Rock Mech Rock Eng DOI 10.1007/s00603-017-1253-8
ORIGINAL PAPER
Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks Debanjan Guha Roy1 • T. N. Singh2 • J. Kodikara3 • Ratan Das2
Received: 25 July 2016 / Accepted: 1 June 2017 Ó Springer-Verlag GmbH Austria 2017
Abstract Fracture and mechanical properties of the water saturated sedimentary rocks have been experimentally investigated in the present paper. Three types of sandstones and one type of shale were saturated in water for different periods of time. They were then tested for their index geomechanical properties such as Brazilian tensile strength (BTS), Young’s modulus (YM), P-wave velocity and all pure and mixed-mode fracture toughness (FT). FT was measured using semicircular bend specimens in a threepoint bend set-up. All the geomechanical and fracture properties of the saturated rocks were compared together to investigate their interrelations. Further, statistical methods were employed to measure the statistical significance of such relationships. Next, three types of fracture criteria were compared with the present experimental results. Results show that degree of saturation has significant effect on both the strength and fracture properties of sedimentary rock. A general decrease in the mechanical and fracture toughness was noticed with increasing saturation levels. But, t-test confirmed that FT, BTS, P-wave velocity and YM are strongly dependent on each other and linear relationships exist across all the saturation values. Calculation of the ‘degradation degree’ (DD) appeared to be a difficult task for all types of sedimentary rocks. While in sandstone, both the BTS and mode-I FT overestimated the DD & Debanjan Guha Roy
[email protected] 1
IITB-Monash Research Academy, Indian Institute of Technology Bombay, Mumbai 400076, India
2
Department of Earth Sciences, Indian Institute of Technology Bombay, Mumbai 400076, India
3
Department of Civil Engineering, Monash University, Clayton, VIC 3800, Australia
calculated by YM method, in shale BTS was found to give a closure value. Keywords Fracture toughness Tensile strength Young’s modulus Saturation P-wave velocity List of symbols KIC Mode-I fracture toughness (mode-I FT) KIIC Mode-II fracture toughness (mode-II FT) Keff Mixed-mode fracture toughness (mixed-mode FT) KIC0 Mode-I fracture toughness of a dry rock KIIC0 Mode-II fracture toughness of a dry rock Keff0 Mixed-mode fracture toughness of a dry rock s Span of support roller rt Brazilian tensile strength (BTS) rt0 Brazilian tensile strength of a dry rock E Young’s modulus (YM) E0 Young’s modulus of a dry rock Vp P-wave velocity Vp0 P-wave velocity of a dry rock CMOD Crack mouth opening displacement Pmax Failure load of the semicircular bend specimen P Failure load of tensile disc t Thickness of the tensile disc D Diameter of the tensile disc a Notch length of fracture toughness specimen R Radius of fracture toughness specimen B Thickness of fracture toughness specimen Y0 Mode-I non-dimensional stress intensity factor Y00 Mode-II non-dimensional stress intensity factor DYM Young’s modulus-based degradation degree DBTS BTS-based degradation degree DFT Fracture toughness-based degradation degree
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1 Introduction Hydraulic fracturing is an important fracture creation technique applicable in a wide variety of mining, civil and reservoir engineering problems. It is used widely in the enhanced geothermal systems (EGS), in the in situ stress measurement, grouting, caving, brown field rejuvenation and shale oil and gas extraction. The creation of the new fractures in any kind of rock is strongly controlled by the strength of the host rock. In the fracture mechanics, the resistance of the rocks against initiation and propagation of new fractures is determined primarily by ‘fracture toughness’. But similar to other geomechanical properties, this material property also gets influenced by the ambient and in situ conditions such as temperature, confining pressure, heterogeneity, anisotropy and fluid saturation. It is necessary to have an understanding on the changing behaviour of the fracture properties of the rocks with changing conditions, so that they can be accurately applied to the subsurface conditions as required. Based on the loading configuration, a rock can fail under three modes of fractures—mode-I, mixed-mode and modeII. Here, mode-I signifies the pure tension, mixed-mode indicates the tearing shear and mode-II indicates the sliding shear- or pure shear-type fractures. So, it is necessary to quantify the critical fracture toughness values for each of the modes of any rock intended to be fractured. Although a lot of research have been done to address the dependencies of the fracture properties on the temperature, confining pressure, loading rate and sample geometry, limited amount of work has been done so far to address the effects of fluid saturation (Al-Shayea et al. 2000; Khan and Al-Shayea 2000; Aliha et al. 2010; Funatsu et al. 2002, 2004; Ayatollahi and Aliha 2007a; Ayatollahi and Aliha 2008; Abd-Elhady 2013; Mahanta et al. 2016; Berto and Gomez 2017; Fayed 2017). Only in the recent past, a few studies attempted to address the effect of humidity and the alkalinity of the reservoir fluid on the fracture properties of the rocks (Nara et al. 2012; Reinhardt and Mielich 2014; Hua et al. 2015). But still, systematic investigation and extensive experimental results on this issue remained an unexplored domain. In contrast to that, the effect of saturation other parameters on the mechanical and other physical properties of the rock have received tremendous attention from the scientific community (Va´sa´rhelyi 2005; Sharma and Singh 2006; Kahraman 2007; Erguler and Ulusay 2009; Karakul and Ulusay 2013; Shukla et al. 2013; Vishal et al. 2015; Guha Roy and Singh 2016). These works confirmed that the strength of the rocks generally reduces with increasing fluid content and predictive empirical relationships can be established between them. Investigations by Zhang (2002) and Jin et al. (2011) also showed that the tensile strength and
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the fracture toughness of the rocks are linearly correlated. So, it is expected that fluid saturation would significantly alter the fracture properties too. Establishing the correlation between the geomechanical and fracture toughness of the sedimentary rocks has a farreaching practical implication. In most of the cases, laboratory measurement of rock fracture toughness is difficult due to the fragility of the rocks, unavailability of large number of specimens and complex instrumental set-up. Therefore, if easily measured properties such as tensile strength, ultrasonic velocities and Young’s modulus are correlated with the toughness, those issues can be easily circumnavigated. Further, fracture toughness of the hydrocarbon bearing rocks can also be accurately estimated based only on the log data. This will reduce both the cost related to extensive coring and experiments, and the time required for well-developed plan implementation. To investigate these issues, a series of strength and fracture property-related experiments were conducted on three different kinds of sandstones and one type of shale. Tensile strength, Young’s modulus, ultrasonic wave velocities and fracture toughness were measured under varying degree of water saturation. These data were then compared together to find trend in the property change and to establish predictive relationships among them. Further, to evaluate the applicability of the mixed-mode fracture criteria for the fracture propagation modelling, all the fracture toughness data have been plotted against the maximum tangential stress criterion (Erdogan and Sih 1963), the maximum energy release rate criterion (Hussain et al. 1974) and the maximum strain energy density criterion (Sih 1974).
2 Laboratory Experiments 2.1 Physical and Mineralogical Properties The specimens used in the present experiments were collected from different places and basins of India. The Dholpur sandstone was collected from state of Rajasthan, white and red sandstones were collected from Jabalpur, Madhya Pradesh, and the shale was collected from the Jharia coal field, Jharkhand. All the specimens were tested for their mineralogical composition and physical properties before the saturation experiments. Each of the rock types was investigated under the petrographic microscope and X-ray diffraction (XRD) analysis for their mineralogical content. The XRD analysis was carried out using the powdered samples at the laboratories of the Department of Earth Sciences, IIT Bombay. While plain-polarized and cross-polarized optical signature of the minerals shows semi-quantitative assessment of the mineralogy, XRD
Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks
provided quantitative description of the whole sample mineralogy. The mineralogy, density and porosity of the rocks are shown in Table 1.
1.2
1
2.2 Sample Preparation and Saturation Experiment All the rock mechanics and fracture mechanical testing were done under different degrees of saturation varying from 0 to 1. First the rock cores were retrieved from the blocks using a diamond core bit. In shale, which shows closely spaced layering, the cores were retrieved perpendicular to the layers. The cores were dried in the room temperature for 24 h prior to any experimentation. Then the specimens were saturated following the recommendations of International Society for Rock Mechanics (ISRM 2007) for the porosity/density measurement using the saturation and buoyancy method. Specimens were saturated with the distilled water to minimize any kind of chemical reaction with the minerals due to water. Throughout the experiment, weights of the specimens were recorded at regular intervals until the weight became constant. At each saturation level, for every geomechanical and fracture toughness test, at least three samples were tested and their average value was taken as the final result. The evolution of degree of saturation as a function of time is shown in Fig. 1. It shows that depending upon the porosity and permeability of the rock, different amounts of time are required for complete saturation. Being highly porous and permeable, Dholpur sandstone takes the minimum time for saturation, followed by the Jabalpur white sandstone, Jabalpur red sandstone and Jharia shale, respectively. 2.3 Fracture Toughness Testing Both the fracture and mechanical tests were performed at the room temperature (25 °C) in the universal testing machine (UTM) as shown in Fig. 2. Fracture toughness of the specimens was tested using the semicircular specimens with geometrical dimensions as prescribed by ISRM standard (Kuruppu et al. 2014) and demonstrated in Fig. 3a. In total, 225 specimens were tested from four types of rocks for three modes of fractures. A steady loading rate of 0.2 mm/min was maintained throughout the experiments. The compressive load, axial displacement and crack mouth opening displacements (CMOD) were measured simultaneously using
Saturation
0.8
0.6 Dholpur sandstone 0.4
Jabalpur white sandstone Jabalpur red sandstone
0.2 Jharia shale 0 0
10
20
30
40
Hours
Fig. 1 Change in water saturation with time
precise instruments to minimize the errors. A monotonically increasing continuous load–displacement curve until the failure was an indication of a successful test. The load was applied using the three-point bend set-up, where the semicircular specimen was placed between a single top roller and two bottom rollers. Positions of the bottom rollers were marked, and the notch plane was aligned with the loading direction. The specimens were investigated post-failure to ensure the validity of the tests. Fore mode-I specimens, it was ensured that fracture plane did not deviate far from the notch plane. The fracture toughness is calculated as follows: pffiffiffiffiffiffi 0 Pmax pa KIC ¼ Y ð1Þ 2RB where KIC is mode-I fracture toughness, Y0 non-dimensional stress intensity factor, Pmax failure load, a notch length, R radius of the specimen and B thickness of the specimen, for mode-I FT calculation, Y0 is calculated as Y 0 ¼ 1:297 þ 9:516ðs=2RÞ ð0:47 þ 16:457ðs=2RÞÞb þ ð1:071 þ 34:401ðs=2RÞÞb2 ð2Þ and b = a/R, a notch length, R radius of the specimen, s span.
Table 1 Mineralogical and physical properties of specimens Rock name
Mineralogy
Density (kg/m3)
Porosity
Dholpur sandstone
Quartz, plagioclase, iron oxide
2114
0.15
Jabalpur white sandstone
Quartz, albite, mica
2515
0.05
Jabalpur red sandstone
Quartz, albite, haematite, mica
2579
0.04
Jharia shale
Quartz, clay minerals, feldspar, iron oxide, organic matter
2212
0.05
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(a)
(b)
(c)
(d)
Fig. 2 Testing set-up a control unit, b compression set-up, c data logger, d three-point bending set-up
As Y0 is strongly influenced by the a/R and s/R ratios, Lim et al. (1993) calculated the values of mode-I and mode-II non-dimensional intensity factors for the entire practical range of geometrical configurations, as shown in Fig. 4a, b. As mode-II is a shear mode fracture with no tensile component, 40° notch angle was chosen where mode-I non-dimensional intensity factor vanishes. At this orientation, the fracture toughness of the specimen is completely dominated by mode-II-type fracture. Required Y00 at this notch angle is then calculated from Fig. 4b. And as mixed-mode incorporates both the tensile and shear fracturing, a notch angle of 30° was chosen for the mixedmode experiments. It is calculated with the following equation:
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Keff ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ K2 KIC IIC
ð3Þ
where KIIC mode-II fracture toughness. 2.4 Geomechanical Tests All the four types of rocks were tested for their destructive and non-destructive mechanical responses as a function of the water saturation. The tensile strength of the rocks was tested following the International Society for Rock Mechanics standard (Hawkes and Bieniawski 1978). As shown in Fig. 3b, the test method involved the placement of a NX size (54.7 mm diameter) tensile disc with diameter-to-length ratio 2:1 inside the Brazilian cage and
Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks
(b)
(a)
a
R
D
B
L
Diameter = D = 54.7 mm Length = 0.5D
Diameter = D = 2R > 10 X grain size Thickness = B = > 0.4 X D Crack length = a = 0.4 ≤ a/R ≤ 0.6 Span length = s = 0.5 ≤ s/2R ≤ 0.8
Fig. 3 a Semicircular bend specimen in three-point load set-up, b tensile disc in Brazilian cage set-up
loading it. The resulting tensile strength is calculated as follows: rt ¼
2P pDt
ð4Þ
where rt is tensile strength (MPa), P failure load (KN), t thickness of disc (mm) and D diameter of the specimen (mm). The Young’s modulus is measured from the slope of the stress–strain curve as acquired by the strain gauges during the compressive strength testing. The linear part of the prefailure curve was considered during the calculation. The P-wave velocity, which shows a strong response to the micro-structure and the fluid content of the medium, was measured using the portable ultrasonic non-destructive digital indicating tester (PUNDIT). This instrument consists of a transmitter and a receiver, and sends longitudinal waves through the specimens. The propagation time was noted, and the P-wave velocity of the specimen is measured using the length of the rock. The compressive strength, tensile strength, P-wave velocity (Vp) and Young’s modulus of the dry rocks are listed in Table 2.
3 Results and Discussion 3.1 Effect of Saturation on Deformational Behaviour The effect of water saturation on the load–displacement (CMOD) curve and stress–strain behaviour of the rocks is shown in Fig. 5a, b. As shown in the graphs, water saturation has a distinct effect on both of the properties. A continuous decrease in the failure load is noticed in Fig. 5a for all the rocks from dry to completely saturated specimens. Similarly, in all the four rocks, the ultimate load and gradient of the stress–strain curves of the saturated rocks are found to be considerably lower than the dry rocks. These changes indicate a decrease in material stiffness with increasing degree of saturation. A close investigation reveals that saturation has a different effect on the Jharia shale specimens than the sandstones. In sandstones, all the stress–strain curves before the failure loads (Fig. 5b) are linear. They indicate that even in completely saturated conditions, all the three sandstones show brittle deformation. But, the stress–strain curve of the
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(a) 5000
(a)
s/R = 0.5 4500
Dholpur sandstone (dry)
10 a/R = 0.3
a/R = 0.7
a/R = 0.5
a/R = 0.1
Y' 0
Dholpur sandstone (sat)
3500
Load (N)
5
4000
-5
Jabalpur white sandstone (dry)
3000
Jabalpur white sandstone (sat)
2500
Jabalpur red sandstone (dry)
2000
Jabalpur red sandstone (sat)
1500
Jharia shale (dry)
1000
Jharia shale (sat)
-10 500
-15
0
0
5
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75
0.1
0.2
0.3
0.4
0.5
CMOD (mm)
Notch angle
(b) 90
3
(b)
s/R = 0.5
Dholpur sandstone (dry)
80 Dholpur sandstone (sat)
70
a/R = 0.5
a/R = 0.1
a/R = 0.3
Y" 1
Axial stress (MPa)
a/R = 0.7 2
Jabalpur white sandstone (dry)
60
Jabalpur white sandstone (sat) Jabalpur red sandstone (dry)
50
Jabalpur red sandstone (sat)
40 Jharia shale (dry)
30
Jharia shale (sat)
20
0
10 0
-1 0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
0
0.0005
0.0015
0.002
0.0025
0.003
0.0035
0.004
Axial strain
Notch angle
Fig. 4 Calculation of non-dimensional stress intensity factors (modified after Lim et al. 1993)
0.001
Fig. 5 a Load–CMOD curve, b stress–strain curve
3.2 Effect of Saturation on P-wave Velocity completely saturated Jharia shale deviates from the linearity and starts showing semi-brittle behaviour. Here, linear elastic region is followed by a small ductile region before failure. Therefore, the concept of linear elasticity is completely valid for all the three sandstones and, overall, can be considered reasonably applicable in the saturated Jharia shale.
Variation of Vp as a function of degree of saturation was recorded for all four types of rocks. Figure 6a shows the change in the P-wave velocity with increasing saturation. It is noticed that all the rocks except the Dholpur sandstone show increasing P-wave velocity with increasing saturation. In Jabalpur white sandstone, P-wave velocity showed
Table 2 Mechanical properties of dry specimens Specimen type
Compressive strength (MPa)
Tensile strength (MPa)
P-wave velocity (Vp) (m/s)
Young’s modulus (GPa)
Dholpur sandstone
44
6.28
3120
12
Jabalpur white sandstone
68
11.18
3410
20
Jabalpur red sandstone
76
10.22
2846
21
Jharia shale
33
5.4
1779
12
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Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks
the compressional wave velocity with increasing degree of saturation. In case of Dholpur sandstone, relatively very high porosity value (*0.15) is considered to be the reason behind the observed change in P-wave velocity.
(a) 4500 4000
P-wave velocity (m/s)
3500
3.3 Effect of Saturation on Tensile Strength
3000 2500 2000 1500 Dholpur sandstone
1000
Jabalpur white sandstone Jabalpur red sandstone
500
Jharia shale 0 0
0.2
0.4
0.6
0.8
1
Saturation
(b)
3.4 Effect of Saturation on Young’s Modulus
60
40
Changes in P-wave velocity (%)
As shown in Fig. 7, depending upon the degree of saturation, a decreasing tensile strength of the sedimentary rocks is observed. From dry to completely saturated situation, the BTS of Dholpur sandstone, Jabalpur white sandstone, Jabalpur red sandstone and Jharia shale decreased by 50.6, 35.2, 31.5 and 77.2%, respectively. These strength reductions are consistent with those observed by other researchers (Ojo and Brook 1990; Lashkaripour 2002; Va`sa`rhelyi and Van 2006; Karakul and Ulusay 2013).
20 Dholpur sandstone 0 Jabalpur white sandstone
Jabalpur red sandstone
Jharia shale
-20
Young’s modulus indicates the deformability of the rock under uniaxial stress conditions. The present experiments show that increasing degree of saturation has a prominent negative effect on the YM of the rocks. As shown in Fig. 8, from dry to completely saturated situation, the YM of the Dholpur sandstone, Jabalpur white sandstone, Jabalpur red sandstone and Jharia shale decreased by 19, 27.1, 20 and 75%, respectively. This trend and amount of decrease are found to be similar to those reported by Kwas´niewski and Oitaben (2009) and Karakul and Ulusay (2013).
-40
-60 14
-80
approximately 20.5% increase from 3410 to 4110 m/s; Jabalpur red sandstone showed approximately 29.8% increase from 2864 to 3693 m/s; Jharia shale showed approximately 67.7% increase from 1779 to 2984 m/s; and Dholpur sandstone showed a 57.6% decrease from 3120 to 1321 m/s, respectively. These changes are illustrated in Fig. 6b. Similar increase in the P-wave velocity with the increasing saturation was reported by Saito (1981), Kahraman (2007), King (2009) and Karakul and Ulusay (2013) for different kinds of rocks. Similar to Dholpur sandstone, decreasing P-wave velocity was also reported for different rocks by Klimentos (1991) and Karakul and Ulusay (2013). These research studies established that high amount of clay content and the porosity causes decrease in
Brazilian tensile strength (MPa)
Fig. 6 a Change in P-wave velocity with saturation, b ultimate change in P-wave velocities from dry to fully saturated rock
Dholpur sandstone Jabalpur white sandstone
12
Jabalpur red sandstone 10
Jharia shale
8
6
4
2
0 0
0.2
0.4
0.6
0.8
1
Saturation
Fig. 7 Change in Brazilian tensile strength (BTS) with saturation
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(a) 1.2
Dholpur sandstone Jabalpur white sandstone Jabalpur red sandstone Jharia shale
Young's modulus (GPa)
20
15
10
Mode-I fracture toughness (MPa.m0.5)
25
1
Dholpur sandstone Jabalpur white sandstone
0.8
Jabalpur red sandstone 0.6
Jharia shale Kushiro sandstone (Nara et al. 2012) Shirahama sandstone (Nara et al. 2012) Berea sandstone (Nara et al. 2012)
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Saturation 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Saturation
Fig. 8 Change in Young’s modulus (YM) with saturation
3.5 Effect of Saturation on Fracture Toughness Figure 9a–c exhibits the change in the pure and mixedmode fracture toughness values with increasing degree of water saturation. A decrease in the fracture toughness values with increasing saturation was observed for all the sedimentary rocks. The overall change in the FT values from dry to saturated rock is shown in Fig. 10. From dry to completely saturated rock, reduction in mode-I FT was between 31.8 and 53%, in mixed-mode the reduction was between 16.2 and 44%, and in mode-II, the reduction was between 10.9 and 45.2%. Results from Nara et al. (2012) are also plotted in Fig. 9a for the comparison purpose. Similar to the present findings, Utagawa et al. (1999) also reported decreasing fracture toughness of Kimachi sandstone, Inada granite and Shin-komatsu andesite when crack tip is saturated. Among all the sedimentary rocks, Jharia shale showed maximum reduction in the fracture toughness in all the pure and mixed-mode fractures. It was noticed that compared to the strength values (BTS and YM), sedimentary rocks showed relatively low reduction in the pure and mixed-mode FT values with increasing saturation. The presence of fluid in a rock can have lubrication effect on the grains. While under stress, grains and inherent microcracks of the saturated rocks are more prone to slide against each other. Also, fluid weakens the bond between the grains, which further makes the rock weak. This lubrication and bond-weakening are thought to be two major causes behind the reduction in the fracture toughness in the present rocks. In mode-I fractures, rock fails under tensile strength, whereas in mixed-mode and mode-II, a significant amount of shear stress at the crack tip plays a major role. As rocks
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1 Dholpur sandstone
0.9
Jabalpur white sandstone Jabalpur red sandstone
0.8
Jharia shale 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
1
Saturation
(c)
1 Dholpur sandstone
Mode-II fracture toughness (MPa.m0.5)
0
Mixed-mode fracture toughness (MPa.m0.5)
(b)
0.9
Jabalpur white sandstone
0.8
Jabalpur red sandstone Jharia shale
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
1
Saturation
Fig. 9 Effect of saturation on fracture toughness (FT) of rocks a mode-I, b mixed-mode, c mode-II
are weaker in tension than under any other kind of stress, in the presence of fluid and tensile load, mode-I specimens are weaker than the other specimens. This is the reason
Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks 1.2
60
1
Mixed-mode 40
Mode-II
Mixed-mode y = 0.5329x + 0.4384 R² = 0.8
0.8
FT/FT0
Decrease in Fracture Toughness (%)
Mode-I 50
30
0.6 Mode-II y = 0.5168x + 0.4578 R² = 0.7
0.4
20
Mode-I y = 0.6513x + 0.3261 R² = 0.78
0.2
10
Mode-I
Mixed-mode
Mode-II
0
0 Dholpur sandstone Jabalpur white sandstone
Jabalpur red sandstone
Jharia shale
0
0.2
0.4
0.6
0.8
1
1.2
BTS/BTS0
Fig. 10 Ultimate decrease in the fracture toughness values from dry to completely saturated rocks
Fig. 11 Normalized FT versus normalized BTS
3.7 Fracture Toughness Versus Young’s Modulus why reduction in mode-I specimens is greater than other specimens. 3.6 Fracture Toughness Versus Tensile Strength Past results showed that fracture toughness and tensile strength of the dry rocks are linearly related to each other. But no work has been done so far to show how this relationship behaves at different degree of saturation. So, the ‘normalized FT’ was plotted against the ‘normalized BTS’ of the rocks having different degree of saturation. The ‘normalized’ value was defined as the ratio of FT or BTS at a certain degree of saturation to the respective FT or BTS of the dry rock. Resulted plot for all the pure and mixedmode fractures are shown in Fig. 11. Results showed that across all the saturations, FT and BTS maintained a linear relationship between them. So, using the linear regression analysis empirical relations can be established between these two parameters with high degree of correlation. The relationships can be expressed as follows:
Similar to BTS, in Fig. 12, ‘normalized FT’ of the saturated rocks has been plotted against the ‘normalized YM’. This plot also yields linear relations between the YM and FT for all the modes. Since measurement of both the static and dynamic YM of the rocks is standard and easy procedure from logs in petrophysics, these empirical equations can also be directly used to estimate the FT of the saturated rocks from their YM. ðKIC =KIC0 Þ ¼ 0:625 ðE=E0 Þ þ 0:3127;
R2 ¼ 0:78 ð8Þ
ðKeff =Keff0 Þ ¼ 0:5174 ðE=E0 Þ þ 0:4232;
R2 ¼ 0:82 ð9Þ
ðKIIC =KIIC0 Þ ¼ 0:5368 ðE=E0 Þ þ 0:4161;
2
R ¼ 0:81 ð10Þ
where KIC0, Keff0, KIIC0 and E0 are the mode-I, mixedmode, mode-II fracture toughness and Young’s modulus (YM) of the air-dried rock.
ðKIC =KIC0 Þ ¼ 0:6513 ðrt =rt0 Þ þ 0:3261; R2 ¼ 0:78
ð5Þ
ðKeff =Keff0 Þ ¼ 0:5329 ðrt =rt0 Þ þ 0:4384; R2 ¼ 0:8
ð6Þ
3.8 Fracture Toughness Versus P-wave Velocity
ðKIIC =KIIC0 Þ ¼ 0:5168 ðrt =rt0 Þ þ 0:4578; R2 ¼ 0:7
ð7Þ
The relationship between the P-wave velocity and the fracture toughness of the sedimentary rocks is shown in Fig. 13a. As the graph shows, except for the Dholpur sandstone, for all the other rocks FT decreased with increasing P-wave velocity. Only, in Dholpur sandstone FT and P-wave velocity showed a nearly linear positive trend. This is possible because P-wave velocity of the rocks is controlled not only by degree of saturation but by the porosity too. The significantly higher porosity value of the
where KIC0, Keff0, KIIC0 and rt0 are the mode-I, mixedmode, mode-II fracture toughness and Brazilian tensile strength of the air-dried rock. One of the major advantages of this plot is that such plot incorporates the degree of saturation within the rock property relation without explicitly stating it in the equation. Hence, it is possible to reliably estimate the FT value of any saturated sedimentary rock if their BTS values are known.
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D. G. Roy et al. 1.2
(a) 1.2
Dholpur sandstone(ModeI) Jabalpur white sandstone(Mode-I) Jabalpur red sandstone(Mode-I) Jharia shale(Mode-I)
1
1
0.8
FT/FT0
FT/FT0
0.8 Mode-II y = 0.5368x + 0.4161 R² = 0.81
Dholpur sandstone (mixedmode) Jabalpur white sandstone(Mixed-mode) Jabalpur red sandstone(Mixed-mode) Jharia shale (Mixed-mode)
0.6 0.4
Mixed-mode y = 0.5174x + 0.4232 R² = 0.82
0.6
0.4
0.2 0 0.5
Mode-I y = 0.625x + 0.3127 R² = 0.78
Mixed-mode
0.2
0.4
0.6
1.1
1.3
1.5
1.7
(b) 1.2
Mode-II
Mode-II y = -0.7412x + 1.7436 R² = 0.88
0 0
0.9
Vp/Vp0
0.2 Mode-I
0.7
0.8
Dholpur sandstone (ModeII) Jabalpur white sandstone (Mode-II) Jabalpur red sandstone (Mode-II) Jharia shale (Mode-II)
1
1
1.2
YM/YM0
R2 ¼ 0:79 ð11Þ
ðKeff =Keff0 Þ ¼ 1:6498 0:6691 ðVp=Vp0 Þ; ðKIIC =KIIC0 Þ ¼ 1:7436 0:7412 ðVp=Vp0 Þ; R2 ¼ 0:88
R2 ¼ 0:82 ð12Þ ð13Þ
where KIC0, Keff0, KIIC0 and Vp0 are the mode-I, mixedmode, mode-II fracture toughness and P-wave velocity of the air-dried rock. 3.9 Statistical Analysis The above analysis of relationship between the key geomechanical properties and the fracture toughness yields mathematically significant empirical equations with high degree of correlation coefficient. As the result suggests,
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FT/FT0
Dholpur sandstone caused the reduction in the P-wave velocity. Porosity of Dholpur sandstone is *66.7% more than the other rocks used in the experiments. Hence, a generalized correlation between these two rock parameters is difficult to postulate without correct knowledge of the rock porosity. But for relatively low porous sedimentary rocks (B0.5), a strong correlation between the P-wave velocities and the fracture toughness of the rocks can be established as shown in Fig. 13b. These correlations are similar to the linear relations between the fracture toughness and acoustic velocities for sandstone and shale as reported by Zhixi et al. (1997). The empirical equations connecting these two parameters are expressed as follows: ðKIC =KIC0 Þ ¼ 1:7901 0:8111 ðVp=Vp0 Þ;
Mixed-mode y = -0.6691x + 1.6498 R² = 0.82
0.8
Fig. 12 Normalized FT versus normalized YM
0.6 Mode-I y = -0.8111x + 1.7901 R² = 0.79
0.4
Mode-I
0.2
Mixed-mode Mode-II 0 0.8
1
1.2
1.4
1.6
1.8
Vp/Vp0
Fig. 13 a Evolution of normalized FT and normalized Vp of all the saturated rocks, b linear regression model of normalized FT vs normalized Vp of rocks (except Dholpur sandstone)
BTS, YM and P-wave velocity of the rocks are strongly correlated with the fracture toughness. The estimated values of the fracture toughness have been plotted with the measured values for all the three relations in Fig. 14a–c. The error in estimation is shown by plotting the data against the 1:1 line, which shows the goodness of the estimation. Next, Student’s t test is performed on the empirical equations to measure their effectiveness and applicability to use them for further prediction. This statistical method uses the t statistic, t distribution and the degree of freedom to compute a p value to test whether population means are different. It is assumed that both the variables are normally distributed and samples are chosen randomly. The t statistic is written as XT XC ts ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varC varT nT þ nC
ð14Þ
Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks
(a)
1
0.9 0.8
Computed FT
0.7 0.6 0.5 0.4 Mode-I
0.3
Mixed-mode
1:1
0.2
Mode-II 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Experimental FT
(b)
1
0.9
Here, the numerator is the difference between the means of the two populations T (tested) and C (computed), and the denominator is an expression to measure the variability of the data points. It is computed by the dividing the variances (varT and varC) of each of the populations by the population size (nT and nC), adding them up and taking the square root. For the present purpose, a two-sample unpaired t test was conducted at 95% confidence level (a = 0.05) to test the null hypothesis that the mean of the two samples is equal. The resulted p values for each of the above equations are tabulated in Table 3. As the table values suggest, p value of each of the empirical equations is significantly higher than the designated a value. That means the null hypothesis that mean of the experimental FT and computed FT is same cannot be rejected. It means a strong correlation exists between the geomechanical properties and the FT of the rocks and these equations can be successfully used to measure the fracture toughness of the reservoir rocks.
0.8
3.10 Degradation Degree
Computed FT
0.7 0.6 0.5 0.4 Mode-I
0.3
1:1
0.2
Mixedmode Mode-II
0.1 0
0
0.2
0.4
0.6
0.8
1
Experimental FT
(c)
1 0.9 0.8
Computed FT
0.7 0.6 0.5 0.4 Mode-I
0.3
Mixed-mode
1:1
0.2
Mode-II 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Experimental FT
Fig. 14 Error in FT estimation from empirical relations a calculation of FT from BTS, b calculation of FT from YM, c calculation of FT from Vp
Reduction in the rock strength properties due to the fluid saturation was noticed for all types of specimen used in the present experiments. But the quantification of the ‘degradation degree’ is not a straight forward task. Depending upon the rock parameter chosen for the degradation calculation, results may vary widely. YM and BTS have been used widely to define the degradation of rock and concrete strength due to thermal treatment, repeated freeze–thaw cycles and repeated wetting–drying cycles (Xian-biao et al. 2009; Li et al. 2012; Gautam et al. 2015; Jia et al. 2015). Recently, fracture toughness has also been used as a measure of ‘degradation degree’ by several researchers (Hua et al. 2015, 2016). In the present research, BTS and YM were used to calculate the ‘degradation degree’ and compared the results with the degradation degree calculated from mode-I FT. These degradation degrees are expressed in the following ways: Esat DYM ¼ 1 100 ð15Þ E0 rt sat 100 ð16Þ DBTS ¼ 1 rt0 KIC sat 100 ð17Þ DFT ¼ 1 KIC0 where DYM, DBTS and DFT are the degradation degrees calculated by YM, BTS and FT, respectively; Esat, rtsat and KICsat are the YM, BTS and mode-I FT at saturation level ‘sat’; and E0, rt0 and KIC0 are the YM, BTS and mode-I FT of air-dried rocks. The evolution of rock degradation
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D. G. Roy et al. Table 3 P values of the empirical equations relating FT and other mechanical properties Equations
P value
Confidence level (a)
ðFT=FT0 ÞI ¼ 0:6513ðBTS=BTS0 Þ þ 0:3261
0.99
0.05
ðFT=FT0 ÞIII ¼ 0:5329ðBTS=BTS0 Þ þ 0:4384
0.99
ðFT=FT0 ÞII ¼ 0:5168ðBTS=BTS0 Þ þ 0:4578 ðFT=FT0 ÞI ¼ 0:625ðYM=YM0 Þ þ 0:3127 ðFT=FT0 ÞIII ¼ 0:5174ðYM=YM0 Þ þ 0:4232
0.99
0.99
ðFT=FT0 ÞII ¼ 0:5368ðYM=YM0 Þ þ 0:4161
0.99
ðFT=FT0 ÞI ¼ 1:7901 0:8111 ðVp=Vp0 Þ
0.99
ðFT=FT0 ÞIII ¼ 1:6498 0:6691 ðVp=Vp0 Þ
0.99
ðFT=FT0 ÞII ¼ 1:7436 0:7412 ðVp=Vp0 Þ
0.99
calculated by all the three methods as a function of water saturation is shown in Fig. 15a, and the final degradation values of all the rocks are shown in Fig. 15b. The results indicate that shale has undergone rapid and maximum degradation than the other three rock types. As suggested by the slope of the degradation curve, all the three sandstones degraded at a relatively slower pace till 80% saturation, and after that, they degraded rapidly. But in case of shale, the degradation rate is higher since the beginning and remained nearly constant throughout the saturation experiment. Figure 15b shows that for all the rocks, BTS method gives higher ‘degradation degree’ than for the rest two methods. In sandstones and shale, the minimum degradation degrees are provided by YM method and FT method, respectively. In sandstones, YM-based degradation degree ranged between 20 and 27% and in shale it was 75%. This is consistent with the results reported in the literature (Wong et al. 2015). It is observed that in sandstones, BTS overestimates the degradation degree by 40–60%, but estimation by the FT method is relatively closer to YM method. Also, in low porosity sandstones, FT- and BTS-based degradation degree values are closer to each other than the higher porosity ones. But in shale, both the BTS and YM methods estimate nearly identical degradation degree, whereas FT method underestimates by about 30%.
0.99
Among them, three criteria—maximum tangential stress (MTS), maximum energy release rate and minimum strain energy density—received maximum acceptance within the scientific community. The results of mixed-mode fractures from all the four types of rocks were plotted with these three criteria in Fig. 16 for comparison and clarity. The results showed that KII/KIC of shale was much higher than the sandstones for any given value of KI/KIC. It was observed that in fully saturated Dholpur sandstone, Jabalpur white sandstone and Jabalpur red sandstone, the MTS criterion showed a good agreement with the experimental value. This result is similar to the findings of (Funatsu et al. 2014). But shale mixed-mode fracture data showed a close agreement with the maximum strain energy density criterion. It should be noted that these three criteria only provide a rough understanding of the possible mixed-mode fracture behaviour. As past research have shown, a more accurate description of the mixed-mode behaviour of the semi-disc specimens must include the higher-order stress terms such as T-stress in the crack-tip stress distribution description (Ayatollahi and Aliha 2007b; Aliha et al. 2008; Aliha and Ayatollahi 2009; Aliha et al. 2012, 2013; Aliha and Ayatollahi 2013, 2014). Research has shown that with the inclusion of the T-stress term in the stress description, generalized MTS (GMTS) criterion has been found to accurately predict both fracture initiation direction and fracturing stress.
3.11 Fracture Criteria 3.12 Effect of Saturation on Fracture Geometry Complete description of the fracture properties of any rock requires a good understanding of the fracture criterion that can be successfully applied to model the mixed-mode fracture propagation. These fracture criteria give a good prediction on the critical stress limit and the propagation direction of the fractures under different loading conditions. Based on the fracture propagation in the metals, polymers, geomaterials, concrete and rocks, eleven fracture criteria have been proposed by different researchers.
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Further the broken specimens were investigated for the influence of water saturation on the fracture initiation point, initiation angle and fracture path. Digital images of the broken specimens show that fracture initiation points vary across different modes of experiments. In mode-I (notch angle 0°) and mixed-mode (notch angle 30°) specimens, all the fracture initiated at the notch tip. But in the mode-II specimens (notch angle 40°), some cracks were
Effect of Water Saturation on the Fracture and Mechanical Properties of Sedimentary Rocks
(a)
1.20
90
MTS
80
Dholpur sandstone (BTS)
70
Jabalpur white sandstone (BTS)
1.00
Max energy release rate Min strain energy density
0.80
Jharia shale (BTS)
KII/KIc
Degradation degree (%)
Jabalpur red sandstone (BTS)
60
Jabalpur red sandstone
Dholpur sandstone (FT)
50 Jabalpur white sandstone (FT)
Jharia shale
0.60 Dholpur sandstone
40
Jabalpur red sandstone (FT)
0.40
Jharia shale (FT)
30 Dholpur sandstone (YM)
20
Jabalpur white sandstone
0.20
Jabalpur white sandstone (YM) Jabalpur red sandstone (YM)
0.00 0.00
10 Jharia shale (YM)
0.20
0.40
0.60
0.80
1.00
1.20
KI/KIc
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Saturation
Fig. 16 Mixed-mode fracture criteria for different types of sedimentary rocks
(b) 90 80
representative broken specimen and schematic diagrams of the fracture paths are shown in Fig. 17a–d.
Degradaon degree (BTS) Degradaon degree (YM)
Degradation degree (%)
70
Degradaon degree (FT)
60
4 Conclusions
50
The present experimental study with different types of sedimentary rocks shows that degree of saturation can have tremendous effect on both the fracture and mechanical properties. These effects are further complicated by various physical properties of the rocks, i.e. porosity and presence of fissile planes. The main findings and significant observations of this study are listed as follows:
40 30 20 10 0 Dholpur sandstone Jabalpur white sandstone
Jabalpur red sandstone
Jharia shale
1.
Fig. 15 a Evolution of degradation degree with increasing water saturation, b Ultimate degradation of the saturated rocks
observed to initiate behind the crack tip. This observation is in accordance with the conclusion of Lim et al. (1994) that with increasing notch angle, crack starts to initiate behind the tip. But in each of the modes, no discernible effect of degree of saturation was observed on the crack initiation point. Similarly, water saturation was found to have no effect on the crack initiation angle and fracture path. On average, across all the saturation levels, the crack initiation angle in mode-I specimen was 0°, 16° and 20°, respectively. In most of the specimens, fracture paths are tortuous and show some deviations from the common plane. But no clear trend in the tortuosity of the crack path can be associated with the degree of water saturation. A
2.
3.
The P-wave velocity, BTS, YM and FT of the sedimentary rocks have been experimentally measured at different levels of water saturation. These mechanical and fracture properties degrade with increasing degree of saturation. Only the behaviour of P-wave is strongly controlled by the initial porosity of the rock. Among all the pure and mixed-mode fractures, mode-I FT experiences maximum reduction with increasing saturation. It is possible to make reliable predictions of FT of saturated rocks from BTS and YM without any exclusive knowledge of degree of saturation. Use of P-wave velocity in prediction requires prior knowledge of the rock porosity. Based on the porosity value, a proper empirical relationship must be chosen. Such equation reduces the number of parameters (e.g. saturation) required and can facilitate faster derivation
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D. G. Roy et al.
(a)
(b)
(c)
(d)
Fig. 17 Fracture trajectories at different degree of saturation a representative broken specimen, b mode-I, c mixed-mode and d mode-II
4.
5.
of FT values directly from the logs. However, these correlations are specific to materials tested. Development of universally applicable and acceptable measurement of ‘degradation degree’ has proved to be a difficult task. Use of different parameters yields different results. But, all types of ‘degradation degree’ measurements show that shale undergoes rapid and maximum degradation compared to the sandstones with increasing saturation. Analysis of FT shows that among all the mixed-mode fracture criteria, sandstones tend to follow MTS criterion, but shale rock shows close agreement with the minimum strain energy density criterion.
The above results highlight that fracture mechanical behaviour of the saturated rocks is significantly different from the dry rocks. With prior knowledge of the saturation, mechanical and physical properties, it is possible to compute the pure and mixed-mode fracture toughness of the rocks. This is particularly helpful for the fragile and weak sedimentary rocks where standard FT sample preparation is difficult and in the situation where the basic log responses, but no core samples are available. But, it is also postulated that very low or high temperatures would have significant influence on the behaviour of saturated rocks. Therefore, this effect must be evaluated carefully before applying the relationships developed here directly to the subsurface conditions.
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