depends on the fracture propagation pressure in fractured formation.
C. Xu et al. / Journal of Natural Gas Science and Engineering 35 (2016) 1216e1227
(2) A new model for fracture propagation pressure accounting for fracture plugging is developed in this paper. Effects of fracture plugging parameters on the fracture propagation pressure are analyzed, including fracture plugging zone length, width and permeability. The larger the plugging zone width and the smaller the plugging zone length and permeability, the higher the fracture propagation pressure. (3) The maximum plugging pressure, total loss volume before sealing and D90 degradation rate are the three indexes for fracture plugging optimization. Experimental results indicate that rigid granule is important to the maximum plugging pressure; fiber plays an important role in improving plugging efficiency and reducing plugging zone permeability; elastic particle is essential for the reduction of D90 degradation rate. The combination of rigid granule, fiber and elastic particle can create a synergistic effect to optimize fracture plugging effect. For the fracture with width 500 mm, the optimal concentrations for rigid granule, fiber and elastic particle are 5.0%, 1.5% and 2.5%, respectively. (4) Application procedure of fracture propagation model to drillin fluid loss control is designed and successfully applied to the field case study in Sichuan basin (China). Effective drill-in fluid loss control is observed.
Pft Pw Qin Qout rw s w W
s v
sH sh x
l m DL DDR FPP LCM
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pressure in the fracture tip, MPa wellbore fluid pressure, MPa inflow rate from wellbore to fracture, mm/s outflow rate from fracture to formation matrix, mm/s wellbore radius, mm the ratio of the distance from wellbore wall to fracture tip to the distance from wellbore center to fracture tip fracture plugging zone width, mm fracture width, mm stress acting on the fracture surface, MPa Poisson's ratio maximum horizontal principal stress, MPa minimum horizontal principal stress, MPa coordinate, mm the ratio of in-situ maximum and minimum horizontal stress fluid viscosity, mPa$s fracture tip length, mm D90 degradation rate fracture propagation pressure loss control material
Appendix A. Derivation of the stress-intensity factor after fracture plugging
Acknowledgments The authors gratefully acknowledge the financial support from the Young Scholars Development Fund of Southwest Petroleum University (Grant No.201599010086), the Open Fund (PLN201614) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), and Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology) (PLC201409, PLC201410).
For the condition in Fig. 4(a), the stress-intensity factor determined by in-situ stress component KI(A) is given by (Tada et al., 2000)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KI ðAÞ ¼ KI sH; h ¼ sh Fl ðsÞ pða þ DLÞ
(A-1)
where
Nomenclature
i h Fl ðsÞ ¼ 0:5ð1 lÞð3 sÞ 1 þ 1:243ð1 sÞ3 i h þ lð1 sÞ 0:5 þ 0:743ð1 sÞ2 þ l
a D90
l¼
sH sh
(A-3)
s¼
a þ DL rw þ a þ DL
(A-4)
D90i D90d E H Id Ki Kz KI KIc KI(A) KI(B) KI(C) KI(C1) KI(C2) L Pi Pfin Pfz
fracture plugging zone length, mm the value of the particle diameter at 90% in the cumulative size distribution, mm the initial D90 before particle size degradation, mm the D90 value after particle size degradation, mm rock Young's Modulus, GPa fracture height, mm distance for fracture tip pressure decays to formation pressure, mm formation matrix permeability, mD fracture plugging zone permeability, mD stress-intensity fractor (I type), MPa*mm0.5 critical stress-intensity factor, MPa*mm0.5 stress-intensity factor determined by in-situ stress, MPa*mm0.5 stress-intensity factor determined by wellbore pressure, MPa*mm0.5 stress-intensity factor determined by fracture pressure, MPa*mm0.5 stress-intensity factor determined by pressure in plugging zone, MPa*mm0.5 stress-intensity factor determined by pressure in fracture tip, MPa*mm0.5 total fracture length, mm initial formation pressure, MPa fracture pressure, MPa pressure in the plugging zone, MPa
(A-2)
where sH is the maximum horizontal effective stress, sh is the minimum horizontal effective stress, rw is wellbore radius, a is fracture plugging zone length, and DL is fracture tip length. For the condition in Fig. 4(b), the stress-intensity factor determined by wellbore pressure component KI(B) is given by (Tada et al., 2000)
KI ðBÞ ¼ KI ðPw Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin io h ¼ Pw pða þ DLÞ 1 þ ð1 sÞ 0:5 þ 0:743ð1 sÞ2 (A-5) where Pw is wellbore pressure. For the condition in Fig. 4(c), fracture pressure tends to increase the stress-intensity factor and results in fracture propagation. As shown in Fig. 5, fracture pressure Pfin includes two parts: one is plugging zone pressure Pfz, which is the pressure in the fracture plugging zone; the other is fracture tip pressure Pft, which is the pressure in the fracture tip (Fig. 5).
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The stress-intensity factor in condition of Fig. 4(c) is expressed as
1 KI ðCÞ ¼ pffiffiffiffiffiffi pL
ZL
Pfin Pw
ffi rLffiffiffiffiffiffiffiffiffiffi þx Lx
L
dx
(A-6)
where KI(C) is stress-intensity factor determined by fracture pressure component, and Pfin is fracture pressure. The fracture tip pressure can increase from the original formation pressure and become constant when the inflow rate Qin is equal to the outflow rate Qout (Fig. 5). According to the principle of mass conservation:
Qin ¼ Qout
(A-7)
Kz WH Pw Pft
ma
¼
2Ki DLH Pft Pi
(A-8)
mId
where Qin is inflow rate from wellbore to fracture, Qout is outflow rate from fracture to formation matrix, W is fracture width, H is fracture height, Pft is fracture tip pressure, Kz is plugging zone permeability, Ki is formation matrix permeability, m is fluid viscosity, Pi is original formation pressure, and Id is the distance for fracture tip pressure decays to formation pressure. According to Eq. (A-8), the fracture tip pressure Pft is given by
Pft ¼
w Kz P þ 2DLP a Ki w Id i w Kz þ 2DL a Ki Id
Pw Pft ðjxj rw Þ a
(A-10)
8 Pw Pft > ðjxj rw Þ Pfz ¼ Pw > > a > > > > > < > > > > > > > > :
w Kz 2DL Pw þ P a Ki Id i Pft ¼ w Kz 2DL þ a Ki Id
rw jxj < a (A-11) a jxj L
Accordingly, the stress-intensity factor determined by fracture pressure component consists of two parts. One is determined by fracture tip pressure:
KI ðC1 Þ ¼
Pft Pw pffiffiffiffiffiffi pL 0 1 Lþ Z DL rffiffiffiffiffiffiffiffiffiffiffi ZL rffiffiffiffiffiffiffiffiffiffiffi L þ x L þ x B C @ dx þ dxA Lx Lx L
Pft Pw B ¼ pffiffiffiffiffiffi @ a pL
Zrw LþDL
LþDL
rw
1 rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi L Z DL Lþx Lþx C dx þ dxA ðx rw Þ ðx rw Þ Lx Lx rw
(A-13) The total stress-intensity factor caused by fracture pressure component in the condition of Fig. 4(c) is given by
KI ðCÞ ¼ KI ðC1 Þ þ KI ðC2 Þ
(A-14)
Pft Pw L DL DL L arccos arccos þp KI ðCÞ ¼ L pffiffiffiffiffiffi L L pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pft Pw qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 r 2w DLð2L DLÞ þ 2L pffiffiffiffiffiffi a pL rw L DL þ rw arcsin arcsin L L (A-15) The total stress-intensity induced by in-situ stress, wellbore pressure and fracture pressure is obtained by means of superposition:
KI ¼ KI ðAÞ þ KI ðBÞ þ KI ðCÞ
(A-16)
References
where Pfz is plugging zone pressure, and x is the distance from wellbore center along the fracture direction. Consequently, the fracture pressure Pfin which includes plugging zone pressure and fracture tip pressure is given by
Pfin ¼
0
1 L Zrw rffiffiffiffiffiffiffiffiffiffi Z DL rffiffiffiffiffiffiffiffiffiffi Lþx Lþx C dx þ dxA Lx Lx
(A-9)
Linear decrease is assumed when the wellbore pressure Pw is reduced to the fracture tip pressure Pft through the fracture plugging zone. Therefore, the pressure in the plugging zone is given by
Pfz ¼ Pw
0 Pfz Pw B KI ðC2 Þ ¼ pffiffiffiffiffiffi @ pL
(A-12)
LDL
The other is determined by fracture plugging zone pressure:
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