Laboratory Simulation of Flow through Single ...

Laboratory Simulation of Flow through Single Fractured Granite

K. K. Singh, D. N. Singh & P. G. Ranjith

Rock Mechanics and Rock Engineering ISSN 0723-2632 Volume 48 Number 3 Rock Mech Rock Eng (2015) 48:987-1000 DOI 10.1007/s00603-014-0630-9

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Author's personal copy Rock Mech Rock Eng (2015) 48:987–1000 DOI 10.1007/s00603-014-0630-9

ORIGINAL PAPER

Laboratory Simulation of Flow through Single Fractured Granite K. K. Singh • D. N. Singh • P. G. Ranjith

Received: 5 September 2013 / Accepted: 25 June 2014 / Published online: 16 July 2014 Ó Springer-Verlag Wien 2014

Abstract Laboratory simulation on fluid flow through fractured rock is important in addressing the seepage/fluidin-rush related problems that occur during the execution of any civil or geological engineering projects. To understand the mechanics and transport properties of fluid through a fractured rock in detail and to quantify the sources of nonlinearity in the discharge and base pressure relationship, fluid flow experiments were carried out on a cylindrical sample of granite containing a ‘single rough walled fracture’. These experiments were performed under varied conditions of confining pressures, r3 (5–40 MPa), which can simulate the condition occurring about 1,000 m below in the earth crust, with elevated base pressure, bp (up to 25 MPa) and by changing fracture roughness. The details of the methodologies involved and the observations are discussed here. The obtained results indicate that most of the data in the Q verses bp plot, fall on the straight line and the flow through the single fracture in granite obeys Darcy’s law or the well-known ‘‘cubic law’’ even at high value of bp (=4 MPa) and r3 (=5 MPa) combination. The Reynolds number is quite sensitive to the bp, r3 and fracture

K. K. Singh (&)  D. N. Singh Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra, India e-mail: [email protected] D. N. Singh e-mail: [email protected] P. G. Ranjith Department of Civil Engineering, Monash University, Clayton, Melbourne, VIC, Australia e-mail: [email protected]

roughness, and there is a critical bp, beyond which transition in flow occurs from laminar to turbulent. It is believed that such studies will be quite useful in identifying the limits of applicability of well know ‘cubic law’, which is required for precise calculation of discharge and/or aperture in any practical issues and in further improving theoretical/numerical models associated with fluid flow through a single fracture. Keywords Rockmass  Single fracture  Fluid flow  Non-linear flow  Confining pressure List of symbols l Dynamic viscosity of water (=1 9 10-3 N s/m2 for pure water at 20 °C) q Fluid density (=998 kg/m3 for pure water at 20 °C) r3 Confining pressure (MPa) bp Base pressure/fluid pressure (MPa) reff. Effective confining pressure (MPa) pf Failure load (kN) l Length of sample (mm) d Diameter of the sample (mm) A Cross sectional area of the loading surface (mm2) g Acceleration due to gravity (=9.81 m/s2) Ag Area of grove (mm2) pi Fluid inflow at the bottom of the sample (MPa) po Fluid outflow at the top of the sample (MPa) Q Discharge (m3/s) e Hydraulic fracture aperture (mm) n Number of sampling points Z Asperity height (mm) Rrms Root mean square roughness (mm) Ra Roughness average (mm) HPTC High pressure triaxial cell

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1 Introduction Investigation on fluid flow properties of fractured rock has quite significant role in addressing the seepage/fluid-inrush related problems that occur during the execution of civil or geological engineering projects, such as for oil or natural gas exploration (Mulder et al. 1992; Gauthier et al. 2000; Camac et al. 2006; Li et al. 2009; Wu et al. 2011) ore minerals and geothermal energy extraction (Grant et al. 1982; Brown 1987; Cook 1992; Min et al. 2009) liquid waste disposal/injection (Hakami and Larsson 1996; Min et al. 2009; Shen et al. 2011) erection of dam foundations and rock caverns (Dhawan et al. 2004a, b) construction of open cast and longwall mines (Wang and Park 2002; Li and Aubertin 2009) and developing suitable design for blasting (Kleine et al. 1997) and grouting activities (Barton and de Quadros 1997). In general, most of these engineering projects are associated with hard or crystalline rock, where fluid flow is mainly governed by the fracture/s and as such, discharge, Q through the fracture becomes much higher than the intact rock. Therefore, the relationship between Q and base pressure, bp becomes non-linear. The work of earlier researchers on non-linear flow behavior through fractured rockmass under different stress conditions are worth mentioning (Lomize 1951; Scheidegger 1957; Tek et al. 1962; Gangi 1978; Witherspoon et al. 1980; Zimmerman and Bodvarsson 1995, 1996; Hakami and Larsson 1996; Pyrak and Morris 2000; Wu 2002; Lucas et al. 2007; Ranjith 2010, Singh et al. 2014). It has been observed that fluid flow behavior through a rockmass depends on several factors such as, the geometry of the fracture (aperture, length, density, orientation), interconnection of void spaces/fracture and fracture in-filling materials, roughness of the fracture, fluid pressure, confining pressure etc. Further, some of the earlier researchers have also observed that due to increase or decrease in bp, the relationship between Q and bp, becomes non-linear (Ruth and Ma 1992; Whitaker 1996; Kohl et al. 1997). Experimental result of air flow tests conducted by Ranjith and Viete (2011) on a fractured granite specimen under triaxial test conditions have shown that the ‘cubic law’ expression used by Snow (1969), Kranz et al. (1979), Witherspoon et al. (1980), Tsang and Witherspoon (1981), Brown and Scholz (1985), Schrauf and Evans (1986), based on the parallel plate theory (which assumes linear flow between two parallel smooth plates without having contact with each other), can be employed to non-Darcian flow case for Re B 3.5 to Re \ 4 depending on the bp. However, most of the time natural fracture surfaces are rough and irregular making contact with each other at discrete points (Rissler 1978; Brown and Scholz 1985;

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Brown 1987; Brown et al. 1998; Indraratna et al. 1999). It has been stated that due to a rough/irregular fracture surface, the error in flow estimation occurs in a 1–2 order of magnitude, if modeled using the parallel plate theory (Iwai 1976; Tsang 1984; Brown 1987; Ranjith 2010). However, Witherspoon et al. (2010) have reported that ‘cubic law’ can be used even in case of rough or irregular fracture surfaces, and the fluid flow results are not dependent on rock type. Further, it has also been reported that the effects of topography on fluid flow are negligible at low fluid pressures and for large joint apertures (Ranjith 2010). However, at elevated fluid pressures, fluid flow properties may not be modelled using conventional ‘cubic law’ because of the development of a turbulent/non-linear flow (Ranjith 2010). Consequently, several non-linear relationships have been proposed to simulate flow through a fracture and the most commonly used ones are, the Forchheimer equation and the Izbash equation. Also, few analytical solutions based on Buckley and Leverett (1942) and Barree and Conway (2004) models were derived for non-Darcy displacement of immiscible fluids in porous media (Wu 2001; Wu et al. 2011). It must be noted that the constants of these equations are mainly based on either numerical simulation (Last and Harper 1990a, b; Harper and Last 1990; McDermott and Kolditz 2004; Cammarata et al. 2007; Fidelibus 2007) or in situ scale testing conditions (Alm 1999). Laboratory scale studies are very limited (Singh et al. 2013). This may be due to the involvement of complicated instrumentations and cumbersome test procedures. Further, most of the laboratory investigations have been concentrated on open and closed fractures, effect of normal stress acting on fracture and influence of fluid pressures. However, to elucidate the understanding of the basic mechanics of fluid flow through a fracture/s, controlled laboratory scale investigations, which take into account, collectively, the effect of extreme elevated fluid pressure, confining pressures, and geometry of fracture (roughness and aperture) are necessary and would be of great interest to the research fraternity. In the present work, fluid flow experiments were carried out on a cylindrical sample of granite containing a ‘single rough walled fracture’. The experiments were conducted under varied conditions of r3 (5–40 MPa) which can simulate the conditions occurring 1,000 m below, in the earth’s crust, with elevated bp (up to 25 MPa) and by changing fracture roughness (different types of granite rock are selected based on their grain size). In addition, 3D laser scanning was conducted on the fracture surfaces of all the samples to characterize the fracture roughness qualitatively and quantitatively. The fracture roughness was correlated with the discharge value obtained from the fluid flow experiments. This was done to quantify

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the source of non-linearity in the discharge and the base pressure relationship. Further, the limits of the applicability of the ‘Cubic law’ are discussed.

2 Methodology 2.1 Sample Preparation In this study, three types of granites were selected (based on visual identification of grain size under in situ conditions from a local quarry at Victoria, Southern Australia) and brought to the laboratory in the form of irregular blocks—&300 mm, in length, 200 mm, in width and 150 mm, in height. Cylindrical rock cores of a diameter to height ratio of 1:2, were obtained according to the recommendation of Indian Society of Rock Mechanics (ISRM) by employing a diamond core driller of (DD 120 Diamond Core System make Hilti USA). These cores were 38 mm, in diameter and 76 mm, in length and were designated as S1-CG-38, S2-MG-38, and S3-FG-38 representing coarse, medium, and fine grained samples respectively. The sample designation, and their geometrical details, along with engineering properties are presented in Table 1. The core samples were grinded and polished accurately according to the American Society of Testing Manual (ASTM) standards for rock testing (ASTM D4543 2008) and oven dried for 24 h. Further a single vertical fracture running all along the length of the core was created, as described in the following. To create a ‘single fracture’ in the rock core sample, a setup to produce a single vertical fracture (based on a Brazilian technique), made up of high carbon (1.20 % C by weight) steel (refer Fig. 1) was developed. This setup consisted of a pair of V-blocks and each of the V-block is sitting on the flat metal plate. The setup consists of four holders to clutch the samples in the center of the V-block. These V-blocks were fixed on the top and bottom platen of the Universal Compression Testing Machine (UCTM) respectively. To ensure proper adjustment and alignment of the top and bottom V-blocks, both the top and bottom platen of the UCTM were brought close to each other and it was ensured that the sharp portions of the top and bottom plate of the V-blocks match each other. After ensuring the alignment of the V-blocks, the top platen of the UCTM was

raised up and the sample was placed in between the V-blocks positioning it horizontally, so that the sharpened portion of the V-block makes contact along the sample length. Before placing the sample in between the V-blocks, a 2 9 1 mm2 groove was made on two sides (at 180°) of the sample along the length. This facilitates proper coupling of the sharpened portions of the V-blocks and the cross sectional area, A (computed by multiplying the l of the sample and the grove thickness or line of contact of the V-block i.e., 2 mm) of the sample on which loading was applied. The computed area of grove, Ag (mm2) for all the samples is mentioned in the Table 1. This also allows equal distribution of the load throughout the sample length. Further, loading was applied perpendicular to the sample length using the UCTM till failure, and details of the failure load Pf are listed in the Table 1. In this way, the rock core was separated into two cylindrical halves, as depicted in Fig. 2a. Subsequently, both cylindrical halves were joined together by means of a silicon adhesive imbibing ‘single fracture’ as depicted in Fig. 2b. 2.2 Roughness Measurement of the Fracture Surfaces Roughness of any surface can be defined by the geometric properties of a surface through a quantitative parameter describing the intuitive notions of ‘‘rough’’ or ‘‘smooth’’ (Cord et al. 2007). The characterization of the geometric properties of roughness of the fracture surface and the aperture, statistical analysis of the asperity height and of the joint aperture are important (Lanaro 2000). Roughness measurement can be made by employing a 3D laser profile scanner. The laser beam can be used to capture the texture of the fracture surface or the topography of the surfaces. The co-ordinates of the scanned surface will be in ASCII or binary files in the X Y and Z format. X Y and Z coordinates represent the width, length and asperity height of the fracture surface respectively. Several earlier researchers have used root means square (RMS) roughness Rrms and Roughness Average Ra along with the basic statistical parameters (mean median mode skewness kurtosis standard deviation etc.) to detect the variation on the fracture surface roughness (Guerrero et al. 2002; Vasconcelos et al. 2006) as depicted in Fig. 3. Rrms is a mathematical representation of the asperities height and depth of the fracture surface and is defined as

Table 1 Engineering properties of the samples and their geometrical details Sample no.

Rock

S1-CG-38

Granite

Grain size

d (mm)

l1 (mm)

l2 (mm)

l2–l1 (mm)

Ag (mm2)

rc (MPa)

E (GPa)

Pf (kN)

Coarse

39.7

75.8

87.7

11.9

152.4

150

42

25

S2-MG-38

Medium

39.5

73.4

85.8

12.4

146.6

172

24

28

S3-FG-38

Fine

39.6

73

77.7

4.7

149.2

249

77

35

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Fig. 1 Schematic diagram of V-Blocks setup a base plate, b top plate, c both top and bottom plate placed over each other, and d sample placed in between V-Blocks setup attached to the top and bottom platen of UCTM

Fig. 2 a Cylindrical halves of the sample, S1-CG-38, b Samples, S1-CG-38, S2-MG-38 and S3-FG-38 with a single fracture

Rrms

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  2 1X ¼ Zi  Z n i¼1

Ra ¼

Fig. 3 The asperity profile and the statistical parameters associated with it

the average between the asperity height deviations and the mean of the line/surface taken over the entire surface. Ra is the mean height as calculated over the entire surface. Rrms and Ra were computed by employing Eqs. (1, 2), respectively (Mellott et al. 2001; Guerrero et al. 2002; Vasconcelos et al. 2006).

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n   1X Zi  Z  n i¼1

ð1Þ

ð2Þ

where n is the number of data points and Z is the asperity height. The above mentioned parameters are scale dependent and their magnitude differs for the same surface depending on the measurement done. For example:-, in measurements done with a profilometer (scale from mm to cm) or with an atomic force microscope (scale from A° to mm). However, these parameters Rrms and Ra help in differentiating the surfaces of different topography quantitatively (Guerrero et al. 2002) and can be used for quantitative measurements of the surface roughness.

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In the present work, roughness measurements were performed on both the cylindrical halves/fracture surfaces, which were produced from direct tensile tests conducted on samples of granites at 0.04 mm point spacing before affixing together. This was done by employing the ROMER 3D laser profile absolute arm scanner, seven axis ‘‘SI’’ series certified to B:89 specifications. The path of the scanning surface was performed manually by rotating the scanner 180° with respect to the vertical axis to avoid problems of shadows and reflection from the Quartz grains. The measurements were collected automatically and the co-ordinates of the scanned surface were further discharged into ASCII or binary files. Statistical analysis was performed to generate classical statistical parameters mean, median, skewness, kurtosis, standard deviation, and the like, on the data generated from the 3D laser topographical inspection scanner and the results are discussed further. Data obtained from the 3D scanner (in X Y Z format) were exported to the SURFER software v.10 to a produce grid file. This was followed by developing the 3D surface profile of both the cylindrical halves/fracture surfaces. Further, by employing ArcGIS software v.9.3 2D roughness profiles for all the sample was generated and analyzed, as elaborated on, ahead. The data obtained in X Y and Z coordinates were converted into a point coverage shape file. Then using this shape file for all the samples, a 3D surface was generated in ArcGIS by employing the Kriging technique of raster interpolation function under the 3D-analyst tool. A section, A–B was taken at the center of the sample whole along the length of the generated 3D surface roughness profile as depicted in dotted line in Fig. 4a, Fracture Surface, FS1. 2D-roughness profile was generated along the section A–B for all the samples and length of flow path l2 and the sample length l1 were computed using the measure tool as depicted in Fig. 5. The computed l1 and l2 for all the samples are listed in Table 1. 2.3 The Fluid Flow Test Setup and Working Principle The fluid flow characteristic of the cylindrical sample of granite imbibing a ‘single fracture’ was determined using a high pressure triaxial cell (HPTC) (B60 MPa) developed by Shukla et al. (2012). The schematic diagram of the experimental setup is depicted in Fig. 6. The test setup consists of a loading system, a loading frame, a triaxial cell, a compressor unit, a hydraulic oil reservoir, a high pressure hydraulic pump, and a water pressure pump. The test setup also consists of a load cell of a capacity of 150 tons to measure the applied load a linear variable differential transducer (LVDT) placed on the top of the cell in vertical direction to measure the eaxial. For further details of

Fig. 4 3-Dimensional surface profile of the fracture surfaces of Sample a S1-CG-38, b S2-MG-38, and c S3-FG-38

the setup and the working principle refer to Shukla et al. (2012). The cylindrical sample with a ‘single rough walled fracture’ was placed on the base pedestal and top cap was placed above the sample. Before keeping the sample on the

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Fig. 5 The computed surface roughness profile from the center of the sample S1-CG-38, showing length l1 and l2

Fig. 6 The schematic diagram of the experimental setup

base pedestal, two porous stones at the top and the bottom were also placed. Later the sample was covered with a rubber membrane and two O-rings and horse-shoe clamps on each top cap and base pedestals were placed over the rubber membrane to avoid leakage and shot circuiting with the applied confining pressure, r3 and the base pressure, bp. Hence the test setup idealizes as a one dimensional leak proof flow condition. Then, the cell was closed and filled with hydraulic oil ensuring that the cell was free of the air. Fluid flow tests were performed on the fractured granite specimen at different combinations of r3 and bp as presented in Table 2. Hydraulic oil was used as the fluid in the cell to apply r3, which was applied using a hydraulic pump. Its magnitude was recorded using a pressure transducer that carries a combined total error band of ±0.75 % of the maximum measurement. Once the sample was exposed to the target r3, water was pumped from the high

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pressure water injection pump (B25 MPa) to the bottom of the sample and collected from an outlet hose, which was connected to the top of the sample. The magnitude of the inflow, pi was also recorded by an independent pressure transducer for more accuracy. The fluid inflow pressure, pi, applied at the bottom of the sample, comes out through the top of the sample under an atmospheric outflow pressure, po, after flowing through the fracture, known as a discharge, Q. This discharge Q was collected in an airtight container to avoid water losses. The container was placed over a weighing balance of 1,200 g capacity along with a computer interface, and the Q was recorded at every 3 s interval. The test was allowed to continue until the flow became stabilized, corresponding to each bp and/or before the confining pressure dropped due to change in temperature or leakage and the time was approximately about 30–45 min into the test. For the sake of brevity, the relation

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Table 2 Different values of r3 and bp for fluid flow tests

Table 3 Statistical parameters of the both surfaces of the fracture/ cylindrical halves (all values in mm)

r3 (MPa)

bp (MPa)

0.40

5

10

15

20

30

40

1

1

2

2

2

2

2

2

4

4

4

4

3

4

6

8

8

8

4

6

8

10

14

14

8

10

14

18

18

14

18

25

25

σ3 - 5 MPa

bp - 4 MPa

0.35

Q (ml/s)

0.30 bp - 3 MPa

0.25

Statistical Parameters

S1-CG-38

S2-MG-38

S3-FG-38

Mean

4.56

5.57

2.36

Median

4.22

5.64

2.27

Standard deviation

1.894

2.229

0.950

Sample variance

3.589

4.970

0.902

Kurtosis

-0.786

-1.017

-0.726

Skewness

0.361

-0.168

0.317

Standard error Range

0.003 9.814

0.004 10.001

0.002 5.070

Minimum

0

Maximum

9.814

10.001

5.070

RMS roughness (Rrms)

1.894

2.229

0.950

Roughness average (Ra)

1.611

1.903

0.799

Confidence level (95.0 %)

0.007

0.008

0.003

0

0

0.20 bp - 2 MPa

0.15 bp - 1 MPa

0.10 0.05 0.00

0

2000

4000

6000

8000 10000 12000

time (sec)

Fig. 7 The variation of Q with t for samples S1-CG-38 corresponding to different bp (=1, 2, 3, and 4 MPa) values

between Q and t of sample S1-CG-38 at r3 = 5 MPa corresponding to varied bp values (=1, 2, 3 and 4 MPa) was plotted as depicted in Fig. 7.

3 Results and Discussion Data obtained from the 3D laser scanner of both the surfaces of fracture for the samples S1-CG-38, S2-MG-38, and S3-FG-38 were analyzed statistically and the following statistical parameters such as mean, median, standard deviation, skewness, kurtosis along with RMS roughness, Rrms and roughness average, Ra were computed and presented in Table 3. It can be observed from the Table 3 that the parameters mean, and median, cannot differentiate the samples noticeably. However the standard deviation and range (Max.–Min.) values for samples S1-CG-38, S2-MG38, and S3-FG-38 are 1.894, 2.229, 0.950, and 9.814, 10.001, 5.070 respectively. It can be inferred that the higher values of standard deviation and range indicate a rough surface, whereas lower values indicate a smooth surface, and the same can be visualized from the 3D profile of the samples S1-CG-38, S2-MG-38, and S3-FG-38 (refer Fig. 4a–c respectively). Figure 4b for sample S1-MG-38

shows more undulations in the topography of the fracture surfaces as compared to Fig. 4a, c of samples S1-CG-38 and S3-FG-38 respectively. It can also observed from the Table 3, that the calculated Rrms and Ra by employing Eqs. (1, 2) respectively are well correlated, and Rrms values are systematically higher than Ra values. The higher value of Rrms and Ra (2.229 and 1.903 respectively) for sample S2MG-38 indicates that the fracture surface is rough with more undulation as compared to the samples S1-CG-38 and S3-FG-38. It should be noted that from the 2-D roughness profile also the computed (l2–l1) of sample S2-MG-38 shows a higher value of 12.4 as compared to their counterparts (refer Table 1) and indicates a more tortuous path of flow. The sample S3-FG-38 shows the least Rrms and Ra values of 0.950 and 0.799 respectively, indicating a smooth fracture surface. Though; the sample S2-MG-38, consists of medium grain the fracture surface developed using the tensile technique was rougher compared to their counterparts. This can be attributed to the difference in mineral composition, grain size distribution, and the presence of discontinuities (micro cracks/flaws) in the sample. However the fracture surfaces developed for samples S1-CG-38 and S3-FG-38, respectively, are moderately rough and smooth and systematically correlated with the grain size (visual identification) of the samples that is, the finer the grain size, the smoother the developed fracture surface. Further experimental results of the samples of rockmass have been analyzed and the relationship between Q and bp, corresponding to different r3 values, have been plotted for samples S1-CG-38, S2-MG-38, and S3-FG-38, as depicted in Fig. 8a–c respectively. It can be observed from these figures that a linear relationship exists between Q and bp for most of the data points, except for the case, when the effective confining pressure, reff.(=r3 - bp), is B2 MPa

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5 15 30

10 20 40

σ3 (MPa)

3.0

5 15 30

2.5 -7 Q (X10 m3/s)

5 -7 Q (X10 m3/s)

3.5

σ 3 (MPa)

6

4 3 2

10 20 40

2.0 1.5 1.0

1 0.5

0

0.0

0

5

10

15

20

25

0

5

10

15

b p (MPa)

b p (MPa)

(a)

(b)

25

σ 3 (MPa)

6

5 15 30

5 -7 Q (X10 m3/s)

20

10 20 40

4 3 2 1 0 0

5

10

15

20

25

b p (MPa)

(c) Fig. 8 The variation of Q with bp for samples a S1-CG-38, b S2-MG-38 and c S3-FG-38

(i.e., r3 = 15 MPa and bp = 14 MPa). In such cases, data deviates from the approximately linear relationship between Q and bp (such data are enclosed in a circle). This deviation in the flow can be a result of an experimental error, such as, side wall leakage (flow enters through the membrane and the specimen). As most of the measured experimental data of the Q and bp plots (refer Fig. 8) follow the linear trend, it can be assumed that the flow through the fracture obeys Darcy’s law and the well-known ‘‘cubic law’’ which assumes that linear flow between two parallel smooth plates (Gangi 1978; Kranz et al. 1979; Tsang and Witherspoon 1981; Schrauf and Evans 1986). Therefore, the flow data can be employed to estimate the hydraulic aperture of the fractures (by assuming the density of water = 997.05 kg/m3 at 25 °C and the dynamic viscosity of water = 8.90 9 10-4 kg/m s for water at 25 °C). Further, it must be noted that it would require highly

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sophisticated instrumentation to measure the hydraulic aperture of the fracture, which decreases due to incremental change in the confining pressure. The variation in slope of bp versus Q with r3 corresponding to samples S1-CG-38, S2-MG-38, and S3-FG-38 was plotted in Fig. 9. It can be observed from the figure that the slope of bp versus Q becomes low with applied r3 and almost becomes constant beyond r3 C 30 MPa, for samples S1-CG-38, S2-MG-38, and S3-FG-38. This indicates resistance in flow through the fracture with increase in r3 and fracture roughness does not contribute to fluid flow beyond certain critical r3 C 30 MPa. It can be inferred that increase in resistance of flow is caused due to almost complete closer of fracture aperture at higher values of r3 C 30 MPa. A closer look at Fig. 9, reveals that at r3 = 5 MPa, slope of bp versus Q is more for coarse grained sample (S1-CG-38) as compared to its

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1.8 S1-CG-38 S2-MG-38 S3-FG-38

1.5

-7

s (X10 )

1.2 0.9 0.6 0.3 0.0 0

5

10

15

20

25

30

35

40

45

σ 3 (mPa) Fig. 9 The variation in slope of bp versus Q with r3 for samples S1CG-38, S2-MG-38 and S3-FG-38

counterparts. However, with increase in r3 (=10, 15 and 20 MPa), slope of bp versus Q is more for fine grained sample (S3-FG-38) as compared to coarse and medium grained sample. It can be inferred that beyond r3 C 10, the resistance in flow for fine grained sample (less rough/ smooth fracture surface, refer Fig. 4c; Table 3) is less as compared to coarse or medium grained sample (S1-CG-38 and S2-MG-38). This is mainly because of less tortuous path of flow in case of sample S3-FG-38, as compared to its counterparts (S1-CG-38 and S2-MG-38). The variation in Q with r3 for samples S1-CG-38, S2MG-38, and S3-FG-38 corresponding to bp = (2, 4, 8, 14, 18, and 25 MPa) has been plotted as depicted in Fig. 10a– f. It can be noticed from these figures that the reduction in Q with r3, is less for the fine grained sample (S3-FG-38) and shows a comparatively higher Q, as compared to its counterparts corresponding to all values of bp. In the contrary, for samples, S1-CG-38 and S2-MG-38, the behavior is very puzzling; crossing over each other and almost showing similar Q corresponding to r3 C 10 MPa. It can be inferred that S3-FG-38 is the fine grained sample with less roughness (refer Ra and Rrms values in Table 3), and hence the less tortuous path of flow [refer (l2–l1) value Table 1] and hence has a higher Q, as compared to the samples S1-CG-38 and S2-MG-38. All these figures (refer Fig. 10a–f) exhibit an almost similar non-linear trend, and asymptote at higher values of r3 C 20 for all the samples. It can be observed from the Fig. 10a–f that there is rapid reduction (&78–99 %) in Q for the initial period of applied r3 (=5–20 mPa), however, in the contrary, reduction in Q in the latter period of applied r3 (=20–40 mPa) is only (&1–12 %). This can be attributed to rapid closer of fracture aperture at initial applied r3 (=5–20 mPa). Further, at higher r3 C 25 MPa, the effect

of fracture roughness and tortuosity are almost negligible and curves becomes asymptotic for all the samples. It indicates almost complete closer of fracture aperture i.e., now fracture aperture changes to residual aperture. In other words, this is the maximum possible joint closer at infinite or higher r3 C 25 (Saeb and Amadei 1992), as all asperities are in close contact or complete matching of two surfaces of fracture with each other and hence, there is no further space or path for fluid to flow. In addition, at higher bp C 14 MPa (refer Fig. 10d), curves diverges for sample, S3-FG-38 even at higher r3 C 30 MPa, indicating that the effect of bp is more pronounced than the applied r3 on the Q, for a smooth fracture surface. It can be attributed to the fact that for a smooth fracture surface, the height of asperities is very less (refer Fig. 4c) as compared to the counterparts; hence the fluid has to travel through less tortuous path thereby, a higher Q can be observed. The variation in Q with respect to an effective confining pressure, reff.(=r3 - bp), for samples S1-CG-38, S2-MG38, and S3-FG-38 has been plotted in Fig. 11. It can be observed from the figure that Q decreases almost non-linearly with reff., and the variation in Q with reff., is less significant for samples S1-CG-38 and S2-MG-38. On the contrary, variation in Q with reff., is quite distinct for the samples S3-FG-38, in this plot, showing a higher Q. It can be assumed that the effect of fracture roughness is quite significant for sample S3-FG-38. Further, the variation in Q with reff., is negligible beyond reff. C 20 MPa, for S1CG-38, S2-MG-38, and S3-FG-38. This indicates that fracture roughness does not contribute to fluid flow beyond certain critical reff. C 20 MPa. It can be inferred that most part of the asperities comes in contact with each other at a higher reff., and fluid flow through such case completely ceases and hence, all curves asymptote at a point, as depicted in Fig. 11. The Reynolds number, Re, which defines the flow as laminar or turbulent is important for understanding the fluid flow mechanism through fractured rockmass computed from the measured Q and sample dimension, by employing the following relationship (Zimmerman et al. 2004; Ranjith 2010; Ranjith and Viete 2011). Re ¼

qQ lW

ð3Þ

where q, Q, and l are the density, discharge, and dynamic viscosity of the fluid, respectively, and W is the fracture width, which is a function of e. The variation of Re with bp for samples S1-CG-38, S2MG-38, and S3-FG-38 is depicted in Fig. 12a–c respectively. For the sake of brevity, a typical data sheet of measured and calculated parameters along with Re for sample, S1-CG-38 is depicted in Table 4. It can be

123

Author's personal copy 996

K. K. Singh et al. 5

7 S1-CG-38 S2-MG-38 S3-FG-38

bp= (2 MPa)

4

bp= (4 MPa)

S1-CG-38 S2-MG-38 S3-FG-38

6

-7 3 Q (x10 m /s)

-7 3 Q (x10 m /s)

5

3

2

4 3 2

1 1

0

0

0

4

5

10

15

20

25

30

35

40

0

45

10

15

20

25

σ 3 (MPa)

(a)

(b)

bp= (8 MPa)

4

S1-CG-38 S2-MG-38 S3-FG-38

3

30

35

40

45

S1-CG-38 S2-MG-38 S3-FG-38

bp= (14 MPa)

3 -7 Q (x10 m3/s)

-7 3 Q (x10 m /s)

5

σ 3 (MPa)

2

1

0

2

1

0

5

10

15

20

25

30

35

40

10

45

15

20

25

30

σ3 (MPa)

σ 3 (MPa)

(c)

(d)

2.0 S1-CG-38 S2-MG-38 S3-FG-38

bp= (18 MPa)

0.9

35

bp= (25 MPa)

40

45

S1-CG-38 S2-MG-38 S3-FG-38

-7 3 Q (x10 m /s)

-7 Q (x10 m3/s)

1.5

1.0

0.5

0.0

15

0.6

0.3

0.0

20

25

30

35

40

45

25

30

35

σ 3 (MPa)

σ 3 (MPa)

(e)

(f)

40

45

Fig. 10 The variation of Q with the r3 corresponding to different bp for the samples S1-CG-38, S2-MG-38 and S3-FG-38

observed from the Table 4 and Fig. 12, that as expected, Re is quite sensitive to bp and a linear relationship exists between Re and bp.

123

Further, a line representing Re [ 10 (Hassanizadeh and Gray 1987) was added to mark the transition in flow from laminar to turbulent. It can be observed from the Fig. 12,

Author's personal copy Laboratory Simulation of Flow through Single Fractured Granite

997

7 S1-CG-38 S2-MG-38 S3-FG-38

6

-7 Q (x10 m3/s)

5 4 3 2 1 0 0

5

10

15

20

σ

(MPa)

eff.

25

30

35

40

Fig. 11 Discharge, Q as a function of effective confining pressure, reff = (r3 - bp)

18 16

σ3 (MPa)

(a)

R = >10 e Turbulent Flow

10

Laminar Flow (Ranjith and Viete, 2011)

R = >4 e Turbulent Flow Laminar Flow

4

Laminar Flow

R = >10 e

8

8 6

σ3 (MPa)

Turbulent Flow

(Hassanizadeh & Gray, 1987)

Re

Re

(Hassanizadeh & Gray, 1987)

(b)

10

5 10 15 20 30 40

14 12

that most of the data points for all the samples falls below the critical Re [ 10, except for some data of S1-CG-38 and S3-FG-38, corresponding to reff. \ 2 MPa (refer Table 4). It should be noted that the sample S1-CG-38 is a coarse grained sample with moderately high Rrms and Ra values (refer Table 3), indicating that a rough fracture surface and larger asperities and hence, a large aperture. In case of a large aperture, the effect of surface roughness on fluid flow is negligible (Ranjith 2010) and hence, higher Q and thereby, higher Re [ 10 can be observed. On the contrary, sample S3-FG-38 has less Rrms and Ra values (refer Table 3), which indicates a smooth fracture surface and a less tortuous path of the flow corresponding to r3 B 5 MPa, and hence, yields a high Q and a high Re ([10). For such cases, transition in-flow takes place from laminar to turbulent and the computation of e employing ‘cubic law’ relationship would be incorrect.

6

5 10 15 20 30 40

(Ranjith and Viete, 2011)

R = >4 e Turbulent Flow

4

Laminar Flow

2

2 0

0 5

10

15

20

25

0

5

10

18

20

25

σ3 (MPa)

(c)

5 10 15 20 30 40

15 (Hassanizadeh & Gray, 1987)

R = >10 e

12

Turbulent Flow

Re

15

bp (MPa)

bp (MPa)

Laminar Flow

9

(Ranjith and Viete, 2011)

6

R = >4 e Turbulent Flow Laminar Flow

3 0 0

5

10

15

20

25

bp (MPa) Fig. 12 The variation of Re with bp corresponding to different r3 for samples a S1-CG-38, b S2-MG-38 and c S3-FG-38

123

Author's personal copy 998 Table 4 Typical data sheet of measured and calculated parameters for sample, S1-CG-38

K. K. Singh et al.

r3 (MPa) 5

10

15

20

30

40

bp (MPa)

reff. = r3 - bp (MPa)

ecal. (910-4 m)

Re

cal.

1

4

8.63

2.65

2

3

20.88

4.48

5.90

3

2

35.84

6.13

10.12

4

1

57.14

7.89

16.14

1

9

2.29

1.70

0.65

2

8

3.35

2.43

0.95

4

6

7.51

4.01

2.12

6

4

13.14

5.53

3.71

8

2

19.88

6.99

5.62

2

13

1.08

1.67

0.31

4

11

2.04

2.60

0.58

6

9

3.39

3.52

0.96

8

7

4.71

4.33

1.33

10

5

6.77

5.26

1.91

14 2

1 18

14.29 0.15

7.55 0.87

4.04 0.04

4

16

0.25

1.28

0.07

8

12

0.99

2.57

0.28

10

10

1.22

2.97

0.34

14

6

2.17

4.03

0.61

18

2

4.12

5.42

1.16

2.44

2

28

0.14

0.85

0.04

4

26

0.20

1.20

0.06

8

22

0.27

1.67

0.08

14

16

0.59

2.61

0.17

18

12

0.88

3.23

0.25

25

5

1.72

4.52

0.49

2

38

0.10

0.74

0.03

4

36

0.10

0.96

0.03

8

32

0.13

1.31

0.04

14 18

26 22

0.20 0.27

1.82 2.17

0.06 0.08

25

15

0.48

2.96

0.14

To further quantify the transition boundary, a line representing Re [ 4 (Ranjith and Viete 2011), was also added in Fig. 12. It can be observed from the Fig. 12a, that sample S1-CG-38 and S2-MG-38, yields a low Re (\4); for most of the data points except for the case where reff. \ 2 MPa (refer Table 4). However, on the contrary, it can also be observed from Fig. 12c, that sample S3-FG-38 indicates a higher Re [ 4, for most of the data points corresponding to r3 B 15 MPa. In such case, the influence of bp is much more pronounced and hence higher Re [ 4 can be observed. Further, it can also observed from the Table 3, that the sample, S3-FG-38 indicates less Rrms and Ra values of 0.950 and 0.799, respectively. On the contrary, with an increase in r3 C 15 MPa, for the sample, S3-FG-38, the influence of bp

123

QExp. (910-8 m3/s)

on Re is less (refer Fig. 12c). It can be inferred that in case of a smooth fracture surface, corresponding to r3 B 15 MPa, flow takes place through a less tortuous path and hence a higher Re [ 4 can be observed, suggesting turbulent flow. However with applied r3 C 15 MPa, Q is less and hence Re \ 4 can be observed, which can be attributed to closure of fracture aperture at higher r3 C 15 MPa. On the contrary, sample S2-MG-38 with a high roughness value (refer Table 3; Fig. 12b) shows low Re \ 4, corresponding to r3 C 10 MPa. This can be mainly attributed to less Q due to rapid reduction in e. Further, with an increase in reff. [ 5 MPa, Q reduces rapidly due to further rapid reduction in e under high reff., and hence indicates Re \ 4 for most of the data points. It can also be attributed to the

Author's personal copy Laboratory Simulation of Flow through Single Fractured Granite

fact that the flow path increases due to high surface roughness as indicated in Fig. 4b and in Table 1, [refer higher l2–l1 value (=12.4) for sample S2-MG-38].

4 Conclusions A simple and easy to adopt methodology to simulate the flow of water through a single fractured granite rock under the laboratory environment has been developed and presented in this manuscript. It has been demonstrated that, samples with a single fracture created with the help of tensile tests, can be used quite satisfactorily to simulate the response of extreme elevated base pressures, confining pressures and fracture roughness collectively. The methodology was found useful in understanding the mechanics of fluid flow through fracture/s in a better way. Based on the study, it has been demonstrated that high precision 3D laser scanning was found to be useful in studying the effect of fracture roughness on the fluid flow. The obtained data can be utilized to compute statistical parameters and to generate a 3D surface profile to differentiate the different fracture surfaces. Fluid flow study has demonstrated that discharge is quite sensitive to confining pressure, base pressure, and fracture roughness. There is a critical base pressure beyond which deviation in linearity occurs. Also the study has clearly demonstrated that the ‘cubic law’ relationship is still valid even at low reff. B 2 MPa and thus can be useful for back computing the hydraulic aperture, e of the fracture from the fluid flow data. This is otherwise quite difficult and too intricate to measure and it requires highly sophisticated instrumentations. It is believed that such studies will be quite useful in understanding the basic mechanism of fluid flow through the single fractured rockmass. However, to study, the phenomenon taking place in the natural real conditions, rock samples containing natural fracture/fractures (single or multiple fractures with different orientations) should be tested following the same testing conditions. Acknowledgments The authors are thankful to the IITB-Monash Research Academy, Indian Institute of Technology Bombay, Mumbai India for the opportunity to work on this project. This work was supported and funded by BHP Billiton Ltd. Melbourne Victoria Australia. The authors are also thankful to Mr. Prakash, Kr. Singh, and Mr. Ravi Shankar (research scholar) for their insightful comments and suggestions to improve the manuscript. Further, the authors are also thankful to the Geological Survey of India, for permission to continue part time research.

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