Fluid friction and heat transfer through a single rough fracture in granitic rock under confining pressure

International Communications in Heat and Mass Transfer 75 (2016) 78–85

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Fluid friction and heat transfer through a single rough fracture in granitic rock under confining pressure☆ Xiaoxue Huang a,b, Jialing Zhu a,b, Jun Li a,b,⁎, Bing Bai c, Guowei Zhang a,b a b c

Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, Ministry of Education, Tianjin University, Tianjin 300072, China Tianjin Geothermal Research and Training Center, School of Mechanical Engineering, Tianjin University, 92 Weijin Road, Nankai District, Tianjin 300072, China State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China

a r t i c l e

i n f o

Available online 11 April 2016 Keywords: Fractured rock Enhanced geothermal systems Surface roughness Flow friction Heat transfer

a b s t r a c t To better understand flow and heat transfer in the fractured rock in enhanced geothermal systems (EGS), experiments were conducted to investigate the single-phase convective heat transfer and pressure drop of water flowing through a single fracture in a cylindrical granite rock. Dimensionless correlations for the Poiseuille number (Po) and the average Nusselt number (Nu) with the Reynolds number (Re) were obtained from the experimental data. It was found that the experimental results significantly deviated from those of rectangular macrochannel flows, and the flow friction is raised and the heat transfer is weakened significantly due to the large relative roughness. As confining pressure is applied, reduction of the aperture raises both Po and Nu. A roughness–viscosity model (RVM) was employed to account for the effects of the surface condition, and a satisfactory agreement between the numerical results based on RVM and the experimental data was achieved. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The ability to provide reliable low-carbon power makes geothermal energy one of the promising renewable energy options to meet the growing energy demand through replacing fossil fuels [1]. However, the limited availability of naturally occurring hydrothermal systems that have both sufficient permeability and in-situ fluids for commercial geothermal energy applications, severely limits its deployment. Alternatively, enhanced geothermal systems (EGS) offer a solution to utilize a much larger fraction of deep geothermal resources [1]. To harness those hot dry rock (HDR) resources, hydraulic stimulation is required to create interconnected fractures for lower flow resistance. Among the studies of EGS, prediction of the artificially fractured reservoir performance encounters considerable challenges due to limited knowledge of the detailed nature of flow and heat transfer in the fractured rocks [2]. Regarding the hydraulic behavior of water flowing through rock fractures, many laboratory studies have been conducted on rocks with a single fracture. Theories for laminar flows through ideal parallel plates demonstrate that the Poiseuille number (Po), which is the product of the Reynolds number (Re) and the friction factor (f), is determined solely by the cross-sectional dimension of the channel [3]. Accordingly, the widely employed cubic law which indicates a linear relationship

☆ Communicated by Dr. W.J. Minkowycz. ⁎ Corresponding author at: Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, Ministry of Education, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (J. Li).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.03.027 0735-1933/© 2016 Elsevier Ltd. All rights reserved.

between the ratio of flow rate to hydraulic gradient and the cube of fracture aperture was derived [4]. Accurate predictions of flow rates can be achieved using the hydraulic aperture, a fitting parameter which is calculated from the measured flow rates and hydraulic gradients [4]. However, fracture surface roughness and aperture variation make the application of the cubic law unsuccessful when, as is often the case in EGS, the experimental data is insufficient to form the basis for fitting a hydraulic aperture [5]. Therefore, it is necessary to carry out fundamental investigations to further study the physics of water flowing through a fracture with an emphasis on the effects of fracture surface conditions, and to obtain an empirical relations between the flow rate and the pressure drop over the rough fracture for predicting the flow. In addition, heat transfer in rock fractures is also of great significance to EGS as thermal energy is extracted through the fluid–rock heat exchange. An understanding of convective heat transfer between the rock fracture surfaces and the circulating fluid plays a key role in estimating heat recovery in fractured rocks. In large-scale reservoir simulations that explicitly represent fractures, the heat transfer coefficient is a critical parameter as it determines the heat exchange between the circulating fluid and the rocks, and the resultant production of hot water. However, only limited experimental data pertaining to heat convection in rock fractures has been reported [6,7]. A series of experiments to study heat convection in rock fractures under various flow velocities was conducted by Zhao and Tso [6,7], and the heat transfer coefficient was correlated with velocity. Moreover, as suggested by the results, the fracture surface roughness plays a dominant role in heat transfer. Therefore, additional efforts should be directed towards exploring

X. Huang et al. / International Communications in Heat and Mass Transfer 75 (2016) 78–85

Nomenclature A Ac H/W L L+ h Nu P Pc Po Q Ra Re Rek Rh b cp f h l

coefficient in Eq. (10) cross-section area, m2 height-to-width ratio fracture length, m non-dimensional heating length Nusselt number fluid pressure, Pa confining pressure, Pa Poisueille number volumetric flow rate, m3/s fracture roughness, m Reynolds number roughness Reynolds number hydraulic radius, m fracture width, m specific heat, J/kg∙°C friction factor heat transfer coefficient, W/m2 shortest distance from the point to the central line of the fracture, m q heat transfer rate, W T fluid temperature, °C Ti fracture surface temperature, °C rock outer surface temperature, °C To fluid inlet temperature, °C T1 fluid outlet temperature, °C T2 u velocity, m/s uave(=Q/δ/b) average velocity, m/s velocity at the top of the roughness element, m/s uk u+(=u(y)/uave) non-dimensional velocity w total uncertainty y,z Cartesian coordinates, m non-dimensional distance from the fracture surface y+ ΔP pressure drop, Pa∙s ΔT mean temperature difference, °C effective viscosity, Pa∙s μeff fluid viscosity, Pa∙s μf roughness viscosity, Pa∙s Μr rock thermal conductivity, W/m∙K λr water thermal conductivity, W/m∙K λw ρ density, kg/m3 δ fracture aperture, m

the relation between the surface conditions and the heat transfer characteristics in the fracture. Due to the poor current understanding of the relationship between heat transfer coefficient and fluid velocity, fracture geometry and thermal properties of the fluid, the heat transfer coefficient is usually assumed to take a constant value [8,9]. Thus, investigation of the forced convection through a rough fracture in terms of the fracture surface conditions and fluid thermal properties is also significant for more accurate simulation models. The objective of this study is to investigate flow and heat transfer in a fractured granitic rock. Relations for dimensionless parameters are obtained based on the measured data for different rock temperatures and confining pressures. To demonstrate the influence of fracture surface, the relative roughness is quantified by measuring the roughness and the fracture aperture. In view of the small dimension of the aperture, a roughness–viscosity model (RVM) for microchannel flows is applied to predict hydraulic and heat transfer performance, and numerical predictions which compare reasonably well with the experimental results are obtained.

79

2. Experiment To simulate the subsurface conditions of granite rocks in EGS, high temperatures were attained for the granite rock in the experiment, and confining pressures (Pc) were applied and varied to change the fracture aperture reflecting the thermo–hydro–mechanical effects in EGS. Fluid–rock heat exchange took place as cold water was pumped through the fracture which was created under tensile stress. To acquire the surface roughness and the average aperture of the fracture, non-contact laser scanning was used for measurement. 2.1. Rock sample preparation A cylindrical rock sample with a diameter of 50 mm and a length of 100 mm was cored from a granite block. It was then split into two halves under tensile stress by sharp wedges loaded in a uniaxial compressive apparatus (Fig. 1), which is a standard procedure known as the Brazilian method. Adhesive was applied to the lateral sides of the fracture to prevent fluid leakage. Mechanical and thermophysical properties of the granite block, along with geometric parameters measured by a digital caliper are listed in Table 1. 2.2. Surface topology measurement A Laser Scanning Microscope (Olympus® LEXT OLS4100) was used to perform non-contact 3D scan of the rough fracture surfaces. The aperture variation was determined by the asperity heights of both surfaces, and an average value of 135 μm was determined. The wavy topology requires a suitable cutoff value for roughness analysis: a large value would include the waviness and a small one would isolate the waviness. The standard value of 0.8 mm for the cutoff wavelength was chosen and the average surface roughness (Ra) was derived as 20 μm. Accordingly, the relative roughness (Ra/2Rh) is 0.148 without confining pressures. 2.3. Experimental apparatus and procedure Fig. 2 shows the experimental apparatus. It consists of five parts: (1) high-pressure cell; (2) water supply unit; (3) temperature and pressure loading unit; (4) sensor unit; (5) data-acquisition unit. The fractured rock wrapped with a thermal shrinkable sleeve was immersed in anti-wear hydraulic oil that fills the high-pressure triaxial cell. Deionized water was pumped to the test section through a Teledyne® Isco syringe pump. An electric heater with temperature control was wrapped around the cell. A pressure loading unit was connected to the cell to apply Pc through oil. The water flow rate and the pressure were recorded automatically by the pump with 0.5% of full-scale accuracy. Pt100 sensors with an accuracy of 0.5 °C and a response time of 50 ms were

Fig. 1. The cylindrical rock before and after the Brazilian split.

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X. Huang et al. / International Communications in Heat and Mass Transfer 75 (2016) 78–85 Table 1 Parameters of the granitic rock. Parameters (unit)

Value

Thermal conductivity λr(W/m K) Densityρr(kg/m3) Elastic modulus Er(N/m2) Poisson's ratio νr(1) Length L(mm) Diameter D(mm)

2.8 2968 1.12 × 1011 0.28 98.44 50.08

used to measure the inlet and outlet fluid temperatures, as well as the rock outer surface temperature (assumed equal to the surrounding oil temperature). Rock displacement was monitored by the attached transducer. The rock bound with two end blocks was first placed on the cell base. Then hydraulic oil was injected to fill the cell. Subsequently, deionized water was supplied to flow through the rock fracture. The flow rate ranged from 5 ml/min to 35 ml/min, with the upper limit constrained to assure that the hydraulic pressure did not exceed Pc. 3. Data reduction Fig. 3. Schematic presentation of the fractured cylindrical rock.

To derive the dimensionless parameters for empirical correlations, the bulk temperatures at the fracture inlet and outlet (T1 and T2), the rock outer surface temperature (To), the volumetric flow rate (Q), and the total pressure drop (ΔPtotal) were recorded. Po and Nu for flow and heat transfer analysis can be quantified based on the measured data. For the pressure-driven laminar flow through the fracture (Fig. 3), the dimensionless parameters based on the hydraulic diameter are derived from the measured ones as follows. The mean fluid temperature of the fracture inlet and outlet is employed as reference temperature for evaluation of the thermophysical properties. Po is the product of f and Re. The characteristic length to calculate Re is 2Rh which equals approximately 2δ as δ b b b 2

Po ¼ f Re¼

4Rh Ac 2 Δp 2ρQRh 4δ3 b Δp 2ρQδ 8δ3 bΔP ¼ ¼ μ f δb μ f LQ ρLQ 2 ρLQ 2 μ f δb

ð1Þ

Note that the measured ΔPtotal includes the entrance and the exit pressure losses of the fracture, the major losses from the pump outlet to the fracture inlet, and the fracture outlet to the cell outlet. Therefore, the net frictional pressure drop over the fracture is calculated by ΔP ¼ ΔP total −ΔP minor −ΔP major

ð2Þ

ΔPminor and ΔPmajor were estimated using the correlations proposed by Streeter [10]. For the Nu calculation, the average heat transfer coefficient over the fracture is defined in the same manner as in Refs. [6,7]: h¼

q ¼ bLΔT

Fig. 2. Experimental apparatus.

ρcpQ ðT2−T1Þ
 T1 þ T2 bL T i − 2

ð3Þ

X. Huang et al. / International Communications in Heat and Mass Transfer 75 (2016) 78–85

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in which ΔT is the mean temperature difference between the water and the fracture surface, and T i is the average temperature of the fracture surface. For calculation of T i , we assume that heat transfer only takes place as conduction in semi-cylindrical rocks on either side of the fracture and neglect heat conduction along the axial direction. When considering an interval of dz, with constant To, the temperature distribution in each semicircular slice can be readily acquired by solving the two-dimensional energy equation according to Carslaw and Jaeger [11]. The heat conducted through the fracture surface can then be obtained as [11]  Z LZ b ∂T  −λr  dxdz q¼2 ∂y y¼δ 0 0 2 Z L   42:32 42:32 λr ðT o −T i ðzÞÞdz ¼ λr L T o −T i ¼ π π 0

ð4Þ

In Eq. (4) equals to the heat flux absorbed by the fluid which can be derived in terms of the inlet and outlet fluid temperatures. Thus, making a heat balance between the fluid and the rock, the average temperature of the fracture surface T i is given by Ti ¼ To−

πρcp uδbðT 2 −T 1 Þ 42:32λr L

ð5Þ 4.2. Heat transfer

Nu can then be calculated by combining Eqs. (3) and (5) ρcp uδ2 ðT 2 −T 1 Þ
 T1 þ T2 λw L T i − 2 42:32λr ρcp uδ2 ðT 2 −T 1 Þ ¼ 42:32λr λw LT o −πρcp uδbλw ðT 2 −T 1 Þ−21:16λr λw LðT 1 þ T 2 Þ

Nu ¼

hDh ¼ λw

Fig. 4. the Poiseuille number vs. the Reynolds number and curves predicted by Eqs. (7a)–(7c) under different confining pressures and rock outer surface temperatures.

ð6Þ

For fully developed laminar flows, the average Nusselt number based on the hydraulic diameter of the cross section with H/W approaching 0 and a constant wall temperature is 7.54 [12]. In contrast, Nu is about two orders of magnitude smaller in the fracture. The experimental results of the average Nusselt numbers versus the + non-dimensional heating length (L+ h ) are plotted in Fig. 5. Lh is employed to illustrate Nu variation in terms of both Re and Pr, and is defined according to Ref. [13] to identify the thermal entrance region.

4. Experimental results By changing the dimensional parameters,Pc, To and Q, the nondimensional parameters Re, Pr and Ra/2Rh were varied. Empirical correlations for Po and Nu in terms of the dimensionless parameters are derived based on 90 experimental data points. The experimental results indicate that Po and Nu are influenced by Re and Ra/2Rh, unlike those macrochannel flows.

Lþ h ¼

L 2Rh RePr

ð8Þ

According to Fig. 5, Nu decreases with L+ h , which is to be expected as heat transfer is enhanced at higher Re and Pr. However, the slope of Nu against L+ h decreases along the positive x axis. A similar trend was

4.1. Flow friction For fully developed laminar flows through rectangular macrochannels, Po is fixed at 96 with height-to-width approaching 0. However, based on the experimental results in this study, the correlation for Po in terms of Re is derived as Eqs. (7a)–(7c), with Ra/2Rh of 0.148 (Pc = 0 MPa), 0.1526 (Pc = 3 MPa) and 0.1539 (Pc = 6 MPa). Fig. 4 shows Po based on the measured data, together with Po predicted by Eqs. (7a)–(7c). It is demonstrated that Po decreases with Re reflecting a decrease in pressure drop normalized by a viscous scaling, and the slope is decreasing with Re, suggesting that Po may approach a fixed value if Re keeps increasing. Increase of Po is demonstrated with increasing Pc, thus the exponent of the constant 96 increases with Ra/2Rh in Eqs. (7a)–(7c). Under each confining pressure, Po is inversely proportional to the rock temperature as viscosity is lower at higher temperatures. Po ¼ 962:3053 Re−0:2902 ; Ra =2Rh ¼ 0:148

ð7aÞ

Po ¼ 962:3728 Re−0:1723 ; Ra =2Rh ¼ 0:1526

ð7bÞ

Po ¼ 962:4268 Re−0:1575 ; Ra =2Rh ¼ 0:1539

ð7cÞ

Fig. 5. the Nusselt number vs. the Reynolds number and curves predicted by Eqs. (9a)–(9b) under different confining pressures and rock outer surface temperatures.

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presented in Ref. [13], and it was explained that the thermally developed flow is approached. However, L+ h for the thermally developing flow is much larger in this study as compared to 0.14 in Ref. [13]. As Ra/2Rh in Ref. [13] is 0.04–0.06, it can be inferred that the

thermal entrance region is lengthened due to higher roughness in this study. Generally, Nu increases with Pc, suggesting that heat transfer is enhanced with smaller apertures which give larger relative roughness. The relation for Nu is concluded as Eqs. (9a)–(9c) with Pc of 0 MPa, 3 MPa and 6 MPa, respectively. Nu ¼ 7:54−3:6107 Re1:2399 Pr0:8583 ; Ra =2Rh ¼ 0:148

ð9aÞ

Nu ¼ 7:54−3:7815 Re1:3113 Pr1:0251 ; Ra =2Rh ¼ 0:1526

ð9bÞ

Nu ¼ 7:54−4:2424 Re1:3632 Pr1:5388 ; Ra =2Rh ¼ 0:1539

ð9cÞ

The curves predicted by Eqs. (9a)–(9c) are also presented in Fig. 5 to allow comparison with the measured data. From the above equations, it can be seen that Nu is much lower than 7.54, and it increases with Re as heat transfer is enhanced at higher velocities. Another point to note is that the exponent of the Prandtl number is over 1, while those in the empirical correlations for macrochannel flows are usually lower than 1 [12], which can be explained by the RVM as μ f used for the Prandtl number is lower than μ eff. 5. Numerical solutions based on RVM According to studies on microchannels, the presence of roughness causes local changes in the velocity profile with increased disturbance relative to that in macrochannels [13–16]. The RVM suggested by Mala and Li was applied to quantify the effects of the surface roughness on laminar flows in microtubes and trapezoidal microchannels [14,15]. The increased momentum transfer is modelled by means of a local roughness viscosity. Thus, flow friction is increased due to the larger effective viscosity (μ eff), which is the sum of the fluid viscosity (μ f) and the roughness viscosity (μ r). According to Ref. [14], μ r is a maximum near the wall, decreases as moving away from the wall and vanishes at a distance large enough compared to the roughness height. The effective viscosity given by Ref. [14] is written as 
2 μ eff μ f þ μ r l Re l ¼ ¼ 1 þ ARek 1− exp − k μf μf Re Ra Ra

ð10Þ

in which the roughness Reynolds number is defined as Rek ¼

uk ρRa 2ρu 2 Ra ¼ μf μfδ

ð11Þ

In a manner similar to the eddy viscosity in turbulent flow, μr is introduced to the momentum equation as an addition to μf. For pressure-driven flows in this study, velocity variation in the x direction can be neglected considering the minute fracture H/W. The momentum equation in the fracture then becomes: 2

∂ u 1 dP ¼ ∂y2 μ eff dz

ð12Þ

The coefficient A in Eq. (10), depends on the details of the roughness elements and has to be determined from experiments. Ref. [14] provided a detailed procedure on determining A based on the experimental data and the momentum equation, together with the boundary conditions. An empirical relation for the fracture surface is found to take a similar form to that in Refs. [13–17]: A¼ Fig. 6. Comparison of experimental data of fluid flow with numerical results based on the roughness–viscosity model.


2:7334 Rh expð0:2235Re−0:0395ReRh =Ra Þ Ra

ð13Þ

With A predicted by Eq. (13), μ eff can be obtained from Eq. (10), the velocity profile from Eq. (12), and finally the flow rate can be computed.

X. Huang et al. / International Communications in Heat and Mass Transfer 75 (2016) 78–85

Fig. 7. Numerical results of non-dimensional velocity profile using the effective viscosity and the parabolic profile using the fluid viscosity.

83

Fig. 6 compares the experimental data for flow rates as a function of Re along with the numerical results by solving Eq. (12) with the RVM predicted viscosity. The agreement between the numerical predictions and the experimental results implies that the RVM can be used to predict flows through a rough fracture with reasonable accuracy. Fig. 7 shows the velocity distribution obtained by using μeff and that obtained by substituting μf for μeff in Eq. (12). The velocity is plotted as the ratio of local velocity u(y) to the average velocity (uave = Q/δ/b) with the non-dimensional distance from the fracture surface (2y/δ). When plotted in non-dimensional variables, the parabolic profile by μfis fixed and the RVM predicted velocity profile is only influenced by Re and Pc. For Re ranging within 8–80 in this study, change in the velocity profile is slight. Thus, we only show Re = 20 in Fig. 7 for clarity. As predicted by RVM in Fig. 7, the existence of surface roughness leads to the thickening of the boundary layer, which raises the total shearing stress, and thus a larger pressure is required to attain the flow rate. However, the near-wall shearing stress, which is proportional to the velocity gradient, is smaller than that of the parabolic profile, and this can be used to account for the Nu reduction in microchannels as the temperature gradient near the wall is also decreased and heat transfer is weakened. Accordingly, it can be inferred that the velocity results may be further used for predicting the heat transfer performance.

Fig. 8. Comparison of experimental data of heat transfer with numerical results based on the roughness–viscosity model.

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X. Huang et al. / International Communications in Heat and Mass Transfer 75 (2016) 78–85

In this study, Nu can be directly obtained from numerical simulations as the energy equation takes a simple form in the fracture. As the conduction term in x direction can be neglected due to the crosssectional dimensions of the fracture, the energy equation for the forced convective heat transfer is then given by

Table A-1 fractional uncertainties in the primary measurements. Parameter

L

b

ΔP

t

Q

Fractional uncertainty

0.01%

0.02%

0.5%

1.25%

0.5%

2

a

∂ t dt ¼ uðyÞ dz ∂y2

ð14Þ

With the velocity distribution obtained from Eq. (12), the temperature distribution can be easily acquired by a numerical solution of the finite difference form of Eq. (14). Nu predicted based on RVM is shown in Fig. 8 to allow comparison with the experimental data. The numerical results exhibit the same trend as the experimental ones. Generally, the agreement is considered satisfactory, in view of the fact that the experimental Nu is two orders of magnitude smaller than that without the roughness effect.

6. Conclusion Flow and heat transfer characteristics of water flowing through a fractured cylindrical rock were studied. For8 ≤ Re ≤ 80, 1.69 ≤ Pr ≤ 3.39 and0.148 ≤ 2Ra/δ≤ 0.1539, the following conclusions are obtained: 1. Po is much larger than the constant for ideal parallel plates flows, and it decreases with Re as the friction factor decreases. The decrease of Po is gentler at higher Re. Nu is approximately two orders of magnitude smaller than that for parallel plate flows. Nu increases rapidly + with decreasingL+ h , especially at smallLh . 2. Empirical correlations for Po and Nu with Re are obtained for laminar water flows through a fracture. The exponent for Pr using fluid viscosity is much larger than those in theories for macrochannels as μeff is larger thanμf. 3. Using the effective viscosity from RVM as a substitute for the fluid viscosity in the governing equations, a reasonable agreement between the numerical results and the experimental data is obtained. The numerical results show that the thickening of boundary layer induced by the roughness raises the pressure drop as a result of the increase of total shearing stress. In addition, the near-wall velocity gradient is decreased, and thus Nu is smaller as the near-wall temperature gradient is lowered simultaneously. For applications in enhanced geothermal system, a wider range of temperature and relative roughness will be further studied while elevating the pressure of the experiments to avoid phase change.

We thank Professor Donald L. Koch at Cornell University for improving the use of English in the manuscript. The authors gratefully acknowledge the support of this work by the National Natural Science Foundation of China (Grant No. 41272263).

Appendix A. Experimental uncertainty According to Holman [18], if R is a given function of the independent variablesx1,x2, x3, … xn, uncertainty in the result of R can be estimated on the basis of the uncertainties in primary measurements of the variables as "


1 ∂R w1 R ∂x1

2


2
2 #1=2 1 ∂R 1 ∂R þ w2 þ … þ wn R ∂x2 R ∂xn

ðA:1Þ

ðA:2Þ

The Reynolds number, friction factor and the Poiseuille number all take a product form of the measured parameters. Thus, the fractional uncertainty can be obtained as " #1 w 2 w 2 w 2
w 2 w 2 2 wPo δ ΔP L Q b ¼ 3 þ 3 þ þ 3 þ Po δ b ΔP Q L

ðA:3Þ

Uncertainty in Nu takes a more complex form, and the maximum of partial derivatives and the minimum of Nu are used for calculation of the maximalwNu. "
2
2
2
2 wNu 1 ∂Nu ∂Nu ∂Nu ∂Nu ¼ wt 1 þ wt 2 wδ þ wb þ Nu Nu ∂δ ∂b ∂t 1 ∂t 2
2
2 #12 ∂Nu ∂Nu þ wt o þ wL ∂t o ∂L ¼

 2  2 1 h ð1022:52wδ Þ2 þ ð1:31wb Þ2 þ 0:002wt1 þ 0:00043wt2 0:36 i12 þð0:0016wt o Þ2 þ ð0:1wL Þ2 ðA:4Þ

Based on the instruments and methods employed in the experiments, the maximal uncertainties of the measured parameters are listed in Table A-1. As forwδ, the aperture without confining pressure (δ0) is determined by image analysis, and the uncertainty depends on the resolution. When a confining pressure is applied, δ is determined by δ0 and the displacement of facture surfaces (δ′) which are measured by the axial strain sensors. Initially, uncertainty in the aperture is wδ0 ¼ 0:6% δ0

ðA:5Þ

With confining pressures applied, the aperture becomes the difference between δ0 and the displacement δ′, then wδ ¼ δ

Acknowledgements

wR ¼ R

R ¼ Rðx1 ; x2 ; x3 ; …; xn Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0062 þ 0:0012 ¼ 6:08%

ðA:6Þ

Accordingly, one has wPo ¼ 18:4% Po

ðA:7Þ

wNu ¼ 2:46% Nu

ðA:8Þ

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